# Turbulence: Subgrid-Scale Modeling

Post-publication activity

Curator: Charles Meneveau

Subgrid-scale modeling refers to the representation of important small-scale physical processes that occur at length-scales that cannot be adequately resolved on a computational mesh. In Large-Eddy Simulation of turbulence, subgrid-scale (SGS) modeling is used to represent the effects of unresolved small-scale fluid motions (small eddies, swirls, vortices) in the equations governing the large-scale motions that are resolved in computer models. The formulation of physically realistic SGS models requires understanding of the physics and the statistics of scale interactions in hydrodynamic turbulence, and is an open research question owing to the fact that turbulence remains an unsolved problem in classical physics.

## Large Eddy Simulation and subgrid-scale (SGS) modeling

Turbulent flows contain a wide range of excited length and time scales, without clearly defined separation of scales.
Figure 1: Unfiltered and filtered turbulent field showing decomposition used in Large-Eddy Simulation. In subgrid-scale modeling, the effects of the difference between the two signals upon the dynamics of the large scale signal must be modeled.
The co-existence of strongly coupled fluctuations at many scales makes turbulence modeling one of the most challenging unsolved problem in science and engineering. Still, turbulence modeling and prediction is of primary importance to many flow processes and applications ranging from aerodynamics, environmental, geophysical and astrophysical fluid dynamics, energy production and transportation, etc. In Large-Eddy Simulation (LES) of turbulence, the time and space dependence of the fluid motions are resolved down to some prescribed length-scale $$\Delta\ .$$ The motion is separated into small and large scales, often by spatially filtering the velocity field with a kernel, $$G_{\Delta}({\mathbf x})$$ (Leonard 1974). The convolution kernel eliminates scales smaller than $$\Delta\ .$$ The eliminated scales are called subfilter or subgrid-scale (SGS) motions (see Figure 1). The spectrally sharp filter, the Gaussian filter, and the box or top-hat filter (see Pope (2000) for precise definitions) are often used. The filtered velocity (denoted by an overline), is thus obtained by convolution $$\overline{u}_i = G_{\Delta}*{u}_i\ .$$ Other approaches to separate between large and small scales exist, such as expanding the velocity field using orthonormal basis functions. Truncating the summation can be used to define the large-scale field (a projection) and the discarded modes represent the range of subgrid-scale motions. For Fourier modes, the projection is equivalent to applying a spectral cutoff filter.

The equations to be solved in Large-Eddy Simulation of incompressible single-phase hydrodynamics are obtained by filtering the Navier-Stokes equations, and together with the filtered continuity equation, they read$\tag{1} \frac{\partial \overline{\mathbf u}}{\partial t} + \overline{\mathbf u}\cdot \boldsymbol{\nabla} \overline{\mathbf u} = - {1\over \rho}\boldsymbol{\nabla}\overline{p} + \nu \nabla^2 \overline{\mathbf u} - {\boldsymbol{\nabla}} \cdot {\boldsymbol{\tau}}, ~~~~~\boldsymbol{\nabla \cdot} \overline{\mathbf u} = 0,$

where $${\overline{(\dots)}}$$ represents a convolution with $$G_{\Delta}({\mathbf x})\ .$$ The SGS stress tensor $${\boldsymbol{\tau}}$$ is defined according to$\tag{2} \tau_{ij} = \overline{u_iu_j} - \overline{u}_i\overline{u}_j.$

The LES equations (or filtered NS) can be discretized numerically by employing a spatial resolution that is on the order of $$\Delta\ .$$ When $$\Delta$$ is chosen to be much larger than the Kolmogorov scale, $$\eta_K\ ,$$ LES is typically far less expensive than Direct Numerical Simulation (DNS) which requires resolutions near $$\eta_K\ .$$ From a modeling viewpoint, however, Eq. (1) is unclosed since $$\tau_{ij}$$ must be expressed in terms of features of the resolved (filtered) velocity field $$\overline{\mathbf{u}}\ .$$ For incompressible flow, only the deviatoric part of the stress tensor, $$\tau^d_{ij}=\tau_{ij}-\tau_{kk}\delta_{ij}/3$$ is relevant since the gradient of its trace may be absorbed into an effective pressure field. In the sections below, $$\tau_{ij}$$ will be used to denote the trace-free part of the SGS stress tensor, unless otherwise noted.

