An accurate Ghost Cell Immersed Boundary Method for compressible flows with heat transfer

Abstract

In the present study, we analyze and develop Ghost Cell Immersed Boundary Method (GCIBM) formulations for heat transfer between compressible flows and isothermal surfaces. In particular, we focus on the accuracy of the temperature gradient (or Nusselt number) at the immersed boundary under Dirichlet boundary conditions. We first examine the influence of the chosen interpolation scheme for image point reconstruction on the solution accuracy. We demonstrate that the Triangular Shaped Cloud (TSC) interpolation can significantly reduce numerical fluctuations induced by spatial localization errors in comparison with the commonly used bi-linear interpolation. Then, we show that for both interpolation schemes, the standard ghost cell extrapolation treatment for Dirichlet boundary conditions achieves a $2^{\mathrm{nd}}$-order solution for the flow variables, but only a $1^{\mathrm{st}}$-order solution for the Nusselt number. To remedy this issue, we suggest a high-order ghost cell extrapolation treatment that involves two image points. We extensively verify our new high-order extrapolation ghost cell treatment against existing benchmarks from the literature, and compare its performance against the standard ghost cell extrapolation approach. We show that our new high-order ghost cell extrapolation treatment combined with the TSC interpolation scheme achieves a $2^{\mathrm{nd}}$-order accurate solution devoid of numerical fluctuations for the Nusselt number, whereas the standard image point extrapolation procedure leads to a $1^{\mathrm{st}}$-order accurate solution for the Nusselt number. Extensive tests demonstrate these properties for multi-dimensional heat conduction problems and subsonic solutions of the compressible Navier–Stokes equations with heat transfer. Furthermore, for supersonic flows that involve shock waves, we show that a higher than $1^{\mathrm{st}}$-order accurate solution for the Nusselt number can be obtained using the high-order image extrapolation procedure. On the contrary, the standard image point extrapolation procedure leads to an order of accuracy lower than unity for the Nusselt number. Finally, we verify our new method against results from the literature for 3-D solutions of the compressible Navier–Stokes equations with the inclusion of heat transfer.