AED27 SUPERSONIC AERODYNAMICS Lecture 15 Treating Nonlinearities OUTLINE

AED-27: SUPERSONIC AERODYNAMICS Lecture 15 – Treating Nonlinearities

OUTLINE Shocks and Nonlinearities Mathematical background Toy problem: inviscid/viscid Burgers’ equations Numerical Treatment of Shocks Numerical Issues Gibbs Phenomenon Shock-Fitting Versus Shock-Capturing Modeling Adaptive Mesh Refinement Artifitial Viscosity Nonlinear methods Godunov Theory Limiter and Adaptive-Stencil Methods Application: Shock Tube

SEMI-LINEAR FIRSTORDER PDE REVISITED

MOC REVISITED

MOC REVISITED

MOC REVISITED Implicit equation for u(x, t)

INVISCID BURGERS’ EQUATION

INVISCID BURGERS’ EQUATION, CONT’D

INVISCID BURGERS’ EQUATION, CONT’D

STRONG AND WEAK FORM

SHOCKS

SHOCKS, CONT’D

RETURNING TO THE BURGERS’ EQUATION

RETURNING TO THE BURGERS’ EQUATION

REAL SHOCKS (RICHARD VON MISSES)

THE LAX-WENDROFF THEOREM

NUMERICAL TREATMENT OF SHOCKS Shock Fitting Shock Capturing Preferable when the location of shocks is known in advance. Do not involve any special treatment of shock waves Shock is treated as a discontinuity and well defined numerically. Shocks appear naturally within the domain as a direct result of the flow field solution Shock is introduced in to the flow field solution by using the exact Rankine-Hugoniot relations. The remainder of the domain is solved using the governing equations. Preferred for complex problems in which the positions of shocks Suffers from the smearing of shocks produced if there is inadequate grid resolution in the regions of shocks Spurious oscillations downstream of the captured discontinuities accuracy degradation (to first order) in the entire shock-downstream region.

MONOTONE SCHEMES

SPURIOUS OSCILLATIONS The numerical representation of discontinuities often presents ringing artifacts that are mathematically described by the Gibbs phenomenon. Its main cause is the inability that finite discretization have of representing high frequencies (as if the discontinuity had passed through a low-pass filter).

SPURIOUS OSCILLATIONS

NONLINEAR SCHEMES Solution: nonlinear schemes Slope/Flux Limiters Minmod Superbee Monotonized Central-difference (MC) Lax-Wendrof Adaptive stencils (ENO/WENO)

ARTIFICIAL VISCOSITY