CHAPTER 1
THE PLANE TURBULENT FREE JET
1.1SOME EXPERIMENTAL OBSERVATIONS Let us consider a jet of water coming from a plane nozzle of large length into a large body of water or a jet of air into a large expanse of air. Let the height (or thickness) of the jet be 2b0 and let Uo be the uniform velocity in the jet. If we use suitable flow visualization techniques, we will find that the jet mixes violently with the surrounding fluid creating turbulence and the jet itself grows thicker. Figure 1-1shows a schematic representation of the jet configuration discussed above, which is known as the plane turbulent free jet. Experimental observations on the mean turbulent velocity field indicate that in the axial direction of the jet, one could divide the jet flow into two distinct regions. In the first region, close t o the nozzle, known commonly as the flow development region, as the turbulence penetrates inwards towards the axis or centerline of the jet, there is a wedge-like region of undiminished mean velocity, equal to Uo. This wedge is known as the potential core and is surrounded by a mixing layer on top and bottom. In the second region, known as the fully developed flow region, the turbulence has penetrated t o the axis and as a result, the potential core has disappeared. For a plane jet, the length of the potential core is about 12bo, and in this chapter we will consider only the fully developed flow region and we will discuss the flow development region in Chapter 5. In the fully developed flow region, the transverse distribution of the mean velocity in the x-direction, i.e. the variation of u with y a t different sections, has the same geometrical shape as shown in Fig. 1-1.At every section, u decreases continuously from a maximum value of urn on the axis t o a zero value at some distance from the axis. Let us now try t o compare the distributions at different sections in a dimensionless form. At each section, let us make the velocity u dimensionless by dividing it by u, at that section and let b represent a typical length for that section. Let us take b as the value of y where u is equal t o half the maximum velocity. Let us now plot u/u, against y / b . We will find that the velocity distributions at different sections fall on one common curve. Figure 1-2(a and b) illustrate this aspect very vividly (Forthmann, 1934). In Fig. 1-2, X denotes the axial distance from the nozzle. The velocity profiles at different sections which could be superposed in this manner are said to be ‘similar’. The two non-dimensionalizing quantities are called, respectively, the velocity scale and the length scale. A very large number of flows in the field of turbulent jets exhibit this property of
2
(0)
,
POTENTIAL COREJ
Fig. 1-1.Definition sketch of plane turbulent free jets.
similarity. In order t o use these similarity profiles for predicting the mean velocity field in any particular problem, we have t o be able to predict the manner of variation of the velocity and length scales.
1.2 EQUATIONS OF MOTION In this section we will develop the equations of motion for the plane turbulent free jet. The Reynolds equations in the Cartesian system are written as (Schlichting, 1968, Chapter 18): -au + u - - - au + v - - + atl wat ax ay
a jP I au"
-~
ax
ay
-+(,-+v-+wav av av at
ax
ay
I
a u = ---+v 1 aP aZ P ax
am)
[I-11
az
a v = - - -1+ uap
aZ
P aY
a2v)
-+2+2 ax2 ay aZ a2v
3
40 A;=O cm = 10 cm A : 20 cm
o
u
=
0
35
cm
-5201
7 10
-
-5
-10
5
Ocm
10
25
-Y-
Fig. 1-2. Velocity distribution for plane turbulent free jets (Forthmann, 1934).
