If you are somewhat familiar with CFD simulations, you will most likely have come across the term y+. This dimensionless term is one of the most critical aspects of designing a mesh for your CFD simulation, particularly for wall bounded turbulent flows. This can make or break your CFD solver’s ability to accurately predict the behavior of a turbulent boundary layer, which in turn dictates prediction of surface forces such as lift and drag, flow separation, and how turbulence is generated, transported, and dissipated. In this blog post, we will describe what is y+, how does it physically relate to a turbulent boundary layer, how it can be calculated, and what are some recommended values of surface y+ that should be used for different turbulence models.
Definitions Of y+ And u+
y+: This is the nondimensional distance away from the wall (normal to the wall), defined as
where,
y is the dimensional distance away from the wall
ν is the kinematic viscosity of the fluid
u+: This is the nondimensional velocity parallel to the wall, defined as
where,
, the friction velocity
u is the fluid velocity
τw is the wall shear stress
ρ is the fluid density
Turbulent Boundary Layer And The Law Of The Wall
To understand the physical (and numerical) meaning of y+, we will first review the behavior of a turbulent boundary layer. As an example, consider the flow over a flat plate at a Reynolds number higher than the critical Reynolds number, such that the flow is fully turbulent. As long as the boundary layer is fully turbulent and the flow is attached, all turbulent boundary layers follow a universal behavior, described by the u+(nondimensional velocity) vs y+ (nondimensional distance from the wall) profile. This profile is shown below (red curve denotes the turbulent boundary layer profile, whereas blue curves denote the viscous sublayer and the log-law layer curves).
The turbulent boundary layer can be divided into four regions:
Viscous sublayer: This is the region of the turbulent boundary layer closest to the wall, up to y+ < 5, and is given by the following equation:
u+ = y+
Viscous shear stress dominates overs turbulent shear stress, and the flow is always laminar within this region.
Log law layer: This region exists between 30 < y+ < 300, and the relationship between u+ and y+ is given by the following equation:
Where k and C are constants, with widely reported values of k ~ 0.4 and C ~ 0.5 for a smooth wall, determined from experimental data of turbulent boundary layers. This is known as the law of the wall.
Buffer layer: This is the region between the viscous sublayer and the log-law region, typically existing between 5 < y+ < 30. Here neither relationship holds, such that u+ is not equal to y+
Outer layer: Beyond y+ > 300, the effects of the freestream become important, and the turbulent boundary layer loses its universality. This region is known as the outer layer.
y+ For A CFD Mesh
In terms of a CFD mesh, the y+ value refers to the distance of the first mesh point (or height of the nearest computational cell/element) from the wall. This is also often referred to as surface y+ or Dy+. Note that in case of a cell-centered finite volume scheme, the half-height of the first cell is used, as flow values are calculated at the cell center.
This surface y+ value is an indication of how well your mesh resolves the turbulent boundary layer.
Low surface y+ values (y+ < 1)
When y+ < 1, the first cell lies within the viscous sublayer, and this results in a ‘wall-resolved’ simulation. This means that there are enough computational cells/elements to fully resolve the turbulent boundary layer, including the viscous sublayer, the buffer layer, and the log-law layer. For simulations requiring high fidelity near the wall, such as Direct Numerical Simulations (DNS), wall-resolved Large Eddy Simulations (LES), or even Reynolds-Averaged Navier-Stokes (RANS) with flow separation, it’s essential and the norm to maintain y+ values below 1.
Moderate surface y+ values (5 < y+ < 30)
The first cell lies in the buffer layer. This is usually not recommended, however, some CFD solvers still provide an accurate solution if the first cell lies within the buffer layer, depending on special wall function treatment.
High surface y+ values (y+ > 30):
The first cell is in the log-law region of the boundary layer, where turbulence is fully developed. In this case, wall functions are often used to approximate the behavior of the near-wall region (viscous sublayer and buffer layer). Using a mesh with surface y+ > 30 is computationally less expensive as fewer computational cells are needed, and a high level of accuracy is still retained using wall functions. However, this is only valid if the flow is fully attached. Wall functions provide an inaccurate prediction when flow separation occurs, and y+ < 1 should be used in such scenarios.
y+ Calculation And Estimation
Most CFD solvers automatically provide the surface y+ value as a postprocessing output, and that can help you decide if your mesh is able to accurately resolve the turbulent boundary layer. However, you can easily estimate the required size of your computational cell near the walls, for a targeted surface y+ value for your CFD simulation and design your initial mesh accordingly.
As an example, let’s consider designing a mesh for flow over an airfoil. Even though airfoils have curved surfaces, it can be approximated as a flow over a flat plate for y+ estimation.
The following steps can be followed to estimate the height of your near wall computational cell for a targeted surface y+ value:
- Calculate the Reynolds’s number,
- Estimate the skin friction coefficient,
(From Prandtl’s one-seventh-power law for a turbulent flat plate) - Calculate the wall shear stress,
- Calculate friction velocity,
- Calculate the height of near wall cell for a targeted surface y+,
For example, if your target surface y+ = 0.8 for a wall-resolved mesh, simply set y+ = 0.8 in the above equation to calculate the height of the near wall computational cell.
A typical schematic of the near wall boundary layer mesh on a flat plate is shown below:
Final Thoughts
y+ is a fundamental aspect of CFD that influences the accuracy of your simulations, particularly in wall-bounded turbulent flows. It is crucial to understand y+ and how it interacts with your turbulence models and mesh, based on which a decision can be made for selecting the correct y+ value for the current fluid flow scenario. This ensures that the complex physics of turbulence and boundary layers are accurately captured in your simulations, ultimately leading to accurate predictions.
If you need help designing an accurate CFD mesh, performing CFD analysis for turbulent flows, or have a fluids product that needs to be analyzed, don’t hesitate to get in touch with our expert team!
