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AERO. ® ASTRO. LIBRARY 


NATIONAL ADVISORY COMMITTEE 
FOR AERONAUTICS 



REPORT 998 

c 3 

AEPO 

FURTHER EXPERIMENTS ON THE FLOW AND HEAT 
TRANSFER IN A HEATED TURBULENT 

AIR JET 




By STANLEY CORRSIN and MAH1NDER S. UBEROI 



1950 


ev 


For sate by the Superintendent of Documents, U. S. Government Printing Office, Washington 25, D. C. Yearly subscription, $3.50; foreign, $4.50; 

single copy price varies according to size - -- -- -- -- Price 20 cents 


L 



1*2 3:7 Y& 
4/5 't 


w 

0 

m 

1 

p 

S 

s„ 

b 

c 

A 

V 

2 
L 
D 
D 0 


D t 


Dv 

C 


AERONAUTIC SYMBOLS 


1. FUNDAMENTAL AND DERIVED UNITS 



Symbol 

Metric 

English 

Unit 

Abbrevia- 

tion 

Unit 

Abbreviation 

Length _ _ _ 

Time 

Force 

l 

t 

F 

meter 

second _ _ _ 

weight of 1 kilogram _ . 

m 

s 

kg 

foot (or mile) 

second (or hour) 

weight of 1 pound-- _ 

ft (or mi) 
sec (or hr) 
lb 

Power 

P 

V 

horsepower (metric) 


horsepower 

hp 

mph 

fps 

Speed. _ 

fkilometers per hour. _ _ 
\meters per second 

kph 

mps 

miles per hour 

feet per second 


2. GENERAL SYMBOLS 


Weight =mg 

Standard acceleration of gravity =9.80665 m/s 2 
or 32.1740 ft/sec 2 
W 

Mass-— 

9 

Moment of inertia =mk 2 . (Indicate axis of 
radius of gyration k by proper subscript.) 
Coefficient of viscosity 


v Kinematic viscosity 

p Density (mass per unit volume) 

Standard density of dry air, 0.12497 kg-m~ 4 -s 2 at 15° C 
and 760 mm; or 0.002378 lb-ft“ 4 sec 2 
Specific weight of “standard* ’ air, 1.2255 kg/m 3 or 
0.07651 lb/cu ft 


3. AERODYNAMIC SYMBOLS 


Area 

Area of wing 
Gap 
Span 
Chord 

Aspect ratio, ^ 

True air speed 
Dynamic pressure, ^ pF 2 

Lift, absolute coefficient Cl== 

Drag, absolute coefficient G D =^ 


Do 


Profile drag, absolute coefficient Cz> 0 ==^g 
Induced drag, absolute coefficient C Di =^ 
Parasite drag, absolute coefficient C Dp == ^ 


C 


Cross-wind force, absolute coefficient G c ~ ^ 


i to 

it 

Q 

fi 

R 


a. 

e 

<*o 

OCi 

<*a 

7 


Angle of setting of wings (relative to thrust line) 
Angle of stabilizer setting (relative to thrust 
line) 

Resultant moment 
Resultant angular velocity 


Reynolds number, p 


— where l is a linear dimen- 


sion (e.g., for an airfoil of 1.0 ft chord, 100 
mph, standard pressure at 15° C, the corre- 
sponding Reynolds number is 935,400; or for 
an airfoil of 1.0 m chord, 100 mps, the corre- 
sponding Reynolds number is 6,865,000) 
Angle of a t tack 
Angle of down wash 
Angle of attack, infinite aspect ratio 
Angle of attack, induced 

Angle of attack, absolute (measured from zero- 
lift position) 

Flight-path angle 


REPORT 998 


FURTHER EXPERIMENTS ON THE FLOW AND HEAT 
TRANSFER IN A HEATED TURBULENT 

AIR JET 

By STANLEY CORRSIN and MAHINDER S. UBEROI 


California Institute of Technology 


i -x-{ 


National Advisory Committee for Aeronautics 


Headquarters , 1724 F Street N. W., Washington 25 , D. C. 

Created by act of Congress approved March 3 , 1915, for the supervision and direction of the scientific study 
of the problems of flight (U. S. Code, title 50, sec. 15). Its membership was increased from 12 to 15 by act 
approved March 2, 1929, and to 17 by act approved May 25, 1948. The members are appointed by the President, 
and serve as such without compensation. 

Jerome C. Hunsaker, Sc. D., Massachusetts Institute of Technology, Chairman 


Alexander Wetmore, Sc. D., Secretary, 

Detlev W. Bronx, Ph. D., President, Johns Hopkins Univer- 
sity. 

John H. Cassady, Vice Admiral, United States Navy, Deputy 
Chief of Naval Operations. 

Edward U. Condon, Ph. D., Director, National Bureau of 
Standards. 

Hon. Thomas W. S. Davis, Assistant Secretary of Commerce. 

James H. Doolittle, Sc. D., Vice President, Shell Union Oil 
Corp. 

R. M. Hazen, B. S., Director of Engineering, Allison Division, 
General Motors Corp. 

William Little wood, M. E., Vice President, Engineering, 
American Airlines, Inc. 

Theodore C. Lonnquest, Rear Admiral, United States Navy, 
Deputy and Assistant Chief of the Bureau of Aeronautics. 


Smithsonian Institution, Vice Chairman 

Donald L. Putt, Major General, United States Air Force, 
Director of Research and Development, Office of the Chief of 
Staff, Materiel. 

Arthur E. Raymond, Sc. D., Vice President, Engineering, 
Douglas Aircraft Co., Inc. 

Francis W. Reichelderfer, Sc. D., Chief, United States 
Weather Bureau 

Hon. Delos W. Rentzel, Administrator of Civil Aeronautics, 
Department of Commerce. 

Gordon P. Saville, Major General, United States Air Force, 
Deputy Chief of Staff — Development. 

William Webster, M. S., Chairman, Research and Develop- 
ment Board, Department of Defense. 

Theodore P. Wright, Sc. D., Vice President for Research, 
Cornell University. 


Hugh L. Dryden, Ph. D., Director 
John W. Crowley, Jr., B. S., Associate Director for Research 


John F. Victory, LL. D., Executive Secretary 
E. H. Chamberlin, Executive Officer 


Henry J. E. Reid, D. Eng., Director, Langley Aeronautical Laboratory, Langley Field, Va. 

Smith J. DeFrance, B. S., Director, Ames Aeronautical Laboratory, Moffett Field, Calif. 

Edward R. Shari*, Sc. D., Director, Lewis Flight Propulsion Laboratory, Cleveland Airport, Cleveland, Ohio 


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II 


REPORT 998 


FURTHER EXPERIMENTS ON THE FLOW AND HEAT TRANSFER IN A HEATED TURBULENT 

AIR JET 


By Stanley Corrsin and Mahinder S. Uberoi 


SUMMARY 

Measurements have been made of the mean-total-head and 
mean-temperature fields in a round turbulent jet with various 
initial temperatures . The results show that the jet spreads more 
rapidly as its density becomes lower than that of the receiving 
medium , even when the difference is not sufficiently great to cause 
measurable deviations from the constant-density , dimensionless , 
dynamic-pressure profile function. Bough analytical considera- 
tions have given the same relative spread. 

The effective U turbulent Prandtl number” for a section of the 
fully developed jet was found to be equal to the true {laminar) 
Prandtl number within the accuracy\of measurement. 

Measurement of turbulence level ( u' y v') 7 temperature fluc- 
tuation level it ' , and temperature-velocity correlation &u permit a 
comparison of their relative magnitudes. 

Direct measurements have been made of the double correlations 
uv and i)v across a section of the fully developed jet , and the 
shear-stress and heat-transfer distributions have been computed 
therefrom. Finally , these last-mentioned measurements have 
permitted a determination of the distribution of turbulent Prandtl 
number across the jet , and these values agree quite well on the 
average with the effective value computed from mean velocity and 
temperature alone. 

INTRODUCTION 

The present work is concerned with two particular prob- 
lems on the flow of round, low-velocity, turbulent jets: (a) 
The effect of mean-density differences upon the rate of spread 
of the jet, which can be examined through the average veloc- 
ity and density fields; (b) the relative rates of transfer of 
heat and momentum in a heated jet, which must involve a 
study of fluctuations in velocity and temperature as well as 
average fields. 

Problem (a) has been investigated experimentally by Pabst 
(references 1 and 2), Von Bohl (reference 2), and others. 
Some British results consist of a few rough measurements 
made in the course of a different investigation and, unfor- 
tunately, the German reports are available only in practically 
illegible form, so that a thorough reading has not been feasi- 
ble. These references arrived when the present investiga- 
tion was in progress and substantiated the result, discussed 
in detail in the body of the paper, that reduction in density 
of jet relative to receiving medium increases the rate of 
spread. 

Theoretical analysis of the two-dimensional, variable- 
density jet has been attempted by Hu (reference 3), using 
the momentum-transfer theory. However, he has begun by 
assuming the constant-density characteristics of linear jet 
spread and inverse parabolic decrease in axial velocity and | 


therefore has not actually solved the variable-density prob- 
lem. 

Goff and Coogan (reference 4) and Abramovich (reference 
5) have solved the two-dimensional, variable-density, single- 
mixing zone, by momentum-transfer and vorticit 3 r -transfer 
theories, respectively. The former analysis involves the 
drastic assumption of a discontinuous density field. 

In general, however, the shortcomings of the theory of 
mixing length and the more recent theory of constant ex- 
change coefficient, discussed briefly by Liepmann and 
Laufer (reference 6), render such analyses useful perhaps 
only for qualitative results. In fact, as illustrated in 
reference 6, equally useful results in shear-flow problems are 
obtainable with appreciably less difficulty by the use of the 
integrated equations of motion, with reasonable guesses for 
the shape of velocity and/or shear profiles. 

A related method has been used by Ribner (reference 7), 
who has given an approximate solution for the variable- 
density round jet in a moving medium by a rather empirical 
generalization of the method used by Squire and Trouncer 
(reference 8) in constant-density jets. However, this 
method employs the shear assumption of the momentum- 
transfer theory, so that the result, though a saving in labor, 
cannot be expected to give greater accuracy than the com- 
plete momentum-transfer analysis. 

In the present report, a very brief and approximate 
analysis is given for the relative jet spread as a function of 
density ratio, obtainable without a complete solution of the 
integral problem. 

The problem just noted is different from that of a jet of 
high subsonic velocity, in which the temperature differences 
arise from frictional heating. However, since the maximum 
rate of dissipation of turbulent energy (into heat) in a jet 
takes place on the jet axis (reference 9), the over -all behavior 
of a fully developed, turbulent, high-speed, subsonic jet may 
not be very much different from a lower-speed heated jet, if 
the high-speed jet starts out with the same temperature as 
the receiving medium. There appear to be no data available 
on turbulent jets with Mach number approaching unity. 
Abramovich (reference 5) has also applied the vorticity- 
transfer theory to an approximate solution of the high-speed 
(subsonic), plane, single-mixing region with the same 
stagnation temperature for moving and stationary mediums. 

The present investigation on problem (a) has been under- 
taken primarily to determine experimentally the effect of 
density difference upon the jet spread. Another matter of 
interest is the possible deviation from simple geometrical 
similarity for large density differences. 


1 


2 


REPORT 998 — NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 


Somewhat more work has been previously done on the 
simple phases of problem (b), the relative rates of heat and 
momentum transfer in a round turbulent jet. The first 
systematic measurements of velocity and temperature 
distribution were apparently made by Ruden (reference 10), 
who found that the temperature distribution is appreciably 
broader than the velocity distribution in the fully developed 
region. A more detailed investigation of the kinematic 
quantities, including turbulence level near the axis, was 
carried out by Kuethe (reference 11). The results of these 
two investigations have been verified and extended by one 
of the present writers (reference 12). 

Since there exists no satisfactory theory of turbulent shear 
(low, naturally there is none for heat transfer in turbulent 
shear flow. Most of the theoretical work has been done on 
channel and boundary-layer flows, with the heat transfer 
set up in direct analogy to the momentum transfer. The 
empirical factor connecting the two is, of course, exactly the 
quantity that a satisfactory theory must predict. 

For turbulent jets, momentum-transfer considerations lead 
to identical curves for velocity and temperature distributions 
(reference 13), whereas the modified vorticity-transfer theory 
leads to the qualitatively correct result (reference 14) that 
heat diffuses more rapidly than momentum. Quantita- 
tively, however, this result is considerably in error (reference 
12 ). 

Clearly, no satisfactory investigation, either experimental 
or theoretical, of a turbulent-flow phenomenon can be made 
without consideration of the fluctuations as well as the mean 
distributions. In the round turbulent jet, fairly detailed 
measurements have been made of the fluctuation kinematic 
quantities (reference 12), but no previous investigations ap- 
pear to have included the significant quantities involving 
temperature fluctuations. Since a hot-wire technique for 
the simultaneous measurement of temperature and velocity 
fluctuations has recently been developed (references 15 and 
16), it is now possible to study turbulent heat transfer from 
the points of view of both mean and fluctuating variables. 

At the present time, a weakness of the new measuring tech- 
nique is the uncertainty of the detailed physical form of 
King’s equation (reference 17) for the heat loss from a fine 
wire to a flowing fluid. A brief discussion is given in ap- 
pendix A of reference 16. The results presented in the 
present report indicate that, at least for variable temperature 
applications, King’s equation may be quantitatively satis- 
factory. 

The first measurements of temperature fluctuation level 
were made in 1946, in connection with the preliminary phases 
of the present investigation. However, deviations from 
rectangularity in the initial jet temperature distribution 
prompted the construction of a completely new hot-jet 
unit, used in all measurements presented herein, and the 
preliminary measurements have not been published. 

Measurements of mean-total-head and mean-temperature 
distributions at various jet temperatures have been made in 
order to find out if there is any appreciable change in the 
relative rates of heat and momentum transfer with absolute 
temperature, over a moderate range. 


Finally, it should be noted that investigations of heat 
transfer in turbulent shear flow are important not only for the 
immediate results but also as a means of studying The - 


turbulent motion itself. 

This investigation was conducted at the California Insti- 
tute of Technology under the sponsorship and with the 
financial assistance of the National Advisory Committee for 

Aeronautics. 

The authors would like to acknowledge the 

assistance of Miss Dorothy Kerns, Mrs. Beverly (Nottingham, 
Mrs. Sally Rubsamen, and Miss Betsy Barnhart in comput- 
ing the results and drawing the final figures for this report. 


SYMBOLS 

d 

X 

diameter of orifice (1 in.) 
axial distance from orifice 

r 

radial distance from jet axis 

U 

axial component of mean velocity 

V 

radial component of mean velocity 

TJ max 

maximum l r at a section on jet axis 

U 0 

maximum U in the jet (i. e., in the potential 


cone) 

! 1 

dynamic pressure (^'pU 2 ^ 

q max 

maximum dynamic pressure at a section 

( U 

maximum dynamic pressure in the jet 

T a 

mean absolute temperature at a point in the 


flow 

T r 

absolute temperature of the receiving medium 

To 

absolute initial jet temperature 

d=T a -Tr 


Q max 

maximum 6 at a section 

So 

maximum 6 in the jet ( T 0 — T r ) 

u 

axial component of instantaneous velocity 


fluctuation 

V 

radial component of instantaneous velocity 


fluctuation 

0 

instantaneous temperature fluctuation 

i= 

\\ 


v' = yiv 2 


r, 

at any section, the value of r for which q 
1 

— 9 (Z max 


at any section, the value of r for which U 

— - V 

— 9 L max 

at any section, the value of r for which d 

r' 


— cy Umax 

P 

total pressure 

P 

air density 

P 

mean density at a point in the flow (at T a ) 

p' 

density fluctuation 

P min 

minimum ~p at a section on the axis 

Po 

minimum p in the jet (at T 0 ) 

P 00 

density of receiving medium (at T r ) 

p 

viscosity coefficient of air 


FURTHER EXPERIMENTS ON THE FLOW AND HEAT TRANSFER IN A HEATED TURBULENT AIR JET 


3 


V 

k 

Cp 

o 

T 

Q 

€ 

Vt 

A 


A! 

A* 


kinematic viscosity (/ u./p) 

thermal conductivity of air 

specific heat of air at constant pressure 

Fraud tl number 

shear stress 

heat transfer per unit area 
turbulent exchange coefficient 
turbulent heat-transfer coefficient 


“turbulent Prandtl number” 

ayerage or effective a t across a section of the 
jet 

“momentum diameter” of jet 

A at a section in a constant-density jet 
“thermal diameter” of hot jet 


R du — du/&'u' 
Rd V =&v/&'v' 
R UD =u7lu'v' 
e 

a, 7,5 



| double correlation coefficients at a point 

instantaneous hot-wire voltage fluctuation 
sensitivity of skew wire to d, u , and v fluctua- 
tions, respectively 


EQUIPMENT 

A E RO DY N A MIC EQ HI PM ENT 


The 1-inch hot-jet unit is shown schematically in figure 1. 
The three-stage axial blower (reference 18) is driven by a 
2-horsepower variable-frequency motor, operated at a frac- 



iary air is not ducted back into the main blower intake, heat 
is discharged from the annulus at an appreciable rate and is 
exhausted outside the building to minimize room-temperature 
rise. 

