Application of Computer Aided Design to the Image ...
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可 視 化 情 報Vo1.11 No.43 (1991年10月)
論文
Application of Computer Aided Design to the
Image Processing of a Two-Dimensional Laminar Jet Flow
G., ZARBI and K., TAKAHASHI
ABSTRACT
This study utilizes computer aided design graphic simulation in the image processing
of visualized streamlines for the jet flow through a nozzle into the atmosphere. The flow
is assumed to be two-dimensional, incompressible, steady and laminar. The combination
of this simulation modelling with the image processing technique proved to increase the
efficiency of data processing and results in many aspects. Distributions of flow velocity,
vorticity and pressure are calculated using a streamline coordinate system for governing
equations. The results of CAD simulation are then compared with those obtained in
applying direct image processing technique (conventional simulation) and with numerical
ones for various Reynolds numbers and channel angles.
Key Words: Internal Flow, Viscous Flow, Computer Aided Design (CAD), Velocity Dis-
tribution, Streamline Coordinates, Image processing, Fluid Mechanics, Two-Dimensional
Flow, Laminar Jet
Introduction
This study deals with the liquid jet flow
through an orifice into the atmosphere. An
example of such a flow can be seen in a
fluid power component, especially around
the restriction of a hydraulic control valve,
which is a basic means for the control of
fluid power.
The flow inside a control valve is very
difficult to solve numerically due to the
geometry of free surfaces being hard to
determine initially. However, taking free
surfaces as a streamline coordinate curve,
a numerical solution can be achieved. Thus
streamline coordinate methods are very
useful for solving such a problem.
A number of authors have approached
their numerical simulation adopting such
coordinates. Takahashi developed stream-
line coordinate representing the free surface
of liquid for a two-dimensional steady flow
to solve the flow with complicated geom-
etries2). Takahashi et al3) used image pro-
cessing of a jet flow employing streamline
coordinates. Tsukiji and Takahashi4) per-
formed another numerical analysis on axi-
symmetric jet, using a streamline coordin-
ate system. Zarbi and Takahashi5) analyzed
flow through a nozzle into the atmosphere,
expanding the velocity distribution across
*Received Feb . 25, 1991; revision received August
30, 1991.**Department of Mechanical Engineering Sophia Uni-
versity, Tokyo, Japan
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234 G. ZARBI and K. TAKAHASHI
the jet flow to a Fourier series in the
streamline coordinates.
In the present study, the velocity field is
obtained from visualized streamlines using
the streamline coordinate system.
Comparisons are made between various
CAD and CAM (computer aided design and
manufacturing) systems available in the
present industrial market, employing the
experiences of five years work. On the
basis of these comparisons, for their range
of applicability and functionability in com-
bination with the image processing tech-
nique, useful results on the visualized flow
fields are obtained.
Nomenclature
L*: Half of the orifice width shown in
Fig. 1.
Q*: Flow rate per unit thickness.
q•‡*: Uniform velocity at an infinite dis-
tance downstream.
q*: Velocity.
q: Normalized velocity (=q*/q*•‡).
Re: Reynolds number=Q*/v*.
u, v: Normalized velocity components of
q in the x- and y-directions, res-
pectively.
ă: Normalized measuring ratio.
v*: Kinematic viscosity.
Į: Angle made by the x-axis and the
velocity (Fig. 1).ƒ¿
: Half of the channel angle.
ƒÓ: Normalized coordinate defined by
Eq. (9) in the Flow direction.
ƒÓ: Normalized coordinate defined by
Eq. (8) normal to the ƒÓ-direction.ƒ¶
: Normalized vorticity.
Superscript*: Dimensionless quantity: Re-
ference quantities L* and q•‡* are used for
mormalizing quantities.
Fig.1 Flow field geometry.
2. Simulation for obtaining velocity field
2.1 Visualization of streak lines
Streak lines used in this study were vis-
ualized by hydrogen bubble technique the
process and application of which were fully
covered by Reference(4). An example of
the results is shown in Fig.17.
2.2 Image processing by conventional
simulation using streamline coordi-
nates
The visualized streak lines were traced
and placed on a common tablet digitizer.
Since the flow was steady, these streak lines
coincide with streamlines. The coordinates
of points on each streamline then was dig-
itized. A third-order polynomial was
chosen to fit with a streamline between two
neighboring streamline points.
A mesh system then was generated using
a series of curves orthogonal to the curves
fitted to streamlines. In choosing the initial
streamlines from the visualized streak lines,
care was taken to coincide the streak line
with the division points on the orthogonal
curve passing through the two sharp points
of the orifice. These mesh node coordinates
were then fed into a computer program for
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Application of Computer Aided Design to the Image Processing of a Two-Dimensional Laminar 235
further processing using streamline coor-
dinates.
A streamline (ƒÓ-curve) and a curve (
•¬ -curve) normal to it are chosen to repre-
sent the streamline coordinate curves. The
governing equations with respect to inde-
pendent variables ƒÓ and •¬ are expressed
as follows2):
The continuity equation•¬
The equation of metric coefficient ă
•¬ The equation of motion in the ƒÓ-direction
•¬ The equation of motion in the •¬-direction
•¬ The definition equation of vorticity
•¬ The vorticity transport equation
•¬ The combination of Eqs. (1) and (2) leads
to•¬
Flow velocity q and metric coefficient ă
are related to •¬ and ƒÓ as follows:•¬•¬
•¬ The following equations7) will be applied
to evaluate the residual on the artificially
inserted streamlines near the boundaries,
for the equations of continuity and metric
coefficient ƒÉ:•¬
•¬ The values of ƒ¶ and q can be calculated
from the difference equations which appro-
ximate Eqs. (5)•`(9).
2. 3 Image processing by CAD Simulation
The visualized streak lines were sketch-
ed by means of a stylus pen on a
common tablet digitizer and then converted
into the CAD drawing elements (see Fig.
2 for the procedure). For the computer
aided design simulation, a series of com-
puter programs were used which are written
in the form of Macros (Lisp language) uti-
lizing the CAD functions. Each of the Mac-
ros used in the programs combines a num-
ber of CAD commands to form one single
command. The use of the programs are
possible without any programming expe-
rience, so that after a brief introduction
anyone can work with the system. Since
only little storage space is necessary, the
programs can be executed on a personal
computer. A common tablet digitizer is
used as the input device, hence the demand
on peripheral devices is kept low. In Mac-
ros used in the programs, the CAD func-
tions which are common to evry available
CAD system are utilized. Hence the trans-
ferability of the programs to other CAD
will not present any problem. Macros which
are combination of CAD functions speed-
up the data processing, increase the flexibil-
ity and accuracy.
Using CAD equipment the whole visua-
lized flow pattern can be sketched and stor-
ed permanently in the form of drawing
units. Once these streak lines are sketched,
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236 G. ZARBI and K. TAKAHASHI
Fig.2 Flow chart for CAD simulation .
recalling, processing and further manipula-
tion all will be done from within the CAD
graphic simulation area without any uncer-tainty of mislocation errors of traced streak
lines on digitizer, or waste of time in choos-
ing wrong streak lines. When we created
the flow pattern, it is then possible to
build-up a surface and mesh system on it
(see Fig.2).
Two surfaces can be generated on the
upper and lower portion of our flow field
and then join them to have a unified sur-
face for the entire flow field . The mesh then can be generated on this surface with any
desired mesh size. Mesh will
be constructed by blending be-
tween sections, corners, or ed-
ges. This is construction of a
set of new lines that lie on the
surface, proportionally parallel
to one of its edges and equally
spread across the surface.
Another possibility is the use
of a program that can scan the
flow field boundaries and create
a mesh system, smoothing them
with spline functions available
within the CAD with high de-
gree of accuracy (CAD can keep
up to 16 or more decimal di-
gits). The coordinate of the
mesh nodes are then extracted
using a program which runs
within the CAD memory stor-
age, utilizing CAD functions
in the form of Macros. The
mesh coordinates are then fur-
ther processed for the calcula-
tion of the flow field, applying
the governing equations within
the CAD storage area.
Applying conventional, CAD
simulation and numerical calculation me-
thod, velocity fields for symmetrical lam-
inar flows were calculated from visualized
pattern at Re=300 and 400 for various
channel angles. The flow parameters, q, Į,
Ħ and ă, were expressed as functions of
ƒÓ and •¬ in streamline coordinates. Since ƒÓ
and •¬ do not represent physical length, it
is necessary to transform ‡™ƒÓ and ‡™•¬ into
‡™s and ‡™n which are physical length (al-
though they are normalized). Knowing the
values for q, ƒÆ, ‡™ƒÓ and ‡™•¬ will lead us to
the values of ‡™s and ‡™n applying Eqs . (8)
and (9), and the shape of free surfaces
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Application of Computer Aided Design to the Image Processing of a Two-Dimensional Laminar 237
could then be easily determined.
