Calculation of turbulent expansion processes


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Calculation of turbulent expansion processes
Series Title:
Physical Description:
19 p. : ill ; 27 cm.
Tollmien, Walter
United States -- National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


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Aerodynamics   ( lcsh )
Turbulent boundary layer   ( lcsh )
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


On the basis of certain formulas recently established by L. Prandtl for the turbulent interchange of momentum in stationary flows, various cases of "free turbulence" - that is, of flows without boundary walls - are treated in the present report.
Includes bibliographic references (p. 16).
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by Walter Tollmien.
General Note:
"Report date September 1945."
General Note:
"Translation of "Berechnung turbulenter ausbreitungsvorgänge." Zeitschrift für angewandte Mathematik und Mechanik, vol. 6, 1926, pp. 1-12."

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By Walter Tollmien

On the basis of certain formulas recently established by
L. Prandtl for t'he turbulent interchange of momentum in sta-
tionary flows ('redfrence 1), various cases of "free turbu-
lence" that is, of flows without boundary walls are treated
in the present report. Prandtl puts the apparent shearing
stress introduced by the turbulent momentum interchange

T = p12 du (1)
xy dy dy


u average velocity in x direction

y coordinate at right angle to x

1 mixing length

The underlying reasoning is as follows: The fluid bodies en-
tering right and left through a fluid layer with the time av-
erage value of the velocity u, at turbulence, have the aver-

age velocity u + I du or u I du while the transversely
dy. Idu dy
directed mixing velocity is I discounting a constant of
proportionality included in the more or less accurately known
I of formula (1); 1 i.s no constant at a wall 1 = 0.
The previously cited report by Prandtl (reference 1) contains
a lucid foundation for formula (1).

The present report deals first with the mixing of an air
stream of uniform velocity with the adjacent still air, then
*"Eerechnung turbulenter Ausbreitungsvorginge." Reprint
from Zeitschrift fur angewandte Mathematik und Mechanik, vol.
6, 1926, pp. 1-12.

2 NACA TM No. 1085

with the expansion or diffusion of an air jet in the surround-
ing air space. Experience indicates that the width of the
mixing zone increases linearly with x, if x is the dis-
tance from the point where the mixing starts. This fact is
taken into account by the formula.

1 = cx (2)

The constant of proportionality c can as yet be determined
only by comparison with experience; only empirical
constant of the theory. In many'instances it will be expedi-
ent to introduce T] = y/x as a second coordinate.



(Two-dimensional problem of the free jet boundary)

By reason of the limiting conditions for the
locity the formula is preferably expressed with

u = f(y/x) = f(M) (3)

Then the stream function i:s

S= Jf(;) dy (4)

= x f(TI) d. =


S' = + TJ F' Q1)

Quantity T is put according to formulas (1) and (2)

..... = c2 xa du du
*:- : p dy dy

The following boundary conditions exist: At the first
boundary T (homogeneous air stream), u = constant or by
introduction of a suitable scale u = 1; that is,

IACA TM No. 1085

Fl (T ) = 1 (5)


-=- 0

a condition -by which the continuous connection is secured -
that is,

F"(11 ) = 0 (6)

Sv(11) = 0

that is, F(TI) = I 7)

at the second boundary T12 (still air) must be u = 0';
that is,

F'( 2) = 0 (8)

and, to assure continuous connection au 0; that is,

M"(0.1) = o (9)

Since 'the pressure, in first approximation, can be
assumed to be constant, the equation of motion reads

u + v a = i y
ax ay p ay

Counting y and T) from the still toward the moving air,
gives, after introduction of the formulas*, .t,he.equation of

FF".+ c2 F" F' =-.0q ,,( 0)

*It is readily apparent at this point, that the formula
u = f(T)),.ecessarily requires an 1 proportional to x.

