Effect of gravity modulation on the stability of a horizontal double-diffusive layer.

摘要

The effect of gravity modulation on the instability onset in an infinite horizontal layer of double-diffusive fluid is investigated in this dissertation. The spectral-Galerkin method is used to transform the linearized perturbation equations to the system of time-periodic ordinary differentiate equations. The Chebyshev expansion method (Sinha and Wu, 1991) is applied to calculate the foundamental matrix which is used to determine the stability of the system according to the Floquet theory. Fluids of Prandtl number Pr = 0.01, 1, and 7 are investigated. The instability onsets in one of three modes, synchronous, subharmonic, and quasi periodic mode. In the synchronous mode, the instability oscillates at the same frequency as the gravity modulation Ω, in the subharmonic mode, the instability oscillates at Ω/2, while in the quasi-periodic mode, the oscillation frequency of the instability is different from the above two. The quasi-periodic mode onsets at the same thermal Rayleigh number, R_T as that of instability onset under steady gravity. The subharmonic mode onsets with wave numbers in the neighboring region of k where the oscillation frequency of the instability onset in the steady-g case, ω, equals to half of the modulation frequency, Ω/2. Similarly, the synchronous mode onsets at the neighboring of k where the ω equals to Ω. The onset R_T for quasi-periodic mode is not changed by the modulation frequency Ω and the relative amplitude of the modulation, h. For the synchronous and subharmonic modes, destabilization increases with increasing h. If h is large enough, the subharmonic mode will be more unstable than the synchronous and quasi periodic mode, so the instability mode will be switched by increasing h. For a given h with varying Ω, the resonance effect occurs in the neighborhood of Ω ≈ 2ω _cr, i.e, twice the critical oscillation frequency of the instability in the steady-g case associated with the critical R_T. The resonant phenomena is found for fluids with Pr = 0.01, 1, and 7, and the effect diminishes as the Prandtl number increases. The effect of gravity modulation is asymptotic to zero when the modulated frequency Ω approaches zero and infinitely large. For the case of Prandtl number, Pr, = 0.01, it is found that the critical thermal Rayleigh number R_T is reduced from the steady-g value of 2183 by 4%, 41%, and 86% as h is increased from 0.01, 0.1 and 0.2. In fact when h = 0.22737, the layer of fluid is destabilized at Ω = 9.1 with R_T = 0, i.e., without heating from below. This is analogous to the case in the research of Gresho and Sani (1970) that a horizontal layer being heated from above can be destabilized by the oscillation of the layer. In this dissertation the stabilization effect caused by the modulation is found at some cases of Ω, which is analogous to the stability of motion of the pendulum with pivot in oscillation as discussed in Gresho and Sani (1970).