3. Results and Discussion
3.1 Flow net generation
For numerical simulation, to facilitate
the plotting of flow net, ‡™s and ‡™n were
expressed in xy coordinates. From the
geometry of the flow field, shown in Fig.1,
the values of the xy coordinates along curve
FOE can be determined, applying the flow
conditions at the inlet to the convergent
channel. Proceeding from these values, the
x-y coordinates can be calculated for the
next position using ‡™s and ‡™n values.
Meshes with nonuniform grid spacing in
both directions were developed, i. e., for
mesh generating, an increase in the num-
ber of grid points in areas where gradients
in the flow are high (i. e., near the orifice
edges), and a decrease in the number of
grid points in the low gradient region (i.
e., near the center of the channel and far
upstream and downstream of the orifice)
were considered. Useful results were ob-
tained for grid numbers as high as 41•~42.
Figures 3 and 4 present meshes generated
by numerical simulation for Re=300 and 400
and for various channel angles.
In CAD application, a mesh size as big
as 5000 was generated to compare the
computing time with those of conventional
and numerical simulations. Figures 5 and
6 show flow nets generated on CAD with
some artificially inserted streamlines for
Re=400 and 300 and for various channel
angles.
After inputting all streamlines data in
the form of drawing units using a computer
aided design package, several ways were
Fig.3 Numerical result of a flow net gen-
erated for Re=400 and half channel-
angle 45•‹
Fig.4 Numerical result of a flow net
generated for Re=300 and half
channel-angle 30•‹.
Fig.5 CAD simulation result of a flow
net generated for Re=400 and
half channel-angle 45•‹.
Fig.6 CAD simulation result of a flow net
generated for Re=300 and half channel-
angle 30•‹.
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238 G. ZARBI and K. TAKAHASHI
employed to form a mesh system on stream-
lines. Various number of divisions on
streamlines were tried to achieve an op-
timum mesh size. The central streamline
was taken as a reference streamline to
work out the node coordinates on the mesh
system (see Fig. 2).
To have a more clear picture of the con-
dition at the sharp edges of orifice, different
numbers of grids were tried for various
streamline being taken as the base calcu-
lating reference streamline. The best re-
ference line was found to be the central
streamline, as the streamlines are more
stable in the central portion of the channel
and less separation occurs.
In applying CAD simulation method, it
was possible to insert artificial streamlines
near the wall to obtain more mesh near
the solid boundaries and free surfaces of
the fluid. This was done by the surface
generating function of the CAD. The re-
sidual for the continuity and ă equations
were considered using Eqs. (9) and (10),
to check the validity of these streams.
These residual values proved to be less
than for some cases equal to 1% for every
mesh point on these artificial streamlines.
However, near the restriction where the
curvature is large, these residuals were
slightly higher. On the average it can be
said that the residuals are about 1 to 2% .
We can conclude that the precision of CAD
applications on the visualized pattern de-
creases as the curvature of the streamline
increases.
For the conventional simulation of image
processing, meshes were generated for var-
ious Reynolds numbers and channel angles
(see Figs. 7 and 8).
For the experimental results, streamlines
passing through the equally spaced points
Fig.7 Conventional simulation result
of a flow net generated for Re
=400 and half channel-angle 45•‹.
Fig.8 Conventional simulation result of a flow
net generated for Re=300 and half chan
- nel-angle 30•‹.
on curve APD (Fig. 1), or very near to
them were chosen for the visualized flow
pattern.
In case of numerical solution we let the
positions of mesh points being denoted by
i in the ƒÓ-direction and j in the •¬-direction
(1•¬i•¬m and 1•¬j•¬n). The variable ƒÓ was
transformed into a new variable R by a
hyperbolic function R=tanh(ƒÓ), where ƒÓ1=
-•‡ corresponds with R1=-1 and ƒÓm=+•‡
with Rm=+1.
For the convenience in calculation, •¬=•}1
were assigned to the streamlines correspond-
ing to the solid walls.
Taking constant step width for Ri in the
ƒÓ -direction and for •¬i in the •¬-direction,
we proceeded with the computation. The
initial values of qif, Ħij and Įij were set
to zero and that of ăij to unity. Then, the
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Application of Computer Aided Design to the Image Processing of a Two-Dimensional Laminar 239
values of ăij, Ħij, qij and Įij are computed
from the difference equations by iteration5).
The entire streamlines are not shown in
the limited plotting scale area for the ve-
locity field. This is to have a more clear
and better understanding of the flow field
and the streamlines behavior.
