ALBERT

Description: The familiar Segré-Silberberg effect of inertia-induced lateral migration of a neutrally buoyant rigid sphere in a Newtonian fluid is studied theoretically for simple shear flow and for two-dimensional Poiseuille flow. It is shown that the spheres reach a stable lateral equilibrium position independent of the initial position of release. For simple shear flow, this position is midway between the walls, whereas for Poiseuille flow, it is 0·6 of the channel half-width from the centre-line. Particle trajectories are calculated in both cases and compared with available experimental data. Implications for the measurement of the rheological properties of a dilute suspension of spheres are discussed. © 1974, Cambridge University Press. All rights reserved.

Description: A detailed study of the intermittency in the outer region of a flat-plate turbulent boundary layer has been carried out using digital sampling and processing techniques. Conditional averages are used to generate mean and fluctuating components for the turbulent and non-turbulent zones of fluid. More particularly, point averages of these variables, taken with reference to the instantaneous position of the turbulent/non-turbulent interface, have been made to show the distribution of various quantities through the turbulent front. The results indicate that significant differences exist at leading and trailing edges of the turbulent bursts and a more complete picture of the motion of an average large eddy is deduced. © 1974, Cambridge University Press. All rights reserved.

Description: Experimental results for dissociating nitrogen flow over a wedge, obtained in a free-piston shock tunnel, are described. Interferograms of the flow show clearly the curvature of the shock wave and the rise in fringe shift after the shock associated with the dissociation. It is shown that the shock curvature a t the tip of the wedge can be used to calculate the initial dissociation rate and that it is a more sensitive indication of the rate than can be obtained from fringe shift measurements under the prevailing experimental conditions. Because the free-stream dissociation fraction can be adjusted in the shock tunnel, the dependence on atomic nitrogen concentration of the dissociation rate can be determined by the shock curvature method. A detailed calculation of the flow field by an inverse method, starting from the measured shock shape, shows good agreement with experiments. © 1974, Cambridge University Press. All rights reserved.

Description: Measurements of the intermittency factor γ and in particular the crossing frequency fγ of the turbulent/non-turbulent interface in the outer regions of various turbulent shear flows depend strongly on the settings of the intermittency meter used. Two methods of calibrating an intermittency meter of conventional design are described. In the first, turbulent and non-turbulent signals are simulated and switched at random times using an analog computer. Particular attention is given to the spectra of the switching and turbulent signals but the non-turbulent signal is assumed to have the same spectrum as the turbulent signal. In the second method, the same switching process is applied to two real signals, obtained in the fully turbulent and irretational flow regions associated with a turbulent jet with a co-flowing external air stream. A rather simple calibration procedure derived using the results of both methods is applied to the measurements of γ and fγ in the same jet. It is suggested that the simulation process adopted here could be useful in inferring properties of intermittent turbulent flows. © 1974, Cambridge University Press. All rights reserved.

Description: The steady streaming generated in a pipe of slowly varying cross-section when a purely oscillatory pressure difference is maintained between its ends is considered. It is assumed that the perturbation of the pipe wall in the r, θ plane is small compared with the characteristic thickness of the Stokes layer associated with the oscillatory motion of the fluid. The first-order steady streaming is evaluated for the cases when this characteristic thickness is large and small compared with a typical radius of the pipe. In both these limits it is found that the geometry of the pipe is crucial in determining the nature of the induced steady streaming. If the ends of the pipe have the same mean radius it is found that the steady streaming consists of regions of recirculation between the nodes of the pipe. Otherwise the steady streaming is of a larger order of magnitude and has a component which represents a net flow towards the wider end of the pipe. © 1974, Cambridge University Press. All rights reserved.

Description: The steady streaming generated in the boundary layer on a cylinder performing simple harmonic motion in a viscous incompressible fluid which is otherwise at rest is investigated in the case where the Reynolds number Rs associated with this streaming is large. Comparison is made between experimental results obtained here and the theories of Riley (1965) and Stuart (1966). This comparison shows good agreement between the theories and the experiment close to the cylinder, but away from the cylinder significant discrepancies are observed. Possible reasons for these discrepancies are discussed. © 1974, Cambridge University Press. All rights reserved.

Description: By making simple assumptions, an analytical theory is deduced for the mean velocity behind a two-dimensional obstacle (of height h) placed on a rigid plane over which flows a turbulent boundary layer (of thickness δ). It is assumed that h ≫ δ, and that the wake can be divided into three regions. The velocity deficit − u is greatest in the two regions in which the change in shear stress is important, a wall region (W) close to the wall and a mixing region (M) spreading from the top of the obstacle. Above these is the external region (E) in which the velocity field is an inviscid perturbation on the incident boundary-layer velocity, which is taken to have a power-law profile U(y) = U∞(y − y1)n/δn, where n ≫ 1. In (M), assuming that an eddy viscosity (= KhU(h)) can be defined for the perturbed flow in terms of the incident boundary-layer flow and that the velocity is self-preserving, it is found that u(x,y) has the form [formula omitted] and the constant which defines the strength of the wake is [formula omitted] where u = uE(x, y) as y → 0 in region (E). In region (W), u(y) is proportional to In y. By considering a large control surface enclosing the obstacle it is shown that the constant of the wake flow is not simply related to the drag of the obstacle, but is equal to the sum of the couple on the obstacle and an integral of the pressure field on the surface near the body. New wind-tunnel measurements of mean and turbulent velocities and Reynolds stresses in the wake behind a two-dimensional rectangular block on a roughened surface are presented. The turbulent boundary layer is artificially developed by well-established methods (Counihan 1969) in such a way that δ = 8h. These measurements are compared with the theory, with other wind-tunnel measurements and also with full-scale measurements of the wind behind windbreaks. It is found that the theory describes the distribution of mean velocity reasonably well, in particular the (x/h)−1 decay law is well confirmed. The theory gives the correct self-preserving form for the distribution of Reynolds stress and the maximum increase of the mean-square turbulent velocity is found to decay downstream approximately as [formula omitted] in accordance with the theory. The theory also suggests that the velocity deficit is affected by the roughness of the terrain (as measured by the roughness length y0) in proportion to In (h/y0), and there seems to be some experimental support for this hypothesis. © 1974, Cambridge University Press. All rights reserved.

Description: Two-species, irreversible, very rapid reactions, with mild heat release, in a turbulent shear flow are shown to be analogous to the transport of two non-reacting species by the same shear field. Expressions for the probability density functions of the reacting species, the product species and the reaction-generated thermal field are obtained in terms of the joint probability density functions of the two nonreacting species. As an example we have constructed, from recent measurements of temperature statistics at a cross-section in a heated jet, the meanand fluctuating concentration fields of the reacting species and the mean concentration of the product. © 1974, Cambridge University Press. All rights reserved.

Description: An integral analysis of the type used to predict the flow of co-flowing jets has been applied to the problem of a sudden enlargement in a pipe (Borda–Carnot expansion). This technique successfully predicts all the overall flow parameters of interest (e.g. reattachment lengths, pressure profile, etc.). The analysis indicates that the downstream conditions (up to reattachment) are insensitive to wall shear and the point of minimum pressure does not coincide with the location of the maximum return-flow velocity. © 1974, Cambridge University Press. All rights reserved.

Description: The equations which govern the flow at high Reynolds number in the vicinity of the trailing edge of a finite flat plate at incidence to a uniform supersonic stream are solved numerically using a finite-difference procedure. The critical order of magnitude of the angle of incidence α* for the occurrence of separation on one side of the plate is α* = O(R−¼) (Brown & Stewartson 1970), where R is a representative Reynolds number for the flow, and results are computed for three such values of α* which characterize the possible behaviour of the flow above the plate. The final set of computations leads to a numerical value for the trailing-edge stall angle α*s, the angle of incidence which just causes the flow to separate at the trailing edge of the plate. Analytic solutions are available in the form of asymptotic expansions near the trailing edge in terms of the scaled variable of order R−⅜. A multi-layer-type of expansion which occurs in the case α* = αs* is presented in detail for comparison with the computed solution. © 1974, Cambridge University Press. All rights reserved.

Description: The properties of finite-amplitude thermal convection for a Boussinesq fluid contained in a spherical shell are investigated. All nonlinear terms are retained in the equations, and both axisymmetric and non-axisymmetric solutions are studied. The velocity is expanded in terms of poloidal and toroidal vectors. Spherical surface harmonics resolve the horizontal structure of the flow, but finite differences are used in the vertical. With a few modifications, the transform method developed by Orszag (1970) is used to calculate the nonlinear terms, while Green's function techniques are applied to the poloidal equation and diffusion terms. Axisymmetric solutions become unstable to non-axisymmetric perturbations at values of the Rayleigh number that depend on Prandtl number and shell thickness. However, even when stable, axisymmetric solutions are not a preferred solution to the full equations; steady non-axisymmetric solutions are obtained for the same parameter values. Initial conditions determine the characteristics of the finite-amplitude solutions, including, in the cases of non-axisymmetry, whether or not a steady state is achieved. Transitions in horizontal flow structure can occur, accompanied by a transition in functional dependence of heat flux on Rayleigh number. The dominant modes in the solutions are usually the modes most unstable to the onset of convection, but not always. © 1974, Cambridge University Press. All rights reserved.

Description: Jets of water and of poly(ethylene oxide) solutions discharging in air were photographed using a novel image-motion compensating camera. Spray droplet formation is inhibited by low concentration polymer solutions. The effect of the polymer is to reduce, dampen, or eliminate small-scale surface disturbances in the jet, while not reducing but even amplifying larger scale motions. The initial laminar zone present in the jet efflux with water is eliminated with trace quantities of polymer. When substantial quantities of polymer are present (200 p.p.m.), the jet breakup is accompanied by filament formation linking all the drops together. © 1974, Cambridge University Press. All rights reserved.

Description: The inviscid stability of swirling flows with mean velocity profiles similar to that obtained by Batchelor (1964) for a trailing vortex from an aircraft is studied with respect to infinitesimal non-axisymmetric disturbances. The flow is characterized by a swirl parameter q involving the ratio of the magnitude of the maximum swirl velocity to that of the maximum axial velocity. It is found that, as the swirl is continuously increased from zero, the disturbances die out quickly for a small value of q if n = 1 (n is the azimuthal wavenumber of the Fourier disturbance of type exp{i(αx + nφ − αct)}); but for negative values of n, the amplification rate increases and then decreases, falling to negative values at q slightly greater than 1·5 for n = −1. The maximum amplification rate increases for increasingly negative n up to n = −6 (the highest mode investigated), and corresponds to q ≃ 0·85. The applicability of these results to attempts at destabilizing vortices is briefly discussed. © 1974, Cambridge University Press. All rights reserved.

Description: The analytical characteristic method is an effective method for computing non-linear effects in inviscid supersonic flow problems. Although only linear equations have to be solved, the results are essentially nonlinear, in the sense that the functional relations between physical state variables and space co-ordinates are nonlinear in the small perturbation parameter introduced, like the thickness ratio or incidence of a wing. This holds even for the first-order approximation of the method. In the case of two-dimensional (plane or axisymmetric) flow the independent variables are characteristic co-ordinates, i.e. they are chosen so as to be constant along corresponding characteristic lines. The space co-ordinates are considered as dependent variables. In three dimensions there is no unique definition of a characteristic co-ordinate system, because the manifold of characteristic surfaces or bi-characteristics is larger than is necessary for defining a co-ordinate system. The success of a three-dimensional analytical characteristic method, however, depends on the proper choice of the co-ordinate system. The present analytical Characteristic method for three-dimensional flow is based on the fact that three-dimensional flow behaves locally like axisymmetric flow if it is considered in the osculating plane. The corresponding ‘distance from the axis’ is a function of space depending on the flow field. No change of pressure occurs normal to the osculating plane and in isentropic flow no change of speed either. Therefore no co-ordinate perturbation is performed in this normal direction. In the osculating plane the analytical characteristic methodis applied locally as in axisymmetric flow. In the large the space co-ordinates are obtained by integration along the main bi-characteristics. As an example the flow field on the suction side of a flat delta wing with sub-sonic leading edges is computed. As a main result one obtains shock waves in the neighbourhood of the leading edges following the expansion. © 1974, Cambridge University Press. All rights reserved.

Description: Stokes' infinitesimal-wave expansion for steady progressive free-surface waves has been extended to high order using a computer to perform the coefficient arithmetic. Stokes’ expansion has been found to be incapable of yielding the highest wave for any value of the water depth since convergence is limited by a square-root branch-point some distance short of the maximum. By reformulating the problem using a different independent parameter, the highest waves are obtained correctly. Series summation and analytic continuation are facilitated by the use of Padé approximants. The method is valid in principle for any finite value of the wavelength and solutions of high accuracy can be obtained for most values of the wave height and water depth. An alternative expansion procedure proposed by Havelock for the computation of waves short of the highest has been reconsidered and found to be defective. © 1974, Cambridge University Press. All rights reserved.

Description: A fluid layer that lines the inner surface of a circular tube has motion induced by axial surface-tension gradients. The lubrication equations for the system are analysed and it is found that even for thin layers the motions differ markedly from those in planar layers. The planar case serves as a class of outer solutions. These approximate solutions are modified by a boundary-layer correction where the mean surface tension is important. © 1974, Cambridge University Press. All rights reserved.

Description: General dimensional and similarity arguments are applied to derive a heat and mass transfer law for fully turbulent flow along a rough wall. The derivation is quite analogous to Millikan's (1939) derivation of a skin-friction law for smooth-and rough-wall flows and to the derivation of the heat and mass transfer law for smooth-wall flows by Fortier (1968a, b) and Kader & Yaglom (1970, 1972). The equations derived for the heat or mass transfer coefficient (Stanton number) Ch and Nusselt number Nu include the constant term β of the logarithmic equation for the mean temperature or concentration of a diffusing substance. This term is a function of the Prandtl number, the dimensionless height of wall protrusions and of the parameters describing the shapes and spatial distribution of the protrusions. The general form of the function β is roughly estimated by a simplified analysis of the eddy-diffusivity behaviour in the proximity of the wall (in the gaps between the wall protrusions). Approximate values of the numerical coefficients of the equation for β are found from measurements of the mean velocity and temperature (or concentration) above rough walls. The equation agrees satisfactorily with all the available experimental data. It is noted that the results obtained indicate that roughness affects heat and mass transfer in two ways: it produces the additional disturbances augmenting the heat and mass transfer and simultaneously retards the fluid flow in the proximity of the wall. This second effect leads in some cases to deterioration of heat and mass transfer from a rough wall as compared with the case of a smooth wall at the same values of the Reynolds and Prandtl numbers. © 1974, Cambridge University Press. All rights reserved.

Description: This paper deals with local flow characteristics of subsonic turbulent jets in the presence of a cross-flow. For the various types of jet considered (a cylindrical jet and coaxial jets) the experimental results concern the axes and the velocity profiles in the plane of symmetry of the flow. In the case of the cylindrical jet, the shape of the universal axial velocity profile is defined, as are the law of velocity decay along the axis and the laws of variation of the thicknesses of the jet. Finally, the existence of a link between the axis equation and the law of axial velocity decay in the zone of similarity of the velocity profiles is established. © 1974, Cambridge University Press. All rights reserved.

Description: Numerical calculations are made of the torque required to sustain a wavy-vortex flow between rotating cylinders. The results are found to agree well with experimental work of Donnelly (1958), and give further confirmation of the validity of Davey, DiPrima & Stuart's (1968) analysis. © 1974, Cambridge University Press. All rights reserved.

Description: The theory describing the swimming mechanism of an elongated body is extended to cover the cases of locomotion through unsteady streams in pipes. Such an extension is essential for the artificial fish ‘Pod’, a medical device which swims in the patient's blood vessel. Two approaches are considered. First, potential theory is considered, and the results achieved show that the main influence of the pipe is on evaluation of the proper virtual mass. Next the flow is assumed to be viscous. The consideration of viscosity is obviously necessary for flows in pipes. In that case the virtual mass is replaced by another equivalent mass depending on the viscosity and on the angular frequency of the lateral motion and in addition new terms appear in the local lift expressions. These are recognized as the viscous damping force. © 1974, Cambridge University Press. All rights reserved.

Description: The response of a Pitot probe in a uniform laminar stream is commonly expressed in the form [formula omitted] where P s is the probe signal pressure, P is the stream static pressure, U is the stream speed and ρ is the fluid density. It has been found that for ordinary sphere-nosed, round-nosed and square-nosed probes [formula omitted] where θ is the angle between the velocity vector and the probe axis, and Un ≡ U sin θ is the transverse velocity component. The parameters m and K are functions of the probe geometry. These formulae also describe the performance in a turbulent stream when the probe is small compared with the turbulence scale. The evaluation of the time-averaged response is treated, and an answer is developed to the question of what it is that a Pitot probe measures in a turbulent stream. In a turbulent shear flow having the properties of a boundary layer, the reference pressure is best taken to be the static pressure at the shear-layer edge. It is shown that round-nosed probes with D i/D≃0·45 and square-nosed probes with D i/D≃0·15 then detect [formula omitted] with good accuracy, where D /D is the ratio of the inside and outside diameters of the Pitot tube. When measurements are made with two probes of dissimilar geometry, the differential response can be used to find the mean-square level of the transverse velocity fluctuations. Turbulence levels so measured agree closely with results from hot-wire anemometry. © 1974, Cambridge University Press. All rights reserved.

Description: The 41st EUROMECH Colloquium on flows with concentrated vorticity was held in Norwich from 17 to 21 September 1973. There were sixty-five participants from nine countries and the author was the chairman of the organizing committee. © 1974, Cambridge University Press. All rights reserved.

Description: The steady flow in and around a deformable liquid sphere moving in an unbounded viscous parabolic flow and subject to an external body force is calculated for small values of the ratio of the Weber number to the Reynolds number in the creeping-flow regime. It is found that, in addition to the drag force, the drop experiences a force orthogonal to the undisturbed flow direction. When the body force is absent (neutrally buoyant drop), this lift force tends to drive the drop inwards to the axis, where the undisturbed flow velocity is maximum, i.e., towards a position of lower velocity gradient. In the case for which the parabolic flow profile is a Poiseuille flow profile, the lift force is given by the expression. [formula omitted] Here a is the radius of the undeformed sphere, R 0 is the radial distance from the position of maximum undisturbed flow U 0 at the profile axis to the position of zero flow, ε is the ratio of the Weber number to the Reynolds number, given by ε=μU 0 T −1, where μ is the external fluid viscosity and T is the surface tension of the drop, α is the ratio of the drop and external fluid viscosities, b is the radial vector from the flow axis to the centre of mass of the drop, and F is a function of α and a dimensionless parameter dependent on the body force that is determined in the analysis. Reasonable agreement is found between the observations by Goldsmith & Mason (1962) of the axial drift of liquid drops in Poiseuille flow and the predictions of the theory herein. © 1974, Cambridge University Press. All rights reserved.

Description: The flow normal to an infinite circular cylinder which is uniformly accelerated from rest in a viscous fluid is considered. The flow is assumed to remain symmetrical about the direction of motion of the cylinder. Two types of solution are presented. In the first an expansion in powers of the time from the start of the motion is given which extends the results of boundary-layer theory by taking into account corrections for finite Reynolds numbers. Physical properties of the flow for small times and finite but large Reynolds numbers are calculated from this expansion. In the second method of solution the Navier-Stokes equations are integrated by an accurate procedure which is a logical extension of the solution in powers of the time. Results are obtained for R2 = 97·5, 5850, 122 × 103 and ∞, where R is the Reynolds number. This is defined as R = 2a(ab)½/v, where a is the radius of the cylinder, b the uniform acceleration and v the kinematic viscosity of the fluid. The methods are in good agreement for small times. The numerical method of integration has been carried to moderate times and various flow properties have been calculated. The growth of the length of the separated wake behind the cylinder for R2 = 97·5, 5850 and 122 × 103 is compared with the results of recent experimental measurements. The agreement is only moderate for R2 = 97·5 but it improves greatly as R increases. The numerical integrations were continued in each case until the implicit method of integration failed to converge, which terminated the procedure. A secondary vortex appeared on the surface of the cylinder for the case R2 = 122 × 103. © 1974, Cambridge University Press. All rights reserved.

Description: The interactions of weak nonlinear disturbances in a compressible fluid including shocks, expansion waves and contact surfaces are investigated by making use of the reductive perturbation method. It is found that the nonlinear waves belonging to different families of characteristics behave almost independently of each other, while those belonging to the same family are governed by either the Burgers equation or the equation of heat conduction. Thus the statistical properties of one-dimensional shock turbulence in a compressible fluid are reduced to those of the solutions of the Burgers equation. In particular, the law of energy decay of shock turbulence is shown to be identical to that of Burgers turbulence. © 1974, Cambridge University Press. All rights reserved.

Description: The body of given volume with the smallest drag in Stokes flow is obtained by making use of theoretical results due to Pironneau. A suitable family of solutions of the Stokes equations is used and the no-slip condition is expressed numerically by a technique of quadratic minimization. We find that the drag on this optimal body is 0·95425 times the drag on the sphere of equal volume. © 1974, Cambridge University Press. All rights reserved.

Description: A solution of the Boltzmann equation is obtained at the upstream and downstream singular points in a shock wave, for the case of Maxwell molecules. The fluid velocity u, rather than the spatial co-ordinate x, is used as the independent variable, and an equation for ∂f/∂u at a singular point is obtained from the Boltzmann equation by taking the appropriate limit. This equation is solved by using the methods of Grad and of Wang Chang & Uhlenbeck; and it is observed that the two methods are the same, since they involve not only an equivalent system of moment equations but also the same closure relations. Because many quantities are zero at a singular point, the problem becomes sufficiently simple to allow the solution to be carried out to any desired order. At the supersonic singular point, the solution converges very slowly for strong shock waves; but a simple modification to Grad's method provides a rapidly convergent solution. The solution shows that the Navier-Stokes relations, or the first-order Chapman-Enskog results, do not apply unless the shock-wave Mach number is unity, and that they are grossly in error for strong shock waves. The solution confirms the existence of temperature overshoot in a strong shock wave; shows that the critical Mach number in Grad's solution increases monotonically with the order of the solution; provides a simple explanation as to why Grad's closure relations fail and shows how they can be improved; and provides exact boundary values that can be used to guide future numerical solutions of the Boltzmann equation for shock-wave structure. © 1974, Cambridge University Press. All rights reserved.

Description: The sublayer of a turbulent boundary-layer flow with and without polymer additives was measured by a simple optical technique having a high degree of spatial resolution. The reduction of skin friction with polymer additives was found to be due to a decrease in the effective viscosity of water at the interface between the mucus (glycoprotein) or polymer (polyethylene oxide) and the water and a thickening of the laminar sublayer. The rate at which mucus diffuses away from a medium-sized fish is almost constant for all swimming speeds. © 1974, Cambridge University Press. All rights reserved.

Description: The linear stability of a basic flow of two homogeneous inviscid incompressible fluids under the action of gravity is treated mathematically. In the basic state, one fluid is at rest below a horizontal plane z = 0; and the other flows above in the x direction, its speed varying slowly with the lateral co-ordinate y. The eigenvalue problem for normal modes is derived; its equation is a partial differential one, the co-ordinates y and z not being separable. The problem is solved approximately by taking the modes locally as if the basic velocity were independent of y, though the lateral wavenumber is allowed to vary slowly with y. This leads to an ordinary differential equation in y which is solved by the JWKB method. Detailed calculations are made for a parabolic profile, representing the blowing of air over water in a wide channel, and for other profiles. © 1974, Cambridge University Press. All rights reserved.

Description: The structure of density interfaces upon which turbulent motions have been imposed is investigated in the laboratory. Particular attention is paid to the profiles of density across such interfaces, and the temporal response of the interface to the imposed turbulence. Interfaces over a range of Péclet numbers and Richardson numbers are examined. It is found that the interface thickness h/l (non-dimensionalized with respect to the length scale l of the turbulence) is a function of the Péclet number but independent of the Richardson number. At low Péclet numbers (Pe ≲ 200) molecular diffusion is important in the determination of the interface structure and a diffusive core (across which all transport occurs by molecular diffusion) is formed in the centre of the interface. At higher Péclet numbers the interface structure appears to be determined by non-diffusive processes and h/l becomes approximately constant at a value of about 1·5. Some information concerning the intermittent nature of the mixing process at high Péclet number is obtained from records of salinity fluctuations measured at a fixed point. Finally, the implications of these data concerning the interpretations of the measurements of entrainment across an interface made by Turner (1968) are discussed. © 1974, Cambridge University Press. All rights reserved.

Description: The mixing layer bounding the exhaust plume associated with hypersonic highaltitude rockets is analysed as a laminar binary mixture of perfect gases with Lewis and Prandtl numbers of unity. A far-field approximation to the undisturbed jet core and a Newtonian pressure balance between the jet and ambient gases are used to construct the mixing-layer location. Longitudinal pressure variations are neglected and resultant errors are evaluated. Boundary conditions at the edge of the mixing layer are evaluated by streamline tracing to shock entry points. The sensitivity of properties in the mixing layer to variations in the plume angle of attack, engine nozzle efficiency and engine thrust are examined, and an approximate density and thrust scaling of mixing-layer overall properties is developed. © 1974, Cambridge University Press. All rights reserved.

Description: This paper deals with the motion of a heavy particle in a turbulent flow in an open channel with a smooth bottom. For the case when the particle stays in suspension in the main body of the flow almost all the time, (a) the probability density function of the projection on a cross-sectional plane of the particle position at any instant, and (b) the mean velocity and longitudinal dispersion coefficient of particles are determined analytically by employing the Eulerian formulation and applying the Aris moment transformations. It is found that the mean particle velocity decreases and the longitudinal dispersion coefficient of particles increases with the fall velocity. © 1974, Cambridge University Press. All rights reserved.

Description: The effect of surface roughness on the flow past spheres has been investigated over the Reynolds number range 5 × 104 〈 Re 〈 6 × 106. The drag coefficient has been determined as a function of the Reynolds number for five surface roughnesses. With increasing roughness parameter the critical Reynolds number decreases. At the same time the transcritical drag coefficient rises, having a maximum value of 0·4. The vortex shedding frequency has been measured under subcritical flow conditions. It was found that the Strouhal number for each of the various roughness conditions was equal to its value for a smooth sphere. Beyond the critical Reynolds number no prevailing shedding frequency could be detected by the measurement techniques employed. The drag coefficient of a sphere under the blockage conditions 0·5 〈 ds/dt 〈 0·92 has been determined over the Reynolds number range 3 × 104 〈 Re 〈 2 × 106. Increasing blockage causes an increase in both the drag coefficient and the critical Reynolds number. The characteristic quantities were referred to the flow conditions in the smallest cross-section between sphere and tube. In addition the effect of the turbulence level on the flow past a sphere under various blockage conditions was studied. © 1974, Cambridge University Press. All rights reserved.

Description: Velocity measurements were made in the flow field behind a circular cylinder at Reynolds numbers from 10 to 80 and results compared with existing numerical solutions. Takami & Keller's solution for the velocity distribution in the wake shows good agreement at low Reynolds numbers and fair agreement at high Reynolds numbers. The drag coefficient of the cylinder and the size of the standing eddies behind the cylinder were also determined. They are compatible with existing experimental and numerical results. Details of the velocity distribution in the standing eddies are clarified. © 1974, Cambridge University Press. All rights reserved.

Description: Attention is drawn to a principal difference between the transfer of a horizontal magnetic field by turbulence and by three-dimensional cell convection. If the motion of the conducting medium in the cells is such that the heated material ascends at the centre while descending along the sides of the cells, then the magnetic tubes of force will be carried downwards by peripheral flows. Discrete ascending flows separated from one another by descending material carry only closed magnetic field loops. Such loops do not transfer net magnetic flux. As a result, the magnetic flux becomes blocked at the base of the convective layer. © 1974, Cambridge University Press. All rights reserved.

Description: The nonlinear Boltzmann equation has been solved for shock waves in a Max-wellian gas for eight upstream Mach numbers M1 ranging from 1·1 to 10. The numerical solutions were obtained by using Nordsieck's method, which was revised for use with the differential cross-section corresponding to an intermolecular force potential following an inverse fifth-power law. The accuracy of the calculations of microscopic and macroscopic properties for this collision law is comparable with that for elastic spheres published earlier (Hicks, Yen & Reilly 1972). We have made comparisons of the detailed characteristics of the internal shock structure in a Maxwellian gas with those in a gas of elastic spheres. The purpose of this comparative study is to find the shock properties that are sensitive as well as those which are insensitive to the change in collision law and to find effective ways to study them. The variation of thermodynamic and transport properties of interest with respect to density and to each other was found to depend only weakly on the change in collision law. The principal effect on the macroscopic shock structure due to the change in intermolecular potential is in the spatial variation of the macroscopic properties. The spatial variation of macroscopic properties may be determined accurately from the corresponding moments of the collision integral, especially in the upstream and downstream wings of the shock wave. The results for the velocity distribution function exhibit the microscopic shock characteristics influenced by a difference in intermolecular collisions, in particular the departure from equilibrium in the upstream wing of the shock and the relaxation towards equilibrium in the downstream wing. The departure of several characteristics of weak shock waves from those of the Chapman-Enskog linearized theory and the Navier-Stokes shock is also insensitive to the change in collision law. The deviation of the half-width of the function ∫fdvyduz from the Chapman-Enskog first iterate at M1 = 1·59 is in agreement with an experiment (Muntz & Harnett 1969). © 1974, Cambridge University Press. All rights reserved.

Description: We use some well-known properties of the Taylor-Goldstein equation to generate a set of stably stratified shear flows with known singular neutral-mode solutions. The novel feature of the analysis is that it includes such solutions for flows in which, proceeding upwards from a rigid boundary, the Brunt-Väisälä frequency and the flow shear do not change sign and are monotonically decreasing functions of height. Such profiles are much closer to the conditions met in work on the atmospheric boundary layer than the more frequently used inflected flow profiles. © 1974, Cambridge University Press. All rights reserved.

