Showing papers in "Journal of Fluid Mechanics in 1967"

Boundary conditions at a naturally permeable wall

Abstract: Experiments giving the mass efflux of a Poiseuille flow over a naturally permeable block are reported. The efflux is greatly enhanced over the value it would have if the block were impermeable, indicating the presence of a boundary layer in the block. The velocity presumably changes across this layer from its (statistically average) Darcy value to some slip value immediately outside the permeable block. A simple theory based on replacing the effect of the boundary layer with a slip velocity proportional to the exterior velocity gradient is proposed and shown to be in reasonable agreement with experimental results.

The structure of turbulent boundary layers

Abstract: Extensive visual and quantitative studies of turbulent boundary layers are described. Visual studies reveal the presence of surprisingly well-organized spatially and temporally dependent motions within the so-called ‘laminar sublayer’. These motions lead to the formation of low-speed streaks in the region very near the wall. The streaks interact with the outer portions of the flow through a process of gradual ‘lift-up’, then sudden oscillation, bursting, and ejection. It is felt that these processes play a dominant role in the production of new turbulence and the transport of turbulence within the boundary layer on smooth walls.Quantitative data are presented providing an association of the observed structure features with the accepted ‘regions’ of the boundary layer in non-dimensional co-ordinates; these data include zero, negative and positive pressure gradients on smooth walls. Instantaneous spanwise velocity profiles for the inner layers are given, and dimensionless correlations for mean streak-spacing and break-up frequency are presented.Tentative mechanisms for formation and break-up of the low-speed streaks are proposed, and other evidence regarding the implications and importance of the streak structure in turbulent boundary layers is reviewed.

The disintegration of wave trains on deep water Part 1. Theory

Abstract: The phenomenon in question arises when a periodic progressive wave train with fundamental frequency ω is formed on deep water—say by radiation from an oscillating paddle—and there are also present residual wave motions at adjacent side-band frequencies ω(1 ± δ), such as would be generated if the movement of the paddle suffered a slight modulation at low frequency. In consequence of coupling through the non-linear boundary conditions at the free surface, energy is then transferred from the primary motion to the side bands at a rate that, as will be shown herein, can increase exponentially as the interaction proceeds. The result is that the wave train becomes highly irregular far from its origin, even when the departures from periodicity are scarcely detectable at the start.In this paper a theoretical investigation is made into the stability of periodic wave trains to small disturbances in the form of a pair of side-band modes, and Part 2 which will follow is an account of some experimental observations in accord with the present predictions. The main conclusion of the theory is that infinitesimal disturbances of the type considered will undergo unbounded magnification if \[ 0 < \delta \leqslant (\sqrt{2})ka, \] where k and a are the fundamental wave-number and amplitude of the perturbed wave train. The asymptotic rate of growth is a maximum for δ = ka.

Long waves on a beach

Abstract: Equations of motion are derived for long waves in water of varying depth. The equations are for small amplitude waves, but do include non-linear terms. They correspond to the Boussinesq equations for water of constant depth. Solutions have been calculated numerically for a solitary wave on a beach of uniform slope. These solutions include a reflected wave, which is also derived analytically by using the linearized long-wave equations.

Internal waves of permanent form in fluids of great depth

Abstract: This paper presents a general theoretical treatment of a new class of long stationary waves with finite amplitude. As the property in common amongst physical systems capable of manifesting these waves, the density of the (incompressible) fluid varies only within a layer whose thickness h is much smaller than the total depth, and it is h rather than the total depth that must be considered as the fundamental scale against which wave amplitude and length are to be measured. Internal-wave motions supported by the oceanic thermocline appear to be the most promising field of practical application for the theory, although applications to atmospheric studies are also a possibility.The waves in question differ in important respects from long waves of more familiar kinds, and in § 1 their character is discussed on the basis of a comparison with solitary-wave and cnoidal-wave theories on customary lines, such as apply to internal waves in fluids of limited depth. A summary of some simple experiments is included at the end of § 1. Then, in § 2, the comparatively easy example of two-fluid systems is examined, again to illustrate principles and to prepare the way for the main analysis in § 3. This proceeds to a second stage of approximation in powers of wave amplitude, and its leading result is an equation (3·51) determining, for arbitrary specifications of the density distribution, the form of the streamlines in the layer of heterogeneous fluid. Periodic solutions of this equation are obtained, and their properties are discussed with regard to the interpretation of internal bores and wave-resistance phenomena. Solutions representing solitary waves are then obtained in the form \[ f(x) = a\lambda^2/(x^2+\lambda^2), \] where xis the horizontal co-ordinate and where aΛ = O(h2). (The latter relation between wave amplitude and length scale contrasts with the customary one, aΛ2 = O(h3)). The main analysis is developed with particular reference to systems in which the heterogeneous layer lies on a rigid horizontal bottom below an infinite expanse of homogeneous fluid; but in § 4 ways are given to apply the results to various other systems, including ones in which the heterogeneous layer is uppermost and is bounded by a free surface. Finally, in §5, three specific examples are treated: the density variation with depth is taken, respectively, to have a discontinuous, an exponential and a ‘tanh’ profile.

The critical layer for internal gravity waves in a shear flow

Abstract: Internal gravity waves of small amplitude propagate in a Boussinesq inviscid, adiabatic liquid in which the mean horizontal velocity U(z) depends on height z only. If the Richardson number R is everywhere larger than 1/4, the waves are attenuated by a factor as they pass through a critical level at which U is equal to the horizontal phase speed, and momentum is transferred to the mean flow there. This effect is considered in relation to lee waves in the airflow over a mountain, and in relation to transient localized disturbances. It is significant in considering the propagation of gravity waves from the troposphere to the ionosphere, and possibly in transferring horizontal momentum into the deep ocean without substantial mixing.

Calculation of boundary-layer development using the turbulent energy equation

Abstract: The turbulent energy equation is converted into a differential equation for the turbulent shear stress by defining three empirical functions relating the turbulent intensity, diffusion and dissipation to the shear stress profile. This equation, the mean momentum equation and the mean continuity equation form a hyperbolic system. Numerical integrations by the method of characteristics with preliminary choices of the three empirical functions compare favourably with the results of conventional calculation methods over a wide range of pressure gradients. Nearly all the empirical information required has been derived solely from the boundary layer in zero pressure gradient.

Instability due to viscosity stratification

Abstract: The principal aim of this paper is to show that the variation of viscosity in a fluid can cause instability. Plane Couette-Poiseuille flow of two superposed layers of fluids of different viscosities between two horizontal plates is considered, and it is found that both plane Poiseuille flow and plane Couette flow can be unstable, however small the Reynolds number is. The unstable modes are in the neighbourhood of a hidden neutral mode for the case of a single fluid, which is entirely ignored in the usual theory of hydrodynamic stability, and are brought out by the viscosity stratification.

Finite amplitude convective cells and continental drift

Abstract: A solution is obtained for steady, cellular convection when the Rayleigh number and the Prandtl number are large. The core of each two-dimensional cell contains a highly viscous, isothermal flow. Adjacent to the horizontal boundaries are thin thermal boundary layers. On the vertical boundaries between cells thin thermal plumes drive the viscous flow. The non-dimensional velocities and heat transfer between the horizontal boundaries are found to be functions only of the Rayleigh number. The theory is used to test the hypothesis of large scale convective cells in the earth's mantle. Using accepted values of the Rayleigh number for the earth's mantle the theory predicts the generally accepted velocity associated with continental drift. The theory also predicts values for the heat flux to the earth's surface which are in good agreement with measurements carried out on the ocean floors.

A variational principle for a fluid with a free surface

Abstract: The full set of equations of motion for the classical water wave problem in Eulerian co-ordinates is obtained from a Lagrangian function which equals the pressure. This Lagrangian is compared with the more usual expression formed from kinetic minus potential energy.

The interfacial stability of a ferromagnetic fluid

Abstract: Magnetization critical level derived for instability onset for ferromagnetic fluid having nonlinear relation with magnetic induction

The stability of finite amplitude cellular convection and its relation to an extremum principle

Abstract: The stability of cellular convection flow in a layer heated from below is discussed for Rayleigh number R close to the critical value Rc. It is shown that in this region the stable stationary solution is determined by a minimum of the integral \[ \int_0^{H_0}R(H)\,dH, \] where R(H) is a functional of arbitrary convective velocity fields which satisfy the boundary conditions. For the stationary solutions R(H) is equal to the Rayleigh number. H0 is a given value of the convective heat transport. In a second part of the paper explicit results are derived for the convection problem with deviations from the Boussinesq approximation owing to the temperature dependence of the material properties.

Steady flows in rectangular cavities

Abstract: This paper deals with the steady flow in a rectangular cavity where the motion is driven by the uniform translation of the top wall. Creeping flow solutions for cavities having aspect ratios from ¼ to 5 were obtained numerically by a relaxation technique and were shown to compare favourably with Dean & Montagnon's (1949) similarity solution, as extended by Moffatt (1964), in the region near the bottom corners of a square cavity as well as throughout the major portion of a cavity with aspect ratio equal to 5. In addition, for a Reynolds number range from 20 to 4000, flow patterns were determined experimentally by means of a photographic technique for finite cavities, as well as for cavities of effectively infinite depth. These experimental results suggest that, within finite cavities, the high Reynolds number steady flow should consist essentially of a single inviscid core of uniform vorticity with viscous effects being confined to thin shear layers near the boundaries, while, for cavities of infinite depth, the viscous and inertia forces should remain of comparable magnitude throughout the whole domain even in the limit of very large Reynolds number R.

Steady free convection in a porous medium heated from below

Abstract: This is an experimental and numerical study of steady free convection in a porous medium, a system dominated by a single non-linear process, the advection of heat. The paper presents results on three topics: (1) a system uniformly heated from below, for which the flow is cellular, as in the analogous Benard-Rayleigh flows, (ii) the role of end-effects, and (iii) the role of mass discharge. Measurements of heat transfer are used to establish further the validity of the numerical scheme proposed by the author (1966a), while the other flows allow a more extensive study of the numerical scheme under various boundary conditions. The results are very satisfactory even though only moderately non-linear problems can be treated at present.The main new results are as follows. For the Rayleigh-type flow, above a critical Rayleigh number of about 40, the heat transferred across the layer is proportional to the square of the temperature difference across the layer and is independent of the thermal conductivity of the medium or the depth of the layer. This result is modified when the boundary-layer thickness is comparable to the grain size of the medium. The investigation of end-effects reveals variations in horizontal wave-number and a pronounced hysteresis and suggests an alternative explanation of some observations by Malkus (1954).

The vortex wakes of vibrating cylinders at low Reynolds numbers

Abstract: The effect of the transverse motion of a cylinder on its natural vortex wake is examined at various driving frequencies giving special attention to the change in wake geometry along the span. Conditions are established for which the vortex wake frequency is controlled by the driving frequency of the cylinder.

On finite amplitude oscillations in laminar mixing layers

Abstract: In the first part of the paper, a mixing layer of tanh y form is considered, and twodimensional solutions of the non-linear inviscid equations are found representing periodic perturbations from the neutral wave of linearized stability theory. To second order in amplitude the solutions are equivalent to the equilibrium state calculated by Schade (1964), who studied the development of perturbations in time and found an evolution towards that equilibrium state. The present calculation, however, is taken to fourth-order in amplitude. It is noted that the solutions presented in this paper are regular, even though viscosity is ignored; and the relationships to the singular (if inviscid) time-dependent solutions of Schade are explained. Such regular, inviscid solutions have been found only for odd velocity profiles, such as tanh y.Although the details of the velocity distributions are not of the form observed experimentally, it is shown that the amplitude ratios of fundamental and first harmonic, for a given absolute amplitude, are comparable with those observed.In part 2 some exact non-linear solutions are presented of the inviscid, incompressible equations of fluid flow in two or three spatial dimensions. They illustrate the flows of part 1, since they are periodic in one co-ordinate (x), have a shear in another (y) and are independent of the third. Included, as two-dimensional cases, are (i) the tanh y velocity distribution for a flow wholly in the x-direction, (ii) the well-known solution for the flow due to a set of point vortices equi-spaced on the axis, and (iii) an example of linearized hydrodynamic (Orr-Sommerfeld) stability theory. The flows may involve concentrations of vorticity. In three-dimensional cases the z component of velocity is even in y, whereas the x component is odd. Consequently, the class of flows represents, in general, small or large periodic perturbations from a skewed shear layer. Time-dependent solutions, representing waves travelling in the x direction may be obtained by translation of axes.

