Boundary Lyer Theory

730 XXIV. Prec turhi~lent. flowu; jcta and wnkes
in tjhe tlownst.ream direction. Concurrently the jet spreads out and its vclocity de-
creases, but the total momentum remains constant*. A comprehensive account of the
problems of free jets was given by S.I. Pai [26]. See also the book by G.N. Abrnmo-
vich [ll.
A wake is formed behind a solid body which is bcing dragged through fluid
at rest, Fig. 24.1 c, or hehind a solid body which has hcen placed in a stream of fluid.
The velocities in a wnkc arc smaller than those in thc main stream and the losses
in the vclocity in thc wakc amount to a loss of momcnta~m which is due to thc drag
on the hody. Thc sprcad of thc wakc increases as tllc distancc from thc body is
increased and the cliffcrenccs between the velocity in the wake and that outside
I)ecomc smaller.
Qnalitativcly such flows resemble similar flows in the laminar region (Chaps. IX
and XI), but thcrc arc large quantitative differences which are due to the very much
larger turbulcnt friction. Free turbulent flows are much more amenable to mathc-
matical analysis than turbulcnt, flows along walls because turbulent friction is much
larger than lnrninar friction in the wholc region under consitlcration. Consequently,
laminar friction may bc wholly neglected in problcms involving free turbulent flows,
which is not tho c.wc in flows along solid walls. It, will be rccalled that in thc lattcr
case, by cont.mst,, laminar frict.ion must always bc taken into account in thc imrnc-
diatc ncigl~bourhood of thc wall (i. c. in thc laminar sub-lnycr), and that causrs great
mathematical tlifficultics.
Furtlirrmorc, it will ho noted that prol)l~ms in frec turbulent flow arc of n
houndmy-hycr nature, mcaning that tho region of space in which a solntion is being
sought docs not cxtcntl far in a transverse direction, as comparctl with the main
dirc~t~ion of flow, and that tho transverse grndjcnts arc large. Conscqucntly it is
permissible to study such prol)lcms with the aid of the boundary-layer equations.
In tlic two-tlimrnsiond iwomprrssit)lc flow tlicsc are
Ilcrc T t1c:notcs t.hc I~url~~~lcnt shmring sl.rcss. l'hc pressure term has bccn droppcd
in the cqrmtion of motion because in all problems to be considered it is permissible
to assume, at Icast to n firsL approximation, that the prcssnrc remains constant. In
the case of wakcs this assumption is satisfictl only from a certain distance from the
tmtly onwartls.
In ordm to be in a po.sit.ion t,o intrgratc the systcrn of equations (2.1.1) and
(24.2), it is necessary to exprem the tnrbulcnt shearing stress in terms of the paramctcm
of tho main ~Iow. ~t present suct~ an rIimin&tiort can only 110 aotticvcd witti
the aid of sorni-cnipiricd ~~snmptions. 'LY~csc liavc already bcen ciiscuwcd in Chap.
XlX. In this conncxion it is possible to make use of I'rantltl's mixir~g lcngtll tl~cory,
eqn. (19.7):
or of ite extension
b. Estimation of tho incre-c in width ntid or tho dccre,wo in vrlocity
where the mixing lcngths 1 and lI are b be rcgartl~rt na purely local functionst. They
must be suitably dcalt with in each particular case. Further, it is possible to use
Prandtl's hypothesis in eqn. (19.10), namcly
t,=o& aa- ~ I L
,ag - e xl b (urn,, - %,in)
731
(24.5)
where h tlenotcs thc width of the mixing zone and x, is an empirical constant. Morc-
over
is the virtual kinematic viscosity, nssumcd constant ovcr thc wholc width and, IICIICC,
independent of y. In adclition it is possible to use von IChrmh's Ilypothcsis, cqn.
(19.19) and that due to G. I. Taylor, cqn. (19.15~~).
Whcn cithcr of thc nssnmpthns (24.3), (24.4) or (24.5) is uscd it is fonncl that
the rcsolts differ from cach otlicr only compnmtivcly littlo. ?'he bcsl rncnsuro of
ngrccmcnt with cxpcrimcntal rcsnlta is furnished by thc awnml)tion in ccln. (24.5)
and, in addition, the resulting cqnations arc morc convenicrit to solvc:. I'or I.l~c:sc:
reasons we shall express a prcfercncc for this hypothcsis. Ncvcrtl~clcss, sonlo cxar~lplcs
will be st~~diccl with the aid of thc l~y~otlicscs in cqns. (24.3) and (24.4) in ortlcr to
cxhibit tho diffcrcnccs in tltc rcsulta whcn clifhrcnt, l~ypoblicscs arc IIRCCI. Morcovt~r.
the mixing Icngth formula, cqn. (24.3), has rcndcrctl such valuablc service in the
theory of pipe flow that it is useful to tcst ib applicability to thc typo of glow under
consideration. It will be recalled that, among others, the universal logarit~limic
velocity dist,ribution law has bcen dcduccd from it.
b. Estimation of the increase in width anal of the clecrcnoc in vclneity
Bcrorc proceeding to intcgratc cqns. (24.1) ant1 (24.2) fnr scvcrd parth11:rr
cascs wc first propose to make estimatiotis of onlcra of magr~it~utlc. In this way wo
shall bc able to form an idea of the typc of law wl~ich govcrns thc increase in the
width of thc mixing zone and of the decrcasc in the 'hcight' of the ~clocit~y ~~roAlo
with increasing distancc x. The followit~g accourlt will bc based on one first givrt~
by I,. Prantltl [27].
When dealing with problems of turbulcrlt jck and wakes it is usu;dly assumed
that the mixing lcngth 1 is proportional to the width of jct, 0, because in this way
wr are led to 11sdu1 rcsults. Hcnce we put
t This extension was not diucu~acd in Chnp. XTX bccnuuc it is usrd only very rnraly.