Boundary Lyer Theory

106 V. JCxact solutions of the Navicr-Stokm cq~~ntiona b. Othrr exact, solutions 107
This gives
or, tlcfining a Reynolds number based on thc radius and tip vclocity,
R'o
R = - -
and int.ro~lttcing thc nnmerical vnll~c - 2 zG'(0) = 3.87, wc obtain finally
Fig. 6.14 shows n plot of this equation, curve (I), and compares it with mcasuremcnta
1391. For RcYnolcls numbcrs up to about R = 3 x LOS there is cxcellcnt
agreement hnt,vecn tltoory nnd exporimcnt. At highor Raynolds numbers the flow
bccorncs turbulent, an11 tho respective case is considered in Chap. XXI.
Curves (2) and (1) in Fig. 6.14 arc ohtainnl from thc turbulent flow thcory. Oldor
mcasuremcnts, carried out hy G. Kernpf [lG] and W. Schmidb [31], show tolerable
ugrrwnrnt with throrotiral resalts. Prior tn Lrsr aol~t~ions, I). Riahoachinsky [2Gj.
1271 estaldiahcd cmpiricnl fonnulac for the turning mon~cet of rotating disks wllich
werc hmcd on vcry carcful mcasurrments. Those formulae showed very good
sorrcmcnt with the thcorctical equations discovcred suhscqucntly.
-0- -
The quantity of liquid which is pumpcd outwards N a result of thc centrifuging
nctdon on tho one sidc of a disk of radins R is
o NACA Report No. 7g.3
v 0.48 lo 1.69
Kernpf
0 NSchmidt Fig. 5.14. Turning mo-
ment on a rohting dink;
crlrvc (1) from eqn.
(LM), hmimr; eurvea
(2) and (3) from eqns.
(21.30) and (21.33). tw-
bulenl
Calculation shows t,hnt
Q = 0-885 n R2 (11 = 0.885 n Rn (1, R-lI2 . (5.67)
,I I he q~tanf.ity of fluid flowing towards thc disk in the axial dircct,iorl is of cqrlsl
~napnitutlc. Jt is, filrther, wortlly or no(,c tht tJtc pressure tlini:rrt~cc ovcr t,hn 1:~yrr
carried by the (lislr is of tho orrior e r1 a), i. c., vcry small for s~unll vi~co~it~ics. TIIC
prcssnm (Iist,ril~ntion tlcpcntls only on tho tlist,nrlc:o from th(1 wall, rinrl (.II(w is r~o
riuli:r,l ~~rcss~trc grntlicwt,.
A generalisctl fnrnl of tl~e prccetling problem has becn stutlictl 11y M. G, Itogers
and G. N. Lance [28] who assumed that the hid moves with an nnnllIar Vl,IO(*i(,V
antl thc sccond boilntlnry contlition for tho function G(() mttat, Ito rrplncwl t)y
C(m) = s . In this conncxion a comparison should hc mndo with thr caso ofrotating
flow ovrr n fixrd disk given in Scc. XTn. Nnmcrical ~olutions for rotatio~~ ilk tl\c
s:nnc srnsc (s > 0) can be found in [20]. Whcn the rotations arc in opposite scnsps
(s < 0). physically meaningful solutions can bc obtained for s < - 0 2 only iTunifc)rrn
suction :it right, :i~~gIrs to the dislr is n(lniittc(1.
The prol~lem of a rotating dislr in a housing is discussed in Chap. XXl.
It, is particularly tiotcwor(hy that the solutior~ for tlrc rotating disk as wcll as
1.llc solutions obtained for the flow with stagnation are, in the first place, exact
solutions of the Nnvicr-Stokes cquntions anti, in the sccond, that thcy are of a
houi~drcry-la?/rr
(me of vcry small viscosity t,hese solntions show that tho influence of viscosit.y
rxl.rntls over a vcry small lnycr in tllc ~~cighl~ot~rl~ootl of. the solid wnll, ~ 1 1 c ~ t . c : : ~ ~
ill 1,llc wl~olo of 1.l1c rcmnining region t.hc flow is, j)rnct,ic;llly spcalring, i(lrnt.i(~:ll
\vit,l~ (.he corrcspontling itlcal (potcnti:~l) cnsc. '~hcsc cxamplcs show Surthor l.l~;~t
the b~~nnilary hyer has a thickness of the order iv . The one-dimensional examples
of flow discussed previonsly display tho snmc bonntlnry-layer character. In this
conr~cx-ior~ the rcatlcr may wish to conwit a pnpcr by G. I