Boundary Lyer Theory

J'ig. 22.17. 1,ovnl lift. cwefhienb. c,, nt. vnriot~~ rntlinl sections on n rot,llt.ing propeller nccordirlg
tn IIII('ARIIRIIICIIL~ 1wrfor111cc1 by 11. lli~ntnolskntl~p [44]
Tho diagrams in Fig. 22.10 show a comparison bctwccn tlmwy and mrnsurcrnrnt
in a flow past an axially symmetric body; the diagrams plot the Itrynoltls number
formed with thc cncrgy tl~icloicss and the modificrl shape factor.
Tn order to take into account correct,ions due to thrcc-di~~rc~~sio~ralit~y r:111sr(1 by
the possible convergence or divergence of streamlines, J.C. Itotta [8F] proposcs to
base the calculntion on an clTccLive radius R(z). Numcrioal valuos for R(n) nrr snmrnnrizctl
in [86] for all ~ncasr~rcmolts catalog~~rcl
mrrils 1)y Mi.\\'. \Villmart,h ct xl. 1122~1 nntl A.M.O. Stnit,lr [IOh].
in 1541; comI)nrc Iirrt: t,lic III~~:~,sIII.(~-
2. Bn~tt~clnry lnycrs on rntnting hndiea. 'J'l~t: calt:ul:~t.io~~ ol' I:intin:~t. 1~)111itl:i1.v
layers oti rotating bodies placcd in an axiril strcn~n was clisoussotl in Scr. S 1 c.
The mctl~od of calculation which maltcs usc of ntomrntt~n~ int.rgl;ll t~~n;ltions,
formulated for the meridional ant1 t:irc~~mfcrent.ial tlirrc:tions rcsprc~tivc4y, hs been cxtc~~cletl by R. Trucltenbrotlt 11121 to inclutlo t.hc ~urbulrnt, c.:~sr. IT(: wn.s,
moreover, fortunato to succcctl in giving convcnicnL intcgr:ils li~r the rn.lt~~tI;tt ion ol'
the parameters of thc boundary layer. JCxperimcntnl and furtltcr t11cwrct.ic::l.l i~~vrs(.igntions
into the boundary layer on rotnting strcamlinc botlics wcrc c:nrrirtl ot~t I)y
0. Parr 1741. Jn this casc, tho bountlary laycr grows rajjitlly wit.11 t,lro rot.;~l.iotr
paramctcr 2 = (11 RIU,,,; hcrc (1) tlcnotcs tho nng111w velocity, It t,l~t: I:~,t.grst, r;uIius
of thc body, nntl (1, is tho axial rofrrcnc:~ vcloc:il,y. 'I'lrc t,~~rln~lt!r~l. I~ott~i(l:~ry 1:iyt:r
on a rotating body of revolution placctl ill an axial st.re:ltn can IN: (::dt:~tl:~.lt:~l witl~
the aid of thc system of equations (11.45) to (11.48), in whicl~ thc shr:wit~g strcss
must bc ass~tmcd to vary with the rotat.ion pnr:mct,rr. 'I'hc tlin.gr11.m ill I'ig. 22.16
compares Lhc onlculatctl nntl n~cas~trt:d vnlt~cs ol tho rnomcntr~nr I.l~ic:lit~c:ssrs A,, :111t1
a , as rcportcd by 0. l'nrr [74] for :I cylintlrical botly provitlctl wit.11 n. sp11vric.nI Ilosr.
'L'hc ngrccrncnt is good. 'L'lrc rcgion of tmnsition l'ron~ laminar to Lnrl)ulcnt flow
moves forward as thc rotation paramctcr incrcascs; its position coincitlos with the
point at n~hiclr thc momentum tlrickncsscs incrcnsc al)r~~j)(Jy. Scc also Scc. X1112.
A mctliocl for thc calculntion ol tll~rrr-tli~~~c~tisionnl bo11ntl;~ry I:~yc:rs or1 st.;~t.io-
. ~
rtnry botlics as \vdl as on rotating orics, suc:lr as pro[)cllors or 1)l:rtlcs of rol,;lr.y cornpressors
and turbinw, was inclicatcd by A. Magcr [Fl]; comparat,ivc mcasurcmenb
are contained in ref. [621. H. Himmclskamp [44] carried out mcasurcmcnts
in thc boundary laycr on a rotating airscrcw ant1 tlctcrminctl iooal lilL cocffioic~rb
of t,hc blade from mcasurcmcnts of prcssurc tlisLribuLions. Somc of his rosulbs arc?
sccn reproduced in Fig. 22.17; they arc given in the form of plots of lthc local lift
cocfficicnt., c,, at various radial seclions, in term^ of thc nnglc of irlciclcnce, or.
Corresponding mcasurc~nents on n stntionary bladc placcd in a wind tnnnnl are
also shown for comparison. Figure 22.17 shows that ~narlrr?tlly itrt:rcasctI lift coefliri~~~f,~
arc obt.aincd near the hub, and the dcct can LC trnoctl Lo soparation 1)cing clc1:~yccl
to larger angles of incidence. For cxamplc, the scc:t,ion closcst to the hub has I,
maximum lift coefficient of 3.2 compnrcd with 1.4 on thc stat,ionary blatlc. The
tlisplacemcnt of s~pirat~ion towards larger anglcs of incidcncc is cxplai~rctl by the
appearance of an additional acceleration which acts in thc flow direction ant1 which
is crcatcd by Coriolis forces; it has tl~c samo cffcct as n favonrablc prnssurtr gmtliont.
111 addition, but to a lesser extent, the ccntrifugd forces acting in the bound;rry
layer carried with the blade exert a bcncficial influcncc with rcspcot Lo soparntion.
Ii'luict p,zr~icles in the hountlarg layer are actctl upon by a centrifugal forcc which