European Journal of Mechanics / B Fluids 79 (2020) 67–73
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Undular bores in a large circular channel✩ ∗
Ion Dan Borcia a , , Rodica Borcia b , Wenchao Xu c , Michael Bestehorn b , Sebastian Richter b , Uwe Harlander c a
Department of Computational Physics, Brandenburg University of Technology (BTU) Cottbus–Senftenberg, Erich–Weinert–Strasse 1, 03046 Cottbus, Germany b Department of Statistical Physics and Nonlinear Dynamics, Brandenburg University of Technology (BTU) Cottbus–Senftenberg, Erich–Weinert–Strasse 1, 03046 Cottbus, Germany c Department of Aerodynamics and Fluid Mechanics, Brandenburg University of Technology (BTU) Cottbus–Senftenberg, Siemens–Halske–Ring 14, 03046 Cottbus, Germany
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Article history: Received 31 January 2019 Received in revised form 9 July 2019 Accepted 3 September 2019 Available online 5 September 2019 Keywords: Undular bores Bore collision Periodical boundary conditions
a b s t r a c t An experimental device previously developed for studying rotating baroclinic flows has been used to investigate undular bores formation, propagation and collision. Up to our knowledge this is the first experimental study of undular bores in a circular channel. For a setup without barriers, this geometry accomplishes in a natural way the periodic lateral boundary conditions, very often used in numerical simulations. An excellent agreement between the experiment and simulation has been achieved. The spatio-temporal structure of bores is well reproduced for the first few reflections or collisions. © 2019 Elsevier Masson SAS. All rights reserved.
1. Introduction Tidal bores are natural phenomena observed in at least 450 river estuaries all around the world from Europe (Baie du Mont Saint Michel — France) to America (Colorado River — Mexico) and Asia (Qiantang River — China) ([1] and references therein). Tidal bores manifest as series of waves propagating upstream in the estuarine zone of a river. They are formed during the flood tide, favorable conditions to observe them are in spring or in autumn. The wave height of a tidal bore can be of some tenth of centimeters but under some specific condition can reach 5 to 9 m. One of the most important tidal bore is Pororoca, at the entrance of the Amazon. The phenomenon that occurs between February and March causes waves up to 4 m high and travel more than 800 km inland upstream on the Amazon and adjacent rivers. As can be expected, bores with large amplitudes cause damages and also loss of human lives (like in 12 − 13 November 1970 on the southern coastal region of East Bengal and East Pakistan [2]). Tidal bores are usually smaller and less dangerous than Tsunamis, but they can have unpredictable development near the river bank. ✩ This document is the results of the research project funded by the German Research Foundation (DFG). ∗ Corresponding author. E-mail addresses:
[email protected] (I.D. Borcia),
[email protected] (R. Borcia),
[email protected] (W. Xu),
[email protected] (M. Bestehorn),
[email protected] (S. Richter),
[email protected] (U. Harlander). https://doi.org/10.1016/j.euromechflu.2019.09.003 0997-7546/© 2019 Elsevier Masson SAS. All rights reserved.
Moreover, they occur twice a day and have a strong impact on sediment transfer and fishery in the river estuary. Undular bores occur in a long open channel when a constant mass flux is fed into the channel, while the water ahead of the bore is initially at rest (in this case we have a tidal bore). It manifests itself as a propagation of the change in water level [3–6]. The traveling train of the waves has a slow time dependence, a property that bores share with solitary waves [6]. Sozer and Greenberg, using a fully nonlinear dispersive free-surface flow method, are the first in modeling undular bores produced by sources or sinks [7]. Positive bores are induced by switching on sources and negative bores by switching on sinks. Another way to produce undular bores in the laboratory is to start from two level regions separated by an abrupt change in height (ideally step function). In practice, this can be realized by the sudden removal of a barrier, a method also used in Scott Russell’s experiments for the generation of solitary waves [8]. If the length of the two zones with different levels is large enough comparing to the bore wavelength, the structure of the generated bores will be similar with those produced by a constant pumping (the case of tidal bores). The difference is the following: in place of only one bore (positive for pumping or negative for sucking as showed in Fig. 1(a) [6]) one obtains a pair of bores, one positive and one negative, propagating in different directions, originated from the middle of the height difference (as one can see in Fig. 1(b)–1(f)). To better understand the bore structure, a simple model is presented in Fig. 2. If we assume that at the onset of the bore formation the surface is still flat outside the abrupt bore
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Fig. 1. Simulations of bores with injection (solid line) or sucking (dot line) (a) of fluid or wall removal (b)–(f) (from [6]). For the injection/sucking only a positive/negative bore is formed. For wall removing a pair of positive and negative bores propagating in opposite directions occur (b)–(d) and then they reflect as they reach the barrier (e),(f).
