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A stochastic projection method for fluid flow. II: Random process. (English) Zbl 1052.76057

Summary: An uncertainty quantification scheme is developed for the simulation of stochastic thermofluid processes. The scheme relies on spectral representation of uncertainty using the polynomial chaos (PC) system. The solver combines a Galerkin procedure for the determination of PC coefficients with a projection method for efficiently simulating the resulting system of coupled transport equations. Implementation of the numerical scheme is illustrated through simulations of natural convection in a 2D square cavity with stochastic temperature distribution at the cold wall. The properties of the uncertainty representation scheme are analyzed, and the predictions are contrasted with results obtained using a Monte Carlo approach.
For Part I, cf. ibid. 173, No. 2, 481–511 (2001; Zbl 1051.76056).

MSC:

76M35 Stochastic analysis applied to problems in fluid mechanics
76R10 Free convection
80A20 Heat and mass transfer, heat flow (MSC2010)

Citations:

Zbl 1051.76056

Software:

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References:

[1] Le Maı̂tre, O. P.; Knio, O. M.; Najm, H. N.; Ghanem, R. G., A stochastic projection method for fluid flow. I. Basic formulation, J. Comput. Phys., 173, 480 (2001)
[2] Wiener, S., The homogeneous chaos, Am. J. Math., 60, 897 (1938) · Zbl 0019.35406
[3] Cameron, R. H.; Martin, W. T., The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals, Ann. Math., 48, 385 (1947) · Zbl 0029.14302
[4] Chorin, A. J., Hermite expansions in Monte-Carle computation, J. Comput. Phys., 8, 472 (1971) · Zbl 0229.65025
[5] Maltz, F. H.; Hitzl, D. L., Variance reduction in Monte Carlo computations using multi-dimensional Hermite polynomials, J. Comput. Phys., 32, 345 (1979) · Zbl 0437.65005
[6] Meecham, W. C.; Jeng, D. T., Use of the Wiener-Hermite expansion for nearly normal turbulence, J. Fluid Mech., 32, 225 (1968) · Zbl 0155.55705
[7] Crow, S. C.; Canavan, G. H., Relationship between a Wiener-Hermite expansion and an energy cascade, J. Fluid Mech., 41, 387 (1970) · Zbl 0191.25603
[8] Chorin, A. J., Gaussian fields and random flow, J. Fluid Mech., 63, 21 (1974) · Zbl 0285.76022
[9] Ghanem, R. G.; Spanos, P. D., Stochastic Finite Elements: A Spectral Approach (1991), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0953.74608
[10] De Vahl Davis, G.; Jones, I. P., Natural convection in a square cavity: A comparison exercice, Int. J. Numer. Methods Fluids, 3, 227 (1983) · Zbl 0538.76076
[11] Le Quéré, P.; Alziary de Roquefort, T., Computation of natural-convection in two-dimensional cavities with Tschebyscheff polynomials, J. Comput. Phys., 57, 210 (1985) · Zbl 0585.76128
[12] Le Quéré, P., Accurate solution to the square thermally driven cavity at high Rayleigh number, Comput. Fluids, 20, 29 (1991) · Zbl 0731.76054
[13] Chenoweth, D. R.; Paolucci, S., Natural convection in an enclosed vertical layer with large horizontal temperature differences, J. Fluid Mech., 169, 173 (1986) · Zbl 0623.76097
[14] Le Quéré, P.; Masson, R.; Perrot, P., A Chebyshev collocation algorithm for 2D non-Boussinesq convection, J. Comput. Phys., 103, 320 (1992) · Zbl 0763.76061
[15] H. Paillere, and, P. Le Quéré, Modelling and simulation of natural convection flows with large temperature differences: A benchmark problem for low Mach number solvers, presented at 12th Seminar, Computational Fluid Dynamics, CEA/Nuclear Reactor Division, Saclay, France, 2000.; H. Paillere, and, P. Le Quéré, Modelling and simulation of natural convection flows with large temperature differences: A benchmark problem for low Mach number solvers, presented at 12th Seminar, Computational Fluid Dynamics, CEA/Nuclear Reactor Division, Saclay, France, 2000.
[16] M. Christon, P. Gresho, and S. Sutton, Computational predictability of natural convection flows in enclosure, in Computational Fluid and Solid Mechanics, edited by K. Bathe, Proceedings of First MIT Conference on Computational Fluid and Solid Mechanics Elsevier, Amsterdam, 2001, pp. 1465-1468.; M. Christon, P. Gresho, and S. Sutton, Computational predictability of natural convection flows in enclosure, in Computational Fluid and Solid Mechanics, edited by K. Bathe, Proceedings of First MIT Conference on Computational Fluid and Solid Mechanics Elsevier, Amsterdam, 2001, pp. 1465-1468.
