A critical-layer analysis of the resonant triad in boundary-layer transition: nonlinear interactions

1993 ◽  
Vol 256 ◽  
pp. 85-106 ◽  
Author(s):  
Reda R. Mankbadi ◽  
Xuesong Wu ◽  
Sang Soo Lee
Keyword(s):  
Growth Rate ◽  
Plane Wave ◽  
Parametric Resonance ◽  
Low Frequency ◽  
Boundary Layer Transition ◽  
The Self ◽  
Critical Layer ◽  
Linear Growth Rate ◽  
Layer Transition ◽  
Oblique Waves

A systematic theory is developed to study the nonlinear spatial evolution of the resonant triad in Blasius boundary layers. This triad consists of a plane wave at the fundamental frequency and a pair of symmetrical, oblique waves at the subharmonic frequency. A low-frequency asymptotic scaling leads to a distinct critical layer wherein nonlinearity first becomes important, and the critical layer's nonlinear, viscous dynamics determine the development of the triad.The plane wave initially causes double-exponential growth of the oblique waves. The plane wave, however, continues to follow the linear theory, even when the oblique waves’ amplitude attains the same order of magnitude as that of the plane wave. However, when the amplitude of the oblique waves exceeds that of the plane wave by a certain level, a nonlinear stage comes into effect in which the self-interaction of the oblique waves becomes important. The self-interaction causes rapid growth of the phase of the oblique waves, which causes a change of the sign of the parametric-resonance term in the oblique-waves amplitude equation. Ultimately this effect causes the growth rate of the oblique waves to oscillate around their linear growth rate. Since the latter is usually small in the nonlinear regime, the net outcome is that the self-interaction of oblique waves causes the parametric resonance stage to be followed by an ‘oscillatory’ saturation stage.

On the catalytic role of the phase-locked interaction of Tollmien–Schlichting waves in boundary-layer transition

2007 ◽  
Vol 590 ◽  
pp. 265-294 ◽  
Author(s):  
XUESONG WU ◽  
P. A. STEWART ◽  
S. J. COWLEY
Keyword(s):  
Exponential Growth ◽  
Perfect Match ◽  
Three Dimensional ◽  
Phase Speed ◽  
Boundary Layer Transition ◽  
Critical Layer ◽  
Back Reaction ◽  
S Wave ◽  
Layer Transition ◽  
Tollmien Schlichting

This paper is concerned with the nonlinear interaction between a planar and a pair of oblique Tollmien–Schlichting (T-S) waves which are phase-locked in that they travel with (nearly) the same phase speed. The evolution of such a disturbance is described using a high-Reynolds-number asymptotic approach in the so-called ‘upper--branch’ scaling regime. It follows that there exists a well-defined common critical layer (i.e. a thin region surrounding the level at which the basic flow velocity equals the phase speed of the waves to leading order) and the dominant interactions take place there. The disturbance is shown to evolve through several distinctive stages. In the first of these, the critical layer is in equilibrium and viscosity dominated. If a small mismatching exists in the phase speeds, the interaction between the planar and oblique waves leads directly to super-exponential growth/decay of the oblique modes. However, if the modes are perfectly phase-locked, the interaction in the first instance affects only the phase of the amplitude function of the oblique modes (so causing rapid wavelength shortening), while the modulus of the amplitude still evolves exponentially until the wavelength shortening produces a back reaction on the modulus (which then induces a super-exponential growth). Whether or not there is a small mismatch or a perfect match in the phase speeds, once the growth rate of the oblique modes becomes sufficiently large, the disturbance enters a second stage, in which the critical layer becomes both non-equilibrium and viscous in nature. The oblique modes continue to experience super-exponential growth, albeit of a different form from that in the previous stages, until the self-interaction between them, as well as their back effect on the planar mode, becomes important. At that point, the disturbance enters a third, fully interactive stage, during which the development of the disturbance is governed by the amplitude equations with the same nonlinear terms as previously derived for the phase-locked interaction of Rayleigh instability waves. The solution develops a singularity, leading to the final stage where the flow is governed by fully nonlinear three-dimensional inviscid triple-deck equations. The present work indicates that seeding a planar T-S wave can enhance the amplification of all oblique modes which share approximately its phase speed.

