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

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:

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:

https://drive.google.com/file/d/1rtvROyksCT_Fj0OyhyjnBTx5guu7Pcoe/view?usp=share_link 

(Alt text: Formula for friction velocity)

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

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. 

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:

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.

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.

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