When treating turbulence with a transported scalar $$\theta(\mathbf{x},t)$$ - temperature and concentration fields are common examples -, the SGS scalar flux vector $$q_i = \overline{u_i\theta} - \overline{u}_i\overline{\theta}$$ is in need of closure, and if the flow is chemically reacting, production terms involving even stronger nonlinearities (e.g. as arising in Arrhenius rate expressions with very stiff dependencies on temperature) require modeling once they are coarse-grained (Poinsot & Veynante, 2005). When turbulence exists near solid boundaries, often even the mean velocity profile exhibits characteristic length-scales that are too small to be resolved. Such SGS boundary layer behavior implies that models are required for the wall stress, or rates of subgrid wall transport processes. Various wall-modeling approaches are reviewed in Piomelli & Balaras (2002).

## 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. 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 $$\Delta\ .$$ Following classical Kolmogorov (1941) scaling, it can be seen that the characteristic magnitude of the SGS stress scales as $$\vert\boldsymbol{\tau}\vert \sim (\epsilon \Delta)^{2/3}\ .$$ Unlike Reynolds stresses, the SGS stress is a fluctuating turbulence quantity, i.e. a complicated time-dependent field $${\boldsymbol{\tau}}(\mathbf{x},t)\ .$$ Certain statistical properties of $${\boldsymbol{\tau}}$$ 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 $$\boldsymbol{\tau}$$ has to do with the evolution of kinetic energy in turbulence. Lilly (1967) and Leonard (1974) showed based on the transport equation of kinetic energy of turbulence that the term $$-<{\tau}_{ij}\overline{S}_{ij}>$$ (where $$\overline{S}_{ij}=(\partial_j \overline{u}_i+\partial_i \overline{u}_j)/2$$ 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 $$<...>$$ 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 $${\boldsymbol{\tau}}^{mod}$$ should correctly reproduce this correlation with the strain-rate of the large scales, i.e. $$<{\tau}_{ij}^{mod}\overline{S}_{ij}>=<{\tau}_{ij}\overline{S}_{ij}>\ .$$ 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, $$L$$) reads (Meneveau, 1994)$\tag{3} <[\overline{u}_L(\mathbf{x}+\mathbf{r})-\overline{u}_L(\mathbf{x})]^3>+6<\tau_{LL}(\mathbf{x})[\overline{u}_L(\mathbf{x}+\mathbf{r})-\overline{u}_L(\mathbf{x})]> = 6 <\tau_{LL}\overline{S}_{LL}> r = -\frac{4}{5} <\epsilon> r.$

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 (Cerutti et al. 2000), for which the correlation $$<{\tau}_{ij}\nabla^2\overline{S}_{ij}>$$ plays a crucial role. A more general statistical condition on $${\boldsymbol{\tau}}^{mod}$$ was formulated by 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, $$<\partial_j \tau_{ij}\vert\overline{\mathbf V}_1,\overline{\mathbf V}_2,\dots,\overline{\mathbf{V}}_N>$$ (where $$\overline{\mathbf{V}}_1,\overline{\mathbf{V}}_2,\dots,\overline{\mathbf{V}}_N$$ are velocities at all points of the flow), in order to reproduce all statistics of the resolved field. 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.. 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 Comte-Bellot & Corrsin 1971, Kang et al. 2003) and channel flow (e.g. 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 $$\tau_{ij}(\mathbf{x},t)$$ and $$\tau_{ij}^{mod}(\mathbf{x},t)\ ,$$ or their statistical moments, which requires data at high spatial resolution that is sufficiently smaller than $$\Delta\ .$$ For such analysis, 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 Clark et al. (1979) and 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. Tao et al. (2002) for laboratory experiments, and Kleissl et al. (2004) for field experiments in atmospheric turbulence).

