Clear water scour development at vertical wall bridge

abutment

A. Cardoso Full Professor and C. Fael Assistant Professor

An experimental study on the time evolution of clear-water scour at vertical-wall abutments

was carried out. The study deals with the dependence of the scouring process on a set of non-

dimensional parameters which includes the flow intensity, U/Uc, the relative abutment length,

L/d, the obstacle Reynolds number, UL/ν, the relative sediment dimension, L/D50, and a non-

dimensional timescale. This dependence was obtained for the principal phase of the process

and a new equation for the time evolution of scour is suggested. A new predictor of time to

equilibrium, Te, was also established. This new predictor renders quite large values of Te for

field conditions and may explain usual discrepancies between predictions and field

measurements of scour depth; it seems to indicate also that, for clear-water scour, the

prediction of time to equilibrium is not a critical issue for the calculation of scour associated

with common flood hydrographs.

NOTATION

a1; a2 coefficients of Franzetti’s equation – equation (10)

B channel width

Dp pier diameter

D50 median size of the bed material

d approach flow depth

ds scour depth at instant t

dse equilibrium scour depth

g acceleration of gravity

Kg channel shape coefficient

Ks abutment shape coefficient

Kθ abutment alignment coefficient

K1; K2 coefficients of Ettema’s equation – equation (9)

L abutment length

p1, … p4 parameters of the polynomium of Bertoldi and Jones – equation (8)

Se slope of the energy grade line

S0 channel bottom slope

Td test duration

Te time to equilibrium

t time

U approach flow velocity

Uc critical velocity for the beginning of motion of riprap blocks or bed material

Us value of U corresponding to the onset of shear failure

u* friction velocity

ν fluid kinematic viscosity

Πt non-dimensional time

ρ fluid density

ρs sediment density

σD gradation coefficient of bed material

1. INTRODUCTION

A major cause for the failure of bridge foundations – piers and abutments – is scour and the

estimation of the scour characteristics at bridge foundation elements continues to be a concern

for hydraulic engineers and researchers. Research on local scour has been addressed, mostly,

by the use of experimental studies. Until two decades ago, many studies, particularly those

dealing with clear-water conditions (without general bed sediment motion), reported

experiments that might not have lasted long enough to reach equilibrium scour depths. Since

then, most studies claim to have achieved equilibrium scour by carrying out long duration

experiments. With few exceptions, such studies use steady flows.

In nature, long lasting steady flows seldom occur. Unsteadiness is particularly significant

during floods. Most bridge failures occur during flood events under live-bed scour conditions

in the main channel (with general sediment movement in its bed). The rapid development of

scour under live-bed conditions leads to the quick establishment of equilibrium scour depth

for such flows, with the rate of scour development of less importance to the designer. In this

case, which is likely to occur at piers placed in the main channel, the knowledge of

equilibrium scour depth is the main concern.

Many bridge abutments, however, cross only part of the flood plain, where scour mostly

occurs under clear-water conditions. Scour holes develop slowly, approaching the ultimate

scour depth asymptotically.

Some authors suggest the approximation of flood hydrographs by sequences of short duration

steps of constant discharge. Assuming that the scour depth, ds, is known at a given instant, t,

e.g. at tinitial, and that the time rate of scour is also known for the corresponding flow

discharge, then, it is possible to calculate the scour depth at instant tfinal. This new ds will then

be the initial value for the next timestep with a new constant discharge and this procedure can

be repeated to simulate the flood hydrograph. In this way, it is possible to consider the time

evolution of the scour depth in the study of local scour close to bridge abutments for unsteady

flow situations. Consequently, it is important to characterize the time development of local

scour under clear water conditions. This may also be useful for abutments inserted in mobile-

bed main channels, if clear water scour conditions prevail.

Since the nineteen fifties, a number of attempts to describe the temporal development of clear-

water scour has been made by various authors, with many methods being suggested for piers.

Mention can be made, among others, of the contributions by Ettema1, Franzetti et al.2,

Whitehouse3, Melville and Chiew4 or Kothyari and Ranga Raju5. These contributions apply to

the principal phase of the scour process; some also deal with the definition of the time to

equilibrium, Te, i.e., the time corresponding to the end of the principal phase, and the onset of

the equilibrium phase.

