12. Hybrid RANS–LES and LES Models

Chapter 11 described two-equation eddy-viscosity models commonly used to represent the effects of turbulence on the mean flow field. The Reynolds-averaged Navier–Stokes (RANS) methodology is used extensively for turbulent-flow calculations primarily because it is computationally affordable. Although RANS models provide reasonable predictions for many attached flows, they can exhibit severe deficiencies in separated flows, especially when the separation is inherently unsteady, as in vortex shedding.

Direct numerical simulation (DNS), which resolves all dynamically relevant scales without a turbulence model, remains practical only at moderate Reynolds numbers and for relatively simple geometries. A full large-eddy simulation (LES) can resolve many of the deficiencies of RANS, but its cost is still prohibitive for many high-Reynolds-number applications. Hybrid RANS–LES methods provide a useful intermediate strategy by combining the two methodologies.

LES directly resolves the large, energy-carrying eddies and models only the smaller scales, which tend to be more isotropic and are therefore more amenable to broadly applicable modeling. RANS, by contrast, models the effects of all turbulent scales. LES can consequently improve the prediction of large-scale unsteadiness and mixing while remaining substantially less expensive than DNS [Geu04, Pop00, Sag00].

Unlike a steady RANS calculation, an LES is inherently unsteady and three-dimensional. It requires both a large number of grid cells and a large number of time steps, and these requirements depend on the Reynolds number. Wall-resolved LES of high-Reynolds-number, wall-bounded flow in complex geometry is therefore not yet a routine design method [Pio99]. Most of its computational cost arises from resolving the near-wall region, where the required number of cells scales approximately with the square of the Reynolds number [PB02]. Away from walls, the number of cells needed to resolve the large eddies depends only weakly on Reynolds number [BJK97].

Hybrid methods exploit this difference by modeling the near-wall region, or other regions where the grid is too coarse for LES, while resolving the larger turbulent structures elsewhere. Approaches of this kind include

• detached-eddy simulation (DES),

• zonal hybrid RANS–LES methods,

• multiscale RANS–LES models, and

• partially averaged Navier–Stokes (PANS) methods

[Gir06, NN03, SJSA97, Str01, TD04]. The remainder of this chapter first develops DES and its delayed and improved variants, and then describes algebraic LES subgrid-scale models.

12.1. Detached-Eddy Simulation

Detached-eddy simulation was proposed by Spalart et al. [SJSA97] and has subsequently been applied to a broad range of turbulent flows and modified in several important ways [SSST08, Spa09, SDS+06, Str01]. It uses one unsteady, three-dimensional turbulence model in two roles: as a subgrid-scale model where the grid is fine enough to support LES, and as a RANS closure where it is not.

The LES mode is intended to become active where the grid spacing in every direction is much smaller than the local turbulent length scale. There, DES reduces the modeled eddy viscosity relative to RANS and permits resolved large-scale instabilities to develop. In other regions, most notably attached boundary layers, the model remains in RANS mode, although the complete solution remains unsteady. The transition between the two modes is controlled through a comparison of turbulence-model and grid length scales.

DES was originally formulated with the Spalart–Allmaras model [SJSA97], but was later extended to the shear-stress transport (SST) model [MKL03, Str01]. For the SST formulation, the RANS turbulent length scale is

(12.1.1)\[l_{\mathrm{RANS}} = \frac{\sqrt{k}}{\beta^*\omega},\]

whereas the LES length scale is formed from a local grid scale \(\Delta\):

(12.1.2)\[l_{\mathrm{LES}} = C_{\mathrm{DES}}\Delta.\]

The DES length scale is the smaller of the two:

(12.1.3)\[l_{\mathrm{DES}} = \min\left(l_{\mathrm{RANS}},l_{\mathrm{LES}}\right).\]

The SST turbulence-kinetic-energy equation is converted to the DES form by replacing its destruction length scale:

(12.1.4)\[\beta^* k\omega = \frac{k^{3/2}}{l_{\mathrm{RANS}}} \quad\longrightarrow\quad \frac{k^{3/2}}{l_{\mathrm{DES}}}.\]

The coefficient \(C_{\mathrm{DES}}\) is empirical. In the SST-DES formulation, the values 0.78 in the wall-model branch and 0.61 in the outer branch are blended by the SST blending function. The modified \(k\) equation can be written

(12.1.5)\[\frac{\partial(\rho k)}{\partial t} + \frac{\partial(\rho u_i k)}{\partial x_i} = \frac{\partial}{\partial x_i} \left[ \left(\mu+\sigma_k\mu_t\right) \frac{\partial k}{\partial x_i} \right] + \widetilde P_k - \rho\beta^*k\omega F_{\mathrm{DES}},\]

with

(12.1.6)\[F_{\mathrm{DES}} = \max\left( \frac{l_{\mathrm{RANS}}}{l_{\mathrm{LES}}}, 1 \right).\]

Equation (12.1.6) is another way of expressing the destruction term based on the limited DES length scale. The source fragment labels \(C_{\mathrm{DES}}\Delta\) as \(l_{\mathrm{RANS}}\) in this relation; it is \(l_{\mathrm{LES}}\), as defined in Eq. (12.1.2).

