AIAA 2003-4240 Uncertainty Propagation for Turbulent, Compressible Flow in a Quasi-1D Nozzle Using Stochastic Methods

This paper describes a fully spectral, Polynomial Chaos method for the propagation of uncertainty in numerical simulations of compressible, turbulent flow, as well as a novel stochastic collocation algorithm for the same application. The stochastic collocation method is key to the efficient use of stochastic methods on problems with complex nonlinearities, such as those associated with the turbulence model equations in compressible flow and for CFD schemes requiring solution of a Riemann problem. Both methods are applied to compressible flow in a quasi-one-dimensional nozzle. The stochastic collocation method is roughly an order of magnitude faster than the fully Galerkin Polynomial Chaos method on the inviscid problem. Introduction An admittedly small but nevertheless noticeably accelerating trend within the Computational Fluid Dynamics (CFD) community is changing the vision of computational aerodynamics from one of producing a solution as accurately as possible to one of producing a solution with acceptable uncertainty bounds. Most of the effort of the traditional CFD community has been focused on discretization error and turbulence modeling error. Some of the early advocates of a broader perspective on CFD uncertainty were Mehta, 1 Roache, 2 Coleman and Stern 3 and Oberkampf and Blottner. 4 In the past half-dozen years there have been several events which have served to increase awareness of the broader perspective on CFD uncertainty: (1) the publication of the American Institute for Aeronautics and Astronautics guidelines on the verification and validation for CFD; 5 (2) the Drag Prediction Workshop sponsored by the AIAA Applied Aerodynamics TC; 6 and (3) the pair of sessions on CFD uncertainty at the January, 2003 AIAA Aerospace Sciences Meeting. 7–19 The work of Oberkampf, Deigert, Alvin and Rutherford 20 and Oberkampf and Trucano 21 is especially notable for proposing a framework for char-22 have provided a tutorial on uncertainty methods aim at a CFD audience. Hemsch, Luckring and Morrison 10 have presented a comprehensive vision for approaching computational aerodynamics uncertainty. (Their choice of title is noteworthy in that their vision is not confined to Euler and Navier-Stokes methods but embraces the use of simpler aerodynamic models.) Zang, et al 23 have outlined the needs and opportunities for uncertainty quantifi-cation and optimization under uncertainty research for aerospace applications. Some, but by no means all of the uncertainties affecting CFD are reasonably described in proba-bilistic terms. (See Oberkampf, Helton and Sentz 24 for a general mathematical framework for describing uncertainties.) Our focus in this …