How Azore Incorporates Wall treatment for Any Mesh
In Azore, we’ve built in wall treatments that do not burden the modeler with unnecessary mesh topology and length scale requirements.
By Dr. Jeff Franklin, P.E.
Wall treatment is an important piece of turbulence modeling because it sets the stage for how turbulence is generated and dissipated. The latest release of Azore includes the k-omega SST turbulence model. Along with this update, Azore’s wall function implementation smoothly accommodates variations in near wall mesh length scales of Y+ (y plus) < 1 up through the fully turbulent Y+ length scales.
Applying a turbulence model to an industrial application requires that the solver use an empirical relationship to correctly capture the influence of walls on the overall flow structures. When wall treatments are implemented in a solver algorithm, we need to ensure that they do not overburden the modeler with unnecessary mesh topology or length scale requirements. Azore’s wall treatment is designed to be mesh-insensitive, so that it is effective for many different mesh sizes near the wall.
Wall treatment in Azore is defined by two main empirical relationships, the near wall shear stress and the near wall heat flux.
Near Wall Shear Stress
A common way of characterizing the flow structure near a wall is through a plot of U+ versus Y+.
The horizontal axis is Y+, which is a dimensionless distance analogous to a Reynolds number near to the wall. This value represents a cell’s proximity to the wall, and is defined by the product of density (ρ), shear velocity (Uτ), and wall distance (Y) divided by the viscosity (μ). The vertical axis is U+, which is a second dimensionless ratio of the velocity away from the wall (U) to the wall shear velocity (Uτ).
For laminar flow (at low values of Y+), we observe that Y+ equals U+, a relationship we can use very reliably. For turbulent flow, it’s not nearly that simple, but many have found that logarithmic relationships like the one shown for high Y+ values can characterize the flow structure out away from the wall. These two empirical relationships span the near and far values of Y+ but we also want to span the entirety of the Y+ range, since the two empirical relationships on their own result in a departure from experimental data in the transition zone. Azore uses blending functions to capture the curvature and provide continuity between the two regions.
The ultimate goal for the CFD solver through these empirical relationships and blending functions is to properly characterize the shear stress in the area near the wall.
Near Wall Heat Transfer
Even more empirical relationships characterize the heat flux near to the wall. Many material properties change with temperature, so when heat transfer is taking place, those relationships get tricky near to the wall. Viscosity, for example, changes with temperature, and viscosity will no longer be constant if you have heat transfer taking place. If the heat flux is high, then your viscosity could be changing very rapidly, and which cell contains the appropriate value to use? The value at the wall, or in the center of the cell? It also may become very dependent on the mesh size in that location. Azore is designed so that variable fluid properties will scale well over large temperature ranges. The basic form of the wall heat flux q is governed by the following relationship.
Here the YT+ and T* values represent a combination of blending functions. YT+ represents a non-dimensional blending between the thermal boundary layers and is connected to the turbulent kinetic energy in the near neighbor cell. T* allows a blending to take place for the temperature-dependent fluid properties.
These mathematical relationships describe the broad picture of how wall treatment is performed in Azore, dictating how shear stress and heat flux are handled in those very first cells adjacent to a wall in your model. Azore uses a mesh-insensitive treatment of the wall. The methods have been intentionally chosen so that the solver gives a reasonable approximation regardless of the size of that first cell.
