Showing papers in "Journal of Fluid Mechanics in 1958"

On convection cells induced by surface tension

Abstract: A mechanism is proposed by which cellular convective motion of the type observed by H. Benard, which hitherto has been attributed to the action of buoyancy forces, can also be induced by surface tension forces. Thus when a thin layer of fluid is heated from below, the temperature gradient is such that small variations in the surface temperature lead to surface tractions which cause the fluid to flow and thereby tend to maintain the original temperature variations. A small disturbance analysis, analogous to that carried out by Rayleigh and others for unstable density gradients, leads to a dimensionless number B which expresses the ratio of surface tension to viscous forces, and which must attain a certain minimum critical value for instability to occur. The results obtained are then applied to the original cells described by Benard, and to the case of drying paint films. It is concluded that surface tension forces are responsible for cellular motion in many such cases where the criteria given in terms of buoyancy forces would not allow of instability.

Water waves of finite amplitude on a sloping beach

Abstract: In this paper, we investigate the behaviour of a wave as it climbs a sloping beach. Explicit solutions of the equations of the non-linear inviscid shallow-water theory are obtained for several physically interesting wave-forms. In particular it is shown that waves can climb a sloping beach without breaking. Formulae for the motions of the instantaneous shoreline as well as the time histories of specific wave-forms are presented.

The equilibrium range in the spectrum of wind-generated waves

Abstract: Consideration of the structure of wind-generated waves when the duration and fetch of the wind are large suggests that the smaller-scale components of the wave field may be in a condition of statistical equilibrium determined by the requirements for attachment of the crests of the waves. A dimensional analysis, based upon the idea of an equilibrium range in the wave spectrum, shows that for large values of the frequency ω, the spectrum Φ(ω) is of the form $\Phi (\omega) \sim \alpha g^2\omega^{-5}$ where α is an absolute constant. The instantaneous spatial spectrum Ψ ( k ) is proportional to k −4 for large wave numbers k , which is consistent with the observed occurrence of sharp crests in a well-developed sea, and the loss of energy from the wave system to turbulence and heat is proportional to $\rho _w u^3_*$ , where ρ w is the water density and u the friction velocity of the wind at the surface. This prediction of the form of Φ(ω) for large ω with α = 7·4×10 −3 , agrees well with measurements made by Burling (1955).

Some properties of boundary layer flow during the transition from laminar to turbulent motion

Abstract: Transition in the boundary layer on a flat plate is examined from the point of view of intermittent production of turbulent spots. On the hypothesis of localized laminar breakdown, for which there is some expermental evidence, Emmons’ probability calculations can be extended to explain the observed statistical similarity of transition regions. Application of these ideas allows detailed calculations of the boundary layer parameters including mean velocity profiles and skin friction during transition. The mean velocity profiles belong to a universal one-parameter family with the intermittency factor as the parameter. From an examination of experimental data the probable existence of a relation between the transition Reynolds number and the rate of production of the turbulent spots is deduced. A simple new technique for the measurement of the intermittency factor by a Pitot tube is reported.

Finite amplitude cellular convection

Abstract: When a layer of fluid is heated uniformly from below and cooled from above, a cellular regime of steady convection is set up at values of the Rayleigh number exceeding a critical value. A method is presented here to determine the form and amplitude of this convection. The non-linear equations describing the fields of motion and temperature are expanded in a sequence of inhomogeneous linear equations dependent upon the solutions of the linear stability problem. We find that there are an infinite number of steady-state finite amplitude solutions (having different horizontal plan-forms) which formally satisfy these equations. A criterion for ‘relative stability’ is deduced which selects as the realized solution that one which has the maximum mean-square temperature gradient. Particular conclusions are that for a large Prandtl number the amplitude of the convection is determined primarily by the distortion of the distribution of mean temperature and only secondarily by the self-distortion of the disturbance, and that when the Prandtl number is less than unity self-distortion plays the dominant role in amplitude determination. The initial heat transport due to convection depends linearly on the Rayleigh number; the heat transport at higher Rayleigh numbers departs only slightly from this linear dependence. Square horizontal plan-forms are preferred to hexagonal plan-forms in ordinary fluids with symmetric boundary conditions. The proposed finite amplitude method is applicable to any model of shear flow or convection with a soluble stability problem.

Non-linear interactions in a viscous heat-conducting compressible gas

Abstract: The linearized equations of motion show that in a viscous heat-conducting compressible medium three modes of fluctuations exist, each one of which is a familiar type of disturbance. The vorticity mode occurs in an incompressible turbulent flow, the entropy mode is familiar as temperature fluctuations in low speed turbulent heat transfer problems, and the sound mode is the subject of conventional acoustics. A consistent higher order perturbation theory is presented with the only restrictions being that the Prandtl number is 3/4 and the viscosity and heat conductivity are monotinic functions of the temperature alone. The theory is based on expansion of the disturbance fields in powers of an amplitude parameter α. The non-linearity of the full Navier-Stokes equations can be interpreted as interaction between the three basic modes; in order to help physical insight the interactions are classed as ‘mass-like’, ‘force-like’, and ‘heat-like’ effects.Besides the amplitude parameter α there is another subsidiary non-dimensional parameter e which indicates the relative importance of viscosity and heat conduction effects as compared to the inertial effects, e is proportional to the ratio of the molecular mean free path and the characteristic length of the flow pattern (Knudsen number). The main contribution of the paper is the outline of a consistent successive approximation for an arbitrary order in α and the presentation of explicit formulae for the second order (bilateral) interactions.A special case of rather general significance is treated in more detail. This is when all three basic modes have intensities and length scales of the same orders of magnitude and in addition to α the parameter e is also small; the second-order interactions are then relatively few and easily identifiable and are shown in table 1.The present analysis also sheds some light on the ‘zero order’ approximation which treats the vorticity and entropy disturbances as a ‘frozen pattern’ and the sound field as propagating nondissipative waves. The interpretation of hot-wire measurements relies heavily on these simplified models and the present paper lends some support to these current hot-wire practices.

On the non-linear mechanics of hydrodynamic stability

Abstract: In most work on the theory of stability of laminar flow, infinitesimal disturbances only have been considered, so that only the initial growth of the disturbance has been determined. It is the object of the present paper to extend the theory to larger amplitudes and to study the mechanics of disturbance growth with the inherent non-linearity of the hydrodynamical system taken into account.The Reynolds stress (where averages are taken with respect to some suitable space coordinate) is the fundamental consequence of the non-linearity, and its effects can be anticipated as follows. Initially a disturbance grows exponentially with time according to the linear theory, but eventually it reaches such a size that the transport of momentum by the finite fluctuations is appreciable and the associated mean stress (the Reynolds stress) then has an appreciable effect on the mean flow. This distortion of the mean flow modifies the rate of transfer of energy from the mean flow to the disturbance and, since this energy transfer is the cause of the growth of the disturbance, there is a modification of the rate of growth of the latter.It is suggested that, in many cases, an equilibrium state may be possible in which the rate of transfer of energy from the (distorted) mean flow to the disturbance balances precisely the rate of viscous dissipation of the energy of disturbance. A theory based on certain assumptions about the energy flow is given to describe both the growth of the disturbance and the final equilibrium state, and application is made to the cases of Poiseuille flow between parallel planes and flow between rotating cylinders. The distorted mean flow in the equilibrium state can be calculated and from this, in the latter case, the torque required to maintain the cylinders in motion. Good agreement is obtained with G. I. Taylor's measurements of the torque for the case when the inner cylinder rotates and the outer cylinder is at rest.

On the propagation of shock waves through regions of non-uniform area or flow

Abstract: This paper refers to the work of Moeckel (1952) on the interaction of an oblique shock wave with a shear layer in steady supersonic flow and the work of Chester (1955) and Chisnell (1957) on the propagation of a shock wave down a non-uniform tube. It is shown that their basic results can be obtained by the application of the following simple rule. The relevant equations of motion are first written in characteristic form. Then the rule is to apply the differential relation which must be satisfied by the flow quantities along a characteristic to the flow quantities just behind the shock wave. Together with the shock relations this rule determines the motion of the shock wave. The accuracy of the results for a wide range of problems and for all shock strengths is truly surprising.The results are exactly the same as were found by the authors cited above. The derivation given here is simpler to perfom (although the original methods were by no means involved) and of somewhat wider application, but the main reason for presenting this discussion is to try to throw further light on these remarkable results.In discussing the underlying reasons for this rule, it is convenient to use the propagation in a non-uniform tube as a typical example, but applications to a number of problems are given later. A list of some of these appears at the beginning of the introductory section.

