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Modeling for fast calculation of fluid flow ensembles based on time relaxation regularization is studied herein. At each time step, the proposed model requires the storage of a single coefficient matrix with multiple right-hands sides, corresponding to each ensemble member. The time relaxation regularization penalizes the deviation of the fluctuations from the ensemble average. The algorithm is shown to be stable under a time step restriction, which holds provided fluctuations are small enough. Also, the numerical tests show that a grad-div stabilization weakened the condition considerably. Finite element convergence results for the time relaxation ensemble algorithm are studied too. Then, 2D and 3D numerical experiments that support the theoretical results are presented.

Global and localized radial basis function (RBF) meshless methods are compared for solving viscous incompressible fluid flow with heat transfer using structured multiquadratic RBFs. In the global approach, the collocation is made globally over the whole domain, so the size of the discretization matrices scales as the number of the nodes in the domain. The localized meshless method uses a local collocation defined over a set of overlapping domains of influence. Only small systems of linear equations need to be solved for each node. The computational effort thus grows linearly with the number of nodes-the localized approach is slightly more expensive on serial processors, but is highly parallelizable. Numerical results are presented for three benchmark problems-the lid-driven cavity, natural convection within an enclosure, and forced convective flow over a backward-facing stepand results are compared with the finite-element method (FEM) and experimental data.

This paper presents a numerical study of the sensitivity of a fluid model known as time relaxation model with respect to variations of the time relaxation coefficient χ. The sensitivity analysis of this model is utilized by the sensitivity equation method and uses the finite element method along with Crank Nicolson method in the fully discretization of the partial differential equations. We present a test case in support of the sensitivity convergence and also provide a numerical comparison between two different strategies of computing the sensitivity, sensitivity equation method and forward finite differences.

We present a numerical study of the sensitivity of the grad-div stabilization parameter for mixed finite element discretizations of incompressible flow problems. For incompressible isothermal and non-isothermal Stokes equations and Navier-Stokes equations, we develop the associated sensitivity equations for changes in the grad-div parameter. Finite element schemes are devised for computing solutions to the sensitivity systems, analyzed for stability and accuracy, and finally tested on several benchmark problems. Our results reveal that solutions are most sensitive for small values of the parameter, near obstacles and corners, when the pressure is large, and when the viscosity is small.