NACA TM No. 1085 *

which is solved by F" = 0 or F + 2c8 F"' = 0. It affords
uniform velocity in the one case, variable velocity in the
other. The latter solution obviously applies between 'i:
and 1g, the former, outside of these limits. In the bound-
ary points the solutions coincide with discontinuity in F?".
In order to determine the velocity distribution in the mixing
zone, the differential equation of the third order

F + 2c2a = 0 (11)

must be solved. For the time being, a new scale for 1 is
advisable, so that formula (11) simplifies to

F + ?" = 0 (lla)

The result then is

F = ci e-n + c2 ef/ cos -3 1 + ca el/2 sin -
2 2

The five boundary conditions define the constants of integra-
tion cl, c2, c3, and, in addition, the still unknown bound-
ary points | and rT themselves.

The calculation is suitably arranged as follows: Put

T = ) T or = 1 + q,

so that

F = d, e- + da e/a cos r + d3 e1/a sin 3
2 2
The boundary conditions are:

F(f) = I1, F'(T) = 1, F(T) = 0 for 1 = 0, F'(T) = 0,

F"(Ti) = 0 for T-


NACA TM No. 1085

From (12a to c), dl, d2, and d3 can be linearly expressed
in I,, equation (12d) yields 7h, expressed by TIa, and
(12e) finally gives a transcendental equation for Tie, solv-
able by successive approximation. It follows that

11 = -3.02, Ti = 0.981, ,2 = -2.04, di = -0.0062,.

de = 0.987, d3 = 0.577

With this F and F' are defined as function of T1. For
comparison with experience, the original scale must be used -
that is, the reduced Tj, employed thus far, must be multi-
plied by Y 2c-, and the reduced F' by U, the velocity of
the homogeneous air stream. The curves of the velocities and
of the streamlines are given in the table and in figures 2 to
4. The streamlines are plotted for equidistant values of the
stream function. The streamline emanating from x = 0 is a
straight line with angle of inclination

-tani (0.19 V42c)

For comparison,a Gottinwen measurement (reference 2) of the
dynamic pressure distribution with an automatic pressure re-
corder at the edge of the nozzle of the big tunnel was em-
ployed. The distance from the nozzle edge was 112 centime.
ters, the dynamic pressure of the undisturbed jet q = 56
kilograms per square meter. It can be presumed that the as-
sumptions of the two-dimensional problem hold good for the
size of the nozzle. In figure 5 the calculated dynamic pres-
sure is shown as a dashed line over the measured dynamic
pressure. The unknown constant of proportionality c follows
from the conversion factor for TI, which is V2c = 0.0845.
The width of the mixing zone is

b = J2c x 3.02m = 0.0845 x 3.02.x = 0.255x

giving a mixing length of

I = 0.0174 x = 0.0682 b

NACA TM No. 1085

The relative smallness of I is unusual. The agreement be-
tween the theoretical and the measured average velocity dis-
tribution is very good.


Visualize a wall with a narrow slot, which for the study
may be retarded as being linear, through which a jet of air
is discharged and mixes with the surrounding still air. As-
suming, for the first, that the pressure in the jet is the
same as outside, the application of the momentum theorem af-
fords a ready separation 6f the variables'- that is, x and
TI. By reason of the constant pressure-the momentum in x
direction must be

u2 dy = constant
Putting u = cp(x) f(7) results in
p((x) x ( f2) dl = constant
Oonsequently cp(x) = 1

u = -- f(Tl) (13a)

= f(T)dy =,-x f(T)dn = /x F (k) (13b)

v = L F(r) + 1 P'(TI) 71 (13c)

The equation of motion can be set up again, which now,
however, can be immediately integrated once. This interme-
diate integral can equally be obtained direct from the momen-
tum theorem. By mar-king off a control surface conforming to
figure 6,the impulse entering through the lower boundary is
pu v, while on the other side the impulse variation

NACA TM.No. 1085

p u- dy occurs. The turbulent shearing stress

T = pc2 xa j a i
|y oy

acts as outside force, hence the relation exists
uv + uj dy = -

From this follows the equation for F(T1):

2c F"2 = FF' (14)

(valid for positive TI, reflected velocity distribution for
negative i). With a suitable scale for 'i the differential
equation is simplified to

F"2 = FF' (14a)

The order of this differential eauation can be lowered by in-
troducing z = In F; that is, F = es as new dependent
variable. Then, (z" + z'2)2 = z' whence, after putting
z' = Z, finally follows the differential equation of the
fir st. order:

Z' = -Z ./Z

The solution of the original equation then requires only
squaring and removal of the logarithms.

The following conditions must be satisfied: ?or T = 0
(center of jet), v = 0 that is, F = ez = 0. Since
u F' = z'es is not to disappear for T1 = 0, z' must be
of the same order of wo for Ti =0' -as ez is of 0. Hence,
for J = 0 by suitable scale determination,

F = 0 (l5a)

FE = 1


8 NACA TM No. 1085.

Thus there are afforded two conditions through which the z,
that satisfies an equation of the second order, is completely
defined~ ':" .-

The boundary point Tir itself then follows from the

u = 0; that is, z' = Z = 0 (16)

for the boundary Tsr.

Integration of equation (14) gives

] = 0 2. in C/Z + 1)- In (Z / + 1)/ 2 / +3 tan-1 2 1i


The constant of integration C follows from the condition
z' = Z = c for T = 0:
:'S a: .v
0 = C- 3 i'T at C =
3 2 3

The condition (16) Z = 0 for Tir yields

7Tr = C 'tan-1(- -) 4 L 2.412
3 3 ^3

Compliance with equation (15) is predictLed on a study of the
behavior of equation (14) for TI = 0, Z = co. As the solu-
tion (17) in this range is inconvenient, a new form of solu-
tion which applies for TI- 0, Z = co is derived.

For Z -->>, obviously

-, that is, = z' -->


z -> In T + c,, so that F' = z' ez = eC1; for T 0 = 0.

Thus the last constant of integration follows from equation

NACA TM No.. 1085

(15b) as ca = 0; and the asymptotic approximation for
7 = 0 follows at

z = 0.4 % + 0.01 T ., z = in 8 3/ 0.01 .
1 3 3

The quality of the asymptotic approximation (18) is easily ap-
praised by a comparison with the exact solution (17) in a
zone in which both forms of solution are appropriate.

The method is as follows: Compute. T)(Z) and hence
Z(TI) = z'(TI) by equation (17), thus obtaining z(T)), possi-
bly by graphical integration, where in the region about Ti = 0,
the previously determined asymptotic approximation (18) is
taken into account and z' = m. Then the desired functions
F = ez and F' = z'ez are obtained by removal of logarithms
and multiplication.

The solution F = constant joins the just-derived solu-
tion with a discontinuity in ?"' toward ,the outside. In the
center (7 = 0), F' acts as 1 0.4 T3/2, which entails a
disappearance of the radius of curvature.* For comparison
with experience, it is necessary to revert from the reduced
to the original quantities as shown in the table. The conver-
sion factor for Ti is 3 2F ; the letter s, in the table,
signifies a characteristic distance from the gap, where the
speed in the center of the jet equals U.. According to (13a)
the speed at jet center distant x from the gap is then

UmW() = Us J
(See table.)


The corresponding rotationally symmetrical problem, in
which a jet of air discharges from a very narrow hole in a
wall, is treated in exactly the same manner as the two-

Prandtl has given a refinement of the theory by which
the disappearance of the radius of curvature in the center
can be avoided. But, since it would lead too far afield, it
is not discussed here.

10 NAOA TM No. 1085

dimensional prob.l'an. ;.-:Firs't, the.-vaniables x. and 71 are.
easily separated again. For, on assuming tha-t the pre.ssu-re
in the jet is constant,

27r uay dy = constant

'whence for u


(fri) 1 dfl= F(Tl)

F' F' F "
u -, v
XTl x xTl

The differential equation for F is again obtained by i.nte-
gration of the equation of motion or by a second application
of the impulse theorem in analogy .to figure 6:
.. I i/
c 2F" F' = FF'1 (19)

":- With the introduction of a suitable scale for T, the
differential equatibo is simplified to

(jF I = T1 (19a)

By substitution:

z ="ln F, 'F = ez

there is afforded

z1 i+ o.z' .2 ,

and lastly, after introducing Z = z', the differential equa-
tion of the first order

NACA TM No. 1085- 11

Z, = 7 z -ZJZ (20)

In addition, the following conditions hold for 1 = 0:
u may not disappear, while v = 0; that is, F(O) = ez()= O,

while = z'e" remains finite and becomes eoual to unity

by appropriate regularization.

Now a series development of Z(iT) for Ti = 0 can be
applied in such a way that these conditions are satisfied; z
must be negative a for fT = 0, in order that ez = 0,
which is like In P2; because F'/T; then assumes precisely
a finite value. The result is the following development in
powers of Ti3/2:

Z = 2 + a7 + bi2 + ci7/2 + d5 + e13/2. (21)

The coefficients are obtained'by introduction of this formula
in the differential equation and comparison of equal.powers:

2 1 IL, 37
a b = 2 d = e = 0.000014
7 245 1715 240100

The convergence is poor on approaching the boundary point
1,r(Z=0), but a development particularly suitable near 1Ir is
as follows: Put
S= r -

Z= a + b T + c' + d + e 1 + f .

and obtain

1 1 3 3
a = =
4 8 Ir 64 1r2' 64 128 Tir3'

e 19 133 0.00278 ...
256 x 5 +Ir 256 X 40 1r -- Ir T .r
256 x 5 *Hr 256 x 40 By4 Tlr3 T~ r 5

NACA TM No. 1085

The unknown constant of integration Tir is obtained by making
the values for Z, as known from the two developments, agree
in a certain junction point. It results in ir = 3.4.

Quantity F'/Ti acts like 1 0.202 3/ in the center;
the outward junction in F again takes place with a discon-
tinuity in F"'. The conversion factor from the reduced to the
actual quantity Ti is I/ca; s signifies in the table a char-
acteristic distance from the discharge hole for which the
speed in the center of the jet is Ug.

The computed velocities were compared with Gottingen test
data. (See reference 2.) The diameter of the discharge noz-
zle was 137 millimeters. The velocity distributions at 100
centimeters and 150 centimeters distance from the nozzle edge
were used for the comparison. This nozzle distance a may
not be put equal to x, in view of the point discharge ori-
fice assumed in the present calculation; x is rather com-
puted from a by addition of a constant quantity e which
results for example from the fact that for greater a,
for which the comparison with these calculations is solely
permissible, the central velocity decreases as 1/x. In the
present case e = 26 centimeters. Figure 13 shows the theoret-
ical and the experimental dynamic pressure for a = 100 centi-
meters, it amounts to 104 kilograms per square meter at the
discharge orifice; the agreement of the average values is good,
aside from a certain asymmetry of the jet which must have had
different reasons. From the conversion factor for I follows

c a= 0.063

The radius r of the jet is

r =3 2 3.4 x = 0.063 x 3.4 x = 0.214 x

The mixing distance I is = ex = 0.0158 x = 0.0729 r.

Zimm (reference 3) has made corresponding experimental
investigations at considerably lower speed. His findings
would yield

V7 = 0.080

with a dynamic pressure of 5.1 kilograms per square meter in
the discharge orifice. According to it, a slight increase in
mixing path by decreasing Reynolds number is likely.

NACA TM No. 1085.


So far, all cases had been premised on constant pressure.
This first approximation can be improved by analysis of the
pressure differ-jr.ce due to impulse variation on the basis of
the computed. s:,cds ar.d stresses. For the first step, start,
say, with the second equation of motion, which in the first
two cases reads

c-v av 1 Sm 80y\ 1 8p
.u -v + v l (a + aay\ lap
a x y p \x y / p ay

V 2v 1 (T 1 y(" ) _G t 1 ap
1 2-- + v --- + aC7 Zf _
ax oy p \ax y ay y p ay

in the rotationallv symmetrical case; Cy and at are nor-
mal stresses, respectively effective in y direction or at
right angles to y and x. Then, integrate with respect to
Y: Y Y

v + d /uvdy- 1 Td y 1 p = .(22)
o Px. p o p L o
0 0
S y y
v + u /vdyv+ iv dy- 1 Td 1
0 ox y p ax. P
0 0 10 0

o o o
Sy +dyG + d y 1 I (23)
F y P y P JO
0 0

which i-s equivalent to applying the impulse theorem. If, as
heretofore, the normal stress, in this case Cy and ut,
are discounted, there is obtained
Fai' 2/F] i 1 .i
2- F F TdF 2 d'l : p
fL the feP j yo

for the free jet boundary,

NACA TM No. 108.5

x L 4o0 P o

for the two-dimensional jet expansion

+Po d I;
0a o T13 P L P o
for axially symmetrical jet expansion. With Pr denoting the
pressure at the jet bhun&ary, pm the pressure at jet center
and of the homogeneous air stream, respectively, particulari-
zation of the above formulas yields

