# Y+ Grid Spacing Calculator

The Y+ Grid Spacing Calculator allows to estimate the required size of the first mesh cell near the wall in order to reach the correct y+ value.

##### Input Values

Ref. Length L

Ref. velocity U∞

Kinematic viscosity ν

Desired Y+

##### Output Values

Reynolds number Re

First Mesh cell size Δs

#### How to Use

Fill in the required parameters in the orange area. The parameters are unit-agnostic, check that all the values are using the same system of units. The required input parameters are reference length (e.g. a pipe’s diameter or a vehicle’s length), a reference free-stream velocity, kinematic viscosity of the fluid and desired y+ value. Output values are shown in the green area.

#### Background

In order to calculate the required first mesh cell’s size, the following parameters are required:

• L: reference length (e.g. a pipe’s diameter or a vehicle’s length);
• U∞: free-stream velocity;
• ν: fluid’s kinematic viscosity;
• y+: desired y+ value.

Kinematic viscosity can also be found from the following formula: ν = μ / ρ where ρ is the density, while μ is the dynamic viscosity of the fluid.
A useful list of kinematic viscosity values for common fluids can be found here.
The desired values of y+ are generally in the range of 1 to 300, according to the type of turbulence model used. As a rule of thumb, y+ values in the order of 1 should be chosen to resolve the sub-layer (e.g. required for high accuracy in force prediction), while values above 10 should be used for log-law treatment.
The calculation of the first mesh cell size near the wall is based on flat-plate boundary layer theory as described in Frank M. White’s Fluid Mechanics 5th edition.

The first step is the calculation of the Reynolds number:

(Eq. 2)

By using the definition of the kinematic viscosity in Eq.2:

(Eq. 3)
The second step is the calculation of an empirical constant Cf, function of Re:

(Eq. 4)
The shear stress rate at wall τw is then estimated according to the following equation:

(Eq. 5)
Knowing the shear stress rate, it is then possible to calculate the frictional velocity Uf:

(Eq. 6)

Finally, the first mesh cell size near wall Δs is then calculated:

(Eq. 7)

Introducing Eq.4, Eq.5 and Eq.6 into Eq.7, it is possible to calculate the first mesh cell size near wall as function of the dynamic viscosity with the following equation:

(Eq. 8)

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