CFD Benchmark - User contributions [en]

Step 3

2020-08-27T13:51:54Z

Coria: /* Kinetics */

= Step 3 description =

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'''.
In this manner, it is possible to assess the accuracy of the numerical models involved in the description of species and heat diffusion
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.

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.
However, there is no variation of the thermodynamic pressure in time, contrary to the next step.

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>.
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.
Again, periodic boundary conditions are used in all three spatial directions.

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>.

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>).
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>.

To avoid any numerical instability, '''the steps at the interface between both regions are smoothed out using hyperbolic tangent functions''' as follows:

<math>R_d(x) = |x-0.5\,L|,</math> (1)

<math>\psi(x) = 0.5\left[1+\mathrm{tanh}\left(\frac{c\,(R_d(x)-R)}{R}\right)\right],</math> (2)

<math>Y_\mathrm{H_2}(x) = Y^0_{\mathrm{H_2}}\left(1-\psi(x)\right),</math> (3)

<math>Y_\mathrm{O_2}(x) = Y^0_{\mathrm{O_2}}\left(\psi(x)\right).</math> (4)

where <math>R = \pi/4</math> mm and <math>c = 3</math> are the half-width of the central slab and stiffness parameter, respectively.
As a consequence, there is initially a small region where both fuel and oxidizer coexist.
Finally, a nitrogen complement is added everywhere using <math>Y_\mathrm{N_2}=1-Y_\mathrm{H_2}-Y_\mathrm{O_2}</math>.

To also test the behaviour of the codes with respect to heat diffusion, '''a non-homogeneous temperature profile is finally imposed''' as follows:

* Compute in each cell the equilibrium temperature for the local mixture for constant pressure and enthalpy by using a dedicated solver.

* Enforce the resulting temperature profile, but keep the species profiles to their initial values (Eqs. (3) and (4)).

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],
[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.

The resulting, initial configuration thus involves '''multiple species at different temperatures''' with steep profiles, just like in a real flame.
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.
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).
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.
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.
These values have been estimated from a separate DNS for a similar configuration using the mixture-averaged diffusion model.
The precise description of the associated thermodynamic state is available on the website.

{| class="wikitable alternance center"
|+ 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
|-
! scope="row" | Species
| <math>H_2</math>
| <math>H</math>
| <math>O_2</math>
| <math>OH</math>
| <math>O</math>
| <math>H_2O</math>
| <math>HO_2</math>
| <math>H_2O_2</math>
| <math>N_2</math>
|-
! scope="row" | Lewis number
| 0.3290
| 0.2228
| 1.2703
| 0.8279
| 0.8128
| 1.0741
| 1.2582
| 1.2665
| 1.8268
|}

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;
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'''.

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.

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>.
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>.

The '''reference time scale''' is here <math>\tau_\mathrm{ref}=L_0/u_0 = 0.25 \; \mathrm{ms}</math>.
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.

[[File:wmag_t0.png|250px|alt text]]
[[File:T_t0.png|250px|alt text]]
[[File:H2_t0.png|250px|alt text]]
[[File:O2_t0.png|250px|alt text]]

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''').

[[File:profiles_3d_zoom.pdf|250px|alt text]]

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''').

= Kinetics =

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.

This mechanism is provided here in the Cantera format:
* [[File:H2_williams_12.xml.zip | ctml ]]
* [[File:H2_williams_12.cti | cti]]

= Transport and thermodynamic properties used for this Step =

\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.}

\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.}

'''Put the transport data here'''

= Aside suggestions =

TBD

= References =

<references>
<ref name="Boivin2011">
<bibtex>
@article{Boivin2011,
author= {P. Boivin, C. Jiménez, A.L. Sanchez, and F.A. Williams},
title= {An explicit reduced mechanism for <math>H_2</math>-air combustion.},
journal={Proc. Combust. Inst.},
year= {2011},
volume={33},
pages={517--523},
doi= {https://doi.org/10.1016/j.proci.2010.05.002}
}
</bibtex>
</ref>
<ref name="pecs">
<bibtex>
@article{Pecs,
author= {R. Hilbert, F. Tap, H. El-Rabii, and D. Thévenin},
title= {Impact of detailed chemistry and transport models on turbulent combustion simulations},
journal={Prog. Energ. Combust. Sci.},
year= {2004},
volume={30},
pages={61--117},
}
</bibtex>
</ref>
</references>

MediaWiki:Sidebar

2020-08-26T13:49:18Z

Coria:

MediaWiki:Sidebar

2020-08-26T13:48:13Z

Coria:

Step 2

2020-08-24T20:40:36Z

Coria: /* Step 2 Description */

= 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 configuration, a reference solution obtained with a pseudo-spectral solver is available in the scientific literature<ref name="vanrees2011"/> and can be used for validation by a direct comparison.

