Uncertainty Quantification in Fluid Flow

This chapter addresses the topic of uncertainty quantification in fluid flow computations. The relevance and utility of this pursuit are discussed, outlining highlights of available methodologies. Particular attention is focused on spectral polynomial chaos methods for uncertainty quantification that have seen significant development over the past two decades. The fundamental structure of these methods is presented, along with associated challenges. We also discuss demonstrations of their use in a number of fluid flow applications covering a range of complexity that is inherent in turbulent combustion.

[1]  George Em Karniadakis,et al.  Noisy inflows cause a shedding-mode switching in flow past an oscillating cylinder. , 2004, Physical review letters.

[2]  Hermann G. Matthies,et al.  Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations , 2005 .

[3]  M. Rosenblatt.  Remarks on a Multivariate Transformation , 1952 .

[4]  Chris L. Pettit,et al.  investigated aeroelastic behaviors arising from variability in three input variables , the initial pitch angle and two stiffness coefficients , 2006 .

[5]  Hisanao Ogura,et al.  Orthogonal functionals of the Poisson process , 1972, IEEE Trans. Inf. Theory.

[6]  Johan Meyers,et al.  Sensitivity analysis of large-eddy simulations to subgrid-scale-model parametric uncertainty using polynomial chaos , 2007, Journal of Fluid Mechanics.

[7]  Nitin Agarwal,et al.  A domain adaptive stochastic collocation approach for analysis of MEMS under uncertainties , 2009, J. Comput. Phys..

[8]  I. Babuska,et al.  Solution of stochastic partial differential equations using Galerkin finite element techniques , 2001 .

[9]  W. Gautschi.  On Generating Orthogonal Polynomials , 1982 .

[10]  Baskar Ganapathysubramanian,et al.  Sparse grid collocation schemes for stochastic natural convection problems , 2007, J. Comput. Phys..

[11]  Roger Ghanem,et al.  Adaptive polynomial chaos expansions applied to statistics of extremes in nonlinear random vibration , 1998 .

[12]  Edwin Thompson Jaynes,et al.  Probability theory , 2003 .

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

[14]  Erich Novak,et al.  High dimensional polynomial interpolation on sparse grids , 2000, Adv. Comput. Math..

[15]  R. Ghanem,et al.  Uncertainty propagation using Wiener-Haar expansions , 2004 .

[16]  Johan Meyers,et al.  Uncertainty Modeling, Error Charts and Improvement of Subgrid Models , 2008 .

[17]  P. Moin,et al.  Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations , 1984 .

[18]  Efstratios Nikolaidis,et al.  Engineering Design Reliability Handbook , 2004 .

[19]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[20]  Knut Petras,et al.  Smolyak cubature of given polynomial degree with few nodes for increasing dimension , 2003, Numerische Mathematik.

[21]  J. Warnatz,et al.  Resolution of gas phase and surface combustion chemistry into elementary reactions , 1992 .

[22]  L Krist Sherrie,et al.  CFL3D User''s Manual (Version 5.0) , 1998 .

[23]  K. Ritter,et al.  High dimensional integration of smooth functions over cubes , 1996 .

[24]  Yoon-ha Lee,et al.  Uncertainty Quantification for Multiscale Simulations , 2002 .

[25]  E. Somersalo,et al.  Statistical and computational inverse problems , 2004 .

[26]  Philip S. Beran,et al.  Airfoil pitch-and-plunge bifurcation behavior with Fourier chaos expansions , 2005 .

[27]  G. Karniadakis,et al.  Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures , 2006, SIAM J. Sci. Comput..

[28]  Thomas Y. Hou,et al.  Wiener Chaos expansions and numerical solutions of randomly forced equations of fluid mechanics , 2006, J. Comput. Phys..

[29]  Ramana V. Grandhi,et al.  Uncertainty Quantification of Structural Response Using Evidence Theory , 2002 .

[30]  Kathleen V. Diegert,et al.  Error and uncertainty in modeling and simulation , 2002, Reliab. Eng. Syst. Saf..

[31]  W. L. Oberkampf,et al.  Validation Methodology in Computational Fluid Dynamics (invited) , 2000 .

[32]  Sean P. Kenny,et al.  Needs and Opportunities for Uncertainty- Based Multidisciplinary Design Methods for Aerospace Vehicles , 2002 .

[33]  A. Chorin.  A Numerical Method for Solving Incompressible Viscous Flow Problems , 1997 .

[34]  Wright-Patterson Afb,et al.  Polynomial Chaos Expansion Applied to Airfoil Limit Cycle Oscillations , 2004 .

