E-proceedings of the 38th IAHR World Congress
September 1-6, 2019, Panama City, Panama

doi:10.3850/38WC092019-0685

ANALYSIS OF FLOWS IN UNDULAR AND BREAKING HYDRAULIC JUMPS BY NONHYDROSTATIC QUASI THREE-DIMENSIONAL MODEL CONSIDERING FLOWEQUATIONS ON BOUNDARY SURFACES (Q3D-FEBS)YOSHIHARU TAKEMURA (1) & SHOJI FUKUOKA (2)(1)

Research and Development Initiative, Chuo University, Tokyo, Japan,
takemura@tamacc.chuo-u.ac.jp(2)

Research and Development Initiative, Chuo University, Tokyo, Japan,
sfuku@tamacc.chuo-u.ac.jp

ABSTRACTThe flow structure and the energy dissipation rate are greatly different between an undular jump and a breaking
jump. A depth-integral model that can calculate both undular and breaking jumps is required for the design ofhydraulic structures in rivers. This paper proposes a new non-hydrostatic quasi three-dimensional model (Q3DFEBS) that introduces the flow equations on boundary surfaces (a free surface and a bottom surface).
Calculation results of Q3D-FEBS well explain water surface profiles and velocity distributions of the previous
experiments of undular and breaking jumps. In addition, based on a numerical experiment, it is shown that Q3DFEBS can calculate the transitional process from an undular jump to a breaking jump by analyzing the flow
separation at the water surface using the flow equations on boundary surfaces.
Keywords: Undular jump; breaking jump; non-hydrostatic pressure; depth-integral model; Q3D-FEBS

1

INTRODUCTION
A hydraulic jump occurs in various forms depending mainly on the Froude number (Fr). The forms can bedivided roughly into following two types. For 1.7 > Fr > 1, a wavy water surface is formed downstream of ahydraulic jump and the jump called an undular jump. For Fr > 1.7, a surface roller develops just below the water
surface of a hydraulic jump and the jump called a breaking jump. Since the flow structure and the energy
dissipation rate of the undular and breaking jumps are greatly different from each other, it is necessary toaccurately predict the forms of the jump under given hydraulic conditions. 3D-RANS and MPS are effective
method for such problems. However, more efficient method based on a depth-integral model is needed forlarge-scale flows varying in time and space such as flows around hydraulic structures in rivers during floods.
Although many researchers have proposed depth-integral models for hydraulic jumps, there is no depth-integral
model that can calculate both undular and breaking jumps including the transitional process between the jumps.
Most of the models for undular jumps have been developed based on the Boussinesq type equation thatconsiders non-hydrostatic pressure due to the curvature of the water surface (e.g. Serre 1953, Iwasa 1955,
Hosoda & Tada 1994 and Castro-Orgaz et al. 2015), and usually neglect the velocity distributions along verticaldirection. In contrast, the models for breaking jumps have been developed based on the strip integral methodthat assumes velocity distributions within the jump using the wall jet theory or polynomials (Narayanan 1975,
McCorquodale & Khalifa 1983, Tsubaki 1949 and Madsen & Svendsen 1983), and neglect non-hydrostaticpressure in many cases. In order to calculate both undular and breaking jumps in the framework of the depthintegral model, it is necessary to model the three-dimensional velocity and the non-hydrostatic pressure
distributions in a generalized manner as much as possible.
This paper proposes a new non-hydrostatic quasi three-dimensional model (Q3D-FEBS) that introduces
the flow equations on the boundary surfaces (free surfaces and bottom surfaces) and investigates theapplicability of Q3D-FEBS to the calculation of undular and breaking jumps including the transitional process
between the jumps by using the previous experiments and a numerical simulation.
2

