Large eddy simulations of high Reynolds number turbulence based on interscale energy transfer among resolved scales

In the present work we show how the subgrid-scale (SGS) energy transfer among resolved scales in large eddy simulations (LESs) can be used to evaluate unknown constants in SGS models and derive new models. The essence of the method is that for a given LES velocity field energy transfers among resolved scales can be computed without reference to a particular SGS model and then used to estimate the total SGS energy transfer for an unknown, full velocity field. The total transfer becomes a physical constraint on any proposed SGS model and can be used to obtain and update model constants at each time step in actual LES, allowing self-contained simulations. The method is evaluated by implementing it in LESs of high Reynolds number isotropic turbulence and for several classical SGS modeling expressions. It is shown that the performance of models depends not only on their ability to capture the total SGS dissipation (which is enforced by the method) but also by distribution of the SGS dissipation among scales of motion (which is enforced by a model). However, the main conclusion is that a broad class of modeling expressions that only qualitatively approximate the SGS dissipation distribution among scales perform very well in LESs as long us the total dissipation constraint is satisfied. We also discuss the relation of the method to the well-known dynamic modeling procedure.

Figure 1

Spectral eddy viscosity functions. Solid line: Chollet-Lesieur, Eq. (27); symbols $\circ$: function $C_2{f}_2$, Eq. (29); dotted line: constant eddy viscosity, Eq. (28); broken line: SVV viscosity, Eq. (30).

Figure 2

Results for classical eddy viscosity models. Line with symbols $\circ$: initial condition; solid line: case “ivis”; broken line: case “iviscl”; dotted line: no-model LESs; broken-dotted line: similarity model (57) with Gaussian filter. In this and all subsequent figures thin straight lines show, as appropriate, $-5/3$ and 2 slopes, and a boundary of the forcing band at $k=3$. For compensated spectra (b) horizontal lines mark expected range of values for the Kolmogoroff constant.

Figure 3

Models with shape functions $f_0$, $f_1$, and $f_3$. Line with symbols $\circ$: initial condition; solid line: case “ceddy”; broken line: case “CLeddy”; broken-dotted line: case “SVVmod.”

Figure 4

Model with shape function $f_2$ (case “CLedk4”) and different values of $D_2$. Line with symbols $\circ$: initial condition; solid line: $D_2=1.10$; broken line: $D_2=0.55$; broken dotted line: $D_2=0.275$; dotted line: $D_2=0$.

Figure 5

Spectral eddy viscosities computed from LESs performed with shape function $f_2$ (case “CLedk4”) and different values of $D_2$. Solid line: $D_2=1.10$; broken line: $D_2=0.55$; broken dotted line: $D_2=0.275$. Dotted line: eddy viscosity from LES data for case “ceddy”; line with symbols $\circ$: analytical expression (29).

Figure 6

A model with a shape function computed from a precursor LES. Line with symbols $\circ$: initial condition; solid line: case “ceddy”; broken line: results using a shape function computed from case “ceddy.”

Figure 7

Energy spectra for the model “ceddy” at different resolutions. Straight diagonal line: $\sim k^{-5/3}$. Line with symbols $\circ$: initial condition; solid line above $k^{-5/3}$ line: resolution ${32}^3$ modes; broken-dotted line: resolution ${64}^3$ modes; solid line below $k^{-5/3}$ line: resolution ${64}^3$ with averaging over thicker shells.

Figure 8

A pulse spectrum initial condition. Line with symbols $\circ$: initial condition; solid line: case “ivis”; broken line: case “ceddy.”

Figure 9

Models with shape functions $f_0$, $f_2$, and $f_3$ using Gaussian filtering. Line with symbols $\circ$: initial condition; solid line: case “ceddy”; broken line: case “CLedk4”; broken-dotted line: case “SVVmod.”