Description: The early and intermediate development of a highly accelerated (or decelerated) turbulent boundary layer is analysed. For sufficiently large accelerations (or pressure gradients) and for total normal strains which are not excessive, the equation for the Reynolds shear stress simplifies to give a stress that remains approximately constant as it is convected along streamlines. The theoretical results for the evolution of the mean velocity in favourable and adverse pressure gradients agree well with experiment for the cases considered. A calculation which includes mass injection at the wall is also given. © 1974, Cambridge University Press. All rights reserved.

Description: In two earlier papers (Gallagher & Mercer 1962, 1964) the results for the first four eigenvalues of the problem of the stability of plane Couette flow were given. The first twelve eigenvalues have been calculated by the same method and the results show that the manner in which the eigenvalues join to form complex pairs depends on α, the wavenumber of the disturbance. They also would appear to indicate that mode-crossing always occurs. © 1974, Cambridge University Press. All rights reserved.

Description: The averaged equations of slow flow in random arrays of fixed spheres are developed as a hierarchy of integro-differential equations, and an iteration procedure is described for obtaining the mean drag in the case of small volume concentration c. The leading approximation is that given by Brinkman's model of flow past a single fixed sphere, in which the effects of all other spheres are treated as a Darcy resistance. The higher approximations take account of the modification to the mean flow, particularly in the near field, due to the localized nature of the actual resistance. Thus the second approximation finds the change due to a second sphere, and averages over all its possible positions. The result for the mean drag confirms Childress’ terms in clogc and c (apart from an arithmetical correction to the latter), but indicates that for practical values of c numerical evaluation of integrals is needed, rather than expansion in powers of c and log c. The last section of the paper develops the corresponding results for flow through random arrays of fixed parallel circular cylinders. © 1974, Cambridge University Press. All rights reserved.

Description: The subject of the present study is the question of how the sound power of a jet of constant exit velocity would vary if the jet exit density were varied. Changes in jet exit density would inevitably be accompanied in a real experiment by changes in the speed of sound (temperature) in the jet, so that both effects must be considered simultaneously. The point of view advanced at the end of the study is that experimentally observed results in this area seem to admit an explanation based on how the radiative efficiency of moving acoustic sources is affected by the shrouding effect of a jet flow whose velocity, temperature and density differ from those of the ambient fluid. This change in efficiency is calculated with the aid of a simple model as follows. We determine the acoustic power output of a convected monopole source, simple harmonic in its own frame of reference, moving along the axis of a plug-flow round jet whose velocity is the same as that of the source. The jet is doubly infinite and the source is assumed to have an infinite lifetime. The density and temperature of the jet are allowed to differ from those of the ambient fluid though the specific-heat ratio of the jet fluid is assumed to be the same as that of the ambient. The requirement of equality of the static pressure inside and outside the jet then calls for a certain restraint on how the jet density and temperature vary. For a specific value of the jet exit velocity, the variation of acoustic power with the ratio of jet to ambient density along with a simple assumption on how the source strength varies with jet density are employed to deduce theoretically the ‘jet density exponent for jets which are subsonic with respect to the ambient speed of sound. The jet density exponent is found to depend both on the jet Mach number and even more strongly on a source frequency parameter. The theoretical results are compared with some experimental studies of this problem. Encouraging agreement is obtained both for the detailed observed effects on the power spectrum and the exponent for the overall power. © 1974, Cambridge University Press. All rights reserved.

Description: Thermal oscillations have been found to occur during crystal growing, thereby introducing undesirable striations in the solid crystal produced. A reason for the existence of such oscillations is given in this paper, and relevant experiments discussed. © 1974, Cambridge University Press. All rights reserved.

Description: Turbulent convection for a rotating layer of fluid heated from below is studied in this paper. The boundaries of the fluid layer are taken to be free. The underlying principle, used is the Malkus hypothesis that the flow tends to transport the maximum amount of heat possible, subject to certain constraints. By taking the Prandtl number to be infinite, a linear differential constraint and an integral constraint are used. The variational problem that follows then depends on two dimensionless parameters, the Taylor number T and the Rayleigh number R. Asymptotic analysis for the turbulent regime shows that the flow arranges itself so as to tend to offset the stabilizing effect of the rotational constraint, at least in so far as the heat flux is concerned. The dimensionless heat flux, or the Nusselt number, has in general different dependence on T and R, depending on the particular region in the parameter space. For T ≤ O(R), the flow is essentially non-rotating. For O(R) ≤T ≤ O(R4/3), the flow will always have finitely many horizontal wavenumbers, though the total number of modes increases as T increases in this region. For O(R4/3) ≤ T ≤ O (R3/2), the Nusselt number has a functional dependence proportional to R3/T2, having essentially infinitely many horizontal modes as both R and T increase indefinitely in this region. The last expression is particularly interesting, as it agrees qualitatively with results in finite-amplitude laminar convection. It is also linearly dependent on the layer thickness, as one might expect from dimensional argument. It is suggested that, in the context of the maximum principle, the result in this region of the parameter space may be applicable as well to the same fluid layer with rigid boundaries through the existence of an Ekman layer that is thinner than the thermal layer. © 1974, Cambridge University Press. All rights reserved.

Description: The low-speed combustion of initially unmixed gaseous reactants under an irreversible two-step chain reaction is examined. Both equilibrium burning, in which two spatially separated flames of zero thickness arise, and near-equilibrium burning, in which two spatially separated flames of small but finite thickness arise, are studied by limit-process expansion techniques. Two time-dependent flows are examined: the first is (one-dimensional) transient mixing flow; and the second is (two-dimensional) transient counterflow. The latter flow, in which there is an impressed finite strain parallel to the flame, such that the flame itself is longitudinally stretched, is discussed as elucidating the characteristics of combustion in non-equilibrium turbulent shear flow. © 1974, Cambridge University Press. All rights reserved.

Description: The purpose of this study is to generalize the experimental method for deciding when a fluid motion can be considered turbulent. The rationale advanced for processing a transducer signal, which indicates the intermittent nature of a turbulent field, is given a probabilistic outlook. In addition an improved detector function is introduced which uses information from longitudinal and lateral fluctuation components. © 1974, Cambridge University Press. All rights reserved.

Description: An experimental investigation of flows of a large number of inert and polyatomic gases in various channels, a non-ideal orifice, flat slits with different surface roughnesses and wall materials, capillary packets with molten walls and a capillary sieve, has been made. The unsteady flow method and a highly sensitive capacitance micromanometer were used (the sensitivity being ∼ 3 × 10−4N/m2Hz). Measurements were made in a range of Knudsen numbers 5 × 104–10−3 at ∼ 293 °K, and some measurements for flow through a non-ideal orifice were carried out at 77.2°K. It was found that, both in the viscous slip-flow and free-molecule regimes for the channels with molten walls, the experimental conductivities were higher (by ∼ 15%) than theoretical ones calculated assuming diffuse molecular scattering by the walls. We have also observed that the channel conductivity essentially depends on the channel surface roughness and on the kind of gas. The larger the roughness height, the lower the conductivity. From the experimental data the tangential momentum accommodation coefficients were calculated. © 1974, Cambridge University Press. All rights reserved.

Description: In an experimental and theoretical study, we model a phenomenon observed in the summer Arctic, where a fresh-water layer at a temperature of 0°C floats both over a sea-water layer at its freezing point and under an ice layer. Our results show that the ice growth in this system takes place in three phases. First, because the fresh-water density decreases upon supercooling, the rapid diffusion of heat relative to salt from the fresh to the salt water causes a density inversion and thereby generates a high Rayleigh number convection in the fresh water. In this convection, supercooled water rises to the ice layer, where it nucleates into thin vertical interlocking ice crystals. When these sheets grow down to the interface, supercooling ceases. Second, the presence of the vertical ice sheets both constrains the temperature T and salinity s to lie on the freezing curve and allows them to diffuse in the vertical. In the interfacial region, the combination of these processes generates a lateral crystal growth, which continues until a horizontal ice sheet forms. Third, because of the T and s gradients in the sea water below this ice sheet, the horizontal sheet both migrates upwards and increases in thickness. From one-dimensional theoretical models of the first two phases, we find that the heat-transfer rates are 5–10 times those calculated for classic thermal diffusion. © 1974, Cambridge University Press. All rights reserved.

Description: Linear stability theory is applied to the natural convection of slightly, elastic, viscous fluids in an infinitely long vertical slot. Travelling, as well as stationary, disturbances are considered. It is found that the elasticity (i) slightly stabilizes the stationary disturbances while strongly destabilizing the travelling disturbances, (ii) strongly increases the wave speed while slightly decreasing the wavenumber and (iii) reduces the transition Prandtl number, which separates the stationary cells from the travelling waves, from its value of 12.7 for Newtonian fluids. Experiments are carried out with a viscoelastic fluid prepared by mixing Separan AP30 with water, giving a Prandtl number of about 30. This fluid is shown to produce wave instability at a Grashof number between 3900 and 4300. Under the same conditions, a Newtonian fluid is shown to remain stable to both stationary and travelling disturbances. © 1974, Cambridge University Press. All rights reserved.

Description: The perturbation of pre-existing surface gravity waves caused by the presence of an internal wave was studied both experimentally and analytically. An extensive series of experiments was performed, and quantitative results were obtained for the one-dimensional monochromatic interaction of internal waves and surface gravity waves. Internal wave-induced surface slope, amplitude and wavenumber modulations were measured for a wide range of interaction conditions. A complementary theoretical analysis, based on the conservation approach of Whitham (1962) and Longuet-Higgins & Stewart (1960,1961), was performed and a closed form solution obtained for the one-dimensional wave interaction. Both the theory and the experiment demonstrate that the effect increases with interaction distance. The maximum interaction effect is found to occur when the phase speed of the internal wave and the group velocity of the surface wave are matched. The phase of the internal wave at which maximum surface-wave modulation occurs is found to be a sensitive and continuous function of the relative wave speeds. The experimental data are in good agreement with the present theoretical analysis. © 1974, Cambridge University Press. All rights reserved.

