Properties of the turbulence in the transition region of a round jet

FROPmrl!2 OF THE tTURBULlIfCE IN THE 'PRANSlTION Rml ON

or

A ROtJm;>

JEr

. by

PROPERTIES OF THE TURBULENCE IN THE TRANSITION REGION OF A ROUND JET

by

B. M. Nayar, T. E. Siddon, W. T. Chu

Manuscript received August, 1968.

FOREWORD

The experimenta1 work described herein was carried out in 1964. Due to the pressure of other matters, preparation of the present report has been stretched o~t over anumber of years.

ACKNOWLEDG EMENTS

+he authors wish to thank Dr. H. S. Ribner for the encouragement and guidanc~ which he lent to this undertaking.

The use of facilities at the Institute for Aerospace Studies is gratefully acknowledged.

This research was supported by the National Research Council of Çanada and by the Ai~Force Qffice of Scientific Research Office of

Aerospace Research, United States Air Force, under AFOSR Grant Nr

67-o672A.

SUMMARY

Experimental data are presented for various properties of the

turbulence in the transition region of a low speed air jet

(4.5

to

9

diameters downstream) •

Previous investigations have been restricted to either the mixing region or the region of fully developed flow, the two regions 0f distinctly different, yet self-preserving turbulence structure. An attempt is made to tie in the present data for the intervening transition region with that of these earlier works

[e.g.

3-11]. The overlap of results is generally q~ite

10 Ilo I I l . IV. TABLE OF CONrENTS NarATIQN INTRODUCTION EXPERIMENTAL TEX::HNIQ,UE 2.1 The Jet Source

2.2 Mean Velocity Measurement

2.3

Fluctuating Velocity - Axial Component

2.4

Correlation Technique

2.5

Frequency Spectra

EXPERIMENTAL AND DERIVED RumLTS

3.1

Mean Velocity Variation

3.2

Turbulence Intensity Profiles

3.3

Longitudinal Space Correlations

3.4

Frequency Spectra

3

.

5

"Macroscale" of the Turbulence Definitions of Length Scale Experimental Evaluation of L Comparison with Data of

Othe~s

3.6

~wo Point Space - Time Correlations

3.6.1

3.6.2

3.6.3

Moving Frame Autocorrelation Eddy Convection Speed

Time Scales and Typical Eddy Frequencies SUMMARY OF RESULTS REF ERENC ES FIGURES APPENDIX 1 2 2 2

3

4

4

4

5

5

6

7

7

9

9

10 11 11 12

13

15

D ,:> ~ L x M R K(~,T) t 'rc U Ui U 0 U U c V Vi X Y ~ <IJ

_(W)

T

W

w_f k, 1'), a Hz rms N<YrATION Diameter of jet nozzle

Frequency - cycles/second or Hertz

Axial scale of turbulence based on u fluctuations Mach number

Radius of ~et nozzle (D/2{

Correlation coefficient (Sec. 3.6) Time

Integral time scale (convected)

Axial (streamwise) velocity fluctuation Root-mean-square value of u

Nozzle exit velocity Local mean velocity

Local eddy convection speed (Sec. 3.6.2) Transverse (tangential) velocity fluctuation Root-mean-square value of v

Downstream distance from nozzle exit Radial di stance from jet centerline Separation of probes in x-direction Spectral density f~nction

Time delay

Radial frequency (2~) Typical ed~y frequency

Parameters in Gaussian velocity profile (Sec. 3 .• 1) Hertz (cycles/second)

I. INTRODOCTION

This work was undertaken as part of a continuing study of jet noise and. the associated turbulence properties of free air. jets presently underway

at Ul'IAS. The initial objective was the accumulation of a bulk of infermation _

concerning properties of turbulence structure in the transition* region of a

four inch low speed air jet. The motivation was the possibility of relating

these properties to the noise producing characteristics of such a jet following

the notions of Ribner [1,2].

Measurements of several characteristic parameters of the turbulence

were made. These included mean and fluctuating (turbulence) components of

velocity, longitudinal space correlations, space-time (i.e. cross) correlations,

and frequency spectra. For the most part, results are based on the streamwise

or u-component of turbulent velocity, in view of the ease with which this

component can be determined by hot-wire anemometr~. Data was recorded for

various cross-stream locations at downstream stations _{ranging from x/D} =

4

.

5

to x/D =

9·0.

From these results it was possible to extract additional properties

including moving frame autocorrelations, eddy convection speeds, integral length

scales, time scales and typical eddy frequencies.

Wherever possible, comp~isohS are matle with the findings of other

a~thors [3-12] who have carried out similar studies of both the mixing region,

and the region of fully developed flow. An attempt is made to tie in the

trans-ition region data with these earlier.results. Agreeme~t is generally goed although marked disparity is· evident in some instances. The inconsistency is

ascribed tb various factors. Chief of these is a suspicion that initial (exit)

conditions exert considerable influence over subsequent development of the

turbulent flow. Hence results obtained for a particular jet wil;L directly

reflect such parameters ~s nozzle Reynolds N~er, boundary layer thickness,

and initial (potentiai·~ore) tûrbulence level. Other possible sources of

disparity are a Mach Number effyct (since our jet, with M z .13 is relatively

low speed) and a diversity in characteristics of theelectronic systems employed.

The la~ter relates particularly to th~ effect of low frequencycut-off on

corr-elatiotJ. resul ts • .

To facilitate extrapolation of data fr om one experimental situation

to others, non-dimensional variables are adhered to throughout.

Ilo EXPERIMENTAL TECHNIQUE

The major portion of the experimental data was obtained with hot-wire

probes in conjunction with a linearized hot-wire anemometer.· Particular

facili-ties and techniques are discussed below.

* The word transition is herein taken to apply to that portion of the jet downstream of the self-preserving mixing region but not sufficiently far

downstream to be considered an asympt.otic or fully developed jet flow •.

It generally lies within the limits

4

~ _x/D ~ 10. This is in contra

-distinction to the familiar region of boundary layer transition, near to the nozzle. (As referred to in

[5].)

2.1 The Jet Source

The UTlAS low-speed air jet facility is described elsewhere in detail. [14] Basically, however, it consists of a screened diffuser and settling chamber and a 12:1 contraction cone leading to a four inch diameter (round) nozzle~ Flow is provided by a one horsepower centrifugal blower.

Exit velocities Uo of up to 140 feetjsecond ( M ~ .13) are available. All measurements presented were made at the maximum velocitr, for which

root-mean-square turbulence level in the potential core (at exit) is less than 1%. Boundary layer thickness is of order 0.10 inches. Reynolds nurnber for this condition is 875,000 per foot.

The jet facility was originally designed to enable the study of

micr~-scale phenomena. (i.e for the purpose of making two-point correlations with very small separations). Thus it was necessary to produce a flow of large macroscale which would afford good spatial resolution; this implied a large diameter, low speed jet. lts use for the present work was incidental to our overall research program. [e.g. see Ref. 25].

2.2 Mean Velocity Measurement

The radial distribution of mean velocity was determined both with a Pitot tube, and with a hot-wire probe. Bo~h sets of readings were corrected approximately for errors due to fluctuating components. The end results agreed fairly well although the hot wire values seemed reasonably low in some instances.

The Pitot tube measurements were considered to be the more reliable in view of the more pronounced scatter in hot-wire data. This was due to slight calibration drift and non-linear response.

A standard round-nosed Pitot-static tube was utilized, yaw error being mirtimized by accurate probe alignment. Probe diameter was sufficiently

small relative to the mixing layer width that .displacement of the effective pitot center was negligible [15].

Corrections for turbulence intensity were made according t~:

Ü

_{corr [}U -2 -

:2 ]

u

~

exp (2.1)

Here mean square valves of the streamwise fluctuation u were ob-tained from hot wire measurements to be described in the next section. For a discussion of turbulence corrections the reader is referred to Hinze [13] where

[2.~ is noted as being appropriate for a round-flosed Pitot tube.

2.3 Fluctuating Velocity-Axial Component

Values of the streamwise (u-component) turbulence velocity were determined with a Hubbard Model llHR Type 3a dual channel hot-wire anemometer. This hot-wire set is of the constant temperature type. lt incorporates a built-in lbuilt-inearizbuilt-ing stage which compensates for the King's Law relation [13] between velocity and hot wire voltage. Overall frequency response of the system is

for the u-component suggests negligible energy content for frequencies above

2000 Hz indicating the adequacy of anemometer response. (See Fig.l).

Probes were of the conventional single-wire variety. The sensing

element was .00013" diameter tungsten wire - copper plated exclusive of a 1/32"

active length.

Calibration was carried out in the laminar exit plane flow of the

jet, both before and af ter each run. A typical calibration curve is shown in

Figure 2. Occasionally a serious sensitivity change was noted over the duration of a run, in which case the affected data was discarded. The drift appeared

to arise primarily from an accumulation of dirt particles on the active element

of the wire. A fil~ering system added to the blower intake greatly alleviated

the problem.

Transverse component - Limited measurements of the cross-stream

vel-ocity component vare reported. These we re obtained with a new type of lift

sensitive aerofoil probe [16,17] and the familiar crossed hot-wire probe.

Manipulation of the probes within the jet flow was accomplished with a special+y designed three-axis traversing gear, with provision for symmetric

separatioh of two probes (see[14] ).

A Flow Corporation Model 12Al true root-mean-square voltmeter seryed

to indicate the anemometer output. This instrument incorporated a

thermo-coupletype integrator to accurately deduce the mean energy content of the

in-coming signal (i.e. ~). The integrator has a 16 second time constant. Overall freq~ency response of the meter is constant from 2 Hz to 250,000 Hz.

2.4 Correlation technique

~ongitudinal two-point space and space-time correlation' coefficients

K(~,T) were determined with the aid of an electronic correlator. A lengthy

description of the system is to be found elsewhere [18] but the basic set-up

is illustrated in Figure 3.

The operation involved simultaneous recording of signals from two

hot-wire probes on two channels of an Ampex Type 351 tape recorder. (Probes

were separated by distance ~ in the streamwise direction). Recorded data was

played back to a Philbrick analogue multiplier. The resulting product signal

was subsequently time averaged using 10 second integration time' 0 A digi tal

voltmeter displayed the integrator output (i.e. the correlation).

The tape recorder was fitted with a pair of heads arranged so that

one could be staggered relative to the other by means of a micrometer screw.

This introduced an adjustable time delay T between the two tracks on playback. The correlations so obtained were functions of T as well as ~.

Repeatability of the correlating system was excellent. Results

indicate a minimum of scatter, even for very low valves of the correlation co-efficient.

Due to limitations imposed by inadequate low frequency response of the tape recorder, curves of longitudinal space correlation are rather

and the Appendix. It is sufficient to say that correlator frequency response

was essentially flat from 25 to 10,000 Hz. (See Figure 4).

2.5 Frequency spectra

Frequency spectra of the velocity fluctuation were determined from

pre-recorded anemometer data. Recordings were played back to a Muirhead

Pame-trada wave analyzer giving 1/3 octave spectral density. True spectrum level

was deduced by correcting the 1/3 octave results for filter bandwidth and tape

recorder frequency response. Spectra span the range from 20 - 16,000 Hz.

111. EXPERIMENTAL AND DERIVED RESULTS

3.1 Mean velocity variation

Figure 5 depicts radial variation of mean stream velocity for

several downstream stations. The left hand set of curves are centered on y R

where y is the radial co-ordinate and R the nozzle radius.

The same data is plotted in more conventional manner (i.e. versus

y) on the right. Spreading of the jet is taken into account by normallizing

y and y-R with downstream distance X.

The parameter (y-R)/x has been widely accepted as a similarity

parameter for radial plots of both mean and turbulent velocities, within the

IIUxJ.ng region. (e .g. see [4 J or [5 J ). However, for position further

down-stream (i.e. in the transition region 4.0 D

<

x

<

10), the degree of similarity

achieved by use of this parameter deteriorates significantly. The left hand

plot of Figure 5 bears this out, suggesting that the flow is no longer

self-preserving.

Recently it has been reported [19J that ~se of an alternate

para-meter (y - Yo.5)/(Yo.l-Yo.9) gives rise to a univeral curve for radial mean

velocity variation within the mixing region. (Subscripted co-ordinates denote

value of y where

Ü

is subscript fraction of centerline velocity). The present

work extends application of this new parameter to transition and fully developed

regions of the jet, as depicted in Figure 6. The degree of similarity is

excellent. We should point out th at although the transition region data arises

from total head measurements, results shown for

X/D

=

3

&

4 were determined with

a hot-wire probe. Both sets of values were corrected for errors due to

turbu-lence fluctuations. (e.g. see Sec. 2.2 and Ref. 13).

Chu has indicated[19J that Gortler's theoretical profile fo~

incompressible jets [20J gives a good fit to the universal curve, within the

mixing region. However the data on which this fit was based had not been

cor-rected for the turbulence component. Corrected data (see Figure 6) appears to

be better described by an empirical Gaussian function of the form:

-k( 1) + a)2

Ü/u

~

=

e where 1) (3.1)

Results of other authors [e.g.

