A long slender axisymmetric body is considered placed at rest in a general linear flow in such a manner that the undisturbed fluid velocity is identically zero on the body axis. Formulae for the total force and torque on the body are found as an expansion in terms of a small parameter κ defined as the radius-to-length ratio of the body. These general results are used to determine the resistance to axial rotation of the body and also the equivalent axis ratio of the body for motion in a shear flow.

European Mechanics Colloquium number 17 was held at Cambridge from 1 to 3 July 1970, when the subject of bluff bodies and vortex shedding was discussed. The following report summarizes some of the papers presented. No formal proceedings of the meeting will be published.

The investigation by Hayes (1964a) of the behaviour of a constant-density inviscid rotational flow in the neighbourhood of a stagnation point on a plane wall has been extended to include the effects of viscosity. The principal effect is the manner in which the singularity in vorticity discovered by Hayes is removed. A solution of only the boundary-layer equatiosn indicates the vorticity decays algebraically from the wall. Application of the method of matched asymptotic expansions, however, shows that the difference between boundary layer and outer vorticity, when carried out to second order in the outer flow, does not contribute to an algebraic decay. These results suggest that an infinite number of higher-order outer terms are generated which match the algebraic terms thereby yielding the conventional exponential decay. Numerical results are presented which also support this conclusion. The main contribution of the wall shear stress in the immediate neighbourhood of the stagnation point is shown to come from the external lateral vorticity.

Further evidence of a universal maximum drag reduction asymptote is presented. In the elastic sublayer model, inferred therefrom, the mean velocity profile during drag reduction is approximated by three zones: the usual viscous sublayer, an elastic sublayer where the mixing-length constant is derived from the maximum drag reduction asymptote, and an outermost region with Newtonian mixinglength constant. Upon integration the model yields a friction factor relation, parametric in elastic sublayer thickness, which properly reproduces the known features of turbulent dilute polymer solution flow. The dependence of elastic sublayer thickness upon flow and polymeric parameters is inferred from experimental data revealing two hitherto unknown relationships: namely that on Prandtl co-ordinates, 1/f½vs. log Re f½ the difference in slope between a polymer solution and solvent is proportional to the square root of molar concentration and to the three-halves power of backbone chain links in the macromolecule. The proportionality constant in the preceding relationship is approximately the same for several different polymer species of carbon-carbon or similar skeletal structure in various thin solvents; there is an indication that this constant further depends upon the product of solvent viscosity times the cube of the effective bond length per chain link of the polymer species. Some recent results regarding the onset of drag reduction are also summarized.

The flow in a pipe of circular cross-section which is coiled in a circle is studied, the pressure gradient along the pipe varying sinusoidally in time with frequency . The radius of the pipe a is assumed small in relation to the radius of curvature of its axis R. Of special interest is the secondary flow generated by centrifugal effects in the plane of the cross-section of the pipe, and an asymptotic theory is developed for small values of the parameter = (2/a2), where is the kinematic viscosity of the fluid. The secondary flow is found to be governed by a Reynolds number , where is a typical velocity along the axis of the pipe, and asymptotic theories are developed for both small and large values of this parameter. For sufficiently small values of it is found that the secondary flow in the interior of the pipe is in the opposite sense to that predicted for a steady pressure gradient, and this is verified qualitatively by an experiment described at the end of the paper.

Drag reduction caused by dilute, distilled water solutions of four polyethylene-oxides and one polyacrylamide, molecular weights respectively 0·1 × 106 to 8 × 106 and 13 × 106, was studied experimentally in one smooth and three sand-roughened pipes, relative roughnesses (R/k) 15, 23, 35, all of about 0·34, in inside diameter. The onset of drag reduction in the rough pipes occurred at the same wall shear stress as in smooth; the onset wall shear stress was essentially independent of polymer concentration, varied inversely as the square of polymer radius of gyration and was unaffected by the flow régime, hydraulically smooth, transitional or fully rough, during which onset occurred. Following onset a flow régime was observed wherein the fractional slip, i.e. fractional increase in mean velocity relative to solvent at a given friction velocity, obtained with a given polymer solution in a rough pipe was the same as the fractional slip in the smooth pipe despite marked differences in the respective rough and smooth friction factors. This ‘effectively smooth’ régime prevailed for values of non-dimensional roughness k+* < k+ < k+es from onset, k+*, to an upper limit given by k+es ∼ 50 for all of the present experiments. For k+ < k+es, the fractional slip in the rough pipes was always less than that corresponding to smooth and was a function of relative roughness as well as flow and polymeric parameters. The maximum drag reduction possible in the rough pipes was limited by an asymptote which was independent of polymeric parameters. Under asymptotic conditions, friction factors in all the rough pipes identically obeyed the smooth pipe friction factor relation for k+ < 12; the onset of roughness at k+ ∼ 12 indicated that the maximum viscous sublayer thickness attained during drag reduction was approximately 2½ times Newtonian.

Laboratory analogues of theoretical models of wind-driven ocean circulation are based on ideas presented by Stommel (1957). A particularly simple demonstration of the applicability of these ideas is contained in a paper by Stommel, Arons & Faller (1958). The present work develops the source-sink laboratory analogue of ocean circulation models to a point where chosen parametric values allow one to simulate the theoretical models of Stommel (1948) and Munk (1950) exactly. The investigation of the flow in a rotating cylinder generated by a source of fluid near the outer wall leads to a detailed description of the roles of the various boundary layers which occur. This knowledge is used to analyse the more complex source-sink flow in a pie-shaped basin. The laboratory analogue to the Stommel circulation model is analyzed in detail. It is shown that the change in the flow pattern brought about by a radial variation of the position of the eastern boundary in the pie-shaped basin is confined to the interior flow and the boundary layer is largely unaffected. When the bottom of the pie-shaped container slopes, the circulation pattern is changed significantly. For the particular case treated, the depth of the basin along the western boundary is unchanged and the maximum depth occurs at the southeast corner. The circulation generated by a source introduced at the apex of the pie has a gyre whose centre is shifted more toward the southwest corner than the corresponding centre of the gyre for a flat-bottomed basin. Two experiments are reported showing that the western boundary may separate because of the effect of bottom topography or because of the pressure of a cyclonic and an anti-cyclonic gyre generated by suitably placed sources and sinks.

Eady's model for the stability of a thermal wind in an inviscid, stratified, rotating system is modified to allow for deviations from hydrostatic equilibrium. The stability properties of the flow are uniquely determined by two parameters: the Richardson number Ri, and the ratio of the aspect ratio to the Rossby number δ. The latter parameter may be taken as a measure of the deviations from hydrostatic equilibrium (δ = 0 in Eady's model). It is found that such deviations decrease the growth rates of all three kinds of instability which can occur in this problem: ‘geostrophic’ baroclinic instability, symmetric instability, and Kelvin–Helmholtz instability. The unstable wavelengths for ‘geostrophic’ and Kelvin–Helmholtz instability are increased for finite values of δ, while the unstable wavelengths for symmetric instability are unaffected. The ‘non-hydrostatic’ effects (δ ≠ 0) are significant for symmetric and Kelvin–Helmholtz instability when δ [gsim ] 1, but not for ‘geostrophic’ instability unless δ [Gt ] 1. Consequently, the first two types of instability tend to be suppressed relative to ‘geostrophic’ instability by ‘non-hydrostatic’ conditions. Figure 3 summarizes the different instability régimes that can occur. In laboratory experiments symmetric instability can be studied best when δ [lsim ] 1, while Kelvin–Helmholtz instability can be studied best when δ [Lt ] 1.

An extension of the StuartWatson technique for examination of the non-linear hydrodynamic instability of time-dependent flows is proposed. An example in Bnard convection is used to illustrate the method. Extensions to more general problems are indicated.

Repeated observations of a uniformly twisted director field in cholesteric liquid crystals are used to motivate an expression for a free energy which is obtained as an expansion about this state. Terms quadratic in the director perturbation and the gradients of this perturbation are retained. Utilizing invariance arguments, it is possible to obtain significant simplification of the coefficients which appear in the expansion. A properly invariant form of the free energy is produced which agrees with the expansion for small excursions about a twisted state, and which assigns arbitrary values to the non-vanishing coefficients.

