Donatella Termini1,
Cristiana Di Cristo2 , Massimo Greco2
1Dipartimento di Ingegneria idraulica ed Applicazioni Ambientali, Università di Palermo, Viale delle Scienze, 90128 - Palermo, Italy- E- mail:dony@idra.unipa.it
2 Dipartimento di Ingegneria Idraulica ed Ambientale “G. Ippolito”, Università degli Studi di Napoli “Federico II”, Via Claudio, 21 – 80125 Napoli, Italy
E-mail:
grecom@unina.it
Address for correspondence: Prof. Massimo Greco, Dipartimento di Ingegneria Idraulica ed Ambientale “G. Ippolito”, Università degli Studi di Napoli “Federico II”, Via Claudio, 21 – 80125 Napoli, Italy – Tel.: 081-7683427, Fax.: 081-5938936 – E-mail: grecom@unina.it
Abstract: Numerical modelling of the flow in a meandering channel is compared with experiments done by one of the authors. A peculiar 2DH model, where bottom shear stresses are computed taking into account also accelerations, has been developed for the comparison. Good results are obtained in some sections of the channel, but, in others, some experimentally observed flow features are not well reproduced. An explanation is offered for the difficulties encountered in modelling the strong secondary current that appears along the bend.
Keywords: bottom shear stress, 2DH numerical models, meandering rivers
To examine the evolution of bed topography and sorting processes local boundary shear stress responsible for sediment erosion and deposition must be determined accurately. In natural meandering channels erosion and deposition patterns are very sensitive to spatial variations in boundary shear stress induced by the complex interplay between the flow and bed and bank topography. The geometric shape of channel curvature produces a complicated three-dimensional flow field. The critical point of models aiming to predict flow and bed topography is how to explain the secondary circulation and its variation with channel curvature and depth. Many models (Zimmerman and Kennedy, 1978; Engelund, 1974) consider the secondary circulation as an essential element to correctly predict sediment transport process, but they may be applied only to fully developed regions where no longitudinal gradient exists.
In order to take adequately into account the influence of secondary flow, in recent years many three dimensional (3D) models have been developed for scientific as well as commercial applications. These models have been applied efficiently in many schemes to predict flow field and bed evolution (Ouillon & Dartus, 1997; Fiorotto & Cividin, 1996, Wang & Jia, 1999). Nevertheless, the use of these model is still not straightforward, requiring long computation times and accurate definition of the boundary conditions. Therefore, 1D and 2DH models are still fashionable tools in some kind of problems. 2DH models, based on vertically averaged flow and transport equations, under shallow water hypothesis, are largely used in river hydraulics when dealing with complex river geometry. Many commercial codes are based on 2DH models (TELEMAC 2D, MIKE 21) and are preferred for their computational efficiency when compared to 3D models. The major limitation of 2DH models comes from the shallow water hypothesis, requiring gradual variations of depth and width along the river. Furthermore, in principle, they cannot resolve secondary currents and other effects depending on the variability of velocity along depth. Despite this, 2DH models seem the only reasonable choice in the design phase of water works or in management problems when a large number of simulations of long river reaches are needed. For this reason 2DH application is attempted also in situations where a three dimensional flow field is expected, as in presence of bends and meanders.
Recent experimental works conducted in meandering channels (Whiting and Dietrich, 1991; Termini, 1996, Termini, 1999) showed that the spatial accelerations produced by the continuously changing channel curvature play the most important role. In these cases, the variation of flow in the streamwise direction is even more important than the secondary current, and both exert an influence on the bed shear stress distribution (Termini, 1999). In this context depth-averaged models may adequately describe the primary flow, but, depending on the system geometry, they can lead to a very poor localization and sizing of the bottom shear stresses, so compromising any serious attempt to simulate bed evolution (Duan et al., 1999)
The writers have developed an innovative quasi-3D model (Carravetta et al., 1999; Carravetta et al., 2000). The model couples a 2DH formulation for flow field resolution to a 3D framework for the sediment transport and includes an additional bottom shear stress, that takes into account the non uniformity of the flow. The computed flow field is, thus, expanded to a complete three dimensional velocity field. This 3D velocity field should exhibit some qualitative aspects of the measured flow, and should be able to account, at least partially, for transverse circulation. In this paper the model is applied to one of the experiments by Termini, and the comparison of the results with the experimental ones allows some insight on the relevance of the secondary currents and convective accelerations, that are often neglected in analytical solutions.
The writers’ model was described in detail in previous papers (Carravetta et al., 1999; Carravetta et al., 2000). Therefore, only a brief description is given herein.
Shallow water equations in conservation form:
where:
(1)
are solved in
a finite volume explicit integration scheme. In (1) g is gravity, x and y are
the horizontal plane coordinates, t is the independent time variable, h
is water depth, zf is bed elevation, U
and V are the x and y components of
the depth averaged local velocity,
.
is the bottom stress, calculated using Chezy formula.
A zero equation model (Rodi, 1980) is
used for the eddy viscosity, assuming
(Elder, 1959), where
is the shear velocity. This formula
is still widely used today (Lambert & Sellin, 1996; Olsen , 1999).
The term
is introduced assuming that the
bottom stress in non-uniform flows should depend on velocity variability in time
and space.
