Saint Venant Equations

A modified and expanded form (Fread, 1988, 1992) of the original one-dimensional Saint-Venant equations [(1) and (2)] consist of the conservation of mass equation, i.e., dQ/dx + dsc(A + A0)/dt -q = 0 (29)

and the momentum equation, i.e., o[%smQ)/dt + d(PQ2/A)/dx] + gA(dh/dx + Sf + S^ + + L + WfB = 0 (30)

where Q is discharge, h is the water-surface elevation, A is the active cross-sectional area of flow, A0 is the inactive (off-channel storage) cross-sectional area, sr and sm are area-weighted and conveyance-weighted sinuosity factors, respectively (DeLong, 1986, 1989), which correct for the departure of a sinuous in-bank channel from the x-axis of the floodplain, x is the longitudinal mean-flow-path distance measured along the center of the waterway (channel and floodplain), t is time, q is the lateral inflow or outflow per lineal distance along the waterway (inflow is positive and outflow is negative), it is a numerical filter (0 < a < 1, usually a = 1) to enable the equations to properly handle mixed subcritical/supercritical flows (Fread et al., 1996) during the numerical solution (see the discussion on subcritical/supercritical mixed flow for more on a later in this chapter), fi is the momentum coefficient for nonuniform velocity distribution within the cross section, g is the gravity acceleration constant, Sf is the boundary friction slope, Scc is the expansion/contraction (large eddy loss) slope, and S, is the viscous dissipation slope for mud/debris flows.

Friction Slope. The boundary friction slope (Sf) is evaluated by rearranging the Manning Eq. (3) for uniform, steady flow into the following form:

in which n is the Manning coefficient of frictional resistance (Chow, 1959; Barnes, 1967; Arcement and Schneider, 1984; Jarrett, 1984; and Fread, 1989), R is the hydraulic radius, p is a units conversion factor (1.49 for U.S. units and 1.0 for SI), and K is the channel conveyance factor. The absolute value of Q is used to correctly account for the possible occurrence of reverse (negative) flows. The conveyance formulation is preferred (for numerical and accuracy considerations) for composite channels having wide, flat overbanks or floodplains in which K represents the sum of the conveyance of the channel (which is corrected for sinuosity effects by dividing by sm), and the conveyances of left and right floodplain areas.

When the conveyance factor (K) is used to evaluate Sf, the river channel/valley cross-sectional properties are designated as left floodplain, channel, and right flood-plain rather than as a composite channel/valley section. Special orientation for designating left or right is not required as long as consistency is maintained. The conveyance factor is evaluated as follows (Fread and Lewis, 1998):

where:

IxAcR{

in which the subscripts /, c, and r designate left floodplain, channel, and right floodplain, respectively.

Sinuosity Factors. The area-weighted and conveyance-weighted sinuosity factors (sc and sm, respectively) in Eqs. (29), (30), and (34) represent the ratio(s) of the flow-path distance along a meandering channel to the mean-flow-path distance along the floodplain. Initially, only one sinuosity factor (sk) is specified as varying only with each Jth depth of flow (J — 1,2,..., J, where J is the number of user-specified tabular top widths (5) versus h values, which describe the cross-section geometry), but then this is recomputed within the model according to the following relations:

in which AA = AJ+X — A,, and sk represents the sinuosity factor for a differential portion of the flow between the Jth depth and the J + 1th depth, and K is the conveyance factor.

4 DYNAMIC ROUTING MODEL 553 Expansion/ Contraction Effects. The term Sec is computed as follows:

in which kec is the expansion/contraction coefficient (negative for expansion/ positive for contraction), which varies from — 1.0/0.4 for an abrupt change in section geometry to —0.3/0.1 for a very gradual, curvilinear transition between cross sections. The A represents the difference in the term (Q/A)2 at two adjacent cross sections separated by a distance Ax. If the flow direction changes from downstream to upstream, kec can be automatically changed (Fread, 1988).

Large floods such as dam-break-generated floods usually have much greater velocities; it is important, especially for nonuniform channels (Rajar, 1978) to include in the Saint-Venant momentum equation (30) the expansion/contraction losses via the Scc term defined by equation (38). The ratio of expansion/contraction action losses (form losses) to the friction losses can be in the range of 0.01 < S^/Sj < 1.0. The larger ratios occur for very irregular channels with relatively small n values and for flows with large velocities (dam-break floods).

Momentum Correction Coefficient The momentum correction coefficient (/?) for nonuniform velocity distribution across the cross section is (Chow, 1959)

P = (Kf/A, + K2c/Ac + K2/Ar)/[{K, +KC+ Krf/(A, +AC+ Ar)} (39)

in which K is conveyance, A is wetted area, and the subscripts /, c, and r denote left floodplain, channel, and right floodplain, respectively. When floodplain properties are not separately specified and the total cross section is treated as a composite section, /? can be approximated as 1.0 < /? < 1.06 in lieu of Eq. (39). Also, in this case, Sc and Sm are set to unity in lieu of Eqs. (36) and (37).

Lateral Flow Momentum. The term L in Eq. (30) is the momentum effect of lateral flows and has the following form (Strelkoff, 1969): (a) lateral inflow, L = —qvx, where vx is the velocity of lateral inflow in the x direction of the main channel flow; (b) seepage lateral outflow, L = —0.5Q/A \ and (c) bulk lateral outflow, L = -qQ/A.

Mud or Debris Flows. The friction loss term (S,) is included (Fread, 1988) in the momentum equation (30) in addition to Sf to account for viscous dissipation effects of non-Newtonian flows such as mud or debris flows. Also, mine tailings dams, where the viscous contents retained by the dam have non-Newtonian properties, are dam-breach flood applications requiring the use of Sj in Eq. (30). This effect becomes significant only when the solids concentration of the flow is greater than about 40% by volume. For concentrations of solids greater than about 50%, the flow behaves more as a landslide and is not governed by the Saint-Venant equations. Si is evaluated for any non-Newtonian flow as follows (Jin and Fread, 1997):

in which y is the fluid's unit weight, r0 is the fluid's yield strength, D is the hydraulic depth (A/B), b = \/m where m is the exponent of the power function that fits the fluid's strcss(rs)-strain(ii'!;/ii'v) properties, and k is the apparent viscosity or scale factor of the power function, i.e., xs = r0 + K(dv/dy)m. The viscous properties, r0 and k, can be estimated from the solids concentration ratio of the mud flow (O'Brien and Julien, 1984).

Wind Effects. The last term (fVfB) in Eq. (30) represents the resistance effect of wind on the water surface (Fread, 1985, 1992); B is the wetted topwidth of the active flow portion of the cross section; and Wj = Vr\ Vr\cw, where the wind velocity relative to the water is Vr = Vw cos w + V, Vw is the velocity of the wind, positive (+) if opposing the flow velocity and negative (—) if aiding the flow, w is the acute angle the wind direction makes with the x-axis, V is the velocity of the unsteady flow, and cw is a wind friction coefficient (1 x 10"6 < cw < 3 x 10~6). This modeling capability can be used to simulate the effect of potential dam overtopping due to wind setup within a reservoir by applying the Saint-Venant equations to the unsteady flow in a reservoir.

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