# Y+ Boundary Layer Thickness

### Key Takeaways

• Y+ is a dimensionless parameter that is a measure of distance from the first grid cell to the surface.

•  The Y+ value determines the accuracy of the boundary layer thickness prediction.

•  A Y+ value between 1 and 30 is considered appropriate for the simulation. Let's consider a common example of fluid flowing over a flat plate. Near the surface of the plate, the fluid slows down due to friction and creates a thin layer called the boundary layer. The thickness of the boundary layer is calculated as the distance between the surface to the point where the fluid flow velocity is equivalent to 99% free stream velocity. The estimation of this boundary layer is important because of its ability to support the prediction of drag and lift, understanding of fluid-structure interactions, and turbulence modeling for fluid-structure design and optimization.

In computational fluid dynamics (CFD), the estimation of boundary layer thickness is made easier with the parameter known as the Y+ value. Y+ boundary layer thickness is useful for enhancing the accuracy and efficiency of the computation for fluid simulation.

In this article, we will dive into the relationship between the Y+ value and boundary layer thickness in fluid system analysis.

## Understanding the Y+ Value

The Y+ value is a dimensionless parameter that represents the distance from the first grid cell to the surface wall. In CFD, the Y+ value is an important parameter for determining the accuracy of the boundary layer thickness. Mathematically, the Y+ value can be calculated as:

### Formula for Y+ value

Here, u_τ is the friction velocity, y is the wall distance, and μ is the kinematic viscosity of the fluid.

The friction velocity can be calculated in terms of wall shear stress τ_w:

### Formula for friction velocity

(Alt text: Formula for friction velocity)

From the above equation, the equation for wall shear stress can be derived into:

### Formula for wall shear stress

The above equation is expressed in terms of the coefficient of skin friction (C_f) and free stream velocity (U_f).

As the value for Y+ in boundary layer thickness analysis is derived from the above equations, there are the following points to note:

• If Y+ <1, the first grid cell is located within the laminar sublayer.
• If Y+ > 30, the first grid cell is located within the viscous sublayer.
• If 1< Y+< 30, the first grid cell is located within the viscous sublayer but not too close to the surface wall.

Too high or too low a  Y+ value can lead to incorrect boundary layer predictions.

 Effects of Different Y+ Values in Simulation Accuracy Too high Y+ value Too low Y+ value When Y+ value is greater than 30The first grid cell is placed outside the viscous sublayer, far from the solid surfaceLack of accuracy when capturing the flow behavior in the boundary layer during simulation When Y+ value is less than 1The first grid cell is placed too close to the solid surface and is located within the laminar sublayerLack of accuracy in simulation of turbulent behavior near the boundary layer

## Calculating Y+ in Boundary Layer Thickness Analysis

The Y+ value needs to be calculated for each cell near the surface to ensure the appropriate placement of the first cell in the boundary layer. In CFD simulation, the Y+ value calculation involves the following steps:

 Y+ Value Calculation in CFD Simulation Define flow conditions Identify fluid parameters like velocity, density, and viscosity. Turbulence modeling Choose an appropriate turbulence model (k-ε, k-⍵, etc. ) based on: Reynolds number of the flow Flow regime Accuracy requirements Define the domain Define the geometry of the problem in the computational domain.Ensure that the mesh is fine enough to capture all complex flow behavior near the boundary layer.Ensure that the initial boundary condition is properly defined. Solve the governing equation Solve the governing equation associated with the fluid, such as the Navier-Stokes equation to solve for velocity, pressure, temperature, etc. at the flow field. Calculate the influencing parameters Calculate wall shear stress and friction velocity (using the equations mentioned above). Calculate the Y+ value Calculate the Y+ value for each cell near the surface.Ensure that the value falls within the desired range.

For a given Y+ value, the grid cell size can be computed to accurately capture boundary layer behavior in CFD simulation.

### Calculate Y+ Boundary Layer Thickness

The boundary layer thickness can be calculated as the sum of displacement thickness and momentum thickness.

### Boundary layer thickness formula

For the calculated boundary layer thickness, check that the Y+ value lies within the range, i.e., between 1 to 30 for turbulent flows.

## Ensure Accuracy of Results With CFD Tools

The right CFD tool will simplify the process of domain creation, turbulence modeling, meshing, and solving governing equations for a given flow field. The ability to visualize and characterize the flow behavior near the solid boundary and interaction with the fluid allows engineers to analyze the parameters such as wall shear stress and friction velocity. The result can be used for calculating Y+ and boundary layer thickness. The ability to choose the appropriate Y+ value for each simulation enhances the level of accuracy and reliability of the CFD simulation.