dissertation in water engineering

UNIVERSITY OF PISA

Faculty of Engineering Department of Civil Engineering

Energy Dissipation in Presence of

Submerged Block Ramps

Stefano Pagliara, Paolo Di Ludovico

Report no 7 PITLAB, novembre 2006

SEU Servizio Editoriale Universitario di Pisa

Index

1. Abstract ………………………………………………………………………. Pag. 1

2. Introduction ………………………………………………………………… Pag. 2

3. Experimental Model

3.1 Description of the experimental model ……………………………………… Pag. 4

3.2 Materials used for channel bed ……………………………………………… Pag. 5

3.3 Materials used for ramp ……………………………………………………... Pag. 6

4. Introduction to Experintal Results

4.1 Dimensional analysis ……………………………………………………….. Pag. 8

4.2 Diagram sketch ……………………………………………………………… Pag. 10

5. Presentation and Elaboration of the Data

5.1 Analysis of results related to free jump condition …………………………... Pag. 11

5.1.1 Influence of channel bed material on energy dissipation ………….…. Pag. 11

5.1.2 Influence of ramp condition on energy dissipation (Er1) …………….. Pag. 12

5.1.3 Influence of ramp condition on energy dissipation (Er2) ………….…. Pag. 17

5.1.3.1 Influence of ramp condition on energy dissipation ……….…. Pag. 17

5.1.3.2 Difference between mobile and fixed channel bed condition .. Pag. 22

5.1.4 An indirect relationship for relative energy dissipation (Er2) ……….. Pag. 26

5.1.5 A direct relationship for relative energy dissipation (Er2) …………… Pag. 30

5.2 Analysis of the data for submerged jump …………………………………… Pag. 31

5.2.1 Influence of channel bed material on energy dissipation ………….…. Pag. 32

5.2.2 Comparison of energy dissipation in submerged jump between

presence and absence of scour ………………………………………..

Pag.

35

5.2.3 Use in the elaboration of additional data …………………………….. Pag. 40

5.2.4 Elaboration of submerged ramp condition data for partial

submergence (L/LT=1/3 and 2/3) ……………………………………...

Pag.

41

5.3 Elaboration to obtain one single relationship for free and submerged jump ... Pag. 47

5.4 Elaboration of submerged ramp condition data for complete submergence

(3/3) …………………………………………………………………………..

Pag.

50

6. Conclusion …………………………………………………………………… Pag. 52

7. Notation …………………………………………………………………….…. Pag. 53

8. Reference …………………………………………………………………….. Pag. 54

9. Appendix ……………………………………………………………………… Pag. 56

1. Abstract Block ramps or rock chutes are stream restoration structures used in many hydraulic projects. They generally produce high energy dissipation. Most of the previous studies were confined in finding the energy profile at the toe of the ramp. But the energy dissipation and flow condition when the ramp is submerged has received relatively little attention, whereas this type of condition can often take place in case of high discharge and slope. The present study shows the result of the experimental investigation conducted at Hydraulic Laboratory of the University of Pisa, Pisa, Italy on ramps of different slopes characterized by different bed materials and different submerged conditions. The experiments were carried out by varying the slopes between 1V:4H and 1V:8H, in different submerged ramp condition. The study shows that the energy loss at various locations of the ramp in submerged condition are functions of relative submergence, the ratio between the critical water depth and ramp height and the ratio between the reduced ramp length in submerged condition and horizontal length of the ramp. The present study also investigates the difference in energy dissipation at the toe of the ramp before the hydraulic jump and after the hydraulic jump for different bed condition such as fixed smooth bed, fixed rough bed and movable bed. The difference was found out to be very little between different conditions. The viscous effects and air entrainment was considered negligible in the experimental range. Key words: Energy dissipation, Submerged, Ramp, Hydraulic jump.

1

2. Introduction Block ramps are often used in natural rivers as river restoration structures. The main advantage of using bock ramp is that it keeps the morphological characters unchanged. Unlike other structures such as check dams and sills it does not create any barrier for the fish passage and also diminishes solid transport by dissipating large amount of energy. Block ramps are characterized by large bed slopes generally varying from 10% to 40% and along which the dissipation of energy is more than in traditional drop structures and this dissipation is due to localized variation of stream slope and high roughness condition exerted by the bed material(Pagliara and Peruginelli, 2000, Pagliara and Dazzini, 2002). In a simple way it can be said that it guarantees the perfect ecological balance and offers significant energy loss. The ramp is characterized by a steep slope followed by a mild one. Generally it is evident that for low downstream tail water condition, critical depth occurs at the beginning of the ramp and at the toe, a hydraulic jump is formed. Different studies have been carried out in order to determine a relationship between the amount of the energy dissipation and the characteristics of the ramp. Many researchers carried out tests on steep slope block ramp. Researchers such as Chamani and Rajaratnam (1999), Diez-Cascon et. al. (1991), Peyras et al. (1992), Stephenson (1991) and Christodoulou (1993) investigated the energy dissipation on stepped channel in which hydraulic behaviour in case of skimming flow is similar to the block ramp one. The hydraulics of stepped channel was early investigated by Essery and Horner (1978), while Chanson (1994) gave the equation for energy dissipation in condition of nappe and skimming flow. Chanson (1994) obtained the equation for energy dissipation between entrance and toe of the ramp and it can be written as follows:

Hhhh

EEE

c

cur +

+−=

Δ=Δ

5.1cos

13

0

11

α (2.1)

Where E0 is energy at the entrance, E1 is energy at the toe of the ramp, ΔE1=E0-E1, ΔEr1 is the relative energy dissipation, hu uniform flow depth, hc is the critical flow depth, H is the ramp height and α is the spillway slope. This is characterized by an unstable wake beyond the gravel or in the cavity of the steps, formed of eddies which increase flow resistance. The study on flow resistance has been carried out by many authors before such as Hey (1979), Bathurst (1978, 1985, 2002), Bathurst et al. (1981), Colosimo et al. (1988), Aguirre Pe and Fuentes (1990), Afzalimehr and Antcil (1998), Rice et al.(1998), Aberle et al. (1999). Bathurst (1985) proposed the following relationship for mountain streams

4log62.58

8410 +⎟⎟

⎠

⎞⎜⎜⎝

⎛=

dh

f (2.2)

where f = Darcy-Weisbach friction factor; h = water depth; d84 = diameter for which 84% of the material is finer. The range of slope in equation (2.2) was 0.004%-4%. Bathurst and also other authors in various studies mentioned the effect of scale roughness in the flow resistance.

According to Bathurst three roughness conditions can be identified for flow in a very rough bed: small scale roughness which occurs for relative submergence greater than 4 (relative submergence is defined as the ratio of the uniform flow depth hu to characteristic particle size d84), intermediate scale which occurs for relative submergence between 1.2 and 4 and large scale roughness which occurs when the relative submergence is less than approximately 1.2. In a recent study Pagliara and Chiavaccini (2006) proposed a relation for energy dissipation at the toe of the ramp and it can be written as follows:

HhiCBr

ceAAEEE /).(

0

11 )1( +−+=

Δ=Δ (2.3)

Where A,B,C= three parameters depending on the scale roughness of the flow over the ramp and i is the slope of the ramp. In this case the bed of the jump was smooth and fixed. Several other authors Ead and Rajaratnam (2002), Hugh and Flack (1979) studied the characteristics of hydraulic jump in

2

the case of fixed bed condition also in presence of roughness element. The effect of the hydraulic jump on the scour mechanism in the case of mobile bed conditions was investigated by Rajaratnam (1965), Wu and Rajaratnam (1995) and Solari (2004). They considered the effect produced by turbulent wall jets.

The study of submerged jump has received relatively less attention. However Kindsvarter (1944), Rajaratnam (1967) and Hager(1992) has studied the phenomenon of hydraulic jump in sloping channels. Kindsvater (1944) classified jumps according to their toe position relative to the bottom kink Figure 2.1.

LT

Q

xy

kD

C

AB

Tailwaterincreasing

Tailwaterdecreasing

Figure 2.1 Types of sloping channel jumps

A-jump for which the toe is at the kink, B-jump is intermediate to A- and C- jumps, C-jump for which the end of the roller is above the kink, and D-jump where the entire roller is on the sloping channel portion. In the present study the energy dissipation in submerged jump is investigated from the results of the experimental study in block ramps. The difference in energy dissipation before and after the jump for different bed condition is also investigated. Runs were carried out for block ramps with a slope range between 1V:8H and 1V:4H. The three different roughness condition: small, intermediate and large scale were reproduced.

3

3. Experimental Model 3.1 Description of the experimental model The experimental tests were carried out in the laboratory flumes of the Hydraulic Laboratory of the Department of Civil Engineering at the University of Pisa, Pisa, Italy. The model employed to carry out the experiments is actually a hydraulic recirculating flume. It is composed of a big tub from where the water is pumped into a smaller tank situated just at the entrance of the main channel as shown in Figure 3.1.1. This small tank is used to reduce the turbulence of the pumped water before entering the main channel.

Figure 3.1.1 Small tank at the entrance of main channel for turbulence dissipation

The water after flowing through the main channel again comes back to the large tub. The main channel is made of plexiglass connected by means of self-tapping lives. The dimension of the channel is 3.5 m long, 0.25 m wide and 0.3 m deep. A broad crested weir with variable height was inserted into the channel, which helped to obtain different desired slopes. The range of slope investigated in the present study is from 1V:8H to 1V:4H. The chute structure was made by means of a 2 mm thick smooth plane with a length between 0.8 and 1.2 m, on which the granular material was glued. Water levels was measured by a point gage, 0.1 mm precise while flow rates were measured by means of a calibrated magnetic flow meter located in the supply line. An adjustable sluice gate at the end of the channel allowed to attain required downstream water level. A view of the main channel is shown in Figure 3.1.2.

4

Figure 3.1.2 View of the entire model with adjustable sluice gate at the end of the channel

During the experiment of submerged jump condition the gate was adjusted accordingly to obtain the desired submergence level. 3.2 Materials used for channel bed The bed material in the present study is not fixed, rather it is mobile and two different homogeneous material were used for the present study. The two bed material were indicated by Mf1(Figure 3.2.2) and Mf2 (Figure 3.2.4). The mean diameter of the material Mf1 and Mf2 (D50) were 3.2mm and 6.8 mm respectively. Both the materials used were homogeneous (i.e. s = (D84/D16)1/2 <1.3). D is the sediment size and he subscript indicates the percentage of passage across a sediment net and σ is the coefficient of uniformity of material. Figure 3.2.1 & 3.2.3 shows the granulometric curve of material Mf1 and Mf2 respectively.

0

50

100

0 4

Equivalent diameter (mm)

% passage

8

Figure 3.2.1 Granulometric curve of material Mf1

5

Figure 3.2.2 Material Mf1

0

50

100

0 6

Equivalent diameter (mm)

% passage

12

Figure 3.2.3 Granulometric curve of material Mf2

Figure 3.2.4 Material Mf2

3.3 Materials used for ramp In the present study experiments were carried out using three different ramp materials. These three ramp materials are denoted as d1, d2, d3. The materials used for the experiments are homogeneous. The d50 of d1, d2 and d3 are 1mm, 8mm and 20mm respectively, where d50 means 50% of bed material is finer. The following figures show the granular material d1, d2 and d3 glued on the ramp respectively.

6

Figure 3.3.1 Ramp with material d1



Data of other researchers who carried out experiments in the same channel and in same hydraulic conditions were also considered for comparison. The ramp and bed material characteristics and hydraulic condition of other author and present study is furnished in Table. 3.3.1 for comparison. Table. 3.3.1

NAME OF RESEARCHERS

HYDRAULIC CONDITION BED MATERIAL RAMP MATERIAL EXPERIMENT

TYPE Di Grigoli-Pagliara-

Chiavaccini

2 l/s<Q<8 l/s 1.2<Fr1<4.2 Smooth Bed

1.1<σ<1.3 d1=1mm, d4=2mm d2=8mm, d3=20mm

Free jump Submerged 1/3 Submerged 2/3

Mazzoncini 2 l/s<Q<8 l/s 1.2<Fr1<4.2

Mobile 1.1<σ<1.3

Mf3, D50=5.7mm,

1.1<σ<1.3 d3=20mm, d5=4mm

Free jump

Di Matto 2 l/s<Q<8 l/s 1.2<Fr1<4.2

Mobile 1.1<σ<1.3

Mf4, D50=1.9mm, Mf5, D50=6.3mm,

1.1<σ<1.3 d6=10mm, d5=4mm

Free jump

Author 2 l/s<Q<8 l/s 1.2<Fr1<4.2

Mobile 1.1<σ<1.3

Mf1, D50=3.2mm, Mf2, D50=6.8mm,

1.1<σ<1.3 d1=1mm, d2=8mm

d3=20mm

Free jump Submerged 1/3 Submerged 2/3 Submerged 3/3

7

4. Introduction to Experimental Results 4.1 Dimensional analysis

The energy dissipation over a block ramp is effected by many variables. To understand how these variables effect the hydraulic behavior two dimensional analisys were carried out: one for the energy dissipation at the toe of the ramp and other for the energy dissipation downstream of the hydraulic jump.

• Dimensional analysis of energy dissipation between upstream section of the ramp and upstream section of the hydraulic jump (in presence of free hydraulic jump):

The energy dissipation and hydraulic functioning of a ramp of height D and horizontal length of the ramp LT, and depth of water at section 1 be h1, can be described by the following functional relation:

( μ,,,,,, 500

10 gdbkLDfE

EET=

− )

)

(4.1.1)

Where E0=1.5k+D is the specific energy at the upstream section of the ramp, is the energy dissipation between upstream section and toe of the ramp, k is the critical height, b is the cross-sectional width, d

1EΔ

50 is the diameter where 50% of the bed material are finer, g acceleration due to gravity and μ is the dynamic viscosity. The main hypothesis taken into account are: - free hydraulic jump at the toe of the ramp; - uniformity of bed material; - wide channel; The first hypothesis allows to separate two case which affect the energy dissipation : one of free hydraulic jump and the other of submerged hydraulic jump; the second allows to consider the roughness of the material constant and equal to d50 and the last obtained from literature considerations. The Buckingham Π theorem allow to give a new functional relationship with the use of three dimensionless group Π 1, Π 2, Π 3:

( 0,, 321 =ΠΠΠf (4.1.2) Where f is a fucntional symbol. The main independent variables of the dimensionless group are D, g, and μ, can make a reference unit frame. The first Π1 group is equal to:

TLgD ⋅⋅⋅=Π γβα μ1 (4.1.3) Writing in terms of dimensions equation 4.1.3 becomes: ( ) ( ) ( ) ( ) ( )LTLLTLTLM γβα 122000 −−= (4.1.4) Equating the powers of M, L and T on each side of the equation we get:

120 +++= γβα γ=0

γβ +⋅−= 20

8

And the solution obtained is : 1−=α

0=β 0=γ

Putting the value of α, β and γ in equation (4.1.3) we get:

DLT=Π1 (4.1.5)

Similarly group Π2 and Π3 becomes: kgD ⋅⋅⋅=Π γβα μ2 (4.1.6)

503 dgD ⋅⋅⋅=Π γβα μ (4.1.7) And finally:

Dk

=Π 2 (4.1.8)

Dd50

3 =Π (4.1.9)

The Π-theorem allows to give new dimensionless groups combining the precedent group instead of (i=1,..,3). iΠ

From eq. (4.1.8) and (4.1.9) the new group obtained:

kd

kD

Dd 5050

2

31 =⋅=

ΠΠ

=Ψ (4.1.10)

And from eq. (4.1.5):

iLD

T

==Π

=Ψ1

21 (4.1.11)

Where i is the slope of the ramp. The functional relationship (4.1.2) becomes:

( 0,, 212 =ΨΨΠ )φ (4.1.12) And putting the dimensionless groups the final functional relationship obtained is:

⎟⎠⎞

⎜⎝⎛=

Δk

di

Dkf

EE 50

0

1 ,, (4.1.13)

Equation 4.1.13 can be used as base of the hydraulic behaviour for free jump.

• Dimensional analysis of energy dissipation between upstream section of the ramp and downstream section of the hydraulic jump:

In this functional relationship the variables for the occurrence of scour must be included:

( 501500

20

0

2 ,,,,,,,,, DhzgdbkLDfE

EEEE

mT μ=−

=Δ ) (4.1.14)

Where is the energy dissipation at the downstream section of the ramp, where the conjugate height h

2EΔ2 is located, h1 is the height at the toe of the ramp, D50 is the characteristic size of the bed

material where the scour is located and zm is the measured medium cross-sectional scour depth at the downstream section of the ramp.

9

From this functional relationschip, using the Buckingham Π-theorem we can find a new relationship that represents the base of the present study for submerged jump:

⎟⎟⎠

⎞⎜⎜⎝

⎛=

Δ

50

150

0

2 ,,,,DhZ

kd

iDkf

EE

m (4.1.15)

where Zm=zm/h1 4.2 Diagram sketch Two diagram sketch of two different condition of experimental set-up is shown in Figure 4.2.1(a,b). Figure 4.2.1(a) shows the diagram sketch for free jump and Figure 4.2.1(b) shows the diagram sketch for submerged jump.

D

Q

xy

1

1

k

h1

0

0LT

h2

2

2

Figure. 4.2.1(a) Diagram sketch for free jump experiments

From the figure it can be seen that h1 is the water depth at the toe of the ramp, h2 is the downstream conjugate depth, D is drop height, Q is discharge, k is the critical depth and LT is the horizontal ramp length.

D

LT

Q

xy

0

0

k

h2(1/3)

h2(2/3)

h2(3/3)

L

Zm

Complete submergence (3/3)

Partial submergence (2/3)

Partial submergence (1/3)

Figure 4.2.1(b) Diagram sketch for submerged jump experiments

It can be seen from Figure 4.2.1(b) that three submerged conditions were investigated L/LT = 1/3, 2/3 and 3/3, where L=horizontal distance of the begining of the jump from the toe of ramp and LT=horizontal length of the ramp.

10

5. Presentation and Elaboration of the Data 5.1. Analyses of the results related to free jump condition In the case of free jump the dissipation of energy between the sections 0 and 2 has been calculated in two methods in indirect way (i.e. first between section 0 and section 1, then between section 1 and section 2) and subsequently in direct way (i.e directly between section 0 and section 2).

Figure 5.1.1 Test of Free Jump with ramp material d2

5.1.1 Influence of channel bed material on energy dissipation The first step of the elaboration was to compare and find out the dependence of energy dissipation on bed material. Two different homogeneous bed materials were used for the mobile bed (Mf1 with D50 equal to 3.2mm and Mf2 with D50 equal to 6.8mm). The dependence of this two bed material were checked to see whether they give different results. For this purpose the curves of (E0-E2)/E0 as a function of k/D were drawn. The comparison was done by keeping the ramp material same for different bed materials in different slope condition. In the graph E0 represents the energy at the entrance of the ramp, section 0, and is calculated by (E0=1.5k+D) while E2 represents the energy at the downstream end of the ramp i.e. at section 2. The following figure shows the curve of (E0-E2)/E0 against k/D.

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

k/D

(E0-

E 2)/E

0

1:8 Mf1 (free jump) 1:8 Mf2 (free jump)1:6 Mf1 (free jump) 1:6 Mf2 (free jump)

1:4 Mf1 (free jump) 1:4 Mf2 (free jump)

Figure 5.1.2 Relationship between (E0-E2)/E0 and k/D for ramp material d1

11

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

k/D

(E0-

E 2)/E

01:8 Mf1 (free jump) 1:8 Mf2 (free jump)1:6 Mf1 (free jump) 1:6 Mf2 (free jump)1:4 Mf1 (free jump) 1:4 Mf2 (free jump)

Figure 5.1.3 Relationship between (E0-E2)/E0 and k/D for ramp material d2

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

k/D

(E0-

E 2)/E

0

1:8 Mf1 (free jump) 1:6 Mf1 (free jump)1:4 Mf1 (free jump) 1:8 Mf2 (free jump)1:6 Mf2 (free jump) 1:4 Mf2 (free jump)

Figure5.1.4 Relationship between (E0-E2)/E0 and k/D for ramp material d3

From the graphs it is evident that in the experimental range the size of bed materials does not effect the energy dissipation. No definite trend for different size of bed material was found. Hence for further elaboration dependence of bed material size on energy dissipation was not taken into account. 5.1.2 Influence of ramp condition on energy dissipation ( )

1rE

From the dimensional analysis we get: ( ) ⎟⎠⎞

⎜⎝⎛=

−=

kd

iDkj

EEE

Er50

0

10 ,,1

. The dependence of energy

dissipation at the toe of the ramp ⎟⎟⎠

⎞⎜⎜⎝

⎛ −

0

10

EEE

on the parameters iDk , and

kd50 are checked.

Experiments and elaborations on this particular phenomenon were carried out previously by Pagliara and Chiavaccini (2006). The same method is followed in the present study to check and validate the data. They classified the scale roughness in the following way as shown in the Table. 5.1.1.

