A SECONDARY FLOW APPROACH TO THE INLET VORTEX FLOW FIELD 
by 
Henri Charles Viguier 
Ing~nieur, Ecole Centrale des Arts et Manufactures, Paris 
( 197 9) 
SUBMITTED IN PARTIAL FULFILLMENT 
OF THE REQUIREMENTS OF THE 
DEGREE OF 
MASTER OF SCIENCE IN 
MECHANICAL ENGINEERING 
at the 
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 
November 1980 (,:e, re.~ f'l{./,1;, /. . ,t.?.' .· © Massachusetts Institute of Technology 1980 
A /~ 
, 
Signature of the Author Signature redacted 
Depa~Mechanical Engineering 
November 14, 1980 
Certified by 
Accepted by 
Signature redacted 
=== T -, r·, .. - , Edward M. Grei tzer 
• . . A . Thesis Supervisor 
Signature redacted., . 
Warren M. Rohsenow 
Chairman, Departmental Graduate Committee 
ARCHIVES 
MASSACHUSETTS INSTITUTE 
OF TECHNOlOG V 
MAY 1 1981 
LIBRARIES 
2A SECONDARY FLOW APPROACH TO THE INLET VORTEX FLOW FIELD
by
Henri Charles Viguier
Submitted to the Department of Mechanical Engineering
on November 14, 1980, in partial fulfillment of the
requirements for the Degree of Master of Science in
Mechanical Engineering
ABSTRACT
A theoretical study is presented of the fluid mechanics of the
inlet vortex (or ground vortex) phenomenon. The vorticity field
associated with the vortex is investigated using a secondary flow
approach. In this approach the flow is assumed to be composed of an
irrotational primary flow and a weak shear flow, with the vortex
filaments associated with the latter being regarded as convected by
the former. The potential flow field induced by the inlet-ground
plane combination is computed using the panel method developed by
Hess, Mack and Stockman. Using the analysis, material lines (which
coincide with vortex lines) can be tracked between a far upstream
location, where the vorticity can be taken as known, and the engine
face location. The deformation of the material lines thus shows
directly the generation and amplification of the streamwise compo-
nent of vorticity, which is responsible for the velocity distortion
at the compressor face. Two representative flow configurations are
considered, one with headwind only and one with the flow at forty-
five degrees to the inlet axis of symmetry. Although the results
so far yield only qualitative information, they appear to provide some
insight into one mechanism associated with the inlet vortex formation.
Thesis Supervisor: Dr. Edward M. Greitzer
Associate Professor of Aeronautics and Astronautics
3ACKNOWLEDGEMENTS
I wish to express my sincere thanks. to Professor E. M. Greitzer
for his guidance and advice throughout the course of this research.
I would also like to thank Dr. C. S. Tan for his valuable suggestions.
In addition, the comments of Dr. N. A. Cumpsty (Cambridge University)
are gratefuly acknowledged. Further, I am very grateful to Alice
Karoghlanian for typing this thesis.
This research was supported by a Fellowship from the Jean Gaillard
Memorial Foundation (during the first two terms of the author's stay
at MIT) and by the Air Force Office of Scientific Research under
Contract No. F49620-78-C-0084.
4TABLE OF CONTENTS
Abstract
Acknowledgements
Table of Contents
List of Figures
List of Symbols
Chapter 1 Introduction
Chapter 2 Background
Chapter 3 Analysis
Chapter 4 Flow Model Description
4-1 Flor Geometry
4-2 Potential Flow Calculation
4-2-1 General Discussion of Panel Methods
4-2-2 Potential Flow about an Inlet at
Prescribed Mass Flow
4-2-3 Potential Flow Computation of Hess,
Mack and Stockman
Chapter 5 Results
5-1 Potential Flow Calculation
5-2 Streamlines
5-3 Material Lines
5-4 Quantitative Calculation of Vorticity
Amplification
Chapter 6 Summary and Conclusions
Chapter 7 Recommendations for Future Work
Page
2
3
4
6
10
11
12
20
27
27
28
28
35
37
39
39
41
47
53
62
65
5TABLE OF CONTENTS (CONTINUED)
Appendix A
Appendix B
Fluid Particle Tracking Computer Program
Qualitative Investigation of the Flow Using
a Simple Two-Sink Model
B-2 Model Description
B-2 Results
Tables
Figures
References
Page
68
92
92
96
99
100
151
6LIST OF FIGURES
1 Flow Geometry
2 Inlet Geometry
3 Inlet Panel Representation
a N-line and Panel Vertices
b Front View (because of Symmetry, only Half the Inlet is Shown)
4 Side.View of Inlet Panel Representation
5 Streamlines in the x, z Plane, Case 1
6 Streamlines in the x, z Plane, Case 1
7 Streamlines in the x, y Plane, Case 1
8 Projections of Streamlines on the y, z Plane, Case 1
9 Projections of the Streamlines Shown in Figure 8 on the x, y Plane,
Case 1
10 Cross Section of the Capture Surface, Four Inlet Heights Upstream
of the Inlet, Front View of the Stagnation Line, Case 1
11 Capture Surface, Case 1
a Trace on the x, z Plane
b Trace on the x, y Plane
12 Perspective View of Stagnation Locations, Case 1
13 Streamlines in the x, y Plane, Case 2
14 Projections of Streamlines on the y, z Plane, Case 2
15 Projections of the Streamlines Shown in Figure 14 on the x, y Plane,
Case 2
16 Projection of Forward Stagnation Streamline on the x, z Plane, Case 2
17 Perspective View of Stagnation Locations, Case 2
718 Trace of Capture Surface on the x, y Plane, Case 2
19 Suggested Behavior of a Far Upstream Horizontal Vortex Line, Case 1
20 Deformations of a Fluid Line which is Vertical and in the x, z
Plane at a Location Four Inlet Heights Upstream of the Inlet,
x, z Plane, Case 1
21 Deformations of a Fluid Line which is Vertical and 0.6 Inlet Heights
from the x, z Plane at a Location Four Inlet Heights Upstream of
the Inlet, Projections on the x, z Plane, Case 1
22 Projections of the Material Lines Shown in Figure 21 on the y, z
Plane
23 Deformations of a Fluid Line which is Vertical and 1.6 Inlet Heights
from the x, z Plane at a Location Four Inlet Heights Upstream of
the Inlet, Projections on the y, z Plane, Case 1
24 Sugggested Deformation of a Far Upstream Uniform Distribution of
Vertical Vortex Lines, Case 1
25 Deformationsof a Far Upstream Vertical Material Line, Projections
on the x, z Plane, Case 2
26 Projections of the Material Lines Shown in Figure 25 on the y, z
Plane
27 "Far Upstream" Location of the Vertical Fluid Line Tracked in
Figures 25 and 26, Common "Far Upstream" Projection of the Horizontal
Material Lines Tracked in Figures 28 through 33, x, y Plane, Case 2
28 Deformations of a Fluid Line which is Initially Horizontal and 0.8
Inlet Heights from the Ground Plane, Projections on the x, y Plane,
Case 2
29 Projections of the Material Lin.es Shown in Figure 28 on the y, z Plane
830 Deformations of a Fluid Line which is Initially Horizontal and
0.4 Inlet Heights from the Ground Plane, Projections on the x, y
Plane, Case 2
31 Projections of the Material Lines Shown in Figure 30 on the y, z
Plane
32 Deformations of a Fluid Line which is Initially Horizontal and
0.16 Inlet Heights from the Ground Plane, Projections on the x, y
Plane, Case 2
33 Projections of the Material Lines Shown in Figure 32 on the y, z
Plane
34 Behavior of Fluid Lines Connecting Neighbouring Particles
35 Front View of Grid Tracked at a "Far Upstream" Location, Case 1
36 Deformation of Grid Shown in Figure 35 at "Engine Face" Location,
Projection on the y, z Plane
37 Vorticity Distribution at the Engine Face Location Due to a Far
Upstream Distribution of Vertical Vorticity, Case 1 (See Key
Table 1 p 99)
38 Enlarged View of a Portion of Figure 37 (See Key Table 1 p 99)
39 Initial (Far Upstream) Location of Grid Tracked, Projection on the
Ground Plane, Case 2
40 Front View of Grid Tracked at a "Far Upstream" Location, Case 2
41 Deformation of Grid Shown in Figure 40 at "Engine Face" Location,
Projection on the y, z Plane
42 Vorticity Distribution at the Engine Face Location Due to a Far
Upstream Distribution of Vertical Vorticity, Case 2 (See Key Table 1
p 99)
943 Enlarged View of a Portion of Figure 42 (See Key Table 1 p 99)
44 Vorticity Distribution at the Engine Face Location Due to a Far
Upstream Distribution of Horizontal Vorticity, Case 2 (See Key
Table 1 p 99)
45 Enlarged View of a Portion of Figure 44 (See Key Table 1 p 99)
B-1 Basic Two-Sink Model
B-2 Streamlines in the x, z Plane
B-3 Streamlines in the x, z Plane
B-4 Streamlines in the x, y Plane
B-5 Streamlines in the Neighborhood of Stagnation Point S2' x, z Plane
B-6 Perspective View of Flow Field (Because of Symmetry, Only Half
the Flow Field is Shown)
10
LIST OF SYMBOLS
D
H
m
PT
Q
S
V
vx vy z
V.
x, y, z
p
OU
Inlet Diameter
Height of Inlet Centerline above Ground
Inlet Mass Flow
Total Pressure
Sink Strength (positive)
Stagnation Point
Velocity
Velocity Components
Average Inlet Velocity (M = D2 Vp)
Wind Velocity
Spatial Coordinates
Flow Angle
Velocity Potential
Density
Vorticity
11
CHAPTER 1
INTRODUCTION
When a gas turbine engine is operated near a ground plane at static
or near static conditions a strong vortex is often observed to form
between the ground and the inlet. This so-called inlet or ground
vortex can be a severe problem. For example it is able to pick up loose
debris from the ground, which, when drawn into the engine, can cause
damage to the latter. More importantly, however, the presence of a
vortex at the engine face is associated with a severe distortion of the
inlet flow which can have serious consequences for engine stability.
Indeed the potential for vortex induced compressor surge has risen with
the advent of wide body aircraft powered by high bypass ratio turbofans
since in these configurations the engines have larger diameters and are
closer to the ground than in the past.
This report presents a theoretical investigation of the fluid
mechanics of the ground vortex phenomenon. A secondary flow approach is
used to study the vorticity field due to an inlet vortex, the purpose of
this being to understand the basic fluid mechanics of the flow that
give rise to such a vortex. After a review of the available literature,
the secondary flow approach taken is presented. This approach requires
the potential flow induced by a jet inlet close to a groundplane to be
calculated. The computational method used to do this is then described.
Finally the-results obtained using the approach selected are presented
and discussed.
12
CHAPTER 2
BACKGROUND
The inlet vortex has been the object of several investigations.
In 1955 Rodert and Garrett [1] conducted an experimental investigation
of the phenomenon using a full scale jet engine. They observed vortices
not only between inlet and ground but also between inlet and fuselage.
It was shown that the vortex could lift up loose material capable of
damaging the engine. The formation of the vortices was found to depend
on engine power (i.e., inlet velocity), wind speed and direction relative
to the aircraft, and engine height above the ground surface. In partic-
ular increasing the inlet velocity resulted in the broadening of the
range of wind velocities over which vortices were observed. Likewise,
tailwind conditions were found to give wider ranges than headwind
conditions.
Klein [2][3] also dealt with the problem of jet engines ingesting
debris from runways. He first tested (with no nominal ambient wind) a
small scale model consisting of an inch-diameter inlet connected to a
vacuum cleaner. The role played by the vortex formed beneath the inlet
in the pick up of loose material was again seen, without the inlet vortex
the airflow into the engine intake was found to be unable to cause
ingestion of foreign objects from the ground. Furthermore Klein deduced
from the collected data the ratio of particle terminal speed to inlet
air speed at the threshold of particle aspiration as a function of the
ratio of centerline height to engine diameter. A twenty percent scale
model was next built to investigate the influence of wind on the
13
threshold line previously determined. A fan added circulation to the
air approaching the inlet. Both the updraft caused by the engine and
the circulation existing in the oncoming air were found to be essential
in the ground vortex formation and increasing the strength of either
resulted in an intensification of the vortex. In addition he carried
out a significant test by simulating the ground plane by means of a
second "image" inlet. In this case the observed vortex, which ran from
one inlet to the other, was apparently similar to the one visualized
with the actual ground surface. This result implies that the thin
boundary layer developed on the ground due to the sink flow effect of
the inlet does not play a significant role in the inlet vortex formation.
(However this does not imply that, in real cases, the ground Qcts only
as a simple symmetry plane. According to Helmholtz's vortex laws,
circulation must be present in the aspirated air if the vortex is to
form. In the reported exporiment it was generated by means of a fan.
On the other hand, in actual inlet flows, the vorticity associated with
the inlet vortex can arise from the (viscous) interaction of the
atmosphere and the ground. The length scale of these shear layers can
be much larger than the inlet diameter). This study also reported tests
of different remedies to prevent ingestion of foreign objects, none of
which were successful. Finally a forty percent scale model was used
to develop a new scheme aimed at easing the ingestion problem. This
remedy consisted of a downward directed jet which induced a local flow
below the Ihlet. This removed the stagnation point present in the
external flow field from the ground surface. The existence of a stagnation
point on the ground plane is often regarded as necessary for an inlet
vortex to form. The basic reasoning for this is that in an inviscid
flow* the vortex filaments must either form closed loops or end on the
fluid boundaries, consequently a stationary vortex can be observed
only if a stagnation point exists on the ground plane where it can
attach.
Glenny [4] developed scaling laws to relate model and full scale
tests of ground vortex induced lifting of particles. Among the
dimensionless groups he derived the following ones are to be noted:
ratio of inlet velocity to wind velocity, ratio of the height of the
inlet centerline to the inlet diameter and ratio of the inlet velocity
to the product of wind velocity gradient and inlet diameter. The last
parameter is directly related to the strength of the ambient vorticity.
Increasing the first ratio or decreasing the second one (called ground
clearance) was found to enhance the risk of foreign objects ingestion.
Furthermore it was shown that the vortex can be blown away by sufficiently
increasing the wind velocity. The so-called blowaway velocity was noted
to rise when the wind direction varied from headwind to tailwind or
the vorticity strength parameter (as defined above) decreased.
Experimental results demonstrated the validity of these laws which can
be used to predict conditions under which vortex ingestion can occur
and the sizes of the particles which can be lifted from the ground.
*In fact viscosity effects cannot be neglected in the vicinity of the
ground surface. The no-slip condition which is satisfied on this
stationary surface makes vortex lines ending on the ground plane
impossible: all vortex lines are either closed loops or extend to infinity.
14
Various methods to prevent debris ingestion were tested but none worked
satisfactorily for the low values of ground clearance of current
interest. Glenny proposed that to reduce the ingestion of material
the ratio of inlet velocity to wind velocity be kept as low as possible
while the aircraft is running on the ground. As he stated, "the worst
thing is to run an engine up to full speed under quiescent conditions".
This last configuration is more likely to be encountered on a
stationary test rig or when the aircraft is starting its take off roll.
Using previously published observations and results of the small
scale tests they ran, Colehour and Farquhar [5] postulated a model of
the flow associated with an inlet vortex. The flow field was divided
into two parts: on the one hand, a viscous flow limited to the vortex
core and the ground plane, on the other hand, an inviscid potential
flow governing the remaining of the flow field. The latter was
calculated for different wind conditions by using the panel method
developed by Rubbert et al. [19], the ground plane being simulated by
means of a second "image" inlet. The behavior of the stagnation point
whose importance has already been underlined was studied; in particular
the analytical study confirmed that a sufficiently high headwind can
remove the stagnation point from the ground. An irrotational vortex
was next added to the flow field. In both cases the calculated velocities
differed from the measured ones, so that the viscous part of the flow
had to be considered. The model could be used to qualitatively explain
the velocity distributions recorded in the vicinity of the stagnation
region. It should be noted here that the model designed by the authors
was aimed at the description of the flow field associated with the ground
15
vortex rather than accounting for the formation of the latter. Different
vortex suppressions techniques were presented, all of them aimed at
preventing the stagnation point from forming on the ground.
Motycka, Walter and Muller 116], who were concerned with the ability
of the inlet vortex to distort the flow sufficiently to produce
compressor surge, carried out an analytical and experimental study of
the phenomenon. A three-dimensional potential flow model using a vortex
panel method to represent the inlet was combined with a viscous vortex
core calculation. The effects of the wind speed and direction, the ground
clearance and the inlet throat velocity on these flow properties were
examined. The stagnation point was assumed to coincide with the foot
of the vortex and the streamline from this point was regarded as defining
the vortex position. The position of these were calculated with varying
wind speeds and directions. The stagnation point blowaway velocity to
inlet velocity ratio was estimated as a function of the ground clearance
and the wind direction. The analytical predictions compared qualitatively
with experimental data. Two important results need to be noticed. Firstly
the ground vortex was observed to produce a velocity and flow angle
distortion at the compressor face rather than a total pressure distortion,
secondly vortex size and strength were found to be proportional to the
inlet diameter.
In a later paper Motycka [7] presented the results of a full scale
test program conducted at Pratt & Whitney to determine the causes of the
engine surges encountered during reverse operation. Comparing the inlet
pressure distortion patterns and the pressure versus time traces
recorded with those observed in the presence of a ground vortex resulted
16
in the identification of the inlet vortex as the dominant cause of these
engine stalls during operation with reverser. The flow from the
reverser was recognized as a major source of vorticity and was shown
to produce an apparent increase in tailwind. Using the forementioned
analytical study, this rise was found to cause the foot of the vortex
assumed to be at the stagnation point to move further upstream and
the stagnation line to shift toward the more distortion sensitive core
air stream. It was also noticed that the pressure distortion patterns
often indicated the existence of two vortices. This investigation was
followed by the testing of a small scale model of a subsonic inlet equipped
with a fan flow reverser. The results were reported by Motycka and
Walter [8]. The authors studied the effects of inlet throat velocity,
inlet height, aircraft ground speed, i.e., wind velocity and reverser
configuration (flow and targeting) on both the ground vortex formation
during reverse operation and the inlet pressure distortion induced
at the compressor face. In addition a three-dimensional potential flow
model similar to the one previously mentioned was employed to examine
the intake flow field during reverse operation.
