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Approximate analytical solution for supercritical flow in rectangular curved channels
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Approximate Analytical Solution for Supercritical Flow in Rectangular Curved Channels Lyes Amara , Ali Berreksi , Bachir Achour PII: DOI: Reference:
S0307-904X(19)30666-3 https://doi.org/10.1016/j.apm.2019.10.064 APM 13131
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
1 April 2019 21 October 2019 30 October 2019
Please cite this article as: Lyes Amara , Ali Berreksi , Bachir Achour , Approximate Analytical Solution for Supercritical Flow in Rectangular Curved Channels, Applied Mathematical Modelling (2019), doi: https://doi.org/10.1016/j.apm.2019.10.064
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Highlights • An asymptotical quasi-2D mathematical model and analytical solution is proposed. • 2D steady shallow water model is converted to 1D unsteady radial flow problem. • Radial wave equation is solved analytically using Laplace transformation. • Solution covers the curved part and downstream straight channel of the transition. • Predicted side-wall water profiles are in satisfactory agreement with experimental data.
Approximate Analytical Solution for Supercritical Flow in Rectangular Curved Channels Lyes Amara (*) 1 Ali Berreksi 2 Bachir Achour 3 1
2
Department of Civil Engineering and Hydraulics, Faculty of Science and Technology, University of Jijel, Algeria,
Laboratory of Applied Hydraulics and Environment (LRHAE), Department of Hydraulics, Faculty of Technology, University of Bejaia, Targa Ouzemour, 06000, Bejaia, Algeria 3
Research laboratory in subterranean and surface hydraulics (LARHYSS), University of Biskra, 07000, Biskra, Algeria
Email: [email protected] ; [email protected] ; [email protected]
Abstract- Asymptotical mathematical model and analytical solution of a supercritical flow in curved rectangular open channels are presented. An original study approach for free-surface configuration and features of the flow in presence of cross shock waves is then proposed. The two-dimensional steady depth-averaged shallow water equations are transformed to an equivalent one-dimensional unsteady flow problem and a first order approximation is then obtained using small perturbation theory. Further, 1D asymptotic model is analytically solved by Laplace integral transformation and the 2D flow field solution is rebuilt according to translating planes. The freesurface profile along the outer chute wall and downstream channel is compared with the available experimental data. Obtained results show a satisfactory agreement for the maximum flow depth, peaks position and wavelength. The proposed approach allows an accurate prediction of flow features and safe design of curved channel transitions. Keywords- channel transitions, curved channels, analytical solution, Laplace transform, supercritical flow, standing waves.
I. INTRODUCTION Open channels flow under supercritical conditions is often required in many hydraulic engineering applications such as spillways chutes, sewers and outlet works. In these conveyance structures, lateral changes of boundary alignments like bends are unavoidable and necessary in many situations. As it is well-known, in supercritical flow any change in wall directions generates cross waves affecting the water surface and velocity field downstream of the channel. Consequently, high disturbances and super-elevations at wall bends are occurred. Special careful must then be given to predict correctly those risky elevations for safe design of curved channels and downstream reach. Supercritical flow in bends was first investigated by Ippen [1] who developed an analytical approach based on the theory of small disturbances and carried out intensive validation experiments. It resulted that the assumption of constant velocity leads to very satisfactory analytical predictions. Von Kármán [2], using shallow water equations and assuming constant energy, gave interesting analytical solution in analogy with gas dynamics theory. In the symposium on supercritical flow, Knapp [3] presented a remarkable synthesis on the analysis of these flows in curved channels, their design and reduction of the super-elevation due to standing waves. Later, Harisson [4] analyzed the supercritical flow in trapezoidal cross section channel. Admitting a linear free-surface, he proposed a simplified model for super-elevation at the edges. A notable jump in the problem resolution approach was made by Lénau [5]. Based on the potential equations theory, the author presented an analytical solution of curved channels (trapezoidal and rectangular cross section) by a subtle approach using perturbation theory and Laplace transform. The solution is valid for an approach Froude number F0 2 and the ratio of approach flow depth to the center radius of the curvature is small. Hager and Altinakar [6] applied the momentum and continuity equations to the analysis of infinitesimal cross waves. For small wave angles, it was shown that the proposed solution is in complete agreement with those of Knapp and Ippen [7] and velocity across the wave remains constant. The computational approach of the aforementioned problem was introduced first by Ellis and Pender [8]. The steady inviscid 2D shallow water equations are solved numerically using the Method of Characteristics for single and S-shaped bends. An identical numerical approach in polar coordinates was successfully used on various tests by Hosoda and Yokosi [9] and Iwasa and Hosoda [10] for the rectangular channels. Dammuler et al. [11] analyzed an unsteady flow on a curved channel following an instantaneous dam break by finite differences method. The same problem was studied by Elliot and Chaudhry [12] using a simplified 1D mathematical model built on wave’s interactions and reflections in two-dimensions. Other numerical works such those of Iwasa et al. [13] and Berger and Stockstill [14] treated trapezoidal channels. Ye et al. [15] undertook an experimental and numerical study on S-shaped spillway using a k turbulence model. Later Hessaroeyeh et al. [16] and Jaefarzadeh et al. [17] used Finite Volume Method with HLLC-TVD and Roe-TVD solvers respectively to simulate supercritical flow in channel bends. However, these advanced tools remain quite complex for engineering practice which requires simple and reliable solutions. The literature review, at authors’ best knowledge, shows a gap in the field of analytical solutions for more than three
decades. The single analytical approach to be noted from Lénau [5] is the work of Hessaroeyeh and Tahershamsi [18]. Extending the idea of Steffler et al. [19] applied to subcritical flow; the authors used the same perturbation technique for supercritical conditions case and solved the linearized 2D differential equations by Laplace transformation. The obtained first order solution
Vr
Vr
Symmetry axis
B
V
h
r re r0
ri
0 z
max Figure 1 Definition sketch of curved rectangular channel and flow field parameters
was confronted with several experimental results and predicts well the maximum super-elevation but with a lag in its position.
In this paper, analytical solution is proposed to a simplified mathematical model for prediction of free-surface flow in rectangular curved channel under supercritical conditions. First, the asymptotic mathematical model based on the classical twodimensional shallow water equations is presented. Then, an analytical solution based on Laplace transformation is obtained by means of an appropriate treatment of the source term. To test and verify the relevance of the analytical solution, results are compared against a set of available experimental data of Poggi [20], Reinauer and Hager [21], Beltrami et al. [22] and also Ippen [1]. The main purpose of this work is to propose a simple and practical analytical model to predict simultaneously water surface profile elevation in channel bends and the downstream reach which has not been so far treated analytically as a whole previously. II. ASYMPTOTICAL MATHEMATICAL MODEL Steady two-dimensional inviscid free-surface flow in curved channels can be described by a set of partial differential equations (PDEs) derived by applying mass and momentum conservation laws. Assuming hydrostatic pressure distribution and uniform velocity distribution in the vertical direction, this mathematical model can be written for a rectangular channel in cylindrical coordinates as follow [23]
h
Vr r
Vr
g Vr
h r
V r
h r
Vr
h V r
Vr r
g h r
V h r
V Vr
r
V V r
Vr h
r
V2 r
Vr V r
(1)
(2)
(3)
where h is the flow depth; V the flow velocity in the curvilinear longitudinal direction; Vr the flow velocity in the radial direction; g acceleration due to gravity;
r and are radial and angular coordinates respectively (Figure 1).
The assumption of frictionless liquid is based on the fact that for fully developed flow in a curved channel, frictional effects on the super-elevation are small [24].
Beside the aforementioned assumption, PDEs set is simplified through the following: (i) the flow field is supercritical in the whole chute bend; (ii) longitudinal gradient effects can be neglected compared to transverse effects as pointed out and affirmed experimentally by Reinauer and Hager [21]. The last assumption states that the change of the streamwise velocity V is very small, that is, Vr r V 0 as the flow is hypercritical. It is then admitted that dominating inertia effect is transversal to the flow streamwise axis. According to the quasi-2D approach developed by Amara et al. [25], if we rewrite equations (1) to (3) for an observer moving along the flow filament or streamline r r0 at constant velocity V s t u 0 , with u 0 as the approach flow velocity and s r the curvilinear coordinate, one obtains the following intrinsic equations in mobile coordinates system
h t
Vr t
h
Vr r
Vr
Vr
Vr r
h r
g
h r
Vr h
(4)
r
V2
(5)
r
V V V h V Vr gh V 2 r r V r h r Vr r
(6)
Dynamic equation (6) in longitudinal axis is therefore reduced to radial velocity profile description V r . In this equation the right hand side factor is the square inverse streamwise Froude number, thus gh V2 F02 . For most practical cases, terms 2
between brackets are of the order of 10 0 . In addition, for supercritical flow with large approach Froude number, F0 has the order of 10
1
or smaller. To a first order approximation, this term can thus be dropped and equation (6) simplifies to
V V 0 r r
(7)
Equation (7) is the well-known differential equation of free vortex in fluid mechanics [26]. Analytical integration is simple and leads to the general solution in the form
V
C
(8)
r
It can be shown for particular solution [27] that the constant C c 2 , where c is the vortex strength. Note that a complete solution of equation (6) leads to a slight deformation only of the free vortex velocity distribution. Under these assumptions the system of equations describing supercritical free-surface flow in curved channel in moving reference is written in vector form
G G E T t r
(9)
where
h V G ; E r Vr g
h ; Vr
Vr h T r2 V r
(10)
This final set of differential equations obtained for a moving observer is then strictly identical to 1D flow. Using the quasi-2D approach [25] one can show that for curved channel under supercritical conditions, the steady 2D shallow water equations tend asymptotically to 1D unsteady flow problem in radial direction. For the observer the problem is then reduced to an analogous 2 problem of unsteady flow in a diverging channel subject to centrifugal acceleration V r . Once the solution of asymptotic time dependent problem model (9) is obtained, the approximate 2D solution flow in space dimension is then easily reconstituted by means of equation (8).
Formally, the passage between the temporal field obtained as 1D solution to the space field required as approximate solution of the 2D flow is ensured by a transformation function noted . This function allows the repositioning of each fluid particle,
obtained in time-dependent problem, into its homologous point in space-dependent problem. The transformation time-space solution can be expressed by the following model
r , r , t
(11)
in which r , and r, t are solutions in space-domain and time-domain respectively. The function can be expressed in term of radial velocity distribution (8) as follows
1 b
V r dr
rr
ri
(12)
where ri and re refer respectively to the inner and the outer channel walls radii. The spatial position associated to each time solution as an image sequences is written simply as the kinematic observer’s law
Xp
t
dt 0
(13)
Xp is the space coordinate of the time solution plane. Hence, the present asymptotical quasi-2D model approach consists on the resolution of 1D hyperbolic time dependent problem combined with a pure advection in space perpendicular to the 1D wave problem plane. III. ANALYTICAL SOLUTION Regarding the initial form of system (9), obviously it consists on nonlinear hyperbolic partial differential equations and does not admit a closed analytical solution except for simplified cases. A numerical integration is therefore needed by classical methods as finite differences [28] or method of characteristics [29, 30]. Nevertheless, analytical solution of (9) for curved channels flow is possible if a linearization is done. For this purpose, let consider a small perturbation of amplitude induced on the initial constant water depth h0 before the transition such as h h0 and h0 . Introducing this condition and tacking into account that for practical cases B / r 1 the system (9) is then reduced to
V h0 r 0 t r
(14)
Vr V2 g t r r
(15)
Equations (14) and (15) are in the form of first order shallow water waves model and admit two eigenvalues gh0 . The combination of these equations leads to the non-homogeneous wave equation 2 3h C ² 2 2 c 04 2 2 t r r
(16)
where c is the celerity of small disturbances. From physical considerations, it is possible to obtain a simple solution of equation (16) using indirect treatment of the source term. Instead of dealing directly with centrifugal acceleration field to which the liquid is subjected in the curvature, it is mathematically convenient to treat the problem with an analogue deceleration field. Indeed, from Einstein’s equivalence principle a dynamic similarity exists if we consider the problem as a liquid flowing in radial direction at a velocity V and gradually decreasing to zero. The hypothetical radial velocity V at which the liquid is initially flowing is computed from dynamic considerations
V2 V r
(17)
in which is the time of complete extinction of the hypothetical radial flow. Here, the deceleration is considered constant and equal to V r0 . Time of radial flow extinction is in reality the time taken by the mobile observer to pass through the curved transition or time in which centrifugal effects hold. Thus 2
sc u0
(18)
with s c max r0 being the curved transition length and max the opening angle of the bend. The main objective is then to solve the homogeneous part of equation (16) for the dependent variables and the hypothetical V subject to initial conditions
V t and 0 . Because of the anti-symmetry of the interference waves along the channel axis, only one-half of the channel needs to be considered. The domain is then limited between radii r0 and re . For commodity of boundary conditions treatment, a local coordinate system transformation is introduced such as r r0 0 and r re L from which 0 B2 L . Introducing Laplace integral transformation of the dependent variables [31]
ˆ , p e pt , t dt
(19)
0
Vˆ , p
0
e ptV , t dt
(20)
The homogeneous wave equation (16) in local coordinates reads then
c2
2ˆ p 2ˆ p t 0 2 t
(21) t 0
Taking into consideration initial conditions of the problem, the subsidiary solution for ˆ in term of hyperbolic functions becomes
p c
p c
ˆ A cosh B sinh
(22)
From dynamic equation (15) written here for V instead of V r the corresponding transformation leads to
ˆ 1 V g
0
pVˆ
(23)
Boundary conditions for decelerating flow in radial direction are 1) At the channel axis 0 , no fluctuation is generated due to negative interferences (node point) thus
0, t 0 so ˆ 0, t 0
(24)
2) At outer boundary wall L , hypothetical radial velocity V decreases linearly to zero. This ensures creation of equivalent dynamic centrifugal effects
t V L, t V0 1 ; V L, t 0t t
(25)
From which
1 1 Vˆ L, t V0 2 p p
; t 1 1 Vˆ L, t V0 2 p p
V0 e p p 2 t
(26)
in which V0 is the initial radial velocity. The constants A and B of equation (22) can be now determined. When conditions (24) and (26) are substituted into equations (22) and (23)
A0
;
cV 1 B 0 g 2 p p cosh c t
and
cV p 1 e B 0 g 2 p p cosh c t
(27)
The subsidiary equations are thus
p sinh cV c ; ˆ 0 g 2 p p cosh c t p sinh cV c 1 e p ˆ 0 g 2 p p cosh c
t
(28)
The inverse Laplace transform is obtained using the Bromwich integrals [32]. Applying the procedure of inversion the freesurface elevation with respect to the undisturbed flow for t gives
, t
cV0 g
8L c c 2
1k 1 sin 2k 1 2 2L k 1 2k 1
cos2k 1
ct
(29) 2 L
For practical application, a convenient and physical meaning solution can be obtained in terms of traveling waves or Heaviside wave alternate. By replacing the hyperbolic functions of equation (28) in the Bromwich integral and using the shifting theorem the solution can be expressed as follows (see appendix 1)
, t
cV0 g
1
k 1
k 1
t
2k 1L u t 2k 1L t 2k 1L u t 2k 1L c
c
c
c
(30)
where u t is the Heaviside step function. For the “after-waves zone” of stage t the inverse Laplace transform leads to
, t
cV0 g
8L 2 c c
cos2k 1
1k 1 sin 2k 1 2 2L k 1 2k 1
ct
cV 0 2 L g
8L 2 c c
1k 1 2 k 1 2k 1
sin 2k 1
2L
cos2k 1
ct 2L
u t (31) t
This holds for downstream straight channel. From practical view point of channel design, it is the side wall water profile which is of interest. Thus for L equation (29) reads
, t The infinite trigonometric series in (29) is replaced by the function expressed as
cV0 L 8L 2 g c c
(32)
cos
1 1 cos 3 2 cos 5 (33) 2 3 5
in which ct 2L . Note that for to 0
(34)
(35)
1 2 2 8
1 2 2 8
and for 0 to
The analytical solution given by equation (32) is a triangular wave of a period 4L / c where the maximum water profile on side wall is reached at 2L / c (see appendix 2). IV. APPLICATION AND MODEL VERIFICATION For the verification of the present model, a comparison with a set of experimental data in a curved rectangular channel conducted by Ippen [1], Poggi [20], Reinauer and Hager [21] and Beltrami et al. [22] is carried out. Analytical calculations are performed to determine the steady state flow solution. Results concern mainly the determination of waterline profiles along the outer wall. A very important feature of this flow, for design purpose, is the first peak location and magnitude of the standing waves. Therefore a particular attention is given to these parameters in comparison with experiments. However the disturbed flow in downstream channel is not of less importance. Thus a part of the model verification is to compare the water surface flow in the whole transition (curved and downstream channel) to check the reliability of the solution to predict interference effects in the straight channel reach. Figures 2 and 3 compare respectively the normalized water surface profile h / h0 for the first wave computed by the proposed analytical model with selected data tests of Poggi [20] and Reinauer and Hager [21] as function of angular location coordinate . One of the experimental investigations conducted by Poggi [20] concerned a bend of rectangular cross-section with deflection angles of 45° and relative curvature B / r 1 / 12 . The bottom slopes were 5 and 10 % and the approach Froude numbers were between 2.27 and 5.1. Reinauer and Hager [21] performed experiments in horizontal, smooth, rectangular channel with a smaller relative curvature deflected at 51°. A range of Froude number of 2.0–12.0 and flow discharge comprised between 21.9 l/s and 70.0 l/s were considered. From these figures, it can be seen that the agreement between the computed and measured water surface profile for the first wave is almost satisfactory. The triangular wave shape predicted is slightly different from the hill-like wave observed as the mathematical model is of first order. For Poggi’s runs (Figure 2) the maximum wave magnitude is well predicted with a maximum deviation of 1.73 %. However a difference in the location is noted for lower Froude number F 3.04 (Figure 2a). The deviation was of 20.35 % in this case but only 3.41 % for F 4.83 (Figure 2b). Similarly, the comparison of computed and measured waterline on side wall by Reinauer and Hager [21] shows a good agreement. For approach Froude numbers 2.5, 3, 4 and 6 respectively (Fig. 3a-3d) the first wave in terms of magnitude and location is reasonably reproduced. For all tests, small overestimation of the super-elevation and a slight shifting in predicting location of the wave were observed. This discrepancy may be attributed to the streamwise momentum effect which is reduced to a pure kinematic motion in the asymptotical present model. The maximum deviation of about 8.7 % is noticed for F 6 . In contrast to Poggi’s tests, the model results show the fact that for horizontal channel some sort of a lag and decay effects proportional to velocity flow appear. As nonlinear terms were not considered, the proposed solution model cannot predict wave damping. Nevertheless, because only the maximum wave is of design interest this offers more safety for side wall sizing. In order to study the performances of the present analytical model to predict wave alternation behind the first shock, additional verification tests were performed. This part of verification concerned the successive wave occurrence in the bend itself and also in the downstream straight channel. In addition to magnitude, wavelength is introduced as a parameter in result analysis. Beltrami et al. [22] conducted an experimental investigation on supercritical flow in 180° bend deflection under different hydrodynamic approach conditions. Two test data for which B 0.2 m and r0 0.8 m were selected. Figures (4a) and (4b) refers to water surface profile comparison for approach flow condition h0 B 0.2 and h0 B 0.26 respectively, under incident Froude number F0 3 .
(a)
2,0
(b)
3,5
Poggi (1956) Present Model
1,8
3,0
1,6
2,5 h/h0
h/h0
Poggi (1956) Present model
1,4
1,2
2,0
1,5
1,0
1,0 0
5
10
15
20
25
30
35
40
0
10
20
Angle (°)
30
40
50
Angle (°)
Figure 2 Comparison of the present analytical model with Poggi’s experimental data for waterline on outer side-wall
(a)
1,6
(b)
1,8
Reinauer and Hager (1997) Present model
Reinauer and Hager (1997) Present model
1,7
1,5
1,6 1,5
1,3
h/h0
h/h0
1,4
1,4 1,3
1,2
1,2 1,1
1,1
1,0
1,0 0
2
4
6
8
10
12
14
16
18
20
22
0
5
10
Angle (°)
(c)
2,4
(d)
4,0
Reinauer and Hager (1997) Present model
2,2
15
20
25
Angle (°)
Reinauer and Hager (1997) Present model
3,5
2,0
3,0 h/h0
h/h0
1,8 1,6
2,5 2,0
1,4
1,5
1,2 1,0
1,0 0
5
10
15
20
Angle (°)
25
30
35
0
5
10
15
20
25
30
35
40
45
50
Angle (°)
Figure 3 Comparison of the present analytical model with Reinauer and Hager’s experimental data for waterline on outer side-wall
(a)
4,0 3,5
3,5
3,0
3,0
2,5
2,5
h/h0
h/h0
(b)
4,0
Beltrami et al. (2007) Present model
2,0
2,0
1,5
1,5
1,0
Beltrami et al. (2007) Present model
1,0 0
20
40
60
80
Angle (°)
100
120
140
0
20
40
60
80
100
120
140
Angle (°)
Figure 4 Comparison of the present analytical model with Beltrami’s et al. experimental data for waterline on outer side-wall
(a)
100
(b)
120 Ippen (1936) Present model
Ippen (1936) Present model
110
90
100
h (mm)
h (mm)
80 70
90 80 70
60
60
Curved section
50
Curved section
Straight section
50 0
1
2
3
4
5
6
7
8
9
10
0
11
1
2
3
5
6
7
8
9
10
11
Distance (m)
Distance (m)
(c)
60
4
(d)
180
Ippen (1936) Present model
Ippen (1936) Present model
160
55
140
50 h (mm)
h (mm)
120
45 40
100 80
35
60
30 0
1
2
3
Curved section
40
Curved section
4
5
6
Distance (m)
7
8
9
10
11
0
1
2
3
4
5
6
7
8
Distance (m)
Figure 5 Comparison of the present analytical model with Ippen’s experimental data for waterline on outer side-wall
As it can be observed, two successive wave crests are formed within the bend reach channel with a water trace shaped like a damped harmonic motion. Results show here similar pace between computed and measured water profiles. Furthermore, first wave in both cases is well reproduced in amplitude and position. The second one however is somewhat overestimated in magnitude and a small lag in position is noticed. In this case the analytical model generates an indefinite triangular wave solution in the bend reach whereas an exponential decay in the real water-profile is produced. As mentioned above, the friction damping plays a major role in this case for discrepancies behind the first wave. A maximum deviation in wavelength of 11.11% is observed in the first test while only 3.45 % deviation is noticed in the second one. In the last and ultimate step, the model is tested for water surface profile prediction in curved and straight downstream reach channel as a whole. As the unique study of this kind available is that carried out by Ippen [1], his experimental data were then used. In his outstanding PhD work, Ippen conducted a number of tests for three channel slopes, three radii of curvature (12.2 m, 6.1 m and 3 m) and two deflection angles (22.5° and 45°). Each geometric configuration was tested for a range of discharges resulting in 93 runs. Results of computed and measured water depth h as function of the curvilinear streamwise coordinate along the outer wall for runs 28, 32, 34 and 52 are given respectively in Figure (5a-5d).
One can see that in overall, computed solution shows trends similar compared with observation. For the curved part of the channel, there is a certain match between the analytical and measured water depth for the first two waves. Nevertheless, an underestimation of the second wave magnitude is noticed for runs 28 and 32 (Figures 5a and 5b). Here, a remarkable experimental fact is that the second wave is of a magnitude larger than its preceding one. This report contradicts the wellestablished assertion that the first wave is always the one to be considered for design interest as stated by many authors [21, 18]. Except the discrepancy for second wave magnitude, the position and wavelength are accurately reproduced with a maximum error deviation of 5.92 % and 11.68 % respectively. In the downstream straight channel the suppression of centrifugal effects creates a symmetrical wave interference phenomenon. Depending upon on the combination of this with flow condition at the bend end, a specific flow pattern configuration is thus created, resulting in a disturbed downstream flow field or not. Such cases are well shown in figure 5. As depicted, computed water surface profiles are relatively compatible with observations at a first order approximation. Whereas a relative difference in magnitude is noticed, a good wave periodicity is however obtained. A phase-angle shift of approximately is observed for run 32 (Figure 5b). It must be pointed out for discrepancies observed that crossing of the waves and reflection at the wall result in some air entrainment in Ippen’s experiments, thereby making the accuracy of the measured results somewhat questionable. Moreover, same difficulties in computing wave amplitude in this case were also noticed even using an advanced CFD model by Brown and Crookston [33]. Comparison of analytical model results and experiments of all test runs is summarized in Table 1. Relative error deviation Table 1 Summary of typical wave characteristics along outer wall and errors analysis of the analytical model
Poggi Reinauer and Hager
Beltrami et al. Ippen
Data Channel 1 2 1 1 1 1 1 1 1 2 2 3
Run 3 5 1 2 3 4 3 5 28 32 34 52
Maximum wave Yobs Ycal 1.77 1.77 2.89 2.94 1.42 1.43 1.60 1.62 2.04 2.11 3.22 3.50 3.25 3.25 3.10 3.25 1.57 1.38 1.81 1.74 1.74 1.77 1.53 1.49
E% 0.00 1.73 0.70 1.25 3.43 8.70 0.00 4.84 12.10 3.87 1.72 2.61
Location θobs/Xobs θcal/Xcal 18.23 22.30 8.83 10.38 13.81 18.77 37.97 41.00 3.63 3.67 3.38 3.79
14.52 23.06 9.93 11.91 15.88 23.82 42.96 42.97 3.58 3.51 3.58 3.75
E%
Wave length Λobs Λcal
20.35 3.41 12.46 14.74 14.99 26.90 13.14 4.80 1.38 4.36 5.92 1.06
1.08 1.16 2.31 2.45 2.14 1.49
1.20 1.20 2.39 2.34 2.39 1.50
E% 11.11 3.45 3.46 4.49 11.68 0.67
from observed values is given for maximum wave magnitude Y hmax h0 , its polar location or streamwise coordinate X and the wavelength where subscripts obs and cal denote observed and calculated values respectively. As mentioned above, analytical computation yields satisfactory results for maximum wave magnitude, with perfect matching for figures (3a) and (4a), but with a relative shifting deviation in peak wave location. Indeed, this is inevitable due to the simplified hypothesis adopted in model formulation. Similar statement of discrepancies was pointed out by Hessaroeyeh and Tahershamsi [18] for their analytical model which is of first order approximation. Moreover, their model has not been so far validated for wave alternation beside the first peak. In contrast, wavelength and interferences in the bend and downstream reach were predicted with a good accuracy in the present model. This is clearly shown for Ippen’s run N°52 where deviate only 0.67% from measurements. It is important here to add a comment about the hypothesis of small deflection in the analytical solution. Indeed, in the linearization process, it was assumed that h0 so that the wave propagation becomes linear i.e. flux function of shallow water model and thus characteristic curves are also linear. By this assumption, no diffusion or dispersion in the wave front propagation is introduced no matter how amplitude is. If the condition of small amplitude is not fulfilled, a slight phase angle shift (i.e. front wave angle) appears but does not affect significantly the maximum wave prediction on the side wall. This is well explained in depth by Cunge et al. [34] by mathematical and physical considerations of shocks and discontinuities. In most cases this order of approximation deviates only slightly from the complete solution using theory of strong shocks and it seems not necessary to resort to such calculation. It is to be pointed out also that, in addition to the model limitation to predicting wave attenuation, no so far reliable solution is expected to be obtained if strong curvatures with bend number
F0 B r greater
than 2 are treated. In this case separation along the inner bend wall occurs. At the limit of very sharp bends, flow blocking due to the reflection effect by lateral walls may occurs as noticed by Valiani and Caleffi [35]. It follows that the first order wave approximation with weak shocks is reliable in this case as far as sharp bends are not considered. V. CONCLUSION
In this paper, approximate analytical model was proposed for application to steady supercritical flow in rectangular curved channel transitions. Using a quasi-2D approach, the two-dimensional depth-averaged equations of motion were transformed to an equivalent 1D time-dependent problem as an asymptotic model for supercritical conditions. A first order approximation was then obtained using small perturbation theory resulting in a non-homogeneous linear wave equation in radial direction. The analytical solution was then obtained using Laplace integral transform. The model validation concerned the prediction of water surface profile along the outer wall in the bend reach and the downstream channel as a novel proposition. For this, results are compared with the available under various geometric and approach flow conditions. The analytical model results obtained for the different test cases indicate a fair good agreement compared with observations. The general physical behavior of the problem is well captured. Accurate results were particularly reached for the first wave magnitude prediction. However a slight lag in predicting the wave crest location was noticed. Particular attention in the verification was paid for the model ability to reproduce wave periodicity. It was found that an almost accurate periodic wave pattern of crests matched well measurements either in the curve or in the downstream straight channel. As the friction terms were neglected in the model some discrepancies in magnitude behind the first wave occurred due to damping effects. Furthermore, the analytical model limits are inherently related to the shallow water model weakness hypothesis and channel’s relative curvature. For engineering application, it is the maximum water depth rise at walls in the curved and the downstream straight channel which is an important design parameter to avoid overtopping. From this view point the proposed analytical solution gives a rather good prediction of flow surface configuration. Improvement of the analytical model can be made to include in particular friction damping effects. Further mathematical complexity in the solution involving special functions (Bessel functions) are then to be expected. VI. APPENDIX
1. Solution in term of Heaviside’s traveling waves
Inversion of equation (28) using Bromwich integral reads
cV0 1 g 2i
e
p sinh c dp p p 2 cosh c
pt
C
(A1)
in which C is the Bromwich contour in the complex plane. Replacing the hyperbolic functions by exponentials and multiplying numerator and denominator by e
sL / c
Since
1 e
cV0 1 g 2i
2 pL / c
p 1 e
e pt e pL / c e p / c e p / c dp 2
C
1 e
2 pL / c
2 pL / c
(A2)
e 4 pL / c e 6 pL / c (A3)
Substituting (A3) in (A2) gives
cV0 2ig
e
L p t c c
e
C
L p t c c
3L p t c c
e p2
e
3L p t c c
dp
(A4)
Noting that
1 2i
e p t k dp 0 for 0 t k C p2 t k for t k
(A5)
Integrating (A4) term by term giving the shifting property
cV0 g
L L L L u t t u t t c c c c
3L 3L t u t c c 3L 3 L t u t c c
(A6)
This can be rewritten in condensed form (30). For the “after waves zone”, a similar procedure gives
cV0 g
1
k 1
k 1
t
2k 1L u t 2k 1L c
c
2k 1L 2k 1L t u t c c
cV0 g
1 k 1
k 1
t
2k 1L u t 2k 1L c
c
2k 1L u t 2k 1L t c c
(A7)
2. Maximum wave magnitude and classical Ippen’s formula
As it was mentioned above, the maximum wave magnitude on the outer wall is reached at the half-period t 2L / c which leads to . Thus, equation (35) becomes
2 8
(A8)
Consequently
L, 2cL max
2 LV0 g
(A9)
From equation (17)
V2 V0 r0
(A10)
Equation (A9) reads after substitution of (A10)
max
2 LV2 gr0
(A11)
max
u 02 B gr0
(A12)
Recalling that L B 2 and V u 0 , one obtain the maximum wave magnitude
which is the classical formula pointed out first by Ippen (1936). It shows that for supercritical flow the increase in depth on the outer side wall is twice than for subcritical case if small curvatures are considered. ACKNOWLEDGMENTS The first author would like to acknowledge the valuable suggestions of Mr. Soufiane Tebache which have improved the quality of the paper.
NOTATION The following symbols are used in this paper
E =Flux jacobian matrix
G = Primitive flow variable vector T = Source term vector B = channel width (m) C = Constant, Bromwich integral contour
c = Small disturbance wave celerity (m/s) F0 = Approach flow Froude number g = Gravitational acceleration (m/s²)
hr , = Flow depth (m) h0 = Approach flow depth (m)
hmax = Maximum flow depth at wall (m)
L = Half-channel width (m) p = Inversion Laplace transform parameter r = Radial coordinates (m)
r0 = Axial radius of curvature (m) ri = Inner radius of curvature (m) re = Outer radius of curvature (m) s = Curvilinear coordinate (m)
s c = Curved channel reach length (m)
t = Time variable (s) u 0 = Approach flow velocity (m/s)
u t = Heaviside step function V = Streamwise depth-averaged velocity component (m/s) V r = Radial depth-averaged velocity component (m/s)
V , t = Hypothetical radial flow velocity (m/s)
V0 = Initial radial flow velocity (m/s) X p = Space-coordinate of the computational plane (m)
Y = Relative flow depth at channel
= Angular coordinates (°)
max = Bend opening angle (°)
c = Vortex strength
r , = Solutions in space-domain r, t = Solution in time-domain =Transformation time-space function = Eigenvalues = Canceling time of hypothetical radial flow (s)
= Centrifugal acceleration (m/s²) = Local radial coordinate (m)
max = Maximum wave magnitude (m)
, t = Small perturbation wave magnitude (m)
ˆ , p = Laplace transform of , t Vˆ r , p = Laplace transform of V , t
= Phase-angle parameter = Trigonometric series function = Wavelength (m)
= Bend number REFERENCES
[1] Ippen, A.T. (1936). An analytical and experimental study of high velocity flow in curved sections of open channels, PhD Thesis, California Institute of Technology. [2] Kármán, T.V. (1938). Eine praktische Anwendung der Analogie zwischen Überschallströmung in Gasen und überkritischer Strömung in offenen Gerinnen. ZAMM ‐ Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 18(1), 49-56. (in German).
[3] Knapp, R.T. (1951). High-Velocity Flow in Open Channels: A Symposium Design of Channel Curves for Supercritical Flow. Transactions of the American Society of Civil Engineers, 116(1), 296-325. [4] Harrison, A.J.M. (1966). Design of Channels for Supercritical Flow. Proceedings of the Institution of Civil Engineers, 35(3), 475-490. [5] Lénau, C.W. (1979). Supercritical flow in bends of trapezoidal section. Journal of the Engineering Mechanics Division, 105(1), 43-54. [6] Hager, W.H., Altirtakar, M.S. (1984). Infinitesimal cross-wave analysis. Journal of Hydraulic Engineering, 110(8), 1145-1150. [7] Knapp, R.T., Ippen A.T. (1938). Curvilinear flow of liquids with free surfaces at velocities above that of wave propagation. Proc. 5th Int. Congress Applied Mech., 531–536. Cambridge Cambridge University Press, NewYork. [8] Ellis, J., Pender, G. (1982). Chute spillway design calculations. Proceedings of the Institution of Civil Engineers, 73(2), 299312. [9] Hosoda, T., Yokosi, S. (1987). Some Considerations on High Velocity Flows Through Curved Open Channels. Doboku Gakkai Ronbunshu, (387), 171-178. [10] Iwasa, Y., Hosoda, T. (1989) Numerical simulation on high velocity flows through curved open channels, The International Conference on Interaction of Computational Methods and Measurements in Hydraulics and Hydrology, Dubrovnik, Yugoslavia, June, 87-96. [11] Dammuller, D.C., Bhallamudi, M.S., Chaudhry, M.H. (1989). Modeling of unsteady flow in curved channel. Journal of Hydraulic Engineering, 115(11), 1479-1495. [12] Elliot, R.C., Chaudhry, M.H. (1992). A wave propagation model for two-dimensional dam-break flows. Journal of Hydraulic Research, 30(4), 467-483. [13] Iwasa, Y., Hosoda, T., Kawamura, N., Yoneyama, N. (1991). High Velocity Flow with Free Boundary in Open Channels. Proceedings of hydraulic engineering, 35, 531-536. [14] Berger, R.C., Stockstill, R.L. (1995). Engineering, 121(10), 710-716.
Finite-element
model for high-velocity
channels. Journal of Hydraulic
[15] Ye, M., Wu, C., Chen, Y., Zhou, Q. (2006). Case study of an S-shaped spillway using physical and numerical models. Journal of Hydraulic Engineering, 132(9), 892-898. [16] Hessaroeyeh, M.G., Tahershamsi, A., Namin, M.M. (2011). Numerical modelling of supercritical flow in rectangular chute bends. Journal of Hydraulic Research, 49(5), 685-688. [17] Jaefarzadeh, R.M., Reza Shamkhalchian, A., Jomehzadeh, M. (2012). Supercritical flow profile improvement by means of a convex corner at a bend inlet. Journal of Hydraulic Research, 50(6), 623-630. [18] Hessaroeyeh, M.G., Tahershamsi, A. (2009). Analytical model of supercritical flow in rectangular chute bends. Journal of Hydraulic Research, 47(5), 566-573. [19] Steffler, P.M., Rajaratnam, N., Peterson, A.W. (1985). Water surface at change of channel curvature. Journal of Hydraulic Engineering, 111(5), 866-870. [20] Poggi, B. (1956). Correnti Veloci nei Canali in Curva. L’energia Elettrica, 33(5), 465–480 (in Italian). [21] Reinauer, R., Hager, W.H. (1997). Supercritical bend flow. Journal of hydraulic engineering, 123(3), 208-218.
[22] Beltrami, G.M., Del Guzzo, A., Repetto, R. (2007). A simple method to regularize supercritical flow profiles in bends. Journal of Hydraulic Research, 45(6), 773-786. [23] M.B., Abbott, Computational Hydraulics; Elements of the Theory of Free Surface Flow. Pitman Publishing Limited, London, 1979. [24] Yen, C.L., Yen, B.C. (1971). Water surface configuration in channel bends. Journal of the Hydraulics Division, 97(2), 303-321. [25] Amara, L., Berreksi, A., Achour, B. (2017). Quasi-2D model for computation of supercritical free surface flow in sudden expansion. Applied Mathematical Modelling, 46, 396-407. [26] H. Rouse, Fluid mechanics for hydraulic engineers. Dover Publications Inc. 1938. [27] J.F., Douglas, J.M., Gasiorek, J.A., Swaffield, L., Jack, Fluid Mechanics, Pearson/Prentice Hall, Harlow, 2005. [28] M.H., Chaudhry, Open-channel flow. Springer Science and Business Media, 2008. [29] Abbott, M.B., Lindeyer, E.W. (1969). Two transients for a radial nearly-horizontal flow. La Houille Blanche, (1), 65-70. [30] Townson, J.M., Al-Salihi, A.H. (1989). Models of dam-break flow in RT space. Journal of Hydraulic Engineering, 115(5), 561575. [31] Churchill, R.V. Operational Mathematics, McGraw- Hill, New York, 1972. [32] Duffy, D.G. Transform methods for solving partial differential equations. Chapman and Hall/CRC, 2004. [33] Brown, K., Crookston, B. (2016). Investigating Supercritical Flows in Curved Open Channels with Three Dimensional Numerical Modeling. In B. Crookston & B. Tullis (Eds.), Hydraulic Structures and Water System Management. 6th IAHR International Symposium on Hydraulic Structures, Portland, OR, 27-30 June, 230-239. [34] Cunge, J., Holly, F.M., Varwey, A. (1980). Practical aspects of computational river hydraulics. Pitman Publishing Ltd. London. [35] Valiani, A., Caleffi, V. (2005). Brief analysis of shallow water equations suitability to numerically simulate supercritical flow in sharp bends. Journal of Hydraulic Engineering, 131(10), 912-916.
Poggi Reinauer and Hager Beltrami et al. Ippen
Data Channel 1 2 1 1 1 1 1 1 1 2 2 3
Run 3 5 1 2 3 4 3 5 28 32 34 52
Maximum wave Yobs Ycal 1.77 1.77 2.89 2.94 1.42 1.43 1.60 1.62 2.04 2.11 3.22 3.50 3.25 3.25 3.10 3.25 1.57 1.38 1.81 1.74 1.74 1.77 1.53 1.49
E% 0.00 1.73 0.70 1.25 3.43 8.70 0.00 4.84 12.10 3.87 1.72 2.61
Location θobs/Xobs θcal/Xcal 18.23 22.30 8.83 10.38 13.81 18.77 37.97 41.00 3.63 3.67 3.38 3.79
14.52 23.06 9.93 11.91 15.88 23.82 42.96 42.97 3.58 3.51 3.58 3.75
E% 20.35 3.41 12.46 14.74 14.99 26.90 13.14 4.80 1.38 4.36 5.92 1.06
Wave length Λobs Λcal 1.08 1.16 2.31 2.45 2.14 1.49
1.20 1.20 2.39 2.34 2.39 1.50
Table 1 Summary of typical wave characteristics along outer wall and errors analysis of the analytical model
E% 11.11 3.45 3.46 4.49 11.68 0.67
Approximate Analytical Solution for Supercritical Flow in Rectangular Curved Channels Lyes Amara , Ali Berreksi , Bachir Achour PII: DOI: Reference:
S0307-904X(19)30666-3 https://doi.org/10.1016/j.apm.2019.10.064 APM 13131
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
1 April 2019 21 October 2019 30 October 2019
Please cite this article as: Lyes Amara , Ali Berreksi , Bachir Achour , Approximate Analytical Solution for Supercritical Flow in Rectangular Curved Channels, Applied Mathematical Modelling (2019), doi: https://doi.org/10.1016/j.apm.2019.10.064
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Highlights • An asymptotical quasi-2D mathematical model and analytical solution is proposed. • 2D steady shallow water model is converted to 1D unsteady radial flow problem. • Radial wave equation is solved analytically using Laplace transformation. • Solution covers the curved part and downstream straight channel of the transition. • Predicted side-wall water profiles are in satisfactory agreement with experimental data.
Approximate Analytical Solution for Supercritical Flow in Rectangular Curved Channels Lyes Amara (*) 1 Ali Berreksi 2 Bachir Achour 3 1
2
Department of Civil Engineering and Hydraulics, Faculty of Science and Technology, University of Jijel, Algeria,
Laboratory of Applied Hydraulics and Environment (LRHAE), Department of Hydraulics, Faculty of Technology, University of Bejaia, Targa Ouzemour, 06000, Bejaia, Algeria 3
Research laboratory in subterranean and surface hydraulics (LARHYSS), University of Biskra, 07000, Biskra, Algeria
Email: [email protected] ; [email protected] ; [email protected]
Abstract- Asymptotical mathematical model and analytical solution of a supercritical flow in curved rectangular open channels are presented. An original study approach for free-surface configuration and features of the flow in presence of cross shock waves is then proposed. The two-dimensional steady depth-averaged shallow water equations are transformed to an equivalent one-dimensional unsteady flow problem and a first order approximation is then obtained using small perturbation theory. Further, 1D asymptotic model is analytically solved by Laplace integral transformation and the 2D flow field solution is rebuilt according to translating planes. The freesurface profile along the outer chute wall and downstream channel is compared with the available experimental data. Obtained results show a satisfactory agreement for the maximum flow depth, peaks position and wavelength. The proposed approach allows an accurate prediction of flow features and safe design of curved channel transitions. Keywords- channel transitions, curved channels, analytical solution, Laplace transform, supercritical flow, standing waves.
I. INTRODUCTION Open channels flow under supercritical conditions is often required in many hydraulic engineering applications such as spillways chutes, sewers and outlet works. In these conveyance structures, lateral changes of boundary alignments like bends are unavoidable and necessary in many situations. As it is well-known, in supercritical flow any change in wall directions generates cross waves affecting the water surface and velocity field downstream of the channel. Consequently, high disturbances and super-elevations at wall bends are occurred. Special careful must then be given to predict correctly those risky elevations for safe design of curved channels and downstream reach. Supercritical flow in bends was first investigated by Ippen [1] who developed an analytical approach based on the theory of small disturbances and carried out intensive validation experiments. It resulted that the assumption of constant velocity leads to very satisfactory analytical predictions. Von Kármán [2], using shallow water equations and assuming constant energy, gave interesting analytical solution in analogy with gas dynamics theory. In the symposium on supercritical flow, Knapp [3] presented a remarkable synthesis on the analysis of these flows in curved channels, their design and reduction of the super-elevation due to standing waves. Later, Harisson [4] analyzed the supercritical flow in trapezoidal cross section channel. Admitting a linear free-surface, he proposed a simplified model for super-elevation at the edges. A notable jump in the problem resolution approach was made by Lénau [5]. Based on the potential equations theory, the author presented an analytical solution of curved channels (trapezoidal and rectangular cross section) by a subtle approach using perturbation theory and Laplace transform. The solution is valid for an approach Froude number F0 2 and the ratio of approach flow depth to the center radius of the curvature is small. Hager and Altinakar [6] applied the momentum and continuity equations to the analysis of infinitesimal cross waves. For small wave angles, it was shown that the proposed solution is in complete agreement with those of Knapp and Ippen [7] and velocity across the wave remains constant. The computational approach of the aforementioned problem was introduced first by Ellis and Pender [8]. The steady inviscid 2D shallow water equations are solved numerically using the Method of Characteristics for single and S-shaped bends. An identical numerical approach in polar coordinates was successfully used on various tests by Hosoda and Yokosi [9] and Iwasa and Hosoda [10] for the rectangular channels. Dammuler et al. [11] analyzed an unsteady flow on a curved channel following an instantaneous dam break by finite differences method. The same problem was studied by Elliot and Chaudhry [12] using a simplified 1D mathematical model built on wave’s interactions and reflections in two-dimensions. Other numerical works such those of Iwasa et al. [13] and Berger and Stockstill [14] treated trapezoidal channels. Ye et al. [15] undertook an experimental and numerical study on S-shaped spillway using a k turbulence model. Later Hessaroeyeh et al. [16] and Jaefarzadeh et al. [17] used Finite Volume Method with HLLC-TVD and Roe-TVD solvers respectively to simulate supercritical flow in channel bends. However, these advanced tools remain quite complex for engineering practice which requires simple and reliable solutions. The literature review, at authors’ best knowledge, shows a gap in the field of analytical solutions for more than three
decades. The single analytical approach to be noted from Lénau [5] is the work of Hessaroeyeh and Tahershamsi [18]. Extending the idea of Steffler et al. [19] applied to subcritical flow; the authors used the same perturbation technique for supercritical conditions case and solved the linearized 2D differential equations by Laplace transformation. The obtained first order solution
Vr
Vr
Symmetry axis
B
V
h
r re r0
ri
0 z
max Figure 1 Definition sketch of curved rectangular channel and flow field parameters
was confronted with several experimental results and predicts well the maximum super-elevation but with a lag in its position.
In this paper, analytical solution is proposed to a simplified mathematical model for prediction of free-surface flow in rectangular curved channel under supercritical conditions. First, the asymptotic mathematical model based on the classical twodimensional shallow water equations is presented. Then, an analytical solution based on Laplace transformation is obtained by means of an appropriate treatment of the source term. To test and verify the relevance of the analytical solution, results are compared against a set of available experimental data of Poggi [20], Reinauer and Hager [21], Beltrami et al. [22] and also Ippen [1]. The main purpose of this work is to propose a simple and practical analytical model to predict simultaneously water surface profile elevation in channel bends and the downstream reach which has not been so far treated analytically as a whole previously. II. ASYMPTOTICAL MATHEMATICAL MODEL Steady two-dimensional inviscid free-surface flow in curved channels can be described by a set of partial differential equations (PDEs) derived by applying mass and momentum conservation laws. Assuming hydrostatic pressure distribution and uniform velocity distribution in the vertical direction, this mathematical model can be written for a rectangular channel in cylindrical coordinates as follow [23]
h
Vr r
Vr
g Vr
h r
V r
h r
Vr
h V r
Vr r
g h r
V h r
V Vr
r
V V r
Vr h
r
V2 r
Vr V r
(1)
(2)
(3)
where h is the flow depth; V the flow velocity in the curvilinear longitudinal direction; Vr the flow velocity in the radial direction; g acceleration due to gravity;
r and are radial and angular coordinates respectively (Figure 1).
The assumption of frictionless liquid is based on the fact that for fully developed flow in a curved channel, frictional effects on the super-elevation are small [24].
Beside the aforementioned assumption, PDEs set is simplified through the following: (i) the flow field is supercritical in the whole chute bend; (ii) longitudinal gradient effects can be neglected compared to transverse effects as pointed out and affirmed experimentally by Reinauer and Hager [21]. The last assumption states that the change of the streamwise velocity V is very small, that is, Vr r V 0 as the flow is hypercritical. It is then admitted that dominating inertia effect is transversal to the flow streamwise axis. According to the quasi-2D approach developed by Amara et al. [25], if we rewrite equations (1) to (3) for an observer moving along the flow filament or streamline r r0 at constant velocity V s t u 0 , with u 0 as the approach flow velocity and s r the curvilinear coordinate, one obtains the following intrinsic equations in mobile coordinates system
h t
Vr t
h
Vr r
Vr
Vr
Vr r
h r
g
h r
Vr h
(4)
r
V2
(5)
r
V V V h V Vr gh V 2 r r V r h r Vr r
(6)
Dynamic equation (6) in longitudinal axis is therefore reduced to radial velocity profile description V r . In this equation the right hand side factor is the square inverse streamwise Froude number, thus gh V2 F02 . For most practical cases, terms 2
between brackets are of the order of 10 0 . In addition, for supercritical flow with large approach Froude number, F0 has the order of 10
1
or smaller. To a first order approximation, this term can thus be dropped and equation (6) simplifies to
V V 0 r r
(7)
Equation (7) is the well-known differential equation of free vortex in fluid mechanics [26]. Analytical integration is simple and leads to the general solution in the form
V
C
(8)
r
It can be shown for particular solution [27] that the constant C c 2 , where c is the vortex strength. Note that a complete solution of equation (6) leads to a slight deformation only of the free vortex velocity distribution. Under these assumptions the system of equations describing supercritical free-surface flow in curved channel in moving reference is written in vector form
G G E T t r
(9)
where
h V G ; E r Vr g
h ; Vr
Vr h T r2 V r
(10)
This final set of differential equations obtained for a moving observer is then strictly identical to 1D flow. Using the quasi-2D approach [25] one can show that for curved channel under supercritical conditions, the steady 2D shallow water equations tend asymptotically to 1D unsteady flow problem in radial direction. For the observer the problem is then reduced to an analogous 2 problem of unsteady flow in a diverging channel subject to centrifugal acceleration V r . Once the solution of asymptotic time dependent problem model (9) is obtained, the approximate 2D solution flow in space dimension is then easily reconstituted by means of equation (8).
