Author: Krishnendu G S

Introduction

Wall Y Plus (y $^+$ ) is a non-dimensional distance used in computational fluid dynamics (CFD) to determine the grid resolution near walls. Accurate y $^+$ calculation is crucial for capturing boundary layer effects and ensuring correct turbulence modelling.

Figure1: Boundary layer growth on a flat plate [1]

This guide outlines the steps to calculate y $^+$ for both internal and external flows.

Concept of Y Plus (y $^+$ )

Accurately modeling near-wall effects in CFD is challenging because boundary layers are the primary source of vorticity and turbulence. Understanding the underlying physics and applying proper meshing techniques is essential for reliable results. For a detailed explanation, refer to the blog “The Y Plus Concept: Key to Boundary Layer Analysis.”

A finer mesh allows for better resolution of the boundary layer near the wall. However, refining the mesh everywhere significantly increases computational cost. This is where the y+ value becomes crucial, it helps determine the appropriate first cell height for meshing the near-wall region. To accurately position the first cell height near the wall, we rely on the y $^+$ value, which is calculated using the following formula:

$y^+ = \frac{y \cdot u_\tau \cdot \rho}{\mu}$

where, y is the first cell thickness, $u_\tau$ is the friction velocity, $\rho$ is the fluid density, and $\mu$ is the dynamic viscosity of the fluid.

Steps to Calculate the 1st cell thickness

Step 1: Identify the Flow Type

Internal Flow: Flow inside pipes, ducts, or enclosed geometries.
External Flow: Flow over objects such as airfoils, vehicles, and flat plates.

Step 2: Required Inputs

Characteristic length or Hydraulic diameter (m)
Free-stream velocity (m/s)
Density (kg/m³)
Dynamic viscosity (kg/ms)
Desired y $^+$ value

Internal Flow Calculation

Step 3.1: Calculate Reynolds Number (Re)

$\text{Reynolds Number,} \quad Re = \frac{\rho \cdot U \cdot D_h}{\mu}$

where $D_h$ is the hydraulic diameter

Hydraulic Diameter ( $D_h$ ) Calculation

Circular Duct,

$\text{Hydraulic Diameter,} \quad D_h=D$

where D is the diameter of the circular duct

Rectangular Duct,

$\text{Hydraulic Diameter,} \quad D_h=\frac{2WH}{W+H}$

where W and H are the Width and Height of the rectangular duct

Square Duct,

$\text{Hydraulic Diameter,} \quad D_h=L$

where L is the side length of the duct

Step 3.2: Estimate Wall shar stress ( $\tau_w$ )

$\tau_w=\frac{\rho \times U^2 \times f}{8}$

where, f is the Darcy friction factor

Darcy friction factor(f) Calculation

According to Blasius approximations for smooth pipes (4000 < Re < 10^5) [2]

$f=0.316 \times \text{Re}^{-1/4}}$

Step 3.3: Compute Friction Velocity ( $u_\tau$ )

$\text{Friction velocity,} \quad u_\tau = \sqrt{\frac{ \tau_w}{\rho}}$

Step 3.4: Calculate First Cell Thickness (y)

$\text{First cell thickness,} \quad y= \frac{y^+ \cdot \mu }{\rho \cdot u_\tau}$

4. External Flow Calculation

Step 4.1: Calculate Reynolds Number

$\text{Reynolds Number,} \quad Re = \frac{\rho \cdot U \cdot L}{\mu}$

where L is the characteristic length

Step 4.2: Compute Wall Shear Stress ( $\tau_w$ )

$\tau_w=\frac{1}{2}\times\rho\times U^2\times C_f$

Where $C_f$ is the skin friction coefficient

Skin Friction Coefficient ( $C_f$ ) Calculation

Using the Schlichting skin-friction correlation for Re $<10^9$ [3]

$C_f = [2\text{log}_{10}(\text{Re})-0.65]^{-2.3}$

Step 4.3: Determine Friction Velocity ( $u_\tau$ )

$\text{Friction velocity,} \quad u_\tau = \sqrt{\frac{\tau_w}{\rho}}$

Step 4.5: Calculate First Cell Thickness (y)

$\text{First cell thickness,} \quad y= \frac{y^+ \cdot \mu }{\rho \cdot u_\tau}$

References

[1] Cengel, Y., & Cimbala, J. (2013). Ebook: Fluid mechanics fundamentals and applications (si units). McGraw Hill.

[2] Blasius, H. (1913). Das aehnlichkeitsgesetz bei reibungsvorgängen in flüssigkeiten. In Mitteilungen über Forschungsarbeiten auf dem Gebiete des Ingenieurwesens: insbesondere aus den Laboratorien der technischen Hochschulen (pp. 1-41). Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-662-02239-9_1

[3] Schlichting, Hermann (1979), Boundary Layer Theory, ISBN 0-07-055334-3, 7th Edition.

[4] White, F. M. (2017). Fluid mechanics 8th edition.

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Introduction

Concept of Y Plus (y)

Steps to Calculate the 1st cell thickness

Step 1: Identify the Flow Type

Step 2: Required Inputs

Internal Flow Calculation

Step 3.1: Calculate Reynolds Number (Re)

Hydraulic Diameter () Calculation

Step 3.2: Estimate Wall shar stress ()

Darcy friction factor(f) Calculation

Step 3.3: Compute Friction Velocity ()

Step 3.4: Calculate First Cell Thickness (y)

4. External Flow Calculation

Step 4.1: Calculate Reynolds Number

Step 4.2: Compute Wall Shear Stress ()

Skin Friction Coefficient () Calculation

Step 4.3: Determine Friction Velocity ()

Step 4.5: Calculate First Cell Thickness (y)

References

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