Immersed Boundary Projection Method for Internal Flows
An extended work on the immersed boundary projection method derived by Taira and Colonius [TK08] and modified by Eldredge [Eld21]. Derivations in this paper closely follow the steps taken in the paper by Eldredge, and the method is applied to internal flows simulations with different boundary conditions: velocity and traction. Masking functions are incorporated into the governing incompressible Navier-Stokes equation by introducing the Heaviside function to allow for the prescription of boundary values at either side of the immersed interface. The discrete system of equations was reformulated into saddle-point form and solved using LU decomposition. Lastly, a prescribed traction term was isolated from the Lagrange multiplier term in the vorticity form of Navier-Stokes so that its value can be assigned for problems with traction (pressure) type boundary conditions. The lid-driven cavity problem and Poiseuille flow were simulated to show application using velocity and pressure conditions respectively. It is shown that the computed results agree well with other studies and exact solutions.