Description: The properties of convective flow driven by an adverse temperature gradient in a fluid-filled porous medium are investigated. The Galerkin technique is used to treat the steady-state two-dimensional problem for Rayleigh numbers as large as ten times the critical value. The flow is found to look very much like ordinary Bénard convection, but the Nusselt number depends much more strongly on the Rayleigh number than in Bénard convection. The stability of the finite amplitude two-dimensional solutions is treated. At a given value of the Rayleigh number, stable two-dimensional flow is possible for a finite band of horizontal wavenumbers as long as the Rayleigh number is small enough. For Rayleigh numbers larger than about 380, however, no two-dimensional solutions are stable. Comparisons with previous theoretical and experimental work are given. © 1974, Cambridge University Press. All rights reserved.

Description: The method of matched asymptotic expansions is used to solve the differential equation describing the shape of a meniscus on the outside of a circular cylinder. Since the perturbation quantity is proportional to the cylinder radius, the solution is valid basically for small oylinders. The predicted meniscus height is compared with numerical data to determine the accuracy of the two-term result; the third term is found but does not improve the estimate. © 1974, Cambridge University Press. All rights reserved.

Description: A linear stability analysis is applied to a stably stratified, thermally radiating shear layer. The grey Milne–Eddington approximation is employed as a radiation model. In contrast to a previously reported optically thin analysis, no inviscid instability exists, in the limit of vanishing horizontal wavenumber, for this selfabsorbing model. The inviscid neutral-stability boundary (Richardson number us. dimensionless wavenumber) for the Milne–Eddington approximation converges to the optically thick limit as the optical depth of the shear layer is increased. As the optical depth of the shear layer is decreased, the inviscid Milne–Eddington neutral-stability boundary approaches the optically thin limit, although not uniformly in the wavenumber. For fixed mean velocity gradient and fluid properties, the inviscid critical Richardson number approaches infinity as the optical depth of the shear layer approaches zero. Viscous effects neutralize this radiative destabilization, and the critical Richardson number eventually returns to zero as the optical depth continues to decrease. A shearlayer thickness exists for which the viscous critical Richardson number is a maximum. For shear depths greater than this thickness, self-absorption effects increase the stability; and for shear depths less than this thickness, viscous effects increase the stability. Results of the analysis are applied to the atmospheres of Venus and the earth. A critical Richardson number somewhat above the non-radiating value of 3 (although below the previously reported optically thin value) is found for the lower troposphere of the earth. No substantial effect is found for the earth's lower stratosphere or for the 100 km level above Venus. © 1974, Cambridge University Press. All rights reserved.

Description: An analytical study is made of simple models of steady fronts in the atmosphere in which the temperature field is subjected to deformation as the fluid moves downstream in a large-scale horizontal flow. One fundamental approximation is made and then a Lagrangian method, in which fluid particles are identified by conservation of entropy and potential vorticity, and by Bernoulli's theorem, enables the steady problem to be solved. Solutions for models of surface fronts and upper tropospheric fronts are compared with those obtained from a model in which there is no variation along the front and the frontogenesis proceeds in time. If the thermal wind is comparable with the basic wind, and the potential vorticity is not negligible in some sense, the frontogenesis is increased where the thermal wind opposes the basic flow but, decreased where it reinforces the flow. © 1974, Cambridge University Press. All rights reserved.

Description: In this paper, the change in energy dissipation due to a small hump on a body in a uniform steady flow is calculated. The result is used in conjunction with the variational methods of optimal control to obtain the optimality conditions for four minimum-drag problems of fluid mechanics. These conditions imply that the unit-area profile of smallest drag has a front end shaped like a wedge of angle 90°. © 1974, Cambridge University Press. All rights reserved.

Description: A laboratory study has been undertaken to measure the energy transfer from two surface waves to one internal gravity wave in a nonlinear, resonant interaction. The interacting waves form triads for which [formulla omitted] σj and κj being the frequency and wavenumber of the jth wave. Unlike previously published results involving single triplets of interacting waves, all waves here considered are standing waves. For both a diffuse, two-layer density field and a linearly increasing density with depth, the growth to steady state of a resonant internal wave is observed while two deep water surface eigen-modes are simultaneously forced by a paddle. Internal-wave amplitudes, phases and initial growth rates are compared with theoretical results derived assuming an arbitrary Boussinesq stratification, viscous dissipation and slight detuning of the internal wave. Inclusion of viscous dissipation and slight detuning permit predictions of steady-state amplitudes and phases as well as initial growth rates. Satisfactory agreement is found between predicted and measured amplitudes and phases. Results also suggest that the internal wave in a resonant triad can act as a catalyst, permitting appreciable energy transfer among surface waves. © 1974, Cambridge University Press. All rights reserved.

Description: Detailed measurements of pressure distributions, mean velocity profiles and Reynolds stresses were made in the thick axisymmetric turbulent boundary layer near the tail of a body of revolution. The results indicate a number of important differences between the behaviour of a thick and a thin boundary layer. The thick boundary layer is characterized by significant variations in static pressure across it and an abnormally low level of turbulence. The static-pressure variation is associated with a strong interaction between the boundary layer and the potential flow outside it, while the changes in the turbulence structure appear to be a consequence of the transverse surface curvature. In order to predict the behaviour of the flow in the tail region of a body of revolution it is not therefore possible to use conventional thin-boundary-layer calculation procedures. © 1974, Cambridge University Press. All rights reserved.

Description: A time-dependent current distribution is abruptly switched on, initiating an unsteady disturbance of a two-phase magnetohydrodynamic configuration. This comprises a magnetically permeated, conducting fluid flow contained by a vacuum magnetic field, from within which the source current radiates. An exact solution to the proposed part-time problem is constructed for the magnetic line distortion of the vacuum field. If the source current is oscillatory, two progressive, non-dissipative waves are normally encountered, these being superposed upon an infinite discrete set of terms obeying separate rules of decay. Propagation occurs longitudinally, parallel to the flow, but with amplitudes dependent on the transverse variable. The waves advance behind two fronts, the faster of which always travels down-stream. Depending on whether the flow speed exceeds or is exceeded by (or equals) √2 × the quadratic mean of both Alfvkn speeds involved, the slower front proceeds, respectively, downstream or upstream (or disappears, in which case, only one wave exists). Along a characteristic (a front path), appropriate contributions depend solely on the transverse co-ordinate, behaving otherwise like Riemann invariants. Contrary to expectation, the net perturbation is continuous across each characteristic. Various steady modes are ultimately attained after an infinite period. The radiation principle is satisfied. Both travelling waves are vibrationally sustained, vanishing with the source frequency. In such an event, the infinite series result is summable to a closed form. From this, the general solution, corresponding to an arbitrary space–time source distribution, is deduced. Certain characteristic-associated equivalence laws are then established. An asymptotic approximation is made. © 1974, Cambridge University Press. All rights reserved.

Description: Measurements of the mean flow, intermittent structure and turbulent fluctuations were made in a cold-wall boundary layer at a stream Mach number of 9·4 and Reynolds number based on momentum thickness of 36 800. For these conditions, the r.m.s. sublayer thickness was 32 times smaller than that of the boundary layer proper, and the interfacial standard deviation of the latter was about three times proportionately smaller than has been found at low speeds. The mean flow data, which extended well into the sublayer, revealed a large increase in static pressure from the layer edge to the wall and a quadratic law relation between the total temperature and velocity. While the transformed velocity profile was in good agreement with the incompressible law of the wake, no indication of a linear variation of velocity in the sublayer was detected. Hot-wire fluctuation data, interpreted with the use of appropriate assumptions concerning the nature of the sound field, indicated that the turbulence is dominated by high-frequency pressure fluctuations whose magnitude at the wall and beyond the layer edge agree with extrapolation of data acquired at supersonic speeds. The static temperature fluctuations agreed with expectations from adiabatic, supersonicdata apparently because they were suppressed by the cooled-wall condition. The fluctuations in the longitudinal velocity component were generally small and differed little from lower Mach number results. The high turbulence Reynolds numbers found generated an inertial-subrange spectral decay, while the longitudinal integral scales were found independent of turbulence mode and about one-fifth the boundary-layer thickness. © 1974, Cambridge University Press. All rights reserved.

Description: A mixing layer is formed by bringing two streams of water, moving at different velocities, together in a lucite-walled channel. The Reynolds number, based on the velocity difference and the thickness of the shear layer, varies from about 45, where the shear layer originates, to about 850 at a distance of 50 cm. Dye is injected between the two streams just before they are brought together, marking the vorticity-carrying fluid. Unstable waves grow, and fluid is observed to roll up into discrete two-dimensional vortical structures. These turbulent vortices interact by rolling around each other, and a single vortical structure, with approximately twice the spacing of the former vortices, is formed. This pairing process is observed to occur repeatedly, controlling the growth of the mixing layer. A simple model of the mixing layer contains, as the important elements controlling growth, the degree of non-uniformity in the vortex train and the ‘lumpiness’ of the vorticity field. © 1974, Cambridge University Press. All rights reserved.

Description: Forced nonlinear acoustic oscillations near the resonant frequency of closed and open tubes are studied experimentally. In particular, the motion in tubes terminated with different orifice plates is studied, and comparison is made with second- and third-order theories of the motion which contain an adjustable end-wall reflexion coefficient. It is found that oscillations at resonance in an open tube exhibit remarkably large amplitudes despite the fact that in some cases shock waves are emitted from the open end. For oscillations at resonance in a closed tube, the effect of substituting an orifice plate for the solid end wall is to reduce the amplitude and thicken the compressive portion of the shock waves which occur under these conditions. In both the open-tube and closed-tube experiments the reflexion coefficients which are evaluated by fitting theory to experiment are found to increase with increasing amplitude, in agreement with the observations of previous investigators (Ingard & Ising 1967). In fact, for the open end the same linear dependence upon amplitude is observed, but the constant of proportionality is different. Qualitative differences are observed between the reflexion coefficients of a given orifice at the open-end and the closed-end resonant frequencies; at the open-end frequency the reflexion from the given orifice is less ideal than at the closed-end frequency. The implications of reflexion coefficients dependent on the wave forms are discussed. © 1974, Cambridge University Press. All rights reserved.

