Abstract
In practical engineering problems, there are always side-frequency components whose frequencies are close to those of the dominant-frequency waves. In this paper, the parabolized stability equations are employed to study the influence of a side-frequency component on the development of a dominant-frequency disturbance and on the transition by resonant-triad interactions. The numerical results are qualitatively consistent with the experimental data and the asymptotic analysis results. It is found that the resonant-triad waves and the mean flow distortion cannot trigger transition by themselves. We identify a new mechanism, which we refer to as the Steady-Spanwise-Waves-Working (SSWW) mechanism, which is necessary to cause transition, in that the steady spanwise waves generated by the nonlinear interaction between the pair of three-dimensional waves play an indispensable role. For the transition caused by resonant-triad interactions with a side-frequency component, the side-frequency wave makes transition occur earlier, and the relative amplitude rather than the absolute amplitude of the side-frequency disturbance plays the essential role in the transition advance. If the relative amplitude reaches the threshold level of 40%, the transition location can be affected substantially. In this kind of transition, the SSWW mechanism still works, and the side-frequency perturbation enhances the effects of the SSWW mechanism such that the transition occurs earlier.
Highlights
The laminar–turbulent transition in a boundary layer is an important problem in fluid mechanics, and it is of great significance to many practical engineering problems
Nayfeh and Bozatli (1980) did asymptotic analysis to investigate the nonlinear interaction between two T–S waves in a flat plate boundary layer and found that the nonlinear effects could result in rapid growth of the difference-frequency mode, even if the difference-frequency wave would decay by itself according to the linear stability theory
The governing equations of parabolized stability equations (PSEs) are derived by four steps: (i) The N–S equations are non-dimensionalized by appropriate reference scales. (ii) The instantaneous quantities in the flow field are decomposed as the sum of the steady basic flow and the disturbance such that one obtains the disturbance equations. (iii) The disturbance quantities are expressed in the form of traveling waves, and one can get the stability equations. (iv) In the stability equations, the second order derivatives of shape functions in the streamwise direction are higher order quantities, which when neglected result in PSE
Summary
The laminar–turbulent transition in a boundary layer is an important problem in fluid mechanics, and it is of great significance to many practical engineering problems. Saric and Reynolds (1980) did an experiment to investigate the nonlinear behavior of two T–S waves in a flat plate boundary layer and noted the evolution of the difference-frequency disturbance. Nayfeh and Bozatli (1980) did asymptotic analysis to investigate the nonlinear interaction between two T–S waves in a flat plate boundary layer and found that the nonlinear effects could result in rapid growth of the difference-frequency mode, even if the difference-frequency wave would decay by itself according to the linear stability theory. The perturbation evolutions and the transition caused by resonance-triad interactions with a side-frequency mode in an incompressible boundary layer are studied by using the nonlinear PSE. IV, the traditional PSE is used to investigate the mechanism of transition caused by resonant-triad, the influence of a side-frequency wave on transition, and the mechanism of transition caused by the resonant-triad with a side-frequency mode
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
