Boundary Lyer Theory

162 VIII. General properties of the boundary-layer equations b. 'Sin~ilnr' solutions or the boundnry-lnycr cquntions
Such a solution is valid for any Reynolds number, provided that the boundary
layer is laminar. In particular, it follows further that the position of the point of
separation is independent of the Reynolds number. The angle wl~ich is formed between
the streamline through the point of separation and the body, Fig. 7.2, simply decreases
in the ratio 1/R1I2 as t,he Reynolds number increases.
Morrovrr, tl~c far!, lht, srpar:ll ion tlors ldtc phcr is prcsrrvrtl wl~c*n tlic- proccss
of passing to the limit R + co is carried out. Tl~us, in the case of body shapes which
cxhibit separation, the boundary-layer theory presents a totally different picture
of the flow pattern than the frictionlcss potential theory, even in the limit of R 400.
This argument confirms the conclusion which was already emphatically stressed
in Chap TV, namely that the proccss of passing to the limit of frictionlcss flow must
not be performed in the differential equations themselves; it may only be undertalren
in the integral solution, if physically meaningful rcsults are to be obtained.
11. 'Similnr* soletions of the boundary-lnyer equations
A sccond, and very important, question arising out of the sol~~t~ion of boundarylayer
equations, is the investigation of the conditions untlcr which two solutions
arc 'similar'. We shall define here 'similar' ~olut~ions as those for which the component
u of the velocity has the propcrty that two velocity profiles u(z, y) locat.ed
at different coordinates x differ only by a scale factor in u and y. Therefore, in the
rase of such 'similar' solutions the velocity profiles u(x, y) at all values of x can
be madr congrnent if they are plotted in coordinates which have been made dimensionless
with reference to the scale factors. Such velocity profiles will also sometimes
be e:llled mifine. The local potential velocity U(x) at section x is an obvious scale
factor for u, because the dimensionless u(x) varies with y from zero to unity at all
sect*ions. The scalc factor for y denoted by g(x), must be made proportional to the
local boundary-layer tl~ickncss. The requirement of 'similarity' is seen to reduce
itself to the requirement that for two arbitrary sections, x, and x,, the components
~(x, y) must satisfy the following equation
't'hc boundary layer along a flat platc at zero incidence considered in the preceding
rl~apter possessed this property of 'similarity'. The free-streani velocity U, was
the scalc factor for u, and the scale factor Sol y was equal to the quantity g = 1/ v x/U,
which was propor(,ionnl to the boundary-layer thickness. All velocity profiles became
- -
it1ent.ica.l in a ~lot of u/IJ,, against y/g = y )/ U,/v x = T] , IFig. 7.7. Similarly,
the rases of t,wo- and threc-clirnerisiorlal stagnation flow, Chap. V, afforded examples
of solutions w11id1 proved to be 'similar' in the present sense.
r 3
I he quest, for 'similar' sol~lt~ions is particulyly imporbant with respect to
t.he mnthomnticnl cl~trrnctnr of the solut.iorl. In cnses when 'similar' soldions exist
it. is pwsiblr, 11s we sl~nll sre in ~norc? drtnil later, to reducc the system of partial
dilt'rrent.in1 equations to onc involving ordinary differential equations, which, evidcntly,
cot-~stit.ntcs a considerable mathematical simplification of the problem.
'i'he ho~~nclary layer along a flat platc can serve as an example in this respect also.
-- --
It will be recallad that with the similarity transformdon T] = y 1 / -/v ~ r,cqn.
(7.24), we ohtained an ordinary differential cquation, eqn. (7.28), for tho strcan~
function /(q), instead of the original partial diKercntial equatior~s.
We shall now concern ourselves with the ty~~os of potential flows for
wl~ich .such 'similar' sol~~l.ions exist. l'11is pro1)lom WILH (IIN(:IIHH(:CI it1 ~ron(, tI(,(.l~.il
fir~l~ by S. (h~ltl~l,oi~~ 1.4j, m t l I I L ~ : by ~ W. Mangler [!)J. ,'1'11~ point or d(:pt~r!,~~rt> is
to consider the boundary-layer equations for plane stdady flow, cqns. (7.10) and
(7.11) together with eqn. (7.5a), which can be written as
au av I
& -t -=o,
ay
the boundary conditions bcirig ?r. =-7 a -- O for y = 0, : ~ r d u - I/ for ?/ --. oo. 'l'ho
cqu:ltion of c:ontinuit1y is it~tc:gratctl by 1,110 introc1uc:tion of the tilrc:r~n func:Ition
y(x, y) wibh
Thus the equation of motion bccon~cs
with the boundary conditions ay/az = 0 and appy = 0 for y = 0, and aypy = IJ
for y = oo. In order to discuss the question of 'similarity', dimensionless quantities
are introduced, as was done in See. VIIIa. All lengths are reduced with the aid
of a suitable reference length, L, and all velocities arc made dimcnsionlcss with
rdference to a suitable velocity, I/,. As a result the Reynolds number
appears in the equation. Simultaneo~~sly the y-coordinate is reforred to the climonsionles~
scale factor q(x), so that we put
proposed by F. Schultz-Grunow [Gn, 15a], ninkes it poasiblc to rcduce uevcrnl problems involving
self-similar solutions to that of bl~e flnt plate at zero incidence. If A = 612 R is chosen
as the curvature parametor, the trnnaformntions can be npplicd to flows nlong longitudinnlly
curved walls with blunt or shnrp lending edges as well ns wit,h blowing or suction (Chnpt. XIV).
The preceding trnnsfnrrnation is exnct to second ordcr in curvnt,urc which mcnns tbnt all t,crms
of the ordcr A hnvr been inclded.
163