Open boundary condition for unsteady open-channel flow K. Mizumura Civil Engineering Department, Kanazawa Institute of Technology, 7-1 Ogigaoka, Nonoichimachi, Ishikawa Pref. 921, Japan Abstract Initial and boundary conditions are needed to solve the unsteady flow equations for flood routing in rivers. As the boundary conditions, inflow discharge hydrograph at the upstream boundary and a rating curve (stage - discharge relationship) at the downstream boundary are employed. But in the computation of long wave propagation the open boundary condition is applied at the boundary and a rating curve is not necessary as in the computation of flood routing. Since the governing equations of the unsteady flow and the long wave propagation are essentially the same, the open boundary condition in the case of the computation of the long wave propagation is applicable to the case of the computation of flood routing. Introduction The computation of the flood routing is the most important subject in flood control of rivers. It is very evident from that EEC 2 model were developed recently in the United States. The usual methods to solve the governing equations for the unsteady flow are the finite difference method or the method of characteristics. During the numerical computation, the initial and boundary conditions are necessary to solve differential equations. This study applies the open boundary condition to the method of characteristics and proposes the method of the flood routing without a rating curve. The first application of the open boundary condition to the computation of the long wave propagation was done by Hino (Hino, 1987; Hini and Nakaza, 1989) for the case of the finite difference scheme. In this study, for given discharge hydrograph at the upstream boundary the rating curve at the downstream boundary is computed during the flood
224 Hydraulic Engineering Software Upstream Boundary Computational Zone => Downstream Boundary Figure 1: Coordinate system routing and the characteristics of the rating curve is investigated. The stationary water depth at the downstream boundary for the flood routing is numerically determined. Problem Formulation The continuity equation of unsteady flow in the uniform channel of rectangular cross section is given by da dx in which A = cross-sectional area; / = time; Q = discharge; and x = the coordinate system along the flow direction. The dynamic equation of unsteady flow is also written by dh dx <, o dx gadt * in which g = the gravitational acceleration; B = top width of cross section; h = water depth; So = the bottom slope; and Sf = the energy slope. The coordinate system is represented in Fig.l. By using the Manning's formula, the energy slope is expressed as in which u = cross-sectionally averaged flow velocity, Q/A; n Manning's roughness coefficient; and R = hydraulic radius. To solve Eqs.(l) and (2) numerically, the method of characteristics is used. Eqs.(l) and (2) are transformed to the following set of ordinary differential equations. Along the curve LM in Figs.2 or 3, we have (1) (2) (3) dx (4a)
Hydraulic Engineering Software 225 XL Figure 2: Grid System of Computation for Subcritical Flow M R XL Figure 3: Grid System of Computation for Supercritical Flow
226 Hydraulic Engineering Software Along the curve RM in the samefigure,we have (46) fga *" = "~Y (5a) du (56) Figs.2 and 3 correspond to the explanation of thefinitedifference scheme for the subcritical and supercritical flow, respectively. These equations are expressed in the form offinitedifference with reference to Figs.2 or 3 as follows: (60) (66) (7a) 2 J <"> When we use the fixed grid system, unknowns to be derived from the above equations are %&, %#, AM, WM- The values of ^, w#, h^, 6^ are computed by the linear interporation of known (computed in the previous step) variables using Eqs.(6a) and (7a). Boundary Conditions To solve Eqs.(l) and (2) numerically, the upstream and downstream boundary conditions are needed in addition to the initial condition if the open boundary condition is not used. First, let us consider the boundary conditions in the case of the subcritical flow. Usually, the input discharge hydrograph at the upstream boundary and the rating curve at the downstream boundary are employed as the boundary conditions. The rating