Description: Earlier measurements of the contribution of four distinct classes of motions, i.e. (u 〈 0, v 〉 0), (u 〉 0, v 〈 0), (u 〈 0, v 〈 0) and (u 〉 0, v 〉 0), to the Reynolds stress [formulla omitted] in the wall region of a bounded turbulent shear flow have been extended. These classes were obtained by truncating the u and v signals about zero. Various statistical properties of the truncated streamwise and normal velocity components u and v and of their product uv have been determined in an attempt to characterize quantitatively the motions in this flow. Average values and probability density distributions both of the truncated and untruncated signals have been taken. © 1974, Cambridge University Press. All rights reserved.

Description: A point-vortex representation is used to study numerically the evolution of an initially plane vortex sheet. By introducing a tip vortex to represent the tightly rolled portion of the vortex sheet, the chaotic motion which was a feature of some earlier studies is eliminated and the details of the outer portion of the spiral are calculated. The rate of rolling up is calculated and is shown to be governed by the analytically predicted similarity law of Kaden during the initial stages of the rolling up. The calculations are continued until 99% of the vorticity has been rolled up, at which stage the spiral displays a marked ellipticity. © 1974, Cambridge University Press. All rights reserved.

Description: The linear stability of the developing flow of an incompressible fluid in the entrance region of a circular tube is investigated. The case of non-axisymmetric small disturbances is considered in the analysis. The main-flow velocity distribution used in the stability calculations is that from the solution of the linearized momentum equation. The eigenvalue problem consisting of the disturbance equations and the boundary conditions is solved by a direct numerical integration scheme along with an iteration procedure. An orthonormalization method is employed to remove the ‘parasitic errors’ inherent in the numerical integration of the coupled disturbance equations. The flow is found to be unstable to non-axisymmetric disturbances with an azimuthal wavenumber of one. Neutral-stability curves and critical Reynolds numbers at various axial locations are presented. A comparison of these results is made with those for axisymmetric disturbances reported by Huang & Chen. It is found that the first instability of the flow is due to non-axisymmetric disturbances and occurs in the entrance region of the pipe with a minimum critical Reynolds number of 19780. © 1974, Cambridge University Press. All rights reserved.

Description: Recent field observations in the polar oceans show that the hollow tubes of ice called ice stalactites form around streamers of cold brine rejected by the growing sea ice. In a laboratory study of this process, we inject cold, dense brine a t a constant salinity, temperature and volume flux into an insulated tank of sea water held at its freezing point, then photograph the resultant stalactite growth. Because the inner wall temperature of the stalactite remains on the salinity-determined freezing curve, as the stalactite grows and the temperature deficit of the brine goes into the growth of ice, the inner wall melts to dilute and cool the adjacent brine back to its freezing point. This melting means that both the inner and outer stalactite radii increase with time. The radius of the stalactite tip, which is constant for each experiment, is shown to be controlled by the onset of a convective instability. If the tip becomes too large, overturning occurs and the sea-water intrusion freezes, reducing the radius of the tip so that the flow leaving the tip is marginally stable. Inside the stalactite, since the inner radius increases with time, both theory and experiment show the interior flow to be convectively unstable. The present study also derives a solution from the constant-heat-flux Graetz solution for the growth in both length and side-wall area of the stalactite. The experiments show that away from the stalactite base and the very beginning of the experiment this solution, with convection accounted for by an adjustable coefficient, describes the experimental growth. Finally, analysis of the experiments shows that as much as 50% of the ice represented by the cold brine does not go into the stalactite, rather the ice goes directly into the ocean as loose crystals. © 1974, Cambridge University Press. All rights reserved.

Description: Solutions are obtained for the baroclinic instability problem for situations in which the static stability and mean shear vary geminately with height. The simple solution given by Eady is shown to be a special limiting case of a class of exact solutions for flows whose basic states have a vanishing interior potential vorticity gradient. The generalized solutions show that the temperature amplitude distribution is particularly sensitive to vertical variations in static stability but that phases and other amplitudes are only slightly influenced by such variations. When the static stability and shear increase (decrease) with height an enhanced temperature maximum occurs at the upper (lower) surface in comparison with the standard Eady solution. The generalized solutions also help to explain the character of annulus waves and predict a short-wave cut-off that is the same as that given by Eady's theory provided that it is based on the vertically averaged gravitational frequency. © 1974, Cambridge University Press. All rights reserved.

Description: A deep surface-piercing wedge-ended hull model was towed through still water. Measurements of the surface wave pattern confirmed earlier findings for ship models, that the measured bow-wave cusp line often lies well forward of the position predicted by thin-ship theory, and that this shift increases with bow water-line angle and with decreasing model speed. Two possible explanations are considered here in terms of changes of wave phase speed with wave convection and steepness. Calculations based on a transformation method due to Guilloton predict more realistic wave profiles than linear theory, but account for less than half the observed shift. Some tentative conclusions are drawn. The singularity in the Green's function double integral is removed by an improved method, which simplifies the numerical integration. The new integrand decays within one oscillation. © 1974, Cambridge University Press. All rights reserved.

Description: It is proposed that a spilling breaker can be regarded as a turbulent gravity current riding down the forward slope of a wave, and can be treated using methods which have been successful in other contexts. The whitecap retains its identity because it is lighter than the water below, owing to the trapping of air bubbles, and information from laboratory experiments is used to estimate the density of the air-water mixture in different circumstances. Entrainment of water from below, at a rate E(Ri 0,) which is a function of the overall Richardson number Ri 0, has two opposing effects. It provides increasing mass and buoyancy fluxes which can produce an accelerating flow and it also gives rise to a drag, because the entrained fluid has upslope momentum (inco-ordinates moving with the wave crest). A similarity solution is obtained under the assumptions that the flow is steady in time, and that the slope and the density difference remain constant. In this solution, the thickness of the whitecap is proportional to the distances measured from the crest of the wave. The tangential velocity is proportional to s ½. Since the velocity in a Stokes 120° angle is also proportional to s ½ this implies that such a flow can start from a small disturbance with zero flux, and propagate with constant acceleration. An important consequence of the analysis is that solutions of this kind are possible only when the slope and the density difference between the whitecap and the water below are sufficiently large; otherwise the upward drag dominates, and a self-sustaining flow cannot form. For a slope of 30°, near the crest of the breaking wave, the theory predicts that a density difference greater than 8% is required to sustain a steady motion, at which point the downslope velocity is 12% of the opposing velocity at the wave surface. A 'starting plume’ model of the advancing front of the breaker is also discussed, which suggests that this too will be accelerating uniformly, but will have a velocity somewhat less than that in the flow behind. A comparison with the laboratory observations of Kjeldsen & Olsen verifies several features of the model, including the order of magnitude of the relative velocities in the whitecaps and the wave beneath. It also reveals the intermittent nature of the flow, which is here explained as due to the intermittent rounding of the wave crest due to damping of the wave by the whitecap on the forward face. © 1974, Cambridge University Press. All rights reserved.

Description: The stability of the oscillatory Stokes layers is examined using two quasi-static linear theories and an integration of the full time-dependent linearized disturbance equations. The full theory predicts absolute stability within the investigated range and perhaps for all the Reynolds numbers. A given wavenumber disturbance of a Stokes layer is found to be more stable than that of the motionless state (zero Reynolds number). The quasi-static theories predict strong inflexional instabilities. The failure of the quasi-static theories is discussed. © 1974, Cambridge University Press. All rights reserved.

Description: This paper examines the features of the flow field off the surface of an oscillating flat-plate airfoil immersed in a two-dimensional supersonic flow Although the exact linearized solution for a supersonic unsteady airfoil has been known for a long time, its expression in the form of an integral is not convenient for a physical interpretation. In the present paper, the quintessential features of the flow field are extracted from the exact solution by obtaining an asymptotic expansion in descending powers of a frequency parameter through the repeated use of the stationary-phase and steepest descent methods. It is found that the flow field consists of two dominant and competing signals: one is the acoustic ray or that component arising from Lighthill's ‘convecting slab’ and the other is the leading-edge disturbance propagating as a convecting wavelet. The flow field is found to be divided into several identifiable regions defined by the relative magnitude of the signals, and the asymptotic expansions appropriate for each flow region are derived along with their parametric restrictions. Such intimate knowledge of the flow field in unsteady, supersonic flow is of interest for interference aerodynamics and related acoustic problems. © 1974, Cambridge University Press. All rights reserved.

Description: A theoretical analysis has been made of turbulent viscous interaction on iso-thermal surfaces at hypersonic speeds. The important parameters governing the effects of incidence and displacement have been obtained under both strong and weak interaction conditions for flat-plate flows. A more general expression relating boundary-layer growth to the external pressure field and effective body shape has been obtained. The method is applied to the wedge compression corner problem and the results compared with some experimental data. © 1974, Cambridge University Press. All rights reserved.

Description: Data are presented for the extensional flow of suspensions containing 0.1−1 % of fibres by volume. The aspect ratio of the fibres was varied from 280 to 1260. The observed stress levels were between one and two orders of magnitude greater than in shearing flow, in agreement with the a priori predictions of Batchelor (1971). © 1974, Cambridge University Press. All rights reserved.

Description: Two-dimensional incompressible motion is generated by a steady external body force varying sinusoidally with a transverse co-ordinate. Such flow is found to be unstable for Reynolds numbers greater than 2½, and under these conditions evolves towards a new steady state. This ‘steady-eddy’ state is itself unstable in a sense, and its breakdown suggests the catastrophic onset of a cascade of turbulence. The mechanics of this cascade can be represented by a kind of recursion system in which the turbulence dynamics of one scale is repeated in the next, and a law of turbulent stress results. The spectrum of kinetic energy generated by a steady input of momentum at a discrete wavelength shows a rapid decrease (as k −5) towards shorter wavelengths but a much slower decrease (as k) towards longer wavelengths. © 1974, Cambridge University Press. All rights reserved.