‘Inactive’ motion and pressure fluctuations in turbulent boundary layers

Abstract: Townsend's (1961) hypothesis that the turbulent motion in the inner region of a boundary layer consists of (i) an ‘active’ part which produces the shear stress τ and whose statistical properties are universal functions of τ and y, and (ii) an ‘inactive’ and effectively irrotational part determined by the turbulence in the outer layer, is supported in the present paper by measurements of frequency spectra in a strongly retarded boundary layer, in which the ‘inactive’ motion is particularly intense. The only noticeable effect of the inactive motion is an increased dissipation of kinetic energy into heat in the viscous sublayer, supplied by turbulent energy diffusion from the outer layer towards the surface. The required diffusion is of the right order of magnitude to explain the non-universal values of the triple products measured near the surface, which can therefore be reconciled with universality of the ‘active’ motion.Dimensional analysis shows that the contribution of the ‘active’ inner layer motion to the one-dimensional wave-number spectrum of the surface pressure fluctuations varies as τ2w/k1 up to a wave-number inversely proportional to the thickness of the viscous sublayer. This result is strongly supported by the recent measurements of Hodgson (1967), made with a much smaller ratio of microphone diameter to boundary-layer thickness than has been achieved previously. The disagreement of the result with most other measurements is attributed to inadequate transducer resolution in the other experiments.

On the oscillatory instability of a differentially heated fluid loop

Abstract: A theoretical discussion is given of the motion of a fluid contained in a tube forming a closed loop that is heated from below and cooled from above. The fluid is assumed to have uniform temperature over each cross-section, and the heat transfer is assumed proportional to the difference between the local temperatures of the fluid and the tube. The latter temperature is prescribed. The system has one steady solution with warm fluid rising in one branch and cold fluid sinking in the other. This solution may, however, become unstable in an oscillatory manner. A weak instability takes the form of pulsations, the motion being always of one sign, while a strong instability takes the form of oscillations with zero mean motion. These oscillations are irregular and do not repeat themselves even over very long times.These unstable motions are associated with thermal anomalies in the fluid that are advected materially around the loop. The anomalies amplify through the correlated variations in flow rate. A warm pocket of fluid creates maximum flow rate going through the upper part and minimum flow rate going through the lower part of the loop. Accordingly it passes quicker through the heat sink than through the heat source, and the latter becomes more effective. Similarly, the heat sink acts more effectively on a cold pocket of fluid.The curve of neutral stability is worked out as a function of the two parameters of the problem, a non-dimensional gravity and a non-dimensional friction coefficient. The instability has also been studied by direct numerical time integration of the model equations.It is suggested that the mechanism of instability found for this model operates also in more complicated systems, and can explain the pulsative type of motions observed recently in certain convection experiments.

Solitary internal waves in deep water

Abstract: A new type of solitary wave motion in incompressible fluids of non-uniform density has been investigated experimentally and theoretically. If a fluid is stratified in such a manner that there are two layers of different density joined by a thin region in which the density varies continuously, this type of wave propagates along the density gradient region without change of shape. In contrast to previously known solitary waves, these disturbances can exist even if the fluid depth is infinite. The waves are described by an approximate solution of the inviscid equations of motion. The analysis, which is based on the assumption that the wavelength of the disturbance is large compared with the thickness, L, of the region in which the density is not constant, indicates that the propagation velocity, U, is characterized by the dimensionless group (gL/U2) In (ρ1/ρ2), where g is the gravitational acceleration and ρ is the density. The value of this group, which is dependent on the wave amplitude and the form of the density gradient, is of the order one. Experimentally determined propagation velocities and wave shapes serve to verify the theoretical model.

