14. Instability of Superposed Fluids

Transcription

1 4. Instability of Superposed Fluids Figure 4.: Wind over water: A layer of fluid of density ρ + moving with relative velocity V over a layer of fluid of density ρ. Define interface: h(x,y,z) = z η(x,y) = 0 so that h = ( η x, η y,). The unit normal is given by h ( η x, η y,) nˆ = = (4.) h ( ) / η x + η y + Describe the fluid as inviscid and irrotational, as is generally appropriate at high Re. Basic state: η = 0, u = φ, φ = V x for z±. Perturbed state: φ = V x + φ ± in z±, where φ ± is the perturbation field. Solve u = φ ± = 0 (4.) subject to BCs:. φ ± 0 as z ± η t. Kinematic BC: = u n, where ( ) φ ± φ ± φ ± u = V x + φ ± = Vxˆ+ xˆ+ ŷ + ẑ (4.3) x y z from which ( ) η φ ± φ ± φ ± = V + ( η x )+ ( η y )+ (4.4) t x y z Linearize: assumeperturbation fields η, φ ± andtheirderivatives aresmall and thereforecanneglect their products. η φ Thus ηˆ ( η ± x, η y,) and = ± Vη x + t z 3. Normal Stress Balance: p p + = n on z = η. Linearize: p p + = (η xx + η yy ) on z = 0. φ ± η η = V on z = 0 (4.5) z t x 55

2 Chapter 4. Instability of Superposed Fluids We now deduce p ± from time-dependent Bernoulli: φ ρ + ρu + p+ ρgz = f(t) (4.6) t φ where u = ± 4 V V x + H.O.T. Linearize: ( ) φ ± φ ± ρ ± + ρ ± V + p ± + ρ ± gη = G(t) (4.7) t x so φ ± φ V φ φ + p p + = (ρ + ρ )gη +(ρ + ρ )+ (ρ + ρ + ) = (η xx + η yy ) (4.8) t t x x is the linearized normal stress BC. Seek normal mode (wave) solutions of the form where φ ± = 0 requires k = α + β. φ ± η η Apply kinematic BC: = V at z = 0 Normal stress BC: z t x iαx+iβy+ωt η = η 0 e (4.9) kz iαx+iβy+ωt φ ± = φ 0± e e (4.0) kφ 0± = ωη 0 iαvη 0 (4.) k η 0 = g(ρ ρ + )η 0 + ω(ρ + φ 0+ ρ φ 0 )+ iαv (ρ + φ 0+ + ρ φ 0 ) (4.) Substitute for φ 0± from (4.): [ ] [ ] k 3 = ω ρ + (ω iαv)+ρ (ω + iαv) + gk(ρ ρ + )+ iαv ρ + (ω iαv)+ρ (ω + iαv) so ( ρ ρ + ) ω + iαv ω α V + k C 0 = 0 (4.3) ρ + ρ + 4 ( ρ ρ + ) g where C + k. 0 ρ +ρ + k ρ +ρ + Dispersion relation: we now have the relation between ω and k ( ) [ ] / ρ+ ρ ρ ρ + ω = i k V ± (k V ) k C0 (4.4) ρ + ρ + (ρ + ρ + ) where k = (α,β), k = α + β. The system is UNSTABLE if Re (ω) > 0, i.e. if ρ + ρ C (k V ) > k (4.5) ρ + ρ + 0 Squires Theorem: Disturbances with wave vector k = (α,β) parallel to V are most unstable. This is a general property of shear flows. We proceed by considering two important special cases, Rayleigh-Taylor and Kelvin-Helmholtz instability. MIT OCW: Interfacial Phenomena 56 Prof. John W. M. Bush

3 4.. Rayleigh-Taylor Instability Chapter 4. Instability of Superposed Fluids 4. Rayleigh-Taylor Instability We consider an initially static system in which heavy fluid overlies light fluid: ρ + > ρ, V = 0. Via (4.5), the system is unstable if ρ ρ + g C0 = + k < 0 (4.6) ρ + ρ k ρ + ρ + k 4π i.e. if ρ + ρ > g = gλ. Thus, for instability, we require: λ > πλc where λ c = J Δρg is the capillary length. Heuristic Argument: Change in Surface Energy: [ f ] λ ΔE S = } Δl = ds λ = 0 4 ǫ k λ. arc length Change in gravitational potential energy: Figure 4.: The base state and the perf λ ( ) ΔE = ρg h h dx = ρgǫ λ. turbed state of the Rayleigh-Taylor system, G When is the total energy decreased? heavy fluid over light. When ΔE total = ΔE S + ΔE G < 0, i.e. when ρg > k, so λ > πl c. The system is thus unstable to long λ. Note:. The system is stabilized to small λ disturbances by. The system is always unstable for suff. large λ 3. In a finite container with width smaller than πλ c, the system may be stabilized by. 4. System may be stabilized by temperature gradients since Marangoni flow acts to resist surface deformation. E.g. a fluid layer on the ceiling may be stabilized by heating the ceiling. Figure 4.3: Rayleigh-Taylor instability may be stabilized by a vertical temperature gradient. MIT OCW: Interfacial Phenomena 57 Prof. John W. M. Bush

4 4.. Kelvin-Helmholtz Instability Chapter 4. Instability of Superposed Fluids 4. Kelvin-Helmholtz Instability We consider shear-driven instability of a gravitationally stable base state. Specifically, ρ ρ + so the system is gravitationally stable, but destabilized by the shear. Take k parallel to V, so (V k) = k V and the instability criterion becomes: Equivalently, g ρ ρ + V > (ρ ρ + ) + k (4.7) k λ π ρ ρ + V > (ρ ρ + )g + (4.8) π λ Note: Figure 4.4: Kelvin-Helmholtz instability: a gravi. System stabilized to short λ disturbances by tationally stable base state is destabilized by shear. surface tension and to long λ by gravity.. For any given λ (or k), one can find a critical V that destabilizes the system. Marginal Stability Curve: ) / ( ρ ρ + g V(k) = + k (4.9) ρ ρ + k ρ ρ + dv d dk dk V(k) has a minimum where = 0, i.e. V = 0. J This implies Δρ + = 0 k c = Δρg =. l cap k The corresponding V c = V(k c ) = ρ ρ + Δρg is the minimal speed necessary for waves. Figure 4.5: Fluid speed V(k) required for the growth of a wave with wavenumber k. E.g. Air blowing over water: (cgs) V c = V c 650cm/s is the mini- 3 mum wind speed required to generate J waves. 0 3 These waves have wavenumber k c = cm, so λ c =.6cm. They thus correspond to capillary waves. MIT OCW: Interfacial Phenomena 58 Prof. John W. M. Bush

5 MIT OpenCourseWare Interfacial Phenomena Fall 00 For information about citing these materials or our Terms of Use, visit: