bluuebunny/crossref_metadata_2025 huggingface dataset:
Wave resistance to vertical motion in a stratified fluid
Authors: Warren F. W. G.
Published: March 2006

When a body moves through a stratified fluid, i.e. one whose density decreases upwards, gravity waves are set up and this causes a resistance to motion. An axisymmetric case is considered in which a body moves steadily and vertically through a fluid whose density decreases exponentially upwards. The fluid is supposed perfect, incompressible, and unbounded in all directions. The equations of motion are linearized, and with a fairly general initial motion of the surrounding fluid, the limit of the solution as t → ∞ is evaluated. Transform methods are used to solve the equation of motion, and the methods of steepest descents and stationary phase are used to obtain approximate solutions.

Streamlines and the distortion of the constant density levels for a spindle-shaped body are shown. The curves of resistance against a function of the velocity for the circular cylinder, the sphere, and a spindle-shaped body are also given. A criterion is given for when the maximum wave resistance for a sphere may be expected, and an estimate of this maximum resistance is made.
Force-momentum fields in a dual-jet flow
Authors: Miller David R. , Comings Edward W.
Published: March 2006

Measurements of mean velocity, mean flow direction, normal turbulent stress in the direction of flow, and mean static pressure are reported for the subsonic flow field generated by identical twin jets of air issuing from parallel slot nozzles in a common wall and mixing turbulently with ambient room air. At the low nozzle velocity employed (72ft./sec), the two-dimensional plano-symmetric flow was effectively incompressible. Since the end walls prevented interjet air entrainment from the surroundings, a region of highly convergent flow was formed near the nozzles. In this region, contour maps clearly reveal (1) the sub-atmospheric static pressure through that accounts for the jet convergence, (2) a free stagnation point on the plane of symmetry, (3) stable symmetrical contrarotary vortices which recycle air on the concave side of each converging jet, and (4) the super-atmospheric static pressure mound that redirects the merging jet streams in a common downstream direction. Comparisons are made between the development of the flow, in both the region of jet convergence and the region of combined jet flow, and that of the single-jet counterpart which was previously reported.
The boundary layer near the stagnation point in hypersonic flow past a sphere
Authors: Herring T. K.
Published: March 2006

When a sphere or other bluff body travels at supersonic speeds, a shock wave is formed close to the front surface. With increase of speed the air behind the shock is further compressed, and the shock wave moves closer to the surface. This paper considers the case where the region close to the stagnation point between the shock and the sphere can be taken to be a steady laminar boundary layer.

The approximate solution of the equations of motion follows closely the classical work of Homann (1936), ideas similar to those of Lighthill (1957) being used to apply it to the problem in hand. It consists mainly in reducing the equations to ordinary differential form by assuming forms of the flow variables which satisfy the boundary conditions, notably at the shock wave. In addition, several transformations are employed in order to simplify the equations and to increase the range of solutions, and also to facilitate the use of the ‘Mercury’ electronic computer in solving them.

The results give an insight into some aspects of hypersonic flows. Included in this paper are a selection of temperature and transverse velocity profiles across the boundary layer and several graphs relating such quantities as the shock stand-off distance and the skin-friction coefficient with Reynolds number. The last two mentioned are the most interesting. The first set gives the surprising result that the shock stand-off distance increases with increase in Reynolds number, whereas it is known that the skin-friction coefficient is inversely proportional to a decreasing power of the Reynolds number when it is lower than order 103, but the indication is that it tends to the expected constant power of ½ when the Reynolds number is of order 104.
A presentation of cnoidal wave theory for practical application
Authors: Wiegel R. L.
Published: March 2006

Cnoidal wave theory is appropriate to periodic waves progressing in water whose depth is less than about one-tenth the wavelength. The leading results of existing theories are modified and given in a more practical form, and the graphs necessary to their use by engineers are presented. As well as results for the wave celerity and shape, expressions and graphs for the water particle velocity and local acceleration fields are given. A few comparisons between theory and laboratory measurements are included.
Motion of a bore over a sloping beach
Authors: Keller H. B. , Levine D. A. , Whitham G. B.
Published: March 2006

The results of numerical calculations are presented for the motion of a bore over a uniformly sloping beach. The shallow water equations are solved in finite difference form, and a technique is developed for fitting in the bore at each step. The results are compared with the approximate formula given by Whitham (1958) and close agreement is found. The approximate theory is considered further here; the main addition is a rigorous proof that, within the shallow water theory, the height of the bore always tends to zero at the shoreline.
Slow steady rotation of axially symmetric bodies in a viscous fluid
Authors: Kanwal R. P.
Published: March 2006

The Stokes flow problem is considered here for the case in which an axially symmetric body is uniformly rotating about its axis of symmetry. Analytic solutions are presented for the heretofore unsolved cases of a spindle, a torus, a lens, and various special configurations of a lens. Formulas are derived for the angular velocity of the flow field and for the couple experienced by the body in each case.
Experiments on power generation from a moving plasma
Authors: Pain H. J. , Smy P. R.
Published: March 2006

Electrical power of 0·32MW has been extracted for a period of 100 μsec from a plasma of shock-ionized argon travelling at 4×105cm per sec through a magnetic field of 10,000 G. Currents of more than 10,000amp are drawn and the resulting modification of the applied field is considered.

Maximum power is obtained when the external load is matched to the plasma generator which has an internal impedance equivalent to its own electrical resistance in series with a resistance arising from its behaviour as a compressible fluid.

Values of the electrical conductivity of the plasma obtained in these experiments (about 3 × 103mho per metre) show that the plasma resistance is controlled by electron motion and these values are in good agreement with those found by other methods.
Gaskinetics and gasdynamics of orifice flow
Authors: Liepmann Hans W.
Published: March 2006

The paper gives the result of a study on the efflux of gases through circular apertures. The problem is considered as an example of a transition from the gasdynamic to the gaskinetic régime.

The mass flow of helium, argon and nitrogen was measured for a range of upstream pressures corresponding to (mean free path)/(aperture diameter) from about 50 to 5 × 10−3; within this range the transition from molecular effusion to inviscid, transonic flow takes place. The theory for the two asymptotic limits is discussed and first-order corrections to the free molecular and inviscid limit formulae are given.
Theory of flame-front stability
Authors: Eckhaus Wiktor
Published: March 2006

A study of the stability of a plane laminar flame front is made. The effects of disturbances on the flame structure are investigated by a small perturbations technique, taking into account the mechanism of diffusion, heat conduction and unsteady combustion. By use of a simplified model of the flame structure, and the assumption that the flame thickness is small compared with the wavelength of disturbances, a formula for the perturbation of the flame propagation velocity is derived. The flame velocity is shown to depend on the curvature of the flame, and on the rates of change of fluid velocities at the flame boundary. From stability analysis it then follows that properties of the mixture, as expressed in terms of the coefficient of heat conductivity and various coefficients of diffusion, play an important role in determining the stability picture. For some estimated values of these parameters the theoretical results are shown to agree with the general trend of the experimentally observed behaviour.
On a momentum-mass flux diagram for turbulent jets, plumes and wakes
Authors: Morton B. R.
Published: March 2006

Many salient features of fully developed turbulent jets, plumes and wakes with steady mean flow are shown clearly by the relationship between the momentum flux and the mass flux in the column of moving fluid. Using a simple model for the flows, this relationship can be found from the solution of a single ordinary differential equation and the character of many related flows can be represented immediately (except for actual distribution in space) on a single momentummass flux diagram.

In this note some approximate solutions based on dimensional arguments are outlined briefly for cases of buoyant and non-buoyant wakes and jets directed along the axis of a uniform main stream, and momentum-mass flux curves are presented.
A continuum theory of the isothermal flow of liquid helium II
Authors: Townsend A. A.
Published: March 2006

Recent work by Hall and Vinen has established that mutual friction between the normal and superfluid components of liquid helium II is caused by interactions between quantized vortex-lines and the normal fluid. If the mean separation of the vortex-lines is small compared with the channel width, the general character of the flow may not depend on the discrete nature of the lines except in so far as this is the cause of the mutual friction. Equations of motion are developed which refer to components of the velocity field with a scale large compared with the line separation, and these are used to discuss the nature of possible turbulent motions. Reasons are given for believing that isothermal flow is very similar to that of a Newtonian fluid, and the theory is developed for turbulent pressure flow along a channel and a circular pipe. The predicted variation of flow rate with pressure gradient is in good agreement with experimental measurements for Reynolds numbers (based on tube diameter and normal fluid viscosity) above 1400, and it is likely that turbulent flow can exist only above this critical Reynolds number. For Reynolds numbers which are not too small, the equations of motion apply to steady ’laminar’ flow and these lead to a relation between flow rate and pressure gradient in reasonable agreement with experiment.
Flow of a non-homogeneous fluid in a porous medium
Authors: Yih Chia-Shun
Published: March 2006

If the viscosity and specific weight of a fluid are variable, the equations governing its flow in a porous medium are non-linear and in general very difficult to solve. It has been found, however, that steady flows of a fluid of variable viscosity but constant specific weight can be reduced to those of a homogeneous fluid by a remarkably simple transformation, which indicates that the flow patterns of the fluid are the same as those of a homogeneous fluid with the same boundary conditions, and that only the speed need be modified. The speed of the actual flow is obtained by dividing the speed of the homogeneous-fluid flow by a factor proportional to the actual viscosity. The transformation is also used to derive the equations governing steady two-dimensional flows and steady axisymmetric flows of a fluid of variable viscosity and specific weight. In a good many cases of practical importance these equations are exactly linear, in spite of the fact that the governing equations obtained without the use of the above-mentioned transformation are non-linear. An exact solution for a steady two-dimensional flow with prescribed boundary conditions is given. Two inverse methods for generating exact solutions for two-dimensional flows are presented, together with two illustrative examples. The theory also applies to Hele-Shaw flows, so that it can be easily verified in the laboratory.
Deposition of a viscous fluid on the wall of a tube
Authors: Taylor G. I.
Published: March 2006

Measurements of the amount of fluid left behind when a viscous liquid is blown from an open-ended tube are described.
The motion of long bubbles in tubes
Authors: Bretherton F. P.
Published: March 2006

A long bubble of a fluid of negligible viscosity is moving steadily in a tube filled with liquid of viscosity μ at small Reynolds number, the interfacial tension being σ. The angle of contact at the wall is zero. Two related problems are treated here.

In the first the tube radius r is so small that gravitational effects are negligible, and theory shows that the speed U of the bubble exceeds the average speed of the fluid in the tube by an amount UW, where
$W \simeq 1\cdot 29(3 \mu U|\sigma)^{\frac {2}{3}}\;\;\; as\;\;\; \mu U|\sigma$
(This result is in error by no more than 10% provided $\mu U |\sigma \; \textless \;5 \times 10^{-3}\rightarrow 0$). The pressure drop, P, across such a bubble is given by
$P \simeq 3\cdot 58(3\mu U|\sigma)^{\frac {2}{3}}\sigma|r \; \; \;as\; \; \; \mu U|\sigma \rightarrow 0$
and W is uniquely determined by conditions near the leading meniscus. The interface near the rear meniscus has a wave-like appearance. This provides a partial theory of the indicator bubble commonly used to measure liquid flowrates in capillaries. A similar theory is applicable to the two-dimensional motion round a meniscus between two parallel plates. Experimental results given here for the value of W agree well neither with theory nor with previous experiments by other workers. No explanation is given for the discrepancies.

In the second problem the tube is wider, vertical, and sealed at one end. The bubble now moves under the effect of gravity, but it is shown that it will not rise at all if
$\rho gr^2| \sigma \; \textless \; 0 \cdot 842,$
where ρ is the difference in density between the fluids inside and outside the bubble. If
$0 \cdot 842 \; \textless \; 1 \cdot 04,$ then $\rho gr^2| \sigma - 0 \cdot842 \simeq 1 \cdot 25 (\mu U|\sigma)^{\frac {2}{9}} + 2 \cdot 24(\mu U|\sigma)^{\frac {1}{3}},$
accurate to within 10%. Experiments are adduced in support of these results, though there is disagreement with previous work.
The wave drag of wind over water
Authors: Stewart R. W.
Published: March 2006

It is shown to be probable that a large proportion of the drag exerted by a water surface on the wind is in the form of wave drag. As a result the usual relation between the wind profile and the surface stress and roughness length are modified. In particular, close to the surface the relation between the transport of momentum and that of heat and water vapour are different from that obtaining over a rough solid surface.
A non-linear theory for oscillations in a parallel flow
Authors: Benney D. J.
Published: March 2006

Three-dimensional periodic oscillations in the shear flow region between two parallel streams are considered up to that second order of the oscillation amplitude. It is shown that, as an integral part of the oscillation, there is a mean secondary flow in the nature of a longitudinal vortex. Despite the dissimilarity in the profile of the basic flows, several of the principal features of the calculated results can be compared with those observed for the Blasius flow by Schubauer and Klebanoff & Tidstrom at the National Bureau of Standards.
Singularities of attached shock waves in steady axially symmetric flow
Authors: Cabannes Henri , Stael Claude
Published: March 2006

An I.B.M. 704 electronic computer was used to integrate the differential equations, which determine the shock singularity, when the velocity after the shock on a body is subsonic. The results are given in twenty-two cases corresponding to three different bodies; the procedure followed was outlined in a previous paper (Cabannes 1952).
A study of the behaviour of a thin sheet of moving liquid
Authors: Brown D. R.
Published: March 2006

An investigation has been carried out on the behaviour of a thin sheet of liquid in connexion with the new method of lacquer application known as ’curtain coating’. A method is described for measuring the velocity of the sheet and an equation deduced for calculating this for a liquid of any viscosity. The minimum liquid flow rate required to maintain a stable sheet is discussed, and some effects of the impingement of the sheet on a rapidly moving surface are described.
Slow flow past ellipsoids of revolution
Authors: Breach D. R.
Published: March 2006

The results of Proudman & Pearson (1957) for a sphere are generalized to apply to all ellipsoids of revolution both prolate and oblate.
The motion of a shock-wave through a region of non-uniform density
Authors: Bird G. A.
Published: March 2006

The method of characteristics is used to calculate numerical solutions for the one-dimensional motion of a plane shock-wave through a stationary gas which contains a region of non-uniform density. These solutions are compared with those given by the Chisnell-Whitham approach which ignores the effects on the shock-wave of the disturbances which are generated in the flow behind it, and also with the asymptotic solution given by the simple theory which regards the nonuniform region as a contact-surface discontinuity. It is concluded that the results of the simplified theories must be applied with caution.
Cavity flow past a slender pointed hydrofoil
Authors: Cumberbatch E. , Wu T. Y.
Published: March 2006

A slender-body theory for the flow past a slender, pointed hydrofoil held at a small angle of attack to the flow, with a cavity on the upper surface, has been worked out. The approximate solution valid near the body is seen to be the sum of two components. The first consists of a distribution of two-dimensional sources located along the centroid line of the cavity to represent the variation of the cross-sectional area of the cavity. The second component represents the cross-flow perpendicular to the centroid line. It is found that over the cavity boundary which envelops a constant pressure region, the magnitude of the cross-flow velocity is not constant, but varies to a moderate extent. With this variation neglected only in the neighbourhood of the hydrofoil, the cross-flow is solved by adopting the Riabouchinsky model for the two-dimensional flow. The lift is then calculated by intergrating the pressure along the chord; the dependence of the lift on cavitation number and angle of attack is shown for a specific case of the triangular plan form.
A theory for the core of a leading-edge vortex
Authors: Hall M. G.
Published: March 2006

In the flow past a slender delta wing at incidence one can observe a roughly axially symmetric core of spiralling fluid, formed by the rolling-up of the shear layer that separates from a leading edge. The aim in this paper is to predict the flow field within this vortex core, given appropriate conditions at its outside edge.

The basic assumptions are (i) that the flow is continuous and rotational, and (ii) that viscous diffusion is confined to a relatively slender subcore. In addition it is assumed that the flow is axially symmetric and incompressible. Together, these admit outer and inner solutions for the core from the equations of motion. For the outer solution the subcore is ignored, and the flow is taken to be inviscid (but rotational) and conical. The resulting solution consists of simple expressions for the velocity components and pressure. For the inner solution, which applies to the diffusive subcore, the flow is taken to be laminar, and certain approximations are made, some based on the boundary conditions and some analogous to those of boundary-layer theory. The solution obtained in this case is a first approximation, and has been computed.

A sample calculation yields results which are in good qualitative and fair quantitative agreement with experimental measurements.
Ship waves in a stratified ocean
Authors: Hudimac Albert A.
Published: March 2006

The velocity potential for a simple source moving in a straight line at constant depth in a two-layer ocean is obtained by the Fourier transform method. It is used to develop a formula for the wave-making resistance of a ‘thin’ ship for both surface and internal waves. An asymptotic expansion is used to delineate quantitatively the internal wave system. It is shown that at speeds less than the critical speed transverse and divergent wave systems are excited, while at speeds greater than the critical internal wave speed only the divergent wave system is excited. Examples of the shape of wave crests and of wave heights are given.
Analytical theory of turbulent diffusion
Authors: Roberts P. H.
Published: March 2006

Recently Kraichnan (1959) has propounded a theory of homogeneous turbulence, based on a novel perturbation method, that leads to closed equations for the velocity covariance. In the present paper, this method is applied to the theory of turbulent diffusion and closed equations are derived for the probability distributions of the positions of marked fluid elements released in a turbulent flow.

Two topics are discussed in detail. The first is the probability distribution, at timet, of the displacement of an element from its initial position. In homogeneous flows, this distribution is found to resemble that for classical diffusion but with a variable coefficient of diffusion which is proportional to$v^2_0 t$for$t \ll l|v_0$and which approaches a constant value [eDot ]lv0fortt[Gt ]l/v0(l= macroscale,v0= r.m.s. turbulent velocity).

The second topic treated is the joint probability distribution of the displacements of two fluid elements. Particular attention is focused upon the probability distribution of relative displacement, i.e. Richardson's distance-neighbour function. This is found to be Gaussian for separationsrwhich are large ([Gt ]l). For smaller separations (r[Lt ]l), its behaviour at high Reynolds numbers is found to be quite well expressed in terms of a variable diffusion coefficientK(r,t), as suggested by Richardson (1926). For all but extremely short times,K(r,t) is found to depend only onrand on the form of the inertial range spectrumE(k). On assuming$E(r) \propto v^2_0 l(kl)^{- \frac {3}{2}}$as results from Kraichnan's approximation (1959), one finds$E(r) \propto v_0 l(r|l)^{ \frac {3}{2}}$. On the basis of similarity arguments of the Kolmogorov type, which give$E(r) \propto v^2_0 l(kl)^{- \frac {5}{2}}$, one finds$E(r) \propto v_0 l(r|l)^{ \frac {4}{3}}$as, in fact, Richardson originally proposed. The dispersionr2is proportional to$l^2(v_0 t|l)^4$on Kraichnan's theory; while$\langle r^2 \rangle \propto l^2 (v_0 t|l)^3$on the similarity theory. This illustrates that the behaviour of$\langle r^2 \rangle$is very sensitive to the spectrum.
On the stability of a heterogeneous shear layer subject to a body force
Authors: Menkes J.
Published: March 2006

The effects of density variation and body force on the stability of a heterogeneous horizontal shear layer are investigated. The density is assumed to decrease exponentially with height, and the body force is assumed to be derivable from a potential; the velocity distribution in the shear layer is taken to be U(y) = tanh y. The method of small disturbances is employed to obtain a family of neutral stability curves depending on the choice of the Richardson number. It is demonstrated, furthermore, that the value of the critical Richardson number depends on the magnitude of the non-dimensional density gradient.
Unsteady, viscous, circular flow
Authors: Sibulkin Merwin
Published: March 2006

In this paper a study of the energy-transfer processes associated with the motion of a viscous, heat-conducting fluid is begun. The class of motions considered are unsteady, two-dimensional, vortical flows. After developing simplified equations of motion and energy appropriate to this type of flow in the low Mach-number limit, general solutions of the momentum equations are presented.

The concept of a line impulse of angular momentum is introduced as an example of this class of motions for which a solution of the energy field is obtainable in closed form. The solution for the line impulse can be viewed as a combination of velocity, pressure, and temperature waves concurrently radiating from the origin of the impulse and decaying with time. Particular examples of the development of the energy field of the impulse in both liquids and gases are presented for selected values of Prandtl number. The energy-transfer processes are discussed in some detail, and the resulting differences in the energy fields for liquid gases are emphasized.
A simplified analysis of spherical and cylindrical blast waves
Authors: Friedman Manfred P.
Published: March 2006

Investigations into the behaviour of the gas flowbehind spherical or cylindrical blasts have shown that secondary shocks arise within the original detonation gases. The secondary shock, at first weak, is carried outward with the expanding gases. Subsequently it strengthens and bends back toward the origin, arriving there with high intensity.

By using some recently developed techniques in shock dynamics and extending them where necessary, a theory is developed by which the motion of the main shock wave, as well as the formation and subsequent motion of the secondary shock, are given by explicit formulae. In addition, a method for determining, also by explicit formulae, the location of the contact surface between the detonation gases and the outside atmosphere is given. The results of a specific problem, which has been solved by numerically integrating the total equations of motion, and has also been checked experimentally, are compared with the results of the present theory.
Propagation of a solitary wave along a magnetic field in a cold collision-free plasma
Authors: Saffman P. G.
Published: March 2006

It is shown that solitary hydromagnetic waves can propagate parallel to a uniform magnetic field in a cold collision-free plasma. These waves are exact solutions of the non-linear equations of motion except for the quasi-neutral approximation. The velocity of propagation lies in a range of values somewhat larger than the Alfvén velocity, and is of the order of 25 times the Alfvén velocity for hydrogen, the precise value depending upon the strength of the wave. Simple expressions exist for the velocities of the ions and electrons and the magnetic field inside the wave. The lines of force are spirals about the direction of propagation. The waves are symmetrical about their middle. The order of magnitude of their width is the geometric mean of the gyro-radii of the ions and electrons when moving with the Alfvén velocity. The maximum value of the magnetic field can be somewhat larger than the value away from the wave.
Measurements of entrainment by axisymmetrical turbulent jets
Authors: Ricou F. P. , Spalding D. B.
Published: March 2006

A new technique is described for measuring the axial mass flow rate in the turbulent jet formed when a gas in injected into a reservoir of stagnant air at uniform pressure. The jet is surrounded by a porous-walled cylindrical chamber, and air is injected through the wall until the pressure in the chamber is uniform and atmospheric, a condition which is taken to signify that the ‘entrainment appetite’ of the jet is satisfied.

Measurements made with the apparatus have allowed the deduction of an entrainment law relating mass flow rate, jet momentum, axial distance and air density, regardless of the injected gas, and including the effects of buoyancy. When the injected gas burns in the jet the entrainment rate is up to 30% lower than when it does not.
The flow of thin liquid films between rollers
Authors: Pitts E. , Greiller J.
Published: March 2006

When rollers, placed horizontally and side by side so that each is half immersed in a tank of liquid, rotate in opposite directions, liquid is carried through the gap between them and divides to form a sheet over each roller. At low speeds the sheets are of uniform thickness across the width of the rollers, but at higher speeds they are regularly ridged owing to alternate increase and decrease in thickness. Preliminary observations led to the development of an approximate theoretical treatment of the even-flow régime and the critical conditions when the ribbed flow is about to begin. Results of this work are in full agreement with detailed experimental results.
Stewartson's compressibility correlation in three dimensions
Authors: Cooke J. C.
Published: March 2006

On the assumption of (a) zero heat transfer, (b) unit Prandtl number, (c) linear viscosity temperature relation, and (d) small cross-flow, a correlation between a compressible laminar boundary-layer flow and an incompressible flow with different external conditions is established.

The correlation is applied to a known approximate method of solution for incompressible flow. For flow over a slender wing it is found that, so far as the effect on limiting streamlines—and hence on separation—is concerned, the fluid behaves as though it were incompressible but with pressure gradients multiplied by 1 + ½(γ − 1)M20 approximately, where M0 is the Mach number, supposed to be moderate in value. The effect on drag is also estimated.

An attempt is made to assess the effect of variable Prandtl number Pr, and it is found that the above multiplying factor must be replaced by
$1+\frac {1}{2}Pr^{\frac{1}{2}}(\gamma - 1)M^2_0$
but that variations in Prandtl number have little effect on the previous estimate of drag.
Linearized slip flow past a semi-infinite flat plate
Authors: Laurmann J. A.
Published: March 2006

Incompressible slip flow past a semi-infinite flat plat at zero incidence is treated in terms of the linearized viscous flow equations. A formal solution is obtained using Fourier transforms and the Wiener-Hopf technique. Explicit inversion of the transform is not possible, but asymptotic expansions are discussed. These reveal the inadequacy of boundary-layer theory in predicting the nature of the solution, even at the plate surface. For example, the local shear forces on the plate are significantly different from boundary-layer values, even far downstream, where slip effects are small. The boundary-layer limit is approached as the Reynolds number based on the mean free path or, equivalently, the free-stream Mach number tends to infinity.
Equilibrium layers and wall turbulence
Authors: Townsend A. A.
Published: March 2006

In turbulent flow past rigid boundaries, there can be distinguished regions close to the wall in which the local rates of energy production and dissipation are so large that aspects of the turbulent motion concerned with these processes are determined almost solely by the distribution of shear stress within the region and are independent of conditions outside it. These regions are here called equilibrium layers because of the equilibrium existing between local rates of energy production and dissipation. Three kinds of equilibrium layer have been studied experimentally, the constant-stress layer, the transpiration layer and the zero-stress layer, but there are other possible forms. One that is of importance in the theory of self-preserving flow in boundary layers and in diffusers is the ‘linear-stress’ layer in which the stress increases linearly with distance from the wall. The properties of these various equilibrium layers are considered and the distributions of mean velocity are derived from the equation for the turbulent kinetic energy and certain assumptions of flow similarity.

The theory of self-preserving wall flow, usually expressed as a combination of the law of the wall and the defect law, assumes compatibility between the outer flow and the equilibrium layer, and the course of development depends on the kind of equilibrium layer. Earlier work by the author, which assumed the defect law, is only valid if the whole of the equilibrium layer is a constant-stress layer and this is not true in strong adverse pressure gradients. A consistent theory is developed for these flows by assuming a ‘linear-stress’ layer, and the solutions show the relation between flows of finite stress and of zero stress and provide a plausible explanation of the phenomenon of downstream instability observed by Clauser. Self-preserving flow in wedges is treated on similar lines.
End effects in inviscid flow in a magnetohydrodynamic channel
Authors: Sutton G. W. , Carlson A. W.
Published: March 2006

The flow of an inviscid, incompressible electrical conducting fluid in a channel of constant rectangular cross-section is considered, when the flow enters a region which contains a magnetic field transverse to the flow and electrodes on opposite sides of the channel. This geometry is typical of a d.c. induction pump or magnetohydrodynamic generator. The conducting fluid external to the magnetic field acts as a shunt and produces a non-uniform electric potential field and hence a non-uniform Lorenz force on the fluid, and causes the fluid velocity profile to be distorted. These effects are calculated theoretically for small magnetic Reynolds number and small magnetic interaction parameter. It is found that the velocity at the centre-line of the channel is retarded and at the walls the velocity is accelerated. The fractional change of velocity at the wall is equal to approximately 0·44 times a modified magnetic interaction parameter.
On the dynamics of unsteady gravity waves of finite amplitude Part 2. Local properties of a random wave field
Authors: Phillips O. M.
Published: March 2006

Expressions in closed form are derived for a number of local properties of a random, irrotational wave field. They are: (i) the mean potential and kinetic energies per unit projected area; (ii) the energy balance among the processes of energy input from the surface pressure fluctuations, rate of growth of potential and kinetic energy and horizontal energy flux; and (iii) the partition between potential and kinetic energy. These expressions are mainly in terms of quantities measured at the free surface, which are therefore functions of only two spatial variables (x, y) and of time t.

Approximations for these expressions can be found simply by subsequent expansion methods; the fourth order being the highest for which the assumption of irrotational motion is appropriate in a real fluid. It is shown that the mean product of any three first-order quantities is of fourth or higher order in the root-mean-square wave slope, and this result is applied in estimating the magnitude of some higher order effects. In particular, the skewness of the surface displacement is of the order of the root-mean-square surface slope, which has been confirmed observationally by Kinsman (1960).
Swimming of a waving plate
Authors: Wu T. Yao-Tsu
Published: March 2006

The purpose of this paper is to study the basic principle of fish propulsion. As a simplified model, the two-dimensional potential flow over a waving plate of finite chord is treated. The solid plate, assumed to be flexible and thin, is capable of performing the motion which consists of a progressing wave of given wavelength and phase velocity along the chord, the envelope of the wave train being an arbitrary function of the distance from the leading edge. The problem is solved by applying the general theory for oscillating deformable airfoils. The thrust, power required, and the energy imparted to the wake are calculated, and the propulsive efficiency is also evaluated. As a numerical example, the waving motion with linearly varying amplitude is carried out in detail. Finally, the basic mechanism of swimming is elucidated by applying the principle of action and reaction.
Experiments on the flow past a circular cylinder at very high Reynolds number
Authors: Roshko Anatol
Published: March 2006