Bissinger and Braun [9] used a small water tunnel and the hydrogen
bubble technique to visualize the flow field around an inlet with a
ground vortex. The inlet vortex was shown to be part of a vortex
system which also included a trailing inlet vortex, secondary vortices
and ground vortices similar to the ones encountered in front of any
obstacle.. The role played in the ground vortex formation by the thin
boundary layer developed by the ground in the sucked flow was again
shown to be insignificant. In addition, the inlet vortex was observed
to be unsteady and unstable, particularly its foot was noted to move
17
18
randomly on the ground plane, This feature which was also noticed
by Rodert and Garrett [1] and Colehour and Farquhar 15] was explained
by the ground vortex feeding on the widely varying ambient vorticity.
A simple potential flow model consisting of a sink and its image in
a uniform flow was used to study the number and location of the
stagnation points which were shown to play a very important role in the
formation of the whole vortex system. An explanation of the phenomenon
based on the stretching of the oncoming vortex filaments was given.
Finally Bissinger and Braun proposed measures of their own to protect
jet engines against foreign objects aspiration.
Recently Newman and Atassi 110] presented a two-dimensional free-
streamline potential flow model of the flow around an inlet close to a
ground plane. The fluid was assumed to be quiescent far from the inlet,
or in other words the wind velocity was zero. Solutions were obtained
only- for ground clearance values: below to a certain limit. Moreover
a physically meaningful solution was always accompanied by an alternate
spurious solution. An experimental investigation was carried out to
examine these two problems. In addition, the amplification of small
three-dimensional vortical disturbances convected by the inlet flow was
studied.
In summary, the characteristics of the ground vortex phenomenon
mentioned in the literature can be described as follows. The inlet
vortex can form only if the fluid drawn into the engine contains vorticity.
The existence of a stagnation point on the ground is believed to be
required for the phenomenon to appear. The inlet vortex which is part
19
of a whole vortex system may be unsteady. Further the thin boundary
layer due to the inlet (sink flow) alone is not essential in the
vortex formation. The following parameters are used to characterize
the phenomenon: inlet throat velocity to wind velocity ratio,
centerline height to inlet diameter ratio, and the ratio of the inlet
velocity to the product of wind velocity gradient and inlet diameter,
and wind direction. In particular increasing the first ratio or
decreasing the two next ones results in the strengthening of the
ground vortex.
There are thus several questions about the inlet vortex phenomenon
that still remain unanswered. Up to now no satisfactory explanation
of the mechanism of the ground vortex formation has been given. In
particular it is still not understood why, while, for any ingested
material line (i.e., vortex line), two stretched "legs" appear to
connect the far upstream portion of the flow to the inlet one, only one
vortex is observed. In addition, even quantitative relations between
the inlet vortex parameters presented above and the engine inlet flow
field are still to be found. Finally, it is highly desirable to know
the engine inlet flow field induced by an inlet vortex in order to
assess the sensitivity of the designed propulsion plant to the
phenomenon we are concerned with. The desire to answer these questions
prompted the present work.
CHAPTER 3
ANALYSIS
The goal of the current investigation is to explain the basic fluid
mechanics of the inlet vortex formation and to predict the flow distortion
induced by the vortex at the engine compressor face. As already empha-
sized, vorticity must exist in the oncoming fluid if the ground vortex
is to appear. Since the shear present in the ambient flow is generally
small, there must be substantial amplification of the ambient vorticity
for a vortex to form. Consequently we intend to study the variations
of the vorticity between far upstream and the engine compressor face.
The following assumptions can be made to investigate the structure
of the flow field created by a gas turbine engine inlet close to a
ground plane. First compressibility effects in the overall inlet flow
are not expected to be of primary importance. While locally high Mach
number regions (~l) exist near the engine inlet lips their influence on
the phenomenon of interest is small and consequently the fluid can be
assumed incompressible. In addition the fluid can also be supposed
inviscid: for example the Reynolds number based on the inlet centerline
height off the ground and the approach velocity is typically of the
order of 106. This means that, though the ambient shear is created by
viscosity (at some "far upstream" location), the vorticity amplification
can be considered essentially inviscid. Further although there are un-
steady effects observed in the presence of a ground vortex, it is not
clear that these play a significant role in the inlet vortex formation.
Accordingly the flow will also be taken as steady. Finally, body forces
can be neglected. Thus the flow we are concerned with is incompressible,
20
21
inviscid and free of body forces. In this situation Helmholtz's
vortex theorems [11] show that the vortex filaments are always composed
of the same fluid particles, i.e., they move with the fluid. Also for
a vortex filament of fixed identity, the ratio of the vorticity to the
length of the filament remains constant as time proceeds. Thus if a
vortex filament is stretched, the vorticity associated with it increases.
The vorticity is also inversely proportional to the cross-sectional
area of the vortex filament. As a result the evolution of the vorticity
between far upstream and the engine compressor face can be deduced from
the deformation undergone by the vortex filaments in their motion toward
the engine.
To calculate the movement of vortex filaments requires the velocity
field to be determined. The flow under consideration is clearly rotational,
not only in thin shear layers, but throughout the flow domain and is
strongly three-dimensional. However, since the basic concept to be stud-
ied is the intensification of vorticity due to the stretching of the
vortex filaments as they are drawn into the inlet, a useful simplification
can be made by viewing the problem as a small shear, large-disturbance
flow, as described by Hawthorne in Reference 12. In this approximation
which is currently utilized to explain the appearance of secondary flows,
the flow is assumed to be composed of an irrotational primary flow and
a weak shear flow. The vortex filaments associated with this shear are
regarded as being deformed by this primary flow only. In other words
the additional influence of the so-called "secondary"-velocities on the
convection of vortex filaments is neglected. The primary flow can be
determined using potential flow theory and, in particular, advantage can
22
be taken of the fact that numerous numerical methods exist to compute
potential flows.
The following approach can be used to study the variations of the
vorticity in the flows we are interested in. First a flow geometry is
selected. The previously introduced overall parameters describing
the inlet vortex formation are given typical values: it is to be
emphasized that our intention is not to perform a parametric study of
the ground vortex phenomenon but rather to investigate flow situations
which are known to present an inlet vortex. Secondly the potential flow
about this geometry is determined. Thirdly material lines are picked
far upstream and tracked as they move toward the inlet. These lines
once coinciding with vortex lines will coincide with them for all time.
Consequently by examining the stretching and tipping they undergo it is
possible to predict the vorticity pattern induced at the compressor
face by a given far upstream vorticity distribution. A further step
(not taken in the present report) would consist of computing the secondary
flow produced at this location by the calculated vorticity distribution.
The far upstream vorticity can be resolved into three components
respectively: 1) streamwise, 2) normal to the ground surface (vertical)
and 3) transverse to the flow stream direction and parallel to the ground
plane (horizontal). The evolution of the.streamwise component can be
predicted very simply*. Let us consider a flow that has, at a far up-
stream location, streamwise vorticity only. For a constant density
inviscid flow, if the vorticity at this location is streamwise, then the
vorticity everywhere will also be in the streamwise direction, i.e., the
flow is a Reltrami flow*. Streamlines and vortex tubes thus coincide,
Since the streamtube area is inversely proportional to the local
velocity (from continuity), the strength of the streamwise vorticity
at any location will thus be directly proportional to the velocity.
Therefore if the flow in the inlet is taken as uniform the ratio of the
streamwise (or Beltrami) vorticity at the engine face location to that
far upstream is equal to the area contraction ratio of any streamtube
between far upstream and the compressor face. As the fluid is in-
compressible, this last ratio equals the inlet throat velocity to wind
velocity ratio. The flow situations we are concerned with are
characterized by large values of this parameter, say minimum 10 or
greater. Hence this Beltrami vorticity can be expected to increase by
an order of magnitude or more between far upstream and the compressor
face.
* This can be seen by examining the circulation about a small closed
material contour which is aligned with the stream far upstream. The
circulation at this location, and thus anywhere downstream is zero.
Since the contour remains aligned with the streamlines. This implies
that all the vorticities is in the streamwise direction. Alternately
one can use the expressions developed by Hawthorne for changes in the
normal and streamwise components of vorticity. If, there is just
streamwise vorticity v x w = VPT = 0, so the total pressure is constant
and since this remains true there can be no vorticity arising which
is not parallel to the velocity.
23
In spite of this, it is felt that this process is not the primary cause
of the ground vortex formation for several reasons. One of these is
that the inlet vortex formation described in the literature appears
to depend on the angle of yaw of the inlet; whereas the mechanism just
described is independent of it. In addition experiments have been carried
out at MIT in which the shear was generated by a honeycomb so there was
no component of Beltrami vorticity far upstream and a vortex formed.
Further the situation that one would generally expect to find in an
engine test is some sort of ambient shear layer associated with a ground
boundary layer, i.e., region of total pressure defect. Such a flow
consists of normal vorticity only at the far upstream location. Finally
the stretching of filaments that one can get due to the non-Beltrami
component can be considerably larger than that due to the one-dimensional
effects. Thus, and as will be shown below, attention should be focussed
on the components of vorticity which are normal to the streamlines at
the far upstream loacation.
Although the amplification of the Beltrami component can be found
by basically one-dimensional considerations, the evolution of the two
remaining vorticity components is an inherently three-dimensional
phenomenon and has to be studied using the procedure previously described.
In particular material lines are picked which are, far upstream, normal
to the ground surface or both transverse to the flow stream direction
and parallel to the ground plane. The selection of those lines that are
most useful to follow can be greatly helped by a preliminary investiga-
tion of the primary flow field.
24
25
The approach which we intend to follow was first applied by
Lighthill [13] to the prediction of secondary flows and uses the
"drift" or total displacement of fluid particles as the quantity
that is studied. In steady flow streamlines and pathlines coincide and
in such a case the trajectory of any particle can be readily deduced
from an integration along the streamlines. Further the way a given
particle describes its trajectory can be determined using the drift
time concept. The drift time of a particle between two points is the
time this particle needs to go from one point to the other. In a
steady flow this quantity depends only on the extremity points which,
of course, must lie on a same streamline. With the help of this
concept it is possible to follow material lines and surfaces as they
are convected by the flow. In particular the deformations of vortex
lines can be easily studied. In order to study the drift of fluid
particles, i.e., to track fluid particles, a computer program has been
written which, from the initial position of a particle, calculates the
trajectory the latter follows in a given time. This code can also
easily be used to trace flow streamlines and thus obtain further insight
into the flow structure. A general description and a listing of this
program can be found in Appendix A.
The three-dimensional flow field created by a jet engine inlet
close to a ground plane is quite complex. A qualitative appreciation
of this flow field can be obtained using a simple model consisting of
a sink and its image in a uniform flow. The results of this investi-
gation are reported in Appendix B. As emphasized there, the description
of the potential primary flow near the inlet given by this model is too
26
approximate for any quantitative prediction of the vorticity
variations to be expected and a more complex model has to be used. In
the next sections such a model is described and the results obtained
using it are discussed.
I..
27
CHAPTER 4
FLOW MODEL DESCRIPTION
4.1 Flow Geometry
The flow geometry- that is investigated can be seen in Figure 1.
It consists of a cylindrical inlet and its image which is introduced
to simulate the ground plane. The inlets are set at an angle of in-
cidence (denoted a) in a uniform'freestream whose direction is parallel
to the plane of symmetry (between the inlet and its image) and whose
magnitude is V.. Further for the cases investigated the inlet center-
line height off the ground to inlet diameter ratio His taken equal toD
1.25, which is a value representative of a modern, high bypass ratio
engine configuration. Under the approximations listed in the previou
section the operating condition for the given inlet is defined by two
non-dimensional parameters: the ratio of freestream velocity to
average inlet velocity, "0 and the flow angle a. (Note that this
implies kinematical similarity only, rather than full dynamic similarity).
The inlet geometry is shown in Figure 2. Notice that the selected
inlet has no centerbody: the internal machinery which controls the
amount of fluid entering the inlet is assumed to be far enough from the
inlet lip for its effect to be lumped into a single parameter, mass
flow through the inlet, and its geometry to be neglected. Additionally
in order to minimize the influence of the fluid exiting the engine on
the flow near and upstream the forward part of the inlet, the inlet
walls have to be extended as far as possible downstream. As will be
seen below, this is also required for good accuracy in the numerical
calculation of the potential flow about the inlet.
28
4.2 Potential Flow Calculation
4.2.1 General Discussion of Panel Nethods
The approach taken requires the potential flow about
the geometry described above to be computed. Different methods exist
to calculate potential flow about three-dimensional bodies. It is
not the purpose of this report to describe them in detail and a
complete coverage .of these matters can be found in the abundant
literature, for example [14][18][19][22]. However a brief description
of the computational method used to solve the present flow will be
given.
Let us consider the irrotational flow of an inviscid incompressible
fluid around a body of arbitrary shape. The fluid velocity V at any
field point can be expressed as the sum of two velocities.
V- + v (1)
The velocity V is called the onset flow and represents the irrotational
velocity field that would exist in the absence of the body. The
velocity v is denoted the disturbance velocity and can be derived from
a potential function y
v = -V_ (2)
The velocity Vj,is supposed known while the disturbance velocity poten-
tialR has to be determined.
The methods we intend to discuss are all based on Green's theorem
29
which states that the velocity potential V at any point in the flow
field can be expressed in terms of the induced effects of source and
doublet sheets distributed on a surface S which contains the body
contour and can extend into the flow as on a wake. The problem is thus
to determine the source and doublet distribution which induce a flow
field satisfying the boundary conditions. These latter are of the
Neumann type (specification of the normal velocity): in particular,
on the body contour they express the absence of through flow. If the
disturbance velocity potential at a field point p is given by
(p) = s ) dS + 1sq )dS (3)
q
where r = distance between p and the surface point q
n = direction normal to S at q
a = source strength
P = doublet strength
dS = surface element
application of the boundary conditions results in the derivation of the
following integral equation, where the left side is known:
(= c(q) - )'rr- dS + f (q) [ (w)]dS (4)
npp6p p
30
The left hand side is, with p being a point of the body contour,
- VCn (5)
6n
p
The problem amounts to solving the above integral equation for the
source and doublet strengths: once these are known, the velocity field
can be determined using Equations (1) and (2). Note that there are
different singularity distributions that produce identical flow fields
outside the body contour. Consequently one must decide what types of
singularity are to be used to solve the particular boundary value
problem and insure the existence of a unique solution. The approach
generally taken is to cover any part of the surface S with only a
single type of singularity. Some investigators limit themselves to
one type of singularity for the whole surface, thereby restricting the
field of application of the methods they develop. For example methods
using sources [14][15] can compute the flows only over arbitrary
non-lifting configurations while the doublet or surface vorticity
methods [16] can handle problems involving lift. The most widely used
methods are those developed by Hess et al. [14][17][18] and by Rubbert
et al. [19][20][21] which all employ source and doublet distributions.
One major difference between these methods is that in the latter doublet
sheets are used both to generate circulation and to describe bodies
while in the former they are utilized only to introduce circulation
around bodies or wake elements, when necessary.
Once the positioning of sigularity sheets on the body contour and,
if necessary, in the flow (wakes) is completed, the problem is reduced
31
to numerically solving the integral equation expressing the boundary
conditions for the distribution of strength of these singularities.
Note that the flow field we intend to study is truly three-dimensional,
consequently the numerical procedure selected to be used to compute
the singularity distributions must be capable of handling arbitrary
three-dimensional bodies (even if advantage can be taken of the geometry
described above presenting two planes of symmetry). This requirement
eliminates the methods that are restricted to axisymmetric bodies
[16][22] .
In the procedures capable of dealing with arbitrary bodies, the
boundary surface S is approximated by a large number of small quadri-
lateral panels, on each of which the singularity strength is assumed
constant (Typical values of the number of panels are between 250 and
1000). In this way the singularity distribution depends on a finite
number N of parameters and the discretization associated with any
numerical procedure is performed. In principle if the number of panels
is made large enough, the solution obtained will converge to the exact
one. Two remarks are in order at this point. First in the methods.
developed by Rubbert et al., doublet sheets, when meeting a closed
surface represented by sources, are extended inside this surface in
order to avoid large source strength gradients. These internal doublets
sheets are also divided in small panels but the relative strengths of
the latter are specified as linear functions of the strengths of the
.Adjoi-i-ng external panels, so that these in'ternal sheets do not introduce
additional unknowns. Secondly using flat surface elements and singularity
densities that are constant on each element is the simplest way of
32
implementing finite strength surface singularity distribution methods.
The code used in the present investigation was of this type. It was
developed by Hess, Stockman and Mack [23] and is based on the methods
developed by Hess and co-workers over the past two decades. The
present formulation was in fact designed to be used on inlet type
geometries. It can be noted, however, that accuracy can be increased
and computing time can be decreased by utilizing so-called higher-order
methods [15][22], which employ curved surface elements and a singularity
density that varies over each element.
The numerical determination of the N singularity strengths Which
describe the whole singularity distribution proceeds as follows. N points
called control points are selected where the boundary conditions are
imposed. A control point is located at the centroid of each source panel
and exterior doublet panel if any. Further for lifting surfaces a suitable
representation of the Kutta condition must also be implemented [17][18].
As the velocity at any point of the flow field can be expressed as a
linear combination of the N singularity strengths, the coefficients of
which are deduced from the geometric relationship of this point and
the panels, application of the boundary conditions at all control points
yields a set of N linear equations for the N unknown singularity strengths.
If V. is the velocity induced at the control point of the ith panel
by a unit value of the singularity density on the jth panel and if n_.
is the unit normal vector to ith panel, the set of linear equations
obtained is as follows
33
N
I A.. a. = -_ - VC+ V .(6)
where A.. = -V- N
a . denotes the singularity density on the jth panel.
Vni denotes the imposed normal velocity at the control
point of the i th panel (Vni = 0 on body contour)
The above matrix equationwhich approximates the integral Equation (4)
is solved numerically. The linearity of this equation is a very
attractive property which is used to save computing time and to handle
:special applications of interest (lifting flows, inlets at prescribed
mass flow). In particular once the time-consuming computation of
the matrix {A. }1 is completed, the determination of the flow for various
onset flows requires only the linear Equation (6) to be solved. Further
solutions can be superposed to yield solutions to more complex flows.
Moreover use can be made of symmetries in the flow geometry to save
computing time: since the relationships between singularity strengths
on symmetrical panels can be used, the number of unknowns can be decreased,
thus reducing the computational task associated with the solution ot the
Equation (6). On the other hand no gain can be made in the evaluation
of the matrix {A.) since the velocities induced at a random point by
two symmetrical panels must be computed separately. Finally in the basic
formulation mentioned above the Kutta condition is taken to be of the
Neumann type, i.e., to consist of specifying the direction of the velocity
at the trailing edge; however References 17 and 18 describe how to
34
circumvent this restriction and use more physically meaningful
expressions of the Kutta condition.
Although the concepts may appear straightforward there are
computational problems encountered in the implementation of the so-
called panel methods. As already mentioned large source strength
gradients can be observed at the meeting of a closed source surface and
a doublet sheet if the latter ends at this location: such gradients
can induce significant numerical errors in the computation of the
tangential surface velocity component and, accordingly, are unacceptable.
Some methods circumvent the above numerical difficulty by placing the
doublet sheet inside the closed surface (Rubbert et al.) while others,
which use only doublet sheets to produce circulation containing onset
flows, extend the doublet sheets so that they cover the body surface
[17][18]. The latter procedure appears to give better results than the
former (based on results in the open literature). It must be noted that
the source strength gradients do not affect the far-field influence of
the source panels: more important from this point of view are the
numerical errors associated with the normal velocity component boundary
condition. In addition the no through flow condition is satisfied only
at the body contour control points, away from these points the body walls
can "leak". This computational problem, which can be severe in three-
dimensional configurations, can be alleviated by increasing the source
paneldensity in regions of high curvature*, using doublet panels which
*The calculated source strengths on curved surfaces are shown to be
accurate only to first order in the angle subtended by the panel because
of the planar panel approximation.
35
are an order of magnitude more tolerant of surface curvature or
implementing previously mentioned higher-order methods.
4.2.2 Potential Flow about an Inlet at Prescribed Mass Flow
In the cases studied the time-saving technique of super-
position of fundamental solutions can be used to obtain solutions at
various flow conditions. The inlet operating condition is defined in
general by four parameters: angle of attack, angle of yaw, freestream
velocity and mass flow through the inlet. Consequently, since the
governing equation and boundary conditions are linear (Laplace's
equation), the solution for arbitrary values of these parameters can be
obtained by linearly combining four independendent basic solutions. The
problem is reduced to defining these four fundamental solutions. Whatever
the onset flow may be, good accuracy in the potential flow calculation
near the inlet mouth may require the inlet to have a long afterbody
V.
especially for cases of high L . The choice of the first three fundamental
solutions to the flow around the inlet is straightforward: they consist
of the flows due to unit value uniform onset flows at 1) 0 angle of
attack and yaw, 2) 90* angle of attack with 0* angle of yaw, 3) 900 angle
of yaw with 00 angle of attack. A forward position (called control station)
is chosen where the mass flows for these flows are evaluated: flows 2) and
3) give mass flows which are near zero. These three fundamental solutions
can be linearly combined to give the flow about the inlet at any angle
of attack, any angle of yaw and any freestream velocity, however one
does not have freedom to choose the mass flow through the inlet.
Consequently a fourth fundamental flow is required to control the amount
of fluid entering the inelt. Different approaches can be taken. In one
36
early approach the so-called closed-duct method, [24][25], the inlet is
extended with a long afterbody which is closed at the rear by a wall
normal to the centerline, and the flow induced by a unit value uniform
onset flow at 0* angle of attack and yaw about the so-defined body is
used as fourth fundamental flow. In this case although the flow may
vary locally near the closing wall it is well behaved near the lips.
Further the distance between the control station and the closing wall
has to be quite large for the accuracy of the results to be good: this
requirement is often very expensive (computing time, storage) to meet in
three-dimensional configurations due to the large number of panels needed.
Thus another possibility is to use as fourth basic flow that for the inlet
in static condition, in which the flow velocity is zero at infinity
while a finite amount of fluid enters the inlet. This flow can be
obtained in the following two ways. First a singularity sheet is placed
across the inlet duct far from the mouth and a non zero normal velocity
is prescribed on this "suction" surface. The flow in the vicinity of
this surface is inaccurate (for the same reasons as before), however it
is well-behaved at the control station where the evaluated mass flow is
again different from the specified one due to "leakage". This procedure
gives good results inside the inlet and near the lip but inaccuracies
appear in the exterior region [22] (region in which we are, in our
application, very interested). These arise from the fluid feeling the
the influence of the suction surface through the-inlet wall. In parti-
cular a "spurious" stagnation pbint appears on the exterior surface,
which makes this way of generating the static solution inacceptable for
our purpose (particularly if it is noticed that due to the large value
of the average inlet velocity to wind velocity ratio, the static
37
solution is the main component of the global solution in the vicinity of
the inlet). Another way of producing the static solution is to use as
onset flow the flow induced by a vorticity distribution on the shroud
surface. This approach yields a more accurate solution for the static
operation. In particular it can be shown that the solution obtained by
means of this method is the correct solution if the afterbody length is
infinite (even with finite dimension elements) [22]. Further the
velocity on the exterior surface decreases as the distance from the lip
increases and it is always in the right direction. All this makes the
solution obtained in the latter way the most attractive fourth basic
solution.
4.k.3 Potential Flow Computation of Hess, Mack and Stockman
The computational method selected to calculate the potential
flow about the configuration of interest can now be outlined. The core
of this procedure is the constant source strength panel method developed
by Hess [14] to.solve flow about arbitrary non lifting bodies. Most of
the development effort arises from adapting this standard procedure to
the peculiar case of arbitrary three-dimensional inlets. In particular
the previously described technique of superposition is used to obtain the
flow about the inlet at various operating conditions. The fundamental
solutions consist of the flows induced by unit value onset flows parallel
to the coordinate axes and'the flow associated with the inlet in static
operation. The latter flow, whose importance in the situation we are
concerned with has already been emphasized, is produced by a distribution
of unit vortices on the shroud surface (actually a dipole distribution
38
is employed: use is made of the relationship between a dipole sheet
and a vortex sheet given in Reference 18). These vortices are placed in
such a way that they form vortex rings, which, for an axisymmetric
inlet, are oriented exactly on the circular cross sections.
Advantage was taken of the availability of a computer program
implementing the above method. A description of this program, as well
as a more thorough presentation of the method, can be found in Reference
23. In the configuration of current interest the flow geometries asso-
ciated with the above described fundamental flows present two planes
of symmetry. Advantage can be taken of the possibility of predicting
the relationships between the singularity strengths on symmetry related
panels to reduce both the computing time and the required storage. The
available code was able to allow for only one plane of symmetry and,
consequently, had to be modified to bring in the two-plane of symmetry
capability. Even with the introduction of this feature solving the
flow about the inlet (i.e., computing the source strengths associated
with the fundamental flows) remains a fairly lengthy task. Particularly,
the amount of storage required is enough to limit the number of panels
used to approximate the inlet. The next section describes the results
obtained using this computational method.
39
CHAPTER 5
RESULTS
5.1 Potential Flow Calculation
As has been already discussed, the computational method used requires
a large amount of storage, and this puts a limitation on the number of
panels which can be utilized to describe the inlet. The present calcu-
lation was conducted using 240 panels on each half inlet. The panels
which are planar quadrilateral elements are input by means of coordinates
of their vertices. The latter which lie on the inlet surface are orga-
nized in so called N-lines [23] which traverse the inlet in the axial
direction at a given dircumferential location*. In our case all N-lines
are similar and lie in planes containing the inlet centerline.
Initial numerical experiments revealed that the number and circum-
ferential positioning of these lines strongly affect the results (leakage,
streamline pattern in the vicinity of the inlet). Based on these initial
calculations the configuration used was chosen to be eleven equally spaced'
N-lines on each half inlet. In .this case the limitation on the number
of panels and the necessity of using elements whose largest dimension
is small compared to inlet diameter lead to the distribution shown in
Figure 3b (front view) and Figure 4 (exterior side view). The locations
of the panel vertices on any N-line are given in Figure 3a. It must be
*An N-line starts at the mest aft location of the inlet, proceeds forward
along the interior surface, around the lip and then along the exterior
surface up to the end of the inlet.
40
noticed that the inlet investigated is not extended as far downstream
as desirable: the input surface is only 1.3 diameter long. However
with such a configuration the leakage (defined as the relative
variation of the volume flux) between two axial locations at respectively
0.05 D and 1.25 D from the lip is equal to 6.7% for the static operation
fundamental flow and 1.2% for the 0* angle of attack and yaw fundamental
flow. These acceptable results indicate that the selected panel density
is adequate at least for the internal flow. Further it was found that
extending the inlet downstream as desirable requires a considerably larger
number of panels to be used since simply enlarging the panels gave poorer
results..
Once the fundamental solutions are determined, the potential primary
flow about the inlet at any operating condition can be studied. We choose
to investigate the following configurations:
Case 1: = 30 , =0*V.
00
Case 2: V = 45 , cc 45*0V
V.
The cross flow configuration - = 30, a = 450, was not used because of
the absence of a stagnation point in front of the lip on the ground plane:
in this case the wind velocity is large enough with respect to the inlet
velocity for the stagnation point to be blown away.
41
5.2 Streamlines
We will first examine the streamlines of the potential flow field
in the neighborhood of the inlet, using the calculation for fluid
particle trajectories described in Appendix A. Since the flow is
three-dimensional, it is important to note at the outset that the
results are presented as projections of these streamlines on the
coordinate planes or as plots of the streamlines in these planes.
Case 1: V30 ,0*
V0
The streamlines for'this case are shown in Figures 6 through 9.
The first two of these, Figures 5 and 6, show streamlines lying in the
x, z plane (the symmetry plane of the inlet). Figure 5 shows the over-
all flow field, while Figure 6 shows a smaller region but presents
more detail. Some of the general features of the flow can be seen in
these figures, although many aspects are shown more clearly in subsequent
figures. However from these two figures one can infer that there are
three locations of stagnation points or lines. The first is a
stagnation point on the ground plane in front of the inlet and is
denoted Sl in Figures 5 and 6. The stagnation streamline is shown by a
dash-dot line in Figure 5. The second is also a stagnation point and
forms on the ground at the rear of the inlet: it is denoted S2 in
Figure 6. This point -is really an artifice of the calculation and
42
would not be present in the actual situation*. The third location is
a line around the inlet which meets the symmetry plane of the inlet at
S. This again is, to some extent, a product of the calculation. In
the "real" flow it is to be expected that this stagnation line should
be on the outer surface of the inlet, however the strong recirculation
connected with the use of the present code in near static operition
shifts this line off the inlet. In a sense, then, this gives us an
inlet configuration which has an "effective" shape with a much thicker
lip and a somewhat different external geometry. (In fact since the
space between the inlet and the ground is entirely filled with
recirculating flow, the inlet external surface is seen by the oncoming
flow as extending to the ground plane). Because of this recirculation
some aspects of the flow field cannot be described adequately using the
present computational method, although others can be seen clearly.
This distinction between what can and cannot be resolved with an
analysis of the present type will be made in more detail in what follows.
Figure 7 gives the streamlines in the x, y plane (the ground plane)
and presents a different view of the overall features seen in Figures 5
and 6, namely the two stagnation points and the stagnation line. Only
half the flow is shown because of the symmetry. The next Figure, 8,
shows the streamlines in the vicinity of the inlet projected on a plane
*It can also be pointed 'out that the first stagnation point would not
exist in a real (viscous) flow either since particles on the ground plane
would separate before reaching the stagnation point.
43
normal to the axis of the inlet. Projections of these streamlines
on the ground plane can be seen in Figure 9. For convenience the
streamlines are numbered and the same type of lines (i.e., dash-dot,
solid etc .. ) are used in both. figures. The traces shown are, of
course, only parts of the streamlines (which can extend to infinity);
in view of the time taken for each calculation, the particles have been
tracked only in the regions of interest. The stagnation line and, to a
smaller extent, the first stagnation point can be seen in these figures.
Figure 10 shows, together with a front view of the inlet, the arch
shaped stagnation line about the latter and a cross section of the
capture surface* four inlet heights** upstream of the inlet. As seen
in Figure 11 which gives the traces of the capture surfaceon both
the ground plane and the symmetry plane this distance is sufficiently
upstream to be regarded as "far upstream". Finally, Figure 12 gives
a perspective view of the three stagnation locations described
above.
The recirculation of part of the flow exiting the engine plays a
very significant role in the formation of the stagnation points or
*The capture surface is defined here as the surface separating the portion
of the flow which enters the inlet for the first time from the portion of
the flow which does not enter or re-enters the inlet (recirculation).
**The inlet height, centerline to ground is the relevant length scale
to characterize this flow.
44
lines observed. As has been already indicated, this shifts the
stagnation line which would be expected to form on the inlet off the
surface of the latter. Further the second stagnation point S2 arises
also from the fact that the fluid particles leaving the engine turn
sharply instead of moving straight aft. The first stagnation.. point
should occur where fluid particles which come from ahead of and below
the inlet meet. However in the potential flow calculation, the particles
that proceed from under the inlet are in fact exiting the engine instead
of coming directly from far upstream. It follows that for the present
calculation the stagnation streamline associated with S1 lies on the
capture surface. As will appear more clearly in section 4 of this
chapter (Figure 36), the portion of the inlet cross section that is filled
with non recirculating flow is almost circular. Its axis is only
slightly above the inlet centerline while its diameter, which can be
regarded as the effective inlet inside diameter, is estimated to be equal
to 0.8 D. Consequently the actual value of the ratio E is roughly 1.6,
V.D
whilst the ratio equals 30 as desired. This may partially explain
why the flow field above the plane which is parallel to the ground plane
and contains the inlet centerline appears not to be strongly affected by
the presence of the ground in the sense that the streamlines lie in
planes including the inlet centerline as in the case of the inlet alone.
Case2: 4 ,45
The features described above are also observed in the 450 cross
flow case, namely stagnation point S1 near the front of the inlet,
45
spurious stagnati.on point S2 on the ground plane at the rear of the
inlet, arch shaped stagnation line about the engine, recirculating
flow. These aspects of the flow field can be seen in Figure 13 which
shows streamlines lying in the ground plane. The effect of the yaw
introduced by the cross wind appears clearly. The first stagnation
point has moved off the inlet symmetry plane in the direction of the
cross flow while the second stagnation point proceeds in the opposite
direction. The points where the stagnation line meets the ground plane
can Also be seen in this figure (they are denoted L and L'). This line
is no longer roughly normal to the inlet axis and is determined by the
combined effects of the., wind and the recirculating flow. The latter
fills, as previously, the space between the inlet and the ground. As
will be confirmed by results presented in section 4 of this chapter,
this flow is stronger on the leeward side than on the windward side of
the inlet. Figure 14 shows the projections, on a plane normal to the
inlet axis, of streamlines near the stagnation streamline associated
with S1. The latter is shown by a dashed line. In Figure 15 are given
the projections on the ground plane of the same streamlines (which, for
convenience, are numbered in both Figures 14 and 15). The flow coming
from ahead of and below the inlet appears clearly to be associated with
the formation of the first stagnation point while the stagnation stream-
line is again found to lie on the capture surface. In Figure 16 is shown
the projection of this line on the inlet symmetry plane. Figure 17 shows
a perspective view of the three stagnation locations observed. Finally
in Figure 18 the trace of the capture surface on the ground plane is
'given. This surface is found, as previously, to end farther forward
46
than expected. Further the closeness of the forward stagnation
streamline and the capture surface must be noted and kept in mind when
studying the deformation of vortex lines. In summary one can say that
the features of the flow field can be explained by combining the effects
of the yaw with the features recognized in case 1.
In both case 1 and case 2the flow field is found to contain several
expected features, namely forward stagnation point on the ground plane,
stagnation line about the inlet and capture surface whose cross-section
varies rapidly in the neighborhood of the inlet. However the quantitative
nature of these features is affected by the existence of a large re-
circulating flow. For example the capture surface is observed to end
farther forward than expected, instead of forming on the inlet the
stagnation line appears off the latter and the forward stagnation
streamline is found to lie on the capture surface. Further the flow
existing actually below the inlet cannot be determined using the present
model and numerous questions which the simple model investigation re-
ported in Appendix B raises about the nature of the flow under the inlet
at the closing of the capture surface remain unanswered. In the present
calculation the capture surface is smaller than expected and the flow it
bounds is only a part of the flow which is actually sucked by the inlet.
However this part of the flow does seem to be adequately predicted, as
seen for example in the forward stagnation streamline. The shortcomings
of the present model lie rather in the inability to describe certain
regions of the flow. Proceeding through the next steps of the approach
described in Chapter 3 will -reveal whether the present model, in spite
of its- limitations, can be used to explain the inlet vortex formation.
47
At the same time it will be possible to find out how the model can be
improved to achieve a more precise description of the flow field.
Note thatthe strong recirculation which is the cause of all the
difficulties encountered results from the very nature of the model.
rather than numerical inaccuracies. In the current model the total
pressure is assumed to be the same everywhere (potential flow). Conse-
quently, due to the large velocity difference between points in the
internal stream and the external stream, the streamlines must turn
sharply when exiting the inlet. In the actual case the internal and
external streams have different total presssures and the fluid particles
leaving the engine move nearly straight aft in a jet with the static
pressure being essentially constant across the jet while the velocities
outside and inside the latter differ largely. Additionally it must
V.
be noticed that due to the high value of the ratio-- the flow very
V0
close to the inlet is not very different from that about the inlet in
static operation. This static solution is (partially) induced by vortex
rings (dipoles) and this accounts for the tendency of the flow to fan
out downstream of the engine. Using a very long afterbody can ease,
to some extent, the difficulties discussed above but this turns out to
be difficult to implement in the present situation.
5.3 Material Lines
In addition to the streamlines we have examined the behavior of
selected material lines as well. As stated previously, since the
flow considered is incompressible, inviscid and without body forces, if
these lines coincide at some time with vortex lines they do so
always. Thus within the approximation considered here the tracking of
48
the material lines shows directly the convection and amplification
of the vortex filaments.
As discussed in Chapter 3, we resolve the far upstream vorticity
into three components. These are the streamwise (or Beltrami)
component, the component perpendicular to the ground plane (vertical)
and the component parallel to the ground plane (horizontal) and per-
pendicular to the mean flow direction far upstream. The last is the
only component that would exist in a two-dimensional boundary layer
flow over the ground. The material lines that are studied represent
the second and third of these components since, as described before,
the amplification of the Beltrami vorticity between two locations can
be seen directly from.the ratio of velocities. As in the case of the
streamlines we will look at the two cases separately.
