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Chapter 3 The Radial Jet
CHAPTER 3
THE RADIAL JET
3.1 SOME E X P E R I M E N T A L O B S E R V A T I O N S Let us consider the radial fluid flow from the space between two closely spaced parallel circular discs of small diameter into a stagnant mass of the same fluid as shown in Fig. 3-1. If we use some means of flow visualization, we will find that the flow becomes turbulent quickly and proceeds in the radial direction with its width in the axial direction increasing with r. Such a phenomenon is referred to herein as the radial turbulent jet or simply the radial jet. Radial jets have been studied theoretically by Squire (1955) and Schwarz (1963). Heskestad (1966) made an experimental study of radial jets and found that if the distribution of the r-direction velocity 74. (or u ) in the z-direction is plotted in the conventional non-dimensional manner, the velocity distribution in the fully developed flow becomes indeed similar as in the other turbulent jet problems. We will predict the manner of variation of the scales and the velocity distribution and present some experimental results in the following sections for the fully developed flow region (see Fig. 3-1).
3.2 EQUATIONS OF MOTION A N D I N T E G R A L MOMENTUM EQUATION With v, , v@ and v, as the mean velocities and v:, v i , v8 as the fluctuations in the three coordinate directions of a cylindrical system, the Reynolds equations for axisymmetric flow have already been given in Chapter 2. For the present case, v@= 0, and v, 3 v, and transverse gradients, i.e., a/& are much larger than longitudinal gradients, i.e., a/&-. With these boundary-layer assumptions and following a simplification procedure similar to that used in Chapters 1 and 2, and for convenience replacing v, by u and vz by v, the equations of motion for the radial turbulent jet become:
where T = - pu'v' and the pressure gradient in the radial direction has been assumed to be zero. Let us multiply [ 3-11 by p r and integrate with respect to z from z = 0 to z = 00. We get:
51 Orifice
PLAN VIEW
Fully Developed Flow
Region
SECTIONAL VIEW
Fig. 3-1. Definition sketch of radial turbulent jets.
d Hence, [3-31 becomes: - r r p u z d z = r dr 0
3-41
Equation [ 3-41 describes the preservation of the radial momentum flux. Let us now use [3-41 to predict the exponents in the scale factor equations assumed as: u, a r p
and
b arq
C3-51
52
Let: u / u ,
= f(z/b) = f ( q )
[ 3-61
where b is the length scale, equal to z where u = um/2. Substituting [3-51 and [3-61 into [3-41: d
- rukb
dr
jf’dq
0
=
0
or:1+2p+q= 0 or: 2p
+q
= -1
[ 3-81
We need one more equation to solve for these exponents. We will develop this equation using the similarity of the equations of motion.
3.3 SIMILARITY OF THE EQUATIONS OF MOTION In addition to the assumptions already made, let:
T/Wk
=
g(v)
u = u,f
[ 3-101
Let us next develop an expression for v
rl
( qf’dq
= ru,b‘
-
J
[3-111
u,b 0
0
‘
u,b v = u,,,b’J” qf’dq - - J = f d q - - L b r o 0 17
au -
az
=
a
-(u,f) aZ
=
U“f’ b
0
rl
Jfds 0
[ 3-12]
53
[ 3-13]
ariaz
[ 3-14]
= pukg'lb
Substituting [ 3-10] to [ 3-14] in [ 3-11 and simplifying: 17
17
g' = -bu; f f 2 - b ' f ' { f d q - - - - f ' ~ bf d q - - f f l S f dbqu k urn
0
0
Urn
17
o
[ 3-151
From [3-151, we could write:
bu&/urn 0: ro,
b' a ro
and
blr a r0
[ 3-16]
Equations [ 3-16] give q = 1.Hence, using [ 3-81, p = - 1. Thus we have:
urn a l l r
and
ba r
[3-17]
Using the continuity equation, we could write:
dQ dr
= 2nr
2ve =
dr
[ 3-181
where ve is the entrainment velocity. Simplifying [ 3-18] : rue = d (urnbr [fdq) dr
[3-191
0
Since the integral in [3-191 is a constant:
r vl?
[ 3-20]
Assuming as before, ve = aeurn,[ 3-20] becomes: 1+p-(p+q+l-l
)otrO
[ 3-21]
or q = 1. We could similarly use the integral energy equation to show that q = 1. 3.4 DIMENSIONAL CONSIDERATIONS Based on our previous experience with plane and circular jets we could write : urn = f l ( M 0 , P ,
r)
[ 3-22]
54
b = f2(Mo, P , r )
[ 3-23]
where M o is the momentum flux in the radial direction. Using the r-theorem, we could reduce [ 3-22] to:
u r n / d w = constantC1 or:%
=
[ 3-24]
1
+aJ--
UO
r/ro r/bo [ 3-25]
where ro is the radius of the discs and b o is half the width between the discs from which the radial jet emerges. Similarly, [ 3-23] reduces to: [ 3-26]
b = C,r
3.5 G O E R T L E R - T Y P E SOLUTION Following Schwarz (1963), we will make the constant eddy-viscosity assumption and develop a Goertler-type solution for the velocity distribution in the radial jet. Let us write: U/um
=
f(E)
[ 3-27]
where 4: = uz/r; u being an unknown constant to be determined experimentally. Introducing the Stokes stream function $ as: ru =
a$/&
and
rv =
--a$/&
umr2 i.e.: $ = -F U
[ 3-28]
[3-291
r;
where: F =
jf d t 0
and: u/u,
= F’
[ 3-30]
Writing the equation for urn as: urn = n/r
[ 3-31]
55
where n is a dimensional factor equal t o - - C , U o a o independent , of r:
n ur
i.e.: v = -- ( F - EF’)
[ 3-32]
V 1 or: - - -- ( F - t F ’ )
[ 3-33]
urn
U
We could show that:
[3-341 [ 3-35]
For the turbulent shear stress, let us assume: 7
au az
where Then : T
[ 3-36]
= pEE
is the eddy viscosity equal t o kurnb;k being an empirical constant.
n r
au
pn2kC20
a2
r2
= pk - C 2 r - =
F”
1 -- n2a2kC2F ”’ P
a.2
[ 3-37] [ 3-38]
r3
Substituting the relevant expressions into [ 3-11 and simplifying, we get:
u2kC2F“’
+ F12 + F”F
[3-39]
= 0
With u2kC2= 1/2, [3-391 becomes:
F”’
+ 2 ( F f 2+ F F ” ) = 0
[ 3-40]
Integrating [ 3-40], we get:
F”+2FF’ = C
[ 3-41]
Let us now construct the boundary conditions for this problem. For z = 0, or
= 0 ; u = urn and F ‘ ( 0 ) = 1 v = OorF-EF’ = 0
orF(0) = 0
(1) (2)
56 T
Forz =
M,
or
= OorF”(0) = 0
0 or F‘(-) = 0 = 0 or F”(.o) = 0
= -; u = T
(3) (4) (5)
Using boundary conditions (1)to (3), C = 0 and [ 3-41] becomes: F“+2FF’ = 0
[ 3-42]
Integrating again:
c
F’+F~ =
[ 3-43]
Using boundary conditions (1)and (2), C = 1 and [3-431 becomes:
F ’ + F2 = 1
[ 3-44]
Equation [3-441 is exactly the same as the corresponding equation in the Goertler-type solution for the plane free jet with identical boundary conditions. Hence we could write the solution simply as:
F = tanh t
[ 3-45]
= F‘ = 1- tanh2 t
u/u,
v 1 and: - = - (t - t tanh’ Urn
t - 0.5 tanh g)
[ 3-46] [ 3-47]
0
Equation [ 3-46] has already been tabulated and plotted in Chapter 1.Using the above solution, we will now develop a theoretical equation for the velocity scale. We have the integral momentum equation as: m
