Three-dimensional mode growth in boundary layers with tuned and detuned subharmonic resonance

1995 ◽  
Vol 352 (1700) ◽  
pp. 453-471 ◽  
Keyword(s):  
Three Dimensional ◽  
Low Frequency ◽  
Phase Speed ◽  
Subharmonic Resonance ◽  
Oblique Waves ◽  
Periodic Disturbances ◽  
Wave Numbers ◽  
Interaction Order ◽  
Tollmien Schlichting ◽  
Low Amplitude

This work involves the use of controlled periodic disturbances to excite a plane Tollmien-Schlichting (TS) wave at one frequency ( f 2D) along with pairs of oblique waves with equal but opposite wave angles at a different frequency ( f 3D) in order to study the resonant growth of 3D modes in a Blasius boundary layer. In our earlier work (Corke & Mangano 1989; Corke 1990), the frequency of the oblique modes was exactly the subharmonic of the plane Tollmien-Schlichting (TS) mode. These modes were also phase-speed locked so that in terms of their streamwise wave numbers, α2D=1/2 α3D. This so-called ‘tuned’ subharmonic resonance leads to the enhanced growth of the otherwise linearly damped oblique waves, as well as the growth of higher harmonic 3D modes with frequencies and wave numbers: (3/2 f2d, 3/2 α2D, ±β3D), (5/2 f2d, 5/2 α2D, ±β3D), (f2d, α2D, ±2β3D) and (0, 0, ±2β3D). Even when the initial 3D oblique waves have frequencies which are close to the TS subharmonic frequency, a ‘detuned’ subharmonic resonance leads to the enhanced growth of the 3D mode. In addition, it promotes the growth of numerous discrete modes produced by successive sum and difference interactions. These interacted modes are also three dimensional, with higher amplification rates that increase with the interaction order. The growth of these modes accounts for the rapid spectral filling, and low-frequency modulation commonly observed in natural subharmonic transition. Starting from a ‘tuned' resonance, this scenario then provides a mechanism for the generation of a broad spectrum at the later stages of subharmonic mode transition. However, the results also suggest that with ‘natural’ transition, starting from low-amplitude broadband disturbances, the most likely 2D/3D resonance will be ‘detuned’.

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.

Three-dimensional acoustic-roughness receptivity of a boundary layer on an airfoil: experiment and direct numerical simulations

2003 ◽  
Vol 478 ◽  
pp. 135-163 ◽  
Author(s):  
W. WÜRZ ◽  
S. HERR ◽  
A. WÖRNER ◽  
U. RIST ◽  
S. WAGNER ◽  
...  
Keyword(s):  
Boundary Layer ◽  
Acoustic Wave ◽  
Three Dimensional ◽  
Low Frequency ◽  
Physical Nature ◽  
Neutral Stability ◽  
Acoustic Frequency ◽  
Stability Point ◽  
Tollmien Schlichting ◽  
Microscopic Surface

The paper is devoted to an experimental and numerical investigation of the problem of excitation of three-dimensional Tollmien–Schlichting (TS) waves in a boundary layer on an airfoil owing to scattering of an acoustic wave on localized microscopic surface non-uniformities. The experiments were performed at controlled disturbance conditions on a symmetric airfoil section at zero angle of attack. In each set of measurements, the acoustic wave had a fixed frequency fac, in the range of unstable TS-waves. The three-dimensional surface non-uniformity was positioned close to the neutral stability point at branch I for the two-dimensional perturbations. To avoid experimental difficulties in the distinction of the hot-wire signals measured at the same (acoustic) frequency but having a different physical nature, the surface roughness was simulated by a quasi-stationary surface non-uniformity (a vibrator) oscillating with a low frequency fv. This led to the generation of TS-wavetrains at combination frequencies f1,2=fac ∓ fv. The spatial behaviour of these wavetrains has been studied in detail for three different values of the acoustic frequency. The disturbances were decomposed into normal oblique TS-modes. The initial amplitudes and phases of these modes (i.e. at the position of the vibrator) were determined by means of an upstream extrapolation of the experimental data. The shape of the vibrator oscillations was measured by means of a laser triangulation device and mapped onto the Fourier space.

