View source for Analysis of 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 <ref name="vanrees2011"/>. 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 <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 <ref name="vanrees2011"/>.

[[File:vx_3d_incomp.png]]
[[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>.

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'''.

[[File:ke_3d_incomp.png]]
[[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"/>.

= References =

<references>
<ref name="vanrees2011">
<bibtex>
@article{Vanrees2011, 
	author= {W.M. van Rees, A. Leonard, D.I. Pullin, and P. Koumoutsakos},
	title= {A comparison of vortex and pseudo-spectral
methods for the simulation of periodic vortical flows at high Reynolds numbers.},
	journal={J. Comput. Phys.},
	year= {2011},
	volume={230},
	pages={2794--2805},
}
</bibtex>
</ref>
<ref name="abelsamie2016">
<bibtex>
@article{Abdelsamie2016, 
	author= {A. Abdelsamie and G. Fru and F. Dietzsch and G. Janiga and D. Thévenin},
	title= {Towards direct numerical simulations of low-Mach number turbulent reacting and two-phase flows using immersed boundaries},
	journal={Comput. Fluids},
	year= {2016},
	volume={131},
	number={5},
	pages={123--141},
}
</bibtex>
</ref>
</references>