This paper uses the method of kinematic waves, developed in part I, but may be read independently. A functional relationship between flow and concentration for traffic on crowded arterial roads has been postulated for some time, and has experimental backing (§2). From this a theory of the propagation of changes in traffic distribution along these roads may be deduced (§§2, 3). The theory is applied (§4) to the problem of estimating how a ‘hump’, or region of increased concentration, will move along a crowded main road. It is suggested that it will move slightly slower than the mean vehicle speed, and that vehicles passing through it will have to reduce speed rather suddenly (at a ‘shock wave’) on entering it, but can increase speed again only very gradually as they leave it. The hump gradually spreads out along the road, and the time scale of this process is estimated. The behaviour of such a hump on entering a bottleneck, which is too narrow to admit the increased flow, is studied (§5), and methods are obtained for estimating the extent and duration of the resulting hold-up. The theory is applicable principally to traffic behaviour over a long stretch of road, but the paper concludes (§6) with a discussion of its relevance to problems of flow near junctions, including a discussion of the starting flow at a controlled junction. In the introductory sections 1 and 2, we have included some elementary material on the quantitative study of traffic flow for the benefit of scientific readers unfamiliar with the subject.

On kinetic waves, II . A theory of traffic flow on long crowded roads

On kinematic waves II. A theory of traffic flow on long crowded roads

A simple analytic model is constructed to elucidate some basic features of the response of the tropical atmosphere to diabatic heating. In particular, there is considerable east-west asymmetry which can be illustrated by solutions for heating concentrated in an area of finite extent. This is of more than academic interest because heating in practice tends to be concentrated in specific areas. For instance, a model with heating symmetric about the equator at Indonesian longitudes produces low-level easterly flow over the Pacific through propagation of Kelvin waves into the region. It also produces low-level westerly inflow over the Indian Ocean (but in a smaller region) because planetary waves propagate there. In the heating region itself the low-level flow is away from the equator as required by the vorticity equation. The return flow toward the equator is farther west because of planetary wave propagation, and so cyclonic flow is obtained around lows which form on the western margins of the heating zone. Another model solution with the heating displaced north of the equator provides a flow similar to the monsoon circulation of July and a simple model solution can also be found for heating concentrated along an inter-tropical convergence line.

Some simple solutions for heat-induced tropical circulation.

Monograph on sound generation by turbulence and surfaces in arbitrary motion, discussing sound and multipole fields and governing equations

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Sound Generation by Turbulence and Surfaces in Arbitrary Motion

Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ``phantom traffic jams'' even though drivers all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction in the volume of traffic cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize into lanes, while similar systems ``freeze by heating''? All of these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to self-driven many-particle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for self-driven many-particle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well.

Traffic and related self-driven many-particle systems

There are two separate but closely interwoven strands of argument in this paper; one mainly mathematical, and one mainly physical. The mathematical strand begins with a method of asymptotically evaluating Fourier integrals in many dimensions, for large values of their arguments. This is used to investigate partial differential equations in four variables, x, y, z and t , which are linear with constant coefficients, but which may be of any order and represent wave motions that are anisotropic or dispersive or both. It gives the asymptotic behaviour (at large distances) of solutions of these equations, representing waves generated by a source of finite or infinitesimal spatial extent. The paper concentrates particularly on sources of fixed frequency, and solutions satisfying the radiation condition; but an Appendix is devoted to waves generated by a source of finite duration in an initially quiescent medium, and to unstable systems. The mathematical results are given a partial physical interpretation by arguments determining the velocity of energy propagation in a plane wave traversing an anisotropic medium. These show, among other facts not generally realized, that even for non-dispersive (e.g. elastic) waves, the energy propagation velocity is not in general normal to the wave fronts, although its component normal to them is the phase velocity. The second, mainly physical, strand of argument starts from the important and striking property of magneto-hydrodynamic waves in an incompressible, inviscid and perfectly conducting medium, of propagation in one direction only—a given disturbance propagates only along the magnetic lines of force which pass through it, and therefore suffers no attenuation with distance. There are cases of astrophysical importance where densities are so low that attenuation due to collisional effects—for example, electrical resistivity—should be negligible over relevant length scales. We therefore ask how far the effects of a non-collisional nature which are neglected in the simple theory, particularly compressibility and Hall current, would alter the unidirectional, attenuation-less propagation of the waves. These effects have been included previously in magneto-hydrodynamic wave theory, but the directional distribution of waves from a local source was not obtained. This problem explains the need for the mathematical theory just described, and gives a comprehensive illustration of its application.