[35]  Thomas Gerstner,et al.  Dimension–Adaptive Tensor–Product Quadrature , 2003, Computing.

[36]  P. Frauenfelder,et al.  Finite elements for elliptic problems with stochastic coefficients , 2005 .

[37]  Fabio Nobile,et al.  An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data , 2008, SIAM J. Numer. Anal..

[38]  Pierre Sagaut,et al.  Sensitivity of two-dimensional spatially developing mixing layers with respect to uncertain inflow conditions , 2008 .

[39]  Jan S. Hesthaven,et al.  Uncertainty analysis for the steady-state flows in a dual throat nozzle , 2005 .

[40]  Jun Shao,et al.  Quantitative V&V of CFD simulations and certification of CFD codes , 2006 .

[41]  Robert H. Kraichnan.  Direct‐Interaction Approximation for a System of Several Interacting Simple Shear Waves , 1963 .

[42]  D. Bell,et al.  Evidence Theory and Its Applications , 1991 .

[43]  J. Witteveen,et al.  Higher Period Stochastic Bifurcation of Nonlinear Airfoil Fluid-Structure Interaction , 2009 .

[44]  Jon C. Helton,et al.  Evidence Theory for Engineering Applications , 2004 .

[45]  A. O'Hagan,et al.  Probabilistic sensitivity analysis of complex models: a Bayesian approach , 2004 .

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

[47]  Roger G. Ghanem,et al.  Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure , 2005, SIAM J. Sci. Comput..

[48]  Jeroen A. S. Witteveen,et al.  Effect of randomness on multi-frequency aeroelastic responses resolved by Unsteady Adaptive Stochastic Finite Elements , 2009, J. Comput. Phys..

[49]  H. Najm,et al.  A stochastic projection method for fluid flow II.: random process , 2002 .

[50]  O. L. Maître,et al.  Protein labeling reactions in electrochemical microchannel flow: Numerical simulation and uncertainty propagation , 2003 .

[51]  G. H. Canavan,et al.  Relationship between a Wiener–Hermite expansion and an energy cascade , 1970, Journal of Fluid Mechanics.

[52]  R. Askey,et al.  Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials , 1985 .

[53]  W. Schoutens.  Stochastic processes and orthogonal polynomials , 2000 .

[54]  P. A. Newman,et al.  Approach for uncertainty propagation and robust design in CFD using sensitivity derivatives , 2001 .

[55]  Muhammad R. Hajj,et al.  Gust Loads with Uncertainty Due to Imprecise Gust Velocity Spectra , 2007 .

[56]  Jeroen A. S. Witteveen,et al.  An unsteady adaptive stochastic finite elements formulation for rigid-body fluid-structure interaction , 2008 .

[57]  E. Jaynes.  Probability theory : the logic of science , 2003 .

[58]  K. Ritter,et al.  On an interpolatory method for high dimensional integration , 1999 .

[59]  Christoph Schwab,et al.  Sparse finite elements for elliptic problems with stochastic loading , 2003, Numerische Mathematik.

[60]  A. Chorin.  Gaussian fields and random flow , 1974, Journal of Fluid Mechanics.

[61]  O P Le Maître,et al.  Spectral stochastic uncertainty quantification in chemical systems , 2004 .

[62]  Roger Ghanem,et al.  Stochastic Finite Element Analysis for Multiphase Flow in Heterogeneous Porous Media , 1998 .

[63]  Steven A. Orszag,et al.  Dynamical Properties of Truncated Wiener‐Hermite Expansions , 1967 .

[64]  Raúl Tempone,et al.  Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations , 2004, SIAM J. Numer. Anal..

[65]  N. Wiener,et al.  Nonlinear Problems in Random Theory , 1964 .

[66]  Fabio Nobile,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2010, SIAM Rev..

[67]  H. Najm,et al.  Uncertainty quantification in reacting-flow simulations through non-intrusive spectral projection , 2003 .

[68]  Xiang Ma,et al.  An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations , 2009, J. Comput. Phys..

[69]  Dongbin Xiu,et al.  High-Order Collocation Methods for Differential Equations with Random Inputs , 2005, SIAM J. Sci. Comput..

[70]  Guang Lin,et al.  Predicting shock dynamics in the presence of uncertainties , 2006, J. Comput. Phys..

[71]  Jefferson W. Tester,et al.  Incorporation of parametric uncertainty into complex kinetic mechanisms: Application to hydrogen oxidation in supercritical water , 1998 .