Numerical investigation of the interaction of the Klebanoff-mode with a Tollmien–Schlichting wave

2002 ◽  
Vol 450 ◽  
pp. 1-33 ◽  
Author(s):  
HERMANN F. FASEL
Keyword(s):  
Boundary Layer ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Low Frequency ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Free Stream Turbulence ◽  
Turbulent Transition ◽  
Tollmien Schlichting

Direct numerical simulations (DNS) of the Navier–Stokes equations are used to investigate the role of the Klebanoff-mode in laminar–turbulent transition in a flatplate boundary layer. To model the effects of free-stream turbulence, volume forces are used to generate low-frequency streamwise vortices outside the boundary layer. A suction/blowing slot at the wall is used to generate a two-dimensional Tollmien–Schlichting (TS) wave inside the boundary layer. The characteristics of the fluctuations inside the boundary layer agree very well with those measured in experiments. It is shown how the interaction of the Klebanoff-mode with the two-dimensional TS-wave leads to the formation of three-dimensional TS-wavepackets. When the disturbance amplitudes reach a critical level, a fundamental resonance-type secondary instability causes the breakdown of the TS-wavepackets into turbulent spots.

scholarly journals Ultrawide 3D Phononic Bandgap Metastructures as Broadband Low Frequency Filter

2020 ◽  
Author(s):  
Muhammad ◽  
C.W. Lim
Keyword(s):  
Acoustic Waves ◽  
Three Dimensional ◽  
Low Frequency ◽  
Mode Separation ◽  
Gap Ratio ◽  
Unique Material ◽  
Structural Configurations ◽  
Mass Spring ◽  
Low Amplitude ◽  
Global And Local

Abstract This present study reports a novel model for the study on three-dimensional phononic metastructures endued with ultrawide three-dimensional complete bandgaps. The phononic structure is made of a unique material without composite material composition. Based on the principal of mode separation and global and local modal masses participation, the well-engineered structural configurations give extremely wide bandgaps with the gap-to-mid gap ratio of 157.6% and 160.1% for two proposed prototypes that produce the widest three-dimensional bandgaps ever reported. The band structures are explained by a modal analysis and the findings are further corroborated by developing an analytical model based on monoatomic mass-spring chain and further verified with well-established FEM numerical simulations. Thanks to additive manufacturing, the prototypes are developed by using 3D printing technology and low amplitude vibration test is performed to access the real-time vibration mitigation characteristics. An excellent agreement is obtained between analytical, numerical and experimental results. The effects of material damping on transmission response is also taken into consideration that eventually merge the separated bandgaps to form a broadband vibration attenuation zone. The results reported are scale independent and the proposed strategy may pave the way for developing novel meta-devices to control the noise and vibration, and underwater acoustic waves at a wide frequency band in all directions.

Interaction of phase-locked modes: a new mechanism for the rapid growth of three-dimensional disturbances

1996 ◽  
Vol 316 ◽  
pp. 335-372 ◽  
Author(s):  
Xuesong Wu ◽  
Philip A. Stewart
Keyword(s):  
Rapid Growth ◽  
Three Dimensional ◽  
Phase Speed ◽  
Subharmonic Resonance ◽  
Subsequent Stage ◽  
Plane Shear ◽  
Quadratic Interaction ◽  
Symmetric Plane ◽  
The Difference ◽  
Planar Mode