Fig. 2. Sketch of the idealized problem. The initial profile (solid line) is a step function. After a short time t the front splits in two step fronts (dashed line). From mass and momentum conservation it follows that the front splits at a h −h height close to the middle of the two initial heights hr = hmin + max 2 min + (hmax −hmin )2 8hmin
+ O((hmax − hmin )3 ).
front, due to mass and √ momentum conservation and the fact that the wave speed ∼ h, the rupture point is given by hr = hmin +
hmax − hmin 2
+
(hmax − hmin )2 8hmin
+ O((hmax − hmin )3 ), (1)
where hmin and hmax are the two initial liquid levels. This means that the rupture point is situated slightly above the half distance between the two water levels at the left and right sides of the barrier. However, the dispersive effects are very small in the cases studied here. For typical parameters, the second order term in (1) yields a correction of less than one percent, which is below the experimental and numerical resolution. Field measurements of tidal bores were extensively performed in the last years. We mention the ones performed on Qiantang River [9] and on Garrone River at different location like Podensac or Arcins channel [10,11]. Also bore collision is of importance for explaining the suspended sediment transport, bed load, scour and erosion. This phenomena can occur for special local river
geometry like that of Garone River splitting in two branches near Arcins [11]. A considerable number of systematic laboratory studies have been performed on bores over the last two decades [12–15]. All the laboratory studies on tidal bores have been realized in linear horizontal rectangular open channels with lateral boundaries. In contrast, we have designed an experiment of bore formation in an open circular channel of a tank which has been previously used for studying small scale secondary instabilities of baroclinic waves. The tank can librate or rotate allowing more complex surface wave generation. However, in the actual setup we keep the tank at rest. The paper addresses two different cases: the case of rigid boundaries (single bore — here a wall will remain in the channel after the bore generation) and the case of a periodic lateral boundaries (two colliding bores — here no wall will remain in the channel after the bore generation). For the second case, up to our knowledge, we have realized for the first time the bore generation in a circular and hence periodic channel. This geometry accomplishes in a natural way the periodic lateral boundary conditions, very often used in numerical simulations. On the other hand this geometry permits a good and efficient deployment of the measurement equipments along the experimental setup. By experiment and computer simulation, we investigate in this paper the structure of an undular bore generated by an initial water step and the interaction between two identical bores. The paper is organized as follows: Section 2 deals with the description of the experimental setup and the properties of the generated bores. The theoretical formulation of the problem in the frame of an inviscid irrotational fluid is given in Section 3. Numerical results are compared with the experimental results. We gather the conclusions in Section 4.
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Fig. 3. MS-GWave rotating tank: (a) The outer gap, filled with the heating fluid in the baroclinic setup, (see red arrow) is now used to study bores, the working fluid being at constant room temperature; (b) Acoustic sensors mounted around the channel measure the liquid height at fixed positions along the channel; (c) Barriers and sensors localization along the channel for single bore case; (d) Barriers and sensors localization along the channel for bore collision.