Case 1
In the absence of cross flow one does not need to track material
lines which are horizontal far upstream, since, within the context of
the present description of the flow, vortex filaments which lie on these
material lines can only give rise to a vorticity distribution at the
-engine face location which is symmetric with respect to the inlet plane
of symmetry. Consequently, even if parts of these material lines near
the stagnation streamline connecting S1 to the inlet, as sketched in
Figure 19, and the corresponding vortex stretching is large, no single
vortex appears because the associated streamwise components are equal
and opposite. In other words, in this case there is no way that these
filaments can give rise to a single "vortex" rather than two symmetrically
placed counter rotating vorticity distributions. Therefore we have only
investigated the deformation of material lines which are vertical far
49
upstream.
Figure 20 shows the shapes into which a vortex line which is
vertical at a location four centerline heights upstream of the inlet
face is deformed. The material lines shown are in the x, z plane,
i.e., the plane of symmetry of the inlet. It can be seen that the fluid
line undergoes substantial stretching as it enters the inlet. However,
perhaps somewhat surprisingly, this stretching appears to be relatively
symmetric in spite of the fact that there is a stagnation streamline
(shown as a dash-dot line) on one side of the inlet only. The capture
surface is also shown in the figure (as a dashed line) to give an idea
of the extent of the recirculation region (which is almost symmetrical
about the inlet centerline). Figures 21 and 22 show two different
views of a vortex line which is vertical far upstream and 0.60 inlet
heights (0.75 inlet diameters) from the x, z plane. Figure 21 shows the
projections on the x, z plane with the numbers corresponding to different
times. The situation appears quite similar to that occuring on the x, z
plane as regards top and bottom symmetry. Figure 22 shows the pro-
jections on a plane normal to the inlet axis (the y, z plane).
Figure 23 shows the projections on the same plane of the positions
of a vortex line that is initially 1.6 inlet heights (2.0 inlet diameters)
from the x, z plane, i.e., near the capture surface. Note that since
vortex lines cannot end in the flow the lines tracked must extend to
infinity. However those portions of the material lines outside the
capture surface, which do not enter the inlet, are not shown. Note
also that both the head (top) and the foot (bottom) of the parts of the
material lines that are shown will lag behind the middle part since
50
particles which are on the ground plane will stay there, while
particles on the capture surface will approach the stagnation line
as described in the previous section. The velocity differential
means that the particles near the center of the fluid line will exit
the engine while those at the foot and head remain near the stagnation
regions*. Once a vortex line enters the inlet, we can no longer track
it entirely and only parts of the legs extending from the stagnation
regions to the inlet can be traced. These legs are both strongly
stretched and, consequently, have high vorticity levels, although a
quantitative appreciation for their stretching is difficult from just
the tracking of material lines since the deformation increases rapidly
for fluid particles closer and closer to the ground plane or the capture
surface. The next section presents an attempt to evaluate the vorticity
levels of both legs.
The study of the deformations of material lines initially off the
x, z plane gives a clue to the possible inlet vortex formation mechanism.
The existence of two stretched legs and the observation of a single
vortex seem difficult to reconcile. However the locations of these
legs must also be examined. While the foot of any vortex line approaches
the stagnation point S1 and the associated leg nears the corresponding
* By the definition of a stagnation point, the closer to such a point
the streamline followed by a fluid particle is, the longer time it
takes to this particle to reach a position downstream of the stagnation
point.
51
stagnation streamline, the location reached by the upper leg which
extends from the inlet up to the vortex line head (defined as the top
of that part of the vortex line which is on the "inside" of the
capture surface) depends on the vortex line initial position. Knowing
the location of the vortex line head far upstream is sufficient to
determine the location reached by the latter after an infinitely long
time. To a close approximation the streamline followed by the fluid
particle which coincides with the vortex line head lies in a plane
containing the inlet centerline. Further since it is on the capture
surface this particle cannot ever pass the. arch-shaped stagnation line.
Eventually using these two pieces of information it is possible to
determine the ultimate position of the vortex line head. Figure 24
thus shows the suggested deformations of vertical material lines. A
far upstream uniform distribution of vertical vortex lines evolves
into a configuration in which the upper legs of the lines are "fanned out"
over the upper part of the inlet while the lower legs are squeezed around
the stagnation streamline associated with S1. (Although the stagnation
regions are never reached in the actual flow, these trends remain valid).
The predicted squeezing goes with an increase of the associated circula-
tion per unit area, the problem remains to compare this rise with that due to
vorticity amplification in the upper legs. Note that if the existence
of stagnation points is admitted (i.e., separation effects are neglected)
as in the current model, the stretching of both legs is infinite. The
difference lies in the spatial distribution of the associated vorticity
amplification as can be seen in Figure 23 where fluid. particles close to
the foot and head of the vortex line are tracked. Note that the trend
52
discussed above would be expected to be even more pronounced, if the
computation did not include the recirculation region, since the arch-
shaped stagnation line would then be on the inlet external surface.
Case 2
Similar conclusions can be reached in this case for the vertical
material lines. Figures 25 and 26 show a vortex line which is vertical
far upstream. The initial position is marked by the small cross in
Figure 27. Figure 25 gives the projections on the x, z plane of the
shapes into which this line is deformed while Figure 26 shows the pro-
jections on a plane normal to the inlet axis. The configuration into
which a far upstream uniform distribution of vertical vortex lines
evolves is no longer symmetrical with respect to the inlet plane of
symmetry. However the basic features are similar to those identified
in the previous case: the upper legs of the vortex lines tracked are
finally "fanned." over the upper part of the inlet while the lower legs
are squeezed around the stagnation streamline associated with S1 in a
manner that is qualitatively the same as in case 1. Due to the
position of the forward stagnation streamline the vorticity concentration
at the engine face location is located to the left of the symmetry
plane of the inlet when looking downstream into the latter.
As already emphasized, the flow in the present case is not symmet-
rical with respect to the inlet plane of symmetry. Accordingly it is
desirable to study the deformations of far upstream horizontal vortex
lines. This is done in Figures 28 through 33. Thd material lines
tracked are far upstream respectively 0.8, 0.4 and 0.16 inlet heights
53
C1.0, 0.5 and 0.2 inlet diameters) from the ground plane. In addition
they initially have the same projection on the ground plane; the
latter is shown in Figure 27. Figures 28, 30 and 32 show projections
on the ground plane for the three material line locations while
Figures 29, 31, 33 give projections on a plane normal to the inlet axis.
As in the previous case the deformations of only those portions of the
material lines inside the capture surface are investigated. The fluid
particles close to this surface are found to lag behind due to the
presence of stagnation locations in the near vicinity of the inlet while
those in the middle of the portion of material line tracked enter the
engine much sooner. The first fluid particles to be drawn into the inlet
appear to enter that half of the latter which does not contain the
forward stagnation streamline. However when very large drift times are
considered, i.e., particles very close to the capture surface are
tracked, the corresponding legs are observed, again somewhat surprisingly,
to be rather symmetrically placed: this appears clearly in Figures 30
and 31. Thus both legs of the far upstream horizontal material lines
are found to be strongly stretched and no clear indication of the forma-
tion of a-definite vortex can be seen from the material lines. It is
thus desirable to study more thoroughly the stretching undergone by
these lines.
5.4 Quantitative Calculation of Vorticity Amplification
Our purpose in the present section is to attempt to obtain a
quantitative appreciation for the stretching undergone by material lines
as they are drawn into the engine and thus to study more thoroughly the
54
amplification of a far upstream vorticity distribution, In particular
we intend to compare the sizes of the regions of high vorticity which
were indicated by the examination of material lines. We will first
describe the calculation procedure used and then discuss the results
obtained in the two flow configurations currently inyestigated.
The location at which we want to determine the vorticity distri-
bution due to a far upstream shear flow is in the vicinity of the engine
face. In our model this is inthe portion of the inlet where the stream-
lines are straight and parallel, say 0.75 D downstream of the inlet lip.
In the (semi-infinite) inlet model chosen the streamlines downstream of
this section would be essentially straight lines. Further if the stream-
lines at an inlet section are parallel straight lines the velocity over
this section is uniform 'nd, by definition, equals V. Consequently we
can extend the computed streamlines downstream of the inlet section of
interest as straight lines with the velocity equal to V . Once the
above assumptions are made, it is possible to determine the stretching
undergone by a small fluid line connecting two neighbouring particles
at a "far upstream" location. To do this it is only necessary to calculate
the drift times of these particles between this location and the inlet.
Let t1 and t2 be the drift times between a "far upstream" location and
the selected inlet section of two particles initially at respectively
P1 and P2. For purposes of discussion assume t1 to be larger than t2.
In this case when the particle initially at P1 attains the cross section
of interest, at P' , the particle initially at P has moved a distance
2
V. (t1 - t2) downstreamdf this section, tofP (sed Figure*34). To a good
approximation the small straight fluid line which initially connects
the particles P1 and P2 can be regarded as being deformed into the
straight material line P P . The stretching undergone by this line
Pi p'2is then given by p 1 -p- (this approximation of course improves as the
1 2
initial distance P1 P2 decreases.).
The flow situations we are concerned with are characterized by
the strong contraction of the streamtubes between far upstream and
the inlet. It follows that for the values of the distance P1 P2
usually used (less than 0.20 H), the distance P.1 P2 is nearly equal to
Vg (t1 - t2) and the fluid line tracked is finally almost parallel to the
inlet axis. In such a case the vorticity at P-1 due to a vortex line
initially aligned with P P2 is almost streamwise, with its orientation
depending on the direction of the vorticity along the vortex line as
shown in Figure 34. If we carry out this process for a number of such
lines the vorticity distribution induced at the inlet section of interest
by a far upstream shear flow can be found. To do this one can compute
the drift times of fluid particles which far upstream are at the nodes
of a regular grid describing the flow field interior to the capture
surface in such a way that one of its two directions is initially
parallel to the vorticity. It remains then to associate those neigh-
bouring particles that define far upstream fluid lines in the direction
of the initial vorticity (horizontal or vertical). One remark is now
in order. The vorticity value determined with the help. of the procedure
described above, which is actually an average over the small vortex line
tracked, is taken to exist at the location where the streamline followed
by the fluid particle with the larger drift time meets the selected inlet
55
56
section. It follows from this convention that among the few particles
of a given material line which are tracked the particle with the
smallest drift time attains the inlet section of interest at a point
which cannot be given a v&lue of vorticity. This explains the existence.
in our results of points without associated vorticity level. We next
discuss the results yielded by the above approach for the two flow
configurations currently considered.
V.
Case 1: =30 , a = 00
Because of the symmetry of the flow with respect to the x, z plane,
the grid defined by the particles to be tracked has to cover only half
the cross section of the capture surface. This grid is initially in a
plane which is normal to the inlet axis and located 4.8 inlet heights
(6 inlet diameters) upstream of the inlet lip. Figure 35 shows a view
of this plane. The description of the cross section of the capture
surface is seen to be quite complete; indeed the latter is everywhere
less than a quarter of a diameter from the grid. It follows that due to
the high streamtube contraction which characterizes this flow, the
streamlines followed by the grid peripheral fluid particles can be
expected to be close to the recirculating flow boundary. Further since
fluid particles initially near the ground plane enter the inlet in the
vicinity of the forward stagnation streamline, it has been found useful
to increase the grid resolution near the ground plane: the lowest
particles in the grid are only 0.008 H far from the ground surface.
The points where these streamlines meet the inlet section of
interest are shown in Figure 36 (S denotes the trace of .the forward
57
stagnation streamline in this section), Since the streamlines are
assumed to extend downstream as straight parallel lines, Figure 36 can
also be regarded as giving the projection on a plane normal to the inlet
axis of the shape into which the upstream particle grid is deformed.
Using this figure it is possible to obtain an appreciation for the
shape of the portion of the inlet cross section that is filled with
recirculating flow. This region appears to be quite circular, its axis
being only slightly above the inlet centerline, and fills about 40 per
cent of the inlet section. Accordingly it must be noticed that roughly
half of the flow field surrounding the forward stagnation streamline
cannot be described using the present calculation procedure since that
portion of the flow field which is between the stagnation streamline
and the inlet wall is filled with recirculating particles. Although
this could be qualitatively expected in view of the results of the flow
field investigation presented in the second section of this chapter
the quantitative extent of the inability of the present computation Lo
predict the whole flow around the stagnation streamline is only now
apparent.
As has been already discussed, only the stretching of far upstream
vertical vortex lines need- be studied for this case. Figure 37 shows
the vorticity distribution at the inlet section of interest due to
a far upstream distribution of vertical vorticity (in fact the
stretching levels undergone by fluid lines linking neighbouring
particles are given). The numbers are all referenced to unity, i.e.,
to a uniform shear flow, however it should be emphasized that they are
58
relative quantities and can be directly applied to any vorticity
distribution. Figure 38 shows an enlarged view of the stagnation
streamline region. Precise values of the vorticity levels are
in fact illusory and overall trends are more important. Therefore
the results are presented using the key given in Table 1. The
presence of a minus sign means that the vorticity vector has the
magnitude indicated and is pointing upstream. Positive values denote
vorticity vectors pointing downstream. High values of stretching
(say more than 100) are observed-in two regions of small extent
close to the recirculating flow boundary, namely a thin ring in the
top of the inlet section and a more compact zone near the stagnation
streamline. It is to be emphasized that if the former presents
lower stretching values than the latter, this is only due to the
difference in grid resolution and distance of the peripheral fluid
particles from the capture surface. In fact since in' the present
calculatibn separation effects are neglected the stretching increases
without bound when the fluid particles tracked get closer to the capture
surface or the ground plane since the particles that are in these
surfaces ultimately' get "stuck" in the stagnation line or point.
Comparing precisely the shapes of these zones of high vorticity is
difficult because of their small extents and in spite of the existence
of these regions no vorticity distribution pattern is observed which
indicates clearly the formation of a vortex. It appears that this is
due to the fact that in the present calculation the high vorticity
level zones are all found close to the recirculating flow boundary
(i.e., near the capture surface) while the high stretching region
59
associated with the forward stagnation point would be expected to be
found inside the flow that enters the inlet from upstream. Accordingly
due to the inability of the present calculation to predict the whole flow
around the forward stagnation streamline, no quantitative conclusion
can be reached about the magnitude of the vorticity amplification at
this point. An understanding of the latter definitely requires the
suppression or at least the reduction of the recirculating flow,
especially below the inlet and this is recommiended as a high priority
item in the extension of this work.
V.
Case 2: 45, o = 45
Similar conclusions can be drawn from the results of the 450 cross
flow case. As previously the grid defined by the fluid particles
tracked is initially in a plane normal to both' the ground plane and
the free stream direction. Figure 39 shows the trace of this plane in
the ground while in Figure 40 is given a front view of the initial shape
of the grid. We see that here again the description of the capture
surface cross section is quite complete. Further, as before, the grid
resolution is increased in the vicinity of the ground plane in order
to study more thoroughly the part of the flow near the stagnation
streamline which the model can describe: the lowest particles are
initially 0.008 H, 0.016 H or 0.08 H far from the ground plane. Figure
41 shows the projection on the inlet section of interest of the "final"
shape ti.e.,, the shape where the streamlines are parallel) into which
the grid considered is distorted. The trace of the stagnation streamline
60
is again denoted S. The effect of the yaw appears clearly in the
hiily asymmetric deformation of the grid. The recirculating flow is
observed to be stronger on the leeward side than on the windward side
of the inlet. Thus the portion of the inlet cross section which is
filled with non-recirculating flow is not circular. Further it now
covers slightly less than half the inlet section: this is believed to
Vi
be due to the use of a higher value of . Finally as in the previous
case roughly only half the flow surrounding the stagnation streamline
can be predicted.
In contrast to the first case far upstream vertical and horizontal
vorticity components must both be considered. Figures 42 and 43 show
the vorticity distribution induced over the inlet section of interest
by a far upstream uniform vertical vorticity distribution (the key to
be used to read this figure can also be found in Table 1). As previously,
high vorticity levels are only observed in local regions close to the
recirculating flow boundary. However, even if the accuracy of the results
given by the present model for the highest particles in the grid can be
questioned, higher stretching values are found for particles entering
the inlet to the left of the stagnation streamline. This can be
explained by the closeness of the capture surface and the stagnation
streamline and the associated combination of the effects of the stagnation
point S1 and the stagnation line. However again we cannot obtain the
required quantitative information relative to the formation of the ground
vortex due to the region-of greatest interest, namely the stagnation
streamline vicinity, being at the edge of the zone which the calculation
describes correctly.
The attempt to examine a far upstream horizontal uniform shear also
61
encounters this problem. Figures 44 and 45 show the vorticity
distribution induced by such a shear. The only conclusion that can be
made is that the vorticity levels found are generally lower than for
an initially vertical shear. Further the contours of equal vorticity
coincide with the projections of those grid lines that are far upstream
vertical, the higher vorticity values being obtained, as expected, for
fluid particles close to the capture surface. In addition the vorticity
level found does not vary significantly when particles closer and closer
to the ground plana are tracked.
62
CHAPTER 6
SUMMARY AND CONCLUSIONS
The vorticity field associated with an inlet vortex (or ground
vortex) has been investigated using a secondary flow approach. In
this method the three-dimensional rotational flows that characterize
this phenomenon are taken to be composed of an irrotational primary,
flow and a weak shear flow, with the vortex filaments associated with
this shear being regarded as convected by the primary flow only. A
computer program was developed to track specified material lines, which
coincide with the vortex lines in this flow, between far upstream,
where the vorticity can be regarded as known, and the engine face
location. The primary flow was computed using an existing code, which
was Adapted to take advantage of the symnetries present in the geometries
studied.
Two flow configurations were investigated in detail, both with high
values of inlet velocity to far upstream (ambient) velocity ratio: one
with the ambient velocity parallel to the inlet axis of symmetry and
one with the ambient velocity at forty-five degrees to this direction.
Both streamlines and material lines were tracked. Vertical (normal to
the ground plane) and horizontal (parallel to the ground plane) material
lines were used so that the 'behavior of each of the different components
of the far upstream vorticity could be seen. In addition the vorticity
distributions which occurred at the compressor face due to the ampli-
fication (stretching) of the far upstream vorticity distributions
were evaluated.
