SEDIMENT TRANSPORT AND EROSION 1071
size range being considered; S e energy slope; i b fraction
of bed sediment in specified range; j hiding factor; Y lift
correction factor; d 65 X; X correction factor for hydrau-
lically smooth flow; d 11.6 n U * ;UgRS ∗ (^) e (12)
X 0.770 if d 1.80; X 1.39 d if d 1.80.
Schen^15 gives an up-to-date review of the modern stochas-
tic approaches to the bed material transport problem.
The Suspended Load
Equations of Motion of the Fluid The flow in natural streams
is almost always turbulent and may be assumed to be incom-
pressible; consequently the applicable equations of motion
for the fluid are the Reynolds^25 equations
rs
∂
∂
∂
∂
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
∂
∂
U
t
U
U
xx
i j i F
jj
() ,ji i
(13)
in which U
i^ ensemble mean point velocity in the direction i;
s ij stress tensor { P
dji mD
ji^ ruiuj} average pressure;
d ji Kronecker delta; m dynamic viscosity; D
ji deforma-
tion tensor; ru i u j turbulence or Reynolds Stresses; r
fluid density ; u i random component of velocity in the i
direction; F
body force in the i direction. The first term in
the stress tensor represents the normal stresses due to the aver-
age pressure at a point; the second term represents the viscous
shear forces; the last term or Reynolds stress has both normal
and tangential components. A common method of simplify-
ing equations involves the introduction of an eddy viscosity,
(^) m such that
rr
mDuuji i jj i()≠. (14)
The requirement that ( i j ) in Eq. (14) eliminates the normal
stresses due to turbulence; in order to account for these
normal stresses an average turbulence pressure P
i is added to
P
thus yielding the simplified stress tensor
sdmrji
()( ).PPt ji (^) m iDji
(15)
The fluid continuity equation is
∂
∂
U
x
i
i
0. (16)
Equations (13), (15), and (16) may be solved in a few
cases by methods developed to solve the Navier-Stokes
equations.
Transport of a Scalar Quantity in Turbulent Flow In an
incompressible turbulent fluid the conservation of a scalar
quantity requires that the rate of change of a scalar (say c
plus the rate of generation of c
at the point or
Dc
Dt x
h
c
x
uc F
ii
i c
∂
∂
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟ (17)
in which c c
c ; c
ensemble average of c; c random
component of c; D Dt substantial derivative; F
c_ is the gen-
eration term; h molecular diffusion coefficient. It is usual
to introduce, into Eq. (17), an “eddy” transport coefficient
(^) j , such that
uc
c
icxi
∂
∂
.
(18)
Since in most practical problems (^) c h, then Eq. (17) can
be reduced to
Dc
tx
c
x
F
i
c
i
c
d
∂
∂
∂
∂
⎛
⎝
⎜
⎞
⎠
⎟.
(19)
Equations (3) and (19) are valid for low sediment
concentrations. A review paper by Vasiliev^26 discusses the
governing equation which account for various levels of sedi-
ment concentrations. For example a first order correction to
the Reynolds equations is
rs
DU
Dt x
rc F
i
i
ji t
∂
∂
()( 1 ).
(20)
The volume continuity equation is the same as Eq. (16) while
the mass continuity equation becomes
Dc
Dt x
cu v
c
i isx
∂
∂
∂
∂
⎧
⎨
⎩
⎫
⎬
⎭
()
3
(21)
in which r ( S s —1); c
average ensemble concentration
at a point (massmass); n s settling viscosity; x 3 vertical
coordinate (opposite to the direction of n s ).
The Vertical Concentration Profile There is no general solu-
tion for Eqs. (3), (16), and (19) or (18), (20), and (21); however
a few special cases, of practical interest, have been solved.
Using the simplifications which result for steady, uni-
form flow in two dimensions (as shown in Figures 9 and 10),
it is possible to obtain a solution for the vertical velocity, and
concentration profiles. The following assumptions are typi-
cal of those required to solve Eqs. (13), (14), and (21):
a) c
1;
b) (^) c b (^) m where b ; 1;
c) F
_
x; grs 0 ;
C019_001_r03.indd 1071C019_001_r03.indd 1071 11/18/2005 11:06:00 AM11/18/2005 11:06:00 AM