Hydraulic Engineering Software 227 curve sometimes forms a loop. But we practically use the empirical relationship between the water depth and the discharge. This is an approximate relationship and the reliability of this relationship is generally not enough. Errors in using the approximate relationship are also dependent upon sedimentation, erosion, or seasonal vegetations of river channels. To exclude the approximate relationship or to obtain high reliability in the unsteady flow computation, the open boundary condition is used instead of a rating curve. The computational procedure is given as follows: First, the water depth h^ is obtained by assuming that the flow velocity is zero at the downstream boundary in Eq.(6b). Thus, the standing waves are formed at the downstream boundary. Thus, if there is no incident wave from far downstream, the water depth HM at the open boundary is given by A«= ^=-^ + Ao (8) in which A* = the stationary water depth. For the long wave propagation the stationary water depth is the water depth where there is no wave or no water velocity. But the stationary water depth for the unsteady flow is not known in advance. The stationary water depth for the long wave propagation is the initial water depth before waves propagate. The water depth for the steady flow is not equal to the stationary water depth. Because there exists flow velocity even for the steady flow. Thus, if the flow velocity is assumed to be zero at the downstream boundary for the steady flow, the resultant water depth is different from the water depth of the steady flow by blocking the flow. This equation (Eq.(8)) is derived from the fact that the progressing wave height is a half of the standing wave height. That is, the wave height of the standing waves is equal to the summation of the wave height of the incident wave and the reflected wave. To obtain the accurate numerical solution at the downstream boundary, the following procedure is used: For the case of the unsteady flow, the stationary water depth, namely the value of ho is not known in advance. This value corresponds to the stationary water depth for the long wave propagation. Thus, the equation which corresponds to Eq.(9) is given by fc,.*^ ( in which /% = the water depth at the downstream boundary by blocking the steady flow; h^ = the water depth of the steady flow at the downstream boundary. The total water depth at the downstream boundary computed by the perfect reflection consists of the water depth of the steady flow, the increment of the water depth of by blocking the steady flow, and the increment of the water depth due to the perfect reflection of the flood wave. Thus, the stationary water depth of the unsteady flow AQ is given
228 Hydraulic Engineering Software by A. = & -/( This condition determines the stationary water depth of the unsteady flow. Assuming that the flow velocity UM is zero at the downstream boundary, the water depth h'^ there is derived from Eq.(6b). The variables with subscript L are calculated by the linear interporation in the previous time step. BM and AM are the channel top width and cross-sectional area when the water depth is h'^. h'^ is iteratively solved using the method of trials and errors. Thus, if the cross-sectional area A is an arbitrary function of the water depth, the water depth at the downstream boundary is expressed by h* which satisfies A(h') = ~ + A(hl) (11) z in which A* = the water depth at the downstream boundary in the stationary condition and A* = the water depth to satisfy Eq.(ll). The water depth h* is the downstream water depth when the boundary is open. In case of the long wave propagation, h* is equal to the stationary water depth AO, but it is unknown and imaginary in the case of the flood routing. It is numerically computed by the condition that the water surface profile after the flood propagation is equal to the initial water surface profile. After the convergence of the iteration processes, the stationary water depth h* is determined and the water depth h* becomes to equal HM- Using the calculated water depth A&f, the velocity at the downstream boundary is computed from Eq.(6b). Thus, it is understood that the downstream boundary condition is unnecessary if the open boundary condition is employed. At the upstream boundary, the discharge is usually given by the function of time. The water depth at the upstream boundary for the given discharge is obtained by Eq.(7b). When the flow is supercritical, the velocity and the water depth at the point M at the boundaries are computed by Eqs.(6a), (6b), (7a), and (7b). The influence of given discharge hydrograph at the upstream boundary is transmitted through the discharges at the points L and R, because of the computational processes of the linear interporation. Thus, the downstream boundary condition is not needed for this computational procedure. The variables at the downstream boundary do not specify the solution in the computational domain for the supercritical flow. The initial conditions are determined from the following equation: Numerical result 3^ + ^ = a(so-sf) (12) To complete the flood routing calculation using the new computational scheme of the downstream boundary condition, the uniform rectangular
