Boundary Lyer Theory

242 XI. Axially nymmct.riral and tl~rrr-tli~r~rt~siot~al 1)onn~l:~ry Inyrra b. Approximate aolutiona for axinlly apmmctric boundary lnyer~ 243
Fig. 11.8. Velocity distribr~bion in thc inlct port.ion of n pipe for the lnn~inar ca.se; mennurements
perfornicrl by Nikuradac and quotrd from Pranc1t.l-Tiet.jen vol. TI. Theory dl10 to
Scliillrr (901
ltns prarf irally tlrcayctl al, :I clistancc of 40 pipe radii whcn thc Rcy~rol& nuntber
has a value of R = 10:' This is in good agreement with experimental results.
3. Rour~clnry layrra on rotating bodies of revolution. The simplest cxarnplc of
e bounclary laycr on a rotaling hotly is tIi:~t considerod in See. Vb 11, namely the
problem of a disk rotating in a fluid at, rest,. The fluid prticlcs which rotate with the
boundmy laycr arc thrown outwards owing to the existence of ccntrifugal forces
('centrifuging') anti are rcplaccd by part,icles flowing towards the boundary layer
in an axial direction. Tlic casc of a disk of mtlius I< rot.nting with an angular vclocity
o in an axi:~l sl.rc:~m of velocity U, :~lTords a simplc cxtcnsion of the previous
problem. In thc lat,t.cr case the flow is govcrnctl by two parameters: thc Rcynoltls
number and the rot,af ion pammetcr, U,/Rw, which is given by the ratio of frecst.rcam
to tip vclocity. An cxact solution to the problem under consiclcration was
given 11y Mi* D. M. Ilannah [46]t and A. N. Tiffortl 1.1 131 for tho case of laminar
flow; IT. Sal~licl~ting and R. Truckcnbrotlt [98] providcd an approximate solution.
E. Truclrcnbrotlt 11 191 investigated the case of turbulent flow. .Figure 11.9 cont,ains
a plot of the torqnc coefficient,, C, = ilf/g e (2 R" in terms of the Reynolds
numbcr and rotation parameter, U,/ll(u, obtained from such calculations. Here M
clrnotcs the t,orque on thc leading side of the dislz only. When the disk rotatcs we
may stmill assumc Ifhat separation occurs at the edge of the disk. 'l'he 'stn.gnant,'
fluid Itchintl the clislr part,ly rot,n.tcs will1 thc,clislz and contributrs lit,l,le to the
torcpc. Any such contribution has been lcft nt of account in (7, in Fig. 11.9.
It is seen that Llie torque increases rapidly wi P 11 U, at constant angular velocity.
t Arl.~tnlly rrf. 1381 solvm n rrl:rlr~l pro1,lrrn in wltirh the! cxtcrnal ficltl in t.l~:ct, rluc to '& source
at infinity.
Pig. 11.9.
Morncnt cocllicient on
a rotating disk in axial
flow, aftor Gchlichting
and Truckenbrotlt
[98, 1 191
Cnr - Mlf Q w' R';
bf - torque on lcnrling sidc
or dirk
W4 2 4 6 WS 2 4 6 106 2 4 6 lo7
Reynolds number R =g$
Thc flow in a circular box provided with a rotating lid shows a markcd rcscmblance
to that between two rotating dislts mentioned in Scc. V b 11. 7'11~ cnse
of the flow inside the box was investigated in deLail by 1). Grohne [44] who discovered
two peculiar features in it: First, the flow in the friction-free core in tlie
interior of the box can only be determined by taking into account the inllncnrc of
tlie boundary layers which form on the wall, in contrast to normal cascs whcn onc
naturally nssymes that t,hc influence of tho flow in a bountlary layer resu1t.s at, most
in a d.isplaccment. Secondly, the boundary laycrs arc unusual in that they join car11
other. Siniilarly, in the arrangement oonsis1.ing of R rota1,ing channrl irivc?stligat,etl
by IT. 1,udwieg [68], it is possiblc to discern two regions of flow when the spcxd of
robation is sufficic?nt.ly high, ttamcly a fricd.ionlcss corc and bottndnry layrrs which
form on the side walls and which givc risc t.o a secondary flow. 'l'hc t.hcory lcads
to a large increasc of thc drag cocfficicnt which is dnc to rotation, ant1 this fact has
been confirmed by experiment.
Blunt bodies, sncli as o. a. a sphere or a slcrdcr body of revolution, placctl in
axial streams, show a marked influcncc of rotation on dmg, as cvitlrncwl I)y tho
measurements performed by C. Wicsclsbergcr 11231, ant1 S. 1~1t.h:~ndcr and
A. Rydberg [69]. Fig. 11.10 contains a plot of thc drag cocfficicnt of n rotating
sphcre in terms of the Rcynolds numbcr. It is secn that Lhc critical Rcynoltls
number, for which the drag coefficient dcrrcascs abrnpt,ly, depends strongly on tlie
rot,at.ion paramcler U,/Rw, and the same is true of the position of tltc point, of snp:~ration.
The effect, of rotary motion on bl~c posilhn of the linc of 1aniirtn.r sc.pnr:~l.io~~ on
a spltcrr is (l(;s(~il~cd by lhc grc~pli in ltig. I I .I I ; 1,Itc (IILIJL ror it IIILVO INWI ~ X ) I I I ~ ~ I I I ~ I ~ I I
by N. E. lloskin [50]. When the rotatmion para.mctcr 11:~s nl,tdinrtl t,hc vn111c:
Q = w R/[J, = 5, the line of sepnm.t,ion will have moved by about lo0 in 1,hc upst,rean~
direction, as compared with a sphere at rcst. 'l.'hc physicnl ren.son for this
bef~aviour is connected with the centrifugal forces &ding on t,hc fluit1 parLiclcs rolat,ing
wit,]i the body in its bour;tlary layer. Thc crt~trifngal forces have tlic sn,mc rni~t n.s an
atltlit.ionnl pressure gratlirnt dircctd towards t,hc plnne of I,ltc erlun.t,or.