https://benchmark.coria-cfd.fr/api.php?action=feedcontributions&user=Lartigue&feedformat=atomCFD Benchmark - User contributions [en]2024-03-29T05:40:56ZUser contributionsMediaWiki 1.26.2https://benchmark.coria-cfd.fr/index.php?title=Step_3&diff=547Step 32024-02-05T14:22:28Z<p>Lartigue: </p>
<hr />
<div>= Step 3 description =<br />
<br />
The purpose of this configuration is to perform a simulation similar to '''Step 2''' but involving now a '''mixture of different gaseous species''' at different temperatures, still '''without any chemical reaction'''. <br />
In this manner, it is possible to assess the accuracy of the numerical models involved in the description of species and heat diffusion<br />
The thermodynamic and transport properties of the 9 species kinetic scheme of Boivin et al.<ref name="Boivin2011"/> should be used in preparation for the next step, even though the reactions are still neglected here.<br />
<br />
It should be noted that in this case, the '''density is variable''' in both '''space''' and '''time''' due to the changing composition, which was not the case in the previous step: this induces additional numerical difficulties that must be taken into account.<br />
However, there is no variation of the thermodynamic pressure in time, contrary to the next step.<br />
<br />
The simulation domain is a cubic box of size <math>[0;L]^3</math> with <math>L = 2 \pi L_0</math> in each direction, where <math>L_0 = 1\; \mathrm{mm}</math>. <br />
Compared to the previous step, this smaller size is needed to ensure a proper resolution of the reaction fronts for hydrogen oxidation that will appear in the next step and are associated to fixed characteristic dimensions. <br />
Again, periodic boundary conditions are used in all three spatial directions.<br />
<br />
The initial velocity field prescribed at <math>t=0</math> is identical to the one used in Step 2 with the reference velocity set to <math>u_0=4\;\mathrm{m/s}</math>.<br />
<br />
In preparation for the next step, the central part of the box is initially filled with a <math>H_2/N_2</math> mixture (fuel region, molar fraction <math>X^0_\mathrm{H_2}=0.45</math>, <math>T=300 K</math>) while the remaining domain is filled with air (oxidizer region, molar fraction <math>X^0_\mathrm{O_2}=0.21</math>, <math>T=300 K</math>). <br />
The corresponding mass fractions are <math>Y^0_\mathrm{H_2} \approx 0.0556</math> and <math>Y^0_\mathrm{O_2} \approx 0.233</math>.<br />
<br />
To avoid any numerical instability, '''the steps at the interface between both regions are smoothed out using hyperbolic tangent functions''' as follows:<br />
<br />
<math>R_d(x) = |x-0.5\,L|,</math> (1)<br />
<br />
<math>\psi(x) = 0.5\left[1+\mathrm{tanh}\left(\frac{c\,(R_d(x)-R)}{R}\right)\right],</math> (2)<br />
<br />
<math>Y_\mathrm{H_2}(x) = Y^0_{\mathrm{H_2}}\left(1-\psi(x)\right),</math> (3)<br />
<br />
<math>Y_\mathrm{O_2}(x) = Y^0_{\mathrm{O_2}}\left(\psi(x)\right).</math> (4)<br />
<br />
where <math>R = \pi/4</math> mm and <math>c = 3</math> are the half-width of the central slab and stiffness parameter, respectively.<br />
As a consequence, there is initially a small region where both fuel and oxidizer coexist. <br />
Finally, a nitrogen complement is added everywhere using <math>Y_\mathrm{N_2}=1-Y_\mathrm{H_2}-Y_\mathrm{O_2}</math>.<br />
<br />
To also test the behaviour of the codes with respect to heat diffusion, '''a non-homogeneous temperature profile is finally imposed''' as follows:<br />
<br />
* Compute in each cell the equilibrium temperature for the local mixture for constant pressure and enthalpy by using a dedicated solver.<br />
<br />
* Enforce the resulting temperature profile, but keep the species profiles to their initial values (Eqs. (3) and (4)).<br />
<br />
It is found by using '''Cantera''' that the peak adiabatic temperature obtained '''at the fuel/air interface''' is <math>T_{ad}=1910.7~K</math>, leading to the profiles shown in the initial fields of [https://benchmark.coria-cfd.fr/images/4/40/Wmag_t0.png vorticity], [https://benchmark.coria-cfd.fr/images/6/6d/T_t0.png temperature], <br />
[https://benchmark.coria-cfd.fr/images/f/fa/H2_t0.png mass fraction of <math>H_2</math>], [https://benchmark.coria-cfd.fr/images/e/ef/O2_t0.png mass fraction of <math>O_2</math>] and the [https://benchmark.coria-cfd.fr/images/b/b4/Profiles_3d_zoom.pdf initial profiles of temperature and of mass fractions]. Users are encouraged to implement the exact initial profiles from the benchmark website to facilitate later comparisons. <br />
<br />
The resulting, initial configuration thus involves '''multiple species at different temperatures''' with steep profiles, just like in a real flame.<br />
However, for the present simulation, the reaction source terms are all still set to zero, as already mentioned: only '''convection''' and '''diffusion processes''' are considered.<br />
In order to enable benchmark computations also with '''DNS codes''' that do not provide advanced diffusion models, only constant values of the Lewis numbers are considered; for the same reason, thermodiffusion (Soret effect) is neglected on purpose (though it would be obviously relevant for such cases involving hydrogen as a fuel). <br />
The authors are fully aware that this is a crude approximation of reality<ref name="pecs"/>. But it must be kept in mind that the focus is set here on the detailed comparison between different codes and algorithms, and not on the resulting flow or flame structure.<br />
Hence, in this and in the next section, the values of the '''Lewis number''' for each species given in the table below shall be used to enable direct comparisons. <br />
These values have been estimated from a separate DNS for a similar configuration using the mixture-averaged diffusion model.<br />
The precise description of the associated thermodynamic state is available on the website.<br />
<br />
{| class="wikitable alternance center"<br />
|+ Approximate Lewis number for the species appearing in the kinetic scheme of<ref name="Boivin2011" />, to be enforced for Step 3 and Step 4 of the benchmark<br />
|-<br />
! scope="row" | Species<br />
| <math>H_2</math><br />
| <math>H</math><br />
| <math>O_2</math><br />
| <math>OH</math><br />
| <math>O</math><br />
| <math>H_2O</math><br />
| <math>HO_2</math><br />
| <math>H_2O_2</math><br />
| <math>N_2</math><br />
|-<br />
! scope="row" | Lewis number<br />
| 0.3290<br />
| 0.2228<br />
| 1.2703<br />
| 0.8279<br />
| 0.8128<br />
| 1.0741<br />
| 1.2582<br />
| 1.2665<br />
| 1.8268<br />
|}<br />
<br />
On the other hand, considering that implementing advanced models for fluid viscosity and thermal conductivity is not difficult and does not increase the computational time significantly, local values of these quantities depending on composition and temperature should be taken into account; <br />
More specifically, and to ease further comparisons, the same models for viscosity and conductivity have been retained in the three participating codes, i.e. the mixture averaged formalism that is used in both the '''Cantera''' and '''Chemkin packages'''.<br />
<br />
The viscosity is variable in the computational domain as well as with time and the '''Reynolds number''' of this configuration can only be estimated using the minimal value of viscosity (which is obtained in the air at 300 K, <math>\nu_\mathrm{min} \approx 1.56 \, 10^{-5} \mathrm{m^2/s}</math>), leading to <math>Re = {u_0 L_0}/{\nu_{min}}=267</math> which guarantees a laminar flow.<br />
<br />
Concerning resolution, it is also suggested to keep a conservative resolution in time for this benchmark, corresponding to <math>CFL \le 0.25</math> and <math>Fo \le 0.15</math>. <br />
In space, each direction should be resolved by approximately 256 grid points, leading to a spatial resolution <math>\Delta x \approx 25 \mu m</math>.<br />
<br />
The '''reference time scale''' is here <math>\tau_\mathrm{ref}=L_0/u_0 = 0.25 \; \mathrm{ms}</math>.<br />
The simulation should be performed for a physical time of at least <math>t=0.5 \; \mathrm{ms} = 2 \tau_\mathrm{ref}</math> as most of the results presented in this article are taken from this instant.<br />
<br />
[[File:wmag_t0.png|250px|alt text]]<br />
[[File:T_t0.png|250px|alt text]]<br />
[[File:H2_t0.png|250px|alt text]]<br />
[[File:O2_t0.png|250px|alt text]]<br />
<br />
Initial fields of vorticity magnitude, temperature, mass fractions of <math>\mathrm{H_2}</math>, and mass fraction of <math>\mathrm{O_2}</math> (from left to right), for the 3-D, non-reacting case ('''Step 3''').<br />
<br />
[[File:profiles_3d_zoom.pdf|250px|alt text]]<br />
<br />
Initial profiles of temperature and of mass fractions at <math>y=0.5 L</math> and <math>z=0.5 L</math> for the 3-D, non-reacting case ('''Step 3''').<br />
<br />
= Kinetics =<br />
<br />
The kinetic scheme of [https://doi.org/10.1016/j.proci.2010.05.002 Boivin et al.]<ref name="Boivin2011"/> which contains 9 species and 12 reactions has been used for this Benchmark.<br />
<br />
This mechanism is provided here in the Cantera format:<br />
* [[File:H2_williams_12.xml.zip | ctml ]]<br />
* [[File:H2_williams_12.cti | cti]]<br />
<br />
If you are using Chemkin, please use the following files:<br />
* [[File:H2_Boivin_transport.txt | transport]]<br />
* [[File:H2_Boivin_therm.txt | thermodynamic]]<br />
* [[File:H2_Boivin_mech.txt | mechanism]]<br />
<br />
= Transport and thermodynamic properties used for this Step =<br />
<br />
\cf{I think we all used the same transport (viscosity) and thermodynamic properties, but we should check again and compare that the implementations used in cantera/Chemkin provide the same values.}<br />
<br />
\gl{Good idea: we could use 3 or 4 relevant reference compositions to quantify the discrepancies between Cantera and Chemkin (on rho, Cp, conductivity, viscosity for example) and also put these on the website.}<br />
<br />
'''Put the transport data here'''<br />
<br />
= Aside suggestions =<br />
<br />
TBD<br />
<br />
= References =<br />
<br />
<references><br />
<ref name="Boivin2011"><br />
<bibtex><br />
@article{Boivin2011, <br />
author= {P. Boivin, C. Jiménez, A.L. Sanchez, and F.A. Williams},<br />
title= {An explicit reduced mechanism for <math>H_2</math>-air combustion.},<br />
journal={Proc. Combust. Inst.},<br />
year= {2011},<br />
volume={33},<br />
pages={517--523},<br />
doi= {https://doi.org/10.1016/j.proci.2010.05.002}<br />
}<br />
</bibtex><br />
</ref><br />
<ref name="pecs"><br />
<bibtex><br />
@article{Pecs, <br />
author= {R. Hilbert, F. Tap, H. El-Rabii, and D. Thévenin},<br />
title= {Impact of detailed chemistry and transport models on turbulent combustion simulations},<br />
journal={Prog. Energ. Combust. Sci.},<br />
year= {2004},<br />
volume={30},<br />
pages={61--117},<br />
}<br />
</bibtex><br />
</ref><br />
</references></div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=File:H2_Boivin_mech.txt&diff=546File:H2 Boivin mech.txt2024-02-05T14:19:52Z<p>Lartigue: Kinetic constants for Boivin et al. kinetic scheme for H2/Air combustion in Chemkin II format.</p>
<hr />
<div>Kinetic constants for Boivin et al. kinetic scheme for H2/Air combustion in Chemkin II format.</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=File:H2_Boivin_therm.txt&diff=545File:H2 Boivin therm.txt2024-02-05T14:19:33Z<p>Lartigue: Thermodynamic properties for Boivin et al. kinetic scheme for H2/Air combustion in Chemkin II format.</p>
<hr />
<div>Thermodynamic properties for Boivin et al. kinetic scheme for H2/Air combustion in Chemkin II format.</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=File:H2_Boivin_transport.txt&diff=544File:H2 Boivin transport.txt2024-02-05T14:19:10Z<p>Lartigue: Transport properties for Boivin et al. kinetic scheme for H2/Air combustion in Chemkin II format.</p>
<hr />
<div>Transport properties for Boivin et al. kinetic scheme for H2/Air combustion in Chemkin II format.</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Main_Page&diff=537Main Page2021-05-16T14:08:46Z<p>Lartigue: </p>
<hr />
<div>{{#customtitle:TGV Benchmark|The Taylor-Green Vortex as a Benchmark - benchmark.coria-cfd.fr}}<br />
<br />
= The Taylor-Green Vortex as a Benchmark for High-Fidelity Combustion Simulations Using Low-Mach Solvers =<br />
<br />
The present web site is a complement of an article that has been accepted in '''Computers and Fluids''' in June 2021: [https://www.sciencedirect.com/science/article/abs/pii/S0045793021001018].<br />
<br />
Verification and validation are crucial steps for the development of any numerical model.<br />
<br />
While suitable processes have been established for commercial Computational Fluid Dynamics (CFD) codes, more difficult challenges must be faced for high-fidelity solvers.<br />
<br />
Benchmarks have been proposed in a series of dedicated conferences for non-reacting configurations.<br />
However, to our knowledge, no suitable approach has been published up to now regarding turbulent reacting flows.<br />
<br />
'''The purpose of this website is to present a full verification and validation chain for high-resolution codes employed to simulate turbulent reacting flows, first for Direct Numerical Simulation (DNS) of turbulent combustion in the limit of low Mach numbers.'''<br />
<br />
The selected configuration builds on top of the Taylor-Green vortex.<br />
Verification takes place by comparison with the analytical solution in two dimensions.<br />
Validation of the single-component flow is ensured by comparisons with published results obtained with a spectral code.<br />
Mixing without reaction is then considered, before computing finally a hydrogen-oxygen flame interacting with a 3-D Taylor-Green vortex. <br />
Three different low-Mach DNS solvers have been used for this study, demonstrating that the final accuracy of the DNS simulations is of the order of 1% for all quantities considered.<br />
<br />
The website is organised in four major parts:<br />
* A presentation of the three [[codes]] used for the Benchmark<br />
* The [[description]] of the test cases<br />
* The [[analysis]] of the results<br />
* An attempt to give a few guidelines on the [[performances]] that could be expected on the 3D test-cases.<br />
<br />
All the raw [[results]] of the 3 codes are available online.</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Performances&diff=533Performances2021-04-01T15:20:07Z<p>Lartigue: /* Methodology */</p>
<hr />
<div>The results presented above were obtained on different architectures, preventing the direct comparison of the performance of the codes.<br />
However, the TauBench pseudo-benchmark~\cite{TauBench,TauBench2} made it possible to assess in a separate test the raw performance of each machine.<br />
The following sections will describe the configuration of the three high performance <br />
computing (HPC) systems that were used to run YALES2, DINO and Nek5000 for the TGV benchmarks. <br />
Subsequently, the TauBench methodology and its results on the target computers will be presented.<br />
Finally, the performance analysis for the 3-D TGV cases is discussed in the last subsection.<br />
<br />
= Presentation of the machines used for the benchmark = <br />
<br />
== Irene Joliot-Curie from TGCC ==<br />
<br />
The YALES2 results were obtained on the Irene Joliot-Curie machine~\cite{irene} operated by TGCC/CEA for GENCI (French National Agency for Supercomputing).<br />
%This machine built by Atos was introduced in Sept. 2018 and is ranked at the 61th position of June 2020 Top500 ranking~\cite{ireneTop500}.<br />
It is composed of 1,656 compute nodes, with 192 GB memory on each node and two sockets of Intel Skylake (Xeon Platinum 8168) with 24 cores each, operating at 2.7 GHz.<br />
Each core can deliver a theoretical peak performance of 86.4 GFlop/s with AVX-512 and FMA activated.<br />
The interconnect is an Infiniband EDR, and the theoretical maximum bandwidth is 128 GB/s per socket and thus 5.33 GB/s by core when all cores are active.<br />
The software stack consists of the Intel Compiler 19 and OpenMPI 2.0.4, both used for all results obtained in this benchmark.<br />
<br />
== SuperMUC-NG from LRZ ==<br />
<br />
All test cases with DINO were simulated on the SuperMUC machine, hosted at the Leibniz Supercomputing Center (LRZ) in Munich, Germany~\cite{supermuc}. <br />
In the course of the project, results have been obtained on three versions of SuperMUC (Phase I, Phase II, and NG), but <br />
only the performance on the most recent system, SuperMUC-NG, will be discussed here. <br />
%This machine was assembled by Lenovo and ranks at the 13th position at June 2020 Top500~\cite{supermucTop500}.<br />
SuperMUC-NG is a combination of 6,336 compute nodes built from bi-socket Intel Skylake Xeon Platinum Processor 8174 with 96 GB of memory and 24 cores. <br />
The theoretical peak performance of a single core is 99.2 GFlop/s with AVX512 and FMA activated at a sustained frequency of 3.1 GHz. <br />
The nodes are interconnected with Intel OmniPath interconnect network. All tests presented here were obtained with the Intel 19 compiler and Intel MPI.<br />
<br />
== Piz Daint from CSCS ==<br />
<br />
The Nek5000 simulations were performed on the XC40 partition of the Piz Daint machine at CSCS in Switzerland~\cite{pizdaint}.<br />
%This machine ranks at the 231th position of the June 2020 Top500 ranking~\cite{pizdaintTop500} and was manufactured by HPE and Cray.<br />
It is composed of 1,813 compute nodes, each containing 64~GB of RAM and two sockets using the Intel Xeon E5-2695~v4 processors (18 cores at 2.1 GHz by socket).<br />
The theoretical peak performance of a single core is 33.6~GFlop/s with AVX2 and FMA activated.<br />
The interconnect is based on the Aries routing and communications ASIC and a Dragonfly <br />
network topology, and the maximum achievable bandwidth (BW) with a single socket is 76.8 GB/s, <br />
which allows 4.27~GB/s transfer to each core in a fully occupied socket.<br />
All tests presented here have been obtained with the Intel 18 compiler.<br />
<br />
<br />
= TauBench performance benchmark = <br />
<br />
The scalable benchmark TauBench emulates the run time behavior of the compressible TAU flow solver~\cite{dlr22421} with respect to the memory footprint and floating-point performance. <br />
Since the TAU solver relies on unstructured grids, its most important property is that all access points to the grid are indirect, as in most modern CFD solvers.<br />
TauBench can therefore be used to estimate the performance of a generic flow solver with respect to machine properties, like memory bandwidth or cache miss/latencies.<br />
It is used to provide a reference measurement of a system on a workload that is more representative of usual CFD codes than the widely used LINPACK benchmark~\cite{linpack}, which is mostly CPU-bound.<br />
Even though some important effects are neglected by using a single-core benchmark (like MPI communications or memory bandwidth saturation that appear on fully-filled nodes), TauBench is still a good indicator of the relative performance of each architecture.<br />
<br />
<br />
{| class="wikitable alternance center"<br />
|+ Single-core performance obtained by TauBench for the three HPC systems employed in the benchmarks<br />
|-<br />
! scope="col" | Machine<br />
! scope="col" | Irene Joliot-Curie<br />
! scope="col" | SuperMuc-NG<br />
! scope="col" | Pitz Daint<br />
|-<br />
! scope="row" | Frequency [GHz]<br />
| 2.7<br />
| 3.1<br />
| 2.1<br />
|-<br />
! scope="row" | Single core TauBench [GFlop/s]<br />
| 2.97<br />
| 3.34<br />
| 4.13<br />
|-<br />
! scope="row" | Single core Peak [GFlop/s]<br />
| 86.4<br />
| 99.2<br />
| 33.6<br />
|-<br />
! scope="row" | Single core BW [GB/s]<br />
| 21.3<br />
| 21.3<br />
| 19.2<br />
|}<br />
<br />
This table presents some important data of the three considered HPC systems: the processor frequency, the result ofTauBench, the theoretical peak performance and the memory bandwidth. The last three results are given for a singlecore. The most striking – though not unexpected – conclusion is that TauBench provides results that are much lowerthan the theoretical peak performance (3% of the peak for Irene Joliot-Curie and for SuperMUC-NG, and 12% forPitz Daint). This is in agreement with the results from the similar and well-known benchmark HPCG [86] that is alsointended as a complement to the High Performance LINPACK (HPL) benchmark, currently used to rank the Top500computing systems. For example, the HPCG benchmark on the Irene Joliot-Curie revealed a peak performanceof 0.66 GFlop/s per core [87], which is approximately only 0.8% of the peak performance. The main reason forthe discrepancy between the LINPACK and the HPCG benchmark is that the first one is purely CPU-bound whileHPCG is mostly limited by the memory bandwidth and cache-miss effects. TauBench is somewhere in between andis probably a good estimate for many CFD codes.Indeed, the theoretical peak performance can only be reached when performing 2 fully-vectorized Fused Multiply-Add (FMA) instructions per cycle. This situation is never achieved in any CFD code. Most of them are usuallylimited to issuing only one non-vectorized non-FMA instruction per cycle; in this situation, the peak performance (inGFlop/s) is simply equal to the processor frequency (in GHz). This can be observed very clearly on SuperMUC andIrene Joliot-Curie. Regarding the Pitz Daint machine, it appears that the TauBench result is actually much betterthere than on the two other machines when compared to theoretical peak performance; this is somewhat unexpectedand might be due to the use of TurboBoost on this machine, or to a better memory/cache performance.It should be pointed out that no effort was made to find the best parameters (tolerances, timestep, etc.) tominimize the computational cost, and the following results should only be considered as indicative of the time-to-solution of the three codes.<br />
<br />
= Methodology =<br />
Several metrics will be used in the following sections to characterize the codes. They are introduced by givingboth a formal definition as well as a few complementary explanations. In order to avoid any confusion between CPUtime and Simulated time, the corresponding data will be indexed by <sub>CPU</sub> and <sub>Sim</sub>, respectively.<br />
<br />
First, the Wall-Clock Time (WCT) is the elapsed time to perform a simulation on a given number of cores Ncores.The product TCPU = Ncores×WCT is thus the total CPU time for the simulation.A more meaningful metric is the so-called Reduced Computational Time (RCT), which is computed as:<br />
<br />
<math><br />
RCT = \frac{TCPU}{Nit \times N_p}<br />
</math></div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Main_Page&diff=482Main Page2020-09-09T15:22:52Z<p>Lartigue: </p>
<hr />
<div>{{#customtitle:TGV Benchmark|The Taylor-Green Vortex as a Benchmark - benchmark.coria-cfd.fr}}<br />
<br />
= The Taylor-Green Vortex as a Benchmark for High-Fidelity Combustion Simulations Using Low-Mach Solvers =<br />
<br />
The present web site is a complement of an article that has been submitted to '''Computers and Fluids''' in September 2020.<br />
<br />
Verification and validation are crucial steps for the development of any numerical model.<br />
<br />
While suitable processes have been established for commercial Computational Fluid Dynamics (CFD) codes, more difficult challenges must be faced for high-fidelity solvers.<br />
<br />
Benchmarks have been proposed in a series of dedicated conferences for non-reacting configurations.<br />
However, to our knowledge, no suitable approach has been published up to now regarding turbulent reacting flows.<br />
<br />
'''The purpose of this website is to present a full verification and validation chain for high-resolution codes employed to simulate turbulent reacting flows, first for Direct Numerical Simulation (DNS) of turbulent combustion in the limit of low Mach numbers.'''<br />
<br />
The selected configuration builds on top of the Taylor-Green vortex.<br />
Verification takes place by comparison with the analytical solution in two dimensions.<br />
Validation of the single-component flow is ensured by comparisons with published results obtained with a spectral code.<br />
Mixing without reaction is then considered, before computing finally a hydrogen-oxygen flame interacting with a 3-D Taylor-Green vortex. <br />
Three different low-Mach DNS solvers have been used for this study, demonstrating that the final accuracy of the DNS simulations is of the order of 1% for all quantities considered.<br />
<br />
The website is organised in four major parts:<br />
* A presentation of the three [[codes]] used for the Benchmark<br />
* The [[description]] of the test cases<br />
* The [[analysis]] of the results<br />
* An attempt to give a few guidelines on the [[performances]] that could be expected on the 3D test-cases.<br />
<br />
All the raw [[results]] of the 3 codes are available online.</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Codes&diff=478Codes2020-08-26T13:39:28Z<p>Lartigue: /* YALES2 */</p>
<hr />
<div>= Code Description =<br />
<br />
This page is dedicated to a short presentation of the three '''low-Mach number''' codes used for the rest of this benchmarl: [https://www.coria-cfd.fr/index.php/YALES2 YALES2], [http://www.lss.ovgu.de/lss/en/Research/Computational+Fluid+Dynamics.html DINO] and [http://nek5000.mcs.anl.gov Nek5000]. <br />
<br />
Since these codes have already been the subject of many publications and are not completely new, '''only the most relevant features are discussed in what follows''', with suitable references for those readers needing more details.<br />
<br />
== YALES2 ==<br />
<br />
YALES2 is a '''massively parallel multiphysics platform'''<ref name="moureau2011"/> developed since 2009 by Moureau, Lartigue and co-workers at CORIA (Rouen, France).<br />
It is dedicated to the high-fidelity simulation of '''low-Mach number flows in complex geometries'''.<br />
It is based on the '''Finite Volumes formulation''' of the '''Navier-Stokes equations''' and it can solve both '''non-reacting''' and '''reacting flows'''.<br />
It can actually solve various physical problems thanks to its original structure which is composed of a main numerical library accompanied with tens of dedicated solvers (for acoustics, multiphase flows, heat transfer, radiation\ldots) which can be coupled with one another.<br />
YALES2 relies on '''unstructured meshes''' and a '''fully parallel dynamic mesh adaptation technique''' to improve the resolution in physically-relevant zones to mitigate the computational cost<ref name="benard2015"/>. <br />
As a result it can easily handle meshes composed of billions of tetrahedra, thus enabling the '''Direct Numerical Simulation''' of laboratory and semi-industrial configurations.<br />
It is now composed of nearly 500,000 lines of object-oriented Fortran and the parallelism is currently ensured by a pure '''MPI''' paradigm, although a hybrid '''OpenMP/MPI''' as well as a GPU version are under development.<br />
<br />
As most low-Mach number codes, the time-advancement is based on a '''projection-correction method''' following the pioneering work of<ref name="chorin1968"/>.<br />
The prediction step uses a method which is a blend between a '''4th-order Runge-Kutta method''' and a '''4th-order Lax-Wendroff-like method'''<ref name="kraushaar2011"/>, combined with a '''4th-order node-based centered finite-volume discretization''' of the convective and diffusive terms.<br />
Moreover, to improve the performance of the correction step, the pressure of the previous iteration is included in the prediction step to limit the '''splitting errors'''.<br />
The correction step is required to ensure '''mass conservation''', using a pressure that arises from a '''Poisson equation'''.<br />
This is performed numerically by solving a linear system on the pressure at each node of the mesh thanks to a dedicated in-house version of the deflated conjugate gradient algorithm, which has been optimized for solving elliptic equations on massively parallel machines<ref name="malandain2013"/>.<br />
<br />
When considering '''multispecies''' and '''non-isothermal flows''', the extension of the classical projection method proposed by Pierce et al.<ref name="pierce2004"/> is used to account for '''variable-density flows'''.<br />
All the thermodynamic and transport properties are provided by the '''Cantera software'''<ref name="goodwin2003"/>, which has been fully re-implemented in '''Fortran''' to avoid any performance issues.<br />
Each species thermodynamic property is specified by '''5th-order polynomials''' on two temperature ranges (below and above 1000 K).<br />
The mixture is supposed to be both thermally and mechanically perfect.<br />
<br />
Regarding transport properties, several type of models are implemented in YALES2: <br />
<br />
1) the default approach is based on the computation of transport properties for each species (using tabulated molecular potentials), then combining those to obtain mixture-averaged coefficients for '''viscosity''', '''conductivity''' and '''diffusion velocities''' (Hirschfelder and Curtiss approximation<ref name="hirschfelder1964"/>; <br />
<br />
2) alternatively, simplified laws (for example the Sutherland law<ref name="sutherland1893"/> can be used for the '''viscosity''', while fixed values for '''Prandtl''' and '''Schmidt numbers''' allow the computation of '''conductivity''' and '''diffusion coefficients'''; <br />
<br />
3) a mix of both approaches, for example computing '''viscosity''' and '''conductivity''' with a mixture-averaged approximation and then imposing a '''Lewis number''' for each species.<br />
This last approach has been retained in the present benchmark.<br />
<br />
Finally, the source terms used in the '''reacting simulations''' are modeled with an '''Arrhenius law''' with the necessary modifications needed to take into account three-body or pressure-dependent reactions.<br />
<br />
From a numerical point of view, it must be noticed that both the '''diffusion''' and '''reaction processes''' occur at time scales which can be orders of magnitude smaller than the convective time scale.<br />
Solving these phenomena with explicit methods would thus drastically limit the '''global timestep''' of the whole simulation and induce an overwhelming CPU cost.<br />
To mitigate these effects when dealing with '''multi-species reacting flows''', the classical operator '''splitting technique''' is used.<br />
The '''diffusion process''' is solved with a fractional timestep method inside each convective iteration, each substep being limited by a '''Fourier condition''' to ensure stability. <br />
This method gets activated only when the mesh is very fine (typically when performing DNS with very diffusive species like <math>H_2</math> or <math>H</math>, as in the present project); otherwise an explicit treatment is sufficient.<br />
The chemical source terms are then integrated with a dedicated stiff solver, namely the '''CVODE''' library from SUNDIALS<ref name="cohen1996cvode"/><ref name="cvodeUG">A.C. Hindmarsh and R. Serban. User documentation for '''CVODE'''. [https://computing.llnl.gov/sites/default/files/public/cv_guide.pdf].</ref><ref name="sundials"/>. <br />
To that purpose, each control volume is considered as an isolated reactor with both constant pressure and enthalpy.<br />
<br />
Using the stiff integration technique results in a very strong load imbalance between the various regions of the flow: the fresh gases are solved in a very small number of integration steps while the inner flame region may require tens or even hundreds of integration steps. <br />
To overcome this difficulty, a dedicated '''MPI''' dynamic scheduler based on a work-sharing algorithm ensures global load-balancing.<br />
<br />
== DINO ==<br />
<br />
DINO is a '''3-D DNS code''' used for '''incompressible''' or '''low-Mach number flows''', the latter approach being used in this project. <br />
The development of DINO started in the group of D. Thévenin (Univ. of Magdeburg) in 2013. <br />
DINO is a '''Fortran-90 code''', written on top of a 2-D pencil decomposition to enable efficient large-scale parallel simulations on distributed-memory supercomputers by coupling with the open-source library 2-DECOMP&FFT<ref name="li2010"/>. <br />
