When the flow approaches solid surface, steep gradient would 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. 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 wall that will resolve this gradient.
In 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 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.
Flow with high Reynolds number or with increase in Reynolds number boundary layer get thinner making the gradient at the wall more 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 resource.
That be the case, having large cells near to the wall lead 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 the 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. When the mesh is not fine then the velocity gradient would be wrong and hence 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 correct value of 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.
Equating both, the wall shear stress computed by a CFD code and the actual value of wall shear stress we can get the modification require 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 Universal Law of the wall.
It is called universal because flow very close to wall is laminar and looks similar. When we write 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.
Graph shown above is actual velocity proﬁle written in it’s non-dimensional form that we are trying to model and are non-linear. This is the real variation of velocity normal to the wall, which were extracted from experimental measurements of fully developed turbulent ﬂow between 2 parallel plates.
U+ & Y+
Empirical Formula for Viscous-Sub Layer
Empirical Formula for Log Layer
Deriving Wall Function
This is the wall function for kinematic viscosity. 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 first cell is in log layer(y+ > 11.067), then viscosity in the cell adjacent to the wall has to be modified using above expression so that incorrect velocity gradient & modified kinematic viscosity will gives correct wall shear stress.
To use wall function approach, cell center adjacent to wall must lies within the logarithmic region of the boundary layer.
Even if first cell is placed properly in logarithmic region, the overall boundary layer resolution must be achieved by having enough number of cells in 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 first cell in log layer and have enough cell 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. Not easy to place first cell center into log-layer & have enough cell to capture overall boundary layer resolution.