Resolution-induced anisotropy in large-eddy simulations

Large-eddy simulation (LES) of turbulence in complex geometries and domains is often conducted with high-aspect-ratio resolution cells of varying shapes and orientations. The effects of such anisotropic resolution are often simplified or neglected in subgrid model formulations. Here, we examine resolution-induced anisotropy and demonstrate that even for isotropic turbulence, anisotropic resolution induces mild resolved Reynolds stress anisotropy and significant anisotropy in second-order resolved velocity gradient statistics. In large-eddy simulations of homogeneous isotropic turbulence with anisotropic resolution, it is shown that commonly used subgrid models, including those that consider resolution anisotropy in their formulation, perform poorly. The one exception is the anisotropic minimum dissipation model proposed by Rozema et al. [Phys. Fluids 27, 085107 (2015)]. A simple model is presented here that is an anisotropic eddy diffusivity extension of the “Kolmogorov expression” eddy viscosity of Carati et al. [A family of dynamic models for large-eddy simulation, in Annual Research Briefs, Center for Turbulence Research (Stanford University and NASA AMES, 1995), pp. 35–40] that depends explicitly on the anisotropy of the resolution. It also performs well and is remarkable because, unlike other LES subgrid models, the eddy diffusivity only depends on statistical characteristics of the turbulence (in this case, the dissipation rate), not on fluctuating quantities. In other subgrid modeling formulations, such as the dynamic procedure, limiting flow dependence to statistical quantities in this way could have advantages.

Figure 1

Diagram illustrating 2D representation of (a) ellipsoidal and (b) Cartesian wave-space domain types.

Figure 2

Diagram of the limiting (a) book (${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2$) and (b) pencil (${\mathrm{\Delta }}_2={\mathrm{\Delta }}_3$) cell types considered here.

Figure 3

Resolution-induced anisotropy in the (a) unresolved and (b) resolved scale contributions to the Reynolds stress for an inertial range energy spectrum, determined as half the trace of (2) and (1), based on the wave-number domains ${\mathcal{D}}^e$ and ${\tilde{\mathcal{D}}}^e$ given by (3). The value of $k_{cc}/k_m$ is 2 *, 4 $\circ$, 8 $▿$, 16 $▵$, 32 $\square$, 64 $◊$, 128 $⊳$, and 256 $⊲$. Subscript “c” indicates the coarse direction, while “f” indicates the fine direction.

Figure 4

Resolution-induced anisotropy in the second moment of the resolved velocity gradient tensor ${\mathcal{G}}_{ijkl}$ determined from (5) based on wave-number domain ${\mathcal{D}}^e$ given by (3). The value of $k_{cc}/k_m$ is 2 *, 4 $\circ$, 8 $▿$, 16 $▵$, 32 $\square$, 64 $◊$, and 128 $⊳$. Subscript “c” indicates the coarse direction, while “f” indicates the fine direction.

Figure 5

One-dimensional energy spectra $E$ from LES with isotropic resolution using the Smagorinsky, AMD, and M43 models, compared with the equivalently filtered ${\left|\mathit{\kappa }\right|}^{-5/3}$ Kolmogorov inertial energy spectra. Shown are spectra (a) without a spherical cutoff filter and (b) with a spherical cutoff filter.

Figure 6

One-dimensional energy spectra $E$, from LES with the standard Smagorinsky model and anisotropic resolution, compared with the equivalently filtered ${\left|\mathit{\kappa }\right|}^{-5/3}$ Kolmogorov inertial range spectra.

Figure 7

One-dimensional energy spectra $E$ from LES with the AMD model and anisotropic resolution, compared with the equivalently filtered ${\left|\mathit{\kappa }\right|}^{-5/3}$ Kolmogorov inertial energy spectra.

Figure 8

One-dimensional energy spectra $E$ from LES with the M43 model and anisotropic resolution, compared with the equivalently filtered ${\left|\mathit{\kappa }\right|}^{-5/3}$ Kolmogorov inertial energy spectra.

Figure 9

One-dimensional energy spectra $E$ from LES for book cells using the Smagorinsky, AMD, M43, and M43 low-k models and anisotropic resolution with $A_r=8$ and varying values of ${\kappa }_{cc}/{\kappa }_m$, compared with the equivalently filtered ${\left|\mathit{\kappa }\right|}^{-5/3}$ Kolmogorov inertial energy spectra. Shown are spectra in both the fine and coarse directions.

Figure 10

One-dimensional energy spectra $E$ from LES with the Vreman model [7] with $C_v=0.07$ and anisotropic resolution of aspect ratio 32, compared with the equivalently filtered ${\left|\mathit{\kappa }\right|}^{-5/3}$ Kolmogorov inertial energy spectra. Shown are spectra in both the fine and coarse directions.

Figure 11

One-dimensional energy spectra $E$ from LES with the Smagorinsky model and the modified Smagorinsky model of Scotti [6], using anisotropic resolution with aspect ratio 32, compared with the equivalently filtered ${\left|\mathit{\kappa }\right|}^{-5/3}$ Kolmogorov inertial energy spectra. Shown are spectra in both the fine and coarse directions.

Figure 12

Normalized values of the gradient direction ${\varepsilon }_{ij}$ (10) eigenvalues as a function of cell aspect ratio in LES with ${\kappa }_{cc}/{\kappa }_m=16$ using the Smagorinsky, AMD, and M43 models. Also shown are values from a filtered DNS with equivalent forcing and ${\mathrm{Re}}_{\lambda }=205$ performed with ${512}^3$ Fourier modes. The filters used on the DNS were consistent with LES resolution with ${\kappa }_{cc}/{\kappa }_m=16$ (DNSa) and ${\kappa }_{cc}/{\kappa }_m=8$ (DNSb). Black lines represent ${\lambda }_c^{\epsilon }$ and blue lines represent ${\lambda }_f^{\epsilon }$.

Figure 13

Normalized values of componentwise ${\varepsilon }_{ij}$ (19) eigenvalues as a function of cell aspect ratio in LES with ${\kappa }_{cc}/{\kappa }_m=16$ using the Smagorinsky, AMD, and M43 models. Also shown are values from a filtered DNS with equivalent forcing and ${\mathrm{Re}}_{\lambda }=205$ performed with ${512}^3$ Fourier modes. The filters used on the DNS were consistent with LES resolution with ${\kappa }_{cc}/{\kappa }_m=16$ (DNSa) and ${\kappa }_{cc}/{\kappa }_m=8$ (DNSb). Black lines represent ${\lambda }_c^{\epsilon }$ and blue lines represent ${\lambda }_f^{\epsilon }$.

Figure 14

Normalized coefficients for the basic M43 model as a function of cell aspect ratio using (A6). All other cell types fall in between the book and pencil limiting cases.