(1)
Bin Li and Chris A. Fleming
Halcrow Group Ltd., Burderop Park, Swindon, Wiltshire, SN4 0QD, UK
Abstract: Using a sigma co-ordinate system a three-dimensional mathematical model for free surface flow based on the Reynolds-averaged Navier-Stokes equations is developed. The time-splitting method is used to separate advection and diffusion terms from the pressure terms in the governing equations. The hydrodynamic pressure equation is solved by a multigrid method, while the hydrostatic pressure equations are solved very efficiently by an Alternating Direction Implicit (ADI) scheme. The model is tested against available analytical solutions and experimental data. Good agreement has been achieved.
Keywords: three dimensional, hydrodynamic model, current flow
Three-dimensional models can be applied to free-surface hydraulic engineering problems with the rapid increase in computer power in recent years. Most three-dimensional models used in industry are based on the assumption of static pressure. Based on assumption of hydrostatic pressure Lin and Falconer (1997) developed a three-dimensional layer-integrated model for estuarine flow. The hydrostatic approximation is no longer valid where there is rapidly varying slopes or large water density gradient over water depth. In this paper a three-dimensional mathematical model for free surface flow based on the Reynolds-averaged Navier-Stokes equations is developed in a sigma co-ordinate system.
The tidal flow
can be described by the three dimensional Reynolds-averaged Navier-Stokes
equations, which can be written as:
(1)
(2)
(3)
(4)
The
kinematic boundary condition on the water surface is
(5)
Separating of the pressure into hydrostatic and hydrodynamic pressures
and applying the sigma transformation, we have
(6)
(7)
(8)
(9)
The
time-splitting method is used to separate terms of hydrostatic pressure from
other terms. In this way the Alternating Direction Implicit (ADI) method can be
used efficiently to solve those wave equations. From time step n to n+1/3, long
wave equations resulted from hydrostatic pressure are:
Application of the ADI method produces the following equations:
and
Thus the long wave equations can be properly and quickly solved with two
steps. At first step the variables with n+1/6 can be obtained by tridiagonal
elimination method while at second step the variables with n+1/3 can be solved
with the same method.
From equation (6) to (8) we compute explicitly u, v and w for the step at
n+2/3 as follows.
Finally, velocities
u, v, w and hydrodynamic pd at step n+1 can be calculated with
Thus four unknown variables un+1, vn+1, wn+1 and pDn+1 can be solved. From above equations we have
This equation is an elliptic partial differential equation and solved by a multigrid method.
First test of
the model is for a case of tidal currents in a model rectangular harbour (Li
& Falconer, 1995). The laboratory model consisted of a vertically distorted
model of an idealized prototype rectangular harbour. The prototype harbour was
assumed to have plan-form dimensions of 432m×432m and an asymmetric entrance of
width 48m and a horizontal bed of mean depth 6m. The basin was also assumed to
experience repetitive sinusoidal tides of period 12.42 hours and range 4 metres.
The model numerical model is set up to replicate the laboratory conditions.
Comparisons of the predicted and measured velocities along two axes across the
centre of the harbour are shown in Figure 1. The computed results are in good
agreement with the experimental measurements.
Second test of
the model is for a case dealing with the geostrophic adjustment of a cylinder of
lower salinity than the ambient seawater. This has been modelled in the
laboratory by Griffiths and Linden (1981) and numerically modelled by James
(1996) and Tartinville et al. (1998). The detail of the parameters for this test
case can be found in the paper of Tartinville et al. (1998).
The test of the
model in this paper reveals that a full 3-D model can generate an order-two
instability at the end of 144 hour simulation, as expected from laboratory
experiment (Griffiths & Linden, 1981), while model with upwind scheme
produces order-four instability (Tartinville et al., 1998). It should be noted
that the model in this paper produces separation of two eddies after 144 hours,
Figure 2, which has been reported by Griffiths & Linden in their physical
experiment. There is no evidence of eddy separation in the other reported
numerical models (Tartinville et al., 1998).
A three-dimensional model base on the Reynolds-averaged Navier-Stokes has been developed for free surface flow. The sigma coordinate
transformation is used to transform the equations and the corresponding boundary
conditions from the physical domain to the calculation domain. The
time-splitting method is used to separate advection and diffusion terms from the
pressure terms in the governing equations. The resulting hydrodynamic pressure
equation is solved by a multigrid method, while the hydrostatic pressure
equations are solved very efficiently by an Alternating Direction Implicit (ADI)
scheme. The case of freshwater cylinder indicates that the model can reproduce eddy
separation that has been observed in the laboratory with physical experiments.
The authors are not aware of any evidence that such eddy separation has been
reproduced before by other mathematical models.
References
[1] Griffiths, R.W. and Linden P.F. 1981. The stability of vortices in a rotating, stratified fluid. Journal of Fluid Mechanics, vol. 105, pp.283-316.
[2] James, I.D. 1996. Advection schemes for shelf
sea models. J. of Marine Systems, 8, pp.237-254.
[3] Li C.W. and Falconer R.A. 1995. Depth integrated
modelling of tide induced circulation in a squre harbour. J. of Hydraulic
Research, 33, 321-332.
[4] Lin B. and Falconer, R.A. 1997.
Three-dimensional layer-integrated modelling of estuarine flows with flooding
and drying. Estuarine, Coastal and Shelf Science, 44, pp.737-751.
[5] Roe, P.L. 1981. Approximate Riemann solvers,
parameter vectors, and different schemes. J. Comp. Phys., 43(2), 357-372.
[6] Tartinville, B., Deleersnijder E., Lazure, P., Proctor, R. Ruddick, K.G. and Uittenbogaard, R.E. 1998. A coastal ocean model intercomparison study for a three-dimensional idealised test case. Applied Math. Modelling, 22, pp.165-182.
Fig. 1
Fig. 2