.

AIAA-90-0692 Time-Accurate Navier-Stokes Computations of Classical Two- Dimensional Edge Tone Flow Fields

B. L. Liu and J. M. O'Farrell Rockwell International Space Transportation Systems Division Huntsville, AL

Jess H. Jones NASA Marshall Space Flight Center Huntsville, AL

and

28th Aerospace Sciences Meeting January 8-1 1, 1990/Reno, Nevada

c

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 370 L'Enfant Promenade, S W., Washington, D.C. 20024

Time -Accurate Navier-Stokes Computations of Classical Two- Dimenslonal Edge Tone Flow Fields

Rockwell International Space Transportation Systems Division Huntsville, Alabama 35806

and Jess H. Jones *'*

NASA Marshail Space Flight Center Huntsville, Alabama 35812

8. L. Liu * and J. M. OFarrell * *

flixkxl

Time-accurate Navier-Stok computations were performed to study a Class II (ac' ic) whistle, the edge tone, and gain knowledge of the vortex-acoustic coupling mechanisms driving production of these tones. Results were obtained by solving the full Navier-Stokes equations for laminar compressible air flow of a two- dimensional jet issuing from a slit interacting with a wedge. Cases considered were determined by varying the distance from the slit to the edge. Flow speed was kept constant at 1750 cmlsec as was the slit thickness of 0.1 cm, corresponding to conditions in the experiments of Brown. Excellent agreement was obtained in all four edge tone stage cases between the present computational results and the experimentally obtained results of Brown. Specific edge tone generated phenomena and further confirmation of certain theories concerning these phenomena were brought to light in this analytical simulation of edge tones.

NOmenClature

Speed of sound in a fluid Lift Coefficient Pressure coefficient Thickness of slit Total internal energy Frequency Distance from slit to wedge apex Pressure Root-Mean-Square Pressure Reference Pressure Sound pressure level j-th Stage constant Vortex lranspori time Acoustic transpori time Velocity component in the 'x' direction Initial jet velocity Effective vortex transport speed Velocity component in the 'y' direction

a Wedge angle Y Specific heat ratio P Density

* '*

*'* Lead Engineer

Member Technical Staff, Member AlAA Supervisor, Simulation/Analysis , Member AlAA

High frequency pressure oscillat amplitudes are of practical concert

igh inv

applications. Discovered by Sondhaus' in 1854. the edge tone is one of the basic high frequency whistle phenomena which today still challenges both theoretical and experimental fluid dynamicists for a full understanding. In 1940, Lenihan and Richardson" noted that "The problem of edge tones is one which continues to form a battle-ground for rival theories, though a complete solution seems as far off as ever."

A typical edge tone configuration is a jet issuing from a slit and flowing against a sharp edge, as shown in Figure 1. Previous theoretical models of the edge tone have used semi-empirical methods and predictions but have not adequately described experimentally obtained results over wide ranges of edge tone producing conditions. The present analytical simulation was performed in the effort to gain more insight into the conditions leading to edge tone production. In order to adequately model the acoustic propagation and vortex transport. and their interaction, it was necessary to perform detailed time-dependent computations of the unsteady flow. The method used for this computation was the Rockweli Science Center-developed Unified Solutions Algorithm (USA) code.' The USA code used in the present analysis has been benchmarked to be time accurate for subsonic flow fields involving vortex shedding4 and is presently being applied to this and other problems that involve vortex-acoustic interactions.

Figure 1. Edgetone configuration of a jet impinging on a wedge.

copyright 0 1990 A W C K ~ ~ inmtute or Acronaulicr and AItrOnaUtiCs. Inc. All w h 1 S reserved. 1

This classical fluid dynamic problem was selected for demonstration of the USA code's computational accuracy with acoustic mechanisms in flows at low Mach numbers. In the literature there were many edge tone studies [5] to [29]. Due to the complicated apparatus- dependent vortex-acoustic interactions. the Simple apparatus with easily reproducible boundary conditions used by Brown3. were chosen for simulation. Brown's well documented experimental conditions were simulated lor eight cases and the reSuitS were compared directly with Brown's data, Table 1. Brown experimentally demonstrated lour stable edge tone stages, an edge tone onset distance of approximately 0.33 cm. and a distance of approximately 2.0 cm where the edge tone phenomena has disappeared. The present computations replicated all four edge tone stages and confirmed the upper and lower limits of their existence.

The vortex-acoustic interaction is the basic mechanism of the edge tone phenomenon. The closed- loop feedback cycle of regular vortex shedding and jet disturbances generated by acoustic feedback from vortex generated pressure oscillations near the wedge apex controls the edge tone phenomenon. These jet flow oscillations and jet disturbances occur at predictable frequencies for a particular range of jet velocities and edge distances from the jet slit.

