3R39. Finite Analytic Method in Flows and Heat Transfer. - Ching Jen Chen (Col of Eng, Florida A&M Univ), RA Bernatz (Luther Col), KD Carlson (Univ of Iowa), Wanlai Lin (Tech Dev of Next Generation Prod, Emerson Elec Corp). Taylor & Francis Publ, New York NY. 2000. 332 pp. ISBN 1-56032-898-3. $69.95.
Reviewed by G de Vahl Davis (Sch of Mech and Manuf Eng, Univ of New South Wales, Sydney 2052, NSW, Australia).
The finite analytic (FA) method is an Eulerian method that solves the differential equations for CFD/HT by representing the solution domain as a series of homogeneous, constant parameter elements. Within each of these elements an algebraic form of the analytic solution of a linearized form of the equations is constructed. The solution at a nodal value in the interior of each element is expressed as the sum of neighboring nodal values weighted by finite analytic coefficients. A system of these finite analytic algebraic equations is then solved to provide a numerical solution for the dependent variable at prescribed discrete locations within the domain.
The method was conceived by CJ Chen (one of the authors of this book) and his then student, Peter Li, in 1977. It was first published in 1981 and has since been developed, extended, and implemented by Chen, his students and others. A global web search for “finite analytic method” yielded about 200 hits, while ScienceDirect® (for Elsevier publications) yielded 96 references between 1980 and 2001. The method has been successfully applied to a range of problems in two- and three-dimensional flow and heat transfer, both laminar and turbulent, in regular and irregular domains. The authors claim that, compared to traditional finite difference (FD) methods, the method is stable and accurate over a much broader range of flow and computational parameters such as Reynolds number and grid spacing. Nevertheless, it has not yet been widely adopted, and several well-known texts on CFD/HT published in recent years do not mention it.
This book is aimed at graduate students and practitioners of CFD/HT. It is presented in five parts totaling 310 pages, comprising 25 chapters and two appendices. Some knowledge of differential equations and of some analytic methods are assumed.
Part I is entitled Introduction to Computational Fluid Dynamics and contains an introduction followed by chapters on Governing Equations (the N-S and energy equations and turbulence modeling); Classification of PDEs; Well-Posed Problems (including existence and uniqueness); Numerical Methods (a brief survey of FD, finite element (FE) and FA methods); and a more extensive chapter on The Finite Difference Method. In the chapter on numerical methods, the authors briefly compare FA with FD and FE, highlighting the advantages and disadvantages of each; they conclude, not surprisingly, that FA wins out.
In Part II, The Finite Analytic Method, the method is explained in detail. The seven chapters cover Basic Principles; One-, Two-, and Three-Dimensional Cases; Stability and Convergence; Hyperbolic PDEs; and what is called the Explicit Finite Analytic Method for the 2D transport equation for convection dominated flows, which is less complex than the implicit formulations developed in the preceding four chapters.
Part III, Numerical Grid Generation, contains an introduction covering algebraic transformations and a summary of differential methods, followed by chapters on Elliptic Grid Generation; Equations in ξ and η Coordinates; Diagonal Cartesian (DC) Method; and FA Method on DC Coordinates.
In Part IV, several Computational Considerations are discussed: Velocity, Pressure and Staggered Grids; Nonstaggered Grid Methods; and Boundary Conditions.
Finally, Part V describes some Applications of the FA Method to Turbulent Flows; Turbulent Heat Transfer; Complex Domain Flows; and Conjugate Heat Transfer. It is this section, more than any other, which will allow teachers and practitioners to decide whether to adopt the method. The examples include turbulent flow past disc valves; the sea breeze phenomena (sic), a turbulent atmospheric boundary layer circulation driven by surface temperature gradients; groundwater flow and solute transport; and the design of a compact heat exchanger.
The authors say that a 2D laminar Fortran code is available at www.finiteanalytic.com. However, when this reviewer went there, he got the message “This site is currently under construction; please check back at a later time” in several languages.
This book should certainly be seriously considered by teachers of CFD/HT, and graduate students should have some exposure to the ideas presented. Time will tell whether the finite analytic method will take a place alongside—or perhaps even surpass—finite difference and finite element methods.