TS Waves Research Articles

Vibration of Thermally Stressed Pretwisted Cantilever Composite Plates

Fig. 5 Variations of the magnitude of the e ow response kk ( (u, v)kk as a function of the suction wave number ® and wave speed cr for Re= 5 ££ 10 3 ; continuous and dashed lines correspond to the symmetric and asymmetric suctions, respectively. eigensolution (TS wave) and a unique asymmetric particular solution of the inhomogeneous problem. A very large response of the e ow is possible in this case but requires a perfect tuning between the neutral TS wave and the surface suction wave. Results presented in Fig. 3 demonstrate that linear theory is inadequate in the neighborhood of (linear) neutral stability points and a nonlineartheorymustbeusedregardlessofthemagnitudeofthesuction. The nonlinear effectmay expand therangeofthe TSinstability and may initiate an instability of the type studied in Ref. 1. Figure 5 shows e ow response to surface suction for the subcritical value of Reynolds number Re = 5 £ 10 3 when all TS waves are stable. The near resonance between the TS and the surface suction waves may lower the critical Reynolds number due to the subharmonic character of the instability, provided that suction with sufe ciently large amplitude is used. IV. Conclusions The preceding results demonstrate that small suction nonuniformities may affect the e owinstability only if they contain the critical or near-critical suction waves, that is, waves with the same or almost the same wave number a and phase speed cr as the neutral TS waves. Suction waves slightly detuned with the neutral TS waves mayaffectinstabilityprovidedthattheamplitudeofsuctionnonuniformitiesislargeenough. Our conclusionscould be readilytestedin a e owwith a fewsuction slots at the wall that could emulate suction waves with the desired wave number and phase speed.

Resonant growth of three-dimensional modes in Falkner–Skan boundary layers with adverse pressure gradients

This work documents the spatial development of a triad of instability waves consisting of a plane TS mode and a pair of oblique modes with equal-opposite wave angles which are undergoing subharmonic transition in Falkner-Skan boundary layers with adverse pressure gradients. The motivation for this study is that for wings with zero or moderate sweep angles, transition is most likely to occur in the adverse pressure gradient region past the maximum thickness point and, starting with low initial amplitudes, subharmonic mode transition is expected to be the predominant mechanism for the first growth of of three-dimensional modes. The experiment follows that of Corke & Mangano (1989) in which the disturbances to produce the triad of waves are introduced by a spanwise array of heating wires located near Branch I. The initial conditions are carefully controlled. These include the initial amplitudes, frequencies, relative phase and oblique wave angles. The basic flow consisted of a Falkner-Skan (Hartree) boundary layer with a dimensionless pressure gradient parameter in the range -0.06 ≤ β H ≤ -0.09. The frequency of the TS wave was selected to be near the most amplified based on linear theory. The frequency of the oblique waves was the subharmonic of the TS frequency. The oblique wave angles were set to give the largest secondary growth ( 60°). Compared to similar conditions in a Blasius boundary layer, the adverse pressure gradient was observed to lead to an extra rapid growth of the two- and three-dimensional modes. In this there was a relatively larger maximum amplitude of the fundamental mode and considerably shortened amplitude saturation region compared to zero pressure gradient cases. Analysis of these results includes frequency spectra, the wall-normal distributions of each mode amplitude, and mean velocity profiles. Finally, the streamwise amplitude development is compared with the amplitude model from the nonlinear critical layer analysis of Goldstein & Lee (1992).

Nonlinear Triple-Deck Studies in Boundary-Layer Receptivity

A numerical study of the generation of Tollmien-Schlichting waves due to the interaction between a small free-stream disturbance and a small localized variation of the surface geometry such as a hump or suction strip has been carried out using both finite difference and spectral methods. The nonlinear steady flow is of the viscous-inviscid interactive type while the unsteady disturbed flow is assumed to be governed by linearized disturbance equations. The numerical solutions illustrate the growth or decay of the TS waves generated by the interaction, depending on the value of a scaled Strouhal number.

Numerical simulations of transition in oscillatory plane channel flow

The effect of flow oscillation on the stability of plane channel flow is studied via numerical simulation. For weak oscillation, the ratio of the Stokes layer thickness to the distance from the wall to the critical layer in steady flow provides the best normalization for the mean-flow frequency. Maximum growth rates occur when the instantaneous velocity profile has large regions of positive curvature. The effect of oscillation is generally stabilizing. However, at low frequencies, TS wave energies may vary by 106 in a cycle and irreversible secondary instability may be produced at the peak.

