A fully pressure-velocity coupled immersed boundary method based on the Lagrange multiplier approach

TL;DR

This paper introduces a fully pressure-velocity coupled immersed boundary method that improves simulation accuracy and efficiency for incompressible flows around immersed bodies by using a Lagrange multiplier approach.

Contribution

The novel method integrates pressure, boundary forces, and heat sources as Lagrange multipliers, eliminating the need for projection steps and enhancing computational efficiency.

Findings

Accurately simulates incompressible isothermal and natural convection flows.

Implicitly enforces boundary conditions without extra correction steps.

Validated through extensive verification tests.

Abstract

A new formulation of the immersed boundary method, which facilitates accurate simulation of incompressible isothermal and natural convection flows around immersed bodies and which may be applied for accurate linear stability analysis of the flows, is presented. The method is based on the fully pressure-velocity coupled approach, implicitly satisfying the divergence-free velocity constraint with no need for an extra projection-correction step, which is a significant advantage for the computational efficiency. The method treats pressure, boundary forces, and heat sources as Lagrange multipliers, thereby implicitly providing the kinematic constraints of no-slip and the corresponding thermal boundary conditions for immersed surfaces. Extensive verification of the developed method for both isothermal and natural convection flows is provided.