Description: Consistency and uniqueness questions raised by both the 1941 and 1962 Kolmogorov inertial-range theories are examined. The 1941 theory, although unlikely from the viewpoint of vortex-stretching physics, is not ruled out just because the dissipation fluctuates; but self-consistency requires that dissipation fluctuations be confined to dissipation-range scales by a spacewise mixing mechanism. The basic idea of the 1962 theory is a self-similar cascade mechanism which produces systematically increasing intermittency with a decrease of scale size. This concept in itself requires neither the third Kolmogorov hypothesis (log-normality of locally averaged dissipation) nor the first hypothesis (universality of small-scale statistics as functions of scale-size ratios and locally averaged dissipation). It does not even imply that the inertial range exhibits power laws. A central role for dissipation seems arbitrary since conservation alone yields no simple relation between the local dissipation rate and the corresponding proper inertial-range quantity: the local rate of energy transfer. A model rate equation for the evolution of probability densities is used to illustrate that even scalar nonlinear cascade processes need not yield asymptotic log-normality. The approximate experimental support for Kolmogorov's hypothesis takes on added significance in view of the wide variety of a priori admissible alternatives. If the Kolmogorov law [formula omitted] is asymptotically valid, it is argued that the value of μ depends on the details of the nonlinear interaction embodied in the Navier–Stokes equation and cannot be deduced from overall symmetries, invariances and dimensionality. A dynamical equation is exhibited which has the same essential invariances, symmetries, dimensionality and equilibrium statistical ensembles as the Navier–Stokes equation but which has radically different inertial-range behaviour. © 1974, Cambridge University Press. All rights reserved.

Description: The onset of convection is observed in a cylindrical annulus which is heated from the outside, cooled from the inside and rotating about its vertical axis of symmetry. The dynamical constraint exerted by the dominating Coriolis force inhibits the instability when the top and bottom boundaries of the annulus are conical so as to make the vertical height vary with distance from the axis. The experimental observations are in good agreement with the theoretical predictions by Busse (1970a). This confirmation indicates the absence of subcritical finite amplitude instabilities and suggests that the annulus experiment provides a close dynamical model for convection in the liquid core of the earth. © 1974, Cambridge University Press. All rights reserved.

Description: Edge waves travelling along a straight coastline are examined in the case when the water depth approaches a constant value at large distances from the coast. Only the fundamental mode is examined in the limit as the ratio of the water depth at infinity to the edge-wave wavelength tends to zero. Two comparison theorems are used to obtain upper and lower bounds for the dispersion relation. A long-wave approximation procedure is used to obtain the leading terms in the dispersion relation for a wide class of bottom topographies. The results obtained are compared with an exact result for the case when the bottom topography is a rectangular step. © 1974, Cambridge University Press. All rights reserved.

Description: We consider the low Rossby number (R) flow of a stratified fluid in a long rotating channel, for which the bottom elevation varies in the downstream direction. The quasi-geostrophic response is shown to be singular at the side walls of the channel, and thus an ageostrophic analysis is necessary even for vanishingly small R. Part of the ageostrophic steady-state response is a modified quasi-geostrophic perturbation trapped near the bottom. A second component which is present even as R approaches zero is an internal Kelvin wave whose vertical wavelength adjusts so that the wave remains stationary with respect to the channel bottom and which propagates energy and momentum to infinite heights in an unbounded channel. The case of a bounded layer of fluid is also considered, and the resonance conditions are given. We also calculate the flow field when the bottom elevation varies in the cross-stream direction. We conclude that stagnation or flow reversal can be caused either by the modified quasi-geostrophic component or by the Kelvin wave and estimate the critical condition by an extrapolation of the perturbation velocity computed from linear theory. © 1974, Cambridge University Press. All rights reserved.

Description: We consider the flow of a slightly viscous homogeneous fluid over a small two-dimensional obstacle perpendicular to the vertical side walls in a channel rotating about a vertical axis. The flow in the channel is obtained from the solution of the quasi-geostrophic potential vorticity equation in the limit ε = D/L→ 0, where D is the obstacle width and L the channel width. The lowest order, interior flow is shown to be a combination of three effects: a rotational flow caused by vortex stretching and Ekman boundary-layer pumping; a significant irrotational flow induced by the magnitude of the former flow at the vertical boundaries; and the interior Ekman drift due to the basic current. The maximum streamline displacement is calculated and compares very well with recent experiments in the identical parameter range by Boyer (1971a, b). This theory explains how the side walls are responsible for the dependence of the maximum streamline displacement on Rossby number. © 1974, Cambridge University Press. All rights reserved.

Description: The model proposed by Smith & Stewartson (1973), to describe the separated boundary layer induced by strong injection over a finite length of a flat plate in a supersonic mainstream, is shown to provide the basis for a fully consistent solution of the Navier–Stokes equations for this problem, valid in the limit of infinite Reynolds number. The solution takes the form of asymptotic expansions in each of a large number of overlapping regions of the flow field, which are consistently matched across areas of common validity. © 1974, Cambridge University Press. All rights reserved.

Description: By measuring the heat dispersion behind a heated wire stretched across a wind tunnel (Taylor 1921, 1935), the Lagrangian velocity autocorrelation was determined in an approximately isotropic, grid-generated turbulent flow. The techniques were similar to previous ones, but the scatter is less. Assuming self-preservation of the Lagrangian velocity statistics in a form consistent with recent measurements of decay in this flow (Comte-Bellot & Corrsin 1966, 1971), a stationary and an approximately self-preserving form for the dispersion were derived and approximately verified over the range of the experiment. Possibly the most important aspect of this experiment is that data were available in the same flow on the simplest Eulerian velocity autocorrelation in time, the correlation at a fixed spatial point translating with the mean flow (Comte-Bellot & Corrsin 1971). Thus, the Lagrangian velocity autocorrelation coefficient function calculated from the dispersion data could be compared with this corresponding Eulerian function. It was found that the Lagrangian Taylor micro-scale is very much larger than the analogous Eulerian microscale (76 ms compared with 6.2ms), contrary to an estimate of Corrsin (1963). The Lagrangian integral time scale is roughly equal to the Eulerian one, being larger by about 25 %. © 1974, Cambridge University Press. All rights reserved.

Description: Vortex shedding from spheres has been studied in the Reynolds number range 400 〈 Re 〈 5 × 106. At low Reynolds numbers, i.e. up to Re = 3 × 103, the values of the Strouhal number as a function of Reynolds number measured by Möller (1938) have been confirmed using water flow. The lower critical Reynolds number, first reported by Cometta (1957), was found to be Re = 6 × 103. Here a discontinuity in the relationship between the Strouhal and Reynolds numbers is obvious. From Re = 6 × 103 to Re = 3 × 105 strong periodic fluctuations in the wake flow were observed. Beyond the upper critical Reynolds number (Re = 3.7 × 105) periodic vortex shedding could not be detected by the present measurement techniques. The hot-wire measurements indicate that the signals recorded simultaneously at different positions on the 75° circle (normal to the flow) show a phase shift. Thus it appears that the vortex separation point rotates around the sphere. An attempt is made to interpret this experimental evidence. © 1974, Cambridge University Press. All rights reserved.

Description: In this paper we study the development of large-scale wavelike eddies, or instability waves, in a turbulent free shear flow. The model is based on splitting the flow into three components: the mean flow, the instability wave and the fine-scale turbulence. The wave is considered to be sufficiently weak so that it is developing in a pre-existing, known turbulent mean shear flow. The basis for the wave development is its time-averaged kinetic energy flux equation in integral form and the wave description is obtained through a shape assumption: the amplitude is determined by the energy equation; the shape function and local characteristics are determined by the local linear stability theory. The wave energy changes as it is convected into a different streamwise position where its instability properties change. The energy balancing mechanisms are production, work done by the wave pressure gradients and the energy transfer between the wave and the fine-scale turbulence via the wave-induced Reynolds stresses. The latter is taken to be dissipative via an eddy-viscosity model, inertia—elastic effects not being considered. According to forceful evidence from observations in turbulent free shear flows, the wave development is taken as being upstream controlled and begins from a distinct origin rather than being the result of local forcing by variations of the fine-scale turbulent Reynolds stresses. The wave energy flux initially grows via energy supplied by the inflexional mean flow when the shear layer is relatively thin but eventually decays through action of the finescale turbulence, directly via the dissipative energy transfer and indirectly via the turbulence-diffused, rapidly thickened mean shear flow, which renders the production mechanism less available. Numerical calculations are carried out for a turbulent mean shear flow, with speed Tie on one side and zero in the ambient region, its distribution being approximated by a sine profile in the Howarth-Dorodnitsyn co-ordinate. The flow develops from an initial boundary layer of finite thickness δ0 to a similar free-mixing layer far downstream. The wave is characterized by a dimensionless frequency parameter β0 formed from the wave frequency β*, u e, and δ0. Convection speeds, in general, increase in the downstream direction. They are subsonic initially for Mach numbers Me 〈 2 and remain subsonic for Me 〈 1.5. For Me 〉 2 peaking in the local intensity levels occurs when convection speeds are supersonic and this may explain the observed supersonic far-field radiation a t the higher jet speeds. Induced wave patterns in the ambient region are determined by the complex instability-wave speed rather than the real convection speed alone, consequently ambient wave patterns exist even a t subsonic convection speeds, but are more heavily damped near the origin and fan out laterally downstream for a given β0. According to the present model, if the waves are given an upstream excitation level about 10−3–10−2 times that of Ue, resembling, for instance, the levels of the upstream wall turbulent boundary-layer fluctuations over a wide, low-frequency, spectrum or other possible disturbances a t the nozzle exit, the development of the calculated near noise field as a function of downstream distance bears striking resemblances to the observed near jet noise field and is thus fully sufficient to explain such observations. This comparison leads to the suggestion that the essential energetics of the large-scale wavelike eddies are that they are formed at the origin, amplified and subsequently decay in a developing mean turbulent free shear flow. Therefore this leads also to a most important indication of shear-layer instabilities and noise control. If their historical evolution can be controlled so can the noise from the damaging wavelike eddies. Methods of control on the basis of this model are discussed. © 1974, Cambridge University Press. All rights reserved.

