Analysis of axisymetrical turbulent pulsating jet

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FOR FLUID DYNAMICS

TECHNICAL NOTE 104

ANALYSIS OF AN AXISYMMETRICAL

TURBULENT PULSATING JET

D. OLIVARI

NOVEMBER 1974

TEC1t'J12C1iE HOGrnC~O'Ol DELfT

vu

EGTUIG SOU\l.IJK UNDE

B=~.L'OTI~E~G(

1-""'1"""''''''('''' 1 - D~' "T

.. 7

~R. \975

~A~

-~013F-

RHODE SAINT GENESE BELGIUM

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TECHNICAL NOTE 104

ANALYSIS OF AN AXISYMMETRICAL TURBULENT PULSATING JET

D. OLIVARI

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SUMMARY

1. INTRODUCTION .

2. THE EXPERIMENTAL MODEL 3. THE MEASUREMENT CHAIN

4.

DEFINITION OF VARIABLES 1 3

4

6

5.

RESULTS OF MEASUREMENTS . . . . . . . .

8

6.

INTERPRETATION OF THE RESULTS . . . . 11

6.1 A simplified theoretical model . . . . 11

6.2 Determination of convection velocity

for large perturbations . . . . . . . 16

6.3

Behaviour of ,the jet near the di~continuity 17

CONCLUSIONS REFERENCES . FIGURES

19

20

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This note describes some experimental results obtained by investigating the behaviour of a pulsated turbulent, circular jet using a synchronous frequency sampling technique. Results are given for the development of mean velocity profiles, decay of centerline velocity,both as function of time inside a period and of the distance from the nozzle. Some results are also

given on the longitudinal turbulent fluctuation.

The geometrical diameter of the nozzle was 3 mm and the

frequency tested

88

and

176

Hz, at various jet velocity.

An attempt 1S made to g1ve a simple theoretical inter-pretation of the flow field which leads to the defimtion of the useful parameters characterizing the pulsating jet, and allows a useful normalization of the results.

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1. INTRODUCTION

The problem of unsteady, or pulsating, turbulent jet flows, notwithstanding their many interesting features and prac-tical application have received very l i t t l e attention both from the experimental and the theoretical standpoint.

Typical practical applications are : the effect of the unsteadiness on the spread of the jet and on its m1x1ng with the surrounding fluid is of interest to burners and

com-bustion chambers. Many fluidic devices operate with unsteady jets, a typical example being the impact amplifier. Furthermore,

ap-plication of jet to circulation control of airfoil is a widely known technique and its extension to unsteady situation, such as helicopter rotor blades or propeller blades will require the use of pulsating jets.

On the other side, from a more fundamental standpoint, pulsating jets are an ideal subject for testing models of

tur-bulence. They are free fr om direct action of viscosity, as opposed for instance to boundary layers, and present a dependence of the convection terms with distance from the nozzle. So they consti-tute a relatively complex test bench, but with the important

feature of having clearly defined and simple boundary conditions.

Under such circumstances, the availability of accurate experimental results, giving detailed informations on the time dependence of the var10US parameters could be considered extremely useful. It 1S the purpose of this note to present a first series of such results, together with a theoretical analysis intended to provide a way for the interpretation of the experimental data.

Previous investigations on the subject, known to the author, are those of Vulis et al. (Ref. 1), Binder, Favre-Marinet and Craya (Refs. 2, 3) and Olivari (Ref.

4).

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In the first work rather rough methods were used to obtain large amplitude variations of the velocity at the nozzle exit. The main purpose of the research was to determine the behaviour of the mean velocity at the centerline and the mean

jet spread, that is the parameters which affect the jet mixing. The objectove was indeed to develop a high performance burner. As aresult, i t was shown that with suitable frequency and am-plitude of the excitation i t was possible to obtain a better efficiency in the mixing with the surrounding fluid. However, the corresponding physical mechanisms were not investigated in detail.

In references 2 and 3 more attention was glven to the basic phenomena. The instantaneous velocity was measured to have a picture of the flow pattern during the oscillations. However, the amplitude of the velocity pulsations was limited because of mechanism used for their production.

Reference

4

deals with a jet submitted to transversal disturbances. It is of interest essentially because i t gives a detailed description of the measurement technique which is used in the present investigation.

