Uniform flow will become unstable when the channel slope is very steep. When this happens, the free surface will form a series of roll waves. This paper, using the Fluent software and applying the volume of fluid model to simulate free surface flow and the kε turbulence model, analyzes conditions for roll wave formation in free surface flow with the aim of predicting their occurrence in chutes. This study describes how the type of cross-section of the chutes influences the formation of roll waves. Hydraulic characteristics of the flow in the physical model of the flood control structure for Azad Dam spillway, which consists of a side spillway, chute, and flip, are three-dimensionally simulated. Numerical predictions were compared against the experimental data and results show that there is a good agreement between numerical and experimental results. Results indicate that an increase in side slope tends towards flow stability and the slower formation of rolling waves. Also, the rolling waves are expected to be generated for sections of the chute when the difference between the maximum dynamic pressure and the minimum dynamic pressure in those sections is above 9,000 Pa.

INTRODUCTION

In chutes, the uniform flow tends to become unstable and to form waves on the water surface. These waves are called roll waves and are typically characterized by amplitude, wavelength, and speed increasing down the channel (Brock 1969). Roll waves in a pulsating flow are formed through a continuous growth of disturbances. The complex structure of roll waves consists of a series of bores separated by smooth variable water depth (James 1999). Roll waves in a chute structure are undesirable, because they may overtop the chute walls and cause surging in the energy dissipater. A stilling pool would not be an effective energy dissipater with this type of flow, because a stable hydraulic jump could not form. These waves generally form in chute channels that are longer than about 60 metres and have bottom grade slopes flatter than about 20°. The maximum wave height that can be expected is twice the normal depth for the slope and the maximum momentary slug flow capacity is twice the normal capacity (Aisenbrey et al. 1978).

Bruce et al. (2001), using Flow 3-D software that applies the volume of fluid model to calculate the free surface movement, numerically modeled the flow on an ogee spillway in two-dimensional states. Results of the numerical modeling were compared with the experimental results, and the pressure profile on the spillway crest was in a good agreement with the experimental model and indicated the acceptable precision of the method. Chen et al. (2002), using Fluent software, simulated the flow on a stepped spillway. The volume of fluid model and turbulence model were used for spillway modeling. A velocity boundary condition for the model inlet and zero pressure boundary condition for the model outlet and the upper space were applied. Results of the numerical analysis including the velocity, water surface profile, and flow pattern were compared with the physical model and showed a good agreement indicating the acceptable precision of the method. Oslen & Kjellesvig (1998) numerically modeled the two- and three-dimensional flow over a spillway for various geometries and solved the Navier–Stokes equations with the turbulence model. They showed good agreement for water surfaces and coefficient of discharge for a limited number of flows. Taha & Cui (2006a, b) simulated roll waves with the Fluent software with a volume of fluid model to study the geometrical effect of a square vertical capillary and a round vertical tube. In both cases, recirculation patterns were reported, though it should be noted that viscosity effects are neglected. Pena et al. (2007), by using the finite volume method, numerically simulated the flow in a vertical slot fish way. Results of the numerical modeling, including water surface profile and depth and flow velocity, were compared with experimental results and indicated the high accuracy of the modeling method.

There is no simple criterion for the prediction of roll waves, since their formation depends upon the magnitude of the discharge, the type of flow (laminar or turbulent), the slope of the channel, the length of the channel, and the nature and frequency of the initial disturbances, which cause the waves to form. The purpose of this article is to analyze the criteria for the occurrence of the roll-wave phenomenon in the supercritical and turbulent flows from the engineering point of view. In this study, water flow over the Azad Dam spillway is simulated by using a numerical model and compared with the physical modeling results. For this purpose, the Fluent software has been used. To determine the free surface flow along the chute the volume of fluid model has been used, and to simulate flow turbulence, the turbulence model has been used.

METHODS

Physical model

The physical model of the Azad Dam spillway that was built in the Hydraulic Structures Department of Iran Water Research Institute (IWRI) with the scale of 1:33.33 was selected for the study. The model was made of Plexiglas, and the input discharge was 800 m3/s and hydraulic characteristics including flow depth, velocity, and pressure were measured along the spillway. A schematic view of the spillway is shown in Figure 1. It was an ogee spillway with a horizontal length of 9.78 m and the spillway sill was located 1,465 m above sea level, with a spillway equation of so that the start of the chute was located 1,458.13 m above sea level. Horizontal lengths of the chutes with longitudinal slopes of 5 and 36.4% with a width of 30 m was 45.4 and 198.31 m, respectively, and the flip has a radius of 15 m, horizontal length of 10.58 m, and width of 30 m. To include the Manning roughness coefficient effect in the model with the scale of 1:33.33, a roughness height of 0.085 mm, which is equivalent to 0.15 mm for the real-scale model was considered. Hydraulic variables including hydrostatic pressure and flow depth were measured at about 50 points along the physical model located near the walls and width midpoints. Flow depth and hydrostatic pressure were measured in the models using a limnimeter (0.1 mm precision) and a piezometer (0.1 mm precision), respectively.
Figure 1

Longitudinal section of the Azad Dam spillway.

Figure 1

Longitudinal section of the Azad Dam spillway.

Numerical model

Chutes modeled in this study are considered as three-dimensional chutes with upstream and downstream lengths of 16 and 8 m, respectively, inclined section length of 30.59 m and longitudinal slope of 20%, and wall height of 3 m, which are fixed values for all the modeled chutes, and the only variables are the side slope, chute width and input discharge. Hydraulic characteristics of the flow in the chutes with a width of 4 m and side slopes of 0, 0.25, 0.5, and 0.75 for input discharges of 15 and 20 m3/s, and for the chutes with input discharge of 15 m3/s and side slopes of 0, 0.25, 0.5, and 0.75 for the widths of 3.6 and 3.4 m were investigated.

Model geometry was first developed in gambit software and unequal hexagonal rectangular elements were used to make model mesh so that fine mesh is used near the bottom and coarse mesh is used where no water exists. Grid dimensions for the models with the scale of 1:33.33 in areas with a wall nearby and the probability of free surface flow formation and in areas with no water were in the range of 0.002–0.08 m.

As the logarithmic velocity relation is valid for in the range of 30–300, a large value was initially obtained for and after corrections and re-meshing into a finer grid and re-running the model, this value of about 210 was obtained for . A steady flow condition for the numerical model was considered when constant changes of the vertical distribution of the velocity at the given point and almost equal input and output flow discharges were observed. A maximum difference of 0.05% between input and output discharges was considered for the steady flow condition. The water surface was considered at the point at which the fluid volume was 0.5 and linear interpolation was applied to find this point (Dargahi 2006). A schematic view of the grid mesh is shown in Figure 2. The standard kɛ turbulence model is used to represent the turbulent flow over the chute. Boundary conditions are given in Table 1.
Figure 2

Schematic view of the grid mesh with the numerical model of the Azad Dam spillway.

Figure 2

Schematic view of the grid mesh with the numerical model of the Azad Dam spillway.

Table 1

Boundary conditions applied in the numerical model

Mass-flow inlet Model input 
Pressure outlet Model output 
Wall Walls 
Wall Bottom 
Mass-flow inlet Model input 
Pressure outlet Model output 
Wall Walls 
Wall Bottom 

Volume of fluid model

The volume of fluid model was first introduced by Hirt & Nicols (1981). It relies on the principle that two or more fluids do not interfere, and a variable called phase volume ratio is defined to the model in the computational cell. In each control volume, the total volume ratio of all phases is equal to unity. Using the volume ratio of each phase, the values of variables and properties would be considered between the phases that show the mean volumetric values. In other words, if the volumetric ration of the fluid q in a cell is indicated as and would be equal to one, the cell is full of fluid q and if would be equal to zero, the cell is empty, and if the volume fraction would be between zero and one, both phases exist in the cell and an interface is formed between fluid q and the other fluid. Therefore, considering the free surface in a given volume fraction, free surface flow can be determined. The general conservation equations governing fluid flow and transport phenomena are as follows (Celik 1999): 
formula
1
In general, the equation of continuity is 
formula
2
In addition to the continuity equation, the other governing equation for incompressible flow is the momentum equation given by 
formula
3
where 
formula
4
The variables ui and uj are velocities in the xi and xj directions. ρ is the flow density, is acceleration due to gravity in i direction and P is defined as the pressure, μ represents the molecular viscosity of the flow, Г is the diffusion coefficient, sji is the mean strain rate tensor, t is the time, and finally xi and xj correspond to x and y coordinates.

kɛ model

The kɛ model is the most applied two-equation model due to a reasonable accuracy for a wide range of flows in which the turbulent field is expressed based on two variables: turbulence kinetic energy (k) and the turbulence kinetic energy loss (ɛ). The kɛ model is divided into three categories: the standard kɛ model, the RNG (renormalization-group) k-ɛ model and the realizable kɛ model. Launder & Spalding (1974) were the first who presented the semi-empirical standard kɛ model. Applied equations in the model are as follows (Fluent 2004): 
formula
5
 
formula
6
 
formula
7
in which μ indicates the dynamic viscosity, is the fluid density, is the eddy viscosity and is an empirical coefficient with the value of 0.09. Coefficients of the mentioned equation are as follows: 
formula
is a constant that is equal to one if the flow direction is consistent with the gravity direction, and is equal to zero if the flow direction is perpendicular to the gravity direction. and are the turbulent Prandtl numbers for k and ɛ, respectively. and are user-defined source terms. 
formula
8
 
formula
9
 
formula
10
 
formula
11

In the above equations, is the oscillating expansion in compressible turbulence to the total loss rate, is the turbulent kinetic energy production term based on the average velocity gradient, is the turbulent kinetic energy production term due to buoyancy effect, a is the speed of sound, b is the coefficient of thermal expansion, and T is temperature. is the turbulent Prandtl number for energy and the variables and are velocities in the and directions, and finally and correspond to x and y coordinates.

Roll-wave formations

Roll waves are characterized by transverse ridges of high velocity. The regions between the crests are quiescent. For roll waves to form, the surface velocity of the undisturbed flow must be less than the wave velocity. This ensures that the breaking of waves is at downstream ends (similar to moving hydraulic drop or expansion waves). Verified criteria of roll-wave formation and conditions of roll-wave formation are discussed.

Froude number criteria

Jeffreys (1925) and Dressler (1949), by conducting experiments on a wide rectangular channel, and Iwasa (1954), by conducting experiments on a channel with arbitrary cross-section, showed that for a Froude number greater than 2, the flow becomes unsteady and roll waves are formed. Boudlal & Liapidevskii (2002), by using the Gas dynamic model, could prove a flow stability criteria proposed by Jeffreys, Dressler, and Iwasa. A Froude number criterion is presented, if the following inequality were true the formation of the roll waves would be as expected. 
formula
12
In Equation (12), Fr represents Froude number, V is the flow velocity, y is flow depth, and g is gravity acceleration.

Vedernikov number criteria

Vedernikov (1945) developed a criterion for the flow instability so that if the Vedernikov number is greater than 1, formation of the roll waves is expected. Craya (1952) conducted experiments on open channels and obtained similar results. Ponce (1991) conducted experiments on fixed bed channels for the waves with various wavelengths, and the results of the location of roll-wave formation were in good agreement with the Vedernikov number criterion. The Vedernikov number criterion is presented as follows: 
formula
13
In Equation (13), represents the Vedernikov number, b is the bottom width of the chute, is the wet perimeter, d is the average water level in the chute, g is gravity acceleration, β is the energy loss slope angle, and V is the flow velocity.

