- Application and assumptions
- Shallow-water equations
- Implementation and settings
- Test cases and examples
- Conclusion
- References

- Solve depth-averaged variants of Navier-Stokes equation based on assumption of large difference in vertical and horizontal length scales.
- Two-dimensional treatment of three-dimensional flows with local depth calculated as a part of the solution.
- Implementation in PHOENICS is based upon the analogy to compressible gas-dynamic flow.

- Open-channel flow:

bends, expansions, contractions, spillways, hydraulic jumps, bores, flumes, dam breaks, wave bracking etc. - Large-scale hydrailics:

river and coastal dispersion, estuarine flows, tidal waves etc.

- Hydrostatic pressure distribution
- Incompressible, homogenous fluid
- Well-mixed-in-depth flows: uniform vertical mixing
- Small vertical scale relative to horizontal

-d(gh

- ghd(Z

-d(gh

- ghd(Z

Where:

- h = total depth (surface to bed), m;
- U,V = depth-averaged velocities, m/s;
- Z
_{b}= elevation of bed above arbitrary horizontal datum, m; - g = gravitational acceleration, m/s
^{2}; - n = effective kinematic viscosity,
m
^{2}/s; - C = Chezy friction coefficient, m
^{½}/s

- Equations solved by analogy to isentropic, compressible gas flow to get
- U1, V1 = depth-averaged velocity components with
- Pressure, P1= g h
^{2}/2, i.e. - Density, RHO1=(2 P1/g)
^{½}, kg/m^{3} - Reference pressure, PRESS0 = gh
_{in}^{2}/2 and *Depth, h = RHO1, meters*

- Pressure, P1= g h
- If the viscous effects are under consideration the variants of the aboves
are thought to be more appropriate:
- Pressure, P1= r
_{°}gh^{2}/2, i.e. - Density, RHO1=(2r
_{°}P1/g)^{½}, kg/m^{3} - Reference pressure, PRESS0 =
r
_{°}gh_{in}^{2}/2 and *Depth, h = RHO1/r*_{°}, meters,

_{°}is a fluid (water) density. - Pressure, P1= r
- Bed-slope effect represented by fixed-flux source of momentum in appropriate direction.
- Bottom stresses are calculated by relating them to the velocities via Chezy's coefficient.
- Boundary conditions:
- Fixed-fluxes of water discharge at river section inlets,
- Fixed-pressure (equivalent to fixed depth) and
- Time-dependent pressure (depth) for tidal variation.

- flow in an open turn-around channel,
- abrupt open-channel expansion,
- flow impingement on a blunt body,
- spread of depth discontinuity,
- merging of streams,
- hydrailic jumps at the merging of streams,
- flows in channels with complex bed shapes, and
- meandering open channel flows.

The good agreement has been achieved both for free-surface elevation and velocity distributions.

Pictorial extracts from the study now follow.

- An abrupt open-channel expansion
- Blunt body in a shallow water stream
- Open channel flow with varying depth (bed shape)
- Velocity distributions in above
- Bend of an open channel

- The shallow-water equations are easily solved using built-in isentropic option;
- All model settings are available in VR-editor;
- Validation studies show fair agreement with observations.

J J Dronker 1969 "Tidal Computations for Rivers, Coastal Areas and Seas", J. Hydraulic Div., ASCE 95

S A Al-Sanea 1981 "Numerical Modelling of Two-Dimensional Shallow-Water Flows", PhD Thesis, Imperial College, CFD/82/6

J V Soulis 1992 "Computation of Two-Dimensional Dam-Break Flood Flows", Int. J. Numerical Methods in Fluids, vol. 14/6

V Casulli and R T Cheng 1992 "Semi-Implicit Finite Difference Methods for Three-Dimensional Shallow Water Flow", Int. J. Numerical Methods in Fluids, vol. 15/6

C.B. Vreugdenhil 1994 "Numerical Methods for Shallow-Water Flow" (Water Science and Technology Library, Vol 13), Kluwer Academic Pub.

L Gidhagen and L Nyberg 1987 "A Model System for Marine Circulation Studies", 2nd International PHOENICS User Conference

SMHI 1990 "Water Exchange and Dispersion Modelling in Coastal Regions: a Method Study", Swedish Meteoroligical nad Hydrological Institute, Vatten 46: 7-17. Lund

svz/331/0201