aw
aw
and:-+u-+v-+wat ax aulw)
aw
ay
avrwr
aw a2
=
1 aP ----+v
awl2
P
[I-31
The continuity equation is written as:
-au+ - - a+v-
ax
ay
aw = 0 az
where the X-axis defines the axial direction of the jet, the Y-axis is normal to the X-axis and is in the direction of the height of the nozzle and the Z-axis
4
is the third axis of the coordinate system; u , v and w and u', v' and w' are the turbulent mean and fluctuating velocities in the X - , Y- and 2-coordinate directions, p is the mean pressure at any point, v is the kinematic viscosity, and p is the mass density of the fluid and t is the time variable. Because the mean flow is two dimensional, w = 0, a/az of any mean quantity is zero; u'w'= 0; v'w' = 0 and since the mean flow is steady aulat = 0 and &/at = 0. Further, since the transverse extent of the flow is small, u is generally much larger than v in a large portion of the jet and velocity and stress gradients in the y-direction are much larger than those in the x-direction. With these considerations, the equations of motion could be shown t o reduce t o the form: ~
-au+ - =av ax
ay
0
[I-71
Integrating [ 1-61 with respect to y from y t o a point located outside the jet, we obtain: -
p = p_ -pd2
where p , is the pressure outside the jet. Differentiating the above equation and substituting in [ 1-51, we get:
u -au + v - au = ax ay
p_ - -1 _d_
-
a2u au'd p d x + v 7ay- - - - (ay u
a r2-
ax
4 3
[I-81
The last term in the above equation is smaller than the other terms and could be dropped. Hence we obtain the reduced equations of motion as:
where p - is simply written as p for convenience. In [l-91, we could rewrite the last two terms as:
where 71and T~ are, respectively, the laminar and turbulent shear stresses and 1-1 is the coefficient of dynamic viscosity. In free turbulent flows, due to the
5
absence of solid boundaries, rt is much larger than r1 and hence it is reasonable to neglect r1and rewrite [l-91 as:
[l-101 Further, because in a large number of practical problems the pressure gradient in the axial direction is negligibly small and also t o study the jet under relatively simpler conditions, let us set dpldx = 0. Then [l-101 and [l-71 become: [ 1-11] [ 1-121
which are the well-known equations of motion for the plane turbulent free jet with a zero pressure gradient in the axial direction. For the sake of convenience, in this book, rt is often written simply as r. 1.3 THE I N T E G R A L MOMENTUM EQUATION For the plane turbulent jet issuing into a large stagnant environment and expanding under zero pressure gradient, since there is no external force involved, it is easy to see that the momentum of the jet in the axial direction is preserved. Let us now derive this criterion in an elegant manner, and this procedure will be helpful when we study more complex situations. Multiplying [l-111 by p and integrating from y = 0 t o y = 03, we have: [ 1-131
Let us now consider the different terms of the above equation. p 0
au i a 1 d u z d y = -rz(pu2)dy= 2dx rpu2dy 2
(by Liebnitz rule*)
0
A general statement of the Liebnitz rule can be given as: db
yx, b ) -dx
For proof of the above rule, see 'Advanced Mathematics for Engineers' by H.W. Reddick and F.H. Miller. Wiley, New York 1962, third edition, p. 265.
6
since for y = 0; u = u,,
v
=0
and for
v
= ve
y
+ 00;
u = 0,
where v, is a finite quantity known as the ‘entrainment velocity’, which we will consider later. Thus:
The left-hand side of [ 1-13] becomes equal to: d
- jpu’dy
dx
0
Considering the remaining term:
since ~ ( 0=) 0 from considerations of symmetry and it is reasonable t o assume is zero. Hence. [l-131 becomes:
T(W)
dx
Jpu’dy
=
0
[l-141
0
Equation [l-141 tells us that the rate of change of the momentum flux in the X-direction is zero; that is the moment flux in the X-direction is conserved (or preserved). If the plane jet is issuing from an orifice of height 2bo with a uniform velocity of U,, for every unit length of the orifice, the momentum flux M , = 2pb0U,2. If we imagine that this momentum flux is emanating from a (fictitious) line source, located at the so-called virtual origin (discussed later) from which x is measured, integrating [ 1-14]: 2 (pu’dy
=
Mo
[ 1-151
0
The momentum flux M , is an important physical quantity controlling the behaviour of the plane jet. It effectively replaces individual values of bo and U,. That is, for a given value of M,, the same jet behaviour is obtained for different combinations of bo and U,.