GORVERNING EQUATIONS OF Q3D-FEBS
The definition sketch of Q3D-FEBS (Ohno et al, 2018) is shown in Figure 1. We define the bottom surface
𝑧𝑏 slightly above the bed surface 𝑧0 . A vertical coordinate is given as 𝜂 = (𝑧𝑠 − 𝑧)/ℎ and approximates vertical
distributions of the horizontal velocities by third-order polynomials.
𝑢𝑖 = ∆𝑢𝑖 (12𝜂3 − 12𝜂2 + 1) + 𝛿𝑢𝑖 (−4𝜂3 + 3𝜂2 ) + 𝑈𝑖
2049

[1]

E-proceedings of the 38th IAHR World Congress
September 1-6, 2019, Panama City, Panama

Where, 𝑖 = 1, 2 (𝑥1 = 𝑥, 𝑥2 = 𝑦), 𝑢𝑖 : velocity in 𝑖 direction, 𝑈𝑖 : depth-average velocity in 𝑖 direction, 𝑢𝑠𝑖 : water
surface velocity in 𝑖 direction, 𝑢𝑏𝑖 : bottom surface velocity in 𝑖 direction, ∆𝑢𝑖 = 𝑢𝑠𝑖 − 𝑈𝑖 , 𝛿𝑢𝑖 = 𝑢𝑠𝑖 − 𝑢𝑏𝑖 , 𝑧𝑠 : waterlevel, ℎ: water depth. The boundary conditions used in the derivation of Eq. [1] are as follows.
𝑢𝑖 = 𝑢𝑠𝑖 ,∂𝑢𝑖 ⁄𝜕𝑧 = 0 𝑎𝑡

𝜂 = 0 (𝑧 = 𝑧𝑠 )[2]

𝜂 = 1 (𝑧 = 𝑧𝑏 )

[3]

1
1 𝑧𝑠
∫ 𝑢𝑖 𝑑𝜂 (= ∫ 𝑢𝑖 𝑑𝑧) = 𝑈𝑖
ℎ 𝑧𝑏
0

[4]

𝑢𝑖 = 𝑢𝑏𝑖

𝑎𝑡Figure 1. Definition sketch of Q3D-FEBS.
Q3D-FEBS solves the equations of motion on a free surface (𝜂 = 0) and a bottom surface (𝜂 = 1) in addition
to the depth-integral flow equations shown by Eq. [5] and Eq. [6] to calculate the unknown quantities of Eq. [1].
𝜕ℎ 𝜕𝑈𝑖 ℎ
+
=0
𝜕𝑡
𝜕𝑥𝑖

[5]

𝜕𝑈𝑖 ℎ 𝜕𝑈𝑗 𝑈𝑖 ℎ𝜕𝑧𝑠 1 𝜕ℎ𝑝̅′ 𝑝𝑏′ 𝜕𝑧𝑏 𝜕ℎ𝑢𝑖′ 𝑢𝑗′𝜕𝜕𝑈𝑖 𝜕𝑈𝑗𝜏̂𝑏𝑖) − 𝑆̂
+
= −𝑔ℎ
−
−
−
+
𝜈𝑡 ℎ (+𝜕𝑡𝜕𝑥𝑗
𝜕𝑥𝑖 𝜌 𝜕𝑥𝑖
𝜌 𝜕𝑥𝑖
𝜕𝑥𝑗
𝜕𝑥𝑗
𝜕𝑥𝑗 𝜕𝑥𝑖𝜌[6]

Where, 𝜌: density of water, 𝑢𝑖′ = 𝑢𝑖 - 𝑈𝑖 , 𝑔: acceleration of gravity, 𝑝′: non-hydrostatic pressure, 𝜈𝑡 : eddy viscosity,𝜏̂𝑏𝑖 : bottom shear stress in 𝑖 direction, 𝑆̂ = √1 + (𝜕𝑧𝑏 ⁄𝜕𝑥𝑖 )2. Overbar “–“ represents a depth-averaged value. Eq.
[6] has the non-hydrostatic pressure terms and they are calculated from Eq. [7] derived from the depth-integral
equation of motion in vertical direction.
𝑝𝑏′
𝜕𝑊𝜏̂𝑏𝑧 𝑝′ 1 𝑝𝑏′ 𝑈𝑖 ℎ 𝜕(𝑤𝑠 − 𝑤𝑏 )= 𝑈𝑖 ℎ
+ 𝑆̂
, =
+
𝜌
𝜕𝑥𝑖
𝜌 𝜌 2𝜌
12
𝜕𝑥𝑖[7]

