In our previous article, What is Large Eddy Simulation (LES) in CFD, we introduced the basic concept of LES. This guide is needed because to truly master ANSYS Fluent LES, we must go deeper into the fundamental physics and mathematics that make it work.
Energy Cascade and Kolmogorov Scales
The core idea behind all turbulence modeling is the Energy Cascade. Turbulence is not simply chaotic motionIt is a structured process of energy transfer. This concept is called the Energy Cascade. Imagine the flow has many “eddies” (swirls) of different sizes.
- Large Eddies: Energy enters the flow through the largest eddies. The size of these eddies depends on the geometry (e.g., the height of a building). They are anisotropic (their shape depends on direction) and contain most of the flow’s kinetic energy.
- Energy Transfer: These large, energy-filled eddies are unstable. They break apart into smaller and smaller eddies, transferring their energy down the line.
- Small Eddies: This process continues until the eddies are extremely small. At this point, they are isotropic (the same in all directions) and universal. Viscosity (fluid friction) dominates, and the kinetic energy is dissipated (turned into heat). The smallest scale of turbulence is known as the Kolmogorov Scale.
This physical separation of eddy behavior is what allows LES simulation to work. We can directly calculate (resolve) the large, geometry-dependent eddies and only model the small, universal ones.
Figure 1: The energy cascade in turbulence theory, showing how LES resolves large eddies and models small ones.
Fundamental Difference: Spatial Filtering vs. Temporal Averaging
This physical process allows us to choose how to simulate the flow. Direct Numerical Simulation (DNS) is too expensive for industry because it resolves everything, down to the smallest Kolmogorov scale. Therefore, we need a shortcut. The difference between RANS and LES is how they take this shortcut.
- RANS (Reynolds-Averaged Navier-Stokes) uses Temporal Averaging. It averages the flow variables over time. This process removes all turbulent fluctuations and models their total effect, giving a time-averaged result. It cannot capture the unsteady, deterministic motion of large eddies.
- LES (Large Eddy Simulation) uses Spatial Filtering. Instead of averaging over time, it applies a filter in space. You can think of the mesh cell itself as the filter.
- Resolved Scales: Any eddy larger than the filter (the mesh cell) is directly calculated. Its unsteady motion is captured.
- Subgrid Scales (SGS): Any eddy smaller than the filter is removed from the direct calculation and its effect is modeled.
Because LES filters in space and not time, it preserves the time-dependent information of the large eddies. This is why LES can capture unsteady flow phenomena that RANS cannot. We resolve the important, geometry-dependent structures and only model the simple, universal ones.
Figure 2: A diagram comparing the concepts of RANS temporal averaging versus LES spatial filtering in CFD.
Mathematics of Filtering: Filtered N-S Equations
In the last section, we established that Large Eddy Simulation (LES) uses spatial filtering, not temporal averaging, to separate large and small eddies. Now, we will examine how this filtering process mathematically transforms the fundamental equations of fluid motion—the Navier-Stokes equations.
The filtering operation is formally defined by a mathematical filter function, G. However, in most modern CFD simulation software like ANSYS Fluent, we use an implicit filter. This means the grid cell volume itself acts as the filter. The filter’s size, Δ, is simply the size of the mesh cell. This is a critical point: your mesh resolution is not just a numerical setting; it physically defines the boundary between the eddies you resolve and the eddies you model.
When we apply this filtering operation to the incompressible Navier-Stokes equations, we get a new set of equations called the Filtered Navier-Stokes Equations. These equations govern the motion of the large, resolved eddies.
Figure 3: The mathematical derivation of the Filtered Navier-Stokes equations used in LES, showing the creation of the unknown SGS stress term.
As the image shows, the new equations look very similar to the original ones, but they contain an additional, unknown term! This new term is called the Subgrid-Scale (SGS) Stress Tensor, written as τij . It is defined as the difference between the filtered product of velocities and the product of filtered velocities:
This SGS stress tensor is not just a mathematical remainder. It has a vital physical meaning: it represents the effect that the small, filtered-out eddies (the subgrid scales) have on the large, resolved eddies we are calculating. It accounts for the transfer of momentum from the unresolved scales. Because this term depends on the very eddies we chose not to solve for, it is unknown. This creates the central closure problem of LES simulation.
To solve the filtered equations, we must find a way to approximate, or “model,” this unknown stress tensor. The entire purpose of an SGS model (like Smagorinsky, WALE, etc.) is to provide a mathematical formula for this term, connecting it to the resolved flow field we do know.
