Encyclopedia of Environmental Science and Engineering, Volume I and II

1072 SEDIMENT TRANSPORT AND EROSION

d)

∂

∂

()

;

PP

x

t

 0

e) tro gSoD and gtrSo (D y^ );

Chang et al. , used the above assumptions to obtain: (a) the

vertical velocity profile,

U

U

xx



 U

2

1

11

1

3

12

*

/

k

1n

⎛

⎝⎜

⎞

⎠⎟

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪ (22)

and (b) the vertical concentration profile

cy ca ya DDa

DDy

z

() ,

/

^

⎛

⎝⎜

⎞

⎠⎟

()−

()−

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

12 2

(23)

in which U *  friction velocity  gDs 0 ;  fcn ( U * d 
 v )  0.4;

 y 
 D; Ú  average velocity in vertical; a  reference height;

Z  v s 
( bU * k ). Using the Keulegan velocity distributions

U

yU

v

x^575

905

.log 10

⎛. ∗

⎝⎜

⎞

⎠⎟

(for smooth boundaries) (24a)

U

y

d

x^575

30 2

10

65

.log

⎛.

⎝⎜

⎞

⎠⎟

(for smooth boundaries) (24b)

Einstein and others 5,1 have obtained a slightly different equa-

tion for c ( y ), i.e.

cy caaD yyD a

z

() ⎡()()−−.

⎣⎢

⎤

⎦⎥

(25)

The Suspended Sediment Load The suspended sediment dis-

charge q s (weight
unit time
unit width) above a reference

level y  a is given by

qUcdys a x

D

g∫ ,

(26)

where U

• and c
  
  
  • are given by Eqs. (22) and (23) or (24) and (25).
    Longitudinal Dispersion Another problem which has
    received some attention is that of longitudinal diffusion and
    dispersion in natural streams and estuaries. Several research-
    ers 16,33,34,35 have sought analytical and numerical solutions for
    the longitudinal variation in the mean concentration in the
    vertical, c
  
  
  
  


• 
  

  .
  Considering two dimensional longitudinal dispersion,
  Eq. (17) can be approximated by^16

  
  

∂

∂

∂

∂

∂

∂

cˆˆ ˆ

t

U

c

t

E

c

x

    x L

2

2 (27)

in which E L ; coefficient of longitudinal diffusion. A typical^16

value for E L is

EUDL59..∗

The solution of Eq. (27) for an initial step change, M o , in

concentration, is

cxtˆ(,) ()/.

M

Et

o e

L

= −−xUtxLEt

4

(^24)
p
(29)
Other treatments of the dispersion problem may be found in the
works of Holley^28 Householder et al. ,^29 Chiu et al. ,^30 Conover
et al. ,^31 Sooky,^32 Fischer,^33 Harleman et al. ,^34 and Sayre.^35
The Total Sediment Load
Einstein^5 developed a unified total bed material formulae by
converting his computed bed load, q B to a reference concen-
tration at y  a  2 d. Inserting into Eqs. (25) and (26) he
obtained an estimate of q sB the suspended bed material load.
Hence the total bed material load per unit width, q TB is
q TB  q B q sB
  i B q B (1 P e I 1 I 2 ) (30)
all size
ranges
S 0
FLOW
T 0 =gDS 0
g(D–y)S 0
1
y
y
D
T
FIGURE 9 Defining sketch for uniform flow.
D
Ux
Ux
Cσ
C
y
S 0
σ
FIGURE 10 Defining sketch for velocity
and concentration profiles.
C019_001_r03.indd 1072C019_001_r03.indd 1072 11/18/2005 11:06:01 AM11/18/2005 11:06:01 AM