Boundary Lyer Theory

Outline of boundary-layer theory
a. Thc boundary-layer concept
tn tho casc of fluitl motions for which the measured pressure distribution nearly
agrcrs with the perfect-fluid thcory, such as the flow past the streamline body
in Fig. 1.12, or the aerofoil in Fig. 1.14, the influence of viscosity at high Reynolds
numbers is confined to a very thin layer in the immediate neighbourhoocl of the
solid wall. If tho condition of no slip were not to be sat,isfit:d in the casc of a real
fluitl there wollltl 1)c no appreciable tliKcrcncc between the field of flow of thc real
fluitl as comparcd with that of a pcrfcct fluitl. The fact thaL at t,hc wnll thc fluid
adlicres to it means, howcvcr, that frictional forces rctarcl the motion of the fluid
in a thin laycr near the wall. In that, thin layer the velocity of the fluid increases
from zero at thc wall (no slip) to its full value which corresponds to external frictionless
flow. The layer under consideration is called the boundary layer, and the concept
is duo to L. Prantltl 1263.
Figurc 2.1 reproduces a picturc of the motihn of water along a thin flat plate
in which the s!,rcamlincs wcrc made visible bjr the sprinkling of particles on the
surfn.cc of thc water. The traces lcft by the particles arc proportional to the velocity
of flow. Tt is scen that there is a very thin laycr near the wall in which the velocity
is' considorably smallcr t,han at a 1n.rgcr distance from it.. The thickness of this
holtntlary laycr incrc,ascs along thc plate in a downstream direction. Fig. 2.2 repre-
~nnb tliagrammatically the vclocity distribution in such a boundary layer at the
a. The hollndary-laycr concept 25
plate, with t>hc tlimensiorls across it considerably cxaggcratctl. In front of the
leading edge of the plate t,he vrlocit,y elistribrttion is rtnifornl. With increasing distattrc
from thc leading edge in the downstrmm direrlion the thiclrness, cf, of t,lle retardetl
layor incrrasrs continrlor~sly, nn ilicrrnsing qunnlitira of hit1 I)oc*onlo t1TTcy-lrtl.
15vitlcr1tly tho lhiclrnrss or the 1)ountl:~ry Inycr t1wrcvw.s wit11 Oc~crrasir~~ viwosity.
Fig. 2.2. Sketch of borlntlnry ---
layer on a flat plate in pnr-
allel flow at zero inciclcnce -
On the other hand, even with very small viscosities (large Reynolds numbcrs) t.hc
frictional shearing strcsses T = /c au/a!j in the 1)oundary laycr arc consitlcrnblc
bccnusc of the Inrgc vclocily gr~diont, across lllo Ilow, wllcrct~s o~tl~sitlo tho I~ou~~tlttry
layer t11cy arc very small. This physical pict~~rc suggcst~n that the field of flow in t.1~
casc of lluids of small viscosil.y can I)c tlivitlctl, for tho purpose or matliornnt,icnl
annlysis, into two regions: thc t.llin boundary laycr near the wnll, in whic:h rriction
must be taken into account, antl the region outside thr boundary layer, whcrc the
forces due to friction are small antl may be ncglcct~cd, and where, thcrcforc, the
perfect-fluid theory offers a very good approximation. Such a division of the field
of flow, as we shall see in more detail It~tcr, brings about a considerable simplification
of the ~nat,l~ematical theory of the motion of fluids of low viscosity. In fact, t,he
t,heoretical study of such motions was only made possible by Prandt.1 whorl he
introclucctl this concept.
We now propose to explain the basic concepts of boundary-layer thcory wit11
the aid of purcly physical ideas antl without the nsc of ~nat~hcmatics. The rnathcrn:~t.ical
bor~ntlary-layer tllcory which forms the main topic of this book will bc tlisc~~sscel
in the following chaptcrs.
The dccrlcratctl fluid pnrticles in thc boundary laycr (lo not, in all cnscs, rrmnin
in the thin lnycr which atlhcrcs to thr I~ody along thc whole wcttcd lc~~glh of ~ I I P
wall. In some cases the boundary layer increases its thickness considerably in the
downstrcarn tlirection and the flow in tho boundary laycr beconics revcrscd. 'l'his
causes the decclcratcd fluid particles to be forced outwards, which rnmns illat
thc boundary hycr is scpnrated from t11c wall. Wc thcn spcalr of boundniy-ltryer
sepalation. This phenomenon is always associatrd with the formation of vortircs
and with largc energy losses in the wake of the body. It o_ccur_sprjmarjly nrar blunt
bodies, such %s circular cylinders ~ncl~sph_c-~~. Behind such a body thcrc exists a region
of strongly dccrleratrtl flow (so-calletl wake), in whicl~ the pressure distribution
deviates considerably from that in a frictionless fluid, as seen from Figs.l.10 arlcl 1 11
in the ~rsprctiw cnscs of a rylindcr and a sphere. The large drag of such bodics can
be explained by the existence of this large deviation in pressure distribution, which
is, in turn, a consequence of boundary-layer separation.