The Orr-Sommerfeld equation is solved numerically using expansions in Chebyshev polynomials and the QR matrix eigenvalue algorithm. It is shown that results of great accuracy are obtained very economically. The method is applied to the stability of plane Poiseuille flow; it is found that the critical Reynolds number is 5772·22. It is explained why expansions in Chebyshev polynomials are better suited to the solution of hydrodynamic stability problems than expansions in other, seemingly more relevant, sets of orthogonal functions.

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Accurate solution of the Orr–Sommerfeld stability equation

This paper considers the nature of a non-linear, two-dimensional solution of the Navier-Stokes equations when the rate of amplification of the disturbance, at a given wave-number and Reynolds number, is sufficiently small. Two types of problem arise: (i) to follow the growth of an unstable, infinitesimal disturbance (supercritical problem), possibly to a state of stable equilibrium; (ii) for values of the wave-number and Reynolds number for which no unstable infinitesimal disturbance exists, to follow the decay of a finite disturbance from a possible state of unstable equilibrium down to zero amplitude (subcritical problem). In case (ii) the existence of a state of unstable equilibrium implies the existence of unstable disturbances. Numerical calculations, which are not yet completed, are required to determine which of the two possible behaviours arises in plane Poiseuille flow, in a given range of wave-number and Reynolds number.It is suggested that the method of this paper (and of the generalization described by Part 2 by J. Watson) is valid for a wide range of Reynolds numbers and wave-numbers inside and outside the curve of neutral stability.

On the non-linear mechanics of wave disturbances in stable and unstable parallel flows Part 1. The basic behaviour in plane Poiseuille flow

The frequency and amplification rates for a disturbance growing with respect to time are compared with those of a spatially-growing wave having the same wave-number. For small rates of amplification it is shown that the frequencies are equal to a high order of approximation, and that the spatial growth is related to the time growth by the group velocity.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112062001184

A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability

An investigation is made of resonant triads of Tollmien-Schlichting waves in an unstable boundary layer. The triads considered are those comprising a two-dimensional wave and two oblique waves propagating at equal and opposite angles to the flow direction and such that all three waves have the same phase velocity in the downstream direction. For such a resonant triad remarkably powerful wave interations take place, which may cause a continuous and rapid transfer of energy from the primary shear flow to the disturbance. It appears that the oblique waves can grow particularly rapidly and it is suggested that such preferential growth may be responsible for the rapid development of three-dimensionality in unstable boundary layers. The non-linear energy transfer primarily takes place in the vicinity of the critical layer where the downstream propagation velocity of the waves equals the velocity of the primary flow.The theoretical analysis is initially carried out for a general primary velocity profile; then, in order to demonstrate the essential features of the results, precise interaction equations are derived for a particular profile consisting of a layer of constant shear bounded by a uniform flow. Some exact solutions of the general interaction equations are presented, one of which has the property that the wave amplitudes become indefinitely large at a finite time. The possible relevance of the present theoretical model to the experiments of Klebanoff, Tidstrom & Sargent (1962) is examined.

Non-linear resonant instability in boundary layers

The theoretical study presented in this paper was inspired by the recent report (Kramer 1960) of experiments showing that considerable reductions in the drag of an underwater solid body were achieved by covering it with a skin of flexible material; apparently this effect was due to the boundary layer being stabilized in the presence of the skin, so that transition to a turbulent condition of flow was prevented or at least delayed. The stability problem for flow past a flexible boundary is here formulated in a general way which allows a full exploration of the possibility of a stabilizing effect without the need to assign specific properties to the flexible medium; the collective properties of possible boundaries are represented by a ‘response coefficient’ α (a sort of ‘effective compliance’) measuring the deflexion of the surface under a travelling sinusoidal distribution of pressure.A remarkably simple analytical connexion is established between the present general problem and the corresponding stability problem for the boundary layer on a rigid plane wall, and hence many details of the existing theory of hydrodynamic stability are immediately useful. However, the presence of the flexible boundary admits possible modes of instability additional to those which already exist when the boundary is rigid, and clearly every mode must be considered with regard to practical measures for stabilization—that is to say, it might be useless to inhibit one mode by a device which lets in another. What is believed to be an essentially complete interpretation of the over-all possibilities is deduced on recognizing three more or less distinct forms of instability. The first comprises waves resembling the unstable waves which can arise in the presence of a rigid boundary, but now being modified by the effects of flexibility. These waves tend to be stabilized when the boundary has a compliant response to them, which means the respective wave velocity has to be less than the velocity of free surface waves on the boundary; but it is found that the effect of internal friction in the flexible medium is actually destabilizing. The second form of instability is essentially a resonance effect and comprises waves travelling at very nearly the velocity of free surface waves. These waves can only be excited when the latter velocity falls below the free-stream velocity; they are scarcely affected by the viscosity of the fluid since the ‘wall friction layer’ is largely cancelled, so that damping due to the medium itself becomes the only stabilizing factor. The third form is akin to Kelvin–Helmholtz instability.This interpretation of the theoretical results seems to point to the essential factors in the operation of a flexible skin as a stabilizing device, and accordingly in the concluding secttion of the paper two alternative sets of criteria are proposed each of which would provide a logical basis for designing such a device. The principle of the first alternative explains the success of Gamer's invention, but the second appears equally promising and the relative advantages of the two can really be proved only by further experiment.

Effects of a flexible boundary on hydrodynamic stability

Calculated Amplified Oscillations in the Plane Poiseuille and Blasius Flows

On Boundary-Layer Equations in Hypersonic Flow

SUMMARY It is shown that the use of Hooke's gas (y = — 1) is inopportune and that it is preferable to limit oneself to Karman-Tsien' s "tangent gas." One observes that the "correction coefficient" of the latter is nearly exact in the. entire flow if Mmax. = ~0.5.

S. F. Shen

Papers

Calculated Amplified Oscillations in the Plane Poiseuille and Blasius Flows

On Boundary-Layer Equations in Hypersonic Flow

Remarks on "Correction for Aspect Ratio of the Indicial Lift Functions of Wagner and Kussner"