Applied Mathematics, 2018, 9, 291-312 http://www.scirp.org/journal/am ISSN Online: 2152-7393 ISSN Print: 2152-7385 DOI: 10.4236/am.2018.93023 Mar. 30, 2018 291 Applied Mathematics Two-Dimensional Simulation of the Navier-Stokes Equations for Laminar and Turbulent Flow around a Heated Square Cylinder with Forced Convection Rômulo D. C. Santos 1 , Sílvio M. A. Gama 1 , Ramiro G. R. Camacho 2 1 Department of Mathematics and CMUP, Faculty of Science, University of Porto, Porto, Portugal 2 Institute of Mechanical Engineering, Federal of University of Itajubá, Itajubá, Minas Gerais, Brazil Abstract Few studies jointly investigate thermal and turbulent effects. In general, these subjects are treated separately. The purpose of this paper is to use the Im- mersed Boundary Method (IBM) coupled with the Virtual Physical Model (VPM) to investigate incompressible two-dimensional Newtonian flow around a heated square cylinder at constant temperature on its surface with forced convection and turbulence. The VPM model dynamically evaluates the force that the fluid exerts on the immersed surface and the thermal exchange between both in the Reynolds numbers (Re) window 3 40 Re 5 10 ≤ ≤ × . For simulations of turbulence the Smagorinsky and Spalart-Allmaras models are used. The first model uses the Large Eddy Simulation (LES) methodology and is based on the local equilibrium hypothesis for small scales associated with the Boussinesq hypothesis, such that the energy injected into the spectrum of the turbulence balances the energy dissipated by convective effects. The second model uses the concept Unsteady Reynolds Averaged Navier-Stokes Equations (URANS), with only one transport equation for turbulent viscosity, being calibrated in pressure gradient layers. The goal of this work is to analyse the combination of the heat-transfer phenomena with the turbulence for the thermo-fluid-structure interaction in a square cylinder. For this, it was devel- oped a C/C++ code that requires low computational costs in regards to mem- ory and computer facilities. It is observed that, with the increase of the Rey- nolds number, an increase of the drag coefficient occurs, as well as reinforces the influence of the pressure distribution downstream of the cylinder, which is strongly influenced by the formation and detachment of vortices on the upper and lower sides of the square cylinder. How to cite this paper: Santos, R.D.C., Gama, S.M.A. and Camacho, R.G.R. (2018) Two-Dimensional Simulation of the Navi- er-Stokes Equations for Laminar and Tur- bulent Flow around a Heated Square Cy- linder with Forced Convection. Applied Mathematics, 9, 291-312. https://doi.org/10.4236/am.2018.93023 Received: February 16, 2018 Accepted: March 27, 2018 Published: March 30, 2018 Copyright © 2018 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access