Three-dimensional Tollmien-Schlichting Waves Research Articles

Effects of the Control Parameters on the Stability of a Laminar Boundary Layer on a Porous Flat Plate

This work is devoted to the analysis of the linear temporal stability of a laminar dynamic boundary layer on a horizontal porous plane plate. The basic flow is assumed to be laminar and two-dimensional. The basic flow velocity profiles are obtained by numerically solving the Blasius equation using the Runge-Kutta method. The perturbations of these basic solutions are expressed in the form of three-dimensional Tollmien-Schlichting waves. The formulation of the stability problem leads to the Orr-Sommerfeld equation modified by the permeability parameter (Darcy number) and the small Reynolds number. This equation is given in a general form which can be applied to the Chebyshev domain and the boundary layer domain and solved numerically using the Chebyshev spectral collocation method. The marginal stability diagrams, the critical Reynolds numbers and the eigenvalue spectra are obtained for different values of the parameters which have modified the stability equation. Numerical solutions indicate the importance of the effect of these parameters on the flow stability characteristics.

Three-dimensional Tollmien-Schlichting waves generated by sound in the boundary layer on an elastic surface at transonic free-stream velocities

Within the framework of the triple-deck theory, the effect of surface elasticity on three-dimensional packets of Tollmien-Schlichting waves generated by acoustic disturbances induced near the boundary layer at transonic free-stream velocities is investigated. It is shown that the elasticity of the surface considerably weakens the most unstable oblique waves but does not change the characteristic horseshoe shape of wave packets with two disturbance peaks propagating at an angle to the incoming flow.

Effect of wave frequency on the nonlinear interaction between Görtler vortices and three-dimensional Tollmien-Schlichting waves

The nonlinear interaction between Görtler vortices (GV) and three-dimensional Tollmien-Schlichting (TS) waves nonlinear interaction is studied with a spatial, nonparallel model based on the Parabolized Stability Equations (PSE). In this investigation the effect of TS wave frequency on the nonlinear interaction is studied. As verified in previous investigations using the same numerical model, the relative amplitudes and growth rates are the dominant parameters in GV/TS wave interaction. In this sense, the wave frequency influence is important in defining the streamwise distance traveled by the disturbances in the unstable region of the stability diagram and in defining the amplification rates that they go through.

Nonlinear interaction of oblique three-dimensional Tollmien-Schlichting waves and longitudinal vortices, in channel flows and boundary layers

The present theoretical article considers the nonlinear interaction of oblique three dimensional Tollmien-Schlichting waves and induced or input longitudinal vortex motion, mainly for channel flow at large Reynolds numbers. Both the waves and the vortices are controlled by viscous-inviscid balancing but their respective flow structures are rather different because of the different typical timescales involved. This leads to the vortex-wave interaction being governed by nonlinear evolution equations on the vortex timescale, even though the wave amplitudes are notably small. The analogue in boundary-layer transition, addressed in a previous paper, is also re-considered here. Computational and analytical properties of the interaction equations for both channel flows and boundary layers are investigated, along with certain connections with companion studies of other vortex-wave interactions in channel flow. The nonlinear interactions in channel flow are found to lead to finitetime blow-up in amplitudes or to sustained vortex flow at large scaled times, depending on the input conditions. In particular, increasing the input amplitudes of the vortex or the wave can readily provoke blow-up even in the linearly stable regime; whereas in the case of sustained vortex flow new physical effects come into play on slightly longer timescales. Again, a very interesting feature is that the blowup response is found to be confined to a small range of wave angles near 45° relative to the original flow direction.

Effect of streamwise vortices on Tollmien–Schlichting waves in growing boundary layers

The effect of counter-rotating streamwise steady vortices on three-dimensional Tollmien–Schlichting (TS) waves in a growing boundary layer on a concavely curved surface has been studied by using the method of multiple scales. The nonlinear evolution of the amplitudes with streamwise position has been obtained analytically. It is shown that the TS waves are amplified by the presence of streamwise vortices. The effect of combination resonance between TS waves and Goertler vortices is much more pronounced than the effect of parametric resonance between the TS waves and the periodic basic flow. The amplification rates increase with Goertler number and spanwise wavenumber.

The effects of resonance interactions between wave perturbations in a boundary layer

The transition to turbulence in a boundary layer can be induced by perturbations of low intensity and is accompanied by a growth in their energy, the development of three-dimensional structures, and a change in the spectral composition of the field. A number of important properties of the process admit interpretation in the framework of nonlinear stability theory and can be due to a resonance interaction. Experiments [1, 2] have revealed a transition accompanied by an appreciable enhancement of pulsations whose period is twice that of the driving vibrating tape. Theoretical investigations [3–9] have revealed the existence of a resonance mechanism capable of strong excitation of three-dimensional Tollmien-Schlichting waves at the frequency of a subharmonic. It has been suggested [4] that the observed transition regime is the result of evolution of triplets of resonantly coupled oscillations forming symmetric triplets [10]. In contrast to the type of transition considered by Craik et al. [10, 11], the leading role is played by subharmonics distinguished parametrically in the background. Experimental confirmations have been obtained [12, 13] of the coupling of the resonances in symmetric triplets with the subharmonic regime. Further investigation of the resonance mechanism is an important topical problem. This paper presents a study on the formation and special characteristics of the initial stage in the nonlinear development of triplets; the collective interaction of a two-dimensional Tollmien-Schlichting wave with a packet of three-dimensional waves is examined; the behavior of the system is analyzed, taking into account the resonance coupling with the harmonic of the main wave. A comparison is made between Craik's model and experimental data.