The consequences of requiring that a free energy achieve a minimum at a twisted state are explored. A commonly used form of the free energy for cholesteric liquid crystals is seen to be rather severely restricted by this requirement. An alternative to this form is proposed which is a special case of the free energy previously produced. The particular form suggested attains a unique absolute minimum at a characteristic uniform twist.

The governing equations for the problem of linearized flow through a normal shock wave in an emitting, absorbing, and scattering grey gas are reduced to two linear coupled integro-differential equations. By separation of variables, these equations are further reduced to an integral equation similar to that which arises in neutron-transport theory. It is shown that this integral equation admits both regular (associated with discrete eigenfunctions) and singular (associated with continuum eigenfunctions) solutions to form a complete set. The exact closed-form solution is obtained by superposition of these eigen-functions. If the gas downstream of a strong shock is absorption–emission dominated, the discrete mode of the solution disappears downstream. The effects of isotropic scattering are discussed. Quantitative comparison between the numerical results based on the exact solution and on the differential approximation are presented.

An analytical solution is developed to describe the steady, closed streamline velocity field within a cylindrical cavity with a uniformly translating wall at low Reynolds numbers. The solution has application for the case of two-phase flow in a tube where regions of fluid are segmented by a moving train of bubbles or plugs, such as in the pulmonary and peripheral capillaries of the body where segments of plasma are trapped between red blood cells. The mathematical approach presented in this study can in principle be useful in the analysis of a wide class of closed-streamline creeping-flow problems.

An experimental study of jet impingement is completed with the presentation of the measured turbulent characteristics of the circular subsonic jet and the heat transfer rates measured when this jet impinges normal to a flat plate. The data suggest that for impingement very close to the stagnation point, the heat transfer can be computed by applying a turbulent correction factor to the laminar value calculated for a flow having the same pressure distribution as that present in the impingement region. The correction factor is found to be a function of the axial distance and not of Reynolds number. Farther away, the measurements agree well with the heat transfer estimated using the method of Rosenbaum & Donaldson (1967). At large distances from the stagnation point, the heat transfer falls off in inverse proportion with the distance.

The documentation of the turbulent jet flow field includes measurements of the radial and axial velocity fluctuations and their spectra, as well as the radial distribution of turbulent shear $\overline{w^{\prime}u^{\prime}}$. In addition, measurements of the turbulence near the stagnation point and the total pressure fluctuation at the stagnation point are presented.

In this paper we present some results concerning the stability of flow in a circular pipe to small but finite axisymmetric disturbances. The flow is unstable if the amplitude of a disturbance exceeds a critical value, the equilibrium amplitude, which we have calculated for a wide range of wave-numbers and Reynolds numbers. For large values of the Reynolds number, R, and for a real value of the wave-number, α, we indicate that the energy density of a critical disturbance is of order c2i, where −ααci is the damping rate of the associated infinitesimal disturbance. The energy, per unit length of the pipe, of a critical disturbance which is concentrated near the axis of the pipe is of order R−2, and the wave-number α is of order R1/3 For a critical disturbance which is concentrated near the wall of the pipe the energy is of order $R^{-\frac{3}{2}}$ and α is of order R½. This suggests that non-linear instability is most likely to be caused by a ‘centre’ mode rather than by a ‘wall’ mode. The wall mode solution is also essentially the solution for the problem of plane Couette flow when αR is large. We compare it with the true solution.

In an appendix Dr A. E. Gill indicates how some of the results of this paper may be inferred from a simple scale analysis.

An analytical solution is developed to describe the unsteady-state heat transfer to a cylindrical cavity with circulating flow induced by a moving wall. The previously derived velocity field at low Reynolds numbers is incorporated into the energy equation, and the case of heat transfer to a fluid segmented by highly conducting plugs flowing in a tube with constant wall temperature is considered. Calculations of temperature distributions, average temperatures, and heat transfer coefficients as functions of time and Péclet number are presented for a specific cavity geometry, and the degree of enhancement in heat transfer caused by the recirculating flow is determined.