The code offers different features and algorithms in order to investigate different physicochemical processes. <br />
Spatial derivative are computed by default using '''sixth-order central finite differences'''. <br />
Time integration relies on '''several Runge-Kutta solvers'''. <br />
In what follows, an explicit '''4th-order Runge-Kutta approach''' has been used. <br />
A '''3rd-order semi-implicit Runge-Kutta integration''' can be activated as needed, when considering stiff chemistry. <br />
In this case, non-stiff terms are still computed with explicit Runge-Kutta, while the '''PyJac package'''<ref>Create analytical jacobian matrix source code for chemical kinetics.</ref> is used to integrate in an implicit manner all chemistry terms with an analytical Jacobian computation. <br />
All thermodynamic, chemical and transport properties are computed using the open-source library '''Cantera 2.4.0'''<ref>Cantera.</ref>. <br />
<br />
The transport properties can be computed based on three different models: <br />
<br />
1) Constant Lewis numbers, <br />
<br />
2) mixture-averaged, <br />
<br />
3) full multicomponent diffusion, by coupling either again with '''Cantera''' or with the open-source library EGlib<ref name="ern1995"/>. <br />
<br />
The '''Poisson equation''' is solved using Fast Fourier Transform ('''FFT''') for '''periodic''' as well as for '''non-periodic boundary conditions''', relying in the latter case on an in-house pre- and post-processing technique. <br />
The I/O operations are implemented using two different approaches: (1) binary '''MPI-I/O''' using 2-DECOMP&FFT for check-points and restart files; (2) HDF5 files used for analysis and visualization.<br />
<br />
'''Multi-phase flows''' can be simulated in DINO using resolved or non-resolved (point) particles and droplets using a '''Lagrangian approach'''<ref name="abdelsamie2019"/><ref name="abdelsamie2020"/>. Complex boundaries are represented by a novel second-order immersed boundary method implementation ('''IBM''') based on a directional extrapolation scheme<ref name="chi2020"/>. <br />
More details about the implemented algorithms can be found in particular in<ref name="abdelsamie2016"/>. <br />
Since DINO has been developed as a multi-purpose code for analyzing many different '''reacting''' and '''non-reacting flows'''<ref name="chi2018"/><ref name="oster2018"/><ref name="abdelsamie2019-2"/>, a detailed verification and validation is obviously essential.<br />
<br />
== Nek5000 ==<br />
<br />
This '''spectral element low-Mach number reacting flow solver''' is based on the highly-efficient open-source solver Nek5000<ref>Nek5000 version v17.0, Argonne National Laboratory, IL, U.S.A.</ref> extended by a plugin developed at ETH implementing a '''high-order splitting scheme''' for '''low-Mach number reacting flows'''<ref name="tomboulides1997"/>. <br />
The spectral element method ('''SEM''') is a '''high-order weighted residual technique''' for spatial discretization that combines the accuracy of spectral methods with the geometric flexibility of the finite element method allowing for accurately representation of complex geometries<ref name="deville2002"/>.<br />
The computational domain is decomposed into <math>E</math> conforming elements, which are '''quadrilaterals''' ('''in 2-D''') or '''hexahedra''' (in '''3-D''') that conform to the domain boundaries. <br />
Within each element, functions are expanded as <math>N</math>th-order polynomials so that resolution can be increased either by decreasing the element size (<math>h</math>-type refinement) or by increasing the polynomial order (<math>p</math>-type refinement; typically <math>N = 7 - 15</math>). <br />
The grids can be unstructured and allow for '''static local refinement''', while '''adaptive mesh refinement''' has been recently developed<ref name="tanarro2020"/>. <br />
By casting the '''polynomial approximation''' in tensor-product form, the differential operators on <math>N^3</math> gridpoints per element can be evaluated with only <math>O(N^4)</math> work and <math>O(N^3)</math> storage.<br />
<br />
The principal advantage of the '''SEM''' is that convergence is exponential in <math>N</math>, yielding minimal '''numerical dispersion''' and '''dissipation''', so that significantly fewer grid points per wavelength are required in order to accurately propagate a turbulent structure over the extended time required in high Reynolds number simulations. <br />
Nek5000 uses locally '''structured basis coefficients''' (<math>N\times N \times N</math> arrays), which allow direct addressing and tensor-product-based derivative evaluation that can be cast as efficient matrix-matrix products involving <math>N^2</math> operators applied to <math>N^3</math> data values for each element. <br />
As a result, data movement per grid point is the same as for '''low-order methods'''. <br />
It uses scalable domain-decomposition-based iterative solvers with efficient preconditioners. <br />
Communication is based on the Message Passing Interface ('''MPI''') standard, and the code has proven scalability to over one million ranks. <br />
Nek5000 provides balanced I/O latency among all processors and reduces the overhead or even completely hides the I/O latency by using dedicated I/O communicators in the optimal case.<br />
<br />
'''Time advancement''' is performed using the '''splitting scheme''' proposed in<ref name="tomboulides1997" /> to decouple the highly non-linear and stiff thermochemistry (species and energy governing equations) from the hydrodynamic system (continuity and momentum).<br />
Species and energy equations are integrated without further '''splitting using the implicit stiff integrator solver CVODE from the SUNDIALS package'''<ref name="cohen1996cvode" /><ref name="cvodeUG" /><ref name="sundials" /> that uses backward differentiation formulas (BDF).<br />
The '''continuity''' and '''momentum equations''' are integrated using a '''second-''' or '''third-order semi-implicit formulation''' (EXT/BDF) treating the non-linear advection term explicitly<ref name="deville2002" />.<br />
The thermodynamic properties, detailed chemistry, and transport properties are provided by optimized subroutines compatible with '''Chemkin'''<ref name="chemkin"/>.<br />
<br />
The reacting flow solver can handle complex time-varying geometries and has been used for instance to simulate laboratory-scale internal combustion engines<ref name="MS"/><ref name="TCC"/>. <br />
It can account for conjugate fluid-solid heat transfer and detailed gas phase as well as surface kinetics<ref name="catalytic"/>.<br />
<br />
== General comments on the codes ==<br />
<br />
The three '''aforementioned solvers''' are all '''unsteady''', '''high-fidelity codes''' based on the '''low-Mach number''' formulation of the '''Navier-Stokes equations'''.<br />
They can perform both '''Direct Numerical Simulations''' or '''Large Eddy Simulations''' of reacting flows, only '''DNS''' being considered here.<br />
However, they differ in a certain number of points, mainly from the numerical point of view.<br />
The aim of this section is to emphasize those major differences.<br />
First, '''YALES2''' is an '''unstructured code''', designed to handle any type of elements; its main application field pertains to '''LES''' of industrially relevant flows, though '''DNS''' is possible as well. <br />
On the other hand, both '''DINO''' and '''Nek5000''' are mostly dedicated to '''DNS''' of configurations found in fundamental research.<br />
Both '''DINO''' and '''Nek5000''' can only deal with '''quads''' or '''hexas''', with the major difference that '''DINO''' is based on a '''structured connectivity''' while '''Nek5000''' can use '''unstructured meshes''' (pavings).<br />
As a consequence, both '''DINO''' and '''Nek5000''' employ '''higher-order numerical schemes''' compared to '''YALES2''', limited at best to a '''4th-order scheme'''.<br />
All codes rely on dedicated libraries to compute the thermo-chemical properties of the flow, either '''Chemkin''', '''Cantera''', or in-house versions of those.<br />
Moreover, they also rely at least to some extent on external software to perform the temporal integration of the stiff chemical source terms.<br />
Regarding the Poisson equation for pressure which must be solved by all codes, '''DINO''' relies on a spectral formulation by performing direct and inverse Fourier transforms, which is possible thanks to its structured mesh.<br />
On the other hand, both '''YALES2''' and '''Nek5000''' use an iterative solver with an efficient preconditioning technique; this method is more versatile and should be computationally more efficient for large and complex geometrical configurations.<br />
'''Nek5000''' employs CVODE to integrate the thermochemical equations without further '''splitting''' of the different terms accounting for convection, diffusion and chemistry.<br />
The main differences between the three codes are summarized in the table below. <br />
Please note that the presented values are those used for the benchmark, even though some other options are available in each codes.<br />
<br />
{| class="wikitable alternance center"<br />
|+ Table 1: Major numerical properties of the three high-fidelity codes as used in this benchmark<br />
|-<br />
! scope="col" | Code<br />
! scope="col" | YALES2<br />
! scope="col" | DINO<br />
! scope="col" | Nek5000<br />
|-<br />
| Connectivity<br />
| Unstructured<br />
| Structured<br />
| Unstructured<br />
|-<br />
| Discretization Type<br />
| Finite Volumes<br />
| Finite Differences<br />
| Spectral Elements<br />
|-<br />
| Grid point distribution<br />
| Regular hexahedra<br />
| Regular hexahedra<br />
| Regular hexahedra with GLL points<br />
|-<br />
| Spatial order<br />
| 4th<br />
| 6th<br />
| 7th - 15th (typically)<br />
|-<br />
| Temporal method<br />
| expl. RK4<br />
| expl. RK4 / semi-impl. RK3<br />
| semi-impl. BDF3<br />
|-<br />
| Pressure solver<br />
| CG with Deflation MPrec.<br />
| FFT-based<br />
| CG/GMRES with Jacobi/Schwartz Prec.<br />
|-<br />
| Thermo-chemistry<br />
| Cantera (re-coded)<br />
| Cantera<br />
| Chemkin interface<br />
|-<br />
| Chemistry integration<br />
| CVODE<br />
| PyJac<br />
| CVODE<br />
|-<br />
| Operator splitting<br />
| Yes<br />
| No<br />
| No<br />
|-<br />
| Parallel paradigm<br />
| MPI<br />
| MPI<br />
| MPI<br />
|}<br />
<br />
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<bibtex><br />
@article{Abdelsamie2016, <br />
author= {B.O. Arani, C.E. Frouzakis, J. Mantzaras, and K. BoulouchosA. Abdelsamie, G. Fru, F. Dietzsch, G. Janiga, and D. Thévenin},<br />
title= {Towards direct numerical simulations of lowMach number turbulent reacting and two-phase flows using immersed boundaries},<br />
journal={Comput. Fluids},<br />
year= {2016},<br />
volume={131},<br />
number={5},<br />
pages={123--141},<br />
}<br />
</bibtex><br />
</ref><br />
</references></div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Analysis_of_Step_2&diff=476Analysis of Step 22020-08-26T13:32:10Z<p>Lartigue: </p>
<hr />
<div>The second step concerns the '''3-D''', '''non-reacting cold flow''', used as validation by comparison with the published<br />
results of a pseudo-spectral solver <ref name="vanrees2011"/>. The main quantities of interest for this comparison are:<br />
<br />
1. Velocity profiles along centerlines of the domain at <math>t = 12.11 \tau_{ref}</math>, as illustrated in Fig. 5.<br />
2. The evolution of kinetic energy (<math>k(t) = \frac{1}{2} <u_i u_i></math>) and of its dissipation rate (<math>\epsilon (t) = −dk/dt = 2 \nu <S_{ij}S_{ij}></math>)<br />
versus time, as shown in Fig. 6, where <math>S_{ij}</math> is the symmetric strain-rate tensor,<br />
<br />
<math>S_{ij} = \frac{1}{2}(\frac {\partial u_i}{\partial x_j} + \frac {\delta u_j}{\delta x_i})</math> ; (11)<br />
<br />
Comparing the '''velocity fields''' at <math>t = 12.11 \tau_{ref}</math> is done on purpose. As known from the '''TGV''' literature, this instant<br />
corresponds to a complex pseudo-turbulent field, before further turbulence decay due to dissipation (see also Figure<br />
9 in <ref name="abelsamie2016"/>). Getting the correct '''velocity field''' in these conditions is challenging, since the obtained results are very<br />
sensitive with regard to the employed algorithms and discretization.<br />
Unlike the '''2-D''' situation, no analytical solution is available there, and the results can only be compared to<br />
other numerical simulations. In the present case, the reference data are taken from a simulation relying on the<br />
pseudo-spectral code RLPK using <math>512^3</math> grid points <ref name="vanrees2011"/>.<br />
<br />
[[File:vx_3d_incomp.png|250px]]<br />
[[File:vy_3d_incomp.png|250px]]<br />
<br />
Velocities along the centerlines of the domain for cold 3-D TGV case ('''Step 2''') at <math>t = 12.11 \tau_{ref}</math>.<br />
<br />
First, it appears on Fig. 5 that no differences can be identified visually from the '''velocity fields''' along the '''centerlines'''<br />
at <math>t = 12.11 \tau_{ref}</math>.<br />
Looking at Figs. 6 it can be observed that the three codes are able to reproduce the evolution of turbulence kinetic<br />
energy without any visible differences, whereas for the dissipation rate minute deviations appear at two instants (see<br />
enlargements in Fig. 6, right): (1) shortly after transition (<math>11 < t/\tau_{ref} < 13.5</math>) for '''YALES2''', and (2) just before flow<br />
relaminarization (<math>16.25 < t/\tau_{ref} ≤ 20</math>) for both '''DINO''' and '''YALES2'''. The results of '''RLPK''' and of '''Nek5000''' coincide<br />
visually at all times.<br />
These two time-instants are very sensitive moments at which the accuracy of the numerical methods and the<br />
resolution in time and space appear to play a major role. These small discrepancies are regarded as minor and<br />
considered acceptable with respect to the validation process of the codes. '''It must also be kept in mind that the data used as a reference have been obtained with a resolution of <math>512^3</math> grid points with a pseudo-spectral solver'''.<br />
<br />
[[File:ke_3d_incomp.png|250px]]<br />
[[File:eps_3d_incomp.png|250px]]<br />
<br />
Here is the evolution with time of turbulent kinetic energy k (left, written KE in vertical axis) and of its dissipation rate <math>\epsilon</math> (right), by comparison with published reference data from <ref name="vanrees2011"/>.<br />
<br />
= References =<br />
<br />
<references><br />
<ref name="vanrees2011"><br />
<bibtex><br />
@article{Vanrees2011, <br />
author= {W.M. van Rees, A. Leonard, D.I. Pullin, and P. Koumoutsakos},<br />
title= {A comparison of vortex and pseudo-spectral<br />
methods for the simulation of periodic vortical flows at high Reynolds numbers.},<br />
journal={J. Comput. Phys.},<br />
year= {2011},<br />
volume={230},<br />
pages={2794--2805},<br />
}<br />
</bibtex><br />
</ref><br />
<ref name="abelsamie2016"><br />
<bibtex><br />
@article{Abdelsamie2016, <br />
author= {A. Abdelsamie and G. Fru and F. Dietzsch and G. Janiga and D. Thévenin},<br />
title= {Towards direct numerical simulations of low-Mach number turbulent reacting and two-phase flows using immersed boundaries},<br />
journal={Comput. Fluids},<br />
year= {2016},<br />
volume={131},<br />
number={5},<br />
pages={123--141},<br />
}<br />
</bibtex><br />
</ref><br />
</references></div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Analysis_of_Step_1&diff=475Analysis of Step 12020-08-26T13:30:32Z<p>Lartigue: </p>
<hr />
<div>The verification involves a direct comparison with the analytical solution. For this purpose, analytic fields for<br />
'''velocity''' (x- and y-components) and '''vorticity''' <math>(\partial_x v - \partial_y u)</math> at <math>t=10 \tau_{ref}</math> are presented in Fig. 3. It should be noted<br />
that both '''YALES2''' and '''DINO''' used 642 '''grid points''' for this test case, while '''Nek5000''' employed 82<br />
'''spectral elements of order 8''', which results in 64 '''discretization points''' in each direction. The velocity profiles along both centerlines of<br />
the domain at <math>t = 10 \tau_{ref}</math> are shown in Fig. 4. It can be observed that the three codes give perfect visual agreement<br />
14<br />
with the '''analytical solution'''. Table 3 present the analytical maximal velocity at <math>t = 10 \tau_{ref}</math> (as computed from Eq. 2)<br />
and the values obtained with the three codes, as well as the associated relative error: it is observed that the maximal<br />
deviation is less than <math>0.03%</math> for the three codes.<br />
<br />
{| class="wikitable alternance center"<br />
|+ Comparison of the peak velocity at <math>t = 10 \tau_{ref}</math> for Step 1 (verification)<br />
|-<br />
|<br />
! scope="col" | Analytical<br />
! scope="col" | YALES2<br />
! scope="col" | DINO<br />
! scope="col" | Nek5000<br />
|-<br />
! scope="row" | <math>V_{max}</math><br />
| 0.987271<br />
| 0.987583<br />
| 0.987565<br />
| ????<br />
|-<br />
! scope="row" | <math>\epsilon_{rel}</math><br />
| 0 [Ref]<br />
| 0.03%<br />
| 0.03%<br />
| -<br />
|}<br />
<br />
This configuration, although quite far from any realistic flame, is nevertheless an excellent manner to verify the<br />
numerical procedure. It can be used to check the obtained discretization order in space and time and to quantify<br />
numerical dissipation, as documented for instance in Figure 5 of <ref name="abelsamie2016"/>.<br />
<br />
[[File:vx_t2.png|250px|alt text]]<br />
[[File:vy_t2.png|250px|alt text]]<br />
[[File:wz_t2.png|250px|alt text]]<br />
<br />
You can see on the figures above the analytic fields of x-velocity, y-velocity, and vorticity, respectively (from left to right), at <math>t = 10 \tau_{ref}</math> for '''Step 1''' (verification step).<br />
<br />
[[File:vx_2d.png|250px]]<br />
[[File:vy_2d.png|250px]]<br />
<br />
Comparing the results of the '''three codes''' with the analytical solution for the '''2-D Taylor-Green vortex''' ('''Step 1'''). Left: xcomponent of velocity. Right: y-component of velocity.<br />
<br />
= References =<br />
<br />
<references><br />
<ref name="abelsamie2016"><br />
<bibtex><br />
@article{Abdelsamie2016, <br />
author= {A. Abdelsamie and G. Fru and F. Dietzsch and G. Janiga and D. Thévenin},<br />
title= {Towards direct numerical simulations of low-Mach number turbulent reacting and two-phase flows using immersed boundaries},<br />
journal={Comput. Fluids},<br />
year= {2016},<br />
volume={131},<br />
number={5},<br />
pages={123--141},<br />
}<br />
</bibtex><br />
</ref><br />
</references></div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Codes&diff=466Codes2020-08-25T21:19:54Z<p>Lartigue: /* General comments on the codes */</p>
<hr />
<div>= Code Description =<br />
<br />
This page is dedicated to a short presentation of the three '''low-Mach number''' codes used for the rest of this benchmarl: [https://www.coria-cfd.fr/index.php/YALES2 YALES2], [http://www.lss.ovgu.de/lss/en/Research/Computational+Fluid+Dynamics.html DINO] and [http://nek5000.mcs.anl.gov Nek5000]. <br />
<br />
Since these codes have already been the subject of many publications and are not completely new, '''only the most relevant features are discussed in what follows''', with suitable references for those readers needing more details.<br />
<br />
== YALES2 ==<br />
<br />
YALES2 is a '''massively parallel multiphysics platform'''<ref name="moureauwebsite">V. Moureau. Yales2 public website [https://www.coria-cfd.fr/index.php/YALES2 www.coria-cfd.fr/index.php/YALES2].</ref><ref name="moureau2011"/> developed since 2009 by Moureau, Lartigue and co-workers at CORIA (Rouen, France).<br />
It is dedicated to the high-fidelity simulation of '''low-Mach number flows in complex geometries'''.<br />
It is based on the '''Finite Volumes formulation''' of the '''Navier-Stokes equations''' and it can solve both '''non-reacting''' and '''reacting flows'''.<br />
It can actually solve various physical problems thanks to its original structure which is composed of a main numerical library accompanied with tens of dedicated solvers (for acoustics, multiphase flows, heat transfer, radiation\ldots) which can be coupled with one another.<br />
YALES2 relies on '''unstructured meshes''' and a '''fully parallel dynamic mesh adaptation technique''' to improve the resolution in physically-relevant zones to mitigate the computational cost<ref name="benard2015"/>. <br />
As a result it can easily handle meshes composed of billions of tetrahedra, thus enabling the '''Direct Numerical Simulation''' of laboratory and semi-industrial configurations.<br />
It is now composed of nearly 500,000 lines of object-oriented Fortran and the parallelism is currently ensured by a pure '''MPI''' paradigm, although a hybrid '''OpenMP/MPI''' as well as a GPU version are under development.<br />
<br />
As most low-Mach number codes, the time-advancement is based on a '''projection-correction method''' following the pioneering work of<ref name="chorin1968"/>.<br />
The prediction step uses a method which is a blend between a '''4th-order Runge-Kutta method''' and a '''4th-order Lax-Wendroff-like method'''<ref name="kraushaar2011"/>, combined with a '''4th-order node-based centered finite-volume discretization''' of the convective and diffusive terms.<br />
Moreover, to improve the performance of the correction step, the pressure of the previous iteration is included in the prediction step to limit the '''splitting errors'''.<br />
The correction step is required to ensure '''mass conservation''', using a pressure that arises from a '''Poisson equation'''.<br />
This is performed numerically by solving a linear system on the pressure at each node of the mesh thanks to a dedicated in-house version of the deflated conjugate gradient algorithm, which has been optimized for solving elliptic equations on massively parallel machines<ref name="malandain2013"/>.<br />
<br />
When considering '''multispecies''' and '''non-isothermal flows''', the extension of the classical projection method proposed by Pierce et al.<ref name="pierce2004"/> is used to account for '''variable-density flows'''.<br />
All the thermodynamic and transport properties are provided by the '''Cantera software'''<ref name="goodwin2003"/>, which has been fully re-implemented in '''Fortran''' to avoid any performance issues.<br />
Each species thermodynamic property is specified by '''5th-order polynomials''' on two temperature ranges (below and above 1000 K).<br />
The mixture is supposed to be both thermally and mechanically perfect.<br />
<br />
Regarding transport properties, several type of models are implemented in YALES2: <br />
<br />
1) the default approach is based on the computation of transport properties for each species (using tabulated molecular potentials), then combining those to obtain mixture-averaged coefficients for '''viscosity''', '''conductivity''' and '''diffusion velocities''' (Hirschfelder and Curtiss approximation<ref name="hirschfelder1964"/>; <br />
<br />
2) alternatively, simplified laws (for example the Sutherland law<ref name="sutherland1893"/> can be used for the '''viscosity''', while fixed values for '''Prandtl''' and '''Schmidt numbers''' allow the computation of '''conductivity''' and '''diffusion coefficients'''; <br />
<br />
3) a mix of both approaches, for example computing '''viscosity''' and '''conductivity''' with a mixture-averaged approximation and then imposing a '''Lewis number''' for each species.<br />
This last approach has been retained in the present benchmark.<br />
<br />
Finally, the source terms used in the '''reacting simulations''' are modeled with an '''Arrhenius law''' with the necessary modifications needed to take into account three-body or pressure-dependent reactions.<br />
<br />
From a numerical point of view, it must be noticed that both the '''diffusion''' and '''reaction processes''' occur at time scales which can be orders of magnitude smaller than the convective time scale.<br />
Solving these phenomena with explicit methods would thus drastically limit the '''global timestep''' of the whole simulation and induce an overwhelming CPU cost.<br />
To mitigate these effects when dealing with '''multi-species reacting flows''', the classical operator '''splitting technique''' is used.<br />
The '''diffusion process''' is solved with a fractional timestep method inside each convective iteration, each substep being limited by a '''Fourier condition''' to ensure stability. <br />
This method gets activated only when the mesh is very fine (typically when performing DNS with very diffusive species like <math>H_2</math> or <math>H</math>, as in the present project); otherwise an explicit treatment is sufficient.<br />
The chemical source terms are then integrated with a dedicated stiff solver, namely the '''CVODE''' library from SUNDIALS<ref name="cohen1996cvode"/><ref name="cvodeUG">A.C. Hindmarsh and R. Serban. User documentation for '''CVODE'''. [https://computing.llnl.gov/sites/default/files/public/cv_guide.pdf].</ref><ref name="sundials"/>. <br />
To that purpose, each control volume is considered as an isolated reactor with both constant pressure and enthalpy.<br />
<br />
Using the stiff integration technique results in a very strong load imbalance between the various regions of the flow: the fresh gases are solved in a very small number of integration steps while the inner flame region may require tens or even hundreds of integration steps. <br />
To overcome this difficulty, a dedicated '''MPI''' dynamic scheduler based on a work-sharing algorithm ensures global load-balancing.<br />
<br />
== DINO ==<br />
<br />
DINO is a '''3-D DNS code''' used for '''incompressible''' or '''low-Mach number flows''', the latter approach being used in this project. <br />
The development of DINO started in the group of D. Thévenin (Univ. of Magdeburg) in 2013. <br />
DINO is a '''Fortran-90 code''', written on top of a 2-D pencil decomposition to enable efficient large-scale parallel simulations on distributed-memory supercomputers by coupling with the open-source library 2-DECOMP&FFT<ref name="li2010"/>. <br />
The code offers different features and algorithms in order to investigate different physicochemical processes. <br />
Spatial derivative are computed by default using '''sixth-order central finite differences'''. <br />
Time integration relies on '''several Runge-Kutta solvers'''. <br />
In what follows, an explicit '''4th-order Runge-Kutta approach''' has been used. <br />
A '''3rd-order semi-implicit Runge-Kutta integration''' can be activated as needed, when considering stiff chemistry. <br />
In this case, non-stiff terms are still computed with explicit Runge-Kutta, while the '''PyJac package'''<ref>Create analytical jacobian matrix source code for chemical kinetics.</ref> is used to integrate in an implicit manner all chemistry terms with an analytical Jacobian computation. <br />
All thermodynamic, chemical and transport properties are computed using the open-source library '''Cantera 2.4.0'''<ref>Cantera.</ref>. <br />
<br />
The transport properties can be computed based on three different models: <br />
<br />
1) Constant Lewis numbers, <br />
<br />
2) mixture-averaged, <br />
<br />
3) full multicomponent diffusion, by coupling either again with '''Cantera''' or with the open-source library EGlib<ref name="ern1995"/>. <br />
<br />
The '''Poisson equation''' is solved using Fast Fourier Transform ('''FFT''') for '''periodic''' as well as for '''non-periodic boundary conditions''', relying in the latter case on an in-house pre- and post-processing technique. <br />
The I/O operations are implemented using two different approaches: (1) binary '''MPI-I/O''' using 2-DECOMP&FFT for check-points and restart files; (2) HDF5 files used for analysis and visualization.<br />
<br />
'''Multi-phase flows''' can be simulated in DINO using resolved or non-resolved (point) particles and droplets using a '''Lagrangian approach'''<ref name="abdelsamie2019"/><ref name="abdelsamie2020"/>. Complex boundaries are represented by a novel second-order immersed boundary method implementation ('''IBM''') based on a directional extrapolation scheme<ref name="chi2020"/>. <br />
More details about the implemented algorithms can be found in particular in<ref name="abdelsamie2016"/>. <br />
Since DINO has been developed as a multi-purpose code for analyzing many different '''reacting''' and '''non-reacting flows'''<ref name="chi2018"/><ref name="oster2018"/><ref name="abdelsamie2019-2"/>, a detailed verification and validation is obviously essential.<br />
<br />
== Nek5000 ==<br />
<br />
This '''spectral element low-Mach number reacting flow solver''' is based on the highly-efficient open-source solver Nek5000<ref>Nek5000 version v17.0, Argonne National Laboratory, IL, U.S.A.</ref> extended by a plugin developed at ETH implementing a '''high-order splitting scheme''' for '''low-Mach number reacting flows'''<ref name="tomboulides1997"/>. <br />
The spectral element method ('''SEM''') is a '''high-order weighted residual technique''' for spatial discretization that combines the accuracy of spectral methods with the geometric flexibility of the finite element method allowing for accurately representation of complex geometries<ref name="deville2002"/>.<br />
The computational domain is decomposed into <math>E</math> conforming elements, which are '''quadrilaterals''' ('''in 2-D''') or '''hexahedra''' (in '''3-D''') that conform to the domain boundaries. <br />
Within each element, functions are expanded as <math>N</math>th-order polynomials so that resolution can be increased either by decreasing the element size (<math>h</math>-type refinement) or by increasing the polynomial order (<math>p</math>-type refinement; typically <math>N = 7 - 15</math>). <br />
The grids can be unstructured and allow for '''static local refinement''', while '''adaptive mesh refinement''' has been recently developed<ref name="tanarro2020"/>. <br />
By casting the '''polynomial approximation''' in tensor-product form, the differential operators on <math>N^3</math> gridpoints per element can be evaluated with only <math>O(N^4)</math> work and <math>O(N^3)</math> storage.<br />
<br />
The principal advantage of the '''SEM''' is that convergence is exponential in <math>N</math>, yielding minimal '''numerical dispersion''' and '''dissipation''', so that significantly fewer grid points per wavelength are required in order to accurately propagate a turbulent structure over the extended time required in high Reynolds number simulations. <br />
Nek5000 uses locally '''structured basis coefficients''' (<math>N\times N \times N</math> arrays), which allow direct addressing and tensor-product-based derivative evaluation that can be cast as efficient matrix-matrix products involving <math>N^2</math> operators applied to <math>N^3</math> data values for each element. <br />
As a result, data movement per grid point is the same as for '''low-order methods'''. <br />
It uses scalable domain-decomposition-based iterative solvers with efficient preconditioners. <br />
Communication is based on the Message Passing Interface ('''MPI''') standard, and the code has proven scalability to over one million ranks. <br />
Nek5000 provides balanced I/O latency among all processors and reduces the overhead or even completely hides the I/O latency by using dedicated I/O communicators in the optimal case.<br />
<br />
'''Time advancement''' is performed using the '''splitting scheme''' proposed in<ref name="tomboulides1997" /> to decouple the highly non-linear and stiff thermochemistry (species and energy governing equations) from the hydrodynamic system (continuity and momentum).<br />
Species and energy equations are integrated without further '''splitting using the implicit stiff integrator solver CVODE from the SUNDIALS package'''<ref name="cohen1996cvode" /><ref name="cvodeUG" /><ref name="sundials" /> that uses backward differentiation formulas (BDF).<br />
The '''continuity''' and '''momentum equations''' are integrated using a '''second-''' or '''third-order semi-implicit formulation''' (EXT/BDF) treating the non-linear advection term explicitly<ref name="deville2002" />.<br />
