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Think Before You Compute: A Prelude to Computational Fluid Dynamics

E. J. Hinch
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Texts in Applied Mathematics
[Reviewed by
Bill Satzer
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This book is aimed at students interested in computing a flow for a fluid. Those students are assumed to have a basic knowledge of fluid mechanics at the undergraduate level and comparable experience with numerical methods. Although aimed at graduate students, it is intended for beginners, for those just venturing into computational work with fluids. The author says he uses the simplest methods appropriate to the questions, and the book is in no sense focused on research questions. The word “prelude” of the title should be taken seriously. This book is intended in the spirit of a “read this first” before the student attempts any computational work.
The expected background in fluid mechanics includes the incompressible Navier-Stokes equation, acquaintance with Reynolds numbers, boundary layers, vorticity, and stream functions. For numerical methods, the prerequisites include simple finite-difference approaches for differential equations and iterative methods of solution for systems of linear equations. 
The book is divided into three unequal parts. The first is a relatively short introduction that is probably the most important component of the author’s treatment. The second is called “Generalities”, and it handles all the details. The third has a collection of special topics.
 The first part of the book introduces a basic problem in fluid dynamics and carries it through from formulation to a complete numerical solution. Although it is relatively short, this is the core of the author’s approach to computation.  Using this exemplary problem the author demonstrates how to begin to think about computations in fluid dynamics. He begins with the setting: a driven cavity with an incompressible fluid flowing in a rectangular box governed by the two-dimensional Navier-Stokes equations with a constant density and viscosity and subject to certain natural boundary conditions. The overall goal is to evaluate the viscous force on the top.  In teaching how to approach this computationally, the author asks the reader to follow him in examining the governing equations piece by piece. Three main issues are addressed: discretization, time-stepping and solving large sparse systems of linear equations. The author emphasizes the importance of being guided by physics. Among other things, this means choosing the mesh size for the computational grid, time-steps for the calculation, and total integration time consistent with the speed of propagation of information across the grid. By the end of the first part the reader should be able to write code to solve the problem numerically. The author supplies MATLAB routines to do that but suggests that students write their own code.
 The second part, the longest, presents a more detailed treatment of the general issues from discretization of partial differential equations on the front end to numerical solution of the resulting system of linear equations and interpretation of the results. Here there are basic treatments of finite difference, finite element, and spectral methods. Several approaches to time integration are also discussed here.   These include forward, backward, and midpoint Euler methods as well as more advanced methods like symplectic integrators.  
The final part then offers a collection of special topics. These include suggestions for useful software packages, brief discussions of how surfaces can be represented, and short treatments of more specialized solution methods. The most valuable topic for students is probably the description of the difficulties of solving hyperbolic equations numerically.
Only the first part has exercises, and there are not many. Some sections have references for further reading but they are pretty sparse. The book would be a useful tutorial for readers with some background in fluid dynamics who are considering attacking computational problems. The first part of the book is particularly good in this respect.


Bill Satzer ([email protected]), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from speech recognition and network modeling to optical films and material science. He did his PhD work in dynamical systems and celestial mechanics.