KUNDU Fluid Mechanics 2 Episode 12 ppsx

Similarly thc y-momentum, z-momentum, and continuity equations for the pcrturbations are (12.70) all a, all) - + - + - = 0. 3.r ay ilc Thc coellicients in the perturbation equations (1 2.69) and ( 12.70) depend only on yI so that thc cquations admit solutions exponential in x, z, and t. Accordingly, we assumc normal modes of the form iu, = [qy), ei(kx-m: keri (12.71) As the flow is unbounded in x and z, thc wavenumber components k and nn must be real. The wave spccd c = e,. + ici may bc complex. Without loss of generality, we can considcr only positive values for k and nt; the sense of propagation is then left open by kccping he sign of cr unspecified. The normal modes represent waves that travel obliqudy io the basic flow with a wavenumber of magnitude dm and have an aniplitudc that varies in timc as cxp(kcit). Solutions are thercforc stable if ci e 0 and unstable if ci > 0. On substitution of thc normal modes, the perturbation equations ( 12.69) and (1 2.70) become 1 Re ik(U - c)il + CU, = -ikj + -[[t,, - (k2 + ni2)iJ. 1 Re ik(U - c)G = -intb + -[Tc,, - (k' + nt2)1iI, ikil + il, + iniG = 0. (1 2.72) where subscripts denote derivatives with respect to y. These are the normal mode cquations for thrcc-dimensional disturbances. Bcforc proceeding further, wc shall first show thal only two-dimensional disturbances need to be considcred. Squire's Theorem A very useful simplification of he nonnal modc equations was achicved by Squire in 1933, showing that ta cucli irrisrable thme-dimerisirmd disturbance there corresponds u imm rmsruhlr nvn-dirnmsi~,nnl one. To provc this theorem, consider the Squire trarisforniutioii PP L-k' (12.73) Tn subslituting these transformations into Eq. (12.72): the iirst and third of Eq. (12.72) are added; the rest are simply transformed. The result is iki + i,, = 0. These equations are exactly the sanc as Eq. (12.72), but with nz = 5 = 0. Thus, to each three-dimeiisional probleni corresponds an cquivalent two-dimensional one. Moreover, Squire‘s translormation (1 2.73) shows that the equivalent two-dimensional problem is associated with a lower Reynolds number as > k. I1 follows hat the critical Reynolds number at which he instability starts is lower for two-dimensional disturbances. Therefore, we only need to coiisidcr a two-dimensional disturbance if we want to determine the minimum Reynolds number for the onset or instability. The three-dimensional disturbance (1 2.71) is a wave propagating obliquely to the basic flow. If we orient he coordinate system with the new x-axis in this direction, the cquations of motion are such that only the component of basic flow in this direction affects the disturbance. Thus, the effective Reynolds number is reduced. An argument without using the Reynolds numbcr is now given because Squirc’s theorem also holds for scveial other problems that do not involve hc Reynolds numbcr. Equation (1 2.73) shows that the growth rate for a two-dimensional disturbance is cxp(kcit), whereas Eq. (12.71) shows that thc growth rate of a three-dimensional disturbance is exp(kcir). The two-dimensional growth rate is therefore larger because Squire’s transformation requires k > k and C = c. We can thercfore say that thc two-dimensional disturbances are more unstablc. OrrSommerfeld Equation Because of Squire’s theorem, we oiily need to consider the set (12.72) with nz = 8 = 0. The two-dimensionality allows the definition of a streamfunction @(x. y, r) for the perturbation field by w fiY il -r u=- , v= We assume normal modes of the fomi (To be consistent, we should dcnote the complex amplitude of II. by 4; wc are using 4 instead to follow the standard notation for this variable in the literature.) Then we must have A single equation in tcrms of 4 can now be found by eliminating the pressure from thc sei (12.72). This givcs wherc subscripts denote derivatives with respect lo y. It is a fourth-order ordinary diffwenlial equation. The boundary conditions at the walls are the no-slip conditions 11 = u = 0, which rcquirc 4 = 4,. = 0 at y = yl and y?. (1 2.75) Equation ( 12.74) is the well-known On Somrnerfeld equation, which govms the stability of nearly parallcl viscous flows such as those in a straight channcl or in a boundary laycr. Tt is essentially a vorticity equatioii bccausc the pressure has been eliminated. Solutions of the OrrSommerkeld equations arc difficult to obtain, and only the results of somc simple flows will be discussed in the latcr sections. However, we shall first discuss ccrtain rcsirlts obtained by ignoring thc viscous Leri in this eq ualion . 