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

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 multiscale, intermittent and non-
homogeneous fluid and ocean flows. The Dynamically Orthogonal (DO) field equations
provide an efficient time-dependent adaptive methodology to predict the probability density functions of such flows. 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 define
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 finite-volumes are employed in physical space with Total Variation Diminishing schemes for the advection terms.
Other results specific 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 in-
stead of quadratic; (ii) symmetric Total Variation Diminishing-based 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. To verify the correctness of our implementation and study
the properties of our schemes and their variations, a set of stochastic flow benchmarks are
defined including asymmetric Dirac and symmetric lock-exchange flows, lid-driven cavity flows, and flows past objects in a confined channel. Different Reynolds number and
Grashof number regimes are employed to illustrate robustness. Optimal convergence under both time and space refinements is shown as well as the convergence of the probability
density functions with the number of stochastic realizations.