Abstract
Landau’s equation and its generalizations considered in Sect. 4.2 represent a particular weakly-nonlinear approach to the study of flow stability, based on the assumption that the disturbance amplitude A is small enough to justify the expansion of solutions of fluid-dynamic equations in powers of A. However this approach has a severe limitation: only the evolution of one isolated mode of disturbance is traced, while its interaction with all other modes is only roughly characterized by the values of real or complex Landau’s constants of various orders.
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Notes
- 1.
It is often convenient to define the fluctuation amplitude as the root-mean-square value (i.e., as the square root of the temporal mean value of squared fluctuations). This definition is widely used, in particular, in studies of turbulent flows.
- 2.
Below, in cases where complex amplitude is not considered, the real amplitude |A| will usually be denoted simple as A.
- 3.
Some other attempts at numerical simulation of the N-regime of boundary layer instability developments were carried out by Spalart and Yang (1987) and Laurien and Kleiser (1989) (one result of the former authors is shown in Fig. 5.15c). However, in both these papers the less-accurate temporal, and not spatial, simulation was performed (see the small-type text below for discussion of the difference between these two approaches and the remarks about this topic in the next footnote 4) and the results found were less complete than those of Fasel et al. Therefore, except for Fig. 5.15b, these results will not be considered here. On the other hand, Rist and Fasel (1995) improved somewhat on the numerical method of Fasel et al.; however, as to the results relating to N-regime, the paper of 1995 contains only the indication that here “the quantitative agreement between numerical results and experiments was at least as good or even better than that achieved by Fasel et al. (1987)”.
- 4.
The simplest way of doing this is based on the supplementation of the N-S equations by an artificial ‘force term’ guaranteeing the existence of a solution describing the plane-parallel Blasius boundary layer with time-dependent thickness δ(t), growing at a rate equal to that registered by an observer who moves streamwise with a reasonably chosen velocity. According to Gaster’s (1962) arguments, the group velocity c g of a packet of T-S waves (which depends only weakly on the vertical coordinate z) may be chosen as such a ‘reasonable velocity’. Then the corresponding time-dependent plane-parallel boundary layer may be considered as a temporal model of the real streamwise-growing boundary layer (cf. a remark in Chap. 4, about a similar method of numerical simulation of the steady plane-parallel model of a Blasius boundary layer). This method of approximate allowance, in temporal numerical simulations, for the spatial (streamwise) growth of a boundary layer was used, in particular, by Spalart and Yang (1987) and later gained great popularity.
- 5.
However, the Squire waves also possibly made some contribution to the secondary disturbances computed by Herbert and his co-authors (such a possibility was explicitly mentioned by Crouch and Herbert (1993) in their study of nonlinear development of secondary disturbances in boundary layers). It is also possible that some small Squire-wave contribution was present even in some of the computational results of Zel’man and Maslennikova; as was pointed out by E. Reshotko (personal communication) Squire waves sometimes appear quite unexpectedly in numerical solutions of the Navier-Stokes equations.
- 6.
Editors addendum: before he died, Prof. Akiva Yaglom was working on Chapter 6.
- 7.
In Sect. 5.4, attempts by Wray and Hussaini (1984); Zang and Hussaini (1985, 1987, 1990). Murdock (1986); Laurien and Kleiser (1989); Kleiser and Zang (1991); Zang (1992) and some others to simulate numerically the boundary-layer instability development were mentioned. These papers contain a number of results relating to the K-regime of such development and almost all of them agree satisfactorily with available experimental data. However, these results are less accurate and less complete than those by Rist and Fasel (1995) and Rist and Kachanov (1995); therefore results of the earlier numerical simulations of the K-regime will not be considered in this book.
- 8.
Therefore instead of the name ‘Benjamin-Ono (or B-O) equation’ the name ‘Benjamin-Davis-Acrivos (or BDA) equation’ or ‘Davis-Acrivos-Benjamin-Ono (or DABO) equation’ is sometimes used.
- 9.
Note that the results of Butler and Farrell (1992) presented in Chap. 3 were obtained for a simplified, strictly plane-parallel model of the Blasius boundary layer. The optimally-growing disturbance structures for the more accurate model of a streamwise-thickening boundary layer were studied by Andersson et al. (1999) and Luchini (2000) but will not be considered here.
