Analysis of Step 2 - Revision history

Lartigue at 13:32, 26 August 2020

2020-08-26T13:32:10Z

Xonnainty at 19:28, 25 August 2020

2020-08-25T19:28:06Z

Xonnainty at 19:25, 25 August 2020

2020-08-25T19:25:52Z

Xonnainty at 19:22, 25 August 2020

2020-08-25T19:22:49Z

Xonnainty at 19:15, 25 August 2020

2020-08-25T19:15:50Z

Xonnainty at 14:21, 24 August 2020

2020-08-24T14:21:18Z

Xonnainty at 14:14, 24 August 2020

2020-08-24T14:14:09Z

Xonnainty at 12:59, 24 August 2020

2020-08-24T12:59:45Z

Xonnainty at 12:57, 24 August 2020

2020-08-24T12:57:53Z

Xonnainty: Created page with "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 qu..."

2020-08-23T19:00:13Z

Created page with "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 qu..."

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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 <math>t = 12.11 \tau_{ref}</math>, as illustrated in Fig. 5.
2. 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)

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
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 <math>t = 12.11 \tau_{ref}</math>.
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 (<math>11 < t/\tau_{ref} < 13.5</math>) for '''YALES2''', and (2) just before flow
relaminarization (<math>16.25 < t/\tau_{ref} ≤ 20</math>) 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'''.

@@ Line 35: / Line 35: @@
 [[File:eps_3d_incomp.png|250px]]
-Here is the evolution with time of turbulent kinetic energy k (left, written KE in vertical axis) and of its dissipation rate � (right), by comparison with published reference data from <ref name="vanrees2011"/>.
+Here is the evolution with time of turbulent kinetic energy k (left, written KE in vertical axis) and of its dissipation rate <math>\epsilon</math> (right), by comparison with published reference data from <ref name="vanrees2011"/>.
 = References =

@@ Line 14: / Line 14: @@
 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 <ref name="vanrees2011"/>.
+pseudo-spectral code RLPK using <math>512^3</math> grid points <ref name="vanrees2011"/>.
 [[File:vx_3d_incomp.png|250px]]
@@ Line 30: / Line 30: @@
 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'''.
+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 <math>512^3</math> grid points with a pseudo-spectral solver'''.
 [[File:ke_3d_incomp.png|250px]]

@@ Line 16: / Line 16: @@
 pseudo-spectral code RLPK using 5123 grid points <ref name="vanrees2011"/>.
-[[File:vx_3d_incomp.png]]
+[[File:vx_3d_incomp.png|250px]]
-[[File:vy_3d_incomp.png]]
+[[File:vy_3d_incomp.png|250px]]
 Velocities along the centerlines of the domain for cold 3-D TGV case ('''Step 2''') at <math>t = 12.11 \tau_{ref}</math>.
@@ Line 32: / Line 32: @@
 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'''.
-[[File:ke_3d_incomp.png]]
+[[File:ke_3d_incomp.png|250px]]
-[[File:eps_3d_incomp.png]]
+[[File:eps_3d_incomp.png|250px]]
 Here is the evolution with time of turbulent kinetic energy k (left, written KE in vertical axis) and of its dissipation rate � (right), by comparison with published reference data from <ref name="vanrees2011"/>.

@@ Line 16: / Line 16: @@
 pseudo-spectral code RLPK using 5123 grid points <ref name="vanrees2011"/>.
-[[File:vx_3d_incomp.pdf]]
+[[File:vx_3d_incomp.png]]
-[[File:vy_3d_incomp.pdf]]
+[[File:vy_3d_incomp.png]]
 Velocities along the centerlines of the domain for cold 3-D TGV case ('''Step 2''') at <math>t = 12.11 \tau_{ref}</math>.
@@ Line 32: / Line 32: @@
 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'''.
-[[File:ke_3d_incomp.pdf]]
+[[File:ke_3d_incomp.png]]
-[[File:eps_3d_incomp.pdf]]
+[[File:eps_3d_incomp.png]]
 Here is the evolution with time of turbulent kinetic energy k (left, written KE in vertical axis) and of its dissipation rate � (right), by comparison with published reference data from <ref name="vanrees2011"/>.

@@ Line 16: / Line 16: @@
 pseudo-spectral code RLPK using 5123 grid points <ref name="vanrees2011"/>.
-[[File:vx_3d_incomp.pdf|250px|alt text]]
+[[File:vx_3d_incomp.pdf]]
-[[File:vy_3d_incomp.pdf|250px|alt text]]
+[[File:vy_3d_incomp.pdf]]
 Velocities along the centerlines of the domain for cold 3-D TGV case ('''Step 2''') at <math>t = 12.11 \tau_{ref}</math>.
@@ Line 32: / Line 32: @@
 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'''.
-[[File:ke_3d_incomp.pdf|250px|alt text]]
+[[File:ke_3d_incomp.pdf]]
-[[File:eps_3d_incomp.pdf|250px|alt text]]
+[[File:eps_3d_incomp.pdf]]
 Here is the evolution with time of turbulent kinetic energy k (left, written KE in vertical axis) and of its dissipation rate � (right), by comparison with published reference data from <ref name="vanrees2011"/>.

@@ Line 6: / Line 6: @@
 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 {\partial u_i}{\partial 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 10: / Line 10: @@
 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
+in <ref name="abelsamie2016"/>). 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].
+pseudo-spectral code RLPK using 5123 grid points <ref name="vanrees2011"/>.
 First, it appears on Fig. 5 that no differences can be identified visually from the '''velocity fields''' along the '''centerlines'''
@@ Line 40: / Line 40: @@
 	volume={230},
 	pages={2794--2805},
+}
 </bibtex>
 </ref>
 </references>

@@ Line 1: / Line 1: @@
 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:
+results of a pseudo-spectral solver <ref name="vanrees2011"/>. The main quantities of interest for this comparison are:
 . Velocity profiles along centerlines of the domain at <math>t = 12.11 \tau_{ref}</math>, as illustrated in Fig. 5.
@@ Line 26: / Line 26: @@
 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'''.