The current work focuses on the development and application of a new finite volume immersed boundary method (IBM) to simulate threedimensional fluid flows and heat transfer around complex geometries. First, the discretization of the governing equations based on the secondorder 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 surfacemounted 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 scaleresolving simulations of turbulence as well.
Computational fluid dynamics (CFD) has reached a development level at which it is routinely applied in the industrial environment, at least for singlephase 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 everincreasing 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 [
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 [
In the
In the second class of IBM, the
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 [
A new cutcell approach is described in this section and implemented in the inhouse code PSIBOIL [
The governing equations are integrated in time using a semiimplicit projection method [
Diagonally preconditioned conjugate gradient method is used to solve (
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 edgetriangle intersections are then used to define the cutting plane for cells
Procedure for immersing a body into a Cartesian grid: (a) intersecting the edges of cell
Since we are using a staggered grid approach, we have four finite volume grids: one for the scalar variables and one each for the three velocity components. To cope with immersed boundaries, we cut all three grids separately, using the procedure outlined in Figure
Each grid is cut separately: (a) scalar cell and (b), (c), and (d) momentum cells in
Once the scalar cells (pressure and temperature cells) are cut, they are classified into three sets:
The classification of momentum (velocity) cells, whether they are ON (in the fluid) or OFF (in the immersed body), is related to the scalar cells. For instance, the
Figure
A 2D computational domain with classified cells: (a) scalar cell centers and (b) sets. The purple shaded area represents immersed body.
Cuts performed on the computational cells change their geometrical properties, and this has to be incorporated in the discretized system of equations. Modification of the inertial terms for the presence of an immersed body is straightforward. For a cell containing an immersed body, the inertial term, in terms of a generalized variable
Advective terms for the immersed body are affected by the changes in cell face areas. For each cell face which is cut, the fraction remaining in the fluid part of the cell is first estimated. As an example, the cell face at
A cell face,
The fraction of the cell face area remaining in the fluid is
The diffusion terms for the case of an immersed body can be written as follows:
For the evaluation of the diffusion terms, modified for the presence of the immersed body (
(a) Both cells are in the fluid. (b) Cell
The above treatment for the diffusion terms is valid for the momentum equations and for the heat equation when fixing the velocity or the temperature to a predefined value is required. Another approach for the diffusion term in the heat equation should be taken in case conduction is allowed inside the solid. For instance, when both cells
The corrections for the presence of the immersed body, defined for the inertial term by (
The discrete form of the velocity projection equation (
It should be noted that the velocity is only updated for the momentum cells which are inside the fluid (i.e., when the pressure scalar cells
As a first step to assess the performance of our immersed boundary method, we solve the heat conduction equation with a source term:
Periodic boundary condition in the
The computational domain used to solve the heat conduction equation. The green cells belong to the “solidsource” domain and the purple cells to the immersed body (“solid” domain).
The thermal conductivity
The diffusion term was discretized in time using CrankNicolson scheme and in space using the secondorder central difference scheme. Equation (
Temperature profile inside the domain. The vertical black line corresponds to the interface between the “solidsource” and body domains.
In this section, we validate our immersed boundary method for natural convection flow in square domain named “fluid.” The computational domain is similar to the one used in Section
Periodic boundary condition for
Noslip boundary condition for
Results of averaged Nusselt number and maximum vertical dimensionless velocity on the horizontal midplane and its location and maximum horizontal dimensionless velocity on the vertical midplane and its location are compared against the benchmark data published in [
Results are presented in Table
Comparison between the IBM and the benchmark (data) solutions for the natural convection simulation in a square cavity.
Ra 






IBM  data  IBM  data  IBM  data  IBM  data  

1.119  1.118  2.249  2.243  4.528  4.519  8.859  8.800 

3.645  3.649  16.220  16.178  34.755  34.730  64.854  64.630 

0.816  0.813  0.824  0.823  0.855  0.855  0.847  0.850 

3.732  3.697  19.766  19.617  68.709  68.590  219.696  219.360 

0.179  0.178  0.119  0.119  0.067  0.066  0.0418  0.0379 
Temperature profile for natural convection flow in square cavity with Ra =
In this section, the simulation of natural convection in a square enclosure of length one with an immersed circular cylinder of radius
The domain (Figure
A section of the mesh used to simulate natural convection of heated cylinder. The green cells belong to the fluid domain and the purple cells to the immersed cylinder.
The governing equations are solved until steady state is reached. The local Nusselt number (
Local Nusselt number distribution along the surface of the square enclosure for Ra =
Comparison of isothermals contours simulated by our IBM approach (coloured figures) and by Kim et al. [
Comparison of velocity streamlines coloured by temperature simulated by our IBM approach (cylinder shown in red) and by Kim et al. [
In this section, we validate our immersed boundary method for isothermal turbulent flow around a surfacemounted matrix of cubes. As reference data, LDA measurements from [
The representation of the problem domain including all relevant dimensions is sketched in Figure
Schematic representation of the matrix of cubes test case with all relevant dimensions. The computational box is represented by the dashed lines, and coordinate system origin is indicated.
The working fluid is air, with a bulk velocity of
The computed mean and fluctuating streamwise and spanwise velocity profiles are compared with measurements from [
Comparison of computed and measured velocity at
Same as in Figure
Same as in Figure
In this paper, a new immersed boundary method (IBM) based on the
The verification tests comprise steadystate heat conduction equation with a source term solved inside a cavity coupled with the solution of the heat conduction equation without source term inside the immersed body. The parabolic temperature profile obtained inside the cavity compares very well to the analytical solution, and temperature profile inside the immersed body was as expected linear. Next simulations of natural convection in cavity with and without immersed cylinder for different Rayleigh numbers were presented. Results in terms of Nusselt number distribution, isothermal contours, and velocity streamlines were in excellent agreement with results published in [
The general benefits of the IBM approach over bodyfitted grid methods are also shared by the method proposed in this work. The structure of the Cartesian grid allows the usage of efficient linear solvers (multigrid) and easier optimization of the computer code for various computational platforms. Furthermore, the implementation of stateoftheart physical models is a straightforward task on a Cartesian grid. Finally, using an IBM rids the analyst from the burden of a bodyfitted mesh generation procedure.
The authors declare that there are no conflicts of interest regarding the publication of this paper.