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Nonlinear resonances in a laminar wall jet: ejection of dipolar vortices

Published online by Cambridge University Press:  24 September 2007

STEFAN WERNZ  and
HERMANN F. FASEL
STEFAN WERNZ
Affiliation:
Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ 85721, USA
HERMANN F. FASEL
Affiliation:
Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ 85721, USA
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Abstract

Nonlinear mechanisms leading to the ejection of dipolar vortices from a laminar wall jet are being investigated using highly accurate Navier–Stokes simulations. With a set of well-defined numerical experiments for a forced Glauert wall jet, the nonlinear resonant interaction between the large-amplitude harmonic disturbance and a small-amplitude wave packet is systematically explored using two-dimensional simulations. Generated by a small-amplitude pulse, the wave packet experiences rapid resonant growth in the subharmonic part of its spectrum resulting in vortex mergings and, ultimately, the ejection of a pair of counter-rotating vortices from the wall jet. This two-dimensional subharmonic instability, if not mitigated by competing three-dimensional instabilities, can lead to the detachment of the entire wall jet from the surface. As shown using three-dimensional direct numerical simulations, vortex ejection still occurs in a forced transitional wall jet if the two-dimensional wave packet can reach a large amplitude level upstream of the region of three-dimensional turbulent breakdown. Movies are available with the online version of the paper.

