An immersed boundary method for simulation of flow with heat transfer A. Mark ⇑ , E. Svenning, F. Edelvik Fraunhofer–Chalmers Research Centre for Industrial Mathematics, Chalmers Science Park, SE-412 88 Göteborg, Sweden article info Article history: Received 22 August 2011 Received in revised form 6 February 2012 Accepted 10 February 2012 Available online 27 October 2012 Keywords: Natural convection Immersed boundary methods Fluid structure interaction abstract In this work, the hybrid immersed boundary method is extended with immersed boundary conditions for the temperature field. The method is used to couple the flow solver with a shell heat transfer solver. The coupling back to the shell is handled by a heat source, calculated from Fourier’s law. Natural convection in a square cavity with and without a hot circular cylinder, and free air cooling of an electrically heated plate are studied. For all cases an excellent agreement with numerical and experimental data is obtained. The proposed method is very well suited for many industrial applications involving natural convection. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction Natural convection occurs in many industrial applications such as free air cooling of electric components, the draft in a chimney and in ovens used for industrial drying processes. These applica- tions often involve natural convection in complex geometries, heat transfer between fluids and solid surfaces and heat conduction in the solid objects. Models are therefore needed that can handle arbitrary geometries and coupling between fluids and immersed solid objects. Immersed boundary methods are well suited for such applications. The original immersed boundary method (IBM) developed by Peskin [1] enforces the boundary condition explicitly with a force. A discrete Dirac delta function is used to distribute a Lagrangian force from the immersed boundary (IB) to the Eulerian grid. Due to the distribution function the resulting method is only first-order accurate in space. Mohd-Yusof [2,3] developed a momentum forc- ing method that enforces the fluid velocity at the IB by introducing an explicit force in the momentum equations. The force is applied onto the cells lying inside but close to the IB generating a reversed velocity field over the IB. The resulting problems with mass conser- vation are solved by Kim and co-workers [4]. The explicitly formu- lated method is second-order accurate in space and frequently used in the literature. Majumdar and co-workers [5] developed an immersed bound- ary method, which implicitly constrains the velocity of the fluid at the IB with an immersed boundary condition (IBC). The method has potential problems with the weighting coefficients in the boundary condition, which may result in oscillations in the resulting solution. To handle these problems Mark and co-workers [6] developed a hybrid IB method. The method mirrors and extrap- olates the velocity field onto the IB and the resulting fictitious velocity field inside the IB is excluded in the continuity equation. The extrapolation points give improved accuracy compared to a pure mirroring method [7]. The implicit hybrid IBM is second-order accurate in space and generates no unphysical oscillations around the IB. Problems involving natural convection in a geometrically sim- ple domain, such as a square cavity, are described extensively in the literature [8–12]. A geometrically more complex problem arises when natural convection in a cold square enclosure with a heated inner circular cylinder is studied. This problem was ana- lyzed by Kim et al. [13], who performed simulations of natural con- vection for Ra 2 [10 3 ,10 6 ] using a second-order accurate finite volume method. The presence of the cylinder was modeled with the immersed boundary method developed by Kim and Choi [4]. The isothermal boundary condition on the IB was enforced with a heat source term in the temperature equation. In the present work, the fluid flow solver IBOFlow [14] is used. IBOFlow is a finite volume based Navier–Stokes solver for incom- pressible flows. It has previously been used to study flow problems with complex boundaries and fluid–structure interaction [6]. IBO- Flow uses the hybrid IBM to account for the presence of immersed objects in the flow. The Navier–Stokes equations are discretized on a Cartesian octree grid with dynamic refinements. Natural convec- tion is accounted for by solving the transport equation for temper- ature and introducing the Boussinesq approximation in the momentum equations to capture density changes resulting from temperature differences. Heat conduction in thin shells is resolved by applying the finite element method on the triangulated surface of the immersed object. The hybrid IBM is extended with IB 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.09.010 ⇑ Corresponding author. Tel.: +46 (0) 31 7724251. E-mail address: andreas.mark@fcc.chalmers.se (A. Mark). International Journal of Heat and Mass Transfer 56 (2013) 424–435 Contents lists available at SciVerse ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt