We propose a data-driven filtered reduced order model (DDF-ROM) framework for the numerical simulation of fluid flows. The novel DDF-ROM framework consists of two steps: (i) In the first step, we use ROM projection to filter the nonlinear PDE and construct a filtered ROM. This filtered ROM is low-dimensional but is not closed (because of the nonlinearity in the given PDE). (ii) In the second step, we use data-driven modeling to close the filtered ROM, i.e., to model the interaction between the resolved and unresolved modes. To this end, we use a quadratic ansatz to model this interaction and close the filtered ROM. To find the new coefficients in the closed filtered ROM, we solve an optimization problem that minimizes the difference between the full order model data and our ansatz. We emphasize that the new DDF-ROM is built on general ideas of spatial filtering and optimization and is independent of (restrictive) phenomenological arguments. We investigate the DDF-ROM in the numerical simulation of a 2D ch...

An efficient algorithm is proposed and studied for computing flow ensembles of incompressible magnetohydrodynamic (MHD) flows under uncertainties in initial or boundary data. The ensemble average of J realizations is approximated through a clever algorithm (adapted from a breakthrough idea of Jiang and Layton [23]) that, at each time step, uses the same matrix for each of the J systems solves. Hence, preconditioners need to be built only once per time step, and the algorithm can take advantage of block linear solvers. Additionally, an Elsässer variable formulation is used, which allows for a stable decoupling of each MHD system at each time step. We prove stability and convergence of the algorithm, and test it with two numerical experiments.

The numerical investigation of magneto-hydrodynamic (MHD) mixed convection flow and entropy formation of non-Newtonian Bingham fluid in a lid-driven wavy square cavity filled with nanofluid was investigated by the finite volume method (FVM). The numerical data-based temperature and nanoparticle size-dependent correlations for the Al${}_2$O${}_3$-water nanofluids are used here. The physical model is a two-dimensional wavy square cavity with thermally adiabatic horizontal boundaries, while the right and left vertical walls maintain a temperature of $T_C$ and $T_H$, respectively. The top wall has a steady speed of $u=u_0$. Pertinent non-dimensional parameters such as Reynolds number ($Re=10,100,200,400$), Hartmann number ($Ha=0,10,20$), Bingham number ($Bn=0,2,5,10,50,100,200$), nanoparticle volume fraction ($\varphi =0,0.02,0.04$), and Prandtl number ($Pr=6.2$) have been simulated numerically. The Richardson number $Ri$ is calculated by combining the values of $Re$ with a fixed value of $Gr$, which is the governing factor for the mixed convective flow. Using the Response Surface Methodology (RSM) method, the correlation equations are obtained using the input parameters for the average Nusselt number ($\overline{Nu}$), total entropy generation ($E_s{)}_t$, and Bejan number ($B{e}_{avg}$). The interactive effects of the pertinent parameters on the heat transfer rate are presented by plotting the response surfaces and the contours obtained from the RSM. The sensitivity of the output response to the input parameters is also tested. According to the findings, the mean Nusselt numbers ($\overline{Nu}$) drop when $Ha$ and $Bn$ are increased and grow when $Re$ and $\varphi$ are augmented. It is found that ${\left(E_s\right)}_t$ is reduced by raising $Ha$, but ${\left(E_s\right)}_t$ rises with the augmentation of $\varphi$ and $Re$. It is also found that the $\varphi$ and $Re$ numbers have a positive sensitivity to the $\overline{Nu}$, while the sensitivity of the $Ha$ and $Bn$ numbers is negative.