8. Turbulence Modeling

Unlike a RANS approach which models most or all of the turbulent fluctuations, LES directly solves for all resolved turbulent length scales and only models the smallest scales below the grid size. In this way, a majority of the problem-dependent, energy-containing turbulent structure is directly solved in a model-free fashion. The subgrid scales are closer to being isotropic than the resolved scales, and they generally act to dissipate turbulent kinetic energy cascaded down from the larger scales in momentum-driven turbulent flows. Modeling of these small scales is generally more straightforward than RANS approaches, and overall solutions are usually more tolerant to LES modeling errors because the subgrid scales comprise such a small portion of the overall turbulent structure. While LES is generally accepted to be much more accurate than RANS approaches for complex turbulent flows, it is also significantly more expensive than equivalent RANS simulations due to the finer grid resolution required. Additionally, since LES results in a full unsteady solution, the simulation must be run for a long time to gather any desired time-averaged statistics. The tradeoff between accuracy and cost must be weighed before choosing one method over the other.

The separation of turbulent length scales required for LES is obtained by using a spatial filter rather than the RANS temporal filter. This filter has the mathematical form

(8.1)\[\overline{\phi(\boldsymbol{x},t)} \equiv \int_{-\infty}^{+\infty} \phi(\boldsymbol{x}',t) G(\boldsymbol{x}' - \boldsymbol{x})\, \mathrm{d}\boldsymbol{x}',\]

which is a convolution integral over physical space \(\boldsymbol{x}\) with the spatially-varying filter function \(G\). The filter function has the normalization property \(\int_{-\infty}^{+\infty} G(\boldsymbol{x})\, \mathrm{d}\boldsymbol{x} = 1\), and it has a characteristic length scale \(\Delta\) so that it filters out turbulent length scales smaller than this size. In the present formulation, a simple “box filter” is used for the filter function,

\[\begin{split}G(\boldsymbol{x}' - \boldsymbol{x}) = \left\{ \begin{array}{l@{\quad:\quad}l} 1/V & (\boldsymbol{x}' - \boldsymbol{x}) \in \mathcal{V} \\ 0 & \mathrm{otherwise} \\ \end{array} \right.,\end{split}\]

where \(V\) is the volume of control volume \(\mathcal{V}\) whose central node is located at \(\boldsymbol{x}\). This is essentially an unweighted average over the control volume. The length scale of this filter is approximated by \(\Delta = V^\frac{1}{3}\). This is typically called the grid filter, as it filters out scales smaller than the computational grid size.

Similar to the RANS temporal filter, a variable can be represented in terms of its filtered and subgrid fluctuating components as

\[\phi = \bar{\phi} + \phi'.\]

For most forms of the filter function \(G(\boldsymbol{x})\), repeated applications of the grid filter to a variable do not yield the same result. In other words, \(\bar{\bar{\phi}} \ne \bar{\phi}\) and therefore \(\overline{\phi'} \ne 0\), unlike with the RANS temporal averages.

As with the RANS formulation, modeling is much simplified in the presence of large density variations if a Favre-filtered approach is used. A Favre-filtered variable \(\tilde{\phi}\) is defined as

\[\tilde{\phi} \equiv \frac{ \overline{\rho\phi} }{ \bar{\rho} }\]

and a variable can be decomposed in terms of its Favre-filtered and subgrid fluctuating component as

\[\phi = \tilde{\phi} + \phi''.\]

Again, note that the useful identities for the Favre-filtered RANS variables do not apply, so that \(\bar{\tilde{\phi}} \ne \tilde{\phi}\) and \(\overline{\phi''} \ne 0\). The Favre-filtered approach is used for all LES models in Nalu.

8.1. Standard Smagorinsky LES Model

The standard Smagorinsky LES closure model approximates the subgrid turbulent eddy viscosity using a mixing length-type model, where the LES grid filter size \(\Delta\) provides a natural length scale. The subgrid eddy viscosity is modeled simply as (Smagorinsky)

(8.2)\[\mu_t = \rho \left(C_s \Delta \right)^2 | \tilde {S} |,\]

The constant coefficient \(C_s\) typically varies between 0.1 and 0.24 and should be carefully tuned to match the problem being solved (Rogallo and Moin, [RM84]). The default value of 0.17 is assigned in the code base.

Although this model is desirable due to its simplicity and efficiency, care should be taken in its application. It is known to predict subgrid turbulent eddy viscosity proportional to the shear rate in the flow, independent of the local turbulence intensity. Non-zero subgrid turbulent eddy viscosity is even predicted in completely laminar regions of the flow, sometimes even preventing a natural transition to turbulence. The model also does not asymptotically replicate near wall behavior without either dampening or a dynamic procedure.

8.2. Wall Adapting Local Eddy-Viscosity, WALE

The WALE model of Ducros el al., [DNP98], properly captures the asymptotic behavior for flows that are wall bounded. In this model, the turbulent viscosity is given by,

(8.3)\[\mu_t = \rho \left(C_w \Delta \right)^2 \frac{\left( S^d_{ij}S^d_{ij}\right)^{3/2}} {\left( S_{ij}S_{ij}\right)^{5/2} + \left( S^d_{ij}S^d_{ij}\right)^{5/4}},\]

with the constant \(C_w\) of 0.325 and a standard filter, \(\Delta\) related to the volume, \(V^{\frac{1}{3}}\). The rate of strain tensor is defined as,

(8.4)\[S_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)\]

while \(S^d_{ij}\) is,

(8.5)\[S^d_{ij} = \frac{1}{2} \left( g^2_{ij} + g^2_{ji}\right) - \frac{1}{3} \delta_{ij} g^2_{kk}.\]

Finally, the velocity gradient squared ters are

(8.6)\[ g^2_{ij} = \frac{\partial u_i}{\partial x_k} \frac{\partial u_k}{\partial x_j}\]


(8.7)\[ g^2_{ji} = \frac{\partial u_j}{\partial x_k} \frac{\partial u_k}{\partial x_i}.\]

8.3. One Equation \(k^{sgs}\)

See \(k^{sgs}\) PDE section.

8.4. SST RANS Model

As noted, Nalu does support a SST RANS-based model (the reader is referred to the SST equation set description).

8.5. Wall Models

Flows are either expected to be fully resolved or, alternatively, under-resolved where wall functions are used. A classic law of the wall has been implemented in Nalu. Wall models to handle adverse pressure gradients are planned. For more information of the form of wall models, please refer to the boundary condition section of this manual.