Laminar Backward-Facing Step at Re = 800

Overview

The backward-facing step is the canonical test of a solver's ability to capture laminar flow separation and reattachment. At a Reynolds number of 800 the flow is steady and two-dimensional: the shear layer leaving the step encloses a primary recirculation bubble in the step corner, and the displacement of the core flow drives a second, weaker separation bubble on the opposite wall further downstream. Gartling (1990) computed this configuration with a finite-element method on an extended channel, and the resulting solution is used here as a numerical benchmark.

This is a code-to-code verification: Luminary's solution is compared directly against Gartling's benchmark finite-element numerical solution at the same Reynolds number. The verified quantities are the horizontal-velocity profiles across the channel at two streamwise stations downstream of the step.

Problem Setup

The geometry is a sudden symmetric expansion of ratio two: the inlet channel has height equal to the step height $S$, and the downstream channel height is $H=2S$. The step face sits at $x=0$ and lengths are reported relative to the channel height $H$ (Gartling's convention), measured from the step. Rather than prescribing the parabolic channel profile directly at the inlet, a long upstream channel (about 1.25 times the laminar hydrodynamic entry length) precedes the step, so a uniform inflow develops into the fully-developed profile on its own before reaching the expansion.

The fluid is treated as constant-density with constant viscosity, and the viscosity is chosen so that the Reynolds number based on the downstream channel height and the mean inlet velocity is $Re=\rho UH/\mu =800$. The inlet is a uniform velocity inlet, the outlet a fixed-pressure boundary placed downstream of the reattachment region, and all channel walls are no-slip. The solve uses a second-order scheme with no limiter and is run to a steady-state density-residual target of ${10}^{-9}$.

Quantities of Interest

The quantity of interest is the horizontal-velocity profile $u(y)$ across the channel, examined at two streamwise stations: $x=7$ (through the shear layer and the primary recirculation bubble) and $x=15$ (where the flow is recovering after reattachment). At each station Luminary's profile is compared against Gartling's benchmark numerical solution. The velocity-magnitude field shows the primary step-corner recirculation bubble and the secondary bubble on the opposite wall.