Geoscience Reference
In-Depth Information
constants that are found empirically in the stand-
ard k-ε model are derived explicitly in the RNG
model, like
C
ε
2
is computed from the turbulent
kinetic energy (
k
T
) and turbulent production (
P
T
)
terms.
In order to prevent unphysical small dissipa-
tion rates, the minimum dissipation is limited by
a maximum length scale, represented by TLEN in
MassFLOW-3D™.
is a multiplication of the velocity of the sediment
in each computational cell, bed load thickness,
the volume fraction of the bed-load layer and den-
sity of the sediment species,
Q
=
u
δρ
f
(13)
bi
,
bedload iibi si
,
,
,
Hindered settling is not important for the low
concentration of the sediments.
min
= 0 085
32
32
k
TLEN
/
/
ε
T
.
(12)
,
Additional notations
ρ
f
- fluid density.
ρ
s
,
i
- density of the sediment species.
ρ - mean density.
f
s
,
i
- volume fraction of sediment species
i
.
d
s
,
i
- diameter for sediment species
i
.
C
D
,
i
- drag coefficient for sediment species
i
.
P
- pressure.
TLEN and the initiation of the turbulent kinetic
energy
k
T
are the two main turbulent parameters
that are defined by the user during the setting up
of a model. Equation (2) is a mass balance equa-
tion for each sediment species
i
, which describes
the motion of the suspended sediment in the sys-
tem by the advection-diffusion equation, with the
addition of the effect of drifting and lifting of the
sediment.
Equation (3) is obtained by subtracting the
momentum balance for fluid-sediment mixture (1)
from momentum balances for each sediment spe-
cies
i
, with the use of relative velocity definition
and assumptions: a) that the motion of sediment is
nearly steady at the scale of the computational
time and that the advection term is very small due
to small gradients in the drift velocity, b) that the
ratio of pressure gradient to mixture density is typ-
ically proportional to the acceleration of gravity, g.
Equation (4) is derived from the definitions for
uu
2
2
2
∂
∂
u
x
∂
∂
v
y
∂
∂
w
z
2
+
2
+
2
µ
ρ
PC
V
=
T
SP
2
2
2
f
∂
∂
v
x
+
∂
∂
u
y
∂
∂
u
z
+
∂
∂
w
x
∂
∂
v
z
+
∂
∂
w
y
+
+
+
- turbulent production term.
C
SP
- turbulent parameter, whose default value is 1.0.
µ
ρ
∂
∂
ρ
∂
∂
p
xy
+
∂
∂
ρ
∂
∂
p
yz
+
∂
∂
ρ
∂
∂
p
z
GC
x
T
=−
- the buoy-
ρ
3
ancy production term.
C
ρ
- has a default value of 0.0 unless the problem
is thermally buoyant, in which case it takes a
value of 2.5.
and
Equation (5) is obtained by combining form
drag and Stokes drag, (Clift
et al
., 1978).
Equation (6) is modelled to calculate the entrain-
ment lift velocity (volumetric flux) of sediment
and the entrainment coefficient
α
i
, based on
Mastbergen & Van Den Berg (2003).
,
u
.
ri
,
drifti
,
∂
∂
vk
x
∂
∂
+
∂
∂
vk
y
∂
∂
+
∂
∂
vk
z
∂
∂
T
D
=
T
T
T
T
T
K
σ
σ
σ
x
y
z
k
k
k
- diffusion term for the turbulent kinetic energy,
σ
k
as a value of 0.72 for the RNG model.
C
ε
1
,
C
ε
2
and C
ε
3
- non-dimensional parameters. The
default value for
C
ε
1
is 1.42.
C
ε
2
is computed based
on the value of
k
T
,
ε
T
and the shear rate.
C
ε
3
has a
value of 0.2.
Bed-load transport and packing of sediments
Bed-load transport describes the movement of
large particles along the surface of the bed without
being entrained into the bulk fluid flow. The
Meyer-Peter & Muller formula predicts the volu-
metric flow rate of sediment per unit width over
the surface of the packed bed.
The mass flux of sediment as computed across
computational cell boundaries in MassFLOW-3D™
∂
∂
v
∂
∂
ε
∂
∂
ν
∂
∂
ε
∂
∂
ν
∂
∂
ε
D
=
TT TT TT
+
+
ε
x
σ
x
y
σ
y
z
σ
z
ε
ε
ε
- diffusion term for the dissipation
where
σ
ε
is equal to 0.72 for the RNG model.
n
s
- outward pointing normal vector to the packed
bed interface.
Search WWH ::
Custom Search