A method for the measurement of mixing properties in a turbulent jet flow

FOR FL UID DYNAMICS

TECHNICAL NOTE 130

A METHOV FOR THE MEASUREMENT OF MIXING

PROPERTIES IN

A TURBULENT JET FLOW

C. BORREGO & D. OLIVARI

JUNE 1979

(fJ:Ht4iSCHE UNIVERSITEIT DElfT LUCfirVAART- EN RUIMTEVAARnECHNIEK

BI BUOTHEEK

K!uyverweg 1 - 2629 HS DELFT

7 JAN.

1988

~A~

-~O~-

RHODE SAINT GENESE BELGIUM

DYNAMICS DEPARTMENT CHAUSSEE DE WATERLOO, 72

B - 1640 RHODE SAINT GENESE, BELGIUM

TECHNICAL NOTE 130

A

METHOV FOR THE MEASUREMENT

OF

MIXING PROPERTIES IN A TURBULENT JET FLOW

C. BORREGO AND D. OLIVARI

JUNE 1979

One of the authors,

c.

Borrego, gratefully acknowledges the receipt of a maintenance grant from the Universidade de Aveiro, Portugal.

ABSTRACT

An experimental investigation of the flow development of a turbulent air jet of circular cross section exhausting into still air has been carried out.

The flow field was examined at locations varying from two to twenty diameters downstream of the nozzle.

A method: for measuring certain parameters relevant

to the mixing process has been tested involving laser equipment in combination with the introduction of a tracer in the jet in the form of oil smoke.

A laser doppler velocimeter and a laser scatter system

were used to measure flow velocity and tracer concentration.

Main parameters recorded at each downstream station were : mean axial velocity, mean concentration, fluctuating velocity and concentration and velocity-concentration correlation.

TABLE OF CONTENTS Abstract . . . . List of figures List of symbols 1. INTRODUCTION 2. BASIC EQUATIONS . 3. MEASUREMENT TECHNIQUE . 3.1 Velocity measurement 3.2 Concentration measurement

4. INSTRUMENTATION AND TEST FACILITIES

4.1 Optical system

_{· · ·}

.

4.2 Test facility

_{· ·}

_·

4.3 Calibrations.

5 . TEST RESULTS

. . .

_·

_{· ·}

5 . 1 Mean velocity and concentration

5.2 Turbulent intensities . . .

·

. . .

.

.

_·

.

.

_·

measurements 5.3 Concentration-velocity correlation 6. CONCLUSIONS . . .

. .

.

i i i i iv 1 2 3 3 3 5 5 6 6 8 8 9 10 12 References . . . 13

Table 1 : Velocity and admixtureprof,iles

half-width ratios . . . . . 15

Number 1 2 3 4 5 6 7 8 9 10 LIST OF FIGURES Title

Diagram of experimental arrangement Concentration calibration curves Mean velocity profiles

Decay of center line velocity Mean concentration profiles

Mean admixture profiles at x/2ro

=

20

Decay of center line concentration Turbulent intensity profiles

Concentration fluctuation profiles Concentration-velocity correlation Page 17 18 19 20 21 22 23 24 25 26

r u

v

x A V

E

v

LIST OF SYMBOLS

fluctuating concentration doppler frequency

incident light frequency scattered light frequency wave number of incident light wave number of scattered light

radial coordinate nozzle radius

fluctuating velocity in x-direction fluctuating velocity in r-direction axial coordinate

area

mean concentration diffusion coefficient photomultiplier output

average number of particles per unit volume average number of partieles per unit time

concentration-velocity correlation coefficient Schmidt number

=

v

v

mean velocity in x-direction mean exit velocity

mean velocity in r-direction particle velocity

1-half width of mean concentration profile: C(oc)

=

2Cmax

- 1

-half width of mean velocity profile: U(ou)

=

~ _Umax incident light wavelength

1. INTRODUCTION

Turbulent transport of momentum, heat and matter

dominates many of the fluid flows found in physics, engineering and the environmental sciences. In a number of these applications, techniques for the measurement of flow properties such as velo-city and concentration of solid or liquid particles, using non intrusive instrumentation are of particular interest for the understanding of mixing processes.

The work described in this paper deals mainly with optical detection methods for such measurements. The technique of light scatter is long-established in the analysis of samples of liquids or gases containing suspended particles, but its application to the determination of the concentration fields is more recent. As indicated above, the technique has the ad-vantage common to purely optical probes of causing minimal disturbance of the flow. The introduction of the laser doppler method has made it possible to measure velocity by optical means also. In the present work, the light-scatter and laser doppler methods were combined to obtain the correlation between fluctua-tions in concentration and velocity, an interesting development which introduces the direct measurement of the turbulent trans-port of material (Ref. 1).

To evaluate this new system, tests were carried out on a circular turbulent jet, discharging in still air, a simple configuration for which a considerable amount of data is avai-lable. The results obtained with the method give some insight into the momentum and scalar transport mechanisms in the near field of this turbulent shear flow.

2. BASIC EQUATIONS

The calculation of turbulent heat and mass transfer requires the solution of the transport equations involving scalar quantities, such as temperature or concentration, as well as the momentum equations. Even for the mean scalar quan-tities, the interaction of scalar fluctuations with velocity fluctuations must be considered.

The equation of transport of a scalar quantity, such as concentration C, in a circular free jet, is (Ref. 2) :

U

a"C" + V a"C" + a(uc) + a(vc) + vc

=

1 a _(V

_.!f.)

_{( 1 )}

ax ar ax ar r r ar ax

where TI and Vare mean velocities in the x- and r-directions, c, u and vare concentration and velocity fluctuations and V is a diffusion coefficient. The momentum equation for the axial direction, af ter applying the boundary layer approximations and

neglecting viscous effects (the Reynolds number is assumed to be sufficiently large for the viscous stresses to be negligible compared to the turbulent stresses) is (Ref. 3) :

TI" aU + V aU = 1

_a

_{(r uv) - ( 2 )}

ax ar r ar

A number of probes have been developed that are capable of fót-lowing turbulent fluctuations and of measuring correlations such as uc and

ve,

such as probes based on electrical conductivity or capacitance, electro-kinetic potentials light absorption, radiant emission and raman scattering. I~ the present work, a new

ap-proach has been followed by applying the light scatter technique i~ combination with the laser doppler velocimeter for the

3. MEASUREMENT TECHNIQUE

The instrumentation consisted mainly of a laser doppler velocimeter (LOV) and a light scatter measuring equipment. Both systems share a laser light source and associated optics of sui-table characteristics and two different detector systems consis-ting of two photomultipliers and optics which respond to light scattered by particles in a defined region of the flow field

(Fig. 1). For such measurements, it is necessary to seed the flow so that the particles introduced in the flow will behave in a manner which is representative of the transport properties.

3.1 Velocity measurement

The laser doppler method, which is based on the detec-tion of the doppler shift of the laser light scattered from small particles moving with the medium, has the potentialof complete linearity between transducer response (doppler frequency shift, fO) and velocity

-+

~

-+ -+ fO ;: _fs

-

f.

=

_{(ks-k i )} 1 À. 1 ( 3 ) -+

where Vp is the particle velocity illuminat~d by a laser beam,

which as a frequency f i and wavelength À_{i ; ki is the wave number}

-+

of the incident light and ks is the wave number of the scattered

light. A detailed description of the LOV technique and associated

bibliograpry can be found in references 4 and 5. 3.2 Concentration measurement

The technique is based on the measurement of the amount of light scattered by particles present in the flow. When the

con-cefltrgtion of partieles is sufficiently small, the intensity of the scattered light is proportional to the number of particles in the control volume. With a suitable optical system, this

It can be shown that the proportionality between the number"of particles in the control volume and the amount of scattered light becomes a linear relation only for uniformly sized particles under conditions of independent scattering. The criterion for effectively independent scatter by monodis-perse spheres is that the center-to-center distance between particles must be larger than 3 radii (Ref. 6), which

corres-oonds to a rather high volume fraction of particles of about 30%. Therefore this condition of independent scattering can be met using reasonable operating conditions. But it is quite difficult to achieve uniformly sized particles (Ref. 7).

The validity of this approximation depends on the

average number of particles

N

in the control volume at any

instant. When N becomes too small, the sampl~ may not be

suf-ficient to be typical of the distribution. This is precisely the condition for which ambiguity noise becomes larger; this noise, also called "marker" shot noise, arises from the random arrival of the particles in the control volume (Refs. 8, 9, 10).

Thus the applicability of the technique depends on the exist

-ence of an average number density small enough for independent scattering to occur and large enough for the particle size distribution in the control volume to be the same everywhere at any time.

In summary, the difficulty of measuring absolute con-centrations should be pointed out because of the difficulty of generating and measuring uniformly sized smoke particles. How-ever, it is much easier to measure relative concentrations and; for the purpose of studying diffusion, only relative concentra-tions are of interest, provided the absolute concentraconcentra-tions are sma 11 .

4. INSTRUMENTATION AND TEST FACILITIES 4.1 Optieal system

A helium-neon laser was used as the light source for all the measurements. For a single wevelength operation at

632.8 nm~ the maximum power output was 15 mW. The beam diameter

was approximately 1.1 mmo To determine the flow velocity field, the forward-scatter arrangement was adopted. The optical system eonsisted of a beam splitter which produced two parallel beams

of light~ separated by 50 mm; focusing was accomplished with a

single lens with a focal length of 180 mmo The control volume was a cylinder having a base diameter of 0.13 mm and a height

of 1 mm approximately. The fringe spaeing was 2.3x10-3 _{mm which}

suggests a pin-hole diameter of about 50 ~m (Ref. 4).

The detector system consisted of a photomultiplier tube (PM) with a photo-cathode spectral response of S-20,

extending from 0.3 ~m to 0.8 ~m and thus centered around the

frequeney of the incident light.

The best signal-to-noise ratio for a dual scatter laser doppler velocimeter (LDV) is obtained with fewer than 10 partieles in the measuring volume (Ref. 11). For concentration measurements, 1000 or so particles are needed in the measuring volume to reduce the fluctuations in light intensity due to non uniform seeding and therefore to make the ambiguity noise ne-gligible (Ref. 9). This precludes the use of the same receiver opties for veloeity and concentration measurements. Since the slit of the photomultiplier and the incident beam together define the control volume, a larger scattering volume for the concentration system was obtained by using a second

photomul-tiplier having a larger slipt aperture (100 ~m) and smaller

magnification. It had a high voltage direct current supply of 500 to 2100 V.

The output from the concentration PM was low-pass filtered at 1 kHz and then divided to obtain voltages

propor-tional to

C

and

~,

and to provide an input to the correlator, a second input being the fluctuating velocity.

4.2 Test facility

The test facility simply consisted of a circular duct supplied with compressed air and discharging to atmospheric conditions through a nozzle of 6 mm diameter. The exit plane

of the free jet served as the reference plane. The Reynolds

number of the test was 5x103 _{approximately, based on the nozzle}

diameter.

The mean velocity distribution in the jet was also determined using a total he ad probe.

4.3 Calibrations

The linearity of the PM output with respect to the input energy is given by the manufacturer as being greater than

99%. More difficult to evaluate is the relationshlp between particle concentration versus PM output. If the concentration of particles is sufficiently small so that both secondary

scattering and absorption are insignificant, the output voltage should be linear with concentration. Under the assumption of

independent scattering, it is possible to show (Ref. 9), that

the response E of the PM to light scattered by particles inside a cylindrical focal volume is a linear measure of the particle concentration.

To check the validity of these assumptions, a calibra-tion was carried out in a wind tunnel having a test seccalibra-tion of

120x120 mm2 _{cross section and 1 m long. The velocity could be}

adjusted to any speed within the range of O.3to 11

mis.

Particles

were injected into the flow before the settling chamber, care being taken to ensure a uniform mixing. Since the rate of par-ticles injected was held constant:

( 4 )

where

N

is the average number of particles injected per unit

time, Vm i.s the m.:.an velocity at the center of the wind tunnel

test section and C is the mean particle concentration. The _m

velocity was measured with the LDV.

Figure 2 shows the values of PM output plotted as a function of 1 for two different injection rates. The results

V m

confirm the assumption of linearity indicated above. Care was taken to avoid saturation of the PM and be in the linear region of the calibration curve (Ref. 12). The results show that an

accuracy of the order of 2.5 % can be obtained in the

5. TEST RESULTS

5.1 Mean velocity and concentration measurements The LDV was used to measure mean axial velocity from approximately two to twenty diameter downstream. The mean velo-city at nozzle exit was measured with a total head tube. The

results. are shown in figure 3. The mean velocity profiles are

found to be similar. A comparison is made in the same figure with two theoretical models for predicting the shape of the mean

velocity profiles (Refs. 13 and 14).

The decay of the centerline velocity is presented in figure 4 and agrees weIl with other experimental results. The virtual origin of the flow is located at about 2 diameters

downstream of the nozzle. The difference observed with Tollmien's

theory (Ref. 13) , can be attributed to the predominant effect

of turbulence at nozzle exit on the location of the virtual origine According to Flora and Goldschmidt (Ref. 15), a slight

increase of turbulence upstream of the contraction may move the virtual origin considerably upstream.

The transverse distributions of mean concentration show distinct similarity of the profiles af ter a distance of 8 nozzle diameters from the nozzle exit.

The data obtained agree with those measured by

Catalano et al. (Ref. 16) and Shaughnessy et al. (Ref. 8),

shown in figure 6 with the theoretical model of Prandtl-GHrtler-Richardt. This figure also includes the mean temperature profile reported by Abramovich (Ref. 13) for the heated free jet. The fact th at the temperature and concentration profiles are sub-stantially similar supports the suggestion made by Hinze (Ref. 2) that there is little difference between the transport of heat and matter in turbulent jet and confirms the experimental results of Timm (Ref. 17).

The variation of the mean concentration along the jet centerline is given in figure 7. The virtual origin of the flow, considering the decay of concentration is located at about 0.8 nozzle diameter downstream of the nozzle exit. This agrees well with the theoretical model of Prandtl-Görtler-Richardt and with

the experimental data of reference 19. The difference in the

location of the virtual origin of the flow when considering the decay of velocity and concentration means that while the rate of growth of the jet is sensitive to conditions of self preser-vation, the normalized velocity and concentration profiles are not (Ref. 19).

Available results in the literature indicate that the spread of heat and matter is greater than of momentum. Table 1 shows the ratio 0c/ou at various locations. For comparison

pur-pose, this ratio was also computed from data of Corrsin (Ref. 20)

and Corrsion and Uberoi (Ref. 21). The present measurements

indicate that the value of the ratio 0c/ou remains constant

with distance from the nozzle exit. This would show that matter

spreads at the same rate as momentum. Such a hypothesis is

sup-ported by the equations governing the two phenomena. 5.2 Turbulent intensities

The axial distribution of the turbulent intensity, is shown in figure 8, plotted versus

The rms concentration fluctuations presented in the

non dimensional form,

.j.~2

are plotted in figure 9 versus....!:.. .

Cmax °c

Measurements in the outer region of the jet are complicated by

the fact that the mean concentration rapidly falls to levels of the the order of instrument noise while the fluctuations remain large.

The occurrance of fluctuation intensities in the 100 % range is

evidence of the highly intermittent nature of the flow field

5.3 Concentration-velocity correlation

The concentration-velocity correlation coefficient is defi ned as

( 6 )

where u and care measured at the same point and time. Ruc is

plotted in figure 10 against the non-dimensional radius r/r

_ü'

since the half-width of the velocity and concentration profiles,

Ou

and 0c' have not the same value.

The value of the concentration-velocity correlation is an indication of how closely related the admixture field is

with the velocity field. A zero correlation would indicate that the fluctuations of the concentration are totally independent of the turbulent velocity fluctuations (or the absence of fluc-tuations). This is the case for the profile taken at

=

2, 0.2.

