The convection in an annular container with heated bottom,cooled top and insulated side walls are studied by both linear instability analysis and direct numerical simulation.The onset of convection is investigated by linear stability analysis and corresponding pattern selection mechanisms are discussed.The nonlinear evolution of different flow patterns and the convective heat transfer are simulated.The transition to oscillatory flow is also given by stability analysis where the base flow is a steady three dimensional flow.The stability predictions are in good agreement with the numerical simulations,including both the growth rate and the dimensionless frequency.
The far-field noise radiated from mixing layers is determined by the near-field flow dynamics which is sensitive to the initial perturbation of instability introduced physically or numerically.This study focuses on the effects of the phase delay in two initial perturbations,one at the fundamental wave number and the other at its subharmonic both calculated from linear instability analysis,on the sound generation in mixing layers.When different phase delaysφ_1 changing from zero to 2πis applied on the fundamental mode,we observe different vortex merging processes(e.g.vortex pairing or tearing).The strong nonlinear interaction in the merging process generates most of the noise from mixing layers.There shows a pattern in a period of 2πfor the response of far-field sound to the change ofφ_1.Similar effects on the dynamics and acoustics can be achieved by adding different phase delaysφ_2 to the subharmonic mode instead,however,the response repeats in a period of onlyπforφ_2.The effects of the combination of different phase delays to other parameters,including the amplitude and wave number for each perturbations,are also investigated.All the results indicate a critical role of nonlinearity in the sound generation mechanism of mixing layers.
Noise is generated in a two-dimensional mixing layer due to the growing of instability waves and vortex pairings. The adjoint-based control methodology has shown to be a robust tool to suppress noise radiation. The mode decomposition algorithms such as the compressible version of proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) are employed to analyze the spatial/spatial-temporal coherent structures for a consecutive data sets of the controlled mixing layer and its uncontrolled counterpart. The analyses of POD indicate that the y-direction body force control mainly modify the most energetic spatial structures, and increase the uniformity of the flow. The analyses of DMD show us prevalent frequencies and corresponding mode structures, and the stability characteristics of each mode can be obtained from DMD-spectrum. The spectral signatures illustrate that a lot of neutral/slightly damping modes emerging in uncontrolled flow within the frequency range (ω < 0.4) are suppressed due to control, relevant spatial-temporal structures are also varied, which is coincident with the change of far-field noise spectra. From the view of mode decomposition, the action of control redistribute the energy for frequency components of ω < 0.4 by weakening nonlinearities and regularizing corresponding dynamic structures in streamwise direction, and thus suppress the noise radiation. Moreover, the PODand DMD-analysis in this study demonstrate that DMD can serve as an important supplement for POD in analyzing a time-resolved physical process.
