The Y Plus Concept: Key to Boundary Layer Analysis

Author: Krishnendu GS

When we think about the sleek design of modern aircraft, our focus often turns to their impressive speed, fuel efficiency, and advanced technology. Yet, there exists an unseen force that significantly influences these attributes: the boundary layer, a thin region in which viscous forces are significant and within which the flow is rotational. This thin layer of air adhering to the aircraft’s surface plays a pivotal role in key aerodynamic factors such as drag, lift, and heat transfer.

The boundary layer theory was introduced by Ludwig Prandtl [1] in 1904, marking a significant breakthrough in fluid mechanics. One of the major accomplishments of boundary layer theory is its capability to predict flow separation in regions with adverse (positive) pressure gradients. Therefore, a comprehensive grasp of the boundary layer concept is indispensable for achieving precise, efficient, and dependable designs. To accurately model and capture the boundary layer behavior, a dimensionless parameter y $^+$ (or yplus) is used in Computational Fluid Dynamics (CFD). A deep understanding of y $^+$ and boundary layer is crucial for engineers and researchers aiming to optimize designs and enhance performance across industries.

In this blog, we will delve deeper into the concept of the boundary layer and y $^+$ , exploring their significance in CFD simulations. Whether you’re a CFD professional or a curious newcomer, this comprehensive guide will provide valuable insights into achieving precise and reliable simulation outcomes.

Boundary layer formation

When a fluid particle contacts a solid surface, its velocity becomes zero; this phenomenon is known as the no-slip condition. The no-slip condition is the primary cause of the formation of a boundary layer in the immediate vicinity of the surface. The fluid particles in contact with the surface slow down the motion of particles in the adjacent fluid layer, causing the velocity of the fluid flow to decrease as it approaches the surface. Starting from 0 m/s at the surface, the velocity increases until it reaches free stream velocity u $_{\infty}$ . The region near the wall where a sharp velocity gradient exists is called the boundary layer (Fig. 2).

Boundary layers can be either laminar, characterized by smooth and orderly layers, or turbulent, marked by chaotic and disordered flow, depending on the Reynolds number. To explore the stages of boundary layer formation, let’s consider a simple example: the boundary layer growth on a stationary flat plate held parallel to a uniform free stream of fluid (Fig. 2).

Figure 1. Boundary layer growth on a flat plate [2]

Due to the no-slip condition, the free-stream velocity u $_{\infty}$ slows down near the surface of the plate, leading to the formation of a boundary layer at the leading edge. Initially, up to a certain length of the plate, the flow remains laminar with a parabolic velocity profile. As the fluid continues to flow downstream, the presence of large shearing stresses and velocity gradients causes the boundary layer thickness $\delta(\text{x})$ to increase, leading to a transition in the flow. Eventually, the flow becomes fully turbulent, characterized by a logarithmic velocity profile.

Let’s have a look into how boundary layer thickness varies:

Boundary layer thickness

Boundary layer thickness refers to the distance from a solid surface to the point in the fluid flow where the flow velocity reaches approximately 99% of the free stream velocity. Paul Richard Heinrich Blasius [3], a PhD student of Prandtl, developed an analytical solution for the shape of a laminar boundary layer over a flat plate with zero pressure gradient. The Blasius solution for boundary layer thickness is:

${\delta(x) \approx 5.0 \frac{x}{Re^{1/2}}$

Due to the unpredictable nature of turbulent flow, accurately determining the boundary layer thickness for a turbulent boundary layer theoretically poses challenges. Therefore, empirical approximations based on experimental results are commonly used. According to the one-seventh power law [2] for the velocity profile, the boundary layer thickness can be estimated as follows:

${\delta(x) \approx 0.16 \frac{x}{Re^{1/7}} }$

Concept of y $^+$

Numerical modelling of near-wall effects is a challenge in CFD modelling, because the boundary layers are the main source of vorticity and turbulence. So, understanding the underlying physics and how to properly resolve it through meshing is imperative to get an accurate result. Using a very fine mesh to resolve the steep gradients in the inner part of the boundary layer remains prohibitively expensive for many industrial CFD simulations. Therefore, accurate near-wall modelling is crucial for most industrial applications. To differentiate the various regions near the wall, the concept of wall distance y $^+$ has been formulated.

${\text{Dimensionless Velocity,} \quad u^+ &= \frac{u}{u_\tau}}$

${\text{Dimensionless Distance,} \quad y^+ &= \frac{y \cdot u_\tau}{\nu}}$

${\text{Friction velocity,} \quad u_\tau &= \sqrt{\frac{2 \tau_w}{\rho}}}$

where, u is the local velocity, $\tau_{\omega}$ is the wall shear stress, $\rho$ is the fluid density, y is the distance from the wall, and $\nu$ is the kinematic viscosity of the fluid.

Understanding the concept of y $^+$ is fundamental for considering the effects of flow near the wall and is crucial in selecting the appropriate turbulence model. Let’s look into “how this concept of y $^+$ will help us and where to use this? Let’s start with the near wall flow profile.

Near Wall Velocity Profile

At the solid surface, the fluid remains stationary, and very close to the wall the behavior of the fluid is dominated by viscous effects. As we know, the behavior of the flow near the wall is a complex phenomenon. Let’s have a look at the different layers of turbulent boundary layer [4,5] near wall (Fig. 3).

