Showing papers in "Journal of Fluid Mechanics in 1973"

Numerical study of slightly viscous flow

Abstract: A numerical method for solving the time-dependent Navier–Stokes equations in two space dimensions at high Reynolds number is presented. The crux of the method lies in the numerical simulation of the process of vorticity generation and dispersal, using computer-generated pseudo-random numbers. An application to flow past a circular cylinder is presented.

Measurements of the structure of the Reynolds stress in a turbulent boundary layer

Abstract: Additional experimental studies of the structure of Reynolds stress which supplement our previous work (Willmarth & Lu 1971) are reported. The velocity at the edge of the viscous sublayer is again used as a detector signal for bursts and sweeps. The signal uv obtained from an X-wire probe at various locations is conditionally sampled and sorted into four quadrants of the u, v plane. Using this method it is found that, when the velocity uw at the edge of the viscous sublayer becomes low and decreasing, a burst occurs. On the other hand, a sweep occurs when uw becomes large and increasing. The convection speeds of the bursts and the sweeps are found to be equal and are about 0·8 times the local mean velocity and 0·425 times the free-stream velocity at a distance y ≈ 0·15δ* from the wall (δ* is the displacement thickness). Throughout the turbulent boundary layer, the bursts are the largest contributors to from different events. Both mean time intervals are approximately equal and constant for most of the turbulent boundary layer. The scaling of the mean time interval between bursts with outer flow variables is confirmed. It is suggested that many of the features of the fluctuating flow revealed by the measurements may be explained by convection past the measuring station of an evolving deterministic flow pattern such as the hairpin vorticity model of Willmarth & Tu (1967).

On transition in a pipe. Part 1. The origin of puffs and slugs and the flow in a turbulent slug

Abstract: Conditionally sampled hot-wire measurements were taken in a pipe at Reynolds numbers corresponding to the onset of turbulence. The pipe was smooth and carefully aligned so that turbulent slugs appeared naturally at Re > 5 × 104. Transition could be initiated at lower Re by introducing disturbances into the inlet. For smooth or only slightly disturbed inlets, transition occurs as a result of instabilities in the boundary layer long before the flow becomes fully developed in the pipe. This type of transition gives rise to turbulent slugs which occupy the entire cross-section of the pipe, and they grow in length as they proceed downstream. The leading and trailing ‘fronts’ of a turbulent slug are clearly defined. A unique relation seems to exist between the velocity of the interface and the velocity of the fluid by which relaminarization of turbulent fluid is prevented. The length of slugs is of the same order of magnitude as the length of the pipe, although the lengths of individual slugs differ at the same flow conditions. The structure of the flow in the interior of a slug is identical to that in a fully developed turbulent pipe flow. Near the interfaces, where the mean motion changes from a laminar to a turbulent state, the velocity profiles develop inflexions. The total turbulent intensity near the interfaces is very high and it may reach 15% of the velocity at the centre of the pipe. A turbulent energy balance was made for the flow near the interfaces. All of the terms contributing to the energy balance must vanish identically somewhere on the interface if that portion of the interface does not entrain non-turbulent fluid. It appears that diffusion which also includes pressure transport is the most likely mechanism by which turbulent energy can be transferred to non-turbulent fluid. The dissipation term at the interface is negligible and increases with increasing turbulent energy towards the interior of the slug.Mixed laminar and turbulent flows were observed far downstream for \[ 2000 < Re < 2700 \] when a large disturbance was introduced into the inlet. The flow in the vicinity of the inlet, however, was turbulent at much lower Re. The turbulent regions which are convected downstream at a velocity which is slightly smaller than the average velocity in the pipe we shall henceforth call puffs. The leading front of a puff does not have a clearly defined interface and the trailing front is clearly defined only in the vicinity of the centre-line. The length and structure of the puff is independent of the character of the obstruction which created it, provided that the latter is big enough to produce turbulent flow at the inlet. The puff will be discussed in more detail later.

