Modeling Uncertainty in Flow Simulations via Generalized Polynomial Chaos

We present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations. The stochastic input is represented spectrally by employing orthogonal polynomial functionals from the Askey scheme as trial basis to represent the random space. A standard Galerkin projection is applied in the random dimension to obtain the equations in the weak form. The resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode. This approach can be considered as a generalization of the original polynomial chaos expansion, first introduced by N. Wiener (1938). The original method employs the Hermite polynomials (one of the thirteen members of the Askey scheme) as the basis in random space. The algorithm is applied to pressure-driven channel flows with random wall boundary conditions, and to external flows with random freestream. Efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by Monte Carlo simulations. It is shown that the generalized polynomial chaos method promises a substantial speed-up compared with the Monte Carlo method. The utilization of different type orthogonal polynomials from the Askey scheme also provides a more efficient way to represent general non-Gaussian processes compared with the original Wiener-Hermite expansions. Corresponding author, [email protected] 1 Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 10 OCT 2002 2. REPORT TYPE 3. DATES COVERED 00-10-2002 to 00-10-2002 4. TITLE AND SUBTITLE Modeling Uncertainty in Flow Simulations via Generalized Polynomial Chaos 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Brown University,Division of Applied Mathematics,182 George Street,Providence,RI,02912 8. PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S) 11. SPONSOR/MONITOR’S REPORT NUMBER(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution unlimited 13. SUPPLEMENTARY NOTES 14. ABSTRACT 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT 18. NUMBER OF PAGES 37 19a. NAME OF RESPONSIBLE PERSON a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18