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Unformatted text preview: 2 More on Conservation Laws In this section, we expand on the traffic flow model of section 1. First we see how an entirely similar model can be applied to river flow. In this context, we see some of the model's limitations, which motivates us to switch from a single conservation law to systems. We do not pursue the mathematical study of such systems any further though, since this would take us too far afield: the use of partial differential equations is only one of many possible approaches to the modeling of natural phenomena, though a very fruitful one; we need to save time and energy for the study of others. 2.1 Flood waves: a kinematic model To see that the analysis developed in section 1 has applications far beyond traffic flow, we extend it to long waves in rivers. Here the principle of car conservation is replaced by volume conservation along the river, which works well under the assumption of negligible evaporation, infiltration and rain. Since the volume between two crosssections of the river at positions x1 and x2 is given by x2 V = S(x, t) dx ,
x1 where S is the crosssectional area of the river up to the free surface of the water, volume conservation takes the form St + Q x = 0 , (21) entirely analogous to (2), with S playing the role of the car density , and Q(x, t) representing the volume flow per unit time through the river's crosssection at position x and time t: Q(x, t) = u(x, y, z, t) dy dz.
S(x,t) Here u is the component of the fluid velocity normal to the cross-section. As for traffic flow, we have the kinematic constraint that Q=SU, where U is the mean waver speed across the section. In order to close the equation (21), we may want to invoke a relation between Q and S. Hydraulic engineers denote such a relation a hydrological law. This is customarily measured in various crosssections of many of the world's main rivers: a vertical stick measures the water height h (a surrogate for the area S, if the geometry of the river bed is wellknown), while a variety of devices are used for measuring the water speed u at various points; integrating these velocity measurements across S yields an accurate estimate for Q. 15 A hydrological law Q(S) 2.5 2 1.5 Q = S2/2 1 0.5 0 0 0.2 0.4 0.6 0.8 1 S 1.2 1.4 1.6 1.8 2 Figure 7: Example (Q = S 2 /2) of a hydrological law for a river cross-section. We shall assume, therefore, that a functional relation Q = Q(S) exists. An important difference with the traffic model, is that U is typically an increasing function of S, and so Q(S) is a convex function (see figure 7). The reason for this is that the mean speed U follows from a balance between two forces: gravity, pushing the water downslope toward the sea, and lateral friction. Since the latter is proportional to the wetted perimeter of the river, while the former is a body force, proportional to the area, the mean speed grows as the water level increases. What dynamical consequences does this upward concavity bring? As for traffic flow, we have characteristic lines, given by dx = Q (S) , dt along which the water height remains constant. Notice that, unlike the traffic case, all characteristics have positive slope; i.e., they move downstream. Moreover, dx U , so information travels faster than the water particles themselves. dt Hence fine suspensions and floating objects are caught by these long waves from behind. Finally, when shocks form, it is regions with higher values of S that catch up with ones with lower water levels, so shock waves arise during floods. Such discontinuous and violent flood waves are often observed in mountain rivers, sometimes with tragic consequences for nearby campers, taken by surprise by these nearly vertical walls of water coming with no notice, typically due to rain or thaw far upstream. The end of a flood, on the other hand, is much more gradual, as it is brought up by a smooth rarefaction wave. 16 1 0.9 t=0 t=4 0.8 t=8 0.7 S t=12 0.6 0.5 0.4 0.3 0 1 2 3 x 4 5 6 7 Figure 8: Discontinuous flood wave produced by the inviscid Burgers equation. A numerical run with Q = S 2 /2 and initial data S = 0.3(2 + sin(x)), in a periodic domain, is shown in figure 8. One sees a sharp discontinuity developing from smooth initial data, and then slowly decaying as characteristics carrying extreme data "die" at the shock. Equation (21) with this particular choice of Q(S) is called the inviscid Burgers, or Hopf equation. 2.2 Insufficiency of the kinematic model The kinematic model developed above is quite useful for fast computations of the main characteristics of large flood waves. Hydraulic engineers have in fact made much use of models of this kind, which in the hydraulic literature go under the name of Muskingham's. However, there is abundant evidence that such simple models cannot capture many important ingredients of real river flows: For very long waves, the water height h(S) and the volume flux Q are definitely related. However, they are not quite functions of each other. Figure 9 sketches typical measurements, which show a hysteresis phenomenon: Q does not depend only on S, but also on its history. Typically, for the same value of the wetted area S, the flux Q will be larger at the beginning of a flood, when S is growing, than at the end, when it is decreasing. The reason is that, as the flood starts, the water level is much higher upstream than downstream, and so gravity acts strongly. Later on, the downstream area is also flooded, while upstream is actually starting to recover. This yields a smaller surface slope, and hence reduces 17 Hydrological law Q(S) with hysteresis 0.7 0.6 0.5 0.4 Q 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 S 1.2 1.4 1.6 1.8 2 Figure 9: Hydrological law with hysteresis. the resulting water speed. All shocks of the kinematic model correspond to flood waves traveling downstream. As mentioned above, these shocks are observed in mountain rivers. However, shocks (bores in hydraulic terminology) are also observed traveling upstream, carrying the information from high tides in the ocean downstream (a famous example is the Mascaret wave of the Seine). Big reversed standing shocks, denoted hydraulic jumps, also occur downstream of a dam's spillway (and, in a much smaller scale, also on dishes under a faucet, though these have circular shape.) This suggests that there is a second characteristic family, giving the problem a certain amount of leftright symmetry. In the following subsection, we construct a