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Chapter 3 - Theoretical considerations
Introducing equations 3.80 in 3.81 and 3.82 and applying the simplifications 3.83 yields
v–
v s
2 ⋅k 1
⋅ d⋅ g ⋅µ
( ρ
------------------------------------------ s
– ρ w
)
= ⋅ ---------------------- =
k 2
⋅ ( C D
+ µ ⋅ C L
) ρ w
4
-
3
ρ s
d ⋅g ⋅ µ ( – ρ
-------------------------------- w
)
⋅
⋅ ----------------------
( C D
+ µ ⋅ C L
) ρ w
(3.84)
ρ w
and ----- C (3.85)
2 D
k 2
d 2 ( v – v s
) 2 v r
– v
⋅ ⋅ ⋅ ⋅ ⋅ ---------------- sr
– ( ρ
v – v s
– ρ w
) ⋅ g⋅ k 1
⋅ d 3 ⋅
∂h
-----
s
∂r
ρ
µ ( ρ s – ρ w ) g k 1 d 3 ----- w
v
⋅ ⋅ ⋅ – ⋅C 2 L ⋅ k 2 ⋅d 2 ⋅( v–
v s ) 2 ------- sr
– ⋅
⋅ = 0
In this equation, the last term - the centrifugal forces - has been neglected since it is an order of
magnitude smaller compared to the other forces acting on the grain. The longitudinal bed slope
has also been neglected.
Combining equation 3.84 with 3.85, KIKKAWA ET AL. finally obtained a relation giving the direction
of the particle movement:
tanδ
v
------- sr
v s
= =
v
----- r
v
v s
∂h
-----
– -------------------------------------------------------------------------------------
∂r
k 2
µ ⋅ C
----------- ------------------------ D v
⋅
2 ⋅ k1
1 µ C ⋅ ----------------------------------
L
+ ⋅ ------
( s – 1) ⋅ g ⋅ d
C D
(3.86)
At stable state, they assumed that v sr
= 0 , which leads to tanδ
= 0 ; therefore
dh
-----
dr
=
k 2 µ ⋅ C
----------- D
⋅ ------------------------
2 ⋅ k1
1 µ C ⋅ ----------------------------------
L
+ ⋅ ------
( s – 1) ⋅ g ⋅ d
C D
v r
(3.87)
At the bottom, v r
is given by equation 3.79. Now KIKKAWA ET AL. introduced a logarithmic
velocity distribution for rough walls ( v = v θ ( d) = A r ⋅ fr () ⋅ V∗ ) at the particle level z = d
with k s
= d and A r
= 8.5 for rough boundaries. The drag and lift coefficients for spherical
sand particles were measured by CHEPIL (1958). They are given for a wide range of shear Reynolds
numbers Re∗ . The friction coefficient µ was measured by IKEDA (1971). The sheltering coefficient
λ 0 , accounts for the sheltering effects due to other particles. A study of IWAGAKI (1956)
showed that the tractive force on a particle is reduced to 35% of the tractive force without sheltering
if sediment transport occurs over the whole cross-section. Since this sheltering effect is
defined for the square root of the tractive force, i.e. for V∗ ⁄ ( s – 1) ⋅ g ⋅ d, λ 0
is:
C L
λ 0
= 0.592 ; ------ = 0.85 ; C D
= 0.4 ; µ = 0.43
(3.88)
C D
By substituting these values in equation 3.86, KIKKAWA ET AL. obtained:
∂h
v
tanδ sr V h
-----
------- f ----------------- s 1
= = – ⋅ ⋅ ---- ⋅ -- ⋅F( 0)
– -------------------------------------------------------------------------------------
∂r
v s A r
⋅ V∗ r κ
k
----------- 2 µ ⋅ C D
f⋅A ⋅ ------------------------ r
⋅λ 0
⋅V∗
2 ⋅ k1
1 µ C ⋅ ----------------------------------
L ( s – 1) ⋅ g ⋅ d
+ ⋅ ------
C D
(3.89)
page 50 / November 9, 2002
Wall roughness effects on flow and scouring