2D Turbulent Flat Plate (Zero Pressure Gradient, SST)

Overview

The 2D zero-pressure-gradient (ZPG) turbulent flat plate is a standard verification case for Reynolds-averaged Navier–Stokes (RANS) turbulence models. A sharp flat plate is aligned with a uniform free stream; a turbulent boundary layer grows along its length with (nominally) no streamwise pressure gradient. Because the case has no geometric complexity, it isolates the turbulence model and the discretization, making it suited to code-to-code verification.

This case is the companion of the Spalart–Allmaras flat plate, run instead with Luminary's $k\text{-}\omega$ SST model (the Menter SST-2003m variant). The objective is to reproduce, on the same grids, the results the NASA Turbulence Modeling Resource publishes for its reference codes CFL3D and FUN3D using their SST model. The TMR publishes its SST flat-plate data as the SST-Vm variant (vorticity-based production term); Luminary runs SST-2003m (strain-based). For this attached, zero-pressure-gradient boundary layer the strain-rate and vorticity magnitudes coincide throughout the layer, so the two variants are equivalent here, the reference series are labeled "SST-Vm" to name the exact published variant.

This is a grid-convergence study: the case is solved on a family of five successively refined structured grids (35×25, 69×49, 137×97, 273×193, 545×385, each a 2× refinement), and the 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 $Re=5\times {10}^6$ per unit length, matching the NASA TMR specification. The plate occupies $0\le x\le 2$ with the leading edge at $x=0$; the domain extends upstream to $x=-0.333$ (a symmetry plane ahead of the plate) and to $y=1$ in the wall-normal direction. The upper boundary uses a characteristic free-stream (Riemann) condition, the inflow a total-pressure characteristic condition, the outflow a static back-pressure, the plate a no-slip wall, and the spanwise planes symmetry (the mesh is one cell thick in $z$).

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 turbulence set to the TMR values (eddy-viscosity ratio ${\mu }_t/{\mu }_{\infty }=0.009$). Each grid is converged in pseudo-time (density residual driven to $𝒪({10}^{-10})$) 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 skin-friction coefficient $C_f$ at $x=0.97$ and of the total drag coefficient $C_D$, each plotted against $h=(1/N{)}^{1/2}$. On the finest 545×385 grid, Luminary reports $C_D=0.0028431$ (within 0.4% of CFL3D and FUN3D) and $C_f=0.0026796$ at $x=0.97$ (within 0.5% of CFL3D and FUN3D).
• The skin-friction distribution $C_f(x)$ along the plate on the finest grid.
• The law of the wall, $u^{+}$ versus $y^{+}$, at $x=0.97$ and $x=1.90$.
• The eddy-viscosity profile ${\mu }_t/{\mu }_{\infty }$ at $x=0.97$, and the streamwise growth of its boundary-layer peak $max({\mu }_t/{\mu }_{\infty })$.