j2nrdzpu2 = M, = 2nr,2b0pU~
-_
or:
7
u2rdz = 2roboU:
[3-48]
[ 3-49]
-m
2
2
or: u,T J.Tt2dt = 2r0b,U,2 (7
with:
[ 3-50]
-ca
J‘ Ft2dE = 1.29
--oo
Equation [ 3-50] becomes: [ 3-51]
57
Further, ublr = 0.88 and hence: b=--
0.88
u
r
[ 3-52]
We will evaluate u later on using the limited experimental results of Heskestad. 3.6 TOLLMIEN-TYPE SOLUTION
In this section we will develop a Tollmien-type solution for the velocity distribution in the radial jet. For the turbulent shear stress let us write: =
[ 3-53]
pz2(au/az)2
where the mixing length is written as: 1 = PC2r
[ 3-54]
0being an unknown constant. Then: T
= pp2cjr2(au/az)2
[ 3-55]
Let p2Cj = u 3 , where a is another unknown constant. Now, let: u/u,
= f(z/ar) =
and we also have u,
f(4)
[ 3-56]
= nlr. Introducing the Stokes stream function $:
v)
$ = jrudz 0 (0
$ = narIfd@
[3-571
0
$ = narF
[ 3-58]
where: F = f f d 0
Then:
UIU,
= F'
Differentiating the stream function with respect to r:
a
rv = --(narF) ar rv = n a ( $ F ' - F ) and: vlau,
= (q5F'-F)
[ 3-60]
58
We could show that: [ 3-61] [ 3-62] [ 3-63]
[ 3-64]
Substituting [ 3-61], [ 3-62] and [ 3-64] in [ 3-11 and simplifying, we get:
2F”F“’
+ F f 2+ FF”
=
+
0
Integrating: F~~~ FF‘ =
[ 3-65]
c
[ 3-66]
where C is the integration constant. Let us now construct the boundary conditions. = 1.0 and F ’ ( 0 ) = 1 o,@F’-F = OorF(0) = 0 and r = 0; F”(.O) = 0 u = 0, or F”(oo) = 0 a n d 7 = O;orFff(-) = 0
Forz = 0, q5 = 0, u / u , ZI
Forz =
m,
=
Using boundary conditions (1) t o (3), C = 0 and we have:
F”*+FFI =
o
[ 3-67]
Equation [3-671 is exactly the same as the corresponding equation for the plane free jet in the Tollmien solution with the same boundary conditions. Hence, for [3-671, we could borrow the earlier solution of Tollmien which has been tabulated and plotted in Chapter 1. To construct a theoretical equation for the velocity scale, using [ 3-48], we could write: [ 3-68]
[ 3-69]
59 RADIAL JET
"
Urn
08
-
07
-
06
-
M J
% a
+ = 36.6 51.0
A ' I
o = 706
890
0 2
0504 -
v
= 108 0
0
= 145 2
03 02
-
01
-
0
I
I
I
I
015
020
025
030
I
0
005
010
-zr Fig. 3 - 2 . Velocity distribution in radial jets - experimental observations (Heskestad, 1966).
0 Points from
--
Heskestad's mean curve
-
Goertler - type Tollmien
02
-
type
0
04
08
12
16
20
24
2
'7'7; Fig. 3-3. Comparison of Heskestad's observations with theoretical distributions.
We find that:
-m
F f 2 d @= 1.370
and [3-691 could be reduced to the form: [ 3-70]
Further, from the Tollmien solution blur = 0.955 and hence: b = 0.955ar
[3-711
60
0
40
80
120
160
200
240
r b,
Fig. 3-4.Velocity and length scales for radial jets (Heskestad, 1966).
3.7 E X P E R I M E N T A L R E S U L T S There appears to be a scarcity of experimental observations on radial turbulent jets. The only available experimental observations are those made by Heskestad for just one orifice and one value of U , for a Reynolds number (= U02b0/v)of 2.5 * lo4. For Heskestad’s experiment r,/bo = 38.4. Heskestad found that the distribution of u/u, at various sections is simiIar for F/bo 2 50, where F is the distance from the nozzle, as shown in Fig. 3-2. The mean curve drawn through the data of Heskestad in Fig. 3-2 is compared with the theoretical curves of Tollmien and Goertler-type solutions in Fig. 3-3. The experimental observations agree more closely with the Goertler-type solution. Figure 3-4 shows that l/u, increases linearly with F and b increases linearly with F, thereby confirming the earlier theoretical predictions. The average position of the virtual origin as given by the velocity and length scale is reasonably close to the orifice and hence for all practical purposes, it could be assumed to be situated at the orifice itself and then F is the same as r. Heskestad also found that u = 7.86. The parameter a in Tollmien’s solution takes a values of 0.12. Based on Heskestad’s results we could write: b = O.llr
[ 3-72]
or: b / b o = O . l l r / b o
[ 3-73]
[ 3-741
looY
61
80
-
0 0
60 -
X
40 -
20-
01 0
and
I
I
I
0 15
0 10
0 05
I
0 20
-1 O I
2L
00
005
010
015
020
-zr
Fig. 3-5a and b. Fig. 3-5c and d o n page 62.
Further experiments should be conducted to recommend with confidence the numerical values in [ 3-73] and [ 3-74]. Heskestad also measured some of the turbulence characteristics and found that the distributions of the velocity fluctuations are similar for r / b , greater than about 70. Figures 3-5(a and of the three veloc-_ b) show __ the distribution __ ity fluctuations in the form u f 2 / u & , v f 2 / u & and wf2/uif,at a section where r / b o = 90. Figure 3-5c shows the distribution of the turbulent shear at r / b o = 90 along with the computed distribution. But there is considerable discrepancy between the theoretical and experimental distributions like some
62
005c
oozli' t
e
e
L:90 b
e
nni - " '
0
e
I
k
0 05
0
0 10
RADIAL JET
10
Y
>-
5 V I-
5
w
0 20
0 I5
.^I r
366
09
2
e
08 07
06 05
04 03
02 01 0
0
01
02
03
04
05
06
L -
r
Fig. 3-5. Turbulence characteristics of radial jets (Heskestad, 1966).
of the other jet flows. Figure 3-5d shows that so far as the intermittency factor is concerned, similarity is not developed even for r / b , as large as 145. Before closing this chapter, it is again pointed out that more experiments should be carried out on radial jets before we could recommend, with confidence, any equation for the velocity and length scales.
THE RADIAL JET
3.1 SOME E X P E R I M E N T A L O B S E R V A T I O N S Let us consider the radial fluid flow from the space between two closely spaced parallel circular discs of small diameter into a stagnant mass of the same fluid as shown in Fig. 3-1. If we use some means of flow visualization, we will find that the flow becomes turbulent quickly and proceeds in the radial direction with its width in the axial direction increasing with r. Such a phenomenon is referred to herein as the radial turbulent jet or simply the radial jet. Radial jets have been studied theoretically by Squire (1955) and Schwarz (1963). Heskestad (1966) made an experimental study of radial jets and found that if the distribution of the r-direction velocity 74. (or u ) in the z-direction is plotted in the conventional non-dimensional manner, the velocity distribution in the fully developed flow becomes indeed similar as in the other turbulent jet problems. We will predict the manner of variation of the scales and the velocity distribution and present some experimental results in the following sections for the fully developed flow region (see Fig. 3-1).
3.2 EQUATIONS OF MOTION A N D I N T E G R A L MOMENTUM EQUATION With v, , v@ and v, as the mean velocities and v:, v i , v8 as the fluctuations in the three coordinate directions of a cylindrical system, the Reynolds equations for axisymmetric flow have already been given in Chapter 2. For the present case, v@= 0, and v, 3 v, and transverse gradients, i.e., a/& are much larger than longitudinal gradients, i.e., a/&-. With these boundary-layer assumptions and following a simplification procedure similar to that used in Chapters 1 and 2, and for convenience replacing v, by u and vz by v, the equations of motion for the radial turbulent jet become:
where T = - pu'v' and the pressure gradient in the radial direction has been assumed to be zero. Let us multiply [ 3-11 by p r and integrate with respect to z from z = 0 to z = 00. We get:
51 Orifice
PLAN VIEW
Fully Developed Flow
Region
SECTIONAL VIEW
Fig. 3-1. Definition sketch of radial turbulent jets.
d Hence, [3-31 becomes: - r r p u z d z = r dr 0
3-41
Equation [ 3-41 describes the preservation of the radial momentum flux. Let us now use [3-41 to predict the exponents in the scale factor equations assumed as: u, a r p
and
b arq
C3-51
52
Let: u / u ,
= f(z/b) = f ( q )
[ 3-61
where b is the length scale, equal to z where u = um/2. Substituting [3-51 and [3-61 into [3-41: d
- rukb
dr
jf’dq
0
=
0
or:1+2p+q= 0 or: 2p
+q
= -1
[ 3-81
We need one more equation to solve for these exponents. We will develop this equation using the similarity of the equations of motion.
3.3 SIMILARITY OF THE EQUATIONS OF MOTION In addition to the assumptions already made, let:
T/Wk
=
g(v)
u = u,f
[ 3-101
Let us next develop an expression for v
rl
( qf’dq
= ru,b‘
-
J
[3-111
u,b 0
0
‘
u,b v = u,,,b’J” qf’dq - - J = f d q - - L b r o 0 17
au -
az
=
a
-(u,f) aZ
=
U“f’ b
0
rl
Jfds 0
[ 3-12]
53
[ 3-13]
ariaz
[ 3-14]
= pukg'lb
Substituting [ 3-10] to [ 3-14] in [ 3-11 and simplifying: 17
17
g' = -bu; f f 2 - b ' f ' { f d q - - - - f ' ~ bf d q - - f f l S f dbqu k urn
0
0
Urn
17
o
[ 3-151
From [3-151, we could write:
bu&/urn 0: ro,
b' a ro
and
blr a r0
[ 3-16]
Equations [ 3-16] give q = 1.Hence, using [ 3-81, p = - 1. Thus we have:
urn a l l r
and
ba r
[3-17]
Using the continuity equation, we could write:
dQ dr
= 2nr
2ve =
dr
[ 3-181
where ve is the entrainment velocity. Simplifying [ 3-18] : rue = d (urnbr [fdq) dr
[3-191
0
Since the integral in [3-191 is a constant:
r vl?
[ 3-20]
Assuming as before, ve = aeurn,[ 3-20] becomes: 1+p-(p+q+l-l
)otrO
[ 3-21]
or q = 1. We could similarly use the integral energy equation to show that q = 1. 3.4 DIMENSIONAL CONSIDERATIONS Based on our previous experience with plane and circular jets we could write : urn = f l ( M 0 , P ,
r)
[ 3-22]
54
b = f2(Mo, P , r )
[ 3-23]
where M o is the momentum flux in the radial direction. Using the r-theorem, we could reduce [ 3-22] to:
u r n / d w = constantC1 or:%
=
[ 3-24]
1
+aJ--
UO
r/ro r/bo [ 3-25]
where ro is the radius of the discs and b o is half the width between the discs from which the radial jet emerges. Similarly, [ 3-23] reduces to: [ 3-26]
b = C,r
3.5 G O E R T L E R - T Y P E SOLUTION Following Schwarz (1963), we will make the constant eddy-viscosity assumption and develop a Goertler-type solution for the velocity distribution in the radial jet. Let us write: U/um
=
f(E)
[ 3-27]
where 4: = uz/r; u being an unknown constant to be determined experimentally. Introducing the Stokes stream function $ as: ru =
a$/&
and
rv =
--a$/&
umr2 i.e.: $ = -F U
[ 3-28]
[3-291
r;
where: F =
jf d t 0
and: u/u,
= F’
[ 3-30]
Writing the equation for urn as: urn = n/r
[ 3-31]
55
where n is a dimensional factor equal t o - - C , U o a o independent , of r:
n ur
i.e.: v = -- ( F - EF’)
[ 3-32]
V 1 or: - - -- ( F - t F ’ )
[ 3-33]
urn
U
We could show that:
[3-341 [ 3-35]
For the turbulent shear stress, let us assume: 7
au az
where Then : T
[ 3-36]
= pEE
is the eddy viscosity equal t o kurnb;k being an empirical constant.