The interaction between a pair of circular cylinders normal to a stream

Abstract: This paper describes how the flows around two circular cylinders, displaced in a plane normal to the free stream, interact as the two bodies are brought close together. Surface pressure measurements at a Reynolds number of 2·5 × 104, based on the diameter of a single cylinder, show the presence of a mean repulsive force between the cylinders. An instability of the flow was found when the gap between the cylinders was in the range between one diameter and about 0·1 of a diameter. Correlation measurements of hot-wire outputs indicate how mutual interference influences the formation of vortex streets from the two cylinders. Spanwise correlation measurements show that the correlation length doubles as the cylinders are brought into contact.

On optimum profiles in Stokes flow

Abstract: In this paper, we obtain the first-order necessary optimality conditions of an optimal control problem for a distributed parameter system with geometric control, namely, the minimum-drag problem in Stokes flow (flow at a very low Reynolds number). We find that the unit-volume body with smallest drag must be such that the magnitude of the normal derivative of the velocity of the fluid is constant on the boundary of the body. In a three-dimensional uniform flow, this condition implies that the body with minimum drag has the shape of a pointed body similar in general shape to a prolate spheroid but with some differences including conical front and rear ends of angle 120°.

A note on tsunamis: their generation and propagation in an ocean of uniform depth

Abstract: The waves generated in a two-dimensional fluid domain of infinite lateral extent and uniform depth by a deformation of the bounding solid boundary are investigated both theoretically and experimentally. An integral solution is developed for an arbitrary bed displacement (in space and time) on the basis of a linear approximation of the complete (nonlinear) description of wave motion. Experimental and theoretical results are presented for two specific deformations of the bed; the spatial variation of each bed displacement consists of a block section of the bed moving vertically either up or down while the time-displacement history of the block section is varied. The presentation of results is divided into two sections based on two regions of the fluid domain: a generation region in which the bed deformation occurs and a downstream region where the bed position remains stationary for all time. The applicability of the linear approximation in the generation region is investigated both theoretically and experimentally; results are presented which enable certain gross features of the primary wave leaving this region to be determined when the magnitudes of parameters which characterize the bed displacement are known. The results indicate that the primary restriction on the applicability of the linear theory during the bed deformation is that the total amplitude of the bed displacement must remain small compared with the uniform water depth; even this restriction can be relaxed for one type of bed motion. Wave behaviour in the downstream region of the fluid domain is discussed with emphasis on the gradual growth of nonlinear effects relative to frequency dispersion during propagation and the subsequent breakdown of the linear theory. A method is presented for finding the wave behaviour in the far field of the downstream region, where the effects of nonlinearities and frequency dispersion have become about equal. This method is based on the use of a model equation in the far field (which includes both linear and nonlinear effects in an approximate manner) first used by Peregrine (1966) and more recently advocated by Benjamin, Bona & Mahony (1972) as a preferable model to the more commonly used equation of Korteweg & de Vries (1895). An input-output approach is illustrated for the numerical solution of this equation where the input is computed from the linear theory in its region of applicability. Computations are presented and compared with experiment for the case of a positive bed displacement where the net volume of the generated wave is finite and positive; the results demonstrate the evolution of a train of solitary waves (solitons) ordered by amplitude followed by a dispersive train of oscillatory waves. The case of a negative bed displacement in which the net wave volume is finite and negative (and the initial wave is negative almost everywhere) is also investigated; the results suggest that only a dispersive train of waves evolves (no solitons) for this case.