## Eddy viscosity models

Traditionally, the effects of SGS motions upon resolved scales are modeled in analogy with molecular degrees of freedom in (e.g.) kinetic theory of gases, in which the momentum fluxes are linearly dependent upon the rate of strain of the large scales. This is written as an eddy-viscosity closure according to$\tag{4} \tau_{ij}^{ev} = - 2 \nu_{sgs} \overline{S}_{ij},$

where $$\nu_{sgs}$$ is the (kinematic) eddy viscosity. This approach complies with the basic requirements of symmetry, Galilean invariance, and tensor transformations from one frame to another. The units of the eddy viscosity are velocity times a characteristic length-scale, or kinetic energy density multiplied by a time-scale. The best known eddy-viscosity model bears the name of Joseph Smagorinsky, an influential meteorologist and first director of the U.S. National Oceanic and Atmospheric Administration (NOAA). In the Smagorinsky (1963) model the eddy viscosity is expressed as the characteristic scale $$\Delta$$ times a velocity scale $$\Delta\vert{\overline{S}}\vert\ ,$$ where $$\vert{\overline{S}}\vert = \sqrt{2\overline{S}_{pq}\overline{S}_{pq}}$$ represents a Galilean invariant estimation of velocity differences over length-scales of order $$\Delta\ .$$ The model contains a dimensionless empirical parameter (the so-called Smagorinsky coefficient $$c_s$$), and can be written as$\tag{5} \tau_{ij}^{smag} = - 2 (c_s \Delta)^2 \vert{\overline{S}}\vert \overline{S}_{ij}.$

Using the requirement that modeled rate of energy transfer from large to small scales, $$-<{\tau}^{smag}_{ij}\overline{S}_{ij}>\ ,$$ be equal to the overall rate of dissipation $$<\epsilon>$$ and exploiting the insights provided by the Kolmogorov (1941) theory to evaluate the moments of strain-rate ($$<\overline{S}_{ij}\overline{S}_{ij}>$$), Lilly (1967) derived $$c_s = (3c_K\pi^{4/3}/2)^{-3/4}\approx 0.16$$ using the known empirical value of the Kolmogorov constant $$c_K \approx 1.6\ .$$ This result is valid for locally isotropic turbulence, when $$\Delta$$ falls in the inertial range of turbulence. Numerical simulations of isotropic turbulence with this value yield very realistic decay rates of kinetic energy and energy spectra with realistic features, except close to the cutoff-wavenumber of the LES.

The Smagorinsky model assumes that an equilibrium exists between kinetic energy flux across scale and the large scales of turbulence. In many applications, such as free stream turbulence impinging on the leading edge of an airplane wing, turbulent water flow through a pipe with changing cross section, or turbulence with strong buoyancy, such equilibrium conditions are not established. In order to model the time development of the small-scale turbulence, which affects the eddy viscosity, so-called kinetic energy models use an eddy viscosity of the form $$\nu_{sgs}=c_e\Delta\sqrt{e(\mathbf{x},t)}\ ,$$ where $$e(\mathbf{x},t)$$ is the SGS kinetic energy. It is defined in terms of the trace of the SGS stress tensor according to $$e(\mathbf{x},t)=\tau_{kk}/2=(\overline{u_k^2}-\overline{u}_k^2)/2$$ and is used to quantify the local velocity scale. In order to determine $$e(\mathbf{x},t)\ ,$$ an additional scalar transport equation, derived from the trace of the transport equation for the SGS stress tensor (Deardorff 1973), is solved. In this equation, diffusion and dissipation terms must be modeled (Schumann 1975)$\tag{6} \frac{\partial e}{\partial t} + \overline{u}_j \frac{e}{\partial x_j} = c_e\Delta\sqrt{e}~\overline{S}_{ij}\overline{S}_{ij} - C_{\epsilon}\frac{e^{3/2}}{\Delta} - \frac{1}{\partial x_j}\left[\left(\nu+\frac{c_e}{\sigma_e}\Delta\sqrt{e}\right)\frac{\partial e}{\partial x_j}\right]$

with two additional empirical coefficients, the SGS Prandtl number $$\sigma_e$$ and the dissipation parameter $$C_{\epsilon}\ .$$ This approach incorporates memory effects and has seen extensive applications especially in simulations of atmospheric flows (Moeng 1984, Mason 1994).

For simulations done in spectral space, a wavenumber-dependent (spectral) eddy viscosity $$\nu(k,k_c)$$ may be prescribed (Chollet & Lesieur 1981, Schilling & Zhou 2002) that has the important advantage that it allows to control explicitly the rate of energy extraction from any given resolved wavenumber $$k\ .$$ Typically, a cusp behavior is enforced, increasing the spectral eddy viscosity near the cutoff wavenumber $$k_c = \pi/\Delta\ .$$ Such models often also include stochastic terms to simulate kinetic energy backscatter, the phenomenon by which part of the energy is transferred from small to large scale motions. Other methods to compute the eddy viscosity have been proposed, among them the structure-function model (Lesieur & Metais, 1996) in which the eddy viscosity is written according to $$\nu_{sgs} = C_F \Delta \sqrt{D_{LL}}$$ and $$D_{LL}(\mathbf{x},t)$$ is the locally evaluated second order velocity structure function and $$C_F$$ a parameter that can be expressed in terms of the Kolmogorov constant. Several versions exist and recent improvements have replaced the velocity structure function with the structure functions of appropriately high-pass filtered velocity fields, in order to selectively sample the smallest of the resolved scales. Such high-pass filtering is of particular usefulness when considering flows near solid walls where the mean shear can overwhelm the fluctuating one, but one does not wish this mean shear to artificially increase the SGS eddy viscosity. As can be found in the detailed account by Sagaut (2006), many other variants of eddy-viscosity closures have been proposed in the literature.

Several more recent proposals have been made to use eddy-viscosity expressions such that they would vanish whenever the resolved flow has some simple structure that is not likely to occur if the flow is locally turbulent and three-dimensional. For instance, the wall-adapting local eddy-viscosity (WALE) model proposed by Nicoud & Ducros (1999) has the property than near solid surfaces in plane shear, the eddy-viscosity vanishes. The model reads $$\nu_{sgs} =(C_w\Delta)^2(W_{ij}^dW_{ij}^d)^{3/2}/[(\overline{S}_{pq}\overline{S}_{pq})^{5/2}+(W_{ij}^dW_{ij}^d)^{5/4})\ ,$$ where $$W_{ij}^d$$ is the traceless part of the tensor $$W_{ij}=\overline{S}_{ik}\overline{S}_{kj}+\overline{\Omega}_{ik}\overline{\Omega}_{kj}\ ,$$ and the model can be shown to lead to vanishing viscosity in the limit of planar flow. Another such model is the Vreman model (Vreman 2004), for which,

$$\tag{7} \nu_{sgs} = C_v \sqrt{\frac{Q_{G}}{\overline{A}_{ij} \overline{A}_{ij}} }, ~~~{\rm where} ~~~ \overline{A}_{ij} = \frac{\partial \overline{u}_i}{\partial x_j} ~~~ {\rm and} ~~~ Q_{G}=\frac{1}{2}[(G_{kk})^2-G_{ij}G_{ji}]$$ are the resolved velocity gradient tensor, and the second invariant of the tensor $$G_{ij}=\Delta^2 \overline{A}_{ik}\overline{A}_{jk}$$, respectively. Note that $$G_{ij}$$ is based on the nonlinear model (see Eq. (12) below), and also note that $$G_{ij}$$ can be generalized for non-isotropic grids or filters (Vreman 2004). A further development of these ideas can be found in the sigma model of Nicoud et al. (2011), in which the eddy viscosity is expressed in terms of the singular values of the tensor $$G_{ij}$$, i.e. the appropriately ordered square-roots of the eigenvalues of $$G_{ij}$$. This approach yields good near-wall scaling and vanishes for 2-dimensional and 2-component flows.

