Wall-Modeled Large Eddy Simulation I

Overview  

As we mentioned in the last edition (Hybrid RANS/LES), one of the main limitations of Large Eddy Simulation is high-level scaling with Reynolds number (especially in wall-bounded flows). This is mainly because - as the name implies - LES is really good at resolving large-scale structures, however, there are no large scales in the near wall region. Instead, you find very tiny structures that get dissipated into heat by viscosity. One of the most inspiring quotes I like in this context is the one from the English physicist Lewis Richardson:

"Big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity." — Lewis Fry Richardson

To overcome this limitation, researchers have developed Wall-Modeled Large Eddy Simulation (WMLES), which combines the benefits of LES with a wall model to accurately predict the near-wall flow while reducing the computational cost. The wall model allows for a coarser mesh in the near-wall region, while still capturing the important features of the flow. Several wall models have been proposed and calibrated for WMLES, such as the logarithmic law of the wall, the one-equation Spalart-Allmaras model, and the two-equation k-omega model. These models provide a set of boundary conditions that relate the flow variables at the wall to the flow variables in the adjacent computational cells.

WMLES has been successfully applied to a wide range of flow problems, including turbulent boundary layers, pipe flows, and aircraft aerodynamics. It has been shown to provide accurate predictions of flow properties such as skin friction and heat transfer, while significantly reducing the computational cost compared to fully resolved LES [1-3].

In this edition, we will explore together the key concepts and techniques used in WMLES, its potential and limitations, and some real-world applications. We will also discuss recent advancements in WMLES and its future directions. Finally, I will showcase a case study on the use of algebraic WMLES (my personal experience), which has led me to develop the Zeta-F model later on.

First things first, what do we really mean by "Wall-Modeled"?

In Wall-Modeled Large Eddy Simulation, the term "wall-modeled" refers to the fact that the near-wall flow is modeled using a wall model, rather than being resolved explicitly in the computational domain. This is because the near-wall region, which is defined as the region of the flow within a few viscous length scales from the solid surface, is usually characterized by high-velocity gradients and turbulent structures that are too small to be captured by the conventional meshes.

Instead, a wall model is used to represent the flow in the near-wall region by providing a set of boundary conditions that relate the velocity, pressure, and other variables at the wall to the flow variables in the adjacent computational cells. This wall model is calibrated using experimental data or high-fidelity simulations (DNS or wall-resolved LES) both of which explicitly resolve the near-wall region.

Key Concepts

To properly use WMLES, there are a couple of things we need to keep in mind:

• Near-wall modeling: In WMLES, the near-wall region is modeled using a wall model, which provides a set of boundary conditions that relate the flow variables at the wall to the flow variables in the adjacent computational cells. The goal of the wall model is to capture the important features of the near-wall flow, such as the wall shear stress and the velocity profiles, while reducing the computational cost compared to fully resolved LES.

• Turbulence modeling: Just like the hybrid RANS/LES, WMLES is a turbulence model-based approach, which means that it requires a turbulence model to close the equations. There are various turbulence models that can be used in WMLES, such as zero equation algebraic models, ODE and PDE models. The choice of turbulence model depends on the specific application and the desired accuracy. It is important to note that the turbulence model is only used to model the unresolved scales, while the resolved scales are directly computed from the governing equations.

• Mesh generation: WMLES requires a relatively high-quality mesh compared to hybrid RANS/LES (the DES-like approach) that can capture the large-scale turbulent structures while modeling the near-wall region with a coarser mesh. Typically, structured or unstructured meshes are used, and the mesh near the wall is refined to capture the near-wall flow. Obviously, the choice of mesh depends on the specific application and the desired accuracy. BUT be careful, generating a mesh here can be quite tricky; you don't want to fall back on WRLES nor a RANS behaviour either. Please have a look on these sources for the best practices that should be followed [4-8].

• Discretization schemes: WMLES uses numerical methods to discretize the governing equations. The most commonly used numerical methods for WMLES are finite volume methods and spectral methods. Here again, the choice of advection method (divergence scheme) depends on the specific application and the desired accuracy - some examples can be found here [8]

• Validation and verification: It cannot be stressed enough that any WMLES model requires validation and verification to ensure its accuracy and reliability. Validation involves comparing the WMLES results with experimental data or DNS, while verification involves assessing the convergence and consistency of the numerical method on the mesh employed. I incorporated some good examples of V&V in this regard into my thesis (e.g. see chapter of Channel Flow) Sayed, M. A. 2022.

The Algebraic Wall-Modeled LES (AWMLES)

In 2008, (Shur et al., 2008) proposed a WMLES model in which most of the turbulent structures are resolved except in the innermost part of the near wall-region. The key parameter in this model was the modified sub-grid length scale, whose dependence on the grid size was extended to include the wall-distance as well. In addition, the model was shown to resolve the known log-layer mismatch problem (also known as the grey-zone problem) which is common in most WMLES models. This was achieved through blending two formulations for computing the turbulent eddy viscosity , between which the solver switches according to the computational cell size and wall normal distance. From wall-modeling view point, the model covers the inner part of the boundary layer in RANS mode using Prandtl Van-Driest mixing length. Since the model’s eddy viscosity is calculated algebraically, it is referred to this model as the Algebraic Wall-Modeled LES (AWMLES). For more details and how to implement the model please have a read on the paper of Shur et al., 2008 - a brilliant one!