The thermodynamic properties, detailed chemistry, and transport properties are provided by optimized subroutines compatible with '''Chemkin'''<ref name="chemkin"/>.<br />
<br />
The reacting flow solver can handle complex time-varying geometries and has been used for instance to simulate laboratory-scale internal combustion engines<ref name="MS"/><ref name="TCC"/>. <br />
It can account for conjugate fluid-solid heat transfer and detailed gas phase as well as surface kinetics<ref name="catalytic"/>.<br />
<br />
== General comments on the codes ==<br />
<br />
The three '''aforementioned solvers''' are all '''unsteady''', '''high-fidelity codes''' based on the '''low-Mach number''' formulation of the '''Navier-Stokes equations'''.<br />
They can perform both '''Direct Numerical Simulations''' or '''Large Eddy Simulations''' of reacting flows, only '''DNS''' being considered here.<br />
However, they differ in a certain number of points, mainly from the numerical point of view.<br />
The aim of this section is to emphasize those major differences.<br />
First, '''YALES2''' is an '''unstructured code''', designed to handle any type of elements; its main application field pertains to '''LES''' of industrially relevant flows, though '''DNS''' is possible as well. <br />
On the other hand, both '''DINO''' and '''Nek5000''' are mostly dedicated to '''DNS''' of configurations found in fundamental research.<br />
Both '''DINO''' and '''Nek5000''' can only deal with '''quads''' or '''hexas''', with the major difference that '''DINO''' is based on a '''structured connectivity''' while '''Nek5000''' can use '''unstructured meshes''' (pavings).<br />
As a consequence, both '''DINO''' and '''Nek5000''' employ '''higher-order numerical schemes''' compared to '''YALES2''', limited at best to a '''4th-order scheme'''.<br />
All codes rely on dedicated libraries to compute the thermo-chemical properties of the flow, either '''Chemkin''', '''Cantera''', or in-house versions of those.<br />
Moreover, they also rely at least to some extent on external software to perform the temporal integration of the stiff chemical source terms.<br />
Regarding the Poisson equation for pressure which must be solved by all codes, '''DINO''' relies on a spectral formulation by performing direct and inverse Fourier transforms, which is possible thanks to its structured mesh.<br />
On the other hand, both '''YALES2''' and '''Nek5000''' use an iterative solver with an efficient preconditioning technique; this method is more versatile and should be computationally more efficient for large and complex geometrical configurations.<br />
'''Nek5000''' employs CVODE to integrate the thermochemical equations without further '''splitting''' of the different terms accounting for convection, diffusion and chemistry.<br />
The main differences between the three codes are summarized in the table below. <br />
Please note that the presented values are those used for the benchmark, even though some other options are available in each codes.<br />
<br />
{| class="wikitable alternance center"<br />
|+ Table 1: Major numerical properties of the three high-fidelity codes as used in this benchmark<br />
|-<br />
! scope="col" | Code<br />
! scope="col" | YALES2<br />
! scope="col" | DINO<br />
! scope="col" | Nek5000<br />
|-<br />
| Connectivity<br />
| Unstructured<br />
| Structured<br />
| Unstructured<br />
|-<br />
| Discretization Type<br />
| Finite Volumes<br />
| Finite Differences<br />
| Spectral Elements<br />
|-<br />
| Grid point distribution<br />
| Regular hexahedra<br />
| Regular hexahedra<br />
| Regular hexahedra with GLL points<br />
|-<br />
| Spatial order<br />
| 4th<br />
| 6th<br />
| 7th - 15th (typically)<br />
|-<br />
| Temporal method<br />
| expl. RK4<br />
| expl. RK4 / semi-impl. RK3<br />
| semi-impl. BDF3<br />
|-<br />
| Pressure solver<br />
| CG with Deflation MPrec.<br />
| FFT-based<br />
| CG/GMRES with Jacobi/Schwartz Prec.<br />
|-<br />
| Thermo-chemistry<br />
| Cantera (re-coded)<br />
| Cantera<br />
| Chemkin interface<br />
|-<br />
| Chemistry integration<br />
| CVODE<br />
| PyJac<br />
| CVODE<br />
|-<br />
| Operator splitting<br />
| Yes<br />
| No<br />
| No<br />
|-<br />
| Parallel paradigm<br />
| MPI<br />
| MPI<br />
| MPI<br />
|}<br />
<br />
= References =<br />
<br />
<references><br />
<ref name="moureau2011"><br />
<bibtex><br />
@article{Moureau2011, <br />
author= {V. Moureau, P. Domingo, and L. Vervisch},<br />
title= {From large-eddy simulation to direct numerical simulation of a lean premixed swirl flame: Filtered laminar flame-pdf modeling},<br />
journal={Combust. Flame},<br />
year= {2011},<br />
volume={158},<br />
number={7},<br />
pages={1340--1357},<br />
}<br />
</bibtex><br />
</ref><br />
<ref name="benard2015"><br />
<bibtex><br />
@article{Benard2015, <br />
author= {P. Benard, G. Balarac, V. Moureau, C. Dobrzynski, G. Lartigue, and Y. D’Angelo},<br />
title= {Mesh adaptation for largeeddy simulations in complex geometries},<br />
journal={Int. J. Numer. Methods Fluids},<br />
year= {2015},<br />
volume={81},<br />
pages={719--740},<br />
}<br />
</bibtex><br />
</ref><br />
<ref name="chorin1968"><br />
<bibtex><br />
@article{Chorin1968, <br />
author= {A.J. Chorin},<br />
title= {Numerical solution of the Navier-Stokes equations},<br />
journal={Math. Computation},<br />
year= {1968},<br />
volume={22},<br />
number={104},<br />
pages={745--762},<br />
}<br />
</bibtex><br />
</ref><br />
<ref name="kraushaar2011"><br />
<bibtex><br />
@article{Kraushaar2011, <br />
author= {M. Kraushaar},<br />
title= {Application of the compressible and low-Mach number approaches to Large-Eddy Simulation of turbulent flows in aero-engines},<br />
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@article{Abdelsamie2019, <br />
author= {A. Abdelsamie and D. Thévenin.},<br />
title= { On the behavior of spray combustion in a turbulent spatially-evolving jet investigated by direct numerical simulation},<br />
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@article{Abdelsamie2020, <br />
author= {A. Abdelsamie and D. Thévenin},<br />
title= {Nanoparticle behavior and formation in turbulent spray flames investigated by DNS},<br />
journal={Direct and Large Eddy Simulation XII, ERCOFTAC Series},<br />
year= {2020},<br />
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}<br />
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@article{Chi2020, <br />
author= {C. Chi, A. Abdelsamie, and D. Thévenin},<br />
title= {A directional ghost-cell immersed boundary method for incompressible flows},<br />
journal={J. Comput. Phys.},<br />
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pages={109122--109142},<br />
}<br />
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@article{Chi2018, <br />
author= {C. Chi, A. Abdelsamie, and D. Thévenin},<br />
title= {Direct numerical simulations of hotspot-induced ignition in homogeneous hydrogen-air pre-mixtures and ignition spot tracking},<br />
journal={Flow Turbul. Combust.},<br />
year= {2018},<br />
volume={101},<br />
number={1},<br />
pages={103--121},<br />
}<br />
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@article{Oster2018, <br />
author= {T. Oster, A. Abdelsamie, M. Motejat, T. Gerrits, C. Rössl, D. Thévenin, and H. Theisel},<br />
title= {On-the-fly tracking of flame surfaces for the visual analysis of combustion processes},<br />
journal={Comput. Graph. Forum},<br />
year= {2018},<br />
volume={37},<br />
number={6},<br />
pages={358--369},<br />
}<br />
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@article{Abdelsamie2019-2, <br />
author= {A. Abdelsamie and D. Thévenin},<br />
title= {Impact of scalar dissipation rate on turbulent spray combustion investigated by DNS},<br />
journal={Direct and Large-Eddy Simulation XI, ERCOFTAC Series},<br />
year= {2019},<br />
volume={25},<br />
}<br />
</bibtex><br />
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@article{Tomboulides1997, <br />
author= {A.G. Tomboulides, J.C.Y. Lee, and S.A. Orszag},<br />
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author= {M.O. Deville, P.F. Fischer, and E.H. Mund},<br />
title= {High-order Methods for Incompressible Fluid Flows},<br />
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}<br />
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title= {Enabling adaptive mesh refinement for spectral-element simulations of turbulence around wing sections},<br />
journal={Flow Turbul. Combust},<br />
year= {2020},<br />
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pages={415--436},<br />
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@article{Chemkin, <br />
author= {R.J. Kee, F.M. Rupley, and J.A. Miller},<br />
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author= {M. Schmitt, C.E. Frouzakis, A.G. Tomboulides, Y.M. Wright, and K. Boulouchos},<br />
title= {Direct numerical simulation of the effect of compression on the flow, temperature and composition under engine-like conditions},<br />
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year= {2015},<br />
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@article{TCC, <br />
author= {G.K. Giannakopoulos, C.E. Frouzakis, P.F. Fischer, A.G. Tomboulides, and K. Boulouchos},<br />
title= {LES of the gasexchange process inside an internal combustion engine using a high-order method},<br />
journal={Flow Turbul. Combust.},<br />
year= {2020},<br />
volume={104},<br />
pages={673--692},<br />
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year= {2017},<br />
volume={36},<br />
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}<br />
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title= {Towards direct numerical simulations of lowMach number turbulent reacting and two-phase flows using immersed boundaries},<br />
journal={Comput. Fluids},<br />
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volume={131},<br />
number={5},<br />
pages={123--141},<br />
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</references></div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Step_3&diff=465Step 32020-08-25T21:14:24Z<p>Lartigue: /* Step 3 description */</p>
<hr />
<div>= Step 3 description =<br />
<br />
The purpose of this configuration is to perform a simulation similar to '''Step 2''' but involving now a '''mixture of different gaseous species''' at different temperatures, still '''without any chemical reaction'''. <br />
In this manner, it is possible to assess the accuracy of the numerical models involved in the description of species and heat diffusion<br />
The thermodynamic and transport properties of the 9 species kinetic scheme of Boivin et al.<ref name="Boivin2011"/> should be used in preparation for the next step, even though the reactions are still neglected here.<br />
<br />
It should be noted that in this case, the '''density is variable''' in both '''space''' and '''time''' due to the changing composition, which was not the case in the previous step: this induces additional numerical difficulties that must be taken into account.<br />
However, there is no variation of the thermodynamic pressure in time, contrary to the next step.<br />
<br />
The simulation domain is a cubic box of size <math>[0;L]^3</math> with <math>L = 2 \pi L_0</math> in each direction, where <math>L_0 = 1\; \mathrm{mm}</math>. <br />
Compared to the previous step, this smaller size is needed to ensure a proper resolution of the reaction fronts for hydrogen oxidation that will appear in the next step and are associated to fixed characteristic dimensions. <br />
Again, periodic boundary conditions are used in all three spatial directions.<br />
<br />
The initial velocity field prescribed at <math>t=0</math> is identical to the one used in Step 2 with the reference velocity set to <math>u_0=4\;\mathrm{m/s}</math>.<br />
<br />
In preparation for the next step, the central part of the box is initially filled with a <math>H_2/N_2</math> mixture (fuel region, molar fraction <math>X^0_\mathrm{H_2}=0.45</math>, <math>T=300 K</math>) while the remaining domain is filled with air (oxidizer region, molar fraction <math>X^0_\mathrm{O_2}=0.21</math>, <math>T=300 K</math>). <br />
The corresponding mass fractions are <math>Y^0_\mathrm{H_2} \approx 0.0556</math> and <math>Y^0_\mathrm{O_2} \approx 0.233</math>.<br />
<br />
To avoid any numerical instability, '''the steps at the interface between both regions are smoothed out using hyperbolic tangent functions''' as follows:<br />
<br />
<math>R_d(x) = |x-0.5\,L|,</math> (1)<br />
<br />
<math>\psi(x) = 0.5\left[1+\mathrm{tanh}\left(\frac{c\,(R_d(x)-R)}{R}\right)\right],</math> (2)<br />
<br />
<math>Y_\mathrm{H_2}(x) = Y^0_{\mathrm{H_2}}\left(1-\psi(x)\right),</math> (3)<br />
<br />
<math>Y_\mathrm{O_2}(x) = Y^0_{\mathrm{O_2}}\left(\psi(x)\right).</math> (4)<br />
<br />
where <math>R = \pi/4</math> mm and <math>c = 3</math> are the half-width of the central slab and stiffness parameter, respectively.<br />
As a consequence, there is initially a small region where both fuel and oxidizer coexist. <br />
Finally, a nitrogen complement is added everywhere using <math>Y_\mathrm{N_2}=1-Y_\mathrm{H_2}-Y_\mathrm{O_2}</math>.<br />
<br />
To also test the behaviour of the codes with respect to heat diffusion, '''a non-homogeneous temperature profile is finally imposed''' as follows:<br />
<br />
* Compute in each cell the equilibrium temperature for the local mixture for constant pressure and enthalpy by using a dedicated solver.<br />
<br />
* Enforce the resulting temperature profile, but keep the species profiles to their initial values (Eqs. (3) and (4)).<br />
<br />
It is found by using '''Cantera''' that the peak adiabatic temperature obtained '''at the fuel/air interface''' is <math>T_{ad}=1910.7~K</math>, leading to the profiles shown in the initial fields of [https://benchmark.coria-cfd.fr/images/4/40/Wmag_t0.png vorticity], [https://benchmark.coria-cfd.fr/images/6/6d/T_t0.png temperature], <br />
[https://benchmark.coria-cfd.fr/images/f/fa/H2_t0.png mass fraction of <math>H_2</math>], [https://benchmark.coria-cfd.fr/images/e/ef/O2_t0.png mass fraction of <math>O_2</math>] and the [https://benchmark.coria-cfd.fr/images/b/b4/Profiles_3d_zoom.pdf initial profiles of temperature and of mass fractions]. Users are encouraged to implement the exact initial profiles from the benchmark website to facilitate later comparisons. <br />
<br />
The resulting, initial configuration thus involves '''multiple species at different temperatures''' with steep profiles, just like in a real flame.<br />
However, for the present simulation, the reaction source terms are all still set to zero, as already mentioned: only '''convection''' and '''diffusion processes''' are considered.<br />
In order to enable benchmark computations also with '''DNS codes''' that do not provide advanced diffusion models, only constant values of the Lewis numbers are considered; for the same reason, thermodiffusion (Soret effect) is neglected on purpose (though it would be obviously relevant for such cases involving hydrogen as a fuel). <br />
The authors are fully aware that this is a crude approximation of reality<ref name="pecs"/>. But it must be kept in mind that the focus is set here on the detailed comparison between different codes and algorithms, and not on the resulting flow or flame structure.<br />
Hence, in this and in the next section, the values of the '''Lewis number''' for each species given in the table below shall be used to enable direct comparisons. <br />
These values have been estimated from a separate DNS for a similar configuration using the mixture-averaged diffusion model.<br />
The precise description of the associated thermodynamic state is available on the website.<br />
<br />
{| class="wikitable alternance center"<br />
|+ Approximate Lewis number for the species appearing in the kinetic scheme of<ref name="Boivin2011" />, to be enforced for Step 3 and Step 4 of the benchmark<br />
|-<br />
! scope="row" | Species<br />
| <math>H_2</math><br />
| <math>H</math><br />
| <math>O_2</math><br />
| <math>OH</math><br />
| <math>O</math><br />
| <math>H_2O</math><br />
| <math>HO_2</math><br />
| <math>H_2O_2</math><br />
| <math>N_2</math><br />
|-<br />
! scope="row" | Lewis number<br />
| 0.3290<br />
| 0.2228<br />
| 1.2703<br />
| 0.8279<br />
| 0.8128<br />
| 1.0741<br />
| 1.2582<br />
| 1.2665<br />
| 1.8268<br />
|}<br />
<br />
On the other hand, considering that implementing advanced models for fluid viscosity and thermal conductivity is not difficult and does not increase the computational time significantly, local values of these quantities depending on composition and temperature should be taken into account; <br />
More specifically, and to ease further comparisons, the same models for viscosity and conductivity have been retained in the three participating codes, i.e. the mixture averaged formalism that is used in both the '''Cantera''' and '''Chemkin packages'''.<br />
<br />
The viscosity is variable in the computational domain as well as with time and the '''Reynolds number''' of this configuration can only be estimated using the minimal value of viscosity (which is obtained in the air at 300 K, <math>\nu_\mathrm{min} \approx 1.56 \, 10^{-5} \mathrm{m^2/s}</math>), leading to <math>Re = {u_0 L_0}/{\nu_{min}}=267</math> which guarantees a laminar flow.<br />
<br />
Concerning resolution, it is also suggested to keep a conservative resolution in time for this benchmark, corresponding to <math>CFL \le 0.25</math> and <math>Fo \le 0.15</math>. <br />
In space, each direction should be resolved by approximately 256 grid points, leading to a spatial resolution <math>\Delta x \approx 25 \mu m</math>.<br />
<br />
The '''reference time scale''' is here <math>\tau_\mathrm{ref}=L_0/u_0 = 0.25 \; \mathrm{ms}</math>.<br />
The simulation should be performed for a physical time of at least <math>t=0.5 \; \mathrm{ms} = 2 \tau_\mathrm{ref}</math> as most of the results presented in this article are taken from this instant.<br />
<br />
[[File:wmag_t0.png|250px|alt text]]<br />
[[File:T_t0.png|250px|alt text]]<br />
[[File:H2_t0.png|250px|alt text]]<br />
[[File:O2_t0.png|250px|alt text]]<br />
<br />
Initial fields of vorticity magnitude, temperature, mass fractions of <math>\mathrm{H_2}</math>, and mass fraction of <math>\mathrm{O_2}</math> (from left to right), for the 3-D, non-reacting case ('''Step 3''').<br />
<br />
[[File:profiles_3d_zoom.pdf|250px|alt text]]<br />
<br />
Initial profiles of temperature and of mass fractions at <math>y=0.5 L</math> and <math>z=0.5 L</math> for the 3-D, non-reacting case ('''Step 3''').<br />
<br />
= Kinetics =<br />
<br />
The kinetic scheme of Boivin et al.<ref name="Boivin2011"/> [https://doi.org/10.1016/j.proci.2010.05.002] which contains 9 species and 12 reactions has been used for this Benchmark.<br />
<br />
This mechanism is provided here in the Cantera format:<br />
* [[File:H2_williams_12.xml.zip | ctml ]]<br />
* [[File:H2_williams_12.cti | cti]]<br />
<br />
= Transport and thermodynamic properties used for this Step =<br />
<br />
\cf{I think we all used the same transport (viscosity) and thermodynamic properties, but we should check again and compare that the implementations used in cantera/Chemkin provide the same values.}<br />
<br />
\gl{Good idea: we could use 3 or 4 relevant reference compositions to quantify the discrepancies between Cantera and Chemkin (on rho, Cp, conductivity, viscosity for example) and also put these on the website.}<br />
<br />
'''Put the transport data here'''<br />
<br />
= Aside suggestions =<br />
<br />
TBD<br />
<br />
= References =<br />
<br />
<references><br />
<ref name="Boivin2011"><br />
<bibtex><br />
@article{Boivin2011, <br />
author= {P. Boivin, C. Jiménez, A.L. Sanchez, and F.A. Williams},<br />
title= {An explicit reduced mechanism for <math>H_2</math>-air combustion.},<br />
journal={Proc. Combust. Inst.},<br />
year= {2011},<br />
volume={33},<br />
pages={517--523},<br />
doi= {https://doi.org/10.1016/j.proci.2010.05.002}<br />
}<br />
</bibtex><br />
</ref><br />
<ref name="pecs"><br />
<bibtex><br />
@article{Pecs, <br />
author= {R. Hilbert, F. Tap, H. El-Rabii, and D. Thévenin},<br />
title= {Impact of detailed chemistry and transport models on turbulent combustion simulations},<br />
journal={Prog. Energ. Combust. Sci.},<br />
year= {2004},<br />
volume={30},<br />
pages={61--117},<br />
}<br />
</bibtex><br />
</ref><br />
</references></div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Step_3&diff=464Step 32020-08-25T21:09:32Z<p>Lartigue: /* Kinetics */</p>
<hr />
<div>= Step 3 description =<br />
<br />
The purpose of this configuration is to perform a simulation similar to '''Step 2''' but involving now a '''mixture of different gaseous species''' at different temperatures, still '''without any chemical reaction'''. <br />
In this manner, it is possible to assess the accuracy of the numerical models involved in the description of species and heat diffusion<br />
The thermodynamic and transport properties of the 12 species kinetic scheme of Boivin et al.<ref name="Boivin2011"/> should be used in preparation for the next step, even though the reactions are still neglected here.<br />
<br />
It should be noted that in this case, the '''density is variable''' in both '''space''' and '''time''' due to the changing composition, which was not the case in the previous step: this induces additional numerical difficulties that must be taken into account.<br />
However, there is no variation of the thermodynamic pressure in time, contrary to the next step.<br />
<br />
The simulation domain is a cubic box of size <math>[0;L]^3</math> with <math>L = 2 \pi L_0</math> in each direction, where <math>L_0 = 1\; \mathrm{mm}</math>. <br />
Compared to the previous step, this smaller size is needed to ensure a proper resolution of the reaction fronts for hydrogen oxidation that will appear in the next step and are associated to fixed characteristic dimensions. <br />
Again, periodic boundary conditions are used in all three spatial directions.<br />
<br />
The initial velocity field prescribed at <math>t=0</math> is identical to the one used in Step 2 with the reference velocity set to <math>u_0=4\;\mathrm{m/s}</math>.<br />
<br />
In preparation for the next step, the central part of the box is initially filled with a <math>H_2/N_2</math> mixture (fuel region, molar fraction <math>X^0_\mathrm{H_2}=0.45</math>, <math>T=300 K</math>) while the remaining domain is filled with air (oxidizer region, molar fraction <math>X^0_\mathrm{O_2}=0.21</math>, <math>T=300 K</math>). <br />
The corresponding mass fractions are <math>Y^0_\mathrm{H_2} \approx 0.0556</math> and <math>Y^0_\mathrm{O_2} \approx 0.233</math>.<br />
<br />
To avoid any numerical instability, '''the steps at the interface between both regions are smoothed out using hyperbolic tangent functions''' as follows:<br />
<br />
<math>R_d(x) = |x-0.5\,L|,</math> (1)<br />
<br />
<math>\psi(x) = 0.5\left[1+\mathrm{tanh}\left(\frac{c\,(R_d(x)-R)}{R}\right)\right],</math> (2)<br />
<br />
<math>Y_\mathrm{H_2}(x) = Y^0_{\mathrm{H_2}}\left(1-\psi(x)\right),</math> (3)<br />
<br />
<math>Y_\mathrm{O_2}(x) = Y^0_{\mathrm{O_2}}\left(\psi(x)\right).</math> (4)<br />
<br />
where <math>R = \pi/4 mm</math> and <math>c = 3</math> are the half-width of the central slab and stiffness parameter, respectively.<br />
As a consequence, there is initially a small region where both fuel and oxidizer coexist. <br />
Finally, a nitrogen complement is added everywhere using <math>Y_\mathrm{N_2}=1-Y_\mathrm{H_2}-Y_\mathrm{O_2}</math>.<br />
<br />
To also test the behaviour of the codes with respect to heat diffusion, '''a non-homogeneous temperature profile is finally imposed''' as follows:<br />
<br />
* Compute in each cell the equilibrium temperature for the local mixture for constant pressure and enthalpy by using a dedicated solver.<br />
<br />
* Enforce the resulting temperature profile, but keep the species profiles to their initial values (Eqs. (3) and (4)).<br />
<br />
It is found by using '''Cantera''' that the peak adiabatic temperature obtained '''at the fuel/air interface''' is <math>T_{ad}=1910.7~K</math>, leading to the profiles shown in the initial fields of [https://benchmark.coria-cfd.fr/images/4/40/Wmag_t0.png vorticity], [https://benchmark.coria-cfd.fr/images/6/6d/T_t0.png temperature], <br />
[https://benchmark.coria-cfd.fr/images/f/fa/H2_t0.png mass fraction of <math>H_2</math>], [https://benchmark.coria-cfd.fr/images/e/ef/O2_t0.png mass fraction of <math>O_2</math>] and the [https://benchmark.coria-cfd.fr/images/b/b4/Profiles_3d_zoom.pdf initial profiles of temperature and of mass fractions]. Users are encouraged to implement the exact initial profiles from the benchmark website to facilitate later comparisons. <br />
<br />
The resulting, initial configuration thus involves '''multiple species at different temperatures''' with steep profiles, just like in a real flame.<br />
However, for the present simulation, the reaction source terms are all still set to zero, as already mentioned: only '''convection''' and '''diffusion processes''' are considered.<br />
In order to enable benchmark computations also with '''DNS codes''' that do not provide advanced diffusion models, only constant values of the Lewis numbers are considered; for the same reason, thermodiffusion (Soret effect) is neglected on purpose (though it would be obviously relevant for such cases involving hydrogen as a fuel). <br />
The authors are fully aware that this is a crude approximation of reality<ref name="pecs"/>. But it must be kept in mind that the focus is set here on the detailed comparison between different codes and algorithms, and not on the resulting flow or flame structure.<br />
Hence, in this and in the next section, the values of the '''Lewis number''' for each species given in the table below shall be used to enable direct comparisons. <br />
These values have been estimated from a separate DNS for a similar configuration using the mixture-averaged diffusion model.<br />
The precise description of the associated thermodynamic state is available on the website.<br />
<br />
{| class="wikitable alternance center"<br />
|+ Approximate Lewis number for the species appearing in the kinetic scheme of<ref name="Boivin2011" />, to be enforced for Step 3 and Step 4 of the benchmark<br />
|-<br />
! scope="row" | Species<br />
| <math>H_2</math><br />
| <math>H</math><br />
| <math>O_2</math><br />
| <math>OH</math><br />
| <math>O</math><br />
| <math>H_2O</math><br />
| <math>HO_2</math><br />
| <math>H_2O_2</math><br />
| <math>N_2</math><br />
|-<br />
! scope="row" | Lewis number<br />
| 0.3290<br />
| 0.2228<br />
| 1.2703<br />
| 0.8279<br />
| 0.8128<br />
| 1.0741<br />
| 1.2582<br />
| 1.2665<br />
| 1.8268<br />
|}<br />
<br />
On the other hand, considering that implementing advanced models for fluid viscosity and thermal conductivity is not difficult and does not increase the computational time significantly, local values of these quantities depending on composition and temperature should be taken into account; <br />
More specifically, and to ease further comparisons, the same models for viscosity and conductivity have been retained in the three participating codes, i.e. the mixture averaged formalism that is used in both the '''Cantera''' and '''Chemkin packages'''.<br />
More details on these models are available on the benchmark repository<ref name="TGV-coria-cfd">[https://benchmark.coria-cfd.fr benchmark.coria-cfd.fr]</ref>, such as reference data for viscosity and conductivity along various cuts at initialization.<br />
<br />
The viscosity is variable in the computational domain as well as with time and the '''Reynolds number''' of this configuration can only be estimated using the minimal value of viscosity (which is obtained in the air at 300 K, <math>\nu_\mathrm{min} \approx 1.56 \, 10^{-5} \mathrm{m^2/s}</math>), leading to <math>Re = {u_0 L_0}/{\nu_{min}}=267</math> which guarantees a laminar flow.<br />
<br />
Concerning resolution, it is also suggested to keep a conservative resolution in time for this benchmark, corresponding to <math>CFL \le 0.25</math> and <math>Fo \le 0.15</math>. <br />
In space, each direction should be resolved by approximately 256 grid points, leading to a spatial resolution <math>\Delta x \approx 25 \mu m</math>.<br />
<br />
The '''reference time scale''' is here <math>\tau_\mathrm{ref}=L_0/u_0 = 0.25 \; \mathrm{ms}</math>.<br />
The simulation should be performed for a physical time of at least <math>t=0.5 \; \mathrm{ms} = 2 \tau_\mathrm{ref}</math> as most of the results presented in this article are taken from this instant.<br />
Data up to <math>t=2.5 \; \mathrm{ms} = 10 \tau_\mathrm{ref}</math> are available on the benchmark repository<ref name="TGV-coria-cfd" />.<br />
<br />
[[File:wmag_t0.png|250px|alt text]]<br />
[[File:T_t0.png|250px|alt text]]<br />
[[File:H2_t0.png|250px|alt text]]<br />
[[File:O2_t0.png|250px|alt text]]<br />
<br />
Initial fields of vorticity magnitude, temperature, mass fractions of <math>\mathrm{H_2}</math>, and mass fraction of <math>\mathrm{O_2}</math> (from left to right), for the 3-D, non-reacting case ('''Step 3''').<br />
<br />
[[File:profiles_3d_zoom.pdf|250px|alt text]]<br />
<br />
Initial profiles of temperature and of mass fractions at <math>y=0.5 L</math> and <math>z=0.5 L</math> for the 3-D, non-reacting case ('''Step 3''').<br />
<br />
= Kinetics =<br />
<br />
The kinetic scheme of Boivin et al.<ref name="Boivin2011"/> [https://doi.org/10.1016/j.proci.2010.05.002] which contains 9 species and 12 reactions has been used for this Benchmark.<br />
<br />
This mechanism is provided here in the Cantera format:<br />
* [[File:H2_williams_12.xml.zip | ctml ]]<br />
* [[File:H2_williams_12.cti | cti]]<br />
<br />
= Transport and thermodynamic properties used for this Step =<br />
<br />
\cf{I think we all used the same transport (viscosity) and thermodynamic properties, but we should check again and compare that the implementations used in cantera/Chemkin provide the same values.}<br />
<br />
\gl{Good idea: we could use 3 or 4 relevant reference compositions to quantify the discrepancies between Cantera and Chemkin (on rho, Cp, conductivity, viscosity for example) and also put these on the website.}<br />
<br />
'''Put the transport data here'''<br />
<br />
= Aside suggestions =<br />
<br />
TBD<br />
<br />
= References =<br />
<br />
<references><br />
<ref name="Boivin2011"><br />
<bibtex><br />
@article{Boivin2011, <br />
author= {P. Boivin, C. Jiménez, A.L. Sanchez, and F.A. Williams},<br />
title= {An explicit reduced mechanism for <math>H_2</math>-air combustion.},<br />
journal={Proc. Combust. Inst.},<br />
year= {2011},<br />
volume={33},<br />
pages={517--523},<br />
doi= {https://doi.org/10.1016/j.proci.2010.05.002}<br />
}<br />
</bibtex><br />
</ref><br />
<ref name="pecs"><br />
<bibtex><br />
@article{Pecs, <br />
author= {R. Hilbert, F. Tap, H. El-Rabii, and D. Thévenin},<br />
title= {Impact of detailed chemistry and transport models on turbulent combustion simulations},<br />
journal={Prog. Energ. Combust. Sci.},<br />
year= {2004},<br />
volume={30},<br />
pages={61--117},<br />
}<br />
</bibtex><br />
</ref><br />
</references></div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Step_3&diff=463Step 32020-08-25T21:05:35Z<p>Lartigue: /* Step 3 description */</p>
<hr />
<div>= Step 3 description =<br />
<br />
The purpose of this configuration is to perform a simulation similar to '''Step 2''' but involving now a '''mixture of different gaseous species''' at different temperatures, still '''without any chemical reaction'''. <br />
In this manner, it is possible to assess the accuracy of the numerical models involved in the description of species and heat diffusion<br />
The thermodynamic and transport properties of the 12 species kinetic scheme of Boivin et al.<ref name="Boivin2011"/> should be used in preparation for the next step, even though the reactions are still neglected here.<br />
<br />
It should be noted that in this case, the '''density is variable''' in both '''space''' and '''time''' due to the changing composition, which was not the case in the previous step: this induces additional numerical difficulties that must be taken into account.<br />
