scholarly journals Interaction of oblique instability waves with weak streamwise vortices

1995 ◽  
Vol 284 ◽  
pp. 377-407 ◽  
Author(s):  
M. E. Goldstein ◽  
David W. Wundrow
Keyword(s):  
Streamwise Vortices ◽  
Rayleigh Instability ◽  
Mean Flow ◽  
Flow Distortion ◽  
Instability Waves ◽  
Instability Modes ◽  
Blasius Boundary Layer ◽  
The Common ◽  
Tollmien Schlichting ◽  
Streamwise Distance

This paper is concerned with the effect of a weak spanwise-variable mean-flow distortion on the growth of oblique instability waves in a Blasius boundary layer. The streamwise component of the distortion velocity initially grows linearly with increasing streamwise distance, reaches a maximum, and eventually decays through the action of viscosity. This decay occurs slowly and allows the distortion to destabilize the Blasius flow over a relatively large streamwise region. It is shown that even relatively weak distortions can cause certain oblique Rayleigh instability waves to grow much faster than the usual two-dimensional Tollmien–Schlichting waves that would be the dominant instability modes in the absence of the distortion. The oblique instability waves can then become large enough to interact nonlinearly within a common critical layer. It is shown that the common amplitude of the interacting oblique waves is governed by the amplitude evolution equation derived in Goldstein & Choi (1989). The implications of these results for Klebanoff-type transition are discussed.

scholarly journals Stability of zero-pressure-gradient boundary layer distorted by unsteady Klebanoff streaks

2011 ◽  
Vol 681 ◽  
pp. 116-153 ◽  
Author(s):  
NICHOLAS J. VAUGHAN ◽  
TAMER A. ZAKI
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Low Frequency ◽  
Finite Amplitude ◽  
Base Flow ◽  
Streamwise Vorticity ◽  
Mean Flow ◽  
S Wave ◽  
Flow Distortion ◽  
Tollmien Schlichting

The secondary instability of a zero-pressure-gradient boundary layer, distorted by unsteady Klebanoff streaks, is investigated. The base profiles for the analysis are computed using direct numerical simulation (DNS) of the boundary-layer response to forcing by individual free-stream modes, which are low frequency and dominated by streamwise vorticity. Therefore, the base profiles take into account the nonlinear development of the streaks and mean flow distortion, upstream of the location chosen for the stability analyses. The two most unstable modes were classified as an inner and an outer instability, with reference to the position of their respective critical layers inside the boundary layer. Their growth rates were reported for a range of frequencies and amplitudes of the base streaks. The inner mode has a connection to the Tollmien–Schlichting (T–S) wave in the limit of vanishing streak amplitude. It is stabilized by the mean flow distortion, but its growth rate is enhanced with increasing amplitude and frequency of the base streaks. The outer mode only exists in the presence of finite amplitude streaks. The analysis of the outer instability extends the results of Andersson et al. (J. Fluid Mech. vol. 428, 2001, p. 29) to unsteady base streaks. It is shown that base-flow unsteadiness promotes instability and, as a result, leads to a lower critical streak amplitude. The results of linear theory are complemented by DNS of the evolution of the inner and outer instabilities in a zero-pressure-gradient boundary layer. Both instabilities lead to breakdown to turbulence and, in the case of the inner mode, transition proceeds via the formation of wave packets with similar structure and wave speeds to those reported by Nagarajan, Lele & Ferziger (J. Fluid Mech., vol. 572, 2007, p. 471).

On the weakly nonlinear development of Tollmien-Schlichting wavetrains in boundary layers

1996 ◽  
Vol 323 ◽  
pp. 133-171 ◽  
Author(s):  
Xuesong Wu ◽  
Philip A. Stewart ◽  
Stephen J. Cowley
Keyword(s):  
Numerical Solutions ◽  
Asymptotic Methods ◽  
Nonlinear Effects ◽  
Mean Flow ◽  
Flow Distortion ◽  
Weakly Nonlinear ◽  
Nonlinear Development ◽  
Finite Distance ◽  
Tollmien Schlichting ◽  
High Reynolds

The nonlinear development of a weakly modulated Tollmien-Schlichting wavetrain in a boundary layer is studied theoretically using high-Reynolds-number asymptotic methods. The ‘carrier’ wave is taken to be two-dimensional, and the envelope is assumed to be a slowly varying function of time and of the streamwise and spanwise variables. Attention is focused on the scalings appropriate to the so-called ‘upper branch’ and ‘high-frequency lower branch’. The dominant nonlinear effects are found to arise in the critical layer and the surrounding ‘diffusion layer’: nonlinear interactions in these regions can influence the development of the wavetrain by producing a spanwise-dependent mean-flow distortion. The amplitude evolution is governed by an integro-partial-differential equation, whose nonlinear term is history-dependent and involves the highest derivative with respect to the spanwise variable. Numerical solutions show that a localized singularity can develop at a finite distance downstream. This singularity seems consistent with the experimentally observed focusing of vorticity at certain spanwise locations, although quantitative comparisons have not been attempted.

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 Low-frequency sound radiated by a nonlinearly modulated wavepacket of helical modes on a subsonic circular jet

2009 ◽  
Vol 637 ◽  
pp. 173-211 ◽  
Author(s):  
XUESONG WU ◽  
PATRICK HUERRE
Keyword(s):  
Acoustic Field ◽  
Low Frequency ◽  
Base Flow ◽  
Matched Asymptotic Expansion ◽  
Sound Waves ◽  
Mean Flow ◽  
Flow Distortion ◽  
Experimental Conditions ◽  
Modulation Equation ◽  
Instability Modes

A possible fundamental physical mechanism by which instability modes generate sound waves in subsonic jets is presented in the present paper. It involves a wavepacket of a pair of helical instability modes with nearly the same frequencies but opposite azimuthal wavenumbers. As the wavepacket undergoes simultaneous spatial–temporal development in a circular jet, the mutual interaction between the helical modes generates a strong three-dimensional, slowly modulating ‘mean-flow distortion’. It is demonstrated that this ‘mean field’ radiates sound waves to the far field. The emitted sound is of very low frequency, with characteristic time and length scales being comparable with those of the envelope of the wavepacket, which acts as a non-compact source. A matched-asymptotic-expansion approach is used to determine, in a self-consistent manner, the acoustic field in terms of the envelope of the wavepacket and a transfer factor characterizing the refraction effect of the background base flow. For realistic jet spreading rates, the nonlinear development of the wavepacket is found to be influenced simultaneously by non-parallelism and non-equilibrium effects, and so a composite modulation equation including both effects is constructed in order to describe the entire growth–attenuation–decay cycle. Parametric studies pertaining to relevant experimental conditions indicate that the acoustic field is characterized by a single-lobed directivity pattern beamed at an angle about 45°–60° to the jet axis and a broadband spectrum centred at a Strouhal number St ≈ 0.07–0.2. As the nonlinear effect increases, the radiation becomes more efficient and the noise spectrum broadens, but the gross features of the acoustic field remain robust, and are broadly in agreement with experimental observations.

Local boundary-layer receptivity to a convected free-stream disturbance

1999 ◽  
Vol 378 ◽  
pp. 291-317 ◽  
Author(s):  
A. J. DIETZ
Keyword(s):  
Experimental Data ◽  
Boundary Layer ◽  
Experimental Verification ◽  
Free Stream ◽  
Linear Stability Theory ◽  
Mean Flow ◽  
Flow Distortion ◽  
Instability Waves ◽  
Free Stream Turbulence ◽  
Local Boundary

An investigation of the local receptivity of a Blasius boundary layer to a harmonic vortical disturbance is presented as a step towards understanding boundary-layer receptivity to free-stream turbulence. Although there has been solid experimental verification of the linear theory describing acoustic receptivity of boundary layers, this was the first experimental verification of the mechanism behind local receptivity to a convected disturbance. The harmonic wake from a vibrating ribbon positioned upstream of a flat plate provided the free-stream disturbance. Two-dimensional roughness elements on the surface of the plate acted as a local receptivity site. Hot-wire measurements in the boundary layer downstream of the roughness confirmed the generation of Tollmien–Schlichting (TS) instability waves by an outer-layer interaction between the long-wavelength convected disturbance and the short-scale mean-flow distortion due to the roughness. The characteristics of the instability waves were carefully measured to ensure that their behaviour was correctly modelled by linear stability theory. This theory was then used to determine the immeasurably small initial wave amplitudes resulting from the receptivity process, from wave amplitudes measured downstream. Tests were performed to determine the range of validity of the linear assumptions made in current receptivity theories. Experimental data obtained in the linear regime were then compared to theoretical results of other authors by expressing the experimental data in the form of an efficiency function which is independent of the free-stream amplitude, roughness height and roughness geometry. Reasonable agreement between the experimental and theoretical efficiency functions was obtained over a range of frequencies and Reynolds numbers.

