5.3 Inviscid instability mechanism of parallel flows

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1 1 Lecture Notes on Flui Dynamics 1.63J/2.21J) by Chiang C. Mei, Invisci instability mechanism of parallel flows We now turn to an oler problem of the instability of parallel flow without stratification an gravity, such as channel flows, jets, wakes an bounary layers Rayleigh s equation In a shear flow a necessary coniton for instablity is that there must be a point of inflection in the velocity profile U), i.e, 2 U 2 = 5.3.1) somewhere in the flow. Let us begin with an invisci uniform flow. U = U i, P = 5.3.2) an a an infinitesimal isturbance so that the total velocity an pressure fiels are u, w, p) =U + u,w,p ) 5.3.3) where primes enote isturbances which are functions of x,, t). The governing equations are u x + w = 5.3.4) ρu t +U + u )u x + w U + u )= p x 5.3.5) ρw t +U + u )v x + w w = p 5.3.6) The bounray conitions are w =, =, 5.3.7) where can be unboune. Lineariing the momentum equations by omitting terms quaratic in isiturbances, we get ρ u t + Uu x + U ) w = p x 5.3.8) ρw t + Uw x )= p 5.3.9) Eliminating p by cross-ifferntiation, ρ u t + Uu x + U ) w = p x = ρw t + Uw x ) x

2 2 Introucing the stream function by u = ψ,w = ψ x,weget ψ t + Uψ x U ) ψ x = ψ xt + Uψ xx ) x Let us consier a wave like isturbance ψ = f)e ikx Ct), where C = ω k 5.3.1) then ikcf + ikuf ik U ) f = ik k 2 Cf + k 2 Uf ) or U) C) f k 2 f ) f = ) which is known as Rayleigh s equation in hyroynamic instability. The bounfay oncitoins are: f =, =, ) Rayleigh s necessary conition for instability Let us rewrite ) as f k 2 f an multiply by the complex conjugate f f ) k 2 ff which can be rewritten by partial integration U C f = ) U C ff = f f ) f 2 k 2 f 2 U C f 2 = Integrating from =to = an applying the bounary conitions we get simply U C f 2 = f 2 + k 2 f 2) ) Now we let C = C r + ic i to be complex, then on the left-han sie, f 2 U C = U C ) f 2 = U C r + ic i ) f 2 U C 2 U C 2 Equation ) becomes U C r + ic i ) f 2 = f U C k 2 f 2) )

3 3 We get from the imaginary part of ), f 2 U C 2 = ) if C i. Thus for instability it is necessary that is positve for some y an negative elsewhere, i.e., must vanish somewhere in the flow, i.e., the velocity profile must have an inflection point insie the flow. This is Rayleigh s necessary conition for instability Physical explanation Lin, 1945) Figure 5.3.1: a) Velocity profile with an inflection point. b) Vorticity profile. c) Effect of exchanging flui parcels on vorticity In Figure we sketch the velocity profile with a inflection point at =anthe corresponing profile of vorticity ζ = U which has a maximum at =. Note the in the right-hane system the y axis points into the paper, therefore the flow of a positive vortex is clockwise. Consier three layers in the region where U) increases with monotonically, i.e., below the inflection point, with ζ 1 ) <ζ 2 ) <ζ 3 ). If a flui parcel escens from level 3 to level 2, it brings with its vorticity without change, accoring the vorticity transport law in an invisci flui. An excess vorticity is create at level 2 which tens to replace the flui on the right in layer 2 by flui with higher vorticity, an the flui on the left in layer 2 by flui with lower voriticity. The net consequence is to force the original excess vortex to return to layer 3. Similarly if a flui parcel ascens from level 1 to level 2, it brings with its

4 4 vorticity, hence creates an excess vorticity efect with tens to replace the flui on the left in layer 2 by flui with higher vorticity, an the flui on the right in layer 2 by flui with lower vorticity. The net consequence is to force the original efect votex to return to layer 3. Thus an acciental iplacement of a flui parcel tens to it original level; the flow is stable. By a similar reasoning, if U) ecreases monotonically in the flow is also stable. However, if there is a level of vorticity extremum, say level at =, then a flui element arriving at this layer is not force back to its origin. A flui element on one sie of level is equally at home on the opposite sie. The flow is unstable Fjortoft s stronger conition Further information can be obtaine from the real part of ), ) U U C r f 2 = f U C k 2 f 2) ) Now let U I be the velocity at the point of inflection, we use ) so that U I C r ) f 2 U C 2 = ) The ifference of ) an ) is U U U I ) f 2 = f U C k 2 f 2) < ) For this to hol it is necessary that th prouct U U I ) < 5.3.2) for all in the flow. For impoications of Fjortoft s theorem, see Figure Jets an wakes satisfy Rayleigth- Fjortoft criterion, but Poiseuille flows in a pipe an bounary layers on a flat plate o not. Why then is Poiseuille flow known to be unstable beyon Re = 21 Reynols)?

5 Figure 5.3.2: Fjortoft Theorm 5

5.3 Inviscid instability mechanism of parallel flows

5.3 Inviscid instability mechanism of parallel flows 1 Lecture Notes on Flui Dynamics 1.63J/2.21J) by Chiang C. Mei, 27 5.3 Invisci instability mechanism of parallel flows We now turn to an oler problem of the instability of parallel flow without stratification