7
Using the integral momentum equation, we will now develop a method of predicting the variation of the velocity and length scales. For the plane turbulent jet, we have seen that the velocity distribution in the fully developed region is similar. That is:
[l-161
uhrn = f(7)
whereq = y / b Let us assume simple forms for u, and b as: u, a x p
[ 1-17]
bax4
[l-181
where p and q are the unknown exponents to be evaluated. Substituting [l-161, [l-171 and [l-181 into [l-141, we get: [ 1-19]
where f 2 stands for f 2 ( q ) . Rewriting [ 1-19] : d --pbu& dx
Jf2dq = 0
[ 1-20]
0
In [l-201, J," f2dq is a constant. Then: d - (buL) = 0 dx
[ 1-21]
From [ 1-21], we can say that bu; is independent of x. That is:
bu2 a x o That is: xq+2pcx x o
and
q
+ 2p
= 0
[l-221
To evaluate p and q , we need one more equation and we will develop this second equation in a number of ways.
8
1.4 SIMILARITY A N A L Y S I S OF EQUATIONS OF MOTION We will once again use the equations of motion already have:
u/u,
= f(q)
with
[l-111 and
[l-121. We
urn a x p , b a x 4
Based on experimental observations and also from dimensional considerations, we could write: T/PUL =
[ 1-23]
g(q)
Let us substitute the above expressions into [ 1-11].We have:
u = umf(V)
au - a _ ax - -(umf) ax
where f ’ = df/dq;
b’ = db/dx;
au
Hence: u - = u m u k f 2 -
ax
u 2 b’ b
u& = du,/dx
vff’
[ 1-24]
To evaluate the second term, we first need an expression for v which we will
obtain by integrating the continuity equation.
au
Y
av v = 1-dy 0 ay
= u,b’
f
0
= -j-dy 0
ax
1 1)
qf‘dq - u k b
fdq
0
[l-251
9
au
ukb'
Hence: v - = -[qff' - f ' aY b
1 a _l a- r - --(pukg) P aY P aY
2
=
Urn
-g b
1 17
fdq) - u,ukf's
0
17
fdq
[l-261
0
r
[ 1-27]
Substituting [ 1-24], [ 1-25] and [ 1-27] into [ 1-11],multiplying right through by b/u& and rearranging:
' - qff ' + f '
s
17
[ 1-28]
fdq)
0
Since the left-hand side of [l-281 is a function of only q , its right-hand side should also be a function of only q. For thisconditions t o be satisfied, bu;/u, and b' should be independent of x. Considering buk/um, it is proportional to x 4 + p - 1 - p , i.e., x w' ; and b' is proportional t o x 4-1. For these two terms to be independent of x,q = 1. Using [l-221, p = - 112. Hence, for the plane turbulent jet, we have: u, a l / d x
and
[ 1-29]
bax
1.5 THE I N T E G R A L E N E R G Y EQUATION Let us multiply the first equation of motion by pu and integrate it with respect t o y from y = 0 t o y = 00. We get:
-
au
au
JPu2 g d y
+J
au spuz--dy ax
= p S u - - da y pu2 ax 2 0
J
a7
[1-30]
puv-dy = u-dy 0 0 ay 0 aY Let E = pu2/2, the kinetic energy per unit volume.
0
= ! u i ) x aE dy
O0
O0
p
1 (g)
\=?p u 2 za ud y f r p u v - da uy = ( u g + v g ) d y = dy 0 aY 0 0 where D / D t stands for the particle derivative and D E / D t is the total rate of change of the kinetic energy.
b
10
r a u / a y is the rate of production of turbulence, by the Reynolds shear stress working on the mean velocity gradient. We have: [l-311 which sags that the rate of decrease of the kinetic energy is equal to the rate at whichturbulence is produced. For our present purposes, we will rewrite the above equation in a slightly different form.
