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A.L. Le Fessant, D. Dartus and J. Chorda
Institute of Fluids Mechanic of Toulouse, UMR 5502 CNRS
Allée
du Professeur Camille Soula, 31400 Toulouse, France
Tel : +33 (0)5 61 28 59 24, fax : +33 (0)5 61 28 58 11, E-mail : fessant@imft.fr
Abstract: The aim of this study is the diagnostic,
through experimental and numerical approaches, of instabilities phenomenon
observed on steep rough slope. Experiments on water flume reveal the occurrence
of undulations of the free surface above a bottom with equidistant barriers
which allows an increase of the roughness. Several configurations of parameters
have been studied: relative barriers spacing, slope, flow rate. It allows to
observing several types of regimes. Then, we could verify the control of the
waves train due to the relative spacing between the barriers, and also the
apparition of others two-dimensional instabilities. In order to understand these
phenomena, we propose a model based on the two dimensional shallow water
equations, where the friction terms are given by the Strickler formula. A
numerical code is developed in order to validate our approach. It is based on a
total variation diminishing (TVD) scheme, allowing the exact resolution of
discontinuities. Boundary conditions are specified by Thomson's method, which is
based on the method of characteristics and the analysis of the different waves
crossing the domain boundary. The final objective of this work to provide a code
that would be able to restitute the development of observed instabilities.
Therefore, we compare numerical and experimental results. This comparison is
based on spectral, correlation and wavelet analysis. It will be possible to
propose theoretical criteria for various configurations and to identify
instabilities triggers.
Keywords: free surface flow, instabilities,
obstacles, signal analysis
This paper deals with flows over open inclined channels. A water flume has been used to study the occurrence of free surface undulations above a smooth ground. These undulations have periodical and stable characteristics, so called roll waves (Dressler, 1949). For industrial requirements, equispaced barriers have been placed to increase the roughness. Trains of solitary waves are generated by bottom macro-roughnesses. The main objective of this work is to understand the mechanism of development of instabilities develop in this kind of channel. Many applications can be found to this study, particularly in environmental management. The main application is fish passage calibration. Dams are modified including ladders to restore the fishes access to their essential habitat (Chorda & Larinier, 1995). These fish ways require an adequate operational flow to make them effective. For each case, the slopes are relatively significant (generally several percents) and the dimension process requires a precise knowledge of such particular regimes. Indeed, the presence of instabilities may occur serious damage. Experiments in a water flume revealed the occurrence of free surface undulations above a steep bottom with equidistant barriers. Several configurations of parameters have been studied: relative barriers spacing, slope, flow rate. It allows to observe several types of regimes. At first sight, one observes a weak draught and a very disturbed free surface, which characterises the steeply sloping regimes above barriers. In order to provide a better understanding of the different observed flows, a numerical code is developed, based on shallow water equations. This paper deals with the study of an important instability observed in this flow: the solitary waves. In the first part, we present the experimental device and the various observed regimes. In the second part, we present the numerical code structure. Then, the numerical results are compared with the experimental data.
The studies were carried out in a glass-wall channel with adjustable slope, of width 0.25 m, height 0.35 m and length 12 m. The maximum slope and flow rate are about 9 % and 50 l/s respectively. The tests are carried out with artificial roughness consisting of barriers of various forms (rectangular, triangular,..). They are laid out with regular intervals on the bottom of the channel. This experimental work is realised in order to reveal the respective influence on the flow due to the relative spacing L/p of the obstacles (L: distance between the barriers, p: height of the barriers), the slope a and the unit flow rate q (Figure 1). The various tested configurations allowed to the observation of a high diversity of flow configurations. Thus, according to the hydraulic and hydrodynamic characteristics of the flow, these instabilities appear on the free surface in different ways.
For small relative spacing and increasing flow rate, patterns of oblique wave type reflecting off the side walls are observed. They are called sinusoids. These waves are more or less out of phase and up to three superimposed patterns were observed. They propagate at a speed superior to average flow speed. The second phenomenon is a periodic train of solitary waves (Figure 2) revealing the same characteristics as sinusoids. A new phenomenon occurs for greater spacing and higher flow rate. They constitute a particular phenomenon and are observed on a transcritic flow (Morris, 1968). In order to characterise and to identify the triggers of these one and two-dimensional instabilities, a numerical code based on the shallow water equations was developed and its structure is presented in the following section. At present, from a numerical point of view, only the solitary waves are studied and constitute the main issue of this paper.
Since the flow tends to become unstable, it results a very deformed free surface, which presents significant fluctuations of level in the majority of the cases. Then, the measurement catches are carried out using resistive probes immersed in the fluid. These sensors allow the acquisition of an analogical signal. via a converter analogic/numeric card, which transcribes it in numerical values then stored in files. In figure 3, one represents the water height measurements for a flow rate Q = 0.75 l/s (Re = 750), a relative spacing L/p = 5 and a slope a = 4 % corresponding to a waves train instability. The determination of the maximum of the cross-correlation function (figure 4-a) between the signals corresponding to two probes spaced by a distance 2L allows the determination of the wave celerity. It is about 0.36 m/s. The Fourier transform of this signal is represented on figure 4-b. Periodic components are put in evidence in the signal with characteristic frequencies f1 = 2Hz (T1 = 0.5 s) and {fn}=n*f1. The treatment of experimental data highlights several characteristics specifically attached to solitary waves. These characteristics constitute the basis of the comparison between experimental and numerical signals developed in the fourth part.
Two-dimensional unsteady water flow in channel is usually described in terms of water depth and discharge. The evolution of these quantities is governed by the shallow water equations here given in their conservative form. They are based on the conservation of mass and momentum:
(1)
where U is the vector of conservative variables, F and G are the fluxes, and S the source term accounting for friction losses and bed slopes. These factors are more precisely given by equation 2 and 3:
et
(2)
(3)
One recalls that h is the depth of water, u and v are the
depth-averaged velocity components in the x and y direction, respectively, g is
the gravity acceleration, Bx and By are the slopes of the
channel bottom in the x end y direction, and Ks is the Strickler
coefficient.
We use a
high-resolution TVD (Total Variation Diminishing) scheme with the approximate
Riemann solver introduced by Roe (Roe, 1981). Harten (Harten, 1983) introduced
the concept of TVD schemes with the aim of improving the Roe's scheme.
Considering some kinds of equations, these algorithms ensure that the total
variation does not increase with time:
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A TVD scheme preserves the monotonicity as it does not introduce new extreme into the solution and does not amplify the extreme already existing. Thus, no spurious oscillations are generated. In our algorithm, we use the less dissipate Yee’s version (Yee, 1987) of Harten’s scheme. Source terms are handled in the algorithm using a standard “strange splitting” method (Le Veque, 1997).
The importance of the boundary conditions in the numerical simulations and their processing is not any more to show. In the present case, the solution depends entirely on the disturbances, which we apply as input of the field. The boundaries at x = xmax and x = xmin are regarded as free input-output of the field. At y = ymin and y = ymax the conditions of wall type corresponding at the edge of the channel are imposed. This approach is the same as the approach developed by Thompson (Thompson, 1987) and generalised by Poinsot & Lele (Poinsot & Lele, 1992) within the framework of the resolution of Navier-Stokes equations. This technique, known under the name of NSCBC (Navier-Stokes Characteristic Boundary Conditions), stands on the representation of the border normal convective terms in their characteristic form.
In this section,
numerical results obtained via simulation with triangular barriers on the bottom
of channel are presented and compared with the experimental data.
The numerical
simulations are realised on a channel width the same width as the experimental
flume, but a larger length. We consider a relative spacing of barriers L/p = 5
corresponding to a great roughness.
In this
configuration, the instability waves overrun definitively all the flow. Example
of numerical model is given in the figure 5. It corresponds to Q = 0.75 l/s, L/p = 5 and a = 4 % (same conditions as previously).
Temporal variations of calculated height water are represented on a
distance of about nine meters downstream from the channel. Similarities with
experimental data are evident. Fourier transform analysis (Figure 6-b) confirms
this impression. Effectively, a 2 Hz-component is put in evidence jointly with
sub-harmonic processes of frequencies. Moreover, the cross correlation function
between two fictitious probes indicates an average celerity about 0.36 cm/s,
which is in accordance with experimental data. These similarities between
experimental and numerical data imply a more precise comparison of the
properties of both. Therefore, analyses and comparison of signals are based one
hand on spectral Fourier and correlation analyses and on the other hand on
multi-resolution time-scale analyses.
To make a more
insight analysis, one uses a time scale analysis called multi-resolution
analysis. This method is described in numerous mathematical texts (Daubechies,
1992), the details of the method itself are not covered in this paper.
Multi-resolution analysis provides a mean to transform signals into time-scale
domain and to extract information at various scales. Then, it allows the
temporal identification of the several sub-components of the signals. In our
case (Figure 7), one presents the Haar multi-resolution analysis of both
experimental data and numerical model results for the same configuration of
parameters (Re = 240, L/p = 5 and a = 6%).
Underneath the signal, the five first decompositions correspond to 0.06 s to 1 s
processes. Both 0.25 s and 0.5 s components appear as very similar including the
presence of beat effect. Finally, the simulation reproduces the four main
components of the experimental signal. A representation of energy repartition
across scales of normalised signal (Figure 8) constitutes a global indicator of
validation of the code. Note that energy is essentially concentrated in the four
smallest scales with a maximum at 0.25 s.
From an experimental point of view, we highlight the appearance of free surface instabilities on a controlled bottom channel corresponding to different configurations of relative barriers spacing, flow rate and slope. A two-dimensional TVD scheme based on finite volume approach has been developed to simulate rapidly varied flows over inclined open channel with obstacles. A series of tests are carried out to validate it and to demonstrate its ability to reproduce strong gradients and discontinuities in open channel flows. This was indispensable since in the majority of the cases under study, the presence of each barrier generates a shock. The numerical code is capable of qualitatively describing the non-linear evolution of localised disturbance to the flow down an inclined open channel with obstacles. The numerical simulations carried out, account well for the physical phenomenon observed in experiments. Therefore, shallow water equations allow the simulation of instabilities. So, instabilities are not related with localised phenomenon. The next step of our work will be the determination of the triggers of observed instabilities.
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Fig. 1 Characteristic dimensions of the bottom with triangular barriers: L - relative barriers spacing, p – barrier height and a - channel slope.
Fig. 2 Passage of a fully developed wave train over the bottom (rectangular barriers) at two different times.
Fig. 3 Experimental results using resistive probes immersed in the fluid.
Fig. 4 Analyses of the experimental
signal:
(a) - velocity estimation by cross-correlation function and
(b) - Fourier transform highlighting characteristic
2 Hz-frequency and sub-harmonics
Fig. 5 Numerical results: water heights obtained by resolution of shallow water equations.
Fig. 6 Analyses of the numerical
signal:
(a) – velocity estimation by cross-correlation function and
(b) - Fourier transform highlighting characteristic
2 Hz-frequency and sub-harmonic.
Fig. 7 Multi-level decomposition of
the signal (Haar multi-resolution analysis)
for Re = 960, L/p = 5 and a = 4%. (a- experimental data and b- numerical results
Fig. 8 Energy repartition across the scales of the normalised signal Figure