Novel Turbulence Length Scale

When production and destruction terms are balanced, the eddy viscosity scales like

$\tilde{\nu}\propto\tilde{d}^{2}$

Accurate estimation of eddy viscosity is crucial as it governs the amount of modeled turbulence and onset of flow separation which are important for lift, drag and pitching moment predictions.

Traditional DDES length scale in LES mode is

$\Delta_{max}=\max(\Delta_{1},\Delta_{2},\Delta_{3})$

Valid for isotropic grid
Airfoil mesh: Highly Anisotropic

Length scale from Scotti et al.¹

$\Delta_{Scotti}=(\Delta_{1}\Delta_{2}\Delta_{3})^{1/3}f(a_{1},a_{2})$

Here
$a_{1}=\Delta_{1}/\Delta_{3}$

and

$\mathrm{a_{2}=\Delta_{2}/\Delta_{3}}$

are the two aspect ratios and f ≥ 1 is a function of a1and a2
Good for general anisotropy in the grid

Length scale for grid with stretching in wall normal direction from Shur et al.²

$\Delta_{Shur}=\min\{\max[C_{w}d_{w},C_{w}\Delta_{max},h_{wn}],\Delta_{max}\}$

Designed for IDDES
Wall-normal dependence
Suitable for highly anisotropic mesh
No effect far from the wall

Proposed Length Scale

$\Delta_{Shur-Scotti-Min}=\min(\Delta_{Shur},\Delta_{Scotti})$

Retains advantage of Shur and Scotti length scales
Shielding is still as effective
No modeled stress depletion

References:

Scotti A., Meneveau, C., and Fatica, M., “Dynamic Smagorinsky model on anisotropic grids”, Center for Turbulence Research Proceedings of the Summer Program, 1996.
Travin, A.K., Shur, M., Spalart, P.R., Strelets, M.K., “Improvement of delayed detached eddy simulation for LES with wall modeling,” European conference on computational ﬂuid dynamics ECCOMAS CFD 2006, TU Delft, 2006.

Length scale from Scotti et al.1

Length scale for grid with stretching in wall normal direction from Shur et al.2

Proposed Length Scale

References:

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Length scale from Scotti et al.¹

Length scale for grid with stretching in wall normal direction from Shur et al.²