Contributions to the theory of the spreading of a free jet issuing from a nozzle

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Title:
Contributions to the theory of the spreading of a free jet issuing from a nozzle
Series Title:
NACA TM
Physical Description:
72 p. : ill. ; 27 cm.
Language:
English
Creator:
Szablewski, W
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Aerodynamics   ( lcsh )
Jet nozzles   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
For the flow field of a free jet leaving a nozzle of circular cross section in a medium with straight uniform flow field, approximate formulas are presented for the calculation of the velocity distribution and the dimensions of the core region. The agreement with experimental results is satisfactory.
Bibliography:
Includes bibliographic references (p. 39).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by W. Szablewski.
General Note:
"Report date November 1951."
General Note:
"Translation of "Zur theorie der ausbreitung eines aus einer düse austretenden freien strahls." Untersuchungen und Mitteilungen Nr. 8003, September 1944."

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University of Florida
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oclc - 95025526
sobekcm - AA00006215_00001
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Full Text
I rvkA-v\M 131(







c -


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1311


CONTRIBUTIONS TO THE THEORY OF THE SPREADTIG

OF A FREE JET ISSUING FROM A NOZZLE

By W. Szablewski


PART I.- THE FLOW FIELD IN THE CORE REGION


ABSTRACT:

For the flow field of a free jet leaving a nozzle of circular cross
section in a medium with straight uniform flow field, approximate formulas
are presented for the calculation of the velocity distribution and the
dimensions of the core region. The agreement with measured results is
satisfactory.

OUTLINE:

I. INTRODUCTION AND SURVEY OF METHOD AND RESULTS
II. CALCULATION OF THE FLOW FIELD
(a) Velocity Distribution in the Core Region
(b) Dimensions of the Core Region
III. COMPARISON WITH MEASUREMENTS
IV. SUMMARY
V. REFERENCES
VI. APPENDICES
No. 1 Calculation of the Transverse Component
No. 2 For Calculation of the Dimensions of the Core Region


I. INTRODUCTION AND SURVEY OF METHOD AND RESULT?


Knowledge of the flow field of a free jet leaving a nozzle is of
basic importance for practical application.

Investigation of such a flow field is a problem of free turbulence.

In theoretical research the following specialized cases of our
problem have already been treated:

(a) The mixing of two plane jets, the so-called plane jet boundary.
These conditions are encountered in the immediate proximity of the nozzle.

*"Zur Theorie der Ausbreitung eines aus einer Duse austretenden freien
Strahls." Untersuchungen und Mitteilungen Nr. 8003, September 1944.








NACA TM 1311


(b) The spreading of a rotationally-symmetrical jet issuing from a
point-shaped slot in a wall, the so-called rotationally-symmetrical jet
spreading. This state defines the conditions at very large distance
from the nozzle.

In considering a free jet leaving a nozzle of circular cross sec-
tion, we may subdivide the spreading procedure, according to an essential
characteristic, into two different regions:

(1) Region where a zone of undiminished velocity is still present
(the so-called jet core). We shall call this range, which extends from
the nozzle to the core end, the core region. For the immediate proximity
of the nozzle the conditions of the plane jet boundary exist.

(2) The region of transition adjoining the core region which is
characterized by a constant decrease of the central velocity. This
region opens into the region of the rotationally symmetrical jet
spreading mentioned above.

So far, there exists only an investigation concerning the core
region (reference 1); it is limited to the case where the surrounding
medium is in a state of rest.


Method and Results

In the present paper, the spreading of a jet in the core region is
treated for the general case where the surrounding medium has a straight
uniform flow field (or, respectively, where the nozzle from which the
jet issues moves at a certain velocity through the surrounding medium at
rest).

The theoretical investigation is based (reference 2) on the more
recent Prandtl expression for the momentum transport


e(x) = Kb(x) uImax umin

One then obtains in the rotationally symmetrical case the following
equations:

Continuity:


+ = (r)








I1ACA TM 1311


Momenti.um transport:

-~ -~ 2
du ou -u 1 ou
u V =-\
u + v = \r2 rEr
X d r \^r2 r Tr)


whe re


E = Kb(x)(ul -u0)

ul = velocity at the jet core

uO = velocity of the surrounding medium


(ul > uO)


With reference to the present problem


Il/////II/i


we introduce, instead of r,


r r
r-- x


as independent variable.

Coint inuity


We obtain:


r

Mlkil77q regq/ion


( ox 7\ x ( \ rox/ r
+ )x x-3 +v+ +x-=- o


*11" I.. ... ...... .


ro77








4 NACA TM 1311


Momentum transport

-1 ou oul ou E(x)/ 1 ou
u x 1 + v ,EW 2 rO
S + --


Velocity distribution in the Core Region

We limit our considerations at first to small disturbances of the
flow field; that is, to relatively small differences in velocity
(ul O small quantity). The partial differential equation for the

momentum transport may then be linearized


-- 2-
o \-
r x = E(x)( + r 2


where

e(x) = Kb(x) 1u
Ul


(It should be noted that by the transformation this equation

is transformed into the equation

2-
2u Ou 1 2u
; + i- x = 0

r r0
which represents a heat conduction equation.) With r = x instead
of r one obtains from the equation of momentum


u +u 1 xx U x
22 + + r0 Tl -(T =
x

if
r0
x 0








NACA TM 1311


This equation is a

of parabolic type.
equation


linear partial differential equation of second order

For the plane case --) oo this results in the
(x


dT2 d- xTix -
rO


with


1

c ul ul)
KC r-----


b(x)
therein c = lim i-x
x
x
--=0
r0


With the
yields


and is to be regarded as a function of


Ul UO
ul


boundary conditions taken into consideration, the integration


"oo e- ( O )2 o ( )
e ( d(a0O)
0


with


1
a C ul uO

2 c(Ul ul)

We now obtain an approximate solution of our problem by generalizing
the plane velocity distribution and setting up the following formulation:


u ( el d a + + 1 +
ul (ul o 1L" 2J 2\ uli


+u O
ul


ro


= ( 1







NACA TM 1311


With

al(x) =

(ui x b
ulO) x


x
1 Kul uO I x b

\rO


we obtain a function which corresponds to the exact solution for small
- as well as for large positive r, thus in boundary zones of the
r0
region of integration as well as in the interior of the region along the
jet I = 0.