9. Tnuisrid Slabili~?- o$l-+u-allel Floius Usetill insights into thc viscous stability of parallel flows can be obtained by first assuming that thc disturbances obey inviscid dynamics. The governjng equation can be found by letting Rc + 30 in the Orr-Sommcrfcld equation, giving (V - C)[f&! - It2#] - U., ,.#= 0, (12.76) which is called the KuyleigIi equriori. If the flow is boundcd by walls at yl and yz where I! = 0, then the boundary conditions are 4 = 0 at y = y1 and y:. (1 2.77) The set [ 12.76) and (1 2.77) defines an eigenvalueproblem, with c(k) as the eigcnvalue and 4 as thc cigcnfunction. As the equations do not involve i, taking the complex conjugate shows that if 4 is an eigenfunction with eigenvalue c for some k, then @* is also an cigenfunction with eigenvalue c* for the same k. Therefore, to each eigenvalue with a positive ci thcrc is a corresponding eigenvalue with a negative ci. Ti1 other words, to euch ginwing triode there is a corresponding decciying made. Stable solutions thcrefore can have only a real e. Note that this is true of inviscid flows only. The viscous tcrm in the fiill On4ommerfeld equation (1 2.74) involves an i, and thc forcgoing conclusion is no longer valid. We sliall now show that certain velocity distributions V(y) art: potentially uiista- blc according to the inviscid Rayleigh equation (12.76). In this discussion it should be notcd thdl we are only assuming that the diufurhances obey iiiviscid dynamics: the hackgrouiid llow V(J) may hc chosen lo be choscn to be any profilc, for example, that of viscous flows such as Poiseuille flow or Rlasius flow. Rayleigh’s Inflection Point Criterion Rayleigh provcd that a necessary (but not suficieiit) criterion for instability of an inviscid paralleljow is that the basic velocity pinjile U (y) has a point of injection. To prove the theorem, rewrite the Rayleigh equation (12.76) in the form and consider the unstable mode lor which c; > 0: and therefore U - c # 0. Multiply this equation by 4*, integrate from yl to yz, by parts if necessary, and apply the boundary condition 4 = 0 at the boundaries. The first term transforms as follows: where the limits on the integrals have not been explicitly written. The Rayleigh equa- tion then gives (1 2.78) Thc first term is real. The imaginary part of the second term can be found by multi- plying the numerator and denominator by (U - c*). The imaginary part of Eq. (12.78) then gives (12.79) For the unstable case, for which ci # 0, Eq. (12.79) can be satisfied only if U,, changes sign at least once in the open interval y~ y e y2. In other words, for instability the background velocity distribution must have at lcast one point of inflection (where U,, = 0) within the flow. Clearly, the existence of a point of inflection does not &&antee a nonzero ci. The inflection point is therefore a nccessary but not sufficient condition for iiiviscid instability. Fjortoft’s Theorem Some seventy years after Rayleigh’s discovery, the Swedish meteorologist Fjortoft in 1950 discovcd a stronger necessary condition for the instability of inviscid parallel flows. He showed that u necessary condition for instability qf inviscid parallelfiws is that U,,,(V - VI) < 0 samewhere in tltejow, where VI is the value of U at the point of inflection. To prove the theorem, take the real part of Eq. ( 12.78): (1 2.80) Suppose that the flow is unstable, so that ci # 0, and a point of inflection does exist according to the Rayleigh criterion. Then it follows from Eq. (12.79) that (12.81) Adding Eqs. (1 2.80) and (1 2.8 I), we obtain so that UJU - UJ) niirst be negative somewhere in thc flow. Some corninon vclocity profiles are shown in Figure 12.21. Only the two flows shown in the bottom row can possibly be unstable, for only they satisfy Fjortofi's thcorcm. Flows (a), (b), and (c) do not have any inflection point: flow (d) does satisfy Rdylcigh's condition but not Fjortoft's bccause U!,.(U - UI) is positive. Note that .::, :: .>. .:: ::- :.::s;:.:-:. (e) 0 Figure 12.21 Fjorltjft's critcrion of' inviscid instahilily. Fiamplcx of panllel flows. Poinls of inflection arc dcnokd by 1. Only (c) and (f) satisfy an alternate way of stating Fjortoft‘s theorem is that the magnitude of vorticit)l aftlze basic.flow must have a nurxinium within the region ufjiow, not at the boundary. In flow (d), the maximum magnitude of vorticity occurs at the walls. The criteria of Rayleigh and Fjortoft essentially point to the importance of having a point of inflection in the velocity profile. They show that flows in jets, wakes, shear layers, and boundary layers with adverse pressure gradients, all of which have a point of inflection and satisfy Fjortoii’s theorem, arc potentially imstable. On the other hand, plane Couette flow, Poiseuille flow, and a boundary layer flow with zero or favorable prcssure gradient have no point of inflection in the velocity profile, and are stable in the inviscid limit. However, ncither of the 1wo conditions is sufficient for instability. An example is the sinusoidal profile U = sin y, with boundaries at y = fh. It has been shown that the flow is stable if the width is restrictcd to 2b < n, although it has an inflection point at y = 0. Critical Layers Tnviscid parallel flows satisfy Howard’s semicircle theorem, which was proved in Section 7 for the more general case oi a stratified shear flow. The theorem states that the phase speed c, has a value that lies betwcen the minimum and thc maximum values of U(y) in the flow field. Now growing and decaying modes are Characterized by a nonzcro ci, whereas ncutral modes can have only a real c = e,. It lbllows that neutral modcs must have U = c somewhere in thc flow field. The neighborhood y around yc at which U = c = e, is called a criticd layer. The point yc is a critical point of the inviscid governing equation (12.76), because thc highest derivative drops out a1 lhis value of y. The solution of the eigcnfunction is discontinuous across this layer. Thc full OrrSommerfeld equation (12.74) has no such critical layer because the highcst-order derivative does not drop out when U = c. It is apparent that in a real flow a viscous boundary layer must form at the location whcm U = c, and the layer becomes thinner as Re -+ cc. The streamline pattern in the neighborhood of thc critical layer where U = c was given by Kclvin in 1888; our discussion here is adaptd froinDrazin and Reid (1981). Consider a flow viewed by an observer inoving with tlie phase velocity c = c,. Then thc basic velocity field seen by this observer is (U - c), so that the streamfunction duc to the basic flow is Q = (U -c)dy. s The total streamfunction is obtained by adding the perturbation: 6 = /(U - c) dy + A#(y) eikx, (1 2.82) whcir: A is an arbitrary constant, and we lmve omitted the time factor on the second term because we are considering only neutral disturbances. Near the critical layer y = yc, a Taylor series expansion shows that Eq. (1 2.82) is approximately 4 = $UYc(y - Y,)~ + A@(y,) COS~X, Figure 12.22 The Kelvin cill's cyc pallcrn nctlr tl critical layer. showing slrcamliiics as sccn by an ohscnw moving with thc wtlvc. where UVc is the value of U, at yc; wc have taken the real part of the right-hand sidc, and thn @(yc) to be real: Thc streamline pattern corresponding to the preceding equation is sketched in Figure 1.2.22, showing the so-called KeAin car's qe pattern. IO. Some lt~sulh of lbrwlld Piscoirx F10u:s Our intuitive expectation is that viscous clTects are stabilizing. The thcrnial and cen- trifugal convections discussed carlicr in this chapter have confirmed this intuitive cxpeclaiion. However, the conclusion that the effect of viscosity is srdbilizing is no1 always me. Consider the Poiscuille Bow and the Blasius boundary layer profles in Figure 12.21, which do not have any inflection point and arc thcrerore inviscidly stable. These flows are known to undergo transition to turbulcncc at some Reynolds numbcr. which suggests that inclusion of viscous efiects may in kt be desrubiliz- hg in these flows. Fluid viscosity may thus have a dual effect in the sense that it can be stabilizing as wcll as destabilizing. This is indeed true as shown by srdbility calculations of parallcl viscous flows. The analytical solution of the OrrSommerleld equation is notorioiisly coin- plicated and will not be presented