The relatively high velocity section between heaters and 
final pressure box promotes adequate mixing behind the 
grid to insure a uniform initial jet-temperature distribution. 
An earlier, simpler unit, mentioned in the introduction, had 
the heating coils in the final (and only) pressure box; as a 
result, the Reynolds number of the coil wire and even that 
of an extra ceramic grid were too low to produce turbulent 
mixing. 


The complete unit is photographed in figure 2 (a), and 
figure 2 (b) is a close-up of the asbestos orifice plate and the 
traversing mechanism. 



tion of rating. Heat is added through two double banks of 
coils of No. 16 Nichrome wire. As can be seen in the sketch, 
a good part of the heated air is ducted around the outside of 
the jet-air pipe in order to maintain a flat initial temperature 
distribution in the jet. A blower is used to help the air 
through this secondary heating annulus. Since this auxil- 



(a) Complete hot-jet unit. 

(b) Close-up of orifice and traverse track. 


Figure 2.— Test setup. 



4 


REPORT 998 — NATIONAL ADVISORY COMMITTEE FOI t AERONAUTICS 


Runs have been made at orifice velocities between 65 and 
115 feet per second and at orifice-temperature differences 
between 0° and 385° C. The initial dynamic-pressure dis- 
tribution (fig. 3) is effectively rectangular, and the initial 
temperature distribution deviates from that only slightly. 



Figure 3.— Initial temperature and total-head distributions. 1-inch round jet. < 7 o = 40 milli- 
meters of alcohol. 

MEASURING EQUIPMENT 

The measuring instruments used were: Hypodermic- 
needle total-head tube, Chromel-Alumel thermocouple, and 
hot-wire anemometer. 

The hot-wires were nominally 0.00025-inch platinum 
etched from Wollaston wire. The etched platinum was 
soft-soldered to the tips of small steel needle supports. The 
anemometer heating circuits and amplifier were designed 
and built by Mr. Carl Thiele in 1941. The circuit is so 
arranged that hot-wire time constants are determined by 
superimposing equal alternating-current voltages at two 
frequencies upon the balanced direct-current bridge. 

The amplifier gain is constant to within ±2 percent over 
a frequency range from below 7 to 7000 cycles. No check 
was possible at less than 7 cycles since that was the lower 
limit of the available oscillators. Compensation for hot- 
wire lag is achieved by a resistance-inductance network 
between two stages of the amplifier. For a normal range of 
wire time constants, the combination of hot-wire and prop- 
erly compensated amplifier is satisfactory over the entire 
fiat range of the uncompensated amplifier. The amplifier 
output was read on an approximately critically damped 
wall galvanometer, with a vacuum thermocouple. 

Mean total head (i. e., dynamic pressure in a free jet) and 
mean temperature were photographically recorded simulta- 
neously, by means of an automatic traversing arrangement, 
used with the total-head tube and thermocouple. The total- 
head-tube pressure line was run into a small copper bellows 
which tilted a mirror, thereby deflecting a narrow light beam 
upon a uniformly moving sheet of sensitized paper. The 
simultaneous recording of temperature on the paper utilized 
directly the light beam reflected from the mirrors of a sensi- 
tive galvanometer. The galvanometer was critically damped 


for all sensitivity settings. The possible errors arising from 
changes in “steady-state” conditions during a continuous 
unidirectional traverse were investigated by means of a few' 
check runs in the opposite direction; these showed no appre- 
ciable difference. 

The automatic traversing w r as accomplished by means of a 
screw-driven carriage running horizontally along a steel 
track (fig. 2). The screw was rotated through a gear, worm, 
and belt drive by a reversible alternating-current motor with 
wide speed range, operating on a continuously adjustable 
transformer. The sensitized-paper holder was mechanically 
connected to the moving carriage, so that the abscissas of 
the recording curves were equal to the true radial distance. 
Paper shrinkage in development was found to be negligible. 
Figure 4 is a typical record. The symmetry axes of total 
head and temperature are offset laterally because of the 
necessary distance between tube and thermocouple. 



PROCEDURES 

MEAN DYNAMIC PRESSURE 

Mean. dynamic pressure is effectively the total head in a 
free jet and was recorded photographically, as described in 
the previous section. The photo record w r as faired and 
traced on cross-section paper. In order to reduce scatter, 
each traverse finally presented in this report has been aver- 
aged from three individual faired runs. Faired curves from 
typical repeated traverses are plotted together in figure 5 



FURTHER EXPERIMENTS ON THE FLOW AND HEAT TRANSFER IN A HEATED TURBULENT AIR JET 


5 


to illustrate the degree of disagreement. These particular 
runs were made with the old hot-jet unit. 

•No correction has been made for the effect of velocity and 
density fluctuations upon the readings, but a brief analysis 
of this correction is given in appendix A. 

MEAN TEMPERATURE 


able unsymmetry, the meter was slightly sensitive to u'lU ; 
appropriate correction has been made. 

I Vu CORRELATION 

The correlation was measured simultaneously with 
u'fU and &'/e, as described in references 15 and 16. 


The temperature was also recorded photographically, as 
described in the previous section. Since the galvanometer 
deflection varied linearly with temperature, there was no 
reading error specifically attributable to the temperature 
fluctuations. 


MEAN VELOCITY 

With the assumption of a perfect gas, velocity was com- 
puted from dynamic pressure and temperature. 


AXIAL COMPONENT OF TURBULENCE 


In the cold jet, the axial component of turbulence was 
measured in the conventional manner with a single hot-wire, 
about 2 millimeters in length, set normal to the x- and 
r-directions. 

In the hot jet, u'lU was measured simultaneously with 
d'/O and du/dU by the technique described in references 
15 and 16. 

Throughout the report, no correction has been made for 
the error (in reading fluctuation levels) due to the fact that 
the fluctuations are not always small, as assumed in the 
theory of turbulence measurement with hot-wires. How- 
ever, some extremely brief check measurements, reported 
in reference 12, indicate that the absolute values may be 
fairly good, even at levels above 50 percent. 

It is important to note that the mean velocity to which 

the hot-wire responds is U R =^U 2j rV 2 rather than simply 
U . Therefore the local turbulence level measured, and plotted 
in the figures, is actually u'IU R . However, this differs 
appreciably from u'lU only at the outer edge of the jet, 
where the flow is not completely turbulent anyway. The 
distributions of u'IU max , however, are just that quantity 


'll! u' l J 

at all points, having been computed from= — ==- . — — 

^ max ^ R l max 

The' mean-velocity distributions plotted in the same figures 


as the fluctuation measurements are all U R IU max and there- 
fore, at the outer edge, are appreciably different from the 
curves obtained from total-head-tube measurements and 
plot ted in other figures. 


TEMPERATURE FLUCTUATION LEVEL 

Temperature fluctuation level was measured directly by 
using a single wire as a pure resistance thermometer. 


RADIAL COMPONENT OF TURBULENCE 

The radial component of turbulence was measured in the 
hot jet with an X-type meter made up of two wires (about 
3 mm long), the voltage fluctuations of which were sub- 
tracted before amplification. Because of almost unavoid- 


uv AND Ov CORRELATIONS 

The uv and correlations were measured simultaneously, 
according to the general procedure given in the foregoing 
references. The extreme slowness of the measurements pre- 
cluded the use of a simple inversion procedure for unsym- 
metry correction (i. e., repetition of readings at a point, with 
the meter rotated through 180°). Instead, the results were 
computed directly with the directional sensitivity calibra- 
tions of the individual wires. 

The mean squares of the fluctuation voltages of the two 
wires are 


et 




9 * UV 

2 y ^ jt2 


e -2 


+w (p) +v (p) + 2 0,71 sr 2 “ A Sr 


9 x UV 
"72^2 -jj2 


Since there are two unknowns, readings must be taken at 
two different hot-wire sensitivities. Then the two pairs of 
differences give 

7?-7?=W-a^ (U + (t, 2 -7 2 *) 

o/ X &U . r»/ * , . % i) , 

2(«i7i— 0L 2 y 2 ) s=+2 (aidi + a 2 0ai} 

QU eu 

• 2( Ti 5, +7 2 S 2 )S 

U* 

(|y+(7/ t -7 2 ' 2 ) (=y+ 


(Sr-S/*) (=y + 2(a,'7 || + 

2(a,'5,' + a 2 '5 2 ') ^1+2( 7 i' 5,'+7 2 V) jjg 
dU u 

Thus, the u, v, and t? sensitivities of each wire must be 
determin ed sep arately. Fr om pre vious measurements, 
(t}fd) 2 , ( u/U ) 2 , tiufol r , and ( vjU ) 2 are known, and the 
foregoing pair of linear equations can be solved simultane- 
ously for the two unknowns dvjdU and uv/U \ 


6 


REPORT 998— NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 


EXPERIMENTAL RESULTS 

The axial distributions of dynamic pressure, tempera- 
ture, and velocity are given in figures 6 and 7, for the 
constant-density jet for an initial density ratio poo/p 0 ~2. 
Since all three variables decrease more rapidly when jet 
density is appreciably less than that of the receiving medium, 
it is to be expected that the lower-density (hot) jet spreads 
at a wider angle. This is verified in figure 8, which shows 



— ■ 









— i — r i — i — i 
e o ^300°c 



V 

\ 









- d 

5 = 

' Ol 




— \ 

— 

\\ 
















V 
















\ 

\ 
















V 
















\ 

\ 

k N 
















\ 

\ 

\ 










































0 4 8 12 16 20 24 28 32 


x/d 



0 4 8 12 16 20 24 28 32 


x/d 



0 4 8 12 16 20 24 28 32 


x/d 

Figure 8.— Spread of heat and momentum. 1-inch round jet. 


the half total-head diameter 2r x and the half temperature 
diameter 2 r' as functions of xjd, for these two cases. 

