There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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

/pdf/immersed-boundary-methods-4id27kda4o.pdf

Immersed boundary methods

A front-tracking method for viscous, incompressible, multi-fluid flows

Level set methods: an overview and some recent results

A front-tracking method for the computations of multiphase flow

Direct numerical simulations of bubbly flows are reviewed and recent progress is discussed. Simulations, of homogeneous bubble distribution in fully periodic domains at relatively low Reynolds numbers have already yielded considerable insight into the dynamics of such flows. Many aspects of the evolution converge rapidly with the size of the systems and results for the rise velocity, the velocity fluctuations, as well as the average relative orientation of bubble pairs have been obtained. The challenge now is to examine bubbles at higher Reynolds numbers, bubbles in channels and confined geometry, and bubble interactions with turbulent flows. We briefly review numerical methods used for direct numerical simulations of multiphase flows, with a particular emphasis on methods that use the so-called "one-field" formulation of the governing equations, and then discuss studies of bubbles in periodic domains, along with recent work on wobbly bubbles, bubbles in laminar and turbulent channel flows, and bubble formation in boiling.

https://assets.cambridge.org/97805217/82401/frontmatter/9780521782401_frontmatter.pdf

Direct Numerical Simulations of Gas-Liquid Multiphase Flows

Preface 1. Introduction: a computational approach to multiphase flow A. Prosperetti and G. Tryggvason 2. Direct numerical simulations of finite Reynolds number flows G. Tryggvason and S. Balachandar 3. Immersed boundary methods for fluid interfaces G. Tryggvason, M. Sussman and M. Y. Hussaini 4. Structured grid methods for solid particles S. Balachandar 5. Finite element methods for particulate flows H. Hu 6. Lattice Boltzmann methods for multiphase flows S. Chen, X. He and L. S. Luo 7. Boundary integral methods for Stokes flows J. Blawzdziewic 8. Averaged equations for multiphase flows A. Prosperetti 9. Point particle methods for disperse flows K. Squires 10. Segregated methods for two-fluid models A. Prosperetti, S. Sundaresan, S. Pannala and D. Z. Zhang 11. Coupled methods for multi-fluid models A. Prosperetti References Index.

/pdf/computational-methods-for-multiphase-flow-22ceu5ihr2.pdf

Gretar Tryggvason

Papers

A front-tracking method for viscous, incompressible, multi-fluid flows

A front-tracking method for the computations of multiphase flow

Direct Numerical Simulations of Gas-Liquid Multiphase Flows

Computational Methods for Multiphase Flow

Computations of boiling flows