The domain size has been set to <math>L = 2 \pi L_0</math> in each direction, where <math>L_0 = 1\; \mathrm{m}</math> 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.

The initial conditions for 3-D TGV are given by the following set of equations:

'''(Eq. 1)''' <math>u(x,y,z,0) = +u_0 \times \sin(x/L_0) \times \cos(y/L_0) \times \cos(z/L_0)</math>

'''(Eq. 2)''' <math>v(x,y,z,0) = -u_0 \times \cos(x/L_0) \times \sin(y/L_0) \times \cos(z/L_0)</math>

'''(Eq. 3)''' <math> w(x,y,z,0) = 0</math>

The reference velocity magnitude is set again to <math>u_0 = 1 \; \mathrm{m/s}</math>.
The desired Reynolds number is again <math>Re = {u_0 L_0}/{\nu}=1,600</math>, as in the previous section, since this value has been used as well in the reference study employed for comparison<ref name="vanrees2011" />.
To obtain the proper value of Re, the kinematic viscosity is set once again to <math>\nu = 6.25 \times 10^{-4} \; \mathrm{m^2/s}</math>.
With this set of parameters, the reference time scale is <math>\tau_\mathrm{ref}=L_0/u_0 = 1 \; \mathrm{s}</math>.
The flow simulation must be carried out for a physical time of <math>t=20 \; \tau_\mathrm{ref}</math>.

Regarding spatial discretization, a resolution of at least '''512 nodes''' (or '''cells''') in each direction is recommended.
This value leads to an over-resolved simulation during the first few <math>\tau_\mathrm{ref}</math> but is necessary to capture properly the '''transition to turbulence''' that should occur '''between 10 and 15''' <math>\tau_\mathrm{ref}</math>.
As a point of comparison, the reference solution from<ref name="vanrees2011" /> has been obtained with a spectral code on a <math>512^3</math> grid.
The timestep should be chosen to ensure that the maximum values of the Courant-Friedrichs-Lewy (CFL) and Fourier (Fo) numbers remain below <math>0.3</math> and <math>0.2</math>, respectively.
This values are actually quite conservative since the focus of this study is set on accuracy comparisons and not on pure computational performance.

The image below gives an indication of the initial fields of the velocity components and of the vorticity obtained thanks to the '''(Eq. 1)''', '''(Eq. 2)''' and '''(Eq. 3)'''.

[[File:fields_step2.png|400px|fields step 2]]

The evolution of the magnitude of the velocity and of the vorticity are presented on the following video:

<div class="infobox" style="width: 700px;">
{| class="floatright" style="border: 1px solid #ccc; margin: 50px;"
|{{#widget:YouTube|id=k7JRrvBOSSY|width=600|height=500}}
|}
</div>

= Aside Suggestion =

Check mesh convergence and time integration independence.

Check that <math>dKE/dt = \int Sij:Sij = \int Enstrophy</math>

= References =

<references>
<ref name="vanrees2011">
<bibtex>
@article{Vanrees2011,
author= {W.M. van Rees, A. Leonard, D.I. Pullin, and P. Koumoutsakos},
title= {A comparison of vortex and pseudo-spectral
methods for the simulation of periodic vortical flows at high Reynolds numbers.},
journal={J. Comput. Phys.},
year= {2011},
volume={230},
pages={2794--2805},
}
</bibtex>
</ref>
</references>

Step 2

2020-08-24T19:14:57Z

Coria: /* Step 2 Description */

= 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 configuration, a reference solution obtained with a pseudo-spectral solver is available in the scientific literature<ref name="vanrees2011"/> and can be used for validation by a direct comparison.

The domain size has been set to <math>L = 2 \pi L_0</math> in each direction, where <math>L_0 = 1\; \mathrm{m}</math> 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.