Transient convection in a porous medium

Abstract: Laboratory and numerical experiments on non-steady convective flows in a porous medium are reported. The main objective is to note the detailed comparison found between the time-dependent solutions and the time-like development of the iterative solutions of the steady equations originally pointed out by Garabedian (1956) and others.Two flows are chosen for study. The first is the flow which develops when a blob of hot fluid is released at the base of a porous slab. The second is the flow which develops when a portion of the base of a porous slab is suddenly heated. The former flow is very simple and ideally suited for establishing the numerical scheme. The latter flow, however, produces several unexpected features. The gross features of the time development, when the motion is strongly non-linear, show an alternation between periods of slow gradual adjustment and periods of rapid change.

The turbulence structure of equilibrium boundary layers

Abstract: Measurements in three boundary layers, one with constant free-stream velocity and two with power-law variations of free-stream velocity giving ‘moderate’ and ‘strong’ adverse pressure gradients, are presented and discussed. Several unifying features of the turbulent motion, expected to appear in all boundary layers not too far from equilibrium, are identified. The intensity spectra at higher wavenumbers follow the Kolmogorov inertial-subrange law, although the Reynolds number is not particularly high even by laboratory standards: in addition the smaller-scale motion in the outer layer is determined entirely by the local shear stress and the boundary-layer thickness. The large eddy motion increases in strength relative to the general turbulence level as the general turbulence level increases, and the limited evidence available suggests that the large eddies are similar to those in the free mixing layer. In all cases the large eddies contribute a significant proportion of the shear stress in the outer layer.

A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid

Abstract: Experiments were conducted to test the linear theory of internal gravity waves produced in a stably stratified liquid by the forced oscillations and the initial impulsive motion of a two-dimensional stationary disturbance. The measurements of the wave configuration in a medium whose density increased linearly with depth were made by means of a Toepler-schlieren system. The agreement between observation and prediction was found to be good.

Non-linear dispersion of water waves

Abstract: The slow dispersion of non-linear water waves is studied by the general theory developed in an earlier paper (Whitham 1965b). The average Lagrangian is calculated from the Stokes expansion for periodic wave trains in water of arbitrary depth. This Lagrangian can be used for the various applications described in the above reference. In this paper, the crucial question of the ‘type’ of the differential equations for the wave-train parameters (local amplitude, wave-number, etc.) is established. The equations are hyperbolic or elliptic according to whether kh0 is less than or greater than 1.36, where k is the wave-number per 2π and h0 is the undisturbed depth. In the hyperbolic case, changes in the wave train propagate and the characteristic velocities give generalizations of the linear group velocity. In the elliptic case, modulations in the wave train grow exponentially and a periodic wave train will be unstable in this sense; thus, periodic wave trains on water will be unstable if kh0 > 1·36, The instability of deep-water waves, kh0 > 1·36, was discovered in a different way by Benjamin (1966). The relation between the two approaches is explained.

The large-scale structure of homogeneous turbulence

Abstract: A field of homogeneous turbulence generated at an initial instant by a distribution of random impulsive forces is considered. The statistical properties of the forces are assumed to be such that the integral moments of the cumulants of the force system all exist. The motion generated has the property that at the initial instant E(kappa) = Ckappa^2 + o(kappa^2) where E(k) is the energy spectrum function, k is the wave-number magnitude, and C is a positive number which is not in general zero. The corresponding forms of the velocity covariance spectral tensor and correlation tensor are determined. It is found that the terms in the velocity covariance Rij(r) are O(r^−3) for large values of the separation magnitude r. An argument based on the conservation of momentum is used to show that C is a dynamical invariant and that the forms of the velocity covariance at large separation and the spectral tensor at small wave number are likewise invariant. For isotropic turbulence, the Loitsianski integral diverges but the integral \[ \int_0^{\infty} r^2R(r)dr = \frac{1}{2}\pi C \] exists and is invariant.

The Toms phenomenon: turbulent pipe flow of dilute polymer solutions

Abstract: Thesis (Sc. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1967.

