Compressible Couette Flow (Viscous Heating)

Overview

This case verifies the coupled momentum and energy solution of the compressible laminar solver against an exact result: plane Couette flow in which frictional dissipation heats the fluid. The shearing motion between two plates does work against viscosity, and for a perfect gas with constant transport properties that dissipated energy produces a temperature distribution known in closed form. The velocity field remains the simple linear Couette profile, while the temperature develops a symmetric parabola peaking at mid-gap, so the case checks not only the viscous momentum balance but the dissipation source term and wall thermal boundary conditions in the energy equation.

Problem Setup

Two parallel plates a distance $h=1m$ apart are both held at $T_w=300K$; the upper plate slides tangentially at $U=300\mathrm{m}/\mathrm{s}$ ($M\approx 0.86$) while the lower plate is stationary, with no imposed streamwise pressure gradient. The domain is a unit square discretised as a single spanwise layer of hexahedra, refined in the wall-normal direction to resolve the temperature parabola. The moving wall is driven by a translating reference frame and both walls are no-slip and isothermal; the streamwise faces are a translational periodic pair and the spanwise faces are symmetry planes. The gas is ideal and laminar with constant viscosity and Prandtl number, so the analytic constant-property solution applies exactly.

Quantities of Interest

The verification compares two wall-normal profiles at mid-channel against their exact forms: the streamwise velocity against the linear Couette profile $u(y)=Uy/h$, and the temperature against the viscous-dissipation parabola $T(y)=T_w+(Pr{U}^2/2c_p)(y/h)(1-y/h)$, whose centreline rise is $\Delta T=Pr{U}^2/8c_p\approx 8K$ at these conditions.