Methodoloav The Unified Solutions Algorithm (USA) CFD code

is a multi-zone compressible Navier-Stokes flow solver. It is a finite volume, upwind-biased, total variation diminishing (TVD) scheme. The TVD discretization eliminates spurious numerical oscillations without introducing additional numerical dissipation. The USA code options used in the present study are perfect gas ( air ) laminar flow, implicit approximate factorization, and full block tridiagonal matrix inversion. The code options used have first-order time accuracy and third- order spatial accuracy. The theoretical formulation is described in the paper by Dougherty".

Figure 2 illustrates the two zones composing the computational geometric domain used in this problem. Zone 1. the inlet channel ( 1 cm long and 1 mm thick), is an H-grid with 41 grid lines along the channel length and 21 grid lines across the channel apertures. Zone 2, an 0-grid surrounding the wedge, was tailored according to cases investigated. Zone 2 was composed of between 1 4 9 to 281 grid lines circumferentially and 41 to 151 grid lines radially. Curvature of the jet was miid in the lower stages the coarser grids were used; however, denser grids were necessary to resolve activity in the higher stages. The grids of both zones are clustered near solid surfaces: the slit aperture and the wedge. The grids are also highly clustered in the active region of the jet between the slit and the wedge, as detailed in Figure 2.A.

There is no other significant sound source inherent in the problem other than the jet-wedge system. To minimize any far-field influence on the activity near the wedge, the boundaries were allowed to surround the wedge at no less than 80 slit widths.

In general. the characteristic distances are the slit- wedge spacing or wedge distance, h, the slit thickness, d = 0.1 cm. and the wedge angle, a = 20.0 degrees. At the channel inlet, a subsonic inflow condition U = 1750.0 cmlsec, is imposed. On the wall surfaces, no-slip adiabatic conditions are used. At the far-field boundaries before and up to the trailing edge of the wedge, outflow conditions are used if the flow is outward, and inflow conditions are used if the flow is inward. Behind the wedge outflow conditions were employed. Subsonic boundary conditions for the USA code require fixing at least one of the variables, pressure or density, on one boundary. A fixed pressure condition was used on the inflow and outflow boundaries of zone 2. The equation for pressure, in terms of the total energy, density, and velocity components ( u and v ) , for a perfect gas is given by:

"

p = ( y - 1.0)'(E,o,-0.5'p'(~2+v2)),

Figure 2. Two-dimensional computational domain and applied boundary conditions.

2

where y is the specific heat ratio for air. In setting outflow boundary conditions, either pressure or density may be lixed. It was found for this simulation fixed pressure conditions gave faster convergence than the fixed density. The boundaly conditions and locations used are shown in Figure 2. -

GiLsfs

Eight edge tone cases were simulated by varying the distance from the slit to the edge in order to induce different stages. According lo Brown's experiments3, Figure 3, when the distance from the wedge lo the slit is less than 0.33 cm. there is inadequate room lor the jet to bend, the flow splits more or less evenly on each side of the wedge, and the edge tone does not occur. In Case 1 , the wedge is within this minimal distance to the slit aperture and the production of an edge tone does not occur. This type of response is called a pre-edge tone case. Case 2 was run at the 0.33 cm edge tone onset distance. Case 3 was a mid-distance Stage I edge tone simulation. Case 4 was a Stage I I simulation. Case 5 was a Stage Ill edge tone simulation, where the wedge was induced to vibrate with a one degree maximum angle of attack, induced according to the expected Stage 111 frequency. Case 6 is a Stage Ill edge tone simulation with a rigid wedge. Case 7 is a Stage IV simulation. Case 8 is a post-edge tone simulation, where the distance to the slit aperture from the wedge is greater than the 2 cm maximum distance at which Brown indicated that edge tones were no longer found. A list of the cases run is found in Table 1. With each case the stages found and the distance lrom the edge to the slit (orifice) are listed. ...

4000 - N r

- 3000 - i. u L

u w e:

5 2000 1

0. 1000 -

0. L A 2 ' ' - ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' -- ' ' ' ' ' , ' 0 0 1 0 2 0 3 0 4 0

I / h . cm-'

Figure 3. Reciprocal wedge distance versus observed and computed frequencies.

Stage 111 calculations for a vibrating wedge gave better frequency results than the values for a rigid wedge. This might be expected since the vibrating wedge more closely approximates the physical situation encountered by a wedge in a flow. Wedge movement was induced in tne "v" ( transverse ) direction a maximum distance corresponding to a one degree angle of attack from an axis running through the wedge apex and the center of the slit aperture. The lrequency driving the wedge corresponds to the expected frequency, derived from Brown's formula, for the 0.82 cm Stage 111 edge tone.