Pm Pr 0.410(2c-8a/a 2 and Pm Pr = 0.248(2c )2/3Um(X)


Pm Pr = -0.316(c2)/ U (x)

Quantity U indicates the speed of the homogeneous air stream,
and Um(x) the central speed at x. In the first and third
case, c has been determined, giving

Pr = 0.00584 P-
and U, ^
andPm Pr = -0.0025 p Umg(x)

It is apparent that the thus computed pressure differences,
being small, do not cause a' substantial .modification of the

When computing the pressure difference with respect to
still air, it should be borne in mind that at the jet boundary
a negative pressure equal to the dynamic pressure of the. ra-
dial inflow speed prevails. With po as the pressure in still

WACA TM No. 1085

m p = 0.338(2c) p U = 0.00482 P U2

.Pm Po = 0.124(2c.)/ p Um (x)

Pm Po = -0.372(c ) p U(x) = -0.00295.- U (x)
1 2

Hence there is positive pressure within the jet in the two-
dimensional cases, but negative pressure in the axially sym-
metricsl case. This surprising result,which also is at vari-
ance with a rough impulse consideration, points to a defect in
the theory. The necessary extension will be given in the fol-


The theorem applied up to now to the stresses introduced
by the turbulent impulse exchange

T = I2 -uI -u 7 = 7, =' t = 0
T = t

is no more than a fist approximation. I n any case, it awn be
easily proved that au/by in tihe cases in point is great with
respect to Cu.. and ad hence the theorem for the mix-
ox ox oy
ing speed I -Cu caused by the speed difference is good.

So, i,n a natural generalization of the previous theorem, the
stress' tenso'r is p'ut equal to

I] (7X X) .

(Vv = affinor of v; v7 is the conjugate affinor.)

LThis relation is important for the calibration of nitot
tubes in a jet discharging from a nozzle.

NACA TM No. .1085

The stresses to be newly added here, are, in general, neglected,

except r = 2 8U aV and d t = 2 12 8U which are used
y ly y y, y
for calculating the pressure differences. This portion cancels
in the model problem worked out for the two-dimensional case
because of the employed boundaries, but not for the axially
symmetrical case. Here the pressure differences are augmented
by the integral

Sy dy

so that

pm Pr = +0.151(c2) p Um 2() = +0.0012 p Uma(X)

m Po = 0.095(c2)/3 p Ua(x) = 0.00075 P Uma(x)

that is, positive pressure within the jet, as in the other

Translation by J. Vanier,
National Advisory Committee
for Aeronautics.


1. Prandtl, L.: Bericht uber Untersuchungen zur ausbebildeten
Turbulenz. Z.f.a.M.M., vol. 5, no. 2, 1925, pp.'136-139.

2. Betz, A.: Velocity and Pressure Distribution behind Bodies
in an Air Current. NACA TM No. 268, 1924.

3. Zimm, Walter: Uber die Str6mungsvorginge im freien Luft-
strahl. Forschungsarbeiten aus dem Gebiete des Ingen-
ieurwesens.,. no. 234, 1921.

NACA TM No. 1085 17

I M co W o M o co4 o M

1 1 1 1 1 1 1
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u ^C e CD C7 Cr Cw c) iN rli I Oi O

Siiodd cooTi doc dopdd

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I IOl l l

NACA TM No. 1085 Figs. 1,2,3,4,5,6,7,8

-- -

\ / .,4. z




Cr4 4

0 ,. cr ,- .

& I

NC t 4 10 N9
a a' C > C at
/ ::; a

C iC / 2

c a
1'se a

L \ i \ i

II 0

--- I\
054 / 1\

NACA TM No. 1085


PC ,

a $4 a W

0 0 *r

.r4 0: 43 *
5=V E-1 CM
a '~

~EI r

Figs. 9,10,11,12,13

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3 1262 08106 276 1

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