Onset of transition in boundary layers

AbstractLaminar‐turbulent transition in boundary layers involves a cascade of weak and strong instabilities. In the model considered here the first instability occurs with respect to two‐dimensional TS waves and causes streamwise, nearly periodic concentrations of vorticity. Linear stability analysis of this periodic flow leads to Floquet systems of equations. These systems support different classes of three‐dimensional disturbances which may initiate different routes to transition. Numerical solutions by use of accurate spectral methods reveal the spectrum of eigenmodes, growth rates and disturbance velocities. The characteristics of this secondary instability are in good agreement with results of experiments and computer simulations of transition. Non‐linear self‐interaction of the rapidly growing three‐dimensional disturbances can sustain or enhance the vital periodic accumulations of spanwise vorticity once their amplitude exceeds some threshold. This feedback loop is considered to be the key to the transition process. Owing to the broad‐band nature of secondary instability, however, the prediction of transition in practice requires additional insight into the ‘natural’ disturbance background. The sensitivity of the transition process to initial data in a broad band of frequencies and spanwise wave numbers poses new challenges for non‐linear theories and numerical simulations.

Effects of compressibility on design of subsonic fuselages for natural laminar flow

Compressible linear boundary-layer stability analyses of two representative axisymmetric fuselage geometries indicate that a favorable effect will be exerted on the characteristics of a fuselage's axisymmetric boundary layer by compressibility. A freestream Mach number increase from 0.6 to 0.8 significantly reduces TS wave growth rates in the laminar boundary layer of the fuselages analyzed. The generally destabilizing effect of increasing length Re number on boundary layer stability can be overpowered by the favorable effects of compressibility on the fluid.

On forced oscillations in the boundary layer at frequencies near the upper branch of the neutral curve

Perturbations introduced into the boundary layer (BL) of an incompressible liquid by a harmonic oscillator in the form of a moving section of the surface are examined. Outside the oscillating part, the streamlined solid is a plane plate. It is assumed that the Reynolds number is large and the oscillation frequency, corresponds, in order of magnitude, to the asymptotic form of the upper branch of the neutral stability curve (NSC). The system of equations for perturbations, at small amplitudes of the oscillator, is linearized and is solved by the Fourier method. In addition, for each Fourier component, the flow field is divided into five sublayers. The amplitude of a Tollmin-Schlichting wave (TS) is calculated and separated from the perturbed background downstream of the oscillator. If the oscillator frequency exceeds the neutral value at the upper branch of NSC with the given Reynolds number, the TS wave amplitude decays. For frequencies below neutral, the wave amplitude increases exponentially downstream. In the final example, the parameters of the TS wave fall within the unstable region, between two NSC branches.

Boundary-layer receptivity to unsteady pressure gradients: experiments and overview

The experimental evidence on the mechanisms of forcing of unstable vorticity waves (the Tollmien–Schlichting–Schubauer or TS waves) of circular frequency ω and wavelength λTS in wall layers by unsteady pressure gradients of amplitude A and frequency ω is reviewed and found to be confused and contradictory. It is proposed that a likely effective receptivity mechanism rests on the fact that under realistic conditions A varies with distance x along any body of finite thickness, A(x), and introduces thereby additional characteristic lengths which can match λTS. Heuristic arguments suggest that through A(x) the pressure gradient infuses vorticity at the wall and forces spatial growth of the TS mean-square vorticity and is proportional to the imaginary part of κTSΔAF(KTS).The proposition is consistent in all currently verifiable respects with one numerical and a series of laboratory experiments. In the laboratory experiments, various configurations of a pulsating pressure source and shielding plates located in the free stream supplied the variable-amplitude pressure gradients over the nearby flat-wall boundary layer. Three of the cases presented here demonstrate that stationary unsteady pressure fields induce Stokes-like sublayers when the boundary layer is stable and self-excited vorticity waves when it is unstable. The results of a fourth experiment suggest that unsteady pressure sources in wakes near the boundary layer can force the growth of unstable wall waves at the wake frequencies even though their propagation speeds differ. Material is also presented on key Soviet experiments and views on receptivity. Finally, these experiments and ours are examined for consistency and complementarity.