An Immersed Boundary Method for Compressible Viscous Flow with Heat Flux and Shear Stress Conditions

A new immersed boundary method for compressible viscous flow based on Peskin's approach is discussed. 

Arbitrary geometry is represented by boundary markers with two types of singular sources: (i) the usual Dirichlet-type sources and (ii) additional Neumann-type sources to incorporate heat flux and shear stress conditions. The formulation is applicable to regular geometries with finite thickness or extent as well as to infinitely thin surface-like geometries (in three-dimensional space). The added singular sources that modify the original Navier-Stokes equations are rigorously related to the jumps of flow variables. The strength of the added singular sources can be solved exactly from the provided boundary conditions under the framework of differential-algebraic equations with the half-explicit Runge-Kutta method without introducing additional stiffness into the system. This includes the Neumann case with two different values of the shear stress or heat flux across an infinitely thin surface. Accuracy is demonstrated analytically and numerically. Two- and three-dimensional examples are carried out to demonstrate convergence. The results are compared with previous numerical and experimental studies and analytical solutions.