However, there is no variation of the thermodynamic pressure in time, contrary to the next step.<br />
<br />
The simulation domain is a cubic box of size <math>[0;L]^3</math> with <math>L = 2 \pi L_0</math> in each direction, where <math>L_0 = 1\; \mathrm{mm}</math>. <br />
Compared to the previous step, this smaller size is needed to ensure a proper resolution of the reaction fronts for hydrogen oxidation that will appear in the next step and are associated to fixed characteristic dimensions. <br />
Again, periodic boundary conditions are used in all three spatial directions.<br />
<br />
The initial velocity field prescribed at <math>t=0</math> is identical to the one used in Step 2 with the reference velocity set to <math>u_0=4\;\mathrm{m/s}</math>.<br />
<br />
In preparation for the next step, the central part of the box is initially filled with a <math>H_2/N_2</math> mixture (fuel region, molar fraction <math>X^0_\mathrm{H_2}=0.45</math>, <math>T=300 K</math>) while the remaining domain is filled with air (oxidizer region, molar fraction <math>X^0_\mathrm{O_2}=0.21</math>, <math>T=300 K</math>). <br />
The corresponding mass fractions are <math>Y^0_\mathrm{H_2} \approx 0.0556</math> and <math>Y^0_\mathrm{O_2} \approx 0.233</math>.<br />
<br />
To avoid any numerical instability, '''the steps at the interface between both regions are smoothed out using hyperbolic tangent functions''' as follows:<br />
<br />
<math>R_d(x) = |x-0.5\,L|,</math> (1)<br />
<br />
<math>\psi(x) = 0.5\left[1+\mathrm{tanh}\left(\frac{c\,(R_d(x)-R)}{R}\right)\right],</math> (2)<br />
<br />
<math>Y_\mathrm{H_2}(x) = Y^0_{\mathrm{H_2}}\left(1-\psi(x)\right),</math> (3)<br />
<br />
<math>Y_\mathrm{O_2}(x) = Y^0_{\mathrm{O_2}}\left(\psi(x)\right).</math> (4)<br />
<br />
where <math>R = \pi/4 mm</math> and <math>c = 3</math> are the half-width of the central slab and stiffness parameter, respectively.<br />
As a consequence, there is initially a small region where both fuel and oxidizer coexist. <br />
Finally, a nitrogen complement is added everywhere using <math>Y_\mathrm{N_2}=1-Y_\mathrm{H_2}-Y_\mathrm{O_2}</math>.<br />
<br />
To also test the behaviour of the codes with respect to heat diffusion, '''a non-homogeneous temperature profile is finally imposed''' as follows:<br />
<br />
* Compute in each cell the equilibrium temperature for the local mixture for constant pressure and enthalpy by using a dedicated solver.<br />
<br />
* Enforce the resulting temperature profile, but keep the species profiles to their initial values (Eqs. (3) and (4)).<br />
<br />
It is found by using '''Cantera''' that the peak adiabatic temperature obtained '''at the fuel/air interface''' is <math>T_{ad}=1910.7~K</math>, leading to the profiles shown in the initial fields of [https://benchmark.coria-cfd.fr/images/4/40/Wmag_t0.png vorticity], [https://benchmark.coria-cfd.fr/images/6/6d/T_t0.png temperature], <br />
[https://benchmark.coria-cfd.fr/images/f/fa/H2_t0.png mass fraction of <math>H_2</math>], [https://benchmark.coria-cfd.fr/images/e/ef/O2_t0.png mass fraction of <math>O_2</math>] and the [https://benchmark.coria-cfd.fr/images/b/b4/Profiles_3d_zoom.pdf initial profiles of temperature and of mass fractions]. Users are encouraged to implement the exact initial profiles from the benchmark website to facilitate later comparisons. <br />
<br />
The resulting, initial configuration thus involves '''multiple species at different temperatures''' with steep profiles, just like in a real flame.<br />
However, for the present simulation, the reaction source terms are all still set to zero, as already mentioned: only '''convection''' and '''diffusion processes''' are considered.<br />
In order to enable benchmark computations also with '''DNS codes''' that do not provide advanced diffusion models, only constant values of the Lewis numbers are considered; for the same reason, thermodiffusion (Soret effect) is neglected on purpose (though it would be obviously relevant for such cases involving hydrogen as a fuel). <br />
The authors are fully aware that this is a crude approximation of reality<ref name="pecs"/>. But it must be kept in mind that the focus is set here on the detailed comparison between different codes and algorithms, and not on the resulting flow or flame structure.<br />
Hence, in this and in the next section, the values of the '''Lewis number''' for each species given in the table below shall be used to enable direct comparisons. <br />
These values have been estimated from a separate DNS for a similar configuration using the mixture-averaged diffusion model.<br />
The precise description of the associated thermodynamic state is available on the website.<br />
<br />
{| class="wikitable alternance center"<br />
|+ Approximate Lewis number for the species appearing in the kinetic scheme of<ref name="Boivin2011" />, to be enforced for Step 3 and Step 4 of the benchmark<br />
|-<br />
! scope="row" | Species<br />
| <math>H_2</math><br />
| <math>H</math><br />
| <math>O_2</math><br />
| <math>OH</math><br />
| <math>O</math><br />
| <math>H_2O</math><br />
| <math>HO_2</math><br />
| <math>H_2O_2</math><br />
| <math>N_2</math><br />
|-<br />
! scope="row" | Lewis number<br />
| 0.3290<br />
| 0.2228<br />
| 1.2703<br />
| 0.8279<br />
| 0.8128<br />
| 1.0741<br />
| 1.2582<br />
| 1.2665<br />
| 1.8268<br />
|}<br />
<br />
On the other hand, considering that implementing advanced models for fluid viscosity and thermal conductivity is not difficult and does not increase the computational time significantly, local values of these quantities depending on composition and temperature should be taken into account; <br />
More specifically, and to ease further comparisons, the same models for viscosity and conductivity have been retained in the three participating codes, i.e. the mixture averaged formalism that is used in both the '''Cantera''' and '''Chemkin packages'''.<br />
More details on these models are available on the benchmark repository<ref name="TGV-coria-cfd">[https://benchmark.coria-cfd.fr benchmark.coria-cfd.fr]</ref>, such as reference data for viscosity and conductivity along various cuts at initialization.<br />
<br />
The viscosity is variable in the computational domain as well as with time and the '''Reynolds number''' of this configuration can only be estimated using the minimal value of viscosity (which is obtained in the air at 300 K, <math>\nu_\mathrm{min} \approx 1.56 \, 10^{-5} \mathrm{m^2/s}</math>), leading to <math>Re = {u_0 L_0}/{\nu_{min}}=267</math> which guarantees a laminar flow.<br />
<br />
Concerning resolution, it is also suggested to keep a conservative resolution in time for this benchmark, corresponding to <math>CFL \le 0.25</math> and <math>Fo \le 0.15</math>. <br />
In space, each direction should be resolved by approximately 256 grid points, leading to a spatial resolution <math>\Delta x \approx 25 \mu m</math>.<br />
<br />
The '''reference time scale''' is here <math>\tau_\mathrm{ref}=L_0/u_0 = 0.25 \; \mathrm{ms}</math>.<br />
The simulation should be performed for a physical time of at least <math>t=0.5 \; \mathrm{ms} = 2 \tau_\mathrm{ref}</math> as most of the results presented in this article are taken from this instant.<br />
Data up to <math>t=2.5 \; \mathrm{ms} = 10 \tau_\mathrm{ref}</math> are available on the benchmark repository<ref name="TGV-coria-cfd" />.<br />
<br />
[[File:wmag_t0.png|250px|alt text]]<br />
[[File:T_t0.png|250px|alt text]]<br />
[[File:H2_t0.png|250px|alt text]]<br />
[[File:O2_t0.png|250px|alt text]]<br />
<br />
Initial fields of vorticity magnitude, temperature, mass fractions of <math>\mathrm{H_2}</math>, and mass fraction of <math>\mathrm{O_2}</math> (from left to right), for the 3-D, non-reacting case ('''Step 3''').<br />
<br />
[[File:profiles_3d_zoom.pdf|250px|alt text]]<br />
<br />
Initial profiles of temperature and of mass fractions at <math>y=0.5 L</math> and <math>z=0.5 L</math> for the 3-D, non-reacting case ('''Step 3''').<br />
<br />
= Kinetics =<br />
<br />
The Boivin scheme [https://doi.org/10.1016/j.proci.2010.05.002] has been used for this Benchmark.<br />
<br />
<br />
<br />
[https://benchmark.coria-cfd.fr/index.php/File:H2_williams_12.xml.zip]<br />
[https://benchmark.coria-cfd.fr/index.php/File:H2_williams_12.cti]<br />
<br />
= Transport and thermodynamic properties used for this Step =<br />
<br />
\cf{I think we all used the same transport (viscosity) and thermodynamic properties, but we should check again and compare that the implementations used in cantera/Chemkin provide the same values.}<br />
<br />
\gl{Good idea: we could use 3 or 4 relevant reference compositions to quantify the discrepancies between Cantera and Chemkin (on rho, Cp, conductivity, viscosity for example) and also put these on the website.}<br />
<br />
'''Put the transport data here'''<br />
<br />
= Aside suggestions =<br />
<br />
TBD<br />
<br />
= References =<br />
<br />
<references><br />
<ref name="Boivin2011"><br />
<bibtex><br />
@article{Boivin2011, <br />
author= {P. Boivin, C. Jiménez, A.L. Sanchez, and F.A. Williams},<br />
title= {An explicit reduced mechanism for <math>H_2</math>-air combustion.},<br />
journal={Proc. Combust. Inst.},<br />
year= {2011},<br />
volume={33},<br />
pages={517--523},<br />
doi= {https://doi.org/10.1016/j.proci.2010.05.002}<br />
}<br />
</bibtex><br />
</ref><br />
<ref name="pecs"><br />
<bibtex><br />
@article{Pecs, <br />
author= {R. Hilbert, F. Tap, H. El-Rabii, and D. Thévenin},<br />
title= {Impact of detailed chemistry and transport models on turbulent combustion simulations},<br />
journal={Prog. Energ. Combust. Sci.},<br />
year= {2004},<br />
volume={30},<br />
pages={61--117},<br />
}<br />
</bibtex><br />
</ref><br />
</references></div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Step_3&diff=462Step 32020-08-25T21:05:06Z<p>Lartigue: /* Kinetics */</p>
<hr />
<div>= Step 3 description =<br />
<br />
The purpose of this configuration is to perform a simulation similar to '''Step 2''' but involving now a '''mixture of different gaseous species''' at different temperatures, still '''without any chemical reaction'''. <br />
In this manner, it is possible to assess the accuracy of the numerical models involved in the description of species and heat diffusion<br />
The thermodynamic and transport properties of the 9 species kinetic scheme of Boivin et al.<ref name="Boivin2011"/> should be used in preparation for the next step, even though the reactions are still neglected here.<br />
<br />
It should be noted that in this case, the '''density is variable''' in both '''space''' and '''time''' due to the changing composition, which was not the case in the previous step: this induces additional numerical difficulties that must be taken into account.<br />
However, there is no variation of the thermodynamic pressure in time, contrary to the next step.<br />
<br />
The simulation domain is a cubic box of size <math>[0;L]^3</math> with <math>L = 2 \pi L_0</math> in each direction, where <math>L_0 = 1\; \mathrm{mm}</math>. <br />
Compared to the previous step, this smaller size is needed to ensure a proper resolution of the reaction fronts for hydrogen oxidation that will appear in the next step and are associated to fixed characteristic dimensions. <br />
Again, periodic boundary conditions are used in all three spatial directions.<br />
<br />
The initial velocity field prescribed at <math>t=0</math> is identical to the one used in Step 2 with the reference velocity set to <math>u_0=4\;\mathrm{m/s}</math>.<br />
<br />
In preparation for the next step, the central part of the box is initially filled with a <math>H_2/N_2</math> mixture (fuel region, molar fraction <math>X^0_\mathrm{H_2}=0.45</math>, <math>T=300 K</math>) while the remaining domain is filled with air (oxidizer region, molar fraction <math>X^0_\mathrm{O_2}=0.21</math>, <math>T=300 K</math>). <br />
The corresponding mass fractions are <math>Y^0_\mathrm{H_2} \approx 0.0556</math> and <math>Y^0_\mathrm{O_2} \approx 0.233</math>.<br />
<br />
To avoid any numerical instability, '''the steps at the interface between both regions are smoothed out using hyperbolic tangent functions''' as follows:<br />
<br />
<math>R_d(x) = |x-0.5\,L|,</math> (1)<br />
<br />
<math>\psi(x) = 0.5\left[1+\mathrm{tanh}\left(\frac{c\,(R_d(x)-R)}{R}\right)\right],</math> (2)<br />
<br />
<math>Y_\mathrm{H_2}(x) = Y^0_{\mathrm{H_2}}\left(1-\psi(x)\right),</math> (3)<br />
<br />
<math>Y_\mathrm{O_2}(x) = Y^0_{\mathrm{O_2}}\left(\psi(x)\right).</math> (4)<br />
<br />
where <math>R = \pi/4 mm</math> and <math>c = 3</math> are the half-width of the central slab and stiffness parameter, respectively.<br />
As a consequence, there is initially a small region where both fuel and oxidizer coexist. <br />
Finally, a nitrogen complement is added everywhere using <math>Y_\mathrm{N_2}=1-Y_\mathrm{H_2}-Y_\mathrm{O_2}</math>.<br />
<br />
To also test the behaviour of the codes with respect to heat diffusion, '''a non-homogeneous temperature profile is finally imposed''' as follows:<br />
<br />
* Compute in each cell the equilibrium temperature for the local mixture for constant pressure and enthalpy by using a dedicated solver.<br />
<br />
* Enforce the resulting temperature profile, but keep the species profiles to their initial values (Eqs. (3) and (4)).<br />
<br />
It is found by using '''Cantera''' that the peak adiabatic temperature obtained '''at the fuel/air interface''' is <math>T_{ad}=1910.7~K</math>, leading to the profiles shown in the initial fields of [https://benchmark.coria-cfd.fr/images/4/40/Wmag_t0.png vorticity], [https://benchmark.coria-cfd.fr/images/6/6d/T_t0.png temperature], <br />
[https://benchmark.coria-cfd.fr/images/f/fa/H2_t0.png mass fraction of <math>H_2</math>], [https://benchmark.coria-cfd.fr/images/e/ef/O2_t0.png mass fraction of <math>O_2</math>] and the [https://benchmark.coria-cfd.fr/images/b/b4/Profiles_3d_zoom.pdf initial profiles of temperature and of mass fractions]. Users are encouraged to implement the exact initial profiles from the benchmark website to facilitate later comparisons. <br />
<br />
The resulting, initial configuration thus involves '''multiple species at different temperatures''' with steep profiles, just like in a real flame.<br />
However, for the present simulation, the reaction source terms are all still set to zero, as already mentioned: only '''convection''' and '''diffusion processes''' are considered.<br />
In order to enable benchmark computations also with '''DNS codes''' that do not provide advanced diffusion models, only constant values of the Lewis numbers are considered; for the same reason, thermodiffusion (Soret effect) is neglected on purpose (though it would be obviously relevant for such cases involving hydrogen as a fuel). <br />
The authors are fully aware that this is a crude approximation of reality<ref name="pecs"/>. But it must be kept in mind that the focus is set here on the detailed comparison between different codes and algorithms, and not on the resulting flow or flame structure.<br />
Hence, in this and in the next section, the values of the '''Lewis number''' for each species given in the table below shall be used to enable direct comparisons. <br />
These values have been estimated from a separate DNS for a similar configuration using the mixture-averaged diffusion model.<br />
The precise description of the associated thermodynamic state is available on the website.<br />
<br />
{| class="wikitable alternance center"<br />
|+ Approximate Lewis number for the species appearing in the kinetic scheme of<ref name="Boivin2011" />, to be enforced for Step 3 and Step 4 of the benchmark<br />
|-<br />
! scope="row" | Species<br />
| <math>H_2</math><br />
| <math>H</math><br />
| <math>O_2</math><br />
| <math>OH</math><br />
| <math>O</math><br />
| <math>H_2O</math><br />
| <math>HO_2</math><br />
| <math>H_2O_2</math><br />
| <math>N_2</math><br />
|-<br />
! scope="row" | Lewis number<br />
| 0.3290<br />
| 0.2228<br />
| 1.2703<br />
| 0.8279<br />
| 0.8128<br />
| 1.0741<br />
| 1.2582<br />
| 1.2665<br />
| 1.8268<br />
|}<br />
<br />
On the other hand, considering that implementing advanced models for fluid viscosity and thermal conductivity is not difficult and does not increase the computational time significantly, local values of these quantities depending on composition and temperature should be taken into account; <br />
More specifically, and to ease further comparisons, the same models for viscosity and conductivity have been retained in the three participating codes, i.e. the mixture averaged formalism that is used in both the '''Cantera''' and '''Chemkin packages'''.<br />
More details on these models are available on the benchmark repository<ref name="TGV-coria-cfd">[https://benchmark.coria-cfd.fr benchmark.coria-cfd.fr]</ref>, such as reference data for viscosity and conductivity along various cuts at initialization.<br />
<br />
The viscosity is variable in the computational domain as well as with time and the '''Reynolds number''' of this configuration can only be estimated using the minimal value of viscosity (which is obtained in the air at 300 K, <math>\nu_\mathrm{min} \approx 1.56 \, 10^{-5} \mathrm{m^2/s}</math>), leading to <math>Re = {u_0 L_0}/{\nu_{min}}=267</math> which guarantees a laminar flow.<br />
<br />
Concerning resolution, it is also suggested to keep a conservative resolution in time for this benchmark, corresponding to <math>CFL \le 0.25</math> and <math>Fo \le 0.15</math>. <br />
In space, each direction should be resolved by approximately 256 grid points, leading to a spatial resolution <math>\Delta x \approx 25 \mu m</math>.<br />
<br />
The '''reference time scale''' is here <math>\tau_\mathrm{ref}=L_0/u_0 = 0.25 \; \mathrm{ms}</math>.<br />
The simulation should be performed for a physical time of at least <math>t=0.5 \; \mathrm{ms} = 2 \tau_\mathrm{ref}</math> as most of the results presented in this article are taken from this instant.<br />
Data up to <math>t=2.5 \; \mathrm{ms} = 10 \tau_\mathrm{ref}</math> are available on the benchmark repository<ref name="TGV-coria-cfd" />.<br />
<br />
[[File:wmag_t0.png|250px|alt text]]<br />
[[File:T_t0.png|250px|alt text]]<br />
[[File:H2_t0.png|250px|alt text]]<br />
[[File:O2_t0.png|250px|alt text]]<br />
<br />
Initial fields of vorticity magnitude, temperature, mass fractions of <math>\mathrm{H_2}</math>, and mass fraction of <math>\mathrm{O_2}</math> (from left to right), for the 3-D, non-reacting case ('''Step 3''').<br />
<br />
[[File:profiles_3d_zoom.pdf|250px|alt text]]<br />
<br />
Initial profiles of temperature and of mass fractions at <math>y=0.5 L</math> and <math>z=0.5 L</math> for the 3-D, non-reacting case ('''Step 3''').<br />
<br />
= Kinetics =<br />
<br />
The Boivin scheme [https://doi.org/10.1016/j.proci.2010.05.002] has been used for this Benchmark.<br />
<br />
<br />
<br />
[https://benchmark.coria-cfd.fr/index.php/File:H2_williams_12.xml.zip]<br />
[https://benchmark.coria-cfd.fr/index.php/File:H2_williams_12.cti]<br />
<br />
= Transport and thermodynamic properties used for this Step =<br />
<br />
\cf{I think we all used the same transport (viscosity) and thermodynamic properties, but we should check again and compare that the implementations used in cantera/Chemkin provide the same values.}<br />
<br />
\gl{Good idea: we could use 3 or 4 relevant reference compositions to quantify the discrepancies between Cantera and Chemkin (on rho, Cp, conductivity, viscosity for example) and also put these on the website.}<br />
<br />
'''Put the transport data here'''<br />
<br />
= Aside suggestions =<br />
<br />
TBD<br />
<br />
= References =<br />
<br />
<references><br />
<ref name="Boivin2011"><br />
<bibtex><br />
@article{Boivin2011, <br />
author= {P. Boivin, C. Jiménez, A.L. Sanchez, and F.A. Williams},<br />
title= {An explicit reduced mechanism for <math>H_2</math>-air combustion.},<br />
journal={Proc. Combust. Inst.},<br />
year= {2011},<br />
volume={33},<br />
pages={517--523},<br />
doi= {https://doi.org/10.1016/j.proci.2010.05.002}<br />
}<br />
</bibtex><br />
</ref><br />
<ref name="pecs"><br />
<bibtex><br />
@article{Pecs, <br />
author= {R. Hilbert, F. Tap, H. El-Rabii, and D. Thévenin},<br />
title= {Impact of detailed chemistry and transport models on turbulent combustion simulations},<br />
journal={Prog. Energ. Combust. Sci.},<br />
year= {2004},<br />
volume={30},<br />
pages={61--117},<br />
}<br />
</bibtex><br />
</ref><br />
</references></div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Step_3&diff=461Step 32020-08-25T21:04:34Z<p>Lartigue: /* Kinetics */</p>
<hr />
<div>= Step 3 description =<br />
<br />
The purpose of this configuration is to perform a simulation similar to '''Step 2''' but involving now a '''mixture of different gaseous species''' at different temperatures, still '''without any chemical reaction'''. <br />
In this manner, it is possible to assess the accuracy of the numerical models involved in the description of species and heat diffusion<br />
The thermodynamic and transport properties of the 9 species kinetic scheme of Boivin et al.<ref name="Boivin2011"/> should be used in preparation for the next step, even though the reactions are still neglected here.<br />
<br />
It should be noted that in this case, the '''density is variable''' in both '''space''' and '''time''' due to the changing composition, which was not the case in the previous step: this induces additional numerical difficulties that must be taken into account.<br />
However, there is no variation of the thermodynamic pressure in time, contrary to the next step.<br />
<br />
The simulation domain is a cubic box of size <math>[0;L]^3</math> with <math>L = 2 \pi L_0</math> in each direction, where <math>L_0 = 1\; \mathrm{mm}</math>. <br />
Compared to the previous step, this smaller size is needed to ensure a proper resolution of the reaction fronts for hydrogen oxidation that will appear in the next step and are associated to fixed characteristic dimensions. <br />
Again, periodic boundary conditions are used in all three spatial directions.<br />
<br />
The initial velocity field prescribed at <math>t=0</math> is identical to the one used in Step 2 with the reference velocity set to <math>u_0=4\;\mathrm{m/s}</math>.<br />
<br />
In preparation for the next step, the central part of the box is initially filled with a <math>H_2/N_2</math> mixture (fuel region, molar fraction <math>X^0_\mathrm{H_2}=0.45</math>, <math>T=300 K</math>) while the remaining domain is filled with air (oxidizer region, molar fraction <math>X^0_\mathrm{O_2}=0.21</math>, <math>T=300 K</math>). <br />
The corresponding mass fractions are <math>Y^0_\mathrm{H_2} \approx 0.0556</math> and <math>Y^0_\mathrm{O_2} \approx 0.233</math>.<br />
<br />
To avoid any numerical instability, '''the steps at the interface between both regions are smoothed out using hyperbolic tangent functions''' as follows:<br />
<br />
<math>R_d(x) = |x-0.5\,L|,</math> (1)<br />
<br />
<math>\psi(x) = 0.5\left[1+\mathrm{tanh}\left(\frac{c\,(R_d(x)-R)}{R}\right)\right],</math> (2)<br />
<br />
<math>Y_\mathrm{H_2}(x) = Y^0_{\mathrm{H_2}}\left(1-\psi(x)\right),</math> (3)<br />
<br />
<math>Y_\mathrm{O_2}(x) = Y^0_{\mathrm{O_2}}\left(\psi(x)\right).</math> (4)<br />
<br />
where <math>R = \pi/4 mm</math> and <math>c = 3</math> are the half-width of the central slab and stiffness parameter, respectively.<br />
As a consequence, there is initially a small region where both fuel and oxidizer coexist. <br />
Finally, a nitrogen complement is added everywhere using <math>Y_\mathrm{N_2}=1-Y_\mathrm{H_2}-Y_\mathrm{O_2}</math>.<br />
<br />
To also test the behaviour of the codes with respect to heat diffusion, '''a non-homogeneous temperature profile is finally imposed''' as follows:<br />
<br />
* Compute in each cell the equilibrium temperature for the local mixture for constant pressure and enthalpy by using a dedicated solver.<br />
<br />
* Enforce the resulting temperature profile, but keep the species profiles to their initial values (Eqs. (3) and (4)).<br />
<br />
It is found by using '''Cantera''' that the peak adiabatic temperature obtained '''at the fuel/air interface''' is <math>T_{ad}=1910.7~K</math>, leading to the profiles shown in the initial fields of [https://benchmark.coria-cfd.fr/images/4/40/Wmag_t0.png vorticity], [https://benchmark.coria-cfd.fr/images/6/6d/T_t0.png temperature], <br />
[https://benchmark.coria-cfd.fr/images/f/fa/H2_t0.png mass fraction of <math>H_2</math>], [https://benchmark.coria-cfd.fr/images/e/ef/O2_t0.png mass fraction of <math>O_2</math>] and the [https://benchmark.coria-cfd.fr/images/b/b4/Profiles_3d_zoom.pdf initial profiles of temperature and of mass fractions]. Users are encouraged to implement the exact initial profiles from the benchmark website to facilitate later comparisons. <br />
<br />
The resulting, initial configuration thus involves '''multiple species at different temperatures''' with steep profiles, just like in a real flame.<br />
However, for the present simulation, the reaction source terms are all still set to zero, as already mentioned: only '''convection''' and '''diffusion processes''' are considered.<br />
In order to enable benchmark computations also with '''DNS codes''' that do not provide advanced diffusion models, only constant values of the Lewis numbers are considered; for the same reason, thermodiffusion (Soret effect) is neglected on purpose (though it would be obviously relevant for such cases involving hydrogen as a fuel). <br />
The authors are fully aware that this is a crude approximation of reality<ref name="pecs"/>. But it must be kept in mind that the focus is set here on the detailed comparison between different codes and algorithms, and not on the resulting flow or flame structure.<br />
Hence, in this and in the next section, the values of the '''Lewis number''' for each species given in the table below shall be used to enable direct comparisons. <br />
These values have been estimated from a separate DNS for a similar configuration using the mixture-averaged diffusion model.<br />
The precise description of the associated thermodynamic state is available on the website.<br />
<br />
{| class="wikitable alternance center"<br />
|+ Approximate Lewis number for the species appearing in the kinetic scheme of<ref name="Boivin2011" />, to be enforced for Step 3 and Step 4 of the benchmark<br />
|-<br />
! scope="row" | Species<br />
| <math>H_2</math><br />
| <math>H</math><br />
| <math>O_2</math><br />
| <math>OH</math><br />
| <math>O</math><br />
| <math>H_2O</math><br />
| <math>HO_2</math><br />
| <math>H_2O_2</math><br />
| <math>N_2</math><br />
|-<br />
! scope="row" | Lewis number<br />
| 0.3290<br />
| 0.2228<br />
| 1.2703<br />
| 0.8279<br />
| 0.8128<br />
| 1.0741<br />
| 1.2582<br />
| 1.2665<br />
| 1.8268<br />
|}<br />
<br />
On the other hand, considering that implementing advanced models for fluid viscosity and thermal conductivity is not difficult and does not increase the computational time significantly, local values of these quantities depending on composition and temperature should be taken into account; <br />
More specifically, and to ease further comparisons, the same models for viscosity and conductivity have been retained in the three participating codes, i.e. the mixture averaged formalism that is used in both the '''Cantera''' and '''Chemkin packages'''.<br />
More details on these models are available on the benchmark repository<ref name="TGV-coria-cfd">[https://benchmark.coria-cfd.fr benchmark.coria-cfd.fr]</ref>, such as reference data for viscosity and conductivity along various cuts at initialization.<br />
<br />
The viscosity is variable in the computational domain as well as with time and the '''Reynolds number''' of this configuration can only be estimated using the minimal value of viscosity (which is obtained in the air at 300 K, <math>\nu_\mathrm{min} \approx 1.56 \, 10^{-5} \mathrm{m^2/s}</math>), leading to <math>Re = {u_0 L_0}/{\nu_{min}}=267</math> which guarantees a laminar flow.<br />
<br />
Concerning resolution, it is also suggested to keep a conservative resolution in time for this benchmark, corresponding to <math>CFL \le 0.25</math> and <math>Fo \le 0.15</math>. <br />
In space, each direction should be resolved by approximately 256 grid points, leading to a spatial resolution <math>\Delta x \approx 25 \mu m</math>.<br />
<br />
The '''reference time scale''' is here <math>\tau_\mathrm{ref}=L_0/u_0 = 0.25 \; \mathrm{ms}</math>.<br />
The simulation should be performed for a physical time of at least <math>t=0.5 \; \mathrm{ms} = 2 \tau_\mathrm{ref}</math> as most of the results presented in this article are taken from this instant.<br />
Data up to <math>t=2.5 \; \mathrm{ms} = 10 \tau_\mathrm{ref}</math> are available on the benchmark repository<ref name="TGV-coria-cfd" />.<br />
<br />
[[File:wmag_t0.png|250px|alt text]]<br />
[[File:T_t0.png|250px|alt text]]<br />
[[File:H2_t0.png|250px|alt text]]<br />
[[File:O2_t0.png|250px|alt text]]<br />
<br />
Initial fields of vorticity magnitude, temperature, mass fractions of <math>\mathrm{H_2}</math>, and mass fraction of <math>\mathrm{O_2}</math> (from left to right), for the 3-D, non-reacting case ('''Step 3''').<br />
<br />
[[File:profiles_3d_zoom.pdf|250px|alt text]]<br />
<br />
Initial profiles of temperature and of mass fractions at <math>y=0.5 L</math> and <math>z=0.5 L</math> for the 3-D, non-reacting case ('''Step 3''').<br />
<br />
= Kinetics =<br />
<br />
The Boivin scheme [https://doi.org/10.1016/j.proci.2010.05.002] has been used for this Benchmark.<br />
<br />
<br />
<br />
[https://benchmark.coria-cfd.fr/index.php/File:H2_williams_12.xml.zip]<br />
[https://benchmark.coria-cfd.fr/index.php/File:H2_williams_12.cti]<br />
<br />
<br />
Reference: <br />
P. Boivin, C. Jiménez, A.L. Sánchez, F.A. Williams,<br />
An explicit reduced mechanism for H2–air combustion,<br />
Proceedings of the Combustion Institute,<br />
Volume 33, Issue 1,<br />
2011,<br />
Pages 517-523,<br />
ISSN 1540-7489,<br />
<br />
= Transport and thermodynamic properties used for this Step =<br />
<br />
\cf{I think we all used the same transport (viscosity) and thermodynamic properties, but we should check again and compare that the implementations used in cantera/Chemkin provide the same values.}<br />
<br />
\gl{Good idea: we could use 3 or 4 relevant reference compositions to quantify the discrepancies between Cantera and Chemkin (on rho, Cp, conductivity, viscosity for example) and also put these on the website.}<br />
<br />
'''Put the transport data here'''<br />
<br />
= Aside suggestions =<br />
<br />
TBD<br />
<br />
= References =<br />
<br />
<references><br />
<ref name="Boivin2011"><br />
<bibtex><br />
@article{Boivin2011, <br />
author= {P. Boivin, C. Jiménez, A.L. Sanchez, and F.A. Williams},<br />
title= {An explicit reduced mechanism for <math>H_2</math>-air combustion.},<br />
journal={Proc. Combust. Inst.},<br />
year= {2011},<br />
volume={33},<br />
pages={517--523},<br />
doi= {https://doi.org/10.1016/j.proci.2010.05.002}<br />
}<br />
</bibtex><br />
</ref><br />
<ref name="pecs"><br />
<bibtex><br />
@article{Pecs, <br />
author= {R. Hilbert, F. Tap, H. El-Rabii, and D. Thévenin},<br />
title= {Impact of detailed chemistry and transport models on turbulent combustion simulations},<br />
journal={Prog. Energ. Combust. Sci.},<br />
year= {2004},<br />
volume={30},<br />
pages={61--117},<br />
}<br />
</bibtex><br />
</ref><br />
</references></div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Step_3&diff=460Step 32020-08-25T21:04:00Z<p>Lartigue: /* References */</p>
<hr />
<div>= Step 3 description =<br />
<br />
The purpose of this configuration is to perform a simulation similar to '''Step 2''' but involving now a '''mixture of different gaseous species''' at different temperatures, still '''without any chemical reaction'''. <br />
In this manner, it is possible to assess the accuracy of the numerical models involved in the description of species and heat diffusion<br />
The thermodynamic and transport properties of the 9 species kinetic scheme of Boivin et al.<ref name="Boivin2011"/> should be used in preparation for the next step, even though the reactions are still neglected here.<br />
