On kinematic waves I. Flood movement in long rivers
10 May 1955-Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences (The Royal Society)-Vol. 229, Iss: 1178, pp 281-316
TL;DR: In this article, the theory of a distinctive type of wave motion, which arises in any one-dimensional flow problem when there is an approximate functional relation at each point between the flow q and concentration k (quantity passing a given point in unit time) and q remains constant on each kinematic wave.
Abstract: In this paper and in part II, we give the theory of a distinctive type of wave motion, which arises in any one-dimensional flow problem when there is an approximate functional relation at each point between the flow q (quantity passing a given point in unit time) and concentration k (quantity per unit distance). The wave property then follows directly from the equation of continuity satisfied by q and k. In view of this, these waves are described as 'kinematic', as distinct from the classical wave motions, which depend also on Newton's second law of motion and are therefore called 'dynamic'. Kinematic waves travel with the velocity $\partial $q/$\partial $k, and the flow q remains constant on each kinematic wave. Since the velocity of propagation of each wave depends upon the value of q carried by it, successive waves may coalesce to form 'kinematic shock waves'. From the point of view of kinematic wave theory, there is a discontinuous increase in q at a shock, but in reality a shock wave is a relatively narrow region in which (owing to the rapid increase of q) terms neglected by the flow-concentration relation become important. The general properties of kinematic waves and shock waves are discussed in detail in section 1. One example included in section 1 is the interpretation of the group-velocity phenomenon in a dispersive medium as a particular case of the kinematic wave phenomenon. The remainder of part I is devoted to a detailed treatment of flood movement in long rivers, a problem in which kinematic waves play the leading role although dynamic waves (in this case, the long gravity waves) also appear. First (section 2), we consider the variety of factors which can influence the approximate flow-concentration relation, and survey the various formulae which have been used in attempts to describe it. Then follows a more mathematical section (section 3) in which the role of the dynamic waves is clarified. From the full equations of motion for an idealized problem it is shown that at the 'Froude numbers' appropriate to flood waves, the dynamic waves are rapidly attenuated and the main disturbance is carried downstream by the kinematic waves; some account is then given of the behaviour of the flow at higher Froude numbers. Also in section 3, the full equations of motion are used to investigate the structure of the kinematic shock; for this problem, the shock is the 'monoclinal flood wave' which is well known in the literature of this subject. The final sections (section section 4 and 5) contain the application of the theory of kinematic waves to the determination of flood movement. In section 4 it is shown how the waves (including shock waves) travelling downstream from an observation point may be deduced from a knowledge of the variation with time of the flow at the observation point; this section then concludes with a brief account of the effect on the waves of tributaries and run-off. In section 5, the modifications (similar to diffusion effects) which arise due to the slight dependence of the flow-concentration curve on the rate of change of flow or concentration, are described and methods for their inclusion in the theory are given.
Citations
More filters
Journal Article•
TL;DR: In this paper, a functional relationship between flow and concentration for traffic on crowded arterial roads has been postulated for some time, and has experimental backing, from which a theory of the propagation of changes in traffic distribution along these roads may be deduced.
Abstract: 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.
3,983 citations
TL;DR: The theory of kinematic waves is applied to the problem of estimating how a ‘hump’, or region of increased concentration, will move along a crowded main road, and is applicable principally to traffic behaviour over a long stretch of road.
Abstract: 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.
3,911 citations
TL;DR: In this paper, a simple representation of traffic on a highway with a single entrance and exit is presented, which can be used to predict traffic's evolution over time and space, including transient phenomena such as the building, propagation, and dissipation of queues.