Where,𝑊: depth-average velocity in vertical direction, 𝑤𝑠 : water surface velocity in vertical direction, 𝑤𝑏 : bottom
surface velocity in vertical direction. 𝑊 is obtained from Eq. [1] and the equation of continuity.
𝑊=

1 𝜕(𝑧𝑠 + 𝑧𝑏 ) 1 𝜕(𝑧𝑠 + 𝑧𝑏 ) 1 𝜕 2 𝛿𝑢𝑖 ∆𝑢𝑖
)
+ 𝑈𝑖
+
ℎ (
+
2
𝜕𝑡
2
𝜕𝑥𝑖
ℎ 𝜕𝑥𝑖
20
10

[8]

𝑤𝑠 and 𝑤𝑏 are given from the kinematic boundary conditions at a water surface and a bottom surface.
𝑤𝑠 =

𝜕𝑧𝑠
𝜕𝑧𝑠
𝜕𝑧𝑏
𝜕𝑧𝑏
+ 𝑢𝑠𝑖
, 𝑤𝑏 =
+ 𝑢𝑏𝑖
𝜕𝑡
𝜕𝑥𝑖
𝜕𝑡
𝜕𝑥𝑖

[9]

Eq. [10] and Eq. [11] are the equations of motion on a water surface (𝜂 = 0) and a bottom surface (𝜂 = 1).
𝜕𝑢𝑠𝑖
𝜕𝑢𝑠𝑖
𝜕𝑧𝑠 1 𝜕𝑝′ 𝜕𝑧𝑠 𝜈𝑡𝑠 𝜕 2 𝑢𝑖
|
|
+ 𝑢𝑠𝑗
= −𝑔
+
+
𝜕𝑡
𝜕𝑥𝑗
𝜕𝑥𝑖 𝜌 𝜕𝑧 𝑠 𝜕𝑥𝑖𝜌 𝜕𝑧 2 𝑠𝜕𝑢𝑏𝑖
𝜕𝑢𝑏𝑖𝜕𝑧𝑠 1 𝜕𝑝𝑏′ 1 𝜕𝑝′ 𝜕𝑧𝑏𝜕𝜕𝑢𝑏𝑖 𝜕𝑢𝑏𝑗𝑆̂ 𝜏̂𝑏𝑖 − 𝜏̂0𝑖
|
)+
+ 𝑢𝑏𝑗
= −𝑔
−
+
+
𝜈𝑡𝑏 (
+
𝜕𝑡
𝜕𝑥𝑗
𝜕𝑥𝑖 𝜌 𝜕𝑥𝑖 𝜌 𝜕𝑧 𝑏 𝜕𝑥𝑖 𝜕𝑥𝑗
𝜕𝑥𝑗
𝜕𝑥𝑖
𝜌 𝛿𝑧𝑏

[10]
[11]

Where, 𝜏̂0𝑖 : bed shear stress in 𝑖 direction, 𝛿𝑧𝑏 = 𝑐𝑧𝑏 ℎ, 𝑐𝑧𝑏 = 0.03. The shear stress and pressure at a water
surface are assumed 0. The gradient of the non-hydrostatic pressure in vertical direction at a water surface anda bottom surface are calculated from Eq. [12] derived from the equations of motion in the vertical direction on awater surface and a bottom surface.
2050

E-proceedings of the 38th IAHR World Congress
September 1-6, 2019, Panama City, Panama