[17] D. M. Christopher, Numerical prediction of natural convection flows in a tall enclosure, in Computational Fluid and Solid Mechanics, edited by K. Bathe, Proceedings of First MIT Conference on Computational Fluid and Solid Mechanics Elsevier, Amsterdam, 2001, pp. 1469-1471.; D. M. Christopher, Numerical prediction of natural convection flows in a tall enclosure, in Computational Fluid and Solid Mechanics, edited by K. Bathe, Proceedings of First MIT Conference on Computational Fluid and Solid Mechanics Elsevier, Amsterdam, 2001, pp. 1469-1471.
[18] G. Comini, M. Manzan, C. Nonino, and O. Saro, Finite element solutions for natural convection in a tall rectangular cavity, in Computational Fluid and Solid Mechanics, edited by K. Bathe, Proceedings of First MIT Conference on Computational Fluid and Solid Mechanics Elsevier, Amsterdam, 2001, pp. 1472-1476.; G. Comini, M. Manzan, C. Nonino, and O. Saro, Finite element solutions for natural convection in a tall rectangular cavity, in Computational Fluid and Solid Mechanics, edited by K. Bathe, Proceedings of First MIT Conference on Computational Fluid and Solid Mechanics Elsevier, Amsterdam, 2001, pp. 1472-1476.
[19] G. Groce and M. Favero, Simulation of natural convection flow in enclosures by an unstaggered grid Finite volume algorithm, in Computational Fluid and Solid Mechanics, edited by K. Bathe, Proceedings of First MIT Conference on Computational Fluid and Solid Mechanics Elsevier, Amsterdam, 2001, pp. 1477-1481.; G. Groce and M. Favero, Simulation of natural convection flow in enclosures by an unstaggered grid Finite volume algorithm, in Computational Fluid and Solid Mechanics, edited by K. Bathe, Proceedings of First MIT Conference on Computational Fluid and Solid Mechanics Elsevier, Amsterdam, 2001, pp. 1477-1481.
[20] Gresho, P. M.; Sutton, S., (Bathe, K., 8:1 thermal cavity problem, in Computational Fluid and Solid Mechanics (2001), Elsevier: Elsevier Amsterdam), 1482-1485
[21] H. Johnston and R. Krasny, Computational predictability of natural convection flows in enclosures: A benchmark problem, in Computational Fluid and Solid Mechanics, edited by K. Bathe, Proceedings of First MIT Conference on Computational Fluid and Solid Mechanics Elsevier, Amsterdam, 2001, pp. 1486-1489.; H. Johnston and R. Krasny, Computational predictability of natural convection flows in enclosures: A benchmark problem, in Computational Fluid and Solid Mechanics, edited by K. Bathe, Proceedings of First MIT Conference on Computational Fluid and Solid Mechanics Elsevier, Amsterdam, 2001, pp. 1486-1489.
[22] S.-E. Kim and D. Choudhury, Numerical investigation of laminar natural convection flow inside a tall cavity using a finite volume based Navier-Stokes solver, in Computational Fluid and Solid Mechanics, edited by K. Bathe, Proceedings of First MIT Conference on Computational Fluid and Solid Mechanics Elsevier, Amsterdam, 2001, pp. 1490-1492.; S.-E. Kim and D. Choudhury, Numerical investigation of laminar natural convection flow inside a tall cavity using a finite volume based Navier-Stokes solver, in Computational Fluid and Solid Mechanics, edited by K. Bathe, Proceedings of First MIT Conference on Computational Fluid and Solid Mechanics Elsevier, Amsterdam, 2001, pp. 1490-1492.
[23] T.-W. Pan and R. Glowinski, A projection/wave-like equation method for natural convection flows in enclosures, in Computational Fluid and Solid Mechanics, edited by K. Bathe, Proceedings of First MIT Conference on Computational Fluid and Solid Mechanics Elsevier, Amsterdam, 2001, pp. 1493-1496.; T.-W. Pan and R. Glowinski, A projection/wave-like equation method for natural convection flows in enclosures, in Computational Fluid and Solid Mechanics, edited by K. Bathe, Proceedings of First MIT Conference on Computational Fluid and Solid Mechanics Elsevier, Amsterdam, 2001, pp. 1493-1496.
[24] A. G. Salinger, R. B. Lehoucq, R. P. Pawlowski, and J. N. Shadid, Understanding the 8:1 cavity problem via scalable stability analysis algorithms, in Computational Fluid and Solid Mechanics, edited by K. Bathe, Proceedings of First MIT Conference on Computational Fluid and Solid Mechanics Elsevier, Amsterdam, 2001, pp. 1497-1500.; A. G. Salinger, R. B. Lehoucq, R. P. Pawlowski, and J. N. Shadid, Understanding the 8:1 cavity problem via scalable stability analysis algorithms, in Computational Fluid and Solid Mechanics, edited by K. Bathe, Proceedings of First MIT Conference on Computational Fluid and Solid Mechanics Elsevier, Amsterdam, 2001, pp. 1497-1500.
[25] S. A. Suslov and S. Paolucci, A Petrov-Galerkin method for flows in cavities, in Computational Fluid and Solid Mechanics, edited by K. Bathe, Proceedings of First MIT Conference on Computational Fluid and Solid Mechanics Elsevier, Amsterdam, 2001, pp. 1501-1504.; S. A. Suslov and S. Paolucci, A Petrov-Galerkin method for flows in cavities, in Computational Fluid and Solid Mechanics, edited by K. Bathe, Proceedings of First MIT Conference on Computational Fluid and Solid Mechanics Elsevier, Amsterdam, 2001, pp. 1501-1504.
[26] K. W. Westerberg, Thermally driven flow in a cavity using the Galerkin finite element method, in Computational Fluid and Solid Mechanics, edited by K. Bathe, Proceedings of First MIT Conference on Computational Fluid and Solid Mechanics Elsevier, Amsterdam, 2001, pp. 1505-1508.; K. W. Westerberg, Thermally driven flow in a cavity using the Galerkin finite element method, in Computational Fluid and Solid Mechanics, edited by K. Bathe, Proceedings of First MIT Conference on Computational Fluid and Solid Mechanics Elsevier, Amsterdam, 2001, pp. 1505-1508.
[27] S. Xin and P. Le Quéré, An extended Chebyshev pseudo-spectral contribution to CPNCFE benchmark, in Computational Fluid and Solid Mehcanics, edited by K. Bathe, Proceedings of First MIT Conference on Computational Fluid and Solid Mechanics Elsevier, Amsterdam, 2001, pp. 1509-1513.; S. Xin and P. Le Quéré, An extended Chebyshev pseudo-spectral contribution to CPNCFE benchmark, in Computational Fluid and Solid Mehcanics, edited by K. Bathe, Proceedings of First MIT Conference on Computational Fluid and Solid Mechanics Elsevier, Amsterdam, 2001, pp. 1509-1513.
[28] Lankhorst, A. M., Laminar and Turbulent Natural Convection in Cavities; Numerical Modeling and Experimental Validation (1991), Delft University of Technology
[29] Loève, M., Probability Theory (1997), Springer-Verlag: Springer-Verlag Berlin/New York
[30] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1970), Dover: Dover New York
[31] O. M. Knio, and, R. G. Ghanem, Polynomial Chaos Product and Moment Formulas: A User Utility; O. M. Knio, and, R. G. Ghanem, Polynomial Chaos Product and Moment Formulas: A User Utility
[32] McKay, M. D.; Conover, W. J.; Beckman, R. J., A comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics, 21, 239 (1979) · Zbl 0415.62011
[33] Ghanem, R.; Dham, S., Stochastic finite element analysis for multiphase flow in heterogeneous porous media, Trans. Porous Media, 32, 239 (1998)
[34] Ghanem, R., Probabilistic characterization of transport in heterogeneous porous media, Comput. Methods Appl. Mech. Eng., 158, 199 (1998) · Zbl 0954.76079
[35] Chorin, A. J., A numerical method for solving incompressible viscous flow problems, J. Comput. Phys., 2, 12 (1967) · Zbl 0149.44802
[36] Kim, J.; Moin, P., Application of a fractional-step method to the incompressible Navier-Stokes equations, J. Comput. Phys., 59, 308 (1985) · Zbl 0582.76038
[37] Eldred, M. S.; Giunta, A. A.; Wojkiewicz, S. F.; van Bloemen Waanders, B. G.; Hart, W. E.; Alleva, M. P., DAKOTA, A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Sensitivity Analysis, and Uncertainty Quantification, Version 3.0 Reference Manual (2002)
[38] Wojtkiewicz, S. F.; Eldred, M. S.; Field, R. V.; Urbina, A.; Red-Horse, J. R., A Toolkit for Uncertainty Quantification in Large Computational Engineering Models, Meeting Paper 2001-1455 (2001)
[39] M. S. Eldred, Optimization Strategies for Complex Engineering Applications; M. S. Eldred, Optimization Strategies for Complex Engineering Applications
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