2. Experimental results 2.1. One bore formation and propagation We have used an already available experimental device (MSGWave tank) that has been developed for studying rotating baroclinic flows. This apparatus is the property of DFG (German Research Foundation) and has been constructed for the experimental part of the DFG Research unit MS-GWave (Multi-scale Dynamics of Gravity Waves). Here, only the outer channel indicated by red in Fig. 3(a), used in the original setup for heating, will be filled with the working fluid (water) to study surface waves. The circumference of the outer channel is about L = 4.76 m and the depth of the gap D = 8.5 cm. The bore is produced by removing a barrier which separates two different liquid levels. As it is suggested in Fig. 3(c) the other barrier is fixed. This of course means that we do not have periodic boundary conditions in this case. If the curvature is neglected, the system is equivalent to a straight channel with two rigid walls at each end. For the investigation of one bore generation only two acoustic sensors (S1, S2) have been placed at about 20 cm above the liquid surface with a separation distance of 81 cm one from the other. The sensors have been used to measure precisely and with sufficient time resolution the fluid height at well defined points (Fig. 3(b), 3(c)). For the study of bore collision six different sensors (S1–S6) have been mounted in the collision zone (Fig. 3(d)). The sensor S5 is placed on the symmetry axis. The sensors S4 and S6 can be used to verify the synchronicity of clockwise and anticlockwise bores. Then, the problem symmetry permits the reconstruction of the water level around the collision point. To assure the formation of two identical bores, the two barriers should be simultaneously removed. Fig. 4 presents the structure of the undular bores excited in the circular channel of the MS-Gwaves tank seen from the
sensors S1 and S2 in Fig. 3(c), for two different heights h0 and two height level differences ∆h. In our case, the difference of water levels on the left and right side of the removable barrier describes the ‘source strength’ of the tidal bore and ∆h/h0 is the equivalent of the Froude number for small surface elevations. In our experiments the ratios ∆h/h0 are kept lower than 0.4 in order to avoid breaking waves (turbulent processes) [6]. On the main part of the experiments we kept this limit even under 0.28, thus assuring the formation of undular bores [17,18]. How one can see from Fig. 4, in deeper water the wavelength λ and the propagation velocity vf are higher. We notice here that, by following the definition from [19], the wavelength of the bore has been calculated as the distance between its two first crests. A systematic study on the propagation velocity and bore wavelength versus the depth of the undisturbed water layer h0 and the difference of water level ∆h (before the removing of the wall) is shown in Fig. 5. As shown in Fig. 5(a) the propagation velocity depends not only on the average liquid depth but also on the initial height difference. The smaller the height difference the slower the wave. However, as can be observed in Fig. 5(b), for ∆h → 0 the measured phase velocity plotted with square √ symbols asymptotically approaches the dependence vf = gh0 with g the gravitational acceleration, characteristic for shallow water [20]. This is in agreement with [16]. In the same plot, the results using equation (2.6) from [16] without base flow are plotted for the same values of h0 and ∆h using the same color but with a triangle symbol. In view of the experimental errors the tendency showed by the experiment is retrieved also in the theoretical results. Although we are at the validity limit of the long-wave approximation (h0 /λ ranging between 0.1 and 0.2), the propagation velocity still obeys the dependence as prescribed by the shallow water limit. The wavelength mainly depends on the average depth and less on ∆h (see Fig. 5(c)). In contrast, Fig. 5(d) shows that the front height Hf depends mainly on the initial height difference ∆h.
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Fig. 4. Undular bores excited in the circular channel of the MS-Gwaves tank for two different heights and height level differences: (a) h0 = 5 cm, ∆h/h0 = 0.2 (b) h0 = 16 cm, ∆h/h0 = 0.375. In deeper water, the bore wavelength and the propagation velocity are higher.
