On the propagation of a disturbance in the bed composition of an open channel

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r·

..

-

_-

_.-On the propagation

of a disturbance in the bed composition of an open channel

K, SUZUKI Report R 1976 /09/ L

I

Vloeistofmechanica

Afd. Weg- en Waterbouwkunde Technische Hogeschool Delft

(2)

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16I-/t-3!t

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_{On the} _{propagation of a disturbance} _{in the}

bed composition of an open channel.

by

Koichi SUZUKI

Augus t 1976

1

I

Delft Univ. of Technology, Dept. of Civil Eng., Lab. of Fluid Mechanics, Report R 1976 /09/ L

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Contents 1. General 2. Theoretical analysis 2"':1General

2-2 Equations of motion and continuity of sand mixture

2-3 Propagation velocity of small disturbance of bed height and composition of sand mixture

2-4 Time variation of composition of bed sand

2-5 Sediment transport function

2-6 Moving layer thickness

3. Experimental analysis

3-1 General

3-2 Experimental apparatus and procedure 3-3 Experimental conditions

3-4 Analysis of experimental results

3-4-1 Equation of sediment transport of sand mixture

3-4-2 Propagation velocity of a small disturbance of bed

height

3-4-3 Propagation velocity of a small disturbance of the

compos ition 4. Conclusion

.

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Acknowledgements References Main symbols page 2 3 3 3 6 9 9 17 22 22 22 22 30 30 30 30 36 37 38 40

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On the propagation of a disturbance in the bed composition of an open channel

1. Genera1

The transport of a sand mixture ;s more complicated than that of uniform sand, because the transport velocity of a grain is varying with the grain size. And in case of sand mixture, the change of composition of mixture in addition to the change of bed height according to the change of boundary conditions occurs.

It is very important from the point of river engineering to know how the bed level or the composition of mixture responds to a change of boundary

'conditions. Far example the change of the water level caused by the

construction of a dam causes a change of bed level and of composition of bed sand both up- and down-stream of the dam.

(5)

3 -2. Theoretical analysis 2.1. General

·

1 I

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The morphological phenomena of river bed can be solved theoretically if the equations of motion and continuity of both water and sand are

,

known. But even if these equations are known, it is difficult to find

out the solutions on some boundary and initial conditions.

Af ter some consideration of these equations, propagation phenomena of small disturbance of bed height and composition of sand mixture will

be discussed.

2.2. Equations of moti on and conti nuity of sand mixture

I

~ation of continuity of sand mixture

In fully developed dunes, we may assume, that the average dune height

(= 6H) is constant with time.

In this study the micro behaviour of individual grains is neglected;

therefore the dune layer may be considered as a continuous moving layer

with a thickness 6.

And if we assume that the dune shape is triangular 6 becomes a half of

the dune height (Fig. 2-1).

':

,

1 .

1 ;

1 I

I

'

I

'

I

moving layer sublayer

Fig. 2-1 Modification of moving layer.

If we consider the continuity of sand on a small distance in the bed,

the accumul ated amount of s and di in Mand duri ng a sma 11 time

a

is

I

(a)

I

!

I

.

,

I

where q . is the discharge of sediment with diameter di per unit time

PSl

and unit width. The accumulated amount of sand di in moving layer 6 in

Mand 6t is

(6)

4

-where 7,l:porosity of moving layer, and

Pi: frequency of sand di in moving layer

The accumulated amount of sand di in sublayer can be written as follows:

where z is the bed level and ~is is the frequency of sand di in sublayer.

But ~is cannot be calculated because ~is is the mean frequency of the

sublayer. But if we consider that only in the hatched part of the

sub-layer (Fig. 2-2) the frequency of sand di changes, we can consider the

accumulated amount of sand di in sublayer by considering the frequency

of the hatched part instead of Pis'

---,,---.,-- - - -

- -

-,.---

-.---;

1 a

1

!

I

ti

I

Fig. 2-2 Control section

The volume of the hatched part is:

:t

rz-à) AZ

~

t

and for the degradation of bed the porosity and frequency of the hatched

part are~o and Pio respectively, where suffix 0 means initial value of

sublayer.

But in case of aggradation of the bed, the frequency of the top of the

hatched part is Pi + ~~llt and the frequency of bottom of the hatched

part is Pi' sa we can write the mean frequency of the hatched part, using the frequency of moving layer.

IJ.

+

L

~t.:

At

r

-

2,""fif"

and as to the mean porosity

/L

+

_f_ ó17l.

LJ.t

2

»t

:

1

So, the accumulated amount of sand di in sublayer can be written as fo 11ows:

(7)

1

1 I

I

-~

I

'

I

,

!

I

1

1 I

1

:

1 I

:,

1 I

'

I

1

! I

:

1 I

I

- 5

-for degradation, if TIO and Pio are constants,

(c)

for aggradation,

(a) should be sum of (b) and (c) (or (c' )). Af ter dividing this equation

((a)

=

(b) + (c)) by ~x~t and putting ~x, ~t + 0, the continuity equation

for sand di reads:

for degradation of bed

(2-1)

for aggradation of bed

The total anount of transported sand per unit time and unit width jst is

(2-3)

and eq. (2-1) and (2-2) become for the total transport as follows,

respectively,

for degradation of bed

:e

I

0(1

-71

)

+

(

Z-d

)(1

~

7L

o

)

J

+ ~~

=

0

(2-4)

for aggradation of bed

(2-5)

Eguation of motion of sand

The transport 'Is; of sand with diameter di should satisfy the conditions:

1. when Pi = 0; PSi

=

0

(8)

- 6

-where fi is the transport function of sand di which coincides with the transport equation of uniform sand when Pi = 1.

The most silllpleform to satisfy the conditions 1 and 2 is

(2-6)

Eq (2-6) should be studied further but for the time being we use eq.

(2-6) for transport equation of sand di.

And as to fi, we can use a Illodifiedsediment transport formula of uniform

sand.

Equa tions of moti on and conti nuity of water

The equations of motion and continuity of water for one dimensional flow

can be written as follows

~

+

U dU

+

q ê)~

+

Cf p~

==

.

ot

~x:.

(J (9X. (f ólZ

·

1 I

I

~

I

,

1 ,

~

I

.

1 '

I

where u (2-7)

+

(2-8)

o

velocity in the x-direction

waterdepth

acceleratton of gravity,

bed shear stress bed height

density of water

hydraul ic radi us

2.3. Propagation velocities of small disturbance of bed height and composition

of sand mixture

I

The response of the sand movement to a change of boundary conditions is

very slow as compared with the response of the watermovement. For example,

when the water depth is changed at the downstream end of the flume, the

flow upstrealllis quickly affected if Froude number is much less than unity

a new quesi-steady state of the flow will be reached in a short time but

it takes a long time before the bed profile reaches a new equilibrium

state over the whole length of the flume. A small disturbance of the

water surface is propagated at the velocity u"!_[9Fï. In the next para

-graph the propagation velocity of a small disturbance of the bed profile

and that of a disturbance in the composition of sand mixture will be

(9)

7

-To arrive at solutions for the six dependent variables, viz. z, h, u,

Pi'~st and t-si' as functions of x and t, six equations are needed:

eqs. (2-1)(or(2-2)), (2-4) (or(2-5)), (2-6), (2-3), (2-7) and (2-8).

If w.e rewrite these equations, using

4 /

!w!<b

=

Wand

cJ.=

1/

{

f

-7t

)

,

and if we assume that

S

and D( are cons tant, we get

f)t.)

+

LA ÖlU

+

q

8l

+

Cl élZ - -w'

~

t

if.:( 0 ~X fj

sx. -

.

F}1z_

+

r;{

-t_u

)

= 0 O>V dx.. (a) (b)

~

st

=

ft

(

=

f

m )

j

si

=

~

fj

(c) (d) (e) (f) .j

I

~;

I

1

I

·

1 I

~

I

O(~

+ ~

88-

+

1<

()Z =0 (fl)

ö?

x.

0

~

t

Ijo~ (for degradation)

when our interest is only in the flow of water, eqs. (c), (d), (e) and

(f) may be oillitted.In this case by using eqs. (a) and (b) we get the

wellknown propagation velocity of small water surface disturbance Cl, 2

(2-9)

De Vries 1) derived the propagation velocity of a small disturbance of the bed for unifOrm sand C3, using eqs. (a), (b), (c ) and (e).

(2-10)

..

In case of a sand mixture, we will use the same analysis as ,done by de

Vries for uniform sand. To avoid complexity we may assume ICl,21~ C3 and C4 (C4 is the propagation velocity of a small disturbance of mixture composition) because the unsteady response of sediment is much slower than

that of water, therefore the unsteady term ~IÁ/ê)t. and

o>IL

/

(j

t

in eqs. (a)

and (b) can be neglected. Putting eq. (e), eqs. (c), (d), into eqs. (e),

(10)

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1

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1

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- 8 -,&U

d-t

-

+

o>U

dx:

=

ct

U ~ êYX

~

d

i

+

~~

dL ;::

dz

(g) (h) (i )

we reduce these equations as follows

0

(3

0 _~ 0 0

é)upt

-vi

0

rxtr1;fuJ

tCO)

0

S

o((

*

f6t.J

é)UfU. 0

0 {)((~l))

1 o

0 0

éJ

z

j

-d

₀

-g-Z-liXX.

-d

t

dx

0 0 0 0

dIJ.

0 0

d

t

d

-x

0 ₀

C9

f2

/

a

t

d

z

0 ₀ 0 0

d

t

_dt..

d~

/

~

dt

( A )

X

-

Y

where

ft

₌

_{u. (1}

_-

_1/

_F/

₎

(2-11) (2-11')

Mathematically, the.characteristic velocities are obtained by putting the

determinant of coefficient (A) in eq. (2-11) to zero, that is This gives

.

di

=0 (2-12)

C

= .

di)

__

ff!!_gft

=

_

1 _

1 _L

rlt_

.

3

(

7D~-

f

~U

1 -;t

1 -1'/ ~

i9u' [..( (2-13) (2-14)

(11)

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Eq. (2-12) means C1,2

=

:.':_c>oascompared with C3 or C4i, that is..the

assumption of this analysis. C3 and C4i describe the propagation velocity of a small disturbance of the bed and that of the frequency p. of

1

composition of sand mixture. Here we must take care about that these results

are based on IC1,2\» C3 or C4 i, therefore this analysis yields meaning

-ful results only if Froude number is much smaller or much larger than

unity. C3 and C4i can be calcula ted if the thickness of the moving layer

and sediment transport function ft or fi are known. So, in the paragraph

2.5 and 2.6 fi and

6

are discussed.

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2.4. Time variation of composition of bed sand

When sand with a different grain size composition is introduced into a

river or flume, the composition of the bed will gradually change. The

velocity_C4i of this disturbance front of input sand has already been

discussed in paragraph 2.3; it can be expressed by eq. (2-14).

If the characteristic velocity of di sand in bed C4i(0)

(=rK/d(-t)-ti~é)f;

~.

))

where (0) means values pertain~ng to the bed)) is larger than that of input

sand C4i(1) and (2) (where (1) or (2) means values concerning to input

sand), the time change of the comp~sition Pi can be obtained along the

characteristic direction as figured in the x (distance along the channel)

- t (time) diagram (Fig. 2-3 (a)). But in case of C4i(0)~ C4i(1), when

the upstream disconti nuity reaches to the downs tream discontinuity, a

shock front is formed and the time change of Pi can be figured as in

Fig. 2-3 (b). Here final velocity of shock front C4i( s) , which should

be between C4i(0) and C4i(2), can be written as follows, using

continuity relations:

C4fS)=(t.t:

~

-

t/"t4t)/ (

rr:

t:

O

))

.

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2.5. Sediment transport function fi

We assumed that the amount ~Si of transported sand di may be expressed

as fo 11ows

(2-6)

and when Pi

=

1 (uniform sand), thenfsi

=

fi. We may use one of modified

transport formula of uniform sand for fi.

Hiran02) and Michiue3) pointed out that if we use critical shear stress

for individual grains of mixture, a transport formula of uniform sand

can be applied to a sand mixture.

(12)

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-input sand

k

~I-.o-) --b-e-d-s-a-n-d-__ _,...-lt~f~t._"'r" --- x

I

,

:

1 ,

I

:j

,:

1

'I_,

I

f

l

I

1

t (a) x t (b)

(13)

- 11

-formula used by Michiue will be used. So, fi can be written as follows,

(

2-

1

5 )

.

where

~

(

=

j/~

-1

)

is the relative submerged density of sand,

~

~è

(

=

~

j/

Ad

)

and ~~are the nondimensional effective shear stress and critical shear

stress, respectively and _,AA is a ripple factor, Rb is hydraulic radius,

i is the energy slope and c and nare "constants". And

(2

-

16)

where suffix m means values concerning to the mean diameter sand.

j

l

'

1 j

I

J

I

Non dimensional effective shear stress

7';;

e

i

(

==

.,k<

!4

i/

.2J.

di

)

To calculate the effective shear stress in eq.

(

2 -15)

we must apply a wall connection for hydraulic radius at first and next estimate, a

ripple factor)Á. (Side wall correction for R).

Einsteinsl method for correction of wall roughness can be used in

calculating the hydraulic radius concerning only to the bed R~), using Chêzy coefficient Cl instead of Manning coefficient n used by Einstein.

That is

P

b

=

/I_ (1

-

2 1<

....,

/

8 )

(2-17)

where suffix b is concerning to the bed and w to the wall, h is waterdepth

and B is the width of the channel.

Hydraulic radius concerning to the wall Rw may be calculated from

:

1 ;

1

j

~

I

:

1 /<w

=

u

2/

Cw ;_

(

2 -

18)

with

Cw

==

_.,/;P

/

f/}..o

_c!

J/

2R !/ by trial and error method, using known wall rough-WTo.3

ness

1<""

and Ó/= f1.6i)/u'}f-:: tf.6z)/lrfRwt' Af ter determining Rw' Rb can be calculated, using (eq.

(2

-

17)

.

The wall roughness kw'of the flume used in our experiment is order of -5

10

m.

(ripp 1e factor).

One may consider the total bed shear stress

2b

(

=

~1b1)

which is already

.. .

corrected for side wall effects, to be the sum of two components, viz. a grain shear stress

7

g and a form drag

2j ;

(

2-

1

9 )

I

(14)

I

1

j

;

1 ;

,

I

·

1 I

I

- 12

-where suffix f is concerning to the form drag and 9 is pertaining to the

grain drag. Mayer-Peter

&

Müller 5) divided energy slope ig into if and ig, keeping the hydraulic radius R unchanged and Einstein ) divided Rb into Rf and Rg' keeping i unchanged.

Meyer-Peter

&

Muller assumed that only the grain drag contribute to the

sand transport and related the grain shear stress to the total bed shear stress by means of a bed form factor

~: effective shea r stress And if we use the Chézy formula for the mean velocity u ,

u.

2 =

C;/

R

~

6

I U2=C/

Rl

!

~ can be written as follows

(2-20)

where C is the Chézy coefficient. A closer fit of the Meyer-Peter

&

Müller transport formula to the experimental data was obtained if the theoretical value of the exponent in eq. (2-20) was changed from 2 to 3/2. Prins7) al so modifi ed ~ in the range

0.4 <

C6/q_

<:

0.8 .'

~ =

1 .

3 (Cb

/

C~

y~/2

fitting his experimental data to the Meyer-Peter

&

Müllers' equation.

It is unlogical, however, to change the exponent 2 because eq. (2-20) is derived using Chézy formula.

From experimenta 1 observa tion it appears that immedi ately downs tream of the dune crest some particles are lifted up by the wake; therefore it seems reasonable to assume that not only the roughness drag but also some

part of the form drag contributes to the sand transport, say,

?'e

=

Zl

+

C{Zj =

10;'+

(1-

a

)

(Zj/Zi. )

f

(b

and so,

(2-21)

But ()(should be determined by experiment. Cb can be calculated by

Cb

=

u/~, and Cg can be calculated by using logarithmic distribution of flow velocity

~ =

U/IRb

i

=

/Jl

ilo;

::::'

?

d

/

(15)

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, t - 13 -1

_-I

I

_,

_'/I .0

fD7

11

• TL . o .

fDfJ,~

- .. ?y<:-~~~I

~

· ~

~

o

/

~

-

'<r

(V

6 )

))

.

I

'OW

~

é

t

"yl

4!

;'

1. I

• 'tl-I-i

17 ,

:

~

!I

jo C{0 I 0.(,/

~~;~ • Shinohara

&

Tsubaki

/ 0 0 de Vrles / (I) Prins j -

o

_d_'_A_g_o_s_tino - .. _Aut_h_or 0.1 0.1

(16)

14

-Fig. (2-4) shows the relation between ~ and Cb/Cg' calculated by Meyer-Peter

&

Müllers' equation of sediment transport and experimental data. Nearly all data obtained at Delft are situated between

ft=

(

4/CfJy~/2.

(Meyer-Peter & MUller) and _.#.::=f.3(Cblq)'Wand fit eq. (2-21) in case of

=

0.2 ...0.3.

Non dimensional critical shear stress ~ci

I

1

1 :

1 !

.

Since the report by Shields, it is generally assumed that the non-dim

en-sional critical shear stress

&c.

for uniform sand is constant in turbulent

flow (say U~d/))>;vfoo), .where L{jtis the shear velocity and J is the

kinetic viscosity); values for this constant range from almost 0.045 to

·about 0.065.

Shields - 0.06, Iwagaki ~ 0.05, Meyer-Peter & MUller 0.03"'"0.05 5).

Because of variation in ~xposure of the flow, the smaller grains of a

sand mixture are less easily moved than grains of the same size in case

of uniform sand; for the larger grains of a mixture the reverse hol ds. Several fO~lulae for the critical shear stress of sand mixture have been

proposed (Kramer8), Sakai9), Tsuchiya10), Gessler11), Egiazanoff12)) ..

Among them Egiazanoff's formula is simple and based on theoretical análysis considering drag and gravity forces on grains of uniform sand and velocity

of the top of individual sand of mixture.

The non-dimensional critical shear stress of individual grain whose diameter

is di in the mi xture 2;4is written as fo11ows

I

,I

( I ,

I

r

;

1

(2-22) , I

:

1 ·,

1

.l ]

~

I

where dm is mean diameter of sand mixture. Coxis drag coefficient, which

is equal to 0.4 for completely turbulent flow and

U

0t7is the non-dimensional

critical shear stress for uniform sand whose diameter is dm. Eq. (2-22)

means tha t if dil

<

dm , then ~ci 7Z,cmand if di'> dm,

Z

~

<Z;cm .

Ashida & Michiue13) observed that in case of di/dm <0.4, Cadoes not

vary with di, then

Z:c~

=

o.~

0,

.

Now, if we use Meyer-Peter &Müller's sediment transport equation ve should use 0,047 for the val ue of ~ and so we get

(2-23)

Fig. 2-5 shows the relation eq. (2-15) (where

f;.==Wt

J'

using eq. (2-23) for

2'k

~

and C = 8 and n = 3/2 which are Meyer-Peter & Müllers cons tants.

(17)

1 I

-I

I

,

I

I ! <,

I

ti

rI

·

't1 en

I

""lor! Pi <, tQ 0'

I

₁₀₀

I

.

I

j

,J

I

10-1

I

( ~

I

10-2

I

i

I

- 15 -\ v(/_

r

i

f-•

Exp, data

11-in _Çhap.3 -

f-

9-W

I

-I -I

-t

-

--

-I

I

J

=-0 o 0_7 " Cl • 'I'---( .oef(

r;;

I

(,0o~ 'r;7l:! '\ 'r;7'1'-(

1 :

!~

~

I

.

i.i

1.4-I--I-

l-j

I

(18)

_ .. _ _ ..

,

.

-

~

..

"

,

..

'

_ iÎÏII

~

"

· '"

,

_ .. ..

'

-

..

.

1 ... -- .. - -:_~_ ! . 'j'

I

' '..

I

'

..

lf

..

'

"

j

'

l

I

.

'T

: .

,:

:r

t

l

~

!:1

:

,

.

:

~

:

_-

:

1~J

(!.

: "'

,

::j.

T

L

I .

l

:~

~>[:

i

· I

;

HW

I1

T

j

:

II

"'

1

1 ':'

1 c:"

11 '

,,

~

=

·

· -

-

è

'''-

:

=

F

'''

c

i

;j'

t

i

l

I

T

-,

n

· -

i'

LU

'

r

,

'

l

'

illZl

-

:

--:::1:-'':0.2

:

-::

.

1

'i ; i I .. , -100 ,':',- .." ..'~; : I AC>" 0. _ .:I: .: :

I'

I

:

I

. ,

I:

..

!:

.

':'

n

"

..

.

~.:

L

_

':":__:_t-·~"

,

I

"

I

' ,

-~

~

_

. :. _-

~

..

----;;r

:

ï

ll :..:~:--=t-c=

=

--=-

I

===t=-=--:

.r: ::1[. ..:

.!

! I ;

I

calculated-- __'.

,

i

:

.:

~

;

-)Jf-~_=--::

~~=-~l~

'.'::: :.. ':: : I I . . .

I

'

observed·'

I

J

;

-

-.::----='

-=t

=

.

...

--==- -:.=Ë:-==-= . ":.'_r::-:-:-_-:- _"_, _-_:'._. -I·...·

-

1 '-

.

-

-':__

'

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2 .; Fig.2-6 Relation between bed sand and .inpu t sand compo sit ion _

~

-,

I-' (J')

(19)

17 -.1

I

The va1ue of

t>

,lt/~

'5

becomes sma11er as di/dm becomes sma11er,

and its difference is fairly large in the vicinity of the critical

shear stress of ~~ but as

Z;e~

becomes larger its difference becomes smaller.

In order to check the eq. (2-15), we can use the data which shows the relation between pi which is the frequency of output (or input) sand and Pi which is the frequency of bed sand.

I'

I

,

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1

\

:

1 j

'1

L

1 '

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(2-24)

By eq. (2-24) we can calculate p~ from known bed frequency p. using

1 ~

eq. (2-18) for fi; in Fig. 2-6 (b) solid lines are calculated from dot

lines. That is, at first we divided sand mixture into 10 groups (dl, d2.. ... d10) each of which occupies 10% in mixture (pI = p2 = = plO =

and next for each group we calculate p~ using eq. (2-24) (Fig. 2-6(a)).

1

From Fig. 2-6, the experimental data fit calculated values fairly well. Experimental conditions are listed in Table 3-2.

From these considerations we may use eq. (2-15) and assumption eq. (2-6) for the sediment transport equation for sand mixture.

0.1)

I

2.6 Moving layer thickness

á

Mathematically C4i indicates the propagation velocity of a small distur-bance of composition p. and physically it seems to indicate the transport

1 .

velocity of sand which has the diameter di.

$0, in case of uniform sand C4 should be equal to C3, because in case of

mixture the front of bed height disturbance should b~ one of the dis -turbance front of composition Pi and if we consider almost uniform sand

di = dm every C4i is almost equal and its value is nearly C3.

From C4 = C3, we.get moving layer thickness

J

which is assumed a half of the dune height ( =.áH/2). Using eqs ..,(2-13) and (2-14), we get

: (-

f

)

-

;-1'() u.~

(

;fmfo

(A)

(2-25)

where suffix m means v~lues concerning to the mean value. (for dml.

We can calculate the dune height by eq. (2-25) using proper transport formula of sediment fm. Of Meyer-Peter & Müllers I equation of sediment

transport (eq. 2-15) is used, eq. (2-25) becomes

(2-26)

(20)

i

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and if we can assume that hydraulic radius Rb is equal to water depth h , eq. (2-26) becornes,

't

=

-:;-(1-

!c

)(1

_~2

)

(2-27)

k

,_g_.L_

where, .1=,,{. ..,j

d

.

Fig. 2-7 shows the eqs. (2-26) and (2-27). It is seen that L'1 H/h is mainly

determined by Froude number Fr and

K

or

Kl

'

which contains the effects

of the slope i and h/d.

P. Mardjikoen14) also derived the relation between dune height and (1 - F;) by using the work of de Vries and experimental data of Shinohara -Tsubaki and Znarnenskaya.

(2-28)

where b is defined by f

=

a.ub (a

=

constant) and by using Meyer-Peter

&

Müllers' equation of sediment transport b is expressed as follows

P

=

3/(I-o.4YJ

where

y

==

t'5

/

I

A

7d>

But in deriving eq. (2-29) he assumed b as independent on

the results eq. (2-29) shows that b is dependent on u.

But if b is kept constant, letting a have dirnensions, eq. (2-28) coinsides

with eq. (2-26) (or(2-27)) in case of neglecting the critical shear stress.

Fig. 2-8 shows the relation between observed dune height and calculated

dune height by eq. (2-26). In calculation, n = 3/2 and ~= 0.047, which

are Meyer-Peter & MUllers' values, are used, _"'«iscalculated by eq. (2-21) and eb is calculated

bY

a

~/1i/

.

Data are quoted from experiments

done by De Vries15), Shinohara-Tsubaki16), Prins4), d'Agostino17) and from author's experiments described in the next chapter.

(2-29) u although

If we ,consider the accuracy of measuring the dune height, be said to describe the dune height fairly well. Also posed an experirnental equation, using a lot of data:

eq. (2-26) rnay

(21)

-- 19

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z:

=0.041 ..te

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0.5 1.0 Fig.2-7 Fr

Relation between relative dun height AH/h

and Froude numberF r

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-oShinohara & Tsubaki ede Vries <DPrins ~dtAgostino

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Author

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0.5 1.0 (LlH/h) observed

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-(hcr: critical flow depth for the beginning of sediment move).

whi eh meansLlH/h can not exceed 1/6 and Nordi n et a119) found tha t

1/3 is better than 1/6. Fig. 2-8 shows that there are even values of

~ H/h which exceed1/3. According to Shinohara-Tsubakis' work16),

4H/h is mainly a function of the non-dimensional effective shear stress

2'e*

.

Eq. (2-26) in which KF; is equal to

ZVe ,

shows that i f the

variation of Fr is not ve,ry large, .A H/h is determined only by ~.

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22 -3. Experimental Analysis. 3.1. General

Some experiments were done to check the equation of sediment transport (eq.(2-l5)) and propagation velocity of small disturbance of the bed form

and composition Pi (eq. (2-i3) and eq.(2-l4)) and the variation of the

composition of sand mixture.

3.2. Experimental apparatus and procedure

The experiment has been done in a flume of the laboratory of Fluid Mechanics of the Delft University of Technology. The flume is outlined

inf Fig. 3.1. This flume has a sand elevator which is raised automatically at a desired constant velocity which may range from 0 to about 10 cmjhour

(this is corresponding to 0.0425 m3jhour in bulk volume).

Water discharge is measured by an orifice plate and controlled by a valve;

the water surface level is adjusted by the weir at the downstream end of the flume. The sand output which is trapped by a settling basin near the end of the flume is measured by weighing. The water surface profile and the

bed profile are measured by a point gage and a bed profile indicator, respec-tively, mounted on a carriage which runs over the flume with a constant

velocity. Bed profiles are registrated by paper recorder and by magnetic re-corder.

Before the beginning of the experiment, an equilibrium state is established; for both experiments this took approximately 70 hours (Fig. 3-2).

The sampling of the sand of the bed is done af ter stopping the flow, keeping the water depth larger than the original flow depth. The surface grain layer on the whole area of upstream side of a dune is scraped off in order to achieve a sample which is representative for the moving layer.

The sample weight is about laag; af ter drying, this sand is sieved using calibrated wire sieves.

3.3. Experimerrtal conditions.

(sand)

The composition of the sand used for input Pi (which is different from the composition' of the bed sand P.) is shown in Fig. 3-3. The porocity~of this

(25)

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Experimental conditions are listed in Table 3-1.

In Exp. 1.1'.3 and 3' uniform sand is used. Exp. l' and 3' are done to

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. 1 Table 3-1 Experimental conditions Exp. No 1 l' ₂ 2' ₃ 3' 4 4' Water discharge ~{cm2/s) 900 bed sand (a) (c (b) (d ) (a) (b) input sand (c) {c) {d) (d}

non-steady conditions lIh LIP. lIh lIh lip. lIh

lIh=l cm 1 1

(summary of experiment)

A summary of the data of experiments 1,2,3 and 4 is listed in Table 3-2. Table 3-2 Summary of experiment Exp. no 1 2 3 4 shape is 1,10 x 10-3 1,10 x 10-3 1,06 x

10-3

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