Quantification of linear and nonlinear energy transfer processes in a plane wake
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The transition to turbulence of plane wakes is characterized by the development of the velocity-fluctuation field from a spectrum of weak random background noise in the initial laminar wake to a nearly featureless broad spectrum of intense fluctuations within the turbulent wake. This transition has also been described as a sequence of instabilities and wave-wave interactions. In the initial small-amplitude stage,. a narrow, but continuous, band of dominant instability modes centered near the most unstable mode, known also as the fundamental mode, grow exponentially at rates that can be calculated from the linearized Navier-Stokes equations. As these modes grow, the nonlinear terms become more important and cannot be neglected anymore. The effect of these terms is to introduce wave-wave interactions that lead to quadratic energy transfer between the different spectral components of the velocity-fluctuation field. While the consequences of these interactions, such as broadening of the power spectra, have been observed in many experiments, the characteristics of these interactions have only been examined in limited cases. Previous measurements of the auto-bispectrum showed that three-wave interaction processes are important in the transitioning wake. However, quantification of these processes can only be obtained from measurement of the nonlinear energy transfer rates resulting from the nonlinear wave-wave interactions. Such quantification is very important for understanding the effects of the different mechanisms involved in the transition and final breakdown to turbulence. An understanding of these mechanisms and their effects can then be used to control the transition by enhancing certain mechanisms and reducing the role of others through external excitation. In this work, quantitative estimates of the auto-bispectrum, linear and quadratic coupling coefficients and the resulting energy transfer rates between the interacting waves at different locations are presented in controlled and natural transitions of the plane wake. The results show that, in both natural and controlled transitions, the underlying nonlinear dynamics are similar. Basically, nonlinear interactions between the instability modes result in energy transfer to harmonic bands as well as low-frequency difference components. These components play an important role in the transfer of energy to the sidebands and the valleys between the peaks. The results also show that, while energy-transfer rates in natural transition are lower than in controlled transition, the random nature of wave excitation in natural transition causes energy transfer to a band of low-frequency components which leads to energy transfer to many sidebands and results in a spectrum that differs dramatically from the one obtained in the controlled case where two instabilities are excited.
- Masters Theses