J O U R N A L O F E N G I N E E R I N G PHYSICS

215

A STUDY OF THE TURBULENT CHARACTERISTICS OF A FREE CIRCULAR J E T O F AN ~ N C O M P R E S S I B L E GAS M. Kh. I b r a g i m o v ,

G. A. P e t r i s h e h e v a ,

Inzhenerno-Fizicheskii

and G. S. T a r a n o v

Z h u r n a l , Vol. 14, No. 3, pp. 4 1 5 - 4 2 2 ,

1968

UDC 5 3 2 . 5 1 7 . 4 Results are presented for the measurement of the velocity, temperature, and intensity of variations in these quantities within a free circular jet. Data are presented an the distribution law for these variations and for their derivatives with respect to time, as well as on the autocerrelation and spectral functions of the fiuxuations in velocity and temperature. We measured the turbulent characteristics for a j e t of a i r d i s c h a r g i n g a t a v e l o c i t y of 35 m / s e e ( R e 7 0 . 10 a) f r o m a n o z z l e 30 m m in d i a m e t e r . T h e s t r e a m of a i r w a s h e a t e d to a m a x i m u m of 60 ~ C o v e r t h e t e m p e r a t u r e of t h e a m b i e n t m e d i u m . W e d e t e r m i n e d t h e v e l o c i t y d i s t r i b u t i o n in t h e j e t f r o m t h e d y n a m i c h e a d , m e a s u r e d b y m e a n s of a P i t o t t u b e 0.35 x 0.07 m m in s i z e , and w i t h t h e a i d of a micromanometero The mean temperatures at various p o i n t s in t h e j e t w e r e m e a s u r e d b y m e a n s of a c h r o m e l atumel thermocouple whose junction exhibited an outs i d e d i a m e t e r of 0~ m m , and a l s o w i t h a P P p o t e n t i o m e t e r . T h e t u r b u l e n t f l u c t u a t i o n s of v e l o c i t y and t e m perature within the jet were determined with a thermoanemometer. A single-filament tungsten-wire sensing element was used as the sensor. For purposes of m e a s u r i n g t h e v e l o c i t y f l u c t u a t i o n s , t h e s e n s o r w a s f a b r i c a t e d o u t of a w i r e 20 p m in d i a m e t e r a n d 4.5 m m in l e n g t h ; t o m e a s u r e t h e f l u c t u a t i o n s in t e m p e r a t u r e , t h e s e n s o r w a s m a d e of a w i r e 5 p m in d i a m e t e r ( w i t h t h e s e n s i n g p o r t i o n 1.5 m m in l e n g t h ) ; t h e r e m a i n i n g p o r t i o n of t h e s e n s o r f i l a m e n t w a s c o a t e d w i t h c o p p e r to a d i a m e t e r of 40 # m . A p o s i t i o n i n g d e v i c e w a s u s e d to m o v e t h e s e n s o r s w i t h i n t h e f l o w , t h u s m a k i n g i t p o s s i b l e to d e t e r m i n e t h e m a g n i t u d e s of t h i s d i s p l a c e m e n t to a n a c c u r a c y of 0.05 r a m . T h e m e a n a n d f l u c t u a t i n g m a g n i t u d e s of t h e v e t o city and temperature were measured along the stream a x i s ( x / d = 0 - 1 0 ) a n d a t t h e l a t e r a l c r o s s s e c t i o n of t h e j e t , a t a d i s t a n c e of 10 d i a m e t e r s f r o m t h e n o z z l e outlet. The velocity characteristics were measured with a ETA-5A electrothermoanemometer which operated a t t h e c o n s t a n t t e m p e r a t u r e of t h e s e n s o r f i l a m e n t . The constant-temperature method is particularly exp e d i e n t in m e a s u r e m e n t s f o r j e t s in w h i c h t h e f l u c t u a lions are particularly intense, thus making it necess a r y to a c c o u n t f o r t h e n o n l i n e a r i t y of t h e v e l o c i t y c h a r a c t e r i s t i c of t h e s e n s o r . M o r e o v e r , u n d e r e o n d i l i o n s of h i g h - i n t e n s i t y j e t - v e l o c i t y f l u c t u a t i o n s , t h e constant-temperature regime provides for a more q u a l i t a t i v e c o m p e n s a t i o n of t h e s e n s o r - f i l a m e n t t i m e c o n s t a n t t h a n t h e r e g i m e of c o n s t a n t f i l a m e n t h e a t i n g . T h e s e n s o r f i l a m e n t w a s h e a t e d to 150 ~ C a b o v e t h e flow t e m p e r a t u r e ; t h e s t e e p n e s s of t h e a n e m o m e t e r

