2D Bump-in-channel (Turbulent Boundary Layer, SST)

Overview

The 2D bump-in-channel is a standard verification case for Reynolds-averaged Navier–Stokes (RANS) turbulence models, one step up in complexity from the zero-pressure-gradient flat plate. A turbulent boundary layer develops along the lower wall of a channel as the flow passes over a shallow, smoothly-contoured bump. The bump accelerates the flow over its windward face and decelerates it over the lee face, imposing a favorable-then-adverse streamwise pressure gradient on the boundary layer, so, unlike the flat plate, the wall experiences both a viscous (skin-friction) and a pressure (form) drag, while the geometry stays smooth enough that the flow remains attached.

This case is the companion of the Spalart–Allmaras bump, run instead with Luminary's $k\text{-}\omega$ SST model (the Menter SST-2003m variant). The goal is code-to-code verification of the turbulence model under a streamwise pressure gradient: to reproduce, on the same grids, the results the NASA Turbulence Modeling Resource publishes for its reference codes CFL3D and FUN3D using the matching SST-2003m model.

This is a grid-convergence study: the case is solved on a family of five successively refined structured grids (89×41, 177×81, 353×161, 705×321, 1409×641, each a 2× refinement), and the key integrated quantities are tracked as the mesh is refined toward the continuum ($h=(1/N{)}^{1/2}\to 0$).

Problem Setup

The free-stream Mach number is $M=0.2$ and the Reynolds number is $Re=3\times {10}^6$ based on the reference length $L=1$, matching the NASA TMR specification. The channel spans $-25\le x\le 26.5$ and $0\le y\le 5$. The lower wall is a no-slip viscous wall over $0\le x\le 1.5$; the bump itself rises over $0.3\le x\le 1.2$ with the profile $y=0.05{\sin }^4(\pi x/0.9-\pi /3)$, reaching its crest $y=0.05$ at $x=0.75$. Upstream ($x<0$) and downstream ($x>1.5$) of the plate the lower wall and the channel top ($y=5$) are symmetry planes, and the far upstream/downstream boundaries use a characteristic free-stream (Riemann) condition. The mesh is one cell thick in $z$ with symmetry on the spanwise planes; the reference area is $A_{ref}=1.5$ (plate length × unit span).

Air is modeled as an ideal gas with Sutherland's law for the dynamic viscosity and a laminar Prandtl number $Pr=0.72$. Turbulence closure is the $k\text{-}\omega$ SST (SST-2003m) model, with the free stream set to the TMR turbulence values (${\mu }_t/{\mu }_{\infty }=0.009$) via direct $k$/$\omega$ inflow. Each grid is converged in pseudo-time so that the remaining error is spatial, what a grid-convergence study isolates.

Quantities of Interest

The verification targets are those published on the NASA TMR SST expected-results page:

• The grid convergence of the integrated force coefficients, lift $C_L$, total drag $C_D$, and its pressure (form) and viscous (friction) parts $C_{D_p}$ and $C_{D_v}$, and of the skin-friction coefficient $C_f$ at the bump crest $x=0.75$, each plotted against the grid spacing $h=(1/N{)}^{1/2}$. On the finest grid (1409×641), every reported coefficient, lift, total drag, its pressure and viscous parts, and $C_f$ at the crest, agrees with both reference codes to within 0.30%; the largest Luminary–reference difference is comparable to the CFL3D-to-FUN3D spread on the same quantity.
• The surface distributions on the finest grid: the skin-friction coefficient $C_f(x)$ and the pressure coefficient $C_p(x)$ along the bump, which show the favorable/adverse pressure-gradient structure the boundary layer is subjected to.
• The eddy-viscosity profile ${\mu }_t/{\mu }_{\infty }$ at $x=0.75$, probing the turbulence model's behavior at the bump crest. The peak ${\mu }_t/{\mu }_{\infty }$ (about ninety) and the wall-normal height at which it occurs coincide with the CFL3D and FUN3D SST-2003m profiles, and the profile decays to the same channel-interior level.