Abstract
This chapter on uniqueness and stability provides a first flavor into the subject of laminar-turbulent transition. Two different theoretical concepts are in use and both assume that the laminar-turbulent transition is a question of loss of stability of the laminar motion. With the use of the energy method one tries to find conditions for the laminar flow to be stable. Energy stability criteria operate with the construction of upper bounds of the rate of the perturbed kinetic energy K(t) of the fluid system, in order to obtain by time integration an inequality of the form \(K(t) < K(0)\mathrm {exp}\,(-t/\tau )\). Here, \(\tau > 0\) guarantees decay and \(\tau < 0\) growth rates of the perturbed energy, \(\tau = 0\) guarantees neutral stability of the perturbation flows. The difficulty of the method is that the condition \(\tau = 0\) generally provides poor, i.e., very safe estimates for stability. More successful for pinpointing the laminar-turbulent transition has been the method of linear instability analysis, in which a lowest bound, is searched for, at which the onset of deviations from the laminar flow is taking place. For plane channel flows the Rayleigh and Orr–Sommerfeld equations with associated boundary conditions for an ideal and viscous fluid, respectively, are derived and the associated eigenvalue problems are discussed, which leads to the stability chart, separating Reynolds number dependent stable and unstable flow regimes.
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Notes
- 1.
There is a large number of books treating stability as a whole subject. Among these we mention S. Chandrasekhar [2], F. Charru [3], P.G. Drazin [4], P.G. Drazin and W.H. Reid [6], C. Godreche and C. Manneville [8], C.C. Lin [12], D.D. Joseph [10, 11], S.S. Sritharan [21], H.L. Swinney and J.P. Gollub [22].
- 2.
E. Foá: L’ Industria, 43, 426 (1929), Milan.
- 3.
See e.g. Handbuch der Physik III/1 ‘Strömungsmechanik (I)’, p. 153 ff.
- 4.
An almost periodic function is uniformly bounded and its value at any position x, y is (for fixed z, t) in a distant point again assumed with almost the same value, see Fig. 14.4 .
- 5.
This statement has to be understood in the sense that if e.g. \(V=0\), then \(v^{\prime }\) is obviously not small in comparison to V, but it may still be small in comparison to another variable of the same dimension, e.g. U.
- 6.
For a short biography of Sommerfeld, see Fig. 14.5 .
- 7.
These coordinates differ from those used in Fig. 14.3 or (14.30) by the translation \(y=y^{\star }-\frac{d}{2}\), for which (14.30)\(_2\) reads
$$\begin{aligned} U(y^{\star }) = \frac{3}{2}U_{0} \left( 4\left( \frac{y^{\star }}{d}\right) -4\left( \frac{y^{\star }}{d}\right) ^{2}\right) . \end{aligned}$$Thus, (14.63) is formally valid. Moreover, the prime, \(\varPsi ^{\prime }\), in (14.63) and consecutive formulae designates now \(\mathrm{{d}}\varPsi /\mathrm{{d}}y\).
- 8.
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Hutter, K., Wang, Y. (2016). Uniqueness and Stability. In: Fluid and Thermodynamics. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-33636-7_14
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