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. -' . . hL Sh "'- dx in which hL is the head loss due to the force and dx is the distance over which the loss occurs. (5-7) (2) Since energy loss. is obtained by integrating force applied over distance, Equation 5-7 estimates an addi- tional energy slope to overcome the force. The added slope produces the correct swell head upstream of the structure. The augmented momentum equation now becomes: aQ + d(QV) dt dX + g A (i + Sf + Sh) '" qL VL d. Subcritical and supercritical flow. The unsteady flow equations are wave equations. Disturbances propa- gate according to the rate (5-8) dx _"'V:i:c dt (5-9) where c = the celerity of a gravity wave c = (gD)~ D = hydraulic depth (1) If V < c, the flow is subcritical, and disturbances move both upstream and downstream. Hence, a distur- bance downstream, such as a rise in stage, propagates upstream. If V> c, the flow is supercritical, and the velocity sweeps all disturbances downstream. Hence, a stage disturbance downstream is not felt upstream. (2) Equation 5-9 has profound implications for the application of the unsteady flow equations. Subcritical flow disturbances travel both upstream and downstream; therefore, boundary conditions must be specified at both the upstream and downstream ends of the routing reach. For supercritical flow, the boundary conditions are only specified at the upstream end. (3) Near critical depth, the location for the boundary conditions is changing; hence, the flow and the numerical solution may become unstable. Instability when the depth is near critical is one of the greatest problems encountered when modeling unsteady flow. Most streams which are modeled with unsteady flow are EM 1110-2.1416 15 Oct 93 subcritical at higher stages but, at lower stages the pool and riffle sequence usually dominates flow. Supercritical flow can occur at the riffles. Because unsteady flow models simulate the full range of flow, the models can become unstable during low flows. e. Numerical models. An unsteady flow model (also called a dynamic wave model) solves the full momentum and continuity equations. Forces from all three sources (gravity, pressure, and friction) and the resulting changes in momentum (local and advective accelerations) are all explicitly considered along with mass conservation. If the assumption of one-dimensional flow is justified, and the discretization of flow variables introduces little error, then the simulation results are as accurate as the input data. Unsteady flow models differ in their underlying physical assumptions, in the way in which the real con- tinuous variation of flow variables with space and time is approximated or represented by discrete sets of numbers, and in the mathematical techniques used to solve the resulting equations. Other differences reflect the range of different steam networks, channel geometries, control structures, or flow situations that the model is designed to simulate. For example, not all dynamic wave models are equipped to handle supercritical flow; a typical indi- cation of failure is oscillating water surface profiles and an aborted execution. There are also differences (which can strongly effect study effort) in input data structure, user operation, documentation, user support, and presen- tation of results. (1) Such a model can accurately simulate flows in which acceleration plays an important role, such as flood waves stemming from sharply rising hydrographs such as a dam break flood; disturbances of essentially still water, for example the drawdown of water in the reservoir behind a ruptured dam; and seiching, which is a long period longitudinal oscillation of water. Another example of a situation that can be modeled only by a dynamic wave model is the reflection of a dam break flood wave from a channel constriction. (2) As the bed slope becomes small, it becomes less important than the water surface slope and the accelera- tion tenns play a greater role. The looped rating curve is an example of this phenomenon. For streams of! a low slope, the rising limb of the hydro graph passes ata lower stage than the falling limb for a particular discharge. The loop for the Illinois River at Kingston Mines during the December 1982 flood is shown in Figure 5-19. The flow and stage hydrographs were shown in Figure 5-8. 5-25