Measurements on a large circular cylinder in a pressurized wind tunnel at Reynolds numbers from 106 to 107 reveal a high Reynolds number transition in which the drag coefficient increases from its low supercritical value to a value 0.7 at R = 3.5 × 106 and then becomes constant. Also, for R > 3.5 × 106, definite vortex shedding occurs, with Strouhal number 0.27.
The development of three-dimensional disturbances in an unstable film of liquid flowing down an inclined plane
Authors: Benjamin T. Brooke
Published: March 2006

On the basis of results from a previous paper, expressions are found for the phase velocity and amplification rate of a wave travelling obliquely to the direction of flow. This wave comprises the general harmonic component of three-dimensional small disturbances, and accordingly a double Fourier integral is introduced to represent a bounded disturbance whose initial distribution over the free surface may be arbitrarily prescribed. Hence an asymptotic approximation is derived for a disturbance which is initially concentrated around a point on the free surface. Several distinctive properties of a localized unstable disturbance are noted: for instance, it lies mainly within an elliptical region whose area increases linearly with time as it moves downstream and which is modulated by long-crested waves. An experimental observation of a growing disturbance on an unstable film is recorded, and its main features are seen to be in agreement with the theory.

In so far as linearized perturbation theory remains applicable, the effects investigated are common to a wide class of parallel and nearly parallel laminar flows. In the final part of the paper the method used to analyse the instability of a film is generalized in order to reveal the connexion between this and other problems; this aim is achieved by demonstrating collective properties of the complete class of flows in question, but particular reference is made to the example of laminary boundary layers and Poiseuille flow between parallel planes.
Hydrodynamic stability and the inviscid limit
Authors: Case K. M.
Published: March 2006

It is shown that for appropriately posed problems, the solutions of the linearized Navier–Stokes equations approach those of the linearized Euler equations as the viscosity tends to zero.
Some mathematical problems in the theory of the stability of parallel flows
Authors: Lin C. C.
Published: March 2006

By applying the method of initial values to the theory of stability of shear flows, Case has recently found certain results which are in apparent conflict with those obtained by the theory of normal modes. It is shown how these differences may be reconciled. Some new features in the theory of normal modes are also brought out. The relative merits of the two theories are compared.
Magnetohydrodynamic flow constructions with fundamental solutions
Authors: Gourdine Meredith C.
Published: March 2006

In this paper steady flows of an incompressible, viscous, electrically conducting fluid are constructed from fundamental solutions of magnetohydrodynamics in which the applied magnetic field is parallel to the velocity at infinity. The flat plate and the sphere are considered as examples, and approximate solutions are presented for the limiting cases of large and small Reynolds and magnetic Reynolds numbers. The effects of currents in the body are also considered, and it is found that unless the magnetic Prandtl number is larger than unity, currents in the body have negligible effect on the flow.
A new approach to the measurement of turbulent fluxes in the lower atmosphere
Authors: Taylor R. J.
Published: March 2006

A method is described for the measurement of turbulent shearing stress and vertical heat flux by way of ‘structure functions’—the mean square velocity and temperature differences between two points a known distance apart. The method is particularly suitable for shipboard use because it does not require the measurement of velocity components relative to a fixed frame of reference. An analysis of the available observations, although they are not ideally suited to the purpose, is encouraging and points the way to further development of the method.
The effect of a magnetic field on the flow of a conducting fluid past a body of revolution
Authors: Chester W.
Published: March 2006

The problem described by the title is investigated when both the magnetic field and the streaming motion of the fluid at infinity are uniform and parallel to the axis of symmetry of the body. The flow pattern depends on three parameters, the Reynolds number R, the magnetic Reynolds number Rm and the Hartmann number M. In this paper it is assumed that M [Gt ] 1, M [Gt ] R, M [Gt ] Rm (no other restrictions on the parameters are imposed, so that R and Rm need not be small). The flow pattern then consists of an undisturbed uniform stream outside a cylinder circumscribing the body with generators parallel to the stream. Inside this cylinder the fluid is at rest. The leading term in the expression for the drag on the body is obtained.
The effect of a magnetic field on the flow of a conducting fluid past a circular disk
Authors: Chester W. , Moore D. W.
Published: March 2006

In the previous paper (Chester 1961) it was shown that, for large values of the Hartmann number, the asymptotic solution for the flow past a body of revolution has a discontinuity on the surface of a cylinder which circumscribes the body. The flow in the region of this discontinuity is now investigated in more detail when the body is a circular disk broadside-on to the flow. It will be shown that there is actually a region of transition whose thickness is O(|x|½/M½), where x is the axial distance from the disk and M is the Hartmann number. This region is thin near the disk, but gradually thickens until it merges into the over-all flow field for x = O(M).