The initial conditions for 3-D TGV are given by the following set of equations:

<math>u(x,y,z,0) = +u_0 \times \sin(x/L_0) \times \cos(y/L_0) \times \cos(z/L_0)</math> (1)

<math>v(x,y,z,0) = -u_0 \times \cos(x/L_0) \times \sin(y/L_0) \times \cos(z/L_0)</math> (2)

<math> w(x,y,z,0) = 0</math> (3)

The reference velocity magnitude is set again to <math>u_0 = 1 \; \mathrm{m/s}</math>.
The desired Reynolds number is again <math>Re = {u_0 L_0}/{\nu}=1,600</math>, as in the previous section, since this value has been used as well in the reference study employed for comparison<ref name="vanrees2011" />.
To obtain the proper value of Re, the kinematic viscosity is set once again to <math>\nu = 6.25 \times 10^{-4} \; \mathrm{m^2/s}</math>.
With this set of parameters, the reference time scale is <math>\tau_\mathrm{ref}=L_0/u_0 = 1 \; \mathrm{s}</math>.
The flow simulation must be carried out for a physical time of <math>t=20 \; \tau_\mathrm{ref}</math>.

Regarding spatial discretization, a resolution of at least '''512 nodes''' (or '''cells''') in each direction is recommended.
This value leads to an over-resolved simulation during the first few <math>\tau_\mathrm{ref}</math> but is necessary to capture properly the '''transition to turbulence''' that should occur '''between 10 and 15''' <math>\tau_\mathrm{ref}</math>.
As a point of comparison, the reference solution from<ref name="vanrees2011" /> has been obtained with a spectral code on a <math>512^3</math> grid.
The timestep should be chosen to ensure that the maximum values of the Courant-Friedrichs-Lewy (CFL) and Fourier (Fo) numbers remain below <math>0.3</math> and <math>0.2</math>, respectively.
This values are actually quite conservative since the focus of this study is set on accuracy comparisons and not on pure computational performance.

'''2D Large-Eddy Simulation''', injection of a '''premixed kerosene/air mixture''' on the left with a high level of turbulence.
Some kerosene droplets are added to this premixing.

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.

[[File:fields_step2.png|400px|fields step 2]]

You can observe above the initial fields of the velocity components and of the vorticity. Those fields are decribed thanks to the equations (1), (2) and (2).

<div class="infobox" style="width: 700px;">
{| class="floatright" style="border: 1px solid #ccc; margin: 50px;"
|{{#widget:YouTube|id=k7JRrvBOSSY|width=600|height=500}}
|}
</div>

Here is a video showing the phenomenon of step 2.

= Aside Suggestion =

Check mesh convergence and time integration independence.

Check that <math>dKE/dt = \int Sij:Sij = \int Enstrophy</math>

= References =

<references>
<ref name="vanrees2011">
<bibtex>
@article{Vanrees2011,
author= {W.M. van Rees, A. Leonard, D.I. Pullin, and P. Koumoutsakos},
title= {A comparison of vortex and pseudo-spectral
methods for the simulation of periodic vortical flows at high Reynolds numbers.},
journal={J. Comput. Phys.},
year= {2011},
volume={230},
pages={2794--2805},
}
</bibtex>
</ref>
</references>

Codes

2020-08-23T20:24:32Z

Coria: /* General comments on the codes */

= 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.php/YALES2 YALES2], [http://www.lss.ovgu.de/lss/en/Research/Computational+Fluid+Dynamics.html DINO] and [http://nek5000.mcs.anl.gov Nek5000].

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.

== YALES2 ==

YALES2 is a '''massively parallel multiphysics platform'''<ref>V. Moureau. Yales2 public website [https://www.coria-cfd.fr/index.php/YALES2 www.coria-cfd.fr/index.php/YALES2].</ref><ref>V. Moureau, P. Domingo, and L. Vervisch. From large-eddy simulation to direct numerical simulation of a lean
premixed swirl flame: Filtered laminar flame-pdf modeling. Combust. Flame, 158(7):1340–1357, 2011.</ref> developed since 2009 by Moureau, Lartigue and co-workers at CORIA (Rouen, France).
It is dedicated to the high-fidelity simulation of '''low-Mach number flows in complex geometries'''.
It is based on the '''Finite Volumes formulation''' of the '''Navier-Stokes equations''' and it can solve both '''non-reacting''' and '''reacting flows'''.
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.
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>P. Benard, G. Balarac, V. Moureau, C. Dobrzynski, G. Lartigue, and Y. D'Angelo. Mesh adaptation for large eddy simulations in complex geometries. Int. J. Numer Methods Fluids, 81:719-740, 2015.</ref>.
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.
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.

