Numerical dispersion effects on the energy cascade in large-eddy simulation

Implicitly filtered large-eddy simulation (LES) is by nature numerically under-resolved. With the sole exception of Fourier-spectral methods, discrete numerical derivative operators cannot accurately represent the dynamics of all of the represented scales. Since the resolution scale in an LES usually lies in the inertial range, these poorly represented scales are dynamically significant, and errors in their dynamics can affect all resolved scales. This Letter is focused on characterizing the effects of numerical dispersion error by studying the energy cascade in LES of convecting homogeneous isotropic turbulence. Numerical energy and transfer spectra reveal that energy is not transferred at the appropriate rate to wavemodes where significant dispersion error is present. This leads to a deficiency of energy in highly dispersive modes and an accompanying pile up of energy in the well-resolved modes, since dissipation by the subgrid model is diminished. An asymptotic analysis indicates that dispersion error causes a phase decoherence between triad interacting wavemodes, leading to a reduction in the mean energy transfer rate for these scales. These findings are relevant to a wide range of LES, since turbulence commonly convects through the grid in practical simulations. Further, these results indicate that the resolved scales should be defined to not include the dispersive modes.

Figure 1

The effective wavenumbers corresponding to the first derivative operators for a Fourier-spectral method ($\_\_\_\_\_$), seventh-order B-spline collocation method ($\_\_\_\_\_$), fourth-order B-spline collocation method ($\_\_\_\_\_$), third-order B-spline collocation method ($\_\_\_\_\_$), second-order B-spline collocation method ($\_\_\_\_\_$), and second-order centered difference method ($\_\_\_\_\_$).

Figure 2

One-dimensional spectra in the convection direction of energy $E_{1\mathrm{D}}$ and energy transfer rate $T_{1\mathrm{D}}$, with convection velocities $U=0$ and 35. Shown are spectra from theory ($\_\_\_\_\_$), Fourier-spectral method ($\_\_\_\_\_$), seventh-order B-spline collocation method ($\_\_\_\_\_$), fourth-order B-spline collocation method ($\_\_\_\_\_$), third-order B-spline collocation method ($\_\_\_\_\_$), second-order B-spline collocation method ($\_\_\_\_\_$), and second-order centered difference method ($\_\_\_\_\_$).