2.2. Bore collision To exemplify the bore collision one case is analyzed in detail. The following initial conditions are considered: h0 = 75 mm and ∆h = 5 mm. The bore collision is experimentally investigated using six sensors. For the beginning, if we concentrate only to compare the change in bore shape before and after the collision, it is appropriate to look at the signal of sensor S1 in Fig. 6. This sensor has a sufficient distance from the symmetry axis to guarantee a separation between the arrival of the anticlockwise bore before the collision and the clockwise bore that arrives somewhat later and after the collision. Considering the experiment as being symmetrical and the fact that the bore shape is changing slowly in time, space and time corresponds. That is, a plot of the bores just before and after the collision, using height and time, is equivalent with a plot using height and the azimuthal direction. Our special geometry permits multiple reflections. For our setup, the odd ones (i.e. the first reflection, the third reflection and so on) can be found under sensor S5 and the even ones (second reflection, fourth reflection etc.) occur in the diametrically opposite region of the circular channel. The upper graph in Fig. 6 shows the first, the lower graph the third collision. One can see that after the collision the shape of the bore is slightly changed for all collisions, although for the first collision one can remark a slightly higher change in shape. 3. Simulations and comparison with experiment Next we wish to compare our experimental findings with direct numerical solutions of the Euler equations for inviscid fluids.
In [6], we considered a 2D irrotational flow of an incompressible fluid. The velocity field of the fluid v ⃗ can be replaced by its flow potential (scalar field) Φ
v⃗ = ∇ Φ (x, z , t) and the incompressibility condition (continuity equation) becomes:
∆Φ = 0.
(2)
With the length and time scaled as in [6]: (x, z) → (h0 · x˜ , h0 · z˜ ),
t→
√
h0 /g · t˜
the Bernoulli equation (momentum equation) reads:
∂Φ 1 =1−h− ∂t 2
[(
∂Φ ∂x
)2
( +
∂Φ ∂t
)2 ]
,
(3)
z =h
and the kinematic boundary condition at the deformable surface has the form:
⏐ ⏐ ∂h ∂ Φ ⏐⏐ ∂ h ∂ Φ ⏐⏐ = − , ∂t ∂ z ⏐z =h ∂ x ∂ x ⏐z =h
(4)
h denoting the surface deflection. Eqs. (1)–(3) have been numerically solved in two spatial dimensions for a laterally closed geometry with impermeable side walls, following the scheme described in detail in [6]. Fig. 7 shows the surface profiles from theory and experiment detected by the sensors S1 and S2 (in Fig. 3(c)), correspondingly. One sees a very good agreement:
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Fig. 5. (a), (b) Propagation velocity vf as functions of the depth of the undisturbed water layer h0 and the difference of water level ∆h before removing the wall. In (b) the squares denote experimental results, the red line corresponds to the shallow water theory for small perturbations around h0 , and the triangles are the calculated velocities for shallow water but with finite bore height ∆h as in [16]. Bore wavelength λ and bore front height Hf versus the depth of the undisturbed water layer h0 and the difference of water level ∆h are plotted in (c) and (d), respectively.
Fig. 6. The signal from the sensor S1 for the first (t from 0.5 s to 6 s) and third (t from 6 s to 11.5 s) collisions. For both cases, the left side of the signal gives the bore shape before collision and the right side the bore shape after collision.
the theoretical curve perfectly overlaps the experimental curve even after the third reflection at the fixed wall. The amplitude decreases with time in the experiment because of the viscous friction inside the liquid which has not been taken into account in the numerical model.
Fig. 7. Comparison between experiment and simulation for bore production and reflection. A very good agreement has been observed.
A comparison of the experiment and numerical simulation has been performed also for the bore collision problem. In Fig. 8 we focus only on the first collision and on the region where the sensors are placed. For a better comparison of experimental (Fig. 8(a)) and numerical results (Fig. 8(b)), in both cases only the data corresponding to the sensor positions have been used (including their symmetrical positions). The agreement of the two
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Fig. 8. Comparison between experimental data (a) and computer simulations (b) in the bore collision region. The position of the sensors are marked with dashed lines. Note that also the symmetry of the problem is taken into account. For a better comparison also for the simulation only the signal corresponding to these positions has been used.
