SHALLOW-WATER FLOWS

Purpose of this article

Contents:

1. Application and assumptions
2. Shallow-water equations
3. Implementation and settings
4. Test cases and examples
5. Conclusion
6. References

APPLICATION and ASSUMPTIONS

• 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

SHALLOW-WATER EQUATIONS

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/s²;
• n = effective kinematic viscosity, m²/s;
• C = Chezy friction coefficient, m^½/s

IMPLEMENTATION and SETTINGS

• Equations solved by analogy to isentropic, compressible gas flow to get
• U1, V1 = depth-averaged velocity components with
  • Pressure, P1= g h²/2, i.e.
  • Density, RHO1=(2 P1/g)^½, kg/m³
  • Reference pressure, PRESS0 = gh_in²/2 and
  • Depth, h = RHO1, meters

• If the viscous effects are under consideration the variants of the aboves are thought to be more appropriate:
  • Pressure, P1= r_° gh²/2, i.e.
  • Density, RHO1=(2r_° P1/g)^½, kg/m³
  • Reference pressure, PRESS0 = r_° gh_in²/2 and
  • Depth, h = RHO1/r_° , meters,
  wherein r_° is a fluid (water) density.
• 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.

TEST CASES and EXAMPLES

Test cases

• 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.

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

Cases for example

1. Free-surface elevation in an open U-bend
2. A whirlpool in a pond
3. Hydraulic jump in supercritical flow

CONCLUSIONS

• 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.

REFERENCES

General:

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.

Phoenics:

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