Subgrid-scale model

A subgrid-scale model represents the influence of unresolved motion on the quantities retained in a large eddy simulation. Its need follows directly from the nonlinear velocity product in the Navier-Stokes equations: averaging a product generally differs from multiplying the averages.

Let a bar denote a spatial averaging filter. With constant mass density $\rho$, constant kinematic viscosity $\nu$, and a time-independent translation-invariant filter on an unbounded or periodic domain, filtering the incompressible momentum equation giveswhere repeated indices use the Einstein summation convention, $p$ is pressure, and the subgrid-scale stress in this sign convention isThe assumed filter commutes with the displayed derivatives. The definition of ${\tau }_{ij}$ is exact; a closure must approximate it using available resolved information.^[ref]

For an elementary example, average two equally weighted values of one velocity component, $a$ and $-a$. The mean is zero but the mean square is $a^2$, so the corresponding stress component is $a^2$. A resolved mean alone has lost the information needed to reconstruct that covariance.

The Smagorinsky model closes the deviatoric part with an eddy viscosity:where ${\delta }_{ij}$ is the Kronecker delta, ${\bar{S}}_{ij}=({\partial }_j{\bar{u}}_i+{\partial }_i{\bar{u}}_j)/2$ is the resolved strain-rate tensor, $C_s$ a dimensionless model coefficient and $\Delta$ a characteristic filter width. The isotropic stress can be included in a modified pressure. This strain-dependent viscosity has its historical origin in Joseph Smagorinsky's atmospheric circulation modelling.^[ref][ref]

For an incompressible flow, the model transfers resolved kinetic energy out at the rate per unit massA fixed nonnegative eddy viscosity therefore permits dissipation but excludes local turbulent backscatter. A dynamic subgrid-scale model can instead estimate its coefficient from resolved stresses at two filter levels. This addresses a modelling choice, rather than changing the exact definition of the unresolved stress.^[ref]

A dynamic subgrid-scale model estimates its closure coefficient from the resolved flow as the calculation proceeds. In a common construction, a second, coarser test filter probes interactions among resolved scales, providing information for a subgrid-scale model below the original resolution.^[ref]

Write a bar for the original spatial averaging filter and a hat for the test filter. DefineDirect subtraction gives the Germano identityThe rightmost expression uses only resolved quantities. The identity is exact for these definitions; extrapolating a closure between the filter levels is the modeling step.^[ref]

For instance, suppose a chosen closure writes the deviatoric stresses as ${\tau }^d=CA$ and $T^d=CB$, and assumes $C$ can be taken outside the test filtering. Then $L^d\approx CM$, where $M=B-\hat{A}$. One way to choose $C$ is to minimize ${\sum }_{ij}(L_{ij}^d-C{M}_{ij}{)}^2$. Differentiation giveswhen the denominator is nonzero. This is a least squares fit of a scalar to tensor data. In a one-component illustrative fit, $L^d=0.012$ and $M=0.3$ give $C=0.04$.

The coefficient need not remain positive. Depending on the closure, a negative value can permit turbulent backscatter, but a small denominator can also make an estimate unreliable. Averaging, bounds and treatment of weak strain are additional choices. Dynamically fitting a coefficient does not remove these assumptions or make unresolved motion observable.

A test filter is an additional spatial averaging filter applied to already resolved data in a dynamic subgrid-scale model. If $\bar{f}=G\ast f$ is the field after the grid filter, a test kernel $H$ givesIt probes interactions within the available resolved motion at a coarser analysis level. It does not measure the original subgrid motion directly.^[ref]

For example, suppose two resolved values of one velocity component are $1$ and $3\mathrm{m}{\mathrm{s}}^{-1}$. A test average gives $\hat{\bar{u}}=2\mathrm{m}{\mathrm{s}}^{-1}$ and $\widehat{{\bar{u}}^2}=5{\mathrm{m}}^2{\mathrm{s}}^{-2}$, so the resolved differenceis nonzero. This quantity is computable from the resolved samples; a dynamic subgrid-scale model uses such information to constrain a closure.

The combined filter is $H\ast G$, so its width cannot always be identified with the width of $H$ alone. A centred box averaging filter of width $\Delta$ has second moment ${\Delta }^2/12$. Combining it with one of width $2\Delta$ gives second moment $5{\Delta }^2/12$, because centred convolution kernels add their second moments. A single box of width $\sqrt{5}\Delta$ has the same second moment, but not the same kernel shape.