[72]  G. Canavan.  Some properties of a Lagrangian Wiener–Hermite expansion , 1970, Journal of Fluid Mechanics.

[73]  Kaizar Amin,et al.  Introduction to Active Thermochemical Tables: Several “Key” Enthalpies of Formation Revisited† , 2004 .

[74]  Stephen D. Unwin,et al.  A Fuzzy Set Theoretic Foundation for Vagueness in Uncertainty Analysis , 1986 .

[75]  M. Yousuff Hussaini,et al.  Uncertainty Propagation for Turbulent, Compressible Flow in a Quasi-1D Nozzle Using Stochastic Methods , 2003 .

[76]  Da Ruan,et al.  Foundations and Applications of Possibility Theory , 1995 .

[77]  George Em Karniadakis,et al.  Predictability and uncertainty in CFD , 2003 .

[78]  Philip S. Beran,et al.  Uncertainty Quantification with a B-Spline Stochastic Projection , 2006 .

[79]  Earl Cox,et al.  The fuzzy systems handbook - a practitioner's guide to building, using, and maintaining fuzzy systems , 1994 .

[80]  N. Zabaras,et al.  Using stochastic analysis to capture unstable equilibrium in natural convection , 2005 .

[81]  Glenn Shafer,et al.  A Mathematical Theory of Evidence , 2020, A Mathematical Theory of Evidence.

[82]  Habib N. Najm,et al.  Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics , 2009 .

[83]  Knut Petras,et al.  Fast calculation of coefficients in the Smolyak algorithm , 2001, Numerical Algorithms.

[84]  Jeroen A. S. Witteveen,et al.  An adaptive Stochastic Finite Elements approach based on Newton–Cotes quadrature in simplex elements , 2009 .

[85]  Thomas Gerstner,et al.  Numerical integration using sparse grids , 2004, Numerical Algorithms.

[86]  Fabio Nobile,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007, SIAM Rev..

[87]  A. O'Hagan,et al.  Bayesian calibration of computer models , 2001 .

[88]  Jeroen A. S. Witteveen,et al.  Modeling physical uncertainties in dynamic stall induced fluid-structure interaction of turbine blades using arbitrary polynomial chaos , 2007 .

[89]  Jon C. Helton,et al.  An exploration of alternative approaches to the representation of uncertainty in model predictions , 2003, Reliab. Eng. Syst. Saf..

[90]  P. Roache.  Code Verification by the Method of Manufactured Solutions , 2002 .

[91]  I. Babuska,et al.  Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation , 2005 .

[92]  R. Ghanem,et al.  Quantifying uncertainty in chemical systems modeling , 2004 .

[93]  W. T. Martin,et al.  The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals , 1947 .

[94]  Thomas A. Zang,et al.  Stochastic approaches to uncertainty quantification in CFD simulations , 2005, Numerical Algorithms.

[95]  Sankaran Mahadevan,et al.  Model Predictive Capability Assessment Under Uncertainty , 2005 .

[96]  C. Schwab,et al.  Sparse high order FEM for elliptic sPDEs , 2009 .

[97]  Ismail Celik,et al.  Assessment of numerical uncertainty for the calculations of turbulent flow over a backward‐facing step , 2005 .

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

[99]  A. O'Hagan,et al.  Predicting the output from a complex computer code when fast approximations are available , 2000 .

[100]  Jeroen A. S. Witteveen,et al.  Probabilistic collocation for period-1 limit cycle oscillations , 2008 .

[101]  Adrian E. Raftery,et al.  Bayesian model averaging: a tutorial (with comments by M. Clyde, David Draper and E. I. George, and a rejoinder by the authors , 1999 .

[102]  J. Boyd.  The rate of convergence of Hermite function series , 1980 .

[103]  Igor Kozine,et al.  Imprecise Probabilities Relating to Prior Reliability Assessments , 1999, ISIPTA.

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

[105]  Fabio Nobile,et al.  A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data , 2008, SIAM J. Numer. Anal..

[106]  Michael S. Triantafyllou,et al.  Parametric study of a two degree-of-freedom cylinder subject to vortex-induced vibrations , 2008 .

[107]  Habib N. Najm,et al.  Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems , 2008, J. Comput. Phys..

[108]  Habib N. Najm,et al.  Natural Convection in a Closed Cavity under Stochastic Non-Boussinesq Conditions , 2005, SIAM J. Sci. Comput..

[109]  T. A. Zang,et al.  Uncertainty Propagation for a Turbulent, Compressible Nozzle Flow Using Stochastic Methods , 2004 .

[110]  Habib N. Najm,et al.  Stochastic spectral methods for efficient Bayesian solution of inverse problems , 2005, J. Comput. Phys..

[111]  BabuskaIvo,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007 .

[112]  Matthew F. Barone,et al.  Measures of agreement between computation and experiment: Validation metrics , 2004, J. Comput. Phys..

[113]  Y. Marzouk,et al.  Uncertainty quantification in chemical systems , 2009 .

[114]  Sébastien Deck,et al.  Large eddy simulation for aerodynamics: status and perspectives , 2009, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[115]  G. Karniadakis,et al.  An adaptive multi-element generalized polynomial chaos method for stochastic differential equations , 2005 .

[116]  Michael J. Hemsch,et al.  Statistical Analysis of Computational Fluid Dynamics Solutions from the Drag Prediction Workshop , 2004 .

[117]  Robert W. Walters,et al.  An Implicit Compact Polynomial Chaos Formulation for the Euler Equations , 2005 .

[118]  Roger G. Ghanem,et al.  On the construction and analysis of stochastic models: Characterization and propagation of the errors associated with limited data , 2006, J. Comput. Phys..

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

[120]  R. Ghanem,et al.  A stochastic projection method for fluid flow. I: basic formulation , 2001 .

[121]  Jeroen A. S. Witteveen,et al.  An alternative unsteady adaptive stochastic finite elements formulation based on interpolation at constant phase , 2008 .

[122]  O. L. Maître,et al.  Uncertainty propagation in CFD using polynomial chaos decomposition , 2006 .

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

[124]  Daniel M. Tartakovsky,et al.  Numerical Methods for Differential Equations in Random Domains , 2006, SIAM J. Sci. Comput..

[125]  Christoph Schwab,et al.  Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients , 2007 .

[126]  Olivier Le Maitre,et al.  Dveloppement en polynmes de chaos d'un modle lagrangien d'coulement autour d'un profil , 2006 .

[127]  J. Cadafalch,et al.  Verification of Finite Volume Computations on Steady-State Fluid Flow and Heat Transfer , 2002 .

[128]  Knut Petras,et al.  On the Smolyak cubature error for analytic functions , 2000, Adv. Comput. Math..

[129]  Nicholas Zabaras,et al.  Variational multiscale stabilized FEM formulations for transport equations: stochastic advection- , 2004 .

[130]  Omar M. Knio,et al.  A stochastic particle-mesh scheme for uncertainty propagation in vortical flows , 2007, J. Comput. Phys..

[131]  John Faragher,et al.  Probabilistic Methods for the Quantification of Uncertainty and Error in Computational Fluid Dynamic Simulations , 2004 .

[132]  Habib N. Najm,et al.  Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes , 2005, SIAM J. Sci. Comput..

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

[134]  Chris L. Pettit,et al.  Spectral and multiresolution Wiener expansions of oscillatory stochastic processes , 2006 .

[135]  K. Ritter,et al.  Simple Cubature Formulas with High Polynomial Exactness , 1999 .

[136]  D. V. Lyubimov,et al.  Thermosolutal convection in a horizontal porous layer heated from below in the presence of a horizontal through flow , 2008 .

[137]  R. Ghanem.  Probabilistic characterization of transport in heterogeneous media , 1998 .

[138]  G. Taylor,et al.  Sixth International Congress for applied Mechanics , 1946, Nature.

[139]  R. Ghanem,et al.  Multi-resolution analysis of wiener-type uncertainty propagation schemes , 2004 .

[140]  A. O'Hagan,et al.  Bayesian inference for the uncertainty distribution of computer model outputs , 2002 .

[141]  Christoph Schwab,et al.  Sparse Finite Elements for Stochastic Elliptic Problems – Higher Order Moments , 2003, Computing.

[142]  Daniel M. Tartakovsky,et al.  Stochastic analysis of transport in tubes with rough walls , 2006, J. Comput. Phys..

[143]  Timothy G. Trucano,et al.  Validation Methodology in Computational Fluid Dynamics , 2000 .

[144]  Michael Frenklach,et al.  Transforming data into knowledge—Process Informatics for combustion chemistry , 2007 .

[145]  G. Karniadakis,et al.  Long-Term Behavior of Polynomial Chaos in Stochastic Flow Simulations , 2006 .

[146]  Habib N. Najm,et al.  Multi-Resolution-Analysis Scheme for Uncertainty Quantification in Chemical Systems , 2007, SIAM J. Sci. Comput..