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Numerical Schemes and Studies for Dynamically Orthogonal Equations of Stochastic Fluid and Ocean Flows

Ueckermann, M.P., P.F.J. Lermusiaux and T.P. Sapsis, 2011. Numerical Schemes and Studies for Dynamically Orthogonal Equations of Stochastic Fluid and Ocean Flows. MSEAS Report-11, August 2011.

The quantification of uncertainties is critical when systems are nonlinear and have uncertain terms in their governing equations or are constrained by limited knowledge of initial and boundary conditions. Such situations are common in multi-scale, intermittent and non-homogeneous fluid and ocean flows, and other non-linear dynamical systems. The Dynamically Orthogonal (DO) fi eld equations provide an efficient time- dependent adaptive methodology to predict the probability density functions of such dynamics. The present work derives efficient computational schemes for the DO methodology applied to unsteady stochastic Navier- Stokes and Boussinesq equations, and illustrates and studies the numerical aspects of these schemes. Semi- implicit projection methods are developed for the mean and for the orthonormal modes that defi ne a basis for the evolving DO subspace, and time-marching schemes of first to fourth order are used for the stochastic coefficients. Conservative second-order nite-volumes are employed in physical space with new advection schemes based on Total Variation Diminishing methods. Other results speci fic to the DO equations include: (i) the definition of pseudo-stochastic pressures to obtain a number of pressure equations that is linear in the subspace size instead of quadratic; (ii) symmetric advection schemes for the stochastic velocities; (iii) the use of generalized inversion to deal with singular subspace covariances or deterministic modes; and (iv) schemes to maintain orthonormal modes at the numerical level. While (i) and (ii) are specifi c to fluid flows, (iii) and (iv) are important for any system of equations discretized using the DO methodology. To verify the correctness of our implementation and study the properties of our schemes and their variations, a set of stochastic flow benchmarks are defi ned including asymmetric Dirac and symmetric lock-exchange flows, lid-driven cavity flows, and flows past objects in a confi ned channel. Di fferent Reynolds number and Grashof number regimes are employed to illustrate robustness. Optimal convergence under both time and space refi nements is shown as well as the convergence of the probability density functions with the number of stochastic realizations.