Oblique transition in a laminar Blasius boundary layer

2002 ◽  
Vol 453 ◽  
pp. 177-200 ◽  
Author(s):  
RUDOLPH A. KING ◽  
KENNETH S. BREUER
Keyword(s):  
Boundary Layer ◽  
Streamwise Velocity ◽  
Wall Region ◽  
Mean Flow ◽  
Flow Distortion ◽  
Velocity Fluctuations ◽  
Stationary Structure ◽  
Transition Mechanism ◽  
Inflection Points ◽  
Blasius Boundary Layer

Oblique transition was experimentally investigated in a Blasius boundary layer formed on a flat plate. This transition mechanism was provoked by exciting a pair of oppositely oriented oblique Orr–Sommerfeld (O–S) modes given by (ω/ωts, ±β/βts) = (1, ±1) in the frequency-wavenumber (spanwise) space. Surface waviness with height Δh and a well-defined wavenumber spectrum that is synchronized with the neutral O–S wavenumber at Branch I, (αw, ±βw) = (αts,I, ±βts,I), was used to provide a steady velocity perturbation in the near-wall region. A planar downstream-travelling acoustic wave of amplitude ε was created to temporally excite the flow near the resonance frequency, ωts(= 2πfo), of an unstable eigenmode corresponding to kts = kw (where k =±[α2+β2]1/2). Possible mechanisms leading to laminar-to-turbulent breakdown were examined for various forcing combinations, εΔh. For small values of εΔh, a peak-valley structure corresponding to a spanwise wavenumber of 2βw was observed. As expected, the maximum r.m.s. narrow-band streamwise velocity fluctuations, ut(fo), occur at peak locations, which correspond to regions with mean streamwise velocity, U, deficits. For the largest value of εΔh, significant mean-flow distortion was observed in the spanwise profiles of U. Large spanwise velocity gradients, [mid ]dU/dζ[mid ], exist between peaks and valleys and appear to generate an explosive growth in the velocity fluctuations. The maximum values of ut no longer occur at peak locations of the stationary structure but at locations of spanwise inflection points. The magnitude of ut scales with [mid ]dU/dζ[mid ]. A nonlinear interaction of two non-stationary modes was conjectured as a possible mechanism for the enhancement of the streak ampliflcation rate.

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.

Natural Transition Phenomena on an Axial Compressor Blade

2000 ◽  
Author(s):  
J. D. Hughes ◽  
G. J. Walker
Keyword(s):  
Guide Vane ◽  
Axial Compressor ◽  
Inlet Guide Vane ◽  
Instability Wave ◽  
Blade Loading ◽  
Instability Waves ◽  
Transition Prediction ◽  
Tollmien Schlichting ◽  
Vane Clocking ◽  
Hot Film

Data from a surface hot-film array on the outlet stator of a 1.5 stage axial compressor are analyzed to look for direct evidence of natural transition phenomena. An algorithm is developed to identify instability waves within the Tollmien Schlichting (T-S) frequency range. The algorithm is combined with a turbulent intermittency detection routine to produce space∼time diagrams showing the probability of instability wave occurrence prior to regions of turbulent flow. The paper compares these plots for a range of blade loading, with free-stream conditions corresponding to the maximum and minimum inflow disturbance periodicity produced by inlet guide vane clocking. Extensive regions of amplifying instability waves are identified in nearly all cases. The implications for transition prediction in decelerating flow regions on axial turbomachine blades are discussed.

scholarly journals Influence of the Atmospheric Surface Layer on a Turbulent Flow Downstream of a Ship Superstructure

2013 ◽  
Vol 30 (8) ◽  
pp. 1803-1819 ◽  
Author(s):  
Luksa Luznik ◽  
Cody J. Brownell ◽  
Murray R. Snyder ◽  
Hyung Suk Kang
Keyword(s):  
Surface Layer ◽  
Atmospheric Surface Layer ◽  
The United States ◽  
Naval Academy ◽  
Mean Flow ◽  
Flow Distortion ◽  
Turbulent Statistics ◽  
Sonic Anemometers ◽  
Flight Deck ◽  
Inflow Conditions

Abstract This paper describes a set of turbulence measurements at sea in the area of high flow distortion in the near-wake and recirculation zone behind a ship's superstructure that is similar in geometry to a helicopter hangar/flight deck arrangement found on many modern U.S. Navy ships. The instrumented ship is a 32-m-long training vessel operated by the United States Naval Academy that has been modified by adding a representative flight deck and hangar structure. The flight deck is instrumented with up to seven sonic anemometers/thermometers that are used to obtain simultaneous velocity measurements at various spatial locations on the flight deck, and one sonic anemometer at bow mast is used to characterize inflow atmospheric boundary conditions. Data characterizing wind over the deck at an incoming angle of 0° (head winds) and wind speeds from 2 to 10 m s−1 obtained in the Chesapeake Bay are presented and discussed. Turbulent statistics of inflow conditions are analyzed using the Kaimal universal turbulence spectral model for the atmospheric surface layer and show that for the present dataset this approach eliminates the need to account for platform motion in computing variances and covariances. Conditional sampling of mean flow and turbulence statistics at the flight deck indicate no statistically significant variations between unstable, stable, and neutral atmospheric inflow conditions, and the results agree with the published data for flows over the backward-facing step geometries.

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