12

Table. 5.1.1

Large Scale roughness (LR) Intermediate Scale roughness (IR) Small Scale roughness (SR)

kd50 >0.4 0.15<

kd50 <0.4 0.02<

kd50 <0.15

As it was mentioned by Pagliara and Chiavaccini (2006) here also in the present study it can be seen from Figure 5.1.5 that energy dissipation does not depend on d50/k.

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

d 50/k

(E0-

E 1)/E

0

SR

IR

LR

Figure 5.1.5 Relation between (E0-E1)/E0 with d50/k separated for relative roughness LR,IR and SR

Therefore the functional relationship reduces to the following: ⎟⎠⎞

⎜⎝⎛=

−i

Dkj

EEE

,0

10

Pagliara and Chiavaccini (2006) from their study proposed the following relationship DkiCBeAA

EEE /).(

0

10 )1( ⋅+⋅−+=−

(5.1.1)

Where the parameters A,B and C depend on the scale roughness as shown in Table 5.1.2. Table. 5.1.2 Roughness condition A B C Large Scale roughness (LR) 0.33 -1.3 -14.5 Intermediate Scale roughness (IR) 0.25 -1.2 -12 Small Scale roughness (SR) 0.15 -1.0 -11.5 From the equation (5.1.1) it can be seen that the energy dissipation is a function of slope, scale roughness and the ratio k/D. Hence to check this phenomenon first the graph of energy dissipation as a function of k/D was drawn keeping the slope constant and varying the scale roughness.

13

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(E0-

E 1)/E

0Author, Di Matto, Mazzoncini S=1V:12H; SR Author, Di Matto, Mazzoncini S=1V:12H; IRAuthor, Di Matto, Mazzoncini S=1V:12H; LReq. Pagliara-Chiavaccini 2006 S=1V:12H SReq. Pagliara-Chiavaccini 2006 S=1V:12H IReq. Pagliara-Chiavaccini 2006 S=1V:12H LR

Figure 5.1.6 Relative energy dissipation as function of k/D for constant slope 1V:12H

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(E0-

E 1)/E

0

Author,Di Matto,Mazzoncini S=1V:8H; SRAuthor,Di Matto,Mazzoncini S=1V:8H; IRAuthor,Di Matto,Mazzoncini S=1V:8H; LReq. Pagliara-Chiavaccini 2006 S=1V:8H SReq. Pagliara-Chiavaccini 2006 S=1V:8H IReq. Pagliara-Chiavaccini 2006 S=1V:8H LR


0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(E0-

E 1)/E

0

Author,Di Matto,Mazzoncini S=1V:6H; SRAuthor,Di Matto,Mazzoncini S=1V:6H; IRAuthor,Di Matto,Mazzoncini S=1V:6H; LReq. Pagliara-Chiavaccini 2006 S=1V:6H SReq. Pagliara-Chiavaccini 2006 S=1V:6H IReq. Pagliara-Chiavaccini 2006 S=1V:6H LR


14

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(E0-

E 1)/E

0Autore,Di Matto,Mazzoncini S=1V:4H; SRAutore,Di Matto,Mazzoncini S=1V:4H; IRAutore,Di Matto,Mazzoncini S=1V:4H; LReq. Pagliara-Chiavaccini 2006 1V:4H SReq. Pagliara-Chiavaccini 2006 1V:4H IReq. Pagliara-Chiavaccini 2006 1V:4H LR


As shown in the figures above the relative energy dissipation increases with the reduction of the parameter k/D. This means that at the same discharge values, the dissipation is directly proportional to the height of the ramp. The dissipation is also a function of the scale roughness with notable differences between large scale (LR) and small scale roughness (SR) conditions, where the last one is less dissipative. The intermediate roughness (IR) conditions points lye between those of the other two conditions. It is important to note that the data of the present work is slightly shifted from the equation proposed by Pagliara and Chiavaccini (2006), this can be attributed to the fact that the point of measurement of h1 for the two study was not same. The h1 measured in the present study is slightly before and is at the edge of the ramp, whereas h1 for Pagliara and Chiavaccini is just before the jump as shown in Figure 5.1.10.

D

Q

xy

1

1

k h1 Pagliara-Chiavaccini

0

0LT

h2

h1 autore,Di Matto,Mazzoncini

1

Figure 5.1.10 Different position of measure of h1 for the calculation of E1

The energy dissipation is also a function of the slope. As illustrated in the figure below, as the slope increase the dissipation decreases, even if the differences tend to disappear with the decrease of the slope.

15

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,2 0,4 0,6 0,8 1 1,2 1,4

k/D

(E0-

E 1)/E

0Author,Di Matto,Mazzoncini S=1V:8HAuthor,Di Matto,Mazzoncini S=1V:6HAuthor,Di Matto,Mazzoncini S=1V:4HAuthor,Di Matto,Mazzoncini S=1V:3Heq Pagliara-Chiavaccini 2006 1V:8Heq Pagliara-Chiavaccini 2006 1V:6Heq Pagliara-Chiavaccini 2006 1V:4Heq Pagliara-Chiavaccini 2006 1V:3H

Figure 5.1.11 Relative energy dissipation as function of k/D for SR

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10%

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30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(E0-

E 1)/E

0

Author,Di Matto,Mazzoncini S=1V:8HAuthor,Di Matto,Mazzoncini S=1V:6HAuthor,Di Matto,Mazzoncini S=1V:4HAuthor,Di Matto,Mazzoncini S=1V:3Heq Pagliara-Chiavaccini 2006 1V:8Heq Pagliara-Chiavaccini 2006 1V:6Heq Pagliara-Chiavaccini 2006 1V:4Heq Pagliara-Chiavaccini 2006 1V:3H

Figure 5.1.12 Relative energy dissipation as function of k/D for IR

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10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(E0-

E 1)/E

0

Author,Di Matto,Mazzoncini S=1V:8HAuthor,Di Matto,Mazzoncini S=1V:6HAuthor,Di Matto,Mazzoncini S=1V:4HAuthor,Di Matto,Mazzoncini S=1V:3Heq. Pagliara-Chiavaccini 2006 1V:8Heq. Pagliara-Chiavaccini 2006 1V:6Heq. Pagliara-Chiavaccini 2006 1V:4Heq. Pagliara-Chiavaccini 2006 1V:3H

Figure 5.1.13 Relative energy dissipation as function of k/D for LR

16

Also in these last graphs a slight shift of the experimental data from the equation already proposed by Pagliara and Chiavaccini (2006) is noticed due to difference in location of measurement of h1. The following graph shows the comparison of energy dissipation between the measured values and the values calculated with the Pagliara and Chiavaccini (2006) equation (Eq. 5.1.1). The experimental points are slightly shifted from the already proposed equation which is due to the reason mentioned before. However this comparison holds quite good and confirms the validity of the data of the present study.

0,0

0,2

0,4

0,6

0,8

1,0

0 0,2 0,4 0,6 0,8 1

(E 0-E 1)/E 0 (meas)

SR

IR

LR

(E 0-E 1)/E 0 (calc)

Figure 5.1.14 Comparison between measured and calculated (Eq. 5.1.1) values 5.1.3 Influence of ramp condition on energy dissipation ( )

2rE

5.1.3.1 Influence of ramp condition on energy dissipation Here the dependence of relative energy dissipation between the section 0 and section 2 on ramp condition is discussed. From the dimensional analysis we get:

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=

−=

Δ=

50

150

0

20

0

2 ,,,,2 D

hZk

di

Dkf

EEE

EEE mr (5.1.2)

The dependence of ΔE2/E0 on k/D, i and d50/k (relative roughness) are investigated in the following elaborations. The classification of relative roughness is given earlier in Table. 5.1.1. In the following figures relative energy dissipation as a function of k/D is plotted, keeping constant the slope and varying the scale roughness to see the dependence of scale roughness. Then again the relative energy dissipation as a function of k/D is plotted but this time keeping the roughness condition constant and varying the slope to check the effect of slope. The dependence of Zm and h1/D50 are not checked here since it represents the bed condition.

17

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30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(E0-

E 2)/E

0Author,Di Matto,Mazzoncini S=1V:12H; SR Author,Di Matto,Mazzoncini S=1V:12H; IRAuthor,Di Matto,Mazzoncini S=1V:12H; LR


0%

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30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(E0-

E 2)/E

0

Author,Di Matto,Mazzoncini S=1V:8H; SRAuthor,Di Matto,Mazzoncini S=1V:8H; IRAuthor,Di Matto,Mazzoncini S=1V:8H; LR


0%

10%

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30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(E0-

E 2)/E

0

Author,Di Matto,Mazzoncini S=1V:6H; SRAuthor,Di Matto,Mazzoncini S=1V:6H; IRAuthor,Di Matto,Mazzoncini S=1V:6H; LR


18

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10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(E0-

E 2)/E

0Author,Di Matto,Mazzoncini S=1V:4H; SRAuthor,Di Matto,Mazzoncini S=1V:4H; IRAuthor,Di Matto,Mazzoncini S=1V:4H; LR


0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,2 0,4 0,6 0,8 1 1,2

k/D

(E0-

E 2)/E

0

Author,Di Matto,Mazzoncini S=1V:8HAuthor,Di Matto,Mazzoncini S=1V:6HAuthor,Di Matto,Mazzoncini S=1V:4HAuthor,Di Matto,Mazzoncini S=1V:3H

Figure 5.1.19 Relative energy dissipation as function of k/D for SR

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10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(E0-

E 2)/E

0

Author,Di Matto,Mazzoncini S=1V:8HAuthor,Di Matto,Mazzoncini S=1V:6HAuthor,Di Matto,Mazzoncini S=1V:4HAuthor,Di Matto,Mazzoncini S=1V:3H

Figure 5.1.20 Relative energy dissipation as function of k/D for IR

19

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10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(E0-

E 2)/E

0Author,Di Matto,Mazzoncini S=1V:8HAuthor,Di Matto,Mazzoncini S=1V:6HAuthor,Di Matto,Mazzoncini S=1V:4HAuthor,Di Matto,Mazzoncini S=1V:3H

Figure 5.1.21 Relative energy dissipation as function of k/D for LR

As shown in the figures above the relative energy dissipation increases with the reduction of the parameter k/D. From the figures it is also evident that the relative energy dissipation between section 0 and 2 (ΔE2/E0), is independent of both relative roughness and slope of the ramp. As it was mentioned earlier that the data of the present study, Di Matto and Mazzoncini are all for mobile bed. Hence it is concluded that for mobile bed relative energy dissipation (ΔE2/E0) does not depend on relative roughness and slope of the ramp. The test performed by Di Grigoli was in smooth and fixed bed condition, so the data of Di Grigoli are also examined in similar method to check the dependence of relative roughness and slope of the ramp between section 0 and 2.

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(E0-

E 2)/E

0

S=1V:8H;SR Di GrigoliS=1V:8H;IR Di GrigoliS=1V:8H;LR Di Grigoli

Figure 5.1.22 Relative energy dissipation as function of k/D for constant slope 1V:8H and for fixed smooth bed

20

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30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(E0-

E 2)/E

0S=1V:6H; SR Di GrigoliS=1V:6H; IR Di GrigoliS=1V:6H; LR Di Grigoli


0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(E0-

E 2)/E

0

S=1V:4H; SR Di GrigoliS=1V:4H; IR Di GrigoliS=1V:4H; LR Di Grigoli


0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6

k/D

(E0-

E 2)/E

0

S=1V:8H Di GrigoliS=1V:6H Di GrigoliS=1V:4H Di Grigoli

Figure 5.1.25 Relative energy dissipation as function of k/D for fixed smooth bed in SR condition

21

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10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(E0-

E 2)/E

0S=1V:8H Di GrigoliS=1V:6H Di GrigoliS=1V:4H Di Grigoli

Figure 5.1.26 Relative energy dissipation as function of k/D for fixed smooth bed in IR condition

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20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(E0-

E 2)/E

0

S=1V:8H Di GrigoliS=1V:6H Di GrigoliS=1V:4H Di Grigoli

Figure 5.1.27 Relative energy dissipation as function of k/D for fixed smooth bed in LR condition

Like the phenomenon for mobile bed also in case of fixed bed no appreciable dependence of relative roughness and slope of the ramp (between section 0 and 2) on the energy dissipation were found. 5.1.3.2 Difference between mobile and fixed channel bed condition In the previous section it has been shown that for both mobile and smooth bed the relative energy dissipation within section 0 and 2 does not depend on slope and relative roughness of the ramp.

In this section the difference in energy dissipation between mobile and fixed bed condition is investigated. The ramp condition in both cases are same but since the bed condition is different an occurrence of scour takes place for mobile bed which is absent for smooth bed.

In the following figure energy dissipation for fixed smooth bed (data Di Grigoli) and mobile bed (data of present study, Di Matto and Di Mazzoncini) as a function of k/D are plotted keeping the slope and relative roughness constant. The equation of relative energy dissipation between section 0 and 1, as proposed by Pagliara and Chiavaccini (2006) is also plotted in the same graph.

22

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30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(ΔE)

/E0

Author,Mazzoncini,Di Matto (E0-E2)/E0 S=1V:8H; SR mobile bedDi Grigoli (E0-E2)/E0 S=1V:8H;SR smooth f ixed bedeq. Chiavaccini-Pagliara 2006 (E0-E1)/E0 S=1V:8H; SR

Figure 5.1.28 Difference between mobile and fixed bed for slope 1V:8H and roughness condition SR

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30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(ΔE)

/E0

Author,Mazzoncini,Di Matto (E0-E2)/E0 S=1V:8H; LR mobile bed Di Grigoli 2002 (E0-E2)/E0 S=1V:8H;LR smooth f ixed bedeq. Chiavaccini-Pagliara 2006 (E0-E1)/E0 S=1V:8H LR

Figure 5.1.29 Difference between mobile and fixed bed for slope 1V:8H and roughness condition LR

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30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(ΔE)

/E0

Author,Mazzoncini,Di Matto (E0-E2)/E0 S=1V:4H; SR mobile bedDi Grigoli 2002 (E0-E2)/E0 S=1V:4H;SR smooth f ixed bedeq. Chiavaccini-Pagliara 2006 (E0-E1)/E0 S=1V:4H SR

Figure 5.1.30 Difference between mobile and fixed bed for slope 1V:4H and roughness condition SR

23

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10%

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30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(ΔE)

/E0

Author,Mazzoncini,Di Matto (E0-E2)/E0 S=1V:4H; LR mobile bedDi Grigoli 2002 (E0-E2)/E0 S=1V:4H;LR smooth f ixed bedeq. Chiavaccini-Pagliara 2006 (E0-E1)/E0 S=1V:4H LR

Figure 5.1.31 Difference between mobile and fixed bed for slope 1V:4H and roughness condition LR

It is noted that the difference between the line of 0

10

EEE −

equation and points of 0

20

EEE −

are very

small. Hence it can be concluded that maximum energy dissipation takes place on the ramp i.e. between sections 0 and 1 and dissipation of energy between sections 1 and 2 is very little.

For the relative energy dissipation between sections 0 and 2 it is evident from the figures that there are difference between smooth fixed bed and mobile bed, however the difference is very small. This difference can be because of the fact that in mobile bed there is a formation of scour which is absent in case of smooth bed. This scour changes the profile of the hydraulic jump that is formed at the toe of the ramp as shown in the definition sketch and also results in more energy dissipation compared to the smooth bed condition. Some more figures are also furnished below where the difference between mobile and fixed smooth bed can be seen more prominently.

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(E0-

E 2)/E

0

Author,Mazzoncini,Di Matto (E0-E2)/E0 S=1V:8H SR mobile bedAuthor,Mazzoncini,Di Matto (E0-E2)/E0 S=1V:8H IR mobile bedAuthor,Mazzoncini,Di Matto (E0-E2)/E0 S=1V:8H LR mobile bedDi Grigoli 2002 (E0-E2)/E0 S=1V:8H SR smooth fixed bedDi Grigoli 2002 (E0-E2)/E0 S=1V:8H IR smooth fixed bedDi Grigoli 2002 (E0-E2)/E0 S=1V:8H LR smooth fixed bed

Figure 5.1.32 Difference between mobile and fixed bed for slope 1V:8H

24

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10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(E0-

E 2)/E

0Author,Mazzoncini,Di Matto (E0-E2)/E0 S=1V:4H SR mobile bedAuthor,Mazzoncini,Di Matto (E0-E2)/E0 S=1V:4H IR mobile bedAuthor,Mazzoncini,Di Matto (E0-E2)/E0 S=1V:4H LR mobile bedDi Grigoli 2002 (E0-E2)/E0 S=1V:4H SR smooth fixed bedDi Grigoli 2002 (E0-E2)/E0 S=1V:4H IR smooth fixed bedDi Grigoli 2002 (E0-E2)/E0 S=1V:4H LR smooth fixed bed

Figure 5.1.33 Difference between mobile and fixed bed for slope 1V:4H

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20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6

k/D

(ΔE)

/E0

Author,Mazzoncini,Di Matto (E0-E2)/E0 S=1V:8H mobile bedAuthor,Mazzoncini,Di Matto (E0-E2)/E0 S=1V:6H mobile bedAuthor,Mazzoncini,Di Matto (E0-E2)/E0 S=1V:3H mobile bedAuthor,Mazzoncini,Di Matto (E0-E2)/E0 S=1V:4H mobile bedDi Grigoli 2002 (E0-E2)/E0 S=1V:8H smooth fixed bedDi Grigoli 2002 (E0-E2)/E0 S=1V:6H smooth fixed bedDi Grigoli 2002 (E0-E2)/E0 S=1V:4H smooth fixed bed

Figure 5.134 Difference between mobile and fixed bed for roughness condition SR

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30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(ΔE)

/E0

Author,Mazzoncini,Di Matto (E0-E2)/E0 S=1V:8H mobile bedAuthor,Mazzoncini,Di Matto (E0-E2)/E0 S=1V:6H mobile bedAuthor,Mazzoncini,Di Matto (E0-E2)/E0 S=1V:4H mobile bedDi Grigoli 2002 (E0-E2)/E0 S=1V:8H smooth fixed bedDi Grigoli 2002 (E0-E2)/E0 S=1V:6H smooth fixed bedDi Grigoli 2002 (E0-E2)/E0 S=1V:4H smooth fixed bed

Figure 5.1.35 Difference between mobile and fixed bed for roughness condition IR

25

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10%

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30%

40%

50%

60%

70%

80%

90%

100%

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

k/D

(ΔE)

/E0

Author,Mazzoncini,Di Matto (E0-E2)/E0 S=1V:8H mobile bedAuthor,Mazzoncini,Di Matto (E0-E2)/E0 S=1V:6H mobile bedAuthor,Mazzoncini,Di Matto (E0-E2)/E0 S=1V:4H mobile bedDi Grigoli 2002 (E0-E2)/E0 S=1V:8H smooth fixed bedDi Grigoli 2002 (E0-E2)/E0 S=1V:6H smooth fixed bedDi Grigoli 2002 (E0-E2)/E0 S=1V:4H smooth fixed bed

Figure 5.1.36 Difference between mobile and fixed bed for roughness condition LR

5.1.4 An indirect relationship for relative energy dissipation ( )

2rE

An indirect method to determine relative energy dissipation at downstream end

In the previous section it has been proved that the dissipation of energy in case of mobile bed is more than the smooth bed and this difference is because of the difference in condition of the bed. From the dimensional analysis we get the following functional relation ship:

⎟⎟⎠

⎞⎜⎜⎝

⎛=

−=

Δ=

50

150

0

20

0

2 ,,,,2 D

hZk

di

Dkf

EEE

EEE mr (5.1.3)

Where 1h

zZ m

m = with zm equal to measured medium cross sectional scour depth and D50 equal to

the diameter of the bed material for which 50% of the material is finer and Zm is the dimensionless medium cross sectional scour depth. Here only the parameter Zm and h1/D50 represents the bed condition. Being the dissipation of energy among the section 0 and the section 1, the same and an increase in dissipation for sections 0 and 2 it is concluded that the increase is due to the difference in bed condition. Hence the significance of different bed condition such as smooth fixed bed, rough fixed bed and mobile bed are investigated. Now the comparison of dissipation among sections 1 and 2 among are carried out.

D

Q

xy

1

1

k

h1

0

0LT

h2

2

2

Figure 5.1.37 Representative of the points of measure (section 0, 1 and 2)

26

The influence of the parameter 1

50

hD

is essentially due to the D50 as h1 is at the beginning of the

region of investigation. Hence the influence of D50 is investigated. For this purpose the energy dissipation is compared between section 1 and 2 for fixed smooth bed and fixed rough bed. The downstream conjugate height h2 was calculated using the corrected Belanger equation proposed by Pagliara and Lotti (2007).