Figure2. Turbulent boundary layers on a smooth surface [6]

Viscous/Laminar Sublayer:

This is the layer closest to the solid surface, where viscous effects dominate. In this region, the flow is significantly influenced by molecular viscosity, resulting in an approximately linear velocity profile (u $^+$ = y $^+$ ). The thickness of this viscous sublayer is very small compared to the overall boundary layer thickness.

Buffer Layer:

Above the viscous sublayer is the buffer layer, where the flow transitions from being dominated by viscous effects to being influenced by turbulence. In this region, turbulence starts to develop, causing the velocity profile to deviate from linearity and showing a gradual increase in velocity away from the wall.

Log law region:

In this region, turbulent kinetic energy production and dissipation are nearly equal, i.e. both viscous and turbulent effects are important here. In this inner region, the shear stress is assumed to be constant and equal to the wall shear stress, while gradually varying with the distance from the wall. The relationship between u $^+$ and y $^+$ in this region is given by Log law:

${\textbf{Log law, } {u^+ = \frac{1}{\kappa} \ln(y^+) + B}}$

Where $\kappa$ , and B are constants; their usual values are $\kappa$ = 0.40 to 0.41 and B = 5.0 to 5.5.

This relationship is crucial for predicting and analyzing turbulent flow behavior near walls in various engineering applications. The logarithmic law is one of the most important results for wall-bounded turbulent flows. Fig. 4 confirms this law, showing the various inner layers of a turbulent boundary layer, including the viscous sublayer, buffer layer, and wake region. The viscous sublayer appears as a curve for y $^+$ < 5, where the velocity increases linearly with distance from the wall. In the buffer layer (5 < y $^+$ < 30), where the velocity distribution is neither linear nor logarithmic and the turbulence production reaches its maximum as both viscous and turbulent stresses are significant. The logarithmic layer begins around y $^+$ ≈ 30, where the velocity profile follows a straight line on a log-linear plot, extending across two decades. For y $^+$ > 10⁴, the wake region begins, and the velocity profiles deviate due to differences in Reynolds numbers.

Figure 3. Mean-velocity profile of a smooth-flat-plate turbulent boundary layer [7]

How to model near wall region?

With a finer mesh, we can capture the boundary layer near the wall. But apart from near to the walls, finer cells will increase the mesh sizes. The knowledge of y $^+$ helps to accurately resolve the B.L and place the cells accordingly. The y $^+$ value determines the first cell height for meshing the near-wall region. Using this calculated first cell height, the near-wall area is meshed with a gradual growth. This approach ensures that the effects near the wall are captured accurately without creating an excessively heavy mesh count.

There are two ways to model near wall region [8]: 1) Wall function method, and 2) Resolving viscous sublayer.

Wall function method

Using empirically derived equations to capture the physics in the near wall region.
Typically used for RANS turbulence model like k- model.
To get accurate results, the first cell centre should be in the log-law region.
Since this method avoids the high viscous region (viscous sub-layer and buffer layer), it significantly conserves computational resources by reducing the mesh size.
For high Reynolds number flows, it is easier to place the first cell centre within the log region because of the high y $^+$ value.
For moderate and low Re flows, y $^+$ value is relatively small. So, it is difficult to place the first cell centre within the log region.
One of the challenges during meshing using the wall function method is, that it is difficult to place the first cell center in the log-law region. If the cell center either falls below or goes above the log region, then in both cases wall function still assumes a logarithmic profile which is incorrect. So, this method limits the grid resolution of moderate and low Re flows.

Resolving viscous sublayer

It offers the most reliable and consistent wall treatment when using viscous sublayer models like k- $\omega$ SST turbulence model.
The first cell center should be in the viscous sublayer, so we should use a y $^+$ value below 5. To place enough number of cells inside the boundary layer, the recommended value of y $^+$ is less than 1.
The lower y $^+$ value leads to a very fine mesh near the wall, which increases the grid size and computation time.
To get an accurate result place a minimum of 10 cells inside the boundary layer (even more for utmost precision).

To wrap up, understanding the concept of y $^+$ is pivotal for accurate boundary layer analysis in computational fluid dynamics. As we’ve explored, appropriate y $^+$ values are crucial for capturing the complexities of turbulent flows and achieving precise boundary layer predictions. In practice, adhering to the recommended y $^+$ range can significantly enhance the accuracy of CFD models, providing valuable insights for engineers and researchers alike. Integrating this concept into your computational practices will enable you to better navigate the challenges of boundary layer analysis and advance the quality of your fluid dynamics studies. As the field of CFD continues to evolve, staying attuned to such fundamental principles will remain essential for achieving excellence in simulation and analysis.

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References:

[1] Prandtl L. Verhandlungen des dritten internationalen Mathematiker-Kongresses. Heidelberg, Leipeizig. 1904:484-91.

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

[3] Hager, W. H. (2003). Blasius: A life in research and education. Experiments in Fluids, 34, 566-571.

[4] Versteeg, H. K. (2007). An introduction to computational fluid dynamics the finite volume method, 2/E. Pearson Education India.

[5] Pope, S. B. (2001). Turbulent flows. Measurement Science and Technology, 12(11), 2020-2021.

[6] Kadivar, M., Tormey, D., & McGranaghan, G. (2021). A review on turbulent flow over rough surfaces: Fundamentals and theories. International Journal of Thermofluids, 10, 100077.

[7] Kundu, P. K., Cohen, I. M., & Dowling, D. R. (2015). Fluid mechanics. Academic press.

[8] Menter, F. R. (2011). Turbulence modeling for engineering flows. Ansys, Inc.

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