### Hourglass Control for the Stabilized Finite Element Solution of Coupled Incompressible Viscous Flows and Heat Transfer

*André L. Rossa, José J. Camata, Alvaro L. G. A. Coutinho*

#### Abstract

This work presents an implementation of a stabilized ﬁnite element formulation for incompressible viscous ﬂow coupled with the advective-diffusive temperature transport equation through the Boussinesq approximation approach. For solving of the incompressible Navier-Stokes equations we use the Streamline Upwind Petrov-Galerkin/Pressure Stabilized Petrov-Galerkin (SUPG/PSPG) formulation and for the advective transport equation the SUPG formulation is employed. A comparison of the computational performance between full Gaussian and reduced (centre of the element) integrations for the isoparametric 8-node hexahedron element is presented. A h-stabilization for both advective and viscous/diffusive terms is used to control the spurious hourglass modes introduced by under integrating the element. The implementation has been performed using the libMesh Finite Element Method (FEM) library (http://libmesh.sourceforge.net) which provides support for adaptive mesh reﬁnement and coarsening (AMR/C) and parallel computations. A veriﬁcation is made using the Kim-Moin problem and the computational performance is evaluated solving a natural convection problem in a parallel machine.

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