p APPROVED: f. J gluta

Unsteady Pressure and Vorticity Fields in

Blade- Vortex Interactions

by

Matthew M. Pesce

Thesis submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Master of Science

in

Engineering Mechanics

p APPROVED: f.

J gluta nis, Chairman

D. T. Mook R. D. Kriz a

December, 1990

Blacksburg, Virginia

Lb 8055

\455 440

Pasa Cr

Unsteady Pressure and Vorticity Fields in

Blade- Vortex Interactions

by

Matthew M. Pesce

D. P. Telionis, Chairman

Engineering Mechanics

(ABSTRACT)

The unsteady interaction of a vortex core with a NACA 0015 airfoil is studied in two dimensions.

A two-component, three-beam Helium-Neon laser-Doppler Velocimetry system is used to take

data in a water tunnel. Ensemble-averaged velocity fields are obtained in the region of the leading

edge of the airfoil. Finite-difference algorithms were written to obtain vorticity and pressure in the

data field. Computer animation of the unsteady vorticity was accomplished first with a Fortran

code written for an Apple Macintosh computer and later with a commercial software package for

a SUN Microsystems graphics terminal.

Dedication

Dedicated to my Parents, Ralph and Mary, my sister, Rachael, and my brothers, Adam and

Thaddeus

Proof that the human spirit can overcome all adversity and persevere

Dedication ili

Acknowledgements

I wish to express my gratitude to Dr. Demetri P. Telionis for the opportunity, guidance, and

support to reach an advanced degree.

Thanks also to Drs. Kriz and Mook for serving on my committee and their criticisms concern-

ing this work. Extra thanks to Dr. Kriz for his assistance in data animation.

The work presented here was funded by Boeing Helicopter; without their support I would not

have been able to eat.

Special thanks to Michael Wilder, Ngoc Hoang, Othon Rediniotis, Ron Swecker and other

notables for their help and friendship.

Finally, thanks to Tracey A. Mitchell for her endurance through the years of our aquaintance.

I will always remember.

Acknowledgements iv

Table of Contents

1. Introduction and Literature Review ........ 0. cc cere eee e eee e ener ete ee eeenes 1

2. Facilities and Equipment .......... 0. ce ccc ee cee eee ee ee renee eee reset eees 5

2.1 The Water Tunnel 2.0.0... . cee eee eee eet eee eee eee 5

2.2 The Models 1.2... . 0... cece eee eee eee eee ee eee eee 6

2.3 Laser-Doppler Velocimetry .......... 0... cece eee eee ees Lee eee eee eee 7

2.3.1 The DISA Laser System 0. ee eee eee eee tence eee 9

2.3.2 The TSI Laser System 2.0... ce ee eee ee eee eee eee e eens 10

3. Data Acquisition, Reduction, and Analysis ........... ccc cece cere recs cen evaee 12

3.1 Data Acquisition .. 6... ce ec ee ee ee eee eee ee ee eee ee eee eee 12

3.2 Vorticity Calculations 0.0... ee eee ene eee ee eee ees 13

3.3 Pressure Distribution 2.0... 00... ec ee ee ee ene ee eee ee eens 14

3.4 Animation 0.6.0... ccc ccc ee ee ee ee ee eee ee eee eee ee eee eee eee eee 17

3.4.1 Macintosh Animation ....... 0... ccc ee ee ee eee eee rete eee eee eee 18

3.4.2 SUN Animation 2.0... . ccc ee ee ee eee eee ee ete eee e ee eens 19

Table of Contents Vv

4. Results and Discussion ........ 0... ccc cee eee cece ee ee ee wee teen eee wees teens 21

4.1 Introduction 2.0... ee ee eee e teen nee e eee enes 21

4.2 Velocity and Vorticity 2.0... eee eee e een e nee e ene 22

4.3 Pressure 2. ee ee ee ee eee eee eee eee ee eee 24

4.4 Conclusions 2.0... 0... cc eee eee ee eee eee eee eens 27

Bibliography 2.0... ce ccc ee ee ee eee eee eee eee eee ee eee ee eee eee ees 29

Appendix A. Vorticity Code 2.0... . ccc ccc ee cee eee tere er eee eee teeter enees 32

Appendix B. Pressure Code... . ccc ccc cee ec cc eee rece eee eee eee eee eee eee eene 39

Appendix C. Macintosh Animation Code... 2.0... . cece cee cc eee eet eet e eee 49

Appendix D. SUN Animation Code 2... 0... cc cece eee eee eens 60

Appendix E. Figures 0.0... ccc cc cc cece eee ee ee tee eee teeta eee eens 64

Vita Cc ee etc ee ee te ee eee eee eee eee eet eee eee eee eter ee eee 216

Table of Contents vi

1. Introduction and Literature Review

The interaction of vorticies with wings, propellers, and helicopter rotor blades comprises an

important class of non-uniformities in unsteady aerodynamics. Concentrated filaments of vorticity

are shed from sharp surfaces, such as the trailing edge of a wing or the tip of a blade. Under normal

flight conditions, the vortex cores dissipate; however, when two aircraft fly in close proximity or

when vorticity remains in the helicopter rotor plane, as in powered decent (Figure 1.1), the vortex

filaments induce rapidly fluctuating airfoil pressures and noise. In helicopter aerodynamics, in

particular, vorticity interactions cause vibrational loadings that can damage rotors and amplify fa-

tigue of materials.

The interactions of vorticies with wings or rotors exist between two extreme conditions, the

vortex perpendicular to the blade axis and the vortex parallel to the blade axis (Figure 1.2). The

perpendicular condition affects only a small region of the blade but presents itself as a three-

dimensional problem. The parallel condition affects the entire blade but is only a two-dimensional

problem. As a vortex comes closer to a blade, the area affected by the vortex, the structural loading,

and the noise increase.

1. Introduction and Literature Review 1

Work has been done in an effort to closely stimulate the real helicopter-blade interaction prob-

lem. Neuwerth and Muller (1985) generated vortices which interact with rotating blades in a wind

tunnel. This arrangement resembles the actual conditions of helicopter flight, but accurate study

of the flow character near the rotors is difficult due to the three-dimensional flowfield and the mo-

tion of the model relative to an inertial reference plane. Simpler, numerical and experimental sim-

ulations of a two-dimensional parallel interaction can provide detailed flow information in the

vicinity of the airfoil’s leading edge. A popular method for studying the 2-D parallel interaction is

to generate vorticies by pitching an airfoil upstream of a second airfoil. A street of alternating

positive and negative vorticies follow the freestream and alters the flow around the second, or target,

airfoil (Figure 1.3). The 2-D, parallel, vortex interaction is generally known as the Blade-Vortex

Interaction (BVI) problem. The initial strength and location of the vortex, with respect to the target

airfoil, vary widely in the literature.

Numerical solutions to Blade-Vortex Interactions include Navier-Stokes and discrete-vortex

dynamics methods. Work by Caradonna (1984), Tang and Sankar (1987), and Rai (1987) are

representative of Navier-Stokes solutions. Unmodified N-S solvers numerically dissipate the strong

gradients across the vorticies. Since numerical dissipation is the order of the grid spacing to the 4th

power, this effect is much worse for the coarse grids used in the outer flow than for the fine grids

used in the airfoil boundary layer. A vortex released upstream of the target airfoil will be numer-

ically dispersed before it interacts unless modifications are made. However, fixing the path or

strength of the vortex, a modification used by many researchers, produces incorrect results for close

interactions. In their unsteady, compressible Navier-Stokes algorithm, Tang and Sankar employ

artificial viscosity terms to a modified velocity field; this allows the vortex to retain its sharp features

until the actual interaction. Rai, however, used a Sth-order, thin-layer solver. Very fine grids were

needed to prevent numerical diffusion of the vortex core.

Work by Mook et al. (1987,1990), Lee and Smith (1987), and Poling et al. (1988) are repre-

sentative of discrete-vortex dynamics. Velocities and pressures induced by clouds of point vortices


are calculated. Mook et al. modeled the flow over a pitching airfoil with a panel method. The

airfoil was represented by a piecewise linear distribution of vorticity. As the airfoil pitches, it sheds

vorticity from the tailing edge. The shed vorticity is modeled by vortex cores or blobs of vorticity.

As the cores move away, they may spread too far apart or move too close together; cores are added

or combined, eliminating discontinuities, and the vorticity is redistributed. Lee and Smith and

Poling et al. modeled thé interaction process of a vortex striking an airfoil. A cloud of ideal vortices,

simulating one of the vortices in the unsteady wake, is convected with the freestream over the target

airfoil. Poling et al. modeled the airfoil with a Joukowski transformation; the circle theorem is

employed to solve the problem in complex coordinates. The cloud is modeled as a Gaussian dis-

tribution of vorticity strength in an effort to more closely resemble a real vortex. Lee and Smith

modeled the target airfoil with a panel method. As the cloud warps and moves closer to the airfoil,

the strengths and sizes of the panels are redistributed. The point vorticies in the cloud begin with

equal strength. Discrete-vortex dynamics yields accurate solutions but does not capture airfoil

separation. Common problems are cloud instability and singularities at the center of the point

vorticies.

Experimental work has also been done on the 2-D parallel interaction problem. Booth and

Yu (1986), Poling (1985), and others have employed flow visualization to study the roll-up of

vorticies behind a pitching airfoil and the interaction of vortex cores with stationary airfoils. Booth

and Yu establish three interaction regions--the collision zone, the distortion zone, and the deflection

zone. A vortex moving within the collision zone, a small band above and below the target airfoil’s

leading edge, will split across the nose of the airfoil. A vortex moving in the deflection zone, the

outermost region, is only displaced along with the freestream. In the distortion zone, the interacting

vortex deforms and displaces. The interaction regions are sensitive to the strength, size, and initial

position of the vortex as well as the loading of the airfoil.

Meier and Timm (1985) created a Karman vortex street with a stationary square cylinder and

single start-up vorticies using an airfoil in a shock-tube chamber. Meier and Timm found that when


a vortex moves in the deflection zone, the suction region on the airfoil is reduced. But as the vortex

gets closer, it mixes with the vorticity of the boundary layer and induces a separation bubble. This

secondary vortex collapses as the vortex moves over the airfoil. In the collision zone, the impinging

vortex core disintegrates, leaving only scattered vorticity. Later work by Booth (1986, 1987), Wilder

et al. (1990), and Mayle et al. (1990) obtained quantitative experimental velocity, vorticity, and

pressure fields in the wake of the pitching airfoil and in the vicinity of a target airfoil placed in the

wake.

In the experimental work presented here, we use a NACA 0015 target airfoil at 10° angle of

attack and vortex cores created with a NACA 0012 airfoil pitching upstream of the target airfoil

WC 2U.,

paring freestream velocity to the pitching airfoil’s frequency and chord length. Laser-Doppler

with a reduced frequency of 2. The reduced frequency, k = , 1S a similarity parameter com-

Velocimetry (LDV) measurements are taken on a fine spacial grid to establish the unsteady,

ensemble-averaged velocity flowfield in the vicinity of the target airfoil. The case of a positive

vortex passing above the target airfoil within the distortion zone is explored. Finite differences are

used to calculate flow derivatives to create vorticity as well as pressure coefficients. Previous ex-

perimental research could not supply large amounts of qualitative data on BVI. The work pre-

sented here provides detailed velocity measurements in the vicinity of target airfoil’s leading edge.

The fundamentals of the interaction process are studied and conclusions are presented.


2. Facilities and Equipment

2.1 The Water Tunnel

The water tunnel (Figure 2.1.1) was built in 1976 by the Engineering Science and Mechanics

Department with the support of the Army Research Office. The basic designs can be found in the

work of Koromilas (1978) and Mezaris (1979). Recent modifications are described by Poling

(1985). The tunnel contains 570 gallons of water and is built of poly-vinyl chloride (PVC) pipe and

plexiglass. The test section and parts of the lower tunnel are made of plexiglass to allow flow vi-

sualization and laser velocimetry studies.

Speed control is accomplished with both a butterfly valve and a variable-speed DC electric

motor driving a centrifugal pump. Speeds can range between 0 and 5 m/s by changing the valve

openings and pump speeds; however, high valve openings cause higher turbulence levels. There is

a temperature control unit that can heat the water and increase the temperature at the rate of 3

degrees Celcius per hour.

2. Facilities and Equipment 5

The settling chamber contracts at a six-to-one ratio into the test section. There are turning

vanes as well as screens and 3 cm? aluminum hexagonal honeycombs to reduce the turbulence level

and turbulence scales. Moreover, a small honeycomb section with 1/4-inch cells and three screens

precede the test section. These further reduce turbulence levels and increase flow uniformity.

The test section has a cross-section of 10 inches by 12 inches and is 28 inches long (Figure

2.1.2). It was constructed with false walls to bleed off the side-wall boundary-layer and reduce end

effects. Flaps at the ends of the false walls control the bleed-flow and limit separation at the leading

edges of the false walls. The flaps are kept at 10° to minimize the boundary layer thickness as

mentioned in Kim (1985) and Poling (1985). The boundary-layer thickness is then at most 4% of

the span of the tunnel. The turbulence levels in the test section are on the order of 0.4% to 0.65%

for u’ and v’ components between + 3 inches from the center of the tunnel.

2.2 The Models

Two symmetrical airfoils are mounted in the test section of the water tunnel. The first is a

NACA 0012 with a 4-inch chord, the other is a NACA 0015 with a 6-inch chord placed 8 inches

downstream of the trailing edge of the first airfoil (Figure 2.2.1). The NACA 0012 pitches about

its quarter chord to generate a street of positive and negative vorticity. A shaft connects the airfoil

to a four-bar linkage system running off a 3/4 hp Morse DC electric motor (Figure 2.2.2). The

motor is connected to a 16:1-ratio gear box and to a 6-inch aluminum flywheel. The four-bar

linkage acts to reduce vibrations entering the flowfield during airfoil oscillations. A push-bar con-

nects the four-bar linkage to the aluminum flywheel. Pitching amplitude can be selected by con-

necting the push-bar to any radial position on the flywheel; an amplitude of 20 degrees is used for

the data collected here. A Datel electric shaft encoder coupled with a photo-diode on the flywheel

triggers the data-aquisition system at the beginning of a new pitching cycle.


The NACA 0015 acts as a target airfoil for the generated vortex street. Connected to a rotating

mount embedded into the inner false wall of the test section, the airfoil can be adjusted to various

angles of attack. The rotating mount is fixed into position with two set screws. A third set screw

pushes the model tight against the water tunnel.

2.3 Laser-Doppler Velocimetry

Velocity data are gathered with a laser-Doppler velocimetry system. When two or more laser

beams intersect, an interference pattern of alternating light and dark fringes occurs (Figure 2.3.1).

Particles travelling in a moving fluid refract light and dark as they pass through this intersection.

The scattered light is received by a photodetector and converted to an electric signal. The signal

contains low- and high-frequency noise, a pedestal voltage, and the Doppler frequency (Figure

2.3.2). After processing, the Doppler frequency, fp, is removed from the signal and directly con-

verted to velocity.

Both forward- and backward-scatter modes are used in LDV systems (Figure 2.3.3). In

forward-scatter the photodetector and light receiving optics are on the opposite side of the beam

intersection from the laser. Forward-scatter requires less laser power for good signal quality but is

difficult to traverse. In backward-scatter the photodetector and optics are on the same side as the

laser. The backward-scatter systems need more power for good signals but traverse much more

easily.

Laser-Doppler velocimetry systems have several desirable advantages. LDV has high-frequency

response and high signal-to-noise ratio, and no calibrations are required since the particle velocity

is directly proportional to the frequency of the signal. Also, the control volume of intersection is

small, thereby, allowing detailed measurements. Finally, the laser beams do not physically interfere


with the flow--this is especially important in flows with vortex structures where hot-wire or 5-hole

probes can alter the vortex path or cause early core breakdown. However, laser systems are initially

expensive and difficult to setup.

With LDV systems, flow reversal can be detected by the frequency shifting of one or more of

the intersecting beams, which adds a known frequency to the signal. Without frequency shifting,

particles moving with the same speed but in opposite directions will scatter light at the same

Doppler frequency, but with frequency shifting, the scattered light will have different Doppler fre-

quencies. Therefore, flow reversal can be measured accurately, and the frequency shift can then

be removed during processing.

Another important consideration in LDV is prevention of particle-lag errors. A particle that is

too heavy or large can fall due to gravity and produce an extraneous component of velocity, or it

may not respond to quick movements of the flow. Particle lag is reduced by using small, neutrally

buoyant particles, i.e. particle density near that of the fluid density, dispersed in the fluid in a way

that prevents aggregation of the particles. The signals scattered from particles that are too large or

too small can be rejected during signal processing.

The experimental setup of the present effort includes DISA and TSI optical systems. The DISA

system consists of a 5 mW laser in forward-scatter mode while the TSI system has a 15 mW laser

in backward-scatter mode. Both systems use He-Ne lasers with 632.8 nm wavelength. The DISA

laser system monitors the free-stream velocity. The laser control volume is fixed and the system

uses tracker-type signal processing. The TSI laser system is used to take the two independent

components of unsteady velocity data. It consists of several optical components to insure high

signal-to-noise ratio and detection of flow reversals. The system uses linear variable direct

transducers (LVDT) for computer-controlled stepping through space, and the two independent

Doppler signals are processed with counter-type signal processors. Neutrally buoyant 1.5 micron

silicon carbide particles are used to seed the flow for better signal quality.


The fluid velocity is proportional to the Doppler frequency fp and the interference-fringe spac-

ing, ay, such that V = d;fp, where d= 5 é K The wavelength of the laser-light and the half-angle

between the two intersecting beams are J and k, respectively.

2.3.1 The DISA Laser System

A DISA laser system (Figure 2.3.4) is used to monitor the freestream velocity, which is then

employed for on-line non-dimensionalizing of the data as well as helping reduce errors due to

low-frequency fluctuations in the freestream. A 5 mW laser beam is split into two equal-intensity

beams by a DISA 55L01 beam splitter. The beams focus through a 300 mm focal-length lense into

the water tunnel, upstream of the pitching airfoil. The scattered light is collected in forward-scatter

through a 0.1 mm pin-hole into the DISA 55L10 photomultiplier. A DISA 55L15 high-voltage

source supplies continuously adjustable DC voltage to the photomultiplier. The signal from the

photomultiplier enters the preamplifier (DISA 55L30). The preamp contains narrow-band filtering,

preset for the selectable Doppler frequency ranges available with the frequency tracker (DISA

55L35). The measured Doppler signals can range from 2.25 kHz to 15 MHz and are tracked with

seven selectable ranges. Velocity fluctuations of up to 70% of the mean in a given range can be

followed. For low density seeding situations, signal drop-out circuitry can be set to keep only the

clearest signals. An ouput voltage proportional to the Doppler frequency is then sent as analog

output to the MINC 11 analog-to-digital converter. The freestream velocity data are accepted at

constant time steps and saved on disk along with the u and v velocity components from the TSI

system.

The laser and photomultiplier are arranged in forward-scatter mode and are manually positioned

on an optical bench. The silicon carbide seeding gives a very high data rate in forward-scatter mode.


This, paradoxically, can lead to higher noise since the photodetector could saturate. Lowering the

power to the photodetector compensates for this.

2.3.2 The TSI Laser System

A TSI system is employed to collect data around the NACA 0015 model (Figure 2.3.5). A 15

mW laser beam is split into 3 beams, and two independent velocity components are measured using

a single photomultiplier in backward-scatter. The 15 mW beam is collimated (TSI 9108) to prevent

beam divergence and to place the beam waist, the thinnest diameter, at the beam intersection. This

decreases the size of the control volume, increases signal to noise ratio, and keeps the fringes parallel

and straight for accurate Doppler frequency refraction. The beam then passes through two

quarter-wave plates (TSI 9178-2) that linearly polarize the light. Next, a polarization rotator (TSI

9102-2) rotates the polarization plane of the incoming beam relative to the two split beams coming

out of the variable-intensity beam splitter (TSI 9216-2). The rotation optimizes the intensity split

to 2/3 to 1/3. The variable-intensity beam splitter offsets the 1/3 beam 25 mm from the optical axis

and leaves the 2/3 beam on-axis to be split again (TSI 9115-2) into two equal-intensity beams.

Both beams are set off the axis by 25 mm. The first 1/3 intensity beam is rotated again (TSI

9103-2) to ensure that it is plane-polarized with the last two beams. Two of the three beams are

now frequency shifted with 40 and 60 MHz Bragg cells (TSI 9182-2 and TSI 9282). This puts the

fringes in motion at 20 MHz and 40 MHz in the two independent directions used for measurements.

The photomultiplier (TSI 9162) follows next in the optic assembly. The refracted light is received

in backward scatter along the optic-axis. The photomultiplier system can sense frequencies up to

200 MHz and amplify weak signals. The beams are then displaced inward (TSI 9114-2) and their

diameters are expanded (TSI 9188A). This decreases the intersection size and increases the signal-

to-noise ratio of the system.


The expanded beams reflect through the mirror tower (Figure 2.3.6) and out through an

achromatic focusing lense (TSI 9118) into the test section. The lense has a focal length of 250 mm.

Scattered light is received through the same optics into the photomultiplier. The signal is then

down-mixed to remove the 20 MHz and 40 MHz frequency shifts and a selectable frequency is left

with the signal. This allows for detection of flow reversals and is removed during data reduction.

The signals are then processed with TSI Model 1990 counters. The counters remove the high- and

low-frequency noise from the Doppler signal and then count the number of times the signal crosses

and re-crosses zero (Figure 2.3.7). Once the signal crosses zero twice, the logic gate is opened and

the zero crossings are counted. When the signal crosses zero a preset number of times, the signal

is considered valid and is counted. The counted signals are converted from digital to analog in the

TSI counters and sent to the MINC 11 laboratory computer.

The analog signal must be converted back to digital to be processed and stored on disk. The

MINC 11 expects data at fixed times during data aquisition; however, the signals arrive randomly

due to the dispersion of particles in the flow. With digital input, if no signal is available at

aquisition time, the computer will substitute zero. To prevent this situation, the digital-to-analog-

to-digital set-up is used. Now, if a new signal is not available, the counter keeps sending the pre-

vious value. Acquisition is triggered at the beginning of each pitching period, and the velocities are

ensemble-averaged. This removes the velocity fluctuations and leaves the mean-velocity compo-

nents.

Traversing is accomplished horizontally by moving the optical bench while vertical traversing

is performed with two mirrors, one mirror moves and the other is fixed to the laser table. The

movement is performed with LVDT’s connected to the MINC-11 with feedback circuitry. The

position of the control volume is monitored and corrected as needed.


3. Data Acquisition, Reduction, and Analysis

3.1 Data Acquisition

After the raw data are obtained, the voltages stored on diskette by the MINC 11 lab computer

are converted to velocity. The voltages are reduced on the IBM 3090 mainframe computer using

programs written by Wilder (1988). The IBM 3090 allows faster computing and greater storage.

Processed data can then be plotted on the Versatec Color and B/W electrostatic plotters available

with the VPI & SU computing facilities.

The u and v components of velocity, those components parallel and perpendicular to the pri-

mary flow direction, and the freestream velocity are stored as voltages at each time level and at each

grid position. The measurement grid is a rectangular, orthongonal mesh with its x-y origin centered

at the leading edge of the target airfoil, where x points downstream and y points up (Figure 3.1.1).

Voltages corresponding to beams blocked by the airfoil are steady with time. Data from these

spacial locations are recognized by this feature and are tossed out. However, to keep the data field

uniform, a weighting parameter is stored for each location. If this parameter is equal to one, the

data are good; for all other cases the data are bad. The region of blockage occurs only near the

3. Data Acquisition, Reduction, and Analysis 12

airfoil surface. The three converging beams encompass more space than does the control volume

itself. When the control volume is outside of the airfoil surface, at least one beam can strike the

airfoil. To diminish the size of the blocked region during data aquisition, the beams are tilted in

the vertical plane (Figure 3.1.2). The beams enter the focusing lense at an angle, although still

parallel to each other. The velocities are now measured out of the x-y plane, but this can be cor-

rected analytically. With this modification data can now be taken closer to the airfoil, thus, re-

ducing the bad data region around the airfoil. The rest of the voltages are converted to velocity

directly from the equations presented in Sections 2.4.1] and 2.4.2, These velocities are smoothed

by using IMSL subroutines to eliminate any sharp discontinuities in the velocities.

3.2 Vorticity Calculations

A finite-difference algorithm was written to calculate the two-dimensional vorticity field from

the measured, unsteady velocity field (Appendix A). The vorticity expression

B=tVxV

simplified for two-dimensional flow

_ ov _ ou o= A ay (3.2.1)

is evaluated in space at each time level. The derivatives are represented by the following 2nd Order

central, forward, and backward finite-difference expressions:

6 i4¢, “Hie Central: of = fis Sat + O(Az’) (3.2.2)


of Sfi-rat Vier Hh 2 a = TAS + O(Az’) (3.2.3) Forward:

@ a Via ti Backward: Ly afr Ha th 2 55 + O(Az’). (3.2.4)

where f represents a dependent variable and z represents any independent variable. The subscripts

i,it+ 1,i+ 2, etc., indicate the grid-node where the quantity is evaluated. The spacing, Az, is con-

stant.

Logical expressions route the program to the correct derivatives. With good data on either side

of the point evaluated, central differences are used, and when a blocked data point exists, a one

sided difference is used (Figure 3.2.1). The derivatives above, though shown in one direction, are

substituted for the two dimensional spacial vorticity derivatives by changing the dependent and in-

dependent variables to u or v and x or y, respectively. There are nine independant conditions to

consider (Figure 3.2.2) and, therefore, nine combinations of the finite-difference expressions to be

used. Figure 3.2.1 shows how the derivatives would be used in one direction, but Figure 3.2.2

shows the actual 2-D decisions to be made. The vorticity is set to zero where correct derivatives

cannot be computed. Vorticity iso-bars are then plotted at each time level of the pitching period.

3.3 Pressure Distribution

The ensemble-averaged velocity data are employed in the 2-D non-dimensionalized Euler

equations to evaluate the convective terms and solve for the pressure gradients,

Op | = = (TH 5 y BH y OU) (3.2.5)

ot Ox oy


and

Op _

éy Wy yy Wy Wy. (3.2.6)

a ot Ox oy

where u and v and x, y, and t are dimensionless velocity components, space, and time, respectively,

and T is a dimensionless parameter, often referred to as the Strouhal Number.

The physical quantities are non-dimensionalized according to the formulae

* * *

x= p= P — u

c p U2 U.,

* * *

y P v y= x P= : =] yo :

c Poo U os

t

t=

T

The dimensional terms x* and y* are the actual distances in the water tunnel while c’ is the chord

length of the target airfoil. Similarly, ¢ is real tume measured while +’ is the pitching period. The

physical pressure p* is non-dimensionalized with the dynamic pressure p’U32. The velocities are

non-dimensionalized with the freestream velocity while the density is eliminated during the non-

dimensionalization. The non-dimensional Euler equations retain the same form as the dimensional

equations except for Strouhal Number in front of the time dependant term, T = = .

The derivatives are evaluated as defined by Equations (3.2.2), (3.2.3), and (3.2.4) in Section 3.2.

Central differences are used when there are no blocked data on either side of the point where the

derivative is calculated; one-sided differences are used on the boundaries and near the airfoil.

Moreover, the non-linear terms are evaluated at the point where the derivative is evaluated. For

example, yOu is represented as u; ot )ij, where Ou )i; is evaluated as a finite difference and the Ox Ox

point (i,j) is an arbitrary grid point in the data field.


Once the values of ee and 2 are known everywhere, the pressure coefficients C,(x,y,!) can

be evaluated with partial integration or algebraic approximations on the derivatives. Partial inte-

gration of the differential pressure terms is accomplished by the trapezoidal-rule. The trapezoidal-

rule provides 2nd-order accuracy and requires two differential pressure values and a starting value

for pressure. Since no experimental or numerical pressure solutions were available, the farthest

point upstream of and above the airfoil is used as a reference pressure, p,,. Although this point is

unsteady, it is assumed constant from one time step to the next; this permits time frame compar-

isons. Also, the vortex only indirectly affects this point, so trends in C, should remain accurate.

The horizontal and vertical integration equations are

PX Yjst) — p(x;_ 1 Dpt) _ [ Op dx

pU,,” Ox xi

and

PAX; Ypst) — P(Xp I; - 4) _ [ Op 5 =

pU,,,

respectively, where U,, is the freestream velocity, and i, j,i— 1, andj — 1 are arbitrary locations in

the data grid. The integrals are then approximated with the trapezoidal-rule,

_Ax| | Op AG j= ax) * - (3.2.7)

and

ay[ a 2 AC,); 2 + | (3.2.8)


Integration starts with the reference pressure, p,,, and continues down into the data field. All the

finite AC, values are summed from the reference location to a general grid location to get total C,

based on the reference pressure.

An important consideration is the direction in which to integrate through space. In the data

presented here, differential pressures are integrated first in the positive x-direction, downstream, and

then in the negative y-direction, toward the upper airfoil surface. This prevents integration through

the vortex except when necessary for C, calculations. Different path directions for integration yield

a variety of results, especially in x-direction integrations. This may be due to the course grid spacing

in this direction or integration through the vortex core.

Other methods were tried and compared to the trapezoidal-rule method. A _ sixth-order

Simpson’s rule integration scheme as well as a finite-difference approximation to the derivatives

were used. Both had the same trends as the trapezoidal-rule solution but had different magnitudes.

Simpson’s rule needed six integrand values preceding the solution point and gave results differing

by as much as 20%. Since no exact comparisons could be made, the simplest algorithm was used.

3.4 Animation

Computer animation is the rapid painting and repainting of graphic images on a computer

screen, creating the impression of moving figures. The general procedure is to take the physical

quantity to be animated and convert it to a discrete pixel representation; the calculations are gen-

erally done in a two-step procedure. First, the data are transferred to the screen by converting

physical quantities to graphic images. The pixels on the computer screen are allocated memory

space; each pixel has a location, color, and brightness associated to it. The screen is then stored

as a bitmap, to be recalled as a block quantity, containing only the information needed to locate


and shade the pixels. A second program reads the bitmaps and displays them. Since there are no

calculations in the second program, animation is faster than when using only the first program.

Animating vorticity has several advantages over animation of velocity or pressure. The vortical

structure of the flow and the separation regions on the airfoil are readily visualized. Also, changes

in the position, size, and strength of the vortex are observable along the vortex path. Available

colors are scaled continuously from positive to negative vorticity maximums. Moreover, small flow

disturbances as well as the large distortions due to the airfoil can be visualized. The entire pitching

cycle can be visualized continuously.

3.4.1 Macintosh Animation

Animation of vorticity was achieved first on the Apple Macintosh IIx computer with an Apple

high-resolution RGB color monitor. The monitor is a 13-inch diagonal with 640 by 480 pixel re-

solution and a vertical refresh rate of 66.7 Hz. The computer is based on the 32-bit NuBus with

a Motorola MC68020 processor and MC68881 Math co-processor. Although the system is built

for speed and expandability, all instructions pass through the NuBus to be processed. Mathemat-

ical and graphics intensive applications must vie for processing time. In animation, where math-

ematics and graphics are combined, the processing is slow.

A Fortran code, Microsoft Fortran Version 2.0, was written to take advantage of the Macintosh

II graphics toolbox utilities. The code (Appendix C) draws color pixels into a graphics window for

levels of vorticity. At each time step, the data are searched and assigned a color based on its

vorticity; this color is drawn onto the screen in locations scaled to the data. The target airfoil is

drawn into the graphics image by available line drawing utilities.


The Fortran code is concise and useful, and provides a first look into the animation process and

knowledge into alternative methods. Software changes could be made easily to the source code;

color and processing ideas could be vented at this level of software. However, since all calculations

and drawings are done at the time of animation, the images are refreshed slowly; screens are not

saved as bitmaps to optimize animation time. Also, resolution is limited by the lack of speed; only

real data of 25 by 41 spacial points could be animated, resulting in chunky images. The data set

is interpolated on the IBM 3090 with a bi-cubic spline to 350 by 250 spacial points for animation

on the SUN computer.

3.4.2 SUN Animation

Animation of vorticity data is done on the SUN Microsystems SPARC color workstation. The

monitor is a 16-inch diagonal with a 1152 by 900 pixel resolution and a vertical refresh rate of 66

Hz. The computer is a 32-bit SUN-4 system with a SPARC architecture. The SUN-4 Central

Processing Unit (CPU) combines an integer unit for basic processing and a floating-point unit for

floating-point arithmetic. The SPARC (Scalable Processor Architecture) combines simple in-

struction formats with delayed instruction control transfers in registers, overlapping registers, and

single-cycle execution of most instructions.

The commercial software PV-Wave from Precision Visuals was used to animate the vorticity

data on the SUN (Appendix D). PV-Wave is a general purpose analysis and visualization system.

Commands and keywords, entered both interactively and in programs, act on variables refering to

collections of data. The scientific data, seen as scalars, vectors, and arrays, are expressed and ma-

nipulated as single entities or as data structures. Error checking and error messages are available,

and PV-Wave uses only the ASCII character set.


Once on the SUN, the vorticity data are converted to pixel-image, pixel-shade, and contour

representations. In a pixel-image, the vorticity range is mapped onto a color table of 256 distinct

colors. Each datum point is assigned a color based on its vorticity level. The data points, for ex-

ample, for vorticity values ranging from -5 to -5.2 might have a color level of 11 which could be a

reference to light blue. The pixel-shade maps the vorticity in and out of the plane of data based

on sign and magnitude; a positive vortex will rise out of the plane as a mountain, the summit re-

presenting the maximum vorticity at the core. Imaginary light strikes the formed surfaces projecting

shadows. Small gradients are readily seen this way, however, the operations are math intensive and

slower. The contour representation are lines of constant vorticity traced thru the data set.

There are many advantages and disadvantages to PV-Wave. The program is concise and pow-

erful; the commands can be quickly altered for picture size, location, and color. Simple animations

and analyses take very little knowledge of the program while the programer can grow into the

program’s power and agility. PV-Wave, however, requires the data to be uniformly spaced. Also,

non-data information cannot be incorporated into the graphics. For example, the target airfoil

could not be drawn on the screen with the data using PV-Wave. Changing all the values of vorticity

between the top and bottom of the airfoil to the same level but different from zero compensated

for the program’s limitation. PV-Wave acted as if this were real data and formed an airfoil of a

constant color. Finally, PV-Wave could only animate one physical quantity at a time; velocity,

vorticity, and pressure could not be animated all at once.


4. Results and Discussion

4.1 Introduction

Taken in the vicinity of a NACA 0015 airfoil at « = 10°, the ensemble-averaged velocity data

0 collected from LDV measurements is presented here as velocity vectors, vorticity contours, — and

2 contours, and pressure coefficient plots. The leading edge of the target airfoil is offset above

the zero-degree axis of the pitching airfoil by y/c = 0.12, where y is the vertical distance from the

leading edge of the NACA 0015 airfoil and c is the chord length (Figure 4.1.1). The freestream

velocity and Reynolds number, based on the target airfoil chord, are 0.128 m/s and 19550, respec-

tively. The pitching schedule is sinusoidal with a period of 1.24 seconds, an amplitude of 20°, and

a reduced frequency, k, of 2. A positive vortex enters the data at y/c = 0, level with the nose of

the airfoil, and is pulled above the blade by the lower pressures; however, it does not collide with

the leading edge. Fifty samples are obtained over one pitching period and ensemble averaged 20

times at each spacial location. Each sample is referred to as a time level between 1 and 50, where

time level | is the first sample taken in the period and time level 50 is the last. The TSI laser beams

are tilted down 4° to get data closer to the airfoil; this allows data to be taken upstream of and

4. Results and Discussion 21

above the target airfoil, but causes excessive blockage of the laser beams in the region below the

airfoil. For this reason, almost no data are taken below the leading edge of the target airfoil.

The velocity data are acquired in two blocks (Figure 4.1.2). Between x/c = -0.3 and x/c =

0.05, vertical columns of data are taken between y/c = -0.1375 and y/c = 0.3625, for 41 rows and

15 columns. Between x/c = 0.075 and x/c = 0.3, vertical columns of data are taken between y/c

= Q and y/c = 0.3625, for 29 rows and 10 columns. The latter data block has fewer rows, having

saved time during acquisition by ignoring the blocked region inside and below the target airfoil.

The grid spacings, Ax/c and Ay/c, are 0.025 and 0.0125, respectively. Better resolution in the

normal direction allows better detail of the boundary layer and airfoil-vortex interaction region.

A vortex affects the behavior of flow around an airfoil by changing the apparent direction of the

freestream velocity. The vortex induces higher velocities and changing angles of attack, a, as it

approaches. For a positive vortex, a positive increase in a occurs. Previous research has proved

the formation of a vortex induced separation bubble as well as rapidly fluctuating velocity and

pressure fields. The following results confirm previous work, but also yield insight into the funda-

mentals of the blade-vortex interaction problem.

4.2 Velocity and Vorticity

The velocity vectors and vorticity contours are shown in Figures 4.2.1 through 4.2.50. The

vortex-induced velocity and apparent angle of attack are already significant in the first time level

of the pitching period. The dimensionless velocity vectors are not parallel to the freestream velocity

anywhere upstream of the airfoil. The velocity profile of the x/c = -0.3 column shows a positive

v-component of velocity (perpendicular to U,,), due to the positive rotation of the vortex, every-

where between y/c = -0.1375 and y/c = 0.3625. The velocity increases below and decreases


slightly above y/c = 0, indicating the center of the approaching vortex. Above the airfoil, the flow

is relatively undisturbed and the boundary layer is thin. The airfoil shields this region from the

unsteady effects of the vortex, while imposing its own flow characteristics. This region remains

steady until the vortex passes above the airfoil and becomes unsteady only in the vicinity of the

vortex.

With the entrance of the vortex at time level 5, Figure 4.2.5, the boundary layer has grown in

size and negative vorticity strength, seen as an enlarging region of vorticity less than -20. This in-

dicates an increase in the induced velocity and induced angle of attack on the airfoil. A small ac-

celerated flow region above the airfoil surface, between x/c = 0 and x/c = 0.15, forms, but the flow

above the blade remains steady, dominated by the freestream velocity. The velocity profiles shorten

and lengthen above and below the vortex position, respectively. Upstream of the vortex, the the

v-components of the velocities decrease reducing the angles of the velocity vectors relative to the

freestream direction, thereby, revealing the very localized influence of the vortex and indicating

freestream dominance in the wake of the passing vortex.

By time level 10, Figure 4.2.10, the vortex has moved to x/c = -0.175 and y/c = 0.025. The

velocities downstream of the vortex are increasing and pointing upward, while upstream the veloc-

ities above the vortex are dominated by the freestream, and those below turn downward, compli-

mented by the vortex rotation. The shortening and lengthening moves along with the vortex while

the profiles at x/c = -0.3 are quickly steadying under the freestream’s influence. A separation

bubble is now fully formed, due to the increasing vortex-induced angle of attack. The vortex ac-

celerates quickly between time levels 11 and 16 (Figures 4.2.11-4.2.16 and 4.2.51), pulled into and

distorted by the low-pressure region above the airfoil. Once there, however, the positive rotation

feeds the separation bubble causing it to grow in strength. Moreover, the low pressure preceding

the vortex pulls the separation bubble above the airfoil surface.


At time level 17, Figure 4.2.17, the separation bubble has grown to cover 20% of the chord.

A high-velocity region exists between the vortex and the separation bubble. Since the two regions

rotate in opposite directions, the intersection creates a fluid jet between the vortex structures. The

fastest velocity and, therefore, the largest pressure gradient is between the two vortex structures.

Due to the uneven pressure distribution, the vortex stretches lengthwise in passing above the airfoil.

The core of the vortex rotates eccentrically within the structure, causing it to move from back to

front by time level 20. The secondary vortex, now attached only to the airfoil near its point of

formation, is pulled by the low pressure preceding the primary vortex and moves downstream.

In the last 40% of the period, tume levels 30-50, little occurs on the first three-tenths of the

chord. The vortex pulls the separation region downstream, leaving the boundary layer to stabilize

upstream. The negative vortex follows the positive vortex but is too far away to influence the

suction surface of the airfoil. The path the vortex follows in one complete period is plotted in

Figure 4.2.51.

4.3 Pressure

Pressure gradient contours are shown in Figures 4.3.1 to 4.3.50, where two concentric circles

are drawn based on the vorticity data to mark the average location of the vortex core. Before the

vortex enters the data field, there is a strong localized pressure gradient on the nose of the airfoil.

The pressure decreases towards the upper, leading-edge surface causing ze to be negative and 2

to be positive. This low pressure region is also seen in the pressure-coefficient plots, Figures 4.3.51

to 4.3.75, where C, becomes more negative near the leading-edge of the airfoil. The lowest pressure

is located at x/c = 0.05, the center of the positive 2 region, with C, = —0.9. A portion of this

suction can be attributed to the vortex-induced velocity and angle of attack; however, this early in


the pitching downstroke, the freestream-airfoil interaction characterizes the flow in this region and,

therefore, dictates the local pressures.

By the time the vortex enters at time levels 6 and 7, Figures 4.3.6-4.3.7 and 4.3.53-4.3.54, a

positive a region forms at x/c = 0.125, while the negative region grows in strength and size. This

indicates a rise in pressure just past x/c = 0.1 while the induced angle of attack increases, decreasing

the pressure near the nose. The positive 2 region grows fuller by this time, confirming the de-

creasing pressure above the airfoil. A large decrease in 2 just below the nose is due to the growing

pressure below the airfoil; the stagnation point is moving away from the leading edge along the

bottom of the airfoil. The minimum C, decreases by 50% to -1.3 at x/c = 0.05, where the lowest

pressure and the center of the positive 2 region are located.

As the vortex moves closer, the pressure gradients increase in magnitude and influence. When

the vortex comes within 0.15c of the leading edge at time level 10, Figures 4.3.10 and 4.3.55, a sharp

discontinuity from negative to positive develops in <P above the airfoil at x/c = 0.1; this is the Ox

center of the separation bubble in the vorticity contours, the center of the positive a region, and

the position of minimum pressure. Also, the <P positive region starts to move above the airfoil oy

surface at this time; this indicates the largest pressure gradients are no longer inside the separation

bubble but are in the fluid jet between the separation bubble and the vortex. The minimum C,

continues to decrease with increasing induced angle of attack and moves from x/c = 0.05 to x/c

= 0.1.

op 0 ; At time 15, the vortex begins to disrupt the shapes of the ax and a contours, which are now

. o . ; . . affecting 20% of the airfoil. The negative a region has pulled in very tightly to the leading edge,

indicating a strong increase in pressure. The pressure coefficients continue to decrease to a mini-

mum at time level 17, where the secondary vortex begins to move both away from the surface and

downstream; then, the pressure coefficients begin to increase as the minimum moves with the sec-


ondary vortex further toward the trailing edge. The decreased local pressure in front of the vortex

pulls the separation bubble free of the airfoil and downstream.

As the vortex passes over the blade, time levels 17-24, Figures 4.3.17-4.3.24, the pressure gradi-

ents are pulled along with it, leaving small regions of irregular flow. This parallels the breakdown

of the secondary vortex in Figures 4.2.17-4.2.24. The velocity ahead of the vortex is still consider-

ably faster than the freestream which causes low pressure; behind the vortex, the flow stabilizes

toward freestream conditions. This effect is also seen in Figures 4.3.59-4.3.62, where the pressure

over the front of the blade, in the wake of the exiting vortex, increases rapidly. Pressure coefficients

vary little once the vortex has passed.

The positive vortex influences the first 20% of the airfoil’s suction surface even when it is 0.3c

away from the leading edge. However, once above the airfoil, it has little effect behind its location

and strong effects ahead. This is due to the positive rotation; fast moving fluid is forced down into

the boundary layer and then up and ahead of the vortex. Anything behind the vortex is imme-

diately settled down by the freestream velocity. Once above the leading edge, the vortex begins to

strengthen the secondary vortex by feeding fluid between the nose and x/c = 0.05. Once the vortex

is fully above the separation bubble, the bubble is at its maximum size and strength. This is indi-

cated by the enlarged regions of e and 2 and the minimum pressure coefficient. Once the

bubble starts to move, it can no longer be fed from the positive vortex; the pressures start to in-

crease. In Figures 4.3.76-4.3.78, pressure coefficients at y/c = 0.0625 are shown together for vari-

ous time samples. The minimum C, stays between x/c = 0.05 and x/c = 0.1 until the secondary

vortex moves downstream, where it follows the bubble. Also, the minimum decreases until this

point and then begins to increase.

The last 40% of the period, time levels 30-50, shows almost no variation in 2. 2 and C,

but is included for completeness. The negative vortex, released upstream by the pitching airfoil, is

too far away to have a significant affect on the top of the airfoil.


4.4 Conclusions

In the non-collision case of blade-vortex interaction, three physical phenomena are identified

and studied (Figures 4.4.1 and 4.4.2). First, the free vortex influences its surroundings by inducing

a decrease and an increase in the local velocity above and below its center, respectively, as well as

by adding a vertical velocity component to the flow. This is seen in the first 20% of the period,

time levels 1-10. The induced velocity and induced angle of attack on the airfoil continues to in-

crease until a separation bubble forms on the suction side of the airfoil. Second, the vortex is pulled

into the low-pressure region above the airfoil and begins to interact with the leading edge and the

upstream end of the separation bubble. This region is dominated by the localized influence of the

vortex. Once above the leading edge of the airfoil, the vortex feeds the boundary layer with fast

moving fluid, decreasing the already low pressures and increasing the size of the separation bubble.

The strong pressure variations pull the vortex unevenly, distorting its shape. Above y/c = 0.2, in

the freestream/airfoil dominated flow, the vortex has little influence due to its stretched form. The

radius of influence depends on the reduced frequency of the vortex-freestream formation. In the last

phenomenon, the vortex continuously feeds fast moving fluid into the wall region, increasing the

size and strength of the secondary vortex, until, finally, the vortex has pushed enough energy into

the wall region that the separation bubble lifts free of the airfoil and starts moving downstream.

The primary and secondary vorticies now interact as independent structures. The flow between

them accelerates, moving the minimum pressure gradient to this point; however, once the sepa-

ration bubble is free of the airfoil, it can no longer grow in strength; the vortex is too far past the

bubble/airfoil attachment point to add fluid, and the minimum pressure starts to increase as the

vortex moves toward the trailing edge. The low pressure region pulls the primary and secondary

vorticies downstream while the freestream velocity re-establishes the steady-flow condition up-

stream.


The results of the near-collision or distortion-zone solution yields important conclusions that

can be applied to both the pure collision and the deflection regimes in order to understand the basic

fundamentals of Blade-Vortex Interactions. A vortex in the deflection zone follows the

freestream/airfoil streamlines. Since the vortex never enters the low pressure region above the blade

while in the deflection zone, it is not distorted. However, a vortex in the collsion zone is pulled

by the low pressure above the airfoil but strikes the leading edge. This leaves two smaller regions

of vorticity moving over the airfoil. In the positive vortex case, the top vortex portion will move

counter to the boundary layer, decreasing the pressure, while the lower vortex portion will rotate

in the same direction as the boundary layer, acting against the freestream, and will increase the

pressure. The net effect is a sharp, temporary increase in lift and vibrational loading on the airfoil.


Bibliography

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Appendix A. Vorticity Code

PROGRAM TO CALCULATE VORTICITY FROM VELOCITY FILES USING FINITE DIFFERENCE APPROXIMATIONS FOR DERIVATIVES

MATTHEW M. PESCE MASTER OF SCIENCE CANDIDATE ENGINEERING SCIENCE AND MECHANICS VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY

September 1990

AAaIAaagQq_qanriaaaararanriaqraanrnaagriadaqandaqaananr~aananana

This code reads in u and v velocities and calculates two-dimensional vorticity. The logic statements allow only good data points to be used in the calculations.

Description of Variables:

DUDY 2nd order derivative of u with respect to y DVDX 2nd order derivative of v with respect to x U Velocity in primary flow direction V Velocity in VFILL Input parameter to supply vorticity at bad data

locations (usually zero)

VORT Vorticity WEIGHT Weighting parameter for bad data locations

DX Discretized differential x

Appendix A. Vorticity Code 32

MQAIAAAIMOAAAAMAANAAAQAAANANRANAQEAIAQaANAAAIAIAIAIAIAAaARIAaQQa

2

DX2 DY DY2 FLAG I ITIME J K Kl K2 KDELTA NI NJ NT NX1 NX2 NXMAX NY1 NY2 NYMAX NTMMAX XD YD UFS UFS1 UFS2

ISAVE ISMPL PERIOD REYNOL RFREQ TBD

DX multiplied by 2 Discretized differential y DY multiplied by 2 Flag to make sure all vorticities are calculated Integer for looping in x direction

Looping variable for reading data Integer for looping in y direction Integer for looping in time direction

First time level to begin evaluating vorticity

Final time level to stop evaluating vorticity Integer step between K1 and K2

Number of data columns in the x direction Number of data rows in the y direction

Number of time levels Columns of data in the first data file

Columns of data in the second data file Total possible data columns in array space

Rows of data in the first data file Rows of data in the second data file Total possible data rows in array space Total possible time levels in array space Array holding x locations Array holding y locations Average of freestream velocities Freestream velocity from first data file Freestream velocity from second data file

Number of time samples saved per period Number of samples taken per period Pitching period (in seconds) Reynolds number Reduced frequency Time between data

PARAMETER (NXMAX=50 , NYMAX=50 , NIMAX=102)

DIMENSION U(NXMAX, NYMAX, NTMAX) , V(NXMAX, NYMAX, NIMAX ) DIMENSION VORT(NXMAX, NYMAX, NTMAX) , WEIGHT(NXMAX, NYMAX) DIMENSION XD(NXMAX) , YD(NYMAX)

DO 60 I = 1,NXMAX DO 60 J = 1,NYMAX WEIGHT(I,J) = 1000. DO 60 K = 1,NTMAX U(I,J,K) = 0.