Description: The inviscid compressible flow stability problem is mathematically similar to that of sound propagation in a sheared flow field. This similarity has been exploited by applying an inner expansion technique to study the effect of finite shear gradients on free parallel flow instabilities. This technique had previously been used to investigate the effect of thin boundary layers on sound propagation in ducts. The expansion, which is applicable to flow profiles involving thin, but finite, shear layers separating regions of uniform flow, offers a significant computational advantage over the numerical methods commonly employed to determine the stability of continuous mean flow profiles. Although equally applicable to three-dimensional and to spatially growing hydrodynamic instabilities, the procedure is demonstrated by application to the eigenvalue problem for temporal instabilities of shear layers and jets in plane inviscid compressible flow. For the case of vanishingly thin shear layers, the eigenvalue equations derived here reduce to those obtained by Miles (1958) for parallel flows bounded by vortex sheets. The series solution of Graham & Graham (1969), valid for linear shear-layer profiles of arbitrary thickness, provides a basis of comparison for the expansion-method results. Unstable-mode eigenvalues obtained using the two methods are found to be in good agreement for a significant range of values of the ratio of shear-layer thickness to axial wavelength. © 1974, Cambridge University Press. All rights reserved.

Description: The method of energy is used to develop two stability criteria for a large class of modulated Bénard problems. Both criteria give stability limits which hold for disturbances of arbitrary amplitude. The first of these, designated as strong global stability, requires the energy of all disturbances to decay monotonically and exponentially in time. Application of this criterion results in a prediction of Rayleigh numbers below which the diffusive stagnant solution to the Bous-sinesq equations is unique. The second criterion requires only that disturbances decay asymptotically to zero over many cycles of modulation, and is a weaker concept of stability. Computational results using both criteria are given for a wide range of specific cases for which linear asymptotic stability results are available, and it is seen that the energy and linear limits often lie close to one another. © 1974, Cambridge University Press. All rights reserved.

Description: Wind-tunnel turbulence behind a parallel-rod grid with jets evenly distributed along each rod is nearly isotropic. Homogeneity improvement over prior related experiments was attained by the use of controllable nozzles. Compared with the ‘passive’ case, the downwind-jet ‘active’ grid has a smaller static pressure drop across it and gives a smaller turbulence level at a prescribed distance from it, while the upwind-jet grid gives a larger static pressure drop and larger turbulence level. ‘Counterflow injection’ generates larger turbulence energy and larger scales, both events being evidently associated with instability of the jet system. This behaviour is much like that commonly observed behind passive grids of higher solidities. If the turbulent kinetic energy is approximated as an inverse power law in distance, the (positive) exponent decreases with increasing (downwind or upwind) jet strength, corresponding to slower absolute decay rates. No peculiar decay behaviour occurs when the jet grid is ‘self-propelled’ (zero net average force), or when the static pressure drop across it is zero. The injection does not change the general behaviour of the energy spectra, although the absolute spectra change inasmuch as the turbulence kinetic energy changes. © 1974, Cambridge University Press. All rights reserved.

Description: The distortion of a gas bubble rising steadily in an inviscid incompressible liquid of infinite extent under the action of surface tension forces is investigated theoretically using an appropriate extension of the tensor virial theorem. A convenient parameter for distinguishing the bubble shape is the Weber number W. The virial method leads to an expression relating W and the axis ratio χ, of the transverse and longitudinal axes of the bubble. To first order in W, this relation agrees with the linear theory established by Moore (1959). Also, comparison of the results with his (1965) approximate theory reveals similar features and excellent agreement up to χ = 2. In particular, it confirms his prediction of the existence of a maximum Weber number. Although the present work does not consider the stability of these bubbles, it is interesting to note that the maximum value of 3.271 attained by W differs only by about 2.8% from the critical Weber number obtained by Hartunian & Sears (1957) for the onset of instability. An approximate method for the study of slightly distorted spheroidal gas bubbles is also formulated and the resulting boundary-value problem solved numerically. The theory is then extended to include gravity. The joint effect of surface tension as well as gravitational forces has not been included in earlier theories. The shapes of the bubbles are traced and compared with the unperturbed spheroids. Comparisons for the velocity of bubble rise are made between the present predictions and some experimental results. In particular the results are compared with recent experimental data for the motion of gas bubbles in liquid metals. © 1974, Cambridge University Press. All rights reserved.

Description: Wind, blowing over a water surface, induces a thin layer of high vorticity in which the wind stress is supported by molecular viscosity; the magnitude of the surface drift, the velocity difference across the layer, being of the order of 3% of the wind speed. When long waves move across the surface, there is a nonlinear augmentation of the surface drift near the long-wave crests, so that short waves, superimposed on the longer ones, experience an augmented drift in these regions. This is shown to reduce the maximum amplitude that the short waves can attain when they are at the point of incipient breaking. Theoretical estimates of the reduction are compared with measurements in wind-wave tanks by the authors and by Mitsuyasu (1966) in which long mechanically generated waves are superimposed on short wind-generated waves. The reductions measured in the energy density of the short waves by increasing the slope of the longer ones at constant wind speed are generally consistent with the predictions of the theory in a variety of cases. © 1974, Cambridge University Press. All rights reserved.

Description: A layer of viscous incompressible fluid is confined between two horizontal plates which rotate rapidly in their own plane with a constant angular velocity. A hemisphere has its plane face joined to the lower plate and when a uniform flow is forced past such an obstacle, a Taylor column bounded by thin detached vertical shear layers forms. The linear theory for this problem, wherein the Rossby number ε is set equal to zero on the assumption that the flow is slow, is examined in detail. The nonlinear modifications of the shear layers are then investigated for the case when ε ∼ E½, where E is the Ekman number. In particular, it is shown that provided that the Rossby number is large enough separation occurs in the free shear layers. The extension of the theory to flow past arbitrary spheroids is indicated. © 1974, Cambridge University Press. All rights reserved.

Description: A general characteristic of rapidly rotating fluids is that accurate experimental measurements can only be made of the main (azimuthal) flow. The secondary flow is then usually deduced from theory, although this is often incomplete in the boundary regions where the secondary flow is of most interest. In this paper we consider the case of source-sink flow between the porous walls of a rapidly rotating annular container and numerically integrate the full equations of motion in order to determine the complete structure of the secondary flow. The results are compared with the (approximate) analytic studies of Hide (1968) and Bennetts & Hocking (1973) to show the differences between the two approaches. A defect of many previous numerical papers has been the inability to check the solution in the nonlinear case. To overcome this, new experimental measurements of the azimuthal velocity profile for a Rossby number of 0·238 have been obtained and these are compared with the numerical results. © 1974, Cambridge University Press. All rights reserved.

Description: The theory initiated by Lighthill (1952) to describe the sound radiated by turbulence embedded in an uniform fluid at rest is here extended to the case where the turbulence exists on the edge of a uniformly moving stream. An exact analogy is developed between the distant real sound field and that which would be radiated by a particular quadrupole distribution adjacent to a vortex sheet positioned in the linearly disturbed flow. The equivalent sources in this analogy are quadrupoles identical in strength with those in Lighthill's model, but the quadrupoles are now shown to convect with the fluid-particle velocity. There is no amplifying effect of shear. The particular case of a plane shear layer is worked out in detail for sound waves of scale large in comparison with the shear-layer thickness. A downstream zone of silence is predicted as is the formation of highly directional beams associated with the interference of sound radiated directly and sound reflected from the fluid interface. A distinct structure results in which the variation of sound with flow velocity, density and angle is not easily accounted for by simple power-law scaling. Finally a comparison is made with some features of jet noise; the modelling of the high frequency jet noise problem by a single shear layer yields some features consistent with experiment. © 1974, Cambridge University Press. All rights reserved.

Description: A Mach-Zehnder interferometer was used to stud two-dimensional Bénard convection cells. The experiments were performed with distilled water and sea water in the region where density is a linear function of temperature. Two-dimensional convection rolls were formed with Rayleigh numbers as great as 23400. Reversal in the temperature profile was obtained for R/Rc ≥ 3·8, and an overshoot of about 6% was observed at R/Rc = 9·2 and 13·8. This agrees with the values predicted theoretically by Veronis (1966) for stress-free boundaries and Royal (1969) for rigid boundaries. This disagrees with the experimental results of Gille (1967), who reports an overshoot of only 1 ½% at R/Rc = 16. Many of the other results agree with those of other experimenters, such as the relation between the cell height-to-width ratio and Rayleigh number, the relation between the Nusselt number and Rayleigh number, and the value of the critical Rayleigh number. © 1974, Cambridge University Press. All rights reserved.

Description: The objective of this paper is to describe approximately the coupled steady-state processes of light propagation and induced laminar incompressible fluid flow in the case of natural convection. For the case of a homogeneous fluid and under the assumptions that light energy is instantaneously transformed into heat and that the induced velocities are not too large, it is reasonable to use the boundary-layer equations to describe the induced natural flow. These equations are augmented by the conservation of energy equation. The velocity, temperature and intensity functions are expected to exhibit similarity properties. A high-intensity light beam with a given rotationally symmetric Gaussian initial intensity distribution is propagating vertically upwards into a fluid initially at rest. The fluid characteristics are assumed to be constant. A stream function is introduced to satisfy the conservation of mass equation. The conservation of momentum equation leads to conditions on the unknown functions involved in the stream function. Additional conditions follow from the conservation of energy equation, which involves the local light intensity as a driving term. Under the assumptions made, self-defocusing (thermal blooming) will occur. The main results are an exponential increase of the boundary-layer thickness and an exponential decrease of temperature and of light intensity due to the blooming effect in addition to the exponential decrease due to absorption. © 1974, Cambridge University Press. All rights reserved.

Description: The stability of Couette flow of stratified salt solutions is investigated in an apparatus with both the inner and outer cylinders rotating. The ratio of the radius of the inner cylinder to that of the outer cylinder is 0·2. The flow is visualized by means of shadowgraph and dye-trace methods. Compared with homogeneous fluids, the effect of the stabilizing density gradient is to increase the critical speed of the inner cylinder and to decrease the critical wavelength for a given angular speed of the outer cylinder. When the cylinders are rotating in the same direction, in the critical state, the instabilities appear along the inner cylinder in a spiral wave form which is itself not very stable. With counterrotating cylinders, the instabilities appear as regularly spaced vortices which, for the most part, are neither symmetric Taylor vortices nor simple spirals. In addition, these vortices rotate as a whole at a speed generally smaller than that of the inner cylinder. From shadowgraph observations, stability curves are constructed for three density gradients. The critical wavelength and the rotational periods of the vortices are also determined. © 1974, Cambridge University Press. All rights reserved.