A numerical solution of the Navier-Stokes equations of motion by means of finite-difference forms of the vorticity and continuity equations is presented. This is applied to the study of the flow of an incompressible fluid produced by the motion from rest of a piston in a cylindrical tube of circular cross-section.

Experiments at high Reynolds number indicated the presence in the starting flow of a ring vortex which was not reproduced by computation. Iteration to determine the stream function was not found to be necessary to achieve 1% accuracy. Omitting iteration is equivalent to only slightly disturbing the flow. An additional random disturbance applied to the flow at each time step was found to result in the production of the ring vortex.

Flow visualization has been used quantitatively to determine the flow relative to a piston and a free surface started from rest. The discharge of water from a cylindrical reservoir was investigated. Flow with a free surface started from rest was found to have a critical Reynolds number (based on tube diameter and surface speed) of about 450 above which a ring vortex was produced just below the surface.

Measurements at Reynolds numbers of 525 and 1200 were compared with computations made by the methods described in Part 1. The computed drift of tracer particles agreed well with observed values. The largest discrepancies occurred in the radial component of the drift in the early stages of the motion and amounted to 2½% of the tube diameter.

An exact periodic solution of the unsteady energy equation for an incompressible fluid with constant properties is derived to illustrate the effect of an oscillation through the viscous dissipation on a temperature field. The flow field used here is a generalization of the well-known Couette flow solution of steady flow, in which one wall is at rest and the other wall oscillates in its own plane about a constant mean velocity. The solution is subject to two boundary conditions that correspond to the heat-transfer and thermometer problems. In order to have some suggestions about the approximate solutions, the solution is compared with its own approximate form. The temperature field consists of a time-mean, first and second harmonic fluctuation. The time-mean temperature profiles show the large influence of oscillation. The time-mean heat flux into or the time-mean temperature of the oscillating wall increases with frequency, and is ultimately proportional to the square root of the frequency. In § 4 the present exact solution of the Couette flow is compared with the formerly obtained approximate solution of the flat plate boundary-layer flow in terms of the wall characteristic values at high frequencies.

The Brinkley–Kirkwood theory (1947) is modified to determine the law of propagation of a blast wave in an arbitrary inhomogeneous medium for spherically and cylindrically symmetric cases. The shock path is obtained in terms of a simple quadrature. The numerical results for the shock path and the entire flow region behind the shock, propagating in an exponential atmosphere, show excellent agreement with the exact numerical solution.

The flow of a rotating homogeneous incompressible fluid over a step is investigated. In the physical system considered the rotation axis is vertical and the step, which is assumed to be infinitely long, is located on a horizontal plane surface. Upstream of the step the fluid is in a uniform free stream motion at an angle α to a line perpendicular to the step axis. The analysis is restricted by the following: E [Lt ] 1, Ro ∼ E½, h/D ∼ E½, H/D ∼ E0, and cos α ∼ E0 where Ro and E are the Rossby and Ekman numbers and h/D and H/D are the step height to step width and water depth to step width ratios respectively. The flow field is shown to consist of interior geostrophic regions, Ekman layers on the horizontal surfaces and vertical shear layers located in the vicinity of vertical planes defined by the edges of the step. In the vertical layers there is a balance between the inertial, Coriolis, and pressure terms in the momentum equations while the effects of viscosity are found to be negligible. Downstream of the step the streamlines are shifted to the right (positive or Northern Hemisphere rotation) of their upstream locations by a distance of S = 2½(h/D) E−½ cos α. Experiments are presented which are in good agreement with the theory advanced.

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.

This paper is the first in a series of investigations having the overall objective of developing a new technique for treating the slow viscous motion past finite assemblages of particles of arbitrary shape. The new method, termed the multi-pole representation technique, is based on the theory that any object conforming to a natural co-ordinate system in a particle assemblage can be approximated by a truncated series of multi-lobular disturbances in which the accuracy of the representation is systematically improved by the addition of higher order multipoles. The essential elements of this theory are illustrated by examining the flows past finite line arrays of axisymmetric bodies such as spheres and spheroids which conform to special natural co-ordinate systems. It is demonstrated that this new procedure converges more rapidly and is simpler to use than the method of reflexions and represents the desired boundaries more precisely than the point-force approximation even when the objects are touching one another. Comparison of these solutions with the exact solutions of Stimson & Jeffery (1926) for the two sphere problem demonstrates the rapidity of convergence of this multipole procedure even when the spheres are touching. Drag results are also presented for flows past chains containing up to 101 spheres as well as for chains containing up to 15 prolate or oblate spheroids. The potential value of the technique is suggested by the rapidity with which the drag calculations were made, the 101 sphere problem requiring about 10 seconds on an IBM 360–65 computer to determine both the fluid flow and the drag coefficient.

