wallfunction

A wallfunction is a semi-empirical relation used in computational fluid dynamics to model the near-wall region of turbulent flows without resolving the viscous sublayer. It provides a link between velocity, distance from the wall, and wall shear stress, delivering a boundary condition that replaces the need for very fine near-wall grids in many simulations.

In the typical formulation, dimensionless wall variables are used: y+ = ρ uτ y / μ and u+ = u / uτ, where uτ is the friction velocity. The classical log-law of the wall expresses u+ as a function of y+ via u+ = (1/κ) ln(y+) + B, with κ ≈ 0.41 and B ≈ 5.0, valid in the logarithmic layer for many turbulent boundary layers. Wallfunctions allow the calculation of the wall shear stress from the velocity at the first near-wall cell center, avoiding the need to resolve the viscous sublayer directly.

There are several variants. Standard (equilibrium) wall functions assume a fully turbulent, attached boundary layer and relate wall shear stress to the near-wall velocity through the log-law. Two-layer or scalable wall functions partition the near-wall region into an inner (viscous/sub-layer) part and an outer (log-law) part, blending the solutions to better accommodate different y+ ranges and roughness effects. Scalable wall functions are designed to remain valid over a wide range of mesh densities and Reynolds numbers, reducing sensitivity to grid spacing.

Limitations include reduced accuracy in flows with strong adverse pressure gradients, flow separation, reattachment, high-pressure or transitional effects, and complex roughness. Wallfunctions are most appropriate when the near-wall region is not fully resolved and the flow remains broadly turbulent and attached.