A unified mathematical framework and an adaptive numerical method for fluid–structure interaction with rigid, deforming, and elastic bodies

Abstract

Many problems of interest in biological fluid mechanics involve interactions between fluids and solids that require the coupled solution of momentum equations for both the fluid and the solid. In this work, we develop a mathematical framework and an adaptive numerical method for such fluid–structure interaction (FSI) problems in which the structure may be rigid, deforming, or elastic. We employ an immersed boundary (IB) formulation of the problem that permits us to avoid body conforming discretizations and to use fast Cartesian grid solvers. Rigidity and deformational kinematic constraints are imposed using a formulation based on distributed Lagrange multipliers, and a conventional IB method is used to describe the elasticity of the immersed body. We use Cartesian grid adaptive mesh refinement (AMR) to discretize the equations of motion and thereby obtain a solution methodology that efficiently captures thin boundary layers at fluid–solid interfaces as well as flow structures shed from such interfaces. This adaptive methodology is validated for several benchmark problems in two and three spatial dimensions. In addition, we use this scheme to simulate free swimming, including the maneuvering of a two-dimensional model eel and a three-dimensional model of the weakly electric black ghost knifefish.

Introduction

Problems involving interactions between fluids and solids lead to coupled systems that require the solution of momentum equations for both the fluid and the solid. Common approaches to such problems include methods that use body-fitted meshes and so-called immersed boundary or immersed body methods. Although body-fitted meshes permit sharp resolution of fluid–solid interfaces, methods that employ such discretizations are expensive because they require frequent remeshing [1], [2]. Such approaches also present significant implementation challenges for immersed structures with complex geometries and can be extremely difficult to incorporate into existing fluid solver implementations. The immersed boundary (IB) method [3] does not suffer from these difficulties. The IB approach to such fluid–structure interaction (FSI) problems specifically avoids the need for body-conforming discretizations by accounting for the effect of the solid on the fluid via an additional body force that is added to the (fluid) momentum equation; the solid then moves according to the velocity field computed by the basic fluid solver. Consequently, it is straightforward to employ an IB approach to FSI within existing incompressible flow solvers. In addition, this approach is equivalent to more standard continuum formulations involving jump conditions at fluid–solid interfaces [4], [5]. A drawback of the conventional IB approach is that it does not sharply resolve fluid–solid interfaces. Instead, such interfaces are regularized over a finite region, and high spatial resolution can be required in the vicinity of such interfaces to resolve fluid boundary layers. The need for such localized regions of high resolution has motivated the development of adaptive IB methods [6], [7], [8], [9].

In this work, we develop a mathematical framework and an adaptive numerical method for FSI problems involving rigid, deforming, or elastic structures. For the parts of the bodies with prescribed velocities or deformational kinematics, we use a constraint formulation based upon distributed Lagrange multiplier methods [10], [11], [12], [13], [14], whereas a conventional version of the IB method [3] is employed for the elastic parts of the immersed bodies. Our basic numerical scheme is a fractional-step method, in which we first solve the equations of fluid–structure interaction without imposing any constraints on the motion or kinematics other than the constraint of incompressibility. We then determine the motion of the constrained parts of the body, update the positions of those parts of the body, and correct the material velocity field to account for these constrained motions. The resulting algorithm is reminiscent of Chorin’s original projection method for incompressible flow [15]. In practice, the present algorithm requires an unconstrained fluid solve followed by an additional Poisson solve to ensure that the corrected velocity field remains discretely divergence free. Fast solvers are available for both systems of equations, and the overall computational cost is less than two fluid solves per time step. Moreover, as we demonstrate empirically, this approach yields good momentum conservation and thereby permits the stable use of relatively large time step sizes.

To reduce the computational expense of this methodology, we discretize the fluid equations via a block-structured adaptive mesh refinement (AMR) approach [16], [17], whereby the computational domain is described as a system of nested Cartesian grid levels, and each grid level is comprised of one or more rectangular Cartesian grid patches. This allows us to deploy localized regions of high spatial resolution where they are most needed, such as in the vicinity of fluid–structure interfaces or flow features shed from the immersed body. The locally refined grid is adaptively updated to track such features. Unlike previous adaptive versions of the IB method [6], [7], [8], [9], however, the present scheme permits the immersed body to span multiple grid levels, so that different parts of the immersed body may be associated with different levels of spatial resolution. We apply this adaptive numerical framework to a variety of benchmark FSI problems. We also use it to simulate aquatic locomotion.

Two versions of the IB method have been previously used to simulate aquatic locomotion. In the first approach, which we refer to as the elastohydrodynamic approach, body motions are driven by models of the elastic properties of the body, models of the muscle activation patterns, and/or models of the neuronal activity and tissue electrophysiology. The elastohydrodynamic approach is important for gaining insight into the relationships between muscle physiology and neuronal activity in active swimming. The earliest work using the elastohydrodynamic approach was done by Fauci and Peskin [18]. High-resolution FSI simulations of lamprey swimming that employ a model of neuromuscular coupling were performed by Tytell et al. [19] using an adaptive version of the IB method [7]. Simulations of jellyfish swimming using the elastohydrodynamic approach have been done by Zhao et al. [20] and by Herschlag and Miller [21]. The second approach, which we refer to as the hydrodynamic approach, bypasses the details of the neuromuscular coupling and the elastic properties of the body and instead uses the observed deformational kinematics to understand the hydrodynamics of the swimming [22], [2], [23], [24], [25], [14], [26], [27]. The second approach is suitable for problems in which the aim is to determine swimming velocities or forces generated during swimming. The deformational kinematics data required by the hydrodynamic approach to simulating aquatic locomotion can be obtained from experiments [28], [29], [30], [31], [32], [33], [34].

By adopting an adaptive discretization strategy, we are able to perform extremely high resolution two- and three-dimensional simulations of aquatic locomotion. Herein, we employ the hydrodynamic approach to model the free swimming of eels and of the black ghost knifefish. We consider problems involving both straight swimming and also various turns and maneuvers. We also develop a simple prototype swimmer composed of a rigid head and a flexible tail. The time-dependent elastic properties of the tail drive forward swimming. For the case of the model knifefish, the deformational kinematics are based on experimental data, and an initial validation of the model is obtained by comparing the computed swimming speeds to experimental measurements.