- 10.
In both papers it was assumed that the boundary layer is plane-parallel but in the treatment of data relating to a given value of x, values of δ* and Re* corresponding to this x were used. (A more precise analysis of some data of Cohen et al., which took into account the streamwise growth of the boundary layer, was developed by Cohen (1994).) Measurements by Cohen et al. and Breuer et al. showed that in their studies the pressure gradient in the boundary layer was slightly negative, and therefore the function U(z) was slightly closer to a Falkner-Skan profile for β≈ 0.01 (see Chap. 2, p. 119) than to the Blasius profile corresponding to β = 0. However, the Blasius approximation was found to be accurate enough to be usable in the analysis of the experimental data.
- 11.
The measurements by Breuer et al. discussed here related to waves excited by an acoustic pulse with a different amplitude from that used in the experiments by Cohen et al. (1991). Therefore the streamwise locations of the three stages of wave-packet development mentioned in our discussion of the results of Cohen et al. are not the same as those in the series of experiments considered here.
- 12.
Note that the growth of the disturbance kinetic energy does not necessary imply the growth of disturbance velocities. For example, in the case of transient growth of localized disturbances in plane shear flows studied by Landahl (1980), the growth of disturbance energy due to the “lift-up effect” described by him is due to elongation of the disturbance increasing its volume, and not to the growth of velocities of individual fluid particles.
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Acknowledgments
Exactly as in the case of all previous chapters, this chapter could not have been accomplished without much help given to me by a number of individuals and institutions. And again I must first of all express here my deep gratitude to the Massachusetts Institute of Technology (MIT) in whose Department of Aeronautics and Astronautics the work on this chapter was carried out; the assistance of the Department Head, Prof. E. F. Crawley, in providing me with all necessities for this work must be especially noted. The Center for Turbulence Research (CTR) at Stanford University and NASAAmes Research Center (Director Prof. P. Moin) on its own initiative arranged the publication of the separate chapters of this work as CTR Monographs; moreover, CTR provided a substantial part of the financial support for my work. The other part of the support was given by MIT alumnus Dr. J. W. Poduska, Sr. and his family, who donated a special sum to MIT as a gift intended for supporting the preparation of my book. I feel myself obliged to express here my deep gratitude to CTR and Dr. Poduska's family.
As in the case of previous chapters I am particularly indebted to Prof. P. Bradshaw of Stanford University, who again read the manuscript of the chapter with great care, corrected the defects of my English and made important suggestions relating to the presentation of some topics in this chapter. I am also very grateful to many colleagues who made some useful remarks and/or sent useful written material on the boundary-layer instability. In particular, P. Schmid sent me an incomplete text of the interesting book by Schmid and Henningson (2001) much before the manuscript was wholly prepared and sent to the publisher, and he also helped me with the preparation of some figures from the mentioned book for reproduction in Chap. 5 below. Moreover, V.V. Kozlov sent me a copy of the recent Russian book by Boiko et al. (1999) which was difficult to get in the USA, while F.H. Busse pointed out a number of misprints in Chap. 4 and noted that results given in some papers cited in Chap. 4 contain an error, corrected in the recent paper by Kaiser and Xu (1998). I also received valuable remarks and written materials from K.S. Breuer, S. Grossmann, M.V. Morkovin, E. Reshotko and many others. Ms. Debra Spinks and Ms. Deborah Michael of CTR again helped me with a number of administrative matters and problems related to arranging and implementation of the present publication, and Ms. Jenny Pelka helped with the preparation of manuscript and figures for publication (many figures were sent by me only in a draft-quality form). Ms. Eileen Dorschner, Aeronautics and Astronautics librarian at MIT libraries, helped me to define more precisely all the data needed for the Bibliography at the end of this chapter. To all the persons and organizations who gave help to me I express here my sincere thanks.
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Yaglom, A., Frisch, U. (2012). Further Weakly-Nonlinear Approaches to Laminar-Flow Stability: Blasius Boundary-Layer Flow as a Paradigm. In: Frisch, U. (eds) Hydrodynamic Instability and Transition to Turbulence. Fluid Mechanics and Its Applications, vol 100. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4237-6_5
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