An implicit immersed boundary method for Robin boundary condition

In this work, a novel boundary condition-enforced immersed boundary method (IBM) for simulating complex thermal–fluid–structure interaction (TFSI) problems with Robin boundary conditions is proposed. In this work, two auxiliary layers of Lagrangian points are introduced and placed within the inner and outer parts of the immersed object to enable the simultaneous evaluation of the temperature and its gradient on the solid surface. The mutual thermal interactions between the immersed object and the surrounding flow are taken into account through the temperature corrections on the Lagrangian points by accurately enforcing the Robin boundary condition on the solid boundary. Hence, a system of linear equations can be formulated to compute the temperature corrections. Subsequently, the temperature corrections are biasedly distributed to the Eulerian points located in the inner auxiliary layer to eliminate the diffusion generated by the smooth delta function on the temperature field outside the solid object. The proposed IBM integrated with the reconstructed thermal lattice Boltzmann flux solver (RTLBFS) is extensively and critically assessed with classical benchmark cases ranging from two-dimensional stationary problems to three-dimensional problems involving moving boundaries to demonstrate the robustness and accuracy of the proposed method for Robin boundary conditions. Results from the present work are extensively corroborated with previous works that adopt different approaches for imposing the Robin boundary condition. It shows that the proposed method not only provides a much more accurate representation of the Robin boundary condition but also introduces a simple implementation for the complex Robin boundary condition within the diffuse interface IBM framework.