The spatial discretization should also reflect the active mode. Upwind differencing, which is useful in a RANS region, can introduce enough numerical dissipation to suppress the resolved turbulent content expected in an LES region. A central-difference or blended central–upwind scheme is therefore recommended where the DES model operates in LES mode [MKL03, Str01].

12.2. Delayed Detached-Eddy Simulation

The original DES formulation was intended to treat an attached boundary layer entirely with RANS and to apply LES to separated regions, with a transition or grey area between them. It does not, however, contain a mechanism that prevents the DES limiter from activating inside an attached boundary layer. This can occur when the wall-parallel grid spacing becomes smaller than the boundary-layer thickness even though the grid is not fine enough to support resolved turbulence within the boundary layer.

Under these conditions the DES limiter reduces the eddy viscosity and hence the modeled Reynolds stress, but no comparable resolved stress appears to restore the stress balance. Spalart et al. called this effect modeled-stress depletion (MSD) [SDS+06]. It may appear during systematic grid refinement, where geometric details require locally fine wall-parallel spacing, or as a boundary layer thickens on approach to separation. The resulting premature transition from RANS to LES can produce grid-induced separation (GIS).

12.2.1. Zonal SST-DES Shielding

Menter and co-workers reduced the grid sensitivity of the DES limiter by using the zonal structure of the SST model to shield the RANS portion of the boundary layer [MK04, MKL03]. The original factor

(12.2.1.1)\[F_{\mathrm{DES}} = \max\left( \frac{l_{\mathrm{RANS}}}{l_{\mathrm{LES}}}, 1 \right)\]

is replaced by

(12.2.1.2)\[F_{\mathrm{DES}} = \max\left[ \left(1-F_{\mathrm{SST}}\right) \frac{l_{\mathrm{RANS}}}{l_{\mathrm{LES}}}, 1 \right], \qquad F_{\mathrm{SST}}=F_1\ \text{or}\ F_2.\]

Setting \(F_{\mathrm{SST}}=0\) recovers the original SST-DES formulation. Using \(F_2\) shields most of the boundary layer and significantly reduces the limiter’s influence there. Because \(F_2\) does not cover the entire boundary layer, this zonal formulation reduces rather than completely eliminates the MSD problem.

12.2.2. Generic DDES Shielding

Delayed detached-eddy simulation (DDES) replaces the zonal switch with a more general shielding function based on eddy viscosity and wall distance [SDS+06]. In the original Spalart–Allmaras-based formulation, the delay function is

(12.2.2.1)\[f_d = 1-\tanh\left[(8r_d)^3\right],\]

where

(12.2.2.2)\[r_d = \frac{\nu+\nu_t} {\kappa^2d_w^2 \max\left[ \left( \sum_{i,j} \left(\frac{\partial u_i}{\partial x_j}\right)^2 \right)^{1/2}, 10^{-10} \right]}.\]

Here \(d_w\) is the distance to the nearest wall. The marker \(r_d\) is close to one in the logarithmic region of a turbulent boundary layer and close to zero in a free shear flow. Consequently, \(f_d\) approaches zero in the log layer and approaches one away from the shielded wall region.

The DDES length scale is

(12.2.2.3)\[l_{\mathrm{DDES}} = l_{\mathrm{RANS}} - f_d\max\left( 0, l_{\mathrm{RANS}}-l_{\mathrm{LES}} \right).\]

Within an attached boundary layer, \(f_d\approx0\) and \(l_{\mathrm{DDES}}\approx l_{\mathrm{RANS}}\) independently of the wall-parallel grid spacing. Outside that region, \(f_d\approx1\), and the formulation reduces to the original DES length-scale limiter.

12.3. SST-DDES Formulation

Gritskevich et al. recalibrated the empirical constants in the Spalart–Allmaras DDES delay function for use with SST-based DDES and IDDES [GGSM12]. The recalibration was designed to preserve shielding against modeled-stress depletion without degrading the resolution of separated turbulence or the wall-modeled LES behavior of SST-IDDES.

The SST-DDES turbulence-kinetic-energy equation is

(12.3.1)\[\frac{\partial(\rho k)}{\partial t} + \boldsymbol{\nabla}\mathbin{\cdot}(\rho\boldsymbol{u}k) = \boldsymbol{\nabla}\mathbin{\cdot} \left[ \left(\mu+\sigma_k\mu_t\right)\boldsymbol{\nabla}k \right] + P_k - \rho\frac{k^{3/2}}{l_{\mathrm{DDES}}},\]

and the specific-dissipation-rate equation is

(12.3.2)\[\frac{\partial(\rho\omega)}{\partial t} + \boldsymbol{\nabla}\mathbin{\cdot}(\rho\boldsymbol{u}\omega) = \boldsymbol{\nabla}\mathbin{\cdot} \left[ \left(\mu+\sigma_\omega\mu_t\right) \boldsymbol{\nabla}\omega \right] + 2(1-F_1)\rho\sigma_{\omega2} \frac{\boldsymbol{\nabla}k\mathbin{\cdot}\boldsymbol{\nabla}\omega}{\omega} + \alpha\frac{\rho}{\mu_t}P_k - \beta\rho\omega^2.\]

The turbulent viscosity is

(12.3.3)\[\mu_t = \frac{\rho a_1 k} {\max\left(a_1\omega,F_2S\right)}.\]

The SST blending functions retain their usual form:

(12.3.4)\[F_1=\tanh\left(\Gamma_1^4\right),\]

(12.3.5)\[\Gamma_1 = \min\left[ \max\left( \frac{\sqrt{k}}{C_\mu\omega d_w}, \frac{500\mu}{\rho d_w^2\omega} \right), \frac{4\rho\sigma_{\omega2}k} {CD_{k\omega}d_w^2} \right],\]

(12.3.6)\[CD_{k\omega} = \max\left[ 2\sigma_{\omega2}\rho \frac{\boldsymbol{\nabla}k\mathbin{\cdot}\boldsymbol{\nabla}\omega}{\omega}, 10^{-10} \right],\]

(12.3.7)\[F_2=\tanh\left(\Gamma_2^2\right),\]

and

(12.3.8)\[\Gamma_2 = \max\left[ \frac{2\sqrt{k}}{C_\mu\omega d_w}, \frac{500\mu}{\rho d_w^2\omega} \right].\]

The production term is limited according to

(12.3.9)\[P_k = \min\left( \mu_tS^2, 10C_\mu\rho k\omega \right).\]

The length scales are

(12.3.10)\[l_{\mathrm{DDES}} = l_{\mathrm{RANS}} - f_d\max\left( 0, l_{\mathrm{RANS}}-l_{\mathrm{LES}} \right),\]

(12.3.11)\[l_{\mathrm{LES}} = C_{\mathrm{DES}}h_{\max},\]

and

(12.3.12)\[l_{\mathrm{RANS}} = \frac{\sqrt{k}}{C_\mu\omega},\]

where \(h_{\max}\) is the maximum edge length of the cell. The DES coefficient is blended between the inner and outer SST branches:

(12.3.13)\[C_{\mathrm{DES}} = C_{\mathrm{DES1}}F_1 + C_{\mathrm{DES2}}(1-F_1).\]

The recalibrated delay function and wall marker are

(12.3.14)\[f_d = 1-\tanh\left[ \left(C_{d1}r_d\right)^{C_{d2}} \right],\]

(12.3.15)\[r_d = \frac{\nu+\nu_t} {\kappa^2d_w^2 \sqrt{\tfrac{1}{2}\left(\Omega^2+S^2\right)}}.\]

Here \(S\) and \(\Omega\) are the magnitudes of the strain-rate and vorticity tensors. The model constants are

(12.3.16)\[\begin{split}\begin{aligned} C_\mu&=0.09, & \kappa&=0.41, & a_1&=0.31,\\ C_{\mathrm{DES1}}&=0.78, & C_{\mathrm{DES2}}&=0.61, & C_{d1}&=20, & C_{d2}&=3. \end{aligned}\end{split}\]

The remaining coefficients are blended between the two SST branches. Their branch values are

(12.3.17)\[\begin{split}\begin{aligned} \alpha_1&=\frac{5}{9}, & \beta_1&=0.075, & \sigma_{k1}&=0.85, & \sigma_{\omega1}&=0.5,\\ \alpha_2&=0.44, & \beta_2&=0.0828, & \sigma_{k2}&=1, & \sigma_{\omega2}&=0.856. \end{aligned}\end{split}\]

For example, \(\alpha=\alpha_1F_1+\alpha_2(1-F_1)\), with the other coefficients blended in the same way.