<br />
It should be noted that in this case, the '''density is variable''' in both '''space''' and '''time''' due to the changing composition, which was not the case in the previous step: this induces additional numerical difficulties that must be taken into account.<br />
However, there is no variation of the thermodynamic pressure in time, contrary to the next step.<br />
<br />
The simulation domain is a cubic box of size <math>[0;L]^3</math> with <math>L = 2 \pi L_0</math> in each direction, where <math>L_0 = 1\; \mathrm{mm}</math>. <br />
Compared to the previous step, this smaller size is needed to ensure a proper resolution of the reaction fronts for hydrogen oxidation that will appear in the next step and are associated to fixed characteristic dimensions. <br />
Again, periodic boundary conditions are used in all three spatial directions.<br />
<br />
The initial velocity field prescribed at <math>t=0</math> is identical to the one used in Step 2 with the reference velocity set to <math>u_0=4\;\mathrm{m/s}</math>.<br />
<br />
In preparation for the next step, the central part of the box is initially filled with a <math>H_2/N_2</math> mixture (fuel region, molar fraction <math>X^0_\mathrm{H_2}=0.45</math>, <math>T=300 K</math>) while the remaining domain is filled with air (oxidizer region, molar fraction <math>X^0_\mathrm{O_2}=0.21</math>, <math>T=300 K</math>). <br />
The corresponding mass fractions are <math>Y^0_\mathrm{H_2} \approx 0.0556</math> and <math>Y^0_\mathrm{O_2} \approx 0.233</math>.<br />
<br />
To avoid any numerical instability, '''the steps at the interface between both regions are smoothed out using hyperbolic tangent functions''' as follows:<br />
<br />
<math>R_d(x) = |x-0.5\,L|,</math> (1)<br />
<br />
<math>\psi(x) = 0.5\left[1+\mathrm{tanh}\left(\frac{c\,(R_d(x)-R)}{R}\right)\right],</math> (2)<br />
<br />
<math>Y_\mathrm{H_2}(x) = Y^0_{\mathrm{H_2}}\left(1-\psi(x)\right),</math> (3)<br />
<br />
<math>Y_\mathrm{O_2}(x) = Y^0_{\mathrm{O_2}}\left(\psi(x)\right).</math> (4)<br />
<br />
where <math>R = \pi/4 mm</math> and <math>c = 3</math> are the half-width of the central slab and stiffness parameter, respectively.<br />
As a consequence, there is initially a small region where both fuel and oxidizer coexist. <br />
Finally, a nitrogen complement is added everywhere using <math>Y_\mathrm{N_2}=1-Y_\mathrm{H_2}-Y_\mathrm{O_2}</math>.<br />
<br />
To also test the behaviour of the codes with respect to heat diffusion, '''a non-homogeneous temperature profile is finally imposed''' as follows:<br />
<br />
* Compute in each cell the equilibrium temperature for the local mixture for constant pressure and enthalpy by using a dedicated solver.<br />
<br />
* Enforce the resulting temperature profile, but keep the species profiles to their initial values (Eqs. (3) and (4)).<br />
<br />
It is found by using '''Cantera''' that the peak adiabatic temperature obtained '''at the fuel/air interface''' is <math>T_{ad}=1910.7~K</math>, leading to the profiles shown in the initial fields of [https://benchmark.coria-cfd.fr/images/4/40/Wmag_t0.png vorticity], [https://benchmark.coria-cfd.fr/images/6/6d/T_t0.png temperature], <br />
[https://benchmark.coria-cfd.fr/images/f/fa/H2_t0.png mass fraction of <math>H_2</math>], [https://benchmark.coria-cfd.fr/images/e/ef/O2_t0.png mass fraction of <math>O_2</math>] and the [https://benchmark.coria-cfd.fr/images/b/b4/Profiles_3d_zoom.pdf initial profiles of temperature and of mass fractions]. Users are encouraged to implement the exact initial profiles from the benchmark website to facilitate later comparisons. <br />
<br />
The resulting, initial configuration thus involves '''multiple species at different temperatures''' with steep profiles, just like in a real flame.<br />
However, for the present simulation, the reaction source terms are all still set to zero, as already mentioned: only '''convection''' and '''diffusion processes''' are considered.<br />
In order to enable benchmark computations also with '''DNS codes''' that do not provide advanced diffusion models, only constant values of the Lewis numbers are considered; for the same reason, thermodiffusion (Soret effect) is neglected on purpose (though it would be obviously relevant for such cases involving hydrogen as a fuel). <br />
The authors are fully aware that this is a crude approximation of reality<ref name="pecs"/>. But it must be kept in mind that the focus is set here on the detailed comparison between different codes and algorithms, and not on the resulting flow or flame structure.<br />
Hence, in this and in the next section, the values of the '''Lewis number''' for each species given in the table below shall be used to enable direct comparisons. <br />
These values have been estimated from a separate DNS for a similar configuration using the mixture-averaged diffusion model.<br />
The precise description of the associated thermodynamic state is available on the website.<br />
<br />
{| class="wikitable alternance center"<br />
|+ Approximate Lewis number for the species appearing in the kinetic scheme of<ref name="Boivin2011" />, to be enforced for Step 3 and Step 4 of the benchmark<br />
|-<br />
! scope="row" | Species<br />
| <math>H_2</math><br />
| <math>H</math><br />
| <math>O_2</math><br />
| <math>OH</math><br />
| <math>O</math><br />
| <math>H_2O</math><br />
| <math>HO_2</math><br />
| <math>H_2O_2</math><br />
| <math>N_2</math><br />
|-<br />
! scope="row" | Lewis number<br />
| 0.3290<br />
| 0.2228<br />
| 1.2703<br />
| 0.8279<br />
| 0.8128<br />
| 1.0741<br />
| 1.2582<br />
| 1.2665<br />
| 1.8268<br />
|}<br />
<br />
On the other hand, considering that implementing advanced models for fluid viscosity and thermal conductivity is not difficult and does not increase the computational time significantly, local values of these quantities depending on composition and temperature should be taken into account; <br />
More specifically, and to ease further comparisons, the same models for viscosity and conductivity have been retained in the three participating codes, i.e. the mixture averaged formalism that is used in both the '''Cantera''' and '''Chemkin packages'''.<br />
More details on these models are available on the benchmark repository<ref name="TGV-coria-cfd">[https://benchmark.coria-cfd.fr benchmark.coria-cfd.fr]</ref>, such as reference data for viscosity and conductivity along various cuts at initialization.<br />
<br />
The viscosity is variable in the computational domain as well as with time and the '''Reynolds number''' of this configuration can only be estimated using the minimal value of viscosity (which is obtained in the air at 300 K, <math>\nu_\mathrm{min} \approx 1.56 \, 10^{-5} \mathrm{m^2/s}</math>), leading to <math>Re = {u_0 L_0}/{\nu_{min}}=267</math> which guarantees a laminar flow.<br />
<br />
Concerning resolution, it is also suggested to keep a conservative resolution in time for this benchmark, corresponding to <math>CFL \le 0.25</math> and <math>Fo \le 0.15</math>. <br />
In space, each direction should be resolved by approximately 256 grid points, leading to a spatial resolution <math>\Delta x \approx 25 \mu m</math>.<br />
<br />
The '''reference time scale''' is here <math>\tau_\mathrm{ref}=L_0/u_0 = 0.25 \; \mathrm{ms}</math>.<br />
The simulation should be performed for a physical time of at least <math>t=0.5 \; \mathrm{ms} = 2 \tau_\mathrm{ref}</math> as most of the results presented in this article are taken from this instant.<br />
Data up to <math>t=2.5 \; \mathrm{ms} = 10 \tau_\mathrm{ref}</math> are available on the benchmark repository<ref name="TGV-coria-cfd" />.<br />
<br />
[[File:wmag_t0.png|250px|alt text]]<br />
[[File:T_t0.png|250px|alt text]]<br />
[[File:H2_t0.png|250px|alt text]]<br />
[[File:O2_t0.png|250px|alt text]]<br />
<br />
Initial fields of vorticity magnitude, temperature, mass fractions of <math>\mathrm{H_2}</math>, and mass fraction of <math>\mathrm{O_2}</math> (from left to right), for the 3-D, non-reacting case ('''Step 3''').<br />
<br />
[[File:profiles_3d_zoom.pdf|250px|alt text]]<br />
<br />
Initial profiles of temperature and of mass fractions at <math>y=0.5 L</math> and <math>z=0.5 L</math> for the 3-D, non-reacting case ('''Step 3''').<br />
<br />
= Kinetics =<br />
<br />
The Boivin scheme has been used for this Benchmark.<br />
<br />
<br />
<br />
[https://benchmark.coria-cfd.fr/index.php/File:H2_williams_12.xml.zip]<br />
[https://benchmark.coria-cfd.fr/index.php/File:H2_williams_12.cti]<br />
<br />
<br />
Reference: <br />
P. Boivin, C. Jiménez, A.L. Sánchez, F.A. Williams,<br />
An explicit reduced mechanism for H2–air combustion,<br />
Proceedings of the Combustion Institute,<br />
Volume 33, Issue 1,<br />
2011,<br />
Pages 517-523,<br />
ISSN 1540-7489,<br />
https://doi.org/10.1016/j.proci.2010.05.002.<br />
(http://www.sciencedirect.com/science/article/pii/S1540748910000039)<br />
<br />
= Transport and thermodynamic properties used for this Step =<br />
<br />
\cf{I think we all used the same transport (viscosity) and thermodynamic properties, but we should check again and compare that the implementations used in cantera/Chemkin provide the same values.}<br />
<br />
\gl{Good idea: we could use 3 or 4 relevant reference compositions to quantify the discrepancies between Cantera and Chemkin (on rho, Cp, conductivity, viscosity for example) and also put these on the website.}<br />
<br />
'''Put the transport data here'''<br />
<br />
= Aside suggestions =<br />
<br />
TBD<br />
<br />
= References =<br />
<br />
<references><br />
<ref name="Boivin2011"><br />
<bibtex><br />
@article{Boivin2011, <br />
author= {P. Boivin, C. Jiménez, A.L. Sanchez, and F.A. Williams},<br />
title= {An explicit reduced mechanism for <math>H_2</math>-air combustion.},<br />
journal={Proc. Combust. Inst.},<br />
year= {2011},<br />
volume={33},<br />
pages={517--523},<br />
doi= {https://doi.org/10.1016/j.proci.2010.05.002}<br />
}<br />
</bibtex><br />
</ref><br />
<ref name="pecs"><br />
<bibtex><br />
@article{Pecs, <br />
author= {R. Hilbert, F. Tap, H. El-Rabii, and D. Thévenin},<br />
title= {Impact of detailed chemistry and transport models on turbulent combustion simulations},<br />
journal={Prog. Energ. Combust. Sci.},<br />
year= {2004},<br />
volume={30},<br />
pages={61--117},<br />
}<br />
</bibtex><br />
</ref><br />
</references></div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Step_3&diff=459Step 32020-08-25T20:57:49Z<p>Lartigue: /* Kinetics */</p>
<hr />
<div>= Step 3 description =<br />
<br />
The purpose of this configuration is to perform a simulation similar to '''Step 2''' but involving now a '''mixture of different gaseous species''' at different temperatures, still '''without any chemical reaction'''. <br />
In this manner, it is possible to assess the accuracy of the numerical models involved in the description of species and heat diffusion<br />
The thermodynamic and transport properties of the 9 species kinetic scheme of Boivin et al.<ref name="Boivin2011"/> should be used in preparation for the next step, even though the reactions are still neglected here.<br />
<br />
It should be noted that in this case, the '''density is variable''' in both '''space''' and '''time''' due to the changing composition, which was not the case in the previous step: this induces additional numerical difficulties that must be taken into account.<br />
However, there is no variation of the thermodynamic pressure in time, contrary to the next step.<br />
<br />
The simulation domain is a cubic box of size <math>[0;L]^3</math> with <math>L = 2 \pi L_0</math> in each direction, where <math>L_0 = 1\; \mathrm{mm}</math>. <br />
Compared to the previous step, this smaller size is needed to ensure a proper resolution of the reaction fronts for hydrogen oxidation that will appear in the next step and are associated to fixed characteristic dimensions. <br />
Again, periodic boundary conditions are used in all three spatial directions.<br />
<br />
The initial velocity field prescribed at <math>t=0</math> is identical to the one used in Step 2 with the reference velocity set to <math>u_0=4\;\mathrm{m/s}</math>.<br />
<br />
In preparation for the next step, the central part of the box is initially filled with a <math>H_2/N_2</math> mixture (fuel region, molar fraction <math>X^0_\mathrm{H_2}=0.45</math>, <math>T=300 K</math>) while the remaining domain is filled with air (oxidizer region, molar fraction <math>X^0_\mathrm{O_2}=0.21</math>, <math>T=300 K</math>). <br />
The corresponding mass fractions are <math>Y^0_\mathrm{H_2} \approx 0.0556</math> and <math>Y^0_\mathrm{O_2} \approx 0.233</math>.<br />
<br />
To avoid any numerical instability, '''the steps at the interface between both regions are smoothed out using hyperbolic tangent functions''' as follows:<br />
<br />
<math>R_d(x) = |x-0.5\,L|,</math> (1)<br />
<br />
<math>\psi(x) = 0.5\left[1+\mathrm{tanh}\left(\frac{c\,(R_d(x)-R)}{R}\right)\right],</math> (2)<br />
<br />
<math>Y_\mathrm{H_2}(x) = Y^0_{\mathrm{H_2}}\left(1-\psi(x)\right),</math> (3)<br />
<br />
<math>Y_\mathrm{O_2}(x) = Y^0_{\mathrm{O_2}}\left(\psi(x)\right).</math> (4)<br />
<br />
where <math>R = \pi/4 mm</math> and <math>c = 3</math> are the half-width of the central slab and stiffness parameter, respectively.<br />
As a consequence, there is initially a small region where both fuel and oxidizer coexist. <br />
Finally, a nitrogen complement is added everywhere using <math>Y_\mathrm{N_2}=1-Y_\mathrm{H_2}-Y_\mathrm{O_2}</math>.<br />
<br />
To also test the behaviour of the codes with respect to heat diffusion, '''a non-homogeneous temperature profile is finally imposed''' as follows:<br />
<br />
* Compute in each cell the equilibrium temperature for the local mixture for constant pressure and enthalpy by using a dedicated solver.<br />
<br />
* Enforce the resulting temperature profile, but keep the species profiles to their initial values (Eqs. (3) and (4)).<br />
<br />
It is found by using '''Cantera''' that the peak adiabatic temperature obtained '''at the fuel/air interface''' is <math>T_{ad}=1910.7~K</math>, leading to the profiles shown in the initial fields of [https://benchmark.coria-cfd.fr/images/4/40/Wmag_t0.png vorticity], [https://benchmark.coria-cfd.fr/images/6/6d/T_t0.png temperature], <br />
[https://benchmark.coria-cfd.fr/images/f/fa/H2_t0.png mass fraction of <math>H_2</math>], [https://benchmark.coria-cfd.fr/images/e/ef/O2_t0.png mass fraction of <math>O_2</math>] and the [https://benchmark.coria-cfd.fr/images/b/b4/Profiles_3d_zoom.pdf initial profiles of temperature and of mass fractions]. Users are encouraged to implement the exact initial profiles from the benchmark website to facilitate later comparisons. <br />
<br />
The resulting, initial configuration thus involves '''multiple species at different temperatures''' with steep profiles, just like in a real flame.<br />
However, for the present simulation, the reaction source terms are all still set to zero, as already mentioned: only '''convection''' and '''diffusion processes''' are considered.<br />
In order to enable benchmark computations also with '''DNS codes''' that do not provide advanced diffusion models, only constant values of the Lewis numbers are considered; for the same reason, thermodiffusion (Soret effect) is neglected on purpose (though it would be obviously relevant for such cases involving hydrogen as a fuel). <br />
The authors are fully aware that this is a crude approximation of reality<ref name="pecs"/>. But it must be kept in mind that the focus is set here on the detailed comparison between different codes and algorithms, and not on the resulting flow or flame structure.<br />
Hence, in this and in the next section, the values of the '''Lewis number''' for each species given in the table below shall be used to enable direct comparisons. <br />
These values have been estimated from a separate DNS for a similar configuration using the mixture-averaged diffusion model.<br />
The precise description of the associated thermodynamic state is available on the website.<br />
<br />
{| class="wikitable alternance center"<br />
|+ Approximate Lewis number for the species appearing in the kinetic scheme of<ref name="Boivin2011" />, to be enforced for Step 3 and Step 4 of the benchmark<br />
|-<br />
! scope="row" | Species<br />
| <math>H_2</math><br />
| <math>H</math><br />
| <math>O_2</math><br />
| <math>OH</math><br />
| <math>O</math><br />
| <math>H_2O</math><br />
| <math>HO_2</math><br />
| <math>H_2O_2</math><br />
| <math>N_2</math><br />
|-<br />
! scope="row" | Lewis number<br />
| 0.3290<br />
| 0.2228<br />
| 1.2703<br />
| 0.8279<br />
| 0.8128<br />
| 1.0741<br />
| 1.2582<br />
| 1.2665<br />
| 1.8268<br />
|}<br />
<br />
On the other hand, considering that implementing advanced models for fluid viscosity and thermal conductivity is not difficult and does not increase the computational time significantly, local values of these quantities depending on composition and temperature should be taken into account; <br />
More specifically, and to ease further comparisons, the same models for viscosity and conductivity have been retained in the three participating codes, i.e. the mixture averaged formalism that is used in both the '''Cantera''' and '''Chemkin packages'''.<br />
More details on these models are available on the benchmark repository<ref name="TGV-coria-cfd">[https://benchmark.coria-cfd.fr benchmark.coria-cfd.fr]</ref>, such as reference data for viscosity and conductivity along various cuts at initialization.<br />
<br />
The viscosity is variable in the computational domain as well as with time and the '''Reynolds number''' of this configuration can only be estimated using the minimal value of viscosity (which is obtained in the air at 300 K, <math>\nu_\mathrm{min} \approx 1.56 \, 10^{-5} \mathrm{m^2/s}</math>), leading to <math>Re = {u_0 L_0}/{\nu_{min}}=267</math> which guarantees a laminar flow.<br />
<br />
Concerning resolution, it is also suggested to keep a conservative resolution in time for this benchmark, corresponding to <math>CFL \le 0.25</math> and <math>Fo \le 0.15</math>. <br />
In space, each direction should be resolved by approximately 256 grid points, leading to a spatial resolution <math>\Delta x \approx 25 \mu m</math>.<br />
<br />
The '''reference time scale''' is here <math>\tau_\mathrm{ref}=L_0/u_0 = 0.25 \; \mathrm{ms}</math>.<br />
The simulation should be performed for a physical time of at least <math>t=0.5 \; \mathrm{ms} = 2 \tau_\mathrm{ref}</math> as most of the results presented in this article are taken from this instant.<br />
Data up to <math>t=2.5 \; \mathrm{ms} = 10 \tau_\mathrm{ref}</math> are available on the benchmark repository<ref name="TGV-coria-cfd" />.<br />
<br />
[[File:wmag_t0.png|250px|alt text]]<br />
[[File:T_t0.png|250px|alt text]]<br />
[[File:H2_t0.png|250px|alt text]]<br />
[[File:O2_t0.png|250px|alt text]]<br />
<br />
Initial fields of vorticity magnitude, temperature, mass fractions of <math>\mathrm{H_2}</math>, and mass fraction of <math>\mathrm{O_2}</math> (from left to right), for the 3-D, non-reacting case ('''Step 3''').<br />
<br />
[[File:profiles_3d_zoom.pdf|250px|alt text]]<br />
<br />
Initial profiles of temperature and of mass fractions at <math>y=0.5 L</math> and <math>z=0.5 L</math> for the 3-D, non-reacting case ('''Step 3''').<br />
<br />
= Kinetics =<br />
<br />
The Boivin scheme has been used for this Benchmark.<br />
<br />
<br />
<br />
[https://benchmark.coria-cfd.fr/index.php/File:H2_williams_12.xml.zip]<br />
[https://benchmark.coria-cfd.fr/index.php/File:H2_williams_12.cti]<br />
<br />
<br />
Reference: <br />
P. Boivin, C. Jiménez, A.L. Sánchez, F.A. Williams,<br />
An explicit reduced mechanism for H2–air combustion,<br />
Proceedings of the Combustion Institute,<br />
Volume 33, Issue 1,<br />
2011,<br />
Pages 517-523,<br />
ISSN 1540-7489,<br />
https://doi.org/10.1016/j.proci.2010.05.002.<br />
(http://www.sciencedirect.com/science/article/pii/S1540748910000039)<br />
<br />
= Transport and thermodynamic properties used for this Step =<br />
<br />
\cf{I think we all used the same transport (viscosity) and thermodynamic properties, but we should check again and compare that the implementations used in cantera/Chemkin provide the same values.}<br />
<br />
\gl{Good idea: we could use 3 or 4 relevant reference compositions to quantify the discrepancies between Cantera and Chemkin (on rho, Cp, conductivity, viscosity for example) and also put these on the website.}<br />
<br />
'''Put the transport data here'''<br />
<br />
= Aside suggestions =<br />
<br />
TBD<br />
<br />
= References =<br />
<br />
TBD<br />
<br />
<references><br />
<ref name="Boivin2011"><br />
<bibtex><br />
@article{Boivin2011, <br />
author= {P. Boivin, C. Jiménez, A.L. Sanchez, and F.A. Williams},<br />
title= {An explicit reduced mechanism for <math>H_2</math>-air combustion.},<br />
journal={Proc. Combust. Inst.},<br />
year= {2011},<br />
volume={33},<br />
pages={517--523},<br />
}<br />
</bibtex><br />
</ref><br />
<ref name="pecs"><br />
<bibtex><br />
@article{Pecs, <br />
author= {R. Hilbert, F. Tap, H. El-Rabii, and D. Thévenin},<br />
title= {Impact of detailed chemistry and transport models on turbulent combustion simulations},<br />
journal={Prog. Energ. Combust. Sci.},<br />
year= {2004},<br />
volume={30},<br />
pages={61--117},<br />
}<br />
</bibtex><br />
</ref><br />
</references></div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=File:H2_williams_12.xml.zip&diff=458File:H2 williams 12.xml.zip2020-08-25T20:52:50Z<p>Lartigue: H2/Air kinetic scheme in Cantera ctml format
Reference:
P. Boivin, C. Jiménez, A.L. Sánchez, F.A. Williams,
An explicit reduced mechanism for H2–air combustion,
Proceedings of the Combustion Institute,
Volume 33, Issue 1,
2011,
Pages 517-523,
ISSN...</p>
<hr />
<div>H2/Air kinetic scheme in Cantera ctml format<br />
Reference: <br />
P. Boivin, C. Jiménez, A.L. Sánchez, F.A. Williams,<br />
An explicit reduced mechanism for H2–air combustion,<br />
Proceedings of the Combustion Institute,<br />
Volume 33, Issue 1,<br />
2011,<br />
Pages 517-523,<br />
ISSN 1540-7489,<br />
https://doi.org/10.1016/j.proci.2010.05.002.<br />
(http://www.sciencedirect.com/science/article/pii/S1540748910000039)</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=File:H2_williams_12.cti&diff=457File:H2 williams 12.cti2020-08-25T20:44:33Z<p>Lartigue: H2/Air kinetic scheme in Cantera cti format
Reference:
P. Boivin, C. Jiménez, A.L. Sánchez, F.A. Williams,
An explicit reduced mechanism for H2–air combustion,
Proceedings of the Combustion Institute,
Volume 33, Issue 1,
2011,
Pages 517-523,
ISSN...</p>
<hr />
<div>H2/Air kinetic scheme in Cantera cti format<br />
Reference: <br />
P. Boivin, C. Jiménez, A.L. Sánchez, F.A. Williams,<br />
An explicit reduced mechanism for H2–air combustion,<br />
Proceedings of the Combustion Institute,<br />
Volume 33, Issue 1,<br />
2011,<br />
Pages 517-523,<br />
ISSN 1540-7489,<br />
https://doi.org/10.1016/j.proci.2010.05.002.<br />
(http://www.sciencedirect.com/science/article/pii/S1540748910000039)</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Main_Page&diff=330Main Page2020-08-23T18:28:49Z<p>Lartigue: </p>
<hr />
<div>{{#customtitle:TGV Benchmark|The Taylor-Green Vortex as a Benchmark - benchmark.coria-cfd.fr}}<br />
<br />
= The Taylor-Green Vortex as a Benchmark for High-Fidelity Combustion Simulations Using Low-Mach Solvers =<br />
<br />
The present web site is a complement of an article that has been submitted to '''Computers and Fluids''' in August 2020.<br />
<br />
Verification and validation are crucial steps for the development of any numerical model.<br />
<br />
While suitable processes have been established for commercial Computational Fluid Dynamics (CFD) codes, more difficult challenges must be faced for high-fidelity solvers.<br />
<br />
Benchmarks have been proposed in a series of dedicated conferences for non-reacting configurations.<br />
However, to our knowledge, no suitable approach has been published up to now regarding turbulent reacting flows.<br />
<br />
'''The purpose of this website is to present a full verification and validation chain for high-resolution codes employed to simulate turbulent reacting flows, first for Direct Numerical Simulation (DNS) of turbulent combustion in the limit of low Mach numbers.'''<br />
<br />
The selected configuration builds on top of the Taylor-Green vortex.<br />
Verification takes place by comparison with the analytical solution in two dimensions.<br />
Validation of the single-component flow is ensured by comparisons with published results obtained with a spectral code.<br />
Mixing without reaction is then considered, before computing finally a hydrogen-oxygen flame interacting with a 3-D Taylor-Green vortex. <br />
Three different low-Mach DNS solvers have been used for this study, demonstrating that the final accuracy of the DNS simulations is of the order of 1% for all quantities considered.<br />
<br />
The website is organised in four major parts:<br />
* A presentation of the three [[codes]] used for the Benchmark<br />
* The [[description]] of the test cases<br />
* The [[analysis]] of the results<br />
* An attempt to give a few guidelines on the [[performances]] that could be expected on the 3D test-cases.<br />
<br />
All the raw [[results]] of the 3 codes are available online.<br />
<br />
<div class="infobox floatright" style="width: 320px;"><br />
{| class="floatright" style="border: 1px solid #ccc; margin: 1px;"<br />
|{{#widget:YouTube|id=k7JRrvBOSSY|width=300|height=250}}<br />
|}<br />
</div></div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Main_Page&diff=214Main Page2020-08-16T18:44:06Z<p>Lartigue: /* The Taylor-Green Vortex as a Benchmark for High-Fidelity Combustion Simulations Using Low-Mach Solvers */</p>
<hr />
<div>= The Taylor-Green Vortex as a Benchmark for High-Fidelity Combustion Simulations Using Low-Mach Solvers =<br />
<br />
The present web site is a complement of an article that has been submitted to '''Computers and Fluids''' in August 2020.<br />
<br />
Verification and validation are crucial steps for the development of any numerical model.<br />
<br />
While suitable processes have been established for commercial Computational Fluid Dynamics (CFD) codes, more difficult challenges must be faced for high-fidelity solvers.<br />
<br />
Benchmarks have been proposed in a series of dedicated conferences for non-reacting configurations.<br />
However, to our knowledge, no suitable approach has been published up to now regarding turbulent reacting flows.<br />
<br />
'''The purpose of this website is to present a full verification and validation chain for high-resolution codes employed to simulate turbulent reacting flows, first for Direct Numerical Simulation (DNS) of turbulent combustion in the limit of low Mach numbers.'''<br />
<br />
The selected configuration builds on top of the Taylor-Green vortex.<br />
Verification takes place by comparison with the analytical solution in two dimensions.<br />
Validation of the single-component flow is ensured by comparisons with published results obtained with a spectral code.<br />
Mixing without reaction is then considered, before computing finally a hydrogen-oxygen flame interacting with a 3-D Taylor-Green vortex. <br />
Three different low-Mach DNS solvers have been used for this study, demonstrating that the final accuracy of the DNS simulations is of the order of 1% for all quantities considered.<br />
<br />
The website is organised in four major parts:<br />
* A presentation of the three [[codes]] used for the Benchmark<br />
* The [[description]] of the test cases<br />
* The [[analysis]] of the results<br />
* An attempt to give a few guidelines on the [[performances]] that could be expected on the 3D test-cases.<br />
<br />
All the raw [[results]] of the 3 codes are available online.<br />
<br />
'''Put some images / videos here'''</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Step_3&diff=213Step 32020-08-16T18:42:29Z<p>Lartigue: /* Step 3 descritpion */</p>
<hr />
<div>= Step 3 description =<br />
<br />
The purpose of this configuration is to perform a simulation similar to Step 2 but involving now a mixture of different gaseous species at different temperatures, still without any chemical reaction. <br />
In this manner, it is possible to assess the accuracy of the numerical models involved in the description of species and heat diffusion<br />
The thermodynamic and transport properties of the 9 species kinetic scheme of Boivin et al.<ref name="Boivin2011">P. Boivin, C. Jim´enez, A.L. S´anchez, and F.A. Williams. An explicit reduced mechanism for H2-air combustion. Proc. Combust. Inst., 33:517–523, 2011.</ref> should be used in preparation for the next step, even though the reactions are still neglected here.<br />
<br />
It should be noted that in this case, the density is variable in both space and time due to the changing composition, which was not the case in the previous step: this induces additional numerical difficulties that must be taken into account.<br />
However, there is no variation of the thermodynamic pressure in time, contrary to the next step.<br />
<br />
The simulation domain is a cubic box of size <math>[0;L]^3</math> with <math>L = 2 \pi L_0</math> in each direction, where <math>L_0 = 1\; \mathrm{mm}</math>. <br />
Compared to the previous step, this smaller size is needed to ensure a proper resolution of the reaction fronts for hydrogen oxidation that will appear in the next step and are associated to fixed characteristic dimensions. <br />
Again, periodic boundary conditions are used in all three spatial directions.<br />
<br />
The initial velocity field prescribed at <math>t=0</math> is identical to the one used in Step 2 with the reference velocity set to <math>u_0=4\;\mathrm{m/s}</math>.<br />
<br />
In preparation for the next step, the central part of the box is initially filled with a <math>H_2/N_2</math> mixture (fuel region, molar fraction <math>X^0_\mathrm{H_2}=0.45</math>, <math>T=300 K</math>) while the remaining domain is filled with air (oxidizer region, molar fraction <math>X^0_\mathrm{O_2}=0.21</math>, <math>T=300 K</math>). <br />
The corresponding mass fractions are <math>Y^0_\mathrm{H_2} \approx 0.0556</math> and <math>Y^0_\mathrm{O_2} \approx 0.233</math>.<br />
<br />
To avoid any numerical instability, the steps at the interface between both regions are smoothed out using hyperbolic tangent functions as follows:<br />
<br />
<math>R_d(x) = |x-0.5\,L|,</math><br />
<br />
<math>\psi(x) = 0.5\left[1+\mathrm{tanh}\left(\frac{c\,(R_d(x)-R)}{R}\right)\right],</math><br />
<br />
<math>Y_\mathrm{H_2}(x) = Y^0_{\mathrm{H_2}}\left(1-\psi(x)\right),</math><br />
<br />
<math>Y_\mathrm{O_2}(x) = Y^0_{\mathrm{O_2}}\left(\psi(x)\right).</math><br />
<br />
where <math>R = \pi/4 mm</math> and <math>c = 3</math> are the half-width of the central slab and stiffness parameter, respectively.<br />
As a consequence, there is initially a small region where both fuel and oxidizer coexist. <br />
Finally, a nitrogen complement is added everywhere using <math>Y_\mathrm{N_2}=1-Y_\mathrm{H_2}-Y_\mathrm{O_2}</math>.<br />
<br />
To also test the behaviour of the codes with respect to heat diffusion, a non-homogeneous temperature profile is finally imposed as follows:<br />
<br />
* Compute in each cell the equilibrium temperature for the local mixture for constant pressure and enthalpy by using a dedicated solver.<br />
<br />
* Enforce the resulting temperature profile, but keep the species profiles to their initial values (Eqs.~\ref{eq:YF} and~\ref{eq:YOx}).<br />
<br />
It is found by using Cantera that the peak adiabatic temperature obtained at the fuel/air interface is <math>T_{ad}=1910.7~K</math>, leading to the profiles shown in Figs.~\ref{fig:temper_h2_o2_3-D} and \ref{fig:profiles_3-D_nonreacting}. Users are encouraged to implement the exact initial profiles from the benchmark website to facilitate later comparisons. <br />
<br />
The resulting, initial configuration thus involves multiple species at different temperatures with steep profiles, just like in a real flame.<br />