In this paper, we have identified a new mechanism which can promote rapid growth of three-dimensional disturbances. The mechanism involves the interaction between a planar mode and an oblique mode, or a pair of oblique modes, which are phase-locked in the sense that they have the same phase speed. This allows a powerful nonlinear interaction to take place within the common critical layer(s). The disturbance is not required to form a subharmonic resonant triad, and hence the mechanism operates under much less restrictive conditions than does subharmonic resonance (although it is somewhat less powerful). We show that the quadratic interaction between the planar mode and the oblique modes drives an exceptionally large forced mode with the difference frequency, which in turn interacts with the planar mode to contribute the dominant nonlinear effect. This interaction can cause the oblique modes to grow super-exponentially provided that their magnitude is sufficiently small. As a result of the super-exponential growth, the oblique mode may soon become strong enough to produce a feedback effect on the planar mode, so that the interactions eventually become fully coupled. This subsequent stage takes slightly different forms depending on whether a single or a pair of oblique modes is present. Both cases are investigated. Particular attention is paid to symmetric plane shear layers, e.g. a planar wake or jet, for which subharmonic resonance of sinuous modes is inactive.

The resonant interaction of disturbances at laminar-turbulent transition in a boundary layer

1984 ◽  
Vol 138 ◽  
pp. 209-247 ◽  
Author(s):  
Yu. S. Kachanov ◽  
V. Ya. Levchenko
Keyword(s):  
Boundary Layer ◽  
Three Dimensional ◽  
Low Frequency ◽  
Wave Interaction ◽  
Resonant Interaction ◽  
Boundary Layer Transition ◽  
Layer Transition ◽  
Turbulent Transition ◽  
Random Priming ◽  
Tollmien Schlichting

The three-dimensional resonant interaction of a plane Tollmien-Schlichting wave, having a frequency f1, with a pair of oblique waves having frequencies ½ f1, was observed and studied experimentally. In the initial stages, the interaction proved to be a parametric resonance, resulting in the amplification of small random priming (background) oscillations of frequency ½ f1, and of a packet of low-frequency oscillations. The resonant interaction of waves in a boundary layer was investigated also by introducing a priming oscillation with frequency f’ = ½ f1 + Δf for different values of the frequency detuning Δf. The importance of the discovered wave interaction in boundary-layer transition is demonstrated. Causes of realization of different types of laminar-flow breakdown are discussed.

Towards long length scale unsteady modelling in turbomachines

2003 ◽  
Vol 217 (1) ◽  
pp. 75-82 ◽  
Author(s):  
L Xu ◽  
T. P. Hynes ◽  
J. D. Denton
Keyword(s):  
Body Force ◽  
Three Dimensional ◽  
Low Frequency ◽  
High Amplitude ◽  
Length Scale ◽  
Fine Mesh ◽  
Cfd Method ◽  
Coarse Meshes ◽  
Low Amplitude ◽  
Potential Interactions

The modelling issue of long length scale unsteadiness in turbomachines is discussed in this paper and a new three-dimensional CFD method with a hierarchy of body force models is proposed. A program has been developed to deal with the long length scale problem using efficient coarse meshes and large time steps, together with body force models. The distributed viscous body force can be calculated either using a very simple viscous wall shear stress model or extracted from a ‘stand-alone’ three-dimensional single passage steady calculation using a fine mesh. By eliminating the need to compute fine viscous scales, the proposed method is several orders of magnitude more efficient than ‘standard’ fine mesh N-S unsteady calculations while maintaining respectable resolution down to the scale of blade-to-blade variation, including wake/potential interactions between blade rows. Two sample cases with high-frequency low-amplitude and low-frequency high-amplitude, respectively, are used to illustrate the accuracy of the model.

scholarly journals On the unstable mode merging of gravity-inertial waves with Rossby waves

2011 ◽  
Vol 29 (8) ◽  
pp. 1377-1381 ◽  
Author(s):  
J. F. McKenzie
Keyword(s):  
High Frequency ◽  
Buoyancy Force ◽  
Rossby Waves ◽  
Unstable Mode ◽  
Low Frequency ◽  
Phase Speed ◽  
Intermediate Band ◽  
Long Wavelength ◽  
Wave Numbers ◽  
Inertial Wave

