  Numerical treatment of the problem of two-dimensional viscous fluid flow in
and around circular porous inclusions is considered. The mathematical model is
described by Navier-Stokes equation in the free flow domain $\Omega_f$ and
nonlinear convective Darcy-Brinkman-Forchheimer equations in porous subdomains
$\Omega_p$. It is well-known that numerical solutions of the problems in such
heterogeneous domains require a very fine computational mesh that resolve
inclusions on the grid level. The size alteration of the relevant system
requires model reduction techniques. Here, we present a multiscale model
reduction technique based on the Generalized Multiscale Finite Element Method
(GMsFEM). We discuss construction of the multiscale basis functions for the
velocity fields based on the solution of the local problems with and without
oversampling strategy. Three test cases are considered for a given choice of
the three key model parameters, namely, the Reynolds number ($Re$), the
Forchheimer coefficient ($C$) and the Darcy number ($Da$). For the test runs,
the Reynolds number values are taken to be $Re = 1, 10, 100$ while the
Forchheimer coefficient and Darcy number are chosen as $C= 1, 10$ and $Da =
10^{-5}, 10^{-4}, 10^{-3}$, respectively. We numerically study the convergence
of the method as we increase the number of multiscale basis functions in each
domain, and observe good performance of the multiscale method.