8,

12J for the fully developed

the analysis of Reichardt [21J appropriate to the fully developed region, leads to a Ga~ssian velocity distribution equivalent to

(3.1).

Experimental con-firmation of Reichardt's results has been reported by Mil1er and Commings [22J among o~hers.

The centerline velocity decays with distance as shown in Figure

7.

Results of various investigations have been plotted to illustrate the effect of exit Mach number in controlling potential core persistence. The core is seen to persist for

4

to

6

diameters before initiation of the hyperbolic fall-off. Curves similar to Figure

7

have been presented by several authors, beginning with Corrsin

[9J.

3.2 Turbulence Intensity Profiles

Radial variation of u-component intensity is depicted in Figure

8.

Here again it is observed that the similarity parameter (Y-R)/x begins to lose its significance for downstream positions beyond the end of the mixing zone: there is a progressive departure from similarity with distance down-stream.

Although most attention was focussed on u-component fluctuations, some results were obtained for the transverse (tangential) or v-component, at the downstream location x/D

=

4.5.

1'0 this end the "aerofoil" type of turbulence probe was utilized

(see Sec. 2.3). v-component velocity distribution is shown for comparison with its r-component counterpart, along with similar curves due to Bradshaw

[5J

,

as F gure

9.

The intensity of both components is found to maxlffilze near the

position y/R = 1. This si~uation is known to hold throughout the mixing layer, being the basis for the (Y-R)/x similarity parameter. However further

down-stream, the position of maximum ~ or

V2

tends toward the jet centerline as a fully developed jet evolves from the transition flow.

3

.3

Longitudinal Space Correlations

The longitudinal correlation of the ~-component is defined in the usual normalized form as:

K (~)

=

(3.2)

Primes denote root-mean-square values. K (~) was evaluated for several positions within the transition region, using the electronic corre-lating system (see Sec. 2.4). Figures 10, 11 and 12 depict the results (where streamwise separation ~ is made non-dimensional by jet diameter D).

The correlation curves go negative for large values of separation. Generally this can be ascribed to a combination of two effects:

i) Frenkiel [23J has shown that for a random variable whose spectral distrib'\,ltion, is very "peaky" (i.e. indicating a range of strongly preferred frequency), the autocorrelation wilk exhibit a shape much like that of a damped harmonic

function. (If we invoke the Taylor relationship between auto and space correlation

[13J

we can expect decaying negative and positive lobes to appear in the space corre-lation, for increasing s).

In codtrast, Frenkiel's theoretical results indicate that for a broad band random variable with na pronounced spectral peakiness the correlation

should not go negative to any significant degree, provided that the spectrum has a non-zero value at zero frequency.

ii) The latter statement hints at a second source of negative correlation. If a random variable posesses zero energy at low frequencies (including f=o), then Fourier transform theory requires that the autocorrelation must eventually go negative such that its integral over infinite limits (in

T)

equals zero. Hence if electronic processing equipment is non-responsive to very low frequencies a misleading negative -going correlation will result.

The characteristic turbulence spectra obtained in the present work do not suggest astrong preferred frequency. Rather they are consistently

smooth and with much the shape of that for which Frenkiel deduced a non-negative correlation. Hence it is inferred that most of the negative-going portion of correlation curves herein reported arises from inadequate low frequency response of the correlation computer.

This frequrncy cut-off effect has been briefly discussed by a few investigators [e.g.

7,

24J. In the majority of earlier works however, its

significance is not brought out. Recently Chu [25J has given a detailed theoretical discussion of the problem. Additionally, the Appendix of this report takes a semi-empirical look at the cut-off effect, particularly with regard to correcting experimental data.

Figure

13

compares curves of longitudinal space correlation as determined by various laboratories

[3

,

6J

including our own. Kolpin showed

that such curves exhibit good similarity if plotted against a non-dimensional separation parameter s/x.

dur

data tends to substantiate this observation. It is puzzling however that Lawrence's results do not collapse on a single curve when plotted in this manner. Possibly the degree of similarity could be improved if a more general parameter s/(x-xo) were employed. (Here x

o allows for the position of the virtual jet origin).

It is likely that the spread among the similarity curves of Fig.

13

is partially due to differences in low frequency response of the various correlati~g systems. Reynolds number believed to exert no significant in-fluence since Kolpin's similarity was found to hold for jets of more than one diameter.

3.4

Frequency Spectra

The spectral distribution of turbulent energy for various transition region locations is presented in Figures 14, 15 and 16. These data are

plotted non-dimensionally - as ~

(w)

/UoD, versus

w

_{D/Uo • A marked degree of}

...

As a cross-check on spectrum corrections, and the u-component r.m.S.

Iata, a tYJlical spectrum was integrated over frequency:

00

ie.

I

<P

(w)

dw = u 2

"2

Comparison with the direct measured value of u gave agreement

within 3%.

In studying spectrum data of Bradshaw [ 5] and of Lawrence [3] it is noted that on the high velocity side of the mixing layer (i.e. for y/R~l.O)

the energy spectra for all three velocity components are characterized by a "peaked" shape. It has been inferred [5] that this range of preferred

frequen-cies indicates some sart of wave motion associated with generation of large

eddies. Following our discussion of 3.3 then, we expect the related correlation

functions to exhibi t negati ve loops and a defini te decaying periodici tY-,. '.

In con4rast it is found that most data for the low velocity outer

extreme of the mixing layer (Y/R

>

1) as well as our results for the coreless transition region are characterized by a broad plateau-like spectral maximum. îhis maximum consistently occurs for values of _wD/Uo less than 1. There is no indication of a narrow preferred range of frequencies.

The general shape of spectral data for the transition region suggests existence of an inertial subrange. Following the Universal Equilibrium Theory

of Kolmogoroff [26], such a rang,e is to be anticipated if the energy c ontaining

and dissipation frequency ranges are largely separated. The subrange is charac-terized by a spectrum

<p(w)

decaying as the -5/3 power of frequency.

Due to the difficulty of producing high Reynolds number turbulent

flows (based on turbulent microscale), absolute confirmation of the Kolmogoroff

hypothesis was for a long time guite evasive. Only recently have the works of Betchov [27J Grant, Stewart & Mólliet[28] and Gibson [11] verified t.he existence

of an inertial subrange. Our work further substantiates these findings as

demonstrated b~ the -5/3 slope of log-log plotted spectrum data. (e.g. see

Fig. 17). The subrange exists ~or 1

<

_WD/Uo

<

100. Furthermore, the Kolmogoroff

constant A (see Ref.

t

26]) was found to equal ::' 1.2. Thi s agree~ reasonably

well with the findi~gs reported in [11].

Figure 17 compares spectra of the u and v component fluctuations.

(The latter being measured with the aforementioned aerofoil probe, and a

crossed-wire probe). The tendency to isotropy of spectrum level at the higher

frequencies is an additional prediction of the Kolmogoroff theory.

3.5 MACROSCALE OF THE TURBULENCE

3.5.1 Definition of Length Scale

The integral length scale or macroscale of the turbulence is usually

defined as:

Lx

=

looK

X (

~

) d

~

o

(See for instance

[13] -

page

37).

Assuming a frozen convected pattern over a certain restricted volume of the turbulence (i.e. invoking

the Taylor hypothesis), spatial correlation K (~) is related to Eulerian or

fixed-frame autocorrelation by:

d T

That is, the convection velocity U is given by the ratio of integral length scale to Eulerian integral time scale:

It is well known (e.g. see

[13])

that the Fourier cosine transform

of the autocorrelation Ke (T) yields the one-dimensional energy spectrum ~(w):

00

i.e. ~(W)

J

o

Hence, on combining

3.4, 3.5,

and

3.6:

L x lim

=

W -7 0 U c

J

oo o = 7rUc ~ (0)

211

K (T )CoswTdT e (3.6)

Equations

(3.4)

and

(3.7)

provided two independent means by which

the tllength scaletl may be deduced. (We deal here only with longitudinal or

streamwise scale, based on the u fluctuation). The first method involves

integration of the space correlation K (~) over ~ _{(or Ke(T) over T if Uc is}

known independently). The second method requires a knowledge of the zero

frequency spectrum intercept ~ (0), as well as U_ç• _{(Methods of deducing Uc}

are discussed in

3.6.2).

Both techniques were f~rst' employed by Lawrence

[3] .

A more empirical approach to determination of Lx is to approximate

the spatial correlation K (~) by some simple analytical shape, say that of

an exponential function (e.g. see Ref.

3).

-~/L 00 x

=J

K (~)

=

e where L K (~) d~

(3.8)

x 0 Thus, when K (~)

=

1/ e, ~

=

L • x

Experimental Evaluation of L x

In theory the methods for deducing L are quite straightforward.

However, practical application of these methods ts a more difficult matter.

Elsewhere, we discuss the problem created if recording and processing equipment is non-responsive to low frequencies. (Sec. 3.3 and Appendix). In this circumstance, the fluctuating turbulence information is influenced such that ~x (0) erroneously equals zero. _{Hence, by (3.7),Lx also equals zero. The}

correlation K

(s)

will misleadingly go negative in a premature manner while its integral over

S

(Eq. 3.4) tends to zero.

In the Appendix it is shown that faulty autocorrelation curves may

be corrected by a semi-empirical technique. Hence longitudinal space correlation

curves can also be corrected if Taylor's hypothesis can be invoked. However

the procedure becomes rather time consuming when large quantities of data must be processed.

It has been popular (e.g. see [4J ) to integrate experimental curves of K

(s)

only up to the first zero crossing. The resulting quantity is of ten

regarded as a length scale, but such a practice is inconsistent with the defi-nition of Eq. 3.4. The value of Lx so deduced is quite dependent on low fre-quency response of the particular electronic system employed. Nevertheless, this ~echnique has been utilized in the present work as one means of estimating

L .

x

The so-called "spectrum-intercept" method, as suggested by Eq. 3.7

was also explored. Application of 3.7 requires that the one-dimensional spectrum

curves (Sec. 3.4) be extrapolated to zero frequency by some reasonable criterion. The plateau-like maximum of ~ seems to persist at constant level for frequen-cies below Uo/2~~ Hz. Hence, in the absense of a more logical alternative it was felt permissible (although not theoretically justified) to linearly

extra-polate the plateau back to f

=

o. Values of ~ (0) so obtained were used to calculate L .

x

Figure 18 presents length scale results obtained by the two fore-going methods, as well as by the empirical approach of Eq. 3.8. Results based on K

(s)

were not corrected for the frequency response error. rhey are consis-tently one-half the order of Lx values from spectrum-intercept • . Magni

tude-wise, the latter are felt to be most reliable.

All three sets of results indicate little variation of L with y near the end of the mixing region (x/n

=

4.5). However for stationsxfurther downstream the scale increases significantly with off-axis position, reaching a maximum near y/R

=

l.O. The uppermost curve of Figure 18 (for x/D

=

7.5) exhibits an anomolous "bump" , possibly due to faulty spectrum extrapolation.

3.5.3 Comparison with Data of Others

The integral length scale has been found relatively independent of Mach number [3J. However dimensional reasoning suggests that it should vary directly with the nozzle dimensions, if different jet flows are regarded as self-similar. In preference to the approach of earlier investigations (e.g. [3J and [4J, length scale data would appear to be more useful~y presented in

non-dimensional form Lx/D versus x/D, y/x, y/D, etc. This scheme is used here. In attempting to establish a degree of consistency among existing length scale estimates, results of Lawrence [3J, Davies et al [4J and the present investigation were prepared in the form of Figures 19 and 20. The consequent confusion is not encouraging.

The most significant conclusion fr om Figure 19 is that the radial distribution of L is relatively flat, not varying by more than 35% over the major turbulent r~gion. A tendency to maximize near to y

=

R is noted. As has been pointed out by Davies et al, the noticeably pronounced peaking of Lawrence's data near to y/R

=

0.7 is erroneous. The exaggerated peaking arose from Lawrence's use of local mean velocity

Ü,

instead of the more correct convection velocity Uc ' in Eq. 3.7. Had Uc been employed, Lawrences curves would have more the shape of our results for x/D

=

4.5. .

The findings of [4J rather surprisingly imply a IDln1mum axial scale near y/R

=

l.O. Lawrence's 'two-wire correlation data suggests a somewhat

similar behaviour. However these results are for the supposedly self-similar mixing layer. It is not immediately clear why our correlations based estimates ofaxial scale do not show a similar trend (Fig. 18), unless L is distributed

differently in the "transition" region. x

Axial variation of Lx is depicted in Figure 20. It is notable that values from the spectrum intercept approach seem to fall somewhere between the least-squares fit to Lawrence's results and a line L~

=

.13x (as proposed by Davies et al). Lawrence found th at two-wire correlat1on estimates deviated

abruptly fr om the least-square fit (for spectrum intercept data) when x/D exceeded 3.0. Our transition-region results appear to substantiate this find-ing. The discrepancy is possibly due to the unrealistic technique of linearly extrapolating the spectral maximum (see 3.5.2).

A single autocorrelation-based estimate ofaxial scale for x/D

=

4.0 is included in Figure 20. This re sult is corrected for low-frequency cut-off and is about 20% higher tharrthe value from integration to zero crossing.

It appears that existing length-scale data are not overly-conclusive. The otily significant feature is that Lx increases with downstream distance x according to something like Lx ~ 0.1 x, and its radial variation is not large. 3.6 TWO POINT SPACE-TIME CORRELATIONS

defined by:

The longitudinal space-time correlation coefficient is normally

u(x + ~72, t +

T)

u(x - ~72,t) ul(x + ~/2) u'(x - ~/2)

K(~,T) was evaluated by introducing a known time delay

T

between pre-recorded signals from two hot-wire probes separated a di stance ~ in the streamwise direction. (E.g. - see discussion of Sec. 2.4). Resulting data is presented as families of curves in the K ,T plane, for discrete values of separation ~. (Figures 21-23).

The K

(S,T)

results were alternatively prepared as isocorrelation

contours iP the space-time (i.e.

S,T)

plane. Figures 24-26 exemplify this.

Such methods of presenting so called cross-correlation data are

widely employed

[4,

6, 18, 24] and provide a means of determining certain

properties of the turbulence in a "convecting" frame of reference. 3.601 Moving-Frame Autocorrelation

It has been demonstrated[29, 30] that an envelope drawn tangent to

the family of iso -

s

curves (as in Figure 21) may be regarded as

theapproxi-mate ~onvected autocorrelation of the unsteady pattern. More specifically,

the ehvelope traces out the locus of maximum for K

(S,T)

as viewed in the

S, T

planeo Quantitatively, it represents the rate of decay of turbulence eddy

structure as seen by an observer moving at the local mean eddy convection speed.

The Fourier cosine transform of the convecting autocorrelation is identically the spectrum function of turbulence velocity fluctuations in the "convected" frame of reference. This convected spectrum function is directly

related to the far-field acoustic noise spectrum. Consequently it is of

co~sidera~le interest to t~ose engaged in study of aerodynamically generated

no~seo

Convected autocorrelation envelopes have been drawn on the cross

-correlation data of Figures 21-23. The correspondin~ path of these envelopes

in the

s

,T plane normally approaches a straight line (for quasi-homogeneous .

turbulence) as depicted in Figures 24-26.

3.60,f Ed,dy Convection Speed

Although each infinitesimal volume withifrthe turbulence is moving

with its own randomly varying velocity, there exists at any specific (space

fixed) position alocal mean "eddy convecti:an speed", U_C' This characteristic

quantity is normally defined such that au observer moving with the local Uc

would be unable t0 detect (e.g. by cross-correlation techniques) any signifi

-cant convection of the turbulent eddy structure relative to himself [30] • Values of Uc were derived from Figure 21-23 using the time delay

for which the curve K

(S,T)

versus

T

coincide with the envelope of all such

curves. Resultant radial variation is presented in Figure 27, for axial

sta~ions wi~hin the transition regiono Corresponding profiles of mean axial

velocity are shown for the sake of comparisono

Near the end of the mixing zone (i.e. x/~

=

4.5) the eddy convection

velocity differs significantly from the local mean stream velocity, except in

the vicinity of y/R

=

100. On the high velocity side of the sh~ar layer

(adjacent to the potential core) , Uc is considerably less than U. In contrast,

for radial positions ~here y/R

>

_{1, the limited data indicates that Uc is}

somewhat larger than U. ~his observation closely corresponds with the findings

of ~avis, Fisher and Barratt [4] who give a satisfactory explanation of the phenpmenon.

-cates a tendency for the convection speed to gradualty match up with the local mean veloci~y

U.

In fact the two profiles are almost coincident at

7.5

dia-meters downstream, for y/R

<

l.O. This behaviour is probably associated with a tendency for the turbulence to approach homogeneity in the near-axis region, once the potential core has disappeared. For extreme radial positions however (Y/R> 1), U is seen to exceed

Ü

just as it did further upstream. This proba-bly reflectsCa consistency in the outer layer mixing process, which is inde-pendent of downstream location.

3.6.3

Time Scales and Typical Eddy Frequencies

It is of interest to determine what we will call a typical eddy frequency wf at any fixed location in the flow. Just as the integral length scale was defined by Eq.

3.4,

we define an integral time scale in the convected frame of reference by:

T

c (3.10)

Here, K (T) is the convected autocorrelation - see Sec.

3.6.1.

Tc may be regarded ~s a measure of the average life-time for an eddy, as viewed by an observer moving with local convection speed.

It is usually impracuical to evaluate KC(T) for time delays large enough that the correlation approaches zero. Therefore rather than integratidg

(3.10) directly, it is more common to assume some empirical form for K (T) . Although more precise fits to K (T) have been demonstrated

[25]

it is gxpedient t o assume a s~mple . exponen t· ~al: c

Such that, Hence, -W T f dT == K (T)

=

l/e when T T c c (3'.11)

It was found that KC(T) envelopes of the present study (Figures

21-23)

fell off very slowly fOT large T , especially for the extreme downstream measurements. This suggests that the larger eddies are quasi-frozen in the transition region (i.e. decaying at a very slow raté). Within the range of T employed, most of the curves did not fall much below K

(T)

=

0.5.

It was felt permissible (and convenient) therefore to fit an expongntial to the experimental curves at the point KC(T) ==

0.5:

K (T

_{c o.}

5)=

e = 0.5 giving T

In any case (3.11) gives a poor fit to the act~al K (T), hence, c

this slight modification is felt to be quite legitimate.

The radial variatinn of T is presented in Figure 28a. The curves indicate a peak value near y/R

=

0.8.c At first glance this seems to contradict the findings of Davies et al [4J who report a minimum time scale near the

middle of the mixing layer (this minimum being associated with maximum shear and consequent maximum generation of new turbulence). Our result for x/D = 4.5 does however show some tendency to minimize in the vicinity of y/R = l.O.

Unfortunately the data did not extend to radial extremeties greater than y/R = 1.2 and is therefore inconclusive.

There is a marked tendency for ~ to decrease with increased prox-imity to the jet centerline. This effect isCparticularly pronounced just downstream of the potential core extremity. It iS .likely that the 0bserved reduction results fr om interaction of opposing portions of the mixing layer. In other words the eddy structures of tangentially separated porti ons of the mixing layer, although self-correlated, do not possess significant mutual correlation. Hence, the interaction zone (i.e. near-axis porti ons of the transition region) cross-correlation and the convected autocorrelation KC(T) suffer considerable deterioration. The more rapid decay of KC(T) is accompanied by a corresponding reduction of T •

c

Dimensional reasoning suggests that the integral time scale should vary as D/U , for a particular value of x/Do When compared on this basis, values of U~Tc/D from [4] were consistentlyon the order of twice those re-ported herein for x/D = 4.5. The discrepancy is not, understood but may be tied in with our differing techniques of fitting curves to 'K (T) or an unanti-cipated Mach nurnber effect. Here again the low frequency cuf-off effect prob-ably had significant influence (e.g. see Appendix of Ref. 14). In aqy case it was noted that both sets of results imply a linear increase of time-scale with x/n particularly in the region of y/R=l.O.

