Jackson M.J. Micro and Nanomanufacturing
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412 Micro-and Nanomanufacturing
Along the Mach lines, flow properties remain constant and they are
therefore called "characteristic lines". There are two distinct types of
characteristic lines, right running and left running, which are de-
noted as C+ and C_, (Figure 8.11). The angle that the lines make
with a streamline at an arbitrary angle 0 to the x - axis is given by
the following equations,
— = tan{d-ju) for C_ characteristics (8.9)
dx
— = tan{d
+ ju)
for Q characteristics (8.10)
dx
f
1 ^
Where //-sin"' — (8.11)
\M )
Equations 8.9 and 8.10 are called the characteristic equations.
When two characteristic lines meet, their directions change. For axi-
symmetric flow the compatibility equations (8.12 and 8.13) describe
the change in flow properties at the characteristic lines after the in-
tersection point.
d{6 + u)-—==== = 0 for C_ characteristics (8.12)
V(M'-l)-cot^ r
1 dr
d{d-u)-h—
=
0 for Q characteristics (8.13)
V(M'-l) + cot^ ^
The quantity
{O
+ u) is not constant along a C_ characteristic
line its value depends on the spatial location in the flow field as in-
dicated by the ^y term in equation 4. The same qualification is
made for {d - o) along the Q characteristic. The process of com-
puting an axi-symmetric flow field can be achieved by replacing dif-
Laser Micro- and Nanofabrication 413
ferential equations 8.12 and 8.13 with a finite difference solution.
Flow properties and their locations are found by a step-by-step solu-
tion of these equations, which when coupled with the characteristic
equations 8.9 and 8.10, are used to construct the characteristic net.
Characteristic line C
Fig. 8.11. Part of the characteristic net showing left- and right-running character-
istic Hnes
414 Micro-
and
Nanomanufacturing
The calculation process starts
at the
throat
of the
nozzle
and is
independent
of
flow conditions upstream
to the
flow.
The
character-
istic equations
are
solved
in the
form
of
iterations using
new
values
of properties
at
intersections,
and
locations obtained from compati-
bility equations,
in the
form
of
a finite difference.
By
calculating
the
gradients
of
new
C_ and Q
characteristic lines,
the
solution
of
the
conditions
at the
nozzle contour
can be
obtained
by
propagating
downstream from
the
initial data line.
By
analyzing
the
intersection
of each Mach line
in the
flow regime,
a
nozzle shape
of
shock-free,
isentropic flow
can be
predicted.
The
intersection point
of
two Mach
lines
are
obtained with
the
Euler-predictor scheme, while
the
correc-
tor scheme iterates
the
predicted property
and
updates
the
intersec-
tion point until
the
desired convergence criterion
is
obtained.
The computation process starts
at the
throat radius, point
A
(Fig-
ure 8.12), with
a
known value
for
^maximum, which
is the
maximum
expansion angle that
can be
obtained using
the
Prandtl-Meyer func-
tion,
V
(M),
for a
required Mach number
and is
given
by the
follow-
ing equation.
v(M) =
7 +
1
tan"^
^^(M'-I)\
-tm-'(M'-lf"
(8.14)
Therefore,
^_,._ =2^^)
(8.15)
n
If
we
assume that
A^ = Av =-^n^
then
we can
assume that
an
N
array, A^, exists whose angular spacing decreases
as the
boundary,
AB,
is
approached.
The
result
is a
highly non-uniform grid near
the
throat
of the
nozzle. Consequently,
the
contour
of the
wall down-
stream
of
point
A is
poorly represented. After
a
small duration
in
time during
the
computation,
the
contour
is
corrected
by a
grid com-
pression scheme. Additional characteristics starting
at
point
A are
inserted between
the
sonic line
and the
first angular
C.
characteristic.
Laser Micro- and Nanofabrication 415
Fig. 8.12. Schematic diagram of the flow field inside the minimum length nozzle
A power law distribution is used, thus,
v,=e^ =
N,
^v
(8.16)
Where, 7 = 1, 2, 3, , Nj, and Mj and Nj are fixed integers
that are not incremented withy. Figure 8.13 shows the effects of
grid compression using the following data: specific heat ratio (y)=
1.4; array size (N)= 10; and exit Mach number
(Mexit)
^ 2.065. Fig-
ure 8.13a shows the basic grid with no compression, while Figure
416 Micro-and Nanomanufacturing
8.13b has intermediate compression assuming Nj = 5 and Mj = 2.
Five additional characteristics are inserted upstream of the first C.
characteristic. A greater degree of compression is used in Figure
8.13c, where Nj andMj are both five, but are positioned much closer
to the sonic line and results in a satisfactory wall contour. Grid
compression provides a better representation of contour location in
the straight section specifically designed to cancel all the expansion
waves generated by the sharp comer. The wall contour is deter-
mined by calculating the boundary streamline AC as shown in Fig-
ure 8.14. This is accomplished by solving an ordinary differential
equation (Eq. 8.17) with initial conditions set as, 0^ =^max' ^^ ^^e
following boundary condition, x = 0. The following equation holds
true,
^ = tan^^ (8.17)
ax
Where r^ is the throat radius and it should be noted that equation
8.17 is solved by iteration using Euler's predictor-corrector scheme.
The iteration procedure terminates when the BC boundary character-
istic is reached and when 0
=
0.
It should be noted that the prediction of a nozzle contour does
not take into account viscosity of the fluid and hence it ignores the
formation of both subsonic and supersonic boundary layers, which
may adversely affect the performance of
a
small diameter nozzle.
The presence of a boundary layer on the nozzle wall lowers the
exit Mach number, compared to inviscid flow predictions that are
based on an area ratio. Four nozzle profiles with throat diameters
ranging from 50 |Lim to 300 jum were generated using the C.F.D.
computer code. To compensate for the formation of a boundary
layer, the contour was displaced by the boundary layer displacement
thickness S*. This displacement thickness was calculated using flat-
plate compressible flow boundary layer theory in association with
the Van Driest compressibility factor. Figure 8.15 shows both cor-
rected and non-corrected minimum length nozzle contours for a noz-
zle with a throat diameter of 300 |Lim and inlet pressure of
8
bar.
Laser Micro- and Nanofabrication 417
Minimum Length Nozzle Con to ur (AIR)
Fig. 8.13. Schematic diagram showing grid compression of the flow field in the
minimum length nozzle
418 Micro-and Nanomanufacturing
1.6 n
1.4 H
•a
2
T3
0
C
o
c
<D
E
c
o
0.8 H
0.6-1
0.4
H
0.2 H
I I I I I
0.2 0.4 0.6 0.8 1
Non-dimensioned axisymmetric nozzle length
'M.O.C.
Contour
Actual Contour
Fig. 8.14. Schematic representation of minimum length contour shown in non-
dimensioned parameters, r/„„e/r^/,^o«/ (non-dimensioned nozzle radius), and x/Xe
(non-dimensioned axisymmetric nozzle length)
Laser Micro- and Nanofabrication 419
18i
16H
14
H
12
H
<
§. 10H
iS
(0 O _|
0) O "I
<
6-1
4H
2H
I I I I I I
5 10 15 20 25 30
Pressure ratio (Po/P atmosphere)
•Predictor scheme
Ideal area ratio
•
Predictor-Corrector scheme
Fig. 8.15. The effect of using different computational techniques on the area ratio
(A/Ai) and pressure ratio
(P(/Patmosphere)
of the minimum length nozzle
)TM
A computer program was developed using MATLAB software
for designing an axi-symmetric, straight, sonic-line M.L.N., with a
420 Micro-
and
Nanomanufacturing
sharp throat comer.
A
design inlet pressure
of
8
bar was
used
for air
(/ = 1.4).
Using
a
simple one-dimensional relationship
the
theoreti-
cal exit Mach number
was
calculated
to be
2.0415. Results obtained
from
the
computer programme gave
an
exit Mach number
of
2.065,
an error
of 1.15%. It
should
be
remembered that
the
exit Mach num-
ber controls
the
expansion ratio. Figure
8.15
shows
the
importance
of using
the
predictor-corrector algorithm.
It
should
be
noted that
the exit Mach number
is
related
to the
nozzle area ratio.
Description
of
the Flow Field Used
for
C.F.D.
Validation
C.F.D. results show that
the
flow field
of
supersonic jets
can be
characterized
by
three distinct regions.
The
core region
of
the
jet is
that region where
the
flow
has
reached
the
desired Mach number,
and where viscous effects
are
negligible.
In
the
perfectly expanded condition,
a
uniform maximum veloc-
ity
and
uniform pressure exists throughout
the
length
of
the core
re-
gion.
The
supersonic region extends beyond
the
inviscid core,
and
flow
in
this region
is
affected
by
viscous forces,
but
remains above
sonic speed. This region surrounds
the
core
and can be
character-
ised
by the
supersonic length,
Ls.
The velocity profile within
the
supersonic mixing region
is
con-
tinuous
and has a
maximum value
on the
centerline.
The
third
re-
gion shown
is the
subsonic mixing region that surrounds
the
super-
sonic
jet. The
outer radial boundary
of
this region
is by
definition,
the boundary
of the jet itself. It is
often defined
as the
radial posi-
tion where
the
axial velocity
is
some specific percentage
of
the cen-
terline velocity.
At a
position beyond Ls
the
velocity profile
is
sub-
sonic
and
becomes fully developed.
In order
to
expand
the
flow
in a
perfect manner,
the
contour
of
the nozzle
has to be
convergent-divergent
in
shape with specific
curvature
in the
expanded section
for the
flow
to
follow
the
stream-
line.
If
the phenomenon does
not
exist then
the
flow will under-
or
over-expand depending
on the
inlet conditions
for a
similar area-
nozzle ratio. C.F.D. results
of
three types
of
nozzles
i.e.,
sonic, con-
vergent-divergent,
and
minimum length nozzle, with throat diame-
ters
of 300 ju m and an
inlet pressure
of 8 bar are
shown
in
Figure
Laser Micro- and Nanofabrication 421
8.16. The flow in the convergent-divergent nozzle is under-
expanded for a given condition, whereas the sonic nozzle will al-
ways be under-expanded for inlet pressures above 1.89 bar pressure.
Non-dimesional distance from nozzle exit (X/De)
IVIinimum length nozzle
- - - Sonic nozzle
^— —Convergent-divergent nozzle
Fig. 8.16. Centreline Mach number distribution for three types of nozzles with a
throat diameter of 300
JJ,
m