In figures 6 and 7 it can be seen that both dynamic and 
thermal potential cones are shorter for the low-density jet. 
Since the annular mixing region outside the potential cone 
is somewhat similar to a single two-dimensional mixing zone, 
the nature of the flow in this latter case, with flowing medium 
less dense than stationary medium, can be inferred. 

Figures 9 and 10 show the dimensionless profiles o: total 
head, temperature, and velocity at a section 15 diameters 
from the orifice, for p w / p 0 ~\ and 2. It is immediately 
evident that the shape of these functions is not appreciably 
changed at x\d — 15 by a doubling of the initial density ratio. 



Figure 9.— Total-head and temperature distributions. 1-inch round jet. x/d=\5. 



Figure 10.— Temperature and velocity distributions. 1 -inch round jet. x/d= 15. 



Figure 11.— Check on similarity of total-head distribution. 1-inch hot jet. x/d= 15; ri = 

at which q—Viq max. 


FURTHER EXPERIMENTS ON THE FLOW AND HEAT TRANSFER IN A HEATED TURBULENT AIR JET 


7 


The local maximum density ratios at the section are 
Poo/pmin^l and 1.3, respectively. A verification of the 
negligible change in total-head profile is given in figure 11, 
in which all the functions have been fitted at the point r—r j. 

For a constant-density jot, it is well-known that effective 
similarity of total-head (or velocity) profiles exists for all 
values of x/d greater than 7 or 8 (references 11 and 12). 
This is verified in figure 12. Figure 13 shows that an initial 



Figure 12.— Check on similarity of total-head distributions. Constant-density jet. 
0 O =15° C. n=r at which g = ^9m*x. 



Figure 13.— Check on similarity of total-head distributions. 1-inch hot jet. 0 o =3OO° C 
ri=r at which q=ttqm*x. 

density ratio of 2 is not sufficient to change this result 
appreciably. The deviation of the total-head distributions 
for x/d = 20 at large values of r/ri is almost certainly due to 
inaccuracy of the traced data at a section where even the 
maximum dynamic pressure gives only about a 1-inch 
deflection on the photo record. 

The fact t hat the density-ratio range covered in the present 
series of experiments is insufficient to lead to measurable 
deviation from simple geometrical similarity is clearly 
shown by the constancy of r f jr x and A fr x in figure 14. 
However, again it should be pointed out that, although the 
initial density-ratio range is from 1 to 2, the range of max- 
imum density ratio at this particular section is only from 
1 to 1.3. 

Although r i has been used as a characteristic jet width in 
most of the previous figures, it is clear that a width definition 
of greater physical significance can be made on the basis of 



Figure 14.— Comparison of characteristic lengths. 1-inch heated jet. x/d~ 15. 


momentum flow. The “momentum diameter' 
at any section 

, 1/2 


of the jet 


.= 2 V 2( r-^rdr)' 

\J 0 max / 


is defined as the diameter of a jetof rectangular density and 
velocity profiles (p = p mfn and U=U max ), whose total rate 
of flow of axial momentum is the same as in the actual jet 
at that section. Since the momentum flow in a free jet is 
effectively constant, this chracteristic diameter is particu- 
larly convenient. It is analogous to the momentum thick- 
ness of a boundary layer, except that the latter is based on 
momentum defect. 

The ratio of hot-jet momentum diameter at x/d= 15 to 
that of the constant-density jet at the same section (fig. 15) 



Figure 15.— Variation of momentum diameter with initial temperature or density ratio. 

Hot jet. x/d= 15. 


shows a definite increase in jet width as the jet fluid becomes 
less dense than the receiving medium. A few points obtained 
with the preliminary hot-jet equipment have been included 
because this simple unit was able to produce an appreciably 


: 


- 904G42 — 51 2 


8 


REPORT 998 — NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 


higher initial temperature. The result of a rough the- 
oretical analysis of the effect, presented in the next section, 
is drawn in the same figure. 

A comparison between the cold-jet axial u r distribution 
(fig. 16) and that for an initial temperature difference of 170° 
C (fig. 17) shows that the sharp rise in turbulence level comes 



Figure 16. — Axial velocity fluctuation levels. 1-inch round jet. ^=15° C. 



a bit earlier for the low-density jet. Of course, this may be 
due to an inaccuracy in King’s equation and thus may not 
be a real effect; however, the difference is certainly in the 
direction that would be anticipated as a result of the more 
rapid spread and development of the mean motion pattern 
in the lower-density case. 

The temperature fluctuation level as measured is every- 
where lower than the velocity fluctuation level. 

Two qualitative aspects of the fluctuations were observed 
during these particular measurements: First, extremely 
regular velocity fluctuations were observed in the potential 
cone, just as illustrated in reference 13; second, the tempera- 
ture fluctuations on the jet axis, roughly between x/d = 3 
and 5, were almost entirely one-sided, although the u fluctu- 
ations in that region apparently showed no such tendency. 
The reason for this behavior is not clear at the time of writing. 

In the process of measuring u'/U in a flow with simul- 
taneous velocity and temperature fluctuations, the du 
correlation results more or less as a byproduct; du/9 0 U 0 and 
the coefficient R du =fru/&'u' are plotted in figure 18. In 
spite of tremendous experimental scatter, which has even 



produced occasional negative values for fru, it appears that 
0 in turbulent shear flow. The sign can be obtained 
by the reasoning (in rough analogy to the kinetic theory of 
gases) that uv has a sign opposite to that of dU/dr in a normal 
turbulent shear flow. 

Since du is proportional to the turbulent heat transfer in 
the main flow direction, it is clearly not of so great interest 
as tlietfy correlation. It is, however, of use in the computa- 
tion of the total-head- tube correction for the effect of velocity 
and temperature fluctuations (appendix A). No attempt 
has been made to draw a curve through the experimental 
points. 



; 


FURTHER EXPERIMENTS ON THE FLOW AND HEAT TRANSFER IN A HEATED TURBULENT AIR JET 


9 


A check measurement of the lateral u' distribution in the 
fully developed constant-density jet, in figure 19, shows 
reasonable agreement with earlier results (reference 12). 

Figure 20 contains lateral traverses of both u' and in 
the hot jet at x/d= 20. Again, &'/0 is everywhere slightly 
less than u f /£/, but it should be noted that /~d ma x/>u' /U max 
in the outer part of the jet. Perhaps associated with this 
relative behavior is the fact that, although &'!0 max has a 
definite local minimum on the jet axis, u' /U max is flat in that 
region, within the accuracy of measurement. 

The coefficient Ro u is apparently more or less constant 
over the central part of the jet (fig. 21). 



o 

E 




3 

'*£ _ 








* 
















IJ/U 


\ 

v \ 


-0/( 
" \ 

'rna, 













rn uj 




X 

















' \ 

















































n 

o 
















o 













o 



o 

o 





> 






















o 
















o 





o 











































o 


o 











o 



o 







o 










> 











o 




















































_o 








-.8 ~4 


.4 


.8 1.2 1.6 2.0 2.4 2.8 

r/r 2 


Figure 21.— The &u correlation. Hot jet. x/d= 20; 9 0 = 170° C. 


L 


Considerably more complete measurements have been 
made at x/d=lb in the hot jet. In addition to u and &u 
distributions (figs. 22 to 24), there is also a traverse of the 
lateral component of turbulence v’/U and v' /U max . It may 
be noticed that, in agreement with figure 24 of reference 12, 
v'^>u' in the immediate vicinity of the axis, and u'^>v' 
elsewhere. 




J.O 
a 

-8 

.6 

%.4 

E 


liS* 


o 

.4 
o 2 

^-.2 


.3 
.2 
3 ./ 

•a 

* 0 

-.2 


























N. v 
















i 

1/U 7 

nax 

s. \ 

\ 


0/6 

'wiQJ 
































' S. 






















