The term “immersed boundary method” was first used in reference to a method developed by Peskin (1972) to simulate cardiac mechanics and associated blood flow. The distinguishing feature of this method was that the entire simulation was carried out on a Cartesian grid, which did not conform to the geometry of the heart, and a novel procedure was formulated for imposing the effect of the immersed boundary (IB) on the flow. Since Peskin introduced this method, numerous modifications and refinements have been proposed and a number of variants of this approach now exist. In addition, there is another class of methods, usually referred to as “Cartesian grid methods,” which were originally developed for simulating inviscid flows with complex embedded solid boundaries on Cartesian grids (Berger & Aftosmis 1998, Clarke et al. 1986, Zeeuw & Powell 1991). These methods have been extended to simulate unsteady viscous flows (Udaykumar et al. 1996, Ye et al. 1999) and thus have capabilities similar to those of IB methods. In this review, we use the term immersed boundary (IB) method to encompass all such methods that simulate viscous flows with immersed (or embedded) boundaries on grids that do not conform to the shape of these boundaries. Furthermore, this review focuses mainly on IB methods for flows with immersed solid boundaries. Application of these and related methods to problems with liquid-liquid and liquid-gas boundaries was covered in previous reviews by Anderson et al. (1998) and Scardovelli & Zaleski (1999). Consider the simulation of flow past a solid body shown in Figure 1a. The conventional approach to this would employ structured or unstructured grids that conform to the body. Generating these grids proceeds in two sequential steps. First, a surface grid covering the boundaries b is generated. This is then used as a boundary condition to generate a grid in the volume f occupied by the fluid. If a finite-difference method is employed on a structured grid, then the differential form of the governing equations is transformed to a curvilinear coordinate system aligned with the grid lines (Ferziger & Peric 1996). Because the grid conforms to the surface of the body, the transformed equations can then be discretized in the

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Immersed boundary methods

The numerical simulation of flows with interfaces and free-surface flows is a vast topic, with applications to domains as varied as environment, geophysics, engineering, and fundamental physics. In engineering, as well as in other disciplines, the study of liquid-gas interfaces is important in combustion problems with liquid and gas reagents. The formation of droplet clouds or sprays that subsequently burn in combustion chambers originates in interfacial instabilities, such as the Kelvin-Helmholtz instability. What can numerical simulations do to improve our understanding of these phenomena? The limitations of numerical techniques make it impossible to consider more than a few droplets or bubbles. They also force us to stay at low Reynolds or Weber numbers, which prevent us from finding a direct solution to the breakup problem. However, these methods are potentially important. First, the continuous improvement of computational power (or, what amounts to the same, the drop in megaflop price) continuously extends the range of affordable problems. Second, and more importantly, the phenomena we consider often happen on scales of space and time where experimental visualization is difficult or impossible. In such cases, numerical simulation may be a useful prod to the intuition of the physicist, the engineer, or the mathematician. A typical example of interfacial flow is the collision between two liquid droplets. Finding the flow involves the study not only of hydrodynamic fields in the air and water phases but also of the air-water interface. This latter part