Formally, the passage between the temporal field obtained as 1D solution to the space field required as approximate solution of the 2D flow is ensured by a transformation function noted . This function allows the repositioning of each fluid particle,
obtained in time-dependent problem, into its homologous point in space-dependent problem. The transformation time-space solution can be expressed by the following model
r , r , t
(11)
in which r , and r, t are solutions in space-domain and time-domain respectively. The function can be expressed in term of radial velocity distribution (8) as follows
1 b
V r dr
rr
ri
(12)
where ri and re refer respectively to the inner and the outer channel walls radii. The spatial position associated to each time solution as an image sequences is written simply as the kinematic observer’s law
Xp
t
dt 0
(13)
Xp is the space coordinate of the time solution plane. Hence, the present asymptotical quasi-2D model approach consists on the resolution of 1D hyperbolic time dependent problem combined with a pure advection in space perpendicular to the 1D wave problem plane. III. ANALYTICAL SOLUTION Regarding the initial form of system (9), obviously it consists on nonlinear hyperbolic partial differential equations and does not admit a closed analytical solution except for simplified cases. A numerical integration is therefore needed by classical methods as finite differences [28] or method of characteristics [29, 30]. Nevertheless, analytical solution of (9) for curved channels flow is possible if a linearization is done. For this purpose, let consider a small perturbation of amplitude induced on the initial constant water depth h0 before the transition such as h h0 and h0 . Introducing this condition and tacking into account that for practical cases B / r 1 the system (9) is then reduced to
V h0 r 0 t r
(14)
Vr V2 g t r r
(15)
Equations (14) and (15) are in the form of first order shallow water waves model and admit two eigenvalues gh0 . The combination of these equations leads to the non-homogeneous wave equation 2 3h C ² 2 2 c 04 2 2 t r r
(16)
where c is the celerity of small disturbances. From physical considerations, it is possible to obtain a simple solution of equation (16) using indirect treatment of the source term. Instead of dealing directly with centrifugal acceleration field to which the liquid is subjected in the curvature, it is mathematically convenient to treat the problem with an analogue deceleration field. Indeed, from Einstein’s equivalence principle a dynamic similarity exists if we consider the problem as a liquid flowing in radial direction at a velocity V and gradually decreasing to zero. The hypothetical radial velocity V at which the liquid is initially flowing is computed from dynamic considerations
V2 V r
(17)
in which is the time of complete extinction of the hypothetical radial flow. Here, the deceleration is considered constant and equal to V r0 . Time of radial flow extinction is in reality the time taken by the mobile observer to pass through the curved transition or time in which centrifugal effects hold. Thus 2
sc u0
(18)
with s c max r0 being the curved transition length and max the opening angle of the bend. The main objective is then to solve the homogeneous part of equation (16) for the dependent variables and the hypothetical V subject to initial conditions
V t and 0 . Because of the anti-symmetry of the interference waves along the channel axis, only one-half of the channel needs to be considered. The domain is then limited between radii r0 and re . For commodity of boundary conditions treatment, a local coordinate system transformation is introduced such as r r0 0 and r re L from which 0 B2 L . Introducing Laplace integral transformation of the dependent variables [31]
ˆ , p e pt , t dt
(19)
0
Vˆ , p
0
e ptV , t dt
(20)
The homogeneous wave equation (16) in local coordinates reads then
c2
2ˆ p 2ˆ p t 0 2 t
(21) t 0
Taking into consideration initial conditions of the problem, the subsidiary solution for ˆ in term of hyperbolic functions becomes
p c
p c
ˆ A cosh B sinh
(22)
From dynamic equation (15) written here for V instead of V r the corresponding transformation leads to
ˆ 1 V g
0
pVˆ
(23)
Boundary conditions for decelerating flow in radial direction are 1) At the channel axis 0 , no fluctuation is generated due to negative interferences (node point) thus
0, t 0 so ˆ 0, t 0
(24)
2) At outer boundary wall L , hypothetical radial velocity V decreases linearly to zero. This ensures creation of equivalent dynamic centrifugal effects
t V L, t V0 1 ; V L, t 0t t
(25)
From which
1 1 Vˆ L, t V0 2 p p
; t 1 1 Vˆ L, t V0 2 p p
V0 e p p 2 t
(26)
in which V0 is the initial radial velocity. The constants A and B of equation (22) can be now determined. When conditions (24) and (26) are substituted into equations (22) and (23)
A0
;
cV 1 B 0 g 2 p p cosh c t
and
cV p 1 e B 0 g 2 p p cosh c t
(27)
The subsidiary equations are thus
p sinh cV c ; ˆ 0 g 2 p p cosh c t p sinh cV c 1 e p ˆ 0 g 2 p p cosh c
t
(28)
The inverse Laplace transform is obtained using the Bromwich integrals [32]. Applying the procedure of inversion the freesurface elevation with respect to the undisturbed flow for t gives
, t
cV0 g
8L c c 2
1k 1 sin 2k 1 2 2L k 1 2k 1
cos2k 1
ct
(29) 2 L
For practical application, a convenient and physical meaning solution can be obtained in terms of traveling waves or Heaviside wave alternate. By replacing the hyperbolic functions of equation (28) in the Bromwich integral and using the shifting theorem the solution can be expressed as follows (see appendix 1)
, t
cV0 g
1
k 1
k 1
t
2k 1L u t 2k 1L t 2k 1L u t 2k 1L c
c
c
c
(30)
where u t is the Heaviside step function. For the “after-waves zone” of stage t the inverse Laplace transform leads to
, t
cV0 g
8L 2 c c
cos2k 1
1k 1 sin 2k 1 2 2L k 1 2k 1
ct
cV 0 2 L g
8L 2 c c
1k 1 2 k 1 2k 1
sin 2k 1
2L
cos2k 1
ct 2L
u t (31) t
This holds for downstream straight channel. From practical view point of channel design, it is the side wall water profile which is of interest. Thus for L equation (29) reads
, t The infinite trigonometric series in (29) is replaced by the function expressed as
cV0 L 8L 2 g c c
(32)
cos
1 1 cos 3 2 cos 5 (33) 2 3 5
in which ct 2L . Note that for to 0
(34)
(35)
1 2 2 8
1 2 2 8
and for 0 to
The analytical solution given by equation (32) is a triangular wave of a period 4L / c where the maximum water profile on side wall is reached at 2L / c (see appendix 2). IV. APPLICATION AND MODEL VERIFICATION For the verification of the present model, a comparison with a set of experimental data in a curved rectangular channel conducted by Ippen [1], Poggi [20], Reinauer and Hager [21] and Beltrami et al. [22] is carried out. Analytical calculations are performed to determine the steady state flow solution. Results concern mainly the determination of waterline profiles along the outer wall. A very important feature of this flow, for design purpose, is the first peak location and magnitude of the standing waves. Therefore a particular attention is given to these parameters in comparison with experiments. However the disturbed flow in downstream channel is not of less importance. Thus a part of the model verification is to compare the water surface flow in the whole transition (curved and downstream channel) to check the reliability of the solution to predict interference effects in the straight channel reach. Figures 2 and 3 compare respectively the normalized water surface profile h / h0 for the first wave computed by the proposed analytical model with selected data tests of Poggi [20] and Reinauer and Hager [21] as function of angular location coordinate . One of the experimental investigations conducted by Poggi [20] concerned a bend of rectangular cross-section with deflection angles of 45° and relative curvature B / r 1 / 12 . The bottom slopes were 5 and 10 % and the approach Froude numbers were between 2.27 and 5.1. Reinauer and Hager [21] performed experiments in horizontal, smooth, rectangular channel with a smaller relative curvature deflected at 51°. A range of Froude number of 2.0–12.0 and flow discharge comprised between 21.9 l/s and 70.0 l/s were considered. From these figures, it can be seen that the agreement between the computed and measured water surface profile for the first wave is almost satisfactory. The triangular wave shape predicted is slightly different from the hill-like wave observed as the mathematical model is of first order. For Poggi’s runs (Figure 2) the maximum wave magnitude is well predicted with a maximum deviation of 1.73 %. However a difference in the location is noted for lower Froude number F 3.04 (Figure 2a). The deviation was of 20.35 % in this case but only 3.41 % for F 4.83 (Figure 2b). Similarly, the comparison of computed and measured waterline on side wall by Reinauer and Hager [21] shows a good agreement. For approach Froude numbers 2.5, 3, 4 and 6 respectively (Fig. 3a-3d) the first wave in terms of magnitude and location is reasonably reproduced. For all tests, small overestimation of the super-elevation and a slight shifting in predicting location of the wave were observed. This discrepancy may be attributed to the streamwise momentum effect which is reduced to a pure kinematic motion in the asymptotical present model. The maximum deviation of about 8.7 % is noticed for F 6 . In contrast to Poggi’s tests, the model results show the fact that for horizontal channel some sort of a lag and decay effects proportional to velocity flow appear. As nonlinear terms were not considered, the proposed solution model cannot predict wave damping. Nevertheless, because only the maximum wave is of design interest this offers more safety for side wall sizing. In order to study the performances of the present analytical model to predict wave alternation behind the first shock, additional verification tests were performed. This part of verification concerned the successive wave occurrence in the bend itself and also in the downstream straight channel. In addition to magnitude, wavelength is introduced as a parameter in result analysis. Beltrami et al. [22] conducted an experimental investigation on supercritical flow in 180° bend deflection under different hydrodynamic approach conditions. Two test data for which B 0.2 m and r0 0.8 m were selected. Figures (4a) and (4b) refers to water surface profile comparison for approach flow condition h0 B 0.2 and h0 B 0.26 respectively, under incident Froude number F0 3 .
