DNS 1-5 Quantification of Resolution

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Quantification of resolution

This section provides details of the solution accuracy obtained by tackling the rounded step 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 of 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

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

The comparison with respect to the Taylor microscale is shown in Fig. 5. 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. 6 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 for which 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 5: HiFi-TURB-DLR rounded step, Re=78490. Relation between the mesh size and the Taylor microscale at midspan using MIGALE with DG P3 (~300 million DoF/eqn).

Figure 6: HiFi-TURB-DLR rounded step, Re=78490. 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 stream (${\Delta x^{+}}$), span (${\Delta z^{+}}$) 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 budget of the Reynolds-stress equations and the turbulent kinetic energy (TKE) equation.

As first step, an assessment of code MIGALE in closing the budgets is performed. Fig. 6 reports the budget of Reynolds-stress and TKE equations in a channel flow at ${Re_{\tau }=180}$ using a DG polynomial approximation of degree 5 on a mesh of $91\times 43\times 48$ hexahedral elements (10.5 million DoF/eqn.). Reference results are the DNS data of Moser-Kim-Mansour

Figure 4: Rounded step case, Re=78490. Relation between the mesh size and the Kolmogorov length scale at midspan using MIGALE with DG P3 (~300 million DoF/eqn).

To this aim, the figures below show the turbulent kinetic energy equation budgets and the Reynolds stress streamwise direction budget in a channel flow at ${Re_{\tau }=180}$. The results are compared with the DNS data of Hoyas and Jimenez (2008)

(in circles) and Moser et al. (1999) (in triangles) that can be found freely available online. As it can be seen, the residuals close well within an order of magnitude with respect to the production. This proves that the methodology used in this entry is sound and that Alya can close the budgets if given the necessary grid resolution and integration time.

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: Francesco Bassi, Alessandro Colombo, Francesco Carlo Massa — Università degli studi di Bergamo (UniBG)

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Description

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Quantification of Resolution

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