— 





















y/C 

-z 

















c 


o 



















0 

— o~ 


O 

o 

° ( 

> 




































o 

o 

o 













































n 




3 















5 

0 

o 

o 

O 

o 

o < 
























o 































0 









.8 L 2 1.6 2.0 2.4 2.8 

r/r z 


and 


-.8 -.4 0 .4 

Fiotjre 24. — Hot-jet tfw correlations. z/d=15;F o =170° C. 

The uv and &v correlations and correlation coefficients 
R u ,=uv/u'v' 

Rd V =5vl$'v' 


10 


REPORT 998 — NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 


are plotted in figure 25. There is a great deal of experimental 
scatter but the general behaviors are clear. Furthermore, it 
is obvious that which is exactly what is to be 

anticipated from the fact that the rate of turbulent heat 
transfer is known (from the mean-velocity and mean- 
temperature curves) to be greater than the momentum 
transfer. 



Figure 25. — Hot-jet double correlations. x/d~ 15; 0 o = 17O° C. 



ANALYTICAL CONSIDERATIONS > 

JET WIDTH AS A FUNCTION OF INITIAL DENSITY RATIO 

As mentioned in the introduction, integrated equations, 
along with dimensional reasoning in transforming the- 
variables, can usually be made to yield solutions describing 
the integral behavior of a shear flow. 

The two-dimensional integrated momentum and # 
mechanical-energy equations have been derived by Liepmann 
and Laufer (reference 6). An identical procedure leads to 
the axially symmetrical forms, omitting bars for mean 
motion. Momentum: 


A f 6 

dxja 1 


,U*r dr-ptWb £+ Pa U a ‘a£+[prUV]i=[rr]i (1) 


Of course, because of the effect of density fluctuations, the 
turbulent shear and heat transfer cannot be expressed simply 
as —~pui) and —c p p& v, respectively. A brief calculation 
(appendix D, reference 16) leads to the approximate expres- 
sions 


t = — puv T p 


% 


&V 


Mechanical energy: 
[rUrY- £ rr ^ dr 


.•g+ip.r.-«g + i[,rc>n‘= 


( 2 ) 


Q=c,p( l-£)dv 


In figure 26, the shear stress and heat transfer are plotted 
in the dimensionless forms 


T r + 6 max UV I Omax ^max) ^ 

' t 7 77 2 ' 7 “ 77 5 77 

1 a C/ m ax a u max u max 1 max 


O U 2 

H min ^ max 


A similar treatment of the heat-transfer equation, with 
specific heat assumed independent of temperature, gives, 
again omitting the bars since all quantities are mean quan- 
tities, 

£ j"pUdrdr-pMb £+pU a 6 a a £+[ P rdVY=j- [rQ]l 

(3) 


Q 


C p P min @max L) m 



dv 

^ max f max 


» Fluctuation terms have been neglected in all of the integral relations used in this report. 
Since the turbulent flows are assumed to have similarity in turbulence characteristics as well 
as in average parameters, this ordinarily causes little error. 


FURTHER EXPERIMENTS ON THE FLOW AND HEAT TRANSFER IN A HEATED TURBULENT AIR JET 


11 


For [lie particular case of a free jet entering an infinite 
medium at rest, these three equations simplify to 


( pU 2 r rfr=Constant==4~ 

J 0 ^ 7T 

-ff.C/V*— jf% » 
dxj o Jo c )/• 

j pUdrdr =Constant= 


,7^ 

N 


(4) 

(5) 

( 6 ) 


where the integrals have been extended to infinity to permit 
the use of asymptotic velocity- and temperature-profile 
assumptions. If finite width functions are used, the inte- 
grals can be extended only to some judiciously chosen b. 
It should also be mentioned that in the integral continuity 
equation, which can be used to find the radial velocity V b , 
the finite limit is essential. 

Equation (4) expresses the condition of constant total flux 
of momentum across all planes perpendicular to the jet axis 
in the absence of a pressure gradient. Equation ( 5 ) equates 
the rate of decrease of mean-flow kinetic energy per unit 
'time to the rate of production of heat (in the laminar case) or 
of turbulent kinetic energy (in the turbulent case). 

Equation ( 6 ), expressing the constancy of heat flow across 
all planes perpendicular to the ar-axis, is of course the thermal 
part of the energy equation, for the case in which heat 
production by viscous action is negligible. When the den- 
sity variation is isothermal and due to the use of a jet fluid 
different from the receiving medium, equation ( 6 ) is replaced 
by what may be considered the conservation of mass-flow 
defect or excess: 

f 00 M' 

(p — p«)?7rdr=Constant==- ( 7 ) 

0 7T 


and, in fact, equation (7) is easily obtainable from equation 
( 6 ) with the equation of state of a perfect gas, assuming con- 
stant pressure. 

If there is introduced an appropriate dimensionless vari- 
able, along with reasonable assumptions for thedimension- 
less velocity and density profiles, equations (4), (5), and (7) 
can be solved for U ma z(x), p™i n (x), and the momentum 
diameter A (a?). The integrated continuity equation can 
be used to solve for radial velocities. Although r can in 
principle be determined from the two assumed functions (see, 
for example, references 6 and 12 ) it is sometimes more con- 
venient to make a reasonable assumption for this distribution 
also. 

The purpose of the present analysis is only to obtain an 
approximate expression for A/d as a function of Poo/Po, for 
a fixed value of x/d; it turns out that this can be done without 
the mechanical-energy equation. 

The velocity and density profiles are represented by 
dimensionless functions 


and 


jj —fW 

^ m nr 



(8) 


where rj is the appropriate similarity parameter. It can be 
shown, by resulting inconsistencies in the two equations to 
be solved, that simple geometrical similarity does not exist. 
In fact, it seems physically evident that the profile functions 
should vary with the local density ratio. Taking 


*7 = 



the form of F is obtained by substituting equation ( 8 ) and 
the definition of momentum diameter 


7rA 2 

4 


P min l max — A/ 


into equation (4). Then, 


2x Jo 


\pco~\~{pmin Pc o)y]( max~f~^p>2 


i 7rA“ jt 2 

V d TJ ^ P min 7 max’ 


which gives 


F=-J 2 




where I x and h are pure numbers dependent upon the 
assumed velocity and density functions 

/i=J o vf 2 (v)dy 

and 

/ 2 =J o »?/ 2 0 i)g(y)dy 

Thus, the proper similarity parameter is 

, = 2V2jr/ 2+( / 1 -/ 2 )^r (9) 

A L Pmin J 

which is a function of x. If the problem were solved for 
Pminix) and A(x), 77 could then be expressed in its most 
suitable form, as a function of r, x , and the initial density 
ratio. 

It is of some interest to inspect the parameter 77. For 

constant-density flow, 77 = a / 2 ^i 2 r/A, and simple geometrical 

similarity exists. For heated turbulent jets it is found that 
/i>/ 2 , since < 7 ( 77 ) 5 ^ 1.0 everywhere. Thus, with p min appre- 
ciably less than a particular value of 77 is reached at 
smaller r (for fixed A) than in the constant-density case. 

Substitution of equation (9) into equation ( 4 ) merely gives 
the definition of A, which is now written with the explicit 
expression for M: 


12 


REPORT 998 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 


p m inU max 2 tf=p„U 2 d 2 (10) 

With equation (9) and the explicit expression for M, equation 
(7) becomes 


I 


3 


P m i n Umnx^ip m i n P oo ) 

I \P 2^Pmtn — Pa) 


= {pO—pa)U„ d 2 


( 11 ) 


where 

/• CO 

/ 3 =Jo vf(v)g(v)dv 

Equations (10) and (11) are to be solved for A/d as a 
function of p 0 /pco, at a particular value of x/d. Since there 
are fewer equations than unknowns, some assumption 
must be made concerning the axial distribution of either 
velocity or density. It is convenient to make the simplifying 
assumption that the dimensionless axial density distribution 

Pmi - — — is independent of the initial density ratio. 

Po~P cc 

Then U m ax/U 0 from equation (10) is substituted into 
equation (11), and all p mi n’s are expressed in terms of 

— — • After further approximation by the use of the 

Po— Pen 

first, two terms of binomial expansions, the following result is 
obtained: 


or 


A I\ I p co | ( d 2 1 Pm in P co\ ( P <> , VI 

d~h yiTo L 1 + V7T V V p 0 -p„ ) \T~ 1 >] 


( 12 ) 


where A! is the momentum width at the same axial distance 
for the constant-density jet, and where the desired x/d is 

obtained by choosing ( — — ) according to temperature 

measurements in the constant-density case. Thus it is 
clear that, as may be intuitively anticipated, the jet spreads 
at a wider angle when the jet fluid is less dense than the 
receiving medium, and vice versa. 

A numerical value for I 2 /I\ is obtained by computation 
from velocity and temperature distributions in the measure- 
ments of the constant-density case. These give / 2 //i = 0.727. 
and equation (12a) is plotted in figure 15 for comparison 
with the experimental results. Because of the rather severe 
approximation, agreement is not good. The complete 
solution of equations (4), (5), and (7), however, is rather 
laborious. 