Helical turbulence and absolute equilibrium

Abstract: The interaction of two pure helical (circularly polarized) velocity waves according to the incompressible Navier–Stokes equation produces modulation products of mixed helicity. In general, the interaction of waves of opposite helicity is stronger than that of waves with the same helicity. The inference is that strong net helicity depresses overall turbulent energy transfer. The conservation laws strongly inhibit energy transfer from higher to lower wavenumbers, when the helicity is large. The absolute equilibrium spectra of velocity and helicity for an inviscid flow system truncated at an upper wavenumber k2 are \[ U(k) = 2\alpha/(\alpha^2-\beta^2k^2),\quad Q(k) = 2\beta k^2/(\alpha^2-\beta^2k^2), \] where the velocity variance and helicity/unit volume are ∫U(k)d3k and ∫Q(k)d3k, respectively. The temperature parameters α and β are constrained by α > 0 and |βk2| < α. There are no analogues of the negative-temperature equilibrium states known for two-dimensional inviscid flow. It is argued that the inertial-range energy cascade in isotropic turbulence driven by helical input should not differ asymptotically from that of non-helical turbulence. The absolute equilibrium distributions suggest that, in contrast to the analogous two-dimensional situation, statistically steady helical input at middle wavenumbers should not produce a significant downward cascade of energy to lower wavenumbers.

On the no-slip boundary condition

Abstract: It has been argued that the no-slip boundary condition, applicable when a viscous fluid flows over a solid surface, may be an inevitable consequence of the fact that all such surfaces are, in practice, rough on a microscopic scale: the energy lost through viscous dissipation as a fluid passes over and around these irregularities is sufficient to ensure that it is effectively brought to rest. The present paper analyses the flow over a particularly simple model of such a rough wall to support these physical ideas.

On the Weis-Fogh mechanism of lift generation

Abstract: Weis-Fogh (1973) proposed a new mechanism of lift generation of fundamental interest. Surprisingly, it could work even in inviscid two-dimensional motions starting from rest, when Kelvin's theorem states that the total circulation round a body must vanish, but does not exclude the possibility that if the body breaks into two pieces then there may be equal and opposite circulations round them, each suitable for generating the lift required in the pieces’ subsequent motions! The ‘fling’ of two insect wings of chord c (figure 1) turning with angular velocity Ω generates irrotational motions associated with the sucking of air into the opening gap which are calculated in § 2 as involving circulations −0·69Ωc2 and + 0.69Ωc2 around the wings when their trailing edges, which are stagnation points of those irrotational motions, break apart (position (f)). Viscous modifications to this irrotational flow pattern by shedding of vorticity at the boundary generate (§ 3) a leading-edge separation bubble, and tend to increase slightly the total bound vorticity. Its role in a three-dimensional picture of the Weis-Fogh mechanism of lift generation, involving formation of trailing vortices at the wing tips, and including the case of a hovering insect like Encarsia formosa moving those tips in circular paths, is investigated in § 4. The paper ends with the comment that the far flow field of such very small hovering insects should take the form of the exact solution (Landau 1944; Squire 1951) of the Navier-Stokes equations for the effect of a concentrated force (the weight mg of the animal) acting on a fluid of kinematic viscosity v and density p, whenever the ratio mg/pv2 is small enough for that jet-type induced motion to be stable.

Experiments on instability and turbulence in a stratified shear flow

Abstract: This is a study of turbulence which results from Kelvin—Helmholtz instability at the interface between two miscible fluids in a two-dimensional shear flow in the laboratory. The growth of two-dimensional ‘billows’, their disruption by turbulence, and the eventual decay of this turbulence and the re-establishment of a gravitationally and kinematically stable interface are described. Continuous measurements of density and horizontal velocity from both fixed and vertically moving probes have been made, and the records obtained are presented, together with photographs showing the simultaneous appearance of the flow, which serve to identify the physical nature of events seen in the records. The measurements show how the fine-structure of the density field described in earlier experiments is related to velocity fluctuations. The vertical length scales of the final mean velocity and density structure are found to be different, and to depend on the Richardson number at which instability first occurred. The eventual Richardson number at the centre of layer is, however, not dependent on the initial Richardson number and has a value of about one third. The implications of these results to the eddy diffusion coefficients, to the energy exchange, and to turbulence in the ocean and the atmosphere are discussed.