63
The results of this work can be summarized as follows. The flow
is found to contain several expected features, namely a stagnation
point on the ground plane near the-front of the inlet, a stagnation
line about the inlet and a capture surface whose cross-section varies
rapidly in the neighborhood of the inlet. However the quantitative
nature of these features is affected by the existence of a strong
recirculating flow, which is due to the nature of the calculation
procedure. Because of this, only that part of the flow which is
bounded by the computed capture surface seems to be adequately pre-
dicted by the model developed. In addition the capture surface found
is smaller than the one that would be expected to exist in the real flow.
It follows that only a portion of the flow sucked into the inlet is
correctly described by the model selected. In particular the flow around
the (very important) forward stagnation streamline cannot be studied in
its entirety since this streamline lies on the computed capture surface.
In spite of these limitations, the qualitative investigation of the
deformations of vortex lines initially normal to the free stream
direction shows, in both cases, the vorticity in both legs of these
filaments to be strongly amplified. Further the vorticity distributions
at the engine compressor face due to horizontal and vertical uniform
shear flows are found to contain very local zones of high vorticity
close to the recirculation region, although no pattern is observed which
quantitatively indicates the formation of a vortex.. (It is felt that,
again, this arises partially from the inability of the computation to
describe the details of the flow around the forward stagnation stream-
line:) This result notwithstanding, the present work still provides
64
useful information about the inlet vortex formation mechanism. In
particular, the tracking of vortex lines which are initially vertical
shows the salient features of the generation of streamwise vorticity
between far upstream and the engine face location. Vertical vortex
lines that are uniformly distributed far upstream are seen to evolve
into a configuration in which the upper legs of the lines are fanned
over the upper part of the inlet, while the lower legs are squeezed
around the forward stagnation streamline. This implies the occurrence
of locally higher levels of circulation per unit area in this latter
location, i.e., the formation at the engine face plane of two regions
of opposite circulation and different spatial extents. One of these is
small and appears around the stagnation streamline while the other is
larger and fills the upper part of the inlet; the former strongly
suggests the occurrence of an inlet vortex. Finally while the study
of the deformations of horizontal vortex lines shows the strong
stretching undergone by both legs of such lines, no pronounced asymmetry
was observed in the cases studied which might be clearly identified
with the formation of a definite single vortex.
65
CHAPTER 7
RECOMMENDATIONS FOR FUTURE WORK
It is clear that the inability to draw more specific conclusions
about the inlet vortex formation mechanism arises from the recirculation
flow introduced by the computational method implemented. Therefore
the next logical step is to improve the computational procedure used to
compute the potential flow field so as to suppress this recirculation
of fluid particles. In particular it is necessary that the forward
stagnation streamline be inside the portion of the flow adequately
described. The value of any modification can be assessed by examining
the extent of the recirculation in the flow between the inlet and the
ground, since this part of the flow seems to be the most sensitive
location to the artifical phenomenon observed. The following remarks
and suggestions may make the selection and design of a new computational
method easier. First it must be noticed that in the flow situations
we are concerned with the value of the inlet velocity to wind velocity
ratio is so high that the flow very close to the inlet is not very
different from that about the inlet in static operation. The code used
is not suitable to compute flow configurations in which the inlet is
nearly in static operation unless a huge amount of storage is available.
This is believed to arise from the fact that the onset flow used to obtain
the static solution (which is one of the four fundamental solutions to
be ljnearly combined) is dominant in the cases of interest-and the
vortex rings used to induce this flow cause the streamlines to turn
sharply downstream of the engine. While lengthening the inlet can ease
66
the problem, the feasible improvement is believed to be limited. Note
also that increasing the number of panels would increase the time taken
by the tracking of material lines (which is already large). It is
preferable to modify the onset flow used tc obtain the static solution.
This flow must give a solution which is independent of the first three
basic solutions and vanishes-far upstream. In view of the difficulties
encountered in the present study, it appears that the fourth onset flow
must be parallel to the inlet axis (or at the most slightly diverging)
downstream of the inlet. A possible choice consists of the following.
Inside the inlet surface and'slightly behind the inlet lip is placed an
axisymmetric intake of smaller thickness which extends farther down-
stream than the inlet studied and the axis of which coincides with the
inlet centerline. The flow about this intake in static operation is
axisymmetric and can be computed using appropriate codes. In particular
advantage can be taken of the fact that the intake surface can be
represented in a much more detailed manner than in a three-dimensional
flow, for example it can be as long as desirable while higher-order
methods can be used. The fourth onset flow is obtained by superposing
two axisymmetric flows: one for the added intake in static solution, one
for its image also in static operation. This flow induces a static
solution and, as desired, diverges only slightly downstream of the inlet
studied. Since a complete understanding of the inlet vortex formation
requires the potential flow calculation to be improved, the testing
of the above modification is recommended in the -extension of the
present work,
67
Furthermore, in the current approach, the vorticity amplification
is found to increase without bound when the portions of vortex lines
tracked get closer and closer to the capture surface or the ground
plane. However in the real flow viscous diffusion will limit the
vorticity intensification in regions of high stretching. Therefore,
it is also suggested that future studies should examine the influence
of viscous effects, although -it should be emphasized that it is expected
that this can be done in an approximate manner since we are seeking
the overall effects rather than the details of the velocity distribution
within the viscous core.
Once an accurate quantitative appreciation for the vorticity
amplification is obtained, it will be possible to calculate the vorticity
distribution induced by any far upstream shear distribution at the
engine face location. Up to -now only uniform shear flows have been
considered and the use of more realistic shear profiles (boundary layer-
like for example) is, however, highly desirable. The final step would
then consist of computing the (secondary) velocities produced at the
compressor face by the calculated vorticity distribution.
68
APPENDIX A
FLUID PARTICLE TRACKING COMPUTER PROGRAM
Since the mean flow is assumed to be steady, the fluid particle
trajectories coincide with the flow streamlines, hence, their equations
are by definition:
dx -=dy - dz
VIJVy Vz
where V, V , Vz are the components of the primary flow velocity field
in respectively the i, j, k directions. These functions which are
determined by the potential flow field calculation procedure are inputs
to the program.
Streamlines are curves and, consequently, can be described by means
of a single parameter, which may be one of the space coordinates. If,
for instance, the x-coordinate can be utilized as independent variable,
the above equations can be rewritten as follows:
dx =y= f(x,y,z)
dx Vx
Looking at this system of two first order ordinary differential
equations reveals that the velocity component V has to be non-zero
for the x-coordinate to be a useful parameter. Therefore a convenient
69
way of describing a streamline is to use as parameter the space
coordinate associated with the locally largest velocity component.
In this case, the streamline equations can always be rewritten as a
system of two ordinary differential equations, whose exact form
depends on which coordinate is the actual parameter. Consequently
calculating the trajectory of a particle amounts to solving successively
different systems of first order ordinary differential equations. Each
system is subject to initial conditions which express the continuity
of the solution at the boundary separating two regions of different
parametric representations or, for the first system considered when
tracking a fluid particle, the necessity for the solution to start at
the given initial point.
The progression along any trajectory is controlled in two ways.
First with the aim of monitoring the description of the followed
streamline, the flow velocity components at each calculated point are
compared to see whether, starting from this point, a different space
coordinate has to be used as parameter. Secondly the time elapsed from
the instant when a particle leaves the given initial point to the
instant when the same particle reaches the last calculated point is
evaluated every three pointsorwhen crossing the bouhdary separating two
regions of different -parametric representations and compared with the
given travel time so as to establish whether it is necessary to keep
on calculating the streamline. If not the position reached by the
tracked particle is determined by interpolation. Thus fluid particle
tracking involves two calculations which are carried out simultaneously,
70
namely streamline computation and drift time evaluation. Let us
now consider them separately.
A predictor-corrector method developed by Hamming [26] is utilized
to solve the previously mentioned systems of first order ordinary
differential equations and thus calculate the streamlines. This non
self-starting fourth order method uses four preceding-points to compute
a new one. Further it has the following advantages over one-step
methods of comparable accuracy, such as Runge-Kutta methods. The local
truncation error can be easily estimated and corrected for. Moreover,
if the system of first order ordinary differential equations of interest
is of the form
dy= F(XY)
where X is a real variable
Y is a n-dimensional vector
and F is a vector valued function
as few as two evaluations of the derivative function F are required
per step against four in the often used fourth-order Runge-Kutta
methods. The latter consideration is particularly important when the
derivative function is complicated and each evaluation needs substantial
computing time. In our case evaluating the function F amounts to
calculating the velocity components at a point of the flow field. This
can be quite time-consuming if a panel method is utilized to determine
71
the latter since then the lengthy computation of the influence
coefficients of all tke panels has to be carried out for each point
where the velocity is searched for. A fourth order Runge-Kutta method
proposed by Ralston [27] is employed to start the calculation at the
very beginning and restart it after adjusting the stepsize or changing
the way the streamline is described.
As previously indicated, the streamlines are divided into parts
whose parametric descriptions differ. Let P and P be two points in
such a part of an arbitrary trajectory. The time a particle needs
to travel from P to P is given by
=P' d
where $ is x, y or z depending on which coordinate is actually used
in the region of interest. Consequently, if the streamline segment
connecting two points cannot be wholly described by the same space
coordinate, the evaluation of the time it takes to a particle to move
from one of these points to the other involves computing and adding
up different integrals. The latter are estimated as follows. The
appropriate quantity , , or is evaluated at each calculated
x y z
trajectory point. The obtained values are interpolated by means of a
succession of cubic splire functions and/or parabolae which are then
integrated between adequate limits to yield the value of the integral
searched for. Generally every cubic spline function is associated
with four consecutive points of the followed streamline, sharing the
extremity points with contiguous functions. However when the role of
12
independent variable shifts from one space coordinate to another,
only one or two points may have been calculated since the last drift
time evaluation. Three points can still define a cubic spline function
while two points can be interpolated by means of a parabola providing
thatthe slope of the preceding interpolating function at the first point
is used as additional boundary condition. The position reached by the
tracked particle in a given time is found in the following way: firstly
the independent variable value is determined using the last derived
interpolating function, secondly the corresponding dependent variable
values are computed with the help of the forementioned fourth order
Runge-Kutta method.
Some comments are now in order. The accuracy arid computational
speed of the solution of any first-order ordinary differential equation
is currently controlled by halving the independent variable step size
when the local truncation error is above a certain limit and doubling
it when the same error is below another limit. Nevertheless in our case
it has been found necessary to bound the possible increase in step size.
If the step size when reaching highly curved segments of the followed
streamline is too large, the necessity for changing the way the trajectory
is described is recognized only at a point where the velocity component
associated with the new describing coordinate greatly exceeds the velocity
component associated with the old describing coordinate. This can
result in inaccuracies in the calculation (in the sense that the computed
points do not lie on the same streamline). Further the program has been
observed to run into difficulties in the vicinity of stagnation points.
The drift time evaluation has proven to be much more sensitive to
the step size than the streamline calculation. This can be explained
by the inability of the spline functions and parabolae to interpolate
correctly the locally high-valued and rapidly varying quantities
r , . A reduction in step size can ease the problem,
nevertheless a limit exists beyond which it is no longer economically
possible to perform the intended tracking. Varying the step size
and comparing the obtained final positions has been found a reliable
way of assessing the value of the results.
Note: for reasons which appear clearly in Section 5.4, a version
of the above program has been developed in which the tracking of a
particle is stopped when the latter reaches a given plane x = x plane
while the independent vatiable is the x-coordinate. This additional
feature is denoted "plane cutoff" in the following listing. When this
alternative is selected, the input drift time is a dummy parameter and
the actual drift time is .found by interpolation.
74
C ***** ** * * ** * ***** ******* *** * ** **** **t* * **** ** * * ***** *** *t****t ****n
C FLUID PARTICLE TRACKING PROGRAM
C
C THE VELOCITY FIELD IS GIVEN BY VXVYVZ FUNCTIONS OF XYZ
C THESE QUANTITIES ARE COMPUTED BY SUBROUTINE VXVYVZ(X,Y,Z,VX,VYVZ)
C
C
DI MEN SION
9 IMENS ION
y (2),oYS( 2P4),PFS 2,3),*STI (4,&4 ),F(2), YSS(2),
EC2),YF(2) YFS C2),AUX(3,3)
NLINEI(10), NLINEN(10), NFILE(4),
XC(1100), YC(1100), ZC(1100),
XN( 1100), YN( 1100),P 7N(1100),
SIGMAS(1100),SIGMA1(1100),SIGMA2(1100),SIGMA3(1100)0
FLUXX(4), FLUXY(4), FLUXZ(4), FLUX(4),
AREA (1100)
LOGICAL ILlIL2,IL3,TYPEATYPEBTYPECTYPED
EQUIVALENCE ( XC(1) , XN(1) , SIGMA1(1) , AREA() )
EGUIVALENCE C YC(l) YN(1) , SIGMA2(1) )
EQUIVALENCE ( ZC() , ZN() , SIGMA3(1) )
CHARACTER CH*1(3),RUNID*4
COMM0N
OMMON
COM;ON
CO MON
COMMON
/I
/I
/
/I
/
G
SUPPI
SUPP2
SUPP3
SUPP4
/
/
/
/I
COMMON / HEIGHT /'
COMMON / INFOV I
COMMON
COMMON
C OMMON
/I
/I
/
ONSETF
ANGLE
ROIN FO
/
/
/I
COMMON / CONFLG I
COMMON / INFORM /
COMMON
COMMON
COMMON
/
/
/
RANGES
IMPORT
NORMAL
/
/
/
MTABCJT,T4,MTP
NEXTNSTORE
JJSLP
IA S T (3,3)
IOPTXPLANE
HGROPZMIN
BB(50),SIGMAS,SIGMA1,SIGMA2,SIGMA3,
CC(4),KFLOW
ONSETX (53),ONSET Y(53 ),ONSETZ(53)
IATACK, ALPHAX(3), ALPHAY(3), ALPHAZ C3)
KOUNT(50), NLINE(50)p NTYPE(50), NLT(500),
ISECT
TITLE(1b), CASE; LIFSEC, LIST, NOFF, MPR,
IOUT, IWIDTH, LASWAKP NSYM, KONTRLf IPERS,
IPRINT, SF
NSORCE(10), NWAKE(1O), NSTRIP(10),
WIDTH(50,10), aIDX-T1R(2,1O)
RHOISO, RHO2SQ
NON. NFLOW, NHALF, NCOUNT
XNT(1)# YNT(1), ZNT(1)
DATA NFILE / 4,22,2729 /
READ RUN IDENTIFICATION
READ(7,2007) RUNID
STORAGE JANTED ?
READ(?.2008) NSTORE
IF ( NSTORE.EQ.0 ) GO TO 500
3PEN ( 35 , ACCESS2"DIRECT" )
NEXTal
DRIFT TIME OR XPLANE CUTOFF ?
&
&
&
&
&
&
C
C
C
C
C
&
&
&
&
&
C
C
C
C
C
C
C
C
75
500 REA(C,2314) IOPTsXPLANE
C
C READ CALCULATION PARAMATERS
READ(7,2006)
READ(7,2913) NMULXLIM,EMAX2
C
C PREPARATION FOR SUBROUTINE VXVYVZ
C a 
-4-em- a* -4w -as -m --A
C THE FOLLOWING STATEMENTS ARE SPECIFIC TO
C THE FLOW FIELD CONSIDERED
C
00 600 1=1,4
OPE'4 ( NFILE(I) , MODE a "IN" )
CONTINUE
KUNIT 29
REWIND 4
REWIND 29
READC (KUNIT )
READC (KUNIT )
READ
READ
READ
READ
C(KUNIT )
C(KUNIT )
C(KUNIT )
C(KUNIT )
READC (KUNIT )
HGRO * 0.
ZMIN 0.
IF c NSYv .NE.
READC (KUNIT )
CONTINUE
READC (KUNIT )
REWIND KUNIT
REWIND NSETS
REWIND NFLUX
TITLE, CASE, LIFSEC, LISTS NOFF, MPR, IOUT,
IWIDTH, LASWAK, NSYM, KONTRL
IATACK, ALPHAX, ALPHAY, ALPHAZ, AREA, NLINE1,
NLINEN, RHO1SQ, RH02SQ, TYPEA, TYPEB, TYPEC,
TYPED
XC, YC, ZC
NSORCE, NWAKE, NSTRIP, WIDTH, WIDXTR, BB, 18
XNs YN, ZN
NOFSAV, NFLUXP NSETS, NV, NSETV, NCSAVE, IUNIT,
NON, NFLOW, NHALF, NCOUNT
KOUNT, NLINE, NTYPE, NLT, ISECT, SF
2 ) GO TO 2400
HGRO, ZMIN
DIAM, AR FP
JRITE(38,2003) (TITLE(I),IA1,7)pCASE,HGRODIAM
N2 a 0
00 2500 JS21,LIFSEC
N2 8 N2 + NSTRIP ( JS
ONSETX (N2+1) = 1.
ONSETY (NZ+1) a 0.
ONSETZ (N2+1) a 0.'
ONSETX (N2+2) a 0.
ONSETY (N2+2) a Os
ONSETZ (N2+2) a 1.
ONSETX (N2+3) a 0.
ONSETY (N2+3) a 1.
ONSETZ (N2+3) a 0.
KFLOW * NFLOW#- 3
)
CALL READ1(NSETSSIGMASKONTRL)
CALL READI(NSETS oS IGMA1 ,KONTRL).
CALL REAC1(NSETSSIGMA2,KONTRL)
600
C
C
S
&
&
240:)
C
C
2500.
C
76
CALL READ1(NSETSSIGMA3,KONTRL)
C NOTE: SUBROUTINE READI IS SPECIFIC TO THE FLOW CONFIGURATION
C CONSIDERED AND #THEREFOREP IS NOT GIVEN
XNT( ) * 0.
YNTC1) 0.
ZNT(1) 20.
C
C READ FLOW DATA
READ(7,2006)
READ(7,2013) ALPHAsVINFvVC
READ(7o,2006)
C
ALPHAR * ALPHA * 3.14159265 / 180.
CC(2) * VINF*COSCALPHAR)
CC(3) * 0.