3.2 Comparison of the CAD results with
conventional ones
In the present study a computer aided
design modelling which is rather new in
engineering field was applied to image pro-
cessing of visualized pathlines. The calcu-
lation and procedure steps are explained
in Fig. 2. In order to confirm the accuracies
of this simulation and the conventional one
numerical computation was carried out. In
numerical computation, we employed the
same procedure as Ref. (5).
By comparing Figs. 3 to 8, it is possible
to assess the smoothness and accuracy of
the shape of the streamlines generated by
CAD. The artificially inserted streamlines
help the concentration of grids near the
wall and restriction areas. This in turn
reduces residuals which are greatly related
to the mesh refinement in these areas.
In conventional modeling, it was not
possible to generate the number of meshes
as large as that generated by CAD on per-
sonal computer. This is because of the
massive calculation involved in treating
the high degree of polynomial fitted to the
streamlines. It would cause memory and
over flow problems.
In Figs. 9 to 11 some main feature of
CAD simulation results are given. In all
the cases presented for the present study,
Fig.9 (a) Velocity profiles obtained by
CAD simulation for Re=400 and
half channel-angle 45•‹.
Fig.9 (b) Velocity profiles obtained by
conventional simulation for Re
=400 and half channel-angle 45•‹.
Fig.10 (a) Velocity profiles obtained by
CAD simulation for Re=300 and
half channel-angle 45•‹.
Fig.10 (b) Velocity profiles obtained by
conventional simulation for Re
=300 and half channel-angle 45•‹.
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240 G. ZARBI and K. TAKAHASHI
Fig.11 (a) Velocity profiles obtained by CAD sim-
ulation for Re=300 and half channel-
angle 30•‹.
Fig.11 (b) Velocity profiles obtained by conventio-
nal simulation for Re=300 and half
channel-angle 30•‹.
Fig.12 Velocity profiles obtained by num-
erical simulation for Re=300 and
half channel-angle 45•‹.
Fig.13 (a) Comparison of velocity profiles
of CAD simulation with nume-
rical ones for Re=300 and half
channel-angle 45•‹.
Fig.13 (b) Comparison of velocity profiles
of CAD simulation with nume-
rical ones from the orifice pos-
ition for Re=400 and half
channel-angle 45•‹.
Fig.13 (c) Comparison of velocity profiles of
CAD simulation with numerical-
ones from the orifice position for
Re=300 and half channel-angle 30•‹.the velocities are normalized using the
uniform velocity q*•‡ at an infinite distance
downstream. These figures show that, con-
siderable improvement in velocity profiles
are obtained applying CAD simulation in
comparison to the conventional image pro-
cessing application.
This improvement together with the well
conformed results of CAD with those of
the direct numerical modeling values, give
support to the accuracy of CAD simulation.
Figures 12 to 16 show these . numerical re-
sults and their comparisons with those ob-
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Application of Computer Aided Design to the Image Processing of a Two-Dimensional Laminar 241
Fig.14 (a) Comparison of velocity profiles of
CAD simulation with conventional
ones for Re=400 and half channel-
angle 45•‹.
Fig.14 (b) Comparison of velocity profiles of
CAD simulation with conventional
ones for Re=300 and half channel-
angle 45•‹.
Fig.14 (c) Comparison of velocity profiles of CAD sim-
ulation with conventional ones for Re=300
and half channel-angle 30•‹.
Fig.15 Comparison of the shape of stream-
lines obtained by CAD simulation
with the numerical ones for Re=300
and half channel-angle 45•‹.
Fig.16 Comparison of the velocity distribution
along the free surface of fluid by CAD and
numerical simulation for Re=300 and half
channel-angle 45•‹.
Fig.17 Visualized streak lines for Re=
400 and half channel-angle 45•B.
tained by CAD and conventional simula-tions.In these figures the shape of velocity profiles are being compared and the mag-nitude of the velocities are not indicated. By looking at Figs.9-14, we can see
that the shape of velocity profiles obtained
in applying CAD simulation are well con-
formed with those of numerical and the
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242 G. ZARBI and K. TAKAHASHI
degree of conformity increases with a de-
crease in channel angle.
The computing time for CAD model
proved to be about 1/4 than that of con-ventional modeling by the same computer
facilities. The simplicity and precision of
CAD modeling is also worth noticing.
3.3 Benefits of CAD application in How
visualization
The followings are some merits in apply-
ing CAD simulation to flow visualization:
i) automatically and graphically data pro-
cessing and their manipulation by means
of Macros in absolute, ralative, and polar
coordinates save a great deal of time and
smooth the operation.
ii) it is possible and very useful to allocate
different layer and identities to individual
streamline, hence all the streamlines and
mesh elements can be treated together or
individually.
iii) Entities can be retrieved graphically
by a simple program for their radius of an
arc, normal vector, location of points along
the entity with any specific direction and
distance, intersection points and tangent on
entities, etc., just by referring to the iden-
tity of entities. These procedures are per-
formed graphically and are not involved
with the massive calculation of polynomial
equations fit to the streamlines as in the
conventional modeling. Hence, save enor-
mous amount of computing time, increase
accuracy (in comparison to iterative simu-
lation) and reduce programming struggle.
iv) with a simple group of developed Mac-
ros, it is easy to extract and store mesh
informations such as coordinates in three
different coordinate systems, relative dis-
tance with all neighboring nodes, absolute
distance from the origin, and vector direc-
tion with respect to the successive node.
v) it is possible to return to operating sys-
tem (OS) from within the CAD and utilize
various application softs without exiting
the CAD system which saves time.
vi) the streamlines are sketched using a
common digitizing tablet and stylus pen
(or mouse) with free-hand drawing. The sketched streamlines which are in the form
of drawing entities can be converted into
binary data file which are common to all
available CAD systems.
vii) relocation of any specific streamline
on the flow pattern and alteration in choos-
ing streamlines or in mesh size can be
done simply, quickly, and without error.
iix) the iterative graphically extracted co-
ordinates and simultaneous storage will
allow the generation of very big sizes of
mesh with little time. This would cause
the over flow and memory problems with
the sacrifice of time in conventional ap-
proach.
4. Conclusions
The two-dimensional, steady, incompress-
ible, laminar liquid jet flow from a nozzle
into the atmosphere was visualized. The
flow field was calculated using computer
aided design graphic simulation and image
processing technique. A streamline coordi-
nate system was adopted. In order to con-
firm the validity of this simulation results,
the governing equations were numerically
solved by finite difference method. The
results are well confirmed with the numer-
ical ones. To compare the CAD simulated
results with the conventional ones, the flow
field calculations were performed convention-
ally and CAD results showed considerable
improvements in velocity profiles, generated
flow nets, and in time saving.
For the present study applying CAD sim-
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Application of Computer Aided Design to the Image Processing of a Two-Dimensional Laminar 243
ulation the following conclusions are drawn:
i) it is much easier to manipulate the
number of divisions on streamlines for
their coordinate values with higher
accuracy than conventional simu-
lation.
ii) since a continuous line will be sketch-
ed rather than digitizing point-to-
point on each streamline, the human error will have less effect on extrac-
ting coordinate of points
iii) reduces other possible sources of error
associated with conventional way of
data processing such as error in di-
viding streamlines into equal number
of divisions, error in relocating grid
points on mesh system in which slight dislocation of digitized points will lead
to a large systematic error.
iv) the combination of CAD and image
processing technique saves computing time by 1/4 compared to convention
al image processing simulation.
Acknowledgment
The authors would like to express their
appreciation to Mr. Yuetsu Kodama, De-
partment of Mechanical Engineering, Koga-
kuin university, for his kind cooperation
in carrying out experiments when he was
a graduate student of Sophia university.
Reference
1) Takahashi, K. and Tsukiji, T.: Numerical analysis of a laminar jet using streamline coordinate system, Trans. CSME., Vol. 9, No. 3 (1985), p. 165
2) Takahashi, K.: A numerical analysis of flow using streamline coordinates, Bull. JSME., Vol. 25, No. 209 (1982), p. 1696.
3) Tsukiji, T. and Takahashi, K.: Numerical analysis of an axisymmetric jet using a streamline coordi-nate system, JSME., Int. J., Vol. 30, No. 267 (1987),
p. 1406.4) Takahashi, K., Tsukiji, T. and Sakagami, T.: Im-
age processing of a jet flow using a streamline coordinate system, Fluid Control Measurement, Pergamon Press, Vol. 2, (1985), p. 711.
5) Zarbi, G. and Takahashi, K.: Prediction of the laminar two-dimensional Jet flow through a conver-
gent channel, JSME International Journal, Vol. 34, No. 311 (1991)
6) Takahashi, K.: Recent development of fluid me-chanics in fluid power engineering, JSME Interna-tional Journal Series II, Vol. 32, No. 2 (1989), p. 147.
7) Hori, H., Tsukiji, T. and Takahashi, K.: Numerical processing of a visualized streak line image using a streamline coordinate system, Trans. Jpn. Soc. Mech. Eng., B, Vol. 56, No. 531 (1990), p. 3292.
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