c i r c u i t w a s 120 A / V . T h e t i m e c o n s t a n t of t h e c e n t e r f o r t h e s e c o n d i t i o n s of a n e m o m e t e r - c i r c u i t operation w a s d i m i n i s h e d by a f a c t o r of 250 r e l a t i v e to i t s v a l u e w i t h t h e c o n s t a n t - c u r r e n t m e t h o d a n d a m o u n t e d to 10 -5 see.

::1 i .L

0.4 0.2

5 Ia08 a04

l

i

Fig. I. Distribution of mean velocity, temperature, and intensity of turbulent fluctuations in the jet cross section x/d = 10: 1) U/Urea x in isothermal jet; 2) UflJmax in heated jet; 3) 0/0max; 4) U/Uma x from thermoanemometer measurement; 5) au/Umax; 6) g0/0max; 7) ffU/Umax according to [4]; 8) a0/0ma x a f t e r [51. T h e c a l i b r a t i o n of t h e s e n s o r s s h o w e d t h a t the h e a t balance equation for the filament can be rather accura t e l y a p p r o x i m a t e d b y a n e x p r e s s i o n of t h e f o r m ~o5 = B + CP.

(1)

E q u a t i o n (1) w a s s u b s e q u e n t l y t a k e n into c o n s i d e r a t i o n in t h e d e t e r m i n a t i o n of t h e m e a n v a l u e a n d of t h e i n t e n s i t y of v e l o c i t y f l u c t u a t i o n f r o m t h e o s e i l l o g r a r n of t h e anemometer-bridge current. F o r p u r p o s e s of r e c o r d i n g o n l y t h e t e m p e r a t u r e v a r i a t i o n s w i t h i n t h e flow, w e e m p l o y e d a t h e r m o a n e m o m e t e r c i r c u i t w h i c h w o r k e d on t h e c o n s t a n t - c u r r e n t m e t h o d , s i n c e t h e a n e m o m e t e r f i l a m e n t ( o p e r a t i n g at c o n s t a n t t e m p e r a t u r e ) r e t a i n s g r e a t e r s e n s i t i v i t y to v e l o c i t y f l u c t u a t i o n s w h e n t h e f l u c t u a t i o n s in t e m p e r a l u r e a r e s u b s t a n t i a l . F o r t h e m e a s u r e m e n t of t h e t e m perature fluctuations, the filament was superheated by ~ 0 . 1 ~ C, and t h e t i m e c o n s t a n t w a s 4 - 1 0 - t s e e . T h e s e n s i t i v i t y of t h e s e n s o r s t o a c h a n g e in t e m p e r a t u r e was determined by calibration. A n M P O - 2 l o o p o s c i l l o g r a p h (loop IV) w a s u s e d to r e c o r d t h e c u r r e n t of t h e t h e r m o a n e m o m e t e r bridge

216

INZHENERNO-FIZICHESKII ZHURNAL

O3

/!f

a

0.2

I!U

a/ -3

-2

-I

a

l

e u/cG

-2

-.I

o

I

z~v

r

Re Q6 b ~

- -

g

16

f

0.2

-3 -2 ,q

-!

0

s

O/d

9- 3

-2

o --.I

-1

1

J i./

O.8 C

0.0 0

_&~

l

i'

,,

0.05

g..t

0.15

_s X+R

Fig. 2. Curves for the probability density of the fluctuations and their derivatives in the jet cross section x / d = 10: a) Pu for r / ( x + a) = O; A =-0.2'72; E = - 0 . 1 1 ; b) Po; 1) r/ (x + a) = 0; A =--0.354; E = 0.52; 2) r / ( x + a) = 0.2; A = 1.13; E = 3.63; c) variation in the asymmetry coefficient in distribution of temperature-fluctuation probabilities; d) PU, f o r r / ( x + a ) = 0 ; e ) PO';1 r / ( x + a ) = 0 ; 2) r / (x+ a) = 0.2.