n r
au
pn2kC20
a2
r2
= pk - C 2 r - =
F”
1 -- n2a2kC2F ”’ P
a.2
[ 3-37] [ 3-38]
r3
Substituting the relevant expressions into [ 3-11 and simplifying, we get:
u2kC2F“’
+ F12 + F”F
[3-39]
= 0
With u2kC2= 1/2, [3-391 becomes:
F”’
+ 2 ( F f 2+ F F ” ) = 0
[ 3-40]
Integrating [ 3-40], we get:
F”+2FF’ = C
[ 3-41]
Let us now construct the boundary conditions for this problem. For z = 0, or
= 0 ; u = urn and F ‘ ( 0 ) = 1 v = OorF-EF’ = 0
orF(0) = 0
(1) (2)
56 T
Forz =
M,
or
= OorF”(0) = 0
0 or F‘(-) = 0 = 0 or F”(.o) = 0
= -; u = T
(3) (4) (5)
Using boundary conditions (1)to (3), C = 0 and [ 3-41] becomes: F“+2FF’ = 0
[ 3-42]
Integrating again:
c
F’+F~ =
[ 3-43]
Using boundary conditions (1)and (2), C = 1 and [3-431 becomes:
F ’ + F2 = 1
[ 3-44]
Equation [3-441 is exactly the same as the corresponding equation in the Goertler-type solution for the plane free jet with identical boundary conditions. Hence we could write the solution simply as:
F = tanh t
[ 3-45]
= F‘ = 1- tanh2 t
u/u,
v 1 and: - = - (t - t tanh’ Urn
t - 0.5 tanh g)
[ 3-46] [ 3-47]
0
Equation [ 3-46] has already been tabulated and plotted in Chapter 1.Using the above solution, we will now develop a theoretical equation for the velocity scale. We have the integral momentum equation as: m
j2nrdzpu2 = M, = 2nr,2b0pU~
-_
or:
7
u2rdz = 2roboU:
[3-48]
[ 3-49]
-m
2
2
or: u,T J.Tt2dt = 2r0b,U,2 (7
with:
[ 3-50]
-ca
J‘ Ft2dE = 1.29
--oo
Equation [ 3-50] becomes: [ 3-51]
57
Further, ublr = 0.88 and hence: b=--
0.88
u
r
[ 3-52]
We will evaluate u later on using the limited experimental results of Heskestad. 3.6 TOLLMIEN-TYPE SOLUTION
In this section we will develop a Tollmien-type solution for the velocity distribution in the radial jet. For the turbulent shear stress let us write: =
[ 3-53]
pz2(au/az)2
where the mixing length is written as: 1 = PC2r
[ 3-54]
0being an unknown constant. Then: T
= pp2cjr2(au/az)2
[ 3-55]
Let p2Cj = u 3 , where a is another unknown constant. Now, let: u/u,
= f(z/ar) =
and we also have u,
f(4)
[ 3-56]
= nlr. Introducing the Stokes stream function $:
v)
$ = jrudz 0 (0
$ = narIfd@
[3-571
0
$ = narF
[ 3-58]
where: F = f f d 0
Then:
UIU,
= F'
Differentiating the stream function with respect to r:
a
rv = --(narF) ar rv = n a ( $ F ' - F ) and: vlau,
= (q5F'-F)
[ 3-60]
58
We could show that: [ 3-61] [ 3-62] [ 3-63]
[ 3-64]
Substituting [ 3-61], [ 3-62] and [ 3-64] in [ 3-11 and simplifying, we get:
2F”F“’
+ F f 2+ FF”
=
+
0
Integrating: F~~~ FF‘ =
[ 3-65]
c
[ 3-66]
where C is the integration constant. Let us now construct the boundary conditions. = 1.0 and F ’ ( 0 ) = 1 o,@F’-F = OorF(0) = 0 and r = 0; F”(.O) = 0 u = 0, or F”(oo) = 0 a n d 7 = O;orFff(-) = 0
Forz = 0, q5 = 0, u / u , ZI
Forz =
m,
=
Using boundary conditions (1) t o (3), C = 0 and we have:
F”*+FFI =
o
[ 3-67]
Equation [3-671 is exactly the same as the corresponding equation for the plane free jet in the Tollmien solution with the same boundary conditions. Hence, for [3-671, we could borrow the earlier solution of Tollmien which has been tabulated and plotted in Chapter 1. To construct a theoretical equation for the velocity scale, using [ 3-48], we could write: [ 3-68]
[ 3-69]
59 RADIAL JET
"
Urn
08
-
07
-
06
-
M J
% a
+ = 36.6 51.0
A ' I
o = 706
890
0 2
0504 -
v
= 108 0
0
= 145 2
03 02
-
01
-
0
I
I
I
I
015
020
025
030
I
0
005
010
-zr Fig. 3 - 2 . Velocity distribution in radial jets - experimental observations (Heskestad, 1966).
0 Points from
--
Heskestad's mean curve
-
Goertler - type Tollmien
02
-
type
0
04
08
12
16
20
24
2
'7'7; Fig. 3-3. Comparison of Heskestad's observations with theoretical distributions.
We find that:
-m
F f 2 d @= 1.370
and [3-691 could be reduced to the form: [ 3-70]
Further, from the Tollmien solution blur = 0.955 and hence: b = 0.955ar
[3-711
60
0
40
80
120
160
200
240
r b,
Fig. 3-4.Velocity and length scales for radial jets (Heskestad, 1966).
3.7 E X P E R I M E N T A L R E S U L T S There appears to be a scarcity of experimental observations on radial turbulent jets. The only available experimental observations are those made by Heskestad for just one orifice and one value of U , for a Reynolds number (= U02b0/v)of 2.5 * lo4. For Heskestad’s experiment r,/bo = 38.4. Heskestad found that the distribution of u/u, at various sections is simiIar for F/bo 2 50, where F is the distance from the nozzle, as shown in Fig. 3-2. The mean curve drawn through the data of Heskestad in Fig. 3-2 is compared with the theoretical curves of Tollmien and Goertler-type solutions in Fig. 3-3. The experimental observations agree more closely with the Goertler-type solution. Figure 3-4 shows that l/u, increases linearly with F and b increases linearly with F, thereby confirming the earlier theoretical predictions. The average position of the virtual origin as given by the velocity and length scale is reasonably close to the orifice and hence for all practical purposes, it could be assumed to be situated at the orifice itself and then F is the same as r. Heskestad also found that u = 7.86. The parameter a in Tollmien’s solution takes a values of 0.12. Based on Heskestad’s results we could write: b = O.llr
[ 3-72]
or: b / b o = O . l l r / b o
[ 3-73]
[ 3-741
looY
61
80
-
0 0
60 -
X
40 -
20-
01 0
and
I
I
I
0 15
0 10
0 05
I
0 20
-1 O I
2L
00
005
010
015
020
-zr
Fig. 3-5a and b. Fig. 3-5c and d o n page 62.
Further experiments should be conducted to recommend with confidence the numerical values in [ 3-73] and [ 3-74]. Heskestad also measured some of the turbulence characteristics and found that the distributions of the velocity fluctuations are similar for r / b , greater than about 70. Figures 3-5(a and of the three veloc-_ b) show __ the distribution __ ity fluctuations in the form u f 2 / u & , v f 2 / u & and wf2/uif,at a section where r / b o = 90. Figure 3-5c shows the distribution of the turbulent shear at r / b o = 90 along with the computed distribution. But there is considerable discrepancy between the theoretical and experimental distributions like some
62
005c
oozli' t
e
e
L:90 b
e
nni - " '
0
e
I
k
0 05
0
0 10
RADIAL JET
10
Y
>-
5 V I-
5
w
0 20
0 I5
.^I r
366
09
2
e
08 07
06 05
04 03
02 01 0
0
01
02
03
04
05
06
L -
r
Fig. 3-5. Turbulence characteristics of radial jets (Heskestad, 1966).
of the other jet flows. Figure 3-5d shows that so far as the intermittency factor is concerned, similarity is not developed even for r / b , as large as 145. Before closing this chapter, it is again pointed out that more experiments should be carried out on radial jets before we could recommend, with confidence, any equation for the velocity and length scales.