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International Journal of Chemical Engineering

Volume 2017 (2017), Article ID 1726519, 14 pages

https://doi.org/10.1155/2017/1726519

## A Three-Dimensional, Immersed Boundary, Finite Volume Method for the Simulation of Incompressible Heat Transfer Flows around Complex Geometries

^{1}Future Cities Laboratory, Singapore-ETH Centre, 1 Create Way, No. 06-01 CREATE Tower, Singapore 138602^{2}Department of Architecture, ETH Zurich, Building HIT, Wolfgang-Pauli Str. 27, 8093 Zurich, Switzerland^{3}Nuclear Energy and Safety Department, Paul Scherrer Institute (PSI), 5232 Villigen, Switzerland

Correspondence should be addressed to Hassan Badreddine

Received 20 December 2016; Revised 3 April 2017; Accepted 30 April 2017; Published 19 June 2017

Academic Editor: Michael Harris

Copyright © 2017 Hassan Badreddine et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The current work focuses on the development and application of a new finite volume immersed boundary method (IBM) to simulate three-dimensional fluid flows and heat transfer around complex geometries. First, the discretization of the governing equations based on the second-order finite volume method on Cartesian, structured, staggered grid is outlined, followed by the description of modifications which have to be applied to the discretized system once a body is immersed into the grid. To validate the new approach, the heat conduction equation with a source term is solved inside a cavity with an immersed body. The approach is then tested for a natural convection flow in a square cavity with and without circular cylinder for different Rayleigh numbers. The results computed with the present approach compare very well with the benchmark solutions. As a next step in the validation procedure, the method is tested for Direct Numerical Simulation (DNS) of a turbulent flow around a surface-mounted matrix of cubes. The results computed with the present method compare very well with Laser Doppler Anemometry (LDA) measurements of the same case, showing that the method can be used for scale-resolving simulations of turbulence as well.

#### 1. Introduction

Computational fluid dynamics (CFD) has reached a development level at which it is routinely applied in the industrial environment, at least for single-phase flows. Grid generation techniques and methods for discretization of governing equations all play important roles in successful application of CFD. Clearly, flow simulation is facilitated by the ever-increasing computational power on the one hand and the development of more efficient numerical methods on the other. Prominent examples of these algorithmic advances are the multigrid methods [1] and direct solvers based on Fast Fourier Transforms (FFTs). The most efficient numerical algorithms are usually defined for structured orthogonal grids, suitable only for problems occurring in simple, canonical geometries. Compared to unstructured grids, structured grids take less computational time and data storage.

Flow simulations around complex geometries can be achieved in two ways: either by using unstructured codes already applicable for complex geometries or by tailoring the existing highly efficient and/or highly accurate structured methods to nontrivial geometries. The former approach has already been adopted by commercial CFD vendors and, as such, represents the state of the art. Indeed, leading commercial CFD solvers are based on unstructured grid methods and can be applied to complex geometries. The latter approach can go in two directions. One possibility is to define the underlying highly efficient/accurate method to boundary fitted grids [2, 3]. The other possibility is to leave the basic numerical method in its original form, that is, to discretize the system of governing equations using the original method on orthogonal grids and then to make the necessary changes to include the effects of a complex surface immersed in the computational domain. Such methods, usually referred to as immersed boundary methods (IBMs), can conveniently be divided into two broad classes (*forcing* and* cut-cells methods*), depending on the way the boundary conditions are imposed on the immersed body.

In the* forcing* methods, one can find the continuous forcing approaches [4–7] where a continuous force is added to the governing equations before discretization, making the formulation of such approaches independent of the spatial discretization. On the other hand, in the discrete forcing approaches [8–12], the governing equations are first discretized on the grid neglecting the presence of the immersed body. Then interpolation schemes are used for the grid points at the vicinity of the immersed body in such a way that the required boundary condition at the interface is satisfied. Compared to continuous forcing approaches, discrete forcing approaches have the advantage of allowing direct control over stability and numerical accuracy.

In the second class of IBM, the* cut-cells* methods, once the body is immersed in the domain, and the intersections between computational cells and immersed body are evaluated and the fraction of the cell volumes and cell face surfaces are computed. This procedure is required in order to modify the discretized system of equations for the truncated cells for the purpose of imposing the boundary conditions on the immersed body. Developments in this area include [13–16]. A comparative overview on IBM approaches is given in [17, 18].

Several works exist in the literature regarding extending IBM approaches to solve the energy equation around complex geometries by imposing Dirichlet and Neumann boundary conditions [19–27]. Conjugate heat transfer simulations, where the energy equation is also solved inside the immersed body, using IBM were also reported in [28–31]. However, to the best of the authors’ knowledge, most of these works are based on the extension of the IBM* forcing* method to heat transfer. In this work, the implementation and development of a new IBM based on a* cut-cell* approach to simulate heat transfer flow will be presented.

#### 2. Numerical Method

A new cut-cell approach is described in this section and implemented in the in-house code PSI-BOIL [32]. In this work, we assume incompressibility of fluid and constant physical properties. The governing equations are the Navier-Stokes equations and the energy balance equation written here in integral formulation:In the above equations, is the density,** u** is the velocity vector, is the kinematic viscosity, is the temperature, is the thermal conductivity, and is the specific heat capacity. represents the Boussinesq approximation. is the cell volume and is the area vector of cell surface.

The governing equations are integrated in time using a semi-implicit projection method [33]. For the spatial discretization, the orthogonal staggered finite volume method is used, with the second-order central difference being used for the diffusive terms and advection terms. Momentum equations are first solved for a tentative velocity field, , which is not divergence-free (i.e., ):Here, , , and represent the advective, viscous, and pressure terms in the momentum equations (see (4)), respectively. The superscript represents current time step and indicates Adams-Bashforth time discretization scheme, whereas indicates an implicit time level. Once the tentative velocity field is available, the algorithm proceeds by the solution of the pseudo-pressure elliptic equation:The final step of the algorithm is projection of the velocity onto a divergence-free field :As argued in [34], the pressure in this method converged only at a first-order rate. To converge at a second-order term, the pressure should be updated as follows:

Diagonally preconditioned conjugate gradient method is used to solve (4) and a variant of the additive correction multigrid method [35] is used to solve (5). However, it should be noted that multigrid methods, in certain circumstances, could be inefficient and require large computational time [22]. Other solvers such as FFT [36] could be also used.

##### 2.1. Modification for an Immersed Body

###### 2.1.1. Cutting the Cells

In the first step of our immersed body approach, the immersed body is imported from a file in stereolithographic (STL) format. STL files define triangulated surfaces by the coordinates of the triangle’s vertices and normal surface vectors. The next step is to construct the intersection between the edges of the Cartesian grid cell with the triangles of the immersed body. The edge-triangle intersections are then used to define the cutting plane for cells (Figure 1).