In studying the stability of the boundary layer with surface mass injection, a generalized version of the Orr–Sommerfeld equation was derived which takes account of the transverse velocity component in the mainflow. The new terms in the generalized Orr–Sommerfeld equation are inversely proportional to the Reynolds number. The resulting eigenvalue problem was solved numerically for a wide range of values of the mass injection intensity. It was found that the critical Reynolds number (based on the distance from the leading edge) decreases with increasing mass injection. The deviations between the critical Reynolds numbers from the generalized and conventional Orr–Sommerfeld equations have a different sign at low injection intensities from that at high injection intensities.

Extensive schlieren studies and yawmeter traverses of the wake behind slender cone-cylinders at large angles of incidence have shown that the flow pattern is generally steady. Under certain flow conditions, however, the wake exhibits an instability which is not understood. For cross-flow Reynolds numbers in the subcritical region the wake can be described in terms of a cross-flow Strouhal number which has a constant value of 0·2 for cross-flow Mach number components (Mc) up to 0·7 and then increases steadily to a value of 0·6 at Mc = 1·6. The strength of the wake vortices varies substantially with Mc, increasing to a maximum at Mc ≈ 0·7 and then decreasing rapidly for higher values of Mc. Schlieren photographs of the wake have been analysed by means of the impulse flow analogy and also by considering the vortices to be part of a yawed infinite vortex street. The impulse flow analogy is shown to be of use in determining the cross-flow Strouhal number but estimates of vortex strength are too high. The Kármán vortex street theory combined with the sweepback principle leads to reliable estimates of vortex strength up to Mc = 1·0.

Information is given on the spacing, path and strength of the vortices shed from the body for flow conditions varying from incompressible speeds up to Mc = 1·0. Finally this information is used to determine the vortex drag of a two-dimensional circular cylinder below Mc = 1·0.

A method is presented for integrating numerically the equations of motion for a compressible free shear layer developing from a boundary-layer profile of arbitrary shape. Sutherland's law is used to determine the coefficient of viscosity and the Prandtl number is taken as 0·72. Calculated results are reported for free-stream Mach numbers ranging from 0 to 10 and for stagnation-enthalpy ratios ranging from 0 to 5·0. The effects of varying the initial boundary-layer profile and of a discontinuity in temperature at the origin are also studied. The results include graphs showing the development of dividing-streamline velocity, of local Nusselt number, and of dividing-streamline location.

Numerical solutions are presented for steady two-dimensional symmetric flow past a parabolic cylinder in a uniform stream parallel to its axis. The solutions cover the range R = 0·25 to ∞, where R is the Reynolds number based on the nose radius of the cylinder. For large R, the calculated skin friction near the nose of the cylinder is compared with known theoretical results obtained from second-order boundary-layer theory. Some discrepancy is found to exist between the present calculations and the second-order theory. For small R, it is possible to obtain a reasonably consistent check with a recent theoretical prediction for the limit of the skin friction near the nose of the cylinder as R → 0.

Hot-wire filament vibration generated by fluctuating aerodynamic loads is found to occur in turbulent flow fields. The motion of the filament relative to the stream can cause errors in the indicated velocity fluctuations. Analysis and experiment show that these errors are probably serious only when the velocity fluctuations are confined within a narrow bandwidth spectrum such as in a Kármán vortex street. Under such conditions the filament goes into a skipping or ‘whirling’ mode of motion and the errors in the velocity perturbation reading are substantial. Errors in normally encountered boundary-layer turbulence are suspected not to be serious but this is not yet certain. A convenient experimental method for detecting these vibrations is outlined.

A study is made of the effect of filament vibration on the direct frequency-response test for hot-wire systems. These tests include shaking the probe at high frequencies in a steady stream and the Kármán vortex street method.