Linear and nonlinear resonance in viscous film flow over topography
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1 Linear and nonlinear resonance in viscous film flow over topography N. Aksel 1, V. Bontozoglou 2, H. Uecker 3, A. Wierschem 4, C. Heining 1 1 University of Bayreuth, Department of Applied Mechanics and Fluid Dynamics, D Bayreuth 2 University of Thessaly, Department of Mechanical and Industrial Engineering, GR Volos 3 Carl von Ossietzky Universität Oldenburg, Institut für Mathematik, D Oldenburg 4 Friedrich-Alexander-Universität Erlangen Nürnberg, Lehrstuhl für Strömungsmechanik, D Erlangen
2 Motivation and aims film flow down an undulated incline steady amplification of the free surface, defined as resonance in the following Experimental observations [1] Aims find solutions for the free surface to predict resonance understand the origin of the free surface amplification [1] Hydraulic jumps and standing waves in gravity-driven flows of viscous liquids in wavy open channels, Wierschem, Aksel, Physics of Fluids 16, 3868 (2004)
3 System and modeling 2D-system periodic in flow direction d: Nusselt film thickness d gravity a ξ : = h System parameters Re Bo 1 δ = 2 π d / λ ξ = a / d Reynolds number inverse Bond number dimensionless film thickness steepness parameter σ
4 Phenomena: resonance effects numerical simulation: finite-element Galerkin method study the amplitude of the free surface, relative to the bottom 1 st harmonic ξ <<1 ξ = ξ = ξ = higher order harmonic resonance bistability with increasing bottom steepness
5 Linear resonance: derivation linear approximation for weak bottom steepness ( 2 ) ( 2 ) ( 2 ) ( 2 ) u = u + ξu + O ξ v = v + ξv + O ξ p = p + ξ p + O ξ f = f + ξ f + O ξ leading order: Nusselt solution with f 0 = 1 a ξ = << 1 d free surface has the same periodicity/wavelength as the bottom. Ansatz for the free surface at first order: f 1 = Ae ix A: relative amplitude steady Navier-Stokes eq. reduce to Orr-Sommerfeld equation with zero phase velocity at first order
6 Linear resonance: results first harmonic second harmonic perfect agreement for weak steepness linear theory cannot predict higher harmonics
7 Linear resonance: physical interpretation consider the Orr-Sommerfeld equation with nonzero phase velocity and k=1 look for the least stable eigenmode follow the eigenvalue with increasing Re Relative Amplitude 2,0 1,6 1,2 0,8 0,4 0,0 1 st harmonic Reynolds Number Re phase velocity changes sign at resonance resonance is associated with capillary-gravity waves travelling against the mean flow direction
8 Nonlinear resonance application of integral boundary-layer model for thin films impose a velocity profile (parabolic profile) reduction of the Navier-Stokes eq. to an evolution equation for the film thickness benefit: higher Fourier modes are included, compared to the linear Orr-Sommerfeld theory
9 Nonlinear resonance: results resonance and higher order harmonic resonance analytic resonance condition for each harmonic n Reres, n = ( cotα + n Bo ) 2 resonance with capillary-gravity waves of a nth wavelength of the bottom contour n ( Ca G ) u = u + u π d λ / n
10 Nonlinear resonance: results nonlinear effects are recovered bistability correspondance of the bistability with that of a driven Duffing oscillator
11 Comparison IBL linear full num. IBL-approximation yields reasonable comparison for small to moderate Reynolds numbers
12 Conclusions free surface in film flow over wavy inclines shows resonance (amplification) numerical solution reveals two main phenomena resonance and higher order harmonic resonance bistable resonance with increasing nonlinearity linear approximation with the Orr-Sommerfeld equation prediction of the leading order harmonic resonance is associated with capillary-gravity waves travelling against the mean flow direction nonlinear approximation by the integral boundary-layer theory prediction of all linear and nonlinear resonance effects analytical resonance criterion
13 Available literature Linear resonance in viscous films on inclined wavy planes, Wierschem, Bontozoglou, Heining, Uecker, Aksel, International Journal of Multiphase Flow, 20, (2008) Nonlinear resonance in viscous films on inclined wavy planes, Heining, Bontozoglou, Aksel, Wierschem, International Journal of Multiphase Flow 34, (2009)
14 Integral boundary-layer theory
15 Experimental setup
16 Resonance
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SIESMIC SLOSHING IN CYLINDRICAL TANKS WITH FLEXIBLE BAFFLES Kayahan AKGUL 1, Yasin M. FAHJAN 2, Zuhal OZDEMIR 3 and Mhamed SOULI 4 ABSTRACT Sloshing has been one of the major concerns for engineers in