## Numerical Schemes and Studies 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 multi-scale, intermittent and non-homogeneous fluid and ocean flows, and other
non-linear dynamical systems. The Dynamically Orthogonal (DO) field 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 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 nite-volumes are employed in physical space with new advection
schemes based on Total Variation Diminishing methods. 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 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 specific 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 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.