Studies on Magneto-Hydrodynamic Waves and other Anisotropic Wave Motions

The laminar boundary layer in two-dimensional flow about a cylindrical body, when the velocity of the oncoming flow relative to the body oscillates in magnitude but not in direction, is analyzed mathematically. It is found that the maxima of skin friction at any point anticipate the maxima of the stream velocity, because the pressure gradient needed to speed up the main stream locally produces a given percentage increase in the slow flow near the wall sooner than it can do so in the main stream itself. For each point on the body surface there is a critical frequency $\omega \_{0}$, such that for frequencies $\omega $ > $\omega \_{0}$ the oscillations are to a close approximation ordinary 'shear waves' unaffected by the mean flow; the phase advance in the skin friction is then 45 degrees. For frequencies $\omega $ < $\omega \_{0}$, on the other hand, the oscillations are closely approximated as the sum of parts proportional to the instantaneous velocity and acceleration of the oncoming stream; the phase advance in the skin friction is then tan$^{-1}$ ($\omega /\omega \_{0}$). The part depending on the instantaneous velocity may be called the quasi-steady part of the oscillations. The coefficient of the acceleration of the oncoming stream in the frictional drag of the body may be called the frictional component of the virtual mass. For a flat plate in a stream of speed V, $\omega \_{0}$ = 0$\cdot $6 V/x at a distance x from the leading edge. If c is the length of the plate, its transient motion parallel to itself is governed solely by quasi-steady forces and this added virtual mass provided that $\omega $c/V < 0$\cdot $6. The frictional component of the virtual mass of a flat plate or any thin obstacle is found to be approximately 0$\cdot $5 times the mass of the fluid in the boundary layer's 'displacement area'; it is suggested that the coefficient may need to be increased to about 0$\cdot $8 for turbulent layers. When the body surface is hot, the maxima in heat transfer from it tend to lag behind those of the stream velocity, as a result of thermal inertia, but this is counteracted to some extent by the effect of convection by the phase-advanced velocities near the wall. For layers with a favourable gradient in the mean flow, one finds that the tendency to lag predominates. For the Blasius layer, however, the two effects appear to cancel out fairly closely; and for layers with adverse pressure gradient in the main stream there seems to be phase advance at the lower frequencies. At frequencies well above $\omega \_{0}$ there is always a phase lag of 90 degrees, but the amplitude of heat-transfer fluctuations is then much reduced, even though that of the skin friction fluctuations is increased. Special attention is paid to the phase lag in the heat transfer from a heated circular wire in a fluctuating stream, in the range of Reynolds number for which a laminar boundary layer exists. Curves for the amplitude and phase of the heat-transfer fluctuations as a function of frequency are given in figure 4, from calculations for the layer of nearly uniform thickness, which covers the front quadrant of the wire, and across which most of the fluctuating part of the heat transfer is believed to occur. For frequencies small compared with $\omega _{0}$ = 20V/d (where d is the diameter), the departure of the heat-transfer fluctuations from their quasisteady form consists essentially of a time lag of the order of 0$\cdot $2d/V.

The response of laminar skin friction and heat transfer to fluctuations in the stream velocity

An approximation to the heat transfer rate across a laminar incompressible boundary layer, for arbitrary distribution of main stream velocity and of wall temperature, is obtained by using the energy equation in von Mises's form, and approximating the coefficients in a manner which is most closely correct near the surface. The heat transfer rate to a portion of surface of length l (measured downstream from the start of the boundary layer) and unit breadth is given as -$\frac{\frac{1}{2}k}{(\frac{1}{3})!}\left(\frac{3\sigma \rho}{\mu ^{2}}\right)^{\frac{1}{3}}\int\_{0}^{l}\left(\int\_{x}^{l}\surd \{\tau (z)\}\,dz\right)^{\frac{2}{3}}$ dT$\_{0}$(x), where k is the thermal conductivity of the fluid, $\sigma $ its Prandtl number, $\rho $ its density, $\mu $ its viscosity, $\tau $(x) is the skin friction, and T$\_{0}$(x) the excess of wall temperature over main stream temperature. A critical appraisement of the formula (section 3) indicates that it should be very accurate for large $\sigma $, but that for $\sigma $ of order 0$\cdot $7 (i.e. for most gases) the constant $\frac{1}{2}$3$^{\frac{1}{3}}$/ ($\frac{1}{3}$)! = 0$\cdot 807$ should be replaced by 0$\cdot $73, when the error should not exceed 8% for the laminar layers that occur in practical aerodynamics. This yields a formula Nu = 0$\cdot $52$\sigma ^{\frac{1}{3}}$(R$\overline{\surd C\_{f}}$)$^{\frac{2}{3}}$ for Nusselt number in terms of the Reynolds number R and the mean square root of the skin friction coefficient C$\_{f}$, in the case of uniform wall temperature. However, for the boundary layer with uniform main stream, the original formula is accurate to within 3% even for $\sigma $ = 0$\cdot $7. By known transformations an expression is deduced for heat transfer to a surface, with arbitrary temperature distribution along it, and with a uniform stream outside it at arbitrary Mach number (equation (42)). From this, the temperature distribution along such a surface is deduced (section 4) in the case (of importance at high Mach numbers) when heat transfer to it is balanced entirely by radiation from it. This calculation, which includes the solution of a non-linear integral equation, gives higher temperatures near the nose, and lower ones farther back (figure 2), than are found from a theory which assumes the wall temperature uniform and averages the heat transfer balance. This effect will be considerably mitigated for bodies of high thermal conductivity; the author is not in a position to say whether or not it will be appreciable for metal projectiles. But for stony meteorites at a certain stage of their flight through the atmosphere it indicates that melting at the nose and re-solidification farther back may occur, for which the shape and constitution of a few of them affords evidence. An appendix shows how the method for approximating and solving von Mises's equation could be used to determine the skin friction as well as heat transfer rate, but this line seems to have no advantage over established approximate methods.