JOURNAL

OF

ENGINEERING

PHYSICS

217

during the m e a s u r e m e n t of the velocity and voltages applied to the s e n s o r in the m e a s u r e m e n t of the t e m p e r a t u r e fluctuations; this device m a d e p r o v i s i o n for motion p i c t u r e film r e c o r d i n g or for the use of a c o r r e l o m e t e r which is a type of h i g h - s p e e d analog c o m p u t e r . The c o r r e l o m e t e r d e s c r i b e d in [1] was used to find the m e a n - s q u a r e values of the t e m p e r a t u r e f l u c tuations, as well as the fluctuations in velocity over the r a n g e in m e a n velocity values ( 2 5 - 3 5 m/sec) in which the r e l a t i o n s h i p between the heating c u r r e n t for the f i l a m e n t and the fl0w velocity is close to the l i n e a r . The e x p e r i m e n t a l data obtained in the f o r m of an o s c i l l o g r a m on the f i l m were converted into p u n c h card code notation and then p r o c e s s e d on a n e l e c t r o n i c digital computer. The computation p r o g r a m made it p o s s i b l e to d e r i v e the following c h a r a c t e r i s t i c s of a s t a t i o n a r y r a n d o m function: the m a t h e m a t i c a l expectation, the s t a n d a r d deviation, the p r o b a b i l i t y d i s t r i b u t i o n for the a m p l i tudes of the fluctuations, the a s y m m e t r y and excess p a r a m e t e r s , the a u t o c o r r e l a t i o n and s p e c t r a l functions, and finally, the d i s t r i b u t i o n of the p r o b a b i l i t i e s for the f i r s t d e r i v a t i v e s of the velocity and t e m p e r a t u r e f l u c tuations. The p r o g r a m m a d e p r o v i s i o n for the [ i n e a r i cation of the a n e m o m e t e r - s e n s o r c h a r a c t e r i s t i c s on the b a s i s of Eq. (1) (the a n e m o m e t e r c a l i b r a t i o n curve), and thus e n s u r e d computational r e l i a b i l i t y for the aboveindicated c h a r a c t e r i s t i c s of the velocity fluctuations for the e n t i r e r a n g e of velocities. The m e a s u r e d velocity and t e m p e r a t u r e p r o f i l e s for the flow at the nozzle outlet w e r e flat. The i n t e n -

6

4 2 /0~ 4

2 Z

$

8 /0-I

2

4

6

/0 ~

2

4

6/{

Fig. 3. E n e r g y s p e c t r u m of velocity and t e m p e r a t u r e f l u c t u a t i o n s on the jet axis at the c r o s s section x/d = 10: 1) EU; 2) E0; 3) c a l c u l a t i o n of E U according to the autoc o r r e l a t i o n curve; 4) E U after [9]; 5) E 0 after [9]; 6) s t r a i g h t line with a n g u l a r coeff i c i e n t of - 5 / 3 . sity of t u r b u l e n c e within the flow at the n o z z l e outlet was i n s i g n i f i c a n t (~u/U < 0.01). The p r o f i l e s for the a v e r a g e values of the velocity and t e m p e r a t u r e in the l a t e r a l c r o s s section of the jet (x/d = 10) a r e w e l l - g e n e r a l i z e d by a r e l a t i o n s h i p of the G a u s s i a n - c u r v e type (Fig. la). The data obtained with the t h e r m o a n e m o m e t e r a g r e e with the Pitot tube measurements.

A c o m p a r i s o n of the p r o f i l e s shows that the heated jet s p r e a d s out over a w i d e r a r e a than the i s o t h e r m a l jet, i.e., the t u r b u l e n t mixing is m o r e intense when we have d i f f e r e n c e s in the gas d e n s i t i e s . M o r e o v e r , ~u,P0

i

a6

.

Q6

0

-/o

20

30

oo

T.[6

Fig. 4o A u t o c o r r e l a t i o n functions for t u r b u l e n t velocity and t e m p e r a t u r e fluctuations in the c r o s s section x/d = 10 (T, in see): 1) R U for r/(x + a) = = 0; 2) R 0 f o r r / ( x + a ) = 0 ; 3) R 0 f o r r / ( x + a ) = = 0.08~ the coefficient of t u r b u l e n t t h e r m a l diffusivity is s o m e what higher than the coefficient of t u r b u l e n t viscosity, thus yielding a t e m p e r a t u r e profile that is somewhat f u l l e r than the velocity profile. The convergence of the c u r v e s in the p e r i p h e r a l zone of the jet shows that the ratio of the coefficients is a p p a r e n t l y v a r i a b l e over the c r o s s section. A c c o r d i n g t o a n a l y s i s [2] and a c c o r d ing to the e x p e r i m e n t s c a r r i e d out in [3] with liquids exhibiting s u b s t a n t i a l d i v e r g e n t physical p r o p e r t i e s , the ratio of the coefficients of t u r b u l e n t t h e r m a l diffusivity and t u r b u l e n t v i s c o s i t y in the core of the flow is close to 1.4. The change in the intensity of the fluctuation in the longitudinal velocity component a c r o s s the flow in the case of i s o t h e r m a l flow and theochange in the f l u c t u a t i o n s of t e m p e r a t u r e in a heated jet a r e quite s i m i l a r (Fig. lb). The intensity m a x i m a a r e found in the zone which the flow exhibits the g r e a t e s values for the g r a dients of the averaged fields. These r e s u l t s a r e in close a g r e e m e n t with the C o r r sin data [4,5]. The existing d i v e r g e n c e in the r e l a t i v e i n t e n s i t i e s of the velocity f l u c t u a t i o n s - - r e a c h i n g 30% in the p e r i p h e r a l p o r t i o n of the jet (r/(x + a) = 0 . 1 2 0.17)--is evidently a s s o c i a t e d with the differing m e a s u r e m e n t methods. According to [2], the c o n s t a n t c u r r e n t method in the a n e m o m e t e r operation employed by C o r r s i n can yield a significant u n d e r s t a t e m e n t of the intensity of velocity fluctuation relative to the constant filament-temperature method, when w e consider the nonlinearity of the calibration curve which w a s employed in these experiments. In measuring the intensities of the temperature fluctuations w e employed a method identical to the one

used in [5], ice., the s e n s o r f i l a m e n t s functioned as r e s i s t a n c e t h e r m o m e t e r s . This portion of the r e s u l t s is in good a g r e e m e n t with the C o r r s i n m e a s u r e m e n t s .

218 A significant d i v e r g e n c e is found for the r e s u l t s o b tained from the data of [6], both in t e r m s of the vetoc i t y - f l u c t u a t i o n i n t e n s i t y and the intensity of the t e m perature variation. According to the e x p e r i m e n t a l data, at a distance of about 8d f r o m the nozzle outlet the d i s t r i b u t i o n of the m e a n v e l o c i t i e s and the d i s t r i b u t i o n of the t e m p e r a t u r e s in the jet become s i m i l a r . F o r the d i s t r i b u t i o n of the fluctuation i n t e n s i t i e s a U / U m a x and ao/Oma x we find no tendency toward s e l f - s i m i l a r i t y in the m e a s u r e m e n t s e g m e n t (x/d = 0-10). The r e l a t i v e i n t e n s i t y of the velocity fluctuations i n c r e a s e s over the e n t i r e segment; the i n t e n s i t y of the t e m p e r a t u r e fluctuations at f i r s t i n c r e a s e s and then, beginning f r o m x/d = 9, we find a tendency toward r e d u c t i o n . F r o m the p r o b a b i l i t y standpoint, the m o s t complete c h a r a c t e r i s t i c of the r a n d o m function is the d i s t r i b u tion law. F o r i s o t r o p i c t u r b u l e n c e , we established that the G a u s s i a n curve s e r v e s as an excellent a p p r o x i m a t i o n of the d i s t r i b u t i o n c u r v e for the components of the velocity fluctuations. Thus a c c o r d i n g to the T o w n send [7] data the d i s t r i b u t i o n is s y m m e t r i c a l and e x h i bits an excess p a r a m e t e r in the r a n g e 2 . 9 9 - 3 . The p r o b a b i l i t y d e n s i t y given in [8] for the t e m p e r a t u r e fluctuations in the wake behind a heat source in an isotropic flow deviates m a r k e d l y from the n o r m a l law. Of p a r t i c u l a r i n t e r e s t is the d e t e r m i n a t i o n of the p r o b a b i l i t y density for t u r b u l e n c e with a l a t e r a l shift, where the effect of the velocity and t e m p e r a t u r e g r a dient m a y make itself felt. By p r o c e s s i n g the d e r i v e d o s c i l t o g r a m s of the velocity and t e m p e r a t u r e f l u c t u a tions, we w e r e able to d e t e r m i n e the d i s t r i b u t i o n d e n s i t i e s at s e v e r a l points of the l a t e r a l c r o s s section of the jet. The d i s t r i b u t i o n s of the fluctuation amplitudes at various points in the core of the flow (0 - r/(x + + a) --< 0.1) (Fig. 2a and b) do not exhibit significant differences. The s t a t i s t i c a l d i s t r i b u t i o n is evened out well by the C h a r l i e r c u r v e . With i n c r e a s i n g d i s t a n c e f r o m the core (rAx + a) > 0.1) the excess coefficient i n c r e a s e s and the sign of the a s y m m e t r y coefficient changes. The change in the sign of the a s y m m e t r y (Fig. 2c) is apparently a s s o c i a t e d with the p r e d o m i n a n t g e n e r a tion of i n t e n s i v e l a r g e - s c a l e v o r t i c e s within a s p e c i fic region of the jet in which the coefficient A is close to zero (r/(x + a) ~ 0.1). The p e n e t r a t i o n of such v o r t i c e s to the c e n t e r of the jet and to the p e r i p h e r y l e a d s to intensive fluctuations of c o r r e s p o n d i n g l y different signs, which is consequently reflected in the sign of the a s y m m e t r y coefficient. The p r o b a b i l i t y d e n s i t i e s of the f i r s t d e r i v a t i v e s with r e s p e c t to t i m e for the fluctuations of the velocity and t e m p e r a t u r e components differ m a r k e d l y f r o m the n o r m a l law (Fig. 2d and e), whereby we note a s u b stantial i n c r e a s e in the p r o b a b i l i t y density of the zero values of the d e r i v a t i v e in the p e r i p h e r a l zone of the jet in c o m p a r i s o n with the core. This type of p r o b a b i l i t y d i s t r i b u t i o n of U and O and their d e r i v a t i v e s is associated with the i n t e r m i t t e n c e of the flow. The energy d i s t r i b u t i o n for the t u r b u l e n t f l u c t u a tions with r e s p e c t to f r e q u e n c i e s , or with r e s p e c t to the wave n u m b e r s , was d e t e r m i n e d by p a s s i n g a signal

INZ HENERNO-FIZ ICHESKII ZHURNAL through the n a r r o w - b a n d f i l t e r (a r e l a t i v e passband of 3%) and by the m e a s u r e m e n t of the m e a n - s q u a r e m a g nitude of this signal after the f i l t e r by m e a n s of the c o r r e l o m e t e r . F r o m the derived values of the i n t e n s i ties we subsequently d e t e r m i n e d the v a r i a n c e density for the chosen q u a s i - r e s o n a n c e frequency. The i n f r a s o n i c - f r e q u e n c y a n a l y z e r developed by the L'vov P o l y technic Institute s e r v e d a s t h e b a n d f i l t e r . The frequency range of the f i l t e r was 0.5-1000 Hz. In d e t e r m i n i n g the s p e c t r a l density of the t e m p e r a t u r e f l u c t u a t i o n s - - w h e n the t h e r m o a n e m o m e t e r was working in a c c o r d a n c e with the c o n s t a n t - c u r r e n t m e t h o d - w e introduced a c o r r e c t i o n factor for the effect of the t i m e constant of the a n e m o m e t e r s e n s o r f i l a ment. To v e r i f y the m e a s u r e m e n t s and the s t a n d a r d ization of the r e s u l t s , we used the r e l a t i o n s h i p gu,o =

z E u ' ~ d[

.