VALIDATION OF NUMERICAL MODEL

In this study, the flow over the spillway, chute, and flip is numerically modeled, and the results are compared with the experimental model. Figures 3 and 4 show comparisons of the depth and static pressure changes, respectively, with the discharge of 800 m3/s over the spillway, chute, and flip of the Azad Dam based on numerical modeling and experimental simulation results. Free surface flow, which was obtained from numerical and experimental modeling, was drawn and a good agreement was observed between the numerical and experimental results. Only a small difference was observed in the connection area of the chutes with 5 and 36.4% slopes between the numerical and experimental results, which is attributed to the centrifugal force and separation of the flow lines from the bed and air entrainment to the flow in the convex flow area in which accurate flow depth measurement is difficult. A good agreement was also observed for the static pressure values between the experimental and numerical simulation results. Exerted applied pressure to the chute bottom was obtained using , where P represents the pressure, y is flow depth, and θ is the angle between the chute bottom and the horizon. Therefore, exerted pressure to the chute with 5% slope, which is less steep than the second part of the chute, is higher than the second part of the chute. After validation and calibration of the numerical model results, these data were used for evaluation of hydraulic properties of the flow and determination of the formation criteria of the rolling waves. Flow pressures and depths were measured at points located near the walls and width midpoints of the experimental and numerical models.
Figure 3

Comparison between the curves of depth changes for the experimental and numerical models of the Azad Dam spillway.

Figure 3

Comparison between the curves of depth changes for the experimental and numerical models of the Azad Dam spillway.

Figure 4

Comparison between the curves of static pressure changes for the experimental and numerical models of the Azad Dam spillway.

Figure 4

Comparison between the curves of static pressure changes for the experimental and numerical models of the Azad Dam spillway.

RESULTS AND DISCUSSION

In the study of all numerical models, results showed that for rectangular chutes with hydraulic depth to chute width ratio of less than 0.16, formation of roll waves is inevitable, and when this ratio is less than 0.18, roll waves are expected in trapezoidal chutes. The results of the comparison of the flow hydraulic depths for all numerical models are presented in Tables 2 and 3.

Table 2

Comparison of the hydraulic depth to chute width ratios for a chute with the width of 3.4 m and a longitudinal slope of 20% and flow discharge of 15 m3/s

Point length (m)
Side slope (z)Hydraulic depth (D) and chute width (b)19202530354045
 D 0.786 0.736 0.589 0.514 0.464 0.434 0.415 
D/b 0.231 0.216 0.173 0.151 0.136 0.127 0.122 
 D 0.745 0.70 0.565 0.496 0.448 0.421 0.403 
D/b 0.219 0.205 0.166 0.145 0.131 0.123 0.118 
 D 0.712 0.670 0.545 0.480 0.436 0.409 0.392 
D/b 0.209 0.197 0.160 0.141 0.128 0.120 0.115 
 D 0.683 0.644 0.528 0.466 0.424 0.399 0.383 
D/b 0.201 0.189 0.155 0.137 0.124 0.117 0.112 
Point length (m)
Side slope (z)Hydraulic depth (D) and chute width (b)19202530354045
 D 0.786 0.736 0.589 0.514 0.464 0.434 0.415 
D/b 0.231 0.216 0.173 0.151 0.136 0.127 0.122 
 D 0.745 0.70 0.565 0.496 0.448 0.421 0.403 
D/b 0.219 0.205 0.166 0.145 0.131 0.123 0.118 
 D 0.712 0.670 0.545 0.480 0.436 0.409 0.392 
D/b 0.209 0.197 0.160 0.141 0.128 0.120 0.115 
 D 0.683 0.644 0.528 0.466 0.424 0.399 0.383 
D/b 0.201 0.189 0.155 0.137 0.124 0.117 0.112 
Table 3

Comparison of the hydraulic depth to chute width ratios for a chute with the width of 3.6 m and a longitudinal slope of 20% and flow discharge of 15 m3/s

Point length (m)
Side slope (z)Hydraulic depth (D) and chute width (b)19202530354045
 D 0.751 0.705 0.561 0.488 0.443 0.412 0.393 
D/b 0.208 0.196 0.155 0.135 0.123 0.114 0.109 
 D 0.716 0.673 0.541 0.472 0.430 0.401 0.383 
D/b 0.198 0.187 0.150 0.131 0.119 0.111 0.106 
 D 0.686 0.647 0.523 0.459 0.419 0.391 0.374 
D/b 0.190 0.179 0.145 0.127 0.116 0.108 0.103 
 D 0.660 0.624 0.507 0.446 0.408 0.382 0.365 
D/b 0.183 0.173 0.141 0.124 0.113 0.106 0.101 
Point length (m)
Side slope (z)Hydraulic depth (D) and chute width (b)19202530354045
 D 0.751 0.705 0.561 0.488 0.443 0.412 0.393 
D/b 0.208 0.196 0.155 0.135 0.123 0.114 0.109 
 D 0.716 0.673 0.541 0.472 0.430 0.401 0.383 
D/b 0.198 0.187 0.150 0.131 0.119 0.111 0.106 
 D 0.686 0.647 0.523 0.459 0.419 0.391 0.374 
D/b 0.190 0.179 0.145 0.127 0.116 0.108 0.103 
 D 0.660 0.624 0.507 0.446 0.408 0.382 0.365 
D/b 0.183 0.173 0.141 0.124 0.113 0.106 0.101 