The methods developed in this study may be useful in obtaining analytical solutions to a variety of closed-streamline heat and mass transfer problems with known velocity fields.

This paper is the report of an experimental study of heat transfer from fine horizontal wires of various lengths to fluids having various Prandtl numbers. Transport characteristics were determined for the full spectrum of processes from natural, through mixed, to forced convection. The experimental results are compared with those of past analysis and with correlations. The very high level of accuracy and reproducibility of the data makes possible the detection of very small effects. The influence of cylinder length on transport is shown for both of the asymptotic rgimes of natural and of forced convection. Transport in the mixed-convection rgime is determined and limits of that rgime are estimated.

Measurements are presented of the wave-number/frequency and wave-number/phase velocity spectrum of wall pressure for a two-dimensional turbulent boundary layer in zero pressure gradient, obtained from a Fourier transform of experimental filtered spatial correlations. This method allows the results to be corrected for acoustic disturbances in the wind tunnel, and for finite transducer size. An empirical form for the pressure field is proposed, based on the measurements, and is used to predict a frequency spectrum correction for transducer size which agrees well with measured values.

The velocity just outside the boundary layer and upstream of the separation ring on a sphere moving along the axis of a slightly viscous, rotating fluid is calculated through a least-squares approximation on the hypothesis of no upstream influence. A reverse flow is found in the neighbourhood of the forward stagnation point for k ≡ 2Ωa/U > k = 2·20 (Ω = angular velocity of fluid, U = translational velocity of sphere, a = radius of sphere) and is accompanied by a forwardseparation bubble, such as that observed by Maxworthy (1970) for k [gsim ] 1. Rotation also induces a downstream shift of the peak velocity; the estimated shift of the separation ring in the absence of forward separation increases with k to a maximum of 24°, in qualitative agreement with Maxworthy's observations.

The least-squares formulation is compared with that given by Stewartson (1958) for unseparated flow (Stewartson did not consider separation). Both formulations require truncation of an infinite set of simultaneous equations, but Stewartson's formulation yields a non-positive-definite matrix that may exhibit spurious singularities. The least-squares formulation yields a positive-definite matrix, albeit at the expense of slower convergence for fixed k, and is especially well suited for automatic computation.

An ad hoc incorporation of a cylindrical wave of strength [Uscr ], such that the maximum upstream axial velocity is [Uscr ]U, is considered in an appendix. It is found that k decreases monotonically from 2·2 to 0 as [Uscr ] increases from 0 to 1.

The transient situation which follows when a plane normal shock wave is reflected (from a coplanar wall) into its own relaxation zone is examined theoretically. Approximate inner and outer solutions for the wall-pressure history are employed to establish the timewise variations in the thermodynamic state of the gas adjacent to the wall. Results for chemically relaxing O2 and vibrationally relaxing CO2 are compared with previous numerical solutions based on the method of characteristics, and the agreement is found to be excellent. The approximate technique is simple and requires only a minimum of computing time.

In this, the first part of a two-part experimental study of the behaviour of impinging jets, the mean properties of the flow field are established. Velocity profiles are given for three types of jet flow issuing from a circular convergent nozzle. Measured distributions of surface pressure are given which result when the jets impinge both normally and obliquely at various distances on several surface shapes. The pressure distributions are used to compute the radial velocity gradient at the impingement stagnation point. It is found that for normal impingement this gradient correlates with the free jet centreline velocity and half-radius at the same axial location. A fall-off in the correlated value is noted as the impingement is made oblique. Measurements of the azimuthal distribution of momentum flux in the resulting wall jet are also given. The general behaviour of all three types of jet is found to be similar at locations downstream of any local effects due to the shock waves present in the under-expanded types. A special study of the close-range impingement of an under-expanded jet containing a normal shock disk reveals a region of separated flow surrounding the stagnation point. This condition results in a negative value of the radial velocity gradient at the centreline.