Revision as of 19:42, 23 August 2020

In this section, the results obtained for each configuration will be discussed by comparing the fields obtained with the three codes involved in the benchmark. It is important to emphasize that even small differences will be highlighted, since the proposed benchmark shall be used for the verification and validation of further high-fidelity codes.

Step 1

The verification involves a direct comparison with the analytical solution. For this purpose, analytic fields for velocity (x- and y-components) and vorticity ${\displaystyle (\delta _{x}v-\delta _{y}u)}$ at ${\displaystyle t=10\tau _{ref}}$ are presented in Fig. 3. It should be noted that both YALES2 and DINO used 642 grid points for this test case, while Nek5000 employed 82 spectral elements of order 8, which results in 64 discretization points in each direction. The velocity profiles along both centerlines of the domain at ${\displaystyle t=10\tau _{ref}}$ are shown in Fig. 4. It can be observed that the three codes give perfect visual agreement 14 with the analytical solution. Table 3 present the analytical maximal velocity at ${\displaystyle t=10\tau _{ref}}$ (as computed from Eq. 2) and the values obtained with the three codes, as well as the associated relative error: it is observed that the maximal deviation is less than ${\displaystyle 0.03\%}$ for the three codes.

This configuration, although quite far from any realistic flame, is nevertheless an excellent manner to verify the numerical procedure. It can be used to check the obtained discretization order in space and time and to quantify numerical dissipation, as documented for instance in Figure 5 of [25].

Step 2

The second step concerns the 3-D, non-reacting cold flow, used as validation by comparison with the published results of a pseudo-spectral solver [65]. The main quantities of interest for this comparison are:

1. Velocity profiles along centerlines of the domain at t = 12.11τref, as illustrated in Fig. 5. 2. The evolution of kinetic energy (${\displaystyle k(t)={\frac {1}{2}}<u_{i}u_{i}>}$) and of its dissipation rate (${\displaystyle \epsilon (t)=dk/dt=2\nu <S_{ij}S{ij}>}$) versus time, as shown in Fig. 6, where S{ij} is the symmetric strain-rate tensor,

${\displaystyle S{ij}={\frac {1}{2}}({\frac {\delta u_{i}}{\delta x_{j}}}+{\frac {\delta u_{j}}{\delta x_{i}}})}$ ; (11)

Comparing the velocity fields at t = 12.11 \tau_{ref} is done on purpose. As known from the TGV literature, this instant corresponds to a complex pseudo-turbulent field, before further turbulence decay due to dissipation (see also Figure 9 in [25]). Getting the correct velocity field in these conditions is challenging, since the obtained results are very sensitive with regard to the employed algorithms and discretization. Unlike the 2-D situation, no analytical solution is available there, and the results can only be compared to other numerical simulations. In the present case, the reference data are taken from a simulation relying on the pseudo-spectral code RLPK using 5123 grid points [65].

First, it appears on Fig. 5 that no differences can be identified visually from the velocity fields along the centerlines at ${\displaystyle t=12.11\tau _{ref}}$. Looking at Figs. 6 it can be observed that the three codes are able to reproduce the evolution of turbulence kinetic energy without any visible differences, whereas for the dissipation rate minute deviations appear at two instants (see enlargements in Fig. 6, right): (1) shortly after transition (${\displaystyle 11<t/\tau _{ref}<13.5}$) for YALES2, and (2) just before flow relaminarization (${\displaystyle 16.25<t/\tau _{ref}20}$) for both DINO and YALES2. The results of RLPK and of Nek5000 coincide visually at all times. These two time-instants are very sensitive moments at which the accuracy of the numerical methods and the resolution in time and space appear to play a major role. These small discrepancies are regarded as minor and 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 5123 grid points with a pseudo-spectral solver.

Step 3

The main difference between this step and the previous one, is that now species and heat diffusion are additionally taken into account in an inhomogeneous environment. Neither analytical nor reference solution are available for this configuration, so that only comparisons between the three codes involved in the benchmark are possible. Compared to Step 2, the presence of high-temperature regions additionally modifies the evolution of the TGV with time, turbulence being locally damped due to dilatation and higher viscosities. As a consequence, the needed resolution for this case is less than in Step 2: YALES2 and DINO used only 256 grid points in each direction while Nek5000 used 36 elements (again with 7 Gauss-Lobatto-Legendre points in each element), i.e. 252 points in each direction.

