Verification and validation for laminar hypersonic flowfields

Abstract

4bst!3@The accuracy .of Mach 8 kiminar flow solutions over a spherical y-blunted cone is verifiect, and the various physical models are validated. Verification of the solution accuracy is demonstrated by monitoring iterative convergence, performing a comprehensive grid convergence study, comparison to benchmark inviscid results, and code-to-code comparisons. Although the numerical scheme is nominally second order accurate in space, the presence of first order accuracy at the shock discontinuity results in first order behavior for the surface pressure distributions as the grid is suftlciently refined. Akernative methods are proposed for analyzing the spatial convergence behavior and determining the order of accuracy for mixed first and second order schemes. Validation of the wmputatiomd model is performed via surface pressure comparisons with high-quality experimental data. Careful attention is paid to the assumptions in the computational and experimental modeling. In particular, the thermodynamic state of the hypersonic wind tunnel nozzle is examined and arguments are made for the presence of a significant amount of thermal nonequilibrimn. Bias errors in the experimental Mach number calibration are discussed which are related to the assumption of thermal equilibrium (-0.23%) and the averaging of probe data over the entire test section (+0.6%). Differences between the simula- tion results for surface pressure and the experimental data are found to be as large as 3.3%. These differences are well outside the experimental 20 error bounds, even after accounting for the experimental bias errors.

A posteriori pointwise error estimation for compressible fluid flows using adjoint parameters and Lagrange remainder

“…For schemes of third and fourth accuracy order, reduction of the convergence rate was demonstrated in [4] for the compression wave. The works of [5][6][7] confirm this effect. Spatial nonuniformity of the grid may also reduce the convergence rate ( [8]).…”

Verification and validation benchmarks

“…Some of the reasons are as follows: (1) a programming error exists in the computer code; (2) the numerical algorithm is deficient is some unanticipated way; (3) there is insufficient grid resolution such that the grid is not in the asymptotic convergence region of the power-series expansion for the particular system response quantity (SRQ) of interest, (4) the formal order of convergence for interior grid points is different from the formal order of convergence for boundary conditions involving derivatives, resulting in a mixed order of convergence over the solution domain; (5) singularities, discontinuities, and contact surfaces are interior to the domain of the PDE; (6) singularities and discontinuities occur along the boundary of the domain; (7) the mesh resolution changes abruptly over the solution domain; (8) there is inadequate convergence of an iterative procedure in the numerical algorithm; and (9) boundary conditions are overspecified. It is beyond the scope of this paper to discuss the reasons listed above in detail; however, some of the representative references in these topics are [1,33,44,[47][48][49][50][51][52][53][54][55][56].…”

Assessment of One- and Two-Equation Turbulence Models for Hypersonic Transitional Flows

“…The results from Ref. 33 If the mesh has been re ned suf ciently where the solution error has second-order behavior (but not yet to the point where the rstorder behavior occurs), then the errors on the four meshes have the following relationship:…”