For each iteration, pressures were monitored at several locations around the wedge , slit aperture. and in the farfield. Additionally. lift , C, , and drag, C,,

Table 1. Summary of primary results.

- CASE

__ 1

2

3

4

5'

6

7

8 -

STAGE

PRE-

I

I

I II

I' II. 111'

I II

111

I II

111 I V

POST-

~

DISTANCE h, cm

0.30

0.33

0.40

0.56 0.58

0.82 0.82 0.82

0.86 0.86 0.86

1.60 1.60 1.60 1.60

4.00

REQUENCY

2533

2452

1830

1281 31 77

670 2007 3532

610 2153 3192

430 1090 1664 2386

880

AMPLITUDE

0.003

0.014

0.077

0.058 0.010

0.051 0.045 0.112

0.010 0.078 0.017

0.011 0.003 0.012 0.009

0.003

:REQUENCY

- 2370

1902

1299 3261

859 2807 3491

762 1957 3220

363 955

1561 2188

-

AMPLITUDE

- 0.0740

0.6800

0.0470 0.0120

0.1700 0.1430 0.1710

0.7010 0.2710 0.0700

0.0966 0.0761 0.0457 0.0719

-

EXPERIMENTAL FREOUENCY

(BROWN)

-

2359

1936

1367 3145

916 2107 3481

871 2003 3309

442 1017 1681 2388

-

* SMALL WEDGE VIBRATION INDUCED AT STAGE 111 FREOUENCY

3

coefficients were calculated lrom pressures developed on the wedge. Pressures are non-dimensional pressure coefficients, C A Fast Fourier Transform ( FFT ) was employed on these time histories to determine the Root- Mean-Square (RMS) spectral characteristics for the time dependent phenomena studied. A representative CL history and an associated C,,,, frequency SpeCtrUm gained from an FFT for Case 4 data are shown in Figure 4. Also calculated were pressure differentials. for points on opposite sides of the jet flow near the orifice and on opposite sides of the wedge apex.

P'

BEultS

The results of these edge tone simulations are broken down into three categories: the jet. wedge, and farfield phenomena. Observed phenomena are presented in this manner for the sake of convenience, as the events fall in the general physical region of one category or another. These categories are not mutually exclusive since each affects the conditions in the others.

JetPhenomena

Jet phenomena are those events that are related to the flow of the air issuing from the slit. Due to the nature of the wedge-flow interaction, this category also includes frequencies generated by the jet, vortex generation in the jet boundary, and vortex transport speeds. The frequencies calculated for each stage are listed in Table 1. When jet speed is greater than approximately 500 cm/sec, Stage I occurs simultaneously with each of other stages, as observed by Brown3. The present numerical simulations indicate that for a particular Stage, each lower frequency Stage also appears. The amplitude of the computed pressure oscillations decay rapidly as distance from the wedge tip increases.

The frequencies obtained in these simulations are compared to the semi-empirical 1 curve-fit ) frequencies obtained by Brown. In Table 1 . Brown's empirically obtained formula was used to generate the data for each case listed as "Experimental Frequency" results. The present simulation results and Brown's original

0 .10

0.05

CL 0 .00

-E. 05

- 0 . 1 0 ~ ~ ~ l p ~ ~ l ~ ~ ~ t l t ~ ~ r 8 . 0 0 0 0.E01 0.002 0.003 0

TIM. SEC

experimental data are plotted, Figure 3. against the Stage curve-fits suggested by 13rown.~ The formula employed by Brown to curve-fit the lrequencies was:

f = 0.466'SO)*( U - 40.0 y( 1 .O/h - 0.07 ),

where S(j) is the j-th Stage constant. The Stage constants used in the above formula are S(1) = 1.0, S(2) = 2.3, S ( 3 ) = 3.8. and S(4) = 5.4.

There are two characteristic transport times in the edge tone problem. One is the vortex transport time, Tv, the time required for the vortex to travel from the jet slit to the wedge. The vortex transport time is calculated lrom the following equation :

W

T,=h/U,

where, U, is the effective ( or average ) vortex transport speed.

According to Brown3, U, is approximately 46.6% of the jet speed. A vortex does not travel at the full transport speed when it first appears. Rather it grows in size as it accelerates from its initial location. The additional time required for this growth to full size is called the vortex growth time.