1 𝜕𝑝′
𝜕𝑤𝑠
| = −𝑢𝑠𝑖,𝜌 𝜕𝑧 𝑠
𝜕𝑥𝑖

1 𝜕𝑝′
𝜕𝑤𝑏
| = −𝑢𝑏𝑖𝜌 𝜕𝑧 𝑏
𝜕𝑥𝑖[12]

The bottom shear stress and the bed shear stress are evaluated by Eq. [13] and Eq. [14], respectively.
𝜏̂0𝑖 = 𝜌𝑐𝑏2 𝑢𝑏𝑖 |𝒖𝒃 |，𝜏̂ 0𝑧 = 𝜌𝑐𝑏2 𝑤𝑏 |𝒖𝒃 |𝜏̂𝑏𝑖 = 𝜈𝑡𝑏

𝑐𝑏 =

[13]

𝜕𝑢𝑖
𝜏̂ 0𝑧
| ，𝜏̂ 𝑏𝑧 =𝜕𝑧 𝑏
𝑐𝑧𝑏 + 1𝐶0
√1 + 𝑐𝑧𝑏 ,1 − 2𝐶0⁄𝜅

[14]

𝑔𝑛2
𝐶0 = √ 1⁄3ℎ[15]

Where, 𝜕 2 𝑢𝑖 ⁄𝜕𝑧 2 |𝑠 = (−24∆𝑢𝑖 + 6𝛿𝑢𝑖 )/ℎ2, 𝜕𝑢𝑖 ⁄𝜕𝑥|𝑠 = (−12∆𝑢𝑖 + 6𝛿𝑢𝑖 )/ℎ, 𝑛: Manning’s roughness coefficient,𝜅 = 0.41.
Q3D-FEBS introduces the one equation turbulence model shown in Eq. [16] and Eq. [17] to analyze theenergy dissipation process due to a hydraulic jump. These equations are derived by approximating the vertical
distribution of the turbulent kinetic energy with the third-order polynomial as in Eq. [1].
𝜕𝐾𝜕𝐾
1 𝜕ℎ𝑢𝑖′ 𝑘 ′ 1 𝜕 ℎ𝜈𝑡 𝜕𝐾 𝜈𝑡𝑏 𝜕 2 𝑘
𝐾 3⁄2
|
+ 𝑈𝑖
=−
+
+
+
𝑃
−
𝑐
𝑘
𝑑
𝜕𝑡
𝜕𝑥𝑖
ℎ 𝜕𝑥𝑖
ℎ 𝜕𝑥𝑖 𝜎𝑘 𝜕𝑥𝑖 𝜎𝑘 𝜕𝑧 2 𝑏
𝑙𝑑

[16]

2

1 𝜕𝑈𝑖 𝜕𝑈𝑗24
) + 2 (4∆𝑢𝑖2 − 7𝛿𝑢𝑖 ∆𝑢𝑖 + 𝛿𝑢𝑖2 )]𝑃𝑘 = 𝑐ℎ 𝜈̅𝑡 [ (+2 𝜕𝑥𝑗 𝜕𝑥𝑖
5ℎ

[17]

⁄

𝜕𝑘𝑠
𝜕𝑘𝑠
𝜕 𝜈𝑡𝑠 𝜕𝑘𝑠 𝜈𝑡𝑠 𝜕 2 𝑘
𝑘3 2| − 𝑐𝑑 𝑠
+ 𝑢𝑠𝑖
=
+
𝜕𝑡
𝜕𝑥𝑖 𝜕𝑥𝑖 𝜎𝑘 𝜕𝑥𝑖 𝜎𝑘 𝜕𝑧 𝑠
𝑙𝑑

[18]

𝛼 2 𝑢∗2𝑘𝑏 = ( )𝑐𝑙 1 + 𝑐𝑧𝑏[19]

Where, 𝐾: depth-average turbulent kinetic energy, 𝑘𝑠 : turbulent kinetic energy at a water surface, 𝑘𝑏 : turbulent
kinetic energy at a bottom surface, 𝑃𝑘 : depth-average production rate of turbulent kinetic energy,𝜎𝑘 = 1.0, 𝑐𝑑 =0.08, 𝑙𝑑 = 𝑐𝑙 ℎ, 𝑐𝑙 =0.07, 𝑐ℎ =0.5. The eddy viscosities are given as 𝜈𝑡 = 𝑙𝑑 √𝐾, 𝜈𝑡𝑠 = 𝑙𝑑 √𝜈𝑡𝑠 , 𝜈𝑡𝑏 = 𝑙𝑑 √𝜈𝑡𝑏 .

3

Application to the previous experiments of undular and breaking jumps
Q3D-FEBS is applied to the previous experiments of an undular jump (Chanson 1993) and a breaking jump
(Chacherea & Chanson 2010). The experimental conditions are shown in Table 1.
Table 1. Conditions of the previous experiments of an undular jump and a breaking jump.

Chanson1993.

Forms ofthe jump
Undularjump

Chacherea& Chanson
2010.

Breakingjump

ChannellengthChannelwidth

Channel
slope

20 m

0.25 m

1/225

3.2 m

0.5 m

Horizontal

Rate offlow
4.96
10-3m3/s
44.6
10-3m3/s

Fr

Re

Aspectratio

1.27

2.1×104

8.7

3.1

8.9×104

11.0

In the undular jump experiment (Chanson 1993), the location of the jump was controlled between 9.5 m
and 15 m from the upstream end of the channel. Water depths and pressure distributions were measured on
the centerline of the channel using a pointer gauge and a Pitot tube. The calculation was performed under thesame conditions as the experiment. In order to investigate the influences of a computational grid size ∆𝑥 on thecalculation results, ∆𝑥 = 0.01 m(∆𝑥/ℎ2 ≅ 0.25, ℎ2 ：downstream sequent depth) and ∆𝑥 = 0.004 m(∆𝑥/ℎ2 ≅0.1) were used. Figure 2 shows the comparison of measured and calculated water depth profiles. The
longitudinal distance and the water depths are normalized by the critical depth ℎ𝑐 , respectively. The calculation
results tend to overestimate an attenuation of the waves seen in the measured data. The calculated water
surface profile of ∆𝑥 = 0.01 m indicated by a black solid line can explain the experimental data up to the firstwave crest. On the other hand, the result of ∆𝑥 = 0.004 m indicated by a red solid line can reproduce themeasured data up to the second wave trough. Figure 3 shows the comparison of the vertical distributions of2051

E-proceedings of the 38th IAHR World Congress
September 1-6, 2019, Panama City, Panama

measured and calculated pressure intensity at the crest and trough of the first and second waves on the channel
centerline. The distributions of the pressure intensity are normalized by the hydrostatic pressure intensity at abottom surface and the black solid lines in Figure 3 indicate hydrostatic pressure distributions. As shown in ○and ×, the pressure intensity of the undular jump is lower at the crest and higher at the trough compared to thehydrostatic pressure distributions. The pressure distributions calculated by Q3D-FEBS using ∆𝑥 = 0.004 m arein good agreement with measured data, but the results using ∆𝑥 = 0.01 m cannot explain the pressure
distribution at the trough of the second wave. The calculation results are sensitive to the size of ∆𝑥. This isbecause the non-hydrostatic pressure caused by the curvature of the water surface is important for analysis of
the undular jump. Q3D-FEBS can explain the water surface profile of the undular jump up to the second wave
by using ∆𝑥 of about 1/10 of the water depth.

Figure 2. Comparison of measured and calculated water depth profiles of an undular jump.

(a) ∆𝑥 = 0.01 m(b) ∆𝑥 = 0.004 m

Figure 3. Comparison of the vertical distributions of measured and calculated pressure intensity at the crestand trough of the first and second waves of the undular jump.
In the breaking jump experiment (Chacherea & Chanson 2010), the location of the jump was fixed about
1.5 m from the upstream end of the channel. The calculation was performed under the same conditions as theexperiment and ∆𝑥 = 0.02 m (∆𝑥/ℎ2 ≅ 0.12) was used. Water depths and velocity distributions were measured
on the centerline of the channel using acoustic displacement meters and phase-detection conductivity probes.
Figure 4 shows the comparison of measured and calculated normalized water depth profiles. The black plots
indicate the measured water level averaged over 10 seconds. And the red and blue plots are the measured
average water level ± the standard deviation. The calculation shows only the average water surface profile
indicated by the black solid line because the significant fluctuations are not seen in the calculation results. As
shown in Figure 4, the water surface profiles calculated by Q3D-FEBS is good agreement with the measured
data. Figure 5 is the comparison of measured and calculated velocity distributions. They are averaged over 45
seconds, respectively. The calculation results roughly explain the measured velocity distributions including the2052

E-proceedings of the 38th IAHR World Congress
September 1-6, 2019, Panama City, Panama

backward velocities near the water surface due to the generation of the surface roller. The length of the surface
roller estimated from Q3D-FEBS is 0.44m. It is slightly shorter than the length estimated from Eq. [20] (Hager
1999) 0.56 m.
𝐿𝑟
𝐹1
[20]
= −12 + 160 tanh ( )ℎ1
20Where, 𝐿𝑟 : length of the surface roller, ℎ1 and 𝐹1 : water depth and Fr of approach flow.
From the above results, Q3D-FEBS can well reproduce the water surface profile and velocity distributionsof the breaking jump including the generation of the surface roller that is important for estimating the energy
dissipation within the jump.

Figure 4. Comparison of measured and calculated water depth profiles of a breaking jump.

(a) 𝑥/ℎ𝑐 =0.43(b) 𝑥/ℎ𝑐 =0.80(c) 𝑥/ℎ𝑐 =1.61(d) 𝑥/ℎ𝑐 =3.21

Figure 5. Comparison of measured and calculated velocity distributions of the breaking jump.

4

Numerical simulation of the transitional process from an undular jump to a breaking jump
Using the experimental conditions of Table 1 (Chacherea & Chanson 2010), the transitional process froman undular jump to a breaking jump is simulated by Q3D-FEBS. First, in order to generate an undular jump, aflow rate of 0.02 m3/s and a water depth of 0.04 m are given at the upstream end and a water depth of 0.054 m
is given at the downstream end. After the stable undular jump is formed, Fr of the approach flow grows by
increasing the flow rate to 0.0446 m3/s over 60 seconds while maintaining the water depth at the upstream end
at 0.04 m. The water depth at the downstream end is increased to 0.172 m over 55 seconds in order to control
the location of the jump. Figure 6 shows the simulation results. As shown in Figure 6(a), (b) and (c), the locationof the jump moves downstream and the flow velocities near the water surface are decelerated at the crest of
the first wave of the undular jump as Fr increases. When Fr exceeds 2.1, a flow separation occurs at the water
surface and it attenuates the wave height as shown in Figure 6(d). After that, the location of the jump moves2053

E-proceedings of the 38th IAHR World Congress
September 1-6, 2019, Panama City, Panama

upstream while expanding the backflow area near the water surface, and finally the breaking jump is formed asshown in Figure 6(e), (f).
Height (m)Horizontal velocity(m/s)

Fr=1.47, Q= 0.02 m3/s, ddw =0.054
Fr: Froud number at 0.8 m point
Q: Rate of flow at the upstream end
ddw:Water depth at the downstream end(a) 0 seconds after stable undular jump is formed.

Height (m)Horizontal velocity(m/s)

Fr：1.85, Q = 0.0253 m3/s. ddw =0.082 m(b) 13 seconds after stable undular jump is formed.

Height (m)Horizontal velocity(m/s)

Horizontal velocity(m/s)

Height (m)Height (m)Longitudinal distance (m)Fr：2.63, Q = 0.0352 m3/s, ddw =0.130 m(e) 37 seconds after stable undular jump is formed.
Horizontal velocity(m/s)

Longitudinal distance (m)Fr：2.10, Q = 0.0278 m3/s, ddw =0.095 m(d) 19 seconds after stable undular jump is formed.
Horizontal velocity(m/s)

Longitudinal distance (m)Fr：2.00, Q = 0.0274 m3/s, ddw =0.093 m(c) 18 seconds after stable undular jump is formed.

Height (m)Longitudinal distance (m)Longitudinal distance (m)Fr：3.14, Q = 0.0446 m3/s, ddw =0.172 m(f) 37 seconds after stable undular jump is formed.

Longitudinal distance (m)Figure 6. Simulation results of transitional process from an undular jump to a breaking jump by Q3D-FEBS.

2054

E-proceedings of the 38th IAHR World Congress
September 1-6, 2019, Panama City, Panama

It was found that Q3D-FEBS can calculate the transitional process from an undular jump to a breaking
jump in the framework of the depth-integral model by analyzing the flow separation at the water surface using
the flow equations on boundary surfaces.
5

CONCLUSIONSThis paper proposes a new non-hydrostatic quasi three-dimensional model (Q3D-FEBS) that introduces
the flow equations on boundary surfaces (a free surface and a bottom surface). Q3D-FEBS is applied to theprevious experiments of undular and breaking jumps and a numerical simulation for the transitional process
between the jumps is performed. These results show that Q3D-FEBS can predict the flows in both undular andbreaking jumps including the transitional process between the jumps in the framework of the depth-integral
model by introducing the flow equations on boundary surfaces.
REFERENCES
Castro-Orgaz, O., Hager, W. H. & Dey, S., (2015). Depth-averaged model for undular hydraulic jump, Journalof Hydraulic Research, 53(3), 351-363.
Chacherea, Y. & Chanson, H., (2010). Free-surface turbulent fluctuations and air-water flow measurements inhydraulics jumps with small inflow Froude numbers, Report CH78/10, School of Civil Engineering, The
University of Queensland.
Chanson, H., (1993). Characteristics of undular hydraulic jumps, Research Report CE146, Department of Civil
Engineering, The University of Queensland.
Hager, W. H., (1992). Energy Dissipators and Hydraulic Jump, Springer Science & Business Media, 13-18.
Hosoda, T. & Tada, A., (1994). Free surface profile analysis on open channel flow by means of 1-D basic
equations with effects of vertical acceleration, Annual Journal of Hydraulics Engineering, JSCE, 38, 457-462(in Japanese).
Iwasa, Y., (1953). Undular jump and its limiting conditions for existence, Proceedings of the 5th Japan National
Congress on Applied Mechanics, 2(14), 315-319.
Madsen, P. A. & Svendsen, I. A., (1983). Turbulent bores and hydraulic jumps, Journal of Fluid Mechanics, 129,
1-25.
McCorquodale, J. A. & Khalifa, A., (1983). Internal flow in hydraulic jumps, Journal of Hydraulic Engineering,
109(5), 684-701.
Narayanan, R., (1975). Wall jet analogy to hydraulic jump, Journal of the Hydraulics Division, ASCE, 101(3),347-359.
Ohno, J., Fukuoka, S., Takemura, Y. & Nigo, S., (2018). Analsis of flood flows in mountain streams with boulders
by using non-hydrostatic quasi-three dimensional model (Q3D-FEBS), Proceedings of the 32nd Symposiumon Computational Fluid Dynamics, JSFM, C01-1.
Serre, F., (1953). Contribution à l'étude des écoulements permanents et vari-ables dans les canaux, La HouilleBlanche, 8(12), 830-887 (in French).
Tsubaki, T., (1949). Theory of hydraulic jump, Reports of the Research Institute for Fluid Engineering, Kyushu
University, Fukuoka, Japan, 5(2), 15-29.

2055