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e e e inlo the measured responses. And if, in fact, the reach does behave sufficiently like the calibration events for the flood being studied, the hydrologic approach may be nearly as accurate as any of the hydraulic routing schemes for determining discharge. The difficulty, of course, is in establishing the storage versus flow relation pertinent to the subject flood. 5-12. Unsteady Flow Model a. Unsteady flow equations. Derivations of the unsteady flow equations are presented in numerous refer- ences. Chow (1959), Fread (1978), and User's Manual for UNET (U.S. Army Corps of Engineers 1991b)] are three of such references. They can be obtained from the two-dimensional equations presented in Chapter 4 by assuming that the dependent variables only change in one direction, x, and that direction is along the river axis rather than being a cartesian coordinate. Common for- mulations of the equations are as follows: Equation of continuity ilQ ilA dS_ dX + dt + dt - qL (5-2) Equation of momentum ilQ iJ(QV) (ilZ ) dt +~ +gA dX +Sf =qL VL (5-3) where Q = flow A = active flow area S = storage area qL = lateral inflow per unit flow distance V = Q I A = average flow velocity g = acceleration of gravity Z = water surface elevation Sf = friction slope V L = average velocity of the lateral inflow x = flow distance t = time (1) The assumptions implicit 10 the unsteady flow equations are essentially the same as those for the steady flow equations: (a) the flow is gradually varied; that is, there are no abrupt changes in flow magnitude or direc- tion; (b) the pressure distribution is hydrostatic; therefore, the vertical component of velocity can be neglected. This means, for example, that the unsteady flow EM 1110-2-1416 15 Oct 93 equations should not be used to analyze flow over a spillway, and (c) the momentum correction factor is assumed to be 1. (2) The magnitude of each of the terms in the momentum equation plays a significant role in the hydraulics of the system. The terms in equation 5-3 are: ~ = local acceleration ~ = advective acceleration ilz dX = wa'er swface slope Sf = friction slope The water surface slope can be expressed as ilz=ilh_S dX dX 0 (5-4) in which h is the depth 3nd iJh dX = pressure term So = bed slope The roles of these terms are discussed below. b. Weaknesses of the unsteady flow equations. (1) Friction slope is the portion of the energy slope which overcomes the shear force exerted by the bed and banks, and it cannot be measured. To quantify the fric- tion slope, the Manning or Chezy formulas for steady flow are used: Manning's Equation S _ QIQln2 f - 2.21A 2R 413 (5-5) where 5-23