# Computation of y+

$y^{+}$ is the dimensionless wall distance for a wall-bounded flow. It can be expressed as:

$y^{+}=\frac{u_{*}y}{\nu}$

where $u_{*}$ is the friction velocity $u_{*} \equiv \sqrt \frac{\tau_{w}}{\rho}$, y is the distance to the nearest wall and $\nu$ is the kinematic viscosity of the fluid. $y^{+}$ is commonly used in boundary layer theory and in defining the law of the wall.

$y^{+}$ plays a relevant role in the treatment of the boundary layer. The subdivision of the near-wall region in a turbulent boundary layer can be summarized as follows:

• $y^{+} < 5$: viscous sublayer region (velocity prole is assumed to be laminar and viscous stress dominates the wall shear).
• $5 < y^{+} < 30$: buffer region (both viscous and turbulent shear dominate)
• $30 < y^{+} < 300$: fully turbulent portion or log-law region (turbulent shear predominates)

Figure 1: Turbulence near the wall.

Figure 2: Law of the wall, horizontal velocity near the wall with mixing length model.

From figure 1, to use a wall function approach for a particular turbulence model with confidence, it is necessary to ensure that the $y^{+}$ values are within a certain range see Flow Over A Flat Plate example for the wall function use.

OpenFOAM presents a utility to calculate and report $y^{+}$ for all the wall patches. Note that as it depends on the local Reynolds number, it can only be obtained in the post-processing. Consequently, only approximate values can be estimated when meshing. When a RAS turbulence model is used, the instruction (written in the terminal) to obtain y+ is:

yPlusRAS

Notes:

• In addition to the concern about having a mesh with y+ values that are too large, it is necessary to be aware that if the y+ is too low then the first calculation point will be placed in the viscous sublayer region and the wall function will also be outside its validity see Flow Over A Flat Plate example for the wall function use.
• If an attached flow is modeled, then generally a wall function approach can be used. If flow separation is expected and the accurate prediction of the separation point will have an impact on the results then it would be advisable to resolve the boundary layer with a finer mesh.