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ABSTRACT
A finite-slope Boussinesq equation is developed to model curved transcritical flow over spillways and broad-crested weirs, even with large slopes. At a number of computational points, finite difference approximations for all derivatives in the differential equation are used to give a system of nonlinear algebraic equations which can be solved by standard optimization methods. A number of laboratory experiments were performed with different transcritical flow problems including changes in channel gradients and a trapezoidal weir. The equation and the numerical model were tested using results from those experiments and from those for a steep and sharply-curved weir structure, with good results. They can be used as a computational flume to determine the head-discharge characteristics of proposed structures. A novel feature of the equation and numerical method is that higher derivatives of the bed topography are best ignored, apparently mimicking the effects of flow separation in smoothing it.
Acknowledgements
The first author acknowledges the support provided by the Ministry of Higher Education, I.R. of Iran for spending a sabbatical leave period in Vienna University of Technology. The writers wish to thank Prof. P. Tschernutter and staff of the Institute of Hydraulic Engineering, Vienna University of Technology, for providing office and laboratory facilities and assistance in carrying out the theoretical, computational, and experimental work.
Notation
| A | = | cross-sectional area |
| = | coefficients for numerical derivatives in Table 1 (−) | |
| B | = | width of rectangular cross-section |
| = | finite difference approxmns to steady forms of Boussinesq eqns (27)–(29) multiplied by | |
| = | finite difference approxmns to boundary conditions, Eqns (34a-c), units obvious in each case | |
| = | Froude number | |
| f | = | notional symbol for resistance force |
| = | bottom and surface values of curvature contributions to pressure, Eq. (17) | |
| g | = | gravitational acceleration |
| H | = | head above horizontal top of weir |
| h | = |
|
| = | total upstream water depth | |
| N | = | number of computational intervals |
| = | unit normal vector | |
| = | wetted perimeter in plane perpendicular to local bed | |
| p | = | pressure |
| = | gravitational and dynamical contributions to pressure | |
| q | = | discharge per unit width, |
| = |
| |
| S | = | local bed slope |
| s | = | curvilinear channel bottom coordinate |
| = | horizontal resistance component | |
| t | = | time |
| U | = | is the time and cross-sectional mean of u |
| u | = | x-component of velocity, |
| = | time mean of u | |
| = | time mean square of turbulent fluctuations | |
| = | fluid particle velocity relative to control surface | |
| V | = | mean velocity parallel to the local bed and alternatively, velocity of a fluid particle |
| = | control volume | |
| w | = | weight for boundary condition contributions to ϵ |
| = | Cartesian co-ordinates | |
| = | Local cartesian co-ordinates parallel and normal to the free surface | |
| Y | = | bed elevation |
| = | Boussinesq momentum coefficient | |
| = | dynamical pressure coefficients, functions of velocity distribution parameter μ | |
| = | element of bottom coordinate | |
| = | computational step in x | |
| = | sum of squared errors in Eqn (35) (mixed units) | |
| = | free surface elevation | |
| = | specified numerical value of upstream surface level | |
| = | surface elevation at equi-spaced computational points | |
| = | local slope of the free surface | |
| = | slope of particle path | |
| = | curvature of free surface | |
| = | curvature of particle path | |
| = | Weisbach resistance coefficient | |
| = | velocity distribution parameter | |
| = | coefficient of kinematic viscosity | |
| = | fluid density, | |
| = | stress on solid boundary |