Traffic as a Fluid Flow

Whenever there is a flow of objects such as pedestrians [1] or cars [2], they can be described on a macroscopic scale (on average) as a flow of a fluid. Fluid flow is described by the Navier-Stokes equations. The equation defines a velocity field \vec{u}(\vec{x},t), which satisfies

\frac{\partial \vec{u}}{\partial t}+\vec{u}\cdot \vec{\nabla}\vec{u}+\frac{1}{\rho}\vec{\nabla}P = \frac{\vec{f}}{\rho}.

Here, \rho is the density of the fluid, P is the pressure and \vec{f} is any force per unit volume acting on the fluid. As a side note, it turns out mathematicians study these equations a good bit as well (perhaps more than physicists). The existence and smoothness of the solutions to these equations are unknown in certain cases. If someone can prove this rigorously, he or she can claim a million dollar prize.

This equation is paired with the continuity equation, which is a statement that fluid cannot pile up. All fluid that comes in somewhere must leave. Mathematically, this is:

\frac{\partial \rho}{\partial t}+\vec{\nabla}\cdot(\rho \vec{u}) = 0.

With both of these, the velocity field of the fluid can be solved for.

Let us consider the case of cars moving across the Bay Bridge. This is a straight path of about 4.5 miles where cars only enter or exit on the ends (assuming no cars, or very few cars, get off on Yerba Buena Island). Traffic can really only flow in one direction, so I will take the one-dimensional forms of the equations above.

The density, \rho is simply the number of cars in a certain volume of space. The forcing term tends to speed up cars toward an optimal speed, \tilde{u}. When there are few cars on the road this optimal speed will be pretty high, most likely a little over the speed limit. However, as the density of cars gets higher, the optimal speed will lower [2]. The pressure can be empirically determined, but as expected it tends to increase with larger densities. With all this taken into account, the traffic Navier-Stokes equations are:

 \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x}(\rho u) = 0

 \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x}+\frac{1}{\rho}\frac{\partial P}{\partial x} = \frac{\tilde{u}-u}{\tau}

where \tau is a relaxation time, which sets the scale of how quickly the fluid can get up to the optimal speed.

I will consider steady state solutions which will be applicable whenever a certain stimulus has been present for a long time. Of course it can be interesting to see what how the system gets there. For example, if there were an accident that occurred, it could be interesting to see how long it takes for the other side of the bridge to feel the effect of the accident. To keep it simple, however, I will consider the case when the accident has occurred and been there a while but not cleaned up yet. This means that all of the time derivative terms will be zero. This is nice because the continuity equation now becomes (after dividing through by \rho)

 \frac{1}{\rho}\frac{\partial \rho}{\partial x} = -\frac{1}{u}\frac{\partial u}{\partial x}.

Now I will put in the speed of sound, which is the derivative of the pressure with respect to \rho. Plugging this into the Navier-Stokes equation gives

 u \frac{\partial u}{\partial x}-\frac{c^2}{u}\frac{\partial u}{\partial x} = \frac{\tilde{u}-u}{\tau}.

Now this is just a first order equation for u without any dependence on \rho. It can easily be solved numerically. For the example on the bridge, I set the target speed \tilde{u} to be 65 mph. The relaxation time \tau should be related to how quickly cars can respond to external stimuli, so I take it to be a few seconds, as this seems like a reasonable time scale for a human driving the car. The speed of sound I took to be a free parameter and adjusted it to get reasonable dynamics. I assumed that once a car leaves the toll booth, it takes about half a mile or half a minute to reach full speed. This happened to make c about 100 mph. The speed as a function of distance along the bridge is pretty uninteresting in this case. The cars move up to maximum speed and stay there.


The total time on the bridge can be found from the definition that u(x) = \frac{dx}{dt}. Integrating this relation gives

 t = \int_0^L \frac{dx}{u(x)}

where L is the length of the bridge. In this model it takes about 4.5 minutes to cross the bridge.

Now there are various modifications that can be made to this model. If there is a slope or curve in the bridge, cars will slow down. This can be factored in by making \tilde{u} a function of position on the bridge. The ideal speed can be lower for the regions one would expect cars to go slower. For a slope, one could even include the gravitational force as the forcing term in the Navier-Stokes equation, though this would reintroduce the dependence on density.

To test this, suppose for the last two miles of the bridge there are fireworks visible over San Francisco. There may be a tendency for people to look at this and slow down. To model this, I take the ideal speed to be 65 mph for the first 2.5 miles but 45 mph for the last 2. This results in the velocity looking like the plot shown. The time of the trip increases to 5.3 minutes.

In a steady state solution like this, we probably wouldn't expect to see a sharp transition like this, as we'd expect the cars on the edge to figure it out earlier and not have to stop so abruptly. Because of this, we need a more sophisticated model of the target speed function. I would rather have it go smoothly from 65 mph to 45 mph. I used a fourth order interpolating polynomial to create a function like this between x=0 and x=2.5. Here is a plot of the more realistic model.

The commute time changes a little bit between the models. It takes about 5.8 minutes to cross the bridge like this.

If the time dependent parts were kept, there might be some sign of evolution from the first kind of plot with the sharp edge to something smoother like the one above. This could be something interesting to look at. Others have studied how traffic jams form in models like this, in particular looking at soliton solutions that the authors have coined jamitons [2]. You can see a live jamiton forming in this video!

Here is the Mathematica file I used for this: Traffic.nb

References:
1. Appert-Rolland, C., Modeling of pedestrians. arXiv:1407.4282.
2. Flynn, M.R., et al., Self-sustained nonlinear waves in traffic flow. Physical Review E, 79:056113.