View source for Turbulence: Subgrid-Scale Modeling

== Fundamental properties of the subgrid scale stress tensor ==

The SGS stress is a symmetric tensor. It is invariant to Galilean transformation, i.e. if the turbulence is viewed from frames moving at different velocities, the SGS tensor should be the same. As a proper tensor quantity, it must transform from one frame to another that may be rotated (or be rotating, see e.g. [[#Speziale85|Speziale, 1985]]) in appropriate fashion. Subgrid-scale models should be consistent with these fundamental mathematical requirements. 

Physically, the SGS stress determines the dynamical coupling between large and small scales in turbulence. Dimensionally, it scales quadratically with turbulent velocity differences at scales of order <math>\Delta\ .</math> Following classical [[#Kolmogorov41|Kolmogorov (1941)]] scaling, it can be seen that the characteristic magnitude of the SGS stress scales as <math>\vert\boldsymbol{\tau}\vert \sim (\epsilon \Delta)^{2/3}\ .</math> Unlike Reynolds stresses, the SGS stress is a fluctuating turbulence quantity, i.e. a complicated time-dependent field <math>{\boldsymbol{\tau}}(\mathbf{x},t)\ .</math>  Certain statistical properties of <math>{\boldsymbol{\tau}}</math> are of particular importance for LES to generate realistic results. The mean value of the SGS stress tensor vanishes in isotropic turbulence and for shear flows it decreases rapidly and becomes negligible compared to the Reynolds stresses as the filter scale becomes much smaller than the turbulence integral scale. The mean SGS stress is thus not considered as very significant in LES of turbulence, at least away from solid boundaries.

The most important statistical constraint on the statistics of <math>\boldsymbol{\tau}</math> has to do with the evolution of kinetic energy in turbulence.  [[#Lilly67|Lilly (1967)]] and [[#Leonard74|Leonard (1974)]] showed based on the transport equation of kinetic energy of turbulence that the term <math>-<{\tau}_{ij}\overline{S}_{ij}></math> (where <math>\overline{S}_{ij}=(\partial_j \overline{u}_i+\partial_i \overline{u}_j)/2 </math> is the resolved rate of strain tensor) represents a production of small-scale kinetic energy, or a drain of kinetic energy associated with the large scales resolved in LES. The bracket <math> <...> </math> represents statistical averaging, which may be ensemble averaging, or spatial averaging for homogeneous turbulence and/or time averaging for statistically steady state turbulence. Thus a SGS model <math>{\boldsymbol{\tau}}^{mod}</math> should correctly reproduce this correlation with the strain-rate of the large scales, i.e.  <math><{\tau}_{ij}^{mod}\overline{S}_{ij}>=<{\tau}_{ij}\overline{S}_{ij}>\ .</math> Additional statistical constraints, such as on two-point statistics, may be formulated. The Kolmogorov ''four-fifths law''  for the filtered velocity field in isotropic turbulence (written in terms of components in a ''longitudinal'' direction, <math>L</math>) reads [[#Meneveau94|(Meneveau, 1994)]]:

<math>\label{filtKolmog}
<[\overline{u}_L(\mathbf{x}+\mathbf{r})-\overline{u}_L(\mathbf{x})]^3>+6<\tau_{LL}(\mathbf{x})[\overline{u}_L(\mathbf{x}+\mathbf{r})-\overline{u}_L(\mathbf{x})]> = 6 <\tau_{LL}\overline{S}_{LL}> r = -\frac{4}{5} <\epsilon> r.  </math>

It shows that the two-point correlation between SGS stress elements and velocity increments, and the rate of dissipation, must be predicted correctly for resolved third-order moments of velocity to be realistic. Similar arguments can be made for the prediction of resolved enstrophy ([[#Ceruttietal00|Cerutti et al. 2000]]), for which the correlation <math><{\tau}_{ij}\nabla^2\overline{S}_{ij}></math> plays a crucial role. A more general statistical condition on <math>{\boldsymbol{\tau}}^{mod}</math> was formulated by [[#LangfordMoser99|Langford & Moser (1999)]] while considering the multi-point probability density function of the resolved velocity field. They show that it is sufficient to reproduce the correct multi-point conditional average of the SGS force, <math><\partial_j \tau_{ij}\vert\overline{\mathbf V}_1,\overline{\mathbf V}_2,\dots,\overline{\mathbf{V}}_N> </math> (where <math>\overline{\mathbf{V}}_1,\overline{\mathbf{V}}_2,\dots,\overline{\mathbf{V}}_N</math> are velocities at all points of the flow), in order to reproduce all statistics of the resolved field. [[#LangfordMoser99|Langford & Moser (1999)]] point out that optimal estimation theory shows that such conditional averages also yield the optimal predictor (model) that produces the smallest square error for given available data.

To evaluate empirically SGS models, predictions from LES that uses the model are compared to available data.  
The data can be from [[direct numerical simulation]] (DNS) or from experiments, typically in the form of
mean velocity and Reynolds-stress distributions, spectra, etc.. [[#Piomellietal88|Piomelli et al. (1988)]] introduced the name '''a-posteriori tests'''
for such comparisons to emphasize that the model is evaluated only after it has been implemented in a simulation. Canonical flows used often for ''a-posteriori tests'' are decaying isotropic turbulence (for data, see [[#ComteBellotCorrsin71|Comte-Bellot & Corrsin 1971]], [[#Kangetal03|Kang et al. 2003]]) and channel flow (e.g. [[#Kimetal87|Kim et al. 1987]]). While from an application point of view a-posteriori tests are considered to be the ultimate test of model performance, due to the integrated nature of results (combining effects of numerical space and time discretization, statistical averaging, etc..) they often do not provide clear insights into the reasons SGS models do, or do not, work. A complementary approach uses direct comparison between <math>\tau_{ij}(\mathbf{x},t)</math> and <math>\tau_{ij}^{mod}(\mathbf{x},t)\ ,</math> or their statistical moments, which requires data at high spatial resolution that is sufficiently smaller than <math>\Delta\ .</math> For such analysis, [[#Piomellietal88|Piomelli et al. (1988)]] coined the name '''a-priori tests''' to emphasize that no actual LES is involved. The data for such studies can be generated using DNS, which allows processing the full three-dimensional velocity field but is limited to moderate Reynolds numbers (for early papers see [[#Clarketal79|Clark et al. (1979)]] and [[#Bardinaetal80|Bardina et al. (1980)]]). The data can also be from experiments, which has practical limitations such as limited spatial resolution, but can examine high Reynolds number flows (see e.g. [[#Taoetal02|Tao et al. (2002)]] for laboratory experiments, and [[#Kleissletal04|Kleissl et al. (2004)]] for field experiments in atmospheric turbulence).