(2)

The difference between the r i g h t - and left-hand m e m b e r s of Eq. ( 2 ) w a s insignificant. In p a r t i c u l a r , i n t h e d e t e r m i n a t i o n of the o n e - d i m e n s i o n a l s p e c t r u m of the velocity fluctuations, the d i v e r g e n c e amounted to 4%. The s p e c t r u m of the longitudinal velocity f l u c t u a tions and of the t e m p e r a t u r e fluctuations on the jet axis was taken for a f r e q u e n c y r a n g e of 4 - 1 0 0 0 Hz (wave n u m b e r s of 1.0zl. 10-2-2.6 cm -1) (Fig. 3). A n a l y s i s of the c u r v e s shows that 75% of the f l u c t u a tion energy is conc_entrated in the w a v e - n u m b e r r a n g e up to 1 cm -1. T h e r e is no significant difference b e tween the s p e c t r a of the velocity and t e m p e r a t u r e f l u c tuationso Both of the s p e c t r a in the r e g i o n of large wave n u m b e r s a r e close to the Kolmogorov s p e c t r a l law (-5/3). In c o m p a r i s o n with the r e s u l t s of C o r r s i n [9], the s p e c t r a l - d e n s i t y m e a s u r e m e n t s which we c a r ried out a r e broadened by an o r d e r of magnitude in the region of lower wave n u m b e r s c o r r e s p o n d i n g to e n e r g y - c o n t a i n i n g v o r t i c e s . In both of the w a v e - n u m b e r r a n g e s , the c o m p a r a b l e s p e c t r a exhibit no s i g n i ficant divergences. P r o c e s s i n g of the o s c i l l o g r a m s gave u s one of the i n t e g r a l c h a r a c t e r i s t i c s which reflected the i n t e r r e lationship of the t i m e v a r i a t i o n s - - t h e n o r m e d a u t o c o r r e l a t i o n function (Fig. 4). The a u t o c o r r e l a t i o n functions of the pulsation c o m ponents of U and 0 at all points of the flow can be approximated by an exponential c u r v e . The t i m e c o r r e l a t i o n s ' f o r the c e n t r a l portion of the jet (r/(x + a) ~< 0.05) a r e close to those r e p r e s e n t e d in the figure by the a u t o c o r r e l a t i o n c u r v e s of the fluctuations on the jet axis. We find some d i v e r g e n c e between the c o r r e l a t i o n functions of the velocity and t e m p e r a t u r e f l u c tuations for values of T < 1 . 5 . 1 0 -3, w h e r e a s the s p e c t r a l d e n s i t i e s of these fluctuations are close to one another. This is a p p a r e n t l y a s s o c i a t e d with the effect of the t h e r m a l i n e r t a of the s e n s o r in m e a s u r i n g the t e m p e r a t u r e fluctuations. The effect of the t i m e constant for the s e n s o r f i l a m e n t in the d e t e r m i n a t i o n of the a u t o c o r r e l a t i o n function of the t e m p e r a t u r e f l u c tuations was not taken into c o n s i d e r a t i o n ; the i n e r t i a was taken into c o n s i d e r a t i o n in the a n a l y s i s of the spectrum.

JOURNAL OF ENGINEERING PHYSICS The t i m e c h a r a c t e r i s t i c s of t u r b u l e n c e can be found f r o m the a u t o c o r r e l a t i o n c u r v e . The magnitude of the l a r g e s t t i m e r e l a t i o n s h i p for the fluctuations ( m a c r o scale) can be a s s u m e d equal to Te = 1 . 2 . 1 0 -3 s e e ; t h e t i m e s e g m e n t ~-e = 0.7 • 10 -3 sec c o r r e s p o n d s to the m o s t rapid changes in the fluctuation m a g n i t u d e s on the axis of the jet (Euler m i c r o s c a l e ) . This s m a l l diff e r e n c e b e t w e e n t h e m a c r o - and m i c r o - s c a l e s indicates the c o m p a r a t i v e l y u n i f o r m s t r u c t u r e of the t u r b u l e n t v o r t i c e s and the t e m p e r a t u r e p e r t u r b a t i o n s . F o r points in the jet r e m o v e d f r o m the axis by m o r e then r j x + a) = 0.05, the f o r m of the c o r r e l a t i o n curve is f l a t t e r and the f r a c t i o n of the l o w - f r e q u e n c y f l u c t u a tion e n e r g y i n c r e a s e s . The s p e c t r a l function can be d e t e r m i n e d by F o u r i e r t r a n s f o r m a t i o n of the a u t o c o r r e l a t i o n function. F i g u r e 3 shows the d i s t r i b u t i o n of the s p e c t r a l density for the velocity f l u c t u a t i o n s on the jet axis, found in this m a n h e r . F o r a wave n u m b e r K = 0 the s p e c t r a l density was 1.71 cm. The v a l u e s of the s p e c t r a l density, c a l culated f r o m the a u t o c o r r e l a t i o n function and derived by m e a s u r e m e n t with the aid of the band f i l t e r , a r e in good a g r e e m e n t . NOTATION d is the nozzle d i a m e t e r ; x is the coordinate along the jet axis; r is the r a d i u s of the m e a s u r e m e n t point a is the d i s t a n c e f r o m the nozzle outlet section to the conditional d i s c h a r g e s o u r c e ; U is the longitudinal velocity; U is the longitudinal component of the f l u c t u a tion velocity; Umax is the velocity at the jet axis; 0- is the excess t e m p e r a t u r e at a point in the flow; 0ma x is the e x c e s s t e m p e r a t u r e at the jet axis; 0 is t h e t e m p e r a t u r e fluctuation; u U is the i n t e n s i t y of the longitudinal velocity f l u c t u a t i o n s ; 0-0 is the i n t e n s i t y of the t e m p e r a t u r e fluctuations; PU and PU' a r e , r e s p e c t i v e l y , the

219 p r o b a b i l i t y d e n s i t i e s for the velocity fluctuations and its t i m e d e r i v a t i v e ; P0 and P0' are, r e s p e c t i v e l y , the p r o b a b i l i t y density for the t e m p e r a t u r e fluctuations and its t i m e d e r i v a t i v e ; E is the excess coefficient; A is the a s y m m e t r y coefficient; R U and R 0 a r e , r e s pectively, the a u t o c o r r e l a t i o n coefficients for the v e l o city and t e m p e r a t u r e fluctuations; T e is the E u l e r t i m e m a c r o s c a l e ; Te is the E u l e r t i m e m i c r o s c a l e ; E U and E 0 a r e , r e s p e c t i v e l y , the s p e c t r a l d e n s i t i e s for the t u r b u l e n t fluctuations in velocity and t e m p e r a t u r e ; f is the frequency; K = 2~//U [s the wave n u m b e r ; I is the f i l a m e n t - h e a t i n g c u r r e n t ; T is the t i m e ; Re is the Reynolds n u m b e r . REFERENCES 1. V. P. Bobkov, Yu. I. Gribanov, M. Kh. I b r a g i -

mov, E. V. Nomofilov, and V. I. Subbotin, Teplofizika vysokikh temperatur, 3, no. 5, ]965. 2. I. O. Hintze, Turbulence [Russian translation], Fizmatgiz, 1963. 3. Z. B. Sakipov and D. Zh. Temirbaev, TepLoi massoperenos, 2, 1965. 4. S. Corrsin, NACA, Rep. W-94, 1943. 5. S. Corrsin and M. S. Uberoi, NACA[in Russian],, Rep. no. 998, 1950. 6. G. So Antonova, Transactions of the Conference on Applied Gasdynamies [in Russian], Alma-Ata, 1959o 7. A. A. Townsend, Proc. Cambridge Phil. Soc. 43, pt. 40, Oct. 1947. 8. M. S. Uberoi and S. Corrsin, NACA,Rep. 1942, 1952. 9. S. Corrsin and M. S. Uberoi, NACA,Rep. 1040, 1941. 20 June 1967

P h y s i c s and P o w e r - E n g i n e e r i n g Institute, Obninsk