Selected Papers by Chia-Shun Yih

The following sections are included:

• Contents

The number of conjugate states for the flow of a fluid system of two layers is investigated by means of the momentum principle. The uniqueness of the conjugate state is proved for the cases in which the modified Froude number for either layer is predominantly large. Specific experimental results for three special cases demonstrate the uniqueness of the state downstream from the hydraulic jump, and show that, for a first approximation, the simple analysis provides a means for determining the downstream depths, with smaller errors for lower jumps.

This paper, which is restricted to the flow of inviscid fluids, is divided into three parts. In the first part existing formulas by A. R. Richardson, Hans Lewy, and K. N. Tong for constructing free-surface flows are identified with one another so that potential flows of one fluid in contact with a stagnant layer of another fluid can be constructed exclusively from one general formula, with proper modification of the gravitational acceleration. Further, a method for constructing potential flows of two fluids having a common interface (which may or may not be prescribed) is devised. It is hoped that this method will be of some use in the investigation of internal gravity waves. In the second part, it is shown that, for flows of a fluid system of discrete layers, Long's equation of motion for two-dimensional flow of an inhomogeneous fluid reduces to the Laplace equation for each layer (if the flow is irrotational) and the usual boundary conditions at the interfaces, so that flows with discontinuous density variations can be properly considered as limiting cases of flows with continuous ones. For the sake of completeness, equations of motion of an inhomogeneous fluid in axisymmetric motion in cylindrical and spherical coordinates are also given. In the third part the stability of a periodic disturbance present in a parallel flow with continuous density variation is discussed. Sufficient conditions for stability are found, and an upper bound for the amplification factor is given (if instability occurs) for a general class of flows.

This paper contains a discussion of the effect of inertia on steady stratified flows of an incompressible and inviscid fluid and an exact solution of the non-linear partial differential equations governing such flows. The latter is a solution for the steady two-dimensional flow of a stratified fluid in a channel toward a line sink, with constant density gradient in the vertical direction at infinity.

The effect of density variation on the flow of an incompressible and inviscid fluid is twofold. On the one hand, the inertia of the fluid changes in direct proportion to the density. On the other hand, the body force acting on a fluid element also changes in direct proportion to the density. Since body force is not the only force acting on the fluid, the inertia effect and the gravity effect of density variation do not cancel each other, and many interesting phenomena occur in the flow of a heterogeneous fluid that do not occur in the flow of a homogeneous fluid.

In this paper it is shown that the inertia effect can be simply evaluated for steady flows. If the velocity in the steady flow of a heterogeneous fluid in the absence of gravity is multiplied by the square root of the density, the result represents a dynamically possible flow of a homogeneous fluid. At the other extreme, when the gravitational effect dominates the flow, it has been shown both analytically and experimentally that the motion of a fluid is confined to the layer at which it originates. As usual, it is when the inertia effect and the gravity effect are comparable that the solutions of stratified flows become difficult, even if the flow is assumed to be steady and the fluid inviscid. From one series of such solutions and the supporting experiments one sees that, on the one hand, infinitely many modes of stationary internal waves of finite amplitude are dynamically possible (apart from the consideration of generation), and, on the other hand, physically significant solutions of stratified flows may involve velocity discontinuities.

The following sections are included:

• Introduction

• The governing differential system

• General considerations

• Surfaces of density discontinuity

• Equipartition of energy

• The case of infinite depth

• Calculation of phase velocities

• Waves generated by a plane wave-maker

• Stability of stratified liquid under vertical vibration

• REFERENCES

The following sections are included:

• The equation governing steady two-dimensional flow of a stratified fluid

• Types of exact solutions

• Class (a): pseudo-potential flows

• Class (b): waves of arbitrary amplitude, with application to flow over a barrier

• Class (c): another class of waves of arbitrary amplitude, with possible application to atmospheric flows

• REFERENCES

The following sections are included:

• Introduction

• The transformation

• Equation governing two-dimensional flows in a gravitational field

• Swirling axisymmetric flows in a gravitational field

• Lee waves of large amplitude

• REFERENCES

The following sections are included:

• Steady seepage flow of a fluid of variable viscosity

• The equation governing steady two-dimensional flows of a nonhomogeneous fluid

• The equation governing steady axisymmetric flows of a nonhomogeneous fluid

• Exact solutions for two-dimensional flows

• Stratified flow in Hele–Shaw cells

• REFERENCES

The following sections are included:

• Introduction

• Steady Flows of a Stratified Fluid Into a Sink

• Steady Flow of a Rotating Fluid Into a Point Sink

• Experimentation

• Discussion of Experimental Results

• Acknowledgment

• References

The following sections are included:

• Gerstner waves in a stratified liquid

• Edge waves in a stratified fluid

• REFERENCES

The following sections are included:

• Introduction

• Some preliminaries

• Derivation of the equations for an incompressible fluid

• Derivation of the equations for a compressible fluid

• Discussion

• REFERENCES

The following sections are included:

• Introduction

• The basic assumptions and their principal consequence

• Shallow-water theory for stratified liquids in steady flow

• Discussion

• REFERENCES

The following sections are included:

• PART I. GENERAL CONSIDERATIONS
  • Introduction
  • The governing equations
  • The inertial effect of density variation
  • The gravity effect of density variation
  • The creation of vorticity and circulation by density variation
  • Imbedment of the vorticity lines

• PART II. WAVES OF VERY SMALL AMPLITUDE
  • The governing differential system
  • Wave motion in a fluid otherwise at rest
  • Two-dimensional wave motion
  • Propagation of disturbance in three dimensions
  • Geophysical applications
  • Stability

• PART III. STEADY FLOWS OF FINITE AMPLITUDE
  • Equations governing steady three-dimensional motion
  • Equation governing steady two-dimensional flow
  • Two-dimensional stratified flow into a sink
  • Two-dimensional flow into a sink at low Froude numbers
  • Prevention of separation at low Froude numbers
  • Flow over a barrier and lee waves
  • Blocking
  • Gerstner waves in a stratified fluid
  • Periodic waves of finite amplitude
  • Internal solitary and cnoidal waves
  • Gravity currents

• ACKNOWLEDGMENT

• LITERATURE CITED

A theorem giving sufficient conditions for stability of stratified flows, which is a natural generalization of Rayleigh's theorem for shear flows of a homogeneous fluid, is given. Sufficient conditions for the existence of singular neutral modes , and consequently of unstable modes, are also presented, and in the development the possibility of multi-valued wave number for neutral stability of the same flow is explained. Finally, neutral waves with a wave velocity outside of the range of the velocity of flow (non-singular modes) are studied, and results concerning the possibility of these waves are given. In addition, Miles' theorem [1961] on the stability of stratified flows for which the Richardson number is nowhere less than ¼, and Howard's semi-circle theorem [1961] are extended to fluids with density discontinuities.

The flow with a free surface of a fluid, homogeneous in density, but with inhomogeneous velocity distribution, is a special case of your class of stratified fluids. Burns [1953] considered this case and I guess his results are comprised in yours. When viscosity is allowed for, the problem becomes much more complicated. It may be of interest to note that Velthuizen and I [1969a, 1969b] studied this problem taking viscosity into account. We obtained results essentially different from Burn's results, which is due to viscous effects…

It is well known that Rayleigh's sufficient condition for stability of inviscid fluids flowing between rigid boundaries is satisfied by a parabolic velocity profile, whereas plane Poiseuille flow, which has this profile, has been found by Heisenberg and Lin to be unstable at sufficiently large Reynolds numbers, when viscous effects are taken into account. Since the present paper is a study of the stability of inviscid fluids, and, in particular, Rayleigh's criterion for stability is generalized in it, Professor van Wijngaarden's position that the consideration of viscosity may force us to modify some of the conclusions in the paper is easily acceptable…

The following sections are included:

• Waves of Small Amplitude
  • The Differential Equation for Normal Modes
  • Boundary Conditions
  • Proof that ω² is Real
  • Existence of Eigenvalues
  • The Variation of ω² and c² with k²

• Waves of Finite Amplitude
  • Lee Waves in a Stratified Stream

• Stability
  • Miles' Theorem
  • Howard's Semicircle Theorem
  • Sufficient Conditions for Stability
  • Sufficient Conditions for Instability
  • Nonsingular Modes

A theory for internal progressive waves of permanent form in any continuously stratified fluid is presented, and a calculation for the flow and the wave velocity is carried out for an exponential stratified fluid. The most important conclusion from this calculation, also valid for other weak stratifications, is that the wave velocity always decreases with the amplitude, provided the density gradient is weak and the wavelength is not too short. This conclusion is significant because it entails the existence or nonexistence of solitary waves in weakly stratified fluids. The validity of the Boussinesq approximation and the significance of the well known exactly linear cases are also discussed.

Exact solutions for finite-amplitude motion of an inviscid, incompressible and stratified fluid corresponding to rectilinear vortices or vortex rings are found. Some of the rectilinear vortex pairs, in circular-cylindrical form, can propagate with a constant horizontal velocity in a surrounding homogeneous fluid, and some of the spherical vortex rings can be placed in a surrounding quiescent homogeneous fluid without translational propagation. Equations governing finite-amplitude two-dimensional motion and axisymmetric motion with swirl in the presence of a magnetic field are derived for a stratified conducting fluid, and the effects of a magnetic field on vortex motion are ascertained.

The following sections are included:

• Summary

• Introduction

• The differential system

• Tree-dimensional waves in circular or elliptic pipes

• Long waves

• Acknowledgments

• References

The following sections are included:

• Introduction

• The differential system

• Asymptotic solutions for the special cases in which one layer is extremely thin

• Algebraic method for the numerical solutions

• Results

• REFERENCES

The following sections are included:

• Similarity of steady stratified flows of an incompressible fluid

• Creation of vorticity in steady stratified flows of an incompressible fluid

• Extension to entropy stratification in gases

• REFERENCES

The differential equation governing steady axisymmetric flow of an incompressible fluid was first derived by Yih [3]. If cylindrical coordinates (r, y, z) are used with z measured vertically upward, and if g denotes the gravitational acceleration, ρ denotes the density, and ρ₀ denotes a constant reference density, the equation is [3, 4, 5]

The following sections are included:

• Summary

• Introduction

• Governing equations

• Generation of vorticity and circulation

• Steady flows into a point sink

• Sink flows when diffusion is taken into account

• REFERENCES

The object of this note is to establish a relationship between the stability of two-dimensional parallel flows for three-dimensional disturbances and that for two-dimensional ones. The special case of confined flow of a homogeneous fluid has been considered by Squire¹. In the present note neither is the upper surface of the fluid necessarily assumed to be fixed, nor are the gravitational force and variations in density and viscosity neglected. The variations in density and viscosity, which for two-dimensional flow can occur only in the direction normal to the plane boundary along which the fluid flows, may be continuous or discontinuous…

The following sections are included:

• Introduction

• Stability of Fluid Between Plane Walls

• Stability Of Fluid In A Tube

• Conclusions

• Acknowledgment

• References

The following sections are included:

• Introduction

• The undisturbed state

• Formulation of the problem

• Sufficient condition for stability

• Solution for small spacings

• Feasibility of experiment

• REFERENCES

The inhibition of instsbility of a viscous fluid contained in a circular cylinder and heated from below by an electric current is investigated. Previous results indicate that, for a thermally nonconducting wall, the critical Rayleigh number is 452.1 for symmetric convection, and 67.9 for the first (and critical) mode of unsymmetric convection. It has been found in this investigation that unsymmetric convections can be delayed or completely inhibited by an electric current, whereas symmetric convection is not at all affected. This indicates a very interesting physical situation at Rayleigh number 452.1, for an electric current just strong enough to inhibit unsymmetric convection. If the current is slightly increased, only symmetric motion will occur. If it is slightly decreased, unsymmetric convection, being more unstable, will prevail. Thus the physically significant solution of a differential system may have a sudden change of behavior at certain critical values of its parameters.

The centripetal acceleration of a rotating liquid film is tantamount to a centrifugal force, which tends to cause the liquid film to form rings around the circular cylinder to which it is attached. The stabilizing factors are surface tension and, presumably, viscosity. But it is shown in this paper that instability occurs even for large values of the surface-tension parameter and at small Reynolds numbers. The critical wave number is shown to depend predominantly on the surface tension. Its dependence on the Reynolds number, R, is slight if R is small, and nil if R is large. The effect of viscosity is therefore essentially to slow down the rate of amplification of the unstable disturbances.

The analysis is carried out for both large and small Reynolds numbers, for various ratios of film thickness to cylinder radius, and for various surface tension parameters. (The calculation for intermediate Reynolds numbers turns out to be unnecessary for the purpose of comparison with the experiments obtained. Enough information is provided by the calculations performed for practical applications.) Numerical results are given. Comparison of results obtained from 65 experiments with pure glycerine, water + glycerine mixture, and water with the analytical results shows satisfactory agreement.

The stability of a viscous fluid between rotating cylinders and with a radial temperature gradient against the formation of axisymmetric disturbances (Taylor vortices) is considered, and it has been found that viscosity has a dual role. If the circulation increases radially outward (so that the flow would be stable in the absence of density variation) but the density decreases with the radial distance, the situation can arise that viscosity actually has a destabilizing effect. In the opposite circumstance, thermal diffusivity is always destabilizing. Detailed results for small spacing of the cylinders and sufficient conditions for stability of a revolving fluid of variable density or entropy also are given.

IN AN EXPERIMENTAL INVESTIGATION sponsored by the Technical Association of Pulp and Paper Industries we found that rings formed in a liquid film attached to the inside of a rotating cylinder, as the cylinder was suddenly stopped or slowed down. In the former case the rings (Fig. 1) lasted a few seconds before gravity pulled the liquid down to the lower part of the drum. (The axis of the drum was horizontal.) In the latter case the rings appeared for a shorter length of time but disappeared again, leaving a smooth film rotating at a smaller speed, so long as the final speed was great enough to withstand the pull of gravity…

A case of free-surface instability closely related to the phenomenon of spouting on a fourdrinier wire is investigated. As a rotating cylinder with a liquid film attached to its inner surface is slowed down or stopped instantaneously, the liquid will be retarded near the surface of the cylinder while farther away the speed remains high. This is an unstable situation, and is similar to the situation prevailing at a stock-carrying fourdrinier wire as it leaves a table roll. In the latter case the fluid is slowed down at the near-stagnation region on the wire as it leaves the table roll, while the free-surface velocity remains high. Experimental data obtained indicate that the critical wavenumber (2π times the film thickness divided by the critical wavelength) of the most unstable mode is 2.7, approximately.

The stability of a liquid layer flowing down an inclined plane is investigated. A new perturbation method is used to furnish information regarding stability of surface waves for three cases: the case of small wavenumbers, of small Reynolds numbers, and of large wavenumbers. The results for small wavenumbers agree with Benjamin's result obtained by the use of power series expansion, and the results for the two other cases are new. The results for large wavenumbers, zero surface tension, and vertical plate contradict the tentative assertion of Benjamin. The three cases are then re-examined for shear-wave stability, and the results compared with those for confined plane Poiseuille flow. The comparison serves to indicate the vestiges of shear waves in the free-surface flow, and to give a sense of unity in the understanding of the stability of both flows. The case of large wavenumbers also serves as a new example of the dual role of viscosity in stability phenomena.

The topological features of the c_i curves for four cases (surface tension = 0 or ≠ 0 and angle of plate inclination = or <½π) are depicted. The effect of variability of surface tension is briefly assessed.

One of the causes of stock instability on a fourdrinier wire after the table rolls perhaps the most important—is the change of acceleration from downward to upward and back to downward as the stock goes over the table rolls. This cause is analyzed by considering a layer of fluid being subject to (1) a sudden change of acceleration and (2) a gradual change of acceleration. In both cases growth of surface waves is found when the conditions are right, and these conditions are always realized on a fourdrinier wire. To verify the theory, an apparatus was constructed to observe the effect of variable acceleration. A rectangular box containing a layer of water was allowed to fall freely to a foam-rubber pad. As soon as it hit the pad, the acceleration quickly became positive and then decreased to a negative value. Waves originally observed in the water were seen to grow, in some cases so violently that they spilled out of the container. The experiments verify the main conclusions of the theory at least qualitatively, and it is now felt that variable acceleration is indeed a very important factor in stock instability.

The following sections are included:

• Introduction

• The governing differential system

• Solution for symmetric convection

• Conclusions

• REFERENCES

A layer of a non-Newtonian liquid, of which the constitutive equation is triply nonlinear, flows down an inclined plane under the action of gravity. The stability of the flow against wave formation is investigated. With M denoting a parameter involving the first and the second viscosities, the critical Reynolds number is given as a function of M and the slope of the plane, for small values of M. The theory presented here shows how free-surface instability of non-Newtonian fluids can be attacked, and provides a basis for stability experiments with non-Newtonian fluids.

The following sections are included:

• Introduction

• The primary flow

• The differential system governing stability

• Solution for the case of moving upper boundary

• Solution for the case of stationary boundaries

• Discussion

• REFERENCES

The following sections are included:

• Introduction

• Primary flow

• General formulation of the stability problem

• Solution of the stability problem for plane Couette flow

• REFERENCES

The stability of the interface in the presence of a periodic electric field is considered. It is shown that the stability is governed by a Mathieu equation, that the interface can be unstable even if the electric field is at all times weaker than that needed for instability in the case of a steady field, and that, when instability occurs, the waves may either be synchronous with the electric field, or have twice its frequency.

The following sections are included:

• Introduction

• Primary flow

• Formulation of the stability problem

• A discussion of the approach of quasi-steadiness

• Extension of the Floquet theorem

• Solution of the problem

• Discussion

• REFERENCES

The following sections are included:

• Preliminary

• An upper bound for c_i

• A sufficient condition for stability

• REFERENCES

The following sections are included:

• Introduction

• Primary temperature distribution

• Differential system governing stability

• Method of approach

• Analysis

• The case T₁ = T₀

• Discussion of results

• REFERENCES

The following sections are included:

• Introduction

• Recapitulation of some known results for plane flows

• New results for unstable and neutral waves in plane flows

• Neutral and some damped modes in plane flows

• Results for Poiseuille flow

• REFERENCES

The stability of stratified flows consisting of a middle layer of homogeneous fluid inlinear shear flow and contiguous upper and lower stratified layers of constant velocities is considered. The density and velocity are continuous throughout. It is shown that there are infinitely many pairs of neutral modes with the phase velocity c_r, in the range of the velocity which, in general, are not represented by the stability boundary, but which merge as the parameter N (the inverse of the Froude number squared) increases. As N increases further the coalesced neutral modes become modes with complex c, one of which is unstable. The instability may be considered to be caused by the resonance of the original pair of separate modes. In addition, instability of stratified flows is considered for general density and velocity distributions without a layer of constant density and linear velocity, and it is concluded that in that case the modes with complex c on the unstable side of a stability boundary do not continue into any neutral normal modes on the stable side.

The following sections are included:

• General Introduction

• Instability of Surface Waves

• Instability of Internal Waves

• ACKNOWLEDGMENTS

• REFERENCES

The following sections are included:

• Introduction

• The primary flow

• Formulation of the stability problem

• Algebraic formulation

• Numerical method

• Results, conclusion and discussion

• REFERENCES

There are many instances of hydrodynamic instability induced by a variation, or stratification, in either a fluid property or a flow property. In this article a new instability is presented. It is shown that when there is a variation in thermal conductivity in the fluid, instability can occur in the presence of a longitudinal gravitational field.

The following sections are included:

• Summary

• Introduction

• The primary temperature distribution

• The energy method

• Determination of R_e

• Results

• Discussion

• Appendix

• Acknowledgment

• REFERENCES

When a vertical temperature gradient is applied to a large solid containing a spherical fluid inclusion, the temperature in the fluid is a function only of height. The stability of this fluid against convection is investigated and it is found that the principle of exchange of stabilities applies. The linear differential system governing stability is then solved; the results show that the thermal conductivity of the surrounding solid is always stabilizing and that the most unstable mode is the first asymmetric mode, for which the critical Rayleigh number is given. The energy method can be applied, with due modifications to account for heat conduction in the surrounding solid. The same mathematical governing differential system would then be obtained, giving the same number for the upper bound of the Rayleigh numbers below which the fluid is stable. This number is then truly critical: The fluid is stable or unstable according to whether the Rayleigh number is below or above it, whatever the magnitude of the disturbance. The results are discussed in the context of the movement of the spherical inclusion in a soluble solid. The greater instability of the asymmetric mode indicates that when instability occurs, the fluid inclusion will have a sidewise component, which is greater for a greater supercritical Rayleigh number. The effect of double diffusion is also discussed.

The following sections are included:

• Introduction

• The primary flow

• Formulation of the stability problem

• A simplification

• Construction of the eigenfunction φ

• Construction of the eigenfunction χ

• Calculation of the growth rate

• Application of the theory to a special case

• Conclusion

• REFERENCES

The following sections are included:

• Contents

Tides in estuaries and around small islands are studied in this paper. Under the assumption that the width and the mean depth of the estuary can be adequately expressed as power functions of the longitudinal distance from a certain point upstream, and that the depth of the ocean varies as a power function of the radial distance from the island, analytical solutions can be found by very simple transformations.

The following sections are included:

• Introduction

• Governing equations

• The semicircle theorem

• The Reynolds stress

• Sufficient conditions for stability

• Singular neutral modes

• Unstable modes contiguous to singular neutral modes

• Non-singular neutral modes

• Appendix

• REFERENCES

Surface waves of a homogeneous liquid and internal waves of a stratified liquid in basins of variable depth are considered. Inequalities involving the frequencies of oscillation are obtained when a container with one size or geometry is compared with another with a different size or geometry, when waves with one wavelength are compared with waves with another wavelength, or when one stratification is compared with another. Since exact solutions for gravity waves in basins of variable depth are so rare, one hopes the comparison theorems presented herein will be useful.

Surface waves created by water flowing in an open channel with vertical side-walls and variable width are considered and analytical solutions given. It is shown that there are infinitely many Froude numbers, depending on the wavenumber of the channel-width variation and on the transverse wavenumber, at which the amplitude of one of the wave components becomes infinite. These critical Froude numbers are interpreted physically. The waves created generally have a diamond pattern.

The case of channels of varibale depth as well as variable width is then investigated and the solutions given. Finally, internal waves are created briefly and some results presented.

Gravity waves in trapezoidal channels and channels with curved bottoms, including sloshing, longitudinal, and combined modes, are treated. Analytical-numerical solutions are given for the wave frequency and the velocity potential for waves in trapezoidal channels, and analytical solutions based on the shallow-water theory are obtained for waves in curved channels.

The following sections are included:

• Introduction

• Formulation of the problem

• A transformation for the meandering

• Solution of the problem

• Meandering channels of variable depth

• Internal waves in a meandering stream

• Explanation for the self-induced waves observed by Binnie

• REFERENCES

A mechanism for creation of edge waves is proposed. It is shown that a longshore current flowing over a ridge in a sloping sea-bed with an angle of inclination γ not greater than π/4 produces edge waves in the lee of the ridge. These edge waves have a wave number equal to gU⁻²sinγ, where g is the gravitational acceleration, U the velocity of the longshore current, and γ the angle of inclination of the sea-bed. The amplitude of the edge waves produced depends on the amplitude and geometry of the ridge as well as on the three variables mentioned above.

The following sections are included:

• Summary

• Introduction

• Surface gravity waves

• Removal of singularities in the solution

• Acknowledgment

• REFERENCES

The following sections are included:

• Summary

• Introduction

• The Cauchy-Poisson solution

• Wave groups produced by an oscillating pressure distribution

• Wave trains of finite length

• Wave groups produced by a moving pressure distribution

• Gravity-wave trains of finite length created by a moving disturbance in deeb water

• Effect of viscosity

• Acknowledgment

• REFERENCE

Patterns of water waves created by a moving disturbance representing a moving body, floating or submerged, can be found by applying (1) the principle of stationary phase, (2) the principle that the phase lines are normal to the wave-number vector, and (3) the perception that the local phase velocity of the waves must be equal to the component of the velocity of the disturbance normal to the phase line. The three equations thus obtained are solved, and formulas for the phase lines are derived, which depend explicitly on the dispersion equation, and on that equation only. These formulas are applied to deep-water surface waves, surface waves in water of finite depth, internal waves, and capillary waves in thin sheets to obtain the wave patterns sufficiently far from the moving disturbance.

Finally, the patterns of the surface waves in deep water created by a moving body are determined, with the nonuniformity of the mean velocity of the fluid in the wake taken into account. The vorticity in the direction along the phase lines is shown to be small, so that the wave motion can still be assumed irrotational in a first approximation. The wave patterns differ from the Kelvin-wave pattern, as a result of the nonuniformity of fluid velocity in the wake.

The following sections are included:

• Introduction

• Analysis

• Procedure of Computation

• Results

• Acknowledgment

• REFERENCES

Groups of gravity waves of permanent form in deep water are investigated. The analysis provides a systematic procedure for determining the form of the group to any order of approximation, and a calculation is carried to the third order of the amplitude at least and, where it matters, to the fourth order. Closed formulas for the phase velocity c of the basic waves and the group velocity c_g, are obtained. Inspection of the analytic procedure reveals that these formulas remain intact for all subsequent calculations to any order of approximation. These formulas are in terms of the group wavenumber ε which, to the attained order of approximation, is found to be proportional to the amplitude a and the square of the basic wavenumber k, but is, for any assigned k, a power series in a. It is found that c increases and c_g decreases with ε, in such a way that 2cc_g = g/k, where g is the gravitational acceleration. The results are compared with the corresponding ones obtained by the cubic-Schrödinger-equation (CBE) approach, and wherever comparison is possible there is agreement. The CBE approach, however, does not give the variation of c_g with the amplitude.

The collision of wave groups with different group velocities is also investigated, and it is found that after the faster group has overtaken the slower one, both groups retain their original forms, without any phase shift for either group. The interaction terms eventually die down everywhere. When a group is reflected by a vertical boundary normal to its velocity, then the reflected group is, in time, just the continuation of its mirror image across the boundary, without any phase shift.

Nonlinear groups of gravity-capillary waves in deep water are investigated by a systematic direct approach that can be applied to nonlinear groups of other dispersive waves. Two formulas in closed form expressing the variations of the phase velocity c of the basic waves and of their group velocity c_g with the amplitude of the waves are obtained. These are in terms of the wavenumber ε of the envelope and ε² can be determined by the present approach as a power series in a², if 2a represents the amplitude of the waves. To the order of approximation achieved here, ε² is determined as a multiple of a². If k is the wavenumber of the basic waves, g is the gravitational acceleration, ρ is the density of the fluid, T is surface tension, and β = Tk²/ρg, then wave groups are possible for

although the analysis is valid only when βis not near ½. The phase velocity increases with the amplitude in the former interval for β and decreases with the amplitude in the latter interval. The group velocity c_g decreases with the amplitude in the former interval for β, or for β < β < 1, but increases with the amplitude if β > 1. When the results of this paper are compared with the results of previous authors, wherever comparison is possible, complete agreement is found. (Previous authors did not give the variation of c_g with the amplitude.)

This paper contains an exact closed solution for the temperature distribution in a preheated air jet when the flow is steady and laminar. Both the two-dimensional and the axially symmetrical cases have been treated. The solution is immediately applicable to other similar problems of diffusion.