CC(4) * VIfF*SIN(ALPHAR)
READ .( NFLUX ) JP
READ C NFLUX ) (FLUXX(K),FLUXY(K)OFLUXZ(K),FLUX(K),K=1,4)
CC(1) (VC*ARFP-CC(2) FLUX(2)-C(3)*FLUX(3)-CC(4)*FLUX(4 )
& FLUX()
C
C WRITE FLOW DATA
WRITE(38,2004) ALPHAVINFVC
C
C END PREPARATION FOR SUBROUTINE VXVYVZ
C-------------------------------------------------------------------------
C
C WRITE CALCULATION PARAMETERS
WRITE(38,2010) NMULXLIMEMAX2
C
C NUMBER OF CALCULATED STREAMLINES
NPCO
C
C STORE GENERAL DATA
IF C NSTORE.EQ.0 ) GO TO 210
WRITE(35'NEXT) RUNIDCASEHGRO
NEXT*NE XT+1
WRITE(35'NEXT) DIAMALPHAVINF
NEX T=NEX T+l
WRITE(35'NEXT) VCNMULXLIM
NEXTuNEXT+1
WRITE (35 'NEXT) EMAX2
NEXT=NEXT+l
WRITE(35'NEXT) NPC
NEX TuNEXT+l
C.
210 CHI()w'X'
CH(2)=$y'-
CH(3)*'Z'
C
250 NPC*NPC+1
c
C READ STARTING POINT DATA
READ(7,2013)XOYOZOKP,KPP,H.TMIPEMAXEMIN
C
C WRITE STARTING POINT DATA
WRITE(38,2000)X0,Y0,ZOH TM,EMAXEMINKP
C'
c INITIALIZATIONS
HI a H
77
4SUP a NMUL * H
T *0.
li pa 0IC 3 1
IJUI4P 2 0
IFIRST 1
C
C STORE STARTING POINT DATA
IF ( NSTORE.EQ.0 ) GO TO 220
4RITE(35'IJEXT) XOYOZO
NEXT=NEXT+l
WRITE(35'NEXT) HTMEMAX
NEXT =NEXT+1
WRITE(35'NEXT) EMIN.KP
NEX T=NEX T+1
NRNPTS NEXT
WRITE(35'NEXT) IC
NEXT=NEXT+1
C
C STARTING TESTS
220 CALL VXVYVZ(X0,Y0,Z0,ABC)
XLIM1 = XLIM + 3*NMUL*H*(KP-1)
CALL SPHERE (ISPKPtA,0'CXOXLIfl1,EMAX2)
IF C KP .NE. 0 ) GO TO 1100
IF ( ISP .NE. 0 ) GO TO 1110
IJUMP a 1
GO TO 1120
1110 ASUP = NMUL * HSUP
H a NMUL * H
GO TO 1120
1100 IF ( ISP .EQ. -1 ) GO TO 1120
HSUP 3 NMUL * HSUP
H a N'UL H
IJUIP a 1
1120 CONTINUE
CALL FVMAX(ABC,RVIW)
I F(RV) 1,1,2
I H(2*KP-1)*H -
GO TO (100,200,300)4W
2 Hz(1-2*KP)*H
GO TO (100O200,300),IW
C
C START TRACKING
C - w- - - -
C
100 . MTI1
SO TO 400
200 MTu2
GO TO 400
300 MTx3
C
400 x IAST(IMT)*XO+IAST(2,MT)*Y0+iAST(3,MT)*ZO
Y(1)SIAST(3,MT)*XO+IAST(1,MT)*Y0+IAST(2,MT)*ZO
Y(2)=IASTC2,MT)*X0+IASTC3v.4T)*YO+IAST(1,MT)*Z0
VS (i,1) YCi)
YS(21)=Y(2)
WRITE(38,2001)CH(MT),ICX0,Y0,Z0oABCH
C
C STORE STARTING POINT
IF(IFIRST.EQ.0) GO TO 230
78
CALL WRPLOT(X0.Y0,ZO)
IFIRST = 0
C
C DRIFT TINE CALCULATION IN TRANSITION FIELD
230 IF(J.EQ.2) GO TO 6
GO TO (3,4),J-2
3 CALL SPLINE2(STI<K,XSYSS)
GO TO 5
4 CALL SPLINE3(STIKKXSYSS)
5 IF(KK.EO.1) GO TO 900
6 STI(1,1)=X
STI(2,1)=1./(IAST(1,MT)*A+IAST(2,MT)*B+
& IAST(3,MT)*C)
STI(3,1)xY(1)
STI(4,1).Y (2)
C
C FIND FOUR ST-ARTING POINTS
H=H+H
MR21
GO TO 50
7 DO 13 1=1,3
x AUX(1,I)
Y (1) a AUX(2,I)
Y(2) = AUX(3,I)
ICxIC+l
YS(1,I+1)=Y(1)
YS(2,I+1)=Y(2)+
CALL TRANS (XY(1),Y(2).XP,YP,ZP)
CALL WPPLOT(XPYP.ZP)
IF(MOD(ICIP).NE.0) GO TO 10
WRITE(3$,2002) ICXPYPZP
10 CALL EQJAT(XYF)
FS(l)=F (l)
FS(2,1)=F(2)
STI C1,J) =X
STI(2,J)=1./(IAST(1,MT)*A+IAST(2,MT)*B+
& IAST(3,MT)*C)
STI(3,J)Y (1)
STI(4,4J)=Y(2)
IF(J.NE.4) GO TO 11
CALL SPLINE4(STIKKXSYSS)
IFCKK.EQ.1) GO TO 900
11 J4J+1
13 CONTINUE
C
C FOUR STARTING POINTS HAVE BEEN FOUND
C START HAMMING ALGORITHM
12 I=3
14 CALL HAMMING(YSFSIHEX)
C
HP U H
CALL TRANS(XYSCI,4),YS(2,4),XP,YP,ZP)
IF ( IJUMP . EQ. 1 ) GO TO 75
CALL SPHERE (ISPrPA,B,C,XPXLIMEMAX2)
IF ( ISP) 75,71,72
71 IJUMP a I
H a 2 * SIGN ( HI a H )
ILl a .TRUE.
IL2a .FALSE.
79
GO TO 21
72 IJUMP s I
ISUP v NMUL * HSUP
C
75 GTEST = ( ABSCE(1)) + ABSCE(2)) )*6.7222222
IL1= GTEST.GT.EMAX
IL2=(GTEST.LT.EMIN).AND.(ABS(H).LT.HSUP)
IL3=IL1.OR.IL2
IFCIL3) GO TO 21
C
C 1. ERROR ACCEPTABLE
IC=IC+1'
CALL WRPLOT(XPYPZP)
IF(MOD(ICPIP).NE.0) GO TO 15
ARITEC3,2002)ICXPYPZP
15 STI(1,J)=X
STIC2,J)=l./(IAST(1,MT)*A+IAST(2,MT)*B+
C IAST(3,MT)*C)
STI(3,J)=YS(1,4)
STI(4,J)=YS(2,4)
IF(J.NE.4) GO TO 16
CALL SPLINE4(STIKK.,XSPYSS)
IF(KK.EG.1) GO TO 900
16 j=J+1
C
C CHANGE OF PARAMETER ? .
CALL FVMAX(A,B,C,RVIW)
IF(IW.NE.MT) GO TO 18
I1=1+1
GO TO 14
18 XOnXP
YO3yp
ZOlP
MTPzMT
IF CRV) 19,19,20
19 Hz(2*KP-1)*ABS(H)
GO TO (1002O0,3O00),IW
20 H=(1-2*KP)*ABS(H)
G0 TO (100,200,300),IW
C
C 2. ERROR UNACCEPTABLE
21 XUX-HP
YS(1,1)=YS(1.3)
YS(2,1)=YS(2,3)
Y(!)=YS(1,3)
Y(2)=YS (2,3)
MR=2
IFILI) GO TO SO
H.2*H
C
C FIND FOUR STARTING POINTS
54 DO 70 1I1,3
C
IFC.NOT.IL2) GO TO 1130
CALL RUNGECX.Y.H)
GO TO 1140
1130 x = AUX(1,II)
Y (1) = AUX( 2, II)
Y(2) * AUX(3,I1)
1140 CONTINUE
80
C
CALL TRANS(X,Y(1),Y(2),XPYPZP)
CALL ELUAT(X,Y,F)
C
IF C IJUMP .EQ. 1 ) GO TO 80
XLIMI = XLIM + 3*HI*(KP-1)
CALL SPHERE (ISPKPP,A,B, CsXPXLI 41, Ev4AX 2)
IF ( ISP .NE. 0 ) GO TO 80
I JUMP a I
X aSTI(1 J-1)
Y(1) : STI(3,J-1)
Y (2) = SU1(4;J-1)
CALL TRANS(XY(1),Y(2),XOY0,ZO)
H =SIGNCHIH)
MTP = MT
GO TO (100,200,300),MT
C
80 IC=IC+1
YS(1.II+1)=Y(1)
YS (2,ll+1)=Y(2)
CALL WRPLOT(XP,YP,ZP)
IF(AOD(lC,IP).NE.0) GO TO 24
ARITE(38,2002)ICPXPYP,ZP
24 FS(1,II)tF(l)
F S (2,II)=F (2)
STI (1,J)zX
STI(2,J)=1./(IASTC1,MT)*A+IAST(2,MT)*B+
& IAST(3,MT)*C)
STI(3,J)iY (1)
STI(4,J)aY(2)
IF(J.NE.4) GO TO 25
CALL SPLINE4(STIKKXS.YSS)
IF(KK.EQ.1) GO TO 900
25 JuJ+l
C
C CHANGE OF PARAMETER ?
CALL FVMAX(A,B.CRV,IW)
IF(IW.NE.MT) GO TO 18
C
70 CONTINUE
C GO BACK TO HAMMING ALGORITHM
GO TO 12
C
C FIND STEPSIZE TO BE USED IN RUNGE-KUTTA CALCULATION
50 YFS(1) Y(1i)
YFS(2) .Y(2)
ICOUNT * 0
51 YF(1) YFS(1)
YF(2) =YFS(2)
Y(1) *YFS()
Y(2) YFS(2)
x1ax
HH/2.
ICOUNT ICOUNT + I
H2H+H
CALL RUNGE(X1,YH)
AUX(1,1) a X1
AUX(2,1) Y(1)
AUXC3,1) * a(2) '
CALL RUNGE(XITH)
81
X1ax
CALL RUIGE(XI,YF,H2)
ERRI=AIS( CY (1) -YF(1)) /30.)
ERR2=AGS((Y(2)-YF(2))/30.)
IL1=(ERR1.LE.E4AX).AND.(ERR2.LE.EMAX)
IF(ILI) GO TO 1150
IF(ICOUNT.GT.40) STOP MORE THAN 40 BISECTIONS IN RUNGE #1
GO TO 51
1150 X1 m AUX(1,1)
YFCI)= AUX(2,1)
YF(2)= AUX(3,1)
ICOUNT 0 $
GO -TO 53
52 YFC1)=YFS(1)
YFC2)=YFSC2)
x1IX
CALL RUNGE(X1,YFH)
AUX(1,1) aX1
AUX(2,1) = TE l)
AUX(3,1) x VF(2)
53 Y(I)=YF()
Y(2)=YF(2)
CALL RUNGE(X1,YPH)
AUX(1,2) a X1
AUX(2,2) = YCI)
AUX(3,2) aY(2)
CALL RUNGECXIPYH)
AUX(1,3) a X1
AUX(2,3) a Y(1)
AUX(3,3) a Y(2)
H2zM+H
XI=XI-H2
CALL RUNGE(X1.YFA2)
ERRiSABSC((Y(1)-YFCl))/30.)
ERR2=ABS (CYC2)-YF(2))/30.)
IL1CERR1.LE.EVAX).AND.(ERR2.LE.EMAX)
JECILI) GO TO (7,54),MR
HuH/2.
ICOUlNT = ICOUNT + 1
IF(ICOUNT.GT.40) STOP ' MORE THAN 40 BISECTIONS IN RUNGE #2
60 TO 52
C
C
C WRITE RESULTS
C - - -----
900 WRlTEC38,20O5)TICXSYSS(1),YSS(2)
C
IF C NSTORE.EQ.0 ) GO TO 360
LPJ aJ*JJ
IF ( M0 CLPJ*60).NE.0 ) GO TO 1160
NPTSsIC
NEXT3NEX T-1
GO TO 1170
1160 IF( LPJ.NE.40 ) GO TO 1180
NPTSUIC-2
NEXTuNEXT-3
GO TO 1170
1180 CONTINUE
NPTSBICI--
SEXTONEXT-2
ON3
/L'O',0,0''O0'0'L/ ISV! 0IVO
CS'E) SVI / ddflS / NOWfrO2
vivo )1019,
ON 3
dOtS
(A) VWWOJ f LO?
(SZLI@u CI 31 0131J Mn ) XVW3 /1*2.
'5?*L3'. -ZWYIX .1,1 H * 'E1'.zdflSH *'/)IVWtiOJ OLO?
(LI'/)IVWtIOJ Bo0z
(V'&///) vvod~J 00?
(/0V~l~~iV~v~j9002
(SZL3 ?
I R1 */'5ZL3',vA 6*'/PSZt3,zX a)(da(*Sf9J1
6,' V'IL 3, 3 WI IJjI a *'/',e:5J-fl53H I *'/) VWbOJA 002
((oHL)6LO/ I
'iCV3tV 1VNOI 23S-SSOJ2 * A. BDVM3AV =MOld 3WI1AQ s *2
OP0Ws= A 3DVU3AV /@*I
'SgZL 30 1 AIN IJN IA /*'S3 31J93 0 &'*?'94',0=aVHdlIV *) VWJOJ 70
C/0(-HL)6L0/S?2LJ, = tb313t-.VlQ 13INI */d'SL3
= sxONfl0bS 3A02V .HOI3H 9*-/'f V '=0 1 BSV2* '/'7VLL' XL'/I) IVWUOdI 00?
"?L3',1 *t" '? L3'. =A *f S'?L3,=X : *'91* .JN IOd s*) VWUOJ ?00?
',~X A,'*XT"'/'5?ZL3', 1=Zv'X'SZL3't sA@.'Xri 'ZL3S
'=x ,'*X~'/'91'. IN IOd :8?3 3QVb'ld S I *'LV, a,'/) IV WJ4LOOU
W(/',(I 0lJ13 1IV JVd I I Qmd4N *0MOIJUINfl02 tuLd)) *I'. d4 * 2
'/0'LJ'f,t90?U3 WfwlWNIW ,S'L3'a~b08b3 fWfWXVW l I
'/'S*2L3',=3W1 I IIQ '/'5?L3',=3ZISdJ S */f'SL3 2
'i7 .JS'?ZL3',nuA 1*2L3,=X : INIod ON I I?JVLIS *//) IVWJO1 000?
3CMI N N0009
CSE ) 3S012
L+LX 3N= IX3K
UdN (JXJ3N.SD311IUM
009 0 09 C CO330 SN ) it
3ImrlNtJO23 001
C(r)3ltJN ) 35012
*VOL=[ 00 00
OS? 01 09 ( 00r3*dd4) Ilr091
IN!I NzI X3N
Lt X3NLIX3N
CI dN (CX3N.aS9U3 I1sM
S id NUN LX 3NJ
IXINriNI N
LtL1X3Nu 1 X3 N
(Z)SSA'(L) SSA*'S X(C X3 NsS i) 311 dM
3flNtJN02 0L L
83
C SUOROUTINES
C * * * ** *
C
C SUBROUTINE VAVYVZ C XYPZVXpVYVZ)
C ------- --- --------------------------------
C
SUBROUTINE TRANS CUVW, XPYP, ZP)
C
COf4MON/G/MTA,BCJTTMMTP
COMMON/ SUPP3/ IAS T(3,3)
C
XPU IAST(-1,4T) *U+IAST (3,MT)*V+IAST (2,.MT)*W
YP=IAS T (2,MT ) *U+IAST1,14T) *V+I ASTC(3,M T) *W
Z P=IAST (3,MT )*U+IAST(2,MT) *V+IAST(1,MT)*W
RETURN
END
C
C
SUBROUTINE EQUAT(XYF)
C
DIMENSION Y(2)hF(2)
COMMON/G/MT,A, B, C, JT, T,MTP
C
GO TO (I,2,6),MT
C Z IS PARAMETER
6 xCY(1)
YCBY(2)
ZC =X
CO TO 3
C X IS PARAMETER
I XCCX
YCY (1)
ZCaY(2)
60 TO 3
C Y IS PARAMETER
2 XC=Y(2)
Y Cux
Z C mY()
3 CALL VXVYVZ(XCYCZCVXVYVZ)
AuVX
eavy
CSVZ
GO TO (A,5.7),MT
7 F(I)uVX/VZ
F(2)nVY/VZ
RETURN
I. F(1)&VY/VX
F(2)uVZ/VX
RETURN
5 F(1)=VZ/VY
F (2) VX/VY
RETURN
END
c
C
SUBROUTINE RUNGE(XDYH)
C
DIMENSION Y(2),F(2),Y9(2),PHI(2),SKI(2),SK2(2)
COMMON/G/MT, AeBCPJVT,TMsMTP
84
C
CALL EQUAT(XYF)
DO 1 I l,2
YB(I)RY(I)
PHI (I )=O.1 7476028*F(I)
Y(1) YB( 1)+0.4*F(I)*H
I SK1(I)zF()
X TBX+0.4*H
CALL EQUAT(XTYF)
DO 2 1al,2
PHI(I)IPHI(I)-0.55148066*F(I)
Y(I)=YB I) +3 29697761*SK1(1) *11+0.15875964*F ()*H
2 SK2(I)UFCI)
X T=X+0.45573725*H
CALL EQUAT(XT,Y,F)
DO 3 ImI,2
PHI( I)=PHIC 1)+1.20553560*F (I)
3 Y(I) =YB(l)+(0.21810040*SKlIC)--3.05096516*sK2(x)+3.832864
76*F(I))*H
XuX+H
CALL EQLAT(XYF)
DO 4 JIu,2
PHl(I)PHIC I)+0.17118578*F(I)
4 Y(I)=Y3(1)+PHI(I)*H
RETURN
E ND
C
c
SUBROUTINE HAMMING(YS, FSIHEX)
C
DIMENSION YS(2,4),FS(2,3),YT1(2),YT2(2),F(2),E(2)
COMMON/G/MT,AB.CJ,T,TM,MTP
C
C PREDICTOR
XzX+H
YT1(1) YS(C1,I)+(2* FS(1.3)-FS(1.2)+2*FS(1 ,I))*H*4/3.
YT1(2)=YS(2,1)+(2*FS(2,3)-FS(2,2)+2*FS(2,1))*H*4/3.
IF(I-3)2,1,2
2 YT2(1)=YT1(1)+E(1)*112/9.
Y T2(2)=YT1(2)+E(2)*112/9.
