Turbulence does tricky things near walls. Boundary layers and laminar sub-layers compact interesting flow patterns into a very small space. Small it may be, but experience proved we cannot ignore it. The boundary layer forms on the body, which is our object of interest, arguably the most critical region. Turbulence is most critical near the wall, and we need to consider near wall effects.
Near wall effects remain synonymous with another phrase: “law of the wall”. The law of the wall showed that boundary layer of any wall divides into three regions, based on the distance from the wall. (Figure 2‑1) (Notice the logarithmic scale for the X-axis in the figure.)
Right next to the wall, we see a laminar sub-layer. Notice how the data points curve upwards, matching an exponential pattern. In this layer, turbulent flow damps out and all flow returns to laminar.
Going farther out, we return to fully turbulent flow. You can identify this on the graph when the data points fit to a straight line (as plotted on a logarithmic X-axis). Moving farther out, the patterns break down. This indicates that we exited the boundary layer and transitioned into the main flow stream. The exact thickness of these different layers depends on the fluid involved and the flow velocities; that is why you see them represented as non-dimensional numbers in Figure 2‑1.
Non-dimensional numbers play an important role in near wall effects, and the most important number is Y+. Y+ considers the CFD mesh. This number is the non-dimensional height of the first cell on the wall. Equation 1 shows the definition for Y+.
Technically, you can extend the definition to measure any non-dimensional distance away from the wall. But unless explicitly stated otherwise, assume that Y+ references the height of the first cell above the wall.
As part of the quality control, the CFD engineer plots out Y+ on the body (Figure 3‑1 shows an example). Y+ varies, due to changes in mesh size and local flow velocities. Notice that some parts of the body have higher Y+ than others. These variations matter because the CFD engineer seeks a target Y+ value, and practically every part of the body should achieve that target value.
What value to target for Y+? It depends on the wall treatment method for your turbulence model. Remember the law of the wall; CFD developers generally approach the laminar sub-layer with two possible solutions:
A wall damping function assumes you defined a very fine mesh that resolved the entire boundary layer, with cells going all the way down into the laminar sub-layer. In this case, the damping function reduces turbulence model near the wall, based on the Y+ distance.
Alternatively, a wall function assumes the entire laminar sub-layer was contained within the thickness of the first cell on the wall. Rather than damping out the turbulence model, it integrates the effect of the laminar sub-layer and applies that effect within that first cell only. All the cells remain fully turbulent.
The meshing strategy for the CFD engineer depends on the selected wall function. The mesh height near the wall must be set to match the selected wall treatment. And we measure mesh height by Y+. Table 4‑1 shows recommended values for Y+ settings. The table shows the Y+ value for the typical break between laminar sub-layer and turbulent boundary layer. But most CFD engineers prefer to include a safety margin and ensure their entire body remains meshed with the correct settings. The third column in the table shows recommended settings with those safety margins.
Table 4‑1: Recommended Values for Y+ Ranges
| Y+ Range | Recommended Setting (with safety margin) |
Wall damping function | 12-30 | 2-5 |
Wall function | 30+ | 40 – 80 |
The boundary layer may be small, but it features as the most critical part of turbulence. Near wall effects determine how turbulence interacts with the body, our object of interest. Accurate modeling of these small regions is critical. The CFD engineer needs to understand the law of the wall, the options for wall treatment of turbulence, and how that effects the meshing strategy. Small details make a big difference.
[1] | Q. Wang, C. Yan and T. Hui, “Mechanism Design for Aircraft Morphing Wing,” Research Gate, <https://www.researchgate.net/figure/y-plus-value-for-the-CFD-model-10-degree-of-extension-angle-of-attack-6_fig4_268478784>, Accessed: 2019, Jan, 01, April 2012. |
[2] | V. R. Raj, “Quadratic Profile Used in QUICK Scheme,” Wikimedia Commons, 12 Nov 2012. . Available: https://commons.wikimedia.org/wiki/File:Quadratic_profile.jpg. . |
[3] | Max Pixel, “Cumulus Storm Turbulence Thunderstorm Cloud Roller,” Max Pixel, 01 Jan 2019. . Available: https://www.maxpixel.net/Cumulus-Storm-Turbulence-Thunderstorm-Cloud-Roller-567678. . |
[4] | S. Wasserman, “Choosing the Right Turbulence Model for Your CFD Simulation,” Engineering.com, 22 Nov 2016. . Available: https://www.engineering.com/DesignSoftware/DesignSoftwareArticles/ArticleID/13743/Choosing-the-Right-Turbulence-Model-for-Your-CFD-Simulation.aspx. . |
[5] | S. Tao, F. Yuqing, L. Graeme and J. Kaixi, “CFD simulation of bubble recirculation regimes in an internal loop airlift reactor,” in 27th International Mineral Processing Congress, <https://www.researchgate.net/publication/289001212_CFD_simulation_of_bubble_recirculation_regimes_in_an_internal_loop_airlift_reactor>, Accessed: 2019, Jan, 01., January 2014. |