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.

### Basic concept

• 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.

### Practical applications

• 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.

### Assumptions

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

### 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:

• h = total depth (surface to bed), m;
• U,V = depth-averaged velocities, m/s;
• Zb = elevation of bed above arbitrary horizontal datum, m;
• g = gravitational acceleration, m/s2;
• n = effective kinematic viscosity, m2/s;
• C = Chezy friction coefficient, m½/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 = ghin2/2 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° ghin2/2 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.
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

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 :
• 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.

### 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.

### 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

svz/331/0201