33.6. Macroscopic Flow Models

If one looks into trac ow from a very long distance, the ow of fairly heavy trac appears like a stream of a uid. Therefore, a macroscopic theory of trac can be developed with the help of hydrodynamic theory of uids by considering trac as an eectively one-dimensional compressible uid.

Macroscopic flow models simplify the understanding of traffic flow by comparing it to fluid dynamics. When we consider heavy traffic from a distance, the flow resembles a fluid stream, allowing for theories from fluid dynamics to apply to traffic. This means that the intricacies of individual vehicle behavior are less important than the overall flow of a large number of vehicles, which can be analyzed similarly to how fluids behave.

Think of a river flowing steadily. Just as we observe how the river moves as a whole rather than the individual drops of water, macroscopic flow models allow us to look at traffic as a large flow, simplifying the complex interactions between vehicles.

The behaviour of individual vehicle is ignored and one is concerned only with the behaviour of sizable aggregate of vehicles. The earliest trac ow models began by writing the balance equation to address vehicle number conservation on a road.

In these models, it is crucial to apply the principle of conservation of vehicles. This principle states that the total number of vehicles must remain constant along a road segment, which means that the number entering must equal the number exiting. This is often described using mathematical equations that represent how vehicle density (the number of vehicles in a certain length of road) and flow (the number of vehicles passing a point in a given time) are related.

Imagine a line of cars at a toll booth. As cars enter the toll lane and pay, cars leave the line at the same rate, illustrating the conservation principle. The number of cars in the toll lane is a reflection of both the entering traffic (cars approaching) and exiting traffic (cars leaving after passing the toll).

Assuming that the vehicles are owing from left to right, the continuity equation can be written as ∂k(x,t) ∂q(x,t)

• =0 (33.15)

The continuity equation is a mathematical representation of the conservation of vehicles. In this equation, k(x,t) represents the density of vehicles at a certain point and time, while q(x,t) represents the vehicle flow. The equation essentially states that any change in vehicle density over time must match the change in flow, ensuring no vehicles are lost. This is crucial for understanding how traffic behaves over time and space.

Picture a funnel. As you pour sand into the funnel (creating density), it pours out at the bottom (flow). If you pour too quickly (increase density), the outflow can't keep up, just as in traffic if too many cars enter a road segment, congestion occurs.

One possible solution is to write two equations from two regimes of the ow, say before and after a bottleneck. In this system the ow rate before and after will be same, or k v =k v (33.16)

When analyzing traffic flow, bottlenecks (like road narrowing or an accident) lead to different flow dynamics before and after the obstruction. By establishing that the flow rate before the bottleneck equals the flow after, it is possible to develop an equation that can help predict and analyze what occurs in traffic as it approaches and passes through these points.

Imagine a busy highway that merges into a single lane. The traffic density increases (more cars in a shorter space) as it approaches the bottleneck and you can think of it as cars lining up to enter a narrow tunnel. The flow of cars entering the tunnel must equal the flow coming out on the other side.

From this the shockwave velocity can be derived as q q 2 1 v(t ) = − (33.17) o p k k 2 1 −

Shockwaves in traffic occur when there's a change in flow conditions, similar to a disturbance in a fluid. The formula captures how quickly the change in density affects flow. When a sudden slowdown occurs (due to an accident or traffic light), the impact will send a 'shockwave' backward through the traffic, affecting vehicles behind the blockage. This equation helps calculate that velocity by looking at the difference in flow before and after the event.

Think of a wave created when a pebble is thrown into a pond. The ripples spread outwards; similarly, a sudden stop in traffic causes a 'traffic wave' that spreads back through the line of vehicles. The formula gives us a way to measure how fast that wave travels as it affects cars further back on the road.

An alternate possibility which Lighthill and Whitham adopted in their landmark study is to assume that the ow rate q is determined primarily by the local density k, so that ow q can be treated as a function of only density k.

Lighthill and Whitham proposed a model where flow rate is dependent solely on local density, simplifying the system by reducing the number of unknowns. This means that if you know the density of vehicles at a certain point, you can directly determine the flow rate without needing additional information. It refines the prediction of how traffic behaves under varying densities.

Imagine a crowded movie theater. The number of people wanting to enter the theater (flow) depends on how many seats are available (density). If you know the number of seats (density), you can predict how many people will be able to enter without needing to count them individually.

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