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<br /> <br />EM 1110-2.1416 <br />15 Oct 93 <br /> <br />455 <br /> <br />450 <br /> <br />~ <br />z <br /> <br />...., <br />-. 445 <br />c <br />.2 <br />...., <br />o <br />> <br />Q) <br />LaJ <br /> <br />440 <br /> <br />435 <br />o <br /> <br />. <br /> <br />20 <br /> <br />40 60 <br />Discharge. 103 cfs <br /> <br />80 <br /> <br />100 <br /> <br />. <br /> <br />Figure 5.19. Looped rating curve for the Illinois River at Kingston Mines, 15 Nov 82 031 Jan 83 <br /> <br />The peak flow always precedes peak stage. The loop can <br />be explained with the help of Figure 5-20. The slope of <br />the water surface is greater on the rising limb than on the <br />falling limb, thus the flow is accelerating on the rise and <br />decelerating on the fall. <br /> <br />(3) If the flow changes rapidly, then the acceleration <br />tenns become important regardless of the slope of the <br />bed. The advective acceleration tenn diffuses the dis- <br />charge downstream; it lengthens and attenuates any rapid <br />change in discharge. Figure 5-21 shows a test of routing <br />a rapidly rising and falling hypothetical hydrograph <br />through a channel of unit width using an unsteady flow <br />model. In 8,000 feet the peak discharge had attenuated <br />by over a third and the hydrograph had lengthened <br />dramatically. This is typical of dam break type waves. <br /> <br />f. Numerical approximations. Discretization, the <br />representation of a continuous field of flow by arrays of <br />/ <br /> <br />5-26 <br /> <br />discrete values, is a major concern in the construction of <br />unsteady flow models. The choice of scheme influences <br />the ease of writing, correcting, and modifying the pro- <br />gram; the speed at which the program executes; accuracy <br />of the solution, including satisfaction of volume conser- <br />vation, momentum conservation, and computation of <br />proper wave velocities; the robustness of the model; and <br />ultimately, its stability~ <br /> <br />(1 ) Explicit solution schemes allow the computation <br />of flow variables at the end of a time step at one point in <br />the channel, independent of the solution for neighboring <br />points. Implicit schemes solve for the flow variables at <br />the end of a time step at all points in the channel simul- <br />taneously. The former are easier to program and main- <br />tain, but require small time steps to avoid computational <br />/ instability. The required size of the time steps for <br />usually much less than that needed to resolve the rates at , <br /> <br />. <br />