Wall-pressure fluctuations associated with subsonic turbulent boundary layer flow

Abstract: Experimental results are given for various statistical properties of the fluctuating wall-pressure field associated with a subsonic turbulent equilibrium boundary layer, developed on a smooth wind tunnel wall after natural transition from laminar to turbulent flow. The statistical quantities of the wall-pressure field investigated were root-mean-square pressure, frequency power spectrum and space-time correlations. Space-time correlation measurements were made in both broad and narrow frequency bands. The experiments were made at flow Mach numbers of 0·3 and 0·5 and covered a Reynolds number range of about 5 to 1. The main conclusion to which the measurements lead is that the wall-pressure field has a structure produced by contributions from pressure sources in the boundary layer with a wide range of convection velocities, and comprises two families of convected wave-number components. One family is of high wave-number components and is associated with turbulent motion in the constant stress layer; the components are longitudinally coherent for times proportional to the times taken for them to be convected distances equal to their wavelengths and laterally coherent over distances proportional to their wavelengths. The other family comprises components of wavelength greater than about twice the boundary-layer thickness, which lose coherence as a group more or less independently of wavelength and are associated with large-scale eddy motion in the boundary layer, outside the constant stress layer. The evolution of the pressure field is discussed in terms of these two wave-number families.

Convection in a box: linear theory

Abstract: The linear stability of a quiescent, three-dimensional rectangular box of fluid heated from below is considered. It is found that finite rolls (cells with two non-zero velocity components dependent on all three spatial variables) with axes parallel to the shorter side are predicted. When the depth is the shortest dimension, the cross-sections of these finite rolls are near-square, but otherwise (in wafer-shaped boxes) narrower cells appear. The value of the critical Rayleigh number and preferred wave-number (number of finite rolls) for a given size box is determined for boxes with horizontal dimensions h, ¼ ≤ h/d ≤ 6, where d is the depth.

Convection in horizontal layers with internal heat generation. Theory

Abstract: A theoretical study has been made of an experiment by Tritton & Zarraga (1967) in which eonvective motions were generated in a horizontal layer of water (cooled from above) by the application of uniform heating. The marginal stability problem for such a layer is solved, and a critical Rayleigh number of 2772 is obtained, at which patterns of wave-number 2·63 times the reciprocal depth of the layer are marginally stable. The remainder of the paper is devoted to the finite amplitude convection which ensues when the Rayleigh number, R , exceeds 2772. The theory is approximate, the basic simplification being that, to an adequate approximation, Fourier decompositions of the convective motions in the horizontal ( x, y ) directions can be represented by their dominant (planform) terms alone. A discussion is given of this hypothesis, with illustrations drawn from the (better studied) Benard situation of convection in a layer heated below, cooled from above, and containing no heat sources. The hypothesis is then used to obtain ‘mean-field equations’ for the convection. These admit solutions of at least three distinct forms: rolls, hexagons with upward flow at their centres, and hexagons with downward flow at their centres. Using the hypothesis again, the stability of these three solutions is examined. It is shown that, for all R , a (neutrally) stable form of convection exists in the form of rolls. The wave-number of this pattern increases gradually with R. This solution is, in all respects, independent of Prandtl number. It is found, numerically, that the hexagons with upward motions in their centres are unstable, but that the hexagons with downward motions at their centres are completely stable, provided R exceeds a critical value (which depends on Prandtl number, P , and which for water is about 3 R c ), and provided the wave-number of the pattern lies within certain limits dependent on R and P .

The nozzle-fluid concentration field of the round, turbulent, free jet

Abstract: The light-scatter technique has been used to study the nozzle-fluid concentration field in an isothermal, turbulent, axisymmetric air/air free jet with the nozzle air marked by an oil smoke. The data on the mean concentration field appear to be the most accurate yet obtained, due to the peculiar advantages of the technique. The turbulent concentration fluctuations have been characterized as to intensity, spectral distribution, and two-point correlation. The intermittency factor has been measured and the properties of the turbulent fluid computed. Comparison with the results of other investigators who used heat to mark the nozzle fluid indicates a close similarity between the concentration and temperature fields.

Computational and experimental study of a captive annular eddy

Abstract: Results of calculations and experiments on the flow of a viscous liquid through an axisymmetric conduit expansion are reported. The streamlines and vorticity contours are presented as functions of the Reynolds number of the flow. The dynamic interaction between the main flow and the captive eddy between it and the walls is analysed, and it is concluded that, for laminar flow, the main role of the eddy is that of shaping the flow with a rather small energy exchange.