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An immersed boundary method on Cartesian mesh with local refinement appliced to heat convection problems

The incompressible Navier-Stokes equations with heat transfer are solved by an implicit pressure correction method on Cartesian mesh with local refinement. A simple immersed boundary method is developed to treat arbitrary solid bodies in the flow field. A direct forcing term was added to cells inside the immersed body to enforce the Dirichlet boundary condition for velocity and temperature. This forcing is also assumed to have an active range of one cell size normal to the immersed boundary, acting on fluid cells external to the immersed body. For those cells within the active range, the forcing term is scaled by the normal distance from the cell center to the body surface. The same pressure Poisson equation is applied to the entire flow field without distinguishing whether it is inside or outside the immersed body. Various tests are computed to verify this simple immersed boundary method, including the steady and unsteady forced convection over a circular cylinder and the natural convection over a heated circular cylinder inside a square cavity. INTRODUCTION Recently numerical methods for solving incompressible flows on fixed Cartesian grids are gaining popularity for their relative ease in treating complex immersed bodies [1-10]. The differences among these methods lie on the different ways the boundary conditions for the immersed bodied are enforced. For example, the immersed boundary method [1] and the virtual boundary method [2] simulate the no-slip boundary condition on the body surfaces by adding appropriate forcing terms to the appropriate cells. The force added to the flow field is spread over several grid cells using a delta function-like distribution normal to the fluid-solid interface. In the direct-forcing method [4, 5] the no-slip condition at the fluid-solid interface is used to interpolate the velocity distribution for cells around the solid body without actually solving the fluid equations. The fictitious domain method [6] is a finite-element method that enforces the no-slip boundary condition using Lagrange multiplier. Very recently, Pan [7,8] viewed the domain inside the solid body as being occupied by the same fluid as outside with a prescribed divergence-free velocity field. In this view a fluidbody interface is similar to a fluid-fluid interface commonly encountered in the Volume of Fluid (VOF) method for the twofluid flow problems. Thus a Volume of Body (VOB) function analogous to the VOF function can be used to identify and track the presence of the immersed body. For the grid cells containing the fluid-body interface, a mixture of the “two fluids” based on the fluid volume is assumed. This volume averaging of the velocity automatically enforces the no-slip boundary condition inside the interface cell, but it also smears the fluidbody interface to the size of one cell width. In this work, we tried to improve the representation of the fluid-body interface by using a scaled forcing term. A direct forcing is added to cells inside the body to enforce the Dirichlet boundary condition for the velocity and temperature. This forcing is further assumed to have an effective range of one cell size normal to the body surface, acting on cells external to the immersed body. For those cells within the active range, the amount of added forcing is scaled by a Delta function based on the normal distance from the cell center to the body surface. Test results of this simple scaled direct forcing method are reported in this paper, including the steady and unsteady forced convection over a circular cylinder and the natural convection over a heated circular cylinder inside a square cavity. IMPLICIT PRESSURE CORRECTION METHOD The incompressible Navier-Stoked equations with heat transfer can be written as 0 = ⋅ ∫ CS S d v v v