12.4. Improved Delayed Detached-Eddy Simulation

Improved delayed detached-eddy simulation (IDDES) grew from attempts to use the original DES model as a wall-modeled LES (WMLES) formulation. That was not the original purpose of DES, and it exposed a logarithmic-layer mismatch (LLM) between the inner RANS region and the outer LES region. IDDES adds blending and shielding functions so that one non-zonal formulation can operate in both DDES and WMLES modes [SSST08].

An early application of DES as a wall model in fully developed channel flow allowed arbitrarily large wall-parallel grid spacing in wall units [NNW+00]. The predicted computational cost then grew logarithmically with Reynolds number rather than with the power-law behavior of wall-resolved LES. The calculation nevertheless produced two logarithmic layers: an inner layer imposed by the RANS closure and an outer layer formed where the LES became adequately resolved. Because the two layers had different intercepts, the resulting LLM caused skin friction to be underpredicted by as much as 15–20 percent [SSST08]. DDES inherits this weakness when it is used as a wall model.

The aim of IDDES is therefore broader than merely delaying the DES switch. It seeks one set of relations suitable both for conventional DES/DDES applications and for WMLES, so that different regions of a complex geometry can be treated automatically by the appropriate branch of the model.

12.5. SST-IDDES Formulation

Like SST-DDES, SST-IDDES borrows its blending and shielding structure from the Spalart–Allmaras-based formulation, with recalibration and simplification for the SST model [GGSM12]. The governing \(k\) and \(\omega\) equations and the turbulent-viscosity relation are those in Eqs. (12.3.1)–(12.3.3), with \(l_{\mathrm{DDES}}\) replaced by \(l_{\mathrm{IDDES}}\).

The IDDES length scale is a blend of RANS and LES scales:

(12.5.1)\[l_{\mathrm{IDDES}} = \widetilde f_d\,l_{\mathrm{RANS}} + \left(1-\widetilde f_d\right)l_{\mathrm{LES}},\]

where

(12.5.2)\[l_{\mathrm{LES}}=C_{\mathrm{DES}}\Delta, \qquad l_{\mathrm{RANS}}=\frac{\sqrt{k}}{C_\mu\omega},\]

and

(12.5.3)\[C_{\mathrm{DES}} = C_{\mathrm{DES1}}F_1 + C_{\mathrm{DES2}}(1-F_1).\]

The source fragment prints a minus sign before the LES contribution in the length-scale blend. The plus sign in Eq. (12.5.1) is required for the expression to interpolate between positive RANS and LES length scales and is the form used by the SST-IDDES formulation.

The LES filter width includes both grid and wall-distance information:

(12.5.4)\[\Delta = \min\left[ C_w\max\left(d_w,h_{\max}\right), h_{\max} \right].\]

The empirical blending function is

(12.5.5)\[\widetilde f_d = \max\left( 1-f_{dt}, f_b \right),\]

with

(12.5.6)\[f_{dt} = 1-\tanh\left[ \left(C_{dt1}r_{dt}\right)^{C_{dt2}} \right],\]

(12.5.7)\[r_{dt} = \frac{\nu_t} {\kappa^2d_w^2 \sqrt{\tfrac{1}{2}\left(\Omega^2+S^2\right)}},\]

(12.5.8)\[f_b = \min\left[ 2\exp\left(-9\alpha^2\right), 1 \right],\]

and

(12.5.9)\[\alpha = 0.25-\frac{d_w}{h_{\max}}.\]

In addition to the SST-DDES constants, this formulation introduces

(12.5.10)\[C_w=0.15, \qquad C_{dt1}=20, \qquad C_{dt2}=3.\]

The source fragment repeats \(C_{dt1}\) in its constants list; the unique constants actually used by the relations above are shown in Eq. (12.5.10).

Note

The current Stream SST-IDDES implementation uses the total kinematic viscosity, \(\nu+\nu_t\), in \(r_{dt}\), matching the robust wall-marker form used by SST-DDES. Equation (12.5.7) preserves the original theory-fragment expression so that the distinction remains explicit.

