8 results
Experimental investigation of absolute instability of a rotating-disk boundary layer
- H. OTHMAN, T. C. CORKE
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- Journal:
- Journal of Fluid Mechanics / Volume 565 / 25 October 2006
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- 28 September 2006, pp. 63-94
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A series of experiments were performed to study the absolute instability of Type I travelling crossflow modes in the boundary layer on a smooth disk rotating at constant speed. The basic flow agreed with analytic theory, and the growth of natural disturbances matched linear theory predictions. Controlled temporal disturbances were introduced by a short-duration air pulse from a hypodermic tube located above the disk and outside the boundary layer. The air pulse was positioned just outboard of the linear-theory critical radius for Type I crossflow modes. A hot-wire sensor primarily sensitive to the azimuthal velocity component, was positioned at different spatial ($r,\theta$) locations on the disk to document the growth of disturbances produced by the air pulses. Ensemble averages conditioned on the air pulses revealed wave packets that evolved in time and space. Two amplitudes of air pulses were used. The lower amplitude was verified to produced wave packets with linear amplitude characteristics. The space–time evolution of the leading and trailing edges of the wave packets were followed past the critical radius for the absolute instability, $r_{c_{A}}$. With the lower amplitudes, the spreading of the disturbance wave packets did not continue to grow in time as $r_{c_{A}}$ was approached. Rather, the spreading of the trailing edge of the wave packet decelerated and asymptotically approached a constant. This result supports previous linear DNS simulations where it was concluded that the absolute instability does not produce a global mode and that linear disturbance wave packets are dominated by the convective instability. The larger-amplitude disturbances were found to produce larger temporal spreading of the wave packets. This was accompanied by a sharp growth in the wave packet amplitude past $r_{c_{A}}$. Explanations for this are discussed.
Resonance in axisymmetric jets with controlled helical-mode input
- T. C. Corke, S. M. Kusek
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- Journal:
- Journal of Fluid Mechanics / Volume 249 / April 1993
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- 26 April 2006, pp. 307-336
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This work involves active control of fundamental two- and three-dimensional amplified modes in an axisymmetric jet by introducing localized acoustic disturbances produced by an azimuthal array of miniature speakers placed close to the jet lip on the exit face. The independent control of each speaker output allowed different azimuthal amplitude and phase distributions of periodic inputs. The types of inputs used in this study consisted of conditions to force helical mode pairs with the same frequency and equal but opposite azimuthal wavenumbers, m = ±1, separately, or with axisymmetric (m = 0) modes. Three forcing conditions were studied in detail. The first consisted of a weakly amplified helical mode pair which was essentially ‘superposed’ with the natural jet instability modes. This provided a reference to the second case which consisted of the same helical mode pairs along with an axisymmetric mode at the harmonic streamwise wavenumber. This combination led to the resonant growth of the otherwise weakly (linear) amplified subharmonic helical modes. A weakly nonlinear three-wave amplitude evolution equation with a coupling coefficient derived from the data was found to model the enhanced growth of the subharmonic helical modes well. The third case consisted of forcing only m = ±1 helical modes at a frequency which was close to the most amplified. This was compared to the results of Corke et al. (1991) who forced an axisymmetric mode at the same frequency and found it to lead to the enhanced growth of near-subharmonic modes, as well as numerous sum and difference modes. The helical modes had effects identical to the previous work and confirmed the resonant amplification of a near-subharmonic mode. The amplitude development was also well represented by the nonlinear amplitude equation, including the dependence of the streamwise amplification rate on the azimuthal change in the fundamental-mode initial amplitude. However, the coupling coefficient in this case was approximately one-third that with exact fundamental-subharmonic resonance. Finally we offer some explanation for the selection of the different mode frequencies in this case.
