2D Coflowing Jet (Compressible Mixing Layer)

Overview

The 2D coflowing jet is a compressible free-shear verification case: two parallel streams of differing speed meet at the sharp trailing edge of a thin splitter plate and develop into a planar mixing layer downstream. With no wall bounding the shear layer and no separation, the case isolates a turbulence model's behavior in free shear, its spreading rate, its centerline velocity decay, and its eddy viscosity. The goal here is code-to-code verification: to reproduce, on the same grids, the results the NASA Turbulence Modeling Resource (TMR) publishes for its reference codes CFL3D and FUN3D with the Spalart–Allmaras (SA) one-equation model.

This is a grid-convergence study: the case is solved on a family of five successively refined structured grids (each a 2× refinement), and the integrated and profile quantities are tracked as the mesh is refined toward the continuum (grid spacing $h=(1/N{)}^{1/2}\to 0$).

Problem Setup

The inner stream enters at $M=0.5$ and the outer (co-flowing) stream at $M\approx 0.254$, giving the velocity ratio that drives the mixing layer. The Reynolds number is $Re=50,000$ based on a unit reference length. The two streams are separated, upstream of the trailing edge, by a thin splitter plate modeled as a zero-thickness two-sided adiabatic no-slip wall occupying $-10\le x\le 0$ at $y=0.5$; the mixing layer develops over the downstream region $0\le x\le 200$, with the centerline ($y=0$) and the far top boundary treated as symmetry planes.

Each inflow is imposed as a total-pressure (subsonic) inlet with the exact TMR stagnation ratios, the inner stream at $P_t/P_{\infty }=1.1862$, $T_t/T_{\infty }=1.05$ and the outer stream at $P_t/P_{\infty }=1.046$, $T_t/T_{\infty }=1.0$, and the downstream boundary as a static back-pressure outlet at $P/P_{\infty }=1.0$. Air is modeled as an ideal gas with a constant molecular viscosity set so that $Re=50,000$, and turbulence closure is the Spalart–Allmaras model. Each grid is converged deeply in pseudo-time (continuity residual driven below ${10}^{-9}$) so that the remaining error is purely spatial, exactly what a grid-convergence study must isolate.

Quantities of Interest

Velocities are reported as $u/a_{\infty }$ (normalized by the free-stream speed of sound), matching the TMR data convention, so the inner core reads $\approx 0.5$. The verification targets are those published on the NASA TMR "SA expected results" page:

• The grid convergence of the splitter drag coefficient $C_D$, plotted against grid spacing $h=(1/N{)}^{1/2}$. On the finest grid the Luminary $C_D=0.0057196$ is within 0.8% of CFL3D ($0.0057658$) and within 1.0% of FUN3D ($0.0056648$).
• The centerline velocity decay $u/a_{\infty }$ along $y=0$, measuring how the mixing layer erodes the high-speed core downstream of the splitter trailing edge.
• The cross-stream velocity profiles $u/a_{\infty }$ versus $y$ at the downstream stations $x=2.72$, $x=29.2$ and $x=95.5$, which capture the spreading of the layer.
• The eddy-viscosity profile ${\mu }_t/{\mu }_{\infty }$ at $x=29.2$.