- Shopping Bag ( 0 items )
NUMERICAL TECHNIQUES FOR FLUID FLOWS WITH MOVING BOUNDARIES
Moving boundary problems arise in a variety of important engineering applications (Crank 1984, Zerroukat and Chatwin 1994). Typical examples are materials processing, fluid- structure interactions, flame propagation and oil recovery. These systems are characterized by internal boundaries or interfaces demarcating regions with different physico-chemical properties. Across these interfaces, compositions, phases, material properties and flow features can vary rapidly. The interfaces move under the influence of the flowfield and in turn affect the behavior of the flow. Under certain conditions, usually characterized by one or more control parameters, the interfaces may experience instabilities. Such an event-typically a cascade of interfacial instabilities on an initially planar or smooth front-often leads to the formation of patterns and morphological structures, due to the interaction of the competing physical mechanisms present in the system. The prediction of the dynamics of such unstable, evolving interfaces is important in many technological applications. For example, the formation of deep cells in the solidification processing of multi-component materials is quite important since the resulting compositional inhomogeneities will affect the performance and properties of doped semiconductors and the structural integrity of alloy materials (Ostrach 1983, Langlois 1985, Brown 1988, Shyy 1994). Another example is the interaction of fluid flows with non-rigid structures, where fluid convection and deformation of the structure strongly interact, resulting in a highly nonlinear system (Marchaj 1979). Contact melting (Sparrow and Myrum 1985, Moallemi et al. 1986, Hirata et al. 1991) is another example; as recently reviewed by Bejan (1994), this topic has received much attention in the area of tribology (Pinkus 1990).
In moving boundary problems, not only are the transport of momentum, heat, and mass coupled, but also the formation, evolution, and dynamics of the interface play major roles in defining the behavior of the system. Apart from the inherent nonlinearity of these diverse phenomena, the interfacial deformation in itself can be a highly complicated and analytically intractable feature. Several physical phenomena involve interfaces that, as a result of instabilities or external impulses, undergo severe deformations, such as dilatation, compression, fragmentation, and collision. The physics involved in such processes impose severe demands on the numerical, analytical, or experimental approaches used in their study.
For example, during crystallization the interface separating the solid from its melt or vapor can undergo morphological changes that can have serious technological implications (Tiller 1991a) as well as aesthetic appeal (Bentley and Humphreys 1962). These changes of morphology can contribute to the inhomogeneous distribution of the solute that is rejected from the solid into the melt. The rejected solute accumulates in the regions around the irregularly shaped phase front and both microscopic and macroscopic irregularities in the solute distribution result. This phenomenon of solute rejection and transport is called micro–or macro–segregation depending on the scale at which such processes occur. It is evident that in producing crystals and alloys for technological applications, such behavior of the solute field is undesirable and thus has to be minimized or eliminated. The understanding and observation of these phenomena are however hampered due to the extremely disparate length scales involved (ranging from a few microns to meters) and the inherent nonlinearity of the problem rendering analysis very difficult. Instabilities during solidification lead to highly deformed geometries, such as cellular and dendritic forms. In the events leading to complete solidification, a series of topological changes occur.
The interconnected solid matrix encloses isolated liquid pools, which solidify upon cooling. In the case of dendritic branch detachment (Sato et al. 1987, Heinrich et al. 1991, Neilson and Incropera 1991), or in equiaxed or globulitic crystal forms (Beckermann et al. 1994, Ni and Beckermann 1993), convection and settling of isolated crystallites significantly influences the final grain structure. In some experiments, periodic wavelength readjustment of cellular phase fronts has been induced by changing the growth conditions (Cladis et al. 1990, de Cheveigne et al., 1990, Flesselles et al. 1991). Macrosegregation resulting from formation of freckles (Felicelli et al. 1991, Flemings 1974), of channels (Neilson and Incropera 1993) and positive and negative segregated bands (Diao and Tsai 1993) can significantly impact on the quality of the alloys. Shrinkage-induced convection can also have noticeable effects on the solidification process (Chiang and Tsai 1992a, b). It is clear that in order to follow the entire spectrum of possible phenomena in the solidification process, it is necessary to cope with a variety of topological changes. A recent review by Prescott and Incropera (1993) has summarized from an engineering point of view the interaction between fluid flows and macrosegregation. Beckermann and Viskanta (1993), Langlois (1985), Samanta et al. (1987), Shyy (1994), and Viskanta (199) have given accounts of mathematical and computational modeling of transport processes during solidification of alloys.
In liquid–gas systems, the formation of sprays (Bogy 1979, Lefebvre 1989), the shattering of droplets, and other surface instabilities (Fromm 1981, Orme and Muntz 1990, Spangler et al. 1995) present challenges in tracking multiple, interacting interfaces. In practical systems such as propulsion devices (Lefebvre 1989) and internal combustion engines (Kuo 1991, Lai and Przekwas 1994), since the phase change process takes place in chemically reacting flows (Rosner 1986, Williams 1985), these problems become extremely difficult to solve. In each of these phenomena, the necessary boundary conditions at the interface are functions of the interface shape, since the solution in the entire domain critically depends on the accurate determination of the interface shape and its derivatives.
