DNS 1-6 Quantification of Resolution

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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. 6. 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. 7 is reported the comparison with respect to the Kolmogorov length scale. It is commonly accepted that DNS requirements are achieved when ${\Delta }/{\eta _{K}}\leq 5$. Current simulation shows above the flat plate upstream the rounded step a ratio ${\Delta }/{\eta _{K}}$ below 5.5, while above the rounded step a ratio lower than 7.5. DNS requirements are thus not fulfilled in this last region. This is the reason why the present study is referred to as under-resolved DNS (uDNS). For future highly resolved simulations of this test case a mesh refinement is advised above and downstream the rounded step.

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

Figure 7: HiFi-TURB-DLR rounded step, Re=78,490. Relation between the mesh size and the Kolmogorov length scale at midspan using MIGALE with DG P3 (~300 million DoF/eqn).

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.

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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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Quantification of Resolution

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