Appendix A. Vorticity Code

60

10

110

150

V(I,J,K) = 0. CONTINUE WRITE(6,*) ‘INITIALIZATION OK'

FILE 1

READ(1,*) TBD, ISMPL, ISAVE, PERIOD READ(1,*) NX1,NY1,NT READ(1,1000) UFS1 WRITE(6,*) UFS1 DO 10 I = 1,NX1 DO 10 J = 1,NY1 READ(1,1010) XD(I),YD(J) DO 10 ITIME = 1,NT,3 READ(1,1020) (U(I,J,K),V(1I,J,K),K=ITIME, ITIME+2)

CONTINUE

READ(1,*) ((WEIGHT(I,J),J=1,NY1),I=1,NX1) WRITE(6,*) 'FILE 1 READING COMPLETE’

FILE 2

READ(2,*) TBD, ISMPL, ISAVE, PERIOD READ(2,*) NX2,NY2,NT READ(2,1000) UFS2 DO 110 I = NX1+1,NX14NX2 DO 110 J = NY1-NY2+1,NY1 READ(2,1010) XD(I),YD(J) - DO 110 ITIME = 1,NT,3 READ(2,1020) (UCI,J,K),V(1,J,K),K=ITIME, ITIME+2)

CONTINUE READ(2,*) ((WEIGHT(1,J) ,J=NY1-NY2+1,NY1), I=NX1+1, NX14NX2)

DO 150 I = NX1+1,NX1+NX2 DO 150 J = 1,NY1-NY2 WEIGHT(I,J) = 1000. DO 150 K = 1,NT U(I,J,K) = 0. V(I,J,K) = 0

CONTINUE

WRITE(6,*) ‘FILE 2 READING COMPLETE'

FIND FLOW VARIABLES


UFS = (UFS1+UFS2)/2. REYNOL = UFS*0.1524/1.007E-06 RFREQ = 2.

C Cane wen nn nn nn en en en en ee we ee een een ee eens

C CALCULATE VORTICITIES Cn we a ee ee eee eee ee ene

C DX = 0.025 DY = 0.0125 DX2 = 2.*DX DY2 = 2.*DY NI = NX1+NX2 NJ = NY1

C WRITE(6,*) 'ENTER VORTICITY FOR BAD DATA POINTS' READ(5,*) VFILL WRITE(6,*) 'ENTER 1ST AND LAST TIME AND STEP DESIRED' READ(5,*) K1,K2,KDELTA

C FLAG = 0. DO 500 K = K1,K2,KDELTA DO 500 J = 1,NJ DO 500 I = 1,NI IF(WEIGHT(I,J).GT.1.) THEN VORT(1,J,K) = VFILL GO TO 500

END IF IF(I.EQ.1.AND.J.EQ.1) THEN DVDX = (-V(I+2,J,K)+4.*V(I+1,J,K)-3.*V(1,J,K)) /DX2 DUDY = (-U(I,J+2,K)+4.*U(1I, J+1,K)-3.*U(I,J,K)) /DY2 VORT(I,J,K) = DVDX - DUDY FLAG = -1.

END IF IF(J.EQ.1.AND.1I.GT.1) THEN IF(WEIGHT(I,J+2).GT.1..OR.WEIGHT(I,J+1).GT.1.) THEN VORT(I,J,K) = VFILL GO TO 500

END IF IF(WEIGHT(I+1,J).GT.1.) THEN DVDX = (V(I-2,J,K)-4.*V(I-1,J,K)+3.*V(I,J,K))/DX2

ELSE DVDX = (V(I+1,J,K)-V(I-1,5,K))/DX2

END IF DUDY = (-U(I, J+2,K)+4.*U(I, J+1,K)-3.*U(I,J,K))/DY2 VORT(I,J,K) = DVDX-DUDY FLAG = -1.

END IF IF(I.EQ.1.AND.J.GT.1.AND.J.LT.NJ) THEN DVDX = (-V(I+2,J,K)+4.*V(I4+1, J,K)-3.*V(1I,J,K)) /DX2


DUDY = ( U(I,J+1,K)-U(I,J-1,K))/DY2 VORT(I,J,K) = DVDX-DUDY FLAG = -1.

END IF IF(I.GT.1.AND.I.LT.NI.AND.J.GT.1.AND.J.LT.NJ) THEN IF(WEIGHT(I+1,J).GT.1.) THEN DVDX = (V(I-2,J,K)-4.*V(I-1,J,K)4+V(1,J (K)) /DX2

ELSE DVDX = (V(I+1,J,K)-V(1I-1,J,K))/DX2

END IF DUDY = (U(I,J+1,K)-UCI,J-1,K))/DY2 IF(WEIGHT(I,J+1).GT.1.) THEN DUDY = (UCI, J-2,K)-4.*U(I, J-1,K)+3.*U(1I,J,K))/DY2

END IF IF(WEIGHT(I,J-1).GT.1.) THEN DUDY = (-U(I,J+2,K)+4.*U(I, J+1,K)-3.*U(I,J,K))/DY2

END IF VORT(I,J,K) = DVDX - DUDY FLAG = -1.

END IF IF(I.EQ.1.AND.J.EQ.NJ) THEN DVDX = (-V(1+2,J,K)+4.*V(I+1,J,K)-3.*V(1,J,K))/DX2 DUDY = ( U(I,J-2,K)-4.*U(I,J-1,K)+3.*U(1,J,K))/DY2 VORT(I,J,K) = DVDX - DUDY FLAG = -1.

END IF IF(I.GT.1.AND.I.LT.NI.AND.J.EQ.NJ) THEN IF(WEIGHT(I,J-2).GT.1..OR.WEIGHT(I,J-1).GT.1.) THEN VORT(I,J,K) = VFILL GO TO 500

END IF DUDY = (U(I,J-2,K)-4.*U(1, J-1,K)+3.*U(I,J,K))/DY2 DVDX = (V(I+1,J,K)-V(I-1,J,K))/DX2 VORT(I,J,K) = DVDX - DUDY FLAG = -1.

END IF IF(I.EQ.NI.AND.J.EQ.NJ) THEN IF(WEIGHT(I,J-2).GT.1..OR.WEIGHT(I,J-1).GT.1.) THEN VORT(I,J,K) = VFILL GO TO 500

END IF sK)~4 AVCT 1) SaK)43-89(T4J4K)) /DK2

K)-4.*U(I, J-1,K)+3.*U(1,J,K))/DY2 X - DUDY

qa<

+ a

Vine

iw

Sak

DUDY VORT( FLAG

END IF IF(I.EQ.NI.AND.J.GT.1.AND.J.LT.NJ) THEN IF(WEIGHT(I,J+1).GT.1..OR.WEIGHT(I-1,J).GT.1..OR.

WEIGHT(I-2,J).GT.1.) THEN VORT(I,J,K) = VFILL

il a

teow

GAR

tl tN

be


aqaantna

500

700

800 801 802 803

1000 1010

GO TO 500 END IF IF(WEIGHT(I,J-1).GT.1.) THEN DUDY = (-U(I,J+2,K)+4.*U(1I, J+1,K)-3.*U(1I,J,K))/DY2

ELSE DUDY = ( U(I,J+1,K)-U(I,J-1,K))/DY2

END IF DVDX = (V(I-2,J,K)-4.*V(1-1,J,K)+3.*VC1,J,K))/DX2 VORT(I,J,K) = DVDX - DUDY FLAG = -1.

END IF

NOTE: THE CASE I=NI, J=1 IS COVERED IN ANOTHER CASE SINCE IT IS KNOWN THAT J=1 TO J=12 ARE BAD DATA POINTS

IF(FLAG.EQ.-1.) THEN FLAG = 0.

ELSE VORT(I,J,K) = VFILL

END IF CONTINUE

WRITE(6,*) 'VORTICITY CALCULATIONS COMPLETE'

WRITE(6,*) ‘WRITING VORTICITY FILE... .'

WRITE(10,802) UFS,REYNOL, RFREQ DO 700 K = K1,K2,KDELTA WRITE(10,801) NI,NJ,K WRITE(10,*) 'A',K DO 700 I = 1,NI WRITE(10,802) (XD(I),YD(J), VORT(I,J,K),J=1,NJ) CONTINUE

WRITE(6,*) ‘WRITING COMPLETE’

FORMAT(3(5X,E14.7)) FORMAT( 3(5X,13)) FORMAT(3(5X,E14.7)) FORMAT(6E13.5) FORMAT( /E15.7) FORMAT(2E15.7)


1020 FORMAT(3(3E13.5)) C

STOP END


Appendix B. Pressure Code

PROGRAM TO CALCULATE PRESSURES USING EULER AND NAVIER-STOKES FINITE DIFFERENCE DERIVATIVES AND USING TRAPEZOIDAL RULE NUMERICAL INTEGRATION (2ND ORDER)


September 1990

OOAaAaMAIagaQq_riaaiaaaaraaranraaqanaqananaananndnaanananna

The code reads in u and v velocities and calculates differential pressures by solving the Euler or Navier-Stokes equations. The logic statements allow only calculations using good data points. The differential pressure is then integrated to get pressure everywhere in the field.


DUDX lst derivative of u with respect to x

D2UDX2 2nd derivative of u with respect to x DUDY Ist derivative of u with respect to y D2UDY2 2nd derivative of u with respect to y DVDX lst derivative of v with respect to x D2VDX2 2nd derivative of v with respect to x

Appendix B. Pressure Code 39

MAAMAANAIANAAIANANIAAAANAANAIAANARAAAAIAANAQIANANAQAAAIANANAQINAANANRQaAaANANaANAAAA DVDY

D2VDY2 DPDX DPDY P PW U UIJ UIOJ UI1J UI2J UIJO UIJ1 UIJ2

VIJ VIOJ VI1J VI2J VIJO VIJ1 VIJ2 WT

DX DX2 DXDX DY DY2 DYDY DT DT2 I ITIME J K Kl K2 KDELTA NAVIER NI NJ

NX1 NX2

NXMAX NY1 NY2 NYMAX NT NTMMAX

Ist derivative of v with respect to y 2nd derivative of v with respect to y Derivative of pressure with respect to x Derivative of pressure with respect to y

Numerically integrated pressure

Pressure on airfoil surface

Velocity in primary flow direction (u) Scalar for U(I,J,K) Scalar for U(I-1,J,K)

Scalar for U(I+1,J,K) Scalar for U(CI+2,J,K) Scalar for U(I,J-1,K) Scalar for U(I,J+1,K) Scalar for U(I,J+2,K) Velocity perpendicular to primary flow (v)

Scalar for V(I,J,K) Scalar for V(I-1,J,K) Scalar for V(I+1,J,K) Scalar for V(I+2,J,K) Scalar for V(I,J-1,K) Scalar for V(I,J+1,K) Scalar for V(I,J+2,K)

Weighting parameter for bad data locations

Discretized differential x DX multiplied by 2 DX squared Discretized differential y DY multiplied by 2 DY squared Discretized differential time DT multiplied by 2 Integer for looping in x direction

Looping variable for reading data Integer for looping in y direction Integer for looping in time direction First time level to begin evaluating pressures Final time level to stop evaluating pressures Integer step between K1 and K2 Parameter whether do Euler or Laminar N-S Number of data columns in the x direction Number of data rows in the y direction Columns of data in the first data file Columns of data in the second data file Total possible data columns in array space Rows of data in the first data file Rows of data in the second data file Total possible data rows in array space

Number of time levels Total possible time levels in array space


IQaAAaAAIAAIAQAAAAARIAdARWaAN

Qa

PSUM XD YD UFS UFS1 UFS2

CHORD ISAVE ISMPL PERIOD RE RFREQ TBD TIMCON

Variable for summing pressures in a loop Array holding x locations Array holding y locations Average of freestream velocities Freestream velocity from first data file

Freestream velocity from second data file

Chord length of the target airfoil Number of time samples saved per period Number of samples taken per period Pitching period (in seconds) Reynolds number Reduced frequency Time between data Non-dimensional Strouhal number

PARAMETER(NXMAX=30,NYMAX= 41,NTMAX=102)

DIMENSION U(NXMAX, NYMAX, NTMAX) , V(NXMAX, NYMAX, NIMAX) DIMENSION WI(NXMAX, NYMAX) , XD(NXMAX) , YD(NYMAX) DIMENSION DPDX(NXMAX, NYMAX, NTMAX) DIMENSION DPDY(NXMAX, NYMAX, NTMAX) DIMENSION P(NXMAX, NYMAX,NIMAX) , PWCNXMAX, NIMAX)

60

DO 60 I = 1,NXMAX DO 60 J = 1,NYMAX WT(I,J) = 1000. DO 60 K = 1,NTMAX U(I,J,K) = 0. V(I,J,K) = 0. DPDX(1I,J,K) = 0. DPDY(I,J,K) = 0

CONTINUE WRITE(6,*) ‘INITIALIZATION OK’

FILE 1

READ(1,*) TBD, ISMPL, ISAVE, PERIOD READ(1,*) NX1,NY1,NT READ(1,1000) UFS1


10

110

150

WRITE(6,*) UFS1 DO 10 I = 1,NX1 DO 10 J = 1,NY1 READ(1,1010) XD(I),YD(J) DO 10 ITIME = 1,NT,3 READ(1,1020) (U(I,J,K),V(1I,J,K),K=ITIME, ITIME+2)

CONTINUE

READ(1,*) ((WT(1,J),J=1,NY1),I=1,NX1) WRITE(6,*) ‘FILE 1 READING COMPLETE'

FILE 2

READ(2,*) TBD, ISMPL, ISAVE, PERIOD READ(2,*) NX2,NY2,NT READ(2,1000) UFS2 DO 110 I = NX1+1,NX1+NX2 DO 110 J = NY1-NY2+1,NY1 READ(2,1010) XD(I),YD(J) DO 110 ITIME = 1,NT,3 READ(2,1020) (U(I,J,K),V(I,J,K) ,K=ITIME, ITIME+2)

CONTINUE READ(2,*) ((WI(1,J), J=NY1-NY2+1,NY1), I=NX1+1,NX1+NX2)

DO 150 I = NX1+1,NX14NX2 DO 150 J = 1,NY1-NY2 WT(I,J) = 1000. DO 150 K = 1,NT U(I,J,K) = 0. V(I,J,K) = 0.

CONTINUE WRITE(6,*) ‘FILE 2 READING COMPLETE'

FIND FLOW VARIABLES

UFS = (UFS1+UFS2)/2. RE = UFS*0.1524/1.007E-06 RFREQ = 2. CHORD = 0.1524 TIMCON = CHORD/(PERIOD*UFS)

DX = 0.025 DY = 0.0125 DT = FLOATC(ISAVE)*TBD/PERIOD NI = NX1+NX2 NJ = NY1

WRITE(6,*) 'ENTER 1ST AND LAST TIME AND STEP DESIRED’ READ(5,*) K1,K2,KDELTA


C CALCULATE DIFFERENTIAL C PRESSURES Cee we en en en ee ee eee eee eee ee eene

C C

DX2 = 2.*DX DY2 = 2.*DY DT2 = 2.¥*DT DXDX = DX*DX DYDY = DY*DY

C NAVIER = 1

C C GET DP/DX AND DP/DY ALONG J=NJ LINE C

J=NJ DO 400 K = K1,K2,KDELTA DO 400 I = 1,NI

C UIJ = UCI,J,K) VIJ = V(I,J,K) IF(I.GT.1) THEN UIOJ = U(I-1,J,K) VIOJ = V(I-1,J,K)

END IF IF(I.LT.NI) THEN UI1J = UCI+1,J,X) VI1J = V(I+1,J,K)

IF(I.LT.NI-1) THEN UI2J = U(I+2,J,K) VI2J = V(I+2,J,K)

END IF c

IF(J.GT.1) THEN UIJO = U(I,J-1,K) VIJO = V(I,J-1,K)

END IF IF(J.LT.NJ) THEN UIJ1 = UCI, J+1,K) VIJ1 = V(I,J+1,K)

END IF IF(J.LT.NJ-1) THEN UIJ2 = U(I,J+2,K) VIJ2 = V(I,J+2,K)

END IF C

DUDY = ( U(I,J-2,K)-4.*UIJO+3.*UIJ) /DY2 DVDY = ( V(I,J-2,K)-4.*VIJO+3.*VIJ) /DY2


D2UDY2 = (UIJ-2.*UIJO+U(I,J-2,K))/DYDY D2VDY2 = (VIJ-2.*VIJO+V(I, J-2,K))/DYDY

C IF(K.EQ.1) THEN DUDT = (-U(I,J,K+2)+4.*U(I,J,K+1)-3.*UIJ)/DT2 DVDT = (-V(I,J,K+2)4+4.*V(I,J,K+1)-3.*VIJ)/DT2

ELSE DUDT = (U(I,J,K+1)-U(I,J,K-1))/DT2 DVDT = (V(I,J,K+t1)-V(1I,J,K-1))/DT2

END IF C

IF(I.EQ.1) THEN DUDX = (-UI2J+4.*UI1J-3.*UIJ) /DX2 DVDX = (-VI2J+4.*VI1J-3.*VIJ) /DX2 D2UDX2 = (UIJ-2.*UI1J+UI2J) /DXDX D2VDX2 = (VIJ-2.*VI1J+VI2J) /DXDX

ELSE IF(I.GT.1.AND.I.LT.NI) THEN DUDX = (UI1J-UIOJ) /DX2 DVDX = (VI1J-VIOJ) /DX2 D2UDK2 = (UI1J-2.*UIJ+UI0J) /DXDX D2VDX2 = (VI1J-2.*VIJ+VIOJ) /DXDX

END IF IF(I.EQ.NI) THEN DUDX = ( U(I-2,J,K)-4.*UI0J+3.*UIJ) /DX2 DVDX = ( V(I-2,J,K)-4.*VIOJ43.*VIJ) /DX2 D2UDX2 = (UIJ-2.*UIOJ+U(I-2,5,K)) /DXDX D2VDX2 = (VIJ-2.*VIOJ+V(1I-2,3,K))/DXDX

END IF END IF

C DPDX(I,J,K) = -(TIMCON*DUDT + UIJ*DUDX + VIJ*DUDY)

, +FLOAT(NAVIER)*(D2UDX2 + D2UDY2)/RE DPDY(1,J,K) = -(TIMCON*DVDT + UIJ*DVDX + VIJ*DVDY)

, +FLOAT(NAVIER)*(D2VDX2 + D2VDY2)/RE 400 CONTINUE

c DO 500 K = K1,K2,KDELTA DO 500 I = 1,NI DO 500 J = NJ-1,1,-1

C UIJ = U(I,J,K) VIJ = V(I,J,K) IF(I.GT.1) THEN UIOS = U(I-1,J,K) VIOJ = V(I-1,J,K)

END IF IF(I.LT.NI) THEN UI1J = U(I+1,J,K) VI1J = V(I+1,J,K)


END IF IF(I.LT. UI2J VI2J

END IF

IF(J.GT. UIJO = VIJO =

END IF IF(J.LT. UIJ1 VIJ1

END IF IF(J.LT. U1J2 VIJ2

END IF

NI-1) THEN U(I+2,J,K) V(I+2,J,K)

1) THEN U(I,J-1,K) V(I,J-1,K)

NJ) THEN UCI, J+1,K) V(I, J+1,K)

NJ-1) THEN UCI, J+2,K) V(I,J+2,K)

WTIJ = WTCI,J)

IF(K.EQ. DUDT DVDT

ELSE DUDT DVDT

END IF

IF(1.EQ. DUDX = DVDX = D2UDX2 D2VDX2

ELSE

1) THEN (-U(I, J, K+2)+4.*U(1,J,K+1)-3.*UIJ) /DT2 (-V(I,J,K+2)4+4.*V(1,J,K+1)-3.*VIJ) /DT2

(UCI, J,K+1)-U(1I,J,K-1))/DT2 (V(I,J,K+1)-V(1I,J,K-1))/DT2

1) THEN (-UI2J+4.*UI1J-3.*UIJ) /DX2 (-VI2J+4.*VI1J-3.*VIJ) /DX2 = (UIJ-2.*UI1J+U1I2J) /DXDX

(VIJ-2.*VI1J+VI2J) /DXDX

IF(I.GT.1.AND.I.LT.NI) THEN DUDX = DVDX = D2UDX2

D2VDX2 IF (WTC

(UI1J-UIOJ) /DX2

(VI1J-VIOJ) /DX2 = (UI1J-2.*UIJ+UI0J) /DXDX

= (VI1J-2.*VIJ+VI0J) /DXDX I+1,J).GT.1..AND.WTIJ.EQ.1..AND.I.GT.2) THEN

DUDX = ( U(I-2,J,K)-4.*UI0J+3.*UIJ) /DX2 DVDX = D2UDX D2VDX

END IF IF (WTC DUDX DVDX D2UDX D2VDX

( V(I-2,3,K)-4.*VI0J+3. *VIJ) /DX2 2 = (UIJ-2.*UI0J+U(I-2,J,K)) /DXDX 2 = (VIJ-2.*VIOJ+V(1-2,J,K)) /DXDX

I-1,J).GT.1..AND.WTIJ.EQ.1..AND.I.LT.NI-1) THEN (-UI2I+4.*UI1J-3.*UIS) /DX2 (-VI2J+4.*VI1J-3.*VIJ) /DX2

2 = (UIJ-2.*UI1J+UI2J) /DXDX 2 (VIJ-2.*VI1J+VI2J) /DXDX


END IF END IF IF(I.EQ.NI) THEN DUDX = ( U(I-2,J,K)-4.*UI0J+3.*UIJ) /DX2 DVDX = ( V(I-2,J,K)-4.*VIOJ+3.*VIJ) /DX2 D2UDX2 = (UIJ-2.*UI0J+U(I-2,J,K))/DXDX D2VDX2 = (VIJ-2.*VIOJ+V(1-2,J,K))/DXDX

END IF END IF

C IF(J.EQ.1) THEN DUDY = (-UIJ2+4.*UIJ1-3.*UIJ) /DY2 DVDY = (-VIJ2+4.*VIJ1-3.*VIJ) /DY2 D2UDY2 = (UIJ-2.*UIJ14UIJ2) /DYDY D2VDY2 = (VIJ-2.*VIJ14VIJ2) /DYDY

ELSE IF(J.GT.1.AND.J.LT.NI) THEN DUDY = (UIJ1-UIJO0)/DY2 DVDY = (VIJ1-VIJO) /DY2 D2UDY2 = (UIJ1-2.*UIJ+U1JO) /DYDY D2VDY2 = (VIJ1-2.*VIJ+VIJO) /DYDY IF(WT(I,J+1).GT.1..AND.WTIJ.EQ.1..AND.J.GT.2) THEN DUDY = ( UCI, J-2,K)-4.*UIJO+3.*UIJ) /DY2 DVDY = ( V(I,J-2,K)-4.*VIJO+3.*VIJ) /DY2 D2UDY2 = (UIJ-2.*UIJO+U(1,J-2,K))/DYDY D2VDY2 = (VIJ-2.*VIJO+V(1I, J-2,K))/DYDY

END IF IF(WT(1,J-1).GT.1..AND.WYIJ.EQ.1..AND.J.LT.NJ-1) THEN DUDY = (-UIJ2+4.*UIJ1-3.*UIJ) /DY2 DVDY = (-VIJ2+4.*VIJ1-3.*VIJ) /DY2 D2UDY2 = (UIJ-2.*UIJ1+UIJ2) /DYDY D2VDY2 = (VIJ-2.*VIJ14+V1IJ2) /DYDY

END IF END IF IF(J.EQ.NJ) THEN DUDY = ( U(I,J-2,K)-4.*UIJ0+3.*UIJ) /DY2 DVDY = ( V(I,J-2,K)-4.*VIJO+3.*VIJ) /DY2 D2UDY2 = (UIJ-2.*UIJO+U(I,J-2,K))/DYDY D2VDY2 = (VIJ-2.*VIJO+V(I,J-2,K))/DYDY

END IF END IF

c DPDX(I,J,K) = -(TIMCON*DUDT + UIJ*DUDX + VIJ*DUDY)

+FLOAT(NAVIER)*(D2UDX2+D2UDY2 ) /RE DPDY(I,J,K) = -(TIMCON*DVDT + UIJ*DVDX + VIJ*DVDY)

, +FLOAT(NAVIER)*(D2VDX2+D2VDY2) /RE 500 CONTINUE

C C em ee eee ee eo me ww ew ee em em oe een eee eee ee en

C CALCULATE PRESSURES


qQaanagqananan

qQqan

TRAPEZOIDAL RULE OF INTEGRATION

DO 600 K = P(1,NJ,K) PSUM = 0. DO 600 I = 2,NI PSUM = PSUM + DX/2.*(DPDXC(I,NJ,K)+DPDXC1I-1,NJ,K)) P(I,NJ,K) = PSUM

600 CONTINUE

K1 3M KDELTA

DO 650 K = K1,K2,KDELTA DO 650 I = 1,NI PSUM = P(I,NJ,K) DO 650 J = NJ-1,1,-1 IF(WT(I,J).GT.1.) THEN P(I,J,K) = P(I,J+1,K)

ELSE PSUM = PSUM - DY/2.*(DPDY(1,J,K)+DPDY(1I,J+1,K)) P(I,J,K) = PSUM

END IF 650 CONTINUE

PRINT OUT DATA AT Y/C = 0.3625, 0.2625, 0.1625, AND 0.0625

DO 700 K = K1,K2,KDELTA WRITE(11,804) K WRITE(11,805) NI DO 675 I = 1,NI

675 WRITE(11,803) XD(I),P(I,NJ,K),PC(I,NJ-8,K),P(I,NJ-16,K), P(I,NJ-24,K)

700 CONTINUE

TIME TRACE ON BLADE

DO 740 I = 17,25,2 WRITE( 12,1000) XD(TI)

NUMPTS = (K2-K1)/KDELTA + 1 WRITE( 12,805) NUMPTS DO 740 K = Ki,K2,KDELTA

740 WRITE(12,803) FLOAT(K),P(1I,1,K)

Appendix B. Pressure Code a7

C FORMAT STATEMENTS

800 FORMAT(3(5X,E14.7)) 801 FORMAT(3(5X,1I3)) 802 FORMAT(3(5X,E14.7)) 803 FORMAT(6E13.5) 804 FORMAT(/3X,13) 805 FORMAT(3X, 13)

1000 FORMAT(/E15.7) 1010 FORMAT( 2E15.7) 1020 FORMAT(3(3E13.5))

STOP END


Appendix C. Macintosh Animation Code

PROGRAM TO ANIMATE DISCRETE FIELD VALUES AT SEQUENTIAL TIMES USING A MACINTOSH II COMPUTER AND MICROSOFT FORTRAN V2.2


September 1990

AQAA aIAaIAAANAAAAANAIARQANQAAaAAIAAAIAAaAANRANaD

This code reads in position X and Y and two-dimensional vorticity Z for each time level of the data period. The vorticity is then drawn to the screen as blocks of colored pixels representing a range of vorticity. The pixels are then replaced with the pixels representing the next time level.