As most low-Mach number codes, the time-advancement is based on a '''projection-correction method''' following the pioneering work of<ref>A.J. Chorin. Numerical solution of the Navier-Stokes equations. Math. Computation, 22(104):745–762, 1968.</ref>.
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>M. Kraushaar. Application of the compressible and low-Mach number approaches to Large-Eddy Simulation of turbulent flows in aero-engines. PhD thesis, PhD, Institut National Polytechnique de Toulouse-INPT, 2011.</ref>, combined with a '''4th-order node-based centered finite-volume discretization''' of the convective and diffusive terms.
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'''.
The correction step is required to ensure '''mass conservation''', using a pressure that arises from a '''Poisson equation'''.
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>M. Malandain, N. Maheu, and V. Moureau. Optimization of the deflated conjugate gradient algorithm for the solving of elliptic equations on massively parallel machines. J. Comput. Phys., 238:32–47, 2013.</ref>.

When considering '''multispecies''' and '''non-isothermal flows''', the extension of the classical projection method proposed by Pierce et al.<ref>C.D. Pierce and P. Moin. Progress-variable approach for large-eddy simulation of non-premixed turbulent combustion. J. Fluid Mech., 504:73–97, 2004.</ref> is used to account for '''variable-density flows'''.
All the thermodynamic and transport properties are provided by the '''Cantera software'''<ref>.G. Goodwin. An open-source, extensible software suite for CVD process simulation. Chemical Vapor Deposition XVI and EUROCVD, 14:2003–2008, 2003.</ref>, which has been fully re-implemented in '''Fortran''' to avoid any performance issues.
Each species thermodynamic property is specified by '''5th-order polynomials''' on two temperature ranges (below and above 1000 K).
The mixture is supposed to be both thermally and mechanically perfect.

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<ref>J. Hirschfelder, R.B. Bird, and C.F. Curtiss. Molecular Theory of Gases and Liquids. Wiley, 1964.</ref>;

2) alternatively, simplified laws (for example the Sutherland law<ref>W. Sutherland. LII. the viscosity of gases and molecular force. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 36(223):507–531, 1893.</ref> can be used for the '''viscosity''', while fixed values for '''Prandtl''' and '''Schmidt numbers''' allow the computation of '''conductivity''' and '''diffusion coefficients''';

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.
This last approach has been retained in the present benchmark.

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.

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.
Solving these phenomena with explicit methods would thus drastically limit the '''global timestep''' of the whole simulation and induce an overwhelming CPU cost.
To mitigate these effects when dealing with '''multi-species reacting flows''', the classical operator '''splitting technique''' is used.
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.
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.
The chemical source terms are then integrated with a dedicated stiff solver, namely the '''CVODE''' library from SUNDIALS<ref name="cohen1996cvode">S.D. Cohen, A.C. Hindmarsh, and P.F. Dubois. CVODE, a stiff/nonstiff ODE solver in C. Computers Phys., 10(2):138–143, 1996 </ref><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">A.C. Hindmarsh, P.N. Brown, K.E. Grant, S.L. Lee, R. Serban, D.E. Shumaker, and C.S. Woodward. SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers. ACM Trans. Math. Soft. (TOMS), 31(3):363–396, 2005.</ref>.
To that purpose, each control volume is considered as an isolated reactor with both constant pressure and enthalpy.

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.
To overcome this difficulty, a dedicated '''MPI''' dynamic scheduler based on a work-sharing algorithm ensures global load-balancing.

== DINO ==

DINO is a '''3-D DNS code''' used for '''incompressible''' or '''low-Mach number flows''', the latter approach being used in this project.
The development of DINO started in the group of D. Thévenin (Univ. of Magdeburg) in 2013.
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>N. Li and S. Laizet. 2DECOMP&FFT - a highly scalable 2D decomposition library and FFT interface. In Cray
User’s Group 2010 conference, Edinburgh, 2010.</ref>.
The code offers different features and algorithms in order to investigate different physicochemical processes.
Spatial derivative are computed by default using '''sixth-order central finite differences'''.
Time integration relies on '''several Runge-Kutta solvers'''.
In what follows, an explicit '''4th-order Runge-Kutta approach''' has been used.
A '''3rd-order semi-implicit Runge-Kutta integration''' can be activated as needed, when considering stiff chemistry.
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.
All thermodynamic, chemical and transport properties are computed using the open-source library '''Cantera 2.4.0'''<ref>Cantera.</ref>.

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<ref>A. Ern and V. Giovangigli. Fast and accurate multicomponent transport property evaluation. J. Comput. Phys.,
120:105–116, 1995.</ref>.

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.
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.