here. Thc viscous term in (12.74) contains the highest-order derivative, and therefore the eigcnrunction may contain regions of rapid variation in which thc viscous effects becomc important. Sophisticated asymptotic tcchniques are therefore nwded to treat these boundary layers. Alteinativcl y, solu- tions can be obtained numerically. For our purposes, we shall discuss only ccrlain Featurcs of these calculations. Additional information can be found in Drazin and Reid (1981), and in the revicw arlicle by Bayly, Orszag, and Herbert (1 988). Mixing Layer Consider a mixing layer with the vclocity profile L Y u = u"otanh A shbility diagrain for solution of the OrrSommcrfcld equation for this velocity distribution is skctched in Figurc 12.23. 1.t is seen that at all Reynolds numbers the flow is unstable to waves having low wavenumbcrs in the rangc 0 c k c k,,, wherc 1.0 t STABLE (Ci < 0) kL UNSTABLE (ci > 0) I I 0 40 Re=- UQL V Figure 12.23 Marginal stability curvc for ;1 shear layer u = Vu tanh(y/f.). the upper limit k,, depends on the Reynolds number Re = U"L/u. For high values of Re, the rangc of unstable wavenuinbers incrcases to 0 < k c 1/L, which corrcsponds to a wavelength range of 00 > A > 25r L. 11 is therefore essentially a long wavelcngth instability. Figure 12.23 implies that the critical Reynolds nuinbcr in a mixing layer is zcro. In fact, viscous calculations for all flows with "inncctional profiles" show a small critical Reynolds number; for example, for a jct of the form zi = Usech'(y/L), it is Re,, = 4. These wall-he shear flows therefore become unstable very quickly, and the inviscid criterion hat these flows are always unstable is a fairly good description. The reason the inviscid analysis works well in describing the stability characteristics of free shcar flows can be cxplained as follows. For flows with inflection points the eigenfunction of the inviscid solulion is smooth. On this zero-order approximation, the viscous term acts as a regular pci-turbation, and the resulting corrcction to thc eigenfunction and eigenvalues can be computed as a perturbation expansion in powcw of the sinall parameter 1 /Rc. This is t~uc even though the viscous term in the On-Sommerfcld equation contains the highest-order dcrivative. The instability in flows with iiiflcction points is observcd to form rolled-up blobs or vorticity, much like in Lhc calculations of Figurc 12.18 or in the photograph of Fipc 12.16. This behavior is robust and insensitive to Ihc detailed experimental conditions. They are therefore easily observed. In contrast, the unstable waves in a wall-hounded shear flow are extrcmely dimcult to obsei-ve, as discussed in the next section. Plane Poisde Flow The flow in a channel with parabolic velocity distribution has no point of in flection and is inviscidly stable. Howcver, linear viscous calculations show that the flow becomes unstable at a critical Rcynolds number of 5780. Nonlinear calculations, which con- sidcr the distortion of the basic profile by the finite amplitude of the perturbations, IO. Sotne !iexul&i cr/lhmlid &ct~ux I.’iouw givc a critical number of 25 IO, which apes better with the obscrvcd transition. In any case, the keresting point is that viscosity is destabilizing for this flow. The solution ol the Orr-Sommcifcld cqualion for the Poiseuillc Row and other parallel flows with rigid boundaries, which do not have an inflcction point, is complicated. In conmst to flows wilh inflection points, thc viscosity here acts as a singulur pcrturbation, and thc cigcnrunction has viscous boundary layers on the channel walls and around crib ical layers where U = cr. Thc waves that cause instability in thcsc flows are called Tollmien~cl~lichtjn~ waves, and their experimental dctcction is discussed in the next section. 477 Plane Couette Flow This is thc flow confined between two parallcl plates; it is driven by the motion of onc of the plates parallel to itsclf. The basic velocity profile is lincar, with U = ry. Contrary to the expcrimcntally observed fact that thc flow does become turbulent at high values of Rc, all linear analyses havc shown that the flow is stable to small disturbanccs. 