Considering such a profile, inside the potential core

Ruc

=

0 near the center, then it becomes negative and finally

pos~tive near the edge. It reaches a maximum at r/ro equal to

approximately one, where also both

J

u2 _and

~

_{attai.n stbéir}

maximum value. The three conditions correspond to:

- Ruc

=

0 : absence of fluctuations near the center. The very

small values measured of both

vi

u2 _and

vi

_c2 _{are probably due}

either to imperfect seeding or the influence of ambiguity noise;

- Ruc

<

0 : within the region of the jet which has not been

contaminated by external air, the consideration developed in eq. 4 on the relationship between concentration and velocity would apply leading therefore to a negative correlation between the fluctuations of velocity and concentration;

- RUC

>

0 : the mlxlng of the entrained air from the unseeded

atmosphere surrounding the jet leads to local variations of jet velocity and concentration, which are expected to be correlated. An input, in the mixing process, from air entrained from the ambient atmosphere would influence local velocity and concentra-tion in the same way (decrease) as would, but in the opposite sense (increase), an input from seeded air from the high veloci-ty, high concentration side of the jet.

The measurement of velocity fluctuations at the edge of the jet by means of the LDV technique needs to be discussed in more detail, specially if comparisons are to be made with results from hot wire measurements. Considering the

intermit-tency of the flow there and recalling that the seeding particles

are only provided by the jet itself (the outside atmosphere is assumed to be free of scattering elements), it is evident that the velocity is measured only when the "jet flow" is present.

Thus the measured values will be biased toward the higher

velocities (both mean and fluctuating) and information will be completely absent outside the turbulent burst. Therefore, the particle averaged quantities deduced from them do not agree

with the time averaged quantities and necessitate correction.

Sy contrast, the measurements should be close to those obtained by conditional sampling techniques.

This does mean that the drop-out timeincreases and is so large near the edge as to make the correlation measurements impossible with the equipment used since it is making analog averages over a long time.

6. CONCLUSIONS

The combined laser doppler velocimeter and laser scatter technique has been shown to permit an accurate descrip-tion of the nozzle fluid concentradescrip-tion and velocity fields for a round turbulent free jet. The velocity and concentration pro-Çiles exhibit self-preserving properties and close agreement among each other.

The centerline velocity and conceetration decay are in general agreement with data from other sources. The velocity and concentration fluctuations show, however, some differences which can be attributed to the different test conditions and

to ambiguity noise, for which no corrections have been made.

The concentration-velocity correlation values are of particular interest as they should bring additional insight into turbulent mixing process.

The equipment and technique which has been described and tested, may be used to collect much needed experimental data in studies concerning turbulent mass diffusion and provide new data to assist in the modeling and the closure of the dif-fusion equations.

REFERENCES

1. LUMLEY, J.L.: Turbulent transport of passive contaminants

and particles : fundamental and advanced methods of numerical model ing. in

11 Pol 1 u t a n t Dis per s a 1 11, V KIL S 1 9 7 8 - 7, ~1 a y 1 9 7 8 .

2. HINZE, J.O.: Turbulence.

McGraw Hill Book Co, New Vork, 1975.

3. RODI, W.: A review of experimental data of uni.form density

free turbulent boundary layers. in

lIStudies in Convectionll

, vol. 1, ed. B.E. Launder.

London, Acade~ic Press, 1975.

4. TROLINGER, J.D.: Laser instrumentation for flow field

diagnostics.

AGARDograph 186, 1974.

5. DURST, F.; MELLING, A.; WHITELAW, J.H.: Principles and practice of laser doppler anemometry.

London, Academic Press, 1976.

6. VAN DER HULST, C.H.: Light scattering by small partieles. New Vork, J. Wiley

&

Sons, 1957.

7. MELLING, A. 1& ~~HITELAW, J.H. : Optical and flowaspects of

partieles .

Proceedings LDA-75 Symposium, Copenhagen, 1975.

8. SHAUGHNESSY, E.

&

MORTON, J.B.: Laser light scattering

mea-surements of partiele concentration in a turbulent jet. J. Fluid Mechanics, Vol. 80, Part 1, 1977.

9. BECKER, H.A.; HOTTEL, H.C.; WILLIAMS, G.C.: On the light-scatter technique for the study of turbulence and mixing.

J. Fluid Mechanics, Vol. 30, Part 2, 1967.

10. LUMLEY, J.L.; GEORGE, W.K.; KOBASHI, Y.: The inf~uence of

ambiguity and noise on the measurements of turbulent spectra by doppler scattering.

Proc. Symp. Turbulence Measurements in Liquids, U. of Missouri-Rolla, 1969.

11. MELLING, A.: A technique for simultaneous velocity and con-centration measurements.

Proceedings LDA-75 Symposium, Copenhagen, 1975.