The large eddies of turbulent motion

Abstract: This paper describes an experimental investigation of the form of the large scale motions in turbulent flow. These motions have been found to be more ordered than has usually been supposed and their origin and dynamics are discussed in terms of physical models of typical eddies.Nine components of the double velocity correlation tensor have been measured at a number of positions in the wake of a circular cylinder and in a ‘flat plate’ boundary layer. These have been supplemented by measurements of correlations with separations in directions other than the axial ones. In the wake, the correlations at large values of the separation are explained in terms of two types of large scale motion. One of these is a pair of vortices, side by side and rotating in opposite directions with axes aligned approximately normal to the plane of the wake. The other typical motion is a series of jets in which turbulent fluid is projected outward from the core of the wake. It is suggested that these are the result of an instability of the turbulent shear stress. A qualitative explanation of the apparent structural equilibrium of the wake is given in terms of this instability. The vortex pair eddies were not found in the boundary layer but there is evidence of jets much like those in the wake.Correlations measured in grid turbulence have been found to be highly anisotropic and consistent with the presence of vortex pair eddies. When a plane strain was applied to grid turbulence, the effect on the correlations suggested the presence of a stress instability similar to that postulated for the wake.

Sound propagation in a fluid flowing through an attenuating duct

Abstract: A study is made of the propagation of sound in both a constant gradient shear flow and a turbulent shear flow above a flat surface. Curves are presented showing how, in the case of downstream propagation, the flow gradient tends to channel the sound energy into a narrow layer next to the wall. These results are used in estimating the effect of a flow on the attenuation of sound in a duct with absorbing side walls.

On displacement thickness

Abstract: Four alternative theoretical treatments of ‘displacement thickness’, and, generally, of the influence of boundary layers and wakes on the flow outside them, are set out, first for two-dimensional, and then for three-dimensional, laminar or turbulent, incompressible flow. They may be called the methods of ‘flow reduction’, ‘equivalent sources’, ‘velocity comparison’ and ‘mean vorticity’.The principal expression obtained for the displacement thickness δ1 in three-dimensional flow may be written if, as orthogonal coordinates (x, y) specifying position on the surface, we choose x as the velocity potential of the external flow, and y as a coordinate, constant along the external-flow streamlines, such that hy dy is the distance between (x, y) and (x, y + dy); and if also δx and δy are the streamwise and transverse ‘volume-flow thicknesses’ z is the distance from the surface, u and v are the x and y components of velocity, and u takes the value U just outside the boundary layer.

Shock waves in a dusty gas

Abstract: The plane steady decelerated flow of a dust-gas mixture is analysed in an approximate manner. The problem, which has a five-parameter family of solutions, is reduced to a form such that the analysis can be completed by the integration of a first-order non-linear differential equation and a quadrature. A few integral curves are given and the characterizing features of the flow field are discussed.

On the disturbed motion of a plane vortex sheet

Abstract: A formal solution to the initial value problem for a plane vortex sheet in an inviscid fluid is obtained by transform methods. The eigenvalue problem is investigated and the stability criterion determined. This criterion is found to be in agreement with that obtained previously by Landau (1944), Hatanaka (1949), and Pai (1954), all of whom had included spurious eigenvalues in their analyses. It is also established that supersonic disturbances may be unstable; related investigations in hydrodynamic stability have conjectured on this possibility, but the vortex sheet appears to afford the first definite example. Finally, an asymptotic approximation is developed for the displacement of a vortex sheet following a suddenly imposed, spatially periodic velocity.

The stability of a shear layer in an unbounded heterogeneous inviscid fluid

Abstract: This paper considers the stabilizing effect of density stratification on the horizontal shear layer between two parallel streams of uniform velocities. A simple continuous velocity distribution, . The relation between these results and Goldstein's derivation of the same critical Richardson number for a flow with discontinuous velocity and density gradients is discussed.

On the melting of a semi-infinite body of ice placed in a hot stream of air

Abstract: A simple mathematical model is proposed to describe the steady melting of a body of ice which presents a plane surface transverse to a stream of hot air; the temperature of the air is such that vaporization does not occur.The analysis takes into account the convection of heat away from the surface by the water released in melting and the results show that the rate of transfer of heat to the body and thus the rate of melting, is reduced by as much as 46% by this convection.Simple approximate expressions are obtained for the rate of melting, the thickness of the water layer, and the thickness of the thermal boundary layer in the ice, in terms of a basic parameter S which can be calculated in terms of known quantities. These results are compared with those obtained by a separate Pohlhausen calculation and are found to be in good agreement.It is also shown that there exists a thermal boundary layer, in the body, of thickness much greater than that of the boundary layer in the air, in which the temperature changes rapidly from its value at the melting surface to its value in the far interior.

Surface waves on water of non-uniform depth

Abstract: Gravity waves occur on the surface of a liquid such as water, and the manner in which they propagate depends upon its depth. Although this dependence is described in principle by the equations of the ‘exact linear theory’ of surface waves, these equations have not been solved except in some special cases. Therefore, oceanographers have been unable to use the theory to describe surface wave propagation in water whose depth varies in a general way. Instead they have employed a simplified geometrical optics theory for this purpose (see, for example, Sverdrup & Munk (1944)). It has been used very successfully, and consequently various attempts, only partially successful, have been made to deduce it from the exact linear theory. It is the purpose of this article to present a derivation which appears to be satisfactory and which also yields corrections to the geometrical optics theory.

Non-equilibrium flow of an ideal dissociating gas

Abstract: The theory of an ‘ideal dissociating’ gas developed by Lighthill (1957) for conditions of thermodynamic equilibrium is extended to non-equilibrium conditions by postulating a simple rate equation for the dissociation process (including the effects of recombination). This equation contains the ‘equilibrium’ parameter of the Lighthill theory plus a further ‘non-equilibrium’ parameter which determines the time scale of the dissociation phenomena.The behaviour of this gas is investigated in flow through a strong normal shock wave and past a bluff body. The assumption is made that the gas receives complete excitation of its rotational and vibrational degrees of freedom in an infinitesimally thin region according to the familiar Rankine-Hugoniot shock wave relations before dissociation begins. The variation of the relevant thermodynamic variables downstream of this region is then computed in a few particular cases. The method used in the latter case is an extension of the ‘Newtonian’ theory of hypersonic inviscid flow. In particular, the case of a sphere is treated in some detail. The variation of the shock shape and the ‘stand-off’ distance with the coefficient Λ, which is the ratio of the sphere diameter to the length scale of the dissociation process, is exhibited for conditions extending from completely undissociated flow to dissociated flow in thermal equilibrium. Results would indicate that significant and observable changes from the undissociated values occur, although values for the non-equilibrium parameter are not, at present, available.

Slow viscous flow past a sphere in a cylindrical tube

Abstract: A theoretical treatment is presented for the slow flow of a viscous fluid through a cylindrical container within which a small spherical particle is confined. The sphere is situated in an arbitrary position within the cylinder and moves at constant velocity parallel to the walls. Approximate expressions are derived which give the frictional drag, rotational couple, and permanent pressure drop caused by the presence of this obstacle in the original Poiseuillian field of flow. The primary parameters involved are the ratio of sphere to cylinder radius and fractional distance of the particle from the longitudinal axis of the cylinder. With appropriate modifications, the results are also applicable to a sphere settling in a quiescent fluid. This yields the necessary boundary corrections to Stokes law arising in connection with devices such as the falling ball viscometer when the sphere is eccentrically located.

Influence of diaphragm opening time on shock-tube flows

Abstract: Shock waves generated in a shock tube by use of hydrogen or helium as a driver gas and air, nitrogen, oxygen or argon as a driven gas have higher velocities than predicted by simple theory when sufficiently large diaphragm pressure ratios are used. Expected shock-tube performance curves have been constructed using the equilibrium Hugoniot for the driven gas, both for the usual model of shock tube flow, which assumes instantaneous diaphragm removal, and for a suggested model based on a finite rupture time for the diaphragm. Agreement between experiment and the latter model is in general good, and the differences are qualitatively accounted for by the pressure waves expected to result from mixing between driver and driven gases at the contact surface. These waves may be either compression or expansion waves, depending on the relative heat capacities of the two gases. The maximum shock strength observed as a shock goes down the tube was found to occur at a distance from the diaphragm which increases with the shock strength, and the strongest shocks were found to be still accelerating at the end of a 42 ft. long shock tube of 3 1/2 in. square cross-section. Diaphragm breaking time has been measured and found to be consistent with the observations on the shock formation distance.

Calculations of unsteady viscous flow past a circular cylinder

Abstract: A numerical solution has been obtained for the starting flow of a viscous fluid past a circular cylinder at Reynolds numbers 40 and 100 The method used is the step-by-step forward integration in time of Helmholtz's vorticity equation The advantage of working with the vorticity is that calculations can be confined to the region of non-zero vorticity near the cylinderThe general features of the flow, including the formation of the eddies attached to the rear of the cylinder, have been determined, and the drag has been calculated At R = 40 the drag on the cylinder decreases with time to a value very near that for the steady flow

Turbulent flow in a stably stratified atmosphere

Abstract: Fluctuations of velocity and temperature which occur in a turbulent flow in a stably-stratified atmosphere far from restraining boundaries are discussed using the equations for the turbulent intensity and for the mean square temperature fluctuation. From these, an equation is derived for the flux Richardson number in terms of the ordinary Richardson number and some non-dimensional ratios connected with the turbulent motion. It is shown that the interaction between the temperature and velocity fields imposes on the flux Richardson number an upper limit of 0·5, and on the ordinary Richardson number a limit of about 0·08. If these values are exceeded, no equilibrium value of the turbulent intensity can exist and a collapse of the turbulent motion would occur. Although the analysis applies strictly only to a homogeneous non-developing flow, it should have approximate validity for effectively homogeneous, developing flows, and the predictions are compared with some recent observations of these flows.