A point source of heat is considered to be situated in an infinite plane above which the atmosphere was originally isothermal and at rest, and the resulting steady temperature and velocity distribution are sought. As can be observed from the behavior of smoke from a burning cigarette, the flow caused by the heat source is laminar at first, then at some height above it becomes unstable, and the subsequent flow is turbulent. The interdependent distributions of temperature and velocity are obtained in the laminar zone by solving a pair of simultaneous differential equations, and in the turbulent zone by systematic experimentation guided by a dimensional analysis. The transition from laminar to turbulent flow is also investigated. The results are applicable to similar problems of diffusion.

Exact closed solutions of the following equation of diffusion

where m and n are considered to be independent, are obtained for the following cases: (1) diffusion from a line-source embedded in a smooth surface; (2) diffusion from a smooth surface; (3) vapor concentration in the wake of an evaporating surface. The solutions for Cases (1) and (2) belong to the similarity-solution category. Dimensional analysis has been used in the systematic search for such solutions in as concise a manner as possible. The solution for Case (3) can be obtained from those for Cases (1) and (2). Diffusion in Couette flow is given as an example.

The differential equation of diffusion when the wind velocity and the vertical and lateral diffusivities are power functions of height is

where x, y, and z are measured respectively in the down-wind, vertical, and cross-wind directions, and D₁ and D₂ are physical constants defined in the text. Exact solution of this equation for the case of a point source and for m = k is presented in this paper. In the systematic search for this solution, dimensional analysis has been utilized to the optimum advantage. Although the solution is restricted to the special case m = k, it shows the important characteristics of atmospheric diffusion from a point source.

Elementary analyses of the mean patterns of free convection from a line source and a point source are presented without regard to the specific means by which the gravitational action is produced. The derived functional relationships are then verified and completed through use of velocity and temperature measurements above sources of heat, the generalized form of the results permitting characteristics of the mean flow to be determined over a considerable range of the primary variables. These results should enable meteorologists to evaluate the role of the basic convective process in the more complex movements of the atmosphere.

The following sections are included:

• INTRODUCTION

• FREE CONVECTION DUE TO A POINT SOURCE OF HEAT
  • Laminar Case
  • Turbulent Case
  • Stability

• FREE CONVECTION DUE TO A LINE SOURCE OF HEAT
  • Laminar Case
  • Turbulent Case

• FREE CONVECTION DUE TO TWO PARALLEL LINE SOURCES

• CONCLUDING REMARKS

• ACKNOWLEDGMENTS

• REFERENCES

Based on the velocity distribution given by Homann in 1936 for the laminar incompressible flow near a stagnation point in axisymmetric flow, solutions for the following cases of forced convection are obtained:

(a) A point source of heat is situated at the stagnation point on an insulated boundary.

(b) The difference between the temperature of the boundary and the ambient temperature varies as an arbitrary power of the distance from the stagnation point.

(c) The temperature rise is a result of viscous dissipation alone.

Solutions for practical cases are obtained by combining the result for case (c) with that for case (a) or (b).

Numerical results are given for all three cases for various parametric values of the Prandtl Number, with special emphasis on the Prandtl Number 0.7, which applies approximately to air under normal conditions.

The velocity and temperature distributions in turbulent buoyant induced by a line source or point source of heat are calculated by assuming the eddy viscosity and eddy diffusivity to be constant in any cross section of the plume. Two solutions in closed forms are obtained for the two-dimensional plume, corresponding to turbulent Prandtl number σ equal to 2/3 and 2. Two such solutions are also obtained for the round plume, corresponding to σ equal to 1.1 and 2. The solution for σ = 2/3 is compared with previous measurements for two-dimensional plumes, and the solution for σ = 1.1 is compared with previous measurements for the axisymmetric plume. The analytical and experimental results agree well in the two-dimensional case, and satisfactorily in the axisymmetric case.

With the rise in energy needs and the consequent proliferation of cooling towers (not to mention smoke stacks) on the one hand, and society's enchanced concern with the environment on the other, the study of buoyant plumes caused by heat sources in a transverse wind has become important. Buoyant plumes may also occur in the ocean, such as when a deeply submerged heat source moves horizontally in it. The fluid mechanics involved in buoyant plumes is very nearly the same, be they atmospheric or submarine.

In this paper a similarity solution for turbulent buoyant plumes due to a point heat source in a transverse wind is presented. By a set of transformations the mathematical dimension of the problem is reduced from 3 to 2. Analytical solutions for the first and second approximations are obtained for the temperature and velocity fields. The solution exhibits the often observed pair of longitudinal counter-rotating vortices. As a result of buoyancy, the point of highest temperature and the “eyes” of the vortices at any section normal to the wind direction continuously rise as the longitudinal distance from the heat source increases.

The following sections are included:

• INTRODUCTION

• TURBULENT PLANE JETS

• TURBULENT HALF JETS

• TURBULENT ROUND JETS

• TURBULENT THREE-DIMENSIONAL JETS IN TRANSVERSE WIND

• TWO-DIMENSIONAL TURBULENT PLUME

• ROUND TURBULENT PLUME

• TURBULENT PLUME IN TRANSVERSE WIND

• ACKNOWLEDGMENT

• APPENDIX I.—REFERENCES

• APPENDIX II.—NOTATION

Two approximate solutions, which become exact for Prandtl numbers 1 and 2, are given for laminar round plumes for arbitrary Prandtl number σ. The first of these gives a very accurate solution for laminar found plumes in air, for which σ = 0.73, and the second provides a good solution for laminar round plumes in water, for which σ = 6.7. Two approximate solutions, which become exact for turbulent Prandtl numbers 1.1 and 2, respectively, are given for turbulent round plumes. The one which becomes exact at σ = 1.1 is used to give a highly accurate solution for σ = 1, which, for an eddy-viscosity coefficient λ equal to 0.0156, provides a remarkably good agreement with the experimental data of Beuther, Capp, and George, Jr.

