Purpose of this article
To outline the implementation of the two-dimensional shallow-water
equation in PHOENICS. This includes the background theory and required
- Application and assumptions
- Shallow-water equations
- Implementation and settings
- Test cases and examples
- Solve depth-averaged variants of Navier-Stokes equation based
on assumption of large difference in vertical and horizontal length
- 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
- 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(hU)/dt + d(hU2)/dx + d(hUV)/dy =
+ nh(d2U/dx2 + d2U/dy2)
- ghd(Zb)/dx - g(U2 + V2)
d(hV)/dt + d(hUV)/dx + d(hV2)/dy =
+ nh(d2V/dx2 + d2V/dy2)
- ghd(Zb)/dy - g(U2 + V2)½V/C2
- 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,
- C = Chezy friction coefficient, m½/s
All model settings are made from within VR-Editor of PHOENICS 3.3.1.
The relationships for bottom stresses are introduced via PLANT menu.
- 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:
wherein r° is a fluid
- Pressure, P1= r°
- Density, RHO1=(2r°
- Reference pressure, PRESS0 =
- Depth, h = RHO1/r° , meters,
- Bed-slope effect represented by fixed-flux source of momentum in
- 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.
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
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
Cases for example
Free-surface elevation in an open U-bend
A whirlpool in a pond
Hydraulic jump in supercritical flow
- The shallow-water equations are easily solved using built-in
- 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
"Water Exchange and Dispersion Modelling in Coastal Regions: a Method Study",
Swedish Meteoroligical nad Hydrological Institute, Vatten 46: 7-17. Lund