12.5.1. Historical Development of DES Variants

The principal stages in the development summarized by the source theory fragment are:

Reference	Main development	Significance
Spalart et al. (1997)	Introduced the motivation and basic DES equations, with \(C_{\mathrm{DES}}\) not yet fully calibrated; included two-dimensional examples.	Established the original DES concept in the context of the Spalart–Allmaras model.
Travin et al. (1999)	Presented a fully three-dimensional DES application, calibrated \(C_{\mathrm{DES}}\), and predicted airfoil forces.	Demonstrated practical three-dimensional DES.
Travin et al. (2000)	Examined grid refinement and the drag crisis.	Refined the practical definition and use of DES.
Nikitin et al. (2000)	Applied DES as a wall model within LES.	Provided an early wall-modeled LES application of DES.
Strelets (2001)	Applied DES broadly, including with models other than Spalart–Allmaras.	Introduced the SST-DES formulation.
Menter and Kuntz (2002)	Demonstrated grid-induced separation and proposed \(F_1\)- or \(F_2\)-based shielding to preserve RANS behavior in the boundary layer.	Established zonal SST-DES and helped motivate the more general DDES shielding function.
Spalart et al. (2006)	Introduced DDES, related GIS to modeled-stress depletion, and defined a generic wall-distance and eddy-viscosity shielding function.	Prevented an undesirable RANS-to-LES switch inside much of an attached boundary layer; the original paper used Spalart–Allmaras, and the idea was later adapted to SST.
Shur et al. (2008)	Combined DDES with a hybrid method aimed at WMLES and added further blending and shielding.	IDDES addresses logarithmic-layer mismatch and supplies DDES and WMLES branches through a subgrid length scale that depends on grid spacing and wall distance.
Gritskevich et al. (2012)	Recalibrated DDES and IDDES for the SST model.	Produced SST-DDES and SST-IDDES formulations suitable for attached boundary layers, separated flow, and wall-modeled LES.

12.6. Nichols–Nelson Hybrid RANS Model

The source theory fragment contains this section heading but no accompanying derivation. The following account completes the section from the hybrid multiscale model used by Stream and the formulation of Nelson and Nichols [NN03].

The model blends an SST RANS viscosity with an LES-like viscosity according to a local measure of whether the grid scale can resolve the turbulent length scale. Define the RANS length and viscosity scales by

(12.6.1)\[l_{\mathrm{RANS}}=\frac{\sqrt{k}}{\omega}, \qquad \nu_{t,\mathrm{RANS}} = \frac{a_1k}{\max\left(a_1\omega,SF_2\right)}.\]

A turbulence length used by the blending sensor is

(12.6.2)\[l_{\mathrm{turb}} = \max\left( 6\sqrt{\frac{\nu_{t,\mathrm{RANS}}}{\Omega}}, l_{\mathrm{RANS}} \right).\]

The resolution indicator and its smooth blending function are

(12.6.3)\[\lambda = \left[ 1+ \left( \frac{0.5l_{\mathrm{turb}}}{\Delta} \right)^{4/3} \right]^{-1},\]

(12.6.4)\[f_d = \frac{1}{2} \left[ 1+ \tanh\left( 2\pi\left(\lambda-\frac{1}{2}\right) \right) \right].\]

The LES branch uses \(k_{\mathrm{LES}}=f_dk\) and

(12.6.5)\[\nu_{t,\mathrm{LES}} = \min\left( 0.0854\Delta\sqrt{k_{\mathrm{LES}}}, \nu_{t,\mathrm{RANS}} \right).\]

The final eddy viscosity is the smooth blend

(12.6.6)\[\nu_t = f_d\nu_{t,\mathrm{RANS}} + (1-f_d)\nu_{t,\mathrm{LES}}.\]

Thus the same transported SST variables support both branches, while the grid-sensitive sensor controls how much of the modeled turbulent viscosity is retained.

12.7. Large-Eddy Simulation Models

In LES, the smallest turbulent scales are spatially filtered while the large, energy-containing structures are resolved directly. At sufficiently small scales the turbulence tends to be more nearly isotropic and less dependent on the particular geometry, which makes the unresolved motion more amenable to a model intended to apply across many flows. The unresolved terms in the filtered equations are called subgrid-scale (SGS) terms because the filter width is normally tied to the local mesh scale.

For any flow variable (\phi), a spatial filter can be written schematically as

(12.7.1)\[\overline{\phi}(\boldsymbol{x},t) = \int_{\mathcal{D}} G\!\left(\boldsymbol{x}-\boldsymbol{r};\Delta\right) \phi(\boldsymbol{r},t)\,d\boldsymbol{r},\]

where (G) is the filter kernel and (\Delta) is its characteristic width. The field is decomposed into resolved and residual parts,

(12.7.2)\[\boldsymbol{u} = \overline{\boldsymbol{u}} + \boldsymbol{u}'.\]

Unlike a Reynolds average, this is a local spatial decomposition, and the filtered field retains the resolved unsteady fluctuations. For clarity, begin with the incompressible equations and assume that the filter commutes with spatial and temporal derivatives. Filtering continuity gives