12. BORREGO, C.: Local measurements of ~elocity and concentration

VKI PR 1978-17, June 1978.

13. ABRAMOVICH, H.: The theory of turbulent jets. Cambridge, MIT Press, 1963.

14. SCHLICHTING, H.: Boundary layer theory. New Vork, McGraw-Hill Baak Co, 1968.

15. FLORA, J.J.

&

GOLDSCHMIDT, V.W.: Virtual origin of a free

plane turbulent jet.

AIAA J., Vol. 7, No 12,1969.

16. CATALANO, G.O.; MORTON, J.B.; HUMPHRIS, R.R.: An experimental investigation of an axisymmetric jet in a coflowing airstream.

AIAA J., Vol. 14, No 9,1976.

17. TIMM, G.K.: Heated and simulated heated jet exhausting into still air at ambient temperature.

VKI PR 1969-240, June 1969.

18. WYGNANSKI, I.

&

FIELDER, H.J.: Some measurements in the

self-preserving jet.

J. Fluid Mechanics, Vol. 35, Part 3, 1969.

19. BECKER, H.A.; HOTTEL, H.C.; WILLIAMS, G.C.: The nozzle fluid concentration field of the round, turbulent free jet. J. Fluid Mechanics, Vol. 30, Part 2, 1967.

20. CORRSIN, S.: Investigation of flow in an axially symmetrical heated jet of ait.

NACA Wartime Report 94, 1943.

21. CORRSIN, S.

&

UBEROI, M.S.: Further experiments on the flow

and heat transfer in a heated turbulent air jet. NA CA TN 1865, 1949.

22. ABISS, J.B.; BRADBURY, L.J.S.; WRIGHT, M.P.: Measurements on an axisymmetric jet using a photon correlator.

Proceedings LDA-75 Symposium, Copenhagen, 1975.

23. DURST, F.: Laser doppler measurements in jet and wake flows. 10 LDA Application Notes, DISA Information; 1975.

24. SAMI, S.: Balance of turbulence energy in the region of jet-flow establishment.

J. Fluid Mechanics, Vol. 29, Part 1, 1967.

25. FORSTAL, W.

&

SHAPIRO, A.: Momentum and mass transfer in

coaxial jets.

OT

_°

x or c Reference 2ro

r

_u

_~

2 1.00 4 1. 11 8 1. 03 Present work 12 1. 03 20 1. 03 2 1. 00 4 1. 19 Reference 16 8 1. 09 5 1. 24 10 1. 42 Reference 20 20 1. 38 15 1. 25 Reference 21 20 1. 35 TABLE 1

~)::====:::::;:I

:

SEEO?o'FLOW CONCENTRA -TION VALUES LASER ~~ ASSEMBLY BEAM-SPLITTER "----1r--' L E N S L.D.v. ELECTRON ICS

50 > E UJ 25

r

,

X -INJECTION RATE

,

,

/

•

INJECTION RATE 2

_,

/

I

I

I I I I

1

/,'

1

)A

I )'1 , ! I I I i I 1 I i

/

j

}

I _I

I ,,'

, *

I

V

~/

i

, , !

/*

I _I I I I I

/

1

'/'

-I _i I , I I

Y

I

I

I I

,f

I

j

_I

J

,

y')

I

,

I i I I , _I I

///

_,

I

/

1 I

J,:/i

• I

i',

.'

~i 0.5

,

1.5 2 2.5

FIG. 2 - CONCENTRATION CALIBRATION CURVES

I

,

,

I I 1

,

J. I T

i

1

I

I

I 1 I I I ,

i

I

_I I I !

I

3 1/-_Um (m/s)-1

· 0

•

0 0 0

•

00 0

•

o - - ---all Ó - - _._ -0 0 0 0 0

•

•

0 0 ₀ 0 N =s «0 ... _~ 0.

-0 q

-o N 0 N ~

....

N

--

_"

<J ." c:.o

"

<> . " ..:r 11 0 ., N 11 0 ., 0

-11 ..:r ('t')

•

--~ ,...; 0 ~ 41 ~

...

