Numerical Analysis and Testing of Efficient Ensemble Eddy Viscosity Algorithms for high-Reynolds-number Stochastic Flow Problems

In this paper, we first propose a continuous Ensemble Eddy Viscosity (EEV) model for stochastic flow problems and then introduce a family of fully discrete, grad-div–regularized, efficient ensemble parameterized schemes for this model. The linearized Implicit-Explicit (IMEX) EEV generic algorithm shares a common coefficient matrix for each realization per time-step, but with different right-hand side vectors, which reduces the computational cost and memory requirements to the order of solving deterministic flow problems. Two family members of the proposed time-stepping algorithm are analyzed and proven to be stable. It is found that one is first-order and the other is second-order accurate in time for any stable finite element pairs. Avoiding the discrete inverse inequality, the optimal convergence of both schemes is proven rigorously for both 2D and 3D problems. For appropriately large grad-div parameters, both schemes are unconditionally stable and allow weakly divergence-free elements. The convergence rates are verified numerically using manufactured solutions with high expected Reynolds numbers E[Re]=103,104, 105 ,and 106. For various high E[Re], the schemes are implemented on benchmark problems and are found to perform well. These include a 2D channel flow over a unit step problem, a 2D channel flow past a cylinder problem, a 2D Regularized Lid-Driven Cavity (RLDC) problem examined using a non-intrusive Stochastic Collocation Method (SCM), and a 3D RLDC problem.