Abstract: This paper presents a simple representation of traffic on a highway with a single entrance and exit. The representation can be used to predict traffic's evolution over time and space, including transient phenomena such as the building, propagation, and dissipation of queues. The easy-to-solve difference equations used to predict traffic's evolution are shown to be the discrete analog of the differential equations arising from a special case of the hydrodynamic model of traffic flow. The proposed method automatically generates appropriate changes in density at locations where the hydrodynamic theory would call for a shockwave; i.e., a jump in density such as those typically seen at the end of every queue. The complex side calculations required by classical methods to keep track of shockwaves are thus eliminated. The paper also shows how the equations can mimic the real-life development of stop-and-go traffic within moving queues.
2,781 citations
TL;DR: In this paper, a simple model that satisfies most of these criteria uses depth-averaged equations of motion patterned after those of the Savage-Hutter theory for gravity-driven flow of dry granular masses but generalized to include the effects of viscous pore fluid with varying pressure.
Abstract: Recent advances in theory and experimen- tation motivate a thorough reassessment of the physics of debris flows. Analyses of flows of dry, granular solids and solid-fluid mixtures provide a foundation for a com- prehensive debris flow theory, and experiments provide data that reveal the strengths and limitations of theoret- ical models. Both debris flow materials and dry granular materials can sustain shear stresses while remaining stat- ic; both can deform in a slow, tranquil mode character- ized by enduring, frictional grain contacts; and both can flow in a more rapid, agitated mode characterized by brief, inelastic grain collisions. In debris flows, however, pore fluid that is highly viscous and nearly incompress- ible, composed of water with suspended silt and clay, can strongly mediate intergranular friction and collisions. Grain friction, grain collisions, and viscous fluid flow may transfer significant momentum simultaneously. Both the vibrational kinetic energy of solid grains (mea- sured by a quantity termed the granular temperature) and the pressure of the intervening pore fluid facilitate motion of grains past one another, thereby enhancing debris flow mobility. Granular temperature arises from conversion of flow translational energy to grain vibra- tional energy, a process that depends on shear rates, grain properties, boundary conditions, and the ambient fluid viscosity and pressure. Pore fluid pressures that exceed static equilibrium pressures result from local or global debris contraction. Like larger, natural debris flows, experimental debris flows of ;10 m 3 of poorly sorted, water-saturated sediment invariably move as an unsteady surge or series of surges. Measurements at the base of experimental flows show that coarse-grained surge fronts have little or no pore fluid pressure. In contrast, finer-grained, thoroughly saturated debris be- hind surge fronts is nearly liquefied by high pore pres- sure, which persists owing to the great compressibility and moderate permeability of the debris. Realistic mod- els of debris flows therefore require equations that sim- ulate inertial motion of surges in which high-resistance fronts dominated by solid forces impede the motion of low-resistance tails more strongly influenced by fluid forces. Furthermore, because debris flows characteristi- cally originate as nearly rigid sediment masses, trans- form at least partly to liquefied flows, and then trans- form again to nearly rigid deposits, acceptable models must simulate an evolution of material behavior without invoking preternatural changes in material properties. A simple model that satisfies most of these criteria uses depth-averaged equations of motion patterned after those of the Savage-Hutter theory for gravity-driven flow of dry granular masses but generalized to include the effects of viscous pore fluid with varying pressure. These equations can describe a spectrum of debris flow behav- iors intermediate between those of wet rock avalanches and sediment-laden water floods. With appropriate pore pressure distributions the equations yield numerical so- lutions that successfully predict unsteady, nonuniform motion of experimental debris flows.
2,426 citations
Cites background from "On kinematic waves I. Flood movemen..."
...In experimental debris flows, larger waves tend to overtake and cannibalize smaller waves, as may be anticipated from kinematic wave theory [Lighthill and Whitham, 1955]....
[...]
TL;DR: It is proposed that the flow of pedestrians over the Jamarat Bridge be improved by appropriate barrier placement, that force an effective global view of the goals.
