Revisiting the Slip Boundary Condition: Surface Roughness as a Hidden Tuning Parameter

TL;DR

This work interrogates how numerical boundary roughness and mesh distortion influence slip-boundary simulations of incompressible flow past cylinders in $2$D and $3$D. Using a GLS-stabilized finite element framework with high-order isoparametric geometry mappings and BDF2 time stepping, the authors show that discretization-induced surface roughness destabilizes potential flow and generates nonzero drag/lift, while minimizing geometry error yields a stable, near-potential flow with vanishing forces. Crucially, both numerical surface roughness and mesh distortion act as tunable control parameters, enabling drag and lift values to vary by $3$–$5$ orders of magnitude, which questions the predictive capability of slip BCs for wall-modeling in turbulent flows. The study further demonstrates in 3D that even small nonuniform mesh distortion can trigger three-dimensional instabilities, with a bifurcation around $2.5\%$ distortion separating potential-like and turbulent regimes. Taken together, the results highlight the sensitivity of slip-boundary simulations to discretization choices and geometry representation, suggesting careful treatment of boundary approximation in wall-modeling contexts.

Abstract

In this paper, we investigate the effect of boundary surface roughness on numerical simulations of incompressible fluid flow past a cylinder in two and three spatial dimensions furnished with slip boundary conditions. The governing equations are approximated using a continuous finite element method, stabilized with a Galerkin least-squares approach. Through a series of numerical experiments, we demonstrate that: $(i)$ the introduction of surface roughness through numerical discretization error, or mesh distortion, makes the potential flow solution unstable; $(ii)$ when numerical surface roughness and mesh distortion are minimized by using high-order isoparametric geometry mappings, a stable potential flow is obtained in both two and three dimensions; $(iii)$ numerical surface roughness, mesh distortion and refinement level can be used as control parameters to manipulate drag and lift forces resulting in numerical values spanning more than an order of magnitude. Our results cast some doubt on the predictive capability of the slip boundary condition for wall modeling in turbulent simulations of incompressible flow.

Figures (9)

• Figure 1: (a) Idealized, curved boundary with outward facing boundary normal ${\boldsymbol n}({\boldsymbol x})$; (b) corresponding approximation ${\boldsymbol n}_h({\boldsymbol x})$ for the case of a boundary approximated with a bilinear mapping. Here, the normal field on each face differs. In order to reconstruct a normal ${\boldsymbol n}$ in the vertex we average the normals coming from all adjacent faces.
• Figure 1: 2D and 3D computational domains and chosen boundary conditions for the validation tests. A cylindrical cutout is centered at the origin with a diameter of $D=0.1$. In the background, computational meshes are shown: the coarse mesh is shown on the top half and a single-refined mesh is shown on the bottom half. For 3D, only a 2D $xy$ cut plane is shown; the actual 3D mesh is obtained by extruding the 2D mesh in the third dimension to a depth of with $L=0.41$ with a total of 4 subdivisions on the coarsest level.
• Figure 2: Usage of a high-order manifold and mapping in deal.II: The curvature information is used for (a) the generation of new mesh points during mesh refinement, and (b) for the placement of support points. Here, exemplified for a cubic finite element with corresponding cubic bilinear mapping.
• Figure 2: Validation: flow past a 2D cylinder at $Re=20$ and $100$ with no-slip boundary conditions. Temporal snapshot of the velocity magnitudes at time $T = 20$ are shown for both steady and unsteady simulations with $Re=20$, and $Re=100$, respectively. A no-slip boundary condition is applied on both the cylinder surface and the channel walls. The shown snapshots are for a simulation using ${\mathbb Q}_1$ finite elements on a coarse grid with refinement level $r=4$, totaling $69k$ degrees of freedom.
• Figure 3: Validation: flow past a 3D cylinder at $Re=3900$ with no-slip boundary conditions. 2D cutout (with $z=0$) of a temporal snapshot of the velocity magnitudes at time $T = 0.5$ are shown. A no-slip boundary condition is applied on the cylinder surface and slip boundary conditions are applied on the channel walls. The simulation is performed using ${\mathbb Q}_2$ finite elements with a total of $53M$ degrees of freedom.