Geoscience Reference
In-Depth Information
(
1
a, b)-6 and seven groups of unknowns
u
and swirling flows and for lower Reynolds num-
bers (
Re
), the RNG model behaves better than the
standard
k
−
ε
which is only valid for high Reynolds
number flows.
,, ,
Pc
u
,
u
,
K
,
u
lifti
(
i
= 1,N; N- total number
si
,
ri
,
drifti
,
i
,
of sediment species):
∂
∂
+⋅∇=−∇
++∇+
u
uu
1
∇⋅ =
u
0
,
P
Short description of RNG model implemented
in MassFLOW-3D
TM
t
ρ
νν
T
(1)
(
)
(
(
)
⋅∇
)
g
u
1
ab
,
The kinematic turbulent viscosity is computed from
∂
c
+⋅∇=∇− ⋅∇
−
si
,
2
u
cDc
u
c
si
,
si
,
lifti
,
si
,
∂
t
k
T
ε
2
x
(2)
ν
T
= 0 085
.
(7)
u
⋅ ∇
c
drifti
,
s
,
i
g
(
)
u
=
ρρ
−
f
(3)
The two-equations renormalisation group (RNG)
k
-
ε
turbulence model is based on the turbulence
viscosity hypothesis and solve two transport
equations for the turbulent kinetic energy,
k
T
and
the turbulent dissipation,
ε
T
.
The turbulent kinetic energy is defined as the
energy of the turbulent velocity fluctuations:
ri
,
si
,
si
,
K
i
−
()
∑
Ni
(
)
u
=−
1
f
u
−
f
u
(4)
drifti
,
si
,
ri
,
sj ri
,
,
j
=
1
µ
ρ
f
d
3
4
(5)
f
si
,
K
=
C
u
+
24
i
Di
,
ri
,
d
si
,
fsi
,
(
)
15
.
1
2
(
)
u
=
α
n
d
03
.
θ θ
− ′′
′
+
′
+
′
2
2
2
k
T
=
u vw
(8)
lifti
,
i s i
*
cri
,
(
)
g
d
ρρ
ρ
−
si
,
si
,
f
(6)
Where
u
′,
v
′
and w
′ are the component of the veloc-
ity fluctuations.
The transport equation for the turbulent kinetic
energy
k
T
is
f
(
)
∑
, , ,
- the mean
velocity of the fluid/sediment mixture (bulk
flow),
P
-pressure.
2.
c
s
,
i
- the concentration of the suspended sedi-
ment, in units of mass per unit volume.
3.
uuu
ri
N
N
1. a, b)
u
1
−
f
u
+
f
u
def
sj
f
s j si
j
=
1
j
=
1
∂
∂
k
t
∂
∂
k
x
∂
∂
k
y
∂
∂
k
z
T
+
u
T
+
v
T
+
w
T
(9)
=++−
PGD
ε
T
T
K
,
−
- the relative velocity.
def
,
si
f
The transport equation of the turbulence dissi-
pation
ε
T
is as follows:
4.
u
def
, ,
−
- the drift velocity to compute
the transport of sediment due to drift.
5.
K
i
- drag function.
6.
u
lift
,
i
-the entrainment lift velocity (volumetric
flux) of sediment.
u
u
drifti
si
∂
∂
ε
∂
∂
ε
∂
∂
ε
∂
∂
ε
T
+
u
T
+
v
T
+
w
T
t
x
y
z
(10)
2
C
⋅
ε
ε
T
k
(
)
+−
=
ε
1
T
PCGDC
+⋅
Now we describe how this system of equations
was derived and what it describes.
Equation (1a) is a mass balance equation for
incompressible fluid-sediment mixture.
Equation (1b) is an Unsteady Reynolds-averaged
Navier-Stokes equation for fluid-sediment mixture
(bulk flow), where ν - molecular viscosity and ν
T
-
turbulent viscosity related to Reynolds stresses and
ν
T
≫ ν. Turbulent viscosity is calculated by the RNG
turbulence model (Lyn, 2008). The RNG
k
−
ε
model
has several advantages over the standard
k
−
ε
model. It is more accurate for rapidly strained flows
ε
ε
ε
2
T
3
T
k
T
The rate of turbulent energy dissipation,
ε
T
, in
one-equation model is related to the turbulent
kinetic energy
k
T
:
= 0 085
32
32
k
TLEN
/
/
ε
T
.
(11)
The RNG model uses equations similar to the
equations for the k-ε model. However, equation
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