Figure 28b depicts the variation of typical frequency

w

_{f across}

the jet. The data is normalized to the value at y/R = l.O. Maximum variation from this value is approximately 35%.

IV. SUMMARY OF RESULTS

Several properties of the turbulence in the transition region of a lo~ speed air jet have been measured.

The profiles of longitudinal turbulent velocity are n0nsimilar:

they show a transition from one type of similarity - that of the mixing regien to another type,that of the fully developed jet.

The mean velocity profiles are likewise transitional in character-acter, but the shape of the half-profile is preserved. Thus the half-profiles can be di splayed on a "pniversal" curve against the ordinate parameter

(y - !0.5) / (YO.l - YO.9)· The universal curve is well fitted by a Gaussian functl.on.

The spectra of the longitudinal turbulent velocity for various positions in the transition region show a marked degree of similarity at high

f'requencies. In particular the spectra show the existence of' an "inertial subrange" that f'ollows Kolmogorof'f"s -

5/3

power law.

Approximate values of' the length scales we re obtained by several dif'f'erent methods. Although.the magnitudes given by the several methods are

dif'f'erent, they all show qualitatively a general increase in length scale with distance downstream. In the transition region, the length scales tend to

increase with radial distance f'rom the jet axis up to a position slightly

greater than y/R

=

1. We do not have data beyond this point. An attempt has been made to compare our results with those of' other workers. In general the comparison is poor for a number of' reasods suggested in the text.

In the transition region the ef'f'ective convection speed of' the turbulence structure obtained f'rom correlation measurements tends to match up locally with the mean speed. In the mixing region, on the other hand, the two speeds are known to match up only in the central zone (y

=

R).

In the transition region the "convected" f'rame integral time scale

continues to increase with x/D, as observed f'or the mixing region. At any particular downstream position, this time scale appears to peak around y/R =

1) Ribner, H. S. 2) Ribner,

H.

S. 3) Lawrence, J. C. 4) Davies, P.O.A.L. Barratt, M. J. Fisher , M. J. 5) Bradshaw, B. A. Ferriss, D. H. Johnson, R. F. 6) Kolpin, M. A. 7) Kolpin, M. A. 8) Liepmann, H. W. Laufer,

J.

9) Corrsin, S. 10) Corrsin, S. Uberoi, M. S. 11) Gibson, M. M. 12) Franklin, R. E. Foxwell, J. H. 13

r-

Hinze, J. O. 14) Chu, W. T. 15) Davies, P.O.A.L. 16) Siddon, T. E. Ribner, H. S. REFERENCES

"The generation of Sound by Turbulent Jets". Vol. 8. Advances in Applied Mechanies pp 103-182 (Academie

Press, 1964).

"Aerodynamic Sound from F1uid Dilations - A theory of the Sound from Jets and other Flows". UTIAS Report 86 July 1962.

"Intensity, Scale & Spectra of Turbulence in Mixing Region of Free Subsonic Jet", NACA Report 1292, April 1956.

"Turbulence in the Mixing Regiofi of a Round Jet" , A. R. C. 23, 728-N.200 F. M. 3181-1962.

"Turbulence in the Noise Producing Region of a Circular

Jet", NPL Aero. Report 1054, January 1963.

"Velocity Correlation Measurements in the Mixing Region of a Jet", MIT - ASRL TR-1008, March 1963.

"The Flow in the Mixing Regi0n of a Jet", J. Fluid Mech. 18 (1963) pg. 529.

"Investigation of Free Turbulent Mixing", NACA TN 1257, 1947.

"Investigation of Flow in an Axial1y Symmetrie Heated .Jet of Air". NACA WR W-94, 1946.

"Spectra and Diffusion in A Round Turbulent Jet". NACA TR 1040, 1951.

"Spectra of Turbulence in a R0ilnd Jet". J. Fluid Mech. 15 (1963) p.g. 161

"Pressure Flp.ctuations Near a Cold,. Smal1-Scale Air Jet". ARC 20, 182 - N. 21 F .M. 2682 May, 1958.

"Turbulence" , 1959 McGraw Hill Book Co., New York. "Hot Wire Investigatio~ of Jet Turbulenee" , Uni versi ty of Toronto, UTIA M.A.Sc. Thesis (1962).

"The Behaviour of a Pi tot Tube in Transverse Shear". J. Fluid Mech. 3, 441 - February 1958.

"An

Aerofoil Probe for Measuring the Transverse Component of 'l!urbulence". Journ. AIAA. April 1965, pp. 747-749.

i7) Sidden, T. E. 18) el. Be+oudi, M. Y. 19) Chu, W. T. 20) GÖrtler,. H. 21) Reichardt, H. 22) Miller,

D.

R. Comings, E. W. 23) Frenkiel, F. N. 24) Wooldridge, C. E. Willniarth, W ~ W. 25) Chu, W. T. 26) Kolmogoroff, A.N. 27) Betchov, R. 28) Grant, H.· L. Stewart, R. W. Molliet, A. 29) Ribner, H. S. 30) Ff-Wil1iams,J.E. '. '" .. . ".-':. I " '. ,

" A. Turbu1ence Probe uti1izing AerodynamicLift",

UTIAS Tech •. N. 88, June 1965.

'~Turbulence-Induced Panel Vibratienu urIAS Report

98,

1964.

"Velocity Profile in the Half-Jet 'Mixing Region of

Turbulent Jets". Journ. AlM ~ April 1965, PP.· 789-79J.

"Berechnung ven AUrgaben der freien Turbulens .auf Grund

eZDes neuen Näherungsansatzes" Z.f .a.M.M.Ve1.22, no.5 October 1942, pp. 244-254.

"On a New Theory of Free 'l'urbu1ence", ,J .R.A.E.S. 47,

:ps. 167, 194

3 .

.

~ Z.~.M,M. ~,no.5. 1941.

"Statie Pressure Dbtributionin a:

Fre~

Turbulent Jet"

J .Fluid Mech. 3 - pg. 1 - October 1957.

"Statistical Study of Turbu1ence Spectra1 Functiens and·

Corre1ation Coefficients (Translation) ~CA TM 1436

-Ju1y 1958.

. . . .