Hydraulic Engineering Software 229 cross section of which width is 20m is employed as an example. Manning's roughness coefficient and the bottom slope are selected to be 0.03 and 0.0010, respectively. For the fixed grid system the space increment As is 20m and the time increment At is Q.lAx/Jghow The water depth h^p is the approximate water depth at the upstream boundary. It determines the discharge for the steady flow. For example, the discharge hydrograph at the upstream boundary is assumed to be -cos^4, foro<f <% %J (13) in which C = the coefficient to determine the peak discharge and Tt the period of flood. The basic discharge Q* is calculated based on the water depth /io,up- But this water depth is dependent upon the downstream water depth. Because the water depth at the upstream boundary is the computational result of the steady flow based on the water depth at the downstream boundary. For this study, h^^p 1.5m and C = 20, 5 are selected. Thus, the water depth ho^p determines the discharge and the upstream water depth h<>^p is not used in the computation of the water surface profile. As the initial water surface profile, MI curve, uniform curve, and M% curve are selected. MI curve, uniform curve, and M% curve correspond to that the water depth at the downstream boundary are 2 m, 1.5 m, and 1.0 m in this study, respectively. Fig. 4 represents the computed rating curve when L = 20 km, C = 5, and Tt = 5000 sec. Since the unsteadiness is not remarkable (the length of the channel reach is long and the amplitude and period of the used flood are small and long, respectively, the computed rating curve forms one- value function. Fig. 5 shows the computed rating curve when L = 20 km, C = 20, and Tt 5000 sec. Since the amplitude of the used flood increases, the computed rating curve forms a small loop. Fig. 6 describes the computed rating curve when L = 2 km, C = 5, and Tt - 1000 sec. Although the amplitude decreases, the computed rating curve forms a small loop, since the period and the length of the channel reach decrease. Fig. 7 gives the computed rating curve when L = 2 km, C = 20, and T/ = 1000 sec. Since the insteadiness is strong, the rating curve forms a loop. These numerical results show that unsteadiness plays a very important role for the loop formation. Concluding Remarks Applying the open boundary conditions in long wave propagation to the computation of the unsteady flow, the following conclusions are obtained: 1. This study applies the treatment of the open boundary condition in case of the long wave propagation to the method of characteristics
230 Hydraulic Engineering Software 200 ho = 2.0 i ho = 1.0 100 1 2 3 4 5 Water depth in m Figure 4: Computed Rating Curve 600 - ho = 2.0 m 500 /io = 1.5 m : ^oo a 3 K 300 o> bc % I Q 200 ho = 1.0 100 2 3 4 5 6 Water depth in m Figure 5: Computed Rating Curve
Hydraulic Engineering Software 231 30 ho = 2.0 i ho = 200 100 2 3 4 Water depth in m Figure 6: Computed Rating Curve
ERRATUM Due to an unfortunate printing error, the text on page 232 is missing. This page is reproduced below. We apologise for any inconvenience caused.,> 1000 A, = 1.5 2-0 m 500 Water depth in m Figure 7: Computed Rating Curve for flood routing. This is a very simple computational scheme and does not include the uncertainty introduced by the determination of empirical rating curves. 2. If the periods of floods are very short, the bottom slopes are steep, and the lengths of the study channels are short, that is, the unsteadiness is remarkable, the computed rating curve clearly describes loops. Most of the relationship between stages and discharges in Japan describe loops. References Hino, M. (1987). "A Very Simple Numerical Scheme of Non-Reflection and Complete Transmission Condition of Waves for Open Boundaries," Technical Report, Dept. of Civil Engrg., Tokyo Institute of Technology, Japan No.38, 31-38 (in Japanese). Hino, M. and Nakaza, E. (1989). "Test of a new numerical scheme on a non-reflective and free transmission open-sea boundary for longwaves," Fluid Dynamic Research, Vol.4, Japan Society of Fluid Mechanics 305-316.