Results of the numerical modeling including the roll-wave formation criterion, when hydraulic depth to chute width ratio in rectangular chutes is more than 0.16 and when this ratio in trapezoidal chutes is more than 0.18 for the rectangular chutes with the widths of 4, 3.6 and 3.4 m, and longitudinal slope of 20%, and flow discharges of 20 and 15 m3/s are presented in Table 4.

Table 4

Numerical model results in the roll-wave formation moment for the rectangular chute with the longitudinal slope of 20%

Discharge (Q),
chute width (b), and
side slope (z)
Point lengthNumerical model criterion
Dynamic pressure (P)Roll wave


 
18.5  No 
19.5  Yes 


 
19  No 
20  Yes 


 
19  No 
20  Yes 
Discharge (Q),
chute width (b), and
side slope (z)
Point lengthNumerical model criterion
Dynamic pressure (P)Roll wave


 
18.5  No 
19.5  Yes 


 
19  No 
20  Yes 


 
19  No 
20  Yes 
For the rectangular chute with the longitudinal slope of 20%, width of 3.4 m, input discharge of 15 m3/s at a section with the length of 19 m, the hydraulic depth to chute width ratio is greater than 0.16, and the maximum dynamic pressure and the minimum dynamic pressure difference in that section is less than 9,000 Pa. Roll waves are formed between the points with the lengths of 19 and 20 m of the chute. With the formation of roll waves we have 
formula
14
Here is the maximum dynamic pressure, and is the minimum dynamic pressure.
The moment of roll-wave formation and water surface profile for the model with the width of 3.4 m and flow discharge of 15 m3/s are shown in Figures 5 and 6, respectively. In the study of other numerical models, results showed that for chutes with hydraulic depth to chute width ratio of less than 0.16, formation of roll waves is inevitable, and when this ratio is more than 0.16 in a chute section, if the maximum dynamic pressure and the minimum dynamic pressure difference in that section is more than 9,000 Pa, formation of the roll waves is expected.
Figure 5

Flow pattern at the rolling and wave formation moment for the rectangular chute with the width of 3.4 m, longitudinal slope of 20%, and flow discharge of 15 m3/s.

Figure 5

Flow pattern at the rolling and wave formation moment for the rectangular chute with the width of 3.4 m, longitudinal slope of 20%, and flow discharge of 15 m3/s.

Figure 6

Water surface profile for the rectangular chute with the width of 3.4 m, longitudinal slope of 20%, and flow discharge of 15 m3/s.

Figure 6

Water surface profile for the rectangular chute with the width of 3.4 m, longitudinal slope of 20%, and flow discharge of 15 m3/s.

The results of the comparison between various criteria and numerical model results in the roll-wave formation moment showed that for the location of the roll-wave formation, Froude number and Vedernikov number criteria are consistent with the numerical model results.

CONCLUSIONS

To simulate the flow in chutes, the volume of the fluid simulation model was used, and to simulate the flow turbulence, the standard kɛ model was used, and it was observed that applying the two models simultaneously can plot the free surface and hydraulic depth of the flow in open channels accurately using time-dependent calculations. The derived results indicate that the increase in side slope tends towards flow stability and the slower formation of rolling waves. In all the modeled rectangular chutes with the hydraulic depth to chute width ratios of less than 0.16, the formation of roll waves is inevitable. 
formula
15
And when this ratio is less than 0.18, roll waves are expected in trapezoidal chutes. 
formula
16
In other words, increasing the side slope (z) of the chute will increase the probability of roll-wave formation and will alleviate the critical flow condition. In all the modeled chutes when hydraulic depth to chute width ratio is more than 0.16 for rectangular chutes, and when this ratio is more than 0.18 for trapezoidal chutes, the rolling waves are expected to be generated for sections of the chute when the maximum dynamic pressure and the minimum dynamic pressure difference in that section is above 9,000 Pa. 
formula
17