An example of this vortex growth period may be seen in the Stage 111 results. The three-dimensional plots of time and position versus density ( and pressure ), Figure 5, show this vortex growth period. The top of each section is 0.5 mm above a centerline between the orifice aperture and the wedge apex, the middle is the centerline, and the lower three-dimensional surface is 0.5 mm below the centerline. The locations of these .Q lines are noted in Figure 2.A. The low density troughs are vortices travelling toward the wedge tip. The curved path of the troughs near the orifice indicate the acceleration of the vortices. Later in time, nearer the wedge, the effective vortex transport speed was calculated directly from these plots to be approximately 785 cmlsec. which is about 45% of the jet speed. This information and the measurement of the distance between vortices in the region of the jet where the

0 . 0 6

0.04

EL-

0 . 0 2

0.00

f. HI

Figure 4. Stage II lift coefficient time plot and FFT analysis result

4

vortices are in the effective vortex transport speed range, allows the frequency of the edge tones to be obtained. acoustic transport time, Ta: From the same plots the distance between vortices is estimated to be 0.23 cm. This leads to a frequency prediction of about 341 3 Hz, which compares favorably,

time signals at the fixed points. with 3481 Hz from Brown's formula. Also from Figure 5, the alternate nature approximately 5 % the effective vortex transport time, Tv. of the vortices is evident. Hills on one side of the jet Vortices travel generally downstream, while acoustic correspond lo valleys on the opposite side. waves are also able to propagate upstream. Acoustic

waves are required to complete an edgetone cycle.

The other characteristic transport time is the

T, = h/a

although not as good as with the FFT analyses of the in this the aCOUS~iC transport time, T ~ , is

M V E TI€ CENTERLINE

% ~

ON T N CENTERLINE

' "Os

BELW 1 K CENTERLINE

A . PRESSULE

Figure 5. Three-dimensional space-time versus pressure

5

W O V E 1HL CiNlERLlNE

, vortex growth

% ON T H CENTERLINE

' 0%

I vortex

6'

@€LOU THE CENlERLlNf

8. DENSITY

(density) surfaces bounding the jet flow

Brownz6 noted two methods of vortex generation. Vortices produced by the wake and those produced by acoustic interaction. We first investigate wake-produced vortices. These vortices are highlighted in the streamline plots, Figure 6, and the enthalpy contour plots, Figure 7.

h. - C. STAGE I1

.-* ,'. ". .. . I. U.

h.

E. STAGE I V

Simulations show two large recirculation regions along !he sides of the jet, Figure 6. These regions are caused from flow entrainment by the jet. Other wake recirculations are the vortices produced in the jet boundaries. The recirculation regions which reside in nearly stationary positions along the side of the wedge will be addressed in the wedge phenomena section. "

..- I).. 23.. r:. r:. L...

h. I 8. S l A n I

I... u.. h. I

D. STAGE 1 1 1

..- I... U.. U.. *.,

Figure 6. Streamline diagrams exhibit vortex development in the jet-wedge system

6

The pre-edge tone case exhibits a frequeriby. 2533 Hz. which corresponds to a frequency that may be obtained using Brown’s stage formula and Stage constant S(1). However, the amplitude associated with this frequency is 3.4% that of a Stage I frequency

A. T C M

c. STAGE II

1

, E. STAGE I V

ampiitlde, as may be seen in Table 1. One notices these small undulations in the jet boundary as the jet flow passes over the apex of the wedge, Figure 6.A and Figure 7.1% These undulations have the aforementioned frequency of 2533 Hz. They are wake produced and are

1

, L - y - ’ ,

a. STAGE I

0. STAGE I l l

F. POST-EOCE Tot&

Figure 7. Enthalpy contours demonstrate flow deformations near the wedge apex

7

precursors to the Stage I vortices. The mechanism here is then indicated to be the same as in a Stage I produced flow perturbation. The pressure differences caused by the forced alternate switching of the jet flow from one side of the wedge to the other remain minimal. The reinforcement of the jet to stay on one side or another causes greater pressure variations on the opposite sides 0 1 the wedge and vortices begin to form in the fiow separations. The formation of vortices is due to this flow distortion and is a contributor to the cause of the flow separations once the vortices have appeared. The amplitude amplification in the flow through the onset of vortex formation causes a jump in average amplitude from the 0.3 cm case to the 0.4 cm Stage I case by more than 25 times ( 0.003 to 0.077 ).

The results of the Case 3 Stage I simulation exhibit the dipole nature of the wedge. The formation, growth, transport. and dissipation of vortices can be highlighted through a sequence of streamline plots. Figure 8 shows a sequence of four representative streamline plots. In these figures a vortex above the jet is rotating in a counterclockwise (CCW) direction and one below the jet is rotating in a clockwise (CW) direction due to their relative locations with respect to the jet. In Figure 8.A.