'''Multi-phase flows''' can be simulated in DINO using resolved or non-resolved (point) particles and droplets using a '''Lagrangian approach'''<ref>A. Abdelsamie and D. Thévenin. On the behavior of spray combustion in a turbulent spatially-evolving jet investigated by direct numerical simulation. Proc. Combust. Inst., 37(3):2493–2502, 2019.</ref><ref>A. Abdelsamie and D. Thévenin. Nanoparticle behavior and formation in turbulent spray flames investigated by DNS. In M. Garcia-Villalba, H. Kuerten, and M. Salvetti, editors, Direct and Large Eddy Simulation XII, volume 27 of ERCOFTAC Series. Springer, 2020.</ref>. Complex boundaries are represented by a novel second-order immersed boundary method implementation ('''IBM''') based on a directional extrapolation scheme<ref>C. Chi, A. Abdelsamie, and D. Thévenin. A directional ghost-cell immersed boundary method for incompressible flows. J. Comput. Phys., 404:109122–109142, 2020.</ref>.
More details about the implemented algorithms can be found in particular in<ref>A. Abdelsamie, G. Fru, F. Dietzsch, G. Janiga, and D. Thévenin. Towards direct numerical simulations of low-Mach number turbulent reacting and two-phase flows using immersed boundaries. Comput. Fluids, 131(5):123–141, 2016.</ref>.
Since DINO has been developed as a multi-purpose code for analyzing many different '''reacting''' and '''non-reacting flows'''<ref>C. Chi, A. Abdelsamie, and D. Thévenin. Direct numerical simulations of hotspot-induced ignition in homogeneous hydrogen-air pre-mixtures and ignition spot tracking. Flow Turbul. Combust., 101(1):103–121, 2018.</ref><ref>T. Oster, A. Abdelsamie, M. Motejat, T. Gerrits, C. R¨ossl, D. Thévenin, and H. Theisel. On-the-fly tracking of flame surfaces for the visual analysis of combustion processes. Comput. Graph. Forum, 37(6):358–369, 2018.</ref><ref>A. Abdelsamie and D. Thévenin. Impact of scalar dissipation rate on turbulent spray combustion investigated
by DNS. In M. Salvetti, V. Armenio, J. Fr¨ohlich, B. Geurts, and H. Kuerten, editors, Direct and Large-Eddy
Simulation XI, volume 25 of ERCOFTAC Series. Springer, 2019.</ref>, a detailed verification and validation is obviously essential.

== Nek5000 ==

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">A.G. Tomboulides, J.C.Y. Lee, and S.A. Orszag. Numerical simulation of low Mach number reactive flows. J. Sci. Comput., 12:139–167, 1997.</ref>.
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">M.O. Deville, P.F. Fischer, and E.H. Mund. High-order Methods for Incompressible Fluid Flows. Cambridge
University Press, 2002.</ref>.
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.
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>).
The grids can be unstructured and allow for '''static local refinement''', while '''adaptive mesh refinement''' has been recently developed<ref>A. Tanarro, F. Mallor, N. Offermans, A. Peplinski, R. Vinuesa, and P. Schlatter. Enabling adaptive mesh refinement for spectral-element simulations of turbulence around wing sections. Flow Turbul. Combust., 105(2):415–436, 2020.</ref>.
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.

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.
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.
As a result, data movement per grid point is the same as for '''low-order methods'''.
It uses scalable domain-decomposition-based iterative solvers with efficient preconditioners.
Communication is based on the Message Passing Interface ('''MPI''') standard, and the code has proven scalability to over one million ranks.
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.

'''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).
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).
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" />.
The thermodynamic properties, detailed chemistry, and transport properties are provided by optimized subroutines compatible with '''Chemkin'''<ref name="chemkin">R.J. Kee, F.M. Rupley, and J.A. Miller. Chemkin-II: A Fortran chemical kinetics package for the analysis of gas-phase chemical kinetics. SAND-89-8009, 1989.</ref>.

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">M. Schmitt, C.E. Frouzakis, A.G. Tomboulides, Y.M. Wright, and K. Boulouchos. Direct numerical simulation of the effect of compression on the flow, temperature and composition under engine-like conditions. Proc. Combust. Inst., 35:3069–3077, 2015.</ref><ref name="TCC">G.K. Giannakopoulos, C.E. Frouzakis, P.F. Fischer, A.G. Tomboulides, and K. Boulouchos. LES of the gasexchange process inside an internal combustion engine using a high-order method. Flow Turbul. Combust.,
104:673–692, 2020.</ref>.
It can account for conjugate fluid-solid heat transfer and detailed gas phase as well as surface kinetics<ref name="catalytic">B.O. Arani, C.E. Frouzakis, J. Mantzaras, and K. Boulouchos. Three-dimensional direct numerical simulations of turbulent fuel-lean H2/air hetero-/homogeneous combustion over Pt with detailed chemistry. Proc. Combust. Inst., 36(3):4355–4363, 2017.</ref>.