Abstract: The equations of motion governing the two-dimensional flow of pedestrians are derived for flows of both single and multiple pedestrian types. Two regimes of flow, a high-density (subcritical) and a low-density (supercritical) flow regimes, are possible, rather than two flow regimes for each type of pedestrian. A subcritical flow always fills the space available. However, a supercritical flow may either fill the space available or be self-confining for each type of pedestrian, depending on the boundary location. Although, the equations governing these flows are simultaneous, time-dependent, non-linear, partial differential equations, remarkably they may be made conformally mappable. The solution of these equations becomes trivial in many situations. Free streamline calculations, utilizing this property, reveal both upstream and downstream separation of the flow of pedestrians around an obstacle. Such analysis tells much about the nature of the assumptions used in various models for the flow of pedestrians. The present theory is designed for the development of general techniques to understand the motion of large crowds. However, it is also useful as a predictive tool. The behavior predicted by these equations of motion is compared with aerial observations for the Jamarat Bridge near Mecca, Saudi Arabia. It is shown that, for this important case, pedestrians, that is pilgrims, aim at achieving each immediate goal in minimum time rather than achieving all goals in overall minimum time. Typical of many examples, this case illustrated the strong dependence of path on the psychological state of the pedestrians involved. It is proposed that the flow of pedestrians over the Jamarat Bridge be improved by appropriate barrier placement, that force an effective global view of the goals.
1,002 citations
References
More filters
Journal Article•
TL;DR: In this paper, a functional relationship between flow and concentration for traffic on crowded arterial roads has been postulated for some time, and has experimental backing, from which a theory of the propagation of changes in traffic distribution along these roads may be deduced.
Abstract: 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.
3,983 citations
TL;DR: The theory of kinematic waves is applied to the problem of estimating how a ‘hump’, or region of increased concentration, will move along a crowded main road, and is applicable principally to traffic behaviour over a long stretch of road.
Abstract: 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.
3,911 citations
626 citations
TL;DR: In this paper, it was shown that discontinuous periodic solutions can be constructed by joining together sections of a continuous solution through shocks (or "bores") and that only one special continuous solution can be used as the basis for constructing discontinuous continuous solutions.
Abstract: The purpose of this paper is to obtain solutions which are periodic with respect to distance, describing the phenomenon called “roll-waves,” for water flow along a wide inclined channel, and to discuss the behavior of the mathematical solutions. The basic idea presented in Part I is that discontinuous periodic solutions can be constructed by joining together sections of a continuous solution through shocks (or “bores”). It is shown first that no continuous solutions can be periodic and that only one special continuous solution can be used as the basis for constructing discontinuous periodic solutions. The analysis is based upon the non-linear partial differential equations of the “shallow water theory,” augmented by the Cheay formula to allow for turbulent resistance. The Bresse profile equation is obtained in a form applicable for progressing wave flows. Shock conditions are derived for the case of an arbitrary continuous channel bed and for a flow subject to a resisting force. The special continuous solution is explicitly obtained and analyzed. Branches of it are then joined together through shocks. It is proved that roll-waves cannot occur either if the resistance is zero or if the resistance exceeds a certain critical value. As the resistance decreases, the size of the waves decreases also; and if the resistance becomes too large, the profiles reverse their direction and can no longer be joined by shocks. This critical value is reached when the (dimensionless) resistance coefficient equals one-fourth the value of the channel slope. The presence of a resistance force which varies merely with velocity is not sufficient to permit the construction of periodic solutions; the resistance must also act in such a manner that it decreases as the water depth increases. The analysis proves that the ratio of wave height to wave length of roll-waves is always independent of the speed of the waves. Explicit expressions for water height and shock height as functions of wave length are derived. The investigation studies the static discharge rate as a function of the wave speed, and asymptotic formulas for the wave speed in terms of the average discharge rate are derived. Twelve sets of curves arc presented, based on the equations obtained here, to illustrate the quantitative behavior of roll-waves; these may be used to check this theory against observed data. For prescribed values of slope, resistance, and wave speed, there is a one-parameter family of roll-wave solutions. If the wave length is also prescribed, the solution will then be unique.
270 citations