Few studies have been carried out on the time evolution of clear-water scour at abutments.

Cardoso and Bettess6 describe experiments measuring scour development for vertical-wall

abutments, under conditions of initiation of sediment motion, and the related scour rate with

the relative abutment length, L/d (L = abutment length and d = approach flow depth).

Santos and Cardoso7 looked at the effects of flow intensity, U/Uc (U = average approach flow

velocity; Uc = critical flow velocity for bed sediment entrainment) and relative abutment

length and sediment size, L/D50 (D50 = median size of the sediment particle size distribution)

on scour development for vertical-wall abutments. The result of their work is a temporal

scour-depth predictor and a time to equilibrium predictor. The authors indicate that the

timescale associated with the scour depends on the flow intensity and the relative abutment

size. Their study covers relative abutment lengths, typically, up to ≈ 7, although four tests

selected in the literature, with 7 < L/d < 17, were also included..

Coleman et al.8 proposed an equation to predict the time evolution of scour, concluding that

the scour depth at a given time mostly depends on flow intensity, while it is relatively

insensitive to the relative depth of flow , d/L. Coleman et al. 8 also suggested an upper bound

predictor of time to equilibrium. This predictor is dependent on both the flow intensity and the

relative depth of flow and independent of the relative sediment size, for L/D50 > 100, as

suggested by Santos and Cardoso7. The upper bound predictor of time to equilibrium covers a

wide span of L/d values, up to about 100, but, according to Coleman et al.8, for L/d > ≈ 7, the

data shows some uncertainty. The definition of time to equilibrium, itself, is a controversial

issue and the suggested predictor necessarily reflects the view of the authors. The contribution

of Coleman et al.8 uses the same data as Lauchlan et al.9, which apparently synthesizes New-

Zealand’s research effort on the characterization of time evolution of scouring at bridge

abutments.

Most of the data reported in the above studies refer to relative abutment lengths, L/d, smaller

than ≈ 7. Thus, extending existing knowledge on the evolution of scouring at vertical-wall

abutments to a wider range of L/d as well as to smaller values of U/Uc is the focus of the

present work. For this purpose, an experimental study has been carried out by Fael10.

2. FRAMEWORK FOR ANALYSIS

Scour depth can be described by the following set of independent variables and parameters:

1 ( ) ( ) ( ) ( )

( ) ( )

e 50 D s ss

0 g

flow d,S ,g ,fluid , , bed material D , , ,abutment L,K ,K ,d f

channel B,S ,K , time t

θ ρ ν σ ρ =

where, apart from variables already defined, Se = slope of the energy line; g = acceleration of

gravity; ρ and ν = fluid density and kinematic viscosity, respectively; σD = geometric

standard deviation of the sediment particle size distribution; ρs = sediment density; Kθ and Ks

= coefficients describing the alignment and the shape of the abutment; B = channel width; S0

= channel bottom slope; Kg = coefficient describing the geometry of the channel cross-

section; t = time. It should be noted that the critical velocity for sediment entrainment, Uc, is

not considered since it is fully defined by g, d, Se, ρ, ν, D50 and ρs. Time to equilibrium, Te, is

not included either because it corresponds to the smallest time, t, for which ds = dse, dse being

the equilibrium scour depth.

For uniform flows in wide rectangular channels, Se = S0 and B and Kg no longer influence

scour. In this context, wide channels are those where the effects of flow contraction due to the

presence of obstacles as well as wall effects are negligible. In the absence of bedforms, Se can

be replaced by both the friction velocity, u* = (gdSe)0.5, or the average approach flow velocity,

U. If the bed material is composed of uniform non-ripple forming sand, which implies D50 > ≈

0.6 mm, σD < 1.5 ~ 1.8 and ρs ≈ constant, σD and ρs can be eliminated from equation (1).