COLORS Color scheme sent to subroutine PIXS COLOR1 Color scheme for positive vorticity only COLOR2 Color scheme for negative vorticity only COLOR3 Color scheme for both positive and negative

vorticity together

Appendix C. Macintosh Animation Code 49

C W Width of pixel block = C H Height of pixel block = C IMAX Dimension of X direction = C JMAX Dimension of Y direction =

C KMAX Dimension in time =

C NCOLOR Number of colors used in pixel image = C NCLEVL Number of vorticity levels =

C YMAXO Value of Y used to convert coordinate =

C system to that of screen =

C X Array holding x locations = C Y Array holding y locations =

C Z Vorticity =

C cc Levels of vorticity to display = C XE Extended x array, four point average = C YE Extended y array, four point average = C ZE Extended vorticity array, four point average = C XMAX Maximum X value =

C XMIN Minimum X value = C YMAX Maximum Y value =

C YMIN Minimum Y value = C XORIG X position of image origin =

C YORIG Y positiion of image origin =

C F Scaling factor for image = C =

C Other variables are part of MS-Fortran V2.2 Toolbox = C routines or are supplemental variables for looping = C and decision control = C =

C =

INTEGER*4 COLORS(8),COLOR1(8),COLOR2(8),COLOR3(8),W,H COMMON /STUFF1/ IMAX, JMAX, KMAX COMMON /STUFF2/ W,H,NCOLOR,NCLEVL, YMAXO COMMON /STUFF3/ X(30),Y(41),2(30,41, 30),CC(10) COMMON /STUFF4/ XE(30),YE(41),ZE(30,41, 30) COMMON /COORD/ XMAX, XMIN, YMAX, YMIN, XORIG, YORIG,F

C C DEFINE THE COLOR SCEMES FOR THE PARTICLES C

DATA COLOR1/205,341,69,409, 273, 137,33, 30/ DATA COLOR2/409, 273,33, 205, 341,69, 137, 30/ DATA COLOR3/409, 273,33, 30,205, 341, 69, 33/

C (Cn ww ne ee en ee ee ee wen een enn ene

C GET THE DATA FROM FILES Cw ww ww ee ee enn en ew en ene eww ew wn w eee

C CALL INPUT

C CEC

C LOOPING TO DO POSITIVE VORTICITY, NEGATIVE VORTICITY,


Ca ew ne nn ee ee eee ew eee eee en cee eee we eee eee enn ee eee eee

C KOUNT =

300 CONTINUE CALL FOIL

C G POSITIVE VORTICITY LOOP C

IFC(KOUNT.EQ.1) THEN ZMIN = 3. ZMAX = 28. NCLEVL = 4 NCOLOR = 3 DO 233 ICOLS =

233 COLORS(CICOLS) END IF

1,8 = COLOR1(ICOLS)

C NEGATIVE VORTICITY LOOP

IF(KOUNT.EQ.2) THEN ZMIN = -28. ZMAX = -4. NCLEVL = 4 NCOLOR = 3 DO 234 ICOLS = 1,

234 COLORSCICOLS) = 8 COLOR2( ICOLS)

END IF c c COMBINATION LOOP c

IF(KOUNT.EQ.3) THEN ZMIN = -28. ZMAX = 28. NCLEVL = 8 NCOLOR = 7 DO 235 ICOLS = 1,8

235 COLORS(ICOLS) = COLOR3(ICOLS) END IF

C C eee eee ee ee ee @ ® e m ee ® oe on on om oe 08 08 On mo moe ne 08 On) ee me es as ee

C SET CONTOUR LEVELS C et ee ee

C R = ABS(ZMAX-ZMIN) /FLOAT(NCLEVEL-1) RR = ZMIN - R DO 440 II = 1,NCLEVL RR = RR+R

440 CC(II) = c


DO 200 K = 1,KMAX-1 CALL PIXS(COLORS, K)

200 CONTINUE

C KOUNT = KOUNT + 1 IF(KOUNT.GT.3) KOUNT=1

C PAUSE GO TO 300 STOP END

Cc C Cc

SUBROUTINE INPUT Cc

INTEGER*4 W,H CHARACTER*5 JUNK COMMON /STUFF1/ IMAX, JMAX, KMAX COMMON /STUFF2/ W,H,NCOLOR,NCLEVL, YMAXO COMMON /STUFF3/ X(30),Y(41),2(30,41,30),CC(10) COMMON /STUFF4/ XE(30),YE(41),ZE(30,41, 30) COMMON /STUFF5/ ILABLE, IAVERG COMMON /COORD/ XMAX, XMIN, YMAX, YMIN, XORIG, YORIG,F

C OPEN(11,FILE='COLOR. INP' , STATUS="OLD' ) OPEN(12,FILE='VORT1.DATA' , STATUS='OLD' )

Cc C sm ae RS ODES Sen eB EDO BRST BSSSHT HEHE Eee BOE Te BEBE eB eT A eee

C READ IN INPUT PARAMETERS: MAKE SURE THERE IS A BLANK Cc AT THE END OF THE FILE C BRO RASS SSS SSS EDO RETAKE SSSS SSS enor ane noes eos eee

C READ(11,*) JUNK READ(11,*) IMAX, JMAX, KMAX, YMAXO, ZMIN, ZMAX READ(11,*) JUNK READ(11,*) NCOLOR,NCLEVL, XORIG, YORIG,F READ(11,*) JUNK READ(11,*) W,H, ILABLE, IAVERG CLOSE(11)

C Ce wn eww wenn we wwe we ewe a se see ew eae eee ETO EEE Dew eee REE EE EEO eee

Cc READ IN FIRST FILE C ewe eee wee wen eB eB Be EMO Oe TZ OBO eT BBO eee wwe TEE RE eee eee

Cc Cc


IF(KMAX.GT.25) THEN KF1 = 25

ELSE KF1 = KMAX

END IF Cc

READ(12,*) JUNK DO 60 K = 1,KF1 READ(12,*) IMAX, JMAX, KDUM DO 50 I = 1, IMAX DO 50 J = 1,JMAX READ(12,*) X(1),Y(J),2(1,J,K)

50 CONTINUE 60 CONTINUE

CLOSE(12) C

IF(KMAX.LE.25) GO TO 68 Cc

eee

C READ IN SECOND FILE C sae Eee ONE REE Dene eee Bee ene eee EE Oe EB Owe eee ee eee

Cc C

OPEN(13,FILE='VORT2.DATA' , STATUS='OLD' ) READ(13,*) JUNK DO 65 K = 26,KMAX READ(13,*) IMAX, JMAX, KDUM DO 55 I = 1, IMAX DO 55 J = 1,JMAX

READ(13,*) X(I),Y(J),2(1,J5,K) 55 CONTINUE 65 CONTINUE

CLOSE(13) Cc

68 CONTINUE Cc C ae ee mm mm om we oe em ae om ms om ne Oe mse me eee em oe ee eee ee

Cc SHIFT COORDINATES TO MATCH SCREEN Cc (0,0 IN UPPER LEFT CORNER) C see tee we ee eee sna EBB TSS Mes eB BOOKS SB eee TET He EE Hee TOT Oe eee ee

Cc Cc

DO 70 J = 1, JMAX Y(J) = YMAXO - Y(J)

70 CONTINUE Cc

XMIN = X(1) XMAX = X(IMAX) YMIN = Y(JMAX)


200

YMAX = Y(1)

IAVERG = 1 IF(IAVERG.EQ.1) THEN DO 200 J = 1, JMAX YE(J) = 0.5*(Y(J)+¥(J+1)) DO 200 I = 1, IMAX IF(I.EQ.IMAX) THEN XE(I) = X(I)

ELSE XE(I) = 0.5*(X(1I)+X(I+1))

END IF DO 200 K = 1,KMAX IF(I.EQ.IMAX) THEN ZE(1I,J,K) = 0.5*(Z(1,J,K)+Z(1, J+1,K))

ELSE ZE(1,J,K) = 0.25*(Z(1,J3,K)+Z(1+1,J,K)+Z(1, J+1,K)+

Z(I+1, J+1,K)) END IF

CONTINUE END IF

RETURN END

SUBROUTINE FOIL

INCLUDE TOOLBX. PAR

CHARACTER*256 LABLE INTEGER*4 FBYTE, BYTCNT, SRCOPY,W,H, TSIZE INTEGER*2 RECT(4) COMMON /COORD/ XMAX, XMIN, YMAX, YMIN, XORIG, YORIG,F COMMON /STUFF2/ W,H,NCOLOR,NCLEVL, YMAXO

DRAW INNER BOX

RECT(1) = INTC YMIN*F+YORIG) -H-1 RECT(2) = INT(XMIN*F+XORIG) -W-1 RECT(3) = INTC YMAX*F+YORIG)+H+1 RECT(4) = INTCXMAX*F+XORIG)+W+1 CALL TOOLBX(FRAMERECT, RECT)


ao

100

110

DRAW OUTER BOX

RECT(1) = INTC YMIN*F+YORIG) -H-6 RECT(2) = INTC XMIN*F+XORIG) -W-6 RECT(3) = INTC YMAX*F+YORIG)+H+6 RECT(4) = INTC XMAX*F+XORIG)+W+6 CALL TOOLBX( FRAMERECT, RECT)

XZERO YZERO Wo

tt ©

CALL TOOLBX(MOVETO, INT( XZERO*F+XORIG) , INT( YZERO*F+YORIG) ) PALF = 10./180.*ACOS(-1.) PALF = -PALF Al = 0.2969

A2 = -0.126 A3 = -0.3516 A4 = 0.2843 A5 = -0.1015 XAF = 0. DAF = 0.0005

YAF = 15./20.*(A1*SQRT(XAF)+( ( (A5*XAF+A4 )*XAF+A3 ) *XAF+A2 ) *XAF ) XX = COS(PALF)*XAF + SIN(PALF)*YAF YY = -SIN(CPALF)*XAF + COS(PALF)*YAF XAFP = XX + XZERO YAFP = YY + YZERO

CALL TOOLBX( LINETO, INT(XAFP*F+XORIG) , INTC YAFP*F+YORIG) ) XAF = XAF + DAF IF (XX. LT. XMAX+FLOAT(W)/F) GO TO 100

CALL TOOLBX(MOVETO, INT(XZERO*F+XORIG) , INTCYZERO*F+YORIG) ) XAF = 0. YAF = 15./20.*(A1*SQRT(CXAF)+( ( (A5*XAF+A4 ) *XAF+A3 ) *XAF+A2 )*XAF ) XX = COS(PALF)*XAF + SINCPALF)*(-YAF)

YY = -SIN(PALF)*XAF + COS(PALF)*(-YAF) XAFP = XX + XZERO

YAFP = YY + YZERO CALL TOOLBX(LINETO, INT(XAFP*F+XORIG) , INTC YAFP*F+YORIG) )

XAF = XAF + DAF IF (XX.LT.XMAX+FLOAT(W)/F) GO TO 110

meawewe ewe BP ees eB Pe | Se eee ee ee oe oe es oo oo oe ow oe oo oe oe oe ee eo ow we eso ee eo oe oe wm oe oe ee ee ee oe oe ee oe we oe oe we


Cc ILABLE = 0 IFC ILABLE.EQ.0) GO TO 800

Cc LABLE = CHAR(24)//'BLADE VORTEX INTERACTION’ CALL TOOLBX(MOVETO, INT(XMAX*F+XORIG)+W+15, INT(YMIN*F+YORIG) ) CALL TOOLBX(DRAWSTRING, LABLE)

Cc LABLE = CHAR(18)//'VORTICITY CONTOURS' CALL TOOLBX(MOVETO, INT(XMAX*F+XORIG)+W+15 , INT(YMIN*F+YORIG)+12) CALL TOOLBX( DRAWSTRING, LABLE)

Cc LABLE = CHAR(24)//'FREESTREAM U = 0.126 M/S' CALL TOOLBX(MOVETO, INT(XMAX*F+XORIG)+W+15 , INT(YMIN*F+YORIG)+36) CALL TOOLBX( DRAWSTRING, LABLE)

Cc LABLE = CHAR(23)//'REYNOLDS NUMBER = 19950' CALL TOOLBX(MOVETO, INT(XMAX*F+XORIG)+W+15 , INT(YMIN*F+YORIG)+48 ) CALL TOOLBX( DRAWSTRING, LABLE)

C LABLE = CHAR(6)//'K = 2.' CALL TOOLBX(MOVETO, INT(XMAX*F+XORIG)+W+15 , INT(YMIN*F+YORIG)+60) CALL TOOLBX( DRAWSTRING, LABLE)

Cc 800 CONTINUE

Cc RETURN END

C Cc Cc

SUBROUTINE PIXS(COLORS, K) Cc

INCLUDE TOOLBX. PAR Cc

INTEGER*4 COLOR, COLORS(8),W,H,W1,H1 COMMON /STUFF1/ IMAX, JMAX,KMAX COMMON /STUFF2/ W,H,NCOLOR,NCLEVL, YMAXO COMMON /STUFF3/ X(30),Y(41),2(30,41, 30) ,CC(10) COMMON /STUFF4/ XE(30),YE(41),ZE(30,41, 30) COMMON /STUFF5/ ILABLE, IAVERG COMMON /COORD/ XMAX, XMIN, YMAX, YMIN, XORIG, YORIG,F

C CALL TOOLBX(PENSIZE,W,H)

Cc C SSS TBST SONS KSTSB FAA SHS SSSA STHSHSSSSSSHKHZFZ BB OKHae BZ eee nananssanwanawauanaan

Cc DRAW FIRST PICTURE Cn ew ee we ee ew we wee eee ewe ee ew eee eee wee eeene


IF(K.EQ.1) THEN

DO 100 I = 1, IMAX DO 100 J = 1,JMAX ICOLOR = 1 ZZ = 2(1,J,K) DO 90 II = 1,NCLEVL-1 IF(II.EQ.4) GO TO 85 IF(ZZ.GE.CC(II).AND.2ZZ.LT.CC(I1+1)) THEN COLOR = COLORS(ICOLOR) CALL TOOLBX(FORECOLOR, COLOR) CALL TOOLBX(MOVETO, INT(X(I)*F+XORIG) -W/2,

INT(Y(J)*F+YORIG)-H/2) CALL TOOLBX(LINE, 1,0)

END IF IF(IAVERG.EQ.1) THEN ZZE = ZE(1,J,K) IF(ZZE.GE.CC(II).AND.ZZE.LT.CC(II+1)) THEN COLOR = COLORS(ICOLOR) CALL TOOLBX(FORECOLOR, COLOR) CALL TOOLBX(MOVETO, INT(XE(1)*F+XORIG)-W/2,

INT(YE(J)*F+YORIG)-H/2) CALL TOOLBX(LINE, 1,0)

END IF END IF

85 CONTINUE ICOLOR = ICOLOR + 1 IF(ICOLOR.GT.7.OR.ICOLOR.GT.NCOLOR) ICOLOR=1


END IF

DO 200 I = 1, IMAX DO 200 J = 1, JMAX

C ERASE PREVIOUS PIXELS AT LOCATION X,Y

COLOR = 30 CALL TOOLBX(FORECOLOR, COLOR) ZZ = Z(1,J,K) DO 180 II = 1,NCLEVL-1 IFC(CII.EQ.4) GO TO 180


IF(ZZ.GE.CC(II).AND.2Z2.LT.CC(II+1)) THEN CALL TOOLBX(MOVETO, INTC X(I)*F+XORIG) -W/2,

INTC Y(J)*F+YORIG) -H/2) CALL TOOLBX( LINE, 1,0)

END IF IF(IAVERG.EQ.1) THEN ZZE = ZE(1I,J,K) IF(ZZE.GE.CC(II).AND.ZZE.LT.CC(II+1)) THEN CALL TOOLBX(MOVETO, INT(XE(1)*F+X0RIG) -W/2,

INT(YE(J)*F+YORIG) -H/2) CALL TOOLBX(LINE, 1,0)

END IF END IF

180 CONTINUE C C PLACE NEW PIXELS AT LOCATION X,Y C

ICOLOR = 1 ZZ1 = Z(I,J,K+1) DO 190 II = 1,NCLEVL-1 IF(II.EQ.4) GO TO 185 IF(ZZ1.GE.CC(II).AND.2Z21.LT.CC(II+1)) THEN COLOR = COLORS(ICOLOR) CALL TOOLBX(FORECOLOR, COLOR) CALL TOOLBX(MOVETO, INT(X(I)*F+XORIG) -W/2,

INT(Y(J)*F+YORIG) -H/2) CALL TOOLBX(LINE, 1,0)

END IF IF(IAVERG.EQ.1) THEN ZZ1E = ZE(1,J,K+1) IF(ZZ1E.GE.CC(I1I).AND.2ZZ1E.LT.CC(II+1)) THEN COLOR = COLORS(ICOLOR) CALL TOOLBX(FORECOLOR , COLOR) CALL TOOLBX(MOVETO, INT(XE(I)*F+XORIG) -W/2,

INT(YE(J)*F+YORIG) -H/2) CALL TOOLBX(LINE, 1,0)

END IF END IF

185 CONTINUE ICOLOR = ICOLOR + 1 IF(ICOLOR.GT.7.OR.ICOLOR.GT.NCOLOR) ICOLOR=1


IF(K.EQ.KMAX-1) THEN


390 400

PAUSE COLOR = 30 CALL TOOLBX(FORECOLOR, COLOR)

DO 400 I = 1, IMAX DO 400 J = 1,JMAX ZZ = Z(1,J,K+1) DO 390 II = 1,NCLEVL-1 IF(II.EQ.4) GO TO 390 IF(ZZ.GE.CC(II).AND.Z2Z.LT.CC(II+1)) THEN CALL TOOLBX(MOVETO, INTC X(I)*F+XORIG)-W/2,

INTC Y(J)*F+YORIG) -H/2) CALL TOOLBX(LINE, 1,0)

END IF IFC IAVERG.EQ.1) THEN ZZE = ZE(I,J,K+t1) IF(ZZE.GE.CC(II).AND.ZZE.LT.CC(II+1)) THEN CALL TOOLBX(MOVETO, INT(XE(I)*F+XORIG) -W/2,

INT(YE(J)*F+YORIG) -H/2) CALL TOOLBX(LINE,1,0)

END IF END IF

CONTINUE CONTINUE

END IF

COLOR = 33 CALL TOOLBX(FORECOLOR, COLOR) W1 = 1 H1 = 1 CALL TOOLBX(PENSIZE,W1,H1)

RETURN END


Appendix D. SUN Animation Code

PROGRAMS TO ANIMATE VORTICITY AS PIXEL-IMAGES, PIXEL-SHADES, AND CONTOURS USING PV-WAVE SCIENTIFIC VISUALIZATION SOFTWARE FOR A SUN MICROSYSTEMS GRAPHIC WORKSTATION


September 1990

MQAaAQAAIAQAAANAANAIQIAAANAaQaANAaAARQAIaAANAIAAMQINMQaAaANgD

The following procedure files read in two-dimensional unsteady vorticity data and convert the data to bit-maps. The bit-maps are then animated with a second procedure file.

Description of Variables and Commands

a Variable holding the picture assoc Associates arrays with variable names ax Set horizontal viewing angle az Set vertical viewing angle begin Begin looping of for statement bytarr Defines the byte array for the bit-map close Close an external file contour Draw 2-D contours

Appendix D. SUN Animation Code 60

AAAI AQAqraaaaqaaaraaagriaaanaanaanan empty

end

endfor

filtarr

for

nlevels

openr

openw

P pro

readf

return

shade_surf

tv

tvrd

tvscl

window

xpos

xsize

ypos

ysize Z

Clears the register memory End of the procedure End of a for loop Defines a floating point array Beginning of time loop Number of contour levels Open a file for reading (only) Open a file for writing (only) Variable holding the picture Precedes procedure filename Read in formatted data Return procedure control to PV-Wave interactive mode Converts data to shaded surfaces Displays gray-scale image Reads gray-scale image to associated byte array Scales image to 0-255 before displaying image Creates a window with designated parameters Horizontal position of lower window corner Width of window Vertical position of lower window corner Height of window Variable containing vorticity data

Hn

neh

wd

dn

ft tnt

ht

tt

nn

tod

tt fo

n on

tt

RIAN R REE RERER RRR ERE ERTRTRER ERE RRR RR RREERRERERRERER

Procedure file to draw the pixel-images. scaled to cover the entire range of the vorticity data.