MEASURE OF TURBULENT PRANDTL NUMBER 

One of the more important and interesting problems in the 
study of turbulent shear flow is the relation between momen- 
tum transfer and heat transfer. 


For laminar flows, the ratio of momentum to heat transfer 
is directly computable from the kinetic theory of gases and 
is conventionally described by the well-known Prandtl 
number 


c p p 

a= ~¥ 


For nearly perfect gases, a is only slowly variable with tem- 
perature over a wide range since c p is nearly constant and p 
and k vary nearly proportionally. For effectively isothermal 
laminar flow, c p , p, and k are individually constant over the 
entire flow field. 

The quantity analogous to <r, for turbulent flow, is 



(13) 


which maybe termed the “turbulent Prandtl number.” The 
“eddy viscosity” and “eddy conductivity,” as defined by 
Dryden (reference 19) are 


_ 5C7 v 

- pUV = e-^- 

y 

— 50 

-C p p#V = P 


(14) 


and if the density fluctuations are negligible, as is the case 
within the accuracy of the present experiments, 


T = € 


5 U 

dr 


Q=P 


50 

5r 


Since e and (3 are themselves rather artificial quantities, in- 
troduced merely in overstrained analogy to laminar flow, it 
is not to be expected that a t has any fundamental physical 
basis, other than the exact definition given in equation (15). 
However, in the absence of a satisfactory analysis of turbu- 
lent shear flow, it serves as an empirical measure of the ratio 
of momentum transfer to heat transfer. Although e and (3 
vary from point to point, even in an isothermal turbulent 
shear flow, the variations in their ratio (a t ) may be of distinct 
interest. An excellent discussion from the point of view of 
the mixing-length theories has been given by Dryden (refer- 
ence 20) and need not be repeated herein. 

From equations (13) and (14), the local turbulent Prandtl 
number in terms of measurable quantities' may be written 


as 

50 

__uv 5 r 

dr 


( 15 ) 


2 In this section all C7’s and 0’s are mean values. 


FURTHER EXPERIMENTS ON THE FLOW AND HEAT TRANSFER IN A HEATED TURBULENT AIR JET 


13 


or, in more convenient computational form, 


/ uv \ 

A 1 


\U m j) 

dr \ 

^9 max / 

at / dv N 

\ & 

( u ) 

\ 9 max f max / 

/ dr 

\ F max) 


(15a) 


On the basis of the experimental results presented in the 
previous section, two different measures of <r t can be com- 
puted: First, the effective turbulent Prandtl number for an 
entire section of the jet a t can be computed from a com- 
parison between the width ratio of mean-velocity and 
mean-temperature profiles in the turbulent jet and the same 
ratio as computed theoretically for a laminar jet. Second, 


the direct measurement of and — — ~ — permits direct 

Umax “max U max 

computation of the <r t distribution, according to equation 
(15a). Of course, this latter result will not be very accurate, 
both because of the uncertainty of King’s equation and 
because of the excessive experimental scatter. 

Effective turbulent Prandtl number. — Effective turbulent 
Prandtl number provides another illustration of the useful- 
ness of the integral equations when a complete solution is 
either not necessary or not feasible. For simplicity, the 
analysis is restricted to the constant-density case. 

In addition to equations (4), (5), and (6), for the round 
jet an equation analogous in form to the mechanical-energy 
equation (2) can be derived by multiplying the heat-transfer 
equation by U before integration: 


d_ r 

dxj o 


U*6rdr= f“rQ 

pCpJo 



VT 


y, \ dr 06) 


and 




dU d9 , 

r -5— sr dr 
dr dr 


(18) 


It is to be expected that a good measure of the Prandtl 
number will be the difference in momentum and thermal 
diameters of the jet. The latter is defined so that 


7T&0 2 

T~ 


P min 


I max@> 


max'' max 


“ 2 , j. 


pU0rdr=N 


Then equations (4) and (6) are replaced by the definition of 
A and the foregoing equation with p=Constant; that is, 


AM 

U m J A 2 =— (19) 

7T p 

AN 

U max e max A e 2 =~- ( 20 ) 

TTpCp 

With constant density, simple geometrical similarity exists, 
and there may be introduced the dimensionless variable 
rj / =2r/A, along with the functional assumptions UIU max = 
JW) and 9/9 max = /t ( 77 7 ) . 

Substitution of 77' and j into equation (17), followed by 
substitution from equation (19) into the result, gives the 
differential equation 


where 



( 21 ) 


The physical significance of equation (16) is not so clear as 
that of equation (5). However, it may be regarded as a 
device for retaining Q (which disappears in the integration 
of the heat-transfer equation). 

The solution for the velocity field of a laminar round jet 
has been given by Schlichting (reference 20) and by Bickley 
(reference 21), but the temperature field apparently has not 
been treated. For laminar flow, 

dU 

and 


are substituted into equations (5) and (16), giving 

<i7) 


This gives the well-known hyperbolic decrease of axial veloc- 
ity in a round isopycnic laminar jet: 


7 max 




\2iry.J \I B J X 


(22a) 


which with equation (19) demonstrates the corresponding 
linear spread of such a jet: 


\ 4pm) i a 


(22b) 


Substitution of 77',/, and h into equation (18), followed by 
substitution of equation (20) leads to another expression for 
d U max . 
dx 


dU m 


dx 


= 4m ( 1+ <t)/c (a) Umaxdn 

J Q ” iTW 


(23) 


where 


14 


REPORT 998— NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 


Id = 



(W 


and similarity has been used in assuming A 0 /A independent 
of x. 

Equating equation (21) to equation (23) arid substituting 
for Umax/Omax from the ratio of equations (19) and (20), 
there results 

5K7f( I +0- 1 (24) 

Since - f 4 is a function of Ae/A, equation (24) will lead to 
Ibjc 

the desired relation between a and Ae/A. The simplest 
procedure seems to be the assumption of reasonable func- 
tional forms for /(??') and h(rj'): 

j =e ~ AW)2 

k= e ~BW) 2 


From the definition of A it turns out that A=}{; from the 

/AV /a\ 2 1 

definition of A# it turns out that B =4 ~ ) — A =4 — ) ~~2 
With these, the F s can be evaluated: 



Ic = 


JML 

2 +(f)' 


whence equation (24) yields 



(25) 


It should be noted that A e /A=^2 corresponds to <7 = 0. 
This results from the form of the original definition of A 0 , 
and suggests that a different characteristic measure of 
thermal diameter (i. e., one which — > °° as cr-^0) may be 
desirable. 

An identical treatment of a constant-density turbulent jet 
on the crude assumption of constant e and 0 results, of course, 
in an identical expression: 



where a t is the over-all effective Prandtl number for the 
turbulent-jet cross section. 


For air, in the normal range of room temperature, the 
considerable scatter in thermal conductivities obtained in 
the various experimental determinations leads to an approxi- 
mate range of possible true (laminar) Prandtl numbers, 
0.70^ <xg 0.76. A tabular and graphical summary of “best” 
measured values up to 1938 has been made by Tribus and 
Boelter (reference 22). They give, for_example, <r = 0.708 
for T a — 302° absolute and 0.701 for T a =348 0 absolute. 
These particular values have been chosen for comparison 
with two computations of the turbulent measurements. 

For the present turbulent jet, at x/d= 15 and initial 
temperature differences of 00=15° and 300° C, the width 
ratios are A e /A= 1.102 and 1.107, respectively. From equa- 
tion (26), these give o-<=0.727 and 0.713. With the two 
orifice temperatures given, the average absolute temperatures 
across the section are about 302° and 348° (these numbers 
are actually T r AV^m a x)- This gives the extremely interest- 
ing result that the average of effective turbulent Prandtl 
number in a round jet is equal to the laminar value, within 
the probable accuracy of the present measurements. 

Turbulent Prandtl number distribution. — Faired curves for 

the measurements of uv/U ma x and &v/d ma zUmax (fig- 25) have # 
been used, along with the dimensionless velocity and tem- 
perature profiles, to compute local values of in accordance 
with equation (15a). The values have been plotted in 
figure 27, along with a heavy line giving c?/ as computed 
from Ae/A. Near the axis and near the jet edge, the com- 
putation becomes more or less indeterminate. 


1.0 


g .8 

><D 

ictT .6 

^ .2 
^ 0 


.8 

.6 

^0.4 

.2 


— ^ 





















s N 
\ 















\U! 


ix..- 

k 



6/0 _ 













rtlULJL 














■v 












































Effective Prandtl number from 
L mean -temperature and mean- 






ve 

/ocity dis trib u tions 


1 
















J 





0 

O " 


0 

















j 


































-.4 0 .4 .8 1.2 1.6 2.0 2.4 2.8 

r/r 2 

Fioure 27.— “Turbulent Prandtl number.” 1-inch hot jet. x/d= 15; 0„=17O° C. 


The agreement between a t and the average value of the a t 
points is extremely good. In fact, it is rather better than 
was anticipated, because of the considerable scatter of the 
fluctuation measurements and the possible inexactness of 
King’s equation. 

DISCUSSION 

ERRORS 

Most of the sources of error and the corrections, both 
applied and not applied, have been covered in the previous 
sections. However, it may be well to outline and extend 
the previous remarks. 


FURTHER EXPERIMENTS ON THE FLOW AND HEAT TRANSFER IN A HEATED TURBULENT AIR JET 


15 


Total-head-tube and thermocouple data. — The total-head- 
tube readings have not been corrected for the effect of 
velocity and density fluctuations (appendix A). Also, since 
the range of possible sensitivities was quite limited and 
since the response is proportional to 77 2 , the deflections near 
the jet edge were small and difficult to obtain accurately. 

A certain amount of experimental scatter in all measured 
quantities resulted from small fluctuations in U 0) due pri- 
marily to the partly stalled condition of the blower, which 
was designed to operate against a somewhat smaller pressure 
rise. 

The temperature distributions for small values of 6 0 and 
at large values of x/d and r/r l were naturally susceptible to 
considerable scatter due to slow temperature fluctuations of 
perhaps a fraction of a degree in the laboratory. 

Hot-wire measurements. — The usual sources of error (see 
discussions by Dry den (reference 23) and Simmons (reference 
24)) for hot-wire measurements were encountered in the 
present investigation. 

Two additional difficulties were attributable to the par- 
ticular flow configuration: First, the extremely high tur- 
bulence level present in a free jet is certainly outside the 
small disturbance assumptions basic to the standard hot- 
wire response calculations. However, some rather brief 
measurements (reference 12) indicate that opposite errors 
may reduce this error considerably. Second, the extremely 
large degree of fluctuation introduces some uncertainty in 
the bridge balancing for the setting of hot-wire resistence. 

The principal additional source of error inherent in the 
hot-jet measurements was the extreme difficulty of deter- 
mining R a , the unheated wire resistance at local ambient 
temperature T a . Since the d' sensitivity of a wire is propor- 
tional to R a —R r (where R r is the unlieated resistance at 
T r ), it is clear that, in regions of relatively small 0, a small 
error in R a may cause a large error in d'/d, and as it turns out, 
also in du/.OU The complexity of the computational pro- 
cedures in these new applications of the hot-wire anemome- 
ter, often involving small differences between two large 
quantities, each with a fairly large amount of experimental 
scatter, has obviously magnified this scatter immensely in 
the final results. 

Because of the foregoing factors, it was felt that hot-wire 
length corrections were not appropriate; the scale of the 
turbulence was of the order of 0.5 to 1 centimeter, and the 
scale of temperature fluctuations was probably of the same 
order of magnitude. 

However, it should be emphasized that free jet measure- 
ments represent the most stringent kind of test for the new 
measuring technique. It is certain that similar investiga- 
tions in channels and boundary layers, for example, will 
show more consistent experimental points. 

VARIATION OF JET SPREAD WITH DENSITY 

As pointed out, the considerable divergence between com- 
puted and measured curves of A/Ai against p«>/p 0 (fig. 15) is 
probably due to the very rough nature of the simplifying 
assumption used. Apparently the curve shapes agree 
reasonably well. This theoretical curve is computed par- 
ticularly for the case in which density difference is obtained 


by heating; the relation between heat and momentum 
transfer determines the relation between the velocity func- 
tion/^) and the density function #(77) and thus determines 
the ratio I 2 /Ii in equation (12). In cases where the density 
difference is obtained by the use of different gases, 1 2 /1 1 may 
be different, so that the curve of A/Aj against p m /p 0 will 
not be the same. However, it does not seem likely that 
material diffusion in turbulent shear flow is much different 
from momentum and heat diffusion, 3 so the final result should 
be fairly close to that computed herein. 

FLUCTUATION LEVELS 

There are two characteristics of the measured fluctuation 
levels that are worthy of note. The first is that u'/U>d'IO 
everywhere in the jet. Although there must be some uncer- 
tainty since the complete physical form of King’s equation 
has never been verified (appendix A, reference 16), the fact 
that u'lU in the heated jet is measured at about the same 
value as in the cold jet seems evidence that the results are 
reasonable. An additional bit of evidence is the result that 

d' lQmax>u r / U max in the outer part of the jet ^ where 

>5r ( TT~ ))’ wlicrcas u> I U mai> d'/d m ,n near the axis (where 

^(T— )>^ (/-))• This is quite consistent with the 

result arrived at by the conventional qualitative phenomeno- 
logical approach which considers a u fluctuation (or a d 
fluctuation) arising from a v fluctuation with a fixed mixing 
length. Thus, if this (purely fictitious, but sometimes 
convenient) mixing length were about the same for velocity 
and temperature fluctuations, u' would be relatively greater 
than d' where the V profile was relatively steeper than the 
0 profile, and vice versa. Of course, this does not necessarily 
account for the slight local minimum in d'/d max on the jet axis. 

RELATIVE TRANSFER OF HEAT AND MOMENTUM 

The a ( distribution shown in figure 27, although agreeing 
in average with a t computed directly from A® /A, does not 
necessarily prove any significant result with regard to the 
function <x*(r). Both the experimental scatter in the uv 
and do measurements and the difficulty of making accurate, 
slope determinations from the mean-velocity and mean- 
temperature profiles permit only the conclusion that a t is 
roughly constant across the main part of a round turbulent 
jet. 

That the effective turbulent ITandtl number might pos- 
sibly be the same as the laminar Fraud tl number was appar- 
ently first suggested by Dryden (reference 19), though not 
with any particular physical justification. In fact, it appears 
that no one has yet put forth any rational hypothesis, either 
rigorous or phenomenological, to predict the relative rates 
of heat and momentum transfer in turbulent shear flow. 

Since the vorticity transport theory is apparently the 
only analytical approach that has led even to the quali- 
tatively correct result, it seems probable that simple momen- 
tum-exchange considerations will be inadequate. 

3 At the Seventh International Congress for Applied Mechanics, B. G. van der Hegge Zijnen 
and J. O. Hinze reported careful measurements showing heat and material diffusion to be 
effectively the same in a round turbulent jet. 




16 REPORT 998 — NATIONAL ADVISORY 

SUMMARY OF RESULTS 

From measurements of mean- total-head and mean- tem- 
perature fields in a round turbulent jet with various initial 
temperatures, the following enumerated results can be stated 
with confidence: 

1. The rate of spread of a round turbulent jet increases 
with a decrease in the density of jet fluid relative to receiving 
medium. When the density difference is obtained by heat- 
ing, the effect is as shown by the experimental points in 
figure 15. 

2. Up to local maximum density ratio p m /p rn in of about 
1.3 simple geometrical similarity still exists in a fully devel- 
oped jet, within the accuracy of the present measurements. 
The total-head and temperature profile functions are effec- 
tively the same as in the constant-density jet. 


COMMITTEE FOR AERONAUTICS 

3. The average or effective “ turbulent Prandtl nuiliber,” 
for a section in the fully developed air jet, is very nearly 
equal to the laminar Prandtl number of air. This does net 
necessarily imply any primary dependence of turbulent 
Prandtl number on laminar Prandtl number. 

4. The local turbulent Prandtl number at a section in 
the fully developed jet, away from axis and edge, is roughly 
constant. In addition, with somewhat less certainty, it has 
been found that d'jtX^u/IU everywhere m the jet and that 
the relative magnitude of d'/O ma x and u' Jl rnax varies roughly 
as the relative magnitude of the slopes of dimensionless 
temperature and velocity profiles. 

California Institute of Technology, 

Pasadena, Calif., August 18, 1947 . 


f* 


APPENDIX A 


EFFECT OF SIMULTANEOUS VELOCITY AND DENSITY FLUC- 
TUATIONS ON READING OF A TOTAL-HEAD TUBE 

Neglecting both static-pressure fluctuations and the effect 
of lateral velocity fluctuations, the tube reading is 


p=\ (p+p')(U+uy 

Therefore, since pUu= 0, 

P=\ p(£7 2 +~ 2 )+£7p“+i TV 


(Al) 


(A 2) 


For small fluctuations the last term can be- neglected. 
Also, since the density fluctuations of interest herein are due 
to temperature fluctuations, 



is substituted into equation (A2), which finally gives for the 
true mean dynamic pressure in terms of the tube reading: 



P 


1+^r— 

U 2 


6 du 

%w 


(A 3) 


It is interesting to note that, since &u is apparently ordi- 
narily positive, the errors due to the velocity and the tem- 
perature fluctuations are in opposite directions. 

The justification for neglecting the effect of lateral fluctua- 
tions in this approximate calculation is that its effect on the 
magnitude of the instantaneous velocity vector is at least 
partially balanced by the directional sensitivity of the tube. 

REFERENCES 

1. Pabst, O.: Die Ausbreitung heisser Gasstrahlen in bewegter Luft. 

Untersuchuiigen und Mitteilungen, Nr. 8004, Aug. 1944. 

2. Pabst, O., and Von Bold, J. G.: Mischungsvorgang eines Leucht- 

gasstrahles mit ruhender Luft. FW-Hausbericht 09 021 I. 

3. Hu, N. : On the Turbulent Mixing of Two Fluids of Different Den- 

sities. Part II of Ph. D. Thesis, C. I. T., 1944. 

4. Goff, J. A., and Coogan, C. H.: Some Two-Dimensional Aspects 

of the Ejector Problem. Jour. Appl. Mech., vol. 9, no. 4, 

Dec. 1942, pp. A-151— A-154. 


5. Abramovich, G. N.: The Theory of a Free Jet of a Compressible 

Gas. NACA TM 1058, 1944. 

6. Liepmann, Hans Wolfgang, and Laufer, John: Investigations of 

Free Turbulent Mixing. NACA TN 1257, 1947. 

7. Ribner, Herbert S. : Field of Flow about a Jet and Effect of Jets 

on Stability of Jet-Propelled Airplanes. NACA ACR L6C1 3, 
1946. 

8. Squire, H. B., and Trouncer, J.: Round Jets in a General Stream. 

R. & M. No. 1974, British A. R. C., 1944. 

9. Liepmann, H. W., Corrsin, S., and Laufer, J.: Some Measurements 

in Free Turbulent Shear Flow. Note presented at Sixth Int. 
Cong. Appl. Mech. (Sept. 1946, Paris). (Unpublished; copy 
available from Corrsin.) 

10. Ruden, P.: Turbulente Ausbreitungsvorgange im Freistrahl. Die 

Naturwissenschaften, Jahrg. 21, Heft 21/23, May 26, 1933, pp. 
375-378. 

11. Kuethe, Arnold M.: Investigations of the Turbulent Mixing Re- 

gions Formed by Jets. Jour. Appl. Mech., vol. 2, no. 3, Sept. 
1935, pp. A-87 — A— 95. 

12. Corrsin, Stanley : Investigation of Flow in an Axially Symmetrical 

Heated Jet of Air. NACA ACR 3L23, 1943. 

13. Goldstein, S., ed.: Modern Developments in Fluid Dynamics. 

Vol. II. The Clarendon Press (Oxford), 1938, pp. 672-673. 

14. Howarth, L. : Concerning the Velocity and Temperature Distribu- 

tions in Plane and Axially Symmetrical Jets. Proc. Cambridge 
Phil. Soc., vol. 34, pt. 2, April 1938, pp. 185-203. 

15. Corrsin, Stanley: Extended Applications of the Hot-Wire Anemom- 

eter. Rev. Sci. Instr., vol. 18, no. 7, July 1947, pp. 469—471. 

16. Corrsin, Stanley: Extended Applications of the Hot-Wire Anemom- 

eter. NACA TN 1864, 1949. 

17. King, Louis Vessot: On the Convection of Heat from Small Cylin- 

ders in a Stream of Fluid: Determination of the Convection 
Constants of Small Platinum Wires with Applications to Hot- 
Wire Anemometry. Phil. Trans. Roy. Soc. (London), ser. A, 
vol. 214, 1914, pp*. 373-432. 

18. Bowen, W. H.: Tests of Axial Flow Fans Designed by Lattice 

Theory. M. S. Thesis, C. I. T., 1938. 

19. Dryden, H. L. : Aerodynamics of Cooling. Vol. VI of Aerody- 

namic Theory, div. T, W. F. Durand, ed., Julius Springer 
(Berlin), 1934. 

20. Schlichting, H.: Laminare Strahlausbreitung. Z.f.a.M.M., Bd. 

XIII, Aug. 1933, pp. 260-263. 

21. Bickley, W. G.: The Plane Jet. Phil. Mag., ser. 7, vol. 23, no. 

156, April 1937, pp. 727-731. 

22. Tribus, Myron, and Boelter, L. M. K.: An Investigation of Air- 

craft Heaters. II — Properties of Gases. NACA ARR, Oct. 
1942. 

23. Dryden, Hugh L.: Isotropic Turbulence in Theory and Experi- 

ment. Theodore von Karm&n Anniversary Volume, C. I. T. 
(Pasadena), 1941. 

24. Simmons, L. F. G.: Note on Errors Arising in Measurements of 

Turbulence. R. & M. No. 1919, British A. R. C., 1939. 

17 




II «; GOVE 9N MFNT PRINTING OFFiCF • JA5 1 L 


j ' ; r / 

1 





■ r r *- T * — • '■r r v*"* / *■■ / » - ri r^ct ^ ■■ " ■■■■»■ * ■»■» T • < CT I '/T^ T 

* 


r \ 


- ■ 

V, - ..>r •*^.> ; . •>• ’■ ■ v 




■ ■ 


S •'»•;• ••• ><■ 

;< * 

.• <- ■■'.£?- r *: 

*»; r' -^* 

C’ V ’ . ’. 


V iV-. : 'i 


WS. /*•;•£? 


>-K - ?.*£j 


AERONAUTIC SYMBOLS 

FUNDAMENTAL AND DERIVED UNITS. 


Vii- :, 


/’ T\\C . < Jr 


*.. '\ A 

: ..;* ••; t «$.* 


**~’7 C 

' "v’o. 

/ • • 

Symbol 

*- - — — — ; 

Metric 

» -* . " ” ■ v_ "■ • ' ’ • - ' •_ V - 

English 

Unit 

Abbrevia- 

tion 

Unit 

v* ; A w ; 

Abbreviation 

Length 

l 

t 

meter., 

-T m 

8 

foot (or mile) _ 
second (or hour) — 

ft (or mi) 
sec (or hr) 
lb 

Force - 

F 

weight of 1 kilogram 

weight of 1 pound. 

Power 

Speed...- — 

P 

V 

horsepower (metric) 

f kilometers per hour 

\meters per second-, ---^- 

Vph 

mps 

horsepower 

miles per hour. 

feet per second 

— ■■■■■■ • • 

hp . 

mph 

fps 




2. GENERAL SYMBOLS 


w 

9 

m 

l 


•:^^ v -v.v.^v-:^ HV' 

' ' ■ ' ••_ V.' - ' \ *-r> .‘ : Wfr 

Weight=m<7 ~ ; - 

Standard acceleration of gravity =9.80665 m/s 2 p 


Kinematic viscosity 
Density (mass per unit volume) 


or 32.1740 ft/sec 2 


V Standard density of dry air, 0.12497 Jtg-m“ 4 -s 2 at 15° C 
W and 760 mm; or 0.002378 lb-ft“ 4 sec 2 

Mass=— Specific weight of “standard” air, 1.2255 kg/m s or 

Moment of inertia=m£ 2 . (Indicate axis of 0.07651 lb/cu ft 
radius of gyration k by proper subscript.) 

Coefficient of viscosity 






Area 

Area of wing 
Gap 

Span y 

Chord 

. 5 2 

Aspect ratio, -g 
True air speed 
Dynamic pressure, ^ pV 2 

Lift, absolute coefficient Czj=J^ 

Drag, absolute coefficient D d =^ 


S ' 

S» 

G 
b 
c 

A 

V 

9 
L 
D 
Do 

D x 

Pv 

VT^- •• • V- -V * *■ — - - , v . ... ... # ^ ^ yy 

C Cross-wind force, absolute coefficient Cc = ^ 


Profile drag, absolute coefficient Dd^=-^ 
Induced drag, aj&tlufe coefficient 

D 

Parasite drag, absolute coefficient C Dv ~ 

C 


3, AERODYNAMIC SYMBOLS / ‘ 

i v Angle of setting of wings (relative to thrust line) 
i t Angle of stabilizer setting (relative to thrust 
line) 

Q Resultant moment 

Q Resultant angular velocity 

VI 

R Reynolds number, p — where l is a linear dimen- 
sion (e.g., for an airfoil of 1.0 ftr chord, 100 
mph, standard pressure at 15° C, the corre- 
sponding Reynolds number is 935,400; or for 
an airfoil of 1.0 m chord, 100 mps, the corre- 
^ ^ sponding Reynolds number is 6,865,000) 

a Angle of attack 

€ Angle of downwash 

otq Angle of attack, infinite aspect ratio 

<x { Angle of attack, induced v_ 

a* Angle of attack, absolute (measured from zero- 

' - lift position) _ . : _ 

Flight-path angle 


Do 








m 


rf. 


y 


i'-si-,