View source for Analysis of 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 <math>t = 2 \tau_{ref} = 0.5 ms</math> 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 <math>t = 2 \tau_{ref} = 0.5 ms</math> 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 <math>t = 2 \tau_{ref} = 0.5 ms</math> 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 <math>O_2</math> and <math>OH</math>)
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 <math>O_2</math> is still smaller than the mass fraction of <math>OH</math> 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 <math>t = 2 \tau_{ref} = 0.5 ms</math> of physical time.
Only the quantities showing visible discrepancies at a resolution of 2563
(heat release, <math>Y_{O_2}</math>
, <math>Y_{OH}</math>) 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 <math>t = 2 \tau_{ref} = 0.5 ms</math> 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.'''