A color table is

Each

vorticity location is then assigned a color pixel, which is then plotted to the screen

pro makemovietv openr,1,'vort.data' openw,2,'vorttv.byt' z=fltarr(350, 250) p=assoc(2,bytarr(350,250)) window, 1,xsize=350, ysize=250

for i=1,50 do begin readf,1,z

tvscl,z

p(i)=tvrd(0,0,350,250) empty

endfor

close,1l

close,2

return

end


Procedure file to animate the pixel-images. The byte file is

read in and painted to the screen, one picture at a time.

pro movietv openr,9, 'vorttv.byt' a=assoc(9, bytarr( 350, 250))

window, 9,xsize=350, ysize=250, xpos=740, ypos=400 for i=1,50 do begin tv,a(i) empty

endfor close,9

return

end

RITE ETRE RTERERRRERER ERE RETCEREREREEREERERERERERERERRRERERR ERE

Procedure file to draw the shade-images. The data is drawn out of the data plane to represent its magnitude.

pro makemovieshade openr,7,'vort.data' openw, 8, 'vortshade. byt’ z=fltarr(350,250) p=assoc(8, bytarr(700,500))

window, 8&,xsize=700, ysize=500, xpos=400, ypos=10 for i=1,50 do begin readf,7,z

shade_surf,z,az=30, ax=45

p(ij=tvrd(0,0, 700,500) empty

endfor close,7 close,8&

return

end

Procedure file to animate the shade-images. The byte file is read in and painted to the screen, one picture at a time.

pro movieshade openr,8, 'vortshade.byt' a=assoc(8,bytarr(700,500)) window, 8,xsize=700, ysize=500, xpos=0, ypos=370


for i=1,50 do begin tv,a(i)

empty

endfor

close,8 return

end

WIRRCTEERERERERIRERERREREERRTRRERRERRERRRERERERRECER ERR ERERERERERRERERREREE

Procedure file to draw the contour images. The levels can be set as with standard contouring software.

pro makemoviecont openr,1,'vort.data' openw,2,'vortcont.byt' 2=fltarr( 350,250)

p=assoc(2,bytarr(450,350))

window, 1,xsize=450, ysize=350, xpos=670, ypos=10

for i=1,50 do begin readf,l,z contour, z,nlevels=20 pCij=tvrd(0,0,450,350) empty

endfor

close,1

close,2

return

end

Procedure file to animate the contour-images. The byte file is read in and painted to the screen, one picture at a time.

pro moviecont

openr,7,'vortcont.byt' a=assoc(7,bytarr(450, 350) )

window, 7,xsize=450, ysize=350, xpos=670, ypos=10

for i=1,50 do begin tv,aci)

empty

endfor

close,7 return

end

FedeteedeAeRRRRRRRERECRTCEREREREERERERREREREETRERREREREERER ERR REE ERR REE EE


Appendix E. Figures

Appendix E. Figures 64

iS YoY Yoy Powered Decent

HL

Path of shed

vortex

\

|

Rotation

I Direction

Figure 1.1 The shedding of vorticity from one blade

interacts with the following blade in the powered descent condition of helicopter flight

65

Figure 1.2 The extreme conditions of blade-vortex interaction, perpendicular and parallel to the target airfoil axis

66

Vortex street from airfoil

Von Karman vortex street

Figure 1.3 Vorticity street from a pitching airfoil as

compared to a Von Karman vortex street

67

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Figure 2.2.2 The four-bar linkage system

71

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mo HH} iin : : ae

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wavefronts o o lase nS. cause an

interference pattern of altern ng light nd dark fringes. Assuming Geussian beams,

the measwrin ng volume *s ellips ae 1 wath

depth and length, dm and lnm, spectively

72

Up (Particle Velocity)

e@

eT

aa aN L. N { _)\ \_ _| \ Jf

i ™~ 7 Lee

V ’

Pedestal Voltage

Doppler Signal

Time

Figure 2.3.2 The signal from the photodetector contains high-frequency noise, the low-frequency

pedestal voltage, and the Doppler frequency

73

Forward Scatter

Oo

Laser System _—— o

/ Control

Volume

Backward Scatter Photodetectors

_

Laser System _——

Control

Volume

Figure 2.3.3 Forward- and backward-scatter arrangements in LDV

74

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buiaiesey =

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| Stepping Motor

va Lead Screw

a Mirror Figure 2.3.6 The mirror tower assembly

77

Doppler Signal

Threshold

{ Hysteresis — FT Tv

it 18 j I}

|

Zero Cross

Detection

N-Cycle

Envelope (N=4) |

| f- Origin ! i rns I

| 1 11 i 11 \]

PUY Jen flu ee eat att

Pot Pet a tt ah ryt ee ee | en

ee ee 0

Pot deel teavie teat dt

1 | tel tea he tea dd

Lot dud tea Wa teat oT Schmitt LLL Trigger 1 | 1 |

| | 1 |

Figure 2.3.7 The counting of signals with counter-type signal processing

wma ystks

aqQeuTpzA00D

sy

pue

uoqT_oeztp

aoTZ

Azeutzad

sy

O02 @4TReTSA

Squseuodu0D

AAToOoTeA

Aa pue

n s9UL

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o/x

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S91

79

K is the half-angle

of the beam intersection

No Tilt

Control

Volume

° Jere

eo Beams Tilted

Figure 3.1.2 Tilting of beams to reduce the size of the blocked region

80

het ~ Q

_

1

jt1 O O O O Oo

1 I

3 1 O O @ Oo O I

]

. I j-l , O Oo O O O

I

j

32 'O OO O08 90 O a ~ x/c

" 4-2 i-1 i itl i+2

O ~=~——- @ -————>- O Central Difference

i-l i itl

Oo ~—- O + @ Backward Difference

i-2 i-l i itl

e-~ O —>- —0O Forward Difference

i-1l i it+l i+2

@ Point where the derivative is evaluated

OQ Other points needed in difference expression

fA Location of blocked/bad data point to be

avoided

Figure 3.2.1 An arbitrary point (i,j) and its corresponding central-, backward-, and forward-difference

representations in the x/c direction

81

Figure

oO

Oo

@ 0

S @®

0 0

@® 0

0 N

Oo ®@

3.2.2

Oo O

Oo O00 @ 4G Z@eo oO

Oo oO

Oo

oO

Oo Oo @ 0 2eo od

O

O

Z2@eoo O00 @ @

oO oO

Oo Oo

Point where the derivative ts evaluated

Other points needed in difference expression

Location of blocked/bad data point to be

avoided

The nine data conditions to consider when

around the airfoil

82

83

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a jeutTpzr00D

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yebzeqy pue

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0.3625, --2 -2 3 st st

2-2? tt eee EE EE EE

\\ \ ‘ nt

HAN

AK

‘ _— 0.2625 -. tt

EZ ae

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Vy

MM LEE SS aT at lt

A.1IS7SL re L } |

| | \ \ ! ! ! , |

\\ | \\ i d Me \ vt MM : \ : \\ | 4 \ ‘ NN AN it

t | | tl UN \ | \ : \ \ : i :

t \ :

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0.3000 -0.2000 «-0.1000 —_ -g..0000 , xe

0.50F

0.20 F

We gaat -10.0

0.00 F

0.10 Dera SiptcteelnteelsrtieS caer heh So So ben enced o

BVI

VECTOR VELOCITY FIELD REFERENCE FRAME FCXED WRT THE ALRFOIL

S(NUSOLDAL PITCHING SCHEDULE PHASE = 0.9 (OEG) TIME LEVEL 1

t « 2.02 Re, - 12967.2

Re» 19436.7 Ul > G.128 ws « « 10.0 (DEC.)

VORTIOTY CONTOURS

13.0

20.0

Figure 4.2.1 Velocity vectors and vorticity contours at time level 1 (1/50 of period, ™=1.24 sec.)

85

t qT qT |

O.3625,00 2-2-3 2 tt tt te et tet os «6 | | BVI a rt et et

a rt re pate rr ht crt at

SEE EEE AES SSS S REFERENCE Pare, FEXEE WRT THE ALRFOLL 2 tet tt tt

EEE SSS eee Regt 0.2626 Ne a Ri Eo ie Rr eRe orn eta | \\ : 2

Se Zs SINUSOIDAL P{TCHING SCHEDULE 22 et ee FEE EEE Bee es SS PHAGE = 7.3 (DEG)

B23 TIME LEVEL 2 \ |

k » 2.02

0.1625 Re, +» 12957.2

Re - 19436.7 Ul O.128 wm

Ye

0.0625

« + 10.0 (0EG.)

0.0375

0.1375

| | | a | | ad. ed.

0.3000. -+-0.2000 «—-0.1000 _--0.0000 sg. 1000 0.2000 0.3000

x/C

0.30

VORTICTY CONTOURS

0.20 +

We asf q

0.00 F.

0.10 Figure 4.2.2 Velocity vectors and vorticity contours at time

level 2 (2/50 of period, 1=1.24 sec.)

86

~ “ “ + 4

0.3625 BVI \ \ N i Nt iM ‘ 4

3 Ea aa eee VECTOR VELOCITY preLo r ha tt REFERENCE FRAME xED THE RFO

DEES ET SEE EE SESE RASS ASS eee (XED WR ALRFOCL eT MF Nn i Tce Vine Ne Np pee eet NN Yr Bie PPB pet

0.2625 i ed NP Rt te! et ee em ae eer ae a a See SINUSOIDAL PITCHING SCHEDULE

SSS S333: ==. PHASE = 14.5 (UEG}

Se TIME LEVEL 3 pone ager na ee BEAR AABASS t . 2m

QO. 162s Re, a 12957 .2

Re - 19436.7 UL > 0.128 ws

v7e « + 10.0 (OBD.) 0.0625

-0.0375

0.1375

0.SFr

VORTIOTY CONTOURS

0.20;

0.00}

0.10

Figure 4.2.3 Velocity vectors and vorticity contours at time

level 3 (3/50 of period, 1T=1.24 sec.)

87

TT ' t tr '

0.3625. a Re kt J BV [ ant) tt et dt et et et et et ee eee

DnB PaO Na Ne iN a Fe Da OP ce Pn Pi RB rr le A

SSE SSS SSS SS SSS SSS REFERENCE PRPE: FERED MRT THE ALRFO a te ; ESS REFERENCE FRAME F(XED THE tL pe Nir ee Fe pNP arr Per Pre a eh

0.2625 Re RD RD 5555332 7 S{NUGO(DAL PITCHING SCHEDULE

Dia et A PHASE = 21.8 <(OEC) BERS BSS SS TIME LEVEL 4 ee NC PR Rly t . 2.a2

Q. 1625 BA Ba Re, » (2967.2 tt et . ‘ . ~ rn nt tat et 7

Co net Rs 7 — . 19436.

Ul» 0.128 ae Ye Prat amen « - 10.0 (DEC.)

0.0625 L Sars 4 tae

0.0375 -

0.1376 L -

___ ee Sf. a | ik. |

a, .2000-—_--0.1000—s -0.. 0000 0.2000 «0.3000 x/C

0.30 VORTICITY CONTOURS

4

0.20 1

4 8D

4D v/e -~52 4

0.10 -10.0 ~13.0 ~20.0

0.00

~4.10

0.50 3.20 0.10

xf


level 4 (4/50 of period, T=1.24 sec.)

88

and vorticity contours

qT qv ' 7 ' i f)

0.3628 - i Ne Pc Rt RBs D at ttt tt rd tt er tt te

Pt ma RE ae ee i i i ee Net Pi Ne tt Ra Rar eB

tS Dr i i iP aor Ba nt

eT Bee ee Oe Mga Fe AF eet ee Pr ep Niet Sr rt Piet et Br Dat Der Lr Rene Rent Se oar >

0.26258. ttt tt ot te tee ed et et te

gg a, sos eg tr Ret CR i Rie Rar

A A AABA BSR Oe pe ine a crn era al ee, EAA pe ee ge Sia

0.16257 ee eee ara a oe

orca Re Sar Rag 3 cael naga Ne RP Rig emp etic

Y/c “4 —

0.0625). a

-0.0375

0.1375 _ ee tt O_O Ot

el, J t _t a! 1 A

4.3000 -0.2000—_ -9.. 1000 0.0000 9.1000 0.2000 0.3000

x/C

0.30+ ‘ - 1

0.20+ 1 } 1

1 5 7

WE go} £3 6

4 4

4

1 71

050 0.10 020 0.0

x/c

Figure 4.2.5 Velocity vectors

level 5 (5/50 of period, ™=1.24

BVI

VECTOR VELOC(TY FIELD REFERENCE FRAME FIXED WRT THE ALRFOIL

SINUGOTDAL PITCHING SCHEDULE PHAGE = 23.0 ‘(OEG} TIME LEVEL S&S

t . 2.02 Re, - 12957.2

Re - 19435.7

U, > 0.128 as « +» 10.0 «OEG.)

VORTICITY CONTOURS

at time

sec.)

89

| 1

Q. 3625 BVI

VECTOR VELOCITY FIELD REFERENCE FRAME F(XED WRT THE ACRFOIL

0.2625 SINUSOIDAL PITCHING SCHEDULE PHAGE = 36.3 (DEG)

TIME LEVEL 6

ks «2.02 0. 1625 Re, » 12957.2

Re « 19435.7 fl

Ul. 0.128 we v7c « + 10.0 (DEG.)

0.0826

~0.0375

0.1375

_l ! __| _| \ \ |

0.3000 -0.2000 0.1000 -0.0000 0.1000 G.2000~ 0.3000 xc

1

| VORTICTY CONTOURS

We <—_

q

1

“= 10 a0 “O40 OD 0.30 Figure 4.2.6 Velocity vectors and vorticity contours at time

level 6 (6/50 of period, t=1.24 sec.)

90

0.3625

0.2625

0. 1625

Y/e

0.0625

0.0375

~0.1376

v/e

4 “4 “ 4 4

\ \ \ \ d : ‘ WN Ni \ \\ 4 2

te |

wel. 1 i | { a | j

0.3000 -0.2000 -c.1000 -0.0000 0.1000 0.2000 g. 3000

x/C

Soe saendinl

70 am 020

0.20

BVI

VECTOR VELOC(TY FELD REFERENCE FRAME FCXED WRT THE ALRFO(L

SCNUGOTOAL PITCHING SCHEDULE PHASE = 43.6 (OEG) TIME LEVEL 7

t » 2.02 Re, + 12957.2

Re. 19436.7 Ul + 0.128 as e «= 10.0 (OEGC.)

VORTIOTY CONTOURS


level 7 (7/50 of period, t =1.24 sec.)

91

4 a 4 +4

0.3625 BVI

VECTOR VELOC(TY F(ELO REFERENCE FRAME FEXED WRT THE ARFOCL

0.2625 SINUGO(DAL PETCHING SCHEDULE PHREE = 50.8 (OEC) TIME LEVEL 8

k » 2.02

Q. 1625 Re, + 12957.2

Re» 19436.7 UL + 0.128 as

“ « «= 10.0 (DEG.)

0.0625

~0.1375

T 0.30

VORTIOTY CONTOURS

v T

=a

9.20

WE a0 -10.9

715.0

9.00

0.10

4.2.8 Velocity vectors and vorticity contours at time Figure

“3 level 8 (8/50 of period, 1=1.24 sec.)

92

qT T T t T t v

0.3626 L + BVI

VECTOR VELOCITY FIELD REFERENCE FRAME FUXED WRT THE ALRFOIL

0.26275 4 SINUSOIDAL PCTCHING SCHEDULE PHAGE = 58.1 (OEG) TIME LEVEL 9 k + 2.02

0. 1625 | Re, + 12967.2 Re. 19436.7 U. + 0.128 ws

Y/e « » 10.0 (OEG.)

0.0625 L 4

-0,0375 J

ne a a aa

4.1375 ae ge TO

wl i J | | i

0.3000 -0.2000 -0.1000 -0.0000 «G.1000 sso. 2aS—=é‘iL. 3000 x/

0.30

YORTICITY CONTOURS

0.20 12.0 80 60

WE gto -5.0 ~10.0 -15.0

? -20.0 0.00

-0.19 Figure 4.2.9 Velocity vectors and vorticity contours at time

level 9 (9/50 of period, ™=1.24 sec.)

93

4 4 -4

0.3625 Pat tt ee ee | BV Rh rN NP er

ne I Ec a IR can i Nt Pe

i BP Pe i i i Pe Rr Rie i cB Part Ser ie Sit Ne P eH SE EI I A ESE IEE TE VECTOR VELOC(TY F{ELD a tt tt dt te rt et tt ee REFERENCE FRAME FIXED WRT THE AIRFO[L

a td tt tt wD Pn Ra PD a ee aE se Pet OS Pir Rr Soap Bir ar Rar ir bern Rar De >

0.2625, —-+-++- 2 Pa I I a Re Ret th 4 2s S(NUSOIDAL PITCHING SCHEDULE

PHASE = 66.3 (OEG) TIME LEVEL 10

ko. 2. 0. 1626 L Re, - 12957.2

Re» 19435.7 Ul: 0.126 ww

ve « + 10.0 OE.) 0.0626 |

0.0376.

0.1375

0.3000 50.2000 0.1000 ss -0.0000~—C—s«*O8". 110000 0.2000 0.3000

x/C

0.30}

y VORTICITY CONTOURS

4.

0.20r

and

Do eres

We a.s0 -3.0 -10.0 15.0

0.00

0.10

Figure 4.2.10 Velocity vectors and vorticity contours at

time level 10 (10/50 of period, 1T=1.24 sec.)

94

, r ' ' q T —=T

0.3625 BVI

VECTOR VELOCITY FIELD REFERENCE FRAME FIXED WAT THE AIRFOIL

0.2625 |. S{NUGOLOAL PITCHING SCHEDULE PHASE = 72.6 <DEG)

TIME LEVEL 11

kos 62.02 0.1625 L Re, » 12957.2

Re» 19435.7

Ul + 9,128 we

“ « + 10.0 (DEC.)

0.0625).

0.0375 |

~G.1375 -

| __ j ot 4 1 4

-0.3000 -0.2000 -0.1000 -0.0000 d.1000 «9.20000. 3000 x/C

| rF v ¥

0.30+ J

r VORROTY CONTOURS bo

o.20}+ 4

L | (BD

r 4 120 1 aD

We a0 40

50 j ~100

-t§.0

0.00 -200

: 2.10 B

1.30

/C


time level 11 (11/50 of period, t=1. 24 sec.)

95

' q 1 T OF v Tt

0.3625 | BVI

VECTOR VELOCITY FIELD REFERENCE FRAME FIXED NRT THE AIRFOIL

0.2606 | SINUSOLOAL PITCHING SCHEDULE PHASE = 79.8 (DEC) TIME LEVEL 12

«62.02 0.1625. Re, » 12957.2

Re» 19436.7

U,+ 0.128 we ve « + 10.0 (OEG.)

0.0625

0.0376

-0.1375L

L ___/ I _I a | J _I

0.3000 0.2000 0.1000 -d.0000 G.1000) 0 «060.2000~=— 0. 3000 x/C

0.30}

r VORTICITY CONTOURS ;

azo}

L.

We ato}

L

o.00/

L

0.10}

Figure 4.2.12 Velocity vectors and vorticity contours at time level 12 (12/50 of period, 1t=1.24 sec.) .

96

F T T t ? t

0.3625 |.

0.2625 |.

0.1625.

y/c

0.0625 |.

0.0375.

-0.1375/.

} | __f i tL { t

-0.3000 -0.2000 -0.1000 -0.0000 «9.1000 -0.2000-—Ss «0. 300 x/C

0.30} :

r 1 0.20} 1

r SAAS : Weasel SSSA ANA : 1

| \ ~ ~ 4

2

. NS ’. 7 1

L ~ y 0.00} SS yf ;

' NANOS 7

a 7

0.10

Ji i. a. L JS. ‘3.10 alice 000 Memon. 0.10 i. dl. 02. 0.

x/C

Figure 4.2.13 Velocity vectors time level 13

(13/50 of period,

BVI

VECTOR VELOCITY FIELO REFERENCE FRAME FCXED WRT THE ACRFOLL

SINUSOLOAL P{TCHING SCHEDULE PHASE = 67.1 TIME LEVEL

k - 2.02 Re, » 12967.2

Re » 19435.7

U,- 0.128 we « « 10.0

VORTICITY CONTOURS

(OEG) i3

(OEG.)

and vorticity contours at T=1.24 sec.)

97

v 1 7. T T TT tT

o. 3625 |. BVI


0.2675 SINUGOCDAL PLTCHING SCHEDULE PHAGE = 94.4 (DEC) TIME LEVEL 14

- 2.2 0. 1625 L Re, » 12957.2

Re» 19436.7 U, > 0.126 a

ve « + 10.0 (OEG.)

0.0625 L

0.0375 |.

-0.1375L

4 ed _t 1 __! = |

0.3000 -0.2000 -d.l000 -0.0@00 «G.1000 «0.2o0)Stié*D. 3000 x/c

; 0.50} 1

r 1 VORTCITY CONTOURS f . 4

0.20} .

¥/C 0.10}

0,00 L.

~O.10b

a)


time level 14 (14/50 of period, t=1.24 sec.)

98

0.3625 BVI

VECTOR VELOC(TY FELD REFERENCE FRAME F(XE0 WRT THE A(RFOCL

0.2628 SINUSOIDAL PITCHING SCHEDULE PHASE = 101.6 <(OEG) TIME LEVEL 15 k « 2.02 Re, + 12957.2

Re - 19435.7

U,> 0.126 ar « «- 10.0 (DEG.)

0.1625

Y/e

0.0625

-0.1375

Th SSCS

rT

0.30

—-

f VORTICITY CONTOURS

0.20 16.0

12.0 T ¥

T

4D

-5.0

10.0

~15.0

YW 30

et

Q.00-

—Q.10F



99

0.3625

0.2625

0. 1625

y/ce

0.0625

0.0375

0.1375

| ;

0.30} J

¢ 4

1 0.20 |

. - < rund SS 1

f é ~ 7 Lo ZK r * Of of Ate x

WC aso} Wy 04 ;

r M4 aa

r

! SX 0.00 § N NY,

° ; . }

—Q.10+ a

00 SCT SS~«CO 0.10

x/C

Figure 4.2.16 time level 16

BVI

VECTOR VELOC(TY FCELO REFERENCE FRAME F(XED WRT THE ATRFOCL

SINUSOIDAL PITCHING SCHEDULE PHASE = 108.9 (DEG) TIME LEVEL 16 k » 2.02 Re, - 12957.2

Re. 19435.7 Ul, > 0.128 os « - 10.0 (OEC.)

VORTICTY CONTOURS

Velocity vectors and vorticity contours at

(16/50 of period, T=1.24 sec.)

100

OBE Z5 a te et

128

0.2625

re ee ee Ener Per ee et, me D

— ha

or ee

Y/e

0.0625

0.0375

0.1375

L

0.30 Fr

}

LL

9.207 |

5

WO gtop

r a

oxol

ant 4 i. i A L a0

BVI

VECTOR VELOC(TY FCELD REFERENCE FRAME FIXED WAT THE A(RFOCL

SC(NUSOLOAL PITCHING SCHEDULE PHASE = 116.1 ‘OEC) TIME LEVEL [7

k - 2.02 Re, - 12957.2

Re + 19436.7 U» 9.120 ws

| «e » 10.0 (DEG.)

VORTIOTY CONTOURS



101

VECTOR VELOCITY FIELD REFERENCE FRAME F(XED WRT THE AIRFOIL

0.2625 SINUSOIDAL PITCHING SCHEDULE PHASE = 123.4 (OEG)

TIME LEVEL 18

bk « 2.02

0.1825 Re,» 12957.2

Re» 1943.7 U,- 0.128

“ « -» 10.0 <OEC.)

0.0625

0.0375

0.1375

0.30} 4

f ] VORTICITY CONTOURS

0.20} 120 a0

j 49

WE sot af

“aa ~13.0 “20

al

L -0.10+

00

x/C


time level 18 (18/50 of period, 1=1.24 sec.)

102

0.3625 BVI

VECTOR VELOCITY FIELD REFERENCE FRAME FIXED WRT THE ACRFOLL

0.2626 SINUSOLDAL PITCHING SCHEDULE PHASE = 130.7 (DEG)

TIME LEVEL 19 l - 2.02

0.1625 |. Re, + 12957.2

Re. 19435.7 Ul: 0.128 a

vee « - 10.0 (O6G.)

0.0625}.

0.0375 |.

0.1375,

__f_~ LL. a a __. li i

0.3000 -0.2000 -0.1000 -0.0000 «0.1000 «0.2000 0.3000 xT

——* . — :

0.30+ 1 ' ] VORTIATY CONTOURS

0.20-

v/C r 1 asof

L

o.oo}

r 0.10

030 0.20


time level 19 (19/50 of period, T=1.24 sec.)

103

Ct OF v Tv t ms rT

0.3625 | 4 BVI

VECTOR VELOC(TY FLELO REFERENCE FRAME FIXED WRT THE ALRFOIL

0.2625 a SINUSO[DAL PITCHING SCHEDULE PHRASE = 137.9 (GEG) TIME LEVEL 20

k « 2.02

0. 1625}. ; Re, + 12957.2

Re. 19436.7

U,+ O.128 ws

ae «e « 10.0 (08G.)

0.0825 a

0.0375 L -

eee eee ee oe eee

0.1375 — eee re

1 ! a | |} i A

-0.3000 -~0.2000 -0.1000 -0.0000 0.1000 9.2000 0.3000

x/E

po 0.50Fr

4

1 VORTIOTY CONTOURS

0.20 m

We G.10F

.

0.00F r

—G.10+

i dll 2. i. i. ‘310 A. A 000° a. A 0.10 i 020 0.30

x/C



104

i 1 T T F T 1

0.3625 BVI


0.2625 | SINUGOIDAL PITCHING SCHEDULE PHASE = 145.2 (0EG) TIME LEVEL 21 ‘ - 2.02

0.1626 | Re, » 12957.2

Re» 19435.7 U, + 0.128 a

ve e » 10.0 (0EC.)

0.0625}

0.0376 |.

0.1375

| j <I __| __ | 1 a |

~0.3000 -0.2000 -0.1000 -0.0000 0.1000 0.2000 0.3000 x/C

ss} 2 v RS

0.20}

We asok

L

0.005

L

0.10


time level 21 (21/50 of period, Tt=1.24 sec.)

105

oC T qT t T ' i

0.3625 BVI

VECTOR VELOCITY F[ELO REFERENCE FRAME F(XED WRT THE AIRFOIL

0.2625 L S(MUSOLOAL PLTCHING SCHEDULE PHASE = 152.4 (OEG) TIME LEVEL 22 k . 2.02

0. 1625, Re, « 12967.2

Re . 19436.7

Ys UL+ 0.128 we

© « + 10.0 (0EC.) 0.0625/

0.0375

— eo oe PR nF Ee

0.1375, pe nn.

__| | __f a | I 1 ___}

-0.3000 -0.2000 -0.1000 -0.0000 G.i00 o.2000)—= (0.3000 x/e

0.50 F

VORTICITY CONTOURS

0.20

We 910 L

Q.00+

0.10; 2.50 Re 0.30

X/C .

Figure 4.2.22 Velocity vectors and vorticity contours at time level 22 (22/50 of period, 1=1.24 sec.)

106

TS r T ~T “T T T

0.3625 - a a ee eee BVI nn ena tng ap — ae

nn a a pa Rg cre i een tla eae

A i RR a ag le he 8 REFERENCE FRAME FCKED WRT THE AIRFOIL i Rr a ns ei ep

0.2625 L 4 S(NUSOIODAL PCTCHING SCHEDULE

eg ge ee ee PHASE = 1995.7 (DEG)

k « 2.02

Os, “sao eee ee eS | Pe, + 12987.2

Pi i hin i Ri ae a a Egy Re + 19436.7 2S UL» 0.128 aw v“e « + 10.0 (0EC.)