However, for the present simulation, the reaction source terms are all still set to zero, as already mentioned: only convection and diffusion processes are considered.<br />
In order to enable benchmark computations also with DNS codes that do not provide advanced diffusion models, only constant values of the Lewis numbers are considered; for the same reason, thermodiffusion (Soret effect) is neglected on purpose (though it would be obviously relevant for such cases involving hydrogen as a fuel). <br />
The authors are fully aware that this is a crude approximation of reality<ref name="pecs">R. Hilbert, F. Tap, H. El-Rabii, and D. Th´evenin. Impact of detailed chemistry and transport models on turbulent combustion simulations. Prog. Energ. Combust. Sci., 30:61–117, 2004.</ref>. But it must be kept in mind that the focus is set here on the detailed comparison between different codes and algorithms, and not on the resulting flow or flame structure.<br />
Hence, in this and in the next section, the values of the Lewis number for each species given in Tab.~\ref{Tab:Lewis} shall be used to enable direct comparisons. <br />
These values have been estimated from a separate DNS for a similar configuration using the mixture-averaged diffusion model.<br />
The precise description of the associated thermodynamic state is available on the website.<br />
<br />
{| class="wikitable alternance center"<br />
|+ Approximate Lewis number for the species appearing in the kinetic scheme of<ref name="Boivin2011" />, to be enforced for Step 3 and Step 4 of the benchmark<br />
|-<br />
! scope="row" | Species<br />
| <math>H_2</math><br />
| <math>H</math><br />
| <math>O_2</math><br />
| <math>OH</math><br />
| <math>O</math><br />
| <math>H_2O</math><br />
| <math>HO_2</math><br />
| <math>H_2O_2</math><br />
| <math>N_2</math><br />
|-<br />
! scope="row" | Lewis number<br />
| 0.3290<br />
| 0.2228<br />
| 1.2703<br />
| 0.8279<br />
| 0.8128<br />
| 1.0741<br />
| 1.2582<br />
| 1.2665<br />
| 1.8268<br />
|}<br />
<br />
On the other hand, considering that implementing advanced models for fluid viscosity and thermal conductivity is not difficult and does not increase the computational time significantly, local values of these quantities depending on composition and temperature should be taken into account; <br />
More specifically, and to ease further comparisons, the same models for viscosity and conductivity have been retained in the three participating codes, i.e. the mixture averaged formalism that is used in both the Cantera and Chemkin packages<br />
More details on these models are available on the benchmark repository<ref name="TGV-coria-cfd">[https://benchmark.coria-cfd.fr]</ref>, such as reference data for viscosity and conductivity along various cuts at initialization.<br />
<br />
The viscosity is variable in the computational domain as well as with time and the Reynolds number of this configuration can only be estimated using the minimal value of viscosity (which is obtained in the air at 300 K, <math>\nu_\mathrm{min} \approx 1.56 \, 10^{-5} \mathrm{m^2/s}</math>), leading to <math>Re = {u_0 L_0}/{\nu_{min}}=267</math> which guarantees a laminar flow.<br />
<br />
Concerning resolution, it is also suggested to keep a conservative resolution in time for this benchmark, corresponding to <math>CFL \le 0.25</math> and <math>Fo \le 0.15</math>. <br />
In space, each direction should be resolved by approximately 256 grid points, leading to a spatial resolution <math>\Delta x \approx 25 \mu m</math>.<br />
<br />
The reference time scale is here <math>\tau_\mathrm{ref}=L_0/u_0 = 0.25 \; \mathrm{ms}</math>.<br />
The simulation should be performed for a physical time of at least <math>t=0.5 \; \mathrm{ms} = 2 \tau_\mathrm{ref}</math> as most of the results presented in this article are taken from this instant.<br />
Data up to <math>t=2.5 \; \mathrm{ms} = 10 \tau_\mathrm{ref}</math> are available on the benchmark repository<ref name="TGV-coria-cfd" />.<br />
<br />
[[File:wmag_t0.png|250px|alt text]]<br />
[[File:T_t0.png|250px|alt text]]<br />
[[File:H2_t0.png|250px|alt text]]<br />
[[File:O2_t0.png|250px|alt text]]<br />
<br />
Initial fields of vorticity magnitude, temperature, mass fractions of <math>\mathrm{H_2}</math>, and mass fraction of <math>\mathrm{O_2}</math> (from left to right), for the 3-D, non-reacting case (Step 3).<br />
<br />
[[File:profiles_3d_zoom.pdf|250px|alt text]]<br />
<br />
Initial profiles of temperature and of mass fractions at <math>y=0.5 L</math> and <math>z=0.5 L</math> for the 3-D, non-reacting case (Step 3).<br />
<br />
= Kinetics =<br />
<br />
The Boivin scheme has been used for this Benchmark.<br />
<br />
'''Put the data here'''<br />
<br />
<br />
<br />
= Transport and thermodynamic properties used for this Step =<br />
<br />
\cf{I think we all used the same transport (viscosity) and thermodynamic properties, but we should check again and compare that the implementations used in cantera/Chemkin provide the same values.}<br />
<br />
\gl{Good idea: we could use 3 or 4 relevant reference compositions to quantify the discrepancies between Cantera and Chemkin (on rho, Cp, conductivity, viscosity for example) and also put these on the website.}<br />
<br />
'''Put the transport data here'''<br />
<br />
= Aside suggestions =<br />
<br />
TBD<br />
<br />
= References =<br />
<br />
TBD</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Main_Page&diff=212Main Page2020-08-16T18:41:43Z<p>Lartigue: </p>
<hr />
<div>= The Taylor-Green Vortex as a Benchmark for High-Fidelity Combustion Simulations Using Low-Mach Solvers =<br />
<br />
The present web site is a complement of an article that has been submitted to '''Computers and Fluids''' in August 2020.<br />
<br />
Verification and validation are crucial steps for the development of any numerical model.<br />
<br />
While suitable processes have been established for commercial Computational Fluid Dynamics (CFD) codes, more difficult challenges must be faced for high-fidelity solvers.<br />
<br />
Benchmarks have been proposed in a series of dedicated conferences for non-reacting configurations.<br />
However, to our knowledge, no suitable approach has been published up to now regarding turbulent reacting flows.<br />
<br />
'''The purpose of this website is to present a full verification and validation chain for high-resolution codes employed to simulate turbulent reacting flows, first for Direct Numerical Simulation (DNS) of turbulent combustion in the limit of low Mach numbers.'''<br />
<br />
The selected configuration builds on top of the Taylor-Green vortex.<br />
Verification takes place by comparison with the analytical solution in two dimensions.<br />
Validation of the single-component flow is ensured by comparisons with published results obtained with a spectral code.<br />
Mixing without reaction is then considered, before computing finally a hydrogen-oxygen flame interacting with a 3-D Taylor-Green vortex. <br />
Three different low-Mach DNS solvers have been used for this study, demonstrating that the final accuracy of the DNS simulations is of the order of 1% for all quantities considered.<br />
<br />
The website is organised in four major parts:<br />
* A presentation of the three [[codes]] used for the Benchmark<br />
* The [[description]] of the test cases<br />
* The [[analysis]] of the results<br />
* An attempt to give a few guidelines on the [[performances]] that could be expected on the 3D test-cases.<br />
<br />
All the raw [[results]] of the 3 codes are available.<br />
<br />
'''Put some images / videos here'''</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Step_3&diff=210Step 32020-08-16T18:35:59Z<p>Lartigue: /* Step 3 descritpion */</p>
<hr />
<div>= Step 3 descritpion =<br />
<br />
The purpose of this configuration is to perform a simulation similar to Step 2 but involving now a mixture of different gaseous species at different temperatures, still without any chemical reaction. <br />
In this manner, it is possible to assess the accuracy of the numerical models involved in the description of species and heat diffusion<br />
The thermodynamic and transport properties of the 9 species kinetic scheme of Boivin et al.<ref name="Boivin2011">P. Boivin, C. Jim´enez, A.L. S´anchez, and F.A. Williams. An explicit reduced mechanism for H2-air combustion. Proc. Combust. Inst., 33:517–523, 2011.</ref> should be used in preparation for the next step, even though the reactions are still neglected here.<br />
<br />
It should be noted that in this case, the density is variable in both space and time due to the changing composition, which was not the case in the previous step: this induces additional numerical difficulties that must be taken into account.<br />
However, there is no variation of the thermodynamic pressure in time, contrary to the next step.<br />
<br />
The simulation domain is a cubic box of size <math>[0;L]^3</math> with <math>L = 2 \pi L_0</math> in each direction, where <math>L_0 = 1\; \mathrm{mm}</math>. <br />
Compared to the previous step, this smaller size is needed to ensure a proper resolution of the reaction fronts for hydrogen oxidation that will appear in the next step and are associated to fixed characteristic dimensions. <br />
Again, periodic boundary conditions are used in all three spatial directions.<br />
<br />
The initial velocity field prescribed at <math>t=0</math> is identical to the one used in Step 2 with the reference velocity set to <math>u_0=4\;\mathrm{m/s}</math>.<br />
<br />
In preparation for the next step, the central part of the box is initially filled with a <math>H_2/N_2</math> mixture (fuel region, molar fraction <math>X^0_\mathrm{H_2}=0.45</math>, <math>T=300 K</math>) while the remaining domain is filled with air (oxidizer region, molar fraction <math>X^0_\mathrm{O_2}=0.21</math>, <math>T=300 K</math>). <br />
The corresponding mass fractions are <math>Y^0_\mathrm{H_2} \approx 0.0556</math> and <math>Y^0_\mathrm{O_2} \approx 0.233</math>.<br />
<br />
To avoid any numerical instability, the steps at the interface between both regions are smoothed out using hyperbolic tangent functions as follows:<br />
<br />
<math>R_d(x) = |x-0.5\,L|,</math><br />
<br />
<math>\psi(x) = 0.5\left[1+\mathrm{tanh}\left(\frac{c\,(R_d(x)-R)}{R}\right)\right],</math><br />
<br />
<math>Y_\mathrm{H_2}(x) = Y^0_{\mathrm{H_2}}\left(1-\psi(x)\right),</math><br />
<br />
<math>Y_\mathrm{O_2}(x) = Y^0_{\mathrm{O_2}}\left(\psi(x)\right).</math><br />
<br />
where <math>R = \pi/4 mm</math> and <math>c = 3</math> are the half-width of the central slab and stiffness parameter, respectively.<br />
As a consequence, there is initially a small region where both fuel and oxidizer coexist. <br />
Finally, a nitrogen complement is added everywhere using <math>Y_\mathrm{N_2}=1-Y_\mathrm{H_2}-Y_\mathrm{O_2}</math>.<br />
<br />
To also test the behaviour of the codes with respect to heat diffusion, a non-homogeneous temperature profile is finally imposed as follows:<br />
<br />
* Compute in each cell the equilibrium temperature for the local mixture for constant pressure and enthalpy by using a dedicated solver.<br />
<br />
* Enforce the resulting temperature profile, but keep the species profiles to their initial values (Eqs.~\ref{eq:YF} and~\ref{eq:YOx}).<br />
<br />
It is found by using Cantera that the peak adiabatic temperature obtained at the fuel/air interface is <math>T_{ad}=1910.7~K</math>, leading to the profiles shown in Figs.~\ref{fig:temper_h2_o2_3-D} and \ref{fig:profiles_3-D_nonreacting}. Users are encouraged to implement the exact initial profiles from the benchmark website to facilitate later comparisons. <br />
<br />
The resulting, initial configuration thus involves multiple species at different temperatures with steep profiles, just like in a real flame.<br />
However, for the present simulation, the reaction source terms are all still set to zero, as already mentioned: only convection and diffusion processes are considered.<br />
In order to enable benchmark computations also with DNS codes that do not provide advanced diffusion models, only constant values of the Lewis numbers are considered; for the same reason, thermodiffusion (Soret effect) is neglected on purpose (though it would be obviously relevant for such cases involving hydrogen as a fuel). <br />
The authors are fully aware that this is a crude approximation of reality<ref name="pecs">R. Hilbert, F. Tap, H. El-Rabii, and D. Th´evenin. Impact of detailed chemistry and transport models on turbulent combustion simulations. Prog. Energ. Combust. Sci., 30:61–117, 2004.</ref>. But it must be kept in mind that the focus is set here on the detailed comparison between different codes and algorithms, and not on the resulting flow or flame structure.<br />
Hence, in this and in the next section, the values of the Lewis number for each species given in Tab.~\ref{Tab:Lewis} shall be used to enable direct comparisons. <br />
These values have been estimated from a separate DNS for a similar configuration using the mixture-averaged diffusion model.<br />
The precise description of the associated thermodynamic state is available on the website.<br />
<br />
{| class="wikitable alternance center"<br />
|+ Approximate Lewis number for the species appearing in the kinetic scheme of<ref name="Boivin2011" />, to be enforced for Step 3 and Step 4 of the benchmark<br />
|-<br />
! scope="row" | Species<br />
| <math>H_2</math><br />
| <math>H</math><br />
| <math>O_2</math><br />
| <math>OH</math><br />
| <math>O</math><br />
| <math>H_2O</math><br />
| <math>HO_2</math><br />
| <math>H_2O_2</math><br />
| <math>N_2</math><br />
|-<br />
! scope="row" | Lewis number<br />
| 0.3290<br />
| 0.2228<br />
| 1.2703<br />
| 0.8279<br />
| 0.8128<br />
| 1.0741<br />
| 1.2582<br />
| 1.2665<br />
| 1.8268<br />
|}<br />
<br />
On the other hand, considering that implementing advanced models for fluid viscosity and thermal conductivity is not difficult and does not increase the computational time significantly, local values of these quantities depending on composition and temperature should be taken into account; <br />
More specifically, and to ease further comparisons, the same models for viscosity and conductivity have been retained in the three participating codes, i.e. the mixture averaged formalism that is used in both the Cantera and Chemkin packages<br />
More details on these models are available on the benchmark repository<ref name="TGV-coria-cfd">[https://benchmark.coria-cfd.fr]</ref>, such as reference data for viscosity and conductivity along various cuts at initialization.<br />
<br />
The viscosity is variable in the computational domain as well as with time and the Reynolds number of this configuration can only be estimated using the minimal value of viscosity (which is obtained in the air at 300 K, <math>\nu_\mathrm{min} \approx 1.56 \, 10^{-5} \mathrm{m^2/s}</math>), leading to <math>Re = {u_0 L_0}/{\nu_{min}}=267</math> which guarantees a laminar flow.<br />
<br />
Concerning resolution, it is also suggested to keep a conservative resolution in time for this benchmark, corresponding to <math>CFL \le 0.25</math> and <math>Fo \le 0.15</math>. <br />
In space, each direction should be resolved by approximately 256 grid points, leading to a spatial resolution <math>\Delta x \approx 25 \mu m</math>.<br />
<br />
The reference time scale is here <math>\tau_\mathrm{ref}=L_0/u_0 = 0.25 \; \mathrm{ms}</math>.<br />
The simulation should be performed for a physical time of at least <math>t=0.5 \; \mathrm{ms} = 2 \tau_\mathrm{ref}</math> as most of the results presented in this article are taken from this instant.<br />
Data up to <math>t=2.5 \; \mathrm{ms} = 10 \tau_\mathrm{ref}</math> are available on the benchmark repository<ref name="TGV-coria-cfd" />.<br />
<br />
[[File:wmag_t0.png|250px|alt text]]<br />
[[File:T_t0.png|250px|alt text]]<br />
[[File:H2_t0.png|250px|alt text]]<br />
[[File:O2_t0.png|250px|alt text]]<br />
<br />
Initial fields of vorticity magnitude, temperature, mass fractions of <math>\mathrm{H_2}</math>, and mass fraction of <math>\mathrm{O_2}</math> (from left to right), for the 3-D, non-reacting case (Step 3).<br />
<br />
[[File:profiles_3d_zoom.pdf|250px|alt text]]<br />
<br />
Initial profiles of temperature and of mass fractions at <math>y=0.5 L</math> and <math>z=0.5 L</math> for the 3-D, non-reacting case (Step 3).<br />
<br />
= Kinetics =<br />
<br />
The Boivin scheme has been used for this Benchmark.<br />
<br />
'''Put the data here'''<br />
<br />
<br />
<br />
= Transport and thermodynamic properties used for this Step =<br />
<br />
\cf{I think we all used the same transport (viscosity) and thermodynamic properties, but we should check again and compare that the implementations used in cantera/Chemkin provide the same values.}<br />
<br />
\gl{Good idea: we could use 3 or 4 relevant reference compositions to quantify the discrepancies between Cantera and Chemkin (on rho, Cp, conductivity, viscosity for example) and also put these on the website.}<br />
<br />
'''Put the transport data here'''<br />
<br />
= Aside suggestions =<br />
<br />
TBD<br />
<br />
= References =<br />
<br />
TBD</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Results&diff=38Results2020-07-29T19:36:45Z<p>Lartigue: Created page with " The results of the Benchmark are organized as follows: This is a link to get the file File:Step2 YALES2 512 CFL0.10.txt"</p>
<hr />
<div><br />
The results of the Benchmark are organized as follows:<br />
<br />
<br />
This is a link to get the file [[File:Step2 YALES2 512 CFL0.10.txt]]</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=File:Step2_YALES2_512_CFL0.10.txt&diff=37File:Step2 YALES2 512 CFL0.10.txt2020-07-29T19:36:12Z<p>Lartigue: </p>
<hr />
<div>Results of Step 2 obtained with YALES2 on a 512^3 grid with CFL=0.10 (the minimum timestep was approximately 0.87ms).<br />
<br />
This file contains 3 columns: <br />
<br />
* 1: simulation time [s]<br />
* 2: volume average of kinetic energy [m^2/s^2]<br />
* 3: volume average of dissipation [m^2/s^3]</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=File:Step2_YALES2_512_CFL0.10.txt&diff=36File:Step2 YALES2 512 CFL0.10.txt2020-07-29T19:33:36Z<p>Lartigue: Results of Step 2 obtained with YALES2 on a 512^3 grid with CFL=0.10 (the minimum timestep was approximately 0.87ms).
This file contains 3 columns:
1: simulation time [s]
2: volume average of kinetic energy [m^2/s^2]
3: volume average of dissipation [...</p>
<hr />
<div>Results of Step 2 obtained with YALES2 on a 512^3 grid with CFL=0.10 (the minimum timestep was approximately 0.87ms).<br />
This file contains 3 columns: <br />
1: simulation time [s]<br />
2: volume average of kinetic energy [m^2/s^2]<br />
3: volume average of dissipation [m^2/s^3]</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Step_4&diff=35Step 42020-07-27T14:19:24Z<p>Lartigue: </p>
<hr />
<div>= Description of Step 4 =<br />
<br />
In this last benchmark, Step 3 is repeated while now activating chemical reactions from the start. To this aim, the kinetic scheme of Boivin et al.~\cite{Boivin2011} involving 9 species and 12 reversible reactions should be used in order to enable direct comparisons.<br />
Figure~\ref{fig:profiles_3-D_nonreacting} shows the initial profiles for temperature, heat release, mass fractions of H$_2$ and of O$_2$ (????).<br />
As it can be observed from this figure, the species profiles are now different along the fuel/oxidizer interface compared to those in the non-reacting case (Step 3), shown in Fig.~\ref{fig:profiles_3-D_nonreacting}. <br />
Now, the local species concentration corresponds to the equilibrium values obtained after initializing composition with Eqs.~\ref{eq:YF} and ~\ref{eq:YOx} at initial temperature $T_0=300$ K and atmospheric pressure; in this manner, the initial conditions already mimic a real flame front, with local composition and temperature in agreement with each other.<br />
<br />
Again, periodic boundary conditions are used in each direction.<br />
There is thus no net fluxes of mass, momentum or energy in any direction: the simulation domain can thus be considered to be closed.<br />
As a consequence, and contrary to Step 3, the thermodynamic pressure evolves in time during the present simulation because of the heat released by the chemical reactions and of the resulting dilatation. <br />
To enable meaningful comparisons, it is necessary to take into account this effect in the energy or temperature equation:<br />
\begin{eqnarray}<br />
\frac{\partial T}{\partial t}+ u_j \frac{\partial T}{\partial x_j} &=& \frac{1}{\rho C_p}\left[\frac{dP_0}{dt}+\frac{\partial }{\partial x_j} \left(\lambda \frac{\partial T}{\partial x_j} \right) -\frac{\partial T}{\partial x_j} \sum_{k=1}^{N_s} \rho C_{p,k} Y_k \theta_{k,j} -\sum_{k=1}^{N_s} h_k \dot{\omega}_k\right]\, \label{eq:temp_energy}.<br />
\end{eqnarray}<br />
All symbols used in this equation are defined below.<br />
Usually, most simulations of low-Mach flow conditions assume thate the thermodynamic pressure $P_0$ does not vary in time in case of open boundaries, considering that $P_0$ corresponds to the surrounding (usually atmospheric) pressure; therefore, $dP_0/dt$ vanishes in Eq.~(\ref{eq:temp_energy}). <br />
On the other hand, for a closed computational domain -- as assumed here --, $P_0$ may change in time due to dilatation and must be evaluated at each timestep as follows~\cite{Motheau2016}:<br />
<br />
\begin{eqnarray}<br />
P_0 = \frac{M_0 \, \mathrm{R}}{\int_\Omega \left( T \sum_{k=1}^{N_s} Y_k/W_k \right)^{-1} dV}\, \label{eq:thermod_p0}.<br />
\end{eqnarray}<br />
<br />
This equation is obtained by integrating the equation of state for perfect gases on the whole computational domain while assuming that the total mass is fixed and that the thermodynamic pressure is homogeneous inside the domain.<br />
In Eqs.~(\ref{eq:temp_energy}) and (\ref{eq:thermod_p0}), $\rho$, $u_j$, $T$, $Y_k$, $W_k$, $N_s$, $C_{p,k}$, $h_k$, $\theta_{k,j}$ and $\mathrm{R}$ are the density of the gas mixture, $j$-th component of flow velocity, gas temperature, $k$-th species mass fraction, $k$-th species molecular weight, number of species, $k$-th species specific heat capacity at constant pressure, $k$-th species enthalpy, $j$-th component of the species molecular diffusion velocity for each species $k$, and ideal gas constant, respectively. <br />
Finally, if the system is closed, the total mass $M_0$ of the mixture inside the computational domain $\Omega$ can be computed as:<br />
\begin{eqnarray}<br />
M_0 = \int_\Omega \rho \, dV\,.<br />
\end{eqnarray}<br />
or an additional ODE on mass can be integrated if some mass fluxes are present at boundaries.<br />
All other physical and numerical parameters are identical for Step 3 and Step 4, the only difference between the two benchmarks being the activation of the reaction terms. <br />
As in Step 3, the simulation should be performed for a physical time of at least $t=0.5 \; \mathrm{ms} = 2 \tau_\mathrm{ref}$ and can be continued up to $t=2.5 \; \mathrm{ms} = 10 \tau_\mathrm{ref}$ for a full comparison with the data available on the benchmark repository~\cite{TGV-coria-cfd}.<br />
<br />
Because of the various time-integration techniques used in each code it is difficult to give recommendations regarding the timestep here.<br />
For example, DINO being fully explicit, it is always limited by the Fourier Condition and thus uses small timesteps.<br />
On the other hand, YALES2 can treat diffusion implicitly and thus uses convective timestep as a limiter.<br />
Nek5000 has no operator splitting and uses implicit schemes: its timestep constraints are again different.<br />
In all cases, the chemical source terms are treated by a stiff integrator which is the only solution to deal with the extremly small timescales present in flames.<br />
The conservative values CFL=0.3 and Fo=0.2 that were proposed in Step 3 can be used as a first attempt.<br />
In any case, it is recommended to check that the timestep is small enough to not influence the final result.<br />
<br />
<br />
= Aside suggestions =<br />
<br />
Should we prove that it is the case for our codes? It could be done easily, for each mesh, on a small number of iterations, say simulate 0.1ms with 2 or 3 different timesteps. This could also be done for Step 2 and 3 actually<br />
<br />
<br />
= References =<br />
<br />
TBD</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Step_4&diff=34Step 42020-07-27T14:16:38Z<p>Lartigue: Created page with "= Description of Step 4 = In this last benchmark, Step 3 is repeated while now activating chemical reactions from the start. To this aim, the kinetic scheme of Boivin et al.~..."</p>
<hr />
<div>= Description of Step 4 =<br />
<br />
In this last benchmark, Step 3 is repeated while now activating chemical reactions from the start. To this aim, the kinetic scheme of Boivin et al.~\cite{Boivin2011} involving 9 species and 12 reversible reactions should be used in order to enable direct comparisons.<br />
Figure~\ref{fig:profiles_3-D_nonreacting} shows the initial profiles for temperature, heat release, mass fractions of H$_2$ and of O$_2$ (????).<br />
As it can be observed from this figure, the species profiles are now different along the fuel/oxidizer interface compared to those in the non-reacting case (Step 3), shown in Fig.~\ref{fig:profiles_3-D_nonreacting}. <br />
Now, the local species concentration corresponds to the equilibrium values obtained after initializing composition with Eqs.~\ref{eq:YF} and ~\ref{eq:YOx} at initial temperature $T_0=300$ K and atmospheric pressure; in this manner, the initial conditions already mimic a real flame front, with local composition and temperature in agreement with each other.<br />
<br />
Again, periodic boundary conditions are used in each direction.<br />
There is thus no net fluxes of mass, momentum or energy in any direction: the simulation domain can thus be considered to be closed.<br />
As a consequence, and contrary to Step 3, the thermodynamic pressure evolves in time during the present simulation because of the heat released by the chemical reactions and of the resulting dilatation. <br />
To enable meaningful comparisons, it is necessary to take into account this effect in the energy or temperature equation:<br />
\begin{eqnarray}<br />
\frac{\partial T}{\partial t}+ u_j \frac{\partial T}{\partial x_j} &=& \frac{1}{\rho C_p}\left[\frac{dP_0}{dt}+\frac{\partial }{\partial x_j} \left(\lambda \frac{\partial T}{\partial x_j} \right) -\frac{\partial T}{\partial x_j} \sum_{k=1}^{N_s} \rho C_{p,k} Y_k \theta_{k,j} -\sum_{k=1}^{N_s} h_k \dot{\omega}_k\right]\, \label{eq:temp_energy}.<br />
\end{eqnarray}<br />
All symbols used in this equation are defined below.<br />
Usually, most simulations of low-Mach flow conditions assume thate the thermodynamic pressure $P_0$ does not vary in time in case of open boundaries, considering that $P_0$ corresponds to the surrounding (usually atmospheric) pressure; therefore, $dP_0/dt$ vanishes in Eq.~(\ref{eq:temp_energy}). <br />
On the other hand, for a closed computational domain -- as assumed here --, $P_0$ may change in time due to dilatation and must be evaluated at each timestep as follows~\cite{Motheau2016}:<br />
<br />
\begin{eqnarray}<br />
P_0 = \frac{M_0 \, \mathrm{R}}{\int_\Omega \left( T \sum_{k=1}^{N_s} Y_k/W_k \right)^{-1} dV}\, \label{eq:thermod_p0}.<br />
\end{eqnarray}<br />
<br />
This equation is obtained by integrating the equation of state for perfect gases on the whole computational domain while assuming that the total mass is fixed and that the thermodynamic pressure is homogeneous inside the domain.<br />
In Eqs.~(\ref{eq:temp_energy}) and (\ref{eq:thermod_p0}), $\rho$, $u_j$, $T$, $Y_k$, $W_k$, $N_s$, $C_{p,k}$, $h_k$, $\theta_{k,j}$ and $\mathrm{R}$ are the density of the gas mixture, $j$-th component of flow velocity, gas temperature, $k$-th species mass fraction, $k$-th species molecular weight, number of species, $k$-th species specific heat capacity at constant pressure, $k$-th species enthalpy, $j$-th component of the species molecular diffusion velocity for each species $k$, and ideal gas constant, respectively. <br />
Finally, if the system is closed, the total mass $M_0$ of the mixture inside the computational domain $\Omega$ can be computed as:<br />
\begin{eqnarray}<br />
M_0 = \int_\Omega \rho \, dV\,.<br />
\end{eqnarray}<br />
or an additional ODE on mass can be integrated if some mass fluxes are present at boundaries.<br />
All other physical and numerical parameters are identical for Step 3 and Step 4, the only difference between the two benchmarks being the activation of the reaction terms. <br />
As in Step 3, the simulation should be performed for a physical time of at least $t=0.5 \; \mathrm{ms} = 2 \tau_\mathrm{ref}$ and can be continued up to $t=2.5 \; \mathrm{ms} = 10 \tau_\mathrm{ref}$ for a full comparison with the data available on the benchmark repository~\cite{TGV-coria-cfd}.<br />
<br />
Because of the various time-integration techniques used in each code it is difficult to give recommendations regarding the timestep here.<br />
For example, DINO being fully explicit, it is always limited by the Fourier Condition and thus uses small timesteps.<br />
On the other hand, YALES2 can treat diffusion implicitly and thus uses convective timestep as a limiter.<br />
Nek5000 has no operator splitting and uses implicit schemes: its timestep constraints are again different.<br />
In all cases, the chemical source terms are treated by a stiff integrator which is the only solution to deal with the extremly small timescales present in flames.<br />
The conservative values CFL=0.3 and Fo=0.2 that were proposed in Step 3 can be used as a first attempt.<br />
In any case, it is recommended to check that the timestep is small enough to not influence the final result.<br />
<br />
<br />
= Aside suggestions <br />
<br />
Should we prove that it is the case for our codes? It could be done easily, for each mesh, on a small number of iterations, say simulate 0.1ms with 2 or 3 different timesteps. This could also be done for Step 2 and 3 actually<br />
<br />
<br />
= References =<br />
<br />
TBD</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Step_3&diff=33Step 32020-07-27T14:05:40Z<p>Lartigue: Created page with "= Step 3 descritpion = The purpose of this configuration is to perform a simulation similar to Step 2 but involving now a mixture of different gaseous species at different te..."</p>
<hr />
<div>= Step 3 descritpion =<br />
<br />
The purpose of this configuration is to perform a simulation similar to Step 2 but involving now a mixture of different gaseous species at different temperatures, still without any chemical reaction. <br />
In this manner, it is possible to assess the accuracy of the numerical models involved in the description of species and heat diffusion<br />
The thermodynamic and transport properties of the 9 species kinetic scheme of Boivin et al.~\cite{Boivin2011} should be used in preparation for the next step, even though the reactions are still neglected here.<br />
<br />
It should be noted that in this case, the density is variable in both space and time due to the changing composition, which was not the case in the previous step: this induces additional numerical difficulties that must be taken into account.<br />
However, there is no variation of the thermodynamic pressure in time,contrary to the next step.<br />
<br />
The simulation domain is a cubic box of size $[0;L]^3$ with $L = 2 \pi L_0$ in each direction, where $L_0 = 1\; \mathrm{mm}$. <br />
Compared to the previous step, this smaller size is needed to ensure a proper resolution of the reaction fronts for hydrogen oxidation that will appear in the next step and are associated to fixed characteristic dimensions. <br />
Again, periodic boundary conditions are used in all three spatial directions.<br />
<br />
The initial velocity field prescribed at $t=0$ is identical to the one used in Step 2 (Eq.~\ref{eq:2-D-tgv_init}) with the reference velocity set to $u_0=4\;\mathrm{m/s}$.<br />
<br />
In preparation for the next step, the central part of the box is initially filled with a H$_{2}$/N$_{2}$ mixture (fuel region, molar fraction $X^0_\mathrm{H_2}=0.45$, $T=300$ K) while the remaining domain is filled with air (oxidizer region, molar fraction $X^0_\mathrm{O_2}=0.21$, $T=300$ K). <br />