AWMLES Case Study: Turbulent Channel Flw

In 2020, I conducted an extensive analysis on the performance of AMLES (which implicitely models the wall through the Prandtl-mixing length zero-equation model) vs the ER-HRL (which explicitely activates a 3-equation RANS Mode in the near wall region). The outcome of the study was that AWMLES is really good at revealing more turbulent structures in moderate shear Reynolds numbers (below Re=1000) commpared to the ER-HRL. This was referred to the relatively higher eddy viscosity values generated by the ER-HRL model which damps out small structures in the wall proximity (see Fig. 3 below)

Here you need to be very careful when comparing the two models; especially that each has it's own area of usage (where the model peroforms best). For example, if you're looking into finer flow details than just first and second moment statistics then AWMLES or one of its derivatives would be more suitable. On the other hand if you're happy with the mean flow for a 1D-dominant flow then something like the ER-HRL will result in a huge CPU saving. For a better understanding of this, have a look at Sayed et al., 2022.

When comparing models for performance assessment or validation, I must point out: mean flow and TKE are NOT enough to look at! this is because you might have an overprediction in one of the velocity fluctuation components and a considerable underprediction in another (See Fig. 4). For this, the RMS values should always be analysed! In this study for example, it was found that TKE is overpredicted by the AWMLES while the wall-normal RMS is missing by almost 50% in the near wall region.

Summary

We talked about the concept of Wall-Modeled LES (WMLES) which involves modeling the near-wall region of the flow using a RANS mode, while applying an LES model in the bulk of the flow.

We specifically talked about the Algebraic WMLES (AWMLES) which is particularly popular for its efficiency and easiness of implementation. The AWMLES we refer to here uses a Prandtl-mixing length zero equation model in the near-wall region, which allows for more accurate predictions of the wall shear stress and wall-normal velocity fluctuations. This approach has been shown to give a relatively good resolution for the flow features. Nonetheless, it still suffers from the Log-Layer Mismatch (LLM) issue - for which it needs extra-empricism in high shear Reynolds numbers and complex geometries.

Overall, WMLES with the algebraic approach has the potential to improve our understanding of turbulent flows (especially with the exponential rise in CPU and GPU power), and provide more accurate predictions for a variety of engineering applications.

That's a wrap for now... see you next week! 😊

References

1. Sayed, M. A., Dehbi, A., Niceno, B., & Mikityuk, K. On the prediction of turbulent kinetic energy in channel flow using wall-modeled large eddy simulations. In: AIAA 2020-1329. 2020. https://doi.org/https://doi.org/10.2514/6.2020-1329.
2. Sayed, M. A., Dehbi, A., Niceno, B., & Mikityuk, K. (2021-b). Particle subgrid scale modeling in hubrid rans/les of turbulent channel flow at low-to-moderate reynolds number. Powder Technology. https://doi.org/https://doi.org/10.1016/j.powtec.2021.11.057.
3. Sayed, M. A., Dehbi, A., Niceno, B., Mikityuk, K., & Krinner, M. (2021-a). Flow simulation of gas cyclone separator at high reynolds number using the elliptic-relaxation hybrid les/rans (er-hrl) model, proceedings of the 6th world congress on momentum, heat and mass transfer (mhmt’21). https://doi.org/10.11159/icmfht21.lx.110
4. Kawai, S., and Larsson, J., 2012, “Wall-Modeling in Large Eddy Simulation: Length Scales, Grid Resolution, and Accuracy,” Phys. Fluids, p. 015105.
5. Larsson, J., Laurence, S., Bermejo-Moreno, I., Bodart, J., Karl, S., and Vicquelin, R., 2015, “Incipient Thermal Choking and Stable Shock-Train Formation in the Heat-Release Region of a Scramjet Combustor. Part II: Large Eddy Simulations,” Combustion and Flame, pp. 907–920.
6. Lee, J., Cho, M., and Choi, H., 2013, “Large Eddy Simulations of Turbulent Channel and Boundary Layer Flows at High Reynolds Number with Mean Wall Shear Stress Boundary Condition,” Physics of Fluids, p. 110808.
7. Menter, F. R. (2015). Best practice: scale-resolving simulations in ansys cfd, ver. 2.00.
8. Sayed, M. A., On the Capability of Wall-Modeled Large Eddy Simulations to Predict Particle Dispersion in Complex Turbulent Flows, PhD Thesis 2022, EPFL.

Golden reads 💎

• MKM, Computation of Turbulent Flows Over a Backward-Facing Step, NASA Technical Memorandum,1983.
• Kim, S. W., A Near-Wall Turbulence Model and Its Application to Fully Developed Turbulent Channel and Pipe Flows, Institute for Computational Mechanics in Propulsion, NASA Technical Memorandum, 1988.  

‍