11 is now believed that thc instability is caused by disturbanccs of finite inagnitudc . Pipe Flow The absence of an inflection point in the velocity profile signifies that the flow is inviscidly stable. All linear stability calculations of the viscous pn)blem have also shown rhal the flow is stablc lo small disturbances. In contrast, most experiments show that the transition to turbulence takes placc at a Reynolds number of about Rc = U,,,,, d/u - 3000. However, careful cxpcriments, some of them pcrformed by Rcynolds in his classic investigation of the onsct or turbulence, have been able to maintain laminar flow until Rc = 50,000. Beyond this thc observed flow is invariably turbulent. The observcd transition has been attributed to one of the following cfkcts: {I> It could bc a finite amplitude effcct; (2) he turbulence may be initiated at the entrance of thc tube by boundary laycr instability (Figurc 9.2); and (3) the instability could be causcd by a slow rotation of rhc inlet flow which, whcn added to the Poiseuillc distribution, has been shown to result in instability. This is still under investigadon. Boundary Layers with Pressure Gradients Rccall from Chaptcr 10, Section 7 that a pressure falling in the direction of flow is said to have a “favorable” pdicnt, and a pressure rising in the direction of flow is said to have an “adverse” gradicnt. It was shown there that boundary layers with an adverse pressure gradient havc a point of inflection in the velocity profile. This has a dramatic :ffect on the stabilily characteristics. A schematic plot of the marginal stability curve Tor a boundary layer with favorable and adversc gradients of prcssure is shown in Figure 12.24. The ordinate in the plot represents the longitudinal wavenumber, and thc abscissa reprcscnts the Reynolds number based on the free-strcam velocity and the displacement thickness S* of the boundary laycr. The marginal stability curvc divides stablc and unstablc rcgions, with thc region within thc “loop” reprcsenting instability. Because the boundary layer thickness grows along he direction of flow, 478 Inxlubility ks* STABLE adverse pressure gradient I I Re, Re, Re, = UG*h Figure 12.24 Skctch of marginal stability curvcs ha boundary hycr with favoniblc and advcrsc pressure gdicots. Rea increases with x, and points at various downstmam distances are reprcsented by larger values of Res. The following features can be noted in the figure. The flow is stablc for low Reynolds numbers, although it is unstable at higher Reynolds numbers. Thc cffect of inmasing viscosity is therefore stabilizing in this range. For boundary laycrs with a zero pressure gradient (Blasius flow) or a horable pressure gradient, the instability loop shrinks to zero as Rea + 30. This is consistent with the fact that these flows do not have a point of inflection in the velocity profilc and are thcrefore inviscidly stable. In contnst, for boundary layers with an adverse pressurc gradient, the instability loop does not. shrink to zero; the uppcr branch of the marginal stability curve now becomcs flat with a limiting value of k, as Rea + 00. The flow is then unstable to disturbanccs of wavelengths in thc range 0 < k k,. This is consistent with hc existence of a point of inflcction in thc velocity profile, and the results of the mixing layer calculation (Figure 12.23). Note also that the critical Reynolds number is lower for flows with adverse pressure gradients. Table 12.1 summarizes thc results of the linear stability analyses of some common parallel viscous flows. The first two flows in the table have points of inflection in the vclocity profile and are inviscidly UnStdblC; the viscous solution shows cither a zero or a small critical Reynolds number. The remaining flows are stable in the inviscid limit. Of thcse, the Blasius boundary layer and the planc Poiseuille flow are unstablc in the prcsence of viscosity, but have high critical Reynolds numbers. How can Vicosity Destabilize a Flow? Let us examine how viscous cffects can be destabilizhg. For this we derive an integral form of the kinetic encrgy equation in a viscous flow. The NavierStokes equation [...]... K,, = 5.365 2 Consikr the centrifugal instability problem of Section 5 Making the narrow-gap approximation, work out the algebra of going from Eiq ( 12. 37) to Eq.(1 2. 38) 3 Consider the centrifugal instability problem of Scclion 5 From Eqs (1 2. 38) and ( 12. 