A theory of turbulent flow round two-dimensional bluff bodies

Abstract: By generalizing the theory of ‘rapid distortion’ of turbulence developed by Batchelor & Proudman (1954) it is shown in this paper that the turbulent velocity around a bluff body placed in a turbulent flow can be calculated outside and upstream of the regions of separated flow, if the incident turbulent flow satisfies the following conditions: (i) if a/Lx [Lt ] 1 or = O(1), Re−1 is the r.m.s. velocity of the homogeneous incident turbulence, a is a transverse dimension of the body (the radius in the case of a circular cylinder), Lx is the integral scale of the incident turbulence and v is the kinematic viscosity.Detailed calculations are given for the flow around a circular cylinder with particular emphasis on the turbulence very close to the surface. (The results can be generalized to other cylindrical bodies.) Mean-square values and spectra of velocity have been found only in the limiting situations where the turbulence scale is very much larger or smaller than the size of the body, i.e. Lx [Gt ] a or Lx [Lt ] a. But, whatever the value of a/Lx, if the frequency is sufficiently large the results for spectra tend to those of the limiting situation where Lx [Lt ] a. The reason why the turbulence velocities have not been calculated for intermediate values of a/Lx is that closed-form solutions cannot be found and that the computing time then required is quite excessive. However, some computed results are used in the paper to suggest the qualitative behaviour of the turbulence when Lx is of order a. An important result of the theory is that it illuminates and distinguishes between the governing physical processes of distortion of the turbulence by the mean flow, the direct ‘blocking’ of the turbulence by the body, and concentration of vortex lines at the body's surface.The results of the theory have many applications, for example in calculating turbulent dispersion and fluctuating pressures on the body, as shown elsewhere by Hunt & Mulhearn (1973) and Hunt (1973).In conclusion the theoretical results are briefly compared with experimental measurements of turbulent flows round non-circular cylinders. A detailed comparison with measurements round circular cylinders will be published later by Petty (1974).

A family of steady vortex rings

Abstract: Axisymmetric vortex rings which propagate steadily through an unbounded ideal fluid at rest at infinity are considered. The vorticity in the ring is proportional to the distance from the axis of symmetry. Recent theoretical work suggests the existence of a one-parameter family, [npar ]2 ≥ α ≥ 0 (the parameter α is taken as the non-dimensional mean core radius), of these vortex rings extending from Hill's spherical vortex, which has the parameter value α = [npar ]2, to vortex rings of small cross-section, where α → 0. This paper gives a numerical description of vortex rings in this family. As well as the core boundary, propagation velocity and flux, various other properties of the vortex ring are given, including the circulation, fluid impulse and kinetic energy. This numerical description is then compared with asymptotic descriptions which can be found near both ends of the family, that is, when α → [npar ]2 and α → 0.

Stability of a circular cylinder oscillating in uniform flow or in a wake

Abstract: The lift and drag forces were measured on both a single circular cylinder and tandem circular cylinders in uniform flow at Reynolds numbers from 40 to 104, to investigate the stability of an oscillating cylinder A cylinder (the downstream one in the tandem case) was made to oscillate in either the transverse or longitudinal direction (perpendicular or parallel to the stream) In the case of a single cylinder, its oscillation causes the so-called synchronization in a frequency range around the Strouhal frequency (transverse mode) or double the Strouhal frequency (longitudinal mode) The aerodynamic damping for transverse oscillation becomes negative in the synchronization range In the case of tandem cylinders, at low Reynolds numbers in the pure Karman range synchronization was observed to occur only when the downstream cylinder oscillated inside the vortex-formation region of the upstream one, and at high (low subcritical) Reynolds numbers synchronization occurred irrespective of the cylinder spacing in either oscillating mode In the tandem case, too, the transverse oscillation of the downstream cylinder becomes unstable in the range of synchronization

The effects of wake splitter plates on the flow past a circular cylinder in the range 10 4 < R <5×10 4

Abstract: Experiments were carried out using models having L/D [les ] 2 and the resulting pressure distributions and vortex shedding characteristics are presented. A simple visualization technique which provides explanations of some of the measured results is described. It is concluded that splitter planes reduce the drag markedly by stabilizing the separation points and produce a wake narrower than that for a plain cylinder, raise the base pressure by as much as 50% and affect the Strouhal number to a lesser degree. Careful measurement techniques have enabled these effects to be presented accurately.

