Boundary Lyer Theory

748 XXIV. Free turbulent flowa; jete and wakes
jet is proportional to x and t h centre-line velocity IJ - x-I. Thus t,he virtual
kinematic visrosit,y t~rcomcs
which means that it ronxins constant over t,he whole of the jet, as it was in the
two-dimensional wake. Consequently, the dirercntial cquation for thc velocity
distribution bccornes formally identical with that for the laminar jet, it bcing only
necessary to rcplacc the kinematic viscosity, v, of laminar flow by the virtaal ltinemat,ic
viscosity, F ~ of , turbulent flow. It is, thercforc, possible to carry ovcr t,he
solution for the Iarnitmr, circular jct, ccps. (1 1.15) to (1 1.17). Introclucing, once more,
the constant, kinematic momentum, K, as a measure of the strength of tllc jett,
we obtain
I
3 K 1
U =- -
Xn cox 1+ 1 ,2 ( T ) ''
The empirical constant is now equal to fl/co. Accortiing to the mcasurement
pcrformctl by IT. Reiclmrdt the width of the jcl is given -- by h,/, =- 04848 X. With
7 = 1.286 at u = ) u, we hnvc hllz -- 5.27 x c ,/1/~, and hence
whrrc, as bcforc, I),,, tlcnotcs half t.11~ width at half dcpth
'I'hc diagram in Fig. 24.0 contains a comparison ,between measured velocit,y
tlist,ribut.ion point,s and the tlleorcti~d results from eqns. (24.46) shown as curve (2).
Cnrvo (1) proviclcs a furthcr cornprison wit.11 t,hc thcory due to W. Tollmicn [52].
The mixing 1cngt.h tllcory lends hcrc also to a vclocity distrihut.ion curve wlticll is
sonicwlmt t.oo pointcti near thc mnximum, whereas eqns. (24.46) givc exccllcnt
agreement ovcr the wf~olc widt.11. 'I'hc pnttcrn of stream-lines is &own plotted in
Fig. 24.10. IL is seen tllat the jet draws in at its,haoundary fluid from the surrounding
mass at, rcst, so tl~at thc mass of fluitl carrictl by the jet incrcascs in a downstream
c. Rxarnplcs
. .
Fig. 24.9. Velocity distxibut,ion in n circolnr, turbulent jot,. Menuuromentn duc t,o ltoicl~nrrlt [2D]
'l'hrory: rurvc (I) dur lo Tullmlcn[6Zl:curve (2) from eqns. (24.48)
Fig. 24.10. Pntttlrn of streamlines
in a circulnr, turbulent free jet
tiirection. The mass of fluid carried at a distmcc x from the orifice can bc ralculat.ed
from eqn. (11.18). Inserting the above valnc for F,, we obtain
Calculat,ions on the velocity and tcmpcraturc distributions in two-tlimc~~sional
and circular jets havc also becn carried out by 1,. IIowartll [21], both on the basis
of I,. Prandtl's and of G. I. Taylor's assumption conrcrning turbnlcnt, mixing. 'l'llr
mechanism which governs thc mising of a jct issuing from a circular nozzlc wit,lr
the fluitl in a large pipe was studied cxpcrirnentally by K. Irikt,orin [%I. 'l'hc
experiments covered a range of values of the velocity ratio in thc pipe to that in
the jet of from 0 to 4. Compared with thc mixing of a free jct wit11 the surro~lnd-
ing fluid it is noticed that the pressure increases in t,hc direction of flow in :I m:rllnrr