Verification of a Compressible CFD Code using the Method of Manufactured Solutions

The computational fluid dynamics code Premo is currently being developed to solve compressible flow problems with a finite-volume approach using unstructured meshes. This code employs both the Green-Gauss and the Least Squares gradient approximations in order to achieve a spatial accuracy that is formally second order for both convection and diffusion. In this paper, the Method of Manufactured Solutions is employed to generate exact solutions to the governing equations along with additional source terms. The exact solutions are then used to accurately evaluate the discretization error in the numerical solutions. The Manufactured Solutions approach is applied to the 2D inviscid Euler equations (both subsonic and supersonic), the 3D Euler equations (supersonic), and the 2D Navier-Stokes equations (both subsonic and supersonic). Through global discretization error analyses, the spatial order of accuracy is observed to be second order for all but one of the ten cases examined. With regards to the other nine cases, a high degree of confidence is achieved that the Premo code is free from coding mistakes in the spatial discretization for uniform meshes. 1  † Senior Member of Technical Staff, MS 0825, E-mail: [email protected], Senior Member AIAA ‡ Senior Member of Technical Staff, MS 0316, E-mail: [email protected], Member AIAA § Principal Member of Technical Staff, MS 0316, E-mail: [email protected], Member AIAA * Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC0494AL85000. This paper is declared a work of the U. S. Government and is not subject to copyright protection in the United States. Nomenclature e energy, m2/s2 f source term fs sine or cosine function h measure of grid spacing k thermal conductivity,W/(m·K) L domain length, m p spatial order of accuracy, or pressure, N/m2 Pr Prandtl number (Pr = 1.0) q heat flux vector, N/(m·s) R specific gas constant (R = 287. Nm/(kg·K)) r grid refinement factor position vector, m t time, s T temperature, K u, v, w Cartesian velocity components, m/s x, y, z Cartesian spatial coordinates, m γ ratio of specific heats (γ = 1.4) μ absolute viscosity, N·s/m2 ρ density, kg/m3 τ shear stress tensor, N/m2 φ general solution variable Subscripts exact exact value k mesh level (k = 1, 2, 3, etc., fine to coarse) m mass equation n flowfield node index x, y, z x-, y-, or z-momentum equation e energy equation