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Conservation laws

Conservation laws - 2 More on Conservation Laws In this...

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2 More on Conservation Laws In this section, we expand on the tra c 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 di ff erential 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 tra c 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 cross–sections of the river at positions x 1 and x 2 is given by V = x 2 x 1 S ( x, t ) dx , where S is the cross–sectional area of the river up to the free surface of the water, volume conservation takes the form S t + Q x = 0 , (21) entirely analogous to (2), with S playing the role of the car density ρ , and Q ( x, t ) representing the volume flow per unit time through the river’s cross–section at position x and time t : Q ( x, t ) = S ( x,t ) u ( x, y, z, t ) dy dz. Here u is the component of the fluid velocity normal to the cross-section. As for tra c flow, we have the kinematic constraint that Q = S U , 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 cross–sections 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 well–known), 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
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 S Q = S 2 /2 A hydrological law Q(S) 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 di ff erence with the tra c 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.
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