Deformation and burst of a liquid droplet freely suspended in a linear shear field

Abstract: A theoretical method is presented for predicting the deformation and the conditions for breakup of a liquid droplet freely suspended in a general linear shear field. This is achieved by expanding the solution to the creeping-flow equations in powers of the deformation parameter epsilon and using linear stability theory to determine the onset of bursting. When compared with numerical solutions and with the available experimental data, the theoretical results are generally found to be of acceptable accuracy although, in some cases, the agreement is only qualitative.

Some further studies on the transition to turbulent convection

Abstract: In a horizontal convecting layer of fluid, several distinct transitions occur at certain distinct Rayleigh numbers R, for a given Prandtl number Pr. The regime diagram has been extended to include the Prandtl-number range2·5 × 10−2 [les ] Pr [les ] 0·85 × 104.In particular it is found that distinct changes in the slope of the heat-flux curve occur even for Pr = 2·5 × 10−2. The flow is steady up to R = Rt = 2·4 × 103. For R > Rt, the period of oscillation is compared with the theoretical values of Busse. For Pr = 0·71 decreases as well as increases in the slope of the heat-flux curve are observed.For R just greater than Rc, the preferred orientation of rolls in various side-wall geometries is investigated. For high Prandtl number, the effect of curvature of the roll axis, forced by curved side walls, upon the second transition at RII is investigated. It is found that curvature, as well as previously discussed effects, leads to a lowering of RII. These results, along with the observed hysteresis, support the view that there are metastable states attainable by finite amplitude instability. Finally the nature of the time-dependent flow at large R and high Prandtl number is investigated in a Hele-Shaw cell. It is shown unequivocally that the observed periodicity at a fixed point is due to hot or cold plumes moving past the point.

Flow past an impulsively started circular cylinder

Abstract: An accurate method is described for integrating the Navier-Stokes equations numerically for the time-dependent flow past an impulsively started circular cylinder. Results of integrations over the range of Reynolds numbers, based on the diameter of the cylinder, from 5 to ∞ are presented and compared with previous numerical, theoretical and experimental results. In particular, the growth of the length of the separated wake behind the cylinder has been calculated for R = 40, 100 and 200 and is found to be in very good agreement with the results of recent experimental measurements. The calculated pressure distribution over the surface of the cylinder for R = 500 is also found to be in reasonable agreement with experimental measurements for the case R = 560.For Reynolds numbers up to 100 the equations were integrated until most of the features of the flow showed a close approximation to steady-state conditions. The results obtained are in good agreement with previous calculations of the steady flow past a circular cylinder. For R > 100 the integrations were continued until the implicit method of integration broke down by reason of its failure to converge. A secondary vortex appeared on the surface of the cylinder in the case R = 500, but for higher Reynolds numbers, including the case R = ∞, the procedure broke down before the appearance of a secondary vortex. In all cases the flow was assumed to remain symmetrical.

The origin of secondary flow in turbulent flow along a corner

Abstract: The mechanisms which initiate secondary flow in developing turbulent flow along a corner are examined on the basis of both energy and vorticity considerations. This is done by experimentally evaluating the terms of an energy balance and vorticity balance applied to the mean motion along a corner bisector. The results show that a transverse flow is initiated and directed towards the corner as a direct result of turbulent shear stress gradients normal to the bisector. The results further indicate that anisotropy of the turbulent normal stresses does not play a major role in the generation of secondary flow. Possible extensions of the present results to other related flow situations are ahstrated and discussed.

