Wall Distance and y+

The y+ calculation concept Is very useful for determining the minimum inflation layer thickness around the wall to accommodate the boundary layer.

Mathematically, y+ defined as:

(3.1)

Where u,tau is the friction velocity around the wall, y is the closest distance to the wall, and  is the shear stress. The mathematical form above is not an equation but an equivalence, as the use of the concept y+ is merely an initial estimate that can have different definitions depending on the method used by the researcher because it’s a non-dimensional parameter.

The calculation is commonly performed using an online calculator by inputting the desired y+ value, free stream velocity, Reynolds number, as well as the density and viscosity of the fluid, to obtain the initial inflation thickness in the mesh we will use.

It’s important to note that flows with high Reynolds numbers, such as those around high-speed aircraft or projectiles, will have very thin boundary layers. Therefore, the use of inflation becomes less significant and can be disregarded.

Another important aspect to note is that y+ doesn’t always have to perfectly capture the boundary layer. In turbulent modeling, there are features called wall functions that will be specifically discussed in the turbulent modeling chapter.

Since y+ value is typically complex to estimate, you can use an online calculator such as this to estimate it quickly: tensorcalculators.com

In that online calculator, we  use simple estimation from flat-plate boundary layer approximation:

With:

Re = Reynold number

U = Free stream velocity (m/s)

Cf = Friction coefficient

s = wall spacing (m)

You can use the s value as the first layer thickness of your mesh to obtain your desired calculation accuracy and speed.

The equation for Cf is an approximation for Re < 1E+9 (From reference: Schlichting, Hermann (1979), Boundary Layer Theory, ISBN 0-07-055334-3, 7th Edition), values beyond this range must be carefully evaluated. The Cf equations are widely available with different conditions and Reynold numbers.

3.3.1. Boundary Layer Resolution:

For accurate boundary layer resolution, the grid spacing should be such that y+ falls within a specific range. This ensures that the near-wall region is resolved correctly and that the turbulence model used in the simulation operates within its valid range.

3.3.2. Typical y+ Ranges:

• Laminar Flow: In laminar boundary layers, the first cell’s y+ should ideally be very small (often y+ < 1).
• Turbulent Flow: For turbulence modeling, different turbulence models require different y+ ranges:
  • Standard Wall Functions: Typically, y+ should be between 30 and 300.
  • Low-Reynolds Number Models: Typically, y+ should be less than 5 to 10, with the grid sufficiently fine to resolve the viscous sub-layer.

3.3.3. Wall Functions:

• High y+: If y+ is high (e.g., greater than 30), wall functions are often used to estimate the near-wall behavior without resolving the viscous sub-layer explicitly.
• Low y+: If y+ is low (e.g., less than 5), the grid must be sufficiently fine to resolve the near-wall region accurately, and the turbulence model can be applied more directly.