N _:>

-

_X a::: _:> 0 _a::: UJ ₀ C.f) :t: UJ LU ~ :t: ....J ... u.. Cf) , 0 <!) (/) a::: _z CL

-~ z :> :t: W ~ U _~ U ....J ....J 0 :t: ....J ...J U 0 UJ Cf)

...

> z

«

LU ~ ('t')

.

<!) u..

Umax 3.0 2.0 1.0

-5

--

--

_

...

-10

_

...

_-

...

FIG. , - DECAY OF CENTRE LlNE VELOCITY :

--

-

_

...

T

15 X/2 ro

• =

PRES.'M)RK_i 0 =REF.16; <:>

=

PASST ( Cl TED REF. 16)j

x

=

REF. 18 j

+

=

RE F. 22 j Do = RE F. 23; \l

=

R E F. 24 i 0

=

RE F 25 j

--- TOLLMIEN'S THEORY (REE 13).

20

N

<> <><J ₀ <>'1> N 0 ~C> N <><lc> 0

_.&

11 ~ [> ., N

~

0

t

0 r -Il 0 <J g. 0 ., Q) 11 <>JC> 0 0 <> q ~. r -11 0 0 0 ~ N 11 00 _C><> 0 ~ ~[> 0 ~ N

-

)( CJl 0 W _....J IJ.. 0

~C>

0::: a.. ~ C>~ 'b Z 0 ~

«

<><J 0 0 _~<J 0::: ~ Z w o 0 0 u ....: 0 z <><\> 0 0 ~ u

~

z o 0

«

w ~

OS

0 0 LIl <><J C> (!)

°

<JC> 0 ü: 0<JC> ₀ 0<J N <> 0 x or- a

lull~

LIl d

C Cmax 0.5 2.0 CIJ IJ A

0J..

~ .~ o Tj ,.r. o • u .~ ~ •

.,

0

~'.

,

0 ,~. d 1.0

o

10 _rlBc

FIG. 6 - MEAN ADMIX TURE PROFILES A T xl 2 re = 20: •

=

PRES. WORK; l:l = REF. 8

c

=

REF. 16

---TEMPERATURE PROFILE (REF. 13)

2.0

N N

0 N \ \ \

...

0 \ N \

-

_x \ \ \ \

•

,

_,

lil ('f')

-,

u.:

,

w 0:::

,

_,

>-\ a::: ₀ \ w \ en ::r: \ z ~ \ 0 u.: (/) \ ~ <{ W _0:::

..

~ \

,

0::: ₁₁ 0 ~ _0::: 0 z _I> <{

-

w ::r: u u z

...

_w 0 _{CD 0:::} U _I 0 W u.: 0::: Z W W 0::: ..J ..J ~ 11 _0::: W :0 0::: 0 <!) ~ ., Z ~ _..J W 0::: ~ U 0 0 lil _{~ Z} lL.. (/) <{ 0 _W 0::: 0::: CL ~ CL U 11 W 0

•

t '

.

<!) lL.. 0

lä

0 0 0 ~

I~ ~

p; N .-0

o ('I') o )( cs E I::::> o N ei o o 0 (/) IJJ -J I.L.. 0 0::: D-:> ~ (/) Z IJJ ~ ~ Z 0 -~ Z IJJ -J ::::> al 0::: ::::> ~ Q) c) q I.L.. N 0 N 11 C> N 0 -11 <l

...

Q) 11 <>

...

...:r 11 0 N 11 0 0 ~ N

-

)(

o N o o

-

.

o - - - - . - - -- -.-.- I o q N u cO

-

"-q

-o q N 0 N JI [> .~ Cf) _N W

-...J 11 u.. _<l 0 n:: Q.. 11

z

<> o .... Z N

o

11 ~ 0 n::

...

Z W 0

u ...

Z N

o

-u x

o 11') cS l> o ..:r o <J o (") ci o N o o o o o

-o I q N q

-0 0

-~ N 0 N 11 C.f) l> z 0

.

..

l - _N

«

-...J 11 W <l Ik: Ik: 0 . ., U CD >- 11 I- 0 U 0 . ., ...J -4' W 11 > 0 Z

.

..

0 N I- 11

«

0 Ik: I-Z W 0 U ~ Z N 0

-U )( 0

-

_<.!) IJ..