( )[ 2

1

21

18115.0 rFhh

⋅+⋅++−⋅= ε ] (5.1.4)

where ahh

ahh

a +⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅−⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅⋅=

1

2

2

1

2 256.1256.0ε (5.1.5)

and 1

6505.0h

Da ⋅−= (5.1.6)

where Fr1 is the Froude number and D65 is material size where 65 % of the material is finer. Here in the above corrected Belanger equation ε depends on the roughness of the bed and it is equal to zero for smooth bed. It is necessary to mention that all the materials used by Paglaira and Lotti (2007) for determining the co-efficient were homogeneous in nature. Then the energy at E1 and E2 were calculated and subsequently the relative energy dissipation between section 1 and 2 were calculated. The energy dissipation was also calculated using the energy dissipation equation (Eq. 5.1.7) of jump for comparison.

( )21

312

21 4 hhhhEE⋅×

−=− (5.1.7)

From the numerical comparison no difference was noticed between the rough and smooth fixed bed. This leads to the conclusion that in the experimental range of D50 (1.8<D50<6.8) there is no difference between smooth fixed and rough fixed bed. This hypothesis was confirmed by other experiments carried out in hydraulic laboratory of the University of Pisa. As for D50<7 (which is similar to the case E2 as shown in figure below) and for Froude number between 1.2-4 the deviation of Belanger equation for different value of ε is negligible.

1

5

9

1 7

Y 2/Y 1

Fr 1

13

Belanger Eq.(1) withE1E2E3E4E5E6E7

e=0

Figure 5.1.38 Comparison among the classical (ε=0) and corrected (ε≠0) Belanger equation

Therefore it can be concluded that in the present experimental range relative energy dissipation is independent of D50.

The next step is to check the dependence of 1h

zZ m

m = on relative energy dissipation between

section 1 and 2. The value of Zm can be determined by the equation proposed by Pagliara and Palermo (2007) and valid for slope range 1V:4H to 1V:12H.

27

8.175.050

58.0 dm FiZ ××= (5.1.8)

where

50

50

d

vFs

d

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

ρρρ

is the densimetric Froude number with v as average flow velocity in

section 1 and sρ , ρ are bed material density and water density respectively. The calculated value of Zm from the aforesaid equation (5.1.8) are plotted as a function with the ratio of relative energy

dissipation for mobile bed ⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −

exp1

21

EEE

( which were obtained from the experimental results) and

relative energy dissipation for fixed bed ⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −

theor

Belanger

EEE

1

21 (obtained by Belanger equation).

0

1

2

0 2 4 6

Z m

((E 1-

E 2)/E

1)ex

p/((E

1-E 2)

/E1)

theo

r

Figure 5.1.39 Influence of Zm on energy dissipation

From the graph it is evident that there is little dependence of Zm on the ratio of relative energy dissipation for mobile bed ( which were obtained from the experimental results) and relative energy dissipation for fixed bed (obtained by Belanger equation). Hence an explicit relation for relative energy dissipation for mobile bed was found.

)(theor

Belangerm E

EEZ

EEE

⎟⎟⎠

⎞⎜⎜⎝

⎛ −⋅+⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −

1

21

exp1

21 31.1074.0 (5.1.9)

Where (5.1.8) 8.175.050

58.0 dm FiZ ××=

The calculated (Eq.5.1.9) and measured value of exp1

21⎟⎟⎠

⎞⎜⎜⎝

⎛ −E

EE is plotted to see the agreement and

confirm the validity of the equation thus derived.

28

0

0,2

0,4

0,6

0,8

1

0 0,2 0,4 0,6 0,8 1

(E 1-E 2)/E 1 (meas)

(E 1-E 2)/E 1 (calc)

Figure 5.1.40 Comparison between calculated (Eq. 5.1.9) and measured values of energy dissipation

A satisfactory agreement can be found which confirms the validity of equation 5.1.9 for mobile bed in the slope range 1V:12H to 1V:4H and bed material size D50<7. Now we have two equation one for relative energy dissipation between section 0 and 1 (Pagliara and Chiavaccini, 2006)

DkiCBeAAE

EE /).(

0

10 )1( ⋅+−+=−

(5.1.1)

Where, Table. 5.1.3 Roughness condition A B C Large Scale roughness (LR) 0.33 -1.3 -14.5 Intermediate Scale roughness (IR) 0.25 -1.2 -12 Small Scale roughness (SR) 0.15 -1.0 -11.5 and other for relative energy dissipation between section 1 and 2 (from the present study)

)(theor

Belangerm E

EEZ

EEE

⎟⎟⎠

⎞⎜⎜⎝

⎛ −⋅+⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −

1

21

exp1

21 31.1074.0 (5.1.9)

29

0

0,2

0,4

0,6

0,8

1

0 0,2 0,4 0,6 0,8 1

(E 0-E 2)/E 0 (meas)

(E 0-E 2)/E 0 (calc)

Figure 5.1.41 Comparison between calculated (indirect method) and measured values

where range of validity of both are same. The value of E1 can be calculated from the first equation

and substituted in the second to get E2 like this the relative energy dissipation ⎟⎟⎠

⎞⎜⎜⎝

⎛ −

0

20

EEE

between

section 0 and 2 can be calculated indirectly. In the following figure a good agreement confirms the

validity of this indirect method of determining ⎟⎟⎠

⎞⎜⎜⎝

⎛ −

0

20

EEE

.

5.1.5 A direct relationship for relative energy dissipation (Er2) A direct method to determine relative energy dissipation at downstream Considering that there is no effect of slope(i), relative roughness condition (d50/k), hi/D50 and also considering the effect of scour for mobile bed as negligible the functional relationship

⎟⎟⎠

⎞⎜⎜⎝

⎛=

Δ=

50

150

0

2 ,,,,2 D

hZk

di

Dkf

EEE mr is reduced to ⎟

⎠⎞

⎜⎝⎛=

ΔDkf

EE

0

2 . Now plotting ⎟⎟⎠

⎞⎜⎜⎝

⎛ −

0

20

EEE

as a

function of k/D we get the following curve.

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,2 0,4 0,6 0,8 1 1,2k/D

(E0-

E 2)/E

0

1V:4H1V:6H1V:8HEq. 5.1.10

Figure 5.1.42 Representation of all energy dissipation data as function of k/D

30

From the figure it is observed that a direct relationship can be obtained and the following relation is proposed:

( ) )(

0

20 1 Dk

BeAA

EEE ⋅

×−+=⎟⎟⎠

⎞⎜⎜⎝

⎛ − (5.1.10)

where A= 0.25 and B= -1.9 This relationship is valid for homogeneous bed material and densimetric Froude number, relative roughness and slope in the range 1.5< <5.5, 0.02<d

50dF 50/k and 1V:8H to 1V:4H. A comparison between measured and calculated [with Eq. (5.1.10)] values of relative energy dissipation between section 0 and 2 are illustrated in Figure 5.1.43.

0,0

0,2

0,4

0,6

0,8

1,0

0,0 0,2 0,4 0,6 0,8 1,0

(E 0-E 2)/E 0 (meas)

(E 0-E 2)/E 0 (calc)

Figure 5.1.43 Comparison between calculated (Eq. 5.1.10) and measured relative energy dissipation values for

free jump between sections 0 and 2 5.2 Analyses of the data for submerged jump In case of submerged jump located at L/LT=1/3, 2/3, 3/3 where LT is the horizontal length of the ramp and L is the distance of the toe of the submerged jump from the toe of the ramp, the dissipation of relative energy has been calculated between section 0 and 2.

Figure 5.2.1 Submerged for L/LT=1/3 with ramp material d1


31


5.2.1 Influence of channel bed material on energy dissipation In case of free jump the dependence of bed material was verified, also in the submerged jump condition the influence of bed material on energy dissipation is investigated in a similar way. As we know two different homogeneous bed materials were used for the mobile bed (D1 with D50 equal to 3.2mm and D2 with D50 equal to 6.8mm) their dependence were checked to see whether they give different results. For this purpose the curve of (E0-E2)/E0 as a function of k/D were drawn. The comparison was done by keeping the ramp material same for different bed materials in different slope condition. The same method is repeated for three submerged condition L/LT=1/3, 2/3, 3/3.

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,2 0,4 0,6 0,8

k/D

(E0-

E 2)/E

0

1:8 Mf1 (sub 1/3)1:6 Mf1 (sub 1/3)1:4 Mf1 (sub 1/3)1:8 Mf2 (sub 1/3)1:6 Mf2 (sub 1/3)1:4 Mf2 (sub 1/3)

Figure 5.2.4 (E0-E2)/E0 as a function of k/D for ramp material d1 and jump submerged (L/LT=1/3 )

32

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,2 0,4 0,6 0,8

k/D

(E0-

E 2)/E

0



0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,2 0,4 0,6 0,8

k/D

(E0-

E 2)/E

0



0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,2 0,4 0,6

k/D

(E0-

E 2)/E

0

0,8



33

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,2 0,4 0,6

k/D

(E0-

E 2)/E

0

0,8



0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,2 0,4 0,6

k/D

(E0-

E 2)/E

0

0,8



0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,2 0,4 0,6 0,8

k/D

(E0-

E 2)/E

0



34

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,2 0,4 0,6 0,8

k/D

(E0-

E 2)/E

0



0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,2 0,4 0,6 0,8

k/D

(E0-

E 2)/E

0



From the figures it is evident that within the range of bed materials used (D50<6.8) there is no dependence of the bed material on relative energy dissipation for submerged jump condition. 5.2.2 Comparison of energy dissipation in submerged jump between presence and absence of scour Now the energy dissipation between section 0 and 2 for submerged ramp is calculated for two different situation. One in absence of scour in bed and other in presence of a scour. Then the values of energy dissipation for presence and absence of scour are compared. The method carried out to collect and compare the data of two situations are described with the help of following figures.

35

2

2

h2

LT0

0

Q k

D yx

L

Tailwater

L

xyD

kQ

0

0LT

h2

2

2

Tailwater

L

D

k

h2h1

LT

xy

0

0

Q

2

2

1

1 Tailwater

L

h2

k

0

0

yx

Q

LT

D

L

h2

k

0

0

yx

Q

LT

D

(a)

(b)

(c)

(d)

(e)

Figure 5.2.13 Diagram sketch for experiments on rough bed (a) submergence L/LT=2/3 (no scour), (b) submergence L/LT=1/3 (no scour), (c) free jump L/LT=0, (d) submergence L/LT=1/3 (with scour) and (e)

submergence L/LT=2/3 (with scour)

At the beginning (when the channel bed material downstream the ramp was levelled), the experiment was carried out for submerged jump where L/LT=2/3 (Figure 5.2.13a) and then for L/LT=1/3 (Figure 5.2.13b) by adjusting the downstream sluice gate. In this two cases no scour occurred in the channel bed for the used bed materials. Then the experiment continued and, adjusting again the sluice gate, a free jump condition was reproduced and a scour occurred in the channel bed, as shown in Figure 5.2.13(c). Then the tailwater was increased gradually. In fact, in presence of scour, the experiments for submerged jump were again carried out for submergence L/LT=1/3 and 2/3 (Figure 5.2.13d-e). In each step all the hydraulic parameters were measured and the discharge was kept constant. The respective ramp submergence configurations were compared to highlight the difference in energy dissipation. The aforesaid comparison is represented in the following diagrams. It should be noted in the following figures that the E2dw indicates the tests where there is no scour whereas E2up indicates the tests with scour on the bed. Each of the following figure compares E2dw with E2up in one submerged situation for one type of ramp roughness (d1, d2 and d3) and varying slope.

36

0

0,02

0,04

0,06

0,08

0,1

0 0,02 0,04 0,06 0,08 0,1

E 0 - E 2up (m)

E 0 - E 2dw (m)

1:8 (sub 3/3)1:6 (sub 3/3)1:4 (sub 3/3)

Figure 5.2.14 Comparison of energy dissipation among two condition of submergence (L/LT=3/3) for ramp

material d1

0

0,02

0,04

0,06

0,08

0,1

0 0,02 0,04 0,06 0,08 0,1

E 0 - E 2up (m)

E 0 - E 2dw (m)

1:8 (sub 2/3)1:6 (sub 2/3)1:4 (sub 2/3)


material d1

0

0,02

0,04

0,06

0,08

0,1

0 0,02 0,04 0,06 0,08 0,1

E 0 - E 2up (m)

E 0 - E 2dw (m)

1:8 (sub 1/3)1:6 (sub 1/3)1:4 (sub 1/3)


material d1

37

0

0,02

0,04

0,06

0,08

0,1

0 0,02 0,04 0,06 0,08 0,1

E 0 - E 2up (m)

E 0 - E 2dw (m)

1:8 (sub 3/3)1:6 (sub 3/3)1:4 (sub 3/3)


material d2

0

0,02

0,04

0,06

0,08

0,1

0 0,02 0,04 0,06 0,08 0,1

E 0 - E 2up (m)

E 0 - E 2dw (m)

1:8 (sub 2/3)1:6 (sub 2/3)1:4 (sub 2/3)


material d2

0

0,02

0,04

0,06

0,08

0,1

0 0,02 0,04 0,06 0,08 0,1

E 0 - E 2up (m)

E 0 - E 2dw (m)

1:8 (sub 1/3)1:6 (sub 1/3)1:4 (sub 1/3)


material d2

38

0

0,02

0,04

0,06

0,08

0,1

0 0,02 0,04 0,06 0,08 0,1

E 0 - E 2up (m)

E 0 - E 2dw (m)

1:8 (sub 3/3)1:6 (sub 3/3)1:4 (sub 3/3)


material d3

0

0,02

0,04

0,06

0,08

0,1

0 0,02 0,04 0,06 0,08 0,1

E 0 - E 2up (m)

E 0 - E 2dw (m)

1:8 (sub 2/3)1:6 (sub 2/3)1:4 (sub 2/3)


material d3

0

0,02

0,04

0,06

0,08

0,1

0 0,02 0,04 0,06 0,08 0,1

E 0 - E 2up (m)

E 0 - E 2dw (m)

1:8 (sub 1/3)1:6 (sub 1/3)1:4 (sub 1/3)


material d3

From the figure it can be seen that there is no appreciable difference whether there is scour in the bed or not i.e. the relative energy dissipation between section 0 and 2 for presence and absence of scour is same in case of submerged jump.

39

5.2.3 Use in the elaborations of additional data Thus now it has been concluded that there is no influence of scour and granulometric material of the bed on energy dissipation for submerged jump. The data of other researchers who carried out experiments on submerged ramp condition in the University of Pisa are also verified if they can be used for the elaboration of the present study. Di Grigoli carried out experiments in submerged block ramp condition for L/LT=1/3 and 2/3 but in his case the bed was smooth and fixed, so his data are compared with the present study to see if there is any difference. A comparison of Di Grigoli and present study data are given in the following figure.

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

k/D

(E0-

E 2)/E

0

Di Grigoli S=1V:8H SR smooth f ixed bedAuthor S=1V:8H SR mobile bedDi Grigoli S=1V:8H IR smooth f ixed bedAuthor S=1V:8H IR mobile bedDi Grigoli S=1V:8H LR smooth f ixed bedAuthor S=1V:8H LR mobile bed

Figure 5.2.23 Energy dissipation as function of k/D in submerged jump (1/3) for ramp slope 1V:8H

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0,0 0,2 0,4 0,6 0,8 1,0 1,2

k/D

(E0-

E 2)/E

0



40

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0,2 0,4 0,6 0,8 1

k/D

(E0-

E 2)/E

0



From the graphs no notable difference can be found between mobile bed (present study) and smooth fixed bed (Di Grigoli data). The same conclusion can be drawn for the case where L/LT=2/3. Therefore the data of Di Grigoli was also utilized for elaboration. 5.2.4 Elaboration of submerged ramp condition data for partial submergence (L/LT=1/3 and 2/3) For submerged jump from dimensional analysis we get the following functional relationship:

⎟⎠⎞

⎜⎝⎛=

−=

kd

iDkf

EEE

Er50

0

20 ,,2

(5.2.1)

To investigate the dependence of k/D, i and d50/k(scale roughness as classified in Table 5.1.1) the value of relative energy dissipation are plotted as a function of k/D. First the slope was kept constant and relative roughness was varied. The following figures shows the significance of parameters k/D and scale roughness.

0%

50%

100%

0,0 0,5 1,0k/D

SRIRLREq.5.2.2 (SR)Eq.5.2.2 (IR)Eq.5.2.2 (LR)

(E 0-E 2)/E 0

Figure 5.2.26 Relative energy dissipation for L/LT=1/3 between sections 0 and 2 and S=1V:8H

41

The relative energy dissipation for submerged ramp where L/LT=1/3 was plotted as a function of k/D with scale roughness as parameter for S=1V:8H as shown in Figure 5.2.26. The dependence of k/D and scale roughness can be seen clearly from the Figure 5.2.26. It is evident that relative energy dissipation decreases with increase in parameter k/D, it means that for the same discharge the energy dissipation is proportional to the height of the ramp and moreover it can be seen from Figure 5.2.26 a trend of the plotted points which depends on scale roughness with maximum energy dissipation takes place for LR followed by IR and SR respectively. It means that having fixed k/D the relative energy dissipation increase when the roughness increases. A relationship of the same form given by Pagliara and Chiavaccini (2006) is proposed here for the submerged ramp.

)/()1( DkB

r eAAE ⋅−+= (5.2.2)

where parameters A and B are functions of scale roughness. The values of A and B for submergence L/LT=1/3 are given in Table 5.2.1a. In Figure 5.2.26 equation (5.2.2) is also plotted for SR, IR and LR to see the fit of the equation with the measured data.

Table 5.2.1a. Values of Parameter of Eq. 5.2.2 for 1/3 Submergence and S=1V:8H Roughness condition A B Small Scale roughness (SR) 0.11 -4.65 Intermediate Scale roughness (IR) 0.15 -4.87 Large Scale roughness (LR) 0.192 -4.65

Similarly the relative energy dissipation for submerged ramp where L/LT=1/3 was plotted as a

function of k/D with scale roughness as parameter for S=1V:6H and S=1V:4H as shown in Figure 5.2.27 and 5.2.28 respectively. Equation 5.2.2 is also plotted by changing the coefficients accordingly for S=1V:6H and S=1V:4H. The coefficients A and B for S=1V:6H and S=1V:4H and for L/LT=1/3 are given in Table 5.2.1b and Table 5.2.1c respectively.

0%

50%

100%

0,0 0,6 1,2k/D


(E 0-E 2)/E 0


42

0%

50%

100%

0,0 0,5 1,0k/D


(E 0-E 2)/E 0


Table 5.2.1b. Values of Parameter of Eq. 5.2.2 for 1/3 Submergence and S=1V:6H Roughness condition A B Small Scale roughness (SR) 0.11 -4.95 Intermediate Scale roughness (IR) 0.16 -5.15 Large Scale roughness (LR) 0.19 -4.8

Table 5.2.1c. Values of Parameter of Eq. 5.2.2 for 1/3 Submergence and S=1V:4H Roughness condition A B Small Scale roughness (SR) 0.08 -5.25 Intermediate Scale roughness (IR) 0.13 -5.3 Large Scale roughness (LR) 0.16 -5

For the same scale roughness almost a same general trend was found for all three different

slopes. The method applied to L/LT=1/3 is again applied to L/LT=2/3 and the values of coefficients A

and B for three slopes 1V:8H, 1V:6H and 1V:4H are furnished in Table 5.2.2a, Table 5.2.2b and Table 5.2.2c respectively.

Table 5.2.2a. Values of Parameter of Eq. 5.2.2 for 2/3 Submergence and S=1V:8H Roughness condition A B Small Scale roughness (SR) 0.055 -8.6 Intermediate Scale roughness (IR) 0.085 -8 Large Scale roughness (LR) 0.11 -7.25

Table 5.2.2a. Values of Parameter of Eq. 5.2.2 for 2/3 Submergence and S=1V:6H Roughness condition A B Small Scale roughness (SR) 0.05 -9 Intermediate Scale roughness (IR) 0.08 -8.5 Large Scale roughness (LR) 0.11 -7.4

43

Table 5.2.2a. Values of Parameter of Eq. 5.2.2 for 2/3 Submergence and S=1V:4H Roughness condition A B Small Scale roughness (SR) 0.05 -9.5 Intermediate Scale roughness (IR) 0.092 -9.1 Large Scale roughness (LR) 0.106 -8

The experimental data obtained for L/LT=2/3 are plotted as a function of k/D with scale roughness as parameter in Figures 5.2.29, 5.2.30 and 5.2.31 for slopes 1V:8H, 1V:6H and 1V:4H. The Equation 5.2.2 valid for submergence L/LT=2/3 with corresponding values of coefficients A and B for scale roughness and slopes are also plotted to show the fitting curve.

0%

50%

100%

0,0 0,5 1,0k/D


(E 0-E 2)/E 0


0%

50%

100%

0 0,6 k/D 1,2


(E 0-E 2)/E 0


44

0%

50%

100%

0,0 0,5 1,0k/D


(E 0-E 2)/E 0


From the observation from Table 5.2.1a-c for L/LT=1/3 it can be seen that the coefficient B for each scale roughness gradually decreases as slope increases. Moreover the coefficient A for respective scale roughness remains almost the same. The same features is also observed for L/LT=2/3 as shown in Table 5.2.2a-c. Hence for coefficient A a mean value is considered for different slopes of respective scale roughness and each value of L/LT. The mean values of coefficient A are shown in Table 5.3.1. The experimental data obtained for L/LT=1/3 are plotted as a function of k/D with slope as parameter in Figures 5.2.32, 5.2.33 and 5.2.34 for different scale roughness SR, IR and LR rrespectively. Equation 5.2.2 is also plotted for corresponding values of coefficient B and with the mean values of coefficient A.

0%

20%

40%

60%

80%

100%

0,0 0,2 0,4 0,6 0,8 1,0 1,2k/D

(E0-

E 2)/E

0

S=1V:4H Author, Di GrigoliS=1V:6H Author, Di GrigoliS=1V:8H Author, Di GrigoliEq.5.2.2 S=1V:4HEq.5.2.2 S=1V:6HEq.5.2.2 S=1V:8H

Figure 5.2.32 Comparison of Eq. 5.2.2 with relative energy dissipation for SR condition for L/LT=1/3

45

0%

20%

40%

60%

80%

100%

0,0 0,2 0,4 0,6 0,8 1,0 1,2k/D

(E0-

E 2)/E

0

S=1V:4H Author, Di GrigoliS=1V:6H Author, Di GrigoliS=1V:8H Author, Di GrigoliEq 5.2.2 S=1V:4HEq 5.2.2 S=1V:6HEq 5.2.2 S=1V:8H

Figure 5.2.33 Comparison of Eq. 5.2.2 with relative energy dissipation for IR condition for L/LT=1/3

0%

20%

40%

60%

80%

100%

0,0 0,2 0,4 0,6 0,8 1,0 1,2k/D

(E0-

E 2)/E

0

S=1V:4H Author, Di GrigoliS=1V:6H Author, Di GrigoliS=1V:8H Author, Di GrigoliEq. 5.2.2 S=1V:4HEq. 5.2.2 S=1V:6HEq. 5.2.2 S=1V:8H

Figure 5.2.34 Comparison of Eq. 5.2.2 with relative energy dissipation for LR condition for L/LT=1/3

From the observation from Table 5.2.1a-c for L/LT=1/3 it is evident that the coefficient B for each scale roughness gradually decreases as slope increases. But from the Figures 5.2.32, 5.2.33 and 5.2.34 it is evident that the influence of slopes is almost negligible, hence also for coefficient B a mean value is considered for different slopes of respective scale roughness (Table 5.3.1). The same features is also observed in case of submergence L/LT=2/3 as can be seen from Figures 5.2.35, 5.2.36 and 5.2.37. Similarly a mean value of coefficient B is considered for different slopes of respective scale roughness in case of submergence L/LT=2/3 (Table 5.3.1).

0%

20%

40%

60%

80%

100%

0,0 0,2 0,4 0,6 0,8 1,0k/D

(E0-

E 2)/E

0


Figure 5.2.35 Comparison of Eq. 5.2.2 with relative energy dissipation for SR condition for L/LT=2/3

46

0%

20%

40%

60%

80%

100%

0,0 0,2 0,4 0,6 0,8 1,0k/D

(E0-

E2)

/E0


Figure 5.2.36 Comparison of Eq. 5.2.2 with relative energy dissipation for IR condition for L/LT=2/3

0%

20%

40%

60%

80%

100%

0,0 0,2 0,4 0,6 0,8 1k/D

(E0-

E2)

/E0

,0


Figure 5.2.37 Comparison of Eq. 5.2.2 with relative energy dissipation for LR condition for L/LT=2/3

5.3 Elaboration to obtain one single relationship for free and submerged jump A mathematical relation is finally obtained which represents the energy dissipation between sections 0 and 2 for free jump and submerged jump as well. The coefficients A and B for free jump (L/LT=0) and submerged jump (L/LT=1/3 and L/LT=2/3) are combined and separated for each scale roughness (small scale, intermediate scale and large scale) and shown in Table 5.3.1. From close observation on the coefficients it can be noted that for free jump the coefficients remains same for all scale roughness and for L/LT= 1/3 and 2/3 they decreases with increasing the ramp submergence values for a fixed scale roughness.

Table. 5.3.1 Coefficients A and B corresponding to L/LT value separated on the basis of roughness condition

Scale roughness Submergence (L/LT) A B 0 0.25 -1.9

1/3 0.1 -4.95 SR 2/3 0.052 -9.033 0 0.25 -1.9

1/3 0.145 -5.106 IR 2/3 0.085 -8.533 0 0.25 -1.9

1/3 0.17 -4.816 LR 2/3 0.109 -7.55

47

The coefficients A and B for each scale roughness are plotted as a function of L/LT in the following figures to see its dependence on the coefficient.

0,0

0,1

0,2

0,3

0,0 0,2 0,4 0,6 0,8L/L T

Coe

ffici

ente

A

Figure 5.3.1 Coefficient A as function of L/LT (Small Scale roughness)

0

2

4

6

8

10

0,0 0,2 0,4 0,6 0,8L/L T

Coe

ffici

ente

B

Figure 5.3.2 Coefficient B as function of L/LT (Small Scale roughness)

0,0

0,1

0,2

0,3

0,0 0,2 0,4 0,6 0,8L/L T

Coe

ffici

ente

A

Figure 5.3.3 Coefficient A as function of L/LT (Intermediate Scale roughness)

48

0

2

4

6

8

10

0,0 0,2 0,4 0,6 0,8L/L T

Coe

ffici

ente

B

Figure 5.3.4 Coefficient B as function of L/LT (Intermediate Scale roughness)

0,0

0,1

0,2

0,3

0,0 0,2 0,4 0,6 0,8L/L T

Coe

ffici

ente

A

Figure 5.3.5 Coefficient A as function of L/LT (Large Scale roughness)

0

2

4

6

8

0,0 0,2 0,4 0,6 0,8L/L T

Coe

ffici

ente

B

Figure 5.3.6 Coefficient B as function of L/LT (Large Scale roughness)

A definite trend of the coefficient A and B for each scale roughness were found and relationships as a function of L/LT of the coefficients for each scale roughness were determined and furnished in Table 5.3.2. Hence the final relationship obtained are as follows: The final equation for relative energy dissipation between section 0 and 2 for free and submerged jump is given by:

DkBeAAE

EE /)(

0

20 )1( ⋅−+=−

(5.3.1)

where A and B are functions of L/LT and is given by:

49

Table. 5.3.2 Expression of coefficient A and B of Eq. 5.3.1 in different scale roughness conditions

Roughness condition A B

Small Scale roughness

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⋅ TLL

e322.2

239.0 ( )729.1/7.10 +⋅− TLL

Intermediate Scale roughness

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⋅ TLL

e618.1

249.0 ( )863.1/95.9 +⋅− TLL

Large Scale roughness

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⋅ TLL

e245.1

253.0 ( )931.1/475.8 +⋅− TLL

By substituting the equations of coefficients A and B from Table 5.3.2 in equation (5.3.1) it is possible to find out the values of relative energy dissipation for three different scale roughness, slopes in the range 1V:8H - 1V:4H, any submergence (L/LT) values in the range 0<L/LT<0.7 and scale roughness SR, IR and LR. In Figure 5.3.7 the comparison between measured and calculated relative energy dissipation is illustrated. It can be seen that the points are generally confined in the range ±20% with respect to the perfect agreement line except for some values of small scale roughness.

0.0

0.5

1.0

0.0 0.5 1.0

(E 0-E 2)/E 0(meas)

SRIRLRPerfect Agreement.

(E 0-E 2)/E 0(calc)

+20%

-20%

Figure 5.3.7 Comparison of measured and calculated relative energy dissipation at various jump location on the ramp

5.4 Elaboration of submerged ramp condition data for complete submergence (3/3)

For a completely submerged ramp by a waved jump located at its top it was not possible to find relationship like the previous ones. In this case the dissipation of energy is not influenced by ramp roughness and slope. But it is only due to the undulating or oscillating jump around critical depth. In fact the experimental data show that energy dissipation varies between 5 and 10 % in all tested roughness and slope conditions of the ramp.

50

0,00

0,20

0,40

0,60

0,80

1,00

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7

k/D

(E 0-E 2)/E 0S=1V:8H SRS=1V:8H IRS=1V:8H LR


0,00

0,20

0,40

0,60

0,80

1,00

0 0,1 0,2 0,3 0,4 0,5 0,6

k/D

(E 0-E 2)/E 0 S=1V:6H SRS=1V:6H IRS=1V:6H LR


0,00

0,20

0,40

0,60

0,80

1,00

0 0,1 0,2 0,3 0,4

k/D

(E 0-E 2)/E 0S=1V:4H SRS=1V:4H IRS=1V:4H LR


51

6. Conclusions

The classical structures used for hydraulic river restorations have many problems. For these reasons in the last few years the understanding of block ramp behaviour has been increased remarkably. This occurrence led to conduct many experimental tests in order to deepen the scour and energy dissipation phenomenon in presence of these type of works and highlight the main parameters on which they depend. In this study the results of some experimental tests on block ramp capacity of dissipating energy are shown. The main results found are summarised in the following points:

• The dependence of energy dissipation on roughness condition and the ratio of critical depth and ramp height (k/D) was confirmed.

• The energy dissipation between entrance of the ramp and downstream section of jump, formed at the toe of the ramp, were investigated in free jump condition and homogeneous ramp material.

• The energy dissipation between entrance of the ramp and downstream section of jump, formed at the toe of the ramp for free jump condition and homogeneous ramp material were investigated separately for mobile (homogeneous material) and fixed smooth channel bed to see the difference.

• The influence of scour on energy dissipation between the toe of the ramp and downstream section of the jump was investigated and a relationship as a function of scour depth was proposed.

• The energy dissipation between entrance of the ramp and downstream section of jump was also investigated in submerged jump condition (L/LT=1/3 and 2/3) and in presence of homogeneous ramp material.

• A new relationship was proposed to calculate the energy dissipation in different submerged condition within a range of submergence from 0<L/LT<0.7, slope from 1V:8H to 1V:4H, k/D ratio between 0<k/D<1.2 and scale roughness SR, IR and LR.

52

7. Notation A, B, C b dxxD50 D, H E0E1 E2

Er1 Er2

50dF

FrFr1Fr2 f g h, huhc, k h1 h2i L LTq Q v zmZmα ΔE ΔE1ΔE2μ ρ σ

Coefficient of equation Width of the channel Ramp material size for which xx% of material is finer Channel bed material size for which 50% of material is finer Ramp height Total energy upstream of ramp (section 0) Total energy at the toe of the ramp (section 1) Total energy at the downstream section of the jump (section 2) Relative energy dissipation (ΔE1/E0) Relative energy dissipation (ΔE2/E0)

Densimetric Froude number

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

50d

v

s

ρρρ

Froude number [=(q/h)/(gh)0.5] Froude number at the toe of the ramp (section 1) [=(q/h1)/(gh1)0.5] Froude number at the downstream section of the ramp (section 2) [=(q/h2)/(gh2)0.5] Darcy-Weisbach friction factor Acceleration due to gravity Uniform flow depth Critical depth Upstream coniugate depth (section 1) Downstream coniugate depth (section 2) Ramp slope Length between toe of the ramp and toe of the submerged jump Horizontal ramp length Unit water discharge Water discharge Average flow velocity in section 1 Measured medium cross sectional scour depth Relative scour depth (zm/h1) Bed slope angle Energy dissipation (E0- E1, E0- E2 and E1- E2) Energy dissipation (E0- E1) Energy dissipation (E0- E2) Water viscocity Water density Coefficient of uniformity of base material (d84/d16)0.5

53

8. Reference

Aberle, J., Dittrich, A., and Nestmann, F. (1999). “Disscussion of ‘Estimation of gravel-bed river flow resistance.’” J. Hydraul. Eng., 125(12), 1315-1317. Afzalimehr, H. and Anctil, F. (1998). “Estimation of gravel-bed river flow resistance”. J. Hydraul. Eng., 124 (10), 1054-1058. Aguirre-Pe, J. and Fuentes, R. (1990). “Resistance to flow in steep rough streams” J. Hydraul. Eng., 116(11), 1374-1387. Bathurst, J.C. (1978). “Flow resistance of large-scale roughness.” J. Hydraul. Div., 104 (12), 1587-1603. Bathurst, J.C. (1985). “Flow resistance estimation in mountain rivers.” J. Hydraul. Eng., 111 (4), 625-643. Bathurst, J.C. (2002). “At-a-site variation and minimum flaw resistance for mountain rivers.” J. Hydrol., 269, 11-26. Bathurst, J.C., Simons, D.B., and Li, R.M. (1981). “Resistance equation for large-scale roughness.” J. Hydraul. Div., 107 (12), 1593-1613. Chamani, M. R., and Rajaratnam, N. (1999). “Onset of skimming flow on stepped spillways.” J. Hydraul. Eng., 125(9), 969-971. Chanson, H. (1994). Hydraulics design of stepped Cascades, Channels, Weirs and Spillways, . Pergamon, Oxford, UK. Colosimo, C.,Copertino, V.A. and Veltri, M. (1988). “Friction factor evaluation in gravel-bed rivers.” J. Hydraul. Eng., 114 (8), 861-876. Christodoulou, G.C. (1993). “Energy dissipation on stepped spillways.” J. Hydraul. Eng., 119(5), 644-650. Diez-Cascon. J., Blanco. J.L., Revilla, J., and Garcia, R, (1991), “Studies on the hydraulic behaviour of stepped spillways.” Int. Water Power Dam Constr., 43(9), 22-26. Di Matto, D. (2004). “Analisi sperimentale del risalto idraulico a valle di rampe in pietrame.” Tesi di laurea, Università Di Pisa, Pisa, Italy. Di Grigoli, M. (2003). “Dissipazione di energia e progettazione di rampe in pietrame.” Tesi di laurea, Università Di Pisa, Pisa, Italy. Essery, I.T.S., and Horner, M. W. (1978). “The hydraulic design of stepped spillways.” Rep. No. 33, CIRIA, London. Ead, S.A. and Rajaratnam, N. (2002) “Hydraulic jump on Corrugated Beds.” J. Hydraul. Eng., 128(7), 656-663 Hager W. H. (1992). Energy dissipators and hydraulic jump, Kluwer Academic Publishers, Dordrecth, The Netherlands. Hey, R. D. (1979), “Flow resistance in gravel-bed rivers.” J. Hydraul. Div., 105 (4), 365-379. Hughes, W. C., and Flack, E.J. (1984). “Hydraulic Jump properties over a rough bed.” J. Hydraul. Eng., 110 (12), 1755-1771. Kindsvarter, C. E. (1944) “The Hydraulic jump in sloping channels.” Trans. ASCE,. Vol. 109, 1107-1154. Mazzoncini, S. (2004). “Analisi sperimentale degli scavi a valle di strutture in pietrame.” Tesi di laurea, Università Di Pisa, Pisa, Italy. Pagliara, S. and Dazzini, D. (2002). “Hydraulics of block ramp for river restoration.” Proc., 2nd Int. Conf. on New Trends in water and environmental engineering for safety and life: eco-compatible solution for aquatic environments, Capri, CSDU, Milano, Italy. Pagliara, S., Chiavaccini, P. (2006) “Energy dissipation on block ramps.” J. Hydraul. Eng., 132(1), 41-48.

54

Pagliara, S., Palermo, M. (2006) “Structural protection downstream of block ramps.” J. Hydraul. Eng., submitted. Pagliara, S., Lotti, I. (2006) “Hydraulic jump on uniform and non homogeneous rough bed.” J. Hydraul. Eng., submitted. Peruginelli, A., and Pagliara, S. (2000). “Energy dissipation comparison among stepped channel, drop and ramp structures.” Proc., Workshop on Hydraulics of stepped Spillways, Balkema, Rotterdam, The Netherlands, 111-118. Peyras, L., Royet, P., and Degoutte, G. (1992). “Flow and energy dissipation over stepped Gabion weirs.” J. Hydraul. Eng., 118(5), 707-717 Rajaratnam, N. (1965). “The hydraulic jump as a wall jet.” J. Hydraul. Div., HY5, 107-132. Rajaratnam, N., (1967). Hydraulic jumps, Department of Civil Engineering, University of Alberta, Edmonton, Canada. Rice, C.E., Kadavy, K.C., and Robinson, K.M. (1998). “Roughness of loose rock riprap on steep slopes.” J. Hydraul. Eng., 124 (2), 179-185. Stephenson, D. (1991). “Energy dissipation down stepped spillways.” Int. Water Power Dam Constr., 43(9), 27-30. Solari, L. (2004). “Scour induced by hydraulic jump: preliminary analysis.” Proc. II Int. Conf. On Scour and Erosion, Singapore. Wu, S., and Rajaratnam, N. (1995). “Free jump, submerged jumps and wall jets.” J. Hydraul. Res., 33(2), 197-212.

55

9. Appendix

Data of free jump (mobile bed)

free jump

(mobile bed)

Scale roughness

Q (m3/s)

K (m)

D (m)

d50 (m)

D50 (m)

i (-)

h1 (m)

h2 (m)

Fr2 (-)

Fd50 (-)

E0 (m)

E1 (m)

E2 (m)

0,005 0,034 0,074 0,001 0,0032 1/8 0,022 0,056 2,005 3,9944 0,1258 0,0648 0,06220,0025 0,022 0,074 0,001 0,0032 1/8 0,017 0,032 1,446 2,5494 0,1068 0,0345 0,03290,008 0,047 0,074 0,001 0,0032 1/8 0,029 0,073 2,076 4,7823 0,1447 0,0908 0,0829

0,0080 0,047 0,074 0,001 0,0068 1/8 0,026 0,058 2,446 4,0759 0,1447 0,1030 0,07350,0054 0,036 0,074 0,001 0,0068 1/8 0,024 0,037 1,885 3,0055 0,1285 0,0656 0,0541

SR

0,0026 0,022 0,074 0,001 0,0068 1/8 0,014 0,022 2,012 2,4601 0,1076 0,0420 0,03320,0025 0,022 0,074 0,008 0,0032 1/8 0,018 0,021 1,294 2,3683 0,1068 0,0334 0,03240,0050 0,034 0,074 0,008 0,0032 1/8 0,025 0,048 1,621 3,4672 0,1258 0,0574 0,05690,0080 0,047 0,074 0,008 0,0032 1/8 0,033 0,062 1,711 4,2026 0,1447 0,0807 0,07510,0080 0,047 0,074 0,008 0,0068 1/8 0,031 0,055 1,888 3,4295 0,1447 0,0853 0,07190,0055 0,036 0,074 0,008 0,0068 1/8 0,025 0,037 1,762 2,8911 0,1291 0,0638 0,0547

IR

0,0026 0,022 0,074 0,008 0,0068 1/8 0,017 0,018 1,477 2,0024 0,1076 0,0357 0,03470,0025 0,022 0,074 0,020 0,0032 1/8 0,025 0,018 0,792 1,7063 0,1068 0,0331 0,03210,0050 0,034 0,074 0,020 0,0032 1/8 0,032 0,038 1,125 2,7172 0,1258 0,0517 0,05050,0080 0,047 0,074 0,020 0,0032 1/8 0,046 0,054 1,026 2,9889 0,1447 0,0703 0,06960,0080 0,047 0,074 0,020 0,0068 1/8 0,038 0,049 1,395 2,8035 0,1447 0,0740 0,07040,0052 0,035 0,074 0,020 0,0068 1/8 0,041 0,048 0,797 1,6719 0,1271 0,0539 0,0505

LR

0,0026 0,022 0,074 0,020 0,0068 1/8 0,026 0,027 0,809 1,3401 0,1076 0,0338 0,03230,0025 0,022 0,099 0,001 0,0032 1/6 0,011 0,033 2,692 3,8582 0,1310 0,0515 0,03780,0054 0,036 0,099 0,001 0,0032 1/6 0,018 0,066 2,892 5,2318 0,1527 0,0920 0,07110,0082 0,048 0,099 0,001 0,0032 1/6 0,027 0,081 2,390 5,2964 0,1700 0,1027 0,08890,0025 0,022 0,099 0,001 0,0068 1/6 0,013 0,044 2,156 2,543 0,1310 0,0430 0,04080,0051 0,035 0,099 0,001 0,0068 1/6 0,020 0,065 2,332 3,398 0,1506 0,0734 0,0702

SR

0,0079 0,046 0,099 0,001 0,0068 1/6 0,028 0,080 2,181 3,7597 0,1683 0,0933 0,08770,0052 0,035 0,099 0,008 0,0032 1/6 0,021 0,058 2,210 4,3183 0,1513 0,0713 0,06480,008 0,047 0,099 0,008 0,0032 1/6 0,028 0,074 2,173 4,9298 0,1689 0,0938 0,0833

0,0026 0,022 0,099 0,008 0,0032 1/6 0,013 0,021 2,269 3,4879 0,1318 0,0458 0,03330,0053 0,036 0,099 0,008 0,0068 1/6 0,020 0,054 2,423 3,5313 0,1520 0,0777 0,06180,008 0,047 0,099 0,008 0,0068 1/6 0,026 0,079 2,468 4,1002 0,1689 0,1037 0,0875

IR

0,0024 0,021 0,099 0,008 0,0068 1/6 0,013 0,017 2,094 2,4601 0,1301 0,0409 0,03270,008 0,047 0,099 0,020 0,0032 1/6 0,033 0,072 1,726 4,2277 0,1689 0,0810 0,0808

0,0051 0,035 0,099 0,020 0,0032 1/6 0,020 0,064 2,332 4,447 0,1506 0,0734 0,06920,0026 0,022 0,099 0,020 0,0032 1/6 0,015 0,024 1,830 3,0228 0,1318 0,0396 0,03340,0051 0,035 0,099 0,020 0,0068 1/6 0,021 0,059 2,167 3,2362 0,1506 0,0694 0,06480,008 0,047 0,099 0,020 0,0068 1/6 0,030 0,076 1,991 3,5535 0,1689 0,0883 0,0851

LR

0,0025 0,022 0,099 0,020 0,0068 1/6 0,016 0,022 1,598 2,0821 0,1310 0,0359 0,03240,0025 0,022 0,146 0,001 0,0032 1/4 0,010 0,018 3,315 4,4329 0,1779 0,0630 0,03350,0042 0,030 0,146 0,001 0,0032 1/4 0,014 0,050 3,291 5,2445 0,1912 0,0884 0,05600,0025 0,022 0,146 0,001 0,0068 1/4 0,010 0,036 3,315 3,3872 0,1779 0,0630 0,0399

SR 0,0051 0,035 0,146 0,001 0,0068 1/4 0,016 0,055 3,051 4,0646 0,1975 0,0932 0,06150,0025 0,022 0,146 0,008 0,0032 1/4 0,010 0,019 3,081 4,2218 0,1779 0,0585 0,03320,0042 0,030 0,146 0,008 0,0032 1/4 0,016 0,034 2,627 4,5134 0,1912 0,0713 0,04620,0025 0,022 0,146 0,008 0,0068 1/4 0,010 0,032 3,171 3,2885 0,1779 0,0602 0,03710,0050 0,034 0,146 0,008 0,0068 1/4 0,016 0,064 2,991 3,9849 0,1969 0,0903 0,0689

IR

0,0072 0,044 0,146 0,008 0,0068 1/4 0,021 0,057 3,093 4,6015 0,2110 0,1189 0,06980,0025 0,022 0,146 0,020 0,0032 1/4 0,013 0,018 2,236 3,4099 0,1779 0,0441 0,03350,0042 0,030 0,146 0,020 0,0032 1/4 0,020 0,041 1,911 3,6506 0,1912 0,0559 0,04950,0025 0,022 0,146 0,020 0,0068 1/4 0,012 0,036 2,522 2,8227 0,1779 0,0487 0,03990,0044 0,031 0,146 0,020 0,0068 1/4 0,017 0,054 2,416 3,3119 0,1927 0,0684 0,0589

Aut

hor

LR

0,0074 0,044 0,146 0,020 0,0068 1/4 0,022 0,076 2,813 4,3592 0,2122 0,1106 0,0836

56


free jump

(mobile bed)

Scale roughness

Q (m3/s)

K (m)

D (m)

d50 (m)

D50 (m)

i (-)

h1 (m)

h2 (m)

Fr2 (-)

Fd50 (-)

E0 (m)

E1 (m)

E2 (m)

0,0068 0,042 0,050 0,004 0,0075 1/12 0,027 0,039 1,964 3,0376 0,1131 0,0788 0,0636SR 0,0082 0,048 0,033 0,004 0,0075 1/12 0,034 0,035 1,676 2,9018 0,1045 0,0810 0,07900,0043 0,031 0,050 0,010 0,0075 1/12 0,023 0,034 1,540 2,213 0,0962 0,0507 0,0467IR 0,0031 0,025 0,050 0,004 0,0075 1/12 0,018 0,033 1,646 2,083 0,0875 0,0424 0,0403

LR 0,0013 0,014 0,050 0,010 0,0019 1/12 0,014 0,021 0,993 2,1549 0,0707 0,0209 - 0,0064 0,040 0,074 0,004 0,0075 1/8 0,022 0,035 2,484 3,4756 0,1350 0,0899 0,06210,0062 0,039 0,074 0,004 0,0075 1/8 0,022 0,036 2,464 3,4155 0,1335 0,0872 0,05960,0041 0,030 0,074 0,004 0,0075 1/8 0,018 0,031 2,203 2,7722 0,1196 0,0610 0,04520,0043 0,031 0,050 0,004 0,0075 1/8 0,021 0,037 1,839 2,4835 0,0958 0,0552 0,0476

SR

0,0081 0,047 0,050 0,004 0,0075 1/8 0,029 0,037 2,102 3,3648 0,1205 0,0924 0,07520,0032 0,026 0,074 0,004 0,0019 1/8 0,016 0,029 2,038 4,7255 0,1129 0,0491 0,03910,0020 0,019 0,074 0,004 0,0019 1/8 0,013 0,023 1,802 3,7157 0,1025 0,0331 0,02910,0043 0,031 0,074 0,010 0,0075 1/8 0,022 0,032 1,635 2,3023 0,1208 0,0521 0,0464

IR 0,0062 0,039 0,074 0,010 0,0075 1/8 0,024 0,033 2,117 3,0868 0,1335 0,0774 0,06090,0015 0,015 0,074 0,010 0,0019 1/8 0,014 0,022 1,180 2,5426 0,0975 0,0234 - LR 0,0023 0,020 0,074 0,010 0,0019 1/8 0,020 0,027 1,023 2,6596 0,1050 0,0306 - 0,0039 0,029 0,146 0,004 0,0075 1/4 0,014 0,026 3,076 3,3833 0,1887 0,0779 0,04380,0064 0,040 0,146 0,004 0,0075 1/4 0,018 0,034 3,410 4,2908 0,2060 0,1213 0,06210,0085 0,049 0,146 0,004 0,0075 1/4 0,025 0,052 2,735 4,0794 0,2189 0,1185 0,07370,0039 0,029 0,146 0,004 0,0075 1/4 0,015 0,024 2,709 3,109 0,1887 0,0691 0,04450,0049 0,034 0,146 0,004 0,0075 1/4 0,016 0,028 3,039 3,6263 0,1959 0,0899 0,0522

SR

0,0044 0,031 0,097 0,004 0,0075 1/4 0,015 0,027 3,093 3,5369 0,1438 0,0850 0,04770,0014 0,015 0,097 0,004 0,0075 1/4 0,008 0,020 2,545 2,1474 0,1194 0,0339 0,02420,0015 0,015 0,146 0,004 0,0019 1/4 0,010 0,020 1,913 3,5089 0,1686 0,0283 0,02460,0052 0,035 0,146 0,010 0,0075 1/4 0,017 0,028 3,010 3,691 0,1984 0,0935 0,05570,0028 0,023 0,146 0,004 0,0075 1/4 0,012 0,034 2,838 2,8953 0,1807 0,0588 0,03940,0020 0,019 0,146 0,004 0,0075 1/4 0,011 0,027 2,110 2,1255 0,1737 0,0368 0,0313

IR

0,0025 0,022 0,146 0,004 0,0075 1/4 0,011 0,026 2,883 2,8129 0,1780 0,0552 0,03350,0011 0,012 0,146 0,010 0,0019 1/4 0,005 0,020 4,345 5,4056 0,1639 0,0480 0,02220,0015 0,016 0,146 0,010 0,0019 1/4 0,006 0,023 3,790 5,5624 0,1689 0,0524 0,02680,0031 0,025 0,146 0,010 0,0075 1/4 0,013 0,029 2,673 2,8636 0,1828 0,0590 0,03790,0021 0,019 0,146 0,010 0,0075 1/4 0,010 0,023 2,788 2,5765 0,1740 0,0469 0,02950,0030 0,025 0,146 0,010 0,0075 1/4 0,012 0,030 3,083 3,1318 0,1824 0,0667 0,0383

LR

0,0020 0,018 0,097 0,010 0,0075 1/4 0,012 0,022 1,980 2,0199 0,1247 0,0346 0,0283SR 0,0061 0,039 0,126 0,004 0,0075 1/3 0,017 0,035 3,458 4,2651 0,1851 0,1193 0,0594

0,0055 0,036 0,126 0,010 0,0075 1/3 0,017 0,029 3,243 3,9411 0,1810 0,1039 0,05790,0039 0,029 0,126 0,010 0,0075 1/3 0,014 0,035 3,097 3,4187 0,1701 0,0794 0,0451IR 0,0020 0,019 0,126 0,004 0,0075 1/3 0,009 0,020 3,237 2,8153 0,1544 0,0530 0,02800,0026 0,022 0,126 0,010 0,0075 1/3 0,011 0,029 2,920 2,8494 0,1593 0,0563 0,0349

Di M

atto

LR 0,0015 0,015 0,126 0,010 0,0019 1/3 0,006 0,023 3,742 5,449 0,1493 0,0504 0,0265

57


free jump

(mobile bed)

Scale roughness

Q (m3/s)

K (m)

D (m)

d50 (m)

D50 (m)

i (-)

h1 (m)

h2 (m)

Fr2 (-)

Fd50 (-)

E0 (m)

E1 (m)

E2 (m)

SR 0,0064 0,040 0,066 0,004 0,0055 1/12 0,026 0,042 1,882 3,2609 0,1270 0,0734 0,06070,0088 0,050 0,066 0,020 0,0059 1/12 0,035 0,046 1,707 2,3275 0,1414 0,0860 0,07540,0091 0,051 0,066 0,020 0,0059 1/12 0,036 0,046 1,692 2,3396 0,1431 0,0875 0,0777IR 0,0098 0,054 0,066 0,020 0,0055 1/12 0,040 0,054 1,530 3,2769 0,1471 0,0879 0,08060,0035 0,027 0,066 0,020 0,0055 1/12 0,027 0,035 0,986 1,7364 0,1071 0,0407 0,04030,0055 0,037 0,066 0,020 0,0055 1/12 0,030 0,038 1,326 2,4572 0,1213 0,0569 0,05490,0070 0,043 0,066 0,020 0,0055 1/12 0,034 0,039 1,423 2,782 0,1305 0,0679 0,0645

LR 0,0053 0,036 0,066 0,020 0,0059 1/12 0,021 0,046 2,165 2,2985 0,1197 0,0709 0,0565

SR 0,0044 0,031 0,099 0,004 0,0055 1/8 0,021 0,030 1,886 2,8762 0,1462 0,0570 0,0471IR 0,0108 0,057 0,099 0,020 0,0059 1/8 0,035 0,060 2,104 2,8636 0,1851 0,1120 0,0861

0,0068 0,042 0,099 0,020 0,0055 1/8 0,027 0,038 1,988 3,4521 0,1623 0,0792 0,06360,0034 0,027 0,099 0,020 0,0055 1/8 0,021 0,031 1,413 2,1907 0,1393 0,0424 0,04070,0050 0,034 0,099 0,020 0,0055 1/8 0,022 0,034 1,906 3,0252 0,1505 0,0626 0,05130,0076 0,045 0,099 0,020 0,0059 1/8 0,024 0,054 2,632 2,9523 0,1669 0,1057 0,0697

LR

0,0051 0,035 0,099 0,020 0,0059 1/8 0,019 0,033 2,563 2,5429 0,1513 0,0794 0,05230,0036 0,027 0,146 0,004 0,0055 1/4 0,013 0,023 3,241 3,8637 0,1867 0,0784 0,04260,0037 0,028 0,146 0,004 0,0059 1/4 0,013 0,026 3,161 2,6316 0,1876 0,0782 0,0424SR 0,0055 0,036 0,146 0,004 0,0059 1/4 0,016 0,031 3,377 3,1262 0,2000 0,1081 0,05570,0022 0,020 0,146 0,004 0,0055 1/4 0,010 0,025 2,943 3,0561 0,1748 0,0507 0,0311IR 0,0015 0,015 0,097 0,004 0,0055 1/4 0,008 0,029 2,529 2,4372 0,1198 0,0344 0,03090,0057 0,037 0,146 0,020 0,0055 1/4 0,025 0,035 1,825 3,0727 0,2016 0,0667 0,05630,0030 0,024 0,146 0,020 0,0055 1/4 0,013 0,030 2,403 2,9723 0,1818 0,0525 0,0379

Maz

zonc

ini

LR 0,0040 0,029 0,146 0,020 0,0055 1/4 0,021 0,035 1,670 2,573 0,1897 0,0501 0,0455

58

Data of free jump (smooth fixed bed)

free jump

(smooth fix bed)

Scale roughness

Q (m3/s)

K (m)

D (m)

d50 (m)

D50 (m)

i (-)

h1 (m)

h2 (m)

Fr2 (-)

Fd50 (-)

E0 (m)

E1 (m)

E2 (m)

0,0067 0,042 0,031 0.001 0 1/8 0,030 0,072 1,658 - 0,0934 0,0705 0,07030,0031 0,025 0,031 0.001 0 1/8 0,017 0,045 1,841 - 0,0682 0,0445 0,04380,0047 0,033 0,050 0.001 0 1/8 0,020 0,047 2,072 - 0,0991 0,0639 0,05510,0018 0,017 0,031 0.001 0 1/8 0,011 0,031 2,032 - 0,0570 0,0331 0,03240,0058 0,038 0,074 0.001 0 1/8 0,023 0,044 2,141 - 0,1310 0,0747 0,05770,0064 0,040 0,074 0.001 0 1/8 0,025 0,083 2,003 - 0,1349 0,0764 0,07560,0012 0,014 0,031 0.001 0 1/8 0,008 0,025 2,371 - 0,0513 0,0290 0,02690,0030 0,024 0,050 0.001 0 1/8 0,016 0,037 1,951 - 0,0862 0,0453 0,04220,0019 0,018 0,050 0.001 0 1/8 0,011 0,024 2,133 - 0,0765 0,0354 0,02870,0014 0,015 0,050 0.001 0 1/8 0,009 0,020 2,006 - 0,0718 0,0280 0,02420,0023 0,020 0,074 0.001 0 1/8 0,012 0,044 2,118 - 0,1048 0,0399 0,03870,0016 0,016 0,074 0.001 0 1/8 0,009 0,025 2,379 - 0,0979 0,0337 0,02830,0067 0,042 0,041 0.001 0 1/6 0,027 0,074 1,935 - 0,1035 0,0770 0,07540,0031 0,025 0,041 0.001 0 1/6 0,015 0,048 2,235 - 0,0783 0,0507 0,05010,0047 0,033 0,066 0.001 0 1/6 0,018 0,050 2,441 - 0,1152 0,0724 0,05720,0018 0,017 0,041 0.001 0 1/6 0,010 0,034 2,351 - 0,0671 0,0369 0,03640,0012 0,014 0,041 0.001 0 1/6 0,007 0,025 2,802 - 0,0614 0,0335 0,02710,0058 0,038 0,099 0.001 0 1/6 0,021 0,079 2,441 - 0,1552 0,0828 0,08000,0064 0,040 0,099 0.001 0 1/6 0,023 0,083 2,280 - 0,1591 0,0839 0,08260,0030 0,024 0,066 0.001 0 1/6 0,013 0,035 2,450 - 0,1023 0,0536 0,04060,0019 0,018 0,066 0.001 0 1/6 0,010 0,022 2,307 - 0,0926 0,0375 0,02800,0014 0,015 0,066 0.001 0 1/6 0,009 0,016 2,256 - 0,0879 0,0305 0,02230,0023 0,020 0,099 0.001 0 1/6 0,011 0,043 2,437 - 0,1290 0,0445 0,04330,0016 0,016 0,099 0.001 0 1/6 0,009 0,026 2,379 - 0,1221 0,0337 0,02870,0067 0,042 0,061 0.001 0 1/4 0,023 0,082 2,466 - 0,1231 0,0921 0,08710,0031 0,025 0,061 0.001 0 1/4 0,012 0,051 2,969 - 0,0978 0,0649 0,05350,0047 0,033 0,097 0.001 0 1/4 0,015 0,055 3,262 - 0,1465 0,0948 0,06060,0058 0,038 0,146 0.001 0 1/4 0,017 0,082 3,363 - 0,2021 0,1118 0,08620,0064 0,040 0,146 0.001 0 1/4 0,019 0,090 3,146 - 0,2060 0,1118 0,09390,0018 0,017 0,061 0.001 0 1/4 0,009 0,037 2,810 - 0,0866 0,0431 0,03860,0012 0,014 0,061 0.001 0 1/4 0,006 0,025 3,298 - 0,0809 0,0393 0,02660,0030 0,024 0,097 0.001 0 1/4 0,012 0,034 3,003 - 0,1336 0,0645 0,04020,0023 0,020 0,146 0.001 0 1/4 0,009 0,036 3,499 - 0,1759 0,0627 0,03940,0019 0,018 0,097 0.001 0 1/4 0,009 0,023 2,649 - 0,1239 0,0421 0,02820,0014 0,015 0,097 0.001 0 1/4 0,007 0,018 2,884 - 0,1192 0,0377 0,02290,0016 0,016 0,146 0.001 0 1/4 0,007 0,028 3,426 - 0,1690 0,0474 0,03030,0065 0,041 0,031 0,002 0 1/8 0,026 0,075 1,987 - 0,0922 0,0767 0,07590,0079 0,046 0,031 0,002 0 1/8 0,034 0,078 1,632 - 0,1007 0,0781 0,07730,0066 0,041 0,031 0,002 0 1/8 0,029 0,069 1,689 - 0,0925 0,0701 0,06980,0048 0,033 0,031 0,002 0 1/8 0,021 0,060 1,977 - 0,0810 0,0625 0,06020,0038 0,028 0,031 0,002 0 1/8 0,018 0,048 1,988 - 0,0737 0,0536 0,0511

Di G

rigol

i

SR

0,0029 0,024 0,031 0,002 0 1/8 0,014 0,044 2,097 - 0,0664 0,0461 0,0455

59


free jump

(smooth fix bed)

Scale roughness

Q (m3/s)

K (m)

D (m)

d50 (m)

D50 (m)

i (-)

h1 (m)

h2 (m)

Fr2 (-)

Fd50 (-)

E0 (m)

E1 (m)

E2 (m)

0,0019 0,018 0,031 0,002 0 1/8 0,011 0,035 2,054 - 0,0579 0,0345 0,03310,0061 0,039 0,074 0,002 0 1/8 0,022 0,079 2,283 - 0,1328 0,0809 0,08080,0044 0,031 0,074 0,002 0 1/8 0,016 0,060 2,735 - 0,1215 0,0761 0,06460,0015 0,015 0,031 0,002 0 1/8 0,010 0,028 1,933 - 0,0538 0,0281 0,02730,0030 0,024 0,074 0,002 0 1/8 0,022 0,048 1,177 - 0,1107 0,0367 0,03490,0019 0,018 0,074 0,002 0 1/8 0,012 0,034 1,797 - 0,1013 0,0318 0,03060,0079 0,046 0,041 0,002 0 1/6 0,029 0,080 1,996 - 0,1108 0,0876 0,08750,0065 0,041 0,041 0,002 0 1/6 0,024 0,078 2,222 - 0,1023 0,0831 0,08400,0066 0,041 0,041 0,002 0 1/6 0,025 0,071 2,112 - 0,1026 0,0804 0,07810,0048 0,033 0,041 0,002 0 1/6 0,019 0,063 2,378 - 0,0911 0,0716 0,06800,0038 0,028 0,041 0,002 0 1/6 0,016 0,051 2,276 - 0,0838 0,0591 0,05560,0029 0,024 0,041 0,002 0 1/6 0,013 0,048 2,547 - 0,0765 0,0537 0,05040,0019 0,018 0,041 0,002 0 1/6 0,010 0,034 2,420 - 0,0680 0,0391 0,03610,0061 0,039 0,099 0,002 0 1/6 0,020 0,085 2,635 - 0,1570 0,0912 0,08920,0015 0,015 0,041 0,002 0 1/6 0,009 0,030 2,196 - 0,0639 0,0307 0,03170,0044 0,031 0,099 0,002 0 1/6 0,016 0,063 2,723 - 0,1457 0,0758 0,06660,0030 0,024 0,099 0,002 0 1/6 0,013 0,041 2,693 - 0,1349 0,0578 0,04490,0019 0,018 0,099 0,002 0 1/6 0,009 0,032 2,966 - 0,1255 0,0470 0,03500,0079 0,046 0,061 0,002 0 1/4 0,026 0,086 2,401 - 0,1303 0,1006 0,09260,0065 0,041 0,061 0,002 0 1/4 0,021 0,085 2,726 - 0,1218 0,0985 0,08970,0066 0,041 0,061 0,002 0 1/4 0,022 0,078 2,543 - 0,1221 0,0931 0,08330,0048 0,033 0,061 0,002 0 1/4 0,016 0,067 2,936 - 0,1106 0,0863 0,07100,0038 0,028 0,061 0,002 0 1/4 0,014 0,055 2,962 - 0,1033 0,0743 0,05890,0029 0,024 0,061 0,002 0 1/4 0,011 0,050 3,120 - 0,0960 0,0648 0,05290,0030 0,024 0,146 0,002 0 1/4 0,047 0,054 0,372 - 0,1818 0,0500 0,05600,0019 0,018 0,061 0,002 0 1/4 0,009 0,034 2,861 - 0,0875 0,0453 0,03680,0061 0,039 0,146 0,002 0 1/4 0,018 0,091 3,206 - 0,2039 0,1099 0,09490,0015 0,015 0,061 0,002 0 1/4 0,008 0,023 2,858 - 0,0834 0,0384 0,02660,0044 0,031 0,146 0,002 0 1/4 0,014 0,064 3,253 - 0,1926 0,0899 0,0680

Di G

rigol

i

SR

0,0019 0,018 0,146 0,002 0 1/4 0,016 0,040 1,212 - 0,1724 0,0274 0,0419

60


free jump

(smooth fix bed)

Scale roughness

Q (m3/s)

K (m)

D (m)

d50 (m)

D50 (m)

i (-)

h1 (m)

h2 (m)

Fr2 (-)

Fd50 (-)

E0 (m)

E1 (m)

E2 (m)

0,0066 0,041 0,050 0,008 0 1/8 0,027 0,071 1,903 - 0,1111 0,0751 0,07760,0060 0,038 0,050 0,008 0 1/8 0,024 0,068 1,981 - 0,1073 0,0723 0,07400,0048 0,034 0,050 0,008 0 1/8 0,021 0,057 1,996 - 0,0999 0,0633 0,06260,0040 0,030 0,050 0,008 0 1/8 0,018 0,051 2,081 - 0,0940 0,0575 0,05630,0057 0,037 0,074 0,008 0 1/8 0,025 0,071 1,887 - 0,1306 0,0683 0,07640,0053 0,035 0,074 0,008 0 1/8 0,023 0,064 1,938 - 0,1275 0,0655 0,06980,0037 0,028 0,074 0,008 0 1/8 0,018 0,050 2,002 - 0,1165 0,0530 0,05460,0025 0,022 0,074 0,008 0 1/8 0,014 0,037 2,000 - 0,1069 0,0409 0,04100,0063 0,040 0,041 0,008 0 1/6 0,027 0,072 1,774 - 0,1010 0,0701 0,07850,0046 0,033 0,041 0,008 0 1/6 0,020 0,057 2,040 - 0,0900 0,0624 0,06200,0066 0,041 0,066 0,008 0 1/6 0,026 0,070 1,952 - 0,1273 0,0763 0,07730,0060 0,038 0,066 0,008 0 1/6 0,024 0,068 2,024 - 0,1235 0,0733 0,07440,0048 0,034 0,066 0,008 0 1/6 0,020 0,060 2,130 - 0,1161 0,0662 0,06490,0040 0,030 0,066 0,008 0 1/6 0,017 0,051 2,317 - 0,1101 0,0622 0,05600,0021 0,019 0,041 0,008 0 1/6 0,012 0,037 2,034 - 0,0700 0,0368 0,03990,0057 0,037 0,099 0,008 0 1/6 0,021 0,073 2,344 - 0,1548 0,0796 0,07820,0017 0,016 0,041 0,008 0 1/6 0,010 0,034 2,000 - 0,0656 0,0309 0,03630,0053 0,035 0,099 0,008 0 1/6 0,020 0,067 2,325 - 0,1517 0,0746 0,07170,0037 0,028 0,099 0,008 0 1/6 0,015 0,045 2,468 - 0,1407 0,0621 0,05050,0025 0,022 0,099 0,008 0 1/6 0,012 0,041 2,569 - 0,1311 0,0497 0,04400,0063 0,040 0,061 0,008 0 1/4 0,024 0,077 2,114 - 0,1205 0,0784 0,08240,0046 0,033 0,061 0,008 0 1/4 0,018 0,060 2,346 - 0,1095 0,0692 0,06500,0066 0,041 0,097 0,008 0 1/4 0,023 0,079 2,396 - 0,1585 0,0886 0,08430,0060 0,038 0,097 0,008 0 1/4 0,021 0,071 2,499 - 0,1547 0,0861 0,07640,0048 0,034 0,097 0,008 0 1/4 0,017 0,056 2,686 - 0,1473 0,0799 0,06220,0040 0,030 0,097 0,008 0 1/4 0,015 0,053 2,812 - 0,1414 0,0736 0,05770,0057 0,037 0,146 0,008 0 1/4 0,018 0,081 3,057 - 0,2017 0,1010 0,08480,0053 0,035 0,146 0,008 0 1/4 0,016 0,072 3,167 - 0,1986 0,0986 0,07580,0034 0,026 0,146 0,008 0 1/4 0,012 0,057 3,277 - 0,1852 0,0764 0,0600

Di G

rigol

i

IR

0,0025 0,022 0,146 0,008 0 1/4 0,010 0,044 3,165 - 0,1780 0,0604 0,0464

61


free jump

(smooth fix bed)

Scale roughness

Q (m3/s)

K (m)

D (m)

d50 (m)

D50 (m)

i (-)

h1 (m)

h2 (m)

Fr2 (-)

Fd50 (-)

E0 (m)

E1 (m)

E2 (m)

0,0021 0,019 0,061 0,008 0 1/4 0,010 0,040 2,798 - 0,0895 0,0477 0,04200,0017 0,016 0,061 0,008 0 1/4 0,008 0,034 2,977 - 0,0852 0,0429 0,03540,0070 0,043 0,050 0,020 0 1/8 0,030 0,079 1,724 - 0,1139 0,0741 0,07390,0059 0,038 0,050 0,020 0 1/8 0,026 0,067 1,758 - 0,1071 0,0669 0,06450,0048 0,033 0,050 0,020 0 1/8 0,022 0,054 1,806 - 0,0996 0,0591 0,05830,0068 0,042 0,074 0,020 0 1/8 0,027 0,077 1,959 - 0,1373 0,0782 0,07500,0036 0,028 0,050 0,020 0 1/8 0,018 0,046 1,938 - 0,0911 0,0512 0,05060,0061 0,039 0,074 0,020 0 1/8 0,025 0,069 1,936 - 0,1328 0,0720 0,07020,0051 0,035 0,074 0,020 0 1/8 0,020 0,056 2,229 - 0,1264 0,0707 0,06260,0043 0,031 0,074 0,020 0 1/8 0,018 0,051 2,196 - 0,1205 0,0621 0,05660,0064 0,040 0,041 0,020 0 1/6 0,031 0,072 1,461 - 0,1015 0,0647 0,06170,0070 0,043 0,066 0,020 0 1/6 0,026 0,078 2,079 - 0,1300 0,0832 0,08070,0059 0,038 0,066 0,020 0 1/6 0,024 0,068 2,055 - 0,1232 0,0737 0,07270,0048 0,033 0,066 0,020 0 1/6 0,020 0,056 2,101 - 0,1157 0,0651 0,06200,0036 0,028 0,066 0,020 0 1/6 0,018 0,047 1,922 - 0,1073 0,0510 0,05010,0061 0,039 0,099 0,020 0 1/6 0,023 0,074 2,229 - 0,1570 0,0795 0,07940,0068 0,042 0,099 0,020 0 1/6 0,024 0,074 2,304 - 0,1615 0,0879 0,08040,0051 0,035 0,099 0,020 0 1/6 0,019 0,061 2,405 - 0,1506 0,0751 0,06680,0043 0,031 0,099 0,020 0 1/6 0,019 0,054 2,134 - 0,1447 0,0608 0,05910,0075 0,045 0,061 0,020 0 1/4 0,030 0,083 1,852 - 0,1279 0,0808 0,08060,0064 0,040 0,061 0,020 0 1/4 0,025 0,072 2,059 - 0,1211 0,0777 0,07640,0070 0,043 0,097 0,020 0 1/4 0,024 0,085 2,334 - 0,1613 0,0907 0,09020,0059 0,038 0,097 0,020 0 1/4 0,021 0,072 2,545 - 0,1545 0,0871 0,07770,0048 0,033 0,097 0,020 0 1/4 0,017 0,059 2,682 - 0,1470 0,0793 0,06450,0068 0,042 0,146 0,020 0 1/4 0,020 0,085 3,085 - 0,2084 0,1140 0,08970,0036 0,028 0,097 0,020 0 1/4 0,013 0,051 3,001 - 0,1385 0,0732 0,05500,0061 0,039 0,146 0,020 0 1/4 0,018 0,076 3,232 - 0,2039 0,1108 0,08090,0039 0,029 0,112 0,020 0 1/4 0,015 0,049 2,640 - 0,1552 0,0679 0,05420,0043 0,031 0,146 0,020 0 1/4 0,014 0,053 3,419 - 0,1916 0,0927 0,0584

Di G

rigol

i

LR

0,0051 0,035 0,146 0,020 0 1/4 0,019 0,059 2,521 - 0,1975 0,0781 0,0648

62

Submerged data (Author down)

Author (dw)

Scale roughness

Q (m3/s)

K (m)

D (m)

d50 (m)

D50 (m)

i (-)

h1 (m)

h2 (m)

Fr2 (-)

Fd50 (-)

E0 (m)

E1 (m)

E2 (m)

0,0050 0,034 0,074 0,001 0,0032 1/8 - 0,093 0,222 - 0,1258 - 0,09560,0026 0,022 0,074 0,001 0,0068 1/8 - 0,073 0,168 - 0,1076 - 0,0736SR 0,0025 0,022 0,074 0,001 0,0032 1/8 - 0,068 0,179 - 0,1068 - 0,06910,0080 0,047 0,074 0,008 0,0032 1/8 - 0,106 0,294 - 0,1447 - 0,11060,0080 0,047 0,074 0,008 0,0068 1/8 - 0,108 0,287 - 0,1447 - 0,11210,0055 0,036 0,074 0,008 0,0068 1/8 - 0,094 0,243 - 0,1291 - 0,09660,0050 0,034 0,074 0,008 0,0032 1/8 - 0,089 0,237 - 0,1258 - 0,09180,0026 0,022 0,074 0,008 0,0068 1/8 - 0,064 0,205 - 0,1076 - 0,0650

IR

0,0025 0,022 0,074 0,008 0,0032 1/8 - 0,070 0,171 - 0,1068 - 0,07090,0080 0,047 0,074 0,020 0,0068 1/8 - 0,106 0,293 - 0,1447 - 0,11080,0050 0,034 0,074 0,020 0,0032 1/8 - 0,084 0,262 - 0,1258 - 0,08660,0026 0,022 0,074 0,020 0,0068 1/8 - 0,067 0,189 - 0,1076 - 0,0685

LR 0,0025 0,022 0,074 0,020 0,0032 1/8 - 0,066 0,187 - 0,1068 - 0,06720,0079 0,046 0,099 0,001 0,0068 1/6 - 0,125 0,228 - 0,1683 - 0,12780,0054 0,036 0,099 0,001 0,0032 1/6 - 0,106 0,198 - 0,1527 - 0,10830,0051 0,035 0,099 0,001 0,0068 1/6 - 0,104 0,192 - 0,1506 - 0,10620,0025 0,022 0,099 0,001 0,0032 1/6 - 0,075 0,155 - 0,1310 - 0,0758

SR

0,0025 0,022 0,099 0,001 0,0068 1/6 - 0,079 0,144 - 0,1310 - 0,07940,0080 0,047 0,099 0,008 0,0032 1/6 - 0,120 0,245 - 0,1689 - 0,12310,0080 0,047 0,099 0,008 0,0068 1/6 - 0,118 0,249 - 0,1689 - 0,12190,0053 0,036 0,099 0,008 0,0068 1/6 - 0,106 0,194 - 0,1520 - 0,10820,0052 0,035 0,099 0,008 0,0032 1/6 - 0,099 0,211 - 0,1513 - 0,10130,0026 0,022 0,099 0,008 0,0032 1/6 - 0,076 0,157 - 0,1318 - 0,0770

IR

0,0024 0,021 0,099 0,008 0,0068 1/6 - 0,073 0,154 - 0,1301 - 0,07390,0080 0,047 0,099 0,020 0,0032 1/6 - 0,119 0,247 - 0,1689 - 0,12260,0080 0,047 0,099 0,020 0,0068 1/6 - 0,121 0,242 - 0,1689 - 0,12400,0051 0,035 0,099 0,020 0,0032 1/6 - 0,101 0,203 - 0,1506 - 0,10270,0051 0,035 0,099 0,020 0,0068 1/6 - 0,100 0,204 - 0,1506 - 0,10210,0026 0,022 0,099 0,020 0,0032 1/6 - 0,082 0,140 - 0,1318 - 0,0828

LR

0,0025 0,022 0,099 0,020 0,0068 1/6 - 0,076 0,150 - 0,1310 - 0,07730,0042 0,030 0,146 0,001 0,0068 1/4 - 0,111 0,144 - 0,1912 - 0,1121SR 0,0025 0,022 0,146 0,001 0,0032 1/4 - 0,097 0,105 - 0,1779 - 0,09750,0050 0,034 0,146 0,008 0,0068 1/4 - 0,113 0,167 - 0,1969 - 0,11460,0042 0,030 0,146 0,008 0,0032 1/4 - 0,114 0,138 - 0,1912 - 0,11510,0025 0,022 0,146 0,008 0,0032 1/4 - 0,100 0,100 - 0,1779 - 0,1003

IR 0,0025 0,022 0,146 0,008 0,0068 1/4 - 0,092 0,114 - 0,1779 - 0,09220,0074 0,044 0,146 0,020 0,0068 1/4 - 0,142 0,175 - 0,2122 - 0,14420,0044 0,031 0,146 0,020 0,0068 1/4 - 0,113 0,147 - 0,1927 - 0,11390,0042 0,030 0,146 0,020 0,0032 1/4 - 0,109 0,148 - 0,1912 - 0,11000,0025 0,022 0,146 0,020 0,0032 1/4 - 0,088 0,121 - 0,1779 - 0,0886

1/3

subm

erge

d

LR

0,0025 0,022 0,146 0,020 0,0068 1/4 - 0,096 0,106 - 0,1779 - 0,0970

63


Author (dw)

Scale roughness

Q (m3/s)

K (m)

D (m)

d50 (m)

D50 (m)

i (-)

h1 (m)

h2 (m)

Fr2 (-)

Fd50 (-)

E0 (m)

E1 (m)

E2 (m)

0,0080 0,047 0,074 0,001 0,0068 1/8 - 0,121 0,241 - 0,1447 - 0,1244SR 0,0050 0,034 0,074 0,001 0,0032 1/8 - 0,110 0,174 - 0,1258 - 0,11150,0080 0,047 0,074 0,008 0,0032 1/8 - 0,126 0,227 - 0,1447 - 0,12900,0080 0,047 0,074 0,008 0,0068 1/8 - 0,132 0,213 - 0,1447 - 0,13450,0050 0,034 0,074 0,008 0,0032 1/8 - 0,105 0,187 - 0,1258 - 0,10640,0026 0,022 0,074 0,008 0,0068 1/8 - 0,084 0,136 - 0,1076 - 0,0844

IR

0,0025 0,022 0,074 0,008 0,0032 1/8 - 0,081 0,138 - 0,1068 - 0,08170,0080 0,047 0,074 0,020 0,0032 1/8 - 0,123 0,235 - 0,1447 - 0,12640,0080 0,047 0,074 0,020 0,0068 1/8 - 0,119 0,248 - 0,1447 - 0,12240,0052 0,035 0,074 0,020 0,0068 1/8 - 0,107 0,188 - 0,1271 - 0,10910,0050 0,034 0,074 0,020 0,0032 1/8 - 0,100 0,200 - 0,1258 - 0,10220,0026 0,022 0,074 0,020 0,0068 1/8 - 0,081 0,144 - 0,1076 - 0,0815

LR

0,0025 0,022 0,074 0,020 0,0032 1/8 - 0,081 0,139 - 0,1068 - 0,08130,0082 0,048 0,099 0,001 0,0032 1/6 - 0,149 0,181 - 0,1700 - 0,15140,0079 0,046 0,099 0,001 0,0068 1/6 - 0,147 0,177 - 0,1683 - 0,14950,0025 0,022 0,099 0,001 0,0032 1/6 - 0,098 0,103 - 0,1310 - 0,0984

SR 0,0025 0,022 0,099 0,001 0,0068 1/6 - 0,099 0,102 - 0,1310 - 0,09920,0080 0,047 0,099 0,008 0,0032 1/6 - 0,149 0,176 - 0,1689 - 0,15170,0080 0,047 0,099 0,008 0,0068 1/6 - 0,148 0,179 - 0,1689 - 0,14990,0053 0,036 0,099 0,008 0,0068 1/6 - 0,131 0,142 - 0,1520 - 0,13200,0052 0,035 0,099 0,008 0,0032 1/6 - 0,123 0,153 - 0,1513 - 0,12430,0026 0,022 0,099 0,008 0,0032 1/6 - 0,108 0,093 - 0,1318 - 0,1086

IR

0,0024 0,021 0,099 0,008 0,0068 1/6 - 0,102 0,093 - 0,1301 - 0,10240,0080 0,047 0,099 0,020 0,0032 1/6 - 0,141 0,191 - 0,1689 - 0,14390,0080 0,047 0,099 0,020 0,0068 1/6 - 0,139 0,195 - 0,1689 - 0,14190,0051 0,035 0,099 0,020 0,0032 1/6 - 0,122 0,152 - 0,1506 - 0,1230

LR 0,0051 0,035 0,099 0,020 0,0068 1/6 - 0,118 0,159 - 0,1506 - 0,11950,0051 0,035 0,146 0,001 0,0068 1/4 - 0,146 0,116 - 0,1975 - 0,14700,0042 0,030 0,146 0,001 0,0032 1/4 - 0,144 0,097 - 0,1912 - 0,14480,0025 0,022 0,146 0,001 0,0032 1/4 - 0,132 0,066 - 0,1779 - 0,1323

SR 0,0025 0,022 0,146 0,001 0,0068 1/4 - 0,127 0,070 - 0,1779 - 0,12730,0042 0,030 0,146 0,008 0,0032 1/4 - 0,146 0,096 - 0,1912 - 0,14620,0025 0,022 0,146 0,008 0,0068 1/4 - 0,131 0,067 - 0,1780 - 0,1312IR 0,0025 0,022 0,146 0,008 0,0032 1/4 - 0,141 0,060 - 0,1779 - 0,14130,0074 0,045 0,146 0,020 0,0068 1/4 - 0,172 0,131 - 0,2125 - 0,17350,0044 0,032 0,146 0,020 0,0068 1/4 - 0,137 0,110 - 0,1929 - 0,13750,0042 0,030 0,146 0,020 0,0068 1/4 - 0,147 0,094 - 0,1912 - 0,14770,0025 0,022 0,146 0,020 0,0032 1/4 - 0,123 0,074 - 0,1780 - 0,1232

2/3

subm

erge

d

LR

0,0025 0,022 0,146 0,020 0,0032 1/4 - 0,130 0,067 - 0,1779 - 0,1307

64


Author (dw)

Scale roughness

Q (m3/s)

K (m)

D (m)

d50 (m)

D50 (m)

i (-)

h1 (m)

h2 (m)

Fr2 (-)

Fd50 (-)

E0 (m)

E1 (m)

E2 (m)

0,0080 0,047 0,074 0,001 0,0032 1/8 - 0,135 0,203 - 0,1447 - 0,13820,0080 0,047 0,074 0,001 0,0068 1/8 - 0,136 0,202 - 0,1447 - 0,13870,0054 0,036 0,074 0,001 0,0068 1/8 - 0,118 0,168 - 0,1285 - 0,11990,0050 0,034 0,074 0,001 0,0032 1/8 - 0,120 0,153 - 0,1258 - 0,12120,0026 0,022 0,074 0,001 0,0068 1/8 - 0,105 0,097 - 0,1076 - 0,1055

SR

0,0025 0,022 0,074 0,001 0,0032 1/8 - 0,105 0,093 - 0,1068 - 0,10570,0080 0,047 0,074 0,008 0,0032 1/8 - 0,136 0,202 - 0,1447 - 0,13890,0080 0,047 0,074 0,008 0,0068 1/8 - 0,137 0,199 - 0,1447 - 0,14000,0055 0,036 0,074 0,008 0,0068 1/8 - 0,124 0,160 - 0,1291 - 0,12560,0050 0,034 0,074 0,008 0,0032 1/8 - 0,120 0,152 - 0,1258 - 0,12140,0026 0,022 0,074 0,008 0,0068 1/8 - 0,103 0,100 - 0,1076 - 0,1035

IR

0,0025 0,022 0,074 0,008 0,0032 1/8 - 0,102 0,097 - 0,1068 - 0,10250,0080 0,047 0,074 0,020 0,0032 1/8 - 0,142 0,190 - 0,1447 - 0,14450,0080 0,047 0,074 0,020 0,0068 1/8 - 0,142 0,189 - 0,1447 - 0,14460,0052 0,035 0,074 0,020 0,0068 1/8 - 0,126 0,148 - 0,1271 - 0,12690,0050 0,034 0,074 0,020 0,0032 1/8 - 0,124 0,145 - 0,1258 - 0,12510,0026 0,022 0,074 0,020 0,0068 1/8 - 0,105 0,096 - 0,1076 - 0,1059

LR

0,0025 0,022 0,074 0,020 0,0032 1/8 - 0,106 0,092 - 0,1068 - 0,10600,0082 0,048 0,099 0,001 0,0032 1/6 - 0,155 0,171 - 0,1700 - 0,15680,0079 0,046 0,099 0,001 0,0068 1/6 - 0,155 0,164 - 0,1683 - 0,15700,0054 0,036 0,099 0,001 0,0032 1/6 - 0,135 0,139 - 0,1527 - 0,13590,0051 0,035 0,099 0,001 0,0068 1/6 - 0,133 0,134 - 0,1506 - 0,13370,0025 0,022 0,099 0,001 0,0032 1/6 - 0,120 0,076 - 0,1310 - 0,1204

SR

0,0025 0,022 0,099 0,001 0,0068 1/6 - 0,119 0,078 - 0,1310 - 0,11890,0080 0,047 0,099 0,008 0,0032 1/6 - 0,160 0,159 - 0,1689 - 0,16160,0080 0,047 0,099 0,008 0,0068 1/6 - 0,159 0,160 - 0,1689 - 0,16100,0053 0,036 0,099 0,008 0,0068 1/6 - 0,139 0,129 - 0,1520 - 0,14060,0052 0,035 0,099 0,008 0,0032 1/6 - 0,136 0,131 - 0,1513 - 0,13740,0026 0,022 0,099 0,008 0,0032 1/6 - 0,126 0,074 - 0,1318 - 0,1262

IR

0,0024 0,021 0,099 0,008 0,0068 1/6 - 0,124 0,070 - 0,1301 - 0,12420,0080 0,047 0,099 0,020 0,0032 1/6 - 0,147 0,179 - 0,1689 - 0,14960,0080 0,047 0,099 0,020 0,0068 1/6 - 0,151 0,173 - 0,1689 - 0,15310,0051 0,035 0,099 0,020 0,0032 1/6 - 0,132 0,135 - 0,1506 - 0,13300,0051 0,035 0,099 0,020 0,0068 1/6 - 0,128 0,141 - 0,1506 - 0,12910,0026 0,022 0,099 0,020 0,0032 1/6 - 0,122 0,077 - 0,1318 - 0,1224

LR

0,0025 0,022 0,099 0,020 0,0068 1/6 - 0,125 0,072 - 0,1310 - 0,12530,0051 0,035 0,146 0,001 0,0068 1/4 - 0,172 0,091 - 0,1975 - 0,17270,0042 0,030 0,146 0,001 0,0032 1/4 - 0,175 0,073 - 0,1912 - 0,17550,0025 0,022 0,146 0,001 0,0032 1/4 - 0,163 0,048 - 0,1779 - 0,1628

SR 0,0025 0,022 0,146 0,001 0,0068 1/4 - 0,159 0,050 - 0,1779 - 0,15870,0072 0,044 0,146 0,008 0,0068 1/4 - 0,188 0,112 - 0,2110 - 0,18920,0050 0,034 0,146 0,008 0,0068 1/4 - 0,167 0,093 - 0,1969 - 0,16780,0042 0,030 0,146 0,008 0,0032 1/4 - 0,180 0,070 - 0,1912 - 0,18040,0025 0,022 0,146 0,008 0,0068 1/4 - 0,172 0,044 - 0,1779 - 0,1722

IR

0,0025 0,022 0,146 0,008 0,0032 1/4 - 0,155 0,052 - 0,1779 - 0,15520,0074 0,044 0,146 0,020 0,0068 1/4 - 0,188 0,115 - 0,2122 - 0,18920,0044 0,031 0,146 0,020 0,0068 1/4 - 0,159 0,088 - 0,1927 - 0,15930,0042 0,030 0,146 0,020 0,0032 1/4 - 0,167 0,078 - 0,1912 - 0,16750,0025 0,022 0,146 0,020 0,0032 1/4 - 0,165 0,047 - 0,1779 - 0,1652

3/3

subm

erge

d

LR

0,0025 0,022 0,146 0,020 0,0068 1/4 - 0,149 0,055 - 0,1779 - 0,1489

65

Submerged data (Author up)

Author (up)

Scale roughness

Q (m3/s)

K (m)

D (m)

d50 (m)

D50 (m)

i (-)

h1 (m)

h2 (m)

Fr2 (-)

Fd50 (-)

E0 (m)

E1 (m)

E2 (m)

0,0050 0,034 0,074 0,001 0,0032 1/8 - 0,092 0,222 - 0,1258 - 0,09440,0025 0,022 0,074 0,001 0,0032 1/8 - 0,072 0,179 - 0,1068 - 0,07300,0080 0,047 0,074 0,001 0,0032 1/8 - 0,103 0,314 - 0,1447 - 0,10790,0080 0,047 0,074 0,001 0,0068 1/8 - 0,103 0,306 - 0,1447 - 0,10780,0054 0,036 0,074 0,001 0,0068 1/8 - 0,086 0,267 - 0,1285 - 0,0890

SR

0,0026 0,022 0,074 0,001 0,0068 1/8 - 0,069 0,168 - 0,1076 - 0,07000,0025 0,022 0,074 0,008 0,0032 1/8 - 0,065 0,171 - 0,1068 - 0,06620,0050 0,034 0,074 0,008 0,0032 1/8 - 0,086 0,237 - 0,1258 - 0,08870,0080 0,047 0,074 0,008 0,0032 1/8 - 0,108 0,294 - 0,1447 - 0,11240,0080 0,047 0,074 0,008 0,0068 1/8 - 0,108 0,287 - 0,1447 - 0,11220,0055 0,036 0,074 0,008 0,0068 1/8 - 0,090 0,243 - 0,1291 - 0,0930

IR

0,0026 0,022 0,074 0,008 0,0068 1/8 - 0,067 0,205 - 0,1076 - 0,06820,0025 0,022 0,074 0,020 0,0032 1/8 - 0,064 0,187 - 0,1068 - 0,06530,0050 0,034 0,074 0,020 0,0032 1/8 - 0,085 0,262 - 0,1258 - 0,08740,0080 0,047 0,074 0,020 0,0032 1/8 - 0,111 0,272 - 0,1447 - 0,11470,0080 0,047 0,074 0,020 0,0068 1/8 - 0,105 0,293 - 0,1447 - 0,10980,0052 0,035 0,074 0,020 0,0068 1/8 - 0,088 0,240 - 0,1271 - 0,0905

LR

0,0026 0,022 0,074 0,020 0,0068 1/8 - 0,067 0,189 - 0,1076 - 0,06820,0025 0,022 0,099 0,001 0,0032 1/6 - 0,075 0,155 - 0,1310 - 0,07550,0054 0,036 0,099 0,001 0,0032 1/6 - 0,099 0,206 - 0,1527 - 0,10160,0082 0,048 0,099 0,001 0,0032 1/6 - 0,118 0,250 - 0,1700 - 0,12220,0025 0,022 0,099 0,001 0,0068 1/6 - 0,075 0,144 - 0,1310 - 0,07610,0051 0,035 0,099 0,001 0,0068 1/6 - 0,099 0,192 - 0,1506 - 0,1008

SR

0,0079 0,046 0,099 0,001 0,0068 1/6 - 0,120 0,228 - 0,1683 - 0,12380,0052 0,035 0,099 0,008 0,0032 1/6 - 0,102 0,211 - 0,1513 - 0,10450,0080 0,047 0,099 0,008 0,0032 1/6 - 0,120 0,245 - 0,1689 - 0,12390,0026 0,022 0,099 0,008 0,0032 1/6 - 0,076 0,157 - 0,1318 - 0,07700,0053 0,036 0,099 0,008 0,0068 1/6 - 0,105 0,194 - 0,1520 - 0,10700,0080 0,047 0,099 0,008 0,0068 1/6 - 0,114 0,249 - 0,1689 - 0,1182

IR

0,0024 0,021 0,099 0,008 0,0068 1/6 - 0,073 0,154 - 0,1301 - 0,07410,0080 0,047 0,099 0,020 0,0032 1/6 - 0,123 0,247 - 0,1689 - 0,12620,0051 0,035 0,099 0,020 0,0032 1/6 - 0,101 0,203 - 0,1506 - 0,10300,0026 0,022 0,099 0,020 0,0032 1/6 - 0,083 0,140 - 0,1318 - 0,08350,0051 0,035 0,099 0,020 0,0068 1/6 - 0,099 0,204 - 0,1506 - 0,10070,0080 0,047 0,099 0,020 0,0068 1/6 - 0,125 0,242 - 0,1689 - 0,1280

LR

0,0025 0,022 0,099 0,020 0,0068 1/6 - 0,082 0,150 - 0,1310 - 0,08300,0025 0,022 0,146 0,001 0,0032 1/4 - 0,086 0,128 - 0,1779 - 0,08640,0042 0,030 0,146 0,001 0,0032 1/4 - 0,102 0,171 - 0,1912 - 0,10340,0025 0,022 0,146 0,001 0,0068 1/4 - 0,090 0,105 - 0,1779 - 0,0908

SR 0,0051 0,035 0,146 0,001 0,0068 1/4 - 0,114 0,156 - 0,1975 - 0,11560,0025 0,022 0,146 0,008 0,0032 1/4 - 0,094 0,100 - 0,1779 - 0,09450,0042 0,030 0,146 0,008 0,0032 1/4 - 0,118 0,138 - 0,1912 - 0,11850,0025 0,022 0,146 0,008 0,0068 1/4 - 0,091 0,114 - 0,1779 - 0,09170,0050 0,034 0,146 0,008 0,0068 1/4 - 0,112 0,167 - 0,1969 - 0,1136

IR

0,0072 0,044 0,146 0,008 0,0068 1/4 - 0,127 0,184 - 0,2110 - 0,12960,0025 0,022 0,146 0,020 0,0032 1/4 - 0,089 0,121 - 0,1779 - 0,08940,0042 0,030 0,146 0,020 0,0032 1/4 - 0,114 0,148 - 0,1912 - 0,11530,0025 0,022 0,146 0,020 0,0068 1/4 - 0,105 0,093 - 0,1779 - 0,10560,0044 0,031 0,146 0,020 0,0068 1/4 - 0,113 0,147 - 0,1927 - 0,1140

1/3

subm

erge

d

LR

0,0074 0,044 0,146 0,020 0,0068 1/4 - 0,136 0,175 - 0,2122 - 0,1386

66


Author (up)

Scale roughness

Q (m3/s)

K (m)

D (m)

d50 (m)

D50 (m)

i (-)

h1 (m)

h2 (m)

Fr2 (-)

Fd50 (-)

E0 (m)

E1 (m)

E2 (m)

0,0050 0,034 0,074 0,001 0,0032 1/8 - 0,115 0,174 - 0,1258 - 0,11600,0025 0,022 0,074 0,001 0,0032 1/8 - 0,096 0,110 - 0,1068 - 0,09650,0080 0,047 0,074 0,001 0,0032 1/8 - 0,120 0,244 - 0,1447 - 0,12390,0080 0,047 0,074 0,001 0,0068 1/8 - 0,119 0,241 - 0,1447 - 0,12260,0054 0,036 0,074 0,001 0,0068 1/8 - 0,113 0,197 - 0,1285 - 0,1144

SR

0,0026 0,022 0,074 0,001 0,0068 1/8 - 0,088 0,118 - 0,1076 - 0,08880,0025 0,022 0,074 0,008 0,0032 1/8 - 0,097 0,107 - 0,1068 - 0,09750,0050 0,034 0,074 0,008 0,0032 1/8 - 0,114 0,163 - 0,1258 - 0,11560,0080 0,047 0,074 0,008 0,0032 1/8 - 0,129 0,227 - 0,1447 - 0,13180,0080 0,047 0,074 0,008 0,0068 1/8 - 0,131 0,213 - 0,1447 - 0,13360,0055 0,036 0,074 0,008 0,0068 1/8 - 0,112 0,181 - 0,1291 - 0,1140

IR

0,0026 0,022 0,074 0,008 0,0068 1/8 - 0,091 0,115 - 0,1076 - 0,09170,0025 0,022 0,074 0,020 0,0032 1/8 - 0,090 0,113 - 0,1068 - 0,09060,0050 0,034 0,074 0,020 0,0032 1/8 - 0,120 0,166 - 0,1258 - 0,12140,0080 0,047 0,074 0,020 0,0032 1/8 - 0,134 0,202 - 0,1447 - 0,13690,0080 0,047 0,074 0,020 0,0068 1/8 - 0,136 0,214 - 0,1447 - 0,13880,0052 0,035 0,074 0,020 0,0068 1/8 - 0,123 0,164 - 0,1271 - 0,1246

LR

0,0026 0,022 0,074 0,020 0,0068 1/8 - 0,101 0,113 - 0,1076 - 0,10150,0025 0,022 0,099 0,001 0,0032 1/6 - 0,096 0,103 - 0,1310 - 0,09610,0054 0,036 0,099 0,001 0,0032 1/6 - 0,119 0,169 - 0,1527 - 0,12080,0082 0,048 0,099 0,001 0,0032 1/6 - 0,143 0,181 - 0,1700 - 0,14540,0025 0,022 0,099 0,001 0,0068 1/6 - 0,098 0,102 - 0,1310 - 0,09840,0051 0,035 0,099 0,001 0,0068 1/6 - 0,123 0,156 - 0,1506 - 0,1245

SR

0,0079 0,046 0,099 0,001 0,0068 1/6 - 0,144 0,177 - 0,1683 - 0,14660,0052 0,035 0,099 0,008 0,0032 1/6 - 0,126 0,153 - 0,1513 - 0,12710,0080 0,047 0,099 0,008 0,0032 1/6 - 0,144 0,176 - 0,1689 - 0,14660,0026 0,022 0,099 0,008 0,0032 1/6 - 0,104 0,093 - 0,1318 - 0,10460,0053 0,036 0,099 0,008 0,0068 1/6 - 0,127 0,142 - 0,1520 - 0,12810,0080 0,047 0,099 0,008 0,0068 1/6 - 0,139 0,185 - 0,1689 - 0,1417

IR

0,0024 0,021 0,099 0,008 0,0068 1/6 - 0,106 0,093 - 0,1301 - 0,10640,0080 0,047 0,099 0,020 0,0032 1/6 - 0,152 0,182 - 0,1689 - 0,15420,0051 0,035 0,099 0,020 0,0032 1/6 - 0,127 0,152 - 0,1506 - 0,12830,0026 0,022 0,099 0,020 0,0032 1/6 - 0,115 0,089 - 0,1318 - 0,11560,0051 0,035 0,099 0,020 0,0068 1/6 - 0,126 0,159 - 0,1506 - 0,12720,0080 0,047 0,099 0,020 0,0068 1/6 - 0,143 0,195 - 0,1689 - 0,1459

LR

0,0025 0,022 0,099 0,020 0,0068 1/6 - 0,111 0,083 - 0,1310 - 0,11170,0025 0,022 0,146 0,001 0,0032 1/4 - 0,142 0,066 - 0,1779 - 0,14220,0042 0,030 0,146 0,001 0,0032 1/4 - 0,144 0,097 - 0,1912 - 0,14420,0025 0,022 0,146 0,001 0,0068 1/4 - 0,132 0,070 - 0,1779 - 0,1323

SR 0,0051 0,035 0,146 0,001 0,0068 1/4 - 0,145 0,116 - 0,1975 - 0,14600,0025 0,022 0,146 0,008 0,0032 1/4 - 0,146 0,060 - 0,1779 - 0,14640,0042 0,030 0,146 0,008 0,0032 1/4 - 0,145 0,096 - 0,1912 - 0,14580,0025 0,022 0,146 0,008 0,0068 1/4 - 0,126 0,067 - 0,1779 - 0,12630,0050 0,034 0,146 0,008 0,0068 1/4 - 0,147 0,114 - 0,1969 - 0,1479

IR

0,0072 0,044 0,146 0,008 0,0068 1/4 - 0,172 0,140 - 0,2110 - 0,17330,0025 0,022 0,146 0,020 0,0032 1/4 - 0,130 0,067 - 0,1779 - 0,13030,0042 0,030 0,146 0,020 0,0032 1/4 - 0,148 0,094 - 0,1912 - 0,14860,0025 0,022 0,146 0,020 0,0068 1/4 - 0,123 0,074 - 0,1779 - 0,12300,0044 0,031 0,146 0,020 0,0068 1/4 - 0,148 0,106 - 0,1927 - 0,1487

2/3

subm

erge

d

LR

0,0074 0,044 0,146 0,020 0,0068 1/4 - 0,177 0,131 - 0,2122 - 0,1784

67


Author (up)

Scale roughness

Q (m3/s)

K (m)

D (m)

d50 (m)

D50 (m)

i (-)

h1 (m)

h2 (m)

Fr2 (-)

Fd50 (-)

E0 (m)

E1 (m)

E2 (m)

0,0050 0,034 0,074 0,001 0,0032 1/8 - 0,120 0,156 - 0,1258 - 0,12090,0025 0,022 0,074 0,001 0,0032 1/8 - 0,101 0,093 - 0,1068 - 0,10160,0080 0,047 0,074 0,001 0,0032 1/8 - 0,137 0,203 - 0,1447 - 0,13930,0080 0,047 0,074 0,001 0,0068 1/8 - 0,134 0,202 - 0,1447 - 0,13690,0054 0,036 0,074 0,001 0,0068 1/8 - 0,116 0,168 - 0,1285 - 0,1177

SR

0,0026 0,022 0,074 0,001 0,0068 1/8 - 0,104 0,097 - 0,1076 - 0,10400,0025 0,022 0,074 0,008 0,0032 1/8 - 0,101 0,097 - 0,1068 - 0,10150,0050 0,034 0,074 0,008 0,0032 1/8 - 0,120 0,152 - 0,1258 - 0,12120,0080 0,047 0,074 0,008 0,0032 1/8 - 0,141 0,202 - 0,1447 - 0,14350,0080 0,047 0,074 0,008 0,0068 1/8 - 0,138 0,199 - 0,1447 - 0,14080,0055 0,036 0,074 0,008 0,0068 1/8 - 0,124 0,160 - 0,1291 - 0,1254

IR

0,0026 0,022 0,074 0,008 0,0068 1/8 - 0,099 0,100 - 0,1076 - 0,09970,0025 0,022 0,074 0,020 0,0032 1/8 - 0,105 0,092 - 0,1068 - 0,10550,0050 0,034 0,074 0,020 0,0032 1/8 - 0,123 0,145 - 0,1258 - 0,12430,0080 0,047 0,074 0,020 0,0032 1/8 - 0,141 0,190 - 0,1447 - 0,14360,0080 0,047 0,074 0,020 0,0068 1/8 - 0,141 0,189 - 0,1447 - 0,14360,0052 0,035 0,074 0,020 0,0068 1/8 - 0,125 0,148 - 0,1271 - 0,1265

LR

0,0026 0,022 0,074 0,020 0,0068 1/8 - 0,106 0,096 - 0,1076 - 0,10600,0025 0,022 0,099 0,001 0,0032 1/6 - 0,116 0,076 - 0,1310 - 0,11660,0054 0,036 0,099 0,001 0,0032 1/6 - 0,133 0,139 - 0,1527 - 0,13420,0082 0,048 0,099 0,001 0,0032 1/6 - 0,150 0,179 - 0,1700 - 0,15270,0025 0,022 0,099 0,001 0,0068 1/6 - 0,119 0,078 - 0,1310 - 0,11960,0051 0,035 0,099 0,001 0,0068 1/6 - 0,129 0,134 - 0,1506 - 0,1305

SR

0,0079 0,046 0,099 0,001 0,0068 1/6 - 0,151 0,164 - 0,1683 - 0,15300,0052 0,035 0,099 0,008 0,0032 1/6 - 0,136 0,135 - 0,1513 - 0,13670,0080 0,047 0,099 0,008 0,0032 1/6 - 0,167 0,159 - 0,1689 - 0,16840,0026 0,022 0,099 0,008 0,0032 1/6 - 0,129 0,074 - 0,1318 - 0,12920,0053 0,036 0,099 0,008 0,0068 1/6 - 0,139 0,129 - 0,1520 - 0,14030,0080 0,047 0,099 0,008 0,0068 1/6 - 0,156 0,160 - 0,1689 - 0,1579

IR

0,0024 0,021 0,099 0,008 0,0068 1/6 - 0,124 0,070 - 0,1301 - 0,12380,0080 0,047 0,099 0,020 0,0032 1/6 - 0,152 0,179 - 0,1689 - 0,15440,0051 0,035 0,099 0,020 0,0032 1/6 - 0,135 0,135 - 0,1506 - 0,13610,0026 0,022 0,099 0,020 0,0032 1/6 - 0,127 0,077 - 0,1318 - 0,12740,0051 0,035 0,099 0,020 0,0068 1/6 - 0,135 0,137 - 0,1506 - 0,13630,0080 0,047 0,099 0,020 0,0068 1/6 - 0,153 0,173 - 0,1689 - 0,1552

LR

0,0025 0,022 0,099 0,020 0,0068 1/6 - 0,125 0,072 - 0,1310 - 0,12480,0025 0,022 0,146 0,001 0,0032 1/4 - 0,158 0,048 - 0,1779 - 0,15800,0042 0,030 0,146 0,001 0,0032 1/4 - 0,171 0,073 - 0,1912 - 0,17110,0025 0,022 0,146 0,001 0,0068 1/4 - 0,153 0,050 - 0,1779 - 0,1527

SR 0,0051 0,035 0,146 0,001 0,0068 1/4 - 0,172 0,091 - 0,1975 - 0,17270,0025 0,022 0,146 0,008 0,0032 1/4 - 0,166 0,044 - 0,1779 - 0,16600,0042 0,030 0,146 0,008 0,0032 1/4 - 0,173 0,070 - 0,1912 - 0,17350,0025 0,022 0,146 0,008 0,0068 1/4 - 0,155 0,052 - 0,1779 - 0,15520,0050 0,034 0,146 0,008 0,0068 1/4 - 0,181 0,088 - 0,1969 - 0,1816

IR

0,0072 0,044 0,146 0,008 0,0068 1/4 - 0,182 0,112 - 0,2110 - 0,18350,0025 0,022 0,146 0,020 0,0032 1/4 - 0,157 0,047 - 0,1779 - 0,15720,0042 0,030 0,146 0,020 0,0032 1/4 - 0,173 0,078 - 0,1912 - 0,17360,0025 0,022 0,146 0,020 0,0068 1/4 - 0,142 0,055 - 0,1779 - 0,14180,0044 0,031 0,146 0,020 0,0068 1/4 - 0,164 0,088 - 0,1927 - 0,1647

3/3

subm

erge

d

LR

0,0074 0,044 0,146 0,020 0,0068 1/4 - 0,189 0,115 - 0,2122 - 0,1899

68

Data of submerged jump (Di Grigoli)

Di Grigoli

Scale roughness

Q (m3/s)

K (m)

D (m)

d50 (m)

D50 (m)

i (-)

h1 (m)

h2 (m)

Fr2 (-)

Fd50 (-)

E0 (m)

E1 (m)

E2 (m)

0,0064 0,040 0,074 0,001 x 1/8 x 0,100 0,258 x 0,1349 x 0,10290,0058 0,038 0,074 0,001 x 1/8 x 0,094 0,254 x 0,1310 x 0,09700,0047 0,033 0,050 0,001 x 1/8 x 0,074 0,295 x 0,0991 x 0,07760,0031 0,025 0,031 0,001 x 1/8 x 0,061 0,261 x 0,0682 x 0,06280,0030 0,024 0,050 0,001 x 1/8 x 0,061 0,254 x 0,0862 x 0,06260,0065 0,041 0,050 0,002 x 1/8 x 0,094 0,288 x 0,1108 x 0,09740,0065 0,041 0,050 0,002 0 1/8 - 0,094 0,288 - 0,1108 - 0,09740,0061 0,039 0,074 0,002 x 1/8 x 0,105 0,227 x 0,1328 x 0,10720,0019 0,018 0,050 0,001 x 1/8 x 0,053 0,198 x 0,0765 x 0,05370,0018 0,017 0,031 0,001 x 1/8 x 0,051 0,200 x 0,0570 x 0,05170,0048 0,033 0,050 0,002 x 1/8 x 0,078 0,279 x 0,0996 x 0,08100,0044 0,031 0,099 0,002 x 1/8 x 0,083 0,235 x 0,1457 x 0,08480,0016 0,016 0,074 0,001 x 1/8 x 0,061 0,129 x 0,0979 x 0,06190,0014 0,015 0,050 0,001 x 1/8 x 0,048 0,171 x 0,0718 x 0,04870,0030 0,024 0,074 0,002 x 1/8 x 0,072 0,196 x 0,1107 x 0,07300,0029 0,024 0,050 0,002 x 1/8 x 0,066 0,215 x 0,0850 x 0,06730,0019 0,018 0,074 0,002 x 1/8 x 0,064 0,148 x 0,1013 x 0,0649

SR

0,0019 0,018 0,050 0,002 x 1/8 x 0,057 0,178 x 0,0765 x 0,05760,0066 0,041 0,050 0,008 x 1/8 x 0,088 0,317 x 0,1111 x 0,09270,0060 0,038 0,050 0,008 x 1/8 x 0,088 0,288 x 0,1073 x 0,09190,0048 0,034 0,050 0,008 x 1/8 x 0,073 0,309 x 0,0999 x 0,0769

IR 0,0040 0,030 0,050 0,008 x 1/8 x 0,073 0,261 x 0,0940 x 0,07500,0070 0,043 0,050 0,020 x 1/8 x 0,088 0,297 x 0,1139 x 0,0933

1/3

subm

erge

d

LR 0,0059 0,038 0,050 0,020 x 1/8 x 0,081 0,301 x 0,1071 x 0,0852

69


Di Grigoli

Scale roughness

Q (m3/s)

K (m)

D (m)

d50 (m)

D50 (m)

i (-)

h1 (m)

h2 (m)

Fr2 (-)

Fd50 (-)

E0 (m)

E1 (m)

E2 (m)

0,0067 0,042 0,041 0,001 x 1/6 x 0,092 0,303 x 0,1035 x 0,09650,0064 0,040 0,099 0,001 x 1/6 x 0,104 0,242 x 0,1591 x 0,10690,0058 0,038 0,099 0,001 x 1/6 x 0,101 0,230 x 0,1552 x 0,10320,0031 0,025 0,041 0,001 x 1/6 x 0,067 0,224 x 0,0783 x 0,06890,0030 0,024 0,066 0,001 x 1/6 x 0,069 0,212 x 0,1023 x 0,07000,0079 0,046 0,041 0,002 x 1/6 x 0,100 0,317 x 0,1108 x 0,10500,0066 0,041 0,041 0,002 x 1/6 x 0,093 0,293 x 0,1026 x 0,09700,0065 0,041 0,066 0,002 x 1/6 x 0,103 0,251 x 0,1269 x 0,10570,0023 0,020 0,099 0,001 x 1/6 x 0,080 0,127 x 0,1290 x 0,08090,0061 0,039 0,099 0,002 x 1/6 x 0,104 0,228 x 0,1570 x 0,10690,0019 0,018 0,066 0,001 x 1/6 x 0,059 0,169 x 0,0926 x 0,05930,0018 0,017 0,041 0,001 x 1/6 x 0,057 0,169 x 0,0671 x 0,05740,0048 0,033 0,066 0,002 x 1/6 x 0,087 0,236 x 0,1157 x 0,08970,0016 0,016 0,099 0,001 x 1/6 x 0,075 0,095 x 0,1221 x 0,07560,0014 0,015 0,066 0,001 x 1/6 x 0,057 0,131 x 0,0879 x 0,05770,0038 0,028 0,041 0,002 x 1/6 x 0,071 0,251 x 0,0838 x 0,07370,0012 0,014 0,041 0,001 x 1/6 x 0,047 0,154 x 0,0614 x 0,04770,0030 0,024 0,099 0,002 x 1/6 x 0,083 0,159 x 0,1349 x 0,08350,0029 0,024 0,066 0,002 x 1/6 x 0,071 0,190 x 0,1011 x 0,07260,0019 0,018 0,099 0,002 x 1/6 x 0,071 0,127 x 0,1255 x 0,07170,0019 0,018 0,066 0,002 x 1/6 x 0,055 0,186 x 0,0927 x 0,0559

SR

0,0015 0,015 0,041 0,002 x 1/6 x 0,050 0,170 x 0,0639 x 0,05030,0066 0,041 0,066 0,008 x 1/6 x 0,100 0,264 x 0,1273 x 0,10320,0060 0,038 0,066 0,008 x 1/6 x 0,093 0,267 x 0,1235 x 0,09610,0057 0,037 0,099 0,008 x 1/6 x 0,108 0,203 x 0,1548 x 0,11060,0053 0,035 0,099 0,008 x 1/6 x 0,104 0,200 x 0,1517 x 0,10560,0048 0,034 0,066 0,008 x 1/6 x 0,082 0,260 x 0,1161 x 0,08500,0040 0,030 0,066 0,008 x 1/6 x 0,081 0,220 x 0,1101 x 0,08320,0037 0,028 0,099 0,008 x 1/6 x 0,095 0,160 x 0,1407 x 0,0963

IR

0,0025 0,022 0,099 0,008 x 1/6 x 0,085 0,130 x 0,1311 x 0,08530,0070 0,043 0,066 0,020 x 1/6 x 0,103 0,268 x 0,1300 x 0,10670,0068 0,042 0,099 0,020 x 1/6 x 0,110 0,235 x 0,1615 x 0,11330,0061 0,039 0,099 0,020 x 1/6 x 0,109 0,213 x 0,1570 x 0,11170,0059 0,038 0,066 0,020 x 1/6 x 0,092 0,269 x 0,1232 x 0,09530,0048 0,033 0,066 0,020 x 1/6 x 0,083 0,254 x 0,1157 x 0,08570,0043 0,031 0,099 0,020 x 1/6 x 0,091 0,196 x 0,1447 x 0,0928

1/3

subm

erge

d

LR

0,0036 0,028 0,066 0,020 x 1/6 x 0,075 0,223 x 0,1073 x 0,0771

70


Di Grigoli

Scale roughness

Q (m3/s)

K (m)

D (m)

d50 (m)

D50 (m)

i (-)

h1 (m)

h2 (m)

Fr2 (-)

Fd50 (-)

E0 (m)

E1 (m)

E2 (m)

0,0030 0,024 0,097 0,001 x 1/4 x 0,077 0,178 x 0,1336 x 0,07810,0023 0,020 0,146 0,001 x 1/4 x 0,101 0,090 x 0,1759 x 0,10120,0019 0,018 0,097 0,001 x 1/4 x 0,062 0,153 x 0,1239 x 0,06320,0018 0,017 0,061 0,001 x 1/4 x 0,054 0,181 x 0,0866 x 0,05500,0044 0,031 0,146 0,002 x 1/4 x 0,100 0,177 x 0,1926 x 0,10110,0016 0,016 0,146 0,001 x 1/4 x 0,079 0,089 x 0,1690 x 0,07900,0014 0,015 0,097 0,001 x 1/4 x 0,056 0,137 x 0,1192 x 0,05620,0038 0,028 0,061 0,002 x 1/4 x 0,084 0,199 x 0,1033 x 0,08520,0012 0,014 0,061 0,001 x 1/4 x 0,047 0,154 x 0,0809 x 0,04750,0030 0,024 0,146 0,002 x 1/4 x 0,089 0,141 x 0,1818 x 0,09010,0029 0,024 0,097 0,002 x 1/4 x 0,075 0,177 x 0,1324 x 0,07610,0019 0,018 0,146 0,002 x 1/4 x 0,084 0,099 x 0,1724 x 0,08450,0019 0,018 0,097 0,002 x 1/4 x 0,064 0,148 x 0,1239 x 0,06480,0015 0,015 0,061 0,002 x 1/4 x 0,052 0,157 x 0,0834 x 0,05290,0030 0,024 0,146 0,001 x 1/4 x 0,098 0,125 x 0,1822 x 0,09880,0060 0,039 0,146 0,001 x 1/4 x 0,128 0,167 x 0,2038 x 0,12980,0078 0,046 0,146 0,001 x 1/4 x 0,156 0,162 x 0,2149 x 0,15800,0118 0,061 0,146 0,001 x 1/4 x 0,188 0,185 x 0,2370 x 0,19120,0070 0,043 0,061 0,001 x 1/4 x 0,110 0,245 x 0,1253 x 0,1133

SR


IR

0,0025 0,022 0,146 0,008 x 1/4 x 0,098 0,104 x 0,1780 x 0,09840,0068 0,042 0,146 0,020 x 1/4 x 0,138 0,168 x 0,2084 x 0,13990,0059 0,038 0,097 0,020 x 1/4 x 0,106 0,218 x 0,1545 x 0,10840,0051 0,035 0,146 0,020 x 1/4 x 0,110 0,178 x 0,1975 x 0,11120,0048 0,033 0,097 0,020 x 1/4 x 0,095 0,208 x 0,1470 x 0,09680,0043 0,031 0,146 0,020 x 1/4 x 0,104 0,162 x 0,1916 x 0,10490,0036 0,028 0,097 0,020 x 1/4 x 0,084 0,188 x 0,1385 x 0,08570,0060 0,039 0,146 0,020 x 1/4 x 0,121 0,182 x 0,2038 x 0,12300,0070 0,043 0,146 0,020 x 1/4 x 0,129 0,193 x 0,2101 x 0,13140,0080 0,047 0,146 0,020 x 1/4 x 0,138 0,199 x 0,2161 x 0,14070,0070 0,043 0,061 0,020 x 1/4 x 0,103 0,270 x 0,1252 x 0,1068

1/3

subm

erge

d

LR

0,0080 0,047 0,061 0,020 x 1/4 x 0,107 0,292 x 0,1313 x 0,1116

71


Di Grigoli

Scale roughness

Q (m3/s)

K (m)

D (m)

d50 (m)

D50 (m)

i (-)

h1 (m)

h2 (m)

Fr2 (-)

Fd50 (-)

E0 (m)

E1 (m)

E2 (m)


SR

0,0030 0,024 0,074 0,002 x 1/8 x 0,095 0,128 x 0,1107 x 0,09620,0066 0,041 0,050 0,008 x 1/8 x 0,103 0,252 x 0,1111 x 0,10600,0060 0,038 0,050 0,008 x 1/8 x 0,097 0,252 x 0,1073 x 0,09960,0040 0,030 0,050 0,008 x 1/8 x 0,083 0,212 x 0,0940 x 0,0850

IR 0,0037 0,028 0,074 0,008 x 1/8 x 0,100 0,150 x 0,1165 x 0,10060,0064 0,040 0,099 0,001 x 1/6 x 0,137 0,159 x 0,1591 x 0,13900,0058 0,038 0,099 0,001 x 1/6 x 0,139 0,141 x 0,1552 x 0,14030,0031 0,025 0,041 0,001 x 1/6 x 0,075 0,188 x 0,0783 x 0,07670,0030 0,024 0,066 0,001 x 1/6 x 0,087 0,149 x 0,1023 x 0,08780,0023 0,020 0,099 0,001 x 1/6 x 0,107 0,083 x 0,1290 x 0,10690,0061 0,039 0,099 0,002 x 1/6 x 0,142 0,143 x 0,1570 x 0,14350,0019 0,018 0,066 0,001 x 1/6 x 0,078 0,110 x 0,0926 x 0,07820,0018 0,017 0,041 0,001 x 1/6 x 0,062 0,147 x 0,0671 x 0,06300,0048 0,033 0,066 0,002 x 1/6 x 0,102 0,186 x 0,1157 x 0,10420,0016 0,016 0,099 0,001 x 1/6 x 0,101 0,061 x 0,1221 x 0,10120,0014 0,015 0,066 0,001 x 1/6 x 0,071 0,096 x 0,0879 x 0,07090,0012 0,014 0,041 0,001 x 1/6 x 0,054 0,127 x 0,0614 x 0,05400,0030 0,024 0,099 0,002 x 1/6 x 0,110 0,104 x 0,1349 x 0,11010,0029 0,024 0,066 0,002 x 1/6 x 0,086 0,144 x 0,1011 x 0,08670,0019 0,018 0,099 0,002 x 1/6 x 0,102 0,074 x 0,1255 x 0,1020

SR

0,0019 0,018 0,066 0,002 x 1/6 x 0,083 0,100 x 0,0927 x 0,08380,0066 0,041 0,066 0,008 x 1/6 x 0,114 0,215 x 0,1273 x 0,11690,0060 0,038 0,066 0,008 x 1/6 x 0,111 0,205 x 0,1235 x 0,11320,0048 0,034 0,066 0,008 x 1/6 x 0,099 0,197 x 0,1161 x 0,10100,0040 0,030 0,066 0,008 x 1/6 x 0,092 0,183 x 0,1101 x 0,09340,0037 0,028 0,099 0,008 x 1/6 x 0,119 0,114 x 0,1407 x 0,1202

2/3

subm

erge

d

IR

0,0025 0,022 0,099 0,008 x 1/6 x 0,113 0,085 x 0,1311 x 0,1129

72


Di Grigoli

Scale roughness

Q (m3/s)

K (m)

D (m)

d50 (m)

D50 (m)

i (-)

h1 (m)

h2 (m)

Fr2 (-)

Fd50 (-)

E0 (m)

E1 (m)

E2 (m)


LR

0,0036 0,028 0,066 0,020 x 1/6 x 0,085 0,184 x 0,1073 x 0,08690,0030 0,024 0,097 0,001 x 1/4 x 0,101 0,119 x 0,1336 x 0,10120,0079 0,046 0,061 0,002 x 1/4 x 0,110 0,276 x 0,1303 x 0,11370,0066 0,041 0,061 0,002 x 1/4 x 0,112 0,234 x 0,1221 x 0,11090,0065 0,041 0,097 0,002 x 1/4 x 0,125 0,187 x 0,1582 x 0,12700,0023 0,020 0,146 0,001 x 1/4 x 0,135 0,058 x 0,1759 x 0,13550,0061 0,039 0,146 0,002 x 1/4 x 0,152 0,129 x 0,2039 x 0,15360,0019 0,018 0,097 0,001 x 1/4 x 0,097 0,079 x 0,1239 x 0,09760,0044 0,031 0,146 0,002 x 1/4 x 0,152 0,094 x 0,1926 x 0,15260,0016 0,016 0,146 0,001 x 1/4 x 0,130 0,042 x 0,1690 x 0,13010,0014 0,015 0,097 0,001 x 1/4 x 0,088 0,068 x 0,1192 x 0,08870,0038 0,028 0,061 0,002 x 1/4 x 0,096 0,162 x 0,1033 x 0,09720,0012 0,014 0,061 0,001 x 1/4 x 0,060 0,107 x 0,0809 x 0,06030,0030 0,024 0,146 0,002 x 1/4 x 0,144 0,069 x 0,1818 x 0,14430,0019 0,018 0,146 0,002 x 1/4 x 0,131 0,051 x 0,1724 x 0,13130,0019 0,018 0,097 0,002 x 1/4 x 0,097 0,080 x 0,1239 x 0,09680,0015 0,015 0,061 0,002 x 1/4 x 0,072 0,098 x 0,0834 x 0,07210,0030 0,024 0,146 0,001 x 1/4 x 0,144 0,070 x 0,1822 x 0,14440,0060 0,039 0,146 0,001 x 1/4 x 0,176 0,104 x 0,2038 x 0,17690,0078 0,046 0,146 0,001 x 1/4 x 0,188 0,122 x 0,2149 x 0,18940,0118 0,061 0,146 0,001 x 1/4 x 0,216 0,150 x 0,2370 x 0,21840,0070 0,043 0,061 0,001 x 1/4 x 0,117 0,223 x 0,1253 x 0,1199

SR


IR

0,0025 0,022 0,146 0,008 x 1/4 x 0,143 0,059 x 0,1780 x 0,14320,0021 0,019 0,061 0,008 x 1/4 x 0,080 0,118 x 0,0070 0,043 0,097 0,020 x 1/4 x 0,141 0,168 x 0,1613 x 0,14250,0068 0,042 0,146 0,020 x 1/4 x 0,166 0,127 x 0,2084 x 0,16730,0061 0,039 0,146 0,020 x 1/4 x 0,168 0,111 x 0,2039 x 0,16940,0059 0,038 0,097 0,020 x 1/4 x 0,127 0,166 x 0,1545 x 0,12870,0051 0,035 0,146 0,020 x 1/4 x 0,150 0,111 x 0,1975 x 0,15090,0048 0,033 0,097 0,020 x 1/4 x 0,117 0,152 x 0,1470 x 0,11830,0043 0,031 0,146 0,020 x 1/4 x 0,150 0,093 x 0,1916 x 0,15060,0060 0,039 0,146 0,020 x 1/4 x 0,170 0,109 x 0,2038 x 0,17100,0070 0,043 0,146 0,020 x 1/4 x 0,169 0,129 x 0,2101 x 0,17040,0080 0,047 0,146 0,020 x 1/4 x 0,179 0,135 x 0,2161 x 0,18060,0070 0,043 0,061 0,020 x 1/4 x 0,108 0,252 x 0,1252 x 0,1114

2/3

subm

erge

d

LR

0,0080 0,047 0,061 0,020 x 1/4 x 0,114 0,265 x 0,1313 x 0,1180

73

dissertation in water engineering

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Transcript of dissertation in water engineering