(a)
2,0
(b)
3,5
Poggi (1956) Present Model
1,8
3,0
1,6
2,5 h/h0
h/h0
Poggi (1956) Present model
1,4
1,2
2,0
1,5
1,0
1,0 0
5
10
15
20
25
30
35
40
0
10
20
Angle (°)
30
40
50
Angle (°)
Figure 2 Comparison of the present analytical model with Poggi’s experimental data for waterline on outer side-wall
(a)
1,6
(b)
1,8
Reinauer and Hager (1997) Present model
Reinauer and Hager (1997) Present model
1,7
1,5
1,6 1,5
1,3
h/h0
h/h0
1,4
1,4 1,3
1,2
1,2 1,1
1,1
1,0
1,0 0
2
4
6
8
10
12
14
16
18
20
22
0
5
10
Angle (°)
(c)
2,4
(d)
4,0
Reinauer and Hager (1997) Present model
2,2
15
20
25
Angle (°)
Reinauer and Hager (1997) Present model
3,5
2,0
3,0 h/h0
h/h0
1,8 1,6
2,5 2,0
1,4
1,5
1,2 1,0
1,0 0
5
10
15
20
Angle (°)
25
30
35
0
5
10
15
20
25
30
35
40
45
50
Angle (°)
Figure 3 Comparison of the present analytical model with Reinauer and Hager’s experimental data for waterline on outer side-wall
(a)
4,0 3,5
3,5
3,0
3,0
2,5
2,5
h/h0
h/h0
(b)
4,0
Beltrami et al. (2007) Present model
2,0
2,0
1,5
1,5
1,0
Beltrami et al. (2007) Present model
1,0 0
20
40
60
80
Angle (°)
100
120
140
0
20
40
60
80
100
120
140
Angle (°)
Figure 4 Comparison of the present analytical model with Beltrami’s et al. experimental data for waterline on outer side-wall
(a)
100
(b)
120 Ippen (1936) Present model
Ippen (1936) Present model
110
90
100
h (mm)
h (mm)
80 70
90 80 70
60
60
Curved section
50
Curved section
Straight section
50 0
1
2
3
4
5
6
7
8
9
10
0
11
1
2
3
5
6
7
8
9
10
11
Distance (m)
Distance (m)
(c)
60
4
(d)
180
Ippen (1936) Present model
Ippen (1936) Present model
160
55
140
50 h (mm)
h (mm)
120
45 40
100 80
35
60
30 0
1
2
3
Curved section
40
Curved section
4
5
6
Distance (m)
7
8
9
10
11
0
1
2
3
4
5
6
7
8
Distance (m)
Figure 5 Comparison of the present analytical model with Ippen’s experimental data for waterline on outer side-wall
As it can be observed, two successive wave crests are formed within the bend reach channel with a water trace shaped like a damped harmonic motion. Results show here similar pace between computed and measured water profiles. Furthermore, first wave in both cases is well reproduced in amplitude and position. The second one however is somewhat overestimated in magnitude and a small lag in position is noticed. In this case the analytical model generates an indefinite triangular wave solution in the bend reach whereas an exponential decay in the real water-profile is produced. As mentioned above, the friction damping plays a major role in this case for discrepancies behind the first wave. A maximum deviation in wavelength of 11.11% is observed in the first test while only 3.45 % deviation is noticed in the second one. In the last and ultimate step, the model is tested for water surface profile prediction in curved and straight downstream reach channel as a whole. As the unique study of this kind available is that carried out by Ippen [1], his experimental data were then used. In his outstanding PhD work, Ippen conducted a number of tests for three channel slopes, three radii of curvature (12.2 m, 6.1 m and 3 m) and two deflection angles (22.5° and 45°). Each geometric configuration was tested for a range of discharges resulting in 93 runs. Results of computed and measured water depth h as function of the curvilinear streamwise coordinate along the outer wall for runs 28, 32, 34 and 52 are given respectively in Figure (5a-5d).
One can see that in overall, computed solution shows trends similar compared with observation. For the curved part of the channel, there is a certain match between the analytical and measured water depth for the first two waves. Nevertheless, an underestimation of the second wave magnitude is noticed for runs 28 and 32 (Figures 5a and 5b). Here, a remarkable experimental fact is that the second wave is of a magnitude larger than its preceding one. This report contradicts the wellestablished assertion that the first wave is always the one to be considered for design interest as stated by many authors [21, 18]. Except the discrepancy for second wave magnitude, the position and wavelength are accurately reproduced with a maximum error deviation of 5.92 % and 11.68 % respectively. In the downstream straight channel the suppression of centrifugal effects creates a symmetrical wave interference phenomenon. Depending upon on the combination of this with flow condition at the bend end, a specific flow pattern configuration is thus created, resulting in a disturbed downstream flow field or not. Such cases are well shown in figure 5. As depicted, computed water surface profiles are relatively compatible with observations at a first order approximation. Whereas a relative difference in magnitude is noticed, a good wave periodicity is however obtained. A phase-angle shift of approximately is observed for run 32 (Figure 5b). It must be pointed out for discrepancies observed that crossing of the waves and reflection at the wall result in some air entrainment in Ippen’s experiments, thereby making the accuracy of the measured results somewhat questionable. Moreover, same difficulties in computing wave amplitude in this case were also noticed even using an advanced CFD model by Brown and Crookston [33]. Comparison of analytical model results and experiments of all test runs is summarized in Table 1. Relative error deviation Table 1 Summary of typical wave characteristics along outer wall and errors analysis of the analytical model
Poggi Reinauer and Hager
Beltrami et al. Ippen
Data Channel 1 2 1 1 1 1 1 1 1 2 2 3
Run 3 5 1 2 3 4 3 5 28 32 34 52
Maximum wave Yobs Ycal 1.77 1.77 2.89 2.94 1.42 1.43 1.60 1.62 2.04 2.11 3.22 3.50 3.25 3.25 3.10 3.25 1.57 1.38 1.81 1.74 1.74 1.77 1.53 1.49
E% 0.00 1.73 0.70 1.25 3.43 8.70 0.00 4.84 12.10 3.87 1.72 2.61
Location θobs/Xobs θcal/Xcal 18.23 22.30 8.83 10.38 13.81 18.77 37.97 41.00 3.63 3.67 3.38 3.79
14.52 23.06 9.93 11.91 15.88 23.82 42.96 42.97 3.58 3.51 3.58 3.75
E%
Wave length Λobs Λcal
20.35 3.41 12.46 14.74 14.99 26.90 13.14 4.80 1.38 4.36 5.92 1.06
1.08 1.16 2.31 2.45 2.14 1.49
1.20 1.20 2.39 2.34 2.39 1.50
E% 11.11 3.45 3.46 4.49 11.68 0.67
from observed values is given for maximum wave magnitude Y hmax h0 , its polar location or streamwise coordinate X and the wavelength where subscripts obs and cal denote observed and calculated values respectively. As mentioned above, analytical computation yields satisfactory results for maximum wave magnitude, with perfect matching for figures (3a) and (4a), but with a relative shifting deviation in peak wave location. Indeed, this is inevitable due to the simplified hypothesis adopted in model formulation. Similar statement of discrepancies was pointed out by Hessaroeyeh and Tahershamsi [18] for their analytical model which is of first order approximation. Moreover, their model has not been so far validated for wave alternation beside the first peak. In contrast, wavelength and interferences in the bend and downstream reach were predicted with a good accuracy in the present model. This is clearly shown for Ippen’s run N°52 where deviate only 0.67% from measurements. It is important here to add a comment about the hypothesis of small deflection in the analytical solution. Indeed, in the linearization process, it was assumed that h0 so that the wave propagation becomes linear i.e. flux function of shallow water model and thus characteristic curves are also linear. By this assumption, no diffusion or dispersion in the wave front propagation is introduced no matter how amplitude is. If the condition of small amplitude is not fulfilled, a slight phase angle shift (i.e. front wave angle) appears but does not affect significantly the maximum wave prediction on the side wall. This is well explained in depth by Cunge et al. [34] by mathematical and physical considerations of shocks and discontinuities. In most cases this order of approximation deviates only slightly from the complete solution using theory of strong shocks and it seems not necessary to resort to such calculation. It is to be pointed out also that, in addition to the model limitation to predicting wave attenuation, no so far reliable solution is expected to be obtained if strong curvatures with bend number
F0 B r greater
than 2 are treated. In this case separation along the inner bend wall occurs. At the limit of very sharp bends, flow blocking due to the reflection effect by lateral walls may occurs as noticed by Valiani and Caleffi [35]. It follows that the first order wave approximation with weak shocks is reliable in this case as far as sharp bends are not considered. V. CONCLUSION
In this paper, approximate analytical model was proposed for application to steady supercritical flow in rectangular curved channel transitions. Using a quasi-2D approach, the two-dimensional depth-averaged equations of motion were transformed to an equivalent 1D time-dependent problem as an asymptotic model for supercritical conditions. A first order approximation was then obtained using small perturbation theory resulting in a non-homogeneous linear wave equation in radial direction. The analytical solution was then obtained using Laplace integral transform. The model validation concerned the prediction of water surface profile along the outer wall in the bend reach and the downstream channel as a novel proposition. For this, results are compared with the available under various geometric and approach flow conditions. The analytical model results obtained for the different test cases indicate a fair good agreement compared with observations. The general physical behavior of the problem is well captured. Accurate results were particularly reached for the first wave magnitude prediction. However a slight lag in predicting the wave crest location was noticed. Particular attention in the verification was paid for the model ability to reproduce wave periodicity. It was found that an almost accurate periodic wave pattern of crests matched well measurements either in the curve or in the downstream straight channel. As the friction terms were neglected in the model some discrepancies in magnitude behind the first wave occurred due to damping effects. Furthermore, the analytical model limits are inherently related to the shallow water model weakness hypothesis and channel’s relative curvature. For engineering application, it is the maximum water depth rise at walls in the curved and the downstream straight channel which is an important design parameter to avoid overtopping. From this view point the proposed analytical solution gives a rather good prediction of flow surface configuration. Improvement of the analytical model can be made to include in particular friction damping effects. Further mathematical complexity in the solution involving special functions (Bessel functions) are then to be expected. VI. APPENDIX
1. Solution in term of Heaviside’s traveling waves
Inversion of equation (28) using Bromwich integral reads
cV0 1 g 2i
e
p sinh c dp p p 2 cosh c
pt
C
(A1)
in which C is the Bromwich contour in the complex plane. Replacing the hyperbolic functions by exponentials and multiplying numerator and denominator by e
sL / c
Since
1 e
cV0 1 g 2i
2 pL / c
p 1 e
e pt e pL / c e p / c e p / c dp 2
C
1 e
2 pL / c
2 pL / c
(A2)
e 4 pL / c e 6 pL / c (A3)
Substituting (A3) in (A2) gives
cV0 2ig
e
L p t c c
e
C
L p t c c
3L p t c c
e p2
e
3L p t c c
dp
(A4)
Noting that
1 2i
e p t k dp 0 for 0 t k C p2 t k for t k
(A5)
Integrating (A4) term by term giving the shifting property
cV0 g
L L L L u t t u t t c c c c
3L 3L t u t c c 3L 3 L t u t c c
(A6)
This can be rewritten in condensed form (30). For the “after waves zone”, a similar procedure gives
cV0 g
1
k 1
k 1
t
2k 1L u t 2k 1L c
c
2k 1L 2k 1L t u t c c
cV0 g
1 k 1
k 1
t
2k 1L u t 2k 1L c
c
2k 1L u t 2k 1L t c c
(A7)
2. Maximum wave magnitude and classical Ippen’s formula
As it was mentioned above, the maximum wave magnitude on the outer wall is reached at the half-period t 2L / c which leads to . Thus, equation (35) becomes
2 8
(A8)
Consequently
L, 2cL max
2 LV0 g
(A9)
From equation (17)
V2 V0 r0
(A10)
Equation (A9) reads after substitution of (A10)
max
2 LV2 gr0
(A11)
max
u 02 B gr0
(A12)
Recalling that L B 2 and V u 0 , one obtain the maximum wave magnitude
which is the classical formula pointed out first by Ippen (1936). It shows that for supercritical flow the increase in depth on the outer side wall is twice than for subcritical case if small curvatures are considered. ACKNOWLEDGMENTS The first author would like to acknowledge the valuable suggestions of Mr. Soufiane Tebache which have improved the quality of the paper.
