Wall Function: Boundary Layer Modelling Approach

When the flow approaches a solid surface, a steep gradient will form due to the no-slip condition at the surface. This is known as Boundary Layer that develops near the solid surface in contact with the flow. The thickness of this boundary layer affects the wall heat flux and wall shear stress. In order to estimate wall heat flux & wall shear stress accurately, we need to resolve the boundary layer accurately. To achieve that, we need many thin cells normal to the wall that will resolve this gradient.  

Reference: ResearchGate

In the Second-Order Finite Volume Method, the variation of any flow quantity across the cell is linear. 

To resolve the gradient near the wall, we need a thin layer of cells normal to the wall. When this gradient is resolved accurately, then the wall heat flux and wall shear stress would be estimated correctly.

Fluid mechanics 101: Fundamental Course 2

Flow with a high Reynolds number or with the increase in Reynolds number, the boundary layer gets thinner, making the gradient at the wall steeper, so we need more thin cells to resolve the gradient.

Sometimes it is not possible to mesh these many thin cells near the wall due to limited computational resources.

That being the case, having large cells near the wall leads to wrong gradient estimation. Wall heat flux and wall shear stress would be wrong as they are the function of velocity and temperature gradient respectively.

Mostly accurate estimation of wall shear stress and wall heat flux is important to engineers. In order to predict wall shear stress accurately, the product of kinematic viscosity and velocity gradient must be correct.

If the mesh is fine enough near the wall, then the wall shear stress would be correct. If the mesh is not fine, then the velocity gradient would be wrong and thus the wall shear stress. 

Now the mesh is not fine, so we can’t have the correct velocity gradient, but if somehow we can modify the kinematic viscosity such that the product of kinematic viscosity and velocity gradient is correct, then we can have the correct value of the wall shear stress.

Modifying the kinematic viscosity is the foundation of near-wall treatment or wall function approach.

Wall Function Approach

In order to correct Wall shear stress (if the cell near the wall is big) or to modify the kinematic viscosity such that wall shear stress is correct, we need to know two things :-

How the wall shear stress is computed in a CFD code. 

What is the actual value of wall shear stress.

By equating both the wall shear stress computed by a CFD code and the actual value of wall shear stress, we can get the modification required in the kinematic viscosity.

Wall shear stress computed by a CFD Code

Calculating Actual Wall Shear Stress

We know how the wall shear stress is computed by a CFD code. Now we need to calculate the actual wall shear stress. Actual wall shear stress can be computed by looking at experimental data known as the Universal Law of the wall.  

It is called universal because flow very close to the wall is laminar and looks similar. When we write a flow parameter in the non dimensional form, that profile holds universal regardless of the speed of the flow or shape of the body. Very close to the wall-flow profile always looks similar.

The graph shown above is the actual velocity profile written in its non-dimensional form that we are trying to model and it is non-linear. This is the real variation of velocity normal to the wall, which was extracted from experimental measurements of fully developed turbulent flow between two parallel plates.

U+ & Y+

Empirical Formula for Viscous-Sub Layer

Empirical Formula for Log Layer

Deriving Wall Function

Summary

This is the wall function for kinematic viscosity. The same process can be used to obtain the wall function for other parameters such as thermal diffusivity, turbulent kinetic energy etc.

When the mesh is coarse near the wall or the centroid of the first cell is in the log layer(y+ > 11.067), then the viscosity in the cell adjacent to the wall has to be modified using the above expression so that the incorrect velocity gradient & modified kinematic viscosity will give the correct wall shear stress.

To use the wall function approach, the cell center adjacent to the wall must lie within the logarithmic region of the boundary layer.

Even if the first cell is placed properly in the logarithmic region, the overall boundary layer resolution must be achieved by having enough cells in the boundary layer.

The thickness of the logarithmic layer depends on the Re number of the flow!

For high Re number, the log layer extends to many thousands of y+ value, easy to place the first cell in the log layer and have enough cells to capture overall boundary layer resolution.

For moderate and low Re numbers, the log layer is very thin. The upper limit can be lower than y+ ~ 150.  It is not easy to place the first cell center into the log layer & have enough cells to capture the overall boundary layer resolution.

References

Fluid Mechanics 101 : Link & Fundamental Course 2

NPTEL: Advance Concepts in fluid mechanics Week-5

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