Finally, assuming a thin vertical-wall abutment (Ks = 1), protruding at right angles from the

channel side wall (Kθ = 1), the scour depth can be written as

2 ( )s * 50d =f d,u ,g,ρ,ν,D ,L,t

One of the possible dimensionless forms of equation (2) is

3

2s * * *

50

d u u L u tL L=f ; ; ; ;

L d gL ν D L

Keeping in mind Shields’ diagram and rearranging, equation (3) can be transformed into

(Fael10):

4 s * * *

*c 50

d u u L u tL L=f ; ; ; ;

L d u ν D L

where u*c = critical shear velocity for sediment entrainment. In the case of fully developed

flow on a flat bed, it can be shown that u*/ u*c = U/Uc, and equation (4) is equivalent to

5 s

c 50

d L U UL L Ut=f ; ; ; ;

L d U ν D L

A particular form of equation (5) is postulated, here, to be

6 s 50

3c

d D νtL U=f ; ;

L d U L

since 50 503

D νt DUt ν=

L UL LL.

Equation (6) applies to thin vertical-wall abutments, protruding at right angles from the side

wall of a fully developed, uniform flow in a wide rectangular channel, on flat bed of uniform,

non-ripple forming sand.

Assuming, finally, that, at least within the scour hole, the flow is rough turbulent and that

L/D50 > ~ 100, such that L/D50 can be claimed to no longer influence the scouring process,

equation (5) reduces to

7 s

c

d L U Ut=f ; ;

L d U L

Equations (6) and (7) constitute a framework for the subsequent analysis. Equation (6)

incorporates the influence of fluid viscosity and relative sediment size, while equation (7)

assumes the influence of these variables to be negligible.

3. EXPERIMENTAL SET-UP AND PROCEDURES

Experiments were carried out in a 28.0 m long, 4.0 m wide, and 1.0 m deep concrete flume.

The right lateral wall is made of glass panels that allow the observation of the flow, namely in

the central reach, where the scour holes develop. This reach includes a 3.0 m long and 0.6 m

deep recess in the bed, to allow for scouring. The bed recess was filled with natural quartz

sand (ρs = 2650 kgm−3; D50 ≈ 1.28 mm; σD ≈ 1.46). Vertical-wall abutments were simulated

by 20 mm thick vertical plates installed at the bottom of the mid cross-section of the recess.

Obstacle lengths ranged from L = 0.64 m to L = 1.86 m. The study was made for reasonably

high flow depths (≈ 0.05 m ≤ d ≤ ≈ 0.07 m), covering different flow velocities in the clear-

water range.

The experimental set-up includes a closed hydraulic circuit where the discharge can be varied

from 0.0 m3s−1 to 0.18 m3s−1. At the downstream end of the hydraulic circuit, at the entrance

of the flume, a diffuser reach evenly distributes the flow across the whole width of the flume.

Another, smaller, circuit exists, discharging up to 5 l/s. This circuit can be used to fill the

flume slowly, thus avoiding the uncontrolled disturbance of the movable bed that could occur

at the beginning of experiments. At the downstream end of the flume, a tailgate allows the

regulation of the water level inside the flume. The water falls into a 100 m3 reservoir, where

the hydraulic circuits start. The flume is equipped with a moving carriage. It allows leveling

of the sand bed and serves as a support to the measuring equipment.

Prior to each test, the sand bed was leveled. Then the area located close to the nose of the

obstacle was covered with a thin metallic plate to avoid uncontrolled scour at the beginning of

the experiment. The flume was filled gradually from the downstream end by using the smaller

hydraulic circuit. In this way, it was possible to impose a high water depth and low flow

velocity. Then, the discharge corresponding to the chosen approach flow velocity was passed

through the larger hydraulic circuit. The flow depth was imposed by adjusting the

downstream tailgate. Once the flow depth was established, the metallic plate was removed

and the experiment started. Scour was immediately initiated and the depth of scour hole was

measured, to an accuracy of the order of ± 1 mm, with an adapted point gauge installed in the

measuring carriage, every ≈ 5 minutes during the first hour. Afterwards, the interval between

measurements increased and, after the first day, few measurements were carried out each day.

Each experiment typically lasted for one week or more. Twenty-seven tests were carried out.

One preliminary test was conducted to evaluate the value of Uc corresponding to the sand

being used. Approximate values were calculated using the equations suggested by Neil11 and

Garde12; the preliminary test allowed the refinement of these calculated values. For the flow

depth d = 0.07 m, Uc was assumed to be equal to 0.355 ms−1, corresponding to a roughness

Reynolds number, u*D50/ν, of about 50, characteristic of the transitional turbulent flow

regime.

4. RESULTS AND DISCUSSION

4.1. Data characterization

As a complement to the data obtained for this study and characterized in Table 1, data from

thirty-two tests were compiled from the literature. Table 2 summarizes the main parameters

associated with the selected experiments. Both tables include the values of d, L, test duration,

Td, time to equilibrium, Te, and parameters U/Uc and L/d.

The tests presented in Table 1 cover values of relative abutment length, L/d, ranging from ≈ 9

to ≈ 36, which are rare in previous studies; U/Uc range from 0.57 to 1.12. In Table 2, data are

identified according to the corresponding authors’ names: S&C stands for Santos and

Cardoso7, C, for Da Cunha13, C&B, for Cardoso and Bettess6 and K, for Kwan14. In all cases,

the obstacles were thin vertical walls placed at the right angle to the flow direction. Non-

ripple forming sandy bed (D50 > 0.6 mm; σD ≤ 1.8) channels with rectangular cross-section

were always used, and clear-water conditions (U/Uc < 1) were observed. In short, both data

sets respect the restrictions underlying equation (6).

The definition of the equilibrium scour depth, dse, and time to equilibrium, Te, is a key issue.

Cardoso and Bettess6 as well as Santos and Cardoso7 assess the equilibrium time as the time

when the principal phase of scouring changes to the equilibrium phase, as determined by

changes in the slope for the scour depth versus logarithm of time plots. This is illustrated in

Fig. 1. For piers, Melville and Chiew4 assumed the time to achieve equilibrium scour to be the

time at which the rate of scouring reduces to 5% of the pier diameter in the succeeding 24-

hour period. Coleman et al.8 defined the equilibrium time as the time at which the rate of

scour reduces to 5% of the smaller of the foundation length (pier diameter or abutment length)

or the flow depth in the succeeding 24-hour period.

In the present study, the onset of equilibrium was assessed by comparing dse values evaluated

by two methods applied to the scour depth records. The first method is based on the proposal

of Cardoso and Bettess6; in the second, equilibrium is assumed to be reached at infinite time

and the equilibrium scour depth is calculated by adjusting the polynomium suggested by

Bertoldi and Jones15

8 s 1 31 2 3 4

1 1d = p 1 +p 1

1+p p t 1+p p t

to the recorded time evolution of the scour depth; in this polynomium, p1, p2, p3 and p4 are

parameters obtained by a regression analysis. The Equilibrium scour depth is obtained for t =

∞, i.e., dse = p1 + p3. The methods suggested by Melville and Chiew4 as well as by Coleman et

al.8 seem less precise.

The equilibrium scour depths predicted by the second method are (3.57 ± 5.77)% (mean ±

standard deviation) higher than those given by the first method; more details can be found in

Fael et al.16. The proximity of these results reinforces the confidence of dse predictions,

namely, by the method of Cardoso and Bettess6. It also seems to show that large changes in

the definition of time to equilibrium may have rather small consequences on the estimated

equilibrium scour depth of long-lasting experiments. The Te values included in Tables 1 and 2

correspond to the application of the method suggested by Cardoso and Bettess6. All tests

lasted longer than Te.

4.2. Time evolution of scouring

As mentioned above, several authors have suggested equations for the time evolution of

scour. Most of them apply to the principal phase of the scour process at piers. One of these

formulas was suggested by Ettema1:

9 s 50

1 23p p

d D tν= k log +k

D D

where Dp = diameter of a circular pier; k1 and k2 = coefficients. Franzetti et al.2 proposed the

exponential function

10

2a

s1

se p

d Ut= 1 exp a

d D

in which a1 = −0.028 and a2 = 1/3. Equations (9) and (10) make use of different scour depth

scales. Recently, Melville and Chiew4 proposed one equation for piers, which is similar to the

following equation of Coleman et al.8 for vertical-wall abutments:

11

1.5s c

se e

d U t=exp 0.07 ln

d U T

In equation (11), both dse and Te depend on L/d and U/Uc.

Other contributions could be mentioned, particularly for piers. Most of them are particular

cases of equations (6) and (7). Those of Ettema1 and Franzetti et al.2 were derived for

practically constant values of U/Uc and, so, do not reflect the influence of this parameter; they

apply for piers, where, naturally, L/d, is comparatively small. The contributions of Melville

and Chiew4 and Coleman et al.8 include the effect of flow intensity, U/Uc, but seem to discard

the effect of the relative abutment length, L/d. It should be noted, however, that their time

scale, Te, and scour depth scale, dse, both depend on L/d. Consequently, L/d is inherently

taken into consideration too.

Comparison of equations (9) and (10) with equations (6) and (7), respectively, suggests that

k1, k2, a1 and a2 might depend on U/Uc as well as on L/d. For this reason, those coefficients

were calculated by regression analysis applied to the data from the principal phase (t < Te) of

each test of Tables 1 and 2, written as ds/L vs. Πt = D50vt/L3 – for purposes of evaluating k1

and k2 – and ds/dse vs. ∏t = Ut/L – for the evaluation of a1 and a2. It should be noted that the

length scale is, naturally, L instead of Dp.

Coefficients a1 and a2 take the values a1 = −0.087 ± 0.049 and a2 = 0.322 ± 0.041, and no

dependence on U/Uc or L/d could be identified. This result clearly differs from the one of

Cardoso and Bettess6 who claimed a1 to vary with U/Uc. The coefficient a2 is similar to the

one suggested by Franzetti et al.2, namely a2 = 1/3. In practice, the large scatter of coefficient

a1 around its mean discourages the use of equation (10) to describe the time evolution of

scouring.

The values of k1 and k2 were grouped into three classes defined by the associated values of

U/Uc and plotted against L/d in Fig(s). 2 and 3. The Fig.s also include the equations suggested

by Cardoso and Bettess6 for U/Uc ≈ 1. It is obvious that the scouring process depends on L/d

as well as on U/Uc: both k1 and k2 decrease as U/Uc decreases and L/d increases, in the ranges

0.54 ≤ U/Uc ≤ 1.12 and 2.01 ≤ L/d ≤ 36.33.

The variation of k1 and k2 with L/d and U/Uc is given by the followings regression equations:

12 1.8810.628

1c

L Uk =0.412

d U

13 2.0320.380

2c

L Uk =1.756

d U

where R2 equals 0.87 and 0.86, respectively. By replacing k1 and k2 in equation (9), the

following new equation for the prediction of the time evolution of scour is obtained:

14 1.881 2.0320.628 0.380

s 503

c c

d D νtL U L U=0.412 log +1.756

L d U d UL

It should be noted that equation (14) inherently takes into consideration the effects of the

obstacle Reynolds number, UL/ν, relative sediment size, L/D50, and the non-dimensional time

parameter, Ut/L. It only applies to the principal phase of scouring; for this reason, a predictor

of time to equilibrium, Te, seems crucial to render the above equation useful for application.

4.3. Time to equilibrium

It can be postulated that, because Te and dse are inherently interdependent, both should depend

on the same set of parameters. Consequently, for the onset of the equilibrium phase, i.e., for t

= Te and ds = dse, equations (6) and (7) can be rearranged to read, respectively,

15 50 e

3c

D νT U L=f ;

U dL

and

16 e

c

UT U L=f ;

U dL

Following the work of Lauchlan et al.9, Coleman et al.8 suggested the following upper bound

predictor of time to equilibrium that applies to the range of L/d values covered in this study:

17

36e

c

UT U d d= 10 3 1.2

L U L L

−

It is obvious that equation (17) is a particular form of equation (16). It was applied to the test

conditions reported in Tables 1 and 2 and the calculated values, Tec, were compared with Te.

For the ensemble of the data, the percent deviation, δ, defined as ( ) ( )[ ]eece TTT −=100%δ , is

−91 ± 130%. Apart from differences that might derive from different definitions of

equilibrium by Coleman et al.8 and by the writers, this result may reflect the fact that equation

(17) was suggested as an upper bound predictor of Te. Still, the influence of U/Uc does not

seem to be properly assessed. In some cases, the equation may under predict Te, particularly

for U/Uc < ≈ 0.75, and loses the nature of envelope curve. This is illustrated in Fig. 4, where

equation (17) is plotted for the central values of each class of data, i.e., for U/Uc = 1.00, 0.79

and 0.62.

Fig. 5 shows the variation of ΠTe = D50vTe/L3 with L/d. Data is grouped into the three

previously defined classes of U/Uc. The Fig. includes the best fit equation to the entire data set

18 -2.301

50 e3

D νT L=0.174

dL

(R2 = 0.94), as well as an error band around the best fit curve whose lower limit is defined as

50% of the predictions while the upper limit is twice the same predictions.

The data suggests that the functional relation implicit in Equation (15) seems to hold. From

Fig. 5, the influence of U/Uc on Te cannot be identified, in disagreement with the finding of

Coleman et al.8. Most of the data fall within the mentioned band, which, a priori, might seem

rather wide. Simply, for clear-water scour close to equilibrium, large variations in time have

small implications in the scour depth, itself, since equilibrium is reached quite slowly.

This fact is illustrated in Fig. 6, where ΠT90 = D50νT90/L3 is plotted against L/d; ΠT90 is

defined with T90 = time when the scour hole reaches 90% of the equilibrium depth (i.e., ds =

0.9dse). It is obvious that, particularly for L/d > ≈ 5, the majority of ΠT90 values fall below the

band defined by Fig. 4. This means that, for the majority of the experiments reported in this

study, the time needed for the scour hole to evolve to a depth of 0.9dse is such that t < 0.5Te.

The new predictor of Te – equation (18) – carries very important practical consequences. Let

us consider a 10.0 m long vertical wall abutment orthogonal to the flow direction, set in a

uniform sandy bed characterized by D50 = 1 mm. Let us also consider a flow depth of 2 m

(L/d = 5) and assume ν = 10−6 m2s−1. According to equation (18), equilibrium would be

reached in practically infinite time (after ≈ 136 years), while the equivalent time for a 0.50 m

long obstacle in a 0.10 m deep flow (L/d = 5) in a flume with the same sand bed, would be ≈

149 h. For U/Uc ≈ 1 – the most unfavourable condition – equation (14) predicts ds = dse ≈ 6.0

m at the 10 m long abutment; an hypothetical flood hydrograph lasting one month (rather long

flood) with U/Uc ≈ 1 leads to ds ≈ 1.1 m << dse. These remarkable differences in scour depth

and equilibrium time may explain why field studies tend to report observed scour depths

smaller than those calculated from equations based on equilibrium laboratory experiments.

This also means that, for the calculation of scour associated with flood hydrographs in flood

plains, equation (14) can be used with no special concern about the possible onset of the

equilibrium phase.

5. CONCLUSIONS

From the previous discussion, the following results on scour at thin vertical-wall abutments,

protruding at right angles from the side wall of a fully developed, uniform flow in a wide

rectangular channel, on flat bed of uniform non-ripple forming sand can be drawn:

i. time evolution of scour depends on the complex combination of non-dimensional

parameters expressed by the proposed new equation (14);

ii. time to equilibrium, Te, seems to depend mostly on L/d, according to equation (18);

iii. the new predictor of Te indicate that equilibrium time can be practically infinite in field

conditions.

iv. as a consequence of conclusion iii, equation (14) would apply with no special concern

with the possible onset of the equilibrium phase for the calculation of scour associated

with common flood hydrographs in flood plains.

6. ACKNOWLEDGEMENTS

The authors wish to acknowledge the financial support of the Portuguese Foundation for

Science and Technology, which funded the present research through the project

POCI/ECM/59544/2004.

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Test

d (m)

L (m)

Td (h)

Te (h)

U/Uc

(-) L/d (-)

F1 0.062 1.86 149.7 59.1 1.12 30.1

F2 0.062 1.86 166.2 87.5 0.90 30.1

F3 0.051 1.86 161.9 115.5 0.81 36.3

F4 0.055 1.71 169.0 160.7 0.75 30.9

F5 0.062 1.71 177.3 72.3 0.90 27.7

F6 0.068 1.71 167.0 127.5 1.02 25.2

F7 0.069 1.55 172.2 163.7 1.00 22.6

F8 0.060 1.55 129.2 54.4 0.92 25.7

F9 0.064 1.55 184.0 54.6 0.65 24.3

F10 0.067 1.40 168.1 105.7 0.62 21.0

F11 0.071 1.40 146.3 125.4 0.79 19.9 F12 0.065 1.40 509.1 150.1 1.06 21.4

F13 0.066 1.25 263.8 75.9 0.63 18.9

F14 0.069 1.25 216.7 91.2 0.80 18.0

F15 0.066 1.25 200.9 49.2 1.04 18.8

F16 0.070 1.09 189.5 129.4 0.58 15.6

F17 0.069 1.09 173.1 78.1 0.81 15.8

F18 0.066 1.09 166.9 70.6 1.05 16.7

F19 0.067 0.94 167.7 90.5 1.04 14.1

F20 0.070 0.94 168.7 91.5 0.79 13.4

F21 0.070 0.94 167.9 111.9 0.59 13.5

F22 0.069 0.79 167.6 88.7 1.01 11.5

F23 0.071 0.79 168.8 134.5 0.79 11.1

F24 0.071 0.79 147.4 96.2 0.57 11.1

F25 0.070 0.64 169.4 143.8 0.99 9.1

F26 0.070 0.64 170.0 112.9 0.58 9.0

F27 0.072 0.64 168.4 83.8 0.78 8.9

Table 1. Characterization of the tests carried out in this study

Test

d (m)

L (m)

Td (h)

Te (h)

U/Uc

(-) L/d (-)

S&C-A1 0.071 0.18 150.0 110.0 1.06 2.5

S&C-A2 0.071 0.18 94.4 94.4 0.96 2.5

S&C-A4 0.069 0.18 125.8 80.2 0.76 2.6

S&C-A5 0.070 0.18 70.6 40.0 0.66 2.6

S&C-B1a 0.071 0.40 237.1 120.0 1.05 5.6

S&C-B2a 0.072 0.40 166.6 110.0 0.93 5.6

S&C-B3a 0.069 0.40 91.7 54.1 0.85 5.8

S&C-B3b 0.069 0.40 91.7 45.0 0.85 5.8

S&C-B4a 0.070 0.40 138.9 52.8 0.75 5.7

S&C-B4b 0.070 0.40 138.9 62.5 0.75 5.7

S&C-B5a 0.069 0.04 114.1 60.9 0.65 5.8

S&C-B5b 0.069 0.04 114.1 55.4 0.65 5.8

S&C-B6a 0.070 0.04 91.7 40.0 0.54 5.7

S&C-B6b 0.070 0.40 91.8 35.0 0.54 5.7

S&C-C1 0.071 0.50 150.1 101.1 1.06 7.0

S&C-C2 0.071 0.50 94.5 72.1 0.96 7.1

S&C-C4 0.069 0.50 126.0 44.7 0.76 7.3

S&C-C5 0.070 0.50 118.7 27.8 0.66 7.1

C1 0.090 0.20 96.0 31.5 0.59 2.2

C2 0.090 0.20 133.0 41.0 0.65 2.2

C3 0.090 0.20 144.0 59.5 0.7 2.2 C4 0.090 0.20 120.0 70.0 0.86 2.2 C9 0.090 0.20 120.0 102.0 1.02 2.2 C&B1 0.031 0.15 100.5 50.0 0.95 4.7

C&B2 0.073 0.15 142.1 120.0 0.93 2.0

C&B4 0.078 0.27 117.5 75.0 0.94 3.5

C&B5 0.028 0.40 104.7 22.0 0.95 14.3

C&B6 0.079 0.40 70.0 55.5 0.92 5.1

K1 0.100 0.16 100.0 90.0 0.92 1.6

K12 0.050 0.52 98.0 44.6 0.93 10.3 K13 0.050 0.72 127.0 37.2 0.90 14.3

K19 0.050 0.87 72.0 34.0 0.90 17.4

Table 2. Characterization of the data selected from the literature

Fig. 1. Definition of the equilibrium phase

Fig. 2. Variation of k1 with L/d and U/Uc

Fig. 3. Variation of k2 with L/d and U/Uc

Fig. 4. Variation of UTe/L with d/L and U/Uc; comparison of measurements with predictions

by equation (17)

Fig. 5. Variation of D50vTe/L3 with L/d and U/Uc