Problems in fluid mechanics of essentially irrotational flow are often so complex that numerical methods must be used in their resolution. A simple, yet generally useful method is the relaxation process, which is based on a network of values of the required function determined by means of the finite-difference theory. The numerical process is simple, and techniques are available for satisfying various types of boundary conditions. The computations are useful in solving problems of efflux and seepage and in analyzing flow through transitions or around submerged bodies. Examples of these various types have been solved, and the results are presented as illustrations of the process.

It is well known^** that, in a domain free from singularities, the maximum speed in irrotational flows of an incompressible fluid occurs on the boundary of that domain. In this note, it will be proved that for two-dimensional irrotational flows of an incompressible fluid, the maximum magnitude of the acceleration in any singularity-free domain D must also occur on its boundary provided the flows are steady.…

In problems of hydrodynamic stability involving axial symmetry, it is sometimes necessary to find the solutions of a differential equation of the type

in which n is a positive integer, and (with D ≡ d/dr) is the Bessel operator of the first order…

The dynamical theory of the motion of a body through an inviscid and incompressible fluid has yielded three relations: a first, due to Kirchhoff, which expresses the force and moment acting on the body in terms of added masses; a second, initiated by Taylor, which expresses added masses in terms of singularities within the body; and a third, initiated by Lagally, which expresses the forces and moments in terms of these singularities. The present investigation is concerned with generalizations of the Taylor and Lagally theorems to include unsteady flow and arbitrary translational and rotational motion of the body, to present new and simple derivations of these theorems, and to compare the Kirchhoff and Lagally methods for obtaining forces and moments. In contrast with previous generalizations, the Taylor theorem is derived when other boundaries are present; for the added-mass coefficients due to rotation alone, for which no relations were known, it is shown that these relations do not exist in general, although approximate ones are found for elongated bodies. The derivation of the Lagally theorem leads to new terms, compact expressions for the force and moment, and the complete expression of the forces and moments in terms of singularities for elongated bodies.

Mr. K. Eggers of the Hamburg Institut fur Schiffbau has pointed out that the term in the expression for the force in Eq. (61) of the paper by Yin and myself (Journal of Fluid Mechanics, Sept, 1956, p. 332) cannot of itself contribute to the force. If it did then a closed body, generated by a source-sink distribution of zero total strength in which a doublet was embedded, in steady motion in an infinite, unbounded fluid, would be subject to a force…

Stream functions exist for general three-dimensional flows of a non-diffusive fluid except unsteady flows of a compressible fluid. Along a streamline, these functions are constant. A simple definition of the velocity in terms of the stream functions has been given whenever the latter exist. This definition includes all the known ones for special flows as special cases, and yields a simple relationship between the volume or mass discharge and the values of the stream functions, from which all the well-known ones for special flows can be immediately deduced. For unsteady flows of a compressible fluid three ≪ path functions ≫ exist in terms of which the density and the products of the density and the velocity components can be expressed — and in such a way that the equation of continuity is identically satisfied. The mass contained in three pairs of surfaces which are time-traces of the material surfaces (or hypersurfaces) corresponding to specific values of the “ path functions ” is shown to be the product of the three differences of these functions. An analogous development for vorticity (instead of velocity) results in a similar relationship between the circulation and the vorticity functions.

The purpose of this note is to show that the maximum speed in a singularity-free region of any steady subsonic flow must occur on its boundary, provided the flow is isentropic and irrotational. The corresponding result for irrotational flows of an incompressible fluid is well known…

The following sections are included:

• Two-dimensional flow into a line sink

• Axisymmetric flow into a point sink

• REFERENCES

The following sections are included:

• Summary

• Introduction

• Effect of rotation

• Effect of gravity

• Effect of a magnetic field

• Effect of an electric current

• Concluding remarks

• Acknowledgment

• BIBLIOGRAPHY

General formulae for generating unsteady potential flows characterized by the presence of a cavity (which can be finite in extent) are presented. The fluid at infinity is either in accelerative motion or at rest, but the two cases are essentially distinct. A class of solution is given explicitly for each case. Steady flows in a gravitational field are also investigated, and a class of solutions representing such flows past a finite cavity is given. Some of the solutions in these three classes have been carried out in detail.

The velocity of a fluid mass imbedded in another fluid, which is of a different viscosity and a different specific weight and flowing in a porous medium under a prevailing uniform pressure gradient, is investigated. The fluid mass may take the form of a circular or elliptic cylinder, a sphere, or an ellipsoid, and the orientation of the fluid mass, if not spherical, is completely arbitrary with respect to both the direction of the pressure gradient and that of gravity. Exact closed solutions are obtained. The results for two-dimensional flows are applicable to Hele-Shaw cells.

A hydraulic jump occurs in a layer of fluid flowing down the inner wall of a rotating cylinder when the downstream conditions are adequate. The theory of these jumps is presented, together with supporting experimental data. The results confirm the similarity between free-surface flows under general rotation (and hence centripetal acceleration) and free-surface flows in the presence of a gravitational field, and indicate that the hydraulic jump in a rotating fluid is just the counterpart of the ordinary hydraulic jump.

