One-side diffuse-interface immersed boundary method for compressible flows

In this work, within the framework of the continuous forcing immersed boundary method (IBM), a novel one-side diffuse-interface IBM is proposed for the simulation of compressible viscous flows around complex geometries, where the flow corrections are implicitly evaluated by solving two linear systems to accurately satisfy the Dirichlet and Neumann boundary conditions. Moreover, the Lagrangian corrections are biasedly distributed on the Eulerian points inside the solid domain through a novel spreading scheme, which effectively eliminates the diffusion effects introduced by the regularization delta function. Consequently, the non-physical pressure jump in the traditional continuous forcing IBM around the immersed object is eradicated. The instantaneous discretization errors of the Dirichlet and Neumann boundary conditions are negligible and close to the machine round-off, indicating that the pressure and thermal boundary conditions are accurately enforced on the immersed object. The proposed diffuse interface IBM integrated with the gas kinetic flux solver (GKFS) can achieve a second-order discretization accuracy in space. The proposed numerical approach is validated with several classical benchmarks, and the results agree well with those of the sharp interface method and body-fitted approach, demonstrating that the proposed diffuse interface IBM can accurately predict the flow characteristics around the immersed object, such as the shockwave locations, the pressure and temperature distributions. Moreover, the proposed diffuse interface IBM is extended to simulate compressible viscous flows around the immersed object with complex geometries, where good agreements between the present results and the body-fitted results are achieved as well.