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arxiv: 2601.10661 · v2 · pith:VCNPVVXPnew · submitted 2026-01-15 · ⚛️ physics.flu-dyn

Boundary treatment algorithms for meshfree RANS turbulence modeling

Mohan Padmanabha , J\"org Kuhnert , Nicolas R. Gauger , Pratik Suchde This is my paper

Pith reviewed 2026-05-16 13:24 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords meshfree methodsRANS turbulence modelingwall functionsnearest-band neighborshifted boundarySpalart-AllmarasNACA 0012 airfoilhigh Reynolds number flows
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The pith

The nearest-band neighbor method enables stable wall-function enforcement in meshfree simulations of turbulent flows over curved surfaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper tests new strategies for applying wall functions inside meshfree RANS solvers for high-Reynolds-number flows. The standard closest-neighbor selection produces noisy results because points move and scatter. The nearest-band neighbor approach instead draws from a thin layer of interior points to keep selections uniform, while the shifted-boundary approach virtually displaces boundary points to a fixed distance. Validation on Couette flow, flat-plate boundary layers, and a NACA 0012 airfoil at high Reynolds numbers shows that the nearest-band neighbor method remains robust on both flat and curved geometries and works reliably with Spalart-Allmaras, k-ε, and k-ω closures.

Core claim

The nearest-band neighbor method, which enforces wall functions on a band of interior points rather than isolated closest neighbors, supplies a stable and flexible treatment that outperforms both the baseline closest-neighbor scheme and the shifted-boundary scheme on practical curved geometries such as the NACA 0012 airfoil.

What carries the argument

The nearest-band neighbor (NBN) method that selects a band of interior points to enforce wall functions, thereby preserving uniform distance sampling in scattered, possibly moving point clouds.

If this is right

  • Meshfree RANS modeling becomes practical for engineering geometries once the nearest-band neighbor treatment is used.
  • The shifted-boundary method delivers perfectly smooth y-plus distributions and high stability on flat plates.
  • All three common closures (Spalart-Allmaras, k-ε, k-ω) produce consistent results when paired with the nearest-band neighbor scheme.
  • The method avoids the numerical diffusion and early separation observed when shifted-boundary points are used on curved surfaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same band-selection idea could be applied directly to other Lagrangian meshfree schemes without requiring new turbulence-model development.
  • Correcting normal vectors inside the shifted-boundary step might remove the curvature-related errors and make that method viable for airfoils as well.
  • Because the approach works with existing turbulence closures, it can be inserted into existing meshfree codes with only local changes to the boundary stencil.

Load-bearing premise

Point distributions and normal vectors remain free of excessive clustering or uncorrected shifting that would distort wall-distance enforcement on curved surfaces under adverse pressure gradients.

What would settle it

Premature flow separation or markedly increased numerical diffusion appearing in NBN results on the NACA 0012 airfoil at Reynolds numbers of order 10^6, matching the shortcomings already seen with the shifted-boundary method.

Figures

Figures reproduced from arXiv: 2601.10661 by J\"org Kuhnert, Mohan Padmanabha, Nicolas R. Gauger, Pratik Suchde.

Figure 1
Figure 1. Figure 1: Illustration of the closest neighbor selection method in 2D, Red points are boundary points, [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Closest neighbor method. Illustration of non-uniform coverage of the selection of the closest [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Nearest-band neighbor method. Selection of neighbor interior points based on specified [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of the shifted-boundary method in 2D with additional shifted points. The [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: 1D Couette Flow schematic illustrating the configuration with a fixed bottom plate with [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: 1D Couette Flow results comparison using the closest neighbor method against experimental [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: 1D Couette Flow results comparison with the shifted-boundary method. Velocity profiles [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Flow over a flat plate geometry setup, illustrating the dimensions of the rectangular channel [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Skin friction coefficient Cf comparison of the closest neighbor (CN) method (green) with the Nearest-band neighbor (NBN) method (red) and the analytical expression (blue). Comparison of Cf shows a large deviation in the prediction of Cf for the CN method due to inadequate points selected to apply the wall function. compared with the CN method, see [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of Skin friction coefficient Cf for different wall function methods using the NBN method. The plot shows a comparison of the standard wall function (green) and Launder–Spalding wall function (red) against the analytical expression (blue) where we see no large difference between the two wall-function methods. We consider several values of the parameter δ, namely δh ∈ {0.3h, 0.5h, 0.7h, 1.0h}. We… view at source ↗
Figure 11
Figure 11. Figure 11: Deletion of points if the points get closer than [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of skin friction coefficient Cf with various selection heights δh using the nearest band neighbor method. The plots show selection heights of δh = 0.3h (green), δh = 0.5h (red), δh = 0.7h (cyan), δh = 1h (pink) against the analytical expression (blue). The zoomed view reveals that δh = 0.5h provides the best match with the analytical solution, followed by δh = 0.3h, while δh = 0.7h and δh = 1h … view at source ↗
Figure 13
Figure 13. Figure 13: Comparison of Skin Friction Coefficient Cf for Various Turbulence Models (SA (Green line), k − ε (Red line), and k − ω (Cyan line)) against the analytical expression using the nearest￾band neighbor method in turbulent flow over a flat plate. The figure indicates that k − ω provides superior accuracy near the wall compared to k − ε and SA mdels. 1 11 1 1 1 1 [PITH_FULL_IMAGE:figures/full_fig_p019… view at source ↗
Figure 14
Figure 14. Figure 14: Comparison of velocity profile at the wall normal direction with respect to [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Comparison of Skin Friction Coefficient Cf for shifted boundary method with Standard wall-function method (green) and Launder–Spalding method (red) against the analytical solution (blue) (von Karman, Theodore). The simulation is carried out using the k − ω turbulence model with resolution h of 0.005, shifted height αh = 0.1h, and momentum thickness of βh = 0.2h. As the wall functions are selected, the nex… view at source ↗
Figure 16
Figure 16. Figure 16: Comparison of Skin Friction Coefficient Cf with different shifted height αh with shifted boundary method for shifted height of αh = 0.05h (green), αh = 0.1h (red), αh = 0.2h (cyan), αh = 0.4h (pink). The comparison shows that the shifted height of αh = 0.1h to 0.2h give the best match to the analytical solution, with 0.2h and 0.1h giving a larger deviation with respect to the analytical expression. 20 [P… view at source ↗
Figure 17
Figure 17. Figure 17: Comparison of Skin Friction Coefficient Cf for momentum height βh = 1.5αh (green), βh = 2.0αh (red), βh = 3αh (cyan) with shifted-boundary method. The sCf comparison shows equidistant spacing between αh and βh gives better results compared to the values. With αh = 0.1h and βh = 0.2h, all three turbulence models (SA, k–ε, k–ω) deliver nearly overlapping Cf curves (see [PITH_FULL_IMAGE:figures/full_fig_p02… view at source ↗
Figure 18
Figure 18. Figure 18: Comparison of Skin Friction Coefficient Cf for various turbulence models (SA (Green line), k − ε (Red line), and k − ω (Cyan line)) against the analytical expression (blue) using the shifted boundary method in turbulent flow Over a flat plate: similar trend can be observed between all the turbulence models and match well with respect to analytical expression with slight variation at the end of the flat pl… view at source ↗
Figure 19
Figure 19. Figure 19: Comparison of velocity profile at the wall normal direction with respect to [PITH_FULL_IMAGE:figures/full_fig_p022_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Flow over a 3D wing with NACA-0012 profile, illustrating the dimensions of the rectangular [PITH_FULL_IMAGE:figures/full_fig_p023_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Refinement strategy for the flow around a NACA-0012 profile. The normalized refinement [PITH_FULL_IMAGE:figures/full_fig_p024_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Comparison of Skin Friction Coefficient Cf along the length of the wing with both shifted￾boundary (green) and nearest-band neighbor method (red) with resolution of h of 0.0035m and 0.003m respectively. The simulation is carried out using the k − ω turbulence model. The reason for the mismatch in skin friction values requires further investigation. Parameter studies show that the effects persist, and this… view at source ↗
read the original abstract

In this paper, we propose improved wall-treatment strategies for meshfree methods applied to turbulent flows. The goal is to enhance wall-function handling in simulations of high-Reynolds-number turbulent flows and to understand the performance of turbulence models within these frameworks. While wall-function techniques are well established for mesh-based methods, their implementation in meshfree methods faces unique challenges. The main difficulties arise from scattered point distributions and dynamic point movement in Lagrangian frameworks. To address these issues, we evaluate a baseline closest-neighbor approach alongside two novel techniques: the nearest-band neighbor (NBN) method and the shifted boundary (SB) method. The NBN method enforces wall functions on a band of interior points, helping to maintain uniform point selection. On the other hand, the SB method virtually moves boundary points to a fixed wall-normal distance, eliminating the spatial noise associated with point movement. We evaluate these methods using turbulence closures: Spalart--Allmaras, $k-\varepsilon$, and $k-\omega$ turbulence models. These methods are validated on 1D Couette flow, a turbulent flat plate, and a 3D NACA 0012 airfoil at high Reynolds numbers. Results demonstrate that both novel methods outperform the standard closest-neighbor approach on flat geometries. For flat plates, the SB method provides stability and perfectly smooth $y^+$ distributions. However, when applied to a curved NACA 0012 profile, the NBN method proves to be robust and flexible. In contrast, the SB method exhibits setbacks in numerical diffusion and premature flow separation on curved geometries. This is due to uncorrected normal-vector shifting and adverse pressure gradients. This work establishes the NBN method as a reliable, robust foundation for simulating turbulent flows over practical geometries using meshfree methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes two novel boundary treatment algorithms—nearest-band neighbor (NBN) and shifted boundary (SB)—for meshfree RANS turbulence modeling to improve wall-function handling in high-Re flows. These are evaluated against a baseline closest-neighbor approach using Spalart–Allmaras, k–ε, and k–ω closures on 1D Couette flow, a turbulent flat plate, and a 3D NACA 0012 airfoil. The central claims are that both new methods outperform the baseline on flat geometries, SB yields perfectly smooth y+ on flat plates, and NBN is robust and flexible on the curved airfoil while SB suffers premature separation from uncorrected normal-vector shifting under adverse pressure gradients.