Adding the above two expressions:
We could now write:
[l-321 "
We see from [l-321 that the rate of decrease of the kinetic energy flux is equal to the rate at which turbulence is produced. Using our earlier assumptions we could rewrite [ 1-32] as:
11
=1 - f3dq = - u& f g f ' d q 2 0 0
d
- bu&
dx
[ 1-33]
where F , and F , are constants, [ 1-33] could be rewritten as: [l-341 d
i.e.:
(buL) dx
i.e.: q
Urn
axo
+ 3p - 1- 3p
= 0
[l-351
Simplifying, q = 1. Then, using [l-221, p = - 1/2.
1.6 ENTRAINMENT HYPOTHESIS If Q is the rate of (forward) flow for unit length at any section of the jet: m
[ 1-36]
Q = 2 Judy 0
If Q is the flow from the nozzle, it is known from experiments that Q/Qo is greater than unity and assumes very large values as x becomes large. That means, the jet entrains a considerable amount of the surrounding fluid as it travels forward. We now write: d dQ _ - 2~ dx
m udy = 2ve
[ 1-37]
where v e is the so-called entrainment velocity. It should be noted that in [l-371 and in the following pages, ve generally represents only the magnitude of the entrainment velocity. From dimensional considerations, we could write:
v, a u ,
or
ve = aeu,
[ 1-38]
where ae is the entrainment coefficient. Thus:
dx
Judy 0
= aeu,
[ 1-39]
12
[ 1-40]
Once again, using [ 1-22], p = - 1/2. The entrainment hypothesis was introduced by Morton et al. (1956) in connection with the analysis of plumes.
1.7 INTEGRAL MOMENT OF MOMENTUM EQUATION Let us multiply the equations of motion by y and integrate from y = 0 t o y = 00. We obtain: [l-411
ruy
0
I d riyuzdy = dy = 2 ax 2 dx
=
s
,,,
0
I d 2 dx
- --
Adding:
au
uy -dy ax
7
- juvdy 0
yu2dy -
r 0
uvdy
13
The integral moment of momentum equation becomes:
4
1 ca
[ 1-42] rpu2ydy = puvdy - rdy dx 0 0 0 Substituting [l-251 for ZI in the first term on the right-hand side of [l-421:
[l-431
J fdrl I)
where
JLv) =
77f-
J fdr) ?)
and
J2(7))
0
=
0
With [ 1-16], [ 1-23] and [ 1-43], [ 1-42] becomes: d dx
- pu$b2 0
qf2dq = pukbb'
1
0
J1(q)dq
[ 1-44]
0
where F 3 , F,, F, and F, are constants, [ 1-44] becomes:
F4
d dx
- (ukb2)- F,&bb'
+ F6umukb2+ F7u&b =
0
[ 1-45]
For [l-451 to be valid:
2p+2q-l= 2p+q = 0
0
[ 1-46] [1-47]
Equation [ 1-47] is the same as [ 1-22]. Solving [ 1-46] and [ 1-47], q = 1 and p = - 1/2. It is interesting to point out that in [l-251, for v/u, to be a function of only q, applicable to sections at different values of x, the exponent q has to be equal to unity.
14
1.8 DIMENSIONAL CONSIDERATIONS In this section we will obtain some useful expressions for the velocity and length scales using the principles of dimensional analysis. From our earlier discussions on the plane jet we may write: urn =
f l ( M 0 2 ~7
[ 1-48]
X )
In free jet problems, if the nozzle Reynolds number ( R e ) = 2b0U0/v is larger than a few thousand, the effect of (molecular) viscosity on [ 1-48] is negligible. Using the n-theorem, we could reduce [ 1-48] to:
u,/dM,/px
= constant
c,
[ 1-49]
Since M , = 2 b o p U i , [l-491 reduces to: _ _ _ _
-
UInlUO = d 2 C l / d X / b , = C,/+/b,
[ 1-50]
The unknown constant C , will be evaluated using the available experimental results in a later section. Regarding the length scale, we could write: [ 1-51]
b = f,(M,,P,X) We could reduce [l-511 to
b/x =
c,
[ 1-52]
or: blb, = C,(x/b,)
11-53]
Once again the coefficient C2 has to be evaluated experimentally. If Q is the forward flow at any section, in a similar manner, we could show that:
Q
=
C3dMoxlp
__
or: Q / Q o = C 3 d x / b o
[l-541
[l-551
where C 3 is an unknown coefficient.