All the above expressions are based on the basic eddy-viscosity closure assumption in Eq. (4). This expression can also be (partly) justified from the linearized evolution equation for the SGS fluctuating velocity $$du_i^\prime/dt = - u_k^\prime \partial \overline{u}_i/\partial x_k\ ,$$ which includes several strong assumptions such as neglect of pressure and nonlinear effects. Coupled with further assumptions of separation of time-scales, the solution can be written in terms of matrix exponentials (Li et al. 2009) and short-time expansion to first order yields the eddy-viscosity closure for the deviatoric part of the SGS stress tensor. The need for the various assumptions illustrates the limitations of the eddy-viscosity closure from a fundamental viewpoint. Still, in practical applications of LES, the eddy-viscosity approach is very popular due to its robustness, general ease of implementation, and low computational cost. Also, provided the eddy viscosity is parameterized realistically, realistic results can be obtained.

## Germano identity and dynamic model

The Germano identity (Germano, 1992) applies quite generally when coarse-graining nonlinear partial differential equations (PDEs) governing the dynamics of, say, a multi-scale field variables $$q(\mathbf{x},t)\ .$$ Suppose the nonlinear term in the PDE is written as $$N(q)$$ and that spatial coarse-graining of the field at scale $$\Delta$$ is denoted by the convolution $$G_{\Delta}*q=\overline{q}\ .$$ Subgrid modeling requires models for the residual $$\overline{N(q)}-N(\overline{q})\ .$$ For the Navier Stokes equations the nonlinear term is $$N(\mathbf{u}) = u_iu_j\ .$$ The Germano identity uses a test-filtering operation at scale $$\alpha\Delta$$ with $$\alpha > 1\ ,$$ denoted by a convolution that acts on a field $$Q$$ according to $$G_{\alpha\Delta}*Q = \widehat{Q}\ .$$ The Germano identity simply reads $$\widehat{\overline{N(q)}}-N(\widehat{\overline{q}})=\left[\widehat{\overline{N(q)}}-\widehat{N(\overline{q})}\right] + \left[\widehat{N(\overline{q})} - N(\widehat{\overline{q}}) \right] \ .$$ The first two terms may be replaced by suitable models that depend upon model parameters, while the last term may be determined from the coarse-grained (resolved) fields. The Germano identity thus provides a self-consistency condition among scales to downscale model parameters determined from the resolved range of scales to the unresolved range. In the case of filtered Navier Stokes and LES, the Germano identity (Germano et al. 1991) is usually written as $\tag{8} T_{ij} = \widehat{\tau}_{ij} + L_{ij}, ~~ {\rm where} ~~ L_{ij} = \widehat{\overline{u}_i\overline{u}_j}-\widehat{\overline{u}}_i\widehat{\overline{u}}_j, ~~{\rm and}~~ T_{ij} = \widehat{\overline{u_iu_j}}-\widehat{\overline{u}}_i\widehat{\overline{u}}_j,$

The tensor $$L_{ij}$$ is called the resolved stress (or Leonard stress) tensor and $$T_{ij}$$ corresponds to the SGS stress tensor arising at a scale that corresponds to the combined hat and bar filters. This relationship is exact and holds at every point in the flow and at every time, for all tensor components. The sketch in Figure 2 illustrates the identity when written in terms of the mean value of the trace of the SGS stresses at various scales (the trace is the area under the spectrum).
Figure 2: Sketch of energy spectrum of turbulence showing the wavenumbers corresponding to filter scales $$\Delta$$ and test-filter scale $$\alpha\Delta\ .$$ The Germano identity states that $$\mathbf{T}=\mathbf{L}+\widehat{\boldsymbol{\tau}}\ .$$ The vertical lines represent the action of the spectrally sharp cutoff filter - for other filters, the separation between scales is less abrupt.

The dynamic Smagorinsky model is based on the Germano identity when the tensors $$\boldsymbol{\tau}$$ and $$\mathbf{T}$$ are written in terms of the Smagorinsky model Eq. (5) at scales $$\Delta$$ and $$\alpha'\Delta\ ,$$ respectively, where $$\alpha'\Delta$$ is the scale that results from repeated application of the two filters. For a spectral filter, $$\alpha'=\alpha$$ while (e.g.) for a Gaussian filter $$\alpha'=\sqrt{\alpha^2+1}\ .$$ Assuming that the Smagorinsky coefficient is the same at scales $$\Delta$$ and $$\alpha'\Delta$$ (scale invariance) and that it does not vary rapidly in directions over which the test filter acts, the result can be written according to$\tag{9} c_s^2 M_{ij} = L_{ij}, ~~{\rm where}~~ M_{ij} = -2\Delta^2\left(\alpha'^2|\widehat{\overline{S}}|\widehat{\overline{S}}_{ij}-\widehat{|\overline{S}|\overline{S}_{ij}}\right).$

Since this equality should hold at every point, time, and for each tensor element, it is an over-determined system which a single scalar parameter, $$c_s^2\ ,$$ is unlikely to satisfy exactly. In turbulence, the expression is typically understood to hold in an average sense only and various formulations of the dynamic model differ in how this condition is understood. Originally motivated by the condition of correct kinetic energy dissipation, Germano et al. (1991) have written the Germano identity by contracting stresses with the strain-rate tensor. When expressing the condition based on energy transfer across scale $$\alpha\Delta\ ,$$ one obtains $$c_2^2 = <L_{ij}\widehat{\overline{S}}_{ij}>/<M_{ij}\widehat{\overline{S}}_{ij}>$$ where the averaging is over appropriate domains available in the simulation. In order to avoid possible divisions by zero, a more robust procedure was subsequently developed by Lilly (1992) and Ghosal et al. (1995). First, an appropriate region over which resolved data will be used to determine the coefficient is chosen. Then one seeks to minimize the mean square error $$<\left(L_{ij}-c_s^2M_{ij}\right)^2>\ .$$ The resulting expression (taking derivative with respect to $$c_s^2$$ and equating to zero) is$\tag{10} c_s^2 = \frac{<L_{ij}M_{ij}>}{<M_{ij}M_{ij}>}.$

The averaging is usually performed over directions of statistical homogeneity in the flow. The dynamic Smagorinsky model has been applied to a large variety of flows. The model can also be applied with averaging over more localized regions, such as volumes containing a few neighboring grid points, temporal domains along the time evolution of fluid particles (the Lagrangian dynamic model Meneveau et al. et al. 1996), or using test-filtering using various scales to capture possible scale-dependencies of the coefficient (Porte-Agel et al. 2000).

Since the Germano identity holds for coarse-graining of any nonlinear operation, further generalizations are possible. Dynamic models have been developed for other SGS models, such as for scalar fluxes and subgrid kinetic energy (Moin et al. 1991) and subgrid scalar variance (Pierce & Moin 1998). Also, the dynamic model can be extended to include additional statistical constraints (Shi et al. 2008). The dynamic approach has also been implemented by You & Moin (2007) to determine the coefficient $$C_v$$ in the Vreman (2004) eddy-viscosity model (see Eq. (7)). Since $$C_v$$ is inherently less variable among different regions of the flow, You & Moin (2007) proposed a global, position independent value based on the condition of overall (volume averaged) kinetic energy balance. However, if the coefficient was indeed independent of flow regime, the need for the dynamic method is not obvious since the result from a single calibration would be expected to work universally. However, as shown in Nicoud et al. (2011), spatial variations as well as resolution dependence have been observed in the model coefficients of these new-generation eddy-viscosity models.

## Similarity, nonlinear and deconvolution models

The similarity model was proposed by Bardina et al. (1980), and in a generalized version reads$\tag{11} \tau_{ij}^{sim} = C_{sim} L_{ij}$

where $$L_{ij}$$ is defined in Eq. (8) and different version of the model differ by the scale of the test-filter $$\alpha\Delta\ .$$ Bardina et al. (1980) used $$\alpha=1\ .$$ Analysis of experimental data (Liu et al. 1994) suggested $$\alpha = 2\ .$$ Already Leonard (1974) noted that this kind of expression was amenable to Taylor-series expansion of $$\overline{\mathbf{u}}(\mathbf{x}_0+\mathbf{x}')$$ around $$\mathbf{x}_0\ .$$ Then evaluating the test-filtering analytically leads to so-called nonlinear models (sometimes also called Clark model (Clark et al. 1979), or tensor diffusivity model)$\tag{12} \tau_{ij}^{nl} = C_{nl} \Delta^2 \frac{\partial \overline{u}_i}{\partial x_k}\frac{\partial \overline{u}_j}{\partial x_k}.$

