K. Stewartson

On three-dimensional packets of surface waves

In this note we use the method of multiple scales to derive the two coupled nonlinear partial differential equations which describe the evolution of a three-dimensional wave-packet of wavenumber k on water of finite depth. The equations are used to study the stability of the uniform Stokes wavetrain to small disturbances whose length scale is large compared with 2π/k. The stability criterion obtained is identical with that derived by Hayes under the more restrictive requirement that the disturbances are oblique plane waves in which the amplitude variation is much smaller than the phase variation.

Self-induced separation

A rational theory is developed to explain the initial pressure rise and consequent separation of a laminar boundary layer when it interacts with a moderately strong shock. In this theory, which is firmly based on the linearized theory of Lighthill (1953), the region of interest is divided into three parts: the major part of the boundary layer, which is shown to change under largely inviscid forces, the supersonic main stream just adjacent to the boundary layer in which the pressure variation is small; and a region close to the wall, on boundary-layer scale, in which the relative variation of the velocity is large but is controlled by the incompressible boundary-layer equations, together with novel boundary conditions. We find that the first two parts can be handled in a straightforward way and the problem of self-induced separation reduces, in its essentials, to the solution of a single problem in the theory of incompressible boundary layers. It is found that this problem has three solutions, one of which corresponds to undisturbed flow and another describes a boundary layer which, spontaneously, generates an adverse pressure gradient and a decreasing skin friction which eventually vanishes and then downstream a reversed flow is set up. The third solution generates a favourable pressure gradient and is not relevant to the present study. Although there has hitherto been no valid numerical method of integrating a boundary layer with reversed flow, we find that an ad hoc method seems to lead to a stable solution which has a number of the properties to be expected of a separated boundary layer. Comparison with experiment gives qualitatively good agreement, but quantitatively errors of the order of 20% are found. It is believed that these errors arise because the Reynolds numbers at which the experiments were carried out are too small.

Multistructured Boundary Layers on Flat Plates and Related Bodies

Publisher Summary The chapter reviews subdivisions of the boundary layers that become necessary under the impact of sudden stream wise changes. If the boundary layer is supersonic, a new phenomenon occurs that appears to have no counterpart in subsonic flow. It leads to a greater ease of study of the flow properties and helps to overcome the barrier of separation that appears to hinder progress in the incompressible studies. The phenomenon is the free-interaction boundary layer, first observed by Ackeret and independently by Liepmann in their study of the interaction between a shock wave and a boundary layer and more extensively studied subsequently by Liepmann, Chapman, Hakkinen, and many others. These studies show that when a shock, sufficiently strong to provoke separation, strikes a laminar boundary layer, the boundary layer actually separates ahead of the foot of the shock and the flow features of the separation region are independent of the characteristics of the shock and depend only on the local properties of the flow. The chapter provides example for incompressible boundary layers when the fluid is compressible and explains the modifications necessary to allow this effect.

A non-linear instability theory for a wave system in plane Poiseuille flow

The initial-value problem for linearized perturbations is discussed, and the asymptotic solution for large time is given. For values of the Reynolds number slightly greater than the critical value, above which perturbations may grow, the asymptotic solution is used as a guide in the choice of appropriate length and time scales for slow variations in the amplitude A of a non-linear two-dimensional perturbation wave. It is found that suitable time and space variables are e t and e ½ ( x + a 1 r t ), where t is the time, x the distance in the direction of flow, e the growth rate of linearized theory and (− a 1 r ) the group velocity. By the method of multiple scales, A is found to satisfy a non-linear parabolic differential equation, a generalization of the time-dependent equation of earlier work. Initial conditions are given by the asymptotic solution of linearized theory.

On the flow near the trailing edge of a flat plate II

Consider an incompressible fluid of density p and kinematic viscosity v in an infinite two-dimensional domain. We assume that the fluid has a uniform velocity U ∞ , in the direction of the positive x *-axis of a rectangular Cartesian coordinate system Ox* y* , at large distances from a fixed flat plate of zero thickness which occupies the interval − l x * Ox *. Of special interest here is the structure of the flow when e ≪ 1 where Re being the Reynolds number of the flow. A first examination was made by Blasius (1908), using Prandtls theory of the boundary layer, who found inter alia that the leading term of the drag D on one side of the plate is given by the numerical factor being determined by Goldstein (1930).

A sufficient condition for the instability of columnar vortices

The inviscid instability of columnar vortex flows in unbounded domains to three-dimensional perturbations is considered. The undisturbed flows may have axial and swirl velocity components with a general dependence on distance from the swirl axis. The equation governing the disturbance is found to simplify when the azimuthal wavenumber n is large. This permits us to develop the solution in an asymptotic expansion and reveals a class of unstable modes. The asymptotic results are confirmed by comparisons with numerical solutions of the full problem for a specific flow modelling the trailing vortex. It is found that the asymptotic theory predicts the most-unstable wave with reasonable accuracy for values of n as low as 3, and improves rapidly in accuracy as n increases. This study enables us to formulate a sufficient condition for the instability of columnar vortices as follows. Let the vortex have axial velocity W(r) , azimuthal velocity V(r) , where r is distance from the axis, let Ω be the angular velocity V / r , and let Γ be the circulation rV . Then the flow is unstable if

On almost rigid rotations. Part 2

V\frac{d\Omega}{dr}\left[ \frac{d\Omega}{dr}\frac{d\Gamma}{dr} + \left(\frac{dW}{dr}\right)^2\right]

On the slow motion of a sphere parallel to a nearby plane wall

The dynamical properties of a fluid, occupying the space between two concentric rotating spheres, are considered, attention being focused on the case where the angular velocities of the spheres are only slightly different and the Reynolds number R of the flow is large. It is found that the flow properties differ inside and outside a cylinder [Cscr ], circumscribing the inner sphere and having its generators parallel to the axis of rotation. Outside [Cscr ] the fluid rotates as if rigid with the angular velocity of the outer sphere. Inside [Cscr ] the fluid rotates with an angular velocity intermediate to the angular velocities of the two spheres and determined by the condition that the flux of fluid into the boundary layer of the faster-rotating sphere is equal to the flux out of the boundary layer of the slower-rotating sphere at the same distance from the axis. The return of fluid is effected by a shear layer near [Cscr ] and we show that it has a complicated structure for it can be divided into three separate layers, two outer ones, of thickness

On the free convection from a horizontal plate

\sim R^{-\frac{2}{7}}

On the stability of a spinning top containing liquid

and ∼ R −¼ , and an inner layer of thickness