5.00

3.00

I .OO

, . 0 0

~3.80

B.EBEE.EE 2.00 4.00 6.80 8 . 0 0 10.8

A 5 . 0 0

3.011

I . 110

- , . o b >

-3.110

- 5 . 8 0

8.000Et00 2.00 4.00 6.00 8.00 10.8

C

Figure 8 Streamline se

the CW vortex under the jet pushes the flow toward the lower side of the wedge apex and the CCW vortex above the wedge pulls the flow away from the upper side of the wedge apex. This flow motion causes the jet to impinge at the lower side of the wedge apex. Consequently, a high pressure center builds up at the lower side of the wedge apex. pressure dipole is pointed upwards. The influence of the pressure dipole has propagated outwards and specifically caused a low pressure center and a new vortex to appear at the upper corner of the orifice.

This new CCW vortex gains energy from the jet and moves downstream at approximately 46% of the jet speed. At this moment in Figure 8.6. the CW vortex rolls under the wedge and causes a high speed and low pressure region at the lower side of the wedge. The influence of the pressure differential across the wedge apex will soon be felt at the corners of the orifice. The pressure field at the orifice will be low at the lower corner and high at the upper corner according to the sense of the dipole pressure at the wedge apex. In Figure 8.C the new vortex just appears in the lower corner of the orifice. There is approximately an 180.0 degree phase shift from Fiqure 8.A. Figure 8.D indicates

The sense of the "

5.00

3.00

1 .BE

- I . R O

-3.00

.< AR . _ _ 8.000E+00 2.00 4.00 6.00 8 . 8 8

B 5 . 0 0

3.110

1.110

- , . e a

-3.110

- 5 nil 8.800E+80 2.00 4.80 6.08 8.110 a D

L 'quence showing dipole Cycle

8

the approach of the CW vortex ( under the jet ) against the wedge and the CCW vortex rolling above the wedge. The direction of the pressure dipole becomes upwards as in Figure 8.A. The pressure wave propagates outwards and causes a low pressure center to form at the upper corner of the orifice. The low pressure center is evident from the initial curl-up of the flow line at the upper corner of the orifice, Figure 8.D. A new vortex will soon emerge at this position thus completing a cycle.

Pressure, as a function of time, at a point under the wedge apex and at a point over the wedge apex are shown in Figure 9.A. The difference in these pressures is shown in Figure 9.B. The four points noted in Figure 9.6 correspond to the four diagrams in Figure 8. The sinusoidal shape of the pressure curves indicate that it is nearly a perfect pressure dipole. The period of the pressure wave estimated from Figure 9 is 0.52 msec.

Pressure signals from the orifice corners (lips) are shown in Figure 10.A. These signals have a frequency more than twice that of the wedge apex signals. The difference between these signals is shown in Figure 10.B. The similarity in pressure differential plots between the wedge apex and orifice implies that these elements have essentially the same frequency and that the orifice must be mainly influenced by the pressure dipole at the wedge apex. As an acoustic wave, the pressure extremes at the wedge affect the orifice pressure field, Figure 10.6, and new vortices are formed at these instances. As low pressure centers, the vortices further modify the pressure field which deviates then from an ideal dipole wave form.

Brownz6 notes that vortices need not originate at the orifice and may form before the edge is reached.

1.007, S I I , I I , , I a I , I I I I

-

...-

This can be confirmed by the Stage IV streamline plot, Figure 6.E and partially by the Stage Ill streamline plot, Figure 6.D where there are vortices which form after the slit and before the wedge apex. These vortices form in the jet boundary upstream of the wedge.

It is also clear from the close-up density plot, Figure t i , that a disturbance at the orifice is associated with the formation of vortices. There are low-density regions travelling downstream in the jet before visible signs of vorticity are evident. Brown" Also noted that the alternate freeing and checking of the stream causes a pulsation in the air supply. In the density contour plot of the channel, upstream of the orifice, Figure 11, pulsations, corresponding to the disturbances noted by Brown, are evident.

The bending of the jet flow is evident in the streamline plots, Figure 6 , for each of the stages. Jet deflections produce vortices which are clearly evident in the associated enthalpy contour sequences. These clearly emulate Brown's smoke pictures from the experiment. The bends in the jet boundary are comparable to the "crests" noted by Brown. These may be seen in the enthalpy contour plots, Figure 7. The relationship of the crests on the jet boundary to the stage number is also evident and follows the formula indicated by Brown:

Number of Crests = Stage Number -1

7 It is generally thought that the periodic

compression and expansion near the wedge apex

0.0158 0.0160 0.0162 0.0164 0.0166 0.0158 0.0160 0.0162 0.0164 0.0166 TIME. SEC TME. SEC

A. Signals above and below apex 6. Differential between points

Figure g Pressure signals at wedge apex

9

generates the edge tones. The regions near the wedge apex and the corners of the slit are crucial areas where relatively large pressure fluctuations and high velocity gradients exist. Several flow-related phenomena are examined in detail, especially at these crucial areas, to produce a coherent picture of the edge tone production.