Turbulent convection in a horizontal layer of water

Abstract: Overall heat transfer and mean temperature distribution measurements have been made of turbulent thermal convection in horizontal water layers heated from below. The Nusselt number is found to be proportional to Ra0·278 in the range 2·76 × 105 < Ra < 1·05 × 108. Eight discrete heat flux transitions are found in this Rayleigh number range. An interferometric method is used to measure the mean temperature distribution for Rayleigh numbers between 3·11 × 105 and 1·86 × 107. Direct visual and photographic observations of the fluctuating interferogram patterns show that the main heat transfer mechanism is the release of thermals from the boundary layers. For relatively low Rayleigh numbers (up to 5 × 105) many of the thermals reach the opposite surface and coalesce to form large masses of relatively warm fluid near the cold surface and masses of cold fluid near the warm surface, resulting in a temperature-gradient reversal. With increasing Rayleigh numbers, fewer and fewer thermals reach the opposite bounding surface and the thermals show persistent horizontal movements near the bounding surfaces. The central region of the layer becomes an isothermal core. The mean temperature distributions for the high Rayleigh number range are found to follow a Z−2 power law over a considerable range, where Z is the distance from the bounding surface. A very limited agreement with the theoretically predicted Z−1 power law is also found.

An inviscid model of two-dimensional vortex shedding

Abstract: An inviscid model of two-dimensional vortex shedding behind a square-based section is developed. The model uses a discrete-vortex approximation for the free shear layers. The motion of the shear layers is computed from the velocities of the discrete vortices, which in turn are derived through a Schwartz-Christoffel transformation of the section. The flow round the body is impulsively started from rest and initially develops symmetrically. The introduction of a small asymmetric disturbance results in asymmetric interaction of the shear layers amplifying into steady vortex-shedding motion.The model is shown to predict the form of vortex shedding, the Strouhal number and some other flow quantities to a good degree of agreement with experimental results.

Relaminarization in highly accelerated turbulent boundary layers

Abstract: The mean flow development in an initially turbulent boundary layer subjected to a large favourable pressure gradient beginning at a point x0 is examined through analyses expected a priori to be valid on either side of relaminarization. The ‘quasi-laminar’ flow in the later stages of reversion, where the Reynolds stresses have by definition no significant effect on the mean flow, is described by an asymptotic theory constructed for large values of a pressure-gradient parameter Λ, scaled on a characteristic Reynolds stress gradient. The limiting flow consists of an inner laminar boundary layer and a matching inviscid (but rotational) outer layer. There is consequently no entrainment to lowest order in Λ−1, and the boundary layer thins down to conserve outer vorticity. In fact, the predictions of the theory for the common measures of boundary-layer thickness are in excellent agreement with experimental results, almost all the way from x0. On the other hand the development of wall parameters like the skin friction suggests the presence of a short bubble-shaped reverse-transitional region on the wall, where neither turbulent nor quasi-laminar calculations are valid. The random velocity fluctuations inherited from the original turbulence decay with distance, in the inner layer, according to inverse-power laws characteristic of quasi-steady perturbations on a laminar flow. In the outer layer, there is evidence that the dominant physical mechanism is a rapid distortion of the turbulence, with viscous and inertia forces playing a secondary role. All the observations available suggest that final retransition to turbulence quickly follows the onset of instability in the inner layer.It is concluded that reversion in highly accelerated flows is essentially due to domination of pressure forces over the slowly responding Reynolds stresses in an originally turbulent flow, accompanied by the generation of a new laminar boundary layer stabilized by the favourable pressure gradient.