"Measurements of the correiation Between the F1uctuating

Ve10cities and Fluctuat1ng. Wal~ Pressure in a Thick

Turbulent Beundary La.yer" '. University of Michigan

Report 02920-2 ..

'1',

(1962).;",."."

...

, '":-.

""lurbu1ence Measurements' Re1eva.nt to Jet Noise" ,UPIAS

Report No. 119, November 1966.

1941 - C. R.·Acad. Sci. rr~~~s.s. 30, 301.

"On the Fine Structure of Turbulent F1ows", J .F1uid

Mech, 3, pg. 205 November 1957.

"Turbu1ence Spectra from a Tidal· Channel" • J. F1uid

Mech. 12, pg 241, 1962.

"A Theory of Sound from Jets and other Flows in Terms

of Simp1e Sources". AFOOR T. N.

60 -

950/1960.

(Sllperceded by UTIAS Report

86,

1962).

"On Convected Turbulence and its Relation to Near Field

1·0

o·e

~

0-6

r

/

X/D

Y/R

=

=

4·5

1·0

U

_o

=

140

fps

·

...

'-: 0·4

0·2

o

15

100

1000

10,000

f-

HZ

•

\

I

o

•

0 0

0

CO 0 CD

o

•

~ _ _ ~ _ _ _ _ ~ _ _ ~ _ _ _ _ - L _ _ _ _ ~ _ _ ~ _ _ _ _ ~ _ _ _ _ L -_ _ - L __ ~

0

o

"

-

0 6 6

f!)

CD •

o

•

a.

....

I )0-~

-0

9

1&1

>

C\J " t!:) H J%.<

HUBBARD HOT WIRE

ANEMOMETER

HUBBARD HOT WIRE

ANEMOMETER

DUAL CHANNEL TAPE RECORDER

WITH STAGGERED HEADS

VOLTAGE

AMPLIFIER

SWITCHING

BOX

VOLTAGE

AMPLIFIER

PHILBRICK ANALOG MULTIPLIER

INTEGRATOR

(ACTIVE)

DIGITAL

VOLT-METER

SWITCHING

BOX

FLOW CORPORATION

TRUE RMS METER

FIG. 3 BLOCK DIAGRAM OF THE VELO CITY CORRELATION MEASUREMENT SYSTEM

26

.a

22

'a I 1&1 Cl)

Z

~

18

-Cl) 1&1

a::

1&1

>

• •

•

.a

•

_•

_•

_•

_{• • •}

•

_•

•

_•

_•

-

..

•

•

-

14

~

_•

.J 1&1

~---a::

10

6

2

20

100

1000

10,000

FREQUENCY - Hz

·08

.04

x

...

0

-

a::

I

>--

_-.04

~08

1.2

.20

.16

., 2

x

"

>-

.08

.04

0

I

_I

_~

" d.

Ó

I

UI

0,4

0.8

1.2

U

o

0

Xjo

=

4.5

9

X/O

=

6.0

•

X/O

=

7.5

8

'YO

=

9.0

_._.- JET C.L.

1,0

1-8-...

."1:1

0-.0'-.~"'èG~ '.~

I>~

't>,O~

•

''''\

0·8

\.\ '1:1

~

,

'.

X/O

0 _

3

'

O

J

MIXING REGION

• -

4·0

• -

4·5

e -

6·0

r

TRANSITION REGION

«> -

7'5

-0,8

-0,6

-0,4

_-0·2

U/Ut.

1·0

0·8

[:] -

FRANKLIN

~

FOXWELL

J

FULL Y

DEVELOPED

a -

LIEPMANN't LAUFER

FLOW

( HALF· .1ET)

.,

1:1

."

".

~,

"

_1:1

o

'e.

o

0·2

0·4

"I.

Y-YO'5

YO"- YO·g

û/

u

_t.

=

e-

I'415('1) +0'7)2 o

---0·6

0·8

0

LAURENCE

MO

= 0·2

® 11

M

₀

=

0·5

1·0 r

•

~ 11

M

= 0·7

0~~Q

0

•

FRANKLIN, FOXWELL Mo=

o·e

~

8~

_

•

UTIAS M = 0·13

0

,

::)

"

.""

.vJ

I::)

0

·

5

o

o

4

8

12

16

20

X/D

( Y-R.Me

·15 .10

.05

o

I

~ .4 -.05~

n

-.\ 0 ....

Ó

1.2

.2'fJx

. '5

.tO

·05

O' I Ó

f I I .

.4

.8

1.2

0

..x.

=

4.5

0

e

~

=

6.0

0

•

-X

=

7.5

0

-.15 FIG.

8

TURBULENCE INTENSITY PROFILES IN TRANSITION REGION

'-2 -2

.16

.14

.12

,

u~

U

_o

El

.10

.06

.04

.02

o

.-U-COMPONENT / HOT WIRE

}

UTIAS

e -

V

COMPONENT / AEROFOIL PROBE (X/D

=

4.5)

. - U COMPONENT}

BRADSH~W

DATA

m-v

COMPONENT

(x/o

=

4.0)

1.0

0.8

t \.

y

0.4

0

-R

Y

0.6

$

-R

y

0.8

•

-0.6

+

'"

R

Y

I.O

'

6t

I

,~

e

-

R

0.4

~

Y

_1.2

~ ... il

-

R

0.2

o

.25

1.0

€/o

-0.2

•

,10

\

•

Fr

y

=

0.2

. 80

~

-0

~

=

_0.4

•

R

y

=

0.6

~

\\

.60.J.

~'"

e

_Fr

y

=

0.8

IR,

I

~~

Q

:i

1.0

R

-. --

--

-.40

.20

8

.25

.50

.75

1.0

f/o

-.20

1.0

0.8

-®

R

Y

₌

_0.2

0

Y

R

=

0.4

0.6

t-'\

""

"'-•

y

₌

0.6

R

e

y

=

0.8

R

~

I

'\.~,

0.4

-Q

:t..

-

_1.0

R

-0.2

o

.25

.50

e/o

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1.0

0.8

0.6

+

R

I

0.4 ...

0.2

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-0.2

'

.

IJ \

\

\ \ \ I!J _\ 0 \

.Y

\ 0 \ _\0

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_\ \ ~m \ \

"

\

r

\

.0

_-a

,

0.1

liJ " ....

_....

'

-0.4

)(.

X/

O

=

0.57

( LAURENCE)

e

XI

o -

-

1.14

lil)

0

X'/O

2.29

( 11 )

---- X/O

=

1,283 (KOLPIN)

EI

>'yO

3.0

(UTlAS)

0

X;o

4.5

( 11 ) ~

~

o -

-

6.0

( 11 )

•

~

o -

-

7.5

( 11 )

0.5

_{_}

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---

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,) -.)