more complex model that includes this second characteristic family, and the associated phenomena. For simplicity though, we do not include in this model the frictional and bottom slope effects that gave rise to the hydrological law Q(S). Hence our kinematic model will not be derivable as a limit of the more complex one; rather, the two models should be thought of as standing side by side, representing different aspects of the rich phenomenology of river flow. 18 2.3 Shallow waters and gas dynamics For simplicity, we consider a rectangular channel with constant width b. Then the wetted area S is given by S = bh, where h represents the water height, and the water flux by Q = bhu. (We switch to lower case for u just for notational homogeneity.) Then the volume conservation equation (21) becomes ht + (h u)x = 0 . Instead of closing this equation with an hydrological law, as before, we take now a more fundamental approach, whereby we write the equation for conservation of momentum: (h u)t + h u2 + P x = 0 . Here bhu is the momentum density (mass times velocity per unit length of the river), where the constant water density and width b have been factored out of the equation. Multiplying this by the speed u yields the momentum flux hu2 , similar to the volume flux Q = hu in the previous equation. The variable P represents the area integral of the pressure forces acting between the fluid parcels at the left and right of a river's crosssection. Unfortunately, in writing down this new equation, we also brought in the new unknown P . In order to close the system, we invoke now the hydrostatic approximation, valid for fluids at rest, whereby the pressure at each point is given by the fluid weight per unit area above it. Under this hypothesis, it follows that the integral of the pressure is given by P = g h2 /2, where g is the gravity constant, and the equations for mass and momentum conservation constitute a closed system, the onedimensional shallow water equations: ht + (h u)x (h u)t + h u + g h2 /2 x 2 = = 0 0. (22) (23) This is an example of a system of conservation laws. Another prototypical example is given by onedimensional isentropic gas dynamics: t + ( u)x ( u)t + u2 + P x = = 0 0. (24) (25) Here is the gas density, u its speed, and P = P () its pressure at constant entropy. As for the kinematic model for rivers, this model for gas dynamics can be shown to fail in the presence of big shocks, for which the isentropic assumption does not hold. When this is the case, the next natural step is to introduce the temperature T , and use the full equation of state P (, T ). Since T 19 is a new unknown, a new equation is required. This is provided by the principle of energy conservation e + u2 /2 t + e u + u3 /2 + P u x = 0 , (26) where e = e(T ) is the internal ("thermal") energy of the gas, u2 /2 the kinetic energy density, and P u the work per unit time performed by the pressure. Notice a common feature of all of our model derivations so far: we start with one equation in two unknowns (typically some kind of mass conservation), and, as we add new equations, they bring in new unknowns. At some point, we decide to call it a day, and introduce a closure assumption, valid under some local equilibrium approximation. For traffic and the kinematic flood models, we did this already at the level of the mass equation, introducing a flux law Q() or Q(S), valid near equilibrium, when the solution is nearly constant in space and time. For the shallow water and the isentropic gas dynamical models, we closed the momentum equation, by introducing a hydrostatic or polytropic law P (S) or P (), again valid under nearly uniform conditions. For real gases, we went one step further, and closed the equations at the energy level, introducing the equation of state P = P (, T ), valid under the assumption of thermodynamical equilibrium. For very dilute gases or very violent shocks, this hypothesis may fail as well, and we may need to move toward a more complex system, such as the Boltzman equations or, in the case of water waves, to the full two dimensional Euler equations. An area of research that has received much recent attention, multiscale simulation, involves replacing the equilibrium closures by more sophisticated statistical representations of the fluxes. We might see more about this developing field by the end of the course. 2.4 General systems of conservation laws The traffic flow equation (2), the kinematic model for river flows (21), the shallow water system (22, 23), and the gas dynamic equations (24, 25, 26), are all systems of conservation laws. In one spatial dimension, these take the general form Ut + F (U )x = 0 , (27) where U = U (x, t) is the n-dimensional vector field of densities of the corre sponding conserved quantities U dx, and F (U ) is a vector function of U with n components, the corresponding fluxes. The associated, more fundamental integral form is (U dx - F dt) = 0 (28) for any domain in the (x, t)-plane, a straightforward generalization of (10). In areas where the solution is smooth, equation (27) can be rewritten in the form Ut + A(U ) Ux = 0 , (29) 20 where A, an n n matrix, is the Jacobian of F : Ai,j = Fi . Uj The next, concluding paragraphs, are intended only for those students very familiar with linear algebra, particularly with eigenvalues and left and right eigenvectors: The system (29) is called hyperbolic when all the eigenvectors of the matrix A are real, and the corresponding eigenvectors span Rn . All the systems that we have written down so far belong to this category under reasonable conditions (for instance, that the water height h be positive.) Hyperbolic systems can be written in a particularly revealing form. To this end, consider a complete set of left eigenvectors of A, lj , with corresponding eigenvalues j :
t t lj A = j lj , where a superindex t denotes the transpose. Multiplying the system (29) on the left by lj , we obtain its characteristic form
t t l j U t + j lj U x = 0 , (30) a scalar equation reminiscent of (5). A further study of systems of conservations laws, beyond the scope of this class unless some of you want to pursue it for a final project, would ask to what extent this analogy carries through to determining the information flow along characteristics, and how much it tells us about the solution U (x, t). 21 ...
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This note was uploaded on 03/25/2011 for the course MATH 250 taught by Professor Tabak during the Spring '11 term at NYU.
- Spring '11