Encyclopaedia Index

SHALLOW-WATER FLOWS

Purpose of this article

To outline the implementation of the two-dimensional shallow-water equation in PHOENICS. This includes the background theory and required settings.

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

Basic concept

Practical applications

Assumptions

SHALLOW-WATER EQUATIONS

Continuity

dh/dt+d(hU)/dx+d(hV)/dy=0

X-momentum

d(hU)/dt + d(hU2)/dx + d(hUV)/dy =
-d(gh2/2)/dx + nh(d2U/dx2 + d2U/dy2)
- ghd(Zb)/dx - g(U2 + V2) ½U/C2

Y-momentum

d(hV)/dt + d(hUV)/dx + d(hV2)/dy =
-d(gh2/2)/dy + nh(d2V/dx2 + d2V/dy2)
- ghd(Zb)/dy - g(U2 + V2)½V/C2

Where:

IMPLEMENTATION and SETTINGS

All model settings are made from within VR-Editor of PHOENICS 3.3.1. The relationships for bottom stresses are introduced via PLANT menu.

TEST CASES and EXAMPLES

Test cases

The number of sub- and super-critical shallow-water flows have been simulated in the frames of ROSA project. Different bed shapes and plane geometries have been considered. They include :

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

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


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