In the present research, the behaviour of a pulsating circular jet, generated by a fluctuating supply pressure, is

investigated. The amplitude of the resulting velocity fluctuations is of the order of twice the mean velocity at the nozzle exit

section. The attention is focused on the details of the instan-taneous velocity and turbulence variation with distance from the nozzle and with time. Measurements are carried out with a

synchronous sampling technique developed at VKI to study periodic phenomena.

An attempt lS made to interprete theoretically the

experimental results and to determine the governing parameters for the description of unsteady turbulent jet flows.

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3.

THE EXPERIMENTAL MODEL

All the experiments were made 1n an axisymmetric jet exhausting from a circular nozzle 3 mm in diameter.

The nozzle, shown in picture 1, consisted 1n a diver-gent section, a settling chamber, where the average and fluctu-ating vaues of the total pressure were measured, and a conver-gent section to uniformize the velocity distribution at the exit section.

The fluctuationsof the total pressure and thus of the exit velocity were obtained by supplying the nozzle with an

existing signal generator . It was this element of the chain

which imposed the small dimensions of the jet.

The working principle of the signal generator (shown

1n picture 2) is that of a back pressure amplifier (Ref.

5)

with its control port opening modulated by the shaped edge of a rotating wheel. The relative position of the control port and of the wheel could be changed in such a way to obtain the desired ratio between the mean value and the fluctuating component of the output signal. The frequency could be adjusted by changing the speed of rotation of the wheel. The output of the generator was connected to the nozzle through a small diameter tube con-taining a T junction connected to a closed pipe of variable length. This length was modified to tune the system at the working frequency so that the reflection waves generated by eventual mismatches of the impedances of the elements of the circuit could be cancelled.

A hot wire probe was used for the measurements 1n the jet. Both the probe and the nozzle were supported by tra-versing mechanisms allowing relative displacements along three orthogonal directions. The symmetry of the jet was checked under steady conditions and proved to be satisfactory.

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3.

THE MEASUREMENT CHAIN

The jet total pressure was monitored, as mentioned before, by measuring its value ln the settling chamber of the nozzle. Measurements were made by means of a variabie reluctance Hydine transducer connected to a carrier frequency bridge and amplifier. The fluctuating pressure was essentially checked tç

ensure that the whole system was correctly tuned, while the

mean pressure was continuously measured to verify the steadiness of the flow condition during the tests.

Measurements of the velocity in~e jet were performed

with a standard DISA constant temperature anemometer and linea-rlser. A miniature gold plated probe was used as sensor. The system was calibrated to enable the continuous measurement of the instantaneous value of the velocity.

The output signal was analyzed by means of the analogue synchronous sampling technique developed at VKI for measurements

in periodic flows (Refs.

4

and

6).

In brief, the output of the

hot wire anemometer (which consists of a periodic fluctuation .

modulated by the turbulence) is periodically sampled for a very

short time

(4

~sec) at each cycle. This sampling corresponds to

a multiplication of the signal by a Dirac function with phase

lag ~. The phase lag is kept constant to a predetermined value

by means of a synchronisation master pulse, derived in this case directly from the rotating wheel of the signal generator. The results are averaged over a time interval long enough in com-parison with the longest period of turbulence fluctuations to

obtain the mean value of the velocity for a given phase ~.

Computing the varlance of the samples with respect to the mean value so obtained, it is possible to de fine a value for the intensity of the turbulent fluctuations at the same phase. By changing the phase lag of the sampling pulse i t is possible to obtain information over a complete cycle of the fluctuations. Thus, the dependence on time of the two quantities which were the object of this investigation, the mean velocity and the

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turbulence level, was accurately determined.

Two modes of operation are possible : either maintain the sampling phase constant and displace the probe across the jet, thus obtaining as aresuit the velocity profiles at a given phase, or keep the probe location fixed and change the phase. By a continuous sweep of the probe it is now possible to obtain the variation of velocity with time at any location in the jet.

A5 a result a resolution of the velocity in terms of

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4.

DEFINITION OF VARIABLES

As a conse~uence of the flow field investigated and of the measurements techni~ue adopted, care should be taken in the definition of the variables.

Thus, with reference to Fig. 3, if

T

=

period of the forced fluctuations t

• =

T

=

sampling phase, and ~

=

360

e.

=

phase angle

u

(x,r,t)

=

instantaneous velocity, we will define

T

U ( ) x,r

=

mean ve I OC1 . t y

=

IT

J

U dor L

(x,r,

o

I

= average velocity at fixed phase = N

where N

=

number of samples

N

L

o

u' (x,r,.)

=

turbulent component of the velocity at a given phase defined as

U ( . ) - (.)

( 2)

It should be noted that if the phenomenon is truly periodic, each cycle can be considered as an independent repetitive event. Thus the defintion of corresponds to the definition of the statistical average of the velocity as intended in the statis-tical analysis of turbulence.

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N

1:

o

U

l2

corresponds to the statistical square average value of the turbulence fluctuations.

The following symbols will also be used

U.

=

velocity at the nozzle exit

J

u

_c

=

velocity at the jet centerline

«

U >

a

=

. c max

- . )

c mln

= relative amplitude of the pulsation

Ó

m

2

U

c

=

thickness of the jet where UU

=

c

1

2

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5.

RESULTS OF MEASUREMENTS

The jet was tested in steady conditions and the results found to be in agreement with the available data from the literature. Thus, the discussion will be limited to the case of fluctuating jets.

The first figures refer to measurements of the mean velocity U, which is representative of the gross features of the jet behaviour.

In figure

4,

the decay of the centerline mean velocity U lS

C

presented as a function of the distance from the nozzle for different frequencies and jet exit velocities.

Figure 5 shows the measured mean velocity profiles 1n the

nor-o U(r) 0 x t

mal1zed form --- at d1fferent values of

d.

It appears tha

U

self similarityCis retained in the case of a pulsating jet. Differences with respect to the steady jet, however, become apparent when the average velocity profiles, <U(r», at a fixed phase are analyzed.

Figure

6

shows, for different distances from the nozzle, a com-parison between two velocity profiles, one for the mean velocity U and the other for the average velocity et a phase which corresponds to the maximum of .

c

In figure

7

are presented, 1n dimensionless form, the profile

of U and two profiles of , measured at the phases corresponding to the maximum and minimum values of .

c

A lack of similarity is apparent 1n particular when the velocity is normalized with the centerline velocity, a phase lag exists at the outer edges of the jet, with a m1n1mum value corresponding to the maximum of , and vice versa for the

c minimum of .

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From this i t may be inferred that the jet lS not only pulsating longitudinally, but also "breathing" radially. This may probably be visualized as a formation of vortex rings, in phase with the velocity fluctuations which propagate along the jet.

As a consequence, it is expected that the entrainment velocity at the edges of the jet depends upon time. Such a

quantity was not measured in this investigation, however, other

measurements (Ref.

4)

tend to confirm this trend.

The following set of results is relative to measurements of the velocity versus time. Figures 8a,b,c,d, show the var1a-tions of the normalized average velocity

c

~ , and of the intensity of turbulence

U.

J

on the axis of the jet,

1<U2>

_c

over a complete U.

J

cycle and for increasing distances from the nozzle.

As it may be seen, the variation of turbulent intensity

lS essentially in phase (at the frequency consid~red) with the

variation of the average velocity. However, their ratio is not constant at the different phases of the cycle.

At the peak value of , the turbulence intensity c

(neglecting the large overshoot) is comparable with the results for a steady jet; at lower values of it is much higher. It

c

is likely that the decrease with time of the velocity acts as a destabilizing factor in the production of turbulence, thus the higher relative levels at lower velocities.

As it may be seen on figure 8, there exists a peak 1n the intensity of turbulence which is located in the region of the maximum slope of the average velocity variation. It seems very unlikely that i t represents true turbulence, because i t is unrelated with the local value of the velocity and with the slope of the velocity distribution. A possible explanation is that the

value of ~ corresponding to the maximum slope of the velocity

m

variation is, within narrow limits, a random function of time, i.e., different from cycle to cycle. Under these conditions,

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the sampling system will measure and compute as turbulence what are in fact phase variations of the average velocity from cycle to cycle.

Such a "phase turbulence" would be reièated to the

local slope of and to the uncertainty of ~ . The latter

c m

effect increases with the distance from the nozzle as a conse-quence of the decrease of coherence of the forced velocity fluc-tuations a trend which can be observed in the experimental

results. Similar peaks were obtained in the experimental inves-tigation of reference 2.

The distribution of turbulence across the jet, measured

at different stations is presented in Fig.