Significance. If the quantitative validation gaps were closed, the work could supply practical boundary treatments for meshfree methods applied to turbulent flows over realistic geometries, directly addressing scattered-point and Lagrangian-movement difficulties. The multi-model, multi-geometry test suite is a constructive element, yet the current absence of absolute error metrics against experiments or established mesh-based RANS solutions limits immediate utility for the community.

major comments (2)
  1. [Abstract] Abstract: the statement that both novel methods outperform the baseline on flat geometries is unsupported by any quantitative metrics (e.g., skin-friction profiles, velocity errors, or y+ statistics) or error bars; only qualitative validation statements are supplied.
  2. [Abstract] Abstract (NACA 0012 results): the claim that NBN constitutes a reliable, robust foundation for practical curved geometries rests solely on relative improvement over closest-neighbor and SB; no absolute accuracy metrics (Cd, Cl, Cf distributions, or y+ profiles) benchmarked against experimental data or mesh-based RANS solutions at the same Re and angle of attack are reported, leaving the central claim without direct verification.
minor comments (2)
  1. The abstract refers to 'high Reynolds numbers' without stating the precise values employed for the flat-plate and airfoil cases.
  2. A summary table collating quantitative performance indicators (drag, lift, separation location, etc.) across all three test cases and all three turbulence models would improve readability and allow direct comparison.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive and detailed review of our manuscript. We address each major comment point by point below, indicating the revisions we will make to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the statement that both novel methods outperform the baseline on flat geometries is unsupported by any quantitative metrics (e.g., skin-friction profiles, velocity errors, or y+ statistics) or error bars; only qualitative validation statements are supplied.

    Authors: We agree that the abstract would benefit from explicit quantitative support. The full manuscript contains quantitative comparisons via skin-friction coefficient distributions, velocity profiles, and y+ statistics for the flat-plate cases that demonstrate the improvements. We will revise the abstract to include specific quantitative statements drawn from these results, such as the reduction in skin-friction deviation from reference values and the uniformity of y+ distributions. revision: yes

  2. Referee: [Abstract] Abstract (NACA 0012 results): the claim that NBN constitutes a reliable, robust foundation for practical curved geometries rests solely on relative improvement over closest-neighbor and SB; no absolute accuracy metrics (Cd, Cl, Cf distributions, or y+ profiles) benchmarked against experimental data or mesh-based RANS solutions at the same Re and angle of attack are reported, leaving the central claim without direct verification.

    Authors: The central objective of the work is to compare boundary-treatment algorithms within the meshfree RANS framework. For the NACA 0012 airfoil we demonstrate that NBN avoids the premature separation and excessive numerical diffusion exhibited by SB under adverse pressure gradients while remaining more stable than the closest-neighbor approach. We acknowledge that absolute metrics against experiments or mesh-based RANS are not included. We will revise the abstract to qualify the robustness claim as relative to the other meshfree treatments and add a brief discussion of this limitation in the conclusions. revision: partial

standing simulated objections not resolved
  • Absolute accuracy metrics (Cd, Cl, Cf distributions, or y+ profiles) for the NACA 0012 airfoil benchmarked against experimental data or established mesh-based RANS solutions at matching Re and angle of attack are not available in the present study.

Circularity Check

0 steps flagged

No circularity: methods validated on independent external benchmarks

full rationale

The paper proposes NBN and SB wall-treatment algorithms for meshfree RANS, then evaluates them comparatively against the baseline closest-neighbor method on three standard external benchmark flows (1D Couette, turbulent flat plate, 3D NACA 0012 at high Re). No equations, parameters, or central claims are shown to reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; performance statements rest on relative numerical outcomes against known reference solutions rather than internal renaming or ansatz smuggling. The derivation chain is therefore self-contained against external data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on standard RANS turbulence model closures and the assumption that the proposed boundary treatments correctly enforce wall functions without introducing new modeling errors beyond those stated.

axioms (1)
  • domain assumption Standard RANS turbulence model assumptions (Spalart-Allmaras, k-ε, k-ω) hold for the tested high-Re flows
    Paper applies these models directly without additional validation of their closures in the meshfree setting.

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