The large quasi-stationary areas of flow recirculation further back on the sides of the wedge are another interesting phenomenon and are highlighted by the streamline plots, Figure 6 . These two recirculating flow centers are responsible for the apparent formation of a vortex street along the side of the wedge. These regions are not actually stationary but their movement is restricted and the two regions are continuously recirculating. The position of these regions is nearer to the apex of the wedge with the higher stages, ending with the interesting Stage IV dual eddy ( rotational center ) case. Another phenomena verified by this analysis and reported by Brown3 , is that as the distance h is increased, the position at which a vortex is formed on the side of the wedge approaches the edge.

The Stage IV case has a region containing a pair of vortices on each side of the wedge, Figure 6.E. The vortices inside this region coalesce as the next vortex from the jet flow moves into the region of influence of the vortex pair. The incoming vortex then becomes the new partner for the dual eddy region, Figure 6.E.

Vortex generation takes place at some distance from the slit wall for the Stage IV edge tone simulation, Figure 6.E. This is also hinted at in the Stage 111 streamline plot, Figure 6.D, where tiny vortices are being formed near the wall. There is a minor disturbance which is produced at the slit aperture for each of these

I 0 0 0 5 I , . OWER LI

I .0002

0.9999 - f a d

0.9996

0.9993

0,9990 0.0158 0.0160 0.0162 0.0164 0.0166

TIME. SEC

A. Signals at upper and lower lips

cases. Figure 6.C show that the orifice, a vortex generating surface, produces these disturbances which are the precursors to vortices - low pressures and densities which sometimes, especially in the lower stages where the wedge is closer to the orifice and its effects are felt more readily, but not always are well defined enough to be considered a vortex. This provides further visual evidence for the phenomenon of vortex generation away from the slit wall3 and yet illustrates a mechanism for their formation which begins at the orifice. The indication is that the corners of the jet aperture influence vortex generation by producing unsteady, low density regions, which are called precursors having the potential to form vortices.

The pressure needed at the slit aperture for the formation of a new vortex precursor is propagated from the wedge apex. The associated time delay is approximately the time duration required by the wave lo travel from the wedge apex back to the slit at the effective sound speed. The slit differential pressure, Figure 10.6, as contrasted with the wedge differential pressure for the Stage II case, shows a frequency in very good agreement ( within 1% ) with Brown's frequency data. Also in Stage 11, Figure 12, the differential pressure pulses at either the wedge or slit have the same frequency but are shifted in phase. This illustrates the fact that the slit pressure is strongly related to the wedge pressure. The alternating pressure extremes at the wedge apex are the most effective pressure source in the flow field. As an acoustic point source, the strength for the signal originating from the wedge decreases in inverse proportion to the radial distance from the apex.15 Amplitudes in the region of the wedge apex have been amplified by a factor of 50 over the amplitude at which they are initially detected at the orifice, Figure 12. v

0.0004

0.0002

z 6- 4).0000 d .a

-0.0002

I I I -OOO04L I ' ' ' ' ' ' ' ' ' ' '

0 0 1 5 8 00160 00162 0 0 1 6 4 0 0 1 6 6 TIME, SEC

E. Differential between points L, Figure 10 Pressure signals at orifice

10

A Stage I case, h = 0.4 cm. was fun with a very fine grid io pick up acoustic properties of the wedge apex. Sound pressure was monitored at several points in the field to study the amplitude and acoustic wave propagation. An amplitude-versus-distance plot, Figure 13, shows that the amplitude varies approximately linearly with distance from the wedge apex. The second point is near the wedge apex. The distance between the second and fourth point is about 0.856 cm and the time delay between these two points is approximately 0.025 msec. This yields a propagation speed of 34,240 cmlsec which is nearly the speed of sound in the stationary medium. It is then deduced that a sound wave is moving radially away from the wedge apex area to the farfield and the orifice.

.. .