Two-dimensional Rayleigh-Benard convection

Abstract: Two-dimensional convection in a Boussinesq fluid confined between free boundaries is studied in a series of numerical experiments. Earlier calculations by Fromm and Veronis were limited to a maximum Rayleigh number R 50 times the critical value R, for linear instability. This range is extended to 1000Rc. Convection in water, with a Prandtl number p = 6·8, is systematically investigated, together with other models for Prandtl numbers between 0·01 and infinity. Two different modes of nonlinear behaviour are distinguished. For Prandtl numbers greater than unity there is a viscous regime in which the Nusselt number . At higher Rayleigh numbers advection of vorticity becomes important and N ∞ R0·365. When p = 6·8 the heat flux is a maximum for square cells; steady convection is impossible for wider cells and finite amplitude oscillations appear instead, with periodic fluctuations of temperature and velocity in the layer. For p < 1 it is also found that N ∞ R0·365, with a constant of proportionality equal to 1·90 when p [Lt ] 1 and decreasing slowly as p is increased. The physical behaviour in these regimes is analysed and related to astrophysical convection.

Experiment on convex curvature effects in turbulent boundary layers

Abstract: Turbulent boundary layers along a convex surface of varying curvature were investigated in a specially designed boundary-layer tunnel. A fairly complete set of turbulence measurements was obtained. The effect of curvature is striking. For example, along a convex wall the Reynolds stress is decreased near the wall and vanishes about midway between the wall and the edge of a boundary layer where there exists a velocity profile gradient created upstream of the curved wall.

On making holes in a sheet of fluid

Abstract: It is suggested in this paper that axisymmetric holes in thin sheets of fluid in which surface tension forces predominate will open out if they are initially large in relation to the thickness of the sheet; but that small holes will close up. No exact criterion has been found for the critical hole size in a free falling sheet, but the behaviour of the sheet may be closely simulated by the suspension of a soap film between coaxial circular rings. Theoretical results and experimental observations on catenoid films so formed are described.For a hole in a sheet standing under gravity on a horizontal plane an equilibrium configuration exists, which is shown to be unstable. It is suggested that in this case the equilibrium position serves to distinguish between holes which open and those which close. Experiments on the behaviour of holes in a mercury sheet reveal a well-defined critical size which is in good agreement with that predicted by the unstable equilibrium.A further series of experiments on holes made in a sheet of water standing on paraffin wax gave no sharp distinction between opening and closing holes, and holes of a wide range of sizes could remain stationary. This behaviour is associated with changes in the angle of contact with the plane. Independent meniscus observations similar to those of Ablett for a steadily moving meniscus show that the angle of contact θa, for a meniscus about to advance is greater than the value θr for a meniscus on the point of receding. It is seen that this difference will produce a range of hole diameters within which a hole will be trapped and remain stationary. Observations on the minimum size of hole on a water sheet which will remain open are reported. But it was found that the largest holes which would remain stationary were too large in relation to the size of the sheet for reliable results to be obtained.

Negative shock waves

Abstract: Negative or rarefaction shock waves may exist in single-phase fluids under certain conditions. It is necessary that a particular fluid thermodynamic quantity Γ ≡ −½δ In (δP/δν)s/δ In ν be negative: this condition appears to be met for sufficiently large specific heat, corresponding to a sufficient level of molecular complexity. The dynamic formation and evolution of a negative shock is treated, as well as its properties. Such shocks satisfy stability conditions and have a positive, though small, entropy jump. The viscous shock structure is found from an approximate continuum model. Possible experimental difficulties in the laboratory production of negative shocks are briefly discussed.

On the origin of meandering and braiding in alluvial streams

Abstract: The paper describes a hydrodynamic stability analysis of the flow in an alluvial channel in which dunes have developed along the bed. The purpose is to develop a mathematical model describing the three-dimensional flow leading to instability of an originally straight channel. The model offers an explanation of the fact that some channels tend to meander, others to braid.

