Quantification of resolution

This section provides details of the solution accuracy obtained by tackling the wing-body junction DNS with MIGALE. After providing details of the mesh resolution in comparison with spatial turbulent scales, a discussion on the closure of the Reynolds stress equations budgets is given.

Mesh resolution

The mesh resolution is quantified by comparing the mesh characteristic length ( ${\Delta }$ ) with the characteristic lengths of the turbulence, i.e., the Taylor microscale ( ${\eta _{T}}$ ) and Kolmogorov length scale ( ${\eta _{K}}$ ). Here, the mesh characteristic length takes into account the degree of the DG polynomial approximation. In particular, it is defined as the cubic root of the ratio between the mesh element volume $\left(V\right)$ and the number of DoFs $\left(N_{DoF}\right)$ within the mesh element per equation

$\Delta ={\sqrt[{3}]{\dfrac {V}{N_{DoF}}}}$

In order to analize the mesh resolution, five planes have been extracted within the highly resolved region of the turbulent flow, see Fig. 1. Planes A and B are parallel to the horizontal solid wall ( $y/T=0$ ) and are placed at $y/T=0.0025$ and $y/T=0.1$ , respectively. Planes C and D are perpendicular to the streamwise direction and are extracted at $x/T=0.75$ (location of maximum wing thickness) and $x/T=4.5$ (behind the wing trailing edge), respectively, being $x/T=0$ the wing leading edge streamwise coordinate. Plane E is the test case geometric symmetry plane ( $z/T=0$ ).

The comparison with respect to the Taylor microscale is shown in Fig. 2. The maximum ratio ${\Delta }/{\eta _{T}}$ within the outer layer of the boundary layer is approximately 0.4. This outcome suggests that the current space resolution is sufficient to capture turbulence scales in the intertial range. In Fig. 3 is reported the comparison with respect to the Kolmogorov length scale. It is commonly accepted that DNS requirements are achieved when ${\Delta }/{\eta _{K}}\leq 5$ . In all planes considered it is clearly visible a region around the wing in which the ratio ${\Delta }/{\eta _{K}}$ is greater than 8, even if lower than 10. This region is characterized by the presence of the horse-shoe vortex. Besides, for plane D and E it can be noticed an additional region of high ${\Delta }/{\eta _{K}}$ ratio downstream the wing trailing edge, close to the symmetry plane. In this region the turbulent boundary layer developed above the wing solid wall is moving downstream and generating a wake. As outcome, the DNS requirements are not fulfilled for the current simulation. For such reason the present study is referred to as under-resolved DNS (uDNS). For future highly resolved simulations mesh refinement is advised in these regions. We want to point out that the accurate simulation of the wake behind the wind away from the horizontal boundary layer is out of the scope of the current computational campaign.

Figure 1: Wing-body junction. Extracted planes for mesh resolution analisys.

Figure 2: Wing-body junction. Relation between the mesh size and the Taylor microscale.

Figure 3: Wing-body junction. Relation between the mesh size and the Kolmogorov length scale.

The average wall resolution in streamwise ( ${\Delta x^{+}}$ ), spanwise ( ${\Delta z^{+}}$ ) and wall-normal ( ${\Delta y^{+}}$ ) directions at different streamwise locations is reported in Tab. 2.

${x/H}$	${\Delta x^{+}}$	${\Delta z^{+}}$	${\Delta y_{1}^{+}}$
${-3.5}$	${17.8}$	${18.6}$	${1.30}$
${3}$	${6.53}$	${8.19}$	${0.55}$
${4}$	${3.90}$	${5.71}$	${0.40}$
${5}$	${9.22}$	${9.28}$	${0.66}$
${6}$	${12.8}$	${13.0}$	${0.93}$

Table 2: Wall space resolution at different streamwise locations

Solution verification

One way to verify that the DNS are properly resolved is to examine the residuals of the Reynolds- stress budget equations. These residuals are among the statistical volume data to be provided as described in Statistical Data section.

Contributed by: Alessandro Colombo (UNIBG), Francesco Carlo Massa (UNIBG), Michael Leschziner (ICL/ERCOFTAC), Jean-Baptiste Chapelier (ONERA) — University of Bergamo (UNIBG), ICL (Imperial College London), ONERA

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Description

Computational Details

Quantification of Resolution

Statistical Data

Instantaneous Data

Storage Format