The corresponding mass fractions are $Y^0_\mathrm{H_2} \approx 0.0556$ and $Y^0_\mathrm{O_2} \approx 0.233$.<br />
<br />
To avoid any numerical instability, the steps at the interface between both regions are smoothed out using hyperbolic tangent functions as follows:<br />
\begin{eqnarray}<br />
R_d(x) &=& |x-0.5\,L|, \\<br />
\psi(x) &=& 0.5\left[1+\mathrm{tanh}\left(\frac{c\,(R_d(x)-R)}{R}\right)\right],\\<br />
Y_\mathrm{H_2}(x) &=& Y^0_{\mathrm{H_2}}\left(1-\psi(x)\right), \label{eq:YF}\\<br />
Y_\mathrm{O_2}(x) &=& Y^0_{\mathrm{O_2}}\left(\psi(x)\right). \label{eq:YOx}<br />
\end{eqnarray}<br />
where $R = \pi/4$ mm and $c = 3$ are the half-width of the central slab and stiffness parameter, respectively.<br />
As a consequence, there is initially a small region where both fuel and oxidizer coexist. <br />
Finally, a nitrogen complement is added everywhere using $Y_\mathrm{N_2}=1-Y_\mathrm{H_2}-Y_\mathrm{O_2}$.<br />
<br />
To also test the behaviour of the codes with respect to heat diffusion, a non-homogeneous temperature profile is finally imposed as follows:<br />
<br />
* Compute in each cell the equilibrium temperature for the local mixture for constant pressure and enthalpy by using a dedicated solver.<br />
<br />
* Enforce the resulting temperature profile, but keep the species profiles to their initial values (Eqs.~\ref{eq:YF} and~\ref{eq:YOx}).<br />
<br />
It is found by using Cantera that the peak adiabatic temperature obtained at the fuel/air interface is $T_{ad}=1910.7$~K, leading to the profiles shown in Figs.~\ref{fig:temper_h2_o2_3-D} and \ref{fig:profiles_3-D_nonreacting}. Users are encouraged to implement the exact initial profiles from the benchmark website to facilitate later comparisons. <br />
<br />
The resulting, initial configuration thus involves multiple species at different temperatures with steep profiles, just like in a real flame.<br />
However, for the present simulation, the reaction source terms are all still set to zero, as already mentioned: only convection and diffusion processes are considered.<br />
In order to enable benchmark computations also with DNS codes that do not provide advanced diffusion models, only constant values of the Lewis numbers are considered; for the same reason, thermodiffusion (Soret effect) is neglected on purpose (though it would be obviously relevant for such cases involving hydrogen as a fuel). <br />
The authors are fully aware that this is a crude approximation of reality~\cite{pecs}. But it must be kept in mind that the focus is set here on the detailed comparison between different codes and algorithms, and not on the resulting flow or flame structure.<br />
Hence, in this and in the next section, the values of the Lewis number for each species given in Tab.~\ref{Tab:Lewis} shall be used to enable direct comparisons. <br />
These values have been estimated from a separate DNS for a similar configuration using the mixture-averaged diffusion model.<br />
The precise description of the associated thermodynamic state is available on the website.<br />
<br />
<nowiki><br />
\begin{table}[H]<br />
\resizebox{\textwidth}{!}{\begin{tabular}{|| l |c|c|c|c|c|c|c|c|c||}<br />
\hline<br />
\hline<br />
Species & H$_2$ & H & O$_2$ & OH & O & H$_2$O & HO$_2$ & H$_2$O$_2$ & N$_2$ \\<br />
\hline<br />
Lewis number & 0.3290 & 0.2228 & 1.2703 & 0.8279 & 0.8128 & 1.0741 & 1.2582 & 1.2665 & 1.8268\\<br />
\hline<br />
\hline<br />
\end{tabular}}<br />
\caption{Approximate Lewis number for the species appearing in the kinetic scheme of~\cite{Boivin2011}, to be enforced for Step 3 and Step 4 of the benchmark}<br />
\label{Tab:Lewis}<br />
\end{table}<br />
</nowiki><br />
<br />
On the other hand, considering that implementing advanced models for fluid viscosity and thermal conductivity is not difficult and does not increase the computational time significantly, local values of these quantities depending on composition and temperature should be taken into account; <br />
More specifically, and to ease further comparisons, the same models for viscosity and conductivity have been retained in the three participating codes, i.e. the mixture averaged formalism that is used in both the Cantera and Chemkin packages<br />
More details on these models are available on the benchmark repository~\cite{TGV-coria-cfd}, such as reference data for viscosity and conductivity along various cuts at initialization.<br />
<br />
The viscosity is variable in the computational domain as well as with time and the Reynolds number of this configuration can only be estimated using the minimal value of viscosity (which is obtained in the air at 300 K, $\nu_\mathrm{min} \approx 1.56 \, 10^{-5} \mathrm{m^2/s}$), leading to Re$={u_0 L_0}/{\nu_{\scriptsize \hbox{max}}}=267$ which guarantees a laminar flow.<br />
<br />
Concerning resolution, it is also suggested to keep a conservative resolution in time for this benchmark, corresponding to CFL $\le$ 0.25 and Fo $\le$ 0.15. <br />
In space, each direction should be resolved by approximately 256 grid points, leading to a spatial resolution $\Delta x \approx 25 \mu$m.<br />
<br />
The reference time scale is here $\tau_\mathrm{ref}=L_0/u_0 = 0.25 \; \mathrm{ms}$.<br />
The simulation should be performed for a physical time of at least $t=0.5 \; \mathrm{ms} = 2 \tau_\mathrm{ref}$ as most of the results presented in this article are taken from this instant.<br />
Data up to $t=2.5 \; \mathrm{ms} = 10 \tau_\mathrm{ref}$ are available on the benchmark repository~\cite{TGV-coria-cfd}.<br />
<br />
<nowiki><br />
\begin{figure}[!ht]<br />
\centering<br />
\includegraphics[scale=0.25]{./figures/Step_3/wmag_t0.png}<br />
\quad<br />
\includegraphics[scale=0.25]{./figures/Step_3/T_t0.png}<br />
\quad<br />
\includegraphics[scale=0.25]{./figures/Step_3/H2_t0.png}<br />
\quad<br />
\includegraphics[scale=0.25]{./figures/Step_3/O2_t0.png}<br />
<br />
\caption{Initial fields of vorticity magnitude, temperature, mass fractions of $\mathrm{H_2}$, and mass fraction of $\mathrm{O_2}$ (from left to right), for the 3-D, non-reacting case (Step 3)}\label{fig:temper_h2_o2_3-D}<br />
\end{figure}<br />
<br />
\begin{figure}[!ht]<br />
\centering<br />
\includegraphics[scale=0.6]{./figures/Step_3/profiles_3d_zoom.pdf}<br />
\caption{Initial profiles of temperature and of mass fractions at $y=0.5 L$ and $z=0.5 L$ for the 3-D, non-reacting case (Step 3)}\label{fig:profiles_3-D_nonreacting}<br />
\end{figure}<br />
</nowiki><br />
<br />
= Kinetics =<br />
<br />
The Boivin scheme has been used for this Benchmark.<br />
<br />
'''Put the data here'''<br />
<br />
<br />
<br />
= Transport and thermodynamic properties used for this Step =<br />
<br />
\cf{I think we all used the same transport (viscosity) and thermodynamic properties, but we should check again and compare that the implementations used in cantera/Chemkin provide the same values.}<br />
<br />
\gl{Good idea: we could use 3 or 4 relevant reference compositions to quantify the discrepancies between Cantera and Chemkin (on rho, Cp, conductivity, viscosity for example) and also put these on the website.}<br />
<br />
'''Put the transport data here'''<br />
<br />
<br />
= Aside suggestions =<br />
<br />
TBD<br />
<br />
= References =<br />
<br />
TBD</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Step_1&diff=32Step 12020-07-27T14:00:36Z<p>Lartigue: /* Notes */</p>
<hr />
<div>= Step 1 Description =<br />
<br />
The 2-D TGV test case has been already well documented in the scientific literature, e.g. <ref name="abelsamie2016"/><ref name="Laizet2009"/>.<br />
Therefore, only a brief description is given here. <br />
This configuration is used as verification step, since an analytic solution exists<ref name="Taylor1937"/>.<br />
<br />
An incompressible flow with constant density and viscosity is simulated.<br />
<br />
The fluid is contained in a square box of dimension <math>[0;L]^2</math> with periodic boundary conditions in both directions.<br />
The value <math>L=2 \pi L_0</math> with <math>L_0 = 1 \; \mathrm{m}</math> has been arbitrarily chosen in the present benchmark for dimensional codes.<br />
The numerical value for the kinematic viscosity is chosen as <math>\nu = 6.25 \times 10^{-4} \; \mathrm{m^2/s}</math> to get suitable conditions, as discussed below.<br />
The initial velocity components are prescribed as follows:<br />
<br />
<math><br />
u(x,y,0) = +u_0 \times \sin(x/L_0) \times \cos(y/L_0)\,,<br />
</math><br />
<br />
<math><br />
v(x,y,0) = -u_0 \times \cos(x/L_0) \times \sin(y/L_0)\,.<br />
</math><br />
<br />
with <math>u_0 = 1 \; \mathrm{m/s}</math>.<br />
The flow is thus initially composed of four vortices, one in each quarter of the box.<br />
The vortex turnover time is defined as <math>\tau_\mathrm{ref} = L_0/u_0 = 1\;\mathrm{s}</math>.<br />
For non-dimensional codes, the relevant parameter for this simulation is the vortex Reynolds number, <math>Re=u_0 L_0/\nu = 1,600</math> in the present setup. <br />
It should be noted that the theoretical solution is only stable for sufficiently small Reynolds numbers, explaining why the particular value <math>Re=1,600</math> has been chosen here, as done in many previous studies using the TGV configuration.<br />
<br />
A major interest of this test-case is that an analytical solution has been derived for such conditions<ref name="Taylor1937"/>:<br />
<br />
<math><br />
u(x,y,t) = u(x,y,0) \times \mathrm{e}^{-2\nu t/L_0^2} = u(x,y,0) \times \mathrm{e}^{-2 \, (t/\tau_{\mathrm{ref}}) / \mathrm{Re}} \,,<br />
</math><br />
<br />
<math><br />
v(x,y,t) = v(x,y,0) \times \mathrm{e}^{-2\nu t/L_0^2} = v(x,y,0) \times \mathrm{e}^{-2 \, (t/\tau_{\mathrm{ref}}) / \mathrm{Re}} \,.<br />
</math><br />
<br />
All physical conditions are now completely defined and only numerical parameters remain to be set.<br />
Based on previously published studies, the recommended parameters for time and space discretization are <math>\Delta t = \tau_\mathrm{ref}/2000 = 5 \times 10^{-4} \; \mathrm{s}</math> and <math>N = 64</math> grid points in each direction.<br />
The flow must be simulated for a physical time of <math>t=10 \; \tau_{\mathrm{ref}}=10\; \mathrm{s}</math>.<br />
<br />
= Aside suggestions =<br />
<br />
This step should clearly not be a big issue for any code.<br />
However, it is a good way to verify the orders of convergence (both in space and time), to check any stability issues, to measure the influence of the convergence criteria of iterative solvers (if any), etc...<br />
<br />
= References =<br />
<references><br />
<ref name="abelsamie2016"><br />
<bibtex><br />
@article{Abdelsamie2016, <br />
author= {A. Abdelsamie and G. Fru and F. Dietzsch and G. Janiga and D. Thévenin},<br />
title= {Towards direct numerical simulations of low-Mach number turbulent reacting and two-phase flows using immersed boundaries},<br />
journal={Comput. Fluids},<br />
year= {2016},<br />
volume={131},<br />
number={5},<br />
pages={123--141},<br />
}<br />
</bibtex><br />
</ref><br />
<ref name="Laizet2009"><br />
<bibtex><br />
@article{Laizet2009, <br />
author= {S. Laizet and E. Lamballais},<br />
title= {High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy},<br />
journal={J. Comput. Phys.},<br />
year= {2009},<br />
volume={228},<br />
pages={5989--6015},<br />
}<br />
</bibtex><br />
</ref><br />
<ref name="Taylor1937"><br />
<bibtex><br />
@article{Taylor1937, <br />
author= {G.I. Taylor and A.E. Green},<br />
title= {Mechanism of the production of small eddies from large ones},<br />
journal={Proc. Royal Soc. A},<br />
year= {1937},<br />
volume={158},<br />
issue={895},<br />
pages={499--521},<br />
}<br />
</bibtex><br />
</ref><br />
</references></div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Step_2&diff=31Step 22020-07-27T14:00:14Z<p>Lartigue: Created page with "= Step 2 Description = The next step in the benchmark is to simulate the Taylor-Green Vortex configuration in 3-D, still for a single-component incompressible flow. For this..."</p>
<hr />
<div>= Step 2 Description =<br />
<br />
The next step in the benchmark is to simulate the Taylor-Green Vortex configuration in 3-D, still for a single-component incompressible flow. <br />
For this configuration, a reference solution obtained with a pseudo-spectral solver is available in the scientific literature~\cite{vanrees2011} and can be used for validation by a direct comparison.<br />
<br />
The domain size has been set to $L = 2 \pi L_0$ in each direction, where $L_0 = 1\; \mathrm{m}$ is an arbitrarily-chosen reference length scale for dimensional codes. As in the previous section, periodic boundary conditions are used in all three spatial directions.<br />
<br />
The initial conditions for 3-D TGV are given by the following set of equations:<br />
\begin{eqnarray}<br />
u(x,y,z,0) &=& +u_0 \times \sin(x/L_0) \times \cos(y/L_0) \times \cos(z/L_0), \nonumber \\<br />
v(x,y,z,0) &=& -u_0 \times \cos(x/L_0) \times \sin(y/L_0) \times \cos(z/L_0), \nonumber \\<br />
w(x,y,z,0) &=& 0,<br />
\label{eq:2-D-tgv_init}<br />
\end{eqnarray}<br />
<br />
The reference velocity magnitude is set again to $u_0 = 1 \; \mathrm{m/s}$.<br />
The desired Reynolds number is again Re$={u_0 L_0}/{\nu}=1,600$, as in the previous section, since this value has been used as well in the reference study employed for comparison~\cite{vanrees2011}.<br />
To obtain the proper value of Re, the kinematic viscosity is set once again to $\nu = 6.25 \times 10^{-4} \; \mathrm{m^2/s}$.<br />
With this set of parameters, the reference time scale is $\tau_\mathrm{ref}=L_0/u_0 = 1 \; \mathrm{s}$.<br />
The flow simulation must be carried out for a physical time of $t=20 \; \tau_\mathrm{ref}$.<br />
<br />
Regarding spatial discretization, a resolution of at least $512$ nodes (or cells) in each direction is recommended.<br />
This value leads to an over-resolved simulation during the first few $\tau_\mathrm{ref}$ but is necessary to capture properly the transition to turbulence that should occur between 10 and 15 $\tau_\mathrm{ref}$. <br />
As a point of comparison, the reference solution from~\cite{vanrees2011} has been obtained with a spectral code on a $512^3$ grid.<br />
The timestep should be chosen to ensure that the maximum values of the Courant-Friedrichs-Lewy (CFL) and Fourier (Fo) numbers remain below $0.3$ and $0.2$, respectively.<br />
This values are actually quite conservative since the focus of this study is set on accuracy comparisons and not on pure computational performance.<br />
<br />
<br />
= Aside Suggestion =<br />
<br />
Check mesh convergence and time integration independence.<br />
<br />
Check that -dKE/dt = \int Sij:Sij = \int Enstrophy<br />
<br />
<br />
= References =<br />
<br />
* van Rees WM, Leonard A, Pullin D, Koumoutsakos P. A comparison of vortex and pseudo-spectral methods for the simulation of periodic vortical flows at high reynolds numbers. J Comput Phys. 2011;230(8):2794-2805.<br />
* Carton de Wiart C, Hillewaert K, Duponcheel M, Winckelmans G. Assessment of a discontinuous galerkin method for the simulation of vortical flows at high reynolds number. Int J Numer Meth Fluids. 2014;74(7):469-493.<br />
* Duponcheel M, Orlandi P, Winckelmans G. Time-reversibility of the euler equations as a benchmark for energy conserving schemes. J Comput Phys. 2008;227(19):8736-8752.<br />
* Bricteux, L, Zeoli, S, Bourgeois, N. Validation and scalability of an open source parallel flow solver. Concurrency Computat: Pract Exper. 2017; 29:e4330. https://doi.org/10.1002/cpe.4330</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Codes&diff=30Codes2020-07-27T13:51:03Z<p>Lartigue: Created page with "= Code Description = This page is dedicated to a short presentation of the three low-Mach number codes used for the rest of this benchmarl: [https://www.coria-cfd.fr/index.ph..."</p>
<hr />
<div>= Code Description =<br />
<br />
This page is dedicated to a short presentation of the three low-Mach number codes used for the rest of this benchmarl: [https://www.coria-cfd.fr/index.php/YALES2 YALES2], [http://www.lss.ovgu.de/lss/en/Research/Computational+Fluid+Dynamics.html DINO] and [http://nek5000.mcs.anl.gov Nek5000]. <br />
<br />
Since these codes have already been the subject of many publications and are not completely new, only the most relevant features are discussed in what follows, with suitable references for those readers needing more details.<br />
<br />
== YALES2 ==<br />
<br />
YALES2 is a massively parallel multiphysics platform~\cite{YALES2-website,moureau2011large} developed since 2009 by Moureau, Lartigue and co-workers at CORIA (Rouen, France).<br />
It is dedicated to the high-fidelity simulation of low-Mach number flows in complex geometries.<br />
It is based on the Finite Volumes formulation of the Navier-Stokes equations and it can solve both non-reacting and reacting flows.<br />
It can actually solve various physical problems thanks to its original structure which is composed of a main numerical library accompanied with tens of dedicated solvers (for acoustics, multiphase flows, heat transfer, radiation\ldots) which can be coupled with one another.<br />
YALES2 relies on unstructured meshes and a fully parallel dynamic mesh adaptation technique to improve the resolution in physically-relevant zones to mitigate the computational cost~\cite{benard2015mesh}. <br />
As a result it can easily handle meshes composed of billions of tetrahedra, thus enabling the Direct Numerical Simulation of laboratory and semi-industrial configurations.<br />
It is now composed of nearly 500,000 lines of object-oriented Fortran and the parallelism is currently ensured by a pure MPI paradigm, although a hybrid OpenMP/MPI as well as a GPU version are under development.<br />
<br />
As most low-Mach number codes, the time-advancement is based on a projection-correction method following the pioneering work of~\cite{chorin1968}.<br />
The prediction step uses a method which is a blend between a 4th-order Runge-Kutta method and a 4th-order Lax-Wendroff-like method~\cite{KraushaarPhD}, combined with a 4th-order node-based centered finite-volume discretization of the convective and diffusive terms.<br />
Moreover, to improve the performance of the correction step, the pressure of the previous iteration is included in the prediction step to limit the splitting errors.<br />
The correction step is required to ensure mass conservation, using a pressure that arises from a Poisson equation.<br />
This is performed numerically by solving a linear system on the pressure at each node of the mesh thanks to a dedicated in-house version of the deflated conjugate gradient algorithm, which has been optimized for solving elliptic equations on massively parallel machines~\cite{malandain2013optimization}.<br />
<br />
When considering multispecies and non-isothermal flows, the extension of the classical projection method proposed by Pierce et al.~\cite{pierce2004progress} is used to account for variable-density flows.<br />
All the thermodynamic and transport properties are provided by the Cantera software~\cite{goodwin2003open}, which has been fully re-implemented in Fortran to avoid any performance issues.<br />
Each species thermodynamic property is specified by 5th-order polynomials on two temperature ranges (below and above 1000 K).<br />
The mixture is supposed to be both thermally and mechanically perfect.<br />
Regarding transport properties, several type of models are implemented in YALES2: 1) the default approach is based on the computation of transport properties for each species (using tabulated molecular potentials), then combining those to obtain mixture-averaged coefficients for viscosity, conductivity and diffusion velocities (Hirschfelder and Curtiss approximation~\cite{hirschfelder1964molecular}); 2) alternatively, simplified laws (for example the Sutherland law~\cite{Sutherland}) can be used for the viscosity, while fixed values for Prandtl and Schmidt numbers allow the computation of conductivity and diffusion coefficients; and 3) a mix of both approaches, for example computing viscosity and conductivity with a mixture-averaged approximation and then imposing a Lewis number for each species.<br />
This last approach has been retained in the present benchmark.<br />
<br />
Finally, the source terms used in the reacting simulations are modeled with an Arrhenius law with the necessary modifications needed to take into account three-body or pressure-dependent reactions.<br />
<br />
From a numerical point of view, it must be noticed that both the diffusion and reaction processes occur at time scales which can be orders of magnitude smaller than the convective time scale.<br />
Solving these phenomena with explicit methods would thus drastically limit the global timestep of the whole simulation and induce an overwhelming CPU cost.<br />
To mitigate these effects when dealing with multi-species reacting flows, the classical operator splitting technique is used.<br />
The diffusion process is solved with a fractional timestep method inside each convective iteration, each substep being limited by a Fourier condition to ensure stability. <br />
This method gets activated only when the mesh is very fine (typically when performing DNS with very diffusive species like H$_2$ or H, as in the present project); otherwise an explicit treatment is sufficient.<br />
The chemical source terms are then integrated with a dedicated stiff solver, namely the CVODE library from SUNDIALS~\cite{cohen1996cvode,cvodeUG,sundials}. <br />
To that purpose, each control volume is considered as an isolated reactor with both constant pressure and enthalpy.<br />
<br />
Using the stiff integration technique results in a very strong load imbalance between the various regions of the flow: the fresh gases are solved in a very small number of integration steps while the inner flame region may require tens or even hundreds of integration steps. <br />
To overcome this difficulty, a dedicated MPI dynamic scheduler based on a work-sharing algorithm ensures global load-balancing.<br />
<br />
<br />
== DINO ==<br />
<br />
DINO is a 3-D DNS code used for incompressible or low-Mach number flows, the latter approach being used in this project. <br />
The development of DINO started in the group of D. Thev\'enin (Univ. of Magdeburg) in 2013. <br />
DINO is a Fortran-90 code, written on top of a 2-D pencil decomposition to enable efficient large-scale parallel simulations on distributed-memory supercomputers by coupling with the open-source library 2-DECOMP\&FFT~\cite{2decompfft}. <br />
The code offers different features and algorithms in order to investigate different physicochemical processes. <br />
Spatial derivative are computed by default using sixth-order central finite differences. <br />
Time integration relies on several Runge-Kutta solvers. <br />
In what follows, an explicit 4th-order Runge-Kutta approach has been used. <br />
A 3rd-order semi-implicit Runge-Kutta integration can be activated as needed, when considering stiff chemistry. <br />
In this case, non-stiff terms are still computed with explicit Runge-Kutta, while the PyJac package~\cite{pyjac} is used to integrate in an implicit manner all chemistry terms with an analytical Jacobian computation. <br />
All thermodynamic, chemical and transport properties are computed using the open-source library Cantera 2.4.0~\cite{cantera}. <br />
The transport properties can be computed based on three different models: 1) Constant Lewis numbers, 2) mixture-averaged, 3) full multicomponent diffusion, by coupling either again with Cantera or with the open-source library EGlib \cite{Ern}. <br />
The Poisson equation is solved using Fast Fourier Transform (FFT) for periodic as well as for non-periodic boundary conditions, relying in the latter case on an in-house pre- and post-processing technique. <br />
The I/O operations are implemented using two different approaches: (1) binary MPI-I/O using 2-DECOMP\&FFT for check-points and restart files; (2) HDF5 files used for analysis and visualization.<br />
<br />
Multi-phase flows can be simulated in DINO using resolved or non-resolved (point) particles and droplets using a Lagrangian approach~\cite{Abdelsamie2019,Abdelsamie2020_dles}. <br />
Complex boundaries are represented by a novel second-order immersed boundary method implementation (IBM) based on a directional extrapolation scheme~\cite{Chi2020}. <br />
More details about the implemented algorithms can be found in particular in~\cite{Abdelsamie2016,Chi2019}. <br />
Since DINO has been developed as a multi-purpose code for analyzing many different reacting and non-reacting flows~\cite{Abdelsamie2015_dles,Oster2018,Abdelsamie2019_dles}, a detailed verification and validation is obviously essential.<br />
<br />
<br />
== Nek5000 ==<br />
<br />
This spectral element low-Mach number reacting flow solver is based on the highly-efficient open-source solver Nek5000~\cite{Nek5000} extended by a plugin developed at ETH implementing a high-order splitting scheme for low-Mach number reacting flows~\cite{tomboulides1997}. <br />
The spectral element method (SEM) is a high-order weighted residual technique for spatial discretization that combines the accuracy of spectral methods with the geometric flexibility of the finite element method allowing for accurately representation of complex geometries~\cite{deville2002}.<br />
The computational domain is decomposed into $E$ conforming elements, which are quadrilaterals (in 2-D) or hexahedra (in 3-D) that conform to the domain boundaries. <br />
Within each element, functions are expanded as $N$th-order polynomials so that resolution can be increased either by decreasing the element size (\emph{h}-type refinement) or by increasing the polynomial order (\emph{p}-type refinement; typically N = 7 - 15). <br />
The grids can be unstructured and allow for static local refinement, while adaptive mesh refinement has been recently developed~\cite{Tanarro2020}. <br />
By casting the polynomial approximation in tensor-product form, the differential operators on $N^3$ gridpoints per element can be evaluated with only $O(N^4)$ work and $O(N^3)$ storage.<br />
<br />
The principal advantage of the SEM is that convergence is exponential in $N$, yielding minimal numerical dispersion and dissipation, so that significantly fewer grid points per wavelength are required in order to accurately propagate a turbulent structure <br />
over the extended time required in high Reynolds number simulations. <br />
Nek5000 uses locally structured basis coefficients ($N\times N \times N$ arrays), which allow direct addressing and tensor-product-based derivative evaluation that can be cast as efficient matrix-matrix products involving $N^2$ operators applied to $N^3$ data values for each element. <br />
As a result, data movement per grid point is the same as for low-order methods. <br />
It uses scalable domain-decomposition-based iterative solvers with efficient preconditioners. <br />
Communication is based on the Message Passing Interface (MPI) standard, and the code has proven scalability to over one million ranks. <br />
Nek5000 provides balanced I/O latency among all processors and reduces the overhead or even completely hides the I/O latency by using dedicated I/O communicators in the optimal case.<br />
<br />
Time advancement is performed using the splitting scheme proposed in~\cite{tomboulides1997} to decouple the highly non-linear and stiff thermochemistry (species and energy governing equations) from the hydrodynamic system (continuity and momentum).<br />
Species and energy equations are integrated without further splitting using the implicit stiff integrator solver CVODE from the SUNDIALS package~\cite{cohen1996cvode,cvodeUG,sundials} that uses backward differentiation formulas (BDF).<br />
The continuity and momentum equations are integrated using a second- or third-order semi-implicit formulation (EXT/BDF) treating the non-linear advection term explicitly~\cite{deville2002}.<br />
The thermodynamic properties, detailed chemistry, and transport properties are provided by optimized subroutines compatible with Chemkin~\cite{chemkin}.<br />
<br />
The reacting flow solver can handle complex time-varying geometries and has been used for instance to simulate laboratory-scale internal combustion engines~\cite{MS, TCC}. <br />
It can account for conjugate fluid-solid heat transfer and detailed gas phase as well as surface kinetics~\cite{catalytic}.<br />
<br />
<br />
== General comments on the codes ==<br />
<br />
The three aforementioned solvers are all unsteady, high-fidelity codes based on the low-Mach number formulation of the Navier-Stokes equations.<br />
They can perform both Direct Numerical Simulations or Large Eddy Simulations of reacting flows, only DNS being considered here.<br />
However, they differ in a certain number of points, mainly from the numerical point of view.<br />
The aim of this section is to emphasize those major differences.<br />
First, YALES2 is an unstructured code, designed to handle any type of elements; its main application field pertains to LES of industrially relevant flows, though DNS is possible as well. <br />
On the other hand, both DINO and Nek5000 are mostly dedicated to DNS of configurations found in fundamental research.<br />
Both DINO and Nek5000 can only deal with quads or hexas, with the major difference that DINO is based on a structured connectivity while Nek5000 can use unstructured meshes (pavings).<br />
As a consequence, both DINO and Nek5000 employ higher-order numerical schemes compared to YALES2, limited at best to a 4th-order scheme.<br />
All codes rely on dedicated libraries to compute the thermo-chemical properties of the flow, either Chemkin, Cantera, or in-house versions of those.<br />
Moreover, they also rely at least to some extent on external software to perform the temporal integration of the stiff chemical source terms.<br />
Regarding the Poisson equation for pressure which must be solved by all codes, DINO relies on a spectral formulation by performing direct and inverse Fourier transforms, which is possible thanks to its structured mesh.<br />
On the other hand, both YALES2 and Nek5000 use an iterative solver with an efficient preconditioning technique; this method is more versatile and should be computationally more efficient for large and complex geometrical configurations.<br />
Nek5000 employs CVODE to integrate the thermochemical equations without further splitting of the different terms accounting for convection, diffusion and chemistry.<br />
The main differences between the three codes are summarized in Table~\ref{Tab:codes}. <br />
Please note that the presented values are those used for the benchmark, even though some other options are available in each codes.<br />
<br />
<nowiki><br />
\begin{table}[H]<br />
\resizebox{\textwidth}{!}{\begin{tabular}{|| l || c || c || c ||}<br />
\hline<br />
\hline<br />
Code & YALES2 & DINO & Nek5000 \\<br />
\hline<br />
\hline<br />
Connectivity & Unstructured & Structured & Unstructured \\<br />
\hline<br />
Discretization Type & Finite Volumes & Finite Differences & Spectral Elements \\<br />
\hline<br />
Grid point distribution & Regular hexahedra. & Regular hexahedra & Regular hexahedra with GLL points \\<br />
\hline<br />
Spatial order & 4th & 6th & 7th - 15th (typically) \\<br />
\hline<br />
Temporal method & expl. RK4 & expl. RK4 / semi-impl. RK3 & semi-impl. BDF3 \\<br />
\hline<br />
Pressure solver & CG with Deflation Prec. & FFT-based & CG/GMRES with Jacobi/Schwartz Prec. \\<br />
\hline<br />
Thermo-chemistry & Cantera (re-coded) & Cantera & Chemkin interface \\<br />
\hline<br />
Chemistry integration & CVODE & PyJac & CVODE \\<br />
\hline<br />
Operator splitting & Yes & ??? & No \\<br />
\hline<br />
Parallel paradigm & MPI & MPI & MPI \\<br />
\hline<br />
\hline<br />
\end{tabular}}<br />
\caption{Major numerical properties of the three high-fidelity codes as used in this benchmark}<br />
\label{Tab:codes}<br />
\end{table}<br />
</nowiki></div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Main_Page&diff=29Main Page2020-07-27T13:33:10Z<p>Lartigue: </p>
<hr />
<div>= The Taylor-Green Vortex as a Benchmark for High-Fidelity Combustion Simulations Using Low-Mach Solvers =<br />
<br />