Interaction of oblique instability waves with a nonlinear plane wave

1994 ◽  
Vol 264 ◽  
pp. 343-372 ◽  
Author(s):  
David W. Wundrow ◽  
Lennart S. Hultgren ◽  
M. E. Goldstein
Keyword(s):  
Growth Rate ◽  
Parametric Resonance ◽  
Wave Interaction ◽  
Adverse Pressure Gradient ◽  
Linear Instability ◽  
Critical Layer ◽  
Spanwise Direction ◽  
Instability Waves ◽  
Resonance Effects ◽  
Nonlinear Plane

This paper is concerned with the downstream evolution of a resonant triad of initially non-interacting linear instability waves in a boundary layer with a weak adverse pressure gradient. The triad consists of a two-dimensional fundamental mode and a pair of equal-amplitude oblique modes that form a subharmonic standing wave in the spanwise direction. The growth rates are small and there is a well-defined common critical layer for these waves. As in Goldstein & Lee (1992), the wave interaction takes place entirely within this critical layer and is initially of the parametric-resonance type. This enhances the spatial growth rate of the subharmonic but does not affect that of the fundamental. However, in contrast to Goldstein & Lee (1992), the initial subharmonic amplitude is assumed to be small enough so that the fundamental can become nonlinear within its own critical layer before it is affected by the subharmonic. The subharmonic evolution is then dominated by the parametric-resonance effects and occurs on a much shorter streamwise scale than that of the fundamental. The subharmonic amplitude continues to increase during this parametric-resonance stage – even as the growth rate of the fundamental approaches zero – and the subharmonic eventually becomes large enough to influence the fundamental which causes both waves to evolve on the same shorter streamwise scale.

The role of nonlinear critical layers in boundary layer transition

1995 ◽  
Vol 352 (1700) ◽  
pp. 425-442 ◽  
Keyword(s):  
Asymptotic Methods ◽  
Nonlinear Effects ◽  
Boundary Layer Transition ◽  
Critical Layer ◽  
Mean Flow ◽  
Layer Transition ◽  
Instability Waves ◽  
Laminar Boundary Layers ◽  
Instability Modes ◽  
Critical Layers

Asymptotic methods are used to describe the nonlinear self-interaction between pairs of oblique instability modes that eventually develops when initially linear spatially growing instability waves evolve downstream in nominally two-dimensional laminar boundary layers. The first nonlinear reaction takes place locally within a so-called ‘critical layer’, with the flow outside this layer consisting of a locally parallel mean flow plus a pair of oblique instability waves - which may or may not be accompanied by an associated plane wave. The amplitudes of these waves, which are completely determined by nonlinear effects within the critical layer, satisfy either a single integro-differential equation or a pair of integro-differential equations with quadratic to quartic-type nonlinearities. The physical implications of these equations are discussed.

scholarly journals From primary instabilities to secondary instabilities in Görtler vortex flows

2019 ◽  
Vol 1 (1) ◽  
Author(s):  
Xi Chen ◽  
Jianqiang Chen ◽  
Xianxu Yuan ◽  
Guohua Tu ◽  
Yifeng Zhang
Keyword(s):  
Boundary Layer ◽  
Growth Rate ◽  
Transformation Process ◽  
Secondary Instability ◽  
Direct Numerical Simulations ◽  
Boundary Layer Transition ◽  
Layer Transition ◽  
Mode Synchronization ◽  
Secondary Instabilities ◽  
Primary Instability

Abstract We have studied the transformation process from primary instabilities to secondary instabilities with direct numerical simulations and stability theories (Spatial Biglobal and plane-marching parabolized stability equations) in detail. First Mack mode and second Mack mode are shown to be able to evolve into the sinuous mode and the varicose mode of secondary instability, respectively. Although the characteristics of second Mack mode eventually lose in the downstream due to the synchronization with the continuous spectrum, second Mack mode is found to be able to trigger a sequence of mode resonations which in turn give birth to highly unstable secondary instabilities. In contrast, first Mack mode does not involve in any mode synchronization. Nevertheless, it can still “jump" to a sinuous mode of secondary instability with a much larger growth rate than that of the first Mack mode. Therefore, secondary instabilities of Görtler vortices are highly receptive to the primary instabilities in the upstream, so that one should consider the primary instability in the upstream and the secondary instability in the downstream as a whole in order to get an accurate prediction of the boundary layer transition.

The Use of Hot Films for Investigation of Boundary-Layer Transition

1993 ◽  
Vol 207 (2) ◽  
pp. 121-126 ◽  
Author(s):  
L Gaudet ◽  
C J Betts ◽  
P R Ashill
Keyword(s):  
Transition Region ◽  
Energy Content ◽  
Low Frequency ◽  
Boundary Layer Transition ◽  
Layer Transition ◽  
Power Spectral ◽  
Time Histories ◽  
The Mean ◽  
Tollmien Schlichting ◽  
Hot Film

The constant temperature hot-film technique has been used to investigate the transition region on a modified flat plate and a large two-dimensional wing. Time histories and r.m.s. values of the output signals together with the power spectral density (PSD) clearly indicate the build up of energy associated with the development of Tollmien–Schlichting waves prior to the onset of transition. A large increase in the low-frequency energy content of the spectrum occurs at the onset of transition. Measurements of skin friction through the transition region have been obtained based on the mean heat loss of the hot film.