== General comments on the codes ==

The three '''aforementioned solvers''' are all '''unsteady''', '''high-fidelity codes''' based on the '''low-Mach number''' formulation of the '''Navier-Stokes equations'''.
They can perform both '''Direct Numerical Simulations''' or '''Large Eddy Simulations''' of reacting flows, only '''DNS''' being considered here.
However, they differ in a certain number of points, mainly from the numerical point of view.
The aim of this section is to emphasize those major differences.
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.
On the other hand, both '''DINO''' and '''Nek5000''' are mostly dedicated to '''DNS''' of configurations found in fundamental research.
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).
As a consequence, both '''DINO''' and '''Nek5000''' employ '''higher-order numerical schemes''' compared to '''YALES2''', limited at best to a '''4th-order scheme'''.
All codes rely on dedicated libraries to compute the thermo-chemical properties of the flow, either '''Chemkin''', '''Cantera''', or in-house versions of those.
Moreover, they also rely at least to some extent on external software to perform the temporal integration of the stiff chemical source terms.
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.
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.
'''Nek5000''' employs CVODE to integrate the thermochemical equations without further '''splitting''' of the different terms accounting for convection, diffusion and chemistry.
The main differences between the three codes are summarized in Table~\ref{Tab:codes}.
Please note that the presented values are those used for the benchmark, even though some other options are available in each codes.

{| class="wikitable alternance center"
|+ Table 1: Major numerical properties of the three high-fidelity codes as used in this benchmark
|-
! scope="col" | Code
! scope="col" | YALES2
! scope="col" | DINO
! scope="col" | Nek5000
|-
| Connectivity
| Structured
| Unstructured
| Unstructured
|-
| Discretization Type
| Finite Volumes
| Finite Differences
| Spectral Elements
|-
| Grid point distribution
| Regular hexahedra
| Regular hexahedra
| Regular hexahedra with GLL points
|-
| Spatial order
| 4th
| 6th
| 7th - 15th (typically)
|-
| Temporal method
| expl. RK4
| expl. RK4 / semi-impl. RK3
| semi-impl. BDF3
|-
| Pressure solver
| CG with Deflation MPrec.
| FFT-based
| CG/GMRES with Jacobi/Schwartz Prec.
|-
| Thermo-chemistry
| Cantera (re-coded)
| Cantera
| Chemkin interface
|-
| Chemistry integration
| CVODE
| PyJac
| CVODE
|-
| Operator splitting
| Yes
| No
| No
|-
| Parallel paradigm
| MPI
| MPI
| MPI
|}

= References =

Main Page

2020-08-23T18:36:38Z

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{{#customtitle:TGV Benchmark|The Taylor-Green Vortex as a Benchmark - benchmark.coria-cfd.fr}}

= The Taylor-Green Vortex as a Benchmark for High-Fidelity Combustion Simulations Using Low-Mach Solvers =

The present web site is a complement of an article that has been submitted to '''Computers and Fluids''' in August 2020.

Verification and validation are crucial steps for the development of any numerical model.

While suitable processes have been established for commercial Computational Fluid Dynamics (CFD) codes, more difficult challenges must be faced for high-fidelity solvers.

Benchmarks have been proposed in a series of dedicated conferences for non-reacting configurations.
However, to our knowledge, no suitable approach has been published up to now regarding turbulent reacting flows.

'''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.'''

The selected configuration builds on top of the Taylor-Green vortex.
Verification takes place by comparison with the analytical solution in two dimensions.
Validation of the single-component flow is ensured by comparisons with published results obtained with a spectral code.
Mixing without reaction is then considered, before computing finally a hydrogen-oxygen flame interacting with a 3-D Taylor-Green vortex.
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.

The website is organised in four major parts:
* A presentation of the three [[codes]] used for the Benchmark
* The [[description]] of the test cases
* The [[analysis]] of the results
* An attempt to give a few guidelines on the [[performances]] that could be expected on the 3D test-cases.

All the raw [[results]] of the 3 codes are available online.

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