The present web site is a complement of an article that has been submitted to '''Computers and Fluids''' in August 2020.<br />
<br />
Verification and validation are crucial steps for the development of any numerical model.<br />
<br />
While suitable processes have been established for commercial Computational Fluid Dynamics (CFD) codes, more difficult challenges must be faced for high-fidelity solvers.<br />
<br />
Benchmarks have been proposed in a series of dedicated conferences for non-reacting configurations.<br />
However, to our knowledge, no suitable approach has been published up to now regarding turbulent reacting flows.<br />
<br />
'''The purpose of this website is to present a full verification and validation chain for high-resolution codes employed to simulate turbulent reacting flows, first for Direct Numerical Simulation (DNS) of turbulent combustion in the limit of low Mach numbers.'''<br />
<br />
The selected configuration builds on top of the Taylor-Green vortex.<br />
Verification takes place by comparison with the analytical solution in two dimensions.<br />
Validation of the single-component flow is ensured by comparisons with published results obtained with a spectral code.<br />
Mixing without reaction is then considered, before computing finally a hydrogen-oxygen flame interacting with a 3-D Taylor-Green vortex. <br />
Three different low-Mach DNS solvers have been used for this study, demonstrating that the final accuracy of the DNS simulations is of the order of 1% for all quantities considered.<br />
<br />
The website is organised in four major parts:<br />
* A presentation of the three [[codes]] used for the Benchmark<br />
* The [[description]] of the test cases<br />
* The [[results]] of the test cases<br />
* An attempt to give a few guidelines on the [[performances]] that could be expected on the 3D test-cases.<br />
<br />
'''Put some images / videos here'''</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Main_Page&diff=28Main Page2020-07-27T05:08:18Z<p>Lartigue: </p>
<hr />
<div>= The Taylor-Green Vortex as a Benchmark for High-Fidelity Combustion Simulations Using Low-Mach Solvers =<br />
<br />
The present web site is a complement of an article that has been submitted to '''Computers and Fluids''' in August 2020.<br />
<br />
Verification and validation are crucial steps for the development of any numerical model.<br />
<br />
While suitable processes have been established for commercial Computational Fluid Dynamics (CFD) codes, more difficult challenges must be faced for high-fidelity solvers.<br />
<br />
Benchmarks have been proposed in a series of dedicated conferences for non-reacting configurations.<br />
However, to our knowledge, no suitable approach has been published up to now regarding turbulent reacting flows.<br />
<br />
'''The purpose of this website is to present a full verification and validation chain for high-resolution codes employed to simulate turbulent reacting flows, first for Direct Numerical Simulation (DNS) of turbulent combustion in the limit of low Mach numbers.'''<br />
<br />
The selected configuration builds on top of the Taylor-Green vortex.<br />
Verification takes place by comparison with the analytical solution in two dimensions.<br />
Validation of the single-component flow is ensured by comparisons with published results obtained with a spectral code.<br />
Mixing without reaction is then considered, before computing finally a hydrogen-oxygen flame interacting with a 3-D Taylor-Green vortex. <br />
Three different low-Mach DNS solvers have been used for this study, demonstrating that the final accuracy of the DNS simulations is of the order of 1% for all quantities considered.<br />
<br />
The website is organised in four major parts:<br />
* A short description of the three codes used for the Benchmark: [https://www.coria-cfd.fr/index.php/YALES2 YALES2], [http://www.lss.ovgu.de/lss/en/Research/Computational+Fluid+Dynamics.html DINO] and [http://nek5000.mcs.anl.gov Nek5000].<br />
* The [[description]] of the test cases<br />
* The [[results]] of the test cases<br />
* An attempt to give a few guidelines on the [[performances]] that could be expected on the 3D test-cases.<br />
<br />
'''Put some images / videos here'''</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Step_1&diff=27Step 12020-07-27T05:05:23Z<p>Lartigue: </p>
<hr />
<div>= Step 1 Description =<br />
<br />
The 2-D TGV test case has been already well documented in the scientific literature, e.g. <ref name="abelsamie2016"/><ref name="Laizet2009"/>.<br />
Therefore, only a brief description is given here. <br />
This configuration is used as verification step, since an analytic solution exists<ref name="Taylor1937"/>.<br />
<br />
An incompressible flow with constant density and viscosity is simulated.<br />
<br />
The fluid is contained in a square box of dimension <math>[0;L]^2</math> with periodic boundary conditions in both directions.<br />
The value <math>L=2 \pi L_0</math> with <math>L_0 = 1 \; \mathrm{m}</math> has been arbitrarily chosen in the present benchmark for dimensional codes.<br />
The numerical value for the kinematic viscosity is chosen as <math>\nu = 6.25 \times 10^{-4} \; \mathrm{m^2/s}</math> to get suitable conditions, as discussed below.<br />
The initial velocity components are prescribed as follows:<br />
<br />
<math><br />
u(x,y,0) = +u_0 \times \sin(x/L_0) \times \cos(y/L_0)\,,<br />
</math><br />
<br />
<math><br />
v(x,y,0) = -u_0 \times \cos(x/L_0) \times \sin(y/L_0)\,.<br />
</math><br />
<br />
with <math>u_0 = 1 \; \mathrm{m/s}</math>.<br />
The flow is thus initially composed of four vortices, one in each quarter of the box.<br />
The vortex turnover time is defined as <math>\tau_\mathrm{ref} = L_0/u_0 = 1\;\mathrm{s}</math>.<br />
For non-dimensional codes, the relevant parameter for this simulation is the vortex Reynolds number, <math>Re=u_0 L_0/\nu = 1,600</math> in the present setup. <br />
It should be noted that the theoretical solution is only stable for sufficiently small Reynolds numbers, explaining why the particular value <math>Re=1,600</math> has been chosen here, as done in many previous studies using the TGV configuration.<br />
<br />
A major interest of this test-case is that an analytical solution has been derived for such conditions<ref name="Taylor1937"/>:<br />
<br />
<math><br />
u(x,y,t) = u(x,y,0) \times \mathrm{e}^{-2\nu t/L_0^2} = u(x,y,0) \times \mathrm{e}^{-2 \, (t/\tau_{\mathrm{ref}}) / \mathrm{Re}} \,,<br />
</math><br />
<br />
<math><br />
v(x,y,t) = v(x,y,0) \times \mathrm{e}^{-2\nu t/L_0^2} = v(x,y,0) \times \mathrm{e}^{-2 \, (t/\tau_{\mathrm{ref}}) / \mathrm{Re}} \,.<br />
</math><br />
<br />
All physical conditions are now completely defined and only numerical parameters remain to be set.<br />
Based on previously published studies, the recommended parameters for time and space discretization are <math>\Delta t = \tau_\mathrm{ref}/2000 = 5 \times 10^{-4} \; \mathrm{s}</math> and <math>N = 64</math> grid points in each direction.<br />
The flow must be simulated for a physical time of <math>t=10 \; \tau_{\mathrm{ref}}=10\; \mathrm{s}</math>.<br />
<br />
= Aside suggestions =<br />
<br />
This step should clearly not be a big issue for any code.<br />
However, it is a good way to verify the orders of convergence (both in space and time), to check any stability issues, to measure the influence of the convergence criteria of iterative solvers (if any), etc...<br />
<br />
= Notes =<br />
<references><br />
<ref name="abelsamie2016"><br />
<bibtex><br />
@article{Abdelsamie2016, <br />
author= {A. Abdelsamie and G. Fru and F. Dietzsch and G. Janiga and D. Thévenin},<br />
title= {Towards direct numerical simulations of low-Mach number turbulent reacting and two-phase flows using immersed boundaries},<br />
journal={Comput. Fluids},<br />
year= {2016},<br />
volume={131},<br />
number={5},<br />
pages={123--141},<br />
}<br />
</bibtex><br />
</ref><br />
<ref name="Laizet2009"><br />
<bibtex><br />
@article{Laizet2009, <br />
author= {S. Laizet and E. Lamballais},<br />
title= {High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy},<br />
journal={J. Comput. Phys.},<br />
year= {2009},<br />
volume={228},<br />
pages={5989--6015},<br />
}<br />
</bibtex><br />
</ref><br />
<ref name="Taylor1937"><br />
<bibtex><br />
@article{Taylor1937, <br />
author= {G.I. Taylor and A.E. Green},<br />
title= {Mechanism of the production of small eddies from large ones},<br />
journal={Proc. Royal Soc. A},<br />
year= {1937},<br />
volume={158},<br />
issue={895},<br />
pages={499--521},<br />
}<br />
</bibtex><br />
</ref><br />
</references></div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Main_Page&diff=26Main Page2020-07-27T04:51:16Z<p>Lartigue: </p>
<hr />
<div>= The Taylor-Green Vortex as a Benchmark for High-Fidelity Combustion Codes =<br />
<br />
Verification and validation are crucial steps for the development of any numerical model.<br />
<br />
While suitable processes have been established for commercial Computational Fluid Dynamics (CFD) codes, more difficult challenges must be faced for high-fidelity solvers.<br />
<br />
Benchmarks have been proposed in a series of dedicated conferences for non-reacting configurations.<br />
However, to our knowledge, no suitable approach has been published up to now regarding turbulent reacting flows.<br />
<br />
'''The purpose of this website is to present a full verification and validation chain for high-resolution codes employed to simulate turbulent reacting flows, first for Direct Numerical Simulation (DNS) of turbulent combustion in the limit of low Mach numbers.'''<br />
<br />
The selected configuration builds on top of the Taylor-Green vortex.<br />
Verification takes place by comparison with the analytical solution in two dimensions.<br />
Validation of the single-component flow is ensured by comparisons with published results obtained with a spectral code.<br />
Mixing without reaction is then considered, before computing finally a hydrogen-oxygen flame interacting with a 3-D Taylor-Green vortex. <br />
Three different low-Mach DNS solvers have been used for this study, demonstrating that the final accuracy of the DNS simulations is of the order of 1% for all quantities considered.<br />
<br />
The website is organised in four major parts:<br />
* A short description of the three codes used for the Benchmark: [https://www.coria-cfd.fr/index.php/YALES2 YALES2], [http://www.lss.ovgu.de/lss/en/Research/Computational+Fluid+Dynamics.html DINO] and [http://nek5000.mcs.anl.gov Nek5000].<br />
* The [[description]] of the test cases<br />
* The [[results]] of the test cases<br />
* An attempt to give a few guidelines on the [[performances]] that could be expected on the 3D test-cases.<br />
<br />
'''Put some images / videos here'''</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Step_1&diff=25Step 12020-07-27T04:45:44Z<p>Lartigue: </p>
<hr />
<div>= Step 1 Description =<br />
<br />
The 2-D TGV test case has been already well documented in the scientific literature, e.g. <ref name="abelsamie2016" /><ref name="Laizet2009"/>.<br />
Therefore, only a brief description is given here. <br />
This configuration is used as verification step, since an analytic solution exists~\cite{Taylor1937}.<br />
<br />
An incompressible flow with constant density and viscosity is simulated.<br />
<br />
The fluid is contained in a square box of dimension <math>[0;L]^2</math> with periodic boundary conditions in both directions.<br />
The value <math>L=2 \pi L_0</math> with <math>L_0 = 1 \; \mathrm{m}</math> has been arbitrarily chosen in the present benchmark for dimensional codes.<br />
The numerical value for the kinematic viscosity is chosen as <math>\nu = 6.25 \times 10^{-4} \; \mathrm{m^2/s}</math> to get suitable conditions, as discussed below.<br />
The initial velocity components are prescribed as follows:<br />
<br />
<math><br />
u(x,y,0) = +u_0 \times \sin(x/L_0) \times \cos(y/L_0)\,,<br />
</math><br />
<br />
<math><br />
v(x,y,0) = -u_0 \times \cos(x/L_0) \times \sin(y/L_0)\,.<br />
</math><br />
<br />
with <math>u_0 = 1 \; \mathrm{m/s}</math>.<br />
The flow is thus initially composed of four vortices, one in each quarter of the box.<br />
The vortex turnover time is defined as <math>\tau_\mathrm{ref} = L_0/u_0 = 1\;\mathrm{s}</math>.<br />
For non-dimensional codes, the relevant parameter for this simulation is the vortex Reynolds number, <math>Re=u_0 L_0/\nu = 1,600</math> in the present setup. <br />
It should be noted that the theoretical solution is only stable for sufficiently small Reynolds numbers, explaining why the particular value <math>Re=1,600</math> has been chosen here, as done in many previous studies using the TGV configuration.<br />
<br />
A major interest of this test-case is that an analytical solution has been derived for such conditions~\cite{Taylor1937}:<br />
<br />
<math><br />
u(x,y,t) = u(x,y,0) \times \mathrm{e}^{-2\nu t/L_0^2} = u(x,y,0) \times \mathrm{e}^{-2 \, (t/\tau_{\mathrm{ref}}) / \mathrm{Re}} \,,<br />
</math><br />
<br />
<math><br />
v(x,y,t) = v(x,y,0) \times \mathrm{e}^{-2\nu t/L_0^2} = v(x,y,0) \times \mathrm{e}^{-2 \, (t/\tau_{\mathrm{ref}}) / \mathrm{Re}} \,.<br />
</math><br />
<br />
All physical conditions are now completely defined and only numerical parameters remain to be set.<br />
Based on previously published studies, the recommended parameters for time and space discretization are <math>\Delta t = \tau_\mathrm{ref}/2000 = 5 \times 10^{-4} \; \mathrm{s}</math> and <math>N = 64</math> grid points in each direction.<br />
The flow must be simulated for a physical time of <math>t=10 \; \tau_{\mathrm{ref}}=10\; \mathrm{s}</math>.<br />
<br />
= Aside suggestions =<br />
<br />
This step should clearly not be a big issue for any code.<br />
However, it is a good way to verify the orders of convergence (both in space and time), to check any stability issues, to measure the influence of the convergence criteria of iterative solvers (if any), etc...<br />
<br />
= Notes =<br />
<references><br />
<ref name="abelsamie2016">A. Abdelsamie and G. Fru and F. Dietzsch and G. Janiga and D. Thévenin , ''Towards direct numerical simulations of low-Mach number turbulent reacting and two-phase flows using immersed boundaries'', '''Comput. Fluids''', 2016 (131-5, pp123--141)</ref>. <br />
<ref name="Laizet2009">S. Laizet and E. Lamballais , ''High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy'', '''J. Comput. Phys.''', 2009 (228, pp5989--6015)</ref>.<br />
</references></div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Step_1&diff=24Step 12020-07-27T04:42:58Z<p>Lartigue: /* Step 1 Description */</p>
<hr />
<div>= Step 1 Description =<br />
<br />
The 2-D TGV test case has been already well documented in the scientific literature, e.g. <ref name="abelsamie2016" /><ref name="Laizet2009"/>.<br />
Therefore, only a brief description is given here. <br />
This configuration is used as verification step, since an analytic solution exists~\cite{Taylor1937}.<br />
<br />
An incompressible flow with constant density and viscosity is simulated.<br />
<br />
The fluid is contained in a square box of dimension <math>[0;L]^2</math> with periodic boundary conditions in both directions.<br />
The value <math>L=2 \pi L_0</math> with <math>L_0 = 1 \; \mathrm{m}</math> has been arbitrarily chosen in the present benchmark for dimensional codes.<br />
The numerical value for the kinematic viscosity is chosen as <math>\nu = 6.25 \times 10^{-4} \; \mathrm{m^2/s}</math> to get suitable conditions, as discussed below.<br />
The initial velocity components are prescribed as follows:<br />
<br />
<math><br />
u(x,y,0) = +u_0 \times \sin(x/L_0) \times \cos(y/L_0)\,,<br />
</math><br />
<br />
<math><br />
v(x,y,0) = -u_0 \times \cos(x/L_0) \times \sin(y/L_0)\,.<br />
</math><br />
<br />
with <math>u_0 = 1 \; \mathrm{m/s}</math>.<br />
The flow is thus initially composed of four vortices, one in each quarter of the box.<br />
The vortex turnover time is defined as <math>\tau_\mathrm{ref} = L_0/u_0 = 1\;\mathrm{s}</math>.<br />
For non-dimensional codes, the relevant parameter for this simulation is the vortex Reynolds number, <math>Re=u_0 L_0/\nu = 1,600</math> in the present setup. <br />
It should be noted that the theoretical solution is only stable for sufficiently small Reynolds numbers, explaining why the particular value <math>Re=1,600</math> has been chosen here, as done in many previous studies using the TGV configuration.<br />
<br />
A major interest of this test-case is that an analytical solution has been derived for such conditions~\cite{Taylor1937}:<br />
<br />
<math><br />
u(x,y,t) = u(x,y,0) \times \mathrm{e}^{-2\nu t/L_0^2} = u(x,y,0) \times \mathrm{e}^{-2 \, (t/\tau_{\mathrm{ref}}) / \mathrm{Re}} \,,<br />
</math><br />
<br />
<math><br />
v(x,y,t) = v(x,y,0) \times \mathrm{e}^{-2\nu t/L_0^2} = v(x,y,0) \times \mathrm{e}^{-2 \, (t/\tau_{\mathrm{ref}}) / \mathrm{Re}} \,.<br />
</math><br />
<br />
All physical conditions are now completely defined and only numerical parameters remain to be set.<br />
Based on previously published studies, the recommended parameters for time and space discretization are <math>\Delta t = \tau_\mathrm{ref}/2000 = 5 \times 10^{-4} \; \mathrm{s}</math> and <math>N = 64</math> grid points in each direction.<br />
The flow must be simulated for a physical time of <math>t=10 \; \tau_{\mathrm{ref}}=10\; \mathrm{s}</math>.<br />
<br />
= Aside suggestions =<br />
<br />
= Notes =<br />
<references><br />
<ref name="abelsamie2016">A. Abdelsamie and G. Fru and F. Dietzsch and G. Janiga and D. Thévenin , ''Towards direct numerical simulations of low-Mach number turbulent reacting and two-phase flows using immersed boundaries'', '''Comput. Fluids''', 2016 (131-5, pp123--141)</ref>. <br />
<ref name="Laizet2009">S. Laizet and E. Lamballais , ''High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy'', '''J. Comput. Phys.''', 2009 (228, pp5989--6015)</ref>.<br />
</references></div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Step_1&diff=23Step 12020-07-27T04:42:31Z<p>Lartigue: /* Step 1 Description */</p>
<hr />
<div>= Step 1 Description =<br />
<br />
The 2-D TGV test case has been already well documented in the scientific literature, e.g. <ref name="abelsamie2016" /><ref name="Laizet2009"/>.<br />
Therefore, only a brief description is given here. <br />
This configuration is used as verification step, since an analytic solution exists~\cite{Taylor1937}.<br />
<br />
An incompressible flow with constant density and viscosity is simulated.<br />
<br />
The fluid is contained in a square box of dimension <math>[0;L]^2</math> with periodic boundary conditions in both directions.<br />
The value <math>L=2 \pi L_0</math> with <math>L_0 = 1 \; \mathrm{m}</math> has been arbitrarily chosen in the present benchmark for dimensional codes.<br />
The numerical value for the kinematic viscosity is chosen as <math>\nu = 6.25 \times 10^{-4} \; \mathrm{m^2/s}</math> to get suitable conditions, as discussed below.<br />
The initial velocity components are prescribed as follows:<br />
<br />
<math><br />
u(x,y,0) = +u_0 \times \sin(x/L_0) \times \cos(y/L_0)\,,<br />
</math><br />
<br />
<math><br />
v(x,y,0) = -u_0 \times \cos(x/L_0) \times \sin(y/L_0)\,.<br />
</math><br />
<br />
with <math>u_0 = 1 \; \mathrm{m/s}$.<br />
The flow is thus initially composed of four vortices, one in each quarter of the box.<br />
The vortex turnover time is defined as <math>\tau_\mathrm{ref} = L_0/u_0 = 1\;\mathrm{s}</math>.<br />
For non-dimensional codes, the relevant parameter for this simulation is the vortex Reynolds number, <math>Re=u_0 L_0/\nu = 1,600</math> in the present setup. <br />
It should be noted that the theoretical solution is only stable for sufficiently small Reynolds numbers, explaining why the particular value <math>Re=1,600</math> has been chosen here, as done in many previous studies using the TGV configuration.<br />
<br />
A major interest of this test-case is that an analytical solution has been derived for such conditions~\cite{Taylor1937}:<br />
<br />
<math><br />
u(x,y,t) = u(x,y,0) \times \mathrm{e}^{-2\nu t/L_0^2} = u(x,y,0) \times \mathrm{e}^{-2 \, (t/\tau_{\mathrm{ref}}) / \mathrm{Re}} \,,<br />
</math><br />
<br />
<math><br />
v(x,y,t) = v(x,y,0) \times \mathrm{e}^{-2\nu t/L_0^2} = v(x,y,0) \times \mathrm{e}^{-2 \, (t/\tau_{\mathrm{ref}}) / \mathrm{Re}} \,.<br />
</math><br />
<br />
All physical conditions are now completely defined and only numerical parameters remain to be set.<br />
Based on previously published studies, the recommended parameters for time and space discretization are <math>\Delta t = \tau_\mathrm{ref}/2000 = 5 \times 10^{-4} \; \mathrm{s}</math> and <math>N = 64</math> grid points in each direction.<br />
The flow must be simulated for a physical time of <math>t=10 \; \tau_{\mathrm{ref}}=10\; \mathrm{s}</math>.<br />
<br />
= Aside suggestions =<br />
<br />
= Notes =<br />
<references><br />
<ref name="abelsamie2016">A. Abdelsamie and G. Fru and F. Dietzsch and G. Janiga and D. Thévenin , ''Towards direct numerical simulations of low-Mach number turbulent reacting and two-phase flows using immersed boundaries'', '''Comput. Fluids''', 2016 (131-5, pp123--141)</ref>. <br />
<ref name="Laizet2009">S. Laizet and E. Lamballais , ''High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy'', '''J. Comput. Phys.''', 2009 (228, pp5989--6015)</ref>.<br />
</references></div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Step_1&diff=22Step 12020-07-27T04:42:15Z<p>Lartigue: </p>
<hr />
<div>= Step 1 Description =<br />
<br />
The 2-D TGV test case has been already well documented in the scientific literature, e.g. <ref name="abelsamie2016" /><ref name="Laizet2009"/><br />
Therefore, only a brief description is given here. <br />
This configuration is used as verification step, since an analytic solution exists~\cite{Taylor1937}.<br />
<br />
An incompressible flow with constant density and viscosity is simulated.<br />
<br />
The fluid is contained in a square box of dimension <math>[0;L]^2</math> with periodic boundary conditions in both directions.<br />
The value <math>L=2 \pi L_0</math> with <math>L_0 = 1 \; \mathrm{m}</math> has been arbitrarily chosen in the present benchmark for dimensional codes.<br />
The numerical value for the kinematic viscosity is chosen as <math>\nu = 6.25 \times 10^{-4} \; \mathrm{m^2/s}</math> to get suitable conditions, as discussed below.<br />
The initial velocity components are prescribed as follows:<br />
<br />
<math><br />
u(x,y,0) = +u_0 \times \sin(x/L_0) \times \cos(y/L_0)\,,<br />
</math><br />
<br />
<math><br />
v(x,y,0) = -u_0 \times \cos(x/L_0) \times \sin(y/L_0)\,.<br />
</math><br />
<br />
with <math>u_0 = 1 \; \mathrm{m/s}$.<br />
The flow is thus initially composed of four vortices, one in each quarter of the box.<br />
The vortex turnover time is defined as <math>\tau_\mathrm{ref} = L_0/u_0 = 1\;\mathrm{s}</math>.<br />
For non-dimensional codes, the relevant parameter for this simulation is the vortex Reynolds number, <math>Re=u_0 L_0/\nu = 1,600</math> in the present setup. <br />
It should be noted that the theoretical solution is only stable for sufficiently small Reynolds numbers, explaining why the particular value <math>Re=1,600</math> has been chosen here, as done in many previous studies using the TGV configuration.<br />
<br />
A major interest of this test-case is that an analytical solution has been derived for such conditions~\cite{Taylor1937}:<br />
<br />
<math><br />
u(x,y,t) = u(x,y,0) \times \mathrm{e}^{-2\nu t/L_0^2} = u(x,y,0) \times \mathrm{e}^{-2 \, (t/\tau_{\mathrm{ref}}) / \mathrm{Re}} \,,<br />
</math><br />
<br />
<math><br />
v(x,y,t) = v(x,y,0) \times \mathrm{e}^{-2\nu t/L_0^2} = v(x,y,0) \times \mathrm{e}^{-2 \, (t/\tau_{\mathrm{ref}}) / \mathrm{Re}} \,.<br />
</math><br />
<br />
All physical conditions are now completely defined and only numerical parameters remain to be set.<br />
Based on previously published studies, the recommended parameters for time and space discretization are <math>\Delta t = \tau_\mathrm{ref}/2000 = 5 \times 10^{-4} \; \mathrm{s}</math> and <math>N = 64</math> grid points in each direction.<br />
The flow must be simulated for a physical time of <math>t=10 \; \tau_{\mathrm{ref}}=10\; \mathrm{s}</math>.<br />
<br />
= Aside suggestions =<br />
<br />
= Notes =<br />
<references><br />
<ref name="abelsamie2016">A. Abdelsamie and G. Fru and F. Dietzsch and G. Janiga and D. Thévenin , ''Towards direct numerical simulations of low-Mach number turbulent reacting and two-phase flows using immersed boundaries'', '''Comput. Fluids''', 2016 (131-5, pp123--141)</ref>. <br />
<ref name="Laizet2009">S. Laizet and E. Lamballais , ''High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy'', '''J. Comput. Phys.''', 2009 (228, pp5989--6015)</ref>.<br />
</references></div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Description&diff=21Description2020-07-27T04:26:26Z<p>Lartigue: </p>
<hr />
<div>= Description of the Benchmark = <br />
<br />
This benchmark is composed of 4 major steps that should be performed carefully one after the other.<br />
<br />
The objective of the following pages is to give a self-contained description of the benchmarks so that anyone can reproduce the [[results]] with its own high-fidelity code.<br />
<br />
* The [[Step 1]] is a simple 2D configuration with a single species at Reynolds number of 1,600.<br />
<br />
* The [[Step 2]] is the extension of the Step 1 to 3D. The Reynolds number is still 1,600. A transition to turbulence is expected to occur.<br />
<br />
* The [[Step 3]] builds on top of the Step 2 by including multiple species with steep gradients of concentrations as well as an inhomogeneous temperature. This situation is clearly closer to a real flame but the chemical reactions must not be activated yet.<br />
<br />
* The [[Step 4]] is a real flame configuration, very similar (but not identical) to Step 3 with reactions enabled.</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Step_1&diff=20Step 12020-07-27T04:24:34Z<p>Lartigue: Created page with "= Step 1 = This step is very easy to set-up: it is a simple 2D box with 4 Green-vortices inside it."</p>
<hr />
<div>= Step 1 =<br />
<br />
This step is very easy to set-up: it is a simple 2D box with 4 Green-vortices inside it.</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Description&diff=19Description2020-07-27T04:23:05Z<p>Lartigue: Created page with "= Description of the Benchmark = This benchmark is composed of 4 major steps that should be performed carefully one after the other. * The Step 1 is a simple 2D configu..."</p>
<hr />
<div>= Description of the Benchmark = <br />
<br />
This benchmark is composed of 4 major steps that should be performed carefully one after the other.<br />
<br />
* The [[Step 1]] is a simple 2D configuration with a single species at Reynolds number of 1,600.<br />
<br />
* The [[Step 2]] is the extension of the Step 1 to 3D. The Reynolds number is still 1,600. A transition to turbulence is expected to occur.<br />
<br />
* The [[Step 3]] builds on top of the Step 2 by including multiple species with steep gradients of concentrations as well as an inhomogeneous temperature. This situation is clearly closer to a real flame but the chemical reactions must not be activated yet.<br />
<br />
* The [[Step 4]] is a real flame configuration, very similar (but not identical) to Step 3 with reactions enabled.</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Main_Page&diff=18Main Page2020-07-27T04:16:29Z<p>Lartigue: </p>
<hr />
<div>= The Taylor-Green Vortex as a Benchmark for High-Fidelity Combustion Codes =<br />
<br />
Verification and validation are crucial steps for the development of any numerical model.<br />
<br />
While suitable processes have been established for commercial Computational Fluid Dynamics (CFD) codes, more difficult challenges must be faced for high-fidelity solvers.<br />
<br />
Benchmarks have been proposed in a series of dedicated conferences for non-reacting configurations.<br />
However, to our knowledge, no suitable approach has been published up to now regarding turbulent reacting flows.<br />
<br />
'''The purpose of this website is to present a full verification and validation chain for high-resolution codes employed to simulate turbulent reacting flows, first for Direct Numerical Simulation (DNS) of turbulent combustion in the limit of low Mach numbers.'''<br />
<br />
The selected configuration builds on top of the Taylor-Green vortex.<br />
Verification takes place by comparison with the analytical solution in two dimensions.<br />
Validation of the single-component flow is ensured by comparisons with published results obtained with a spectral code.<br />
Mixing without reaction is then considered, before computing finally a hydrogen-oxygen flame interacting with a 3-D Taylor-Green vortex. <br />
Three different low-Mach DNS solvers have been used for this study, demonstrating that the final accuracy of the DNS simulations is of the order of 1% for all quantities considered.<br />
<br />
The website is organised in three major parts:<br />
* The [[description]] of the test cases<br />
* The [[results]] of the test cases<br />
* An attempt to give a few guidelines on the [[performances]] that could be expected on the 3D test-cases.<br />
<br />
'''Put some images / videos here'''</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Main_Page&diff=16Main Page2019-08-30T13:11:48Z<p>Lartigue: </p>
<hr />
<div>= THE CFD BENCHMARK PUBLIC WIKI =<br />
<br />
The aim of this website is to gather various Benchmarks to validate CFD codes.<br />
<br />
== Organisation ==<br />
<br />
This website is dedicated to providing the '''DESCRIPTION''' of the various Benchmarks.<br />
<br />
The '''RESULTS''' of the Benchmarks are available ont [https://benchmark_private.coria-cfd.fr/index.php this website].<br />
<br />
== Benchmark List ==<br />
<br />
* [Taylor Green Vortex]<br />
** [[Taylor Green Vortex 1.1 (2D)]]<br />
** [[Taylor Green Vortex 1.2 (3D)]]<br />
** [[Taylor Green Vortex 1.3 (3D multispecies non-reactive)]]<br />
** [[Taylor Green Vortex 2 (3D multispecies + reactive)]]</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Main_Page&diff=15Main Page2019-08-30T13:11:13Z<p>Lartigue: </p>
<hr />
<div>= THE CFD BENCHMARK PUBLIC WIKI =<br />
<br />
The aim of this website is to gather various Benchmarks to validate CFD codes.<br />
<br />
== Organisation ==<br />
<br />
This website is dedicated to providing the '''DESCRIPTION''' of the various Benchmarks.<br />
<br />
The '''RESULTS''' of the Benchmarks are available ont [https://benchmark_private.coria-cfd.fr/index.php this website].<br />
<br />
== Benchmark List ==<br />
<br />
* Taylor Green Vortex<br />
** [[Taylor Green Vortex 1.1 (2D)]]<br />
** [[Taylor Green Vortex 1.2 (3D)]]<br />
** [[Taylor Green Vortex 1.3 (3D multispecies non-reactive)]]<br />
** [[Taylor Green Vortex 2 (3D multispecies + reactive)]]</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Main_Page&diff=14Main Page2019-08-30T13:09:22Z<p>Lartigue: </p>
<hr />
<div>= THE CFD BENCHMARK PUBLIC WIKI =<br />
<br />
The aim of this website is to gather various Benchmarks to validate CFD codes.<br />
<br />
== Organisation ==<br />
<br />
This website is dedicated to providing the '''DESCRIPTION''' of the various Benchmarks.<br />
<br />
The '''RESULTS''' of the Benchmarks are available ont [https://benchmark_private.coria-cfd.fr/index.php this website].<br />
<br />
== Benchmark List ==<br />
<br />
* Taylor Green Vortex<br />
** Taylor Green Vortex 1.1 (2D)<br />
** Taylor Green Vortex 1.2 (3D)<br />
** Taylor Green Vortex 1.3 (3D multispecies non-reactive)<br />
** Taylor Green Vortex 2 (3D multispecies + reactive)</div>Lartiguehttps://benchmark.coria-cfd.fr/index.php?title=Main_Page&diff=9Main Page2019-08-30T12:22:17Z<p>Lartigue: </p>
<hr />
<div>= THE CFD BENCHMARK PUBLIC WIKI =<br />
<br />
The aim of this website is to gather various Benchmarks to validate CFD codes.<br />
<br />
== Organisation ==<br />
<br />
This website is dedicated to providing the '''DESCRIPTION''' of the various Benchmarks.<br />
<br />
The '''RESULTS''' of the Benchmarks are available ont [https://benchmark_private.coria-cfd.fr/index.php this website].</div>Lartigue