The results that will be compared involve:

1. Velocity profiles at ${\displaystyle t=2\tau _{ref}=0.5ms}$ along the centerlines of the domain, as shown in Fig. 7;

2. Profiles of H2 and O2 mass fractions and profile of temperature at ${\displaystyle t=2\tau _{ref}=0.5ms}$ along the y-centerline of the domain, as illustrated in Figs. 8 and 9;

3. Evolution of maximal temperature in the domain vs. time, as depicted in Fig. 10.

Looking at the results of velocity (Fig. 7) at time ${\displaystyle t=2\tau _{ref}=0.5ms}$ along the centerlines of the computational domain, it is observed that the three codes deliver the same velocity profiles; the agreement is visually perfect. The results for the two main species mass fractions (${\displaystyle Y_{H2}}$ and ${\displaystyle Y_{O2}}$) are also in excellent agreement among the three participating codes, as it can be observed from Fig. 8.

Regarding temperature, Figure 9 shows along the centerline two peaks and one valley, as expected. Very small deviations are revealed in the inlaid enlargements shown in Fig. 9. Finally, the evolution with time of maximum temperature inside the computational domain is presented in Fig. 10. Here again, no differences are observed at all concerning this parameter. As a conclusion concerning this step, the three codes are able to reproduce numerically the behavior of a complex multi-species, non-isothermal flow with excellent agreement, and are thus strong candidates for high-fidelity simulations of turbulent flames, as considered in the next and final step.

Step 4

Step 4 is definitely the most complex but also the most interesting case to compare high-fidelity codes for turbulent reacting flows, since it contains all the features relevant for turbulent combustion. Therefore, a more detailed analysis is useful. The comparisons will involve:

1. The evolution of maximum temperature versus time, as depicted in Fig. 10;

2. Velocity fields at ${\displaystyle t=2\tau _{ref}=0.5ms}$ along the centerlines of the domain, as shown in Fig. 11;

3. Profiles of temperature, heat release and mass fractions of H2, O2 and OH at ${\displaystyle t=2\tau _{ref}=0.5ms}$ along the centerline of the domain, as illustrated in Figs. 12 and 13.

The simulations of YALES2, DINO and Nek5000 are presented in the following subsections for two different resolutions in space (2563 and 5123 ), in order to check the impact of the spatial resolution on the results. Additional data for other grids are also available (3843 for both YALES2 and DINO, and 7683 only for DINO). They are not discussed at length in the text and in separate figures in the interest of space, but the corresponding values are included in the Tables 4 and 5 summarizing all results of Step 4. Additionally, all results at all grid resolutions are available online in the benchmark repository [1]. Starting with the evolution of maximum temperature versus time, a perfect visual agreement between all three codes is observed at all resolutions, as shown in Fig. 10 (with a resolution of 5123 ). This quantity does not appear to be difficult to predict correctly, as already observed previously for the non-reacting flow in Step 3, provided that the pressure variation due to the heat release is correctly taken into account.

Comparing results at spatial resolution of ${\displaystyle 256^{3}}$

The results shown in this section have been obtained for the same grid size than in Step 3, i.e. 2563 for YALES2 and DINO and 2523 for Nek5000. The corresponding results for velocity (Fig. 11) and temperature (Fig. 12, left) at time ${\displaystyle t=2\tau _{ref}=0.5ms}$ along the centerlines of the domain show visually a perfect agreement. Nevertheless, the three codes show slight differences concerning heat release and some mass fractions profiles (in particular ${\displaystyle O_{2}}$ and ${\displaystyle OH}$) around the center of the domain, as it can be observed from Figs. 12 (right) and 13. These differences – though small – are larger than those experienced in the non-reacting case. Note that there is originally no oxygen in this region, explaining why the mass fraction of ${\displaystyle O_{2}}$ is still smaller than the mass fraction of ${\displaystyle OH}$ there. One reason behind these discrepancies might be the well-known stiffness of the chemical source terms, inducing different non-linear effects as a function of the underlying algorithms employed for integration in time. Another possible source of error is the employed spatial discretization, which might still be insufficient to perfectly capture the reaction front; in the present case, the typical cell size is approximately 25 µm. To check this last point, the simulations have been repeated with a finer spatial resolution, as discussed in the next subsection.