Contributions to the Theory of Heat Transfer through a Laminar Boundary Layer

For a train of gravity waves of finite amplitude in a frame of reference in which they are stationary, it is suggested that, in addition to the volume flow per unit span Q and the total head R , one may usefully study a third constant S , the rate of flow of horizontal momentum(corrected for pressure force, and divided by the density). The values of Q , R and S probably determine the wave-train uniquely; this is proved explicitly for long waves in a new presenta­tion of the ‘cnoidal wave’ theory (§3). The combinations of Q , R and S which are possible in the general case are illustrated by a diagram (figure 2) in which the co-ordinates are r = R / R c and s = S / S c ; here suffix c refers to ‘critical’ flow at the volume rate Q . There are two barriers on this diagram beyond which no stationary waves, or other steady flows, are possible; at the left-hand barrier (corresponding to uniform subcritical flows) the amplitude tends to zero; at the right-hand barrier (corresponding to uniform supercritical flows) the wave-length tends to infinity (case of the solitary wave). The diagram needs to be completed by a third barrier corresponding to ‘waves of greatest height’. This starts at the point marked Z and moves to infinity within the unshaded region without ever intersecting the left-hand barrier. The diagram may be used to determine what losses of momentum and energy a given stream can undergo, without departing from the condition of steady flow. Thus, one can determine the maximum wave resistance on a cylindrical obstacle athwart a given subcritical stream, or on a two-dimensional ‘step’ in the bed, in the absence of energy dissipation. Again, one can study the transition to a wave-train behind a bore. Such a wave-train (present in all but very strong bores) is shown to be capable of absorbing almost the whole energy which according to classical theory is liberated at the bore, although some minute residual dissipation of energy still appears to be necessary.

On cnoidal waves and bores

When any boundary layer on a straight wall is subjected to an expansive steady disturbance (either an incident wave, or departure of the wall shape from straight), or when a turbulent layer is subjected to a fairly weak compressive one, an interaction between the main stream (if supersonic) and the boundary layer is set up but the layer does not separate. Such an interaction is here treated mathematically by an extension of the author’s method (Lighthill 1950) of perturbing a parallel flow and neglecting the disturbances of the viscous forces. It is shown that these are non-negligible only in an ‘inner viscous sublayer’, in which the Mach number remains small, and which for the turbulent layer is inside the so-called laminar sublayer. Further, if the disturbances are Fourier-analyzed longitudinally, then the effect of the inner viscous sublayer on the behaviour of each harmonic component outside it is exactly as if there were a solid wall at a certain position in the stream, with no flow across it and inviscid flow outside it. This position depends on the wave number k and on the skin friction in the undisturbed layer. This new boundary condition is now inserted into the old theory. Solutions are obtained for large and small k and used to deduce, respectively, (i) the principal local features of both the wall-pressure distribution and the flow outside the boundary layer, and (ii) the extent of upstream influence. An interpolation between them is used in an attempt to predict the complete form of the reflected wave when a shock is incident upon a turbulent layer. Reasonable agreement with experiment is obtained. The paper ends with a discussion of the skin friction distribution and how it influences the onset of non-linearity.

Michael James Lighthill

Papers

Studies on Magneto-Hydrodynamic Waves and other Anisotropic Wave Motions

The response of laminar skin friction and heat transfer to fluctuations in the stream velocity

Contributions to the Theory of Heat Transfer through a Laminar Boundary Layer

On cnoidal waves and bores

On boundary layers and upstream influence II. Supersonic flows without separation