Direct numerical simulation of free-surface and interfacial flow

A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries

This study describes the development and validation of Computational Fluid Dynamics (CFD) methodology for the simulation of dispersed two-phase flows. A two-fluid (Euler-Euler) methodology previously developed at Imperial College is adapted to high phase fractions. It employs averaged mass and momentum conservation equations to describe the time-dependent motion of both phases and, due to the averaging process, requires additional models for the inter-phase momentum transfer and turbulence for closure. The continuous phase turbulence is represented using a two-equation k − ε−turbulence model which contains additional terms to account for the effects of the dispersed on the continuous phase turbulence. The Reynolds stresses of the dispersed phase are calculated by relating them to those of the continuous phase through a turbulence response function. The inter-phase momentum transfer is determined from the instantaneous forces acting on the dispersed phase, comprising drag, lift and virtual mass. These forces are phase fraction dependent and in this work revised modelling is put forward in order to capture the phase fraction dependency of drag and lift. Furthermore, a correlation for the effect of the phase fraction on the turbulence response function is proposed. The revised modelling is based on an extensive survey of the existing literature. The conservation equations are discretised using the finite-volume method and solved in a solution procedure, which is loosely based on the PISO algorithm, adapted to the solution of the two-fluid model. Special techniques are employed to ensure the stability of the procedure when the phase fraction is high or changing rapidely. Finally, assessment of the methodology is made with reference to experimental data for gas-liquid bubbly flow in a sudden enlargement of a circular pipe and in a plane mixing layer. Additionally, Direct Numerical Simulations (DNS) are performed using an interface-capturing methodology in order to gain insight into the dynamics of free rising bubbles, with a view towards use in the longer term as an aid in the development of inter-phase momentum transfer models for the two-fluid methodology. The direct numerical simulation employs the mass and momentum conservation equations in their unaveraged form and the topology of the interface between the two phases is determined as part of the solution. A novel solution procedure, similar to that used for the two-fluid model, is used for the interface-capturing methodology, which allows calculation of air bubbles in water. Two situations are investigated: bubbles rising in a stagnant liquid and in a shear flow. Again, experimental data are used to verify the computational results.

Computational fluid dynamics of dispersed two-phase flows at high phase fractions

Volume-of-Fluid Interface Tracking with Smoothed Surface Stress Methods for Three-Dimensional Flows

This advanced-leveltext describes several computational techniques that can be applied to a variety of problems in thermo-fluid physics, multi-phase flow, and applied mechanics involving moving flow boundaries. Step-by-step discussions of numerical procedures include examples that employ algorithms to solve problems. 1990 edition.

Computational Fluid Dynamics with Moving Boundaries

In this work a mixed Eulerian–Lagrangian technique is devised, hereinafter abbreviated as ELAFINT (Eulerian–Lagrangian Algorithm For INterface Tracking). The method is capable of handling fluid flows in the presence of both irregularly shaped solid boundaries and moving/free phase boundaries. The position and shape of the boundary are tracked explicitly by the Lagrangian translation of marker particles. The field equations are solved on an underlying fixed grid as in Eulerian methods. The interface passes through the grid lay-out and details regarding the treatment of the cut cells so formed are provided. The issues involved in treating the internal boundaries are dealt with, with particular attention to conservation and consistency in the vicinity of the interface. The method is tested by comparing with solutions from well-tested body-fitted co-ordinate methods. Test cases pertaining to forced and natural convection in irregular geometries and moving phase boundaries with melt convection are presented. The capability developed here can be beneficial in solving difficult flow problems involving moving and geometrically complex boundaries.

Elafint: a Mixed Eulerian-Lagrangian Method for Fluid Flows with Complex and Moving Boundaries

ELAFINT - A mixed Eulerian-Lagrangian method for fluid flows with complex and moving boundaries

Accurate numerical calculations have been conducted for buoyancy-driven two-dimensional flow of air between two vertical parallel isothermal plates, of aspect ratio 1, placed inside a rectangular enclosure. The Grashof number based on the channel half width is 105 . Insights have been obtained regarding the structure of the transient flow and thermal fields in a configuration of particular interest to electronics cooling. In the early stages of the transient the flowfield was found to be highly vortical and the net volume flow rate in the channel displayed an oscillatory behavior, periodically reversing its direction. However, the velocity profile adjacent to the heated plates maintained the same direction throughout the process, and hence the Nusselt number was relatively insensitive to the periodic flow reversal in the channel. Detailed studies of the transient flow field and the heat transfer results are presented.

Madhukar M. Rao

Papers

Computational Fluid Dynamics with Moving Boundaries

Computational Fluid Dynamics with Moving Boundaries

Elafint: a Mixed Eulerian-Lagrangian Method for Fluid Flows with Complex and Moving Boundaries

ELAFINT - A mixed Eulerian-Lagrangian method for fluid flows with complex and moving boundaries

Simulation of Transient Natural Convection Around an Enclosed Vertical Channel