Read time: 10 minutes          

Target audience: CFD Researchers/ Automobile Engineers/ Thermal-Fluid Industry/ Aero Industry

Written by: Dr. Tabish Wahidi

1. Background:

As CFD has become an indispensable tool in engineering and scientific fields, used to simulate and analyse fluid flow, heat transfer, and other related phenomena. One of the critical aspects of CFD simulations is accurately representing the fluid behaviour near solid surfaces, known as the “wall region.” To achieve this, two key concepts are frequently discussed: Y+ and wall functions. Understanding both is essential for ensuring the accuracy of turbulence modeling in CFD, particularly in simulations of high-Reynolds-number flows where turbulent boundary layers play a significant role.

This article covers the introduction to Y+ & Wall Functions, and its approaches in CFD simulation to handle CFD simulation in turbulence zone.

2. Introduction:

The Y+ value is a dimensionless measure commonly used to assess the resolution of a volumetric mesh in relation to the flow pattern. It plays a crucial role in turbulence modeling, helping to determine the appropriate size of mesh cells near no-slip walls.

Two key points to note from the definition of Y+ are:

– It is only relevant for turbulent flow.

– It applies specifically to the mesh cells that are directly adjacent to a no-slip wall.

The Y⁺ value is defined as:

Where:

– y+ is the distance of the first grid point from the wall (in the wall normal direction),

– Uτ is the friction velocity (representing the velocity scale near the wall),

– ν is the kinematic viscosity of the fluid.

Wall function is a mathematical model that approximates the behaviour of a turbulent boundary layer near a wall, without requiring an extremely fine grid resolution in that area. Wall functions are typically used when the mesh is too coarse to directly capture the viscous sublayer, especially in high-Reynolds-number flows and The wall functions are based on the empirical date.

3. Y+ and its Implications for Grid Resolution:

The Y+ value plays a crucial role in determining the grid resolution near the wall. In CFD simulations, the region near the wall, known as the boundary layer, can exhibit highly complex behaviour due to the large velocity gradients and turbulence effects. To accurately capture these effects, a high grid resolution is required. However, achieving fine enough resolution in the near-wall region without increasing computational cost excessively can be a challenge.

Low Y+ (Y+ < 1): In this case, the first grid point lies within the viscous sublayer of the turbulent boundary layer. For accurate simulations, the grid must be fine enough to resolve the velocity gradients directly in this region. Directly resolving the viscous sublayer requires a very fine grid, which can significantly increase computational cost.

Moderate Y+ (Y+ ≈ 30 to 300): A Y+ value in this range typically lies within the buffer region of the boundary layer. This region is transitional between the viscous sublayer and the fully turbulent region. The grid resolution is usually coarse enough to resolve the turbulent flow while avoiding excessive computational cost. Most turbulence models like k-ε or k-ω can be applied effectively in this range.

High Y+ (Y+ > 300): In this case, the first grid point lies outside the viscous sublayer and within the fully turbulent region. For high Y+ values, the wall functions (discussed later) are often employed to approximate the behaviour near the wall without requiring an extremely fine grid.

The Y+ value, therefore, helps determine whether it is necessary to resolve the near-wall turbulence structures directly or if the use of wall functions is a more efficient approach.

4. Purpose of Wall Functions:

The goal of wall functions is to avoid resolving the entire viscous sublayer, which is computationally expensive, while still accurately capturing the flow behaviour in the turbulent boundary layer. Wall functions provide a way to bridge the gap between the wall (where the velocity is zero) and the fully turbulent region of the flow. Essentially, they help approximate the velocity profile near the wall based on assumptions about the nature of the turbulence and the wall’s influence.

Wall functions are typically used in turbulence models, such as the k-ε or k-ω models, to avoid the need for resolving the viscous sublayer. They work by imposing a logarithmic profile for the velocity near the wall (known as the log-law), based on the assumption that the flow near the wall behaves in a predictable manner. This profile is derived from empirical observations and theoretical analyses of turbulent boundary layers.

Velocity Profile and Log-Law: In regions of the flow where the Y+ value is high (typically greater than 30), wall functions are used to model the velocity profile, which follows a logarithmic relationship in the turbulent region. The logarithmic law for velocity near the wall can be expressed as:

Where:

– u⁺ is the dimensionless velocity,

– κ is the von Kármán constant (typically around 0.41),

– y⁺ is the dimensionless wall distance,

– C is a constant that depends on the flow conditions.

a. Key Characteristics of Wall Functions:

Logarithmic Velocity Profile: Wall functions assume that the velocity in the turbulent boundary layer follows a logarithmic pattern as a function of the distance from the wall. This is based on the observation that in turbulent flows, the velocity profile near the wall behaves in a predictable logarithmic manner for high Reynolds numbers.

Use in Turbulent Flow: Wall functions are specifically designed for turbulent flow, especially in situations where the turbulence is fully developed.

Grid Resolution: Wall functions are applied when the mesh is not fine enough to resolve the viscous sublayer. They enable larger grid spacing near the wall, reducing computational effort while still providing an accurate approximation of the wall region flow.

Types of Wall Functions: There are standard wall functions and enhanced versions. Enhanced wall functions provide more accuracy in cases where the near-wall turbulence is not fully developed.

b. Types of Wall Functions:

There are two main types of wall functions, depending on the flow conditions:

Standard Wall Functions: These are typically used for high-Reynolds-number turbulent flows where the first grid point lies outside the viscous sublayer. The standard wall function assumes a fully developed turbulent boundary layer and is relatively simple to implement.

Enhanced Wall Functions: These are designed to handle more complex flows with greater accuracy, especially in cases where the turbulence near the wall is not fully developed or where the flow conditions differ significantly from typical high-Reynolds-number turbulence. Enhanced wall functions can improve accuracy in capturing near-wall flow behaviour without the need for excessive grid refinement.

c. Wall Functions and Grid Resolution:

Wall functions are used in simulations where the grid is not fine enough to resolve the viscous sublayer directly. When Y+ values are high (above 30), the first grid point is placed in the logarithmic region of the boundary layer, where wall functions are applied. For low Y+ values, it is often more accurate to resolve the wall region directly by placing the first grid point within the viscous sublayer, and wall functions are not necessary.

5. Conclusion:

In conclusion, Y+ and wall functions are essential tools in CFD for dealing with turbulence near solid boundaries. Y+ provides a way to assess the resolution of the grid in the wall-normal direction, determining whether wall functions are necessary. Wall functions, in turn, allow for accurate and efficient simulations of high-Reynolds-number turbulent flows without the need to resolve every detail of the boundary layer. Balancing the need for accuracy and computational efficiency, especially in wall-bounded flows, requires a careful understanding of these two concepts and their interplay.