# Prediction of Stability/Instability and Transition in High-Speed Flows

In this section we wish to acquaint the reader with the possibilities to actually predict stability/instability and transition in high-speed flows. No review is intended, but a general overview is given with a few references to prediction models and criteria. Because many developments in non-local and non-linear instability prediction were mainly made for transonic flows, these developments will be treated too in order to show their potential also for the high-speed flows of interest here. Transition-prediction theory and methods based on experimental data (ground-simulation facility or free-flight data) are treated under the headings “semi-empirical” and “empirical” transition prediction in Sub-Section 8.4.2. Transition prediction for high-speed flows generally has made much progress in the last decade, although the ultimate goal still is to be achieved. The review paper [101] and the overview paper on AFOSR-sponsored research in aerothermodynamics [79] are good introductions to the present state of the art. For a review of flight data regarding laminar-turbulent transition in high-speed flight see, e. g., [106].

8.4.1 Stability/instability Theory and Methods

Theory and methods presented here are in any case methods for compressible flow but not necessarily for flow with high-temperature real-gas effects. The thermal boundary conditions are usually only the constant surface temperature or the adiabatic-wall condition. The radiation-adiabatic wall condition is implemented only in few methods, although it is in general straight forward to include it into a method. Likewise, it is not a problem to include adequate high-temperature real-gas models into stability/instability methods.

Linear and Local Theory and Methods. The classical stability theory is a linear and local theory. It describes only the linear growth of disturbances (stage 2—see the footnote on page 284—in branch IIa, Fig. 8.1). Neither the receptivity stage is covered, nor the saturation stage and the last two stages of transition.[146] Extensions to include non-parallel effects are possible and have been made. The same is true for curvature effects. However, the suitability of such measures appears to be questionable, see, e. g., [107].

Linear and local theory is, despite the fact that it covers only stage 2, the basis for the semi-empirical en transition prediction methods, which are discussed in Sub-Section 8.4.2.

Linear and local stability methods for compressible flows are for instance COSAL (M. R. Malik, 1982 [108]), COSTA (U. Ehrenstein and U. Dall – mann, 1989 [109]), COSMET (M. Simen, 1991 [110], see also E. Kufner [34]), CASTET (F. Laburthe, 1992 [111]), SHOOT (A. Hanifi, 1993 [112]), LST3D (M. R. Malik, 1997 [113]), COAST (G. Schrauf, 1992 [114], 1998 [115]), LILO (g. Schrauf, 2004 [116]).

Non-local Linear and Non-linear Theory and Methods. Non-local theory takes into account also the downstream changes of the mean flow as well as the changes of the amplitudes of the disturbance flow and the wave numbers. Non-local and linear theory also describes only stage 2 in branch IIa, Fig. 8.1. However, non-parallel effects and curvature effects are consistently taken into account which makes it a better basis for en methods than local linear theory.

Non-linear non-local theory on the other hand describes all five stages, in particular also stage 1, the disturbance reception stage, however not in all respects. Hence, in contrast to linear theory, form and magnitude of the initial disturbances must be specified, i. e., a receptivity model must be employed, Sub-Section 8.3.

Non-local methods are (downstream) space-marching methods that solve a system of disturbance equations, which must have space-wise parabolic character. Hence such methods are also called “parabolized stability equations (PSE)” methods. We do not discuss here the parabolization and solution strategies and refer the reader instead to the review article of Th. Herbert [117] and to the individual references given in the following.

Non-local linear stability methods for compressible flow are for instance xPSE (F. P. Bertolotti, linear and non-linear (the latter incompressible only), 1991 [118]), PSE method (linear and non-linear) (C.-L. Chang et al., 1991 [119]), NOLOS (M. Simen, 1993 [120]), PSE-Chem (H. B. Johnson et al., 2005 [121]), STABL-3D (H. B. Johnson et al., 2010 [122]).

Non-local non-linear stability methods for compressible flow are for instance, COPS (Th. Herbert et al., 1993 [123]), NOLOT/PSE (M. Simen et al., 1994 [124], see also S. Hein [43]), CoPSE (M. S. Mughal and P. Hall, 1996 [125]), PSE3D (M. R. Malik, 1997 [113], with chemical reactions also see [126]), xPSE with rotational and vibrational non-equilibrium (F. P. Bertolotti, 1998 [63]), NELLY (H. Salinas, 1998 [127]), LASTRAC (C.-L. Chang, 2004 [128]), JoKHeR (J. J. Kuehl et al., 2012 [129]).

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