(12.7.3)\[\boldsymbol{\nabla}\mathbin{\cdot}\overline{\boldsymbol{u}}=0.\]

Filtering momentum before introducing a closure gives

(12.7.4)\[\frac{\partial\overline{\boldsymbol{u}}}{\partial t} + \boldsymbol{\nabla}\mathbin{\cdot} \overline{\boldsymbol{u}\boldsymbol{u}} = -\frac{1}{\rho}\boldsymbol{\nabla}\overline p + \nu\nabla^2\overline{\boldsymbol{u}}.\]

Filtering does not commute with the nonlinear product in the sense required to close this equation:

(12.7.5)\[\overline{\boldsymbol{u}\boldsymbol{u}} \ne \overline{\boldsymbol{u}}\, \overline{\boldsymbol{u}}.\]

Adding and subtracting the resolved product exposes the unclosed part:

(12.7.6)\[\boldsymbol{\nabla}\mathbin{\cdot} \overline{\boldsymbol{u}\boldsymbol{u}} = \boldsymbol{\nabla}\mathbin{\cdot} \left( \overline{\boldsymbol{u}}\, \overline{\boldsymbol{u}} \right) + \boldsymbol{\nabla}\mathbin{\cdot} \left( \overline{\boldsymbol{u}\boldsymbol{u}} - \overline{\boldsymbol{u}}\, \overline{\boldsymbol{u}} \right).\]

Define the SGS stress by

(12.7.7)\[\boldsymbol{\tau}_{\mathrm{SGS}} = \overline{\boldsymbol{u}\boldsymbol{u}} - \overline{\boldsymbol{u}}\, \overline{\boldsymbol{u}}.\]

The filtered momentum equation can then be written entirely in terms of the resolved velocity and the SGS closure:

(12.7.8)\[\frac{\partial\overline{\boldsymbol{u}}}{\partial t} + \boldsymbol{\nabla}\mathbin{\cdot} \left( \overline{\boldsymbol{u}}\, \overline{\boldsymbol{u}} \right) = -\frac{1}{\rho}\boldsymbol{\nabla}\overline p + \nu\nabla^2\overline{\boldsymbol{u}} - \boldsymbol{\nabla}\mathbin{\cdot} \boldsymbol{\tau}_{\mathrm{SGS}},\]

The SGS stress represents the effect of the filtered scales on the resolved field and therefore requires closure, much as the Reynolds-stress tensor does in RANS. Using the decomposition above, it can be separated into Leonard, cross, and residual contributions:

(12.7.9)\[\begin{split}\begin{aligned} \boldsymbol{\tau}_{\mathrm{SGS}} ={}& \underbrace{ \left( \overline{ \overline{\boldsymbol{u}}\, \overline{\boldsymbol{u}} } - \overline{\boldsymbol{u}}\, \overline{\boldsymbol{u}} \right) }_{\boldsymbol{L}} \\ &+ \underbrace{ \left( \overline{\overline{\boldsymbol{u}}\boldsymbol{u}'} + \overline{\boldsymbol{u}'\overline{\boldsymbol{u}}} \right) }_{\boldsymbol{C}} + \underbrace{ \overline{\boldsymbol{u}'\boldsymbol{u}'} }_{\boldsymbol{R}}. \end{aligned}\end{split}\]

The Leonard stress \(\boldsymbol{L}\) contains resolved quantities and can be computed. The cross stress \(\boldsymbol{C}\) represents interaction between resolved and unresolved scales, while the residual stress \(\boldsymbol{R}\) represents interaction among unresolved scales. The latter two contributions require modeling.

Many LES closures approximate the deviatoric SGS stress with an algebraic, or zero-equation, model based on the Boussinesq hypothesis:

(12.7.10)\[\boldsymbol{\tau}_{\mathrm{SGS}} - \frac{1}{3} \operatorname{tr}\!\left( \boldsymbol{\tau}_{\mathrm{SGS}} \right)\boldsymbol{I} = -2\nu_{\mathrm{SGS}}\overline{\boldsymbol{S}},\]

where \(\nu_{\mathrm{SGS}}\) is the subgrid-scale eddy viscosity and \(\overline{\boldsymbol{S}}\) is the resolved strain-rate tensor. The isotropic SGS stress is absorbed into the filtered pressure. The Smagorinsky and wall-adapting local eddy-viscosity models are described next.

LES-Filtering Quick Reference

The nonlinear convective product does not close under spatial filtering. Its difference from the product of filtered velocities is the SGS stress. The divergence of that stress enters the filtered momentum equation with a minus sign, and an eddy-viscosity model supplies its deviatoric part. On nonuniform filters, additional filter–derivative commutation errors may appear; they are outside the homogeneous-filter derivation above.

