Approximate analysis of free-mixing flows
| Autor: | Seymour L. Zeiberg, Brent Kaplan |
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| Rok vydání: | 1965 |
| Předmět: | |
| Zdroj: | AIAA Journal. 3:552-554 |
| ISSN: | 1533-385X 0001-1452 |
| DOI: | 10.2514/3.2911 |
| Popis: | AS outgrowths of the recent interest in the wakes of hyper-f"^sonic re-entry vehicles, detailed numerical solutions of the axisymmetric and two-dimensional free-mixing equations have been developed.' 2 Although such methods are necessary to predict flow properties in chemically reacting nonequilibrium flows, their use for less complex flow problems may be prohibited by high computer expense; it is then desirable to have available means to develop accurate approximate solutions. Two commonly employed methods to generate approximate solutions to the boundary-layer equations are the integral method and linearization.The integral method provides rapid solutions with minimum effort; however, the form of the assumed profiles generally puts undesirable constraints on the applicability of the results. Solutions of the linearized free-mixing equations can be generated for arbitrary initial conditions; these solutions demonstrate the proper "downstream" behavior, but are of limited use in the "upstream" regions. The purpose of the present work is to elaborate on a technique advanced by Libby and Schetz for the "improvement" of a linearized solution. The method is based upon a stretching of the coordinates, the stretching factors being found by forcing the linearized solution to satisfy certain aspects of the nonlinear problem. As an extension of ideas put forth by Carrier and Lewis and Carrier, Libby and Schetz solved the two-dimensional laminar boundary-layer problem of slot injection into a constant pressure moving stream by first determining a linearized solution, then replacing the streamwise coordinate x by a function %(x), and then forcing the linearized solution in terms of £ to satisfy the "exact" momentum integral equation. Since d/dx = (d%/dx)(d/d£), the operations just described yielded a simple differential equation for the stretched coordinate £. The method proved to be successful for the problem originally treated, and has since been applied to other situations. Two-dimensional symmetric jets (in a moving ambient stream) were analyzed with streamwise coordinate stretching being accomplished by forcing the linearized solution in terms of £ to satisfy the exact equation written on the symmetry axis. Another approach used by Schetz is pointwise collocation of a linearized solution. In this case, parameters are introduced into the argument of the linearized solution; these parameters are evaluated by forcing the resulting expression to satisfy the exact nonlinear differential equation at discrete points. The present note describes a scheme, applicable to freemixing flows, according to which the forenoted ideas are combined; the momentum integral equation is used to stretch the radial coordinate, whereas the differential equation, written on an appropriate line, is used to stretch the streamwise coordinate. A comparison of the results of an illustrative calculation with an exact numerical solution shows excellent agreement. |
| Databáze: | OpenAIRE |
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