Analysis - Revision history

Xonnainty at 19:02, 23 August 2020

2020-08-23T19:02:02Z

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Xonnainty at 18:56, 23 August 2020

2020-08-23T18:56:47Z

Xonnainty at 18:54, 23 August 2020

2020-08-23T18:54:53Z

Xonnainty: /* Step 2 */

2020-08-23T18:45:18Z

‎Step 2

Xonnainty: /* Step 2 */

2020-08-23T18:43:54Z

‎Step 2

Xonnainty: /* Step 2 */

2020-08-23T18:43:05Z

‎Step 2

Xonnainty: /* Step 2 */

2020-08-23T18:42:33Z

‎Step 2

Xonnainty: /* Quantitative comparisons at the center point at t = 0.5 ms */

2020-08-23T18:40:42Z

‎Quantitative comparisons at the center point at t = 0.5 ms

Xonnainty: /* Quantitative comparisons at the center point at t = 0.5 ms */

2020-08-23T18:39:25Z

‎Quantitative comparisons at the center point at t = 0.5 ms

Xonnainty: /* Step 4 */

2020-08-23T18:38:11Z

‎Step 4

@@ Line 4: / Line 4: @@
 codes'''.
-The analysis is composed of the '''4 steps'''.
+The objective of the following pages is to give the analyses of the different steps, such as:
-The objective of the following pages is to give the analyses of the different steps.
+* The [[Analysis of Step 1]].
-* The [[Step 1]].
+* The [[Analysis of Step 2]].
-* The [[Step 2]].
+* The [[Analysis of Step 3]].
-* The [[Step 3]].
+* The [[Analysis of Step 4]].
-−
-* The [[Step 4]].
 = Step 1 =

@@ Line 3: / Line 3: @@
 highlighted, since the proposed benchmark shall be used for the '''verification''' and '''validation''' of further '''high-fidelity
 codes'''.
 = Step 1 =

@@ Line 48: / Line 48: @@
 results of a pseudo-spectral solver [65]. The main quantities of interest for this comparison are:
-. Velocity profiles along centerlines of the domain at t = 12.11τref, as illustrated in Fig. 5.
+. Velocity profiles along centerlines of the domain at <math>t = 12.11 \tau_{ref}</math>, as illustrated in Fig. 5.
-. 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>)
+. 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>)
 versus time, as shown in Fig. 6, where <math>S_{ij}</math> is the symmetric strain-rate tensor,
-<math>S{ij} = \frac{1}{2}(\frac {\delta u_i}{\delta x_j} + \frac {\delta u_j}{\delta x_i})</math> ; (11)
+<math>S_{ij} = \frac{1}{2}(\frac {\delta u_i}{\delta x_j} + \frac {\delta u_j}{\delta x_i})</math> ; (11)
 Comparing the '''velocity fields''' at <math>t = 12.11 \tau_{ref}</math> is done on purpose. As known from the '''TGV''' literature, this instant

@@ Line 54: / Line 54: @@
 <math>S{ij} = \frac{1}{2}(\frac {\delta u_i}{\delta x_j} + \frac {\delta u_j}{\delta x_i})</math> ; (11)
-Comparing the '''velocity fields''' at '''t = 12.11 \tau_{ref} is done on purpose. As known from the '''TGV''' literature, this instant
+Comparing the '''velocity fields''' at <math>t = 12.11 \tau_{ref}</math> 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
 in [25]). Getting the correct '''velocity field''' in these conditions is challenging, since the obtained results are very

@@ Line 50: / Line 50: @@
 . Velocity profiles along centerlines of the domain at t = 12.11τref, as illustrated in Fig. 5.
 . 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>)
-versus time, as shown in Fig. 6, where S{ij} is the symmetric strain-rate tensor,
+versus time, as shown in Fig. 6, where <math>S_{ij}</math> is the symmetric strain-rate tensor,
 <math>S{ij} = \frac{1}{2}(\frac {\delta u_i}{\delta x_j} + \frac {\delta u_j}{\delta x_i})</math> ; (11)

@@ Line 179: / Line 179: @@
 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
+'''DINO''' (7683) as a reference, all corresponding relative errors have been computed.
 ) 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''').
+• 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''').
 • 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
+than 2563, overall differences of the order of 1% are observed. Often, using a finer resolution leads to a better
 , 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

@@ Line 119: / Line 119: @@
 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
+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
+), 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
+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
+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.
+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
+The results shown in this section have been obtained for the same grid size than in '''Step 3''', i.e. 2563
-for YALES2
+for '''YALES2'''
-and DINO and 2523
+and '''DINO''' and 2523
-for Nek5000. The corresponding results for velocity (Fig. 11) and temperature (Fig. 12, left)
+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 O2 and OH)
+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 O2 is still smaller than the mass fraction of OH there. One reason behind these
+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
+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.
@@ Line 152: / Line 152: @@
 The present results have been obtained on a grid size of 5123
-for YALES2 and DINO, while Nek5000 relies on
+for '''YALES2''' and '''DINO''', while '''Nek5000''' relies on
-similar discretization size of 5143
+similar '''discretization''' size of 5143
-(57 spectral elements of order 9 in each direction). To reduce computational costs,
+(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, YO2
+(heat release, <math>Y_{O_2}</math>
-, YOH) are discussed here
+, <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
@@ Line 169: / Line 169: @@
 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
+(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
+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.
@@ Line 179: / Line 179: @@
 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
+'''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
+• 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).
+'''turbulent combustion''' ('''temperature''', '''mass fractions''', '''heat release''').
-• Compared to the differences observed in the previous verification step (errors below 0.03%), the variations
+• 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
+configuration, with additional physicochemical complexity, stiffer profiles, highly non-linear processes in '''space'''
-and time.
+and '''time'''.
-• Increasing further the spatial resolution (which is also connected to a reduction of the timestep) does not seem
+• 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
+• 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.
+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
+• 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
+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.
+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
@@ Line 207: / Line 207: @@
 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
+After this detailed analysis of uncertainty, '''it is necessary to quantify the corresponding numerical costs needed
-to get this level of accuracy.
+to get this level of accuracy.'''