GO TO 3
1 YT2(1)&YTI(I)
Y T2(2) rY TI(2)
3 CALL EQUAT(X.YT2ZF)
C
C CORRECTOR
YT2(1)=(9*YS(1,4)-YS(1,2)+3*H*(F(1)+2*FS(1,3)-FS(1,2)))?
& * 8.
YT2(2)=(9*YS(2,4)-YS(22)+3*H*(F(2)+2*FS(2,3)-FS(2,2)))/
& 8.
E(1)c(YT2(1)-YTC11))*9/121.
E C?) (YT2(2)-YT1(2))*9/121.
YT20 )VYT2(I)-E(1)
YT2 (2)aYT2(2)-EC2)
00 4 ia1.2
DO 5 IS1,3
5 SCIJPIS)YSCIJISI+)
DO 4 ISSR1,2-
4 FS (1JI SS)Pa FS(I J,'iSS+1)
YS(1,4)aYT2C1)
85
YS(2,4)AY T2(2) -
CALL EQUAT(X*YT2*F)
FS (1o,3)aF(1)
FS (2 .3 ) F(2)
RETURN
END
C
C
SUBROUTINE FVMAX(AB,C,RVIW)
C
DA=ABSCA)
DSaABS (B)-
DC'tABS( C)
IF(DA-DB) 1,2,2
1 IF(D3-DC)3,4p4
4 IW=2
R VaB
RETURN
3 IW83
RVuC
RETURN
2 IF (DA-DC)5,6,6
5 1w=3
R V=C
RETURN
6 IWs1
RV=A
RETURN
END
C
c
C SPLINE SUBROUTINES
C
C
SUBROUTINE SPLINE2(STIKKXSYSS)
C
DIMENSION STi(44),YSS(2),Y(2)
COMMDN/G/MT.,-A8,,CJ, TTMPMTP
C0lMON/SUPP2/JJSLP
COMM4N/SUPP3/IAST(3,3)
COMMON /SUPP4 /IOPTXPLANE
C
I va0
Xl=STI(1,1)
X2ZSTI(1,2)
P AnS LP* (X2-X1I)+S T I(2,)-ST I(2,2)
PA--PA/(X1-X2)**2
PB=SLP-2 *PA*X1
PCUSTI(2,1)-(PA*X1+PB)*XI
C
TPuT
If ( IOPT.EQ.0 ) GO TO 300
XCOORD a IAST(1,MTP)*STICI,2)+IAST(3,MTP)*STI(3,2)+
& IAST(2,MTP)'STI(4,2)
IF ( XCOORD.GE.XPLANE ) GO TO 200
IvaI
300 X RaX2
LABELE2
60 TO 2
86
C
1 XR=(X1+X2)/2
C
2 X INT=PA* (XR** 3-XIF.**3)/3+(P9*(XR+X1 F) /2+PC)*( XR-X1F)
T=TP+XINT
IF ( IV.EQ.1 ) GO TO 10
IF ( IDPT.EQ.1 ) RETURN
DI FFO=ABS(T)-TM
D I FFD IFFO/ TM
IF(ASS(DIFF)-0.00001) 6,3,3
3 IF(DIFFO) 4,4.5
4 X1=XR
GO-TO (1,10),LABEL
5 X2=XR
L ABELal
GO TO 1
IF(LABEL.NE.2) GO TO 200
X =S TI (1,2)
Y (1)2STI (3,2)
Y (2)=ST I (4,2)
GO TO 210
IF ( IOPT.E2.1 ) XR=XPLANE
HH=X R-STI(1,1)
X=STI (1,1)
Y(1)=STI (3,1)
Y(2)STI (4.1)
MT MTP
CALL RUNGE(XY.HH)
CONTINUE
KKwI
4=20
GO TO (7,8.9)
K Sa y
YSS (2)=Y(2)
IF ( IOPT.EQ.0
GO TO 2
RETURN
XSY(2)
YSS C1)X
YSS (2)y(l)
R E T URN
XSBY(l)
YSS(1)=Y(2)
YSS(2)=X
RETURN
KKu0
RETURN
END
MTP
) GO TO 11
SUBROUTfNE SPLINE3(STIKKXSYSS)
DIMENSION STI(4,4),YSS(2),Y(2),H(2),SIG(2)
.REAL*4 M(3)
COMMON/G/MTA.BCJ.T.TMMTP
C
6
200
210
C
7
11
8
9
10
C
C
C
87
COMMOf4/SUPP2/JJ ,SLP
COMMON/S UPP3/IAST (3,3)
COMMON/SUPP4/1OPTXPLANE
C
H01)=STI(1,2)-STI(1,1l)
H(2) STl(l,3)-STI(1,2)
XLB=H(2)/ (Ht)+H (2))
XMU=1-XLB
SIG 1()=(ST1 (2,2)-ST 1(2,1)) /H( 1)
SIG(2)=(STI (2,3)-STI( 2,2))/H(2)
DI16*(SIG(2)-SIG(1))/(H C1)+H(2))
14 C(3)z0.
M(2 =1/2
C
JJ=&
I jJJqJJ+1
TPzT
IF C IOPT.EQ.0 ) GO TO 40
X COORD = IAS T (1,MT P) *S TI (1,JJ+1)+I AST(3,MTP)*STI(3,JJ+1)+
& IAST (2,MTP)*STI(4,JJ+l)
IF ( XCDORD.GE.XPLANE ) GO TO 12
40 XINT=(STI(2,JJ)+STI(2,JJ+1))*HCJJ )/2-(M(JJ)+M(JJ+1))*
& (H(JJ)**3)/24.
T=T+XINT
IF ( IOPT.EQ.1 ) GO TO 2
IF(AES(T)-TM) 2,3,4
2 IF(JJ-2) 1,5,5
5 SLP=H(2)*M(2)/6+(STIC2,3)-STI(2,2))/H(2)
Jul
KKCO
RETURN
3 KK1l
1=30
GO TO (2C,21,22)., T
20 XS=STI(1,JJ+1)
Y SS( 1).= ST I (3,JJ+1)
YSS(2)=STI(4,JJ+1)
RETURN
21 XS=STI(4,JJ+1)
YSS(1 )=STI(1,JJ+1)
YSS(2)=STIC3,JJ+1)
RETURN
22 XSZSTI(3,JJ+1)
YSS (1 )=STI(4JJ+1)
YSS(2)=STIC1,JJ+1)
RETURN
4 KKNI
J=30
X1=STI (1,JJ)
X2=STI (1,JJ+1)
11 XRU(X1+x2)/2
41 XIX=-(JJ)*CXR-STIC1,JJ+1))**4/(24*HCJJ))+M(JJ+1)*(XR-ST
& IClJJ))**4/(24*H(JJ))+(M(JJ)*H(JJ)**2-6*STI(2,JJ))*(XR-
& STI(1,JJ+1))**2/(12*H(JJ))+(6*STI(2,JJ+1)-M(JJ+1)*H(JJ)*
& *2)*(XR-STI(1pJJ))**2f(12*H(JJ))+(M(JJ)*fl(JJ)**3)/24.+(6
& *STI C2,JJ)-M(JJ)*H(JJ)**2)*HCJJ)/12.
TuTP+XIX
IF C IOPT.EQ.1 ) RETURN
DIFFO=ABS(T)-TM
88
25
26
27
12
15
12,25,25
DIFFDIFF0/TM
IF (ABS(DIFF)-0.00001)
IF(DIFFO) 26,26f27
X1BXR
GO TO 11
X 2=XR
GO TO 11
IF IOPT.EQ.1 ) XR=Xl
HH=XR-STI(1.44)
X=STI(1,JJ)
Y (1)=STI(3,J)
Y (2)=STI(4,J3)
MT XMT P
CALL RUNGE(XY,HH)
GO TO (15,16,17) MTP
XS=x
Y SS(1)=Y (l)
YSS(2)=Y (2)
IF ( 10PT.EQ.0 ) GO TI
KKal
J=33
GO TO 41
RETURN
XSaY(2)
Y SS (1)ax
YSS (2)&Y(1)
RETURN
XSY(1)
YSS(1)=Y(2)
Y SS(2)=x
RETURN
END
0 1.8
SUBROUTINE SPLINE4(STIKKXSYSS)
REAL*4 M(4)
DIMENSION STI(4,4),H(3),XLB(2),XMU(2),SIG(3),(2),
& YSS(2).,Y(2)
COMMDN /G/4T, ABCoJoT, T4M,lTP
COM'40N/ SUPP 2/JJSLP
COM4ON/SUPP3/IAST(3&3)
COMMON/ SUPP4/IOPTXPLANE
DO 1 I=1,3
. ( CI)=STIC1,I+1)-STI(,1)
XLB(1)=H(2)/CH(1)+H(2))
XLV32)=HC3)/CHf 2)+H4(3))
XMU(1)=1-XLB(1)
X MU(2)=1-XLB (2)
DO 2 1=1,3
SIGCI)C5STI(2,1+1)-STIC2,I))/I)
DO 3 I=1.2
D( )=6* (SIG(I1+1 )-SIG (I )/(H (I )+H(I+l)
M (I )O.
M(4)nO.
DELTA=XLBC1)*XMU(2)-4.
M (2)= CXLP(1) 'D(2)- 2*.D(1)) /.ELTA
M(3)=(XMU(2)*D(1)-2*D (2))/DELTA
J40 '
PLANE
18
16
17
C
C
C
C
2
3
89
7 JJ=JJ+1
T P=T.
IF ( IOPT.EQ.0 ) GO TO 40
XCOORD = IAST(1,MT)*STI (1,JJ+1)+IAST(3,MT)*STI(3,JJ+1)+
& IAST(2,MT)*STI(4,JJ+1)
IF ( XCOORD.GE.XPLANE ) GO TO 12
40 XINTz(STI(2,JJ)+STI(2,JJ+1)) H(JJ)/2-(M(JJ )+M(JJ+1))*CH(
& JJ).**3)/24.
T=T+XINT
IF ( IOPT.E2.1 ) GO TO 4
IF(A9S(T)-TM) 4,5,6
4 XF(JJ-3) 78;8
8 SL-Pc1(3)*M(3)/6 + (STI(2,4)-STI(2.3))/H(3)
DO 9 1=1,4
9 STI(C,1)=STI(I,4)
Jul
KK80
RE TURI
5 KK.=l
J=40
GO TO (20,21,22),MT
20 XS=STI(1,JJ+1)
-iSS(1)=STI(3,PJJ+1)
YSS(2)=STI(4,JJ+1)
R E T UR N
21 XSxSTI(4,JJ+1)
YSS(1)=STI(1,JJ+1)
YSS( 2)STI (3,JJ+1)
RETURN
22 XS=STI (3,JJ+1)
YSS(1)=STI(4,JJ+1)
YSS(2)=STI( 1,JJ+1)
RETURN
6 KKAI
4J=40
X1=S TI (1PJJ)
X2=STI (1,JJ+1)
11. XR=CXI+X2)/2.
41 XIX=-M(JJ)*(XR-STI(1,JJ+1))**4/(24*HIJJ))+M(JJ+1)*(XR-ST
& I (1,JJ))**4/(24*H(JJ))+(M(.JJ)*H(JJ)**2-6*STI(2pJJ))*(XR-
& STI (l,JJ+1))**2/(12*HCJJ))+(6*STI(2,JJ+1)-M(JJ+1)*H(JJ)*
e. *2)*(XR-STIC1,JJ))**2/(12*H(jj))+(M(JJ)*H(JJ)**3)/24.+(6
& *STIC2,JJ)-4(JJ)*H(JJ)**2)*H(JJ)/12.
T=TP+X IX
IF C IOPT.EQ.1 ) RETURN
DIFFOfA US (T)-TM
01FFn1 FF0/TM
IF(A3S(DIFF)-0.00001) 12.25,25
25 IF(DIFFO) 26.26,27
26 X1WXR
GO TO 11
27 X2=XR
GO TO 11
12 IF ( IOPT.EQ.1 ) XRSXPLANE
HH=XR-STI(1,JJ)
XaSTI(1,JJ)
Y (1)=STI(3,JJ)
.Y(2)=STI(4.JJ)
CALL RUNGE(XYHR)
GO TO (15#16,17),MT
90
18
16
1 ?
COMMON / INFOV / BB(5O),SIGMAS(1100),SIGMA1(1100)v
SIGMA2(1100),SIGMA3(1100),CC(4),KFLOW
C
C
C
C
C
C
C
C
C
C
C.
C,
C
C
C
C
2
3
4
C
) / CC(2) )
) / CC(4) )
DELTVY.GT.EMAX )) GO TO 10
THIS SUBROUTINE STORES THE CALCULATED POINT IN FILE35
XSaX
YSS (1)uY (l)
YSS(2)=Y(2)
IF ( IOPT.EQ.O ) GO TO 18
KKaI
Jm40
GO TO 41
RETURN
XSnT(2)
YSS(1)sX
YSS(2)y(l)
RETURN
X s.y5 (1)
Y ss(l)sY(2)
YSS(2)*X
RETURN
END
SUBROUTINE SPHERE (ISPKPVX,VY,VZ,X0,XLIMEMAX)
THIS SUBROUTINE DETERMINES WHETHER THE CALCULATED POINT IS
IN THE FAR F-IELD OR IN THE NEAR FIELD
IF THE POINT IS IN THE FAR FIELD , THEN
HSUP z ( NMUL ** 2 ) * NI
IF THE POINT IS IN THE NEAR FIELD , THEN
HSUP NMUL * HI
WHERE :
HSUP IS THE LARGEST VALUE THE STEP SIZE ABSOLUTE VALUE CAN TAKE
HI IS THE STEP SIZE INITIAL VALUE C INPUT )
NMUL IS INPUT
IF ( XO .GT. XLIM ) GO TO 10
IF C CCC2) .NE. 0 ) GO TO 1
DELTVX a ABS C VX )
GO TO 2
DELTVX a ABS CVXa - CC(2)
CONTINUE
IF ( CC(4) .NE. 0 ) GO TO 3
DELTVY a ABS C VY )
60 TO 4
DELTVY a ABS C C VY - CC(4)
CONTINUE
IF (C DELTVX.GT.EMAX ).OR.(
FAR FIELD
ISP s 3* KP - 2
RETURN
NEAR FIELD
ISP * eKP
RETURN
END
SUBROUTINE WRPLOT(XeYsZ)
&
C
10
C
C
'C
C
91
C
COMMON/SUPP1/NEXTNSTORE
C
IF C NSTORE.EQ.O ) GO TO 10
WRITE(35 'NEXT) XYPZ
NEXT=NEXT+1
10 RETURN
END
92
APPENDIX B
QUALITATIVE INVESTIGATION OF THE FLOW USING
A SIMPLE TWO-SINK MODEL
B-1 Model Description
The most basic model of the flow to be studied consists of a sink
and its image in a uniform free stream. In the configuration shown in
Figure B.A the flow far away from the sinks is uniform and parallel to
the x-axis, its magnitude is V.. The two sinks, which have the same
strength Q (positive), are located on the z-axis at respectively
z = H and z = -H.
The velocity potential is given by
whe+
7T +
where
lit
it. Vx+~
+
AJ-= I ZrO 9 ! (CH
Therefore the components of the flow velocity at a point M(x,y,z) are
v6 ... _ _ .4.. TT ( x .
+ JL. )
93
- b4' -3
Xk+
-3
+ )L -
vi t GL sbH + + H )
The velocity potential can be rewritten in the following
nondimensional form:
H 4Mr xt
H H
The nondimensional velocity is then given by
V
where the operator V i s equal to
.9
H H H
Examining the above equations results in the recognition of the
ratio . 2 as the dimensionless group quantifying the relative
V H
influence of sink strength, sink height from plane of symmetry and
wind velocity. Its value can be chosen by following the below reasoning.
If a sink of strength Q is used to modelize a jet engine, then
we have
7T
4-
Voo
94
where D and V. are respectively the inlet diameter and the inlet
throat velocity.
If y and u are equal to respectively the inlet velocity to wind
V.
velocity ratio -OO and the inlet centerline height to inlet diameter
ratio H then the above equation can be rewritten as followsD
4 4e
Substituting for y and n representative values of 50 and 1.5 gives
Q- 17.45
V00 H.
The results discussed in the next section correspond to
-_20
V H
As previously mentioned the flow situations we are concerned with
V.
are characterized by high values of the ratio L . One can also say
that they are characterized by high values of the ratio A0  ,
-r
where A is the far upstream cross-sectional area of the jet engine
95
capture surface*, for, if the flow is assumed to be incompressible,
V.
this ratio is equal by continuity tow-. In the simple two-sink
model, the inlet diameter D is not relevant, however, the altitude
of the sink above ground, i.e., plane of symmetry, can be utilized to
evaluate the capture surface size. If the latter is supposed to have
far upstream a half-circular cross-section of radius r , we have
A00  D-V
H H 2Vc*
A 0 
- 2
Substituting again for y and n typical values of 50 and 1.5 yields
H
This result, which does not depend on the inlet model used, gives
an indication of the dimension of the capture surface at infinity.
*The capture surface of a sink-type device (i.e., inlet) is defined as
the surface separating the portion of the flow which enters the sink
(inlet)from the portion of the flow which does not enter.
96
B-2 Results
Using the simple model previously described it is possible to
obtain a qualitative appreciation for the three-dimensional flow induced
by a jet engine inlet close to a ground plane. To do this, some
judiciously chosen particles are followed with the help of the fluid
particle tracking program. The results to be shown consist of pro-
jections of these streamlines on the coordinate planes.
The x, z and x, y planes are both plane of symmetry of the flow
field. The peculiar role played by the x, z plane was recognized
early. FiguresB.2 and B.3 show some of the streamlines running in this
plane. For the particular value of the dimensionless sink strength
used the flow above or in the ground plane is observed to contain
three stagnation points, lying all on the x, z plane. The first one
S1 is formed inside the capture surface on the ground pJpne and is
associated with a converging flow as can be seen in Figures B.2 and
B.4. Figure B.4 shows the projections on the x, y plane (ground plane)
of streamlines in "wrap around" the streamline connecting S1 to the-
sink. The second stagnation point S2 lies on both the ground plane and
the capture surface. Note that the fluid particles only appear to
leave this point either to go inside the capture surface and be finally
sucked into the sink or to move downstream, as is shown in Figures B.3
and B.5. This feature can be explained by streamlines close to the
capture surface and the x, y plane meeting the x, z plane tangentially
97
(i.e., in a cusp*), thus providing the flow leaving S2 in the
x, z plane with fluid particles. In other words the flow observed
about S2 in the x, z plane may be regarded as arising from fluid
particles moving along the capture surface near or in the x, y plane
impinging on the x, z plane. Figure B.4 also indicates this feature.
