Explicit finite difference methods: Some selected applications to inviscid and viscous flows

Abstract

In this chapter we round-out our introductory treatment of computational fluid dynamicsby discussing some applications of explicit finite difference methods to selectedexamples for inviscid and viscous flows. These examples have one thing incommon-they are results obtained by either the present author and/or some of hisgraduate students over the past few years. This is not meant to be chauvinistic; ratherthis choice is intentionally made to illustrate what can be done by uninitiated studentswho are new to the ideas of CFD. These examples demonstrate the power andbeauty of CFD in the hands of students much like yourselves who may have little orno experience in the field. Moreover, in all cases the applications are carried out withcomputer programs designed and written completely by each student. This is followingthe author's educational philosophy that each student should have the experienceof starting with paper and pencil, writing down the governing equations, developingthe appropriate numerical solution of these equations, writing the FORTRANprogram, punching the program into the computer, and then going through all thetrials and tribulations of making the program work properly. This is an importantaspect of CFD education. No established computer programs ('canned' programs)are used; everything is 'home-grown', with the exception of standard graphics packageswhich are used to plot the results. Therefore, by examining these examples, youshould obtain a reasonable feeling for what you can expect to accomplish when youfirst jump into the world of CFD applications.Before we discuss some examples, it is important to describe the mechanismof explicit finite-difference calculations. The distinction between explicit and implicitapproaches was made in Sect. 5.3, which should be reviewed before progressingfurther in this chapter. In the next few sections, we will describe two ratherstraightforward and popular explicit methods. The treatment and application of implicitmethods is given by other lectures in this course, and hence will not be discussedhere.Finally, the examples discussed in this chapter all incorporate the time-dependentmethod, i.e. forward marching in steps of time. The historic break-through madeby this method in the 1960s is discussed in Chap. 1. The vast majority of timedependentsolutions have as their objective the solution of a steady-state flow fieldwhich is approached by the solution at large times; here, the time-dependent mechanismis simply a means towards achieving that end. In other applications, the timedependentmethod is used to calculate the actual transients in an unsteady flow ofinterest. Examples of both are given here. We note, however, that although the followingsections deal with marching forward in time, the same techniques are easilyapplied to a steady flow calculation where spatial marching is done along some coordinateaxis.We have seen in Chap. 4 that such forward marching (in time or space)is appropriate when the governing equations are hyperbolic or parabolic. © Springer-Verlag Berlin Heidelberg 2009.