Inviscid Isentropic Vortex Transport

Overview

This case verifies the unsteady compressible solver against an exact solution of the Euler equations: an isentropic vortex riding on a uniform stream. Because the vortex convects downstream without changing shape, the exact field at any later time is the initial field shifted by the distance the freestream has travelled, so the solver's error can be measured directly. With no physical viscosity in the problem, the discretization error is the only error present, and uniform mesh refinement isolates the spatial accuracy of the convective discretization and gradient reconstruction. Time integration uses the second-order implicit scheme with a small fixed step.

Problem Setup

A single isentropic vortex of core radius $0.01$ is placed at $(0.04,0.05)$ in a uniform $M_{\infty }=0.5$ stream ($p_{\infty }=100\mathrm{k}\mathrm{P}\mathrm{a}$, $T_{\infty }=300K$) on the rectangle $[0,0.15]\times [0,0.1]$, discretised as a single spanwise layer of hexahedra and refined uniformly through $48\times 32,96\times 64,192\times 128,384\times 256$ cells. The streamwise boundaries are farfield, placed far enough that the localized vortex never reaches them during the run; the transverse boundaries form a translational periodic pair; the spanwise faces are symmetry planes. The gas is treated as inviscid and ideal, and the flow is advanced from the exact vortex field to a fixed final time using the second-order implicit scheme.

Quantities of Interest

The verification metric is the discretization error of each conservative equation, continuity, the in-plane momentum components, and energy, measured in the $L_2$ norm at the final time and reported by the solver on each grid. The error is plotted against the representative cell size $h=(1/N{)}^{1/2}$, and the observed order of accuracy between successive grids is compared with the design order of two.