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_:J

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'

...

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__

.

...-.,.

,

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.

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L ~

-30

4

-40!

0 •

y -

_0.4

•

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.

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R

y=

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e

0.6

-

R

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c-•

~

=

0.8

0 ...J 0

_x.::J...

_1.0

R

-60!

_/j.2...=

_1.2

R

I I

0.1

1.0

IlO

160

wo/ua

-10

' = ' : .. : ".: ': ":-:':=- i

-20

-

-

-

--

.

----;8

~: ~

-30

0 ::l _Y

--40+

~

0

R"-

0.2

3'

-

y

-•

...

"R-

0.4

0' 0

•

:L-- 50

l

..J

R

0.6

0 _X

y =

R

0.8

-60t-

A

R-

Y -

1.0

0.1

1.0

W

D/U

O

10

100

-10

- - - -tJa i J .

-- - - - --- -

:-~Y5:=====~,.

__ _

-_ . . . ~ .... ä 7::.--=~JZ..E ~'!

0 __

-20

-30

-40

t

' j o 0

1.-

_{R -}

_{Or 2}

~

y

--

e

3

R

-

0.4

-...

-50t :

•

1.. -

_0.6

R

-...J

Y

-'!. 0,8 0

R

6. Y

_1.0

-60+

R

0.1

1.0

FIG. 16 u-COMPONENT SPECTRA, X/D = 7.5

w

D/U

o

-20

.-~

-30

o

:::>

,

-

3

-I

-Er,

C'

-40

o

o

-50

• -

U-COMPONENT / HOT WIRE

® -

V-COMPONENT / CROSSED HOT WIRES

® -

v-COMPONENT / AEROFOIL PROBE

X/O

=

4.5

y/R=I.O

U

_o

=

138 fps

..

-60!1---:r---~---1ï---~---~----~~

.3

10

100

300

wD/

Uo

o

"'-JC .J

·8

·6

·4

·2

o

o

X/D

=

7·5 ___

-.~

/

____

•

/

~.':6.0

.-.

---/

.

...---.

_./"'

0 8 .

B6D.

AAl.

SPECTRUM

INTERCEPT

INTEGRATION

OF R(€>

EXPONENTlAL FIT

TO CORRELATION

~

0

=4-5

0---_ 0---_ 0---_ 0 0

... _--- ... _---.... ---=

7.5

0 _

... _---- __ --- -7 5

_A,A~':-.~bl'..:'

-&.-

'"

-

•

.t.

=

6·0

---

_-~IiI'''' ~1iI- ~

~

~8-

~

---~

8

B_---_-_-_--_-_-

~

_{_8}

_B

=

6-0

_=4·5

_{_8}

_t:J

_=4-5

_-~-r:1

₈

-·02

·04

·06

·08

y/x

FIG. 18 VARIATION OF LONGITUDINAL SCALE L ACROSS JET

o

.

CD

-

c

'"

a( .,,\ 0 .• . 0

ex»

-;

\

c:

~

I

I

N C \

c·

-e8N

~

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\

'MI

c,

,~

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,lIJ

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/

'M

~\

~

f:

/

/

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~

ei'

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\

oe;,

\/

,

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X,I

\. 0 ,/

e

.

,/·1

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<;>

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rIl/I'

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,

,/

..Ja( \ / /, a(

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~

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1

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V

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0

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1,1

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/i

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~

0

/'

//

9

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en

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,

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,\.\.

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:

/

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o

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v

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--I N

-I

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Je ..J

·8

·6

·4

·2

o

o

o

~ EI 181

•

UTIAS - SPECTRUM INTERCEPT

n -

EXPONENTlAL FIT TO

LONG. CORRELATION

LAURENCE - SPECTRUM INTERCEPT

J

y/R· I

ft

I:R({)d{

LAURENCE - LEAST SQUARES LINE

DAVIES

et ol y/R. I

~ /.. . / .0-

Y /

El

0·1)&

..

\~

/ /

" \ /

/ / / /

/~

~

y/Rc 1·0

/ / ; / 0

~e

IXI-•

/ " El ' .

-I;:::::SS

e -

y/R·

0·6

/

~~e--/ I!!I- .~ _______

~~;:::~8

__

---e

~rr:Cted

Autocorrelotion (Appendix)

y/R

I:

0·4

~LJt'

0·13 X

~

I.OT.

\

.61

.4

.2

o

\ \ \ \

, ,

/'

A

.25

.50

X - 4 5

o -

. , -

Y

₌

I

_.0

.75

U

_o

T/

_O

R

U

_c

= 0.56 U

_o

1.0

~D

-wo.75

.25

1.25

lR.

lp ",

.. 9

.8

.7

.6

.5 , ' , ( .4

><

'> < >k::: 1.0

.9

.8

.1

.6

.x

=4.5, Y =.4

o

R Uc

=.80

Ua

x

y

- =4.5 , - =.6

o

R

U

_c

=.74 Ua .5 (" 'I < ), I

>

>, .1

.25

.50

.75

1.0

UoY'O

.50

UoT/O

~

- 45 Y

o -

.

'R

=

.8 U

_c

=·64 U

_o

t

=

4.5 ,

::f..

-=

I.~

R U_t

=

.48 U

_o

;75 1.0

..

1.25

~

.8

.6

.4

.2

o

.25

.50

.75

UoT/

_O

FIG. 22a SPACE-TIME CORRELATIONS OF u

~

=

6.0 ,

o

y

R

=

1.0

U

_e

=

.57 U

_o

1.0

C/

_o

.75 .25 1.25

•

IR.

1.0 .9

_{0 - ·}

-x -

60

_{f f - ·}

Y - 02

.8

_U

_c

₌

_{.78 U}

_o

.7

.6

.~ 1.0 0' _I

\

.9 • ,

t

=

6.0 ,

;l-

=

0.4

.8

_U

C

="74

U

o

.7

_6

.5

.25 .50 .75 1.0

UoT/O

~

1.0

.9

.8

.7

.6

.5

1.0

.9

.8

.7

.6

.5

.25

.50

.75

U

o

T/O

FIG. 22c SPACE-TIME CORRELATIONS OF u

~

-

6.0 ,

~

=

0.6

U

_c

=

·69

_Uo

~

=

6.0,

~

=

0.8

u

_c

=·63

U_o

1.0

.8

.6

·4

.2

o

rR.

\ \ \

, ,

" ·04

x

y

o

=

7.5 ,

R

=

1.0

U

_c

=.54 U

_o

~o

·19

.75

.25

.25

.50

.75

1.0

lo25

UoYo