Comparing results at spatial resolution of ${\displaystyle 512^{3}}$

The present results have been obtained on a grid size of 5123 for YALES2 and DINO, while Nek5000 relies on similar discretization size of 5143 (57 spectral elements of order 9 in each direction). To reduce computational costs, the simulation is conducted only for the first ${\displaystyle t=2\tau _{ref}=0.5ms}$ of physical time. Only the quantities showing visible discrepancies at a resolution of 2563 (heat release, ${\displaystyle Y_{O_{2}}}$ , ${\displaystyle Y_{OH}}$) are discussed here in the interest of brevity, since all other quantities already revealed a perfect agreement for the previous resolution. It can be observed in Figs. 14 and 15 that doubling the spatial resolution in each direction did not improve the comparisons in a clear way; marginal differences still exist between the codes, and a convergence towards a unique solution is not really visible. To discuss this issue in more detail, a refined analysis is necessary, as discussed in the next subsection.

Quantitative comparisons at the center point at t = 0.5 ms

In Table 4 the values of different variables at the center of the numerical domain at time ${\displaystyle t=2\tau _{ref}=0.5ms}$ are presented and analyzed. These values have been obtained for the three different codes involved in the benchmark (from left to right, YALES2, DINO, Nek5000), for an increasing spatial resolution from left to right, but also with different timesteps. The controlling time-limiter (as a condition on maximum CFL or Fourier number with corresponding value) is also listed in the table; it depends on the retained criteria and on the explicit or implicit integration of the corresponding terms in the equations.

Looking separately at the values obtained by each code, it is not always easy to recognize the convergence toward a single value that would be expected for a grid-independence analysis. By a comparison between the last column for each code, a good agreement is overall observed, in spite of differences regarding algorithms, resolution in space and in time. Nevertheless, the agreement is never perfect, and trends can better be seen by computing differences. This is why, choosing arbitrarily the results of the implicit time integration at the highest spatial resolution with DINO (7683) as a reference, all corresponding relative errors have been computed.

Analyzing in detail all the values, the following intermediate conclusions can be drawn:

• The overall agreement between the three completely independent high-resolution codes employed in the benchmark is very good, with typical relative differences of the order of 1% for the essential quantities used to analyze turbulent combustion (temperature, mass fractions, heat release). 23

• Compared to the differences observed in the previous verification step (errors below 0.03%), the variations are obviously much larger, typically by two orders of magnitude. This is a result of the far more challenging configuration, with additional physicochemical complexity, stiffer profiles, highly non-linear processes in space and time.

• Increasing further the spatial resolution (which is also connected to a reduction of the timestep) does not seem to increase much the observed agreement between the codes. For all considered grids in the analysis finer than 2563, overall differences of the order of 1% are observed. Often, using a finer resolution leads to a better agreement for most of the indicators, but to a worse comparison for some other ones.

• Though this has been attempted, it was impossible to obtain meaningful predictions using the Richardson extrapolation [69, 70], since the results of all codes are non-monotonic when increasing resolution in space.

• Somewhat unexpectedly, the observed uncertainty is in the same range for temperature, mass fractions of main species or of radicals, and heat release. Quantities that are typically considered more sensitive (radicals, heat release) do not lead to larger discrepancies in the analysis.

Finally, the central finding is that all codes employed in the benchmark deliver suitable results for this configuration, and this already at a typical grid resolution of 2563 for this particular case. An irreducible uncertainty of the order of 1% is observed for all quantities relevant for turbulent combustion. This uncertainty, noticeably larger than for cold flows, is apparently the result of stiff non-linear processes, of different splitting schemes, and of the different libraries/library versions employed for computing thermodynamic, diffusion, and reaction parameters. After this detailed analysis of uncertainty, it is necessary to quantify the corresponding numerical costs needed to get this level of accuracy.