The leading terms in the expression for the drag are given by
$\frac{D}{D_s} = \frac{M \pi }{8} \left( 1 + \frac{2}{M} \right) $,
where Ds is the Stokes drag.
The action of Vlasov waves on the velocity distribution in a plasma
Authors: Dungey J. W.
Published: March 2006

A one-dimensional model with no magnetic field is considered. It is supposed that the plasma starts in thermal equilibrium and then a current is forced to grow. Instability leads to the growth of waves, which are shown to stir the distribution in phase space, but only over a limited range of velocity. It is concluded that in order to restore stability the energy in the wave must become comparable to the energy of drift.
On the theory of the wind-driven ocean circulation
Authors: Carrier G. F. , Robinson A. R.
Published: March 2006

A surface distribution of stress is imposed on an ocean enclosed by two continental boundaries; the resulting transport circulation is studied between two latitudes of zero surface wind-stress curl, within which the curl reaches a single maximum. Under the assumption that turbulent transfer of relative vorticity has a minimum effect on the mean circulation, inviscid flow patterns are deduced in the limit of small transport Rossby number. Inertial currents, or naturally scaled regions of high relative vorticity, occur on both the eastern and the western continental coasts. Limits on the relative transports of the currents are obtained and found to depend on the direction of variation of the wind-stress curl with latitude, relative to that of the Coriolis accelerations. The most striking feature of the inviscid flow is a narrow inertial current the axis of which lies along the latitude of maximum wind-stress curl. All eastward flow occurs in this midlatitude jet.

A feature of the flow which cannot remain essentially free of turbulent processes is the integrated vorticity relationship, since the imposed wind-stress distribution acts as a net source of vorticity for the ocean. Heuristic arguments are used together with this integral constraint to deduce the presence and strength of the turbulent diffusion which must occur in the region of the mid-latitude jet. It is further inferred that the turbulent meanders of the jet must effect a net meridional transport of relative vorticity.
The structure of a hydromagnetic front in a cold collision-free plasma
Authors: Saffman P. G.
Published: March 2006

A one-dimensional steady solution of the equations of motion of a cold plasma in a magnetic field is obtained. The plasma is of semi-infinite extent, bounded by a plane interface which separates it from a vacuum or medium at rest. The particles approach from infinity, are reflected at the front, and return to infinity in the opposite direction. At infinity, the magnetic field is parallel and anti-parallel to the plasma streams, and is inclined at an angle to the normal to the interface. The front is a current sheet across which the lines of force are bent, with the component of the magnetic field in the plane of the front changing direction. The inertia of the electrons is neglected, and the characteristic frequency associated with the front is the ion gyro-frequency.
The solitary wave on a stream with an arbitrary distribution of vorticity
Authors: Benjamin T. Brooke
Published: March 2006

The theoretical work reported herein makes a departure from the many previous analyses of the solitary wave which have treated the wave as an example of irrotational fluid motion. The present analysis is of more general scope in that it covers the whole category of examples in which the wave may propagate in either direction on a horizontal stream whose primary velocity distribution U(y) is an arbitrary function (i.e. there is no restriction on the extent of the variations of U(y)). An approximate form of the wave profile is found in general to be a sech2 {(x − ct)/b}, as it is according to previous theories applicable to the wave on a uniform stream, but the relationships amongst the wave amplitude a, the length scale b, and the two propagation velocities c (positive downstream and negative upstream) depend in complicated fashion on the form of U(y).
The behaviour of a laminar compressible boundary layer near a point of zero skin-friction
Authors: Stewartson K.
Published: March 2006

It is shown, with a high degree of certainty, that a general compressible laminar boundary layer can develop a singularity at a point of zero skin-friction only if the heat transfer at that point is zero.
Mass, momentum and energy flux in water waves
Authors: Whitham G. B.
Published: March 2006

This paper gives a direct derivation of some results obtained by Longuet-Higgins & Stewart (1960, 1961) on the amplitude variation of waves propagating on a non-uniform stream. The derivation raises a number of subtle points in the use of the equations of conservation of mass, momentum and energy. These points are of general interest and are discussed in detail with other applications, since they do not seem to have been pointed out previously.
Unsteady, viscous, circular flow Part 2. The cylinder of finite radius
Authors: Sibulkin Merwin
Published: March 2006

The problem considered is that of the two-dimensional motion of the fluid in a cylinder of finite radius after the outer portion of the fluid is given an initial uniform velocity. The primary purpose of the investigation is the study of the changes in the energy distribution in the fluid as the initial motion decays. The appropriate flow equations are developed and then approximated by finite-difference equations. Numerical solutions of these equations are presented, and the energy-transfer processes are discussed in some detail. During the early stages of the flow, it is found that the spatial distribution of energy depends strongly on the Prandtl number. During the later stages, however, there is a net outward flow of energy for the case of a liquid and a net inward flow for a gas.