If we now consider larger disturbances, the solutions obtained for
small disturbances are to be regarded as a first approximation.
u1 u
For the plane case the solution for arbitrary already
ul
exists, compare GBrtler (reference 3). It is found that, purely with
respect to shape, even the first approximation represents a very good
approximation. The velocity distribution calculated by Gortler still
shows an uncertainty insofar as u( + a), with u(|), also represents
a solution. This uncertainty here may be eliminated, because for the
jet core vanishing of the transverse component v is required. There-
with the initial profile of the velocity distribution for arbitrary
ul uO is then unequivocally fixed.
ul

If we limit ourselves, with respect to shape, to the first approxi-
mation, the initial profile is


u 1 eu-\ t2 + 1 ( uO)
.. -l Ie d++- 1+
ul fl \uu I Jo 2

where

S= Gn 0.36 (u -
( ulU








NACA TM 1311


and


1

a rc Ul uo


For the further development of the profiles starting from this initial
profile the regularity found for small disturbances is then taken as a
basis


Se-* 2
e-^
0


1+
dj* + 1 1
2


where


Cl lr UO) +
= a1' 0.36 (u ) + u2


ul UO
For -- O>, this function is transformed into the approxi-
U1l
mation function constructed for small disturbances. How far it may be
considered an approximation in the region for arbitrary disturbance is
not investigated in more detail.

The functions appearing in the integral al(x), a2(x) result from
the approximation calculation for the dimensions of the core region,
carried out on the basis of the momentum theorem.


Calculation of the Transverse Component


The transverse component v of the flow is determined from the
continuity equation


v = r dr
~S~jx )p


(r / 0)


respectively, with our


approximate function being substituted for u.


r- 1
or v


ulV -J ( -
ul (r \ul ~


/ rO^+ 1 xu bu n'rC
+ --i xAx x 71y id
x








8 NACA TM 1311


The integration constant is determined from the requirement that
at the jet core the transverse flow component vanishes.

In order to avoid complication of the calculation, rectilinear
course of the mixing width b(x) is assumed. This assumption proves
approximately correct as results from the calculation of the dimensions
of the core region.


Dimensions of the Core Region

The dimensions of the core region (jet core and width of the
mixing zone) are calculated according to a formulation of the momentum
theorem


(u u) x u( UO)r dr (= rrxy)
r


= r(b(x)(ui U

indicated by Tollmien (reference 4).

The occurring integrals as well as the defining the shearing
stress are determined approximately with the course of the velocity
distribution assumed rectilinear

u uO ul uO r



Then there result for the limiting curve d(x) of the jet core and the
width b(x) of the mixing zone two ordinary differential equations of
the first order which can be reduced to one equation


dx
d f(x) y =f f(x) dx)

This integral can be represented with the aid of elementary functions;
however, for simplicity its calculation here is performed by graphical
method.

K appears as the only empirical constant which results by compari-
son with measurements given by Tollmien (reference 4) as K = 0.01576.








NACA TM 1311


Comparison with Measurements

In order to carry through a comparison between theory and experi-
ul UO
ment, a measurement for the case = 0.5 was performed with the
Ul
test arrangement described in reference 5.

The comparison with the theory offers satisfactory results if one
takes into consideration that the effective radius of the nozzle flow
referring to a rectangular velocity distribution is different from the
geometrical radius.


II. CALCULATION OF THE FLOW FIELD

(a) Velocity Distribution in the Core Region


We base the theoretical investigation on the more recent Prandtl
expression for the turbulent momentum transport


e(x) = Kb(x) u/ax- umin

where K = dimensionless proportionality factor, b = measure for width
of the mixing zone, and u = temporal mean value of the velocity.

We have at our disposal, for calculation of the flow field, the
continuity equation and the momentum equation for the main direction of
motion, which read in rotationally-symmetrical rotation

Continuity

(ru) 6 (ryv)
T-.-+ =0


Momentum transport

ou 6u 2u 1 6-
u + v r = e(x) +
ax br ,2 r r

where

e(x) = Kb(x) (ul uO







NACA TM 1311


U1 = velocity of the issuing jet

uO = straight uniform velocity of the surrounding medium

Ul >uo

We may integrate the continuity equation by introduction of a flow
potential *


ru=a


The momentum equation then is transformed into


Tr T- Vr
o? ox orJ


x -* r2
7x 'Vr


+1 o'ot V((/ J o 2'
+ = x (X) r --
+r 7x 7r ( r2 6r2


+1r or
r ir )


where e(x) = Kb(x) U l UO if we make the velocity dimensionless by
division by ul. According to a method applied by G'rtler (reference 3)
we set up for J the expression


Sul uO 2
/ u l__o


(/Ul U 1
t=^(-\ L ir


*2 + -


developing in powers of the parameter _.
the potential of an undisturbed flow (ul = uo); thus


ar= ru1


Therein V0 is


0=
0
OX


rv = T






NACA TM 1311

If we enter with this formulation into the differential equation, we
obtain


Sor
?~ +


ul u-O-+ *


/Ul UO 21
+ --ul---- ; -7 +
\ ui /d or


*


(- u- o


1 0
r +


r(x r


2 0
'r


ul uO *1
+



ul ()r U


S1 r 2


If one arranges according to powers of
of differential equations for *1, 2j2


(120 )
T5r-Fl


+ 2r


-1 (2
~F^[ r 2


= e(x)(r


r -r


" r2


[2 O
_r2


S1


r + (i,/ uo) *1
"lul r


6r3

+1 j0 ur uO) 1
r or \ u-l o


iul uO one obtains a series
\uli


( o)


2r2
Sr2


*0
koi-;


I 61
+ 7x-


(0


1
+ -
r


For *i


ul UO) 1
ul or2


63*1
6r3


62*0
. S7t







12 NACA TM 1311


or, taking

*0 *0
r = rul -- = O

into consideration


a2 *1 1 G W 13* 62 + 1 *
2x;r / r33 or2 r 7r-

etc.


On the Theory of Small Disturbances

In the following, we shall limit ourselves at first to small
disturbances of the flow field; that is, relatively small differences

in velocity (Ul small quantity).

The velocity field is then defined by the flow potential *1.
6* -
Since = ru, the above equation for V1 may be written as
follows:

r = e() + r



e(x) = Ib(x)( l )


Therewith we have attained for small disturbances a linearization of
the equation of motion.

(It should be noted at this point that by the transformation
r
r- our equation is transformed into


2-
2(x)


0 u lu 1 au
--+ -x=o 2








IJACA TM 1311


With reference to Reichardt's discussions (reference 6), it is of
interest to point out that this equation is of the type of a heat
conduction equation.)

In view of the conditions existing in our problem


x = distance


jet axis), we introduce instead of
coordinate transformation yields


(,r=const


from the nozzle in direction of the
r rO
r the variable n -. This
x


= )n=const


7u I x = x u x


thus the equation


(xT + rO) = c(x) x + (xTr + ro) i2


r
or, resFectively, for -q +- r 0
x


(rO = nozzle radius,


ou = ur ou 1
Tr !F r 7n x







NACA TM 1311


2- 6
u 1 x u x2 -

uX


e(x) = Kb(x)( ul ")


This equation is a linear partial differential equation of the second
order of parabolic type.

The solution of this differential equation is fixed unequivocally
x
by the initial condition that for -- --0 the velocity distribution
ro
of the plane jet boundary appears.

We first derive (for small disturbances) the velocity distribution
of the plane jet rim.

For --)0 we obtain with the expression u(n) the equation
ro


d2u du x
2 d x x
r0

with


r ]r=_ u 1 Ud

ro Ul

b(x) u_ UO
Therein c = lim b( and is to be regarded as a function of u.
x x u1
---=0
r0
With the boundary conditions

ul for i -- -o
u----
uO for rT--- +0







NACA TM 1311


taken into consideration, the integration yields


ul Uo
Ul \~u


4 COT


- 1)


- e (01)


d(a0o) + 1( +


1
00
2Kc ul )
ul

Turning now to our problem, we can expect great difficulties in con-
structing the exact solution. We limit ourselves therefore to forming
an approximate solution. For this purpose we generalize the plane
velocity distribution (the initial profile) and set up the following
expression


u 1 uO l(x)+a2 E ( 2 (x) ia2(x] (] +.
1 vi ( ul U


This formulation insures at the outset a reasonable shape of the
approximation solution.

For al,T2, there immediately result, because of the initial
condition, the requirements


lim rl(x) = C0
x __ 0
r0
'O


lim a2(x) = 0
r 0
O


Now the following equation is valid:


1
a u
ul


= lim
x _
ro


1

-uol I x


with










Accordingly, we put


al(x) = 1




x _xu=uol

x -[ul ;x 2012


NACA TM 1311


Furthermore we take care that -our approximation statement for small -
yields the exact solution. This will be the case when the (x x
r0
of the approximation statement agrees with the (x to be calculated
x
from the differential equation for -= 0.
ro
According to the differential equation:


_u e(x
S2


Thus


(ro\
x o
r0


lim E(x) Lu +u 1 +1
= lim + 2 -- + C 1
o x2 2 x r T


or, with


e(x) 1
x 2i12


= lim 1 i + + N, 0
x 2a~2 1+
r0 V


ro r-o


x
TI X71

C T








IJACA TM 1311


We now enter into this equation with our approximation expression; that
is, we put (except for a common factor)


eU CllTIY+CT
F = e y01


2 e 2


r0


+ 2)T2]


(1'T + (2 )


We then consider the relations


a1 = ao


lim
x
-=0
rO


furthermore, we assume


lim a1
x
X-->0
r


= 0


The last relation signifies that the width b of the mixing zone is,
in the proximity of the nozzle, of rectilinear character, an assumption
which seems justified considering the fact that we approach, in the
proximity of the nozzle, the conditions of the plane jet boundary.


We then obtain for the left side of the equation


lim u e- (0o) 2 (0)
xO (O
x
ro-^ r


lim
x
-=0
rO0


c2 = 0







NACA TM 1311


for the right side
2,j

1 6- + -0(00 2)
F 6 u dU 2 2
lim + X2 20 a (0 e
x 20 1 + ( 0 3- 2a 1
---jO rO ro
r0

Equating yields the equation


a2'(0) = E 2a022'(0)

or respectively,


a2'(0) =

This results in 02 = 0 for small r-

This guarantees first of all that our approximation expression
for -x- 0 represents the exact solution.
rO

If we enter with the approximation expression thus constructed into
the differential equation, we recognize immediately that the latter (due

to the factor el11+a 2 is satisfied also for --- o (and
arbitrary I.
rO/
Thus our approximate expression with cl, a2 fixed in the above
manner yields a function which corresponds in boundary zones of the
region to the exact solution.

As to the behavior of our function in the interior of the region,
it is found that the function in case of suitable "continuation" into
the interior of the region satisfies the differential equation along n = 0.

For i = 0 the differential equation reads


ax) (x u
x x 2 (To OJ
ro







NACA TM 1311


If one enters with the approximation expression and considers

E(x) 1
x 2a12

one obtains

2 2' %2'12 + a1Cri


or

1 1
a2' + a2 o=


As solution one obtains

x
1 1 rOx 1

r 0

For small one has again
ro


a2=1O r)

We may also write


12 f.(ul u) 1 fror x, d.x



Therewith we. have obtained for small disturbances the following
approximation function


I. 1 UO 1) e-alTI*1 dai +2 ( 1 ) uO
S e -- 1 J+ 1 + + -I
ul 0 + ul








20 NACA TM 1311


where

al(x) = 1

J2(UI uO)b


x
1 uO 1 rx
a (x) = 2K( d _
2 2 u x rx r
ro

To sum up: This function satisfies the differential equation with
the initial conditions prescribed for small as well as for large
rO
positive n; in the interior of the region it satisfies the differential
equation along the jet Tr = 0. Therewith we have constructed an approxi-
mate function which in boundary zones of the region of integration and
in its interior along the jet = 0 is to be regarded as exact solution.


On the Theory of Larger Disturbances

U1 Uo
Let us now consider larger disturbances not a small
U1
quantity

First, we shall treat the problem of the initial profile.

Gortler's calculations (reference 3) showed that even the first

approximation for small -u1 represents, purely with respect to
shape, a very good approximation. This applies, however, only to the
shape of the distribution curve not to its position. The velocity
distribution calculated by Gortler is unequivocally fixed by the

arbitrary requirement that U(0) = -2 However, Gortler
points out that with U(rI), V(il), the equations u* = U(q + a),
v* = V(q + a) au(Tr + a) also represents a system of solution. But
this remaining uncertainty is here eliminated by the fact that for the
jet core the transverse component v must vanish as follows from the
continuity.

If U((), 'V(n) is the velocity distribution calculated by Gortler

which is characterized by (O) = 2 the quantity a must there-
fore be determined in such a manner that v1 aul = 0 which







NACA TM 1311


V1
yields a = -. Taking Gortler's calculations as a basis, one obtains
u1
in first approximation

V1 1ul uO` u0.3
ul a u1 U,


thus


aa = 0.36
u ,ul


uI u0
Therewith the initial profile for all -
determined. If we base the shape representation on
mation, the initial profile is


is unequivocally
the first approxi-


1" 1- 1 +
Ul ~Ui 0 I


where


uI u0
S0 0.36 Ul


and


1
U0 = uO)

( ul10


For the further development of the profiles in the core region,
starting from this initial profile, we take as a basis the regularity
found for small disturbances.


U 1 uO Tj*
ul F(F 0


e d + 1 + -O
u







22


where

ul uO
n* = aln 0.36 l---- + a2

with the terms al(x), a2(x) determined before.


NACA TM 1311


This function therefore yields the initial profile in first approxi-
mation. How far it may be regarded as approximation in the region is
Ul UO
not investigated in more detail here. For -- 0 it is trans-
Ul
formed into the approximate function found for small disturbances.

Our approximate function generalized to arbitrary disturbances
therefore reads


Ui= ll 1 e *2
ul \UIl )0


2\ ul


where


tI* = a 01 -


6 ul u
0.36 l + a
U l ,


with


al(x) = 1

l, X


x
1 2K ul u) 1 ro
a2(x) = 2(u i 1
( -0


The coordination to n is obtained by

ul uO
,* + 0.36 u O
1 a1


S1x dx
\rO x








NACA TM 1311


where



a2 1 1 o _ro x _ro

1i 2 a12 x0)J


Thus the curves result from one another by similarity transformations.

Calculation of the curves requires, furthermore, knowledge of the
functions al(x), a2(x). and, respectively, of the mixing width b(x)
and the constant r. These quantities result from the approximate
calculation (carried out with the aid of the momentum theorem) for the
dimensions of the core region.

Figure 1 contains for the parameter values

ul UO
Ul-- = 1.0; 0.8; 0.6; 0.4; 0.2
ul


the velocity distributions -
ul


calculated for = 0
r0


and the core end.


In figure 2 the functions al(x) and
parameter values named above, as functions


Calculation of the Transverse Component


a2(x) are plotted for the
x
of up to the core end.
r0


The transverse component V of the flow is determined from the
continuity equation


+ = 0


and, respectively


v= -r (r dr


r 0







NACA TM 1311


r r0
Transformation of r into =- results in
x


v = r) 1 + -
n1


x. 1 di
ao ') ar l


ro
+
x


The integration constant is determined from the requirement that in
the jet core the transverse component v must vanish.

As the lower limit we .choose accordingly the T determined by the
bounding of the jet core (concerning the dimensions of the core region,
compare next paragraph).


In order to avoid complicating the
of the mixing width b(x) was assumed.
correct. (Compare fig. 11.)


calculation, a rectilinear course
This assumption is approximately


u
For the velocity distribution we substitute our approximate
Ul
function. The performance of the calculation (appendix no. 1) yields
the following final formula.


S1 1 uO 1 1 x
- t 1' I +
ul + ) ul 2 2 rO 2 c1
x


1 1U( uO 1 1 r1
rO u1 2 12 x 2 a,
1 + -1 al
x


where


S e 2


+ e-[. 2


- (2 0.36


+ F


= Fl 1


) 0


U i 10i C


+ F 0







NACA TM 1311 25



iI. ={-[ e- 2 [0 e- 02 + 2 (2 0.36 Ul UO)( e-2 + e-3[02



S + (a2 -0.36 U Uo) (F[ 1

Therein
ul 7
[] = l 0.36 Uluo + c2)



S('ik 0.36 u1 UO + )



F' 2- r e-]2 dC[


and FI'[] and F' -[0, respectively, signify the values of the error
integral taken at the points [] and -C]0, respectively.

In figures 3 to 7 the distributions of the transverse component for
a section )= 0.1 near the nozzle and a section of 3/4 of the core
VO ) uI UO
length are plotted for the parameter values 1.0, 0.8, 0.6, 0.4,
0.2. Ul
Ul uO
In the case = 1.0 there are shown, moreover, the distri-
U1
butions for the sections 1/4 of the core length and the core end itself.

(Remark: The transverse components calculated for the core end seem
to yield too small values of the approach flow; the reason is that the
poor approximation of the velocity distribution, an essential charac-
teristic of the Prandtl expression, in the boundary zones takes the more
effect in the calculation of the v component the more one approaches
the core end.)
ul UO
For small the transverse component becomes very small
Ul
(note the different scales in the various representations).







NACA TM 1311


(b) The Dimensions of the Core Region


The dimensions of the core region are defined by the limiting curve
of the jet core d(x) and the width of the mixing zone b(x) or,
respectively, the outer limiting curve of the latter b(x) + d(x).

