Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
458 XVI. Origin of turbulence I b. Prinriplr~ of the throry of stnhility of l~i~nir~nr flows 459<br />
to thc approprial.~ rlilTrrrntinl cqr~ntions. 'rliis is tlic mrlhod of small dislur6nnces.<br />
This scroritl nirtliotl has led to complctr success and will, for this rcason, be described<br />
with somr tlrtail.<br />
Wc s1i:ill now considor a two-tlimonsional incornprcssiblc mean flow and an<br />
cq~~aliy hwo-tlimcnsiond distnrl)anco. 'L'hc rcsulting motion, clcscribcd by eqns.<br />
(16.2) a.ntl (16.3). sat.isfios tlio two-tlirucnsion:~l form of thc Navicr-Stokes equations<br />
a,. givcn in rqris. (4.4a, b, c). Wc shall further simplify the problem by stipulating<br />
that tlrc mean vclocit,y TJ dcpcntls only on y, i. c., IJ = U(y), whcreas the remaining<br />
two componct~i~ n.rc supposed to be xoro cvcrywhorc, or V W z Of. Wc havc<br />
cr~c:ount~crctl s~~oli flows cnrlicr, tlcscril~ing thcrn :IS pwdlel &OWR. In thc casc of :L<br />
chnnncl with parallel walls or a pipe, such a flow is reproduced with great accuracy<br />
at a sufficient dishncc from tho inlet section. The flow in the bountlary layer can<br />
also he reg.anlcd ns k good approximation to parallel flow because the dependence<br />
of tlic velocity U in the main flow on the x-coordinate is very much ~matlcr than<br />
that on y. As far as thc prcssurc in the main flow is conccrncd, it is obviously nccessnry<br />
to assume a dcpcntlcncc on x as well as on y, i. e., P(x,y), becausc the prcssurc gradient<br />
i)P/ax maintains the Ilow. Thus we assume a mean flow with<br />
Upon the mcan flow wc nssnmc snperimposcd a two-tlirnensional disturbance which<br />
is a function of time and space. Its velocity componcnts and pressure arc, rcspec-<br />
tivcly,<br />
?~'(x,y,t) , vl(x,y,t) , pl(%yJ) . (16.6)<br />
IIenco tlio rcsnltant motion, according fn eqns. (16.2) and (16.3), is described by<br />
It is assumctl that t,lic nmLn flow, cqn. (16.4), is a solution of thc Navicr-Stokes<br />
equations, and it is required that thc resultnnt motion, eqn. (16.6), must also satisfy<br />
tho Navicr-Stokes equations. The supcrimposerl fluctuating vclocities from eqn.<br />
(16.5) arc takwi t,o 11e "small" in thc scnsc that all quadratic tcrrns in the fluctuating<br />
cornponcnts may be ncglccted with respcct to thc lincar terms. The succcctling<br />
section will cont,ain a morc dct,ailctl tlcscription of the form of the clistarbance.<br />
Now, tho task of tho stability t,hcory consists in clcbrmining whether the tlisturhancc<br />
is amplified or whotlicr it clccays for a givcn mean motion; the flow is cynsitlrrctl<br />
nnst.:~l~lc or stahlo tlcpcnding on wlicther the former or the latter is the case.<br />
Substituting cqns. (1 6 6) into tlic Navicr-Stokcs equations for a two-dimensional,<br />
incomprcssiblc, non-steady flow, cqns. (4.4a, b, c), and nrglcrting quadratic terms<br />
in the tlisturbance velocity components, wc obhin<br />
'J'hcw arc rramns In nopponr, a4 sl~n~vri hy (:. 1%. Scli~rba~~~r nntl P. S. IClcbsnofT [831, that<br />
t,l~mc t:or~~pot~c~rl~ nrt: ~I\V:LYR prtwmIr in rcnl IIOWR, particularly in flow^ pmt flat plnte~. I'hcir<br />
r~~ngnit.r~tlo is rwgligil~lc for tno~L lwrl)oscs, l ~ thcy t mern Lo play n part, not yct fully clucidatecl.<br />
ill thc proocas of trnn~ition; RCO nl~n foo1.1iotc on p. 468.<br />
wherc V2 dcnotes thc Laplacinn opcrator a2/i)22 + a2/&/2.<br />
If it is considcrctl that the mrnn flow it,sclf sntisfics the Navicr-Stokcs qua-<br />
tions, the above equations can bc simplified to<br />
Wc havc ol~tainotl throe cqn:~i,ions for IL', 11' and pl. 'l'11c I)o~~ntl~~ry contlitio~~s spwify<br />
that the turbulent velocity components IL' aid v' vanish on thc wtills (no-sli11 coridition).<br />
The pressure p' can be easily climinatcd from the two equations, (10.7) ant1<br />
(1F.8), so that together with thc cqi~nt~ion of continuil.y IJicro arc t\vo cqn:~i,ions for<br />
u' and v'. It is possilh to criticize the a.ssnrnrtl form of tl~c: rnctLn Ilow, ccp. (I(i..l), or1<br />
tlic ground that the variation of thc coniporicnt lI of t.hc vclocity with x as wdl :I,S<br />
the normal component V havc hccn ncglccbtl. Jn this conncxion, howcvcr, .l. I'rctsch<br />
[44] proved that the rcsult.ing terms in the eqnations arc unimportant for the<br />
stability of a boundary layer (see also S. J. Cheng 171).<br />
3. The Orr-Sommerfeld equation. The mean laminar flow in the 2-direction<br />
with a velocity U (y) is assnmcd to be influenced by a disturbance which is composetl<br />
of a number of discretc psrtlial fluctuations, cnch of which is said to consist of n wave<br />
which is propagated in tho x-direction. As it, has already bccn assrirnctl i,li:tt tlic<br />
perturbation is two-dimensional, it is possible to introduce a stream function yi(z, y,t)<br />
thus integrating the equation of contin11it.y (10.9). Thc stream function reprcscnting<br />
a single oscillation of the distnrbance is assumcd to bc of the form<br />
Any arbitrary two-dimensional disturhancc is assumctl cxpantlcd in a I'ouricr<br />
scries; cach of its terms represent8 such a partial osc:illation. In cqn. (16.10) a is n real<br />
quantity and A = 2 x/a is the wavelcngth of t.lrc clis1urb;uicc. The q~iant,ity is<br />
complex,<br />
P = P, I- i PI ,<br />
where p, is the circular frequency of the pnrt.inl oscillation, wlicrcas P, (amplification<br />
factor) determines tho clcgrec of amplificnt.ion or damping. The t1isturl)anc.n~ arc<br />
tlanipocl if P, < 0 ant1 tslic laminar mcan flow is st,al)lc, wlicmas for PI :. 0 i~~st,:~l~ilii.y<br />
~cts in. Apart from a a.nd it is convcnicnl to introtlucc tlic4r ratio<br />
t Tho convenient coniplc?r noLrrt.ion in 11ort1 Iwr. l'hysivnl mr~u~ing is nt.trwl~t.tl only to I.hc rral<br />
part or the ~tmani unction, tlwn
Change language