0.0625

0.0375

0.1378

a wel, j i oben i d

0.3000 «—-0.2000 «—-0.1000 ~-0.0000—=—s—70. 1000 0.2000 Q. 3000

xX/T

r T r Tr ]

0.30F 1 . | VORTICITY CONTOURS

0.20F . 124

r 3.0 44

w/e Q.10- -5.0 .

-10.0

r -15.0 r -208

0.005

~0.10+ r

0.30 0.20



107

, T “4 T T “T T

0.3625 BVI

VECTOR VELOCITY FIELD REFERENCE FRAME FUXEO WAT THE ACRFOCL

0.2625 SINUSOIDAL PLTCHING SCHEDULE PHASE = 1665.9 <0EC) TIME LEVEL 24 1 .» 2.02

0. 1625 Re, + 12957.2

Re» 19436.7 Ul. 0.128 a

we « ¢ 10.0 (O8G.)

0.0625

0.0375

0.1375

| = —— .

0.30

] YORTICITY CONTOURS .

0.20} L

v/e 9.10!

io

} Q.00}

L

-0.10}

oH =8.20



108

0.3625. a tl el eee ad BVI

a ae VECTOR VELOCITY FELD Th eet el enna a ae atte REFERENCE FRAME FIXED WRT THE AIRFOIL

0.2628 4 = SINUSOIDAL PITCHING SCHEDULE

PHASE = 174.2 (DEG) a er el te ne cet ee TIME LEVEL 25

k » 2.02 Re, + 12957.2

Re» 19436.7 Ul. 9,128 ae « » 10.0 «OEG.)

0. 1625.

v/e

0.0625

-0.0375

0.1375

0.3000 -0.2000 -0.1000 -0.0000 90.1000 0.2000 0.3000

x/C

0.07

VORTQTY CONTOURS

O.20r

¥/C 0.10

Q.00r

Q.10F i ici tours at

Figure 4.2.25 Velocity vectors and vorticity con

3 time level 25 (25/50 of period, 1t=1.24 sec.)

109

tT T T T T T T

0.3625 L nk a a el ne = BVI man ln a lage Rr cet

glk hgh gh gh hgh elapse et a Rl nate _ VECTOR VELOCITY FIELD

att REFERENCE FRAME FCXED WRT THE ACRFOCL

ott st et et rl tt — te ee 0.2625 LL

ae “| - SENUSOIDAL PITCHING SCHEDULE a rt ad et gr gr lc eR ee PHASE = 181.5 (OEG)

ral lh ene ee k » 2.02

0.1625. eS ee ee See 4 Re, - 12967.2

Res 19438.7

meen ha Rie Ba Bee Ph a SS 2 SSS Sa eS « + 10.0 (OEG.)

0.0625.

-0.0376 |

~0. 1375}.

al _t. 1 ah _| 1

0.3000 -0.20000 0.100 -0.0000)—ss*0.. 1000 0.2000 0.3000

x/CT

0.305

VORTICITY CONTOURS

¥ v

We ato

ret

_ TT

0,00

Yr


time level 26 (26/50 of period, 1t=1.24 sec.)

110

en ence th rh a rt tlle

Rr rer ane a maa manatee eine pa il Da eel ee en VECTOR VELOC(TY FIELD

a A le ee eet REFERENCE FRAME F(XED WRT THE ACRFOIL

0.2625

at tt PHASE = 168.7 (OEG) ————* TIME LEVEL 27

L » 2.02

0.1626{ SSS SSS Sat foe, - 12967.2

= Re» 19436.7 Y/c ule QO. 128 ws

Gee te PR « » 10.0 (OED.} Zee

0.0625 | SSS |

een e eae

Sao

0.0375). =“

-~d.1 375 a ng eg ag ey as, NN ateasteee, J

j al { —l, | 1 |

0.5000 «=-0.2000) «—-0.1000 -0.0000 «=0.1000 0.2000 Q. 3000

x/C

03

VORTICITY CONTOURS

0.20

I

tT

tT

y/c a.tof

Q.0GF



111

t v t t t T

0.3825 th meh —h —alt . be

4

a BVI

pak ee” 3323 22 re rt a - TE eee eT erent = VECTOR VELOCITY FIELD a kk ore REFERENCE FRAME F(XED WRT THE ALRFO(L Ta ae A hme = 2 tt ot te et tt te ge ne ele mel etn aad 0.2625 a hte eee ee - 4 BRR ht hk etre ete SINUSOIDAL PETCHING SCHEDULE Ta ae Te eer at a ae re ne ee eT at ah at PHASE = 196.0 (DEC)

kh 8k ditccramstita regsstl.manpal, a & . 2.02 ee a ee ee ee eee ee ee ee ee ee hehehe eel 0.1625. lh alpacas ee es eee ee _ Re,» 12967.2

cm RnR I a Re Prt Re ren er ee rl ne Ne Re > 19435.7

Ul, + 0.128 mw 53-23-2233 SS ot « + 10.0 (DEC.) SSS ee ee ee St ot tt tt .

St tet ee

~~ > a eC eee ee ees ee ee ed

-0,0375 . +

ye

. bo De See ee eee ee ee ane Dee ee ee ee

el | _t. al. I

! 0.3000 -0.2000 «-0.1000_ -0.0000-—s 0. 1000 0.2000 G. 3000

x/T

O.30;

VORTIOTY CONTOURS

0.20 ]

{ ¥/¢ 0.10

<B

r fa

0.00

~O.1D >



112

0.3625. a rat a ee eee BVI

a rc lal et th VECTOR VELOCITY F{ELO

TSSSSS SSS Se SS eee REFERENCE FRAME FIXED WAT THE ALRFOLL 0.2625 lll Dn mele a _

Se SINUSOIDAL PITCHING SCHEDULE a el ll rt rr cet el el ll PHASE = 203.2 (0EG) —t ot tt 8 Re rt a et eee ee

TS 33S a a SS SS ee ee Se Se TIME LEVEL 29 er lel atm ok » 2.02

Qe “SS SS ss soe ee | Re - 12957.2 oa rt

Re» 19436.7 ; a BRE a PR RR Rr ND rare ap Ul- O.128 as

¥/C at i Bian i Be i NE eR ii Ri i eR ri RN Pat . 10.0 (QEG.) Dh aE et ae ole “os . . a

ee ee ee ee ee er eR Pt el a mnt -O625 . na a BD Pie Fe tte

Q ———a te Se ey se

0.0375 |. = —— 4

Jy er

ae | —_—d | _ a | ____-t, 1

0.3000 0.2000 0.1000 -0.0000 0.1000 0.2000 0.3000 x/C

i 4

ost 1 VORTICITY CONTOURS

a.20+ - 1 L 1 49

1 L 5.0 WE atot -10.0

L -150 f -20.0

0.00}

0.10}

al ple L. i Ad mr) 0

x/C



113

of C q t v Tt 4

0.3628L a a cele p 5 BVI et ot 8 th rain a cr ee cere nt ttt teeter a Rt ta a lt er? VECTOR VELOCITY FIELD wr a ld epee REFERENCE FRAME FIXED NRT THE AIRFOIL

0.2625, —2--t = | , S(NUBOIOAL PITCHING SCHERRLE

at ttt tt ee ee el TIME LEVEL 30

ke - 2.02 odode

0. 1625 J Re, + 1297.2 ett

Re» 19436.7 a i Ee i PD BD Rie De Bs mae mege ce UL- G.128 ws

ye a SSS SSS eee e »- 10.0 COEG.} et tr et i RS a

te Pee a lg

-0.0375 | = ~ “

gy ee

Or ret OSS te ee

O.1375 re es “7

he. a I 1 __b | |

0.3000 0.2000 «—-0.1000 —_ -0.0000—s—«so0). 1000 0.2000 0.3000

xX/C

L.

o.s0F 1

1 VORTICITY CONTOURS

0.205

We atot

r 0.00+

0.10 \

ai. fh ell 1 ‘ &. “10 0D L i 0.10 . 0.20 0.50



114

0.3626 |

0.2625 |

0.1625 .

y/C

4 4 4 4

-0.

0.30

0.20

WC aso

9.00

Figure

3000 0.2000 -0.1000 | -0.0000 x/C

Lr

a ee Re ee ee

¥

4.2.31 time level 31

BVI

VECTOR VELOCITY FELD REFERENCE FRAME FCXED WRT THE ARFOCL

SINUGOIDAL PITCHING SCHEDULE PHASE = 217.6 (DEG) TIME LEVEL 31 & « 2.02

Re, + 12967.2

Re - 19435.7 Ul+ O.128 ws ae +» 10.0 (0EG.)


(31/50 of period, ™=1.24 sec.)

115

0.3626 4 BVI

cll lr el ene eh REFERENCE FRAME FIXED WRT THE AIRFOIL

0.2626

ia ar Bt Rte ate SC(NUSOLOAL PITCHING SCHEDULE TR tt ae eR Rt cea rn et et lal PHAGE = 225.0 (DEG)

TIME LEVEL 32 k - 2.02

0.1625] Sa Sa | Re, + 12957.2

— Re - 19436.7 Ul. 0.128 Bt

S333 3 23S See ee eee « + 10.0 (OEG.}

¥/Cc

3.0625 | 7

Y

0.03765 ES

——#,

OI ee

a wel, at 1 1 al

-0.3000 -0.2000 « -+0.1000 0.0000: 0. 1000 0.2000 0.3000

X/E

0.30} r VORTIATY CONTOURS

0.20}

We asf

0.00

~2.10 . 2s t

Fiqure 4.2.32 Velocity vectors and vorticity contours a

3 time level 32 (32/50 of period, T=1l. 24 sec.)

116

Fr Te T 7 i a tT

0.3625| ato ae eee ene | BVI ee et ttt rt ee

a a en Tat ne at. eR hl rt aR Re a eh ee ec VECTOR VELOCITY FIELD A i GI Eee TARA REFERENCE FRAME FIXED WAT THE ALRFOIL

Ni I er Ra Re ra eh le a 0.2625 ne Ri a a BE lt are Ne all gl a a

SDS SS ot ot we et 7 SINUSOIDAL PITCHING SCHEDULE

an ma PHASE = 232.3 (DEG) at —— a ee ee TIME LEVEL 33

eS SSS NN a Re el ED eget may eat Dt i » 2.02 - re a ae et elt Oe OSS SSeS SGeas | Re, + 12987.2

i Et S33 — Re - 19435.7 oe ot tt

we gD Re Pa RR i Po i Rin Rr DE ete UL 0.128 we y/e Eee 2 10.0 (OEG.)

Ge See Ph colt ae as . . wt a

.0625 RP rc eg st Bt halt ach ag =

a 8 te et

EES a“ a

-0.0375 LEE 4 . be Tp ag,

SE et ep gy ee ee reanan ena Ft nga:

a = ran aaa oe ae 4 pg sy

~ S SS — ~~ Sess O37. 27 EEE 4

__I 1 | l 1 ad !

~0.3000 -0.2000 -0.1000 -0.0000 J.1000 0.2000 0.3000

x/C

yr —

0.30- 1 L VORTITY CONTOURS

ozo ]

T ee |

ie

We ono

0.00

9.10



117

q | 1 T — Fs re" —

0.3626 | a al a na +» 4 BVI ta tt te et

a a ra et ea mr a tl al ee

SS eee eee VECTOR VELOCITY FIELD oe ee ET a ee eh a rane REFERENCE FRAME FIXED WRI THE AIRFOCL ee eR all meee 0.2678 9 2 se tt ae eth et eek th 4 —_—s ~ RR tl an en S(NUGSOITOAL PITCHING SCHEDULE

35s 32 2 ee PHASE = 239.5 (O&C)

ee ea Se TIME LEVEL 34

te k » 2.02 ee R {2957.2 QO.1625- —2 7 s= Prt 4 .° .

OO a tt tt = Tee oe R 19435.7 x eh hh ht tt ee .

FREES SSS Ul- C.128 wr v/e ae soe en ee TO ee le ie ie ee ee ee ae ee eT aT

se ne ee ae ee ee i es ie re es ee a a - 10.0 (OEG.) pa Path ct me Rg gl eet ey

0.0625 eee Att te ee, 2th ee ng ee

LEE eee aan - So”

2

EEE St eng -0.0375 EGE a

a gage,

a ee

ESS

0.1376 tt ny pe rs |

l 1 1 f ] wt |

-0. 3000 -0.2000 -O. 1000 -0.0000 Q. 1000 0.2000 0.3000

x/C

0.30} 1

r VORTICITY CONTOURS

r | azol q

80 L 1 40

WC a0 ] ~50 10.0

18.0

fone 00000 1 Lo9090000 b 00000009 4

~0.10 90000000 pon O00 cC ON

opoo9 00? all i. L i L all rs A i. Sam 4 i

-0.K0 0.20 0.10 0.00 0.10 0.20 0.30


time level 34 (34/50 of period, %t=1.24 sec.)

118

0.3625,

0.2625

0.1625.

ve

0.0625

| !

| | | |

:

|

\

:

:

\ | | |

\ : |

‘ |

: i | ne Pte TE a ep eg nenge——p ~~" —

\ : :

SS Ot ent nt aaa ee ae ae ee

| I J j |

0.300 «2-0.2000 «~-0.1000 3=-0.0000 Q. 1000

x/T

BVI

FIELO VECTOR VELOC(TY REFERENCE FRAME F{XED WRT THE ATRFOIL

S{NUSOLDAL PITCHING SCHEDULE PHASE = 246.8 (OED) TIME LEVEL 36

- 2.02 Re, » 12967.2

19436.7 0.128 ws 10.0 (0EC.)

Re - u..

L

5 ] VORTOTY CONTOURS

a.20f ;

ve aol L

a.not

0.1016 359009000 eoocoo0000

O,°o oof, 4. i i i ry i i 4 ai. .

a ae 0.00 0.40 om 0.0

xf



119

4 4

a

0.3625, a wd BVI ord Rt

—— TR EE RE lana ae VECTOR VELOCITY FELD

REFERENCE FRAME FIXED WRT THE ALRFOIL

0.2625)

ee

a et ee re en er re el et SINUSOIDAL PITCHING SCHEDULE

8 et et tt dt te PHASE = 254.0 (DEC)

to 2 G.iezs

\ : \ \ :

| t

: \

Re, - 12957.2

Re » 19436.7 U,- 0.128 aw « » 10.0 (QBG.)

y/c

0.0625 L

-0.0375 |. EEE

tee,

os

oO ee

i |

. i \\ Mi AM \ |

0.1575). a

tL | a 1 _ a _—_! 0.3000 -0.2000 ——-0.1000 -0.0000)—si0. 1000 0.2000 0.3000

x/C

T

0.30

VORTICITY CONTOURS

7 r

v q

0.20

WC a0

Lr

re

ns a

es

ee ee eee oe

9.00

900900090

coo000~0on9 4 OF 009009 L i. alk Ft i A | a | A. a.

AD 0.20 0.10 0.00 0.10 0.20 0.30



120

0.3625). ae rt en rE A a eee BVI

| a RD a hl ea nat ea tt VECTOR VELOCITY F(ELD ee Se ee eh ee ok oe et et et REFERENCE FRAME FIXED WRT THE ALRFOCL

SINUSOIDAL PITCHING SCHEDULE PHASE = 261.3 (DEC)

TIME LEVEL 37

k » 2.02

Re, > 12957.2

Re» 19435.7 RA ee ot Ul - 0.128 as

« + 10.0 (0EG.)

0.2625 L i ce Rie orl a eel ee espa

le

Mt : \ \ \ b \\\ WK | : \

g.162s1 et eden ee eth

i 2? tt eae, ah. ie ae

\ | y/c

‘NS

wi Nt iu ye :

baat Ute

0.0625

| \

Kl \

WAAA

Y Hi

HA Vi HAH wi : !

~ oe wo sy Oe Saaiinenmeamete a

LEE ee i a

LE a OI. 7 tp a ene

1 _i _ Ll __t ! -0.3000 ~Q.2000 -0. 1000 0.0000 «0.1000 G.2000 0.3000

x/E

a0 1 1 VORTICITY CONTOURS

0.20+ 1

We at

rr

9.00

eT

a

>d



121

F 4 T TC T of 1

O.SGZS tn ee er JI BVI a et a _® =

ee eee a ek > VECTOR VELOCITY FIELO S33 Pig e BR RRR rt oh Re eee REFERENCE FRAME FIXED WRT THE ALRFOIL 2 a eS

0.2, Cees eS asesa SS a eer SSS | SINUSOIDAL PLTCHING SCHEDULE a ese mgenae te

et a ee ee PHASE = 268.6 ‘O0E&C)

EERE he ee TIME LEVEL 38 RS RT Pet et eB Reheat coe = tN aS AG a Rat eR A arch k » 2.02

0.1625. Ae a ee J se, © 12957. ee Bo et ao.

i oN ed, RnB A ge cele the, Re - 19435.7 ee et het eh ahd ea eg etme

oe ee eer nraraam Ul + 0.126 aw “ FEES Saat « + 30.0 (DEG.) 0.0625, AEE

~0.0376 L

a aT Pee ee, O37 Ee EE

J l | al j | |

0.300 =-0.2000 -0.1000 -0.0000 9.1000 Q.2000 0.3000

Xx/C

VORTICITY CONTOURS

~35.00

onl

} 7 0.20 Fr ]

WC gaat 1

a. b r | 9.00}

be | .

L.

L 1 ~0.10+ ]

.

30 ‘a0 OD 0.30



122

0.3626,

0.2625 {.

0.1625,

y7e

G.06265 |. LED

0.0375,

-0.1375 wal

\ \\

i he hn ere ie ca Rc cre er ol oe ee ecal am

i BR RR Rls gl ee \ \ \ \ \ \ \ \ AN \ r Z Us \ ~ = : |

\ \

\ | \ } | ! | :

uN i ae

4

\ L L : . } L |

Py TI

wl,

4 BVI

VECTOR VELOC(TY FLELO REFERENCE FRAME FIXED WRT THE ACRFOCL

SINUSOIDAL PITCHING SCHEDULE PHAGE = 275.8 (OE) TIME LEVEL 39

k » 2.02 Re,» 12957.2 Re» 19435.7

U.» G.128 on = « 10.0 (O8G.)

0.3000 -0.2000 -0.1000-- -0.0000 0. 1000 0.2000 0.3000

x/C

0.50

G.20

Y/¢ @.to

9.00

0.10

—

Figure

time level 39

VORTIOTY CONTOURS

5.00

~10.00

~15.00

20.00


(39/50 of period, t=1.24 sec.)

123

. eB Ri aR eh 8 nl ed esl ee a Vv

0828 SSS SS SSS SSS See BVI te TD el ee rt wee

ct rl ran ren eee eee ape ae ee eee eee VECTOR VELOCITY FLELO

DETERS SS = te REFERENCE FRAME FIXED WRT THE ACRFO[L nh et nt ee et cet =

Ty Ea Dt oececeste es «| §{NUBOCOAL PITCHING SCHEDULE et ete tee oem Rent te So Rian Rar Re aaa rath eam ZEEE a PHAGE = 283.1 (OEG) = Rt re TIME LEVEL 40

a eR aR el rt met tte —

Zp | 1 0.1625. AEE ESS | Re, + 12957.2 Fe ere Ri im hr pl egret etl me

SSS SSS eee eee Re - 19435.7 we QPP rh ngs coe eet meant U, » 0.128 a

55 35S 33 See oo « = 10.0 (OEG.) ee henge tet

0.0625. ee et oh eet - <——_—— a

— >

0.0376 ae

——— | Lt ey .

Fenty, ey

“0.1376, ee a

a | JI ft | a | |

-0.3000 -0.2000 -0.1000 0.0000 0.1000 G.z009 0.3000 x/C

be : - - _ 7

axl

0.20}

4.00

WiC coos

15.00 ~20.00



124

' q T 5 t rt T

. a nn RD

Tae iad eee eee een a SS AF eee VECTOR VELOC(TY FLELO NE ALRFOLL ae * REFERENCE FRAME FIXED

a.2ez6 Sa ae ZEEE SASS 3333S ae “| SINUSOIDAL PITCHING SCHEDULE ZEEE £33 SS eee oe PHASE = 290.3 (OEC)

RR Se ert TIME LEVEL 41 een t « 2.02

a a et eth eh 0. 1625 Aa pe ts ee ee Re, + 12967.2 een Ratt —2 4% bp 7

eS Re» 19435.7 y Baa ee eo Us 0.128 we “ LEB PRBS omo en «| + 10 ODL 0.0625. LE ———— 4

> >

-0.0376 4

“0. 1375 /- tt RS -

iL Jj i 1 1 a | —_

-0.3000 -0.2000 -0.1000 -0.0000 0.1000 0.2000 0.3000 x/o

a0} 1

r ] VORTIATY CONTOURS

r q i

0.20} Ss

L

We oro} L

0.90} &

.

y ~0.10+

a



125

=~ “ 4 4

0.3625. a2 st \ , | ! ! l | , | | BVI \ d i WN

aa et sos ine ne Ne RM,

<a 5S AR ES VECTOR VELOC(TY FreLo MAT THE AERFOL

et

0.2625 *s eee ene =

ZEEE BOSE EES : S(NUGO(DAL PITCHING SCHEDULE BOR Se BS eee PHASE = 297.6 ‘OEG)}

Bae BS TIME LEVEL 42 lt + 2.02

0.1625 Re, - [2957.2

Re - 19436.7 UL- 0.128 we

ve « + 10.0 (DEG.) 0.0625}.

4

EP, ~

aed | | ml wl. |

-0.3000 -0.20000 -0.1000 -0.0000 d.!000 0.2000 0.3000

X/C

VORTIOTY CONTOURS

rm

m=

G20

rT

¥/C otot

9.00



126

4 BVI 0.3625, -—¢-4- 2 te eee

Zw, a

7,

EEE EES ASS Sa VECTOR VELOCITY FIELD EZ ROSES S33 2S Soe REFERENCE FRAME FIXED WRT THE ALRFOIL

. CD

Be eS Ss 32223 2 Eee SINUSOIDAL PITCHING SCHEDULE ere PHAGE = 304.9 (OEG) AAA SSS 33 _—— TIME LEVEL 43

SSS SS et pt eg & « 2.02

a 0. 1625 2533S 1 Re, - 12957.2 rr ae et et rt ee

LLE BAA Re + 1943.7 2S eee U. + 0.128 a v7e « +» 10.0 (OEC.?

0.0826 L.

0.0375 |

LLELE

0.1375, ~ —

ede —red j lL __4- eel 1

~0.3000 -0.2000 -0.1000 -0.0000 0.1000 o.2000 0.3000 xe

L

0.50} 4 wi

r ] VORTIITY CONTOURS L

0.20}

We arnt 1

|

_

a § 4

r

@.00F

a

-0.10F

beereienel A 2. it.

4.30 2



127

T T T T T T T

. a et cet eter etn Vv 0.3625 EB $3333 4 BVI

= NR NR rr lame etn

EES SR et VECTOR VELOCITY FIELD FEES ES = ———+ REFERENCE FRAME FIXED WRT THE ACRFOCL

i i Pp Reg Re De Ik eae

0: - ZEEE “| SCNUBOIDAL PITCHING SCHEDULE te

4 PES SAAS eo oe PHAGE = 312.1 (DEC) a pte et eee ee <n Dee et TIME LEVEL 44

Basa ee k » 2.02 ee

0.1625). [ZB tee OR 12987.2 £23333 eee Re. 1943.7

grr eat ney UL - 0.128 as ¥/C 232>2————-—— e - 10.0 cOEC.?)

ee nS . bee CN tl eet ete =f 0.0625 IEE

0.0375). 4

LEZ te

ey at at ___y att _

QO. 1375 — en 4

_i i eel i a awl ee

0.3000 -0.2000 -0.1000 -0.0000 g.1000 0.2000 0.3000

x/C

i a

0.30} 1

r ] VORTICTY CONTOURS P |

O20F 4

L 4 -$.00

We gal ; ~10.00 i oO ~15.00

} 2 7 20.00 .

Q.00 F

L

~Q.10}

i. iL i al. L. 3. 0.30

) ity contours at

Fiqure 4.2.44 Velocity vectors and vortici

5 time level 44 (44/50 of period, 1t=1.24 sec.)

128

1 1

0. 3625 BVI

VECTOR VELOCITY FCELO REFERENCE FRAME FIXED WRT THE ALRFOL

0.2625 SINUSOLOAL PITCHING SCHEDULE

PHASE = 319.4 (0&0) TIME LEVEL 45

t . 2.02 Q. 1625 Re, - 12867.2

Re» 19435.7 U,+ 0.128 ws

ee « + 10.0 (DED.)

