An extension of the immersed boundary method based on the distributed Lagrange multiplier approach

Abstract

An extended formulation of the immersed boundary method, which facilitates simulation of incompressible isothermal and natural convection flows around immersed bodies and which may be applied for linear stability analysis of the flows, is presented. The Lagrangian forces and heat sources are distributed on the fluid–structure interface. The method treats pressure, the Lagrangian forces, and heat sources as distributed 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 2D flows is provided. Strategies for adapting the developed approach to realistic 3D configurations are discussed.

Introduction

Since the immersed boundary (IB) method was first introduced by Peskin [1], the IB method and its modifications have become very popular numerical tools for describing the flow around moving or deformable bodies with complex surface geometry [2], [3]. An arbitrary immersed object, whose geometry does not, in general, have to conform to an underlying spatial grid, is typically determined by a set of Lagrangian points. At the Lagrangian points, appropriate volumetric (or surface) forces are applied to enforce no-slip velocity boundary conditions on the body surface. These forces appear as additional unknown variables, whose values – along with those for the pressure and velocity fields – are obtained by solving the Navier Stokes (NS) equations. Since the location of the Lagrangian boundary points does not necessarily coincide with the underlying spatial discretization, interpolation and regularization operators must be defined to convey information to and from the body surface.

An accurate calculation of the Lagrangian forces, precisely enforcing the no-slip constraint on the surface of the immersed body, is the key issue in any IB formulation. Lagrangian forces acting on rigid bodies (as well as on bodies with a prescribed surface motion) can be treated explicitly or implicitly. Historically, explicit treatment of Lagrangian forces has received the most attention, giving rise to the direct forcing approach, introduced by Mohd-Yusof [4] and coauthors [5], and to the immersed interface method (IIM), introduced by Lee and LeVeque [6] and revisited by Linnick and Fasel [7]. The direct forcing approach has recently been extended to thermal flow problems, see e.g. [8], [9], [10], [11], by adding an energy equation along with the appropriate volumetric heat sources at the Lagrangian points. The direct forcing approach is not a standalone solver; rather, it may be viewed as a feature that can be easily plugged into an existing time marching solver, typically developed for the solution of NS equations on structured grids in rectangular domains. The procedure does not require any significant modifications to the existing time marching solver, which explains why the direct forcing approach is so popular. However, the direct forcing approach has a number of drawbacks. First, the no-slip condition is explicitly enforced on the intermediate non-solenoidal velocity field, whereas the divergence-free velocity field is calculated afterwards, after a projection–correction step. Second, it should be stressed that even if the NS equations are exactly solved by the projection method, resulting in a solenoidal velocity field on the Eulerian grid, the velocity interpolated to the Lagrangian points is not necessarily divergence free, which may result in a local mass leakage through the boundaries of the immersed body. Third, a pointwise local calculation of the Lagrangian forces and heat sources does not take into account their mutual interaction, which contradicts the elliptic character of the NS equations. A number of techniques have been developed in the past decade to improve the accuracy of the direct forcing approach. Worth mentioning here are the works of Ren et al. [9], [10], who proposed an implicit evaluation of all the Lagrangian forces and heat sources by assembling them into a single system of equations. Another approach is due to Kempe et al. [12], [13], who introduced additional iterations to enhance Euler–Lagrange coupling, thereby providing a substantially more accurate imposition of the boundary conditions on the immersed body surface.

A coupled scheme in which the momentum equations are implicitly coupled with the Lagrangian forces and heat sources and simultaneously solved as a whole system offers an alternative to the direct forcing approach. The closure of this new system is achieved by adding equations interpolating the Eulerian velocity and the temperature fields on the surface of the immersed body to enforce the prescribed boundary conditions. In this setup, the Lagrangian forces and heat sources distributed on the fluid–structure interface can be seen as distributed Lagrange multipliers, enforcing velocity and temperature constraints on the surface of the immersed body similarly to the pressure comprising distributed Lagrange multiplier that acts to enforce the solenoidal constraint on the velocity field. The power of the coupled Lagrange multiplier approach is that it can be straightforwardly adapted to various numerical methods and applications in fluid mechanics, providing accurate and physically substantiated results. Chronologically, the idea was first expressed in the distributed Lagrange multiplier method (DLM) of Glowinski et al. [14], who used a variational principle framework for discretization of the NS equations and applied it for the simulation of 2D flow around moving disc [14]. More recently, the method was successfully extended to the simulation of particulate flows [15], [16], [17] and to the simulation of fluid/flexible-body interactions [18]. An additional impact on the active development of the coupled Lagrange multiplier approach was due to the work of Taira and Colonius [19], who combined the coupled IB method with a projection approach to satisfy the divergence-free and no-slip kinematic constraints. That study was further successfully implemented for the investigation of steady blowing into separated flows behind low-aspect-ratio rectangular wings [20]; for prediction of the natural convection heat transfer and buoyancy of a hot air balloon [21]; for the simulation of rigid-particle-laden flows [22]; for investigation of the forces and unsteady flow structures associated with harmonic oscillations of an airfoil [23]; and recently for simulating the dynamic interactions between incompressible viscous flows and rigid-body systems [24]. The latest theoretical developments of the coupled Lagrange multiplier approach can be found in two recent studies: Kallemov et al. [25] developed a novel (IB) formulation for modeling flows around fixed or moving rigid bodies suitable for a broad range of Reynolds numbers, including steady Stokes flow, and Stein et al. [26] established immersed boundary smooth extension (IBSE) method, which demonstrates fourth- and third-order pointwise convergence for Dirichlet and Neumann problems, respectively.

The present paper reports on our ongoing effort aimed at extension of the coupled Lagrange multiplier approach to problems involving buoyancy-driven flows, steady-state non-Stokes flows and linear stability analysis of the flows in the presence of immersed bodies of arbitrary shape. To demonstrate the new capabilities of the coupled Lagrange multiplier approach, we utilize the previously developed fully pressure–velocity coupled direct solver (FPCD) [27] as a computational platform. The idea is similar to that established by Taira and Colonius [19], the only differences being that the pressure–velocity coupling is implemented by LU-decomposition of the full Stokes operator instead of by the projection approach. This allows us to formulate a full Jacobian operator to compute the steady-state solution and then to conduct a linear stability analysis by a shift-invert Arnoldi iteration. To the best of our knowledge, to date the only available approach embedding IB functionality into a linear stability analysis is that due to Giannetti and Luchini [28], who utilized an adjoint NS operator (in addition to the direct one) to couple between the immersed body and the surrounding isothermal flow. The present approach does not involve an adjoint NS operator, which is an advantage for computational efficiency.

The paper is organized as follows. In section 2, the numerical formulation of the developed methodology is presented. The section includes an introductory description of the previously developed FPCD solver (section 2.1), the concepts of IB formalism, based on the Lagrange multipliers approach (section 2.2), a detailed description of the time marching solver developed in this work (section 2.3), the steady-state solver (section 2.4) and the linear stability solver (section 2.5). An extended discussion of the pros and cons of the presented approach, including the general strategies for the further enhancement of the established methodology, is presented in section 2.6. Section 3 presents a detailed verification of all the developed solvers for incident and natural convection incompressible 2D flows. The final section presents a summary and the main conclusions of the study.