On critical-layer and diffusion-layer nonlinearity in the three-dimensional stage of boundary-layer transition

1993 ◽  
Vol 443 (1917) ◽  
pp. 95-106 ◽  
Keyword(s):  
Boundary Layer ◽  
Parametric Resonance ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Boundary Layer Transition ◽  
Critical Layer ◽  
Transition To Turbulence ◽  
Maximum Magnitude ◽  
Nonlinear Interactions ◽  
Mean Flow

This paper considers nonlinear interactions in the three-dimensional stage of transition to turbulence, taking an accelerating boundary layer as a prototype flow. Attention is focused on transition via subharmonic resonance in the upper-branch scaling régime. It is shown that the (weakly) nonlinear instability of the flow is described by a seven-zoned structure, cf. the five-zoned structure for the linear problem. The dominant nonlinear interactions take place both in a critical layer and in ‘diffusion layers’. The nonlinearly generated mean flow in turn interacts with the wall to attain a maximum magnitude near the wall. It is emphasized that both the nonlinear mechanism and the flow structure are generic for three-dimensional disturbances. And there is some similarity with the work in the context of wave/vortex interaction. Numerical solutions of the amplitude equations indicate that if the oblique modes initially have a small amplitude, they first experience a rapid growth caused by parametric resonance. Following this the cubic interactions of the oblique modes inhibit the growth and lead to a wavelength shortening. However, if the initial amplitudes of the oblique modes are sufficiently large, the parametric resonance can be completely bypassed. Numerical solutions also suggest that oblique modes with unequal initial amplitudes evolve to an equal-amplitude state.

Numerical and experimental investigations of oblique boundary layer transition

1999 ◽  
Vol 393 ◽  
pp. 23-57 ◽  
Author(s):  
STELLAN BERLIN ◽  
MARKUS WIEGEL ◽  
DAN S. HENNINGSON
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Transition ◽  
Experimental Investigations ◽  
Two Dimensional ◽  
Layer Transition ◽  
Oblique Waves ◽  
Starting Point ◽  
Incompressible Boundary ◽  
Tollmien Schlichting ◽  
Streamwise Streaks

A transition scenario initiated by two oblique waves is studied in an incompressible boundary layer. Hot-wire measurements and flow visualizations from the first boundary layer experiment on this scenario are reported. The experimental results are compared with spatial direct numerical simulations and good qualitative agreement is found. Also, quantitative agreement is found when the experimental device for disturbance generation is closely modelled in the simulations and pressure gradient effects taken into account. The oblique waves are found to interact nonlinearly to force streamwise vortices. The vortices in turn produce growing streamwise streaks by non-modal linear growth mechanisms. This has previously been observed in channel flows and calculations of both compressible and incompressible boundary layers. The flow structures observed at the late stage of oblique transition have many similarities to the corresponding ones of K- and H-type transition, for which two-dimensional Tollmien–Schlichting waves are the starting point. However, two-dimensional Tollmien–Schlichting waves are usually not initiated or observed in oblique transition and consequently the similarities are due to the oblique waves and streamwise streaks appearing in all three scenarios.

scholarly journals A Frequency Domain Analysis: Turbine Pressure Side Heat Transfer

1998 ◽  
Author(s):  
D. G. Holmberg ◽  
T. E. Diller ◽  
W. F. Ng
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Surface Heat Flux ◽  
Low Frequency ◽  
Surface Heat ◽  
Boundary Layer Transition ◽  
Pressure Side ◽  
Layer Transition ◽  
Measured Velocity ◽  
Flow Energy

Simultaneous time-resolved surface heat flux and velocity measurements have been made along the pressure side of an engine-similar turbine blade in a linear cascade. Direct heat flux was measured using inserted Heat Flux Microsensors at three axial locations along the high turning transonic blade. Miniature hot-wires measured velocity above these locations. Grids produced two turbulence fields with different inlet length scales at constant turbulence intensity. This work allows a unique look at fluctuating heat transfer on the blade and its relationship to the fluctuating velocity field above. While fluctuating free-stream flow energy and length scale decays continuously in the passage, surface heat flux energy increases continuously. Low frequency flow energy is attenuated in the constricted passage, while growth of low frequency energy in the heat flux is attributed to boundary layer transition activity. Coherence between heat flux and velocity is seen at all frequencies near the leading edge of the blade but only at higher frequencies farther downstream on the pressure side. Existing mean heat transfer correlations do not perform well in this complicated flow.

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