According to Kuethe's procedure (reference 1) we take as a basis
the theorem of momentum in Tollmien's formulation (reference 4).







NACA TM 1311 27

If one marks off a control area in the indicated manner, one
obtains in the rotationally symmetrical case

U U vr- uu uO r dr = (rTXy)


u0 = velocity of the medium surrounding the jet.

According to the more recent Prandtl expression


Txy = Kb(x) (ul U)

Thus we obtain, if we, furthermore, take the limits of the mixing
zone into consideration


U uo) vr 7x I U(u uO) r dr = rab u U O


According to the existing conditions we transform (according to Kuethe)
with
r d(x)
S= b(x)

Then
r d(x)
S-- b(x)

r = bq + d

S d' b'
fx b b

If we make, in addition, the assumption that u depends only on T,
not on x, there follows


rd+b 1 [( u uO 2bb' T(bd' + b'd) dd]d







28 NACA TM 1311


For v we finally insert the continuity equation


v = r dr
r 1

or


V= (b + d) a 2bbl i(bd' + b'd) dd d


For approximate calculation, we write for the velocity distribution the
sample expression

u uO ul )-
ul ul


r d(x)
S= b(x)

This expression, which may be regarded as a first rough approximation
for the velocity distribution, will probably lead to not too large errors
for the integral calculation. The value br determining the shearing
stress also will probably result in a usable approximation for the
central region of the mixing zone.

The result is
u
ul 1 ul uo
8r~ b ul


d u- u 2ul 2 ( 1- U
TLI j 2u( ul) -uul )

(1) We now put r = 0. The momentum theorem is then transformed by
integration into the form of the theorem of conservation of momentum

b+d
u u UO)r dr = const
0







NACA TM 1311


or


fd+b
ulUl- uO)r dr +


2
(U UO)r dr = ul(ul u) ro


2
Ul(U"- O )-


*1o
+J


u(u uo)(lb + d)b d = ul (ul UO)


IUl -u0)(d2
2 21 i-


2) b2 1 uu +
- r + ----- dsj + bd
0 / J Q 1\ U0 / .


1 -U 'O) 0
I- u


If one inserts
one obtains


S r2 2)
2 d2- r )


U
Ul


-(l -) (I _) and carries out the integration,
Ul )


+ b2 ul" --
ulb 1- ul E


+ bdul
ul


or


1 ul uo
- ul


+ bd 1( 1 1j
M i. i^ S


(2) In order to obtain a second equation between b and d, we
put r = rO.

If one performs the somewhat lengthy elementary calculation, one
obtains finally (compare appendix no. 2)

bbl uO 1 rO d' 1 + (rO d3
bb -- + ro +
ul 7 3 b 3 +

ul u 1 rO- d +3 + 1 (rO d 2 1
(b'd + bd') l u rO d)3 + 2 +-

ddtro d 1] r .ul uo
a'L- -1 ul


1d(x)
O


2
rg
2


b


2
+ d = r


+ = 0
ul 2)







30 NACA TM 1311

The theorem of conservation of momentum reads in differentiated
form. (Compare (1).)


bb' uVl + (b'd + bd') ( u O] + dd = 0


By addition of the two equations one obtains


bb Ob d + (b'd + bd')L ob d -



\(u- uo)(ro- d + dd' = rOK([I U -u )
D\ bl I b Y % u1 )


(3) We now proceed to determine b
obtained. We replace b in the second
the function which we obtain by solving
to b.


and d from the two equations
equation by the expression for
the first equation with respect


b d )\ ---2
ro a(0- + 1a a2
r0 ko)r/ V1 "


where


1 ul Uo
a = --
0 2 1 ul u10



3a,3
aul Uo
A U1


3
a2 =
2k-ul


1 1 3 u -

4 1 ul UO
2 ul







NACA TM 1311



b' a d'


Substitution then yields

d' = i 1uf ul

with


K (I2R +


0-1 [)+ al a2(d)]


([ a23o
6) 3

a d


a/d d 2
a2 -1 1 -
!\ro -o)
+ 1
~,2 O 0


Sro 2


x d
We obtain as a function of --
rO rO


d(x) 1 1
K ul Uo)
\ ul /


d


\ u1


SUl u f2) d(d)





ul UO f d
ul rO-1


The evaluation of the integral could, in itself, be carried out
by analytical method since the integrand is built rationally in and
ro


fl=
1


2 =







32 NACA TM 1311


a square root. However, the breaking up into partial fraction which
has to be done in this procedure is very troublesome. Hence it is
advisable to perform the evaluation graphically.


For = 1 the integrand f -UO f
nite expression The limiting value is
nite expression -. The limiting value is


(f l uO )


1
2


assumes the indefi-


1

(a,)
ao


d
- was determined, analytically or graphically, as a function
rx
b(x) results from


b d+ a, ( 2
80 --O 0+ -al 8r2


The relation


K (ul u aO + a2
= --- a,- + --- ---
lim f1
o--1
r0


1
(Ul U)f


(which by comparison with measurements on the plane jet boundary may
serve for the determination of K) also is of interest.

The symbol K appears as the only empirical constant.

With the measured results on the plane jet boundary with zero outer
velocity (given by Tollmien (reference 4)) as a basis, there results
with


(db)
Wj=_o
r^A


= 0.255


lim
d
--pl
rO


If
x
of --,
r0


ro








IACA TM 1311


1
0.255 = K(-3) 0.185


K = 0.01576

Examples:

In figure 8 the dimensions of the corresponding core region are
represented for the parameter values

ul uo
= 1.0, 0.8, 0.6, 0.4, 0.2
ul

xk
Figure 9 contains the core lengths -, figure 10 the mixing
bk rO ul U
widths at the core end as functions of
ro Ul


Figure 11 shows the mixing widths b- for the various
r0
ul Uo x
values of as functions of -.
ul ro


parameter


Figure 12 represents the angle of spread of the respective mixing

region c = x
r 1


Figure 13 represents


UO = --as a function of
Ul u
Ul


ul uO
with Tollmien's value
defining quantity.
defining quantity.


c = 0.255


ul UO
for = 1i
uI


being the


Figure 14 shows for the medium at rest Ul = ) the quantity K
as a function of c = bx Figure 15 shows Or = as a

f-u-0
(db rO
function of c b *dro
rO








NACA TM 1311


Figure 16, finally, contains the limiting value


lim lul UO

r

necessary for calculation of the integrand in

d
x 1 1 fro ul UO f d
= ____ r fl U]_U\ d\--
rO ul uo 1 u)l r0
ul


III. COMPARISON WITH MEASUREMENTS


Measurements on a free jet issuing from a nozzle and spreading in
moving air of the same temperature do not exist so far.

In order to test the theory by experiment, a measurement for the
ul uO
case = 0.5 was performed at the Focke-Wulf plant.
Ul

The measurements were carried out with the test arrangement with
the 5 millimeter nozzle described in reference 5. A certain experi-
mental difficulty was experienced in producing temperature equality in
the two jets; it was achieved by regulation of the combustion chamber
temperature with the test chamber pressure pk and the probe pres-
sure p kept constant. However a perfect agreement of the jet tem-
peratures could not be accomplished inasmuch as the temperature measure-
ment performed with a thermoelement is rather inaccurate in this low
region.

The test data were:

Outer jet:

Static pressure pk = -100 mm Hg
(Measured relative to atmospheric pressure)

Room temperature tO = 200


Barometer reading


PO 754.5 mm Hg







NACA TM 1311 35


Inner jet:

Total pressure Ps = 340 mm Hg
(Measured relative to atmospheric pressure

Stagnation temperature t, = 590

The evaluation of the measured values was made according to the
adiabatic


T1 =T2(


and the efflux equation



ul K gRT2 1 (- 2; ]


with constant static pressure assured in the mixing region.

Due to the imperfect readability of the thermoelement which, as
mentioned before, is too rough for smaller temperature differences, it
was impossible to measure the distribution of the stagnation tempera-
tures over the mixing region. For the evaluation a linear drop of the
stagnation temperatures along the mixing width was assumed.

For the outer jet there results
O -
tA = 9 uA = 151 meters per second

for the jet issuing from the inner nozzle
o -
ti = 13 ui = 302 meters per second

The inner jet therefore has, compared to the outer jet, an excess
o UI uO
temperature of 4 For the velocity ratio the result = 0.5
uI
was obtained.
P- P+-
Ps + Pk
PB PB
Figure 17 shows the total pressure distribution ,
s central
B + PB Ientral







NACA TM 1311


made dimensionless with the central value, for the various test sec-
tions. The section near the nozzle which still shows the character of
a turbulent pipe flow is represented in figure 18. Figure 19 shows,
in addition, the variation of the total pressures along the jet axis.

Figures 20 to 22 contain the corresponding representations for the
velocities made dimensionless by the velocity ul of the jet issuing
from the nozzle.

As to the comparison with the theory, it must be noted that the
velocity distribution at the exit from the nozzle is not rectangular,
as assumed in the theory, but that it represents the profile of a
turbulent pipe flow. (Compare fig. 21.) Hence it proves necessary to
introduce the conception of the "effective diameter" in contrast to the
geometric diameter.

We define the effective nozzle diameter as the width of the
rectangular velocity distribution of the amount ul which is equiva-
lent to the existing momentum distribution. That is, we calculate the
effective nozzle diameter from the equation

2
r effect. 0 uo)r dr
l u0) 2 u0)r dr
0

with the integral, which according to the theorem of conservation of
momentum represents a constant, to be extended over an arbitrary cross
section.

In our case the integration over the cross section near the
nozzle yields

effect. = 0.95rgeom.


Whereas the plotting over n = r- geom, lets the test points appear
x
r effect.
as still lying on one curve, the plotting over T = eff
x
results in a stagger of the velocity distributions with increasing -
ro
toward negative T. This stagger toward negative q expresses the
immediately obvious fact that the isotacs of the flow field are curved
toward negative n (toward the jet axis).

x Xk
Figure 23 contains the theoretical curves for 0 and -
ro r
(the core end); in addition, the test points of the sections x = 10 mil-
limeters and x = 45 millimeters were plotted. The agreement appears







NACA TM 1311 37


to be good as far as the velocity gradient and the orientation in
space in the central mixing region are concerned; the agreement in the
transitions toward the jet core and the surrounding medium is less
satisfactory. Deviations in these transitions are essential charac-
teristics of the more recent Prandtl expression, but are caused here
probably mainly by the approximation character of our developments.

For the core length there results according to the theory a value
of xk = 22.0reffect, whereas the measurements along the jet axis
(compare fig. 22) result in about xk = = 20. 1rffct.
x K0.945- effect.
It has to be noted that the experimental determination of the core end
is affected by some uncertainty.


IV. SUMMARY


The spreading of a free rotationally symmetrical jet issuing from
a nozzle represents a turbulent flow state.

The theoretical investigation is based on the more recent Prandtl
expression = Kb lmax uminl for the momentum transport. The
continuity equation and the equation of momentum are at disposal for
calculation of the velocity distribution. In case of limitation to

small disturbances Iul- O small quantity, where ul is jet exit

velocity, u0 velocity of the surrounding medium the equation of
momentum may be linearized


r x = E(x)r + r -L


An approximate solution is constructed which is characterized by
the fact that in boundary zones of the region as well as along the jet
n = 0 in the interior of the region it has to be regarded as exact
solution.

(ul u0 initial
For arbitrary disturbances arbitrary > 0 the initial
profile which corresponds to the velocity distributions of two mixing
plane jets is determined by the fact that the transverse component in
the jet core vanishes. The regularity found for small disturbances is
taken as a basis for the further development of the profile from this
initial profile.







38 NACA TM 1311


The transverse component of the flow is determined from the con-
tinuity equation, with the use of the approximate function for the
velocity component in the main flow direction. For simplicity a linear
course of mixing width is assumed.

The dimensions of the mixing region (limiting curve of the jet
core d(x) and mixing width b(x)) are approximately calculated from
the theorem of momentum


u u(vr (u uo) r dr (= rxy)
r


= rKb(x)(ul UO) r

under assumption of a rectilinear course of the velocity distribution
r- d
over T, where -t = b

In order to test the theory by experiment, a measurement was
ul uO
performed for = 0.5 with a 5 millimeter nozzle. In order to
U1
carry out the comparison with the theory, the conception of the effec-
tive nozzle diameter is introduced which complies with the deviation
of the effective velocity distribution for an issuing jet from the
rectangular velocity distribution

2
( u \r effect.
ul ul "O) 2- = u2 uO)r dr
0

The agreement between theory and experiment is satisfactory.








NACA TM 1311


V. REFERENCES


1. Kuethe, Arnold M.: Investigations of the Turbulent Mixing Regions
Formed by Jets. Jour. Appl. Mech., vol. 2, No. 3, 1935,
pp. A87-A95.

2. Prandtl, L.: Bemerkungen zur Theorie der freien Turbulenz.
Z.f.A.M.M., Bd. 22, 1942.

3. GSrtler, H.: Berechnung von Aufgaben der freien Turbulenz auf Grund
eines neuen NNherungsansatzes. Z.f.A.M.M., Bd.. 22, Nr. 5,
Oct. 1942, pp. 244-254.

4. Tollmien, Walter: Berechnung turbulenter Ausbreitungsvorgange
Z.f.A.M.M., Bd. 6, Heft 6, Dec. 1926, pp. 468-478. (Available
as NACA TM 1085.)

5. Pabst: Die Ausbreitungheisser Gasstrahlen in bewegter Luft.
UM 8003, 1944.

6. Reichardt, H.: Uber eine never Theorie der freien Turbulenz.
Z.f.A.M.M., Bd. 21, 1941.







40 NACA TM 1311


APPENDIX

NO. 1. CALCULATION OF THE TRANSVERSE COMPONENT


(1 )(X


1 d
n d


r0O
1 + -f
x


= + -

\ x


1T dTl
r.


S1] 2da
_g'p dT1


+ a-
-3k r0


d] +


I-Tk
-l k


We substitute


C = (an1





-1u -

7u/u1
"srr


0- Uo
- 0.36 --
ul


+ 2)


e -]2 l' ( + a) + a2]


1(/uo
vr.- uj


- i) e]2


p[]
Jo


-[]2
e


- 1)


d[l +


- = (1 ro ^k

X ) I i c


LIl-







NACA TM 1311


If we assume a rectilinear course of the mixing width b(x), we
have ai' = 0.

We then obtain


SV 1 1-

x


e- 2


- 1)2 T
-l) 2.. f Ir


TI
T 1 d -i + '
-k


e-2 d +


1 _l/uO
1 + -
x


1 1) 1
S' _-


2
_- 2 ro 2l
e I
x -


The evaluation of the integrals yields


e
Tnk


d le 2
dlI -- i e-2 d(ail)
alJ


e 2
1 fe-1 d[J
a1


[] = (a


- 0.36 ul uO + c2)
uI


Se dTI = 1] e
k [0



[]o ~TI ) 0.36 Ul


2


d[a


e 2 1
e dq


e- 2


thus






NACA TM 1311


po .p0[
0H O


I e"2


d[] = [[- 1


- []2
e daC
error integral


d- -(F1 +F
1 2 I 1 .41


J e-112
k


ul uo
C] = al 0.36 u + 2
U1


Ul- u0
Tr (2


U + 0.36


This results in


Ul 10
1 r+-2 0] + 0.36 U a
e1 d(alqi
a1 f Q1


= 12 1e-' 2
*=


f2 0.36 u u l a 2
[I 2U1 r e-0 /


I C
[]0


where


F 1 2 ro
F = -L
1:K 10


thus


-[]2
e


re-2 e & =
J


-11'"0
0o


+ "
Jo











NACA TM 1311


0




















. H
rr-




r4





















0


Cu








HI4
iO
I

II


CU















,n 0
0



Cu










CI



--0
JCU






0


So
II




OJ


O

I
-4


+



H-l


tIcyi


Cu

4





(O
I





0


rC
'0




I
b






-,O


I
Cu
-n
4) 0


|m, cu
t- --
Ib
bt


cu

CO


I


(1





o
- -



cu
0


ID









r-- ICU





cu



F-


'^



F F~


Cuj


Cu I


HI

H I


CU


4)
HI



II





CU
Cu









44 NACA TM 1311





4-

+ 4;




Sa
+ a) r4


cu
Sao

u cu



cu --- +1
cu b Io




ri lOJ m b ,
+ 4d 0
cu n
Cu o
c u fcu 0cu

r--4 + (D C n


b 0 4



CWu + U
C- I0 -, + 0 0
Cu r cu m

-r -

+ P


HI-u 4,










1 0

u
-pY ~-







NACA TM 1311

APPENDIX

NO. 2. FOR CALCULATION OF THE DIMENSIONS OF THE CORE REGION

The theorem of momentum with r = rO reads


U u0
r=r0


drb

Lr0


r=r0


r r


^ul ul


Sul- d
Wth the coordinate transrmtion b e obtain
With the coordinate transformation r = we obtain


(r=rO
-O


d+b
O )=


ro-d
1 b
ro 0


d u/ul 2b
dT lbb


- T(bd' + b'd) dd] d-


Sbb' T(bd' + b'd) dd] dT


ro-d
b


We substitute


u- O O (u Tu0







= ul'Ul)






NACA TM 1311


d [u uOJ]
dl ul ul uJ


= 2(1 U
ul i


(i n) -


uOc uO)
uIV ul


1(u1 kuo)


This results in


( U_ -uo(-r
Ul Ul r=r0


ro = (l uo (
S- 1 (


ro -U ) uo [b
b ul


TI T)2
Sd +
0


(bd' + b'd) i dn + dd'
"0 "0


1

r0-d
b


di]


(1 q) +


2 ulO /


2
= rO ul
) 7 = (i}


UA ul /


Sd+b
Sro







NACA TM 1311


If one evaluates the integrals, one obtains


u uO )r
ul U1r=r


S(ul u~O 2 ( 0 bb' ,ro d 3
ul b 3


bd' + b'd
.2 )


= bb'


(ro d 2


dd( ro d
+ t^_^


1
ro-d
b


2(1 T)dT +


(b'd + bd')


1
r d (1 i))dT +

b


UO1 uo\ 1
U1\ ul r0o-d
b


- 11 -



1(rO d



1 ro- d 2
4\o^ul


1
dTb
O-d
b


-bb' 2
7l1 2 [r


z


1
fro-d
b


.(1 T)dT +


L (i d




+ (b'd + bd'


+ (r -Ud + ui
b 7ui I j 2


u i 1 0 10 d lur -- /
dd'tul 2 -b 2u"1 uu


uOl -U
Ul \ U1 1


1 dT +


1
Jr0-d
b


. 47


S- uUr
uI ul )


TI dh +


X jr d+b







4b NACA TM 1311

If we insert these expressions into the equation of momentum and
order, we obtain


bb 'O r d 3 ul uO ro- d
b ul b u b


3'\ u ) ,kb I


1 1 uO
Ill d


2 /r d 4
b


- (u u 2
'6 ui ^


1 u(ul UOrO d3
3ul ul b d


2 l2 UO
1-1 uOI
2\ ul


r d
b )


1(u+1 Uo or d)2
+ (bd' + b'd)- -r )


2 2 -
u u o O b d


\ Uo
3( ul )


2(u1 uO ro d3
3\ b
3\ ul b


UO (Iul UO


1 uoi uu- u(r d 2
2 u\ U A \


au v u ) [


2 (ul- U \2r d
ul b (


(ul -uO)(rO d2 uO(u uo
u b ul ul


UO lui u0)


(rob- d
-b ^


/ l Uo .


ul o 2
u )


- r (


u1 1o uoC
3 ui\ ul )






NACA TM 1311


which finally leads to


1 1 (r d -
+ + -b 3


(b'd + bd') l -O)o Ld) + .1


dd' r-d) = r- l Uo- )


Translated by Mary L. Mahler
National Advisory Committee
for Aeronautics


+ Iro d


i71






NACA TM 1311


o9



- c i


I 17 1-.A I I I I 1i i


0






ci


o /
0


* U2







ci


loo


> J




T tLo
a 0---/

^zz~z ^5^_





NACA TM 1311


t

K. -*-* ___ .4.













0 10 20 30 40 so 60


Sax-
-- -- -- a





-- --- -
0 f0 r0 00 40 SO 60
TO










---__ H----- --- ---- -- -- -- -- -- -- -- --
oo


-- -






_/^
9o / / dzz
/ / o.,
/jy t-
/


o 10 20 30


40 SO 60
x


Figure 2.- Auxiliary quantities for calculation of the velocity
distribution.








NACA TM 1311































0
-4



C1I

(-


o

0
o





__



.r-1






NACA TM 1311






54 NACA TM 1311








NACA TM 1311 55























C-4







0
le









*r





------------ -_----- -- -
-I----- I











o O
a

N

o 0
,3 ____ _.__








56 NACA TM 1311










C"








t.
0





















.0
S-1 1 U



Sa
d ,UD



4->
U'







NACA TM 1311





-- -II








lot.
--- ---- I- s











0
"i ,,




r) 0
I I
I o



S\





S i c
---- --- --- -- ---- ---_ --- ---- --- /








I I .I







NACA TM 1311


0.2 04 0.6 0. 1.0 ---- --


Core lengths.


Figure 9.-








NACA TM 1311 59









































o ---
0
0 0.2 0.4 0.6 0.8 1.0 -l-
v


Figure 10.- Mixing widths at the core end.







60 NACA TM 1311





















- -- -- --- ---
0











-
--\ ---------------

*





__ V _L
--- ^^ --S ^ -3 :-








NACA TM 1311


0 0.2 0.4 0.6 0.8 1.0 U-
U1


Figure 12.- Angle of spread of the mixing region.









NACA TM 1311


Figure 13.- Parameter value of the plane jet rim.


I8.0




6.0




4.0




2.0




0








NACA TM 1311 63


0 0. 0.2 0.3

,o


Figure 14.- K as a function of c.









64 NACA TM 1311



















24 ---------- -

S1.0


20 t--




16 -




12 -




8-------------------------I-
0 0.1 0.2 0.3

%-0
ro


Figure 15.- w0 as a function of c.








NACA TM 1311 65











S a0


o



o
-- -- -- ----- --- -- 0






0:
_--------------4---- 2










o





/ "
-4
__ 0










-4















C
-I
I i-f


0
Lb~








NACA IM 1311







co



E










f).











o-











NA



IF
:i















'i.
*i;


E
E





SII



























0
i ------ ------

1









































-o
i|
0
--- -- -_ -


















'I
--__ ------- --- ---












N x






I- I I
--------- -- -- ---------- -- -- -


_ __ __ __ x__ __ __ __ __ __ __ __ _

~iP~lQ sx


CA


TM 1311


I?






'0





I'-)


C3



N3








NACA TM 1311


O


















CD


G-


e'IZR









NACA TM 1311


Ii
K"-


cs
















E _










o
d
ii






---- ------ -




x




x
Ix



II
--- ---i-----------------------





x

- -r-- --
| X







_- I,-


NACA TM 1311






9I







NACA TM 1311


ca


0D


0


ziz5 2







NACA TM 1311


- I -


- -I I-


EE __
Eco

IK )K





Lo



iS







1 41
_ ^0 __ _
5


SI I ___11


r- -i



CI



u,




0


U
o
c,


NACA-Langley 11-6-51 1000


-------.^

x I..

-i-







Ri s I iN
vie an|5 \ S gI
ca
C3S .'' SS i S4 \ 4)'--S''

'63 M M

U2Zw A 1 1 3
is^is .4 y5y s!sI



0 0 P
M Z




S I r 0 ) 4 *il))


0 01 A.20 NU
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