|
|
| (19 intermediate revisions by the same user not shown) |
| Line 1: |
Line 1: |
| − | In this section, the results obtained for each configuration will be discussed by comparing the fields obtained | + | 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 | + | 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 | + | highlighted, since the proposed benchmark shall be used for the '''verification''' and '''validation''' of further '''high-fidelity |
| − | codes. | + | codes'''. |
| | | | |
| − | = Step 1 =
| + | The objective of the following pages is to give the analyses of the different steps, such as: |
| | | | |
| − | The verification involves a direct comparison with the analytical solution. For this purpose, analytic fields for | + | * The [[Analysis of Step 1]]. |
| − | velocity (x- and y-components) and vorticity (∂xv − ∂yu) at t=10 τref 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 t = 10τref 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 t = 10 τref (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 0.03% for the three codes.
| + | |
| | | | |
| − | This configuration, although quite far from any realistic flame, is nevertheless an excellent manner to verify the
| + | * The [[Analysis of Step 2]]. |
| − | 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 [[Analysis of Step 3]]. |
| | | | |
| − | The second step concerns the 3-D, non-reacting cold flow, used as validation by comparison with the published | + | * The [[Analysis of Step 4]]. |
| − | 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 (k(t) = 1
| + | |
| − | 2
| + | |
| − | huiuii) and of its dissipation rate (�(t) = −dk/dt = 2 ν hSijSij i)
| + | |
| − | versus time, as shown in Fig. 6, where Sij is the symmetric strain-rate tensor,
| + | |
| − | Sij =
| + | |
| − | 1
| + | |
| − | 2
| + | |
| − | �
| + | |
| − | ∂ui
| + | |
| − | ∂xj
| + | |
| − | +
| + | |
| − | ∂uj
| + | |
| − | ∂xi
| + | |
| − | �
| + | |
| − | ; (11)
| + | |
| − | Comparing the velocity fields at t = 12.11τref 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 t = 12.11 τref.
| + | |
| − | 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 (11 < t/τref < 13.5) for YALES2, and (2) just before flow
| + | |
| − | relaminarization (16.25 < t/τref ≤ 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 t = 2τref = 0.5 ms along the centerlines of the domain, as shown in Fig. 7;
| + | |
| − | 2. Profiles of H2 and O2 mass fractions and profile of temperature at t = 2τref = 0.5 ms 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 t = 2τref ms 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 (YH2
| + | |
| − | and YO2
| + | |
| − | ) 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 t = 2τref = 0.5 ms 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 t = 2τref = 0.5 ms 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 <math>256^3</math> ==
| + | |
| − | | + | |
| − | 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 t = 2τref = 0.5 ms 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 O2 and 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 O2 is still smaller than the mass fraction of 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 <math>512^3</math> ==
| + | |
| − | | + | |
| − | 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 2τref = 0.5 ms of physical time.
| + | |
| − | Only the quantities showing visible discrepancies at a resolution of 2563
| + | |
| − | (heat release, YO2
| + | |
| − | , YOH) 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 t = 2τref = 0.5 ms 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.
| + | |
The objective of the following pages is to give the analyses of the different steps, such as:
