Ozen, H. Cagan and Bal, Guillaume 2017. A dynamical polynomial chaos approach for long-time evolution of SPDEs. Journal of Computational Physics, Vol. 343, p. 300.
Zhang, Zhongqiang Yang, Xiu and Lin, Guang 2016. POD-Based Constrained Sensor Placement and Field Reconstruction from Noisy Wind Measurements: A Perturbation Study. Mathematics, Vol. 4, Issue. 4, p. 26.
Tamellini, L. Le Maître, O. and Nouy, A. 2014. Model Reduction Based on Proper Generalized Decomposition for the Stochastic Steady Incompressible Navier--Stokes Equations. SIAM Journal on Scientific Computing, Vol. 36, Issue. 3, p. A1089.
Choi, Minseok Sapsis, Themistoklis P. and Karniadakis, George Em 2014. On the equivalence of dynamically orthogonal and bi-orthogonal methods: Theory and numerical simulations. Journal of Computational Physics, Vol. 270, p. 1.
Cho, H. Venturi, D. and Karniadakis, G. E. 2013. Adaptive Discontinuous Galerkin Method for Response-Excitation PDF Equations. SIAM Journal on Scientific Computing, Vol. 35, Issue. 4, p. B890.
Cordier, Laurent Noack, Bernd R. Tissot, Gilles Lehnasch, Guillaume Delville, Joël Balajewicz, Maciej Daviller, Guillaume and Niven, Robert K. 2013. Identification strategies for model-based control. Experiments in Fluids, Vol. 54, Issue. 8,
Cho, H. Venturi, D. and Karniadakis, G.E. 2013. Karhunen–Loève expansion for multi-correlated stochastic processes. Probabilistic Engineering Mechanics, Vol. 34, p. 157.
Yang, Xiu and Karniadakis, George Em 2013. Reweighted minimization method for stochastic elliptic differential equations. Journal of Computational Physics, Vol. 248, p. 87.
Cheng, Mulin Hou, Thomas Y. and Zhang, Zhiwen 2013. A dynamically bi-orthogonal method for time-dependent stochastic partial differential equations I: Derivation and algorithms. Journal of Computational Physics, Vol. 242, p. 843.
Venturi, D. Tartakovsky, D.M. Tartakovsky, A.M. and Karniadakis, G.E. 2013. Exact PDF equations and closure approximations for advective-reactive transport. Journal of Computational Physics, Vol. 243, p. 323.
Wen, Bin and Zabaras, Nicholas 2012. A multiscale approach for model reduction of random microstructures. Computational Materials Science, Vol. 63, p. 269.
Yang, Xiu Choi, Minseok Lin, Guang and Karniadakis, George Em 2012. Adaptive ANOVA decomposition of stochastic incompressible and compressible flows. Journal of Computational Physics, Vol. 231, Issue. 4, p. 1587.
El-Amrani, Mofdi Seaid, Mohammed and Zahri, Mostafa 2012. A stabilized finite element method for stochastic incompressible Navier–Stokes equations. International Journal of Computer Mathematics, Vol. 89, Issue. 18, p. 2576.
Venturi, D. Choi, M. and Karniadakis, G.E. 2012. Supercritical quasi-conduction states in stochastic Rayleigh–Bénard convection. International Journal of Heat and Mass Transfer, Vol. 55, Issue. 13-14, p. 3732.
Venturi, D. and Karniadakis, G.E. 2012. New evolution equations for the joint response-excitation probability density function of stochastic solutions to first-order nonlinear PDEs. Journal of Computational Physics, Vol. 231, Issue. 21, p. 7450.
We present a new compact expansion of a random flow field into stochastic spatial modes, hence extending the proper orthogonal decomposition (POD) to noisy (non-coherent) flows. As a prototype problem, we consider unsteady laminar flow past a circular cylinder subject to random inflow characterized as a stationary Gaussian process. We first obtain random snapshots from full stochastic simulations (based on polynomial chaos representations), and subsequently extract a small number of deterministic modes and corresponding stochastic modes by solving a temporal eigenvalue problem. Finally, we determine optimal sets of random projections for the stochastic Navier–Stokes equations, and construct reduced-order stochastic Galerkin models. We show that the number of stochastic modes required in the reconstruction does not directly depend on the dimensionality of the flow system. The framework we propose is general and it may also be useful in analysing turbulent flows, e.g. in quantifying the statistics of energy exchange between coherent modes.
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