The third stagnation point S3 is formed off the ground plane and is
associated with the meeting of particles coming from far upstream,
which either proceed downstream or are aspirated by the sink, and
particles leaving the second stagnation point.
The features described above are peculiar to the flow in the x, z
plane. None of the flow patterns observed about S2 is present in the
flow off this plane. Off the x, z plane the fluid particles either
flow past the capture surface or are sucked by one of the sinks, in
this case, describing trajectories which are wrapped about the so-called
stagnation streamline connecting Sl to the sink. A global view of the
flow is sketched in Figure B.6.
Difficulties are encountered when trying to understand the
structure of the flow, especially about S2. These are basically due
to the use of the point sink. Although the sink is suitable for an
investigation of' the far field, where the engine is perceived only as a
*A cusp is the only manner in which a streamline off the x, z plane can
meet the latter since the velocity at any point of a plane of symmetry
is parallel to this plane.
98
sucking device, in the present study we are concerned with the flow
field near the engine as well. The inlet vortex, when present, is
observed to "attach" to the ground within a distance of one or two
inlet diameters from the inlet lip; thus, as the intake height from the
ground to intake diameter ratio is typically equal to 1-2, the length
scales in the problem are all of order of the inlet diameter. There-
fore a geometrical representation of the inlet which is adequate on
this scale is required. There are other drawbacks to the simple sink
model. Firstly the direction of the wind relative to the engine cannot
be intrdduced. Secondly the first stagnation point, where the inlet
vortex, if present, seems to attach, is always found on the positive
part of the x-axis (see Figure B.l), i.e., after the sink projection
when marching along the x-axis in the free stream direction: this is
not consistent with the experimental results reported in the literature.
(The first limitation will obviously vanish if the engine geometry
is taken into account, since then the suction in the absence of
wind will no longer be axisymmetric.) It is therefore seen that a more
exact description of the potential flow near the inlet is needed.
However the use of the simple sink model was able to yield some qualita-
tive appreciation for the three-dimensional flow as well as enable us
to define some of the requirements an acceptable model must meet.
A further use of the sink model was to give, as a test case for the
fluid particle tracking program, some valuable results about the behavior
of this program.
99
TABLE 1: KEY TO FIGURES 37,38,42,43,44,45
Amplitude between key
0 and 5 1
5 - 10 2
10 - 15 3
15 - 20 4
20 - 25 5
25 - 30 6
30 - 40 7
40 - 50 8
50 - 60 9
60 - 70 10
100 - 110" 14
350 - 360 39
* 500 -510 54
For amplitude values larger than.30, the key equals x if the amplitude
is between lOx - 40 and lOx - 30.
100
INLET
D
x
Y
lei
H
FIGURE 1: Flow Gecmetry
00
GROU
CK/
IND
/a I
101
r
FIGURE 2: Inlet Geometry
O.IID
A .0 - -
A V-
Ar11%
102
FIGURE 3a: N-line and Panel Vertices
t
FIGURE 3b: Front View (because of Symmetry, only Half the Inlet is Shown)
FIGURE 3: !nlet Panel Representation
103
FIGURE 4: Side View of Inlet Panel Representation
SWgnoion
Sfreamline
GROUNC
s,
FIGURE 5: Streamlines i the x, z Plane, Case 1
S
INLET
)
x
I
p0
INLET
GROUND
FIGURE 6: Streamlines in the x, z Plane, Case 1
105
S
ci
x .s, S1
mpr
20
fA
3, 'S x
FIGURE 7: Streamlines in the x, y Plane, Case 1
Ijljjll 141111 ljlj3jm0ELNjmJjjjjRm R I'mill I I.Py Romil's ____necomown,
107
%117
>1|6
INLET
14 13 10
- . 10
88
7772
GROUND
Y
Projections of Streamlines on the y, z Plane, Case 1
wp I maoi
FIGURE 8:
40 V
-7
6 / -
le. IRKo.p . M-..e... - -
/10
-
-
- - --. 1 /
4/
3o 4.a
/
/
/
/
/
Y
FIGURE 9: Projections of the Streamlines Shown in Figure 8 on the x, y Plane, Case I
C
CAPT
GROUND
FIGURE 10: Cross Section of the Capture Surface, Four Inlet Heights Upstream of
Front View of the Stagnation Line, Case 1
the Inlet,
URE SURFACE
ISTAGNATION
LINE
INLET
-~J
0)
to
110
CAPTURE SURFACE
INLET
c-:
GROUND z
Si
FIGURE Ila: Trace on the x, z Plane
CAPTURE SURFACE
INLET
x S,
FIGURE 11b: Trace on the x, y Plane
FIGURE 11: Capture Surface, Case 1
111
S[oqnahon Line
I,
/
/
/
'p
1~
/
--V
I
1.
Y
.3
INLET
olp
At/
/7/
Slognolion Streomline
FIGURE 12: Perspective View of Stagnation Locations, Case 1
dia 4 IK
112
FIGURE 13: Streamlines in the x,.y Plane, Case 2
u I 
mom Mmism
L
IL -=4
x
.1
%'
113
INLET
15 z GROUND
I 3S.
FIGURE 14; Projections of Streamlines on the y, z Plane, Case 2
-4P
36 7
9.
FIGURE 15: Projections of the Streamlines Shown in Figure 14 on the x, y Plane, Case 2
115
INLET
'A-/
/
/
/
III
I /GROUND
xS1.
FIGURE 16: Projection of Forward Stagnation Streamline on the x, z
Plane, Case 2
'00
t~ --yti~
116
Z4
Stagnahon Line
/.
I
I
S
Y
INLET
IX&OW Si
R
. Stagnotion Streamline
FIGURE 17: Perspective View of Stagnation Locations, Case 2
-K
117
CAPTURE SURFACE
INLET
x
CAPTURE SURFACE
Trace of Capture Surface on the x, y Plane, Case 2FIGURE 18;
118
zi
INLET
Y
VORTEX LINE
/
I.
/
7/
X
Sragnaion
Streamline
FIGURE 19: Suggested Behavior of a Far Upstream Horizontal Vortex Line,
Case 1
I A
I
/
Si
FIGURE 20: Deformations of a Fluid' Line which is Vertical and in the x, z Plane at a Location
Four Inlet Heights Upstream of the Inlet, x, z Plane, Case 1
x
CAPTURE SURFACE
INLET
Slognaoion
St-eomline
GROUND
__j
"*1 4
.I
x
FIGURE 21: Deformations of a Fluid Line which is Vertical and 0.6 Inlet Heights from the x, z Plane
at a Location Four Inlet Heights Upstream of the Inlet, Projections on the x, z Plane,
Case 1
6
9 INLET
S7
9 .
S 6. GROUND
__j
0~.
121
0,
9
8 INLET
6
. 5
7 8 9
6 z GROUN JD
Y
FIGURE 22; Projections of the Material Lines Shown in Figure 21 on the
y, z Plane
I
122
INLET
GROUND
Y
FIGURE 23: Deformations of a Fluid Line which is Vertical and 1.6 Inlet
Heights from the x, z Plane at a Location Four Inlet Heights
Upstream of the Inlet, Projections on the y, z Plane, Case 1
123
CAPTURI
7*
Shqgnohon Line
E SURFACE
VORTEX LINES
INLET
SWcnafion
Sireamline
FIGURE 24; Suggested Deformation of a Far Upstream Uniform Distribution
of Vertical Vortex Lines, Case 1
INLET
SIognahon
Streomline
GROUND Z
S, x
FIGURE 25: Deformations of a Far Upstream Vertical Material Line, Projections on the x, z Plane,
Case 2
2 INLET
Stognchon
Sftremline
I 1-
GROUND
FIGURE 26: Projections of the Material Lines Shown in Figure 25 on the y, z Plane
N)
CT'
I
s Y
,Ia
126
CAPTURE SURFACE
INLET
x
Vertcal Fluid Line
Horizontal Fluid Lines
CAPTURE SURFACE
FIGURE 27; "Far Upstream" Location of the Vertical Fluid Line Tracked
in Figures 25 and 26, Common "Far Upstream" Projection of
the Horizontal Material Lines Tracked in Figures 28 through
33, x, y Plane, Case 2
CAPTURE SURFACE 
-- INLET
' Sagnohon
'Sreamline. S,
2
FIGURE 28: Deformations of a Fluid Line which is Initially Horizontal and 0.8 Inlet Heights from
the Ground Plane, Projections on the x, y Plane, Case 2
INLE
Shogncho
Streamline
T
GROIUND
sly
Projections of the Material Lines Shown in Figure 28 on the y, z PlaneFIGURE 29:
CAPTURE SURFACE -Sagnohon
Sreamline
3_ _ _ _ _ _ _ __ _ _ _ _ _ _Y
2
3 5 INLE T
FIGURE 30: Deformations of a Fluid Line which is Initially Horizontal and 0.4 InletiHeights from
the Ground Plane, Projections on the x, y Plane, Case 2
.40do
Slognolion
Streamline
INLET
/ ~tz
-dd3
GROUND
S, Y
FIGURE 31: Projections of the Material Lines Shown in Figure 30 on the y, z Plane
0-
aCAPTURE SURFACE
-V
-V
'P
Saqnoion
5 Streamline
p.'
3
INLET
FIGURE 32: Deformations of a Fluid Line which is Initially Hori ontaI and 0.6 Inlet Heights
from the Ground Plane, Projections on the x, y Plane, Case 2
0-A)
INLET
S1'agnoion
SIreamline
GROUND t I
'S, y
FIGURE 33: Projections of the Material Lines Shown in Figure 32 on the y, z Plane
PQ
I
133
d0
STREAMLINES
-OO
I/ e -
I/It
op S
-0-
-00-
- 0 - p l
- -
-OF
INLET41/
2.
,.0
AM
5Iogntion
Sreamline
FIGURE 34: Behavior of Fluid Lines Connecting Neighbouring Particles
Y
09 i 7F
rl
I
CAPTURE SURFACE
. GRID
INLET
a-&
LOa
GROUND
FIGURE 35: Front View of Grid Tracked at a "Far Upstream" Location, Case 1
135
Inlet Internal
Surfoce
GRID
-T-+--I
FIGURE 36: Deformation of Grid Shown in Figure 35 at "Engine Face"
Location, Projection on the y, z Plane
136
Inlet Internal
Surface
-1 0 -to -11 -13
.7 -7 - -8S 1
- .-6 -6 -7 -7 -8 -i
- -
- -5 -6 --7 - -
.i -z -1 - -L -3 -3 -7
-. -i -i -a - -a -I - l -t
I I I I I I I I
-4
377 385 3 1194 I
. +
Ft
Figure 38
FIGURE 37: Vorticity Distribution at the Engine Face Location Due to
a Far Upstream Distribution of Vertical Vorticity, Case 1
(See Key Table 1 p 99)
137
0.1 D
5 5 5 5 5 4 7 7 5 I
7 7 7 S I
5 5 1 SO I , 1440
~G #6 7 1517
FIGURE 38: Enlarged View of a Portion of Figure 37 (See Key Table 1
p 99)
138
CAPTURE SURFACE
IL
x
INLET
N)
GRID
CAPTURE SUIRFACE
FIGURE 39: Initial (Far Upstream) Location of Grid Tracked, Projection
on the Ground Plane, Case 2
139
D
CAPTURE SURFACE
I I I I I I I IIII1
1 = 1=
GROUND
FIGURE 40: Front View of Grid Tracked at a "Far Upstream" Location,
Case 2
140
Inter Internal
Surface
S-
GRID
FIGURE 41: Deformation of Grid Shown in Figure 40 at "Engine Face"
Location, Projection on the y, z Plane
141
Inlet Internal
Surface
-10 13
-50 0g-10-
-8s 
.-
8--9
-L I-7 -7
-5o - -, -- ,- 7
* , - - * - .
-43 -17 -,. - -' -6 -6
. . .. g - t f- - 6 -s
. .. - . -'
-10 -* - 1 -4 - -s5
- I-5 " - -4
-s-s i . ' -4 - -
I, -7 -5 -- - 3-1 -.
-15 - - L - - - 4 i - - -2 - -
- t ..l -t- -
S-.-~ -7 1[, i I-
I-.--~ -1 .. . -
4-
.176 65 '- t t -|~
Figure 43
FIGURE 42:, Vorticity Distribution at the Engine Face Location Due
to a Far Upstream Distribution of Vertical Vorticity,
Case 2 (See Key Table 1 p 99)
142
O.ID
241
0.
7
71 6 7. 6
060 
-.
*to
p 99)1-10316 9
70 -13 3.
71 .a -31
67 * . . 20-
64g -t
57/1
57 55 54
FIGURE 43: Enlarged View of a Portion of Figure 42 (See Key Table I
p 99)
143
Inlet Internal
Surfoce
-31
-30.- -
-304 
. -
.
- t10-15
8 8 -7 - -68
3 ---- - -
ax $ 6 -- -6 -l
. . - 8 - 3 . .3 - -
. 7 -5
14 i - -15 1 S l - - -9
148 - . -1 -7
. 7 - - -3 - .
10 . - 3*-
14 ' -g 6 -- 5 -7 '
13'? . -- -3- . -
13 10 8 6 .
I3 1o 8 6 4 i -I -.3 -5 -7 -9 -I
0 ~3 
-1 -3 -5 -7..3
s ,, . I - -. . ...
10 I-55 3 -is 
-- 3 S-7..7
- 5 31-
-
- .. -4.-s . -I
- - . .- 
- .-
-'..1
-7 9
Figure 45
FIGURE 44: Vorticity Distribution at the Engine Face Location Due to
a Far Upstream Distribution of Horizontal Vorticity, Case 2
tSee Key Table 1 p 99)
144
0.1D
114
U- 3
34 16
-5
I .. a
1 .. 
. .--
1 -0
10 6 - - 5
1- 4-1 -
I .- 5
I 1L
L1
FIGURE 45: Enlarged View of a Portion of Figure 44 (See Key Table 1
p 99)
145
z
SINK
H
V
/
/
/
/
/
7/
/
/
FIGURE B-1:
SINK
Basic Two-Sink Model
x
Y
/
/
146
--- yr~r T p7
Z5
0
mom 1,5
mxm% 3 1 N KMae
ex i::::Ow5'm
4
0.
x
3 0.5 .o 1.5
x
FIGURE B-2: Streamlines in the x, z Plane
I
.0 -i.S -t. -0.5 2.
I I a I
147
AZ
2.5
2.0
1.5
SINK
0.5- S3
S
- .0 -05 0. .0.5 1.0 1.5 2.0 2.5 a
X
FIGURE B-3: Streamlines in the x, z Plane
148
Y
3.5
3.0
-go
-1.0 -0.5
I -I- I I
-1__-
.10.5 i.0 Sl 2.
x
FIGURE B-4: Streamlines in the x, y Plane
.0 -1.5
I I I Jolf jpov I
I II I I
I I I
&.J
0
149
.L2
SINK
10
0.8
0.6
0.4 - x
0.2 
-
0 S, 0.5 .1.0 S, 1.5 0 .
x
FIGURE B-5: Streamlines in the Neighborhood of Stagnation
Point S2' x, z Plane
CAPTURE SURFACE -z 7
It4
INK
FIGURE B-6: Perspective View of Flow Field' (Because of Symmetry, Only HaPF the Flow Field is Sh
own)
C-"
C
\0
dpX
151
REFERENCES
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Objects into Turbine Engines by Vortices", NACA TN3330,
February 1955.
2. Klein, H., "Small Scale Tests on Jet Engine Pebble Aspiration",
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3. Klein, H., "An Aerodynamic Screen for Jet Engines", Institute
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4. Glenny,- D. E., "Ingestion of Debris into Intakes by Vortex
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7. Motycka, D. L., "Ground Vortex - Limit to Engine/Reverser
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8. Motycka, D. L., and Walter, W. A., "An Experimental Investigation
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10. Newmann, W. H., and Atassi, H., "A Two-Dimensional Potential Flow
Model for Ground-Induced Effects on Jet and Fan Inlets", AIAA
paper No. 80-0388, January 1980.
11. Prandtl, L., and Tietjens, 0. G., "Fundamentals of Hydro - and
Aeromechanics", Dover Publications, New York, 1957.
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Laboratory, Report No. 88, 1966.
13. Lighthill, M. J., "Drift", Journal of Fluid Mechanics, Vol. 1, 1956,
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14. Hess, 3. L., and Smith, A. M. 0., "Calculation of Potential Flow
about Arbitrary Bodies", Progress in Aeronautical Sciences, Vol. 8,
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152
15. Hess, J. L., "Review of Integral-Equation Techniques for Solving
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Vol. 5, 1975, pp 145-196.
16. Hill, V. P., "A Surface Vorticity Theory for Propellers Ducts
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17. Hess, J. L., "The Problem of Three-Dimensional Lifting Potential
Flow and its Solution by Means of Surface Singularity Distribution",
Computer Methods in Applied Mechanics and Engineering, Vol. 4, 1974,
pp 283-319.
18. Hess, J. L., "Calculation of Potential Flow about Arbitrary Three-
Dimensional Lifting Bodies", Final Technical Report, McDonnell
Douglas Report No. MDC 35679-01, October 1972.
19. Rubbert, P. E. and Saaris, G. R., "A General Three-Dimensional
Potential Flow Method Applied to V/STOL Aerodynamics", SAE paper
No. 680304, 1968.
20. Rubbert, P. E., and Saaris, G. R., "Review and Evaluation of a
Three-Dimensional Lifting Potential Flow Computational Method
for Arbitrary Configurations", AIAA paper No. 72-188.
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Coefficients Methods Applied to Subsonic Flows", AIAA paper No.
75-50.
22. Hess, J. L., and Martin, R. P., "Improved Solution for Potential
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Order Surface Source Method", McDonnell Douglas Report J6627-01,
also NASA CR 134694, July 1974.
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Oriented Method for Calculating Compressible Flow in and about
Three-Dimensional Inlets", McDonnell Douglas Report No. MDC J7733,
also NASA CR-159578, April 1979.
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Propulsion System Inlets", NASA TMX-71799.
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153
27. Ralston, A., "Runge-Kutta Methods with Minimum Error Bounds",
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