Figure 4: An energy spectrum graph demonstrating how an SGS model provides eddy viscosity to dissipate energy at the grid cut-off, ensuring numerical stability in an LES simulation.
SGS Stress Tensor & The Closure Problem
In the last section, we applied a spatial filter to the Navier-Stokes equations and found that a new, unknown term appeared: the Subgrid-Scale (SGS) Stress Tensor, . This term represents the physical effect of the small, unresolved eddies on the large, resolved ones we are calculating. Because this term is unknown, we cannot solve the filtered equations directly. This is the central challenge in all LES simulation, known as the “closure problem”.
To solve, or “close,” the equations, we must create a model to approximate the SGS stress tensor. This is the entire purpose of an SGS model. Most SGS models in commercial CFD codes like ANSYS Fluent are based on the Boussinesq hypothesis. This is a simple but powerful idea that assumes the momentum transfer caused by the small eddies behaves similarly to the momentum transfer caused by molecular viscosity. This allows us to define a new, artificial viscosity called the Subgrid-Scale (SGS) Viscosity or Eddy Viscosity (). This eddy viscosity is not a real fluid property; it is a mathematical tool used to model the effect of the unresolved turbulence.
Figure 5: A comparison graph showing the unphysical pile-up of energy at the grid cut-off without an SGS model, versus the correct dissipation with an SGS model.
The physical job of this SGS eddy viscosity is critical. In the real energy cascade, energy is transferred from large eddies to small eddies and is finally dissipated into heat by molecular viscosity at the tiny Kolmogorov scales. In an LES CFD simulation, our mesh is not fine enough to reach the Kolmogorov scales. The SGS model must provide the necessary dissipation at the grid cut-off limit. It must remove the correct amount of energy from the smallest resolved eddies to mimic the effect of the real energy cascade.
If an SGS model is not used, or if it is not strong enough, energy is not removed correctly from the resolved flow. This energy will accumulate, or “pile up,” at the smallest resolved scales (the highest wave numbers), leading to an unphysical solution and often causing the simulation to become unstable and diverge. The different SGS models available in ANSYS Fluent—like Smagorinsky, WALE, and the Dynamic models—are all different mathematical methods for calculating this SGS eddy viscosity.
Figure 6: Airplane LES simulation using WALLE SGS model
Comparison of Subgrid-Scale (SGS) Models in ANSYS Fluent
Now that we understand why a Subgrid-Scale (SGS) model is necessary, the next step is to choose the right one for your CFD simulation. In ANSYS Fluent, several SGS models are available, each with a different approach to calculating the SGS eddy viscosity (). The choice of model involves a trade-off between computational cost and physical accuracy.
The following table compares the most common SGS models available for LES simulation.
| Feature | Smagorinsky-Lilly | WALE (Wall-Adapting Local Eddy-Viscosity) | Dynamic Smagorinsky | Dynamic k-eq Transport |
| Model Type | Algebraic (0-Equation) | Algebraic (0-Equation) | Algebraic (Dynamic Procedure) | 1-Equation Transport |
| How it Works | Calculates from the local strain rate and a fixed user-defined constant (). | Calculates using a formula that correctly handles near-wall and shear regions without a constant. | Calculates the Smagorinsky constant () “dynamically” based on the resolved flow field at two different scales. | Solves a transport equation for the subgrid kinetic energy () to find . |
| Advantages | – Very simple and computationally cheap. – A classic, well-understood model. |
– Computationally cheap. – Automatically handles near-wall damping effects correctly. – Gives zero eddy viscosity in laminar flow. |
– No user-tuning required for the constant. – Adapts to different flow regimes (laminar, turbulent, near-wall). |
– Accounts for the history and transport of subgrid energy (non-equilibrium effects). – The most physically complete model. |
| Disadvantages | – The constant is not universal and needs tuning. – Produces incorrect eddy viscosity in laminar/transitional regions. – Requires ad-hoc wall damping. |
– Does not account for the transport/history of turbulence. Assumes local equilibrium. | – More computationally expensive than static models due to the second filter operation. | – The most computationally expensive option as it solves an extra transport equation. |
| Best For… | Rarely recommended today. Largely replaced by WALE for general use. | A great starting point and often the best choice for general industrial flows with complex geometries. | Flows with transitional regions or where a more accurate local dissipation is needed without the cost of a transport equation. | Highly non-equilibrium flows like combustion or flows with strong transients where the transport of subgrid energy is critical. |
This concludes the theoretical foundation of Large Eddy Simulation. We have journeyed from the physical concept of the energy cascade to the mathematical derivation of the filtered equations and the practical comparison of models used to solve the closure problem. In our next blog, we will move from theory to practice.