Spatial Averaging Combined with a Perturbation/Iteration Procedure
Authors: Culick F. E. C. 
Published: September 2012

A matter of increasing interest is finding the best way to integrate the use of powerful computational facilities with the traditional practices of analysis and related disciplines. This is largely a problem in research and development, rather than fabrication of machines, or even development of codes. There is little question that modern manufacturing, for example, cannot be accomplished competitively without computing machinery – in fact, the more the merrier. However, there is a fairly widespread sense that in areas depending on the development and applications of new ideas, and perhaps especially in education, the general emphasis has been skewed too much to rely excessively on computers.

This paper has been prepared partly to make the point with examples taken from the author's experiences with unsteady combustion. My claim here is that in many cases, analytical methods (necessarily approximate) offer a path often initially preferable to that presented by numerical methods, which in the cases at hand, mean computational fluid mechanics. The first simple example treated here is the Rijke tube, well known primarily for two reasons: The physical behavior is easy to produce, and for which data may be relatively easily collected; and the necessary analysis seems quite simple, at first glance. Some recently published experimental results will be cited, with an approximate theory based on the known differential equations for one-dimensional motions.

The next section is a brief historical summary of Galerkin's method, followed by several sections summarizing the manner in which it may be combined with a perturbation/iteration method to give an effective approximate method. The general approach has been widely used to analyze practical problems of combustion instabilities arising in development of operational systems. Thus a large part of the paper is a review of previously published material, but with considerable clarifications of points that have caused some confusion.

The paper ends with a brief discussion answering a serious criticism, of the method, nearly fifteen years old. The basis for the criticism, arising from solution to a relatively simple problem, is shown to be a result of an omission of a term that arises when the average density in a flow changes abruptly. Presently, there is no known problem of combustion instability for which the kind of analysis discussed here is not applicable. The formalism is general; much effort is generally required to apply the analysis to a particular problem.

A particularly significant point, not elaborated here, is the inextricable dependence on expansion of the equations and their boundary conditions, in two small parameters, measures of the steady and unsteady flows. Whether or not those Mach numbers are actually ‘small’ in fact, is really beside the point. Work out applications of the method as if they were! Then maybe to get more accurate results, resort to some form of CFD. It is a huge practical point that the approach taken and advocated here cannot be expected to give precise results, but however accurate they may be, they will be obtained with relative ease and will always be instructive. In any case, the expansions must be carried out carefully with faithful attention to the rules of systematic procedures. Otherwise, inadvertent errors may arise from inclusion or exclusion of contributions.

I state without proof or further examples that the general method discussed here has been quite well and widely tested for practical systems much more complex than those normally studied in the laboratory. Every case has shown encouraging results. Thus the lifetimes of approximate analyses developed before computing resources became commonplace seem to be very long indeed.