Wall shear stress for turbulence models

Shear stress on a resolved wall

The wall viscous stress tensor on a resolved wall in turbulent flows is defined as follows:

where:

• μ is the specified dynamic viscosity of the fluid.
• μ_t is the turbulent viscosity calculated from the specified turbulence model.
• k is the turbulent kinetic energy.
• δ_ij is the Kronecker delta function defined as:

The viscous stress vector applied on the wall boundary is then obtained as:

where n_j is the unit vector normal to the wall boundary.

The shear stress vector is calculated by removing the normal component from the viscous stress vector as follow:

Shear stress with applied wall functions

When wall functions are applied, the flow solver computes the magnitude of the wall shear stress τ_w = |τ_w,i| from the wall function formulation. It is assumed that the wall shear stress vector is aligned with the tangential component of the velocity at the centroid of the wall-adjacent control volume. The tangential component of the velocity is given by:

where is the vector of the relative velocity at the centroid of the wall-adjacent control volume. The flow solver computes the relative velocity with respect to the wall.

The wall shear stress vector is then calculated as follows:

Wall functions provide the relation between U⁺ = |U⁺_t,i| and y⁺ at centroids of wall-adjacent control volumes. Since both U and y at the centroid are available from the solution of the conservation equations and the model geometry, respectively, the only unknown left is the wall shear velocity, U^* The shear velocity, which is directly related to the magnitude of the wall shear stress vector, τ_w, via , is used to convert U and y to corresponding dimensionless forms of U⁺ and y⁺.

Due to the complexity of wall function relations, an iterative method is required to solve for the wall shear stress.

The calculation of τ_w depends on the type of the wall function: standard wall function or hybrid wall function.

Standard wall functions

When you use the standard wall function, instead of an iterative method, the flow solver uses the turbulent kinetic energy, k, which is computed from the solution of turbulence conservation equations, to calculate the wall shear velocity as follows:

The shear velocity is used to convert the wall distance, y, to dimensionless form y⁺. U⁺ is then computed by plugging y⁺ into the wall function relation and converted to U using the shear velocity. The wall shear stress is calculated from the following relation:

The calculation of the shear velocity from the turbulent kinetic energy makes this procedure more sensitive to y⁺ values and less accurate when y⁺ ≪ 1 in comparison to the iterative procedure of hybrid wall functions. At separation points, the wall shear stress and y⁺ diminish. The shear velocity, according to the definition, should follow the shear stress and vanish. However, the turbulent kinetic energy coming from the solution of turbulence conservation equations maintains non-zero values at separation locations above the wall. Any non-zero turbulent kinetic energy leads to a finite shear velocity when standard wall function are used. The flow solver reverts back to an iterative procedure to calculate the wall shear stress when the employed turbulence model involves no equation for the turbulent kinetic energy. This is the case for the mixing length turbulence model.

Hybrid wall functions

When hybrid wall functions are employed, the iterative method described by Knopp et al. [36] is used in the flow solver to calculate the magnitude of the wall shear velocity and the wall shear stress.