Understanding the physical behavior of turbulence requires a review of the underlying mathematics. Following the conceptual overview provided in What is Large Eddy Simulation (LES) in CFD, this blog details the mathematical derivation of LES filtering and the closure of the LES governing equations.
Energy Cascade and Kolmogorov Scales
The basis for turbulence modeling relies on the energy cascade. Turbulence consists of a structured process of energy transfer across multiple scales. Energy enters the flow field through the largest eddies. The dimension of these eddies depends on the global geometry. They are anisotropic and contain the majority of the kinetic energy. These large structures become unstable and break apart, transferring kinetic energy to smaller scales.
This physical mechanism continues until the eddies become isotropic. At the smallest scale, known as the Kolmogorov scale, molecular viscosity dissipates the kinetic energy into heat. This physical separation of scales enables spatial filtering. The solver resolves the large, geometry-dependent eddies directly and models the universal, small-scale structures.
Figure 1: The energy cascade in turbulence theory, showing how large eddies are resolved and small ones modeled
Spatial Filtering vs. Temporal Averaging
Resolving the entire turbulent spectrum down to the Kolmogorov scale requires Direct Numerical Simulation (DNS). Mathematical averaging or filtering reduces the computational requirements.
Reynolds-Averaged Navier-Stokes (RANS) utilizes temporal averaging. This operation averages the flow variables over time, removing all turbulent fluctuations to provide a steady-state approximation.
The alternative approach utilizes LES filtering in space. The computational mesh cell acts as the spatial filter. Any eddy larger than the grid scale is resolved, preserving its time-dependent motion. Any structure smaller than the grid scale is classified as a subgrid scale, and its physical effect is modeled.
Figure 2: A diagram comparing the concepts of RANS temporal averaging versus spatial filtering in CFD
Mathematics of Filtering: Filtered N-S Equations
Applying a spatial filter mathematically transforms the fundamental equations of fluid motion. The filtering operation utilizes a filter function, G. In applied computational fluid dynamics, solvers utilize an implicit filter where the cell volume defines the filter width, Δ. The mesh resolution physically dictates the boundary between resolved eddies and modeled scales.
Applying this operation to the incompressible Navier-Stokes equations generates the filtered navier-stokes equations. These equations govern the unsteady motion of the large resolved scales.
Figure 3: The mathematical derivation of the equations, showing the creation of the unknown SGS stress term
The filtering operation produces an unclosed mathematical system containing an unknown term, the Subgrid-Scale Stress Tensor, τ_ij. This tensor is defined as the difference between the filtered product of velocities and the product of filtered velocities:
\tau_{ij} = \overline{(u_i u_j)} - \bar{u}_i \bar{u}_j
The tensor τ_ij represents the momentum transfer from the unresolved subgrid scales to the resolved large eddies. Because this variable depends on the unresolved velocity field, the mathematical system requires a closure model. An SGS model (like Smagorinsky, WALE, etc.) provides an approximation for this unknown stress tensor, correlating it with the known resolved flow field.
Figure 4: An energy spectrum graph demonstrating eddy viscosity dissipating energy at the grid cut-off limit
SGS Stress Tensor & The Closure Problem
Closing the filtered navier-stokes equations requires calculating the momentum exchange caused by the subgrid scales. Most subgrid models utilize the Boussinesq hypothesis. This hypothesis assumes the momentum transfer from small eddies acts similarly to molecular viscosity.
This formulation introduces a calculated parameter known as the Subgrid-Scale Eddy Viscosity, μ_sgs. In a filtered numerical domain, the mesh resolution cannot resolve the Kolmogorov scale. The subgrid scale model generates the required μ_sgs to dissipate kinetic energy at the grid cut-off limit.
Without an adequate model, kinetic energy accumulates unphysically at the smallest resolved scales, causing numerical divergence. The mathematical formulations available in numerical solvers calculate this required eddy viscosity using varying methodologies.
Figure 5: comparison graph showing the unphysical pile-up of energy at the grid cut-off without proper dissipation
Figure 6: Airplane simulation using WALLE SGS model
Comparison of Subgrid-Scale Models
Multiple mathematical formulations exist to calculate μ_sgs. Selecting a turbulence formulation requires evaluating the balance between computational cost and mathematical accuracy.
The table below outlines the primary models utilized to close the governing equations.
| Feature | Smagorinsky Model | WALE Model (Wall-Adapting Local Eddy-Viscosity) | Dynamic Smagorinsky Model | Dynamic k-eq Transport |
|---|---|---|---|---|
| Model Type | Algebraic (0-Equation) | Algebraic (0-Equation) | Algebraic (Dynamic Procedure) | 1-Equation Transport |
| Formulation | Calculates μ_sgs from the local strain rate and a fixed constant (C_s). | Calculates μ_sgs using a formula that handles near-wall regions without a constant. | Calculates C_s dynamically based on the resolved flow field using a secondary test filter. | Solves a transport equation for the subgrid kinetic energy (k_sgs) to find μ_sgs. |
| Advantages | Low computational cost. Standard algebraic formulation. | Computes zero eddy viscosity in laminar shear flows. Adapts to near-wall physics without user-defined damping functions. | Eliminates the need for a fixed constant. Adapts to transitional flow regimes. | Accounts for the transport of subgrid energy. Represents non-equilibrium effects accurately. |
| Disadvantages | The constant C_s requires tuning. Requires ad-hoc wall damping near boundaries. | Assumes local equilibrium between energy production and dissipation. | Higher computational cost due to the secondary filtering operation. | Highest computational cost due to the additional transport equation. |
Understanding the derivation of the filtered navier-stokes equations and the function of the SGS model is a prerequisite for numerical configuration. To apply these mathematical formulations to a computational domain, review the specific mesh resolution parameters and solver settings detailed in our guide on LES Implementation in ANSYS Fluent.