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.

[1]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[2]  William L. Oberkampf,et al.  Uncertainty and error in computational simulations , 1997 .

[3]  Patrick Roache,et al.  Error Bars for CFD , 2003 .

[4]  D. Xiu,et al.  Modeling uncertainty in flow simulations via generalized polynomial chaos , 2003 .

[5]  G. Sod.  A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws , 1978 .

[6]  Roger Ghanem,et al.  Ingredients for a general purpose stochastic finite elements implementation , 1999 .

[7]  D. Xiu,et al.  Stochastic Modeling of Flow-Structure Interactions Using Generalized Polynomial Chaos , 2002 .

[8]  N. Wiener.  The Homogeneous Chaos , 1938 .

[9]  Patrick J. Roache,et al.  Verification and Validation in Computational Science and Engineering , 1998 .

[10]  Roger W. Logan,et al.  Validation, Uncertainty, and Quantitative Reliability at Confidence (QRC) , 2003 .

[11]  William L. Oberkampf,et al.  Issues in Computational Fluid Dynamics Code Verification and Validation , 1997 .

[12]  Jon C. Helton,et al.  Mathematical representation of uncertainty , 2001 .

[13]  Timothy G. Trucano,et al.  Verification and Validation in Computational Fluid Dynamics , 2002 .

[14]  Edwrd Tinoco,et al.  Minimizing CFD Uncertainty for Commercial Aircraft Applications (Invited) , 2003 .

[15]  Robert W. Walters,et al.  Uncertainty analysis for fluid mechanics with applications , 2002 .

[16]  Gusheng Hu,et al.  FURTHER REFINEMENT AND BENCH MARKING OF A SINGLE-GRID ERROR ESTIMATION TECHNIQUE , 2003 .

[17]  Jun Shao,et al.  Statistical Approach to CFD Code Certification , 2003 .

[18]  Robert E. Childs,et al.  Best Practices for Reduction of Uncertainty in CFD Results , 2003 .

[19]  Michael J. Hemsch,et al.  Statistical Analysis of CFD Solutions from the Drag Prediction Workshop , 2002 .

[20]  Christopher J. Roy,et al.  Discretization Error Estimates Using Exact Solutions to Nearby Problems , 2003 .

[21]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[22]  M. Yousuff Hussaini,et al.  Exploitation of Sensitivity Derivatives for Improving Sampling Methods , 2003 .

[23]  François M. Hemez,et al.  UNCERTAINTY QUANTIFICATION OF SIMULATION CODES BASED ON EXPERIMENTAL DATA , 2003 .

[24]  Hugh W. Coleman,et al.  Uncertainties and CFD Code Validation , 1997 .

[25]  L. Mathelin,et al.  A Stochastic Collocation Algorithm for Uncertainty Analysis , 2003 .

[26]  Joseph H. Morrison,et al.  Uncertainty in Computational Aerodynamics , 2003 .

[27]  M. Yousuff Hussaini,et al.  Uncertainty Quantification in CFD Simulations: A Stochastic Spectral Approach , 2003 .

[28]  U. B. Mehta,et al.  Some Aspects of Uncertainty in Computational Fluid Dynamics Results , 1991 .

[29]  Perry A. Newman,et al.  Some Advanced Concepts in Discrete Aerodynamic Sensitivity Analysis , 2001 .

[30]  Raymond Cosner.  The Importance of Uncertainty Estimation in Computational Fluid Dynamics (Invited) , 2003 .