Typically, the combination of interfacial dynamics and physico-chemical transport processes is at the heart of moving boundary problems. Due to this interaction, as well as to the presence of other complicating mechanisms mentioned above, moving boundary problems often are very difficult to analyze. The difficulty exists even though the governing laws describing interface characteristics are well established from the fluid dynamic and thermodynamic viewpoints. The main difficulty is that the internal boundary position and shape must be determined as part of the solution of the transport equations of mass continuity, momentum, and energy. Within each domain, the field equations must be solved with the location of the internal boundary being determined simultaneously. Under certain conditions the interface may undergo a succession of deformations leading to very complicated shapes. Finally, from the continuum mechanics viewpoint, the interface often is treated as a discontinuity in the flowfield. Within the inevitable limitation of finite grid resolution, this discontinuity needs to be accurately tracked both in time and in space.
1.1.2 Overview of the Present Work
In this work, several issues related to devising solution techniques for moving boundary problems are presented. Therefore, it is appropriate first to provide a brief survey of some existing techniques concerned with tracking highly distorted fronts, including both Lagrangian and Eulerian methods. A variety of methods is available, and as will be evident from the applications detailed later in this work, the choice of an efficient and robust method will depend on the physical problem under investigation.
To illustrate the computational and theoretical issues involved, examples arising from high temperature materials processing and fluid-structure interaction are chosen to give detailed coverage. Of course, there are many more physical examples involving moving boundaries to which we can not do justice within the scope of this book. In the context of each physical topic discussed, we have made an effort to systematically account for the entire scope of the computational task, including formulation, nondimensionalization and scaling, algorithm implementation, and physical interpretation. In particular, besides the development of numerical algorithm, other aspects also will be discussed to help highlight the issues encountered. For example, multiple scaling is a well-known feature in many mechanics problems (Ciarlet and Sanchez-Palencia 1987); it is also a prominent feature of moving boundary problems. This aspect will be addressed theoretically as well as computationally.
In order to supply necessary information to enable the reader to follow the detailed numerical development, the pressure-based algorithm for fluid flow equations is summarized in Chapter 2. This chapter serves mainly to highlight some of the important computational aspects that have not been covered in detail in many texts on computational fluid dynamics. However, most of the computational techniques for handling moving boundaries presented in this book can be directly employed in other fluid flow solution algorithms and are not restricted to any particular fluid flow equation solver employed here. The formulation presented in Chapter 2 is applicable to generalized curvilinear coordinates, with a moving, adaptive grid for the discretization of the physical domain. The Cartesian, fixed grid formulation then reduces to a special case of this more general procedure. To prepare for the later chapters, formulation and computations of Marangoni convection (Chen 1987, Koschmieder 1993, Levich 1962, Probstein 1989, Shyy 1994) also are discussed in Chapter 2. Marangoni convection is associated with flows that arise due to interfacial tension gradients and density gradients. These flows play a pivotal role in the processing of single crystals from a melt and therefore significantly influence crystal quality. The interaction of steady-state Marangoni convection and the material properties is rich in structure and pattern.
In Chapters 3 and 4, a moving grid technique is presented along with a curvilinear grid- based pressure correction algorithm. In this approach, the interface is explicitly tracked at every instant, with the individual domains mapped by curvilinear grids. Grid remeshing is conducted at each time step to conform to the moving boundary. Within each domain, the pressure-based algorithm is utilized to solve the fluid flow equations, in combination with the boundary conditions pertaining to interfacial behavior. To illustrate the performance of this approach, two physical problems are presented, namely a phase change problem at the morphological scale involving the balance between capillarity and heat conduction, and the interaction between a viscous flow and a flexible membrane wing. For problems involving interaction of fluids and structures, high Reynolds number but still laminar flows surrounding a flexible membrane wing are investigated. This grid mapping technique carries with it the advantage of using the explicit interface information in applying the boundary conditions in each phase. However, it encounters difficulties when the interface becomes multiple valued or geometrically complicated. For a curvilinear grid, this presents difficulties in conforming to the interface, since a highly contorted boundary can lead to excessive grid skewness. When the interface exhibits topological changes such as breakups and mergers, a mapping technique faces fundamental difficulties. Massive grid restructuring and reordering may be called for, leading to an insurmountable algorithmic burden.
