View source for Analysis of Step 1

The verification involves a direct comparison with the analytical solution. For this purpose, analytic fields for
'''velocity''' (x- and y-components) and '''vorticity''' <math>(\partial_x v - \partial_y u)</math> at <math>t=10 \tau_{ref}</math> 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 <math>t = 10 \tau_{ref}</math> 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 <math>t = 10 \tau_{ref}</math> (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 <math>0.03%</math> for the three codes.

{| class="wikitable alternance center"
 |+ Comparison of the peak velocity at <math>t = 10 \tau_{ref}</math> for Step 1 (verification)
 |-
 |
 ! scope="col" | Analytical
 ! scope="col" | YALES2
 ! scope="col" | DINO
 ! scope="col" | Nek5000
 |-
 ! scope="row" | <math>V_{max}</math>
 | 0.987271
 | 0.987583
 | 0.987565
 | ????
 |-
 ! scope="row" | <math>\epsilon_{rel}</math>
 | 0 [Ref]
 | 0.03%
 | 0.03%
 | -
 |}

This configuration, although quite far from any realistic flame, is nevertheless an excellent manner to verify the
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 <ref name="abelsamie2016"/>.

[[File:vx_t2.png|250px|alt text]]
[[File:vy_t2.png|250px|alt text]]
[[File:wz_t2.png|250px|alt text]]

You can see on the figures above the analytic fields of x-velocity, y-velocity, and vorticity, respectively (from left to right), at <math>t = 10 \tau_{ref}</math> for '''Step 1''' (verification step).

[[File:vx_2d.png|250px]]
[[File:vy_2d.png|250px]]

Comparing the results of the '''three codes''' with the analytical solution for the '''2-D Taylor-Green vortex''' ('''Step 1'''). Left: xcomponent of velocity. Right: y-component of velocity.

= References =

<references>
<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>