1.9 TOLLMIEN SOLUTION Let us now solve the equations of motion t o obtain theoretically the form of the velocity distribution. We have three unknowns, u , ZI and T ; but we have only two equations. Hence, we need one more equation. For this missing equation, let us use the Prandtl mixing length formula: 7 =
pP(du/dy)2
[l-561
where 1 is the so-called mixing length. At any section, from dimensional considerations we could write:
15
1 = pb
or
l ab
or
1 = pC2x
[l-571
where p is a constant. Let us first consider the shear stress term.
[l-581 where a 3 = 2(/3C2)2,a being another constant. To evaluate the left-hand side of 11-11]: [ 1-59]
where 4 = y/ax and the reason for adopting this form is to get a simpler final equation. Wehave: u,
=
C,U,d/b,/fi
= n/fi
[ 1-60]
where n is a dimensional factor, independent of x. Hence: n
[l-611
= --f
fi
In order t o develop a relation for v, let us bring in the stream function $ defined as: = ali//ay,
li/
=
=
a$
--
ax
= --+/ax
n
J' udy
= -jfaxd@
fi
where: F =
v
v
[ 1-62] = u n f i j f d @ = an+F
[ 1-63]
J' fd@
-
_ _a ( u n f i F ) ax
= -an
1 F [ 1-64]
where: F ' = dF/d@
16
and: u a u = -7 n 2 (;FI2
ax
+4F'Fj
X
(1-651
We have:
[ 1-66]
which could be reduced to the form: [ 1-67]
[ 1-68] [ 1-69]
d
2F"F"'+-(FF') = 0 d4 Let us construct the boundary conditions:
[ 1-70]
y = O ; $ = O;u/u, = F'(0) = 1 y = m ; G = o o ; u / ~ , = F'(oo) = 0 y = O ; @ = 0 ; v = 0;from [l-641, F ( 0 ) = 0 y = 0 ; r = O;F"(O) = 0 y = oo;r = O;F"(oo) = 0
Let us now integrate [ 1-70]. We get:
F " ~ + F F '=
c
[ 1-71]
To evaluate the constant of integration C, let us apply the boundary conditions for 4 = 0. We could see that C = 0. Hence, [l-711 becomes:
F"~+FF'=
17
o
[ 1-72]
This non-linear second-order ordinary differential equation was first obtained by Tollmien (1926) who solved it numerically and the results are given in Table 1-1 with q!I versus u/u, =F'(q!I) and shown plotted in Fig. 1-3a. We could easily construct a table with u / u , versus 7) = y / b as also given in Table 1-1and this plot is shown in Fig. 1-3b. TABLE 1-1 Tollmien solution for the velocity distribution in the plane turbulent free jet (Adapted from Abramovich, 1963)
0 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40
1.00 0.979 0.940 0.897 0.842 0.782 0.721 0.660 0.604 0.538 0.474 0.411 0.357 0.300 0.249 0.200 0.165 0.125 0.095 0.067 0.046 0.030 0.020 0.009 0
0 0.105 0.209 0.314 0.419 0.524 0.628 0.733 0.838 0.942 1.048 1.150 1.255 1.360 1.465 1.570 1.675 1.780 1.880 1.990 2.100 2.200 2.300 2.400 2.510
For Tollmien's solution, from Fig. 1-3b, we could see that D/ax = 0.955. If we find the coefficient a experimentally, then we could evaluate C2 in the equation for the length scale. This constant a also comes into the relation for the velocity scale as shown below. We have: m
2 j'pu2dy = M , = 2bopUi 0
We could rewrite [ 1-73] as:
[ 1-73]
18
($1
m
sum 2 (F ' )2 axd@ = b o U i 0
i.e.:
5 UO
2
=
1
Jo
1
b0
(F')2dq3 ax
1 -1 ___
[ 1-'741
4J; ( F ' ) 2 d q 3 & i m o
Using Tollmien's solution: Hence:
-
uQ
1.21
--__ f
i
d m @= 1/0.685
1
a
[ 1-75]
0
\
urn 0 04 0
8
0
04
08
8
12
16
,
20
24
Y -
b
Fig. 1-3. Tollmien solution for plane turbulent free jets.
1.10 GOERTLER SOLUTION For the turbulent shear stress, Goertler (1942) used the second equation of Prandtl written as:
r
= PE-
au aY
[l-761
where E is known as the coefficient of kinematic eddy viscosity. Goertler assumed that:
19
~au,b
or
E
= ku,b
where k is a constant. Let us assume with Goertler: [ 1-77]
where u is a constant. We could show that: n
[ 1-78]
u =' -F'(t;)
@
[l-SO] and:
T
= pkC2
n2 -aF"
[ 1-81]
X
Substituting these expressions in a suitable form into the equation of motion, we obtain
4FF'
+ kC2u2F" =
Letting: u =
[l-821
0
1
___
BJLC,
[l-821 becomes: 2FF'
+ F"
= 0
Integrating: F 2 + F' = C
[ 1-83]
[ 1-84]
Let us construct the boundary conditions for this problem. y = 0; t = O;u/urn = F ' ( 0 ) = l;i.e. F ' ( 0 ) = 1 y = o ; E = O ; T = O;i.e.Fff(0) = 0 y = 0 ; t = 0 ; v = 0; F(0) = 0 F'(w) = 0 y = G o ; ( = w;u = 0 ; F"(co) = 0 y = co;E = w ; = ~ 0; Using boundary conditions (1)and (3), [l-841 reduces to: [ 1-85]
F ~ + F '= 1 The solution of [ 1-84] could be written as: 1- e-2$ F = tanh [ = 1 ee2[
+
and: F' = .1-tanh2(t)
[1-86]
[l-871
20
TABLE 1-11 Goertler solution for the velocity distribution in the plane turbulent free jet
1.000 0.990 0.961 0.915 0.855 0.786 0.711 0.635 0.558 0.486 0.420 0.302 0.218 0.149 0.102 0.070 0.048 0.021
0 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.50
0
04
08
0 0.114 0.227 0.341 0.455 0.568 0.682 0.795 0.909 1.022 1.136 1.362 1.590 1.820 2.045 2.270 2.500 2.840
12
16
20
24
Y -
b
Fig. 1-4. Goertler solution for plane turbulent free jets.