Note

The Smagorinsky model is included because it is part of the source theory chapter and provides the natural introduction to algebraic LES closure. The current Stream pure-LES model selector exposes the WALE model.

12.7.1. Smagorinsky Model

In the Smagorinsky model the resolved strain-rate tensor is

(12.7.1.1)\[\overline{\boldsymbol{S}} = \frac{1}{2} \left[ \boldsymbol{\nabla}\overline{\boldsymbol{u}} + \left( \boldsymbol{\nabla}\overline{\boldsymbol{u}} \right)^T \right].\]

The subgrid-scale eddy viscosity is

(12.7.1.2)\[\nu_{\mathrm{SGS}} = \left(C_s\Delta\right)^2 \left|\overline{\boldsymbol{S}}\right|,\]

where \(C_s\) is the Smagorinsky constant and

(12.7.1.3)\[\left|\overline{\boldsymbol{S}}\right| = \left( 2\overline{\boldsymbol{S}}: \overline{\boldsymbol{S}} \right)^{1/2}.\]

The filter width is commonly related to the cell volume:

(12.7.1.4)\[\Delta = \left( \Delta_x\Delta_y\Delta_z \right)^{1/3} = V_{\mathrm{cell}}^{1/3}.\]

Equation (12.7.1.2) has the form of a mixing-length closure with

(12.7.1.5)\[l_m=C_s\Delta.\]

Near a wall, the undamped model can overpredict a nonzero subgrid viscosity. A van Driest damping function can be used to force the viscosity toward zero:

(12.7.1.6)\[\nu_{\mathrm{SGS}} = \left[ C_s\Delta \left( 1-e^{-y^+/25} \right) \right]^2 \left|\overline{\boldsymbol{S}}\right|,\]

where \(y^+\) is the dimensionless wall-normal distance. The damping form follows the near-wall scaling introduced by van Driest [vD56].

The Smagorinsky model is widely used because of its simplicity. It has well-known deficiencies in wall-bounded flows but can be adequate for free shear flows. Reported values of \(C_s\) range roughly from 0.1 to 0.35 and are not universal: values near 0.1 are commonly used for wall-bounded flows, 0.18 for isotropic turbulence, and 0.24 for shear-free flows. The original calibration and Lilly’s isotropic-turbulence analysis give a value near 0.18 [Lil67, Sma63].

12.8. Wall-Adapting Local Eddy-Viscosity Model

The wall-adapting local eddy-viscosity (WALE) model was developed by Nicoud and Ducros to overcome several deficiencies of the Smagorinsky model without substantially increasing its complexity [ND99]. Rather than depending only on invariants of the strain-rate tensor, WALE uses invariants of the complete velocity-gradient tensor. It has the correct near-wall scaling, vanishes in laminar shear flow, and can represent transitional flow.

The WALE subgrid-scale viscosity is

(12.8.1)\[\nu_{\mathrm{SGS}} = \left(C_w\Delta\right)^2 \frac{ \left(S_{ij}^{d}S_{ij}^{d}\right)^{3/2} }{ \left(S_{ij}S_{ij}\right)^{5/2} + \left(S_{ij}^{d}S_{ij}^{d}\right)^{5/4} }.\]

The tensor \(S_{ij}^{d}\) is the symmetric, traceless part of the square of the resolved velocity-gradient tensor. With

(12.8.2)\[g_{ij} = \frac{\partial\overline u_i}{\partial x_j},\]

it is

(12.8.3)\[S_{ij}^{d} = \frac{1}{2} \left( g_{ik}g_{kj} + g_{jk}g_{ki} \right) - \frac{1}{3}\delta_{ij}g_{kl}g_{lk}.\]

Writing the velocity gradient as the sum of symmetric and antisymmetric parts,

(12.8.4)\[g_{ij}=S_{ij}+W_{ij}, \qquad S^2=S_{ij}S_{ij}, \qquad \Omega^2=W_{ij}W_{ij},\]

gives the invariant relation

(12.8.5)\[S_{ij}^{d}S_{ij}^{d} = \frac{1}{6} \left[ \left(S^2\right)^2 + \left(\Omega^2\right)^2 \right] + \frac{2}{3}S^2\Omega^2 + 2S_{ik}S_{kj}W_{jl}W_{li}.\]

Both strain and rotation therefore contribute to the WALE viscosity. The combined invariant structure avoids the singular behavior that could arise if the closure depended on only one of these measures and produces the desired near-wall decay without an explicit wall-damping function.