Mode selection and resonant phase locking in unstable axisymmetric jets
- T. C. Corke, F. Shakib, H. M. Nagib
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- Journal:
- Journal of Fluid Mechanics / Volume 223 / February 1991
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- 26 April 2006, pp. 253-311
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This paper presents experimental results on the nonlinear phase locking present in the resonant growth of unstable modes in the shear layer of an axisymmetric jet. The initial instability modes scale with the exiting shear layer and grow convectively with downstream distance. Because of the special condition at the exit lip of the jet, the initial growth of modes is very sensitive to local unsteady pressure fields. A part of the unsteady field is stochastic in nature. To a larger extent, the pressure field at the lip of the jet contains the imprint of the downstream-developing instability modes, in particular the first unstable axisymmetric mode and its subharmonic. These are felt at the lip of the jet as a result of the energetic processes of the first vortex rollup and vortex pairing. As a result, a resonant feedback exists which under special conditions makes the initial region of this flow in some sense absolutely unstable. The features of this process are brought out by the normalized crossbispectrum or cross-bicoherence between the instantaneous unsteady pressure at the lip of the jet and velocity time series measured at the same azimuthal position for different downstream locations. These give a measure of the nonlinear phase locking between the principle modes and their sum and difference modes. Analysis of these show a perfect nonlinear phase locking at the fundamental axisymmetric and subharmonic frequencies between the pressure field at the lip and the velocity field at the downstream locations corresponding to the energy saturations of the fundamental and subharmonic modes. This resonance process can be suppressed or enhanced by low-amplitude axisymmetric mode forcing at the natural preferred frequency of slightly detuned cases. Contrasted to this is the behaviour of the fundamental m = ± 1 helical mode. This mode was found to have the same spatial growth rate as the axisymmetric mode and a streamwise frequency approximately 20 % higher, in agreement with theoretical predictions. However, short-time spectral estimates showed that these two fundamental modes do not exist at the same time or space. This suggests that each is a basin of attraction which suppresses the existence of the other. The apparent non-deterministic switching observed between these modes is probably the result of the response of the jet to stochastic input of axisymmetric or non-axisymmetric disturbances. This scenario may lead to a low-dimensional temporal model based on the interaction between these two modes which captures most of the early random nature seen in our experiments.
Resonant growth of three-dimensional modes in Falkner–Skan boundary layers with adverse pressure gradients
- T. C. Corke, S. Gruber
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- Journal:
- Journal of Fluid Mechanics / Volume 320 / 10 August 1996
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- 26 April 2006, pp. 211-233
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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 [les ] βH [les ] -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).
Resonant growth of three-dimensional modes in trnsitioning Blasius boundary layers
- T. C. Corke, R. A. Mangano
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- Journal:
- Journal of Fluid Mechanics / Volume 209 / December 1989
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- 26 April 2006, pp. 93-150
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By carefully controlled phase-coupled input of simultaneous two- and three-dimensional disturbances, the nonlinear evolution and breakdown of the laminar flow in a boundary layer was examined. This involved the generation of plane Tollmien–Schlichting waves and pairs of oblique waves so as to promote nearresonance conditions which have been theoretically shown to lead to the rapid development of three-dimensionality in unstble boundary layers. Special emphasis is placed on the two prominent mechanisms, namely resonant-triads of Orr–Sommerfeld modes and the secondary instability of the streamwise periodic flow to spanwise periodic three-dimensional disturbances. The sensitivity of these mechanisms on the amplitudes and wavenumbers of the input disturbances was of special focus.
The simultaneous two- and three-dimensional wave generation was accomplished using a spanwise array of line heaters suspended just above the wall at the approximate height of the critical layer in the laminar boundary layer. These were operated to produce, through local heating, time-periodic spanwise-phase-varying velocity perturbations. Of primary emphasis in this paper are conditions obtained by the combined forcing of fundamental plane waves with wavenumbers (α, 0) and pairs of subharmonic oblique waves (½α, ± β). The reslults document resonant growth of energy in the subharmonic modes, the formation of staggered lambda vortex patterns with a cross-stream scale commensurate with the seeded ± β condition, and their subsequet transition to turbulence. Complete documentation of the flow field at these various stages is presented using smoke-wire flow visualization and through phase-conditioned hot-wire surveys measuring all three velocity components in three space dimensions.
