The CFD Pre-Processing – Part 4

One of the lasts procedures of the CFD pre-processing is about the boundary layer mesh. This is critical, because in a race car there are several components, which if the mesh is generated without a proper concern on the boundary layer height, the results of the simulations would be incorrect. This article presents some methods and configurations for the boundary layer mesh.

Near-wall treatment

Since the flow simulated in race car CFD analysis is the turbulent one, it is important to understand the effects of walls on these. Actually, walls are the main sources of mean vortices and turbulence in a flow. The near-wall treatment is proposed, because it is closed to walls where there are the strong gradients, fluxes, momentum and transport of the flow. There are two approaches commonly used to model the flow in the near wall region, the Low Reynolds Number (LRN) and the Wall Function (WF).

Low Reynolds Number

Basically, it solves directly the turbulent boundary layer, which means that v⁺ (shear speed) is equal to the convective length scale δ_V. LRN resolves the details of the boundary layer profile, which are thin inflation layers. It requires a low Reynolds grid with the distance of the first node above the wall y⁺(1) = 1. This method is critical for flows with an adverse pressure gradient, which is what characterize flow separation. LRN is solved by Navier-Stokes equations (NSE).

Wall Function

This method uses empirical formulas to impose specified conditions near the wall without resolving the boundary layer. It requires that the distance of the first node above the wall lies into the Log-law region, which is a fully turbulent one. In this method the macroscopic effects of the laminar and the buffer layers give a solution in terms of the shear velocity from this point to above.

LRN and WF differences

The difference between these methods is that LRN solves exactly all and completely the boundary layer, while WF solves equations from above the log-law region since a simplification of a laminar and buffer layer is made. These methods are important, because when it is requested to solve all the boundary layer, it is required to create many small cells to start from y⁺ = 1. Another approach is to create wall functions that start from log-law region. These approaches are different in terms of accuracy. WF predict a much worse separation of the flow.

The inflation layers

FIGURE 1 – y⁺ is a reference layer which is defined as y⁺ = y/δ_V, a non-dimensional distance from the wall. The distance normal to the wall with respect to the convective length scale which refers to the viscosity and the shear velocity related to the local shear.

The boundary layer mesh is required, because there is a relative velocity between walls and the external flow. In cases of automotive computational fluid dynamics (CFD) simulations, the car is standstill and the air is moving, thus V_car = 0 and U_∞ = 50 m/s (common value). The relative velocity between the car and the air will be the physical boundary layer. The physical boundary layer is defined as the area where there are very strong gradients running through the wall and gradients on the surface. In the first case, the gradients that are running through the wall are very strong and concentrated close to the surface. The boundary layer is what is guiding the flow in terms of separation, re-attachment and acceleration, thus the accuracy required for the boundary layer is very high. Hence, the boundary layer mesh is designed to match the physical boundary layer. For this, some assumptions are required. The main assumption is that the boundary layer is fully turbulent. Hence it is adopted the RANS model, which is the main hypothesis for fully turbulent flows. There is no transition from laminar to turbulent flows. Although there are models that can be added, the Shear Stress Transport (SST) and K-ε models are always accounted for turbulent conditions (Figure 2).

FIGURE 2

The boundary layer accounted is the Log-Law layer. These approaches are related to the turbulence model, because in some cases it is possible to choose which of these are better, even though there are cases which only one model is suitable. K-ε and K-ω are based in the turbulent kinetic energy equations. ω is defined also in the volume at the wall, while ε is undefined at the wall, infinite. When using K-ε it is not possible to choose between these two methods. Actually, it should be carefully selected the mesh to not include regions which are out of the turbulent zone. This is the log-law layer, which the mesh is a way to find a solution for this one. Hence, when creating a boundary layer mesh, it is should be carefully done to not fall inside the separation region.

Strategies for inflation layers determination

FIGURE 3

A good process is the one that, at the first run y⁺ is defined at the estimation of the boundary layer, then the simulation is performed. After this, the control of the aero map of the car is done according to y⁺ at each area of the car. This is done to guarantee that y⁺ is always out of the separation region, thus the target is y⁺ 50. If the solution presents y⁺ between 15 and 20, this simulation is not well modelled. It is strictly recommended to not accept y⁺ lower than 30 (Figure 3). However, if a simulation results in some parts of the car with y⁺ equal to 30 and the turbulent model is K-ε, then the mesh should be re-generated to have a correct first layer. For cases which y⁺ = 1, no Wall Function and there are wide areas which y⁺ is different than 1, the model is not valid. When the turbulence model is SST – K-ω, all kinds of solutions are available. In this case, the solver detects y⁺ values and adopts the correct approach, LRN when y⁺ is 1 and WF for cases which y⁺ is higher than 1. The major concern is to avoid wide areas of y⁺ about 15 or 20, which is strictly not recommended.

FIGURE 4

Actually, the height of the first layer is constrained by y⁺, since the near-wall approach is adopted. y_wall⁺ depends on the method used, for LRN it is below 5 while for WF is between 30 and 80. The total height of the prism cells should cover the inner region (δ_IR⁺) of the physical boundary layer in order to match with the numerical one. It is important to ensure that on the boundary layer cells, in case of agglomeration, the volume uniformity in the transition between these two types of cells need to be preserved as much as possible. In other words, the volumes of the first and the last prims should be approximately the same.

Sizing the entire boundary layer mesh

First step is to define which y⁺ is desired, it is possible to translate y⁺ into a non-dimensional height (y⁺ = y/δ_V), which is the one of the elements added. y⁺ is basically the solution, thus it is valid when it is referred to the solution. However, the solution is at the center of the cell and the volume mesh generator asks for the height of the layer. In other words, the y⁺ target is at the center of the cell, while the height of the first layer (h₀) is two times the estimation based on y⁺. The grow rate (GR) factor is related to the height of the previous and the next cells. First, the height of the first cell (h₀) is obtained, then GR is used to adjust the height of the next cell, which in this case is:

h₁ = GR¹∙h₀

Usually GR goes from 1.1 and 1.3, which the first layer is added to GR at a power of the number of the layer times h₀.

h_n = GRⁿ∙h₀

This number is chosen according to the boundary layer height. However, there are some strategies to define the first layer h₀, these are the first aspect ratio (FAR), the last aspect ratio (LAR) and the first height (FH) methods.

First aspect ratio

In this method, FAR is the input not only for h₀, but also for the surface mesh. The results are based on the surface mesh. Hence, if these change 1 – 5 mm, a difference will be generated in h₀.

FAR = s₀/h₀ → h₀ = s₀/FAR

GR = h_i+1/h_i

FAR constraints are:

FAR = FAR(h₀)

h_tot = Σ_i=0^N-1GRⁱ∙h₀ ; h_tot ≥ H_tot-INPUT = H_tot-INPUT(y_IR⁺)

The software, that in this case is the Fluent, asks for FAR, GR and N, which this one is the number of layers to cover the total boundary layer height.

FIGURE 5 – The problem with FAR method is that the boundary layer mesh is squeezed, but the physical boundary layer is not well resolved.

Graphically, it is no so attractive (Figure 5), but it is a very robust method to accomplish complex geometries. For instance, in sections which are far away from the other, but suddenly these get closed by another surface. This method helps the mesh to accomplish a squeeze of the boundary layer mesh without going into the volume mesh. However, this method is not recommended for sections which their total height decrease, while their boundary layer height increases. For instance, in trailing edge of wing profiles (Figure 5).

Last aspect ratio

This method consists in the definition of the last layer height through the last aspect ratio LAR and the surface discretization (s₀), together with the first height and the number of layers. GR varies according to the area.

LAR = s₀/h_N → h_N = s₀/LAR

The constraints are:

h₀ = h₀(y⁺_wall)

h_N = GR∙^N-1h₀

h_tot = Σ_i=0^N-1GRⁱ∙h₀ ; h_tot ≥ H_tot-INPUT = H_tot-INPUT(y_IR⁺)

Another characteristic of this method is that it preserve the y⁺ uniformity of the first height and improves the prism-tetra volume transition. The total height still remains a function of the surface mesh discretization (s₀).

First height

The first height (FH) method is the best one. Its inputs are h₀, GR and N and the first one is hold fixed. Actually, this is the target method. However, sometimes it is even better to use different methods where they are more suitable.

h₀ = h₀(y_wall⁺)

GR = h_i+1/h_i

The constraints are summarized below:

h_N = GR^N-1∙h₀

h_tot = Σ_k=0^N-1 GRⁱ∙h₀ ; h_tot ≥ H_tot-INPUT = H_tot-INPUT(y_IR⁺)

However, this method is less robust than FAR one, because it is not suitable for proximities areas, especially for complex geometries.

FIGURE 6 – First height method holds h₀ in order to provide an uniform mesh.

FH fixes the first layer height h₀ accordingly to the y_wall⁺ requirements, while the total height h_tot is based on the surface discretization (Figure 6). An interesting approach is to apply FH on particular PID and FAR on the remaining sections of the watertight model.

References

• This article was based on the lecture notes written by the author during the Industrial Aerodynamics course taken at Dallara Academy;
• ANSYS Fluent Theory Guide 12.