A Stage IV detailed investigation of the acoustic effects inherent in the boundary condition specification was also done. The acoustic waves have a small amplitude in the farfield but disturbances show up very readily in pressure and density computations. A Stage 1V case was restarted to issue a step pressure pulse and the ensuing pressure disturbances were closely monitored in time. A wave, Figure 14, leaving from the outer boundary of the problem domain moved toward the wedge at the speed of sound, a = 34729 cmlsec. The wave and its reverberations were followed for several iterations until the impulse wave phenomenon began to dissipate. The reverberation of this wave was imposed by the fixed boundary conditions on the inflow and outflow boundary sections of Zone 2. The inflow/outflow boundary conditions reacted similar to a reflecting and sense-inverting wall, i.e. rarefactions became compressions and vice versa. The wave induced by the edge tone, Figure 15, was found to be starting at the orifice and travelling outward along the wedge simultaneously with the boundary induced wave. Waves probably interfered with each other to exhibit an effective means of amplifying the disturbances within the jet.

-

18.8

6.88

?.be

- 2 . 8 8

-6.68

The sound pressure levels ( SPL ) were calculated in the farfield each case, Table 2. The sound pressure level is defined by:

SPL = 20.0'log,,(P,Ms/P,,,)

where P,,, is the Root-Mean-Square amplitude of the sound pressure signal and PRB, is the reference sound pressure, 2.0'10-5 Nlm'. The 0.4 cm case has the grid which is much more dense in the farfield than other cases and is considered the most accurate SPL calculation.

- 1 0 0 0 0000 0 001: 0 0024 0 0036

t sec

Figure 12 Pressure extrema at wedge cause pressure surge at orifice.

----I 0.00-

0.0 20 4 0 6 0 -16 .8 -6.88 -?.OB 2.88 6 .88 18.8

y, cm

Figure 11 Density pulsations inside the orifice. Figure 13 Pressure coefficient versus distance from wedge apex

11

As each case progressed in time, a farfield acoustic pattern began to emerge. The boundary conditions imposed affected these patterns. Peaks and valleys in farfield pressures yielded significant information. In particular. the distance between the compression ridges in the farfield. Figure 16, corresponds to the expected wavelength for each particular stage frequency. For example the distance between the two peaks in Figure 16 is about 10.1 cm. Assuming that this is an acoustic wavelength, a frequency of 3438 Hz is obtained, which is in good agreement with the Case 5, Stage 111 edge tone frequency obtained from the time signals of pressure and density in the jet region, Figure 5.

CISE

STEP PRESSUIE

. DiSlANCE SOJhD PRESSLRE LEVEdPREDOM.1IIII.T SIACC rrn dB,ROI 2 . 1 0 . 5 N U ~ ,

Table 2. Maximum sound pressure level at 200 cm from wedge apex

2 0.33 96.59 I 3 0.4 80.87 I 4 0.56 lW.60 I

0.82 11 2.M Ill 7 1.M 89.32 111

' SMALL WEDGE VIBRATION INDUCED AT STAGE 111 FREOUENCV

The computed results indicate excellent time accuracy of the computational method as demonstrated by close agreement to experimental results obtained under the conditions of Brown's experiments. The computations made in the range of data generated experimentally by Brown were very close to the experimental frequencies confirming the accuracy of Brown's empirical Stage constants for all four edge tone stages. We believe that constructive interferences between waves and reverberations amplify the jet instabilities which, in turn, cause the alternate vortex impingements on the wedge and the emission of edge tones. The details of the moving jet flow-field, production of vortices, and acoustic wave propagations in this analytical simulation yielded a wealth of detail to enhance understanding of the edge tone generation mechanism.

Figure 14 Acoustic wave propagating from outer boundary (Stage W).

,-REFLECTED

8.08 €188 88.8 168. 248. 328. 488 T

Figure 15 Edgetone induced wave interaction with boundary induced wave (Stage IV).

12

I): 0

0:o

406.0 mm STAGE I STAGE II 0:o 4 o l . o

rnm

STAGE Ill 7

220.0

mm STAGE I V 0:o

Figure 1 6 . Far-Field acoustic wave patterns.

400.J mm

13

REFERENCES 1. Sondhaus, C., Ann Physik (Leipzig). Vol. 91, p. 216,

2. Chakaravarthy. S. R., Szema, K.-Y., Goldberg, U. C., Gorski, J. J., and Osher. S.. "Application of a New Class of High Accuracy TVD Schemes to the Navier-Stokes Equations," AlAA Paper No. 85- 0165, January 1985.

3. Brown. G. E., "The Vortex Motion Causing Edge Tones," Proc. Physical Society (London), vol. 49, pp. 493-507, 1937.

4. Dougherty. N. S., Holt, J. B., Liu, B. L., and O'Farreil. J. M.,' "Time- Accurate Navier-Stokes Computations of Unsteady Flows: The Karman Vortex Street," AIAA Paper No. 89-0144, presented at the AIAA 27th Aerospace Sciences Meeting, Reno, Nevada, January 9-12.

1854

1989.

