Page 187 - Bird R.B. Transport phenomena

§5.6 Turbulent Flow in Jets 171

in which C is the third constant of integration. Substitution of this into Eqs. 5.6-12 and 13
3
then gives

v, = 2 2 2 (5.6-21)
[1 + \(C r/z) ]
3
(
: г " (Сзг/z) - - (C r/z) 3
3
4
3
v = (5.6-22)
2 2
r 2 [1 + \{C r/z) }
3
When the above expression for v z is substituted into Eq. 5.6-2 for /, we get an expression for
the third integration constant in terms of /:
C 3 = J 1 (5.6-23)
The last three equations then give the time-smoothed velocity profiles in terms of /, p, and v^\
A measurable quantity in jet flow is the radial position corresponding to an axial velocity
one-half the centerline value; we call this half-width b . From Eq. 5.6-21 we then obtain
U2
v (b ,z)
z ]/2 1 (5.6-24)
2 2
^, ax(2) 2 [1 + {(C b /Z) ]
m
3
m
6
Experiments indicate that b U2 = 0.0848z. When this is inserted into Eq. 5.6-24, it is found that
C 3 = 15.1. Using this value, we can get the turbulent viscosity v u) as a function of / and p from
Eq. 5.6-23.
Figure 5.6-2 gives a comparison of the above axial velocity profile with experimental
data. The calculated curve obtained from the Prandtl mixing length theory is also shown/
Both methods appear to give reasonably good curve fits of the experimental profiles. The

• x = 20 cm
• = 26 cm
о = 45 cm

2.5

Fig. 5.6-2. Velocity distribution in a circular jet in turbulent flow [H. Schlichting, Boundary-Layer Theory,
McGraw-Hill, New York, 7th edition (1979), Fig. 24.9]. The eddy viscosity calculation (curve 1) and the
Prandtl mixing length calculation (curve 2) are compared with the measurements of H. Reichardt [VDI
Forschungsheft, 414 (1942), 2nd edition (1951)]. Further measurements by others are cited by S. Corrsin
["Turbulence: Experimental Methods," in Handbuch der Physik, Vol. VIII/2, Springer, Berlin (1963)].

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H. Reichardt, VDI Forschungsheft, 414 (1942).
W. Tollmien, Zeits. f. angew. Math. u. Mech., 6, 468^178 (1926).
7