Choosing a width convention is therefore part of the model. The Germano identity holds for the filters actually defined; an assumed relation between modeled stresses at those levels is an additional approximation. Near boundaries, the test operation also needs a specified treatment of data outside the available region.

The Germano identity is an exact relation between stresses at two filtering levels and a stress computable from the resolved field. Let a bar denote a grid filter and a hat a test filter. With the subgrid-scale stress conventionthe identity statesIt follows by substituting ${\hat{\tau }}_{ij}=\widehat{\overline{u_i{u}_j}}-\widehat{{\bar{u}}_i{\bar{u}}_j}$ and cancelling the common term. Massimo Germano and collaborators used this relation to construct a dynamic subgrid-scale model.^[ref]

A scalar block-average example makes every term visible. Start with four velocity values grouped as $(0,2)$ and $(2,4)$. Averaging within each group gives $\bar{u}=(1,3)$ and $\overline{u^2}=(2,10)$, hence $\tau =(1,1)$. A test average over the two groups then givesThus $L=T-\hat{\tau }$ reads $1=2-1$. The resolved group averages contain enough information for $L$, although they alone do not determine the original within-group variances.

The identity has no fitted coefficient. To estimate a missing subgrid-scale stress, a closure assumes some model form at both filtering levels and tries to make their difference agree with the known $L_{ij}$. Extracting a varying coefficient from a filter, choosing a filter width and stabilizing a fit are modeling decisions; they do not enter the cancellation proof.

The Smagorinsky model approximates the deviatoric subgrid-scale stress using an eddy viscosity determined by the resolved strain-rate tensor:Here ${\bar{S}}_{ij}=({\partial }_j{\bar{u}}_i+{\partial }_i{\bar{u}}_j)/2$, $\Delta$ is the filter length, $C_s$ is dimensionless, and repeated indices use the Einstein summation convention. The Kronecker delta selects the isotropic part. The construction estimates a missing effect from resolved deformation rather than calculating each unresolved eddy.^[ref]

For simple shear flow $\bar{u}=(\gamma y,0,0)$, the only nonzero strain components are ${\bar{S}}_{12}={\bar{S}}_{21}=\gamma /2$, giving $|\bar{S}|=|\gamma |$. Taking $C_s=0.1$, $\Delta =0.02\mathrm{m}$ and $\gamma =10{\mathrm{s}}^{-1}$ yieldsThese are illustrative parameter choices, not a universal fitted constant. The dimensions follow from ${\Delta }^2|\bar{S}|$.

For an incompressible flow, the resolved-to-unresolved kinetic energy transfer isA nonnegative eddy viscosity thus removes resolved kinetic energy and cannot represent local turbulent backscatter. Coefficients and near-wall treatment require physical and numerical judgment.

Joseph Smagorinsky's 1963 atmospheric-circulation experiment used a lateral exchange coefficient based on horizontal deformation and a length scale. The modern three-dimensional expression above extends that strain-dependent modeling idea; it should not be mistaken for the complete original atmospheric model.^[ref]

The subgrid-scale stress records velocity-product information missing from a filtered velocity field. For constant mass density and a specified spatial averaging filter, use the conventionIt enters the filtered momentum equation through $-{\partial }_j{\tau }_{ij}$. This ${\tau }_{ij}$ has units of squared speed; multiplication by mass density gives units of mechanical stress. Some sources use the opposite stress sign, so the momentum equation fixes the convention.^[ref]

For a normalized nonnegative kernel, define $v_i(r)=u_i(x-r)-{\bar{u}}_i(x)$. Expanding the product givesIt is therefore a weighted covariance at the chosen point. For every real vector $b$, $b_i{\tau }_{ij}{b}_j=\int G_{\Delta }(b_i{v}_i{)}^{2}dr\ge 0$, so the stress matrix is positive semidefinite. This argument depends on a nonnegative kernel; it cannot be applied unchanged to filters with negative weights.

For example, average equally weighted velocities $(a,0,0)$ and $(-a,0,0)$. The mean velocity vanishes but $\tau =\mathrm{diag}(a^2,0,0)$. With $a=2\mathrm{m}{\mathrm{s}}^{-1}$, ${\tau }_{11}=4{\mathrm{m}}^2{\mathrm{s}}^{-2}$. A zero mean has hidden nonzero motion.

The definition is exact. A subgrid-scale model instead approximates this tensor using available resolved data. In an incompressible flow, the isotropic part ${\tau }_{kk}{\delta }_{ij}/3$ can be absorbed into a modified pressure; the remaining deviatoric stress controls the modeled directional transfer.