Boundary Lyer Theory

136 VII. lloundnry-layer equations for two-dimensionnl flow; boundary laycr on n plnte e. The houndnry layrr along a flat, p1at.e 137
ces x can be made identiral by selectling suitable scale factors for u and yt. The
scale fact,ors for u and y appear quite naturally as the free-stream velocity, U,
ant1 the bountlary-layer thickness, S(x), rcspcctivcly. It will be noted that the latter
increases with tho current distance x. Ilcnce the principle of similarity of velocity
profilrs in the boundary layer can be written as u/lJw = 4(?//6), where the func-
tion 6) must be thc same at all clistanccs x from the lcatling rtlgc.
We can now estimatc the thickncss of the boundary layer. From the exact
solut,ions of tho Navier-Stokes equations considered previously (Chap. V) it was
found, c. g. in t,hc case of a suddenly accelerated plat2c, that (1 - I/yE , where t
clcnotctl tho time from the start of the motion. In relation to the problem under
consideration wc may sub~t~itute for 1 the time which a fluid particle consumes while
travelling from the Icading edge to the point x. For a parbiclc outeide the boundary
layer this is t - x/lJ,, so that we may put S - 1/ v x/lJ, . We now introduce the
new tlimcnsionless coordinate 77 - y/S so that
'I'hc equation of continuity, as already tliscusscd in S~L. VIId, can be integrated
by introducing a stream function y~ (x, y). We put
v = I/VZ~/(T), (7.25)
where J(7) tlcnotcs
poncnLs become :
the dimensionless stream function. Thus the velocity corn-
.,‘=" = 3!?!=Um~~(,,), (7.26)
ay a! ay
the primc denoting differentiation with respect to q. Similarly, the transverse
vclocitv com t~onent is
Writing down t.hc further tcrms of eqn. (7.22), and inserting, wc have
Afkr simplification, the following ordinary differential equation is obtained:
J J" + 2 /"' = 0 (Blaaius's equation). (7.28)
As seen from eqns. (7.23), as well ns (7.26) and (7,.27), the boundary conditions are:
71-0: /=O, /'=O; T=W: /'==I. (7.29)
t Tho prohlem of a//inity or similarity of velocity proflr~ will be considered from n more general
po~nt of view in Chnp. VJII. The more exnct theory shows that the region immediately behind
tho lending eclgo miwt bo excluded; ROC p. 141.
In this cxamplc both partial clifferential equations (7.21) and (7.22) have bccn
transformed into an ordinary different,ial cquation for thc stream fnnclrion by the
~imilarit~y transformation, eqns. (7.24) and (7.25). The resulting diffcrcnLial equation
is non-lincar and of the third ordcr. Thc Llrrce 1)orrnd:wy conditions (7.29) arc,
thcrcforc, sufficient to ~let~crminc the so111tion complctdy.
'I'ht? nnalyl,ic: cvrdlr:kl,ion of Iho sol:rl,iort of lho tlifi:ro~lLinl c!tllrr~l,ior~ (7.28) is
quite t,cdious. 11. Ulssius obtained this solution in thc form OK a series expansion
:wound 71 = 0 and an asymptotic expansion for 71 very large, the two forms being
matchcd at a suitablc valuc of 7. The resulting proccdurc was described in detail,
1)y 1,. Prandtl [22]. Subscqucnt to t,hal,, I,. Bairstow [I] and S. Coldstcin I131 solvcd
thc sanlc cquation but with the aid of a slightly modified procedure. Somewhat,
rarlicr, C. Tocpfer [27] solvcd the Rlasius equation (7.28) numerically by thc
:ipplic:ation of thr mcthod of Runge and I