For rational analogy with the
modelling of unsteady flow in pressure pipes (Brunone et al., 1995) and
following an approach based on the extended irreversible thermodynamics (Axworthy
et al., 2000), this
term is evaluated according to the formula:
(2)
where Ku is a coefficient to be found by calibration.
Then, the bidimensional flow field is
expanded to a fully three dimensional flow field. Logarithmic wall law is
assumed in vectorial form to get the horizontal components of the local velocity
vector
; k being the von Karman
constant:
(3)
where the
shear velocity vector
is computed taking into
account both the
and
terms.
The different direction of the
and
vectors, due to the
additional bottom stresses implied by the
term, which is not parallel to the
depth averaged velocity
, can lead to gradual rotations with depth of the horizontal component of
the local velocity, as experimentally observed by many authors, and implied by
the presence of secondary currents.
The vertical component of the local velocity can then be derived by the divergence equation.
The meandering laboratory flume, shown
in Fig. 1, follows the sine-generated curve with a deflection angle of 110°.
The channel axis is two wave lengths long and the channel cross section is
rectangular with a width of 0.50 m. The channel banks are rigid and made by
clear 0.2 cm thick Plexiglas strips, 0.25 m high. At the upstream end and at
downstream end of the meandering channel two straight channels, called
respectively inflow channel and outflow channel, are built. The inflow channel
is 3 m long and the outflow channel is 2 m long. The bed of the channel is of
quartz sand, with
,
and
,
where the subscript designates the percent finer. The longitudinal bed slope
along channel axis is 0.371%.
The experiment selected for the comparison, among the many performed, was with fixed bed and was carried out with a flow rate equal to 0.0065 m3/s, at which corresponds a normal depth of about 3 cm and a width/depth ratio greater than 10. During the experiment, the water was discharged from the outflow channel into a pumping tank and, then, was pumped back to the inflow channel through a return pipe. The details of the experimental apparatus are shown in previous works (Termini, 1996; Termini and Bonvissuto, 1997).
During the experiment the flow velocities, both near the bed and near the free surface, and the water depths were measured. The measurements were made along a channel reach one wavelength long, in nine transverse abscissas selected, for each measurement section, symmetrically to the channel axis. The measurement sections are spaced 50 cm apart (see Termini, 1999). The velocities were measured by a propeller, flow direction by a direction-finder and the water depth was measured by a point gauge, opportunely attached to a movable carriage that traversed the channel.
Fig. 1 Experimental apparatus
After having applied the necessary coordinate transformation in (1), the experimental test described before was also computed with the model. As shown in Fig. 1, O is the origin for the local coordinate system, s indicates the curvilinear abscissa and n the transverse coordinate. The comparison between the experimental and the computed results are shown in three measurements sections: 13, 16 and 19. Fig. 2 compares experimental and predicted depth at section 13, that is one of the inflection sections of the channel. The model prediction is within a small tolerance from the observed values.
At section 16, midway between the inflection point and the meander apex (see Fig. 1), the model computed values agree even more closely with the observations. Fig. 3 shows the comparison of depth h and Fig. 4 plots the discharge per unit of width. Fig. 5 compares the bottom shear velocities deduced by the measurements against the model prediction.
The last section considered, section 19, is at the apex of the meander. As it can be seen by Fig. 6 predicted and observed depths agree very closely. But, looking at Fig. 7, a complete mismatch between predicted and observed transverse distributions of the longitudinal velocity Us is found. This result may appear surprising, after the good agreement found in the other sections. It should be noted that the computation predicts the shifting of the maximum velocity near the outer bank fairly downstream of the apex, while the experiment found it to happen just upstream of it (Termini, 1999).
Fig. 2 Depth h at section 13 Fig. 3 Depth h at section 16
Fig. 4 Discharge per unit of width Us×h
Fig. 5 Shear velocity at section 16
at section 16
Fig.
6 Depth h at section
19
Fig. 7 Longitudinal velocity Us
at section 19
An explanation for this behavior may
be found observing that, as shown in Fig. 8, at the apex a strong secondary
current has developed. It moves the low momentum water near the bottom toward
the inner bank, while flow closer the surface, which possesses an higher
momentum, is adverted to the outer bank. Depth averaging the secondary
circulation an almost null transverse velocity is obtained. So, the numerical
model, which computes momentum transfer by the depth averaged quantities, is
unable to account adequately for the process outlined before. In Fig. 8 the
comparison between the vertical distribution of the computed transverse velocity
at section 19, n=0.02 m, and the measured data is reported. As it is shown, a
qualitative agreement exists between computed and measured vertical distribution
of transverse velocities, due to the introduction of the additional
term, but the circulation is still
largely underestimated.
Fig. 8 Vertical distribution of transverse velocity vn at section 19
Modelling flow and bottom shear stress in a meandering channel is not easy. A 2DH model, which should in principle be able to reproduce quite correctly the flow, fully accounting for velocity and depth gradients both in streamwise and transverse direction, still misses the migration of the transverse position of the maximum velocity. The explanation given is connected to the presence of a strong secondary current, detected in the experiments, whose effects on the streamwise flow are not reproduced in a 2DH model. The introduction of an additional shear stress in the model brings to predicting such a current, but still underestimates its effects.
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