The dynamics of wind-generated water waves in the principal stage of the Phillips theory (Phillips, J. Fluid Mech., vol. 2, 1957, pp. 417–445) is investigated by a combined numerical and analytical approach. We perform a number of high-resolution direct numerical simulation (DNS) of turbulent wind over initially calm water to capture the multistage generation of water waves. Detailed analyses are conducted to evaluate the Phillips theory in both physical space and wavenumber space. Numerical evidence is obtained for the existence of a principal stage when the surface elevation variance grows linearly with time. We further propose a random sweeping turbulence pressure–wave interaction model by introducing the random sweeping hypothesis of air pressure fluctuations to the Phillips theory, and obtain an asymptotic solution of the mean square of surface wave elevations over time. This asymptotic analysis captures the temporal oscillations of surface elevation variance in the principal stage, which is also confirmed by our DNS results. The wavenumber spectrum of surface wave elevations is analysed using a time-dependent norm to elucidate the role of the resonance mechanism on wave generation. In physical space, we use the random sweeping turbulence pressure–wave interaction model to obtain a quantitative prediction of the growth rate of surface elevation variance in the principal stage, which is found to agree with the DNS results better than the original Phillips model.
Applications of carbon nanotube (CNT) and graphene in thermal management have recently attracted significant attention. However, the lack of efficient prediction formula for heat transfer coefficient between nanomaterials and gas environment limits the further development of this technique. In this work, a kinetic model has been established to predict the heat transfer coefficient of an individual CNT in gas environment. The heat dissipation around the CNT is governed by molecular collisions, and outside the collision layer, the heat conduction is dominant. At nanoscales, the natural convection can be neglected. In order to describe the intermolecular collisions around the CNT quantitatively, a correction factor 1/24 is introduced and agrees well with the experimental observation. The prediction of the present model is in good agreement with our experimental results in free molecular regime. Further, a maximum heat transfer coefficient occurs at a critical diameter of several nanometers, providing guidelines on the practical design of CNT-based heat spreaders.
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