0.0625

0.0375

0.1376

0.3000 -0.2000 -0.1000 0.0000 0.1000 0.2000 0.3000 x/C

0.30} 1

; ] VORTICTY CONTOURS

/ 4

0.20} 1

| L 4 -$.00

We asl 10.00 L -15,00

# 7 ~20.00

0.00} 4

L ~a.s0}

L 0.30 4 “a 2 S05" pout 3.00 4. a0 0.20 7.30



129

OE TT TT Tt T T "T

. 3625 a i i eR AR ER eet oleh hl al

VECTOR VELOCITY FIELO REFERENCE FRAME FIXED WRT THE AIRFOIL

0.2626 L = S{NUSOLDAL PITCHING SCHEDULE PHAGE = 326.6 (0EG) TIME LEVEL 46 k + 2.02

0.1625 4 Re, « 12957.2

Re. 19436.7 Ul, + 0.128 ws

ve e + 10.0 (OED.) 0.0625 4

T

0.0375 L 1

0.1375 5

__! i al a | wt t J

0.3000 -0.2000 -0.1000 -0.0000 0.1000 O.2000 0.3000 X/C

.

0.30}

r VORTICITY CONTOURS

0.20} L L

YC ato

(2) q a

ool 1 4

r 1

. , ~O.10F 1

a0 tO 0.20 7.30

x/C

Figure 4.2.46 Velocity vectors and vorticity contours at time level 46 (46/50 of period, t=1.24 sec.)

130

0. 3625 |

0.2625

0. 1626 |

y7e

0.0625 L.

0.1375

x

“a Ol alata -0.3000 -0.2000 -0.!1000 -9g.0000

0.30

9.20

v/¢c

9.00

4.10

Figure

oe ee ee ee ee 4

apt, 4

a i

‘

G.10

time level 47

BVI

VECTOR VELOCITY FIELO REFERENCE FRAME FIXED WRT THE ALRFOLL

SINUSOIDAL PITCHING SCHEDULE PHASE = 333.9 (OG) TIME LEVEL 47

k » 2.02 Re, - 12957.2

Re. 19436.7

UL> 0.128 ww « +» 10.0 (OEG.)

VORTICTY CONTOURS


(47/50 of period, 1=1.24 sec.)

131

T T TT T T T T

0, 3625 |. Ot rte eter ee 4 BVI Tn ed

SP et VECTOR VELOCITY FIELO a rh ete REFERENCE FRAME F(XED WRT THE ALRFOIL

7 tt et rt 0.2625 a ee ee ee

Zs 623 See eee 7 SINUSOIDAL PITCHING SCHEDULE et et tt ed ee PHAGE = 341.2 (OEG)

BSS See TIME LEVEL 48

3 3SSe_———oo & « 2.02 ee ee

0.1628 <A tt ee el Re, + 12967.2 eet te S3543-—— Re + 19435.7

y. nn U. + 0.128 “ BS « + 10.0 (EG)

a NG 0.0625 Grete 4

-0.0375 4

0.1375 4

i i

0.2000 g.3000

0.30 F 4

r 1 YORTICITY CONTOURS -

b 5

on 1 »

L J L 1 -i.0

a >» 1 = 4 -

° Qt Fy -w

aml

} 1 ~Q.10F 4

050 O20 0S 0.10 0.20 0.50



132

T t r 4

0.3625 TOE ee <j BY I

~

VECTOR VELOCITY FIELD REFERENCE FRAME FLXED WRT THE AIRFOIL

| STNUSOIDAL PITCHING SCHEDULE PHASE = 348.4 (UEC) TIME LEVEL 49 k . 2.02 Re, - 12957.2

Re. 19436.7

Ul+ 0.128 we e » 10.0 (O0EC.)

al, al. al, —L wel —_l,

0.3000 -0.2000 -9.1000 -0.0000 4.1000 Q.2000 0.3000

x/T

0.305

} VORTICITY CONTOURS

y Le

0.20 ‘

40

5.0

Ye 0.10 -10.0 L. 15.9

r -70.0

9.00F

5

~4.10F -

bed L Figure 4.2.49 Velocity vectors and vorticity contours at

time level 49 (49/50 of period, tT=1.24 sec.)

133

F fp tT ' T F T

0. 3625 - 4 BVI a a a ee Oe td et ot te et

VECTOR VELOCITY FIELO ie Siok Sia ee ia are NaS REFERENCE FRAME F{XED WRT THE ALRFO[L nh eet ee nt tt gt

0. BSS eee 7] STNUBOIDAL PITCHING SCHEDULE Son tte pa a nc

BES ASS PHAGE = 355.7 (DEC) a TIME LEVEL &0 a SD ap

ZEBRA a k - 2.02 Pe leh

0. 1625 ZZ Zea ae Re, « 12957.2 Ame

gS a tee Re. 19436.7 oot RE eal eth nly er . ’ BAAS U,+ 0.128 ae

vc 23533 a + 10.0 (0EG.) 0.0625

0.0375

~0. 1375

0.3000 -0.2000 -0.1000_ -0.0000 «0.100000. 2000~—St—«O0=. 3000 xo

7 “=r

] VORTICITY CONTOURS g “

. 1

azo} ] -

1 1

¥/C a.10F 4

FA ool 7

- q

r ho

~O.10+ L

“<a a —S~« 0.10 0.20 0.30



134

0.35 pa a a .

fd,

0.25

0.15

pe

et

a

y/o]

0.05 -

-0.05 +

rrerrprrrryprrrryprr

re eprreyT

-0.15 } ee -0.3 -0.2 -0.1 0 0.1 0.2 0.3

0 35 i allen: wall. l 4. 7 wall _L aul wall. wail, _L wh. i. ah i wl email ool.

oak

1 oe ee

0.25

+h t

0.15

y/c

0.05

Bh

ee eee

eh

aieo

d,

-0.05 -

TT

-0.15 fr eet 5 9 1 17 21 25

Time Level

Figure 4.2.51 The average location of the vortex in the x-y

plane, relative to the airfoil, and the time-y plane

135

0.30} 1

OPO CONTOURS , j THE LEL |

0.20} 4 m0

r 1 13.00 10.00

v/c 5.00 0.507 o0o00 4

I 09° | -5.00 fo} . -

. ° ow e rou} 4 iso

P -20.90 0.00}

r 4 4

L q © VORTEX POSTION 0.10 ° ° 4

10 0 O10 0.00 “O10 020 OD

x/C

F 1

0.30 Fr

P ] OPUY CONTOURS } ‘ mena |

0.20 f 20.00

15.00 r 10.00 r 5.00 We a0} sw

10.09

-13.00

0.00 -

| O «urs 4,10}

00 2 0.10

Kt

Figure 4.3.1 Pressure gradient contours, dp/dx and p/dy '

at time level 1 (1/50 of period, T=1.24 sec.)

136

O.30-

QPDX CONTOURS q 1 TRE (EME 2

0.20} 4

P q

f 15.00 10.00

200

We ool e006 ‘ 4 300 f ° ] 5.00

° / -1900 } oO : =

aS ~15.00

0.00} q “me

0.10} ° o ’ L o

aii. s 3 4 i A Ak 4 aml ST, A i. 1

~0.50 020 =0.10 0.00 0.10 0.20 72

x/C

L . q

0.30} 4

DPOY CONTOURS f ’ TE LEVEL 2

0.20 4 ] 20.00

r 15.00 | 10.00

Y/e 5.00 0.10 ] r | ~5.00 L | -10.90 f -15.00

9.00 +

L O VORTEX POSITION

0.105

anced gaol enn a el edie 0.50 0.20 0.10 0.00 0.10 0.20 0.50

x/C

Figure 4.3.2 Pressure gradient contours, Op/dx and op/dy '


137

TT

0.30-

0.20

WE 10

0.00 r

0.30r

0.207

y/e

Q.00+

—0.10+ 0.50 0 CSCS

x/C

Figure 4.3.3 Pressure gradient contours, dp/ox and op/dy '


138

4

Q.350-

0.20} L

vf r 0.10F

0.00 -

0.10};

ih. = i‘ 1 al

x/c

v/c

0.50 0.20 =0.16 0.00 0.10 0.20 0.30

xc

Figure 4.3.4 Pressure gradient contours, 9P/0x and op/dy , at time level 4 (4/50 of period, t=1.24 sec.)

139

axl 5 q

r 7

020+

P 4

r 0000 we 0.10} o@go0°0

° ° r o ° r a> 4

0.00}

. ° 4

~G,10 r o> 1

ntl nth. i 4 A. Bh ealinmabie A i. is i. .% A A. i

0.0 0.20 0.10 0.00 0.10 0.20

x/C

1.3 r 4

P q

r 1 r 4

Q.20- 4

d

p 4

We a.tot

> 4

0.00F» q

| ~0.10} 1

r ee ae dcsemmadhyrempardnnarrenemnen tion eae 2. a i. dhrenelivrnrenes

0.0 ~0.20 0.10 0.00 0.10 0.20

Figure 4.3.5

GPUX CONTOURS We Ue. §

20.00

13.00

10.00 RNS

3.0 TIT

-10.00 P-22-0 -13.00 20.09 LLL

20.00

13.00

10.00

3.00

~10.00

-13.00

Pressure gradient contours, Op/dx anda Ip/dy ’ at time level 5 (5/50 of period, T=1.24 sec.)

140

0.30 F

0.20 F

We onto ~0.10+

.

dh dell L Deenend rn

40 0 2.10 000 0.10 020 0.0

x/C

0.30

TF

We at

¥ T

¥

0.00

Pressure gradient contours, op/dx and op/dy ' Figure 4.3.6 at time level 6 (6/50 of period, T=1.24 sec.)

141

0-3

020 20.00

r . ] 10.00 ] 5.00

“5.00. 10.00 15.00 70.00

Wc 0.10}

aot ©

iM? Fe

08 —=0.10 0.00 0.10 0.20 6.30

x/c ;

0.30 ; ~ _

~ q WE

on 1 20.00 15.00 10.00

Y/c 3.00

-10.90

-13.00

Figure 4.3.7 Pressure gradient contours, @p/dx and op/dy , at time level 7 (7/50 of period, t=1.24 sec.)

142

ee lee

O.F

r 1 DPOX CONTOURS 1 Te (EME 8

0.20 4

We aso}

r omy

“0.10 L

050

x/C

—— . ——

0.30F

r 1 OPOY

. 0.20+ 272.00

L 15.00 L 10.00 r 1 5.00 Y/c

0.10} -5.00 ’ ~10.00 L -1500

mt O ) r J r 4 O VORTEX. POSITION

0.10 1

0a 08 “O10 0.20 0.30

Figure 4.3.8 Pressure gradient contours, op/dx and Op/dy '


143

O.3-

0.20F

¥/c

O.00r

ox}

0.20F

We sol

0.00F

Figure 4.3.9 Pressure gradient contours, dp/dx and op/dy ' at time level 9 (9/50 of period, T=1.24 sec.)

144

0.30 Fr

0.20

We ono

0.00

0.20-

We oto

0.00;

Figure 4.3.10

20.0

13.00

10.00

~10.00

135.00

-20.00

Pressure gradient contours, dp/dx and Op/dy '

at time level 10 (10/50 of period, t=1.24 sec.)

145

L OFUX CONTOURS ’ THE LEVEL 11

0.20} ] | 20.00

r 13.00 10.00

We asot $00 r - 5.00 j -10.00

O an aco -20.00

-0.10}

050 020 010 000 ai 0.20 0.50

x/C

0.30 F 4

UPOY CONTOUES } 7 Tae (EEL 11 } 1

0.20} 4

| L —e 4

WARNS, We ast KS ~~

. NUE N i yY, => |

oO -

r 000

0.00} 00 q a0

r Qo

-0.10} ]

Mann dl 0.50 0.0 O10 0.00 0.10 020 0.30

x/t

Figure 4.3.11 Pressure gradient contours, 9P/0x and dp/dy , at time level 11 (11/50 of period, 1=1.24 sec.)

146

re ee ee ee

a 4

0.30} 4

; ] DPOX CONTOURS 1 THE (EMEL 12

F 1 0.20 . »

q 4 220 ERX

| 18.00 $55 / 19.00 SSS ~

\

WE oto} ao ; -1900 #2 2200

r -15.00 ; 20 tiit

0.00+

L O WORTEX POSTON

0.10} L.

i s Ba A . + hh — oo | i A. 77 A A A 4 A Ad

0.30 “2.0 0.10 0.00 0.10 0.20 0.30

x/C

J 0.30

] OPUY CONTOURS L 4 THE LEVEL 12 - 7

0.20} L.

i.

bo

We 0.10}

Q.a0

0.10

ra fi al A L at fs A a A 2

25 0 3.10 0.00 0.10 020 0.30

Figure 4.3.12 Pressure gradient contours, dp/ox and op/dy '


147

0.30}

0.20+

We O.10F

0.00 +

0.30;

0.20};

¥/C 0.10

0.00 F

0.50 0.20 0.10 00 0.10 0.2) 0.30

Pressure gradient contours, dp/dx and op/ody t Figure 4.3.13 at time level 13 (13/50 of period, T=1.24 sec.)

148

TT

L 0.30}

q

r

0.20}

i

We oso}

L

ano}

~0.10}

030 0.20 B10 a0 SCA 020 am

x/C

4

| q

4

020 ’

| : . SN SAAN A 4

Y¥/c 0.10} ‘ ” 7. ~ 1

} as ) > |

L

- _

0.00 r > 7 L 4

-0.10} |

oa -.10—SC~—MS—S~S~N 020 0.0

xe

Figure 4.3.14

13.0

10.00

3.0

-10.00

-15.00

-270.00

Pressure gradient contours, 0P/dx and op/oy , at time level 14 (14/50 of period, T=1.24 sec.)

149

0.90;

r 1 DPOX CONTOURS . 7 TE LEVEL 15

0.20F

Y/c 0.10 }

0.00 F

1)

0.20F

We gsoh te 5

0.00

0.16 1

Figure 4.3.15 Pressure gradient contours, dp/dx and op/dy ' at time level 15 (15/50 of period, 1=1.24 sec.)

150

eT

0.530 Fr

0.20

Wo att

0.00 F

~Q.10F

050 0.0 ~O.10 0.00 0.10 0.20 0.50

0.30}

020}

We oto

cao}

hintaan bil 4 anil. 2. eenlies a des A. ln areien a. 4. d. 4. athe A. diceeell i.

O.K 0.20 0.10 0.00 0.10 0.20 a3

Figure 4.3.16 Pressure gradient contours, dp/dx and dp/dy ’

at time level 16 (16/50 of period, tT=1.24 sec.)

151

0.50F

0.20F

We a0 © of 2 \

0.00;

-O0.10F 0.50 B20 0.10 000° OO 020 0.30

I

oso

0.20F

We osol

L

om

—0.10F

Figure 4.3.17 Pressure gradient contours, Op/dx and dp/dy ’

at time level 17 (17/50 of period, 1=1.24 sec.)

152

b/ 4a)

0.50-

0.20}

we 0.10F

0.00 F

0.50 0.20 0.10 0.00. 0.10 “O20 0.0

Pressure gradient contours, dp/dx and op/dy 1 Figure 4.3.18 at time level 18 (18/50 of period, T=1.24 sec.)

153

a

0.30

_—_————OT

TT

0.20

TT

We a0

0.00

Areal nether farmed Ieeptemedeeendh Jorma Srflagoe

DX 0.20 0.10 0.00 0.10 0.20 0.3

0.30

a : G20;

YC 0.10

oe

es eee

ee 7

0.00 F

oe hee erence nese Saree enediaeenenash

0.50 "TB.20 10 0.00 0.10 0.20 0.0

Figure 4.3.19 Pressure gradient contours, Op/dx and Op/oy , at time level 19 (19/50 of period, T=1.24 sec.)

154

5 d

0.30 F

L.

0.20; d

b 4

r oo 1 We 0.10+ ° O90 4

L 0000 oo

. 6 a

0.00/ 4

L d

L

—~G.10- ° 4 ww

a) a “Ta. 0.00 — 0.10 * 0.20 —

x/C

0.50 q - J

r 4 . 4

020+ 4

We oso }

OOF 3

P 1 L

-0.10¢ ;

0350 mami alban. —— a = “10 000 A. A 0.10 Nem i a 020 i wii.

Figure 4.3.20

20.00

13.00

10.00

3.00

-10.90 ft -13.00

Pressure gradient contours, dp/dx and op/dy ’ at time level 20 (20/50 of period, 1=1.24 sec.)

155

0.

0.20F

VW a

9.00

4.10

TF

0.50F

0.207

r

0.007

L

~0.10 Foner fhn et rec grt elt reece

0.50 0.30 =.10 000 0.10 0D “0. " L dhecesnedl eae Deed, A 4

x/c

Figure 4,3.21 Pressure gradient contours, Op/dx and op/dy ' at time level 21 (21/50 of period, t=1.24 sec.)

156

0.30} q

0.20}

Ww 0.10

0.00F

0.20F

¥/c

0.00+

0.50 0.20 0.10 0.00 0.10 0.20 0.50

Figure 4.3.22 Pressure gradient contours, dp/dx and op/oy , at time level 22 (22/50 of period, t=1.24 sec.)

157

0.

Q20-

Y/e Q.10F

0.00 F

—0.10}

0.S0F

0.205

We anh

0.00F

Figure 4.3.23 Pressure gradient contours, dp/dx and op/dy '


158

0.30}

P DPOX CONTOURS q TE LEVEL 24

Be

0.20 20.00 P 15.00 L 10.00

¥/c 5.00 .10 F

0.10 r 5 -8.00 00 -19.00

. oo

} ° ~-15.00

0.00} “20.0

0.10} J

059-020 0.10 ——«0.00 “710. 020° oD

x/C

0.30}

r PUY CONTOURS L 4 TWE LEVEL 24

0.20} a

¥/C 0.10

0.00

~Q,10 r

0 02

Figure 4.3.24 Pressure gradient contours, dp/dx and dp/dy 1 at time level 24 (24/50 of period, T=1.24 sec.)

159

q

0.30 1

azo

We asot

| L

o.oo L r

i.

-G,10F

r

~0.50 0.20 —ait0

x/t

i.

0.30+

; ] DPOY CONTOURS } THE LEVEL 25 ’ 3

ozo ;

L 1

We osot L ‘ 4

r _ Lr 4

0.00 F |

} 1

-osol | ene

O50 0.20 =0.10 0.00 0.10 020 0”

“fc



160

r q “mt

] DPOX CONTOURS r TME LEEL 26

1 0.20} 4

L ] 20.00 I ] 15.00

, 10.09

We aot 300 : -3.00

L -10.00 L “15.90

0.00} ~-270.00

L

0.10} .

450 020 0.10 0.00 0.10 02 00

x/C

0.30 a 4 OPUY CONTOURS } 4 THE LEVEL 26 > 4

0.20} 1

L ] 5.00 7 a0} -3.00

L -10.00

o.oo}

[ O VORTEX POSTION

~a.10t

area Co | 0.50 0 2:10 0.00 0.10 0.20 0.3

Figure 4.3.26 Pressure gradient contours, Op/dx and op/oy '


161

L os} d

1 DPOX CONTOURS } 1 Ta iEVEL 27 > q

0.20}

r 18.09

| 10.00

WE oto} é a r -10.00

- 13.00 a J -20.0

qj

L 1 O VORTEX POSITION -0.10 1

10 020 010 000 ©1020 @.30

x/C

4

0.30

L OPUY CONTOURS THE LEVEL 37

020+

WC ost ]

0.00} ] Po 4

L 4 O VORTEX POSTION 0,10} J

L

0.5 0.20 0.10. 000 0.10 020 0.30

xe

Figure 4.3.27 Pressure gradient contours, °P/0x and Op/dy , at time level 27 (27/50 of period, T=1.24 sec.)

162

0.20

WE ast

0.00F

0.107

= 4. sdceenahenseenleee eel denn a a Sheena L rl 4 i A 4. deme i —_ i.

0.0 0.20 0.10 0.00 0.10 0.20 0.30

x/C

0.30

rT

T

a ‘4

+

0.20}

WE asl

G.00F+

-0.10}

0.50 20 TO 000 10 S—SOD 0.30

x/C

Figure 4.3.28 Pressure gradient contours, dp/dx and op/dy ' at time level 28 (28/50 of period, t=1.24 sec.)

163

Bs 4

a.soF

L ‘ DPOX CONTOURS TE LEMEL 29

020+ 4 . J 2.0

15.00 r 10.00

WE aot +” T 5.00 r -10.00

15.00

0.00}

I Oo VORTEX POSITION -0.10+

A. all celine s af A abhi i. ail = i. A i. edie aticmrnill AL A .

0.50 O20 =O:10 0.00 0.10 020 0.30

se

o.so} 1

1 OPUY CONTOURS F ’ Te LEVEL 29 a q

0.20 q 4

Y/c 0.10}

0.00}

0.10}

00 020 010 0.00 010 O20 0.30

x/C

Figure 4.3.29 Pressure gradient contours, dp/dx and op/dy ' at time level 29 (29/50 of period, 1=1.24 sec.)

164

0.30} ]

L DPOX CONTOURS 1 TWEE LEVEL 3D

0.20F 1

] 13.00 L 1 10.00

We ato} | 5.00

10.00

-13.90 Q.00F

L

oO

-0.10F L

35 s dt. i A ‘10 SL “0.00 Bed “S10 ‘ A 2 020 ‘ ai A 0.30

x/C

4

0.30 F a

r OPOY CONTOURS r 1 Te LEVEL 30 a q

0.20} 4

We 0.19 r 4

r om 4

0.00} be 4

q Oo VORTEX POSTION

-0.10} 4

heretical Derren A al Meedh. itera erin eelieced setts

7) 0.20 0.10 0.00 0.10 020 0.30

x/C

Figure 4.3.30 Pressure gradient contours, 9p/dx and op/dy ' at time level 30 (30/50 of period, t=1.24 sec.)

165

0.30F

820r

we 0.107

Q20F

0.30 0.20 10

xX/C

0.30} 4

0.20;

We oso

0.00»



166

OF

ri i

020+

We at

0.00}

~0.10}

a A 0° . “a 1O i

x/C

<

0.30 4

| ] DPUY CONTOURS } ; THE LEVEL 52 - q

0.20} ’ L. d

SESS ¥/c 0 10 4 9.000

, r ~5.000 ore

/ _ ]

0.20 L :

; O VORTEX POSITION ~0.10} ;

a eh nedl a A A A i ‘a A i ‘ A A a Dl i. A Deemansclemmendl i

0.90 0.20 0.10 0.00 0.10 020 7.0

x/C

Figure 4.3.32 Pressure gradient contours, dp/dx and Op/oy '


167

0.0F

r DPOX CONTOURS r 1 AME LEVEL 33

0.20F 4

L 4 10.00 - 4 5.00

Wo aso} ] 8.00 -10.00

-13.00

Q.00;-

~0.10}

0.30

0.20 F

We atot |

Stk

0.00F

celica feed Daerah

0.0 0.20 0.10 0.00 9.10 0.20 0.

Figure 4.3.33 Pressure gradient contours, dp/dx and op/oy , at time level 33 (33/50 of period, 1=1.24 sec.)

168

0.30-

0.29

¥/c 0.10;

9.00 fF

0.30

0.207

We ano

0.00 F

~0.10F

Figure 4.3.34 Pressure gradient contours, Op/dx and dp/dy '

at time level 34 (34/50 of period, 1T=1.24 sec.)

169

be : 4

al

OPOX CONTOURS } 7 THE LEVEL 3S } 4

9.207 4

be: a 10.00

L 5.00 We ato} -$.00

-10.00 L -13.00

0.00}

L O VORTEX POSTION

~0.10}

a Ga 0.50 0.20 =a.70 0.00 G10 020 0.50

xf

b q

O.30F 4

P 1 OPOY CONTOURS , 1 Tae LEVEL 35

ozo! | L d

10.00

wc | 3.00 0.10} sa

y ] -10.00

> o00r

.

be q

L d

L ] © VORTEX POSTION

-Q.10F p

i

0 SSC oD

Ye



170

I , a.sot

b 4

L DPDX CONTOURS 1 THE (EVEL 36

0.20 Fr 4

L 4 10.00 L 4 5.00

We ast | 8.00 L 10.00

-15,00

0.00}

O VORTEX POSTION 0.10 F

00 A ‘0 a. ‘30

x/C

0.30}

r 1 OPUY CONTOURS TME LENEL. 36

0.20}

w/e O.AOF

L

a.00+ L.

0.10

gt ag a

Figure 4.3.36 Pressure gradient contours, dp/dx and dp/dy ' at time level 36 (36/50 of period, t=1.24 sec.)

171

O.30F

ri t

o20f

10.00

We asp ; -8.00 -10.00

-13.00

0.00;

0.30

Tr

Deen

0.20-

We ato}

0.00 .

10/7

050 02 0.10 0.00 “O10 020 0.30

Figure 4.3.37 Pressure gradient contours, Op/dx and op/dy '


172

030

0.20

We oto

a

0.00 ¢

0.50 ~O.20 “3.10

O.0F

Phe

heen

4, 4

0.20-

obi

WE orot p 4 -3.00

0.00

0,10 +

fee ncereeeeeenthieeaadineesteehe each dneoenaheasecmil ‘ so che 0.0 0.20 0.10 0.00 9.10 0.20 0.30

x/C

Figure 4.3.38 Pressure gradient contours, dp/dx and op /dy ,


173

+ ; 0.30 1

F 4

L ’ OMe LEVEL 3 4

0.20+

We ast 4 L

oo

r an SO

0.00} Go J } ;

“0,10 .

' > eat ps ,, SPL.

0.50 0.20 ~0.10 0.00 0.10 0.20 0.0

x/¢

0.30} 4

P DPUY CONTOURS - TWE LEVEL 39

0.20}

we oxo 4

ry ot ~ 1

o.oo} ; L q

4 Oo VORTEX POSITION

0.10 4 L =>

05 mS OTSS—ODSSC~CI 0.20 0.0

x/C

Figure 4.3.39 Pressure gradient contours, dp/dx and op/dy 1


174

L ; 0.30}

P OPOX CONTOURS q TWE LEVEL 40

0.20} 4

r 1 15.00

L 10.00 Ye 5.00 0,10} ]

r -3.00 ~10.00

L -13.00

0.00}

L O VORTEX POSITION 0.10

0.50 0.20 0.10 0.00 0.10 O20 OD

x/c

4

0.30} 4

r OPUY CONTOURS } 1 Te LEVEL 40

0.20}

We asot

0.00}

be

aol

050 0.0

Figure 4.3.40 Pressure gradient contours, 9P/dx and op/dy , at time level.40 (40/50 of period, T=1.24 sec.)

175

0.30}

r POX CONTOURS , 1 TE LENEL 41

0.20} ’ q q

L : 10.00 be 5 $.00

We anol 4 8.00 4 -10.00 r e009

-13.00

0.00

0.10

O.F

0.20 ¢

r ] <= §,000 ao SA

a.0F YC

0.00 fF

Figure 4.3.41 Pressure gradient contours, Op/dx and op/oy '

at time level 41 (41/50 of period, 1t=1.24 sec.)

176

0.30 F 1

P DPOX CONTOURS } TE LEVEL 42 > 4

6.20 be q

. 4 10.00

5 4 §.00

We atot 4 8.00 L 10.00 L -15.00

0.00}

L O VORTEX POSTION ~0.10+

L

050 0.0 0.10

w/e

} ; 0.30} ]

P OPOY CONTOURS } TE LEVEL 42 - q

0.20- A 1

ws yo

Y/C oto} ] 3.000

. -$.000

0.00 +

~0.10+

00 An. “a0. i.

Figure 4.3.42 Pressure gradient contours, dp/dx and op/dy ' at time level 42 (42/50 of period, T=1.24 sec.)

177

020r

We asot 4

0.00 F

0.10}

i.

ago tet ott

x/C

ODF

] DPUY CONTOURS r 7 WME LEVEL 43

q

0.20-

r 4

r 1 SSS h >

We ato} 4 5.000 -$.000 roeruye

1 22292

0.00F

=0.10-

0.50 0.20

Figure 4.3.43 Pressure gradient contours, dp/dx and Op/oy ,


178

Ree

Tr T ee

eT

TT

TT

Ts

10.00

3.00

-10.00

—15.00

Pressure gradient contours, dp/dx ana op/dy ' (44/50 of period, t=1.24 sec.)

Figure 4.3.44 at time level 44

179

oso}

0.20}

We aso}

0.00}

0.10>

0.50 8.20 6.10 0.00 S—SO 10 0.20 0.”

x/C

—— T T T —

0.30F

0.20-

r ] 19.00

Y/c | 5.00

L q 4 -10.00

0.00 F

4.20 0.50 0.20 0.10 000° 0.10 “0.20 0.30

Figure 4.3.45 Pressure gradient contours, dp/dx and op/dy , at time level 45 (45/50 of period, T=1.24 sec.)

1 180

0.30}

0.20}

P

We asol L

0.00

4.10

TT

9

0.20

We 0.10

ee

ee

0.00

TTT

0.10}

Dino

nah

‘. Ae

10.00

3.0

-10.00 =

Figure 4.3.46 Pressure gradient contours, dp/dx and op/dy ’

at time level 46 (46/50 of period, 1T=1.24 sec.)

181

030 r

0207

¥/c aso

G.00r

0.10

0.0 0.20 0.10 0.00 0.10 0.20 0.30

x/C

a

O.30F

4 4

2

0.20F

WC r | 0.10F |

0.00 -

0.105

090 0.20 0.10 000 0. —S—SO20 0.50

x/C

Figure 4.3.47 Pressure gradient contours, Op/dx ana Ip/dy '

at time level 47 (47/50 of period, 1=1.24 sec.)

182

0.30

020+

we 0.10 F

0.00 F

~O.10F

Pt Bane saeeed ene eh

0.50 020 0.10 0.00 0.10 0.20 7.30

0.207r

We got 4

0.00 F

0.50 0.20 0.10 00 0.10 020SSSC~«SD

Figure 4.3.48 Pressure gradient contours, 9P/0x and op/dy ' at time level 48 (48/50 of period, t=1.24 sec.)

183

0.530

0.20 4

we 0.107 oo0o°0 4 o00gQ0 ]

0.00;

0.30

0.20+

WC ] 0.10}

ed |

0.10}

ot Dern cted wel i be 2

0.50 70.20 0.10 “3.00 0.10 0.20 0.30 ah reehenehenneed a deed A La

x/C

Figure 4.3.49 Pressure gradient contours, dp/dx and dp/dy ’ at time level 49 (49/50 of period, 1T=1.24 sec.)

184

} BO 4

0.30+

o.20}

-

We at

0.00+ .

0.10

0 cH seen 0 ah.

x/C

0.350} 7

r 1 DPUY CONTOURS L 1 TME LEVEL 50

0.20} 1 20.00

[ 13.00

10.90 r 5.00 VC d 0.10+ “5.00 r -10.00 / -13.00

0.00}

O VORTEX POSITION

~0.10+

25 0.0 0.10

Figure 4.3.50 Pressure gradient contours, 9P/0x and op/dy , at time level 50 (50/50 of period, T=1.24 sec.)

185

-2.5 4 E 1 c Time Level 1

-2 4 E

1 2 y/c=0.3625 -1.5 4 Po —--7y/e=0.2625

Cc 1 r — — ~y/c=0.1625 Pp y/c=0.0625

-0.5 4 E

0 - '

0.5 4 ; aa et -0.3 -0.2 -0.1 0 0.1 0.2 0.3

x/Cc

-3 4 1 1 to mt a

F "2.5 + E Time Level 2

25 poof eee y/c=0.3625 ; —----y/c=0.2625

-1.5 4 E — — ~y/c=0.1625 Cc E y/c=0.0625

Pp : -4 E

E

pha

ee di

0.5

o 4

0.5 Ft q "Ty * Tp Pe br oF +

0.3 -0.2 -0.1 0 O01 #02 0.3

Figure 4.3.51 Pressure coefficients at four y/c locations,

at time levels 1 and 2 (1/50 and 2/50 of

period, t=1.24 sec.)

186

-3 1 april a tL a i 4 Lo. 1 |

-2.5 E Time Level 3

-2 4 Pf ee y/c=0.3625 F | —----y/e=012625

-1.5 4 ro — — ~y/c=0.1625 Cc y/c=0.0625

Pp 7 .

“14 E

0 5 Tt T I F T 7 t T v T J v v | v 7 TT

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

x/Cc

-3 1 rs Leow ae | 1 1 sl a_i tk

-2.5 4 7 Time Level 4

-2 4 Po ee eee y/c=0.3625 F | —----y/e=0.2625 EF | — — -y/e=0.1625

C, y/c=0.0625

erytTt

Figure 4.3.52 Pressure coefficients at four y/c locations, at time levels 3 and 4 (3/50 and 4/50 of

period, tT=1.24 sec.)

187

- 3 . hhh de po at 4 dg

r

"2.5 4 - Time Level 5

24 Ff wees y/c=0.3625 1 r —---~y/c=0.2625

-1.5 = P — — ~y/c=0.1625 C. 1 t y/c=0.0625

E

0.3

-3 pe

2.5 4 C Time Level 6 J r

-2 4 PP tt y/c=0.3625 1 r —----y/c=0.2625

-1.5 4 E — — ~y/c=0.1625 C., 1 E y/c=0.0625

-1 2 L 1

-0.5 4 E

Kee TI Tere ee 0 fee F

0.5 ne ee TT T TT —+

-0.3 -0.2 -0.1 0 0.1 0.2 0.3


period, 7=1.24 sec.)

188

-3 pe a a

]

2.5 4 r Time Level 7

-2 4 FO eee. y/c=0.3625 F | —----y/e=0.2625

-1.5 4 p — — ~y/c=0.1625 C, y/c=0.0625

“14 r

-0.5 4 : 4

9 | : 0.5 +> TTF 77 T TT

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

x/C

-3 oo Deane iy ! a tg

-2.5 4 E Time Level 8

-2 + Fo ee eee y/c=0.3625 1 r —---~y/o=0.2625

-1.5 4 f — — ~y/c=0.1625 Cc : y/c=0.0625

P -1 2 a

-0.5 4 F

7 fb

Oo -

: 0.5 ee +--+

-0.3 -0.2 -0.1 0 0.1 0.2 0.3



period, T=1.24 sec.)

189

-3 bo a tl i 1 I nm

3 r

-2.5 + E Time Level 9

-2 4 Fo eee y/c=0.3625 —----y/c=0.2625

-1.5 4 E — — -y/c=0.1625 C, y/c=0.0625

-3 4 bapa ty i ! oa

; -2.5 ~ E Time Level 10

-2 4 Fo eee y/c=0.3625 r —-+-~y/c=0.2625 b — — ~y/c=0.1625

C. E y/c=0.0625

' rrrryprvrryrrrryprrr



190

-3 4 ia L i nm Li L

-2.5 1 fF «636s Time Level 11

-2 4 fF Od eee y/c=0.3625 1 F —----y/c=0.2625 r — — -y/c=0.1625 C. y/c=0.0625

-3 I I L ait 1 1

-2.5 ; fF Time Level 12

-2 4 2 oo y/c=0.3625 —----y/c=0.2625 — — -y/c=0.1625

y/c=0.0625




191

-3 i

-2.5 4 c Time Level 13

-2 4 Po} wees y/c=0.3625 ] —----y/e=0.2625

-1.5 4 E — — -y/c=0.1625 C, : y/c=0.0625 - 1 4 E

-0.5 4 .

0 4 E

0.5 +— -0.3 0.3

-3 + 1 nad do i

q t -2.5 7 EC Time Level 14

7 b

-2 7 TP ts y/c=0.3625

1 ; —----y/c=0.2625 -1.5 4 C — — ~y/c=0.1625

C, r y/c=0.0625

-1 4 E

-0.5 4 _~---->=~~t

0 4 E r

0.5 eee aa -0.3 -0.2 -0.1 0 0.1 0.2 0.3

x/Cc



192

Figure

arpa tarra

li,

4

rrryt

TT

TTT

rryy

rr

rr

rr

Trt

at -0.3 -0.2 -0.1 0 0.1 0.2 0.3

x/Cc

jdt.

bh

rrryprvrrytrvrryrryryr

4.3.58 Pressure coefficients

at time levels 15 and 16

period, ™=1.24

sec.)

Time Level 15

y/c=0.0625

Time Level 16

Fetes y/c=0.3625 —----y/c=0.2625 — — -y/c=0.1625

y/c=0.0625

at four y/c locations, (15/50 and 16/50 of

193

-3 a Jin 1 foo 1 oo

"2.5 4 F Time Level 17 -

F | eee y/c=0.3625 § —---~y/c=0.2625

P — — ~y/c=0.1625

C, y/c=0.0625

0.5 ST

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

x/Cc

-3 —in 1 Pour ane Ge 1 _t aaa 4

“2.8 1 7 Time Level 18

en eee y/c=0.3625 —----y/c=0.2625

-1.5 4 — — -y/c=0.1625 C., y/c=0.0625

-1 4 1

-0.5 4

0 4

0.5 4 a .

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

x/Cc



194

-3 ao tl ld

r


verte y/c=0.3625 —----y/c=0.2625 — — -y/c=0.1625

C, y/c=0.0625

0.5 4 —T nae ee a To -0.3 -0.2 -0.1 0 0.1 0.2 0.3

x/c

-3 ut oo a i ed +

-2.5 + fF «6836s Time Level 20

24 pf y/c=0.3625 : F | —----y/e=0.2625

-1.5 4 C — ~ ~y/c=0.1625 C, y/c=0.0625



195

-3 1 it 4 t 4 Fare ae rea eer es

4 r .

-2.5 4 F Time Level 21

-2 4 Foo eee y/c=0.3625 —----y/c=0.2625

-1.5 4 F — — ~y/ce=0.1625 C. j r y/c=0.0625

7 be

-1 74 r 4 5

-0.5 = C c

0 - r

3 ; 0.5 T

-0.3 0.3

-3 tan a a po pda 4


1 E -2 4 Po tte y/c=0.3625

q r —----y/e=0.2625 -1.5 4 E — ~ ~y/c=0.1625

C, : y/c=0.0625 1 3 ;

J r -0.5 7

j meet 0 4 bee ce ee ee ro

; : 0.5 ta -0.3 -0.2 -0.1 0 0.1 0.2 0.3

x/C

Figure 4.3.61 Pressure coefficients at four y/ce locations,



196

-3 + at rn a a ee mn Cs Ps sate

"2.5 5 C Time Level 23 4 }

-2 4 2 io y/c=0.3625 . —---~y/c=0.2625

-1.5 4 E — ~ ~y/c=0.1625 Cc ] t y/c=0.0625

-3 a

-2.5 4 C Time Level 24

-2 4 F OU] eee y/c=0.3625 | —----y/c=0.2625

-1.5 + * — — -y/c=0.1625 C., 1 F y/c=0.0625



period, T1=1.24 sec.)

197

-3 4 fe }

7 r

-2.5 4 rf «86 Time Level 25

-2 4 2 y/c=0.3625 1 —----y/c=0.2625

-1.5 - E — — -y/c=0.1625 C. ; c y/c=0.0625

-1 4 r q r

0 eee Cn ne Fe RT ET \E F

oO 5 TT t v “rT { FOF , Te e fot Fr TO TOT

-0.3 -0.2 -0.1 0 0.1 02 0.3

x/Cc

-3 oa a

-2.5 E Time Level 26 é

-2 + rf eee y/c=0.3625 : F | 9/62 012625

-1.5 4 — — ~y/c=0.1625

C. y/c=0.0625

rerryprrerprrrrpryvry

|

0.5 7 T IIIT

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

x/Cc

Figure 4.3.63 Pressure coefficients at four y/c locations, at time levels 25 and 26 (25/50 and 26/50 of period, T=1.24 sec.)

198

-3 an i. een esshinnedh ah de i heehee senate Ll poe

2.5 4 E Time Level 27

"24 F | eee y/c=0.3625 —----y/c=0.2625

-1.5 3 L — — -y/c=0.1625 C, y/c=0.0625

-1 43 ¢

: 0.5 4 T T TTT rT T — f

-0.3 -0.2 -0.1 0 01 O02 03 x/Cc

-3 1 _ to bau a L 4 1 l 4

“2.5 : Time Level 28

-2 4 Ff eee y/c=0.3625 1 r —----y/c=0.2625

-1.5 4 b — — ~y/c=0.1625 C. } y/c=0.0625

-1 4 r



199

-3 pd ll

-2.5 4 fF «6h os Time Level 29

-2 - Po ee eee y/c=0.3625 1 t —----y/c=0.2625

1.5 - E — — -y/c=0.1625 c Tb y/c=0.0625

P H -1 4 C

-0.5 4 E

0 4

0.5 T TT TTI

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

x/c

-3 4 _L fa 1 too 1 L

-2.5 4 rf Time Level 30 +

2p ee y/c=0.3625 —----y/c=0.2625

C -1.5 4 — — ~y/c=0.1625 p j y/c=0.0625

-1 - 1

-0.5 7

0+

0.5 + -0.3




200

-3 fa

Tee

yy

TT

28 C Time Level 31

° E mote y/c=0.3625

f —----y/c=0.2625

18 3 EF 6 | — — ~y/c=0.1625

3 t y/c=0.0625

Pp 1 |

-1 ~~

E

} E

-0.5 4

:

3 —t

0 4 FE rome

EE

3 0.5 + ae —_

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

x/c

-3 Dab ] 1 Lo 4 | : . -

r

28 F Time Level 32

r

“3 pp y/c=0.3625

:

—---~y/c=0.2625

1.5 - r | — — -y/e=0.1625 ¥/e=0.0625



201

-3 1 ase apr tanner tLiyyypy typ pan ti ap pe Lan

2.5 4 r Time Level 33

“2 3 FO, y/c=0.3625 1 c —---~y/c=0.2625

-1.5 4 P — — ~y/c=0.1625 C 1 C y/c=0.0625

Pp 1 ; -1 7 .

-0.5 - r

1 r 04

0.5 +. -0.3

-3 1 ae Do it 1 _

1 r -2.5 + r «6s Time Level 34

a Po free y/c=0.3625 ] —----y/c=0.2625

-1.5 4 L — — -y/c=0.1625

Pp : ye=0.0625

-1 4 a


at time levels 33 and 34 (33/50 and 34/50 of period, T7=1.24 sec.)

202

-2.5 q r Time Level 35

-2 4 Fo woe y/c=0.3625 1 t —---~y/c=0.2625

-1.5 5 E — — ~y/c=0.1625 co | y/e=0.0625 -1 4 E

-0.5 1 E

0 4

0.5 . oe ee rey t T TT oT FOF T ¥ 7° T T roy rere qT T rr T Tr

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

x/Cc

-3 Pane a a a

: c

-2.5 - r Time Level 36 4 e

°2 4 nn y/c=0.3625 t —----y/c=0.2625

-1.5 4 E — — -y/c=0.1625 C 1 t y/c=0.0625

P 1 t “14 C

1 r -0.5 + E

a: O Ree — &

0.5 +. I qt tr ta

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

x/c




203

-3 Pia t tos i +

2.5 q fr «=86Time Level 37

-2 4 a oo y/c=0.3625 1 r —----y/c=0.2625

rT — — ~y/c=0.1625 om y/c=0.0625

E

F E

0.3

-3 po a kk a a

H 2.5 1 E Time Level 38

-2 4 poof ee eee y/c=0.3625 1 r —---~y/c=0.2625

-1.5 4 E — — -y/c=0.1625 C y/c=0.0625

TTT TT rr

T vrvryt


period, 1=1.24 sec.)

204

-3 2 1 he L bn 1 l tt he

3 : -2.5 4 f Time Level 39

-2 4 Fo eee y/c=0.3625 —----y/c=0.2625

-1.5 4 cE — — ~y/c=0.1625 C y/c=0.0625

Pp ; 4 C J r

-0.5 7

0 4 4

1 0.5 T vr TY — * Tf T T T T T TT vey T rT UT | as TF 7 } - T v T

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

x/C

-3 : pe a

1 r -2.5 4 r Time Level 40

-2 4 Poof eee y/c=0.3625 fF | ++ y/e=0.2625

-1.5- E — — ~y/e=0.1625

C E y/c=0.0625 p 7 47 :

C



205

3 Joh hl ”

Time Level 41

-2.5 7 r

1 ne ae y/c=0.3625 -2 7 C —---~y/c=0.2625

] t — — ~y/c=0.1625

-1.5 7 . y/c=0.0625

-3 pec A. ponte, elt pent ene ade

-

95 4 E Time Level 42

po fee y/c=0.3625 "2 4 F | + y/o=0.2625

: — — ~y/c=0.1625 -1.5 7 - y/c=0.0625

Cc } Pp


period, tT=1.24 sec.)

206

3 eh i. ra | _ a It + 1 ehh i

E Time Level 43

2 ae y/c=0.3625 r —----y/c=0.2625 t — — ~y/e=0.1625 E y/c=0.0625

:

: 0.5 — Ty YT? T a ee ee ne eee ee ne a ee ee ee ee ee ee ae r

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

x/C

- 3 L 1. 1. Pas ee ee Ge 4 }

o54 : Time Level 44

| f 2 1 pf tt y/c=0.3625

"ey r —---~y/c=0.2625

} — — ~y/c=0.1625 -1.5 4 E y/c=0.0625

o

’ = i

ey

0.5 ee a rT mT

-0.3 -0.2 -0.1 0 0.1 0.2 0.3


period, 1T=1.24 sec.)

207

f 954 + Time Level 45

Ff wees /c=0.3625 -24 | 9 7e=02625 F | = = ~y/c=0.1625

y/c=0.0625

-3 fe a a

Time Level 46

-2.5 1

Po ee /c=0.3625 "25 —---- 4630-2625

— — -y/c=0.1625 y/c=0.0625

t oh du.

PU TPrrprrreprrrryprvyrryprrrr

yrrryprere


period, 1t=1.24 sec.)

208

Figure

T

VT

rrr

Tr

4.3.74 Pressure coefficients

at time levels 47 and 48


Time Level 47

mts y/c=0.3625 —----y/c=0.2625 — — ~y/c=0.1625

y/c=0.0625

Time Level 48

an y/c=0.3625 —---~y/c=0.2625 — — ~-y/c=0.1625

y/c=0.0625


209

dp pe

Trey rrr

ry rrr ry

rt

To TTT T1774 TTT 171-4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

x/C

-3 ot tou. 2 Il 1 1 i

-2.5 4 L

2:

0.5 TF TT ¢ F 7 1 TT T vr qt v TT T Lm

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

x/C

Figure 4.3.75 Pressure coefficients at time levels 49 and 50

period, 1=1.24 sec -)

Time Level 49

y/c=0.3625 --~y/c=0.2625

— — -y/c=0.1625 y/c=0.0625

Time Level 50

y/c=0.3625 --y/c=0.2625

— — -y/c=0.1625 y/c=0.0625


Time Levels

| -2.5 4 E

; F -24 BP t=2 25 r —-"-~t=6

' — — -t=10 Po t=12 r t=14

ryrT

4 <q +

—_ 4 “4

4 “

— + <q

4 “4

-

4 ~ 4 + 4 4

Time Levels

Figure 4.3.76 At y/c = 0.0625, pressure coefficients for

time levels 2, 6, 10, 12, and 14 are combined. Also, the same curves are shifted

by an integer constant.

2ii

-3 an ita ba fn A. J Pech hole

Time Levels

-2.5 4 rE 2 io t=15

—----t=17 — — -t=19 a---- t=21

t=23

4

rerryprerryprrrvyrrrryrrrryyr

‘ Oo a) ' Oo Ph ' oO

a. oQ oO _— oO NO

oO Ww

x/ec

-2.5 2 a a a dl at a

1 F 1.54 C Time Levels

| a ‘\ r nn t=15 -0.5 4 NN pp LE i res a \ F | DUTIES

0.5 - 7 _ Mee | tcc t=21 Cc +¢ ¢=l cece _" 7 ~ E t=23

P 4.54 Z ‘o£ { _¢22 2 7 oT

2.54 wecese - “AE

3.5 4 E

45 re et

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

x/c

Figure 4.3.77 At y/c = 0.0625, pressure coefficients for

time levels 15, 17, 19, 21, and 23 are combined. Also, the same curves are shifted

by an integer constant.

212

-3 : lien ined dt i a phe he q

-2.5 4 E Time Levels

: Pop eee t=24 . “25 F | + t= 26

J f — — -t=30 -1.5 4 PP eee t=40

C., 7 f t=45

14 : -0.5 $

1 r 0 + E + ;

0.5 : a -0.3 -0.2 -0.1 0 0.1 0.2 0.3

x/C

-2.5 L bus i a pet ou

1.54 b Time Levels

1 eee ee — O54f 000 rete B | oar teed J ¢=0 Lene ; = Jo. v eee eee eee - — — -t=30

0.5 4 RP Up tea t=40

Cc +¢ 1 eee ng ~ t=45 Pp 4 .

1.5 3

2.5 - 3.5 4

4.5 ort Ty * TF Ty CTT T T T T TT T TT F TT TT TT FT

' -0.3 -0.2 -0.1 0 0.1 0.2 0.3

x/Cc

Figure 4.3.78 At y/c = 0.0625, pressure coefficients for time levels 24, 26, 30, 40, and 45 are combined. Also, the same curves are shifted by an integer constant.

213

‘

Velocity

Pattern

Upstream Vortex

y

= Ole

_ ee

J) => in

Figure 4.4.1 The affect of the vortex on the airfoil when

it is far away and in the nose region

214

[TtOszzte DyQ

saoqe

ATINZ St

AY usym

xeqzTOA |yQ

FO Yooezze

YL

Zipp

sandStg

SOUNWESNS

ee

g1qqng uolyesedas

215

Vita

At the age of 25, the author saw a light in the darkness. He strove toward that light only to find

that it was a star on the other side of the void of ignorance. He has built ladders and stairs to reach

that star, all failing to reach the height. He is currently using a pogo-stick. That star is no less

alluring today.

Matthew Michael Pesce, son of Ralph and Mary, is committed to touching that star, although

most of the time he 1s stuck in the void.

Vita 216

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