9,

where a comparison

1S made with fue results obtained for a steady jet. It is seen that

the turbulence intensity is higher at the jet edges this

indi-cates a more complete mixing of the jet with the surrounding fluid and should have as a consequence a non negligible effect on the entrainment velocity.

FigureslO to 14 present the distribution of the center~

line average velocity versus phase angle at increasing

c

distances fr om the nozzle, for all jet velocities and frequencies which were investigated. All the results show a common trend : i.e., a gradual change with increasing distance from the nozzle from a sinusoidal wave form to a saw-tooth wave form, as if something analogous to a shock wave front was developing.

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6.

INTERPRETATION OF THE RESULTS 6.1 A simplified theoretical model

A relevant similarity parameter in the study of un-steady flows is the Strouhal number. For the case of jets, it 1S usually defined as

fod

St

=

U.

J

where f 1S the frequency of the pulsations, d the diameter of the jet nozzle, and U. the mean velocity at the nozzle exit

J

section.

( 6 )

In the present study, the Strouhal number varied from 0.00825 to 0.022. These values are extremely low, but this 1S partially due to the definition used for this parameter.

It is the op1n10n of the author that the definition g1ven above is not fully representative of the flow field under consideration. Indeed, the amplitude of the fluctuations is not

taken into .account. As will be shown later, a more significant

parameter, which depends on the distance x from tne nozzle exit

section, can be defined as (making use of Eq. 5)

a

St

=

at

U dU ~ a2f

ax

u

c x

( 7 )

From the experimental results of Fig.

4,

a law for the decay of

the centerline jet velocity with x can be established as

u

x

c

U.d

J

=

6

which, combined with

(7),

gives

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St

=

3U.d

J

By selecting for x a va1ue at the midd1e of the range covered

by the experiments, the va1ue of St as defined by

(9)

now lies

1n the range 0.142 to 0.38.

The flow field of a jet can be described by the usua1 boundary layer equations. For an incompressib1e turbulent flow they can be written in radial coordinates as

a 1 a<V>r

- - - - +

=

0

ax r ar

a a a

at + a x + <V> ar 1 a +

1

1 a «T> r)

o

=

1 _p ap _ay

<T>

= -

P <u'v'>

with the boundary conditions

r

=

0 <V>

₌

0 <T>

₌

r

₌

co

=

₀ <T>

=

and <u. > = u. (l+a sen wt)

J J

The convent i ona1 separat ion

=

_p _ax _p _{r ar}

0 0

bet ween mean and f1uctuating

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(11)

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com-ponents of the velocity has been made fo11owing the definitions of section 4.

The equations (11) are valid if the wave1ength of the f1uctuations is large in comparison to the jet dimensions, as is the case in the present experiment. Under this condition, the

two terms

t;

and

~

can be neglected if the jet is discharging

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By uS1ng the continuity equation and integrating over

the cross section of the jet, eq.

11

becomes

f~

a

f~

2

f~

a<UV>r dr + ar dr + dr r

₌

at ax ar 0 0 0

- l

f~

l

aTr rdr p r ar

(13)

o

with the given boundary conditions the last two terms are iden-tically equal to zero, thus

a

at r dt +

f

~

(14 )

o

It has been shown 1n section

5

that similarity exists for the

mean velocity profiles U(r). This is also true,although to a much lesser extent, for the average velocity profiles <U(r». Nevertheless, such an assumption will be made in the following analysis, as a first order approximation, so that one may write

=

·f(~)

c where

r ~

=

_b

b being the jet diameter. With this assumption, eq. 14 can be rewritten as

at • C + ax

=

0 wi t

hf""o

f(~)~d~ C

= - - - =

const.

f""

f2(~)~d~

o

(16)

Equations

16

could be averaged over a number N of complete cycles

to lead to

a

(U2b2)

=

0

(22)

or

(18)

That lS the conservation of the mean momentum flux of the jet.

It should be noted that the averaging procedure which leads to eq. 18 is equivalent to assuming that the similarity of velocity profiles is only required for the mean velocity U. Thus, its validity lS more general than for eq.

14.

Making use of the experiment al relation

(8),

from eq. 18 i t can be derived that

b x

d

=

k 1 d

Equation

14

could be simplified making use of dimensionless variables defined as t T = T x Z = _ijc _.d J Y x 2 = 2TU.d J to become az 2Z az 0 + = aT

c

ay

with inlet conditions x

=

0

<U.>

=

U. (l+a sinwt).

J J

(20)

(21)

(22)

(23)

Equation 23 shows that the solution for Z (which 1S proportional

to x) is unique in the (T,y) plane. In other words, typical c

data points from all the experiments must collapse on a single line.

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An example 1S given 1n Fig.15 where the relative position with

respect to ~ (= 360

0 T)

of the maximum value of i s plotted

c as a function of y. ~ c Z U c

Combining equations 18 and 21, one obtains

(24)

Thus Z has -the meaning of a shape factor which characterizes

the time dependence of . It may be seen, from eq. 23, that

c

c U

c

is a function of y only. In other words, for identical values of y the curves

c

versus T (or $) should be identical

U

c

provided that the factor a 1S the same at the jet nozzle exit section.

This conclusion is in excellent agreement with the <Uc>

data shown in Fig.16 where is plotted versus ~ for different

U c

val es of the jet velocity and of the frequency, at diff~rent

distances from the nozzle, and for the same value of y.

It should be noted that the condition of equal y and equal a at the nozzle exit are a different way of expressing the equality of the Strouhal number as defined at the beginning of the section:

St

=

=~

3U.d 3

J

This result strongly support the use of such a fundament al

parameter for pulsating jets.

Equation 23 1S the Burgers equation for the inviscid case, whose characteristics are given by

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dT dy =

2Z

C

Equation 23 can easily be solved. Because of the convergence of some of the characteristic lines, one must expect a discon-tinuity 1n Z with increasing intensity to occur : this repre-sents the significant weak solution.

A first order approximation solution of eq. 23 for one of the experimental cases considered is presented 1n Fig~

17

to

19.

<;U >

It uses as starting condition the measured variation of . c

u

_c

versus T at the nozzle exit section. As i t may be seen, the agreement with the experimental results lS fair.

6.2

Determination of convection velocity for large perturbations

The term

;c

represents in the plane y,T the convection velocity of the fluctuations. The experimental value

.9L~64 C

dT - .

U

c

determined from Fig.

8

has been used for the computation of Fig. 10. It corresponds to the best fit to experimental data.

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The convection velocity 1n the physical plane x,t can be determined from the definition of Zand from

(25).

Writing

.9L

=

2Z

=

d C

2 <U*>x

C U.d J

where <U*> is the convection velocity 1n the physical plane; by using

(8)

and

(26)

and a value

C

=

1.94

computed from eq.

17

and the universal profiles for the jets, one obtains

<U*>

=

1.03 '" 1

(25)

This means that within the experiment al accuracy, the convection velocity at the centerline of the jet is equal to the local jet velocity. This result is somewhat unexpected because convection velocities, in turbulent flows, are generally lower, typically

0.6

of the centerline velocity. A possible explanation is that

the wavelength of the fluctuations is large in relation to jet dimensions, minimum values computed for the "worst case" being as large as 2 to 3 cm.

6.3

Behaviour of the jet near the discontinuity

A solution invOlving a discontinuity is clearly un-acceptable. Furthermore, experimental results show a sharp

velocity change, but clearly no visible discontinuity. It could be expected that the action of the viscosity, through the action of the turbulence will act as a "damper" to smooth out the

sharpest gradients.

Although it lS not the purpose of this paper to analyze

1n detail the behaviour in this region, a few comments will be made.

Two phenomena could be expected to act as to prevent the formation of the discontinuity, involving respectively the action of the normal stresses and of the tangential stresses.

In the first case eq.

14

could be rewritten as

a<u

>b 2 c

at

C +

ax

a

2

_<u

_{>b 2}

=

e: ___ c

ax

2

where e: lS the real viscosity. This equation reduces to the

Burgers equation for a viscous flow. It is easy to demonstrate 1n this case that if e: is assumed to be the molecular viscosity,

v,

the thickness of the discontinuity region is proportional

to the molecular mean free path. If e: is assumed to be an equi-valent turbulent viscosity, the thickness will be increased by

a factor proportional to ~ . At the Reynolds number corresponding

v

(26)

of 100, and such an increase with respect to the mean free path could be expected for the thickness of the discontinuity. But even this value is extremely low in comparison to the

measured one.

The second, and probably more important factor, lS

the action of the shear stress as shown in eq. 11. In proximity of the discontinuity, where the time derivatives become rather large, the resulting effect will be a distorsion of the velocity profiles as indicated by the experiment al results. Thus, the

shape factor C as defined in eq. 17 is not constant near the discontinuity. This willof course invalidate the structure of eq.

16

and 23, which is responsible for the formation of the discontinuities obtained theoretically. A solution involving a variation of C, related to the mechanism of production of

turbulent shear stresses, should thus provide the correct solution in agreement with the experimental results.

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CONCLUSIONS

Measurements have been made ln a turbulent pulsating jet, for fluctuations of velocity which were large compared to its mean value. A synchronous sampling technique was used for the experiments.

The results showed that a large coherence of the forced fluctuations was maintained over distances from the

nozzle exit of at least 25 ~

Self similarity existed for the mean velocity profiles but not so weIl for the instantaneous average velocity.

Mixing between the jet and the surrounding fluid was increased by a mirimal amount with respect to steady jets, although

measurements showed a substantial increase of turbulence inten-sity.

The convection velocity of the perturbation at the jet centerline was found to be nearly equivalent to the local jet velocity.

A theoretical analysis was made to determine the most important parameters which de fine the flow field. A first order

ap-proximation solution of the equation for the propagation of the perturbation has been obtained which helped the analysis of the experimental results.

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LIST OF REFERENCES

1. VULIS, I.A., MIKHASENKO, Yu.I., KHITVIKOV, V.A.: Effective control of propa~tion of a free turbulent jet.

Mekhaniko Zhidkoshi i Gaza, Vol. 1, No 6, 1966.

2. BINDER, G., FAVRE-MARINET, M.: Mixing improvement 1n pulsating turbulent jets.

ASME - Fluid Mechanics of Mixing; The Fluid Engineering Division, Atlanta, 1973.

3. FAVRE-MARINET, M., BINDER, G., CTAYA, A., Te VEUG HAE: Jets instationnaires.

Universite de Grenoble, Lab. de Mecanique des Fluides,

1972.

4.

OLIVARI, D.: Investigation of the dynamic response of

proportional fluidic amplifiers. Part 1 : The behaviour of an oscillating jet.

US Army EU7~pe&m -R~search Office - Final Technical Report, 1974.

5.

FOSTER, K.& PARKER, G.: Fluidics. Wiley.

6. COLIN, P.E. & OLIVARI, D.: Three applications of hot W1re techniques for fluid dynamic measurements.

(29)

(30)

«

a::

«

>

U-a

z

a

t-z

U-w

o

(31)

Üc

6

5

4

3

2

0

0 e

88 A

176

+

88

f;l

88

• 176 23·7 m

Isec

23·7

22·5

V

32

16·1 31·5

_{. /}

/v

--y

v/

~.

/

vlt

/~

/ ç

~

,r_e

.

~

1 :/

o

5

10

15

20

25

30

35 x/d

40

(32)

-

.

-

~----·-T----~--- --or-- ~:I: ... ooor--CX) X_C'l('I')<"'lcO

-1\

L--

-

~

-

~

-

~

-

:

-

--

--.

-

--

-

---

- --- --- -_+_ I

:::J

I

r

I

+

LC'I

-

-+

-

- -

- + - -

- -

40

~~---

__

~

________

~

______

~

______

~

__

---~---~o o a.n

-

°

(33)

U Uc

Q

·

8

Q.6 Q·4

Q·2

1 U

Ut

0·8

Q·6

0·4

Q·2

~

t ~ 1·5

•

-v ) 6

,..

fA

..

;

~

•

~

•

---- - -Cllt.

i

-

~. ..

-~

•

0 [)

X

• ~

8 _.

0·5

o

0·5

• i

0

,

[)

•

8

• _~

~ _qI

• _•

0

•

6

8

~

e

)

0·5

o

0·5

FIG.

6 VELOCITY PROFILES

[)<Uc>max ·<Uc>min

à=12.S

f

= 88 Hz

&

•

8

•

₀ [)

• ~

~ [) 0

•

tl

1 rIl> m/2 1·5

0 _U tl < Uc>max • < Uc> min

x/d= 16

·

6

f

=

88 Hz

<4 a 0 l::J ~ tl 0 6 1 r/Óml2 1·5

(34)

21.

m/sec

20 --,

\

16

\

•

\ \

12

8

4

0

12~--~+---r---1---~ 4~----4---~~---1----~

0~0----~2---~---r~~--~8

28

m/sec

_,

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