The interaction of a vortex ring with a sharp density interface: a model for turbulent entrainment

Abstract: The interaction of a vortex ring with a sharp density interface is investigated in the laboratory. Attention is restricted to the case where the Froude number based on the density difference across the interface, the velocity of propagation of the ring normal to the interface and the diameter of the ring is less than unity. It is found that the depth of maximum penetration of the ring, and the diameter of the region of contact between the ring and the interface, are functions of the Froude number. A simple model of the ring-interface interaction which accounts for the observed motion is proposed. This model is then used to calculate the volume rate of entrainment produced by the vortex rings. It is found that this rate of entrainment is proportional to the cube of the Froude number, a result which agrees with measurements of entrainment across density interfaces caused by grid-generated turbulence (Turner 1968) and by a plume incident on the interface (Baines 1973). Thus the vortex ring would appear to be a good approximation to a turbulent eddy in these situations. The main feature of the model is that it identifies the way in which the kinetic energy of the turbulence is converted into potential energy by entraining fluid across the interface. In particular, it indicates that the essential force balance is inertial, and that it is possible to discuss entrainment across a sharp density interface without explicitly invoking either viscosity or molecular diffusion.

Natural convection in a sloping porous layer

Abstract: This paper describes an experimental and theoretical study of thermal convection in a sloping porous layer. The saturated layer is bounded by two parallel impermeable planes maintained at different temperatures. Several types of flows were observed: a unicellular movement and a juxtaposition of longitudinal coils or of polyhedral cells.A theoretical analysis has been made using the standard bases of the linear theory of stability and by taking into account some assumptions suggested by experimental observations. The critical conditions for the transition between unicellular and polycellular flows has been determined. For flow in longitudinal coils or with polyhedral cells the average heat transfer depends mainly on the filtration Rayleigh number and on the slope of the layer.The experimental study was made in a Rayleigh number range 0–800 and for various slopes (0–90°). For both the transition criterion and the heat transfer, a good fit was observed between the experimental and theoretical results. For maximum slope, i.e. 90°, a correlation which connects the Nusselt number with both the Rayleigh number and the vertical extent of the model is proposed.

Laminar flow over a small hump on a flat plate

Abstract: A boundary layer flows over a flat plate which has on it a small hump situated downstream of the leading edge. The description of the boundary-layer flow, based upon a triple-deck structure, shows how the presence of the hump generates an interaction between the inviscid region just outside the layer and the viscous region near the hump. The pressure force dominant in the boundary layer and the connexion of the local flow with the main stream develop together and are self-perpetuating, and both remain of primary significance for a wide range of hump sizes, even for a hump buried well inside the boundary layer. By consideration of the limiting cases of very small and very large humps, a consistent account of the nature of the disturbances due to the various sizes of hump is produced. The forces and couples on the hump are also evaluated.

On the stratified Taylor column

Abstract: We analyse the effects of small, circularly symmetric topography on the slow flow of an inviscid, incompressible, diffusionless, horizontally uniform, baroclinic current and show that the vertical influence depends primarily on three parameters: a stratification measure S (the square of the ratio of buoyancy frequency times height scale to Coriolis parameter times length scale), a topographic parameter β (ratio of scaled topographic height multiplied by scaled bottom current to Rossby number e) and the scaled upstream shear u′0(z) (the dimensional upstream shear divided by the ratio of the r.m.s. upstream flow speed to height scale).Investigating a linear stratification model we find that the topographic effect is depth independent if S [lsim ] e and a Taylor column, as indicated by the appearance of closed streamlines above the bump, exists when β > 2. Moderate stratification (S ∼ 1) causes the flow to be fully three-dimensional and the Taylor column to be a conical vortex whose height depends on β S and u′0). The results are compared with Davies's (1971, 1972) experiments.Our results tend to support the Taylor column theory of Jupiter's Great Red Spot but effects due to variations in the Coriolis parameter with latitude have been (unjustifiably) ne glected. Using typical values for the earths oceans we find that Taylor columns of significant height could be found there. Some pertinent observations from the ocean are discussed.