NOTATION The following symbols are used in this paper
E =Flux jacobian matrix
G = Primitive flow variable vector T = Source term vector B = channel width (m) C = Constant, Bromwich integral contour
c = Small disturbance wave celerity (m/s) F0 = Approach flow Froude number g = Gravitational acceleration (m/s²)
hr , = Flow depth (m) h0 = Approach flow depth (m)
hmax = Maximum flow depth at wall (m)
L = Half-channel width (m) p = Inversion Laplace transform parameter r = Radial coordinates (m)
r0 = Axial radius of curvature (m) ri = Inner radius of curvature (m) re = Outer radius of curvature (m) s = Curvilinear coordinate (m)
s c = Curved channel reach length (m)
t = Time variable (s) u 0 = Approach flow velocity (m/s)
u t = Heaviside step function V = Streamwise depth-averaged velocity component (m/s) V r = Radial depth-averaged velocity component (m/s)
V , t = Hypothetical radial flow velocity (m/s)
V0 = Initial radial flow velocity (m/s) X p = Space-coordinate of the computational plane (m)
Y = Relative flow depth at channel
= Angular coordinates (°)
max = Bend opening angle (°)
c = Vortex strength
r , = Solutions in space-domain r, t = Solution in time-domain =Transformation time-space function = Eigenvalues = Canceling time of hypothetical radial flow (s)
= Centrifugal acceleration (m/s²) = Local radial coordinate (m)
max = Maximum wave magnitude (m)
, t = Small perturbation wave magnitude (m)
ˆ , p = Laplace transform of , t Vˆ r , p = Laplace transform of V , t
= Phase-angle parameter = Trigonometric series function = Wavelength (m)
= Bend number REFERENCES
[1] Ippen, A.T. (1936). An analytical and experimental study of high velocity flow in curved sections of open channels, PhD Thesis, California Institute of Technology. [2] Kármán, T.V. (1938). Eine praktische Anwendung der Analogie zwischen Überschallströmung in Gasen und überkritischer Strömung in offenen Gerinnen. ZAMM ‐ Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 18(1), 49-56. (in German).
[3] Knapp, R.T. (1951). High-Velocity Flow in Open Channels: A Symposium Design of Channel Curves for Supercritical Flow. Transactions of the American Society of Civil Engineers, 116(1), 296-325. [4] Harrison, A.J.M. (1966). Design of Channels for Supercritical Flow. Proceedings of the Institution of Civil Engineers, 35(3), 475-490. [5] Lénau, C.W. (1979). Supercritical flow in bends of trapezoidal section. Journal of the Engineering Mechanics Division, 105(1), 43-54. [6] Hager, W.H., Altirtakar, M.S. (1984). Infinitesimal cross-wave analysis. Journal of Hydraulic Engineering, 110(8), 1145-1150. [7] Knapp, R.T., Ippen A.T. (1938). Curvilinear flow of liquids with free surfaces at velocities above that of wave propagation. Proc. 5th Int. Congress Applied Mech., 531–536. Cambridge Cambridge University Press, NewYork. [8] Ellis, J., Pender, G. (1982). Chute spillway design calculations. Proceedings of the Institution of Civil Engineers, 73(2), 299312. [9] Hosoda, T., Yokosi, S. (1987). Some Considerations on High Velocity Flows Through Curved Open Channels. Doboku Gakkai Ronbunshu, (387), 171-178. [10] Iwasa, Y., Hosoda, T. (1989) Numerical simulation on high velocity flows through curved open channels, The International Conference on Interaction of Computational Methods and Measurements in Hydraulics and Hydrology, Dubrovnik, Yugoslavia, June, 87-96. [11] Dammuller, D.C., Bhallamudi, M.S., Chaudhry, M.H. (1989). Modeling of unsteady flow in curved channel. Journal of Hydraulic Engineering, 115(11), 1479-1495. [12] Elliot, R.C., Chaudhry, M.H. (1992). A wave propagation model for two-dimensional dam-break flows. Journal of Hydraulic Research, 30(4), 467-483. [13] Iwasa, Y., Hosoda, T., Kawamura, N., Yoneyama, N. (1991). High Velocity Flow with Free Boundary in Open Channels. Proceedings of hydraulic engineering, 35, 531-536. [14] Berger, R.C., Stockstill, R.L. (1995). Engineering, 121(10), 710-716.
Finite-element
model for high-velocity
channels. Journal of Hydraulic
[15] Ye, M., Wu, C., Chen, Y., Zhou, Q. (2006). Case study of an S-shaped spillway using physical and numerical models. Journal of Hydraulic Engineering, 132(9), 892-898. [16] Hessaroeyeh, M.G., Tahershamsi, A., Namin, M.M. (2011). Numerical modelling of supercritical flow in rectangular chute bends. Journal of Hydraulic Research, 49(5), 685-688. [17] Jaefarzadeh, R.M., Reza Shamkhalchian, A., Jomehzadeh, M. (2012). Supercritical flow profile improvement by means of a convex corner at a bend inlet. Journal of Hydraulic Research, 50(6), 623-630. [18] Hessaroeyeh, M.G., Tahershamsi, A. (2009). Analytical model of supercritical flow in rectangular chute bends. Journal of Hydraulic Research, 47(5), 566-573. [19] Steffler, P.M., Rajaratnam, N., Peterson, A.W. (1985). Water surface at change of channel curvature. Journal of Hydraulic Engineering, 111(5), 866-870. [20] Poggi, B. (1956). Correnti Veloci nei Canali in Curva. L’energia Elettrica, 33(5), 465–480 (in Italian). [21] Reinauer, R., Hager, W.H. (1997). Supercritical bend flow. Journal of hydraulic engineering, 123(3), 208-218.
[22] Beltrami, G.M., Del Guzzo, A., Repetto, R. (2007). A simple method to regularize supercritical flow profiles in bends. Journal of Hydraulic Research, 45(6), 773-786. [23] M.B., Abbott, Computational Hydraulics; Elements of the Theory of Free Surface Flow. Pitman Publishing Limited, London, 1979. [24] Yen, C.L., Yen, B.C. (1971). Water surface configuration in channel bends. Journal of the Hydraulics Division, 97(2), 303-321. [25] Amara, L., Berreksi, A., Achour, B. (2017). Quasi-2D model for computation of supercritical free surface flow in sudden expansion. Applied Mathematical Modelling, 46, 396-407. [26] H. Rouse, Fluid mechanics for hydraulic engineers. Dover Publications Inc. 1938. [27] J.F., Douglas, J.M., Gasiorek, J.A., Swaffield, L., Jack, Fluid Mechanics, Pearson/Prentice Hall, Harlow, 2005. [28] M.H., Chaudhry, Open-channel flow. Springer Science and Business Media, 2008. [29] Abbott, M.B., Lindeyer, E.W. (1969). Two transients for a radial nearly-horizontal flow. La Houille Blanche, (1), 65-70. [30] Townson, J.M., Al-Salihi, A.H. (1989). Models of dam-break flow in RT space. Journal of Hydraulic Engineering, 115(5), 561575. [31] Churchill, R.V. Operational Mathematics, McGraw- Hill, New York, 1972. [32] Duffy, D.G. Transform methods for solving partial differential equations. Chapman and Hall/CRC, 2004. [33] Brown, K., Crookston, B. (2016). Investigating Supercritical Flows in Curved Open Channels with Three Dimensional Numerical Modeling. In B. Crookston & B. Tullis (Eds.), Hydraulic Structures and Water System Management. 6th IAHR International Symposium on Hydraulic Structures, Portland, OR, 27-30 June, 230-239. [34] Cunge, J., Holly, F.M., Varwey, A. (1980). Practical aspects of computational river hydraulics. Pitman Publishing Ltd. London. [35] Valiani, A., Caleffi, V. (2005). Brief analysis of shallow water equations suitability to numerically simulate supercritical flow in sharp bends. Journal of Hydraulic Engineering, 131(10), 912-916.
Poggi Reinauer and Hager Beltrami et al. Ippen
Data Channel 1 2 1 1 1 1 1 1 1 2 2 3
Run 3 5 1 2 3 4 3 5 28 32 34 52
Maximum wave Yobs Ycal 1.77 1.77 2.89 2.94 1.42 1.43 1.60 1.62 2.04 2.11 3.22 3.50 3.25 3.25 3.10 3.25 1.57 1.38 1.81 1.74 1.74 1.77 1.53 1.49
E% 0.00 1.73 0.70 1.25 3.43 8.70 0.00 4.84 12.10 3.87 1.72 2.61
Location θobs/Xobs θcal/Xcal 18.23 22.30 8.83 10.38 13.81 18.77 37.97 41.00 3.63 3.67 3.38 3.79
14.52 23.06 9.93 11.91 15.88 23.82 42.96 42.97 3.58 3.51 3.58 3.75
E% 20.35 3.41 12.46 14.74 14.99 26.90 13.14 4.80 1.38 4.36 5.92 1.06
Wave length Λobs Λcal 1.08 1.16 2.31 2.45 2.14 1.49
1.20 1.20 2.39 2.34 2.39 1.50
Table 1 Summary of typical wave characteristics along outer wall and errors analysis of the analytical model
E% 11.11 3.45 3.46 4.49 11.68 0.67