Table 1 Comparison between the maximal value of the centrifugal acceleration amax and cf the gravity g during the bore propagation. As measure for the effect of the curvature we consider the liquid height difference in radial direction due to the centrifugal force liquid height h0 .
amax cf g
D (with D = Rmax − Rmin , D ≪ Rmin ) relative to the mean
Case Most favorable Most disadvantageous Typical
Fig. 9. First six bore collisions. Odd collisions happen around the origin of the position axis. In this region ultrasound sensors are placed and the plotted water level corresponds to the measurements. For the even collisions no sensors are placed, the plotted water level are provided by computer simulations. Blue points are maxima from the numerical simulation. The red and black lines are the fitting of the bore front position before and after collision.
amax cf
h0
∆h
umax
amax cf
(mm)
(mm)
(m/s)
(m/s2 )
(%).
30 160 80
5 60 20
0.12 0.7 0.32
0.019 0.65 0.14
0.5 3.5 1.5
g
·
D h0
figures is very good in particular in the central region where the collision takes place. The stripes appearing at the bottom of Fig. 8(a) are due to the unavoidable errors in the experimental data. We should mention again that, due to the fact that the numerical results are for inviscid fluids, after a few reflections the amplitude is higher than the experimental one. Nevertheless, the wave front position still corresponds to the experimental one. To show this, we constructed a composite from the numerical and the experimental data and combined both data set in one space–time diagram (see Fig. 9). The experimental data cover the place where the odd reflections take place whereas the numerical results cover the regions of even reflections. Because of the very good agreement obvious in Fig. 9, we can in the following focus on the numerical data that are available over the whole length of the tank and with a good resolution (1000 points). The blue points show the position of a number of local maxima. By focusing on the points corresponding with the bore fronts, we can find out whether a bore collision has an influence on the speed and phase of the bore wave. The solid lines are the prolongation of the zones with constant speed of the bore front before (black) and after (red) the first and the third collision. We observe that only at the first reflection a tiny change in the bore velocity can be noticed, for the next reflections the velocity remain practically unchanged. This fact can be correlated with the bore shape (Fig. 6) which changes during the first collision more than for the following ones. Also a small delay in the bore propagation can be observed as a sign that the non-linear effects are not too high for the considered experiment. 4. Conclusion We have designed an experiment for bore formation and propagation in a circular channel with a circumference of about 5
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m. There are generated bores with wavelengths ranging between 10 mm and 80 mm for height level differences ranging between 5 cm and 15 cm. The propagation wave velocities are estimated between 6 cm/s and 14 cm/s. The removal of one or two of the barriers leads to the formation of either one positive bore or two bores. By synchronizing the barriers removal, two identical bores have been achieved. Due to our special geometry we were able to use the symmetry and periodicity of the problem to study multiple reflection processes. An excellent agreement between the experiment and simulation has been achieved. The structure of the bores is perfectly reproduced by the theoretical model presented in Section 3 even after several reflections at the wall or after several collisions. Only the wave amplitudes are damped in time since friction has not been included in the theoretical model. At this point we should make some remarks about the validity of our model. We experimentally observed an exponential decay of the bore amplitude, which is not the case for all surface waves (for example for a class of capillary gravity solitary waves [21]). Considering this, we evaluate the viscous characteristic damping time for a Poiseuille flow in a channel finding a value of about 10 min. During the bore propagation the front moves with the same velocity except for thin boundary layers near the lateral walls. For this particular situation, the viscous characteristic time becomes one order of magnitude smaller. This is in agreement with the experimental results showing damping times between 0.5 and 1 min and it explains why the model works well for times smaller than half a minute. We also note that the centrifugal forces can be neglected. In Table 1 we show the maximal velocities in azimuthal direction umax (achieved only inside the wave front). The corresponding centrifugal forces can lead for the typical case to a height difference of about one percent of the mean liquid height h0 . For forthcoming studies where we plan to consider longer time spans or a periodic pumping to generate bores, the actual model is not suitable anymore and a phenomenological damping should be considered. It might turn out that we need to use the full Navier–Stokes equation (or to include dissipation in reduced models like it was made for solitons in [22]) in order to model these processes. In the future we will improve the theoretical model taking into account a viscous fluid in a fully 3D description and thereafter we will study experimentally and numerically long time processes, for example libration like parametric excitation of flows with ground topography to study resonance phenomena. Moreover, horizontal parametric instabilities for more viscous fluids form another research topic that can be addressed with the present experimental setup.