In Chapters 5, 6 and 7, we address fixed grid techniques. First, we discuss an Eulerian method, known in the current heat transfer literature as the enthalpy formulation. Specifically, the content of the physical problem discussed in Chapter 4 is broadened here; interaction among convection, conduction, and capillarity is stressed. Both free and internal boundaries are addressed. Furthermore, modeling issues associated with scale disparities arising from geometrical complexity and fundamental physical mechanisms also are presented. In Chapter 5, the interface is handled implicitly with the two-phase region modeled as a porous medium. The liquid fraction varies smoothly across this porous, so-called "mushy region". In such a formulation, the interface occupies a finite region instead of being defined as a discontinuity. This allows a unified set of governing equations to represent the transport processes in the whole domain. The mushy zone, or interface region, is modeled via the phase fractions, which are incorporated into the source terms in the governing equations to account for the phase change phenomena. Obviously, both the unified mathematical structure and the fixed grid computational procedures make this approach attractive. However, lack of precise definition and details of the interface makes this approach unsuitable for certain classes of problems. For example, when the capillary effect becomes important, estimation of the interface curvature is of paramount importance, as dictated by the Gibbs-Thomson effect. Under such circumstances the fixed grid approach is not the method of choice.
To combine the strengths of the moving grid and fixed grid techniques, advances have been made recently in the area of combined Lagrangian-Eulerian methods, which are described in Chapters 6 and 7. In this approach, a set of markers is employed to define and follow the interface in a Lagrangian framework. To facilitate the solutions of the field equations, a fixed grid is utilized. On the fixed grid system, the markers advance in time, causing the computational cells in the interface regions to become irregularly shaped. Special treatment is needed to enable accurate computations of the mass, momentum, and energy fluxes and to cast the discretized forms within a pressure-based, control volume framework. In the context of solidification problems, both the morphological evolution subject to the Gibbs-Thomson effect (i.e., when capillarity and heat conduction are dominant), and the macroscopic interface dynamics under the influence of convection and conduction are discussed. Also presented are direct comparisons between the fixed grid, cut-cell approach and the curvilinear grid algorithm for solving highly convective fluid flow problems in an irregularly shaped domain.
The techniques presented here can be used for a wide variety of fluid flow problems with moving boundaries. Many aspects associated with these techniques have yet to be addressed to their full extent; it is our hope that this book can help stimulate progress in this exciting area.
1.2 NUMERICAL METHODS APPLIED TO GENERAL MOVING BOUNDARY PROBLEMS
Several techniques exist for tracking arbitrarily shaped interfaces, each with its own strengths and weaknesses (Crank 1984, Floryan and Rasmussen 1989, Shamsundar and Rooz 1988, Wang and Lee 1989). These techniques may be classified under two main categories: (a) surface tracking or predominantly Lagrangian methods (Harlow and Welch 1965, Vicelli 1969, Chen et al. 1995) and (b) volume tracking or Eulerian methods (Hirt and Nichols 1981, Ashgriz and Poo 1991). The main features of the two types are presented in Fig. 1.1.
In the Lagrangian methods, the grid is configured to conform to the shape of the interface, and thus it adapts continually to it. The Eulerian methods usually employ a fixed grid formulation, and the interface between the two phases is not explicitly tracked but is reconstructed from the properties of appropriate field variables, such as fluid fractions. Based on these basic differences in approach of the two classes of methods, the following comparisons can be made:
1. Interface Definition
The Lagrangian methods maintain the interface as a discontinuity and explicitly track its evolution. If detailed information regarding the interface location is desired, Eulerian methods may need elaborate procedures to deduce the interface location based on the volume fraction information, and uncertainty corresponding to one grid cell is unavoidable (Ashgriz and Poo 1991, Hirt and Nichols 1981, Lafaurie et al. 1994, Liang 1991). In the Lagrangian case, the interface can be tracked as an (n—1)-dimensional entity for an n- dimensional space (De Gregoria and Schwartz 1985, Glimm et al. 1988, 1986, Miyata 1986, Wand and McLay 1986). No modeling is necessary to define the interface or its effect on the flow field. In the case of Eulerian schemes, modelling or the solution of additional equations is required to obtain information regarding fluid fractions or other functions yielding information in the two-phase regions.
Excerpted from COMPUTATIONAL FLUID DYNAMICS WITH MOVING BOUNDARIES by Wei Shyy, H. S. Udaykumar, Madhukar M. Rao, Richard W. Smith. Copyright © 1996 Wei Shyy, H. S. Udaykumar, Madhukar M. Rao, Richard W. Smith. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
|1||Numerical Techniques for Fluid Flows with Moving Boundaries||1|
|2||Governing Equations and Solution Procedure||21|
|3||Moving Grid Techniques: Fluid Membrane Interaction||61|
|4||Moving Grid Techniques: Modeling Solidification Processes||95|
|5||Fixed Grid Techniques: Enthalpy Formulation||135|
|6||Fixed Grid Techniques: ELAFINT - Eulerian-Lagrangian Algorithm For INterface Tracking||195|
|7||Assessment of Fixed Grid Techniques||249|