Abstract. We recapitulate the results of the combined theory of gravity-inertial-Rossby waves in a rotating, stratified atmosphere. The system is shown to exhibit a "local" (JWKB) instability whenever the phase speed of the low-frequency-long wavelength westward propagating Rossby wave exceeds the phase speed ("Kelvin" speed) of the high frequency-short wavelength gravity-inertial wave. This condition ensures that mode merging, leading to instability, takes place in some intermediate band of frequencies and wave numbers. The contention that such an instability is "spurious" is not convincing. The energy source of the instability resides in the background enthalpy which can be released by the action of the gravitational buoyancy force, through the combined wave modes.

Acoustic receptivity and evolution of two-dimensional and oblique disturbances in a Blasius boundary layer

2001 ◽  
Vol 432 ◽  
pp. 69-90 ◽  
Author(s):  
RUDOLPH A. KING ◽  
KENNETH S. BREUER
Keyword(s):  
Boundary Layer ◽  
Surface Roughness ◽  
Acoustic Wave ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Surface Waviness ◽  
S Wave ◽  
Boundary Layer Instability ◽  
Blasius Boundary Layer ◽  
Tollmien Schlichting

An experimental investigation was conducted to examine acoustic receptivity and subsequent boundary-layer instability evolution for a Blasius boundary layer formed on a flat plate in the presence of two-dimensional and oblique (three-dimensional) surface waviness. The effect of the non-localized surface roughness geometry and acoustic wave amplitude on the receptivity process was explored. The surface roughness had a well-defined wavenumber spectrum with fundamental wavenumber kw. A planar downstream-travelling acoustic wave was created to temporally excite the flow near the resonance frequency of an unstable eigenmode corresponding to kts = kw. The range of acoustic forcing levels, ε, and roughness heights, Δh, examined resulted in a linear dependence of receptivity coefficients; however, the larger values of the forcing combination εΔh resulted in subsequent nonlinear development of the Tollmien–Schlichting (T–S) wave. This study provides the first experimental evidence of a marked increase in the receptivity coefficient with increasing obliqueness of the surface waviness in excellent agreement with theory. Detuning of the two-dimensional and oblique disturbances was investigated by varying the streamwise wall-roughness wavenumber αw and measuring the T–S response. For the configuration where laminar-to-turbulent breakdown occurred, the breakdown process was found to be dominated by energy at the fundamental and harmonic frequencies, indicative of K-type breakdown.

scholarly journals Evolution of a narrow-band spectrum of random surface gravity waves

2003 ◽  
Vol 478 ◽  
pp. 1-10 ◽  
Author(s):  
KRISTIAN B. DYSTHE ◽  
KARSTEN TRULSEN ◽  
HARALD E. KROGSTAD ◽  
HERVÉ SOCQUET-JUGLARD
Keyword(s):  
Three Dimensional ◽  
Low Frequency ◽  
Three Dimensions ◽  
Band Spectrum ◽  
Nls Equation ◽  
Random Surface ◽  
Narrow Bandwidth ◽  
Quasi Stationary State ◽  
The Stability ◽  
Three Dimensional Simulations

Numerical simulations of the evolution of gravity wave spectra of fairly narrow bandwidth have been performed both for two and three dimensions. Simulations using the nonlinear Schrödinger (NLS) equation approximately verify the stability criteria of Alber (1978) in the two-dimensional but not in the three-dimensional case. Using a modified NLS equation (Trulsen et al. 2000) the spectra ‘relax’ towards a quasi-stationary state on a timescale (ε2ω0)−1. In this state the low-frequency face is steepened and the spectral peak is downshifted. The three-dimensional simulations show a power-law behaviour ω−4 on the high-frequency side of the (angularly integrated) spectrum.

Sign in / Sign up

Export Citation Format

Share Document