In recent years these models have been shown to be special cases of the more general so-called deconvolution methods (Stolz & Adams, 1999). For the so-called soft deconvolution problem, one assumes that $$G_{\Delta}$$ is an invertible filter, i.e. with inverse $$G_{\Delta}^{-1}\ .$$ Then formally one may obtain the full velocity field from its filtered version using $$\mathbf{u}=G_{\Delta}^{-1}*\overline{\mathbf{u}}\ .$$ When this is replaced into the definition of the SGS stress Eq. (2), a closure is obtained. In practice, the filtering and inverse filtering can be facilitated by various iterative methods based on polynomial approximations of the filter kernel (Adams & Stolz 2001, see also extensive discussion in Sagaut 2006). Further relating such approaches with terms in a Taylor series expansion has shown that the nonlinear model Eq. (12) can be regarded as the first non-trivial term in such an expansion.

While similarity, nonlinear, or deconvolution models display many realistic features when compared with real SGS stress fields in a-priori comparisons, when implemented in simulations in conjunction with non or minimally dissipative numerical schemes, they tend to produce unphysical behavior that most often can be traced to the fact that by themselves the models are not dissipative enough. In the context of similarity and nonlinear models, this has led to so-called mixed models, in which an eddy-viscosity term is added, e.g. $\tag{13} \tau_{ij}^{mix} = \overline{\overline{u}_i\overline{u}_j}-\overline{\overline{u}}_i\overline{\overline{u}}_i ~~ - ~~ 2(c_s\Delta)^2|\overline{S}|\overline{S}_{ij}.$

Such a model was proposed and used by Zang et al. (1993) and Armenio et al. (2000) in conjunction with the dynamic approach to determine $$c_s\ .$$ In the context of deconvolution models, the need for additional dissipative mechanisms has been traced to the fact that in LES, in addition to filtering, there is irreversible loss of information when representing fields, even filtered ones, onto a discrete grid. For additional information about these issues, see section 2 of Domaradzki et al. (2002), where perfect deconvolution is shown to be equivalent to under-resolved DNS. The deconvolution procedure cannot recover all of the small-scale fields and the lost information must be modeled. This has led to so-called regularized" deconvolution models, in which a damping term (numerical or through an eddy viscosity term) provides additional extraction of kinetic energy (Sagaut 2006).

## Other approaches to SGS stress modeling

Other classes of SGS models have been proposed, such as solving additional transport equations for the full SGS stress tensor (Deardorff, 1973), solving equations for the probability density function (Filtered Density Function methods, see Gicquel et al., 2002), or explicitly constructing subgrid velocity fields based, e.g., on small-scale vortices (Misra & Pullin, 1997), fractal signals (Scotti & Meneveau, 1997), or one-dimensional models of turbulence (Kerstein, 2002).

There have been recently various attempts to develop LES approaches based not on coarse-graining the Navier-Stokes equations, but starting from slightly different momentum equations that have more benign mathematical properties. Examples include the so-called Leray regularization or the Lagrangian-averaged Navier-Stokes (LANS) alpha model (see e.g. Guerts and Holm, 2006 for a discussion of both). Careful tests in decaying isotropic 3D turbulence (Guerts et al. 2008) show diminishing accuracy at high grid Reynolds number, i.e. when $$\Delta>>\eta_K\ .$$

Another, fundamentally different, approach is based on the concept of implicit LES (ILES). It is motivated by arguing that it is irrelevant whether the dissipation of resolved kinetic energy is accomplished by using functionally expressed SGS models, or by numerical dissipation implicit in the numerical method utilized. Since there exists an abundance of well-tested numerical codes that have been used in a number of engineering (e.g. Fureby & Grinstein, 2002) or astrophysical (Porter et al. 1994) applications, this approach has been argued to provide a practical alternative to SGS modeling-based approaches (for several applications, see Grinstein et al. 2007).

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