Roy. SOC. (London), A216, 412, 1953. 5. Curle. N.. "The Mechanics of Edge Tones," Proc.

6. Nyborg, W. L., "Self- Maintained Oscillations of the Jet in the Jet-Edge System," J. Acoust. SoC. Am., vol 26, p 174, 1954.

7. Powell, A., "On the Edge Tone," J. Acoust. SOC. Am., vol 33, p. 395, 1961.

8. Brown, G. 8.. "On Vortex Motion in Gaseous Jets and the Origin of Their Sensitivity to Sound," Proc. Physical Society (London), vol. 47, pp. 703-732. 1935.

9. Karamcheti, K., and Bauer, A., "Edge Tone Generation," SUDAER Report 162, Department of Aeronautics and Astronautics, Stanford University, 1963.

10. Shield, W. L. and Karamcheti, K., "An Experimental Investigation of the Edge Tone Flow Field," SUDAAR Report No. 304, Stanford University, Feb. 1967.

11. Lenihan, J. M. A. and Richardson. E. G. "Observations on Edge tones," Phil. Mag. (7), vol. 29, p. 400, 1940.

16. Rockwell, D., Knisely. C. and Ziada. S. , "Methods for Simultaneous Visualization of Vortex Impingement and PressurelForce Measurement," in Merzkirch, W. (editor), Flow Visualization '4 11, Proc. 2nd International Svmo. on Flow Visualization, Bochum, West Germany, pp. 349 353, Sept. 9-12, 1980.

17. Rockwell, D. and Naudascher, E., "Self-sustaining Oscillations of Impinging Free Shear Layers." Ann. Rev. of Fluid Mech., Vol. 11 , 67-94, 1979.

18. Rockwell, D. and Knisely, C., "The Organized Nature of Flow Impingement Upon a Corner,'' J. Fluid Mech.. Vol. 93, No. 3, 413-432. 1979.

19. Rockwell, D. and Knisely. C., "Vortex-edge Interaction: Mechanisms for Generating Low Frequency Components," Physics of Fluids, Val.

20. Ohring, S.. "Calculations Pertaining to the Dipole Nature of the Edge Tone," J. Acoust. SOC. Am. Vol. 83, NO. 6, pp. 2074-2085. 1988.

21. Ohring, S.. "Calculations of Self-excited Impinging Jet Flow," J. Fluid Mechanics, Vo1.163, 69-98, 1986.

22. Smith, C. A. and Karamcheti, K., "Aerodynamic Sound Generation Due to the Interaction of an Unsteady Wake with a Rigid Surface.' AIAA- 75-0454, AlAA 2nd Aero-Acoustics Conference, Hampton, VA, March 24-26, 1975.

23. Morse, P. M. and Ingard. K. U.. Theoretical Acoustics, McGraw-Hill, 1968.

24. Staubii, T and Rockwell. D.. "Interaction of an

23, NO. 2,239-240.1980.

v

. Unstable Planar Jet ' with an Oscillating Leading Edge," J. Fluid Mech.. vol. 176, pp. 135- t 67, 1 987.

25. ~ouyOUcos. J., and Nyborg, W.. "Oscillations of the Jet in a Jet-Edge System". J. Acoust. Sac. Am. No. 26, p. 51 1,1954.

26. Brown, G. El., "The Mechanism of Edge Tone Production," Vol. 49-5, pp. 508-521,1937.

12. Stegen. G. R. and Karamcheti. K., "On the Structure of an Edge tone Flow Field," SUDAAR 27. Brown, G. 8.. "Organ Pipes and Edge Tones." Report No. 303, Stanford University, Feb. 1967. Nature, Vol. 141, pp. 11-13. 1938.

13. Lighthill. M. J.. "On Sound Generated 28. Chanaud, R. c., "Aerodynamic Whistles," Scientific Aerodynamically. 1. General Theory." Proc. R. American, pp. 40-46, June 1970. Soc. London Ser. A 21 1,564, 1952.

29. Dougherty, N. S., Holt, J. B. , Nesman. T. E., and Farr, R. A.. " Time-Accurate Navier-Stokes Computations of Self-Excited Two-Dimensional Unsteady Cavity Flows," AlAA Paper No. 90-0691, presented at the AlAA 28th Aerospace Sciences Meeting, Reno, Nevada, January 8-1 1,1990.

14. Holger. D. K.. Wilson, T. A. and Beavers, G. S.. "Fluid Mechanics of the Edge lone," J. Acoust. SOC. Am., 62, 11 16, 1977.

15. Holger, D. K., Wilson, T. A. and Beavers, G. S.. "The Amplitude of Edge Tone Sound," J. Acoust. SOC. Am., 67, 1507, 1980.

W

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