Three-dimensional-mode resonance in far wakes
- T. C. Corke, J. D. Krull, M. Ghassemi
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- Journal:
- Journal of Fluid Mechanics / Volume 239 / June 1992
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- 26 April 2006, pp. 99-132
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This work is aimed at understanding mechanisms which govern the growth of secondary three-dimensional modes of a particular type which feed from a resonant energy exchange with the primary Kármán instability in two-dimensional wakes. Our approach was to introduce controlled time-periodic three-dimensional (oblique) wave pairs of equal but opposite sign, simultaneously with a two-dimensional wave. The waves were introduced by an array of v-component-producing elements on the top and bottom surfaces of the body. These were formed by metallized electrodes which were vapour deposited onto a piezoelectrically active polymer wrapped around the surface. The amplitudes, streamwise and spanwise wavenumbers, and initial phase difference are all individually controllable. The initial work focused on a fundamental/subharmonic interaction, and the dependence on spanwise wave-number. The results include mode eigenfunction modulus and phase distributions in space, and stream functions for the phase-reconstructed flow field. Analysis of these shows that such a resonance mechanism exists and its features can account for characteristic changes associated with the growth of three-dimensional structures in the wake of two-dimensional bodies.
Acoustic receptivity of the boundary layer over parabolic bodies at angles of attack
- O. M. HADDAD, E. ERTURK, T. C. CORKE
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- Journal:
- Journal of Fluid Mechanics / Volume 536 / 10 August 2005
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- 26 July 2005, pp. 377-400
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The effect of angle of attack on the acoustic receptivity of the boundary layer over two-dimensional parabolic bodies is investigated using a spatial solution of the Navier–Stokes equations. The free stream is decomposed into a uniform flow with a superposed periodic velocity fluctuation of small amplitude. The method follows that of Haddad & Corke (1998) and Erturk & Corke (2001) in which the solution for the basic flow and linearized perturbation flow are solved separately. Different angles of incidence of the body are investigated for three leading-edge radii Reynolds numbers. For each, the angle of attack ranges from $0^{\circ}$ to past the angle where the mean flow separates. The results then document the effect of the angle of incidence on the leading-edge receptivity coefficient ($K_{{\hbox{\scriptsize{\it LE}}}}$), and in the case of the mean flow separation, on the amplitude of Tollmien–Schlichting (T-S) waves at the linear stability Branch II location ($K_{II}$). For angles of attack before separation, we found that the leading-edge receptivity coefficient, $K_{{\hbox{\scriptsize{\it LE}}}}$, increased with angle of incidence which correlated with an increase in the pressure gradient at the physical leading edge. When a separation zone formed at larger angles of incidence, it became a second site of receptivity with a receptivity coefficient that exceeded that of the leading edge. This resulted in dramatic growth of the T-S waves with Branch II amplitudes more than 100 times larger than those at angles just before separation, and 1000 times more than those at $0^{\circ}$ angle of attack.
Stationary travelling cross-flow mode interactions on a rotating disk
- T. C. CORKE, K. F. KNASIAK
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- Journal:
- Journal of Fluid Mechanics / Volume 355 / 25 January 1998
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- 25 January 1998, pp. 285-315
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This work involves the study of the development of Type 1 stationary and travelling cross-flow modes in the three-dimensional boundary layer over a rotating disk. In order to control the characteristics of the stationary modes, we utilized organized patterns of roughness which were applied to the disk surface. These consisted of ink dots which were equally spaced in the azimuthal direction at a fixed radius in order to enhance particular azimuthal wavenumbers. Logarithmic spiral patterns of dots were also used to enhance azimuthal wave angles. Velocity fluctuation time series were decomposed into the components corresponding to the stationary and travelling modes using the instantaneous disk position as a reference. Their development was documented through the linear and nonlinear stages leading to turbulence. The linear stage agreed well with linear stability predictions for both modes. In the nonlinear stage we documented a triad coupling between pairs of travelling modes and a stationary mode. The strongest of these was a difference interaction which led to the growth of a low-azimuthal-number, stationary mode. This mode had the largest amplitude and appeared to dominate transition. In retrospect, we can observe the signs of this mechanism in past flow visualization (Kobayashi, Kohama & Takamadate 1980), and it can account for the ‘jagged’ front normally associated with cross-flow-dominated transition on swept wings.