40), the cigcnvalue problem for determining the marginal stale (a = 0) is (D2 - k2)’i, = (1 +ax)&, ( D2 - k2)2he = -Fd k%,: (1 2. 87) (1 2. 88) with... substitution of Eq ( 12. 85) into the equalions of motion, Lorenz finally obtained x = Pr(Y - X), Y= -xz + r X - Y? ( 12. 86) Z = XY - bZ, + and where Pr is the Prandtl number, r = Ra/Wr, h = 4n2/(7r2 k2) Equations ( 12. 86) rcpresent a set of nonlinear equations with t h e degrces 0 fkcedom, which 1 means that the phase space is thrce-dimensional Equations ( 12. 86) allow lhe steady solution X = Y = 2 = 0, repmenting... f Mmn Flow 5 12 7 Kine& l 'nerg.Hun'@ o f lwhlenl Flow 5 14 8 Thilence 1 3 r h d o n and Cilscade 517 9 SInxhm o lidmlcnce iri Itwrhd f Subrartgy 520 10 ~ L L - ~Sh(xW HOW ~ w 522 Intrmrittmcy 522 Entiairmerit 524 Sdf-Thjmruti(m 524 (hruiqucnco or Self-t'mscrwition in a Phuie Jet 525 Turbulent Kirwtic Knergy Budpi iri uJet 526 I1 Mill-Bauruhi Shmr F h j 528 -I hmr 1.a p... sinrnrx ( 12. 89) ftrl Inserting this i Eq ( 1 2. 87), obtain an equation Cor h,., and arrange so that the solution n satisfies the four remaining conditions on i, With f i r dctermined in this manner and ho given by Eq.( 12. 89), Eq.( 12. 88) leads to an eigenvalue problem lor Ta(k) Following Chandrasekhar (1961, p 300) show that the minimum Taylor number is givenbyEq.( 12. 41)andisreachedatkC, =3. 12 4 Consider... equation in the form s w 2 + g 2 p 2 / p ~ N 2 ) d= - uwU,dV V 2 dt (As in Figurc 12. 25, the integration in x takes place over an intcgral number or wavelengths.) Discuss Lhc physical meaning of each term and Lhc mechanism of instability ‘d/(u’+ l l i ~ e r a ~ w Citt!d u! Bayly, H.J S A Orszag, andT Hcrbcrl(1988) “Instability mechanisms in shear-flow transition.”Annunl Review ofFluid Mechanics tu: 3.59-391... Pliy,sics 1Y 25 - 52 Cirahcwski, W.J (1 980).”Nonpamllcl stability analysis of axisymrnclricslagnation point f o Physicr lw” oj’F!uids U: 1 4 1 960 95Howard, 1 N (1 %I) “Kotc on a papcr of John R?Miles.” lournu/ oJFluid iLfechunics1 3 158-160 Huppcrt, H E and J S Turncr (1981).”Double-difrusivcconvcclion.” Journul ( $Fluid Mechanics 106 2! -2 9?39 Klehanotl: P.S., K I) Tidstmm, and I H.Sargcnt (19 62) .T h... layer.” Journd < #Fluid Mechunics 5 2 499- 528 2 Stcm, M E (1960) ‘The salt fountain and thcrmohaline convection:' 7ellus 1 : 1 72- 175 Stornmcl, H., A B Arons, and D Blanchad (1956) “An oceanographic curiosiry: Thc pcrpclual salt fountain.” Deep-Sea Research 3 1 52- 1 53 Thorpe S A (1971) "Experiments on Ilic instability of skitified shear flows:Miscible fluids.” Jmrnnl oJ F!rrid Mrihnics 4 6 29 9-31 9 Turncr,... thosc of parallel flow 4 82 1rulabilikj- 4x 104 (OV z 2x10‘ :\\ Schlichting I I I I I I I I \ I I 520 0 loa0 2ooo Re, = U,~*/V Figurc 12. 26 Marginal stability curvc for a UIasius boundary layer Thcorclical solulions of Shen w d Schlichting m compared with cxperimenlal data of Schubauer and Shimstad stability theory obtained from solutions of the OrrSommcrfeld equation Reshotko (20 01) provides a rcview... number is givenbyEq.( 12. 41)andisreachedatkC, =3. 12 4 Consider an infinitely deep fluid of density PI lying over an infinikly deep fluid of dcnsity pz > pi By selling U = U = 0, Eq.( 12. 5 1) shows that 1 z ( 12. 90) 494 lndabiii~y Arguc that if the whole system is given an upward vertical acceleration a, thcn g in Fq (I 2. 90) is replaccd by g’ = g a It follows that there is instability if g’ < 0, that... real flows (see Figures 7 and 8 in Bayly et nl., 1988) 13 cm ! ribbon P spacer L 2 0 X Figre 12. 27 Tbdimenuional unstablc waves initiated hy vibrating ribbon Measurcd distributions of intensity of the u-Huctuaticin ut two dislunccs from thc rihhon arc shown P S KlehtmolTer el., Journal o f Fluid Mechnnicr 1 2 1-34, 19 62 and reprintcd w t thc permission of Cambridge Univcrsity Press ih 14 Lktwniiriktic . Squire trarisforniutioii PP L-k' ( 12. 73) Tn subslituting these transformations into Eq. ( 12. 72) : the iirst and third of Eq. ( 12. 72) are added; the rest are simply transformed 1.7(Y/S) I Y/S 2 1, 4 82 1rulabilikj- 4x 104 (OV z 2x10‘ 0 .I I I I :\ Schlichting I I . I I I I I I 520 loa0 2ooo Re, = U,~*/V Figurc 12. 26 Marginal stability. ( 12. 81) Adding Eqs. (1 2. 80) and (1 2. 8 I), we obtain so that UJU - UJ) niirst be negative somewhere in thc flow. Some corninon vclocity profiles are shown in Figure 12. 21.