21
Thus we have: u / u ,
= 1- tanh2(oy/x)
[ 1-88] [ 1-89]
Table 1-11 shows the solution of Goertler in the form of u/u, versus o y / x and u / u , versus y/b and these two distributions are also shown plotted in Figs. 1-4 (a and b). The velocity distribution could also be predicted using other models of turbulence. Goldstein (1938) has discussed the application of the ‘vorticity transfer theory’ to jet problems. For a discussion of Reichardt’s model, see Schlichting (1968). For an excellent discussion of different turbulence models including some recent ones, the reader is referred t o ‘Mathematical Models of Turbulence’ by Launder and Spalding (1972). 1.11EXPERIMENTAL RESULTS The earliest systematic experiments on the plane turbulent jet were conducted by Forthmann (1934). The height of the jet was 3 cm with a length of 65 cm, having an aspect ratio of 21.7, thus ensuring plane flow in the central region of the jet for considerable distance from the nozzle. The experimental results of Forthmann regarding velocity distribution have been presented in Fig. 1-2, where it is also compared with the theoretical curve of Tollmien. Further careful measurements on the plane jet have been made by Albertson et al. (1950), Zijnen (1958a), Heskestad (1965) and others. Zijnen’s experimental results are compared with the theoretical curves of Tollmien and Goertler in Fig. 1-5. (Reichardt (1942, see Schlichting, 1968) obtained another mean velocity distribution from his turbulence model and also performed some experiments.) We find that near the axis of the jet, the Goertler solution appears to be slightly superior, whereas in the outer region the Tollmien solution is generally preferred. It has been experimentally found that the velocity distribution could also be represented satisfactorily by a Gaussian curve. Zijnen (1958a) found that: u/u,
= exp(-GA’)
[l-901
where 6 was found to vary from 70.7 t o 75.0 and X = y/x. Another useful expression is: u/u,
= exp (- 0 . 6 9 3 ~ ’ )
[ 1-91]
Before we begin to discuss the evaluation of the equations for the scale factors, some comments have to be made regarding the virtual origin from which the axial distance x is measured. The virtual origin located from geometrical (or length scale) consideration does not coincide with that located from kinematic or velocity consideration. Further, its location based on either
22
EXPERIMENTAL
8 crn 9 crn
O L 0.18
0.14
0.10
0.06
0.02 0-0.02
-0.06 -0.10
-0.14
-0.18
Fig. 1-5. Dimensionless velocity distribution of plane turbulent free jets (Zijnen, 1958a).
of the above criteria, while insensitive to nozzle design, appears t o be very sensitive t o the turbulence level in the nozzle (Flora Jr. and Goldschmidt, 1969). It is generally found that the virtual origin is located behind the actual nozzle even though some experiments have located it in front of the nozzle. In view of the uncertainty involved in its precise location, for practical purposes, the virtual origin could be located at the nozzle itself, and as a result the distance from the nozzle I becomes identical with x . Using the experimental results of Forthmann (1934) and others, Abramovich (1963) found that the experimental coefficient a varied from 0.09 t o 0.12 and adopted a simple average value of 0.10. With this value for a, the velocity scale equation derived from the Tollmien solution becomes:
u,/u,
= 3.78/&ro
[ 1-92]
Using the experimental results of Reichardt, Goertler (1942) found that his coefficient u = 7.67. Newman (1961) has reported that the above value is satisfactory for large values of x/bo but for smaller values of xlb,, u has been found t o be as large as 12. Using u = 7.67, we could obtain the equation: u,/Uo
=
3.39/m0
[l-931
Zijnen (1958a) found that for his 0.5 x 10 cm nozzle: u,/U,
=
3.52/&r0
= 3.52/J(I
and for his 1cm x 25 cm nozzle:
u,/Uo
= 3.12/Jjs/b,
= 3.12/J(I
-t 1.2b0)/b0
[ 1-94]
+ 2.40bo)/bo
[ 1-95]
From their experiments, Albertson et al. (1950) found: u,/Uo
=
3.24/m0
[ 1-96]
23 0 06
I
I
_ _ - -JET
I
I
I
I
ISSUING F R O M A N O R I F I C E OF 0 5 c m x I O c m JET I S S U I N G F R O M A N O R I F I C E OF 1 cm x 2 5 c m -
1 0 02
0
(a 1
0
I
I
I
002
004
006
0.004I
A; 1
I
OL
1
0
0.02
008
I
I
010
012
I
I
(b)
I
I
I
0.04
014
I
0.06
A.1
0.08
I
I
I
0.10
0.12
0.14
Fig. 1-6. Mixing length and kinematic eddy viscosity for plane turbulent free jets (Zijnen, 1958a).
