Duan C.G., Karelin V.Y. Abrasive Erosion and Corrosion of Hydraulic machinery
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134
Abrasive Erosion and Corrosion of Hydraulic Machinery
dprt) I 0(U
2
4>) I 0(UJ) _ d
+
■
<5£
d_
dr]
% %
JaT
at
+-
d_
Ja
2
T
dr]
Ja
3
T
X.
+
{s
d
+JS,) (3.68)
where S^ is the source term indicated on the right side of governing equations.
S
d
is the extra source term resulting from non-orthogonality, that is,
S
d
=
Ja
4
T
0n.
Ja
4
T
Q
%.
+-
d
Ja
5
r^
N
d
(
dn.
dr,
Ja
5
T-
+
■
Ja
6
T
+
-
d_
JaT
cty_
where U
u
U
2
and £/
3
are contravariant velocity components in the body-
fitted coordinates.
U
2
=j(lJ
x
U
+ TJ
y
V + 1J
z
w)
U
3
=j(£
x
u
+
C
y
v
+
£
t
w)
where cc
u
a
2
, a
3
, a
4
, a
5
and
cc
6
are transformation coefficients.
<**=?xTi
x
+?
y
ri
y
+t;\ri\
a
6
=
Z'
x
£'
x
+Z'
y
C
y
H\C\
and
Analysis
and
Numerical Simulation of Liquid-Solid Two-Phase Flow 135
hy
~ Jby ~
X
C
Z
1
~~
X
n
Z
C
rfx=Jv
x
=yc
z
s-yt
z
c
V'
y
=Jri
y
=x
4
z
c
-x
c
z
4
n'z
=Jriz
=
x
ty
i
-
x
(
y4
C=J<^
x
=y4
z
r,-y,i
z
4
h y
=
J h y
~
X
?i
Z
£
~
X
4
Z
n
C =JCz
=
x
4yr,-
x
r,y
4
And
J
is
the
Jacobia of the coordinate transformation.
J
=
0(x,y,z)/0(54,0
=
y
s
y,
y
(
=
x4-?x+ye-?y+*r?
* Discretization
of
equations
Discretization
of
governing equation (3.66)
is
performed
by
using finite
difference approximations
in the
transformed domain.
The
second order
central differencing
is
used
for
approximating
the
diffusion
and the
source
terms.
For the
convective terms,
the
hybrid differencing scheme
is
employed,
that
is,
using central differecing
for
cell Peclet number less than
or
equal
to
two
and
switching
to
upwind differencing when
the
cell Peclet number
is
greater than
two. The
finite difference equation
is
arranged
by
collecting
terms according
to the
grid nodes around
a
central volume
as
shown
in
Figure
3.1.The final expression
is
given
by
equation (3.67)
in
which
A
represents
the
link coefficients between
the
grid nodes
P, E, W, N, S, T, B, WE, NW, NT,
NB,
SE, SW, ST,
SB, ET, EB, WT and WB.
Abrasive Erosion
and
Corrosion of Hydraulic Machinery
Figure 3.1 Three-dimensional grid system
P
=
A
E<t>E
+
A
W<I>W
+
A
Nr>N
+
A
S^S +
A
TT>T
+
A
B0B
+ S
1
(3.69)
S
1
=S
+ A
m
<p
NE
+ AJW <p
m
+ A
m
(/>
m
+ A
NB
(j)
NB
"*"
^SETSE
"*"
™sw
rsw
"■"
^sr rsr
~*~ ASBr
SB
+
/Igj.
^^ + ylgg
^
£S
+ ^4^
(f)
WT
+ A
WB
(p
WB
Ap
=
Ag +
y4^
+ A^ +
/i^
+
^4j
+ Ag
■
D
-[-t-
D
-
■Vt
A
w
-
D„
2
l/
+
^k
A
N
—
D.(-f-D.
1
An
2
Analysis and Numerical Simulation of Liquid-Solid Two-Phase Flow 137
A
s
=
JH.J- —
,D,
D..
V-f
F
•A(-f
=
A-
.1
F
u
DA+^-
= D
b
•>^b
P.
■*
At
2
V^f
where F is the convective coefficients and D the conductive coefficients as
follows,
F
e
= (pU^\
F
s
= (pVArjl
F
w
= (pU&Zl
F
t
= {pWAC\ F
b
= {pWACl
F F F F F F
p _ e_ p _ w p _ n_ p _ £_ p _ t_ p _ b
r
Ae ~ r^
■>
r
Aw ~ r. '
T
An ~ ^ '
r
As ~ r. '
r
At ~ r, '
T
Ab ~
D. D.. D. D. D. D
u
e
. s
X
J
2
X
w
, A =
X
—-a,
J
3
X
a
*
_ _i
D.
=
D =
p
where the o> represents taking the maximum values,
A
is the grid Reynolds
number. The hybrid-differencing scheme can be indicated as follows,
= 0
P > 2
E
/
■2
< P < 2
E
/ =
1
-
—
P
"P,
Ae
138 Abrasive Erosion and Corrosion of Hydraulic Machinery
SIMPLEC solver for incompressible flow
The SIMPLEC algorithm is applied to drive the pressure field and the
velocity field to be divergence free and convergence.
At first, the finite difference momentum equations for velocity
components u, v, w can be written as
i
A
l
K
=
£
A
i
v
*
- K
+
S
*
(3.70)
i
A;W;
=
^ATW:-P;+S
W
i
where u*, v* and w* are the solutions of equation (3.70), in order to satisfy
the continuity equation, the velocity and the pressure are corrected according
to the following relations:
u-u* +u'
* i
V = V +V
* , (3.71)
w = w +w
* t
p = p +p
Then a new set of momentum equations can then be approximately
constructed using the divergence-free flow field u, v, w and/?
^>„
=
2», -P
X
+S
U
i
A
v
v
=Y A
v
v-P +S (3.72)
v v ' '
s i
y
v
i
A;W
P
=
Y
J
A:W,-P
Z
+S
W
Analysis and Numerical Simulation of Liquid-Solid Two-Phase Flow
139
By subtracting equation (3.70) from equation (3.72), respectively, the
following equations are resulted
i
A
v
v
=YA
v
v -P'
p p Z-i ^ > y
i
(3.73)
According to the SIMPLEC algorithm, equation (3.73) are rearranged by
subtracting terms, ^A"u
p
from both sides to be as,
4-Z4
4-Z4
A
',-Z4
U
P
=E4
:
(«,- -
u
p)-p*
i
V
P=H4(
V
i-
V
p)-Py
i
W
p
=y
Z
A
i{
W
i ~
W
p)-Pz
(3.74)
The first terms on the right-hand sides of equation (3.74) are neglected to
simplify the formulation. Thus
1
i J
P
X
=
~D
U
P
X
1
4-Z4
v
V i J
P
y
= -D..P.
v y
(3.75)
140
Abrasive Erosion and Corrosion of Hydraulic Machinery
W
P
\
A;-T
A
?
P, =
~D
W
P
2
J
By equation (3.72), the continuity equation can be written as
«x
+
V
y +
W
z =
fc
+
V
y
+
W
l)+k
+
V
y
+
M
0 = 0 (3.76)
Substituting equation (3.75) into equation (3.76), the following pressure
correction equation can be obtained:
-h^l+(D
v
P
y
l+{D
w
P
2
\]=-{<+K
+
<)
(3.77)
Solution Procedures of Two-Phase Flow
The present numerical method for solving liquid-particulate two-phase flow
contains the following steps:
1) Estimate the initial velocity and the pressure field of a single liquid
phase flow in the computational domain.
2) Solve for the velocity, the pressure and the turbulent characters k and s
in the single phase flow, by using the SIMPLEC method, to get coarse
convergence solution of the flow.
3) Based on the single phase flow solution, we solve for the particulate
turbulent flow field, by using the algebraic particulate phase turbulence
model.
4) Calculate the interaction terms between the liquid- and the particulate-
phases to get some source terms in the control equations for the liquid
flow.
5) Solve again the momentum equations, the turbulent energy and its
dissipation rate equation for single phase flow, but including the
interaction terms.
6) Set the new variables and return to step 3); repeat the process until it
converges.
Analysis and Numerical Simulation of Liquid-Solid Two-Phase Flow
141
3.3.2 Calculated Examples of Two-Turbulent Flow by Using Two-
Fluid Model
Numerical Simulation of Liquid-Particle (or Gas) Two-Phase Turbulent
Flow through a Centrifugal Pump Impeller
The numerical approaches and the computer program has been applied to
calculate the single-phase incompressible turbulent flow through a rectangular
region with backward-facing step with comparison to the measured data
[3.25],
where the maximum differences between calculated- and measured-
data is less than 7%, and through a centrifugal impeller with comparison to
the experimental results by the PIV technique
[3.26].
On the bases of the theory of two-fluid model for multiphase flow, the
dispersed phase in these flow, at dilute concentration conditions, can be
assumed to be a pseudo-fluid, which occupies a certain fraction volume in the
whole flow domain shared with the continuum phase, i.e., the real fluid phase.
According to this assumption, the simulation of two-dimensional two-phase
flow through a centrifugal pump impeller has been conducted by using the
k
—
s-A turbulence model and the SIMPLEC algorithm at dilute
concentration conditions, for example, Cv (volume concentration) is less than
5%.
Boundary Conditions for Calculations of Liquid-Particle Two-Phase
Turbulent Flow through a Centrifugal Pump Impeller
* Flow in liquid phase
(a) Inlet boundary. The inlet values of the radial velocity component u
r]
and the circumferential one u
m
=
co
r
\
are directly specified and easily altered.
The inlet value of turbulent kinetic energy k
in
adopts 0.5 ~ 1.5% of the inlet
mean kinetic energy of flow, that is,
k
in
= 0.005tt
r
2
,
//„„= puj /100.0
where / is the specific length of
the
inlet boundary, /u
t
the eddy viscosity. The
values of k
jn
, ju
tin
and s
in
do not severely affect on the final results.
142
Abrasive Erosion and Corrosion of Hydraulic Machinery
(b) Outlet boundary. The velocity components at the outlet of the
calculated domain are deduced from their immediate upstream values by
adding a fixed overall mass, whose amount is calculated to ensure overall
mass conservation. The outlet values of k and £ are unimportant, because
limited effects of the upstream are known at high Reynolds number and zero
gradient is specified.
(c) Solid walls. In the near wall regions, to avoid the need for detailed
calculations the wall functions are introduced and are used in the finite
difference calculations. That is, the total tangential velocity V
B
at the point B
near the wall is corrected by velocity profile
V
+
Y
B
YB
y;<n.63
i>K*)
B
(3.78)
Y; > 11.63
where X = 0.41 and E
=
9.0. And
Z=V
B
lu
T
Y
B
+
=Y
B
u
T
/v=pC^
4
k
B
%/
M
where Y
B
is the normal distance from the point B to the wall, r
B
is an
approximation for wall stress r
w
near the wall and is formulated by
neglecting the convection and the diffusion of the turbulent kinetic energy
equation in this region.
T
B
=C"
2
/*
B
Then, we obtain
r.=^c;:x
2
X
n
(
£y
;)
<3
-
79)
From the foregoing equation, then
Analysis and Numerical Simulation of Liquid-Solid Two-Phase Flow
143
**/ =Cr
U2
/p
=
constant
which implies that the wall flux of k is zero. Accordingly, zero normal
gradient prescription rate near the wall, the length scale is assumed to be
proportional to the normal distance from the wall, then the turbulent energy
dissipation rate s
B
and the generation term of turbulent kinetic energy G
w
in
this region is as
/i3/4j.
3/2
XYB
(3.80)
(
T
= T
^w ''w /-, \B
(d) Boundary condition of pressure. The correction values p' of
SIMPLEC scheme possess zero gradient specification every where except at a
point is then required to all the pressure to be calculated.
(e) Periodic condition. On the pairs of upstream and downstream
boundaries extending from the blade leading edge and the trailing edge
respectively, the period conditions must satisfied, that is,
bright
= Kfi
* Particulate phase
Variables in the particulate phase also possess zero normal gradient
specification everywhere except the inlet. At the inlet particle velocity
components are assumed to have the same values as those in liquid flow for
simplicity. And the bulk density of particle phase (or particle number) at the
inlet can be easy determined according to the known volumetric condition.
The velocity components and the bulk density (or the particle number) at the
outlet must be calculated to ensure overall mass conservation also by adding
fixed numbers from their immediately upstream values. The particle (or
bubble) number flux is defined as