REFERENCES

REFERENCES
Aisenbrey
A. J.
Hayes
R. B.
Warren
H. J.
Winsett
D. L.
Young
R. B.
1978
Design of Small Channel Structures
.
Bureau of Reclamation
,
Denver, CO, USA
.
Boudlal
A.
Liapidevskii
V. Y.
2002
Stability of roll waves in open channel flows
.
Comptes Rendus Mecanique
330
(
4
),
291
295
.
Brock
R.
1969
Development of roll-wave trains in open channel
.
Journal of the Hydraulics Division, ASCE
95
(
HY4
),
1401
1427
.
Bruce
M.
Savage
B. M.
Johnson
M. C.
2001
Flow over spillways: physical and numerical model
.
Journal of Hydraulic Engineering, ASCE
127
(
8
),
640
649
.
Celik
I. B.
1999
Introductory Turbulence Modeling
.
Lecture notes
,
Western Virginia University
,
Morganatown, USA
.
Chen
Q.
Dai
G.
Liu
H.
2002
Volume of fluid model for turbulence numerical simulation of stepped spillway over flow
.
Journal of Hydraulic Engineering. ASCE
128
(
7
),
683
688
.
Craya
A.
1952
The Criterion for Possibility of Roll-Wave Formation, Gravity Waves
.
Circular 521
,
National Bureau of Standards
,
Washington, DC, USA
.
Dargahi
B.
2006
Experimental study and 3D numerical simulation for a free-overflow spillway
.
Journal of Hydraulic Engineering. ASCE
132
(
9
),
899
907
.
Dressler
R. F.
1949
Mathematical solution of the problem of roll-waves in inclined open channels
.
Communications on Pure and Applied Mathematics
2
(
3
),
149
194
.
Fluent Inc. Fluent 6.2 User Guide
2004
.
Hirt
C. W.
Nichols
B. D.
1981
Volume of fluid method for the dynamics of free boundaries
.
Journal of Computational Physics
39
(
1
),
201
225
.
Iwasa
Y.
1954
The criterion for instability of steady uniform flows in open channels
.
Memoirs of the Faculty of Engineering
16
(
6
),
264
275
.
James
C. Y.
1999
Roll waves in high gradient channel
.
Water International
24
(
1
),
1
7
.
Jeffreys
H. J.
1925
The flow of water in an inclined channel of rectangular section
.
Philosophical Magazine
49
(
293
),
793
807
.
Launder
B. E.
Spalding
D. B.
1974
The numerical computation of turbulent flows
.
Computer Methods in Applied Mechanics and Engineering
3
(
2
),
269
289
.
Oslen
N. R. B.
Kjellesvig
H. M.
1998
Three dimensional numerical flow modeling estimation of spillway capacity
.
Journal of Hydraulic Research
36
(
5
),
775
784
.
Pena
L.
Puertas
J.
Cea
L.
Vazquez
C. M. E.
Pena
E.
2007
Application of several depth-averaged turbulence models to simulate flow in vertical slot fish ways
.
Journal of Hydraulic Engineering. ASCE
133
(
2
),
160
172
.
Ponce
V. M.
1991
New perspective on the Vedernikov number
.
Water Resources Research
27
(
7
),
1777
1779
.
Taha
T.
Cui
Z. F.
2006a
CFD modeling of slug flow in vertical tubes
.
Journal of Chemical Engineering Science
61
(
2
),
676
687
.
Taha
T.
Cui
Z. T.
2006b
CFD modeling of slug flow inside square capillaries
.
Journal of Chemical Engineering Science
61
(
2
),
665
675
.
Vedernikov
V. V.
1945
Conditions at the front of a translation wave disturbing a steady motion of a real fluid
.
Comptes Rendus (Doklady) de l' Académie des Sciences de l'URSS'
48
(
4
),
239
242
.