246 Computational Modelling in Hydraulic and Coastal Engineering
Furthermore two of the remaining challenges are
1. To estimate the drag on each bubble and the vertical steady-state
buoyancy velocity
2. To correlate the drag force on each bubble to the force exercised by the
surrogate particle used in the simulation procedure to represent a number of air bubbles and to correlate such a particle with a gas volume
The drag on a bubble is given by the formula
F
C U R
d
d
b
b
=
1
2
2
2
ρ π
(8.57)
where C d is the drag coefficient related to the local Reynolds number
R
U D
e
b b
= ν
by the relation
C
R
R
d
e
e
=
+
+ +
0 4
24
6
1
.
(8.58)
From the equilibrium of buoyancy force and the drag force on the bubble, the final U b velocity is given by
U
gR
C
b
b
d
=
8
3
(8.59)
Equations 8.57 to 8.59 can only be solved iteratively.
Given the two main air-bubble flow parameters, q g and R b , the number
of bubbles, n b , entering the flow field is estimated as
n
q
R
b
g
b
= 4
3
3
π
(8.60)
By assuming a large total number of surrogate particles, i p , the number of
particles, n p , entering the solution domain per unit time (second) is
n
i
n t
p
p
t
= ∆
(8.61)
where n t is the total number of time steps. Then, the equivalent drag force
divided by the density, acting on each particle, is