Values of \(C_w\) between approximately 0.3 and 0.6 are common. Stream uses \(C_w=0.325\) by default and evaluates the filter width as \(\Delta=V_{\mathrm{cell}}^{1/3}\).

12.9. References

[BJK97]

J.S. Baggett, J. Jimenez, and A.G. Kravchenko. Resolution requirements in large-eddy simulations of shear flow. In Center for Turbulence Research Annual Research Briefs 1997, 51–66. 1997.

[Geu04]

B.J. Geurts. Elements of Direct and Large-Eddy Simulation. Edwards, Inc., 2004.

[Gir06]

S. Girimaji. Partially averaged navier-stokes model for turbulence: a rans to dns bridging method. Journal of Applied Mechanics, 73(3):422–429, 2006.

[GGSM12] (1,2)

M.S. Gritskevich, A.V. Garbaruk, J. Schutze, and F.R. Menter. Development of DDES and IDDES formulations for the k-omega shear stress transport model. Flow, Turbulence and Combustion, 88:431–449, 2012. doi:10.1007/s10494-011-9378-4.

[Lil67]

D.K. Lilly. The representation of small-scale turbulence in numerical simulation experiments. In Proceedings of the IBM Scientific Computing Symposium on Environmental Sciences, 195–210. 1967.

[MK04]

F.R. Menter and M. Kuntz. Adaptation of eddy-viscosity turbulence models to unsteady separated flows behind vehicles. In R. McCallen, F. Browand, and J. Ross, editors, The Aerodynamics of Heavy Vehicles: Trucks, Buses and Trains. Springer, 2004.

[MKL03] (1,2,3)

F.R. Menter, M. Kuntz, and R. Langtry. Ten years of industrial experience with the sst turbulence model. In K. Hanjalic, Y. Nagano, and M. Tummers, editors, Turbulence, Heat and Mass Transfer 4, pages 625–632. Begell House, Inc., 2003.

[NN03] (1,2)

C.C. Nelson and R.H. Nichols. Application of hybrid rans/les turbulence models. In AIAA Paper. 2003.

[ND99]

F. Nicoud and F. Ducros. Subgrid-scale stress modelling based on the square of the velocity gradient tensor. Flow, Turbulence and Combustion, 62:183–200, 1999. doi:10.1023/A:1009995426001.

[NNW+00]

N.V. Nikitin, F. Nicoud, B. Wasistho, K.D. Squires, and P.R. Spalart. An approach to wall modeling in large-eddy simulations. Physics of Fluids, 12(7):1629–1632, 2000. doi:10.1063/1.870414.

[Pio99]

U. Piomelli. Large-eddy simulation: achievements and challenges. Progress in Aerospace Sciences, 35:335–362, 1999.

[PB02]

U. Piomelli and E. Balaras. Wall-layer models for large-eddy simulations. Annual Review of Fluid Mechanics, 34:349–374, 2002.

[Pop00]

S.B. Pope. Turbulent Flows. Cambridge University Press, 2000.

[Sag00]

P. Sagaut. Large Eddy Simulation for Incompressible Flows. Springer-Verlag, 2000.

[SSST08] (1,2,3)

M.L. Shur, P.R. Spalart, M.K. Strelets, and A.K. Travin. A hybrid rans-les approach with delayed-des and wall-modelled les capabilities. International Journal of Heat and Fluid Flow, 29:1638–1649, 2008.

[Sma63]

J. Smagorinsky. General circulation experiments with the primitive equations: i. the basic experiment. Monthly Weather Review, 91(3):99–164, 1963.

[Spa09]

P.R. Spalart. Detached-eddy simulation. Annual Review of Fluid Mechanics, 41:181–202, 2009.

[SDS+06] (1,2,3)

P.R. Spalart, S. Deck, M.L. Shur, K.D. Squires, M.K. Strelets, and A. Travin. A new version of detached-eddy simulation, resistant to ambiguous grid densities. Theoretical Computational Fluid Dynamics, 20:181–195, 2006.

[SJSA97] (1,2,3)

P.R. Spalart, W.H. Jou, M. Strelets, and S.R. Allmaras. Comments on the feasibility of les for wings and on a hybrid rnas/les approach. In 1st AFOSR Conference on DNS/LES. August 1997.

[Str01] (1,2,3,4)

M. Strelets. Detached eddy simulation of massively separated flows. In AIAA Paper. 2001.

[TD04]

P.G. Tucker and L. Davidson. Zonal k-l based large eddy simulations. Computers and Fluids, 33:267–287, 2004.

[vD56]

E.R. van Driest. On turbulent flow near a wall. Journal of the Aeronautical Sciences, 23(11):1007–1011, 1956. doi:10.2514/8.3713.