Joukowski Airfoil (Turbulent, SA-QCR2000-R)

Overview

This case reproduces Test Case 1 of the High-Fidelity CFD Verification Workshop 2024, turbulent flow over a smooth symmetric airfoil generated by the Joukowski conformal map. The workshop isolates discretization error in turbulence-model implementations: each participating code solves the same Spalart–Allmaras (SA-neg) model with the QCR2000-R rotation/curvature correction on a common family of systematically refined grids, so the spread between codes and their approach to a common grid-converged answer reflects numerical accuracy rather than modeling choices. Comparing our drag coefficient and its grid convergence against the workshop's finite-volume participants and its adapted high-order asymptotic value verifies that our SA-QCR2000-R discretization converges to the workshop answer.

Problem Setup

The airfoil is held at zero incidence in a free stream at Mach $0.15$ and a chord Reynolds number of $R{e}_c=6\times {10}^6$. The flow is steady and fully turbulent; the wall is treated as a no-slip adiabatic surface and the outer boundary, placed 100 chord lengths away, uses a characteristic (Riemann) far-field condition. Because the geometry is symmetric and the angle of attack is zero, the lift vanishes identically and the drag is the sole force of interest.

The verification is a grid-convergence study over six nested "Classic" quadrilateral grids generated from the Joukowski map, from $48\times 16$ up to $1536\times 512$ cells, each level a uniform $2\times$ refinement of the previous one. The grids cluster points at the leading-edge stagnation point and at the trailing edge, where the skin friction is singular, and open out toward the wake. The solver uses its accurate, low-dissipation discretization with no limiter, and every grid is converged to a deep density-residual floor so the remaining error is spatial.

Quantities of Interest

The reported quantity is the drag coefficient $C_D$, split into its pressure (form) component $C_{Dp}$ and viscous (friction) component $C_{Dv}$, with $C_D=C_{Dp}+C_{Dv}$. Each is plotted against the grid spacing $h=(\mathrm{D}\mathrm{O}\mathrm{F}{)}^{-1/2}$ and compared to the workshop's finite-volume codes FUN3D and CFD++ and to the adapted high-order asymptotic value ($C_D=0.007872$, $C_{Dp}=0.001244$, $C_{Dv}=0.006627$). As the grid is refined the coefficients march toward the asymptotic value; the finest-grid values are tabulated beside the reference codes. On the finest $1536\times 512$ grid the total $C_D$ is within $0.2\%$ of the FUN3D and CFD++ values and within $0.2\%$ of the asymptotic value. The pressure-drag component is the most grid-sensitive, as it is dominated by the trailing-edge singularity.