Boundary treatment algorithms for meshfree RANS turbulence modeling

TL;DR

This work addresses the challenge of wall treatment in meshfree RANS turbulence modeling at high Reynolds numbers. It introduces three wall-function strategies for meshfree collocation—closest neighbor (CN), nearest-band neighbor (NBN), and shifted boundary (SB)—and evaluates them with SA, $k$-$ ext{ε}$, and $k$-$ ext{ω}$ models on 1D Couette flow, flow over a flat plate, and flow around a NACA0012 wing. The results show CN performs poorly due to nonuniform wall-function coverage, while NBN improves accuracy with careful selection height and LB strategies; SB provides the most robust performance, allowing coarser grids and substantial computational savings despite higher per-boundary costs. Overall, SB emerges as the most promising approach for robust, accurate, high-Re wall-bounded simulations with meshfree collocation, enabling efficient industrial-scale applications and laying groundwork for future investigations into adverse pressure gradients and flow separation.

Abstract

In this paper, we propose improved wall-treatment strategies for meshfree methods applied to turbulent flows. The goal is to improve wall-function handling in simulations of high-Reynolds-number turbulent flows, and to better understand the performance of turbulence models when used with meshfree methods. While wall-function techniques are well established for mesh-based methods, their inclusion in meshfree methods faces unique challenges that have not been fully explored. The main difficulties arise from the lack of connectivity between points and from point movement in Lagrangian frameworks, which can complicate consistent wall treatment. To address these issues, we explore three wall-treatment techniques. We highlight the drawbacks of the standard closest neighbor approach. We then introduce two novel approaches: the nearest-band neighbor method and the shifted boundary method. We evaluate these methods using first-order turbulence closures: Spalart--Allmaras, $k-\varepsilon$, and $k-ω$. These methods are tested numerically on 1D Couette flow, turbulent flow over a flat plate, and flow around a NACA 0012 airfoil in 3D. The results show that both novel methods outperform the closest neighbor approach. The shifted boundary method achieves higher accuracy, but is more computationally expensive than the nearest-band neighbor method. However, by using smaller shift distances, we can achieve lower $y^+$ values with the same resolution. All turbulence models work well with the shifted boundary method, with only minor differences between them. In contrast, the nearest-band method shows variation in the behavior of the turbulence models, where the Spalart--Allmaras model yields better results, especially further downstream along the plate. This work establishes a robust foundation for simulating wall-bounded turbulent flows at high Reynolds numbers using meshfree collocation methods.

Figures (24)

• Figure 1: Illustration of the closest neighbor selection method in 2D, Red points are boundary points, blue points are interior points, and the solid line is the boundary wall. The arrow represents the identification of the closest interior neighbor point for each boundary point.
• Figure 2: Closest neighbor method. Illustration of non-uniform coverage of the selection of the closest neighbor points in the interior points. Each boundary point (red dot) looks for the closest neighbor interior point (blue dot), leading to regions where there are no interior points selected, causing numerical instabilities.
• Figure 3: Nearest-band neighbor method. Selection of neighbor interior points based on specified distance $\delta h$. Due to non-uniformity of the point distribution, the selection of interior points (gray dot) for application of wall functions is made based on the distance $\delta h$ (shown by dotted line) to the boundary, allowing for uniform selection of points. Other interior points are marked with blue dots.
• Figure 4: Illustration of the shifted-boundary method in 2D with additional shifted points. The boundary points (red dots) are virtually shifted (orange dots) to a distance of $\alpha h$ and are treated as interior points but fixed at a height of $\alpha h$ (dotted line) in the normal direction during the simulation. The momentum thickness approximation is carried out at the height of $\beta h$ (gray dotted line).
• Figure 5: 1D Couette Flow schematic illustrating the configuration with a fixed bottom plate with velocity $U_{\mathrm{w}} = 0 \, \frac{\text{m}}{\text{s}}$ and a moving top plate. The flow is driven by the top plate $U_{\mathrm{m}}$, including the height $H$ and the moving wall velocity $U_{\mathrm{m}} = 12.8 \, \frac{\text{m}}{\text{s}}$. The 1D simulation is carried out along the channel height $H$.