Further space-time correlations of velocity in a turbulent boundary layer

Abstract: This paper describes the results of further experimental investigation of the turbulent boundary layer with zero pressure gradient. Measurements of autocorrelation and of space-time double correlation have been made respectively with single hot-wires and with two hot-wires with the separation vector in any direction. Space-time correlations reach a maximum for some optimum delay. In the case of two points set on a line orthogonal to the plate, the optimum delay Ti is not zero. In the general case it is equal to the corresponding delay Ti, increased by compensating delay for translation with the mean flow. Taylor's hypothesis may be applied to the boundary layer at distances from the wall greater than 3% of the layer thickness. Space-time isocorrelation surfaces obtained with optimum delay have a large aspect ratio in the mean flow direction, even if they are relative to a point close to the wall (0·03δ); the correlations along the mean flow then retain high values on account of the large scale of the turbulence.

The minimum Reynolds number for a turbulent boundary layer and the selection of a transition device

Abstract: In the case of turbulent flow in a pipe there is a lower experimental number to the Reynolds limit for which fully developed turbulent flow occurs. From the similarity and close agreement of the curves showing the coefficient of skin friction cf as a function of the Reynolds number Rθ (based on the momentum thickness θ) for the circular pipe and flat plate, it is suggested that there should be a lower limit to Rθ for fully developed turbulent flow on a flat plate. Rather limited experimental data confirm this and place the lower limit at Rθ = 320. The choice and size of transition device is examined in relation to this minimum Rθ and an approximate theory leads to a ‘wire’ Reynolds number in fair agreement with experience.

Flow in shock tubes with area change at the diaphragm section

Abstract: This paper describes theoretical and experimental studies of the effects on shock tube flows of a monotonic convergence at the diaphragm section. Systematic flow equations are developed for tubes of uniform bore and tubes having either a monotonic convergence or a convergence-divergence in the diaphragm section. Except across the shock front itself, isentropic processes and ideal-gas behaviour have been assumed. Simplified procedures are presented for predicting the ideal-flow parameters over a wide range of operating conditions, as well as for comparing straight and convergent tubes. Such comparisons made by other investigators are found to be incomplete or in error. The experiments described utilize a very simple device for altering the diaphragm section convergence and a multi-station measurement of shock velocity. The expected effect of convergence is verified over a wide range of Mach numbers. Even at Mach numbers where the processes of shock formation can no longer be ignored, it is found that the relative performance between a uniform and convergent tube is preserved.

The stability of a two-dimensional laminar jet

Abstract: This paper deals with the stability of a two-dimensional laminar jet against the infinitesimal antisymmetric disturbance. The curve of the neutral stability in the (α, R)-plane (α, the wave-number; R, Reynolds number) is calculated using two different methods for the different parts of the curve; the solution is developed in powers of (αR)−1 for obtaining the upper branch of the curve and in powers of αR for the lower branch.The asymptotic behaviour of these branches is that for branch I, for α → 0. Some discussion is given on the validity of the basic assumption of the stability theory in relation to the numerical result obtained here.

A model of supersonic flow past blunt axisymmetric bodies, with application to Chester's solution

Abstract: A simple approximate model is set forth for the flow field between the nose of a blunt body of revolution and its detached shock wave. The model tends to explain the poor convergence of Chester's solution, which is based on an improvement of the Newtonian approximation. It suggests a modification of his series for the body shape which appears to improve its convergence considerably.

Effects of compressibility on a laminar wall jet

Abstract: The effects of compressibility on a radial laminar wall jet are investigated. On the assumption that the coefficient of viscosity is proportional to the temperature, it is shown that a similarity solution for the velocity distribution exists, which is expressible directly in terms of the corresponding solution for an incompressible wall jet. For arbitrary Prandtl number the energy equation is studied in detail and solutions are obtained for a variety of temperature conditions.

The effects of a sound wave on the compressible boundary layer on a flat plate

Abstract: The boundary layer on a hot flat plate which is fixed at zero incidence in a slow stream carrying a progressive sound wave is investigated. Formulae are obtained for the skin friction and the heat transfer in the extreme cases when the frequency is very low and very high. In addition, different methods of simplifying the boundary layer equations in unsteady compressible flow are briefly compared.