Shock Waves and Continuum Flow Models

The following excerpt is taken from Chapter 5 (pp. 1-4) of the Transportation Research Board Special Report on Traffic Flow Theory, published on the website http://www.tfhrc.gov/its/tft/tft.htm.

Since the conservation equation describes flow and density as a function of distance and time, one can immediately see that continuum modeling is superior to input-output models used in practice (which are only one dimensional, because they essentially ignore space). In addition, because flow is assumed to be a function of density, continuum models have a second major advantage, (e.g. compressibility). The simple continuum model referred to in this text consists of the conservation equation and the equation of state (speed-density or flow density relationship). If these equations are solved together with the basic traffic flow equation (flow equals density times speed), then we can obtain speed, flow, density at any time and point of the roadway. Knowing these basic traffic flow variables we know the state of the traffic system and can derive measures of effectiveness, such as delays stops, total travel, travel time, and others that allow engineers to evaluate how well the system is performing. . . .

A shock wave is a discontinuity of flow or density, and has the physical implication that cars change speeds abruptly without time to accelerate or decelerate. This is an unnatural behavior that could be eliminated by considering high order continuum models. These models add a momentum equation that accounts for the acceleration and inertia characteristics of traffic mass. In this manner, shock waves are smoothed out and the equilibrium assumption is removed. . . . In spite of this improvement, the most widely known high order models still require an equilibrium speed-density relationship. . . .

. . . where u_f represents the free flow speed and k_j the jam density . . . is a first order quasi-linear, partial differential equation which can be solved by the method of characteristics. . . . In practical terms, the solution . . . suggests that:

• The density k is a constant along a family of curves called characteristics or waves; a wave represents the motion (propagation) of a change in flow and density along the roadway.
• The characteristics are straight lines emanating from the boundaries of the time-space domain.
• The slope of the characteristics is:
  
• This implies that the characteristics have slope equal to the tangent of the flow-density curve at the point representing the flow conditions at the boundary from which the characteristic emanates.
• The density at any point x,t of the time space domain is found by drawing the proper characteristic passing through that point.
• The characteristics carry the value of density (and flow) at the boundary from which they emanate.
• When two characteristic lines intersect, then density at this point should have two values which is physically unrealizable; this discrepancy is explained by the generation of shock waves. In short, when two characteristics intersect, a shock wave is generated and the characteristics terminate. A shock then represents a mathematical discontinuity (abrupt change) in k, q, or u.
• The speed of the shock wave is:

  

  . . . where k_d, q_d represent downstream and k_u, q_u upstream flow conditions. In the flow concentration curve, the shock wave speed is represented by the slope of the line connecting the two flow conditions (i.e., upstream and downstream).

It should be noted that when u_w is positive, the shock wave moves downstream with respect to the roadway; conversely, when u_w is negative, the shock is moving upstream. Furthermore, the mere fact that a difference exists in flow conditions upstream and downstream of a point does not imply that a shock wave is present unless the characteristics intersect. Generally this occurs only when the downstream density is higher than upstream. When density downstream is lower than upstream, we have diffusion of flow similar to that observed when a queue is discharging. When downstream density is higher than upstream, then shock waves are generated and queues are generally being built even though they might be moving downstream.

Figure 5.2 (not included here), taken from Gerlough and Huber (1975), demonstrates the use of traffic waves in identifying the occurrence of a shock wave and following its trajectory. The process follows the steps of the solution of the conservation equation as outlined above. The top of the figure represents a glow-concentration curve; the bottom figure represents trajectories of the traffic waves. On the q-k curve, point A represents a situation where traffic flows at near capacity implying that speed is well below the free-flow speed. Point B represents an uncongested condition where traffic flows at a higher speed because of the lower density. Tangents at points A and B represent the wave velocities of these two situations. The areas where conditions A and B prevail are shown by the characteristics drawn in the bottom of Figure 5.2. This figure assumes that the faster flow of point B occurs later in time than that of point A; therefore, the characteristics (waves) of point B will eventually intersect with those of point A. The intersection of these two sets of waves has a slope equal to the chord connecting the two points on the q-k curve, and this intersection represents the path of the shock wave shown at the bottom of Figure 5.2.

It is necessary to clarify that the waves of the time-space diagram of Figure 5.2 are not the trajectories of vehicles but lines of constant flow and speed showing the propagation of conditions A and B. The velocities of individual vehicles within A and B are higher because the speed of the traffic stream is represented by the line connecting the origin with A and B in the q-k curve.