The following sections are included:

• Preliminaries

• Steady magnetically reinforced flows

• Establishment of irrotational and current-free flows

• A current-induced jet

• REFERENCES

• Appendix

In his Kinematics of Vorticity (p. 183), TRUESDELL [1] called attention to the EARNSHAW conjecture [2] of 1837, which states that in a circulation-preserving motion, a particle once in complex-lamellar motion remains ever in complex-lamellar motion. TRUESDELL stated that he doubted the truth of this conjecture. A disproof of it, still lacking in the literature, is supplied in this brief note…

Peristaltic pumping (viscous fluid flow induced by a sinusoidal traveling wave motion of the walls of a tube) at moderate amplitudes of motion is analyzed in the two-dimensional case. The nonlinear convective acceleration is considered and the nonslip condition is applied on the wavy wall (rather than on the mean position) in order to account for the mean flow induced by the wall motion. In the case in which there is no other cause of flow, the mean flow induced by the peristaltic motion of the wall is proportional to the square of the amplitude ratio (wave amplitude/half width of channel). The velocity profile depends on the mean pressure gradient. In this paper only those cases in which the pressure gradient will produce a flow of the same order of magnitude as that induced by the peristaltic motion are considered. If the pressure gradient is positive and equal to a certain critical value, then the velocity is zero on the center line. Pumping against a positive pressure gradient greater than the critical value would induce a backward flow (reflux) in the core region of the stream. There will be no reflux if the pressure gradient is smaller than the critical value. The velocity profile and the value of the critical pressure gradient are presented in this paper.

Fluid motion in a long straight channel induced by longitudinally varying surface tension has been discussed by Levich. This problem is re-examined and a different solution is given. In addition, the stability of laminar flows involving surface-tension variation is briefly discussed, and a correction of a previous result [C.-S. Yih, J. Fluid Mech. 28, 493 (1967)] is made.

Steady flows of a thin layer of viscous liquid on a horizontal plane induced by the nonuniformity of surface tension at its free surface are treated. If the film is very thin, surface-tension effects dominate gravity effects. Under that circumstance and away from vertical boundaries, a binomial of depth h of the liquid layer is a harmonic function of the Cartesian coordinates x and y in a horizontal plane, and the surface tension is a function of h. Near any vertical boundary there is a velocity boundary layer whose thickness is of the order of h. The velocity distribution in this boundary layer is given explicitly. The diffusion of the surface material affecting the surface tension is considered. Steady flows of a liquid film induced by gravity are also discussed. Simple solutions are possible if the film flows over a horizontal plane.

Exact solutions for free surface flows are rare, especially if gravity is taken onto account. In this note an exact solution is given for a flow of which every streamline can be a free streamline…

As usual, u and v denote the velocity components in the directions of increasing x and y, respectively, x and y being Cartesian coordinates, with y increasing in the direction of the vertical. The gravitational acceleration, assumed constant in this note, will be denoted by g, and the density and the pressure will be denoted by ρ and p, respectively…

The flow of air between a plate at rest and another one falling onto it either vertically or by folding is studied, and the infrequency of breakage of glass plates colliding in this way is explained. The falling plate may be two-dimensional, circular, or elliptic, and the results for an elliptic plate give bounds for the motion of a falling rectangular plate.

Some simple flows with condensation are considered and their solutions given. For the vapor phase, the nonlinearity of the equations of motion and of heat diffusion, and of the equation of state, and the dissipation due to shear and volume viscosities are taken into account. For the liquid phase the density is assumed constant. At the interface, where condensation takes place, the velocity, the stress, and the temperature gradient are all discontinuous. The same approach can be used for flows with evaporation.

A two-parameter family of exact axially symmetric solutions of the Navier–Stokes equations for vortices contained within conical boundaries is found. The solutions depend upon the same similarity variable, equivalent to the polar angle φ measured from the symmetry axis, as flows previously discussed by Long and by Serrin, but are distinct from the cases they treated. The conical bounding stream surfaces of the present solution can be located at any angle φ = φ₀, where 0<φ₀<π. The flows in all of these cases, when solutions exist, are finite everywhere except at the cone vertex which is a source of axial momentum, but not of volume. Solutions are of three types, flow may be (a) towards the vertex on the axis and away from the vertex at the conical boundary, (b) towards the vertex both on the axis and at the cone, or (c) away from the vertex on the axis and towards it at the bounding cone. In the first and second case, strong shear layers form on the cone walls for high Reynolds numbers. In case (c), a region of strong axial shear and strong axial vorticity forms near the axis, even for low Reynolds numbers. The qualitative nature of the possible solutions is deduced, using methods of argument due to Serrin, and examples of flows are numerically computed for cone half-angles of π/4, π/2 (flows above the plane z = 0), and 3π/4. Regions of the parameter space where solutions are proven not to exist are given for the cone half-angles given above, as well as regions where solutions are proven to exist.

The following sections are included:

• Introduction

• The two-dimensional case

• Geometrical significance of the integral I₁

• Examples of fluid drift

• The three-dimensional case

• A proof of Darwin's theorem without calculation

• Connection between Darwin's theorem and Taylor's theorem

• Appendix

• REFERENCES

The temperature distribution in an ellipsoidal liquid inclusion in a soluble solid, with a constant gradient far away from the liquid, and the movement of the liquid inclusion as a whole, which results as a consequence, are investigated. Since the solid is soluble and its concentration in solution is temperature dependent, any temperature variation in the liquid induces a concentration variation, which will transfer mass by diffusion, eroding the wall where the temperature is high and depositing solid material at the wall where the temperature is lower. This erosion or deposition will cause the liquid inclusion to move, and will, through absorption or release of latent heat, in turn affect the temperature distribution. From the result obtained for the general ellipsoid, specific results for prolate and oblate ellipsoids of revolution, the sphere, and circular and elliptic cylinders are obtained.

The following sections are included:

• Preface

• REVIEWS

• —BOOK REVIEW SECTION—