Modeling of stochastic dynamics of time-dependent flows under high-dimensional random forcing

In this numerical study the effect of high-dimensional stochastic forcing in time-dependent flows is investigated. To efficiently quantify the evolution of stochasticity in such a system, the dynamically orthogonal method is used. In this methodology, the solution is approximated by a \emph{generalized} Karhunen-Loeve (KL) expansion in the form of $\mathbf{u}(\mathbf{x},t;\omega) = \overline{\mathbf{u}}(\mathbf{x},t) + \sum_{i=1}^{N} \mathbf{y}_i(t; \omega) \mathbf{u}_i(\mathbf{x},t)$, in which $\overline{\mathbf{u}}(\mathbf{x},t) $ is the stochastic mean, the set of $\mathbf{u}_i(\mathbf{x},t)$'s is a deterministic orthogonal basis and $\mathbf{y}_i(t; \omega)$'s are the stochastic coefficients. Explicit evolution equations for $\overline{\mathbf{u}}$, $\mathbf{u}_i$ and $\mathbf{y}_i$ are formulated. The elements of the basis $\mathbf{u}_i(\mathbf{x},t)$'s remain orthogonal for all times and they evolve according to the system dynamics to capture the energetically dominant stochastic subspace. We consider two classical fluid dynamics problems: (1) flow over a cylinder, and (2) flow over an airfoil under up to one-hundred dimensional random forcing. We explore the interaction of intrinsic with extrinsic stochasticity in these flows.