Computational Methods for Fluid Dynamics

Abstract

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Simple immersed boundary matrix‐cut‐method for cartesian grids using FV and FD discretizations

“…We next describe the application of the Matrix-Cut methodology for a finite volume (FV) framework applied to a two-dimensional Cartesian grid. To generalize the description, we consider a generic transport equation in an integral form with respect to a conserved variable : [31] S u · n dS = S grad · n dS + q d…”

“…where is the density, u is the velocity vector, is the diffusion coefficient, and q is the source term. The details on FV discretization of this equation can be found in the books by Ferziger and Peric [31] or Date. [32] Using the 5-point discretization stencil for the finite volume method shown in Figure 3, Equation (22) can be written in the following form:…”

“…where [A] is the square sparse coefficient matrix, [ ] is a vector of unknown variables at the CV centres, and [q] is the vector containing the terms on the right-hand side of Equation (23). The details on expressions of coefficients AW, AE, AS, and AN for Equation (23) can be found in the books by Ferziger and Peric [31] or by Date. [32] To solve Equation (25), the boundary conditions must be defined at the outer boundaries of a domain, e.g., see Figure 4a.…”

“…after which, the value of A E must be set to zero (A E = 0). [31] This condition can be obtained after approximating the Neumann boundary condition and inserting it into Equation (23).…”

“…Adopted from: Ferziger and Peric. [31] Implementation of Boundary Conditions for an Immersed Object Before we proceed with the formulation of boundary conditions for an immersed object it should be noted that the way of implementing all types of boundary conditions for domain boundaries is well illustrated in classical CFD books. [31][32][33] However, the differences between direct implementation of immersed boundary conditions and domain boundaries have not been described in the literature, except the Dirichlet boundary condition described in works by Kuipers at al.…”