In general, we could represent all these equations by the equation:
[l-971 where C , varies from 3.12 t o 3.78 and C* from 0 t o 2.40. For all practical purposes, C , could be given an average value of 3.50 and C , a value of zero, giving thereby:
uJU0 = 3 . 5 0 / m 0
[ 1-98]
Concerning the length scale b , we have b = C , x . Tollmien’s solution with a = 1.10 gives C , = 0.097. Goertler’s solution with u = 7.67 gives C2 = 0.114. Tollmien’s value of 0.097 is generally found to be better than the Goertler value and we could make C , = 0.10 for convenience. We could now evaluate the entrainment coefficient ae.From [ 1-37] :
v, =
d
-
dx
” d I u d y = -umbJ 0
dx
00
fdq
0
wherein [l-911 has been used to find that C , as 0.1, we find that:
or,
=
v,/um = 0.053
[l-991
JF fdr) = 1.065. Using the value of
24
I
I
I
I
I
1
I
I
I
0
20
40
60
80
100
120
I40
160
180
X 1 2 b, 10
08 06
Y
04 02
0
0
005
lo
010
015
r
0 (C)
(d) Fig. 1-7a--d.
005
010
x
020
015
x
x
025
020
4
030
025
0.35 035
030
25
X
X
(f
1
x
Fig. 1-7. Turbulence characteristics of plane turbulent free jets (Heskestad, 1965).
Regarding the forward flow 4 we already have:
4/40
= C3J./bo
Albertson et al. (1950) found that C 3 = 0.44. If E is the kinetic energy of the flow at any section, we could show that: [ 1-1001
26
where E , is the kinetic energy at the nozzle. Using [l-911 for f , C1 = 3.50 and C, = 0.10:
E / E , = 2.64/Jxlb,
[ 1-1011
which is essentially the same as that obtained by Albertson et al. (1950). Using his experimental results, Zijnen (1958a) studied the variation of the mixing length and the kinematic eddy viscosity and the results are shown in Fig. 1-6. From Fig. 1-6a, we find that for X 2 0.07, Zlx = 0.0229, but for smaller values of X, the assumption of the constancy of the mixing length appears to be in error. Similarly, Fig. 1-6b shows that E/u,x is constant for X 5 0.04, but decreases slowly for larger values of X. This probably accounts for the discrepancy between the Goertler curve and the experimental results in the outer region of the jet. In the expression 1 = Cx,the following values have been obtained for C. Forthmann (1934), C = 0.0165; Reichardt (1942, see Schlichting, 1968), C = 0.0164; Zijnen (1958a), C = 0.0223 and C = 0.0234. 1.12 SOME TURBULENCE M E A S U R E M E N T S Some measurements regarding the turbulence in the plane jet have been made by Zijnen (1958b), Miller and Comings (1957) and by Heskestad (1965). Heskestad, from his measurements, suggested that the turbulence is fully developed or self-preserving for Z / b , greater than about 130. Figure 1-7a shows the variation of P / u , on the axis with 1/2b0. For 1/2b0 greater than about 40, the relative turbulence on the axis increases linearly with the axial distance. Figure 1-7b shows the variation of the intermittency factor y. Figures 1-7 (c to e) show respectively the distribution of the three turbulent velocity fluctuations in a dimensionless manner at f / 2 b 0 = 101. Figure 1-7f shows the distribution of the turbulent shear stress and it is seen that there is appreciable difference between the measured and calculated curves. This difference has been observed in certain other turbulent shear flows also and the blame is generally placed on the measuring instrument. Miller and Comings (1957) from a study of the pressure distribution in the plane jet found that the static pressure in the jet is below the surrounding fluid pressure and Aplpuk becomes equal to about 0.06 at Z/2bo = 40 where A p is the pressure defect. For X/2b0 greater than about 30, the pressure distribution at the different sections was found to be similar. A study of the energy balance for the fluctuating flow has also been done by Heskestad (1965).