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Acknowledgments This work benefits from the DFG, German Research Foundation (DFG BO 3113/4-1, DFG BO 3120/7-1, DFG BE 1300/25-1, and DFG HA 2932/8-1, 8-2). The authors would like to thank Peder Tyvand (Norwegian University of Life Sciences, Ås), Patrice le Gal (University d’Aix-Marseille), Miklos Vincze (Eötvös Loránd University Budapest) for fruitful scientific discussions. The authors are also indebted to the technicians of the Department of Aerodynamics and Fluid Dynamics at Brandenburg University of Technology Cottbus–Senftenberg, namely: Robin Stöbel and Ludwig Stapelfeld. References [1] H. Chanson, Tidal Bores, Aegir, Eagre, Mascaret, Proroca: Theory and Observations, World Scientific, Singapore, 2012. [2] A. Sommer, W.H. Mosley, Epidemiol. Rev. 27 (2005) 1029. [3] M. Landrini, P.A. Tyvand, J. Eng. Math. 39 (2001) 131. [4] M.D. Su, X. Xu, J.L. Zhu, Y.C. Hon, Internat. J. Numer. Methods Fluids 36 (2001) 205. [5] C. Miahr, J. Hydraul. Res. 43 (2005) 234. [6] M. Bestehorn, P.A. Tyvand, Phys. Fluids 21 (2009) 042107. [7] E.M. Sozer, M.D. Greenberg, J. Fluid Mech. 284 (1995) 225–237. [8] F. Marin, Encyclopedia of Complexity and Systems Science, Vol. 9, Springer Science+ Business Media, LLC, New York, 2009, p. 8464, [9] J. Huang Jing, C.H. Pan, C.P. Kuang, J. Zeng, G. Chen, J. Hydrodyn. 25 (2013) 481. [10] K. Martins, P. Bonneton, F. Frappart, G. Detandt, N. Bonneton, C.E. Blenkinsopp, Remote Sens. 9 (2017) 462. [11] C.E. Keevil, H. Chanson, D. Reungoat, Earth Surf. Process. Landforms 40 (2015) 1574–1586. [12] C. Donnelly, H. Chanson, Environ. Fluid Mech. 5 (2005) 481. [13] P. Lubin, H. Chanson, S. Glockner, Environ. Fluid. Mech. 10 (2010) 587. [14] H. Chanson, N. Docherty, Eur. J. Mech. B Fluids 32 (2012) 52. [15] G. Rousseaux, J.M. Mougenot, L. Chatellier, L. David, D. Calluaud, Eur. J. Mech. B Fluids 55 (2016) 31. [16] A. Ali, H. Kalisch, Anal. Math. Phys. 2 (2012) 347–366. [17] T.B. Benjamin, M.J. Lighthill, Proc. R. Soc. Lond. Ser. A 224 (1954). [18] D.H. Peregrine, J. Fluid Mech. 25 (1966) 321–330. [19] M.V. Berry, New J. Phys. 20 (2018) 053066. [20] P. Kundu, I.M. Cohen, D.R. Dowling, Fluid Mechanics, Vol. 361, Academic Press, Heidelberg, 2016. [21] G. Iooss, P. Kirrmann, Arch. Ration. Mech. Anal. 136 (1996) 1–19. [22] M.G. Velarde, A. Nepomnyashchy, Lecture Notes in Phys. 751 (2008) 29–49.