COMSOL 6.3 - The Algebraic yPlus Turbulence Model

The algebraic yPlus turbulence model is an algebraic turbulence model based on the distance to the nearest wall. The model is based on Prandtl’s mixing-length theory and is suitable for internal flows. It is less mesh sensitive than transport-equation models like Spalart–Allmaras or the k-ε model. In what follows, let y be the coordinate normal to the wall, and U the velocity parallel to the wall. Using a mixing length formulation, the balance for the shear stress in the wall layer may be approximated as

which, expressed on dimensionless form, reduces to

where y+ = yuτ/ν, u+ = U/uτ and

is the friction velocity. Close to the wall, the mixing length must be zero such that u+ = y+, and far away from the wall,

such that u+ = (1/κ)log(y+) + B. To obtain the correct behavior, the mixing length is chosen to be

where y* is to be determined. Inserting Equation 3-77 into Equation 3-76

Integrating Equation 3-78

For large values of y+, Equation 3-79 reduces to

Comparing the above expression with the logarithmic law of the wall

the value of the constant y* is obtained as

The dimensionless effective viscosity can be evaluated from

This requires the local value of y+, which is obtained from the Reynolds number

The nonlinear algebraic Equation 3-81 has to be solved at each node point to evaluate the effective viscosity from Equation 3-80. The local Reynolds number Re = Uy/ν is formed with the local absolute value of the velocity and the distance to the nearest wall. This implicitly assumes that the main flow direction is parallel to the wall.

The wall distance, y is provided by a mathematical Wall Distance interface that is included when using the Algebraic yPlus model. The solution to the wall distance equation is controlled using the parameter lref. The distance to objects larger than lref is represented accurately, while objects smaller than lref are effectively diminished by appearing to be farther away than they actually are. This is a desirable feature in turbulence modeling since small objects would get too large an impact on the solution if the wall distance were measured exactly.

The most convenient way to handle the wall distance variable is to solve for it in a separate study step. A Wall Distance Initialization study type is provided for this purpose and should be added before the actual Stationary or Time Dependent study step.

The Algebraic yPlus turbulence model is consistent with a no slip boundary condition, that is u = 0. Since the turbulence model is algebraic, no additional boundary condition is needed. This boundary conditions is applied for equal to .

The low-Reynolds-number formulation of the Algebraic yPlus model can be considered to be well resolved at a wall when the wall resolution in viscous units

where the resolution height is based on the distance between the first computational node inside the domain and the wall. uτ is the friction velocity which is calculated from the wall shear-stress τw,

and is available as the postprocessing variable

(u_tauWall). Also, the boundary variable ,

(lplus_cc), is available to ensure that the mesh is fine enough.

is the distance, measured in viscous units, from the wall to the center of the wall adjacent cell, thus

and according to Equation 3-82

should be about 0.5. Observe that very small values of

can reduce the convergence rate.

The postprocessing variable ,

(Delta_wPlus), is available.

The default boundary treatment for Algebraic yPlus is the wall treatment. The automatic wall treatment assumes that there is a small gap, δw = hw/2, between the computational domain and the physical wall. Here, hw is the height of the mesh cell adjacent to the wall (see Figure 3-7).

The boundary conditions for the velocity is a no-penetration condition u ⋅ n = 0 and a shear stress condition

The postprocessing variable for friction velocity uτ is based on the effective traction at the wall and is available as uτw (u_tauWall). The wall traction using uτw is available as postprocessing variable T_trac. The automatic wall treatment tends to the low-Reynolds-number formulation when hw tends to zero, and it becomes a wall function formulation when the resolution in viscous units

increases. The resolution in viscous units Δw+ is available as a postprocessing variable, Delta_wPlus. The resolution height is based on the distance of the first computational node (which is on the wall) from the imaginary lift-off wall.