Semi-implicit direct forcing immersed boundary method for incompressible viscous thermal flow problems: a Schur complement approach

TL;DR

This paper introduces a semi-implicit immersed boundary method using a Schur complement approach for simulating incompressible viscous thermal flows with high accuracy and adaptability to existing solvers.

Contribution

It presents a novel semi-implicit direct forcing immersed boundary method that enforces boundary conditions via a Schur complement, adaptable to standard pressure-velocity solvers.

Findings

Accurately enforces boundary conditions across a wide Rayleigh number range.

Successfully simulates complex 3D natural convection flows.

Results compare favorably with existing literature.

Abstract

An extended immersed boundary method utilizing a semi-implicit direct forcing approach for the simulation of confined incompressible viscous thermal flow problems is presented. The method utilizes a Schur complement approach to enforce the kinematic constraints of no-slip and the corresponding thermal boundary conditions for immersed surfaces. The developed methodology can be straightforwardly adapted to any existing incompressible time marching solver based on a segregated pressure-velocity coupling. The method accurately meets the thermal and the no-slip boundary conditions on the surfaces of immersed bodies for the entire range of Rayleigh numbers $10^3\leqslant Ra\leqslant10^6$. Strategies for further increasing the computational efficiency of the developed approach are discussed. The method has been extensively verified by applying it for the simulation of a number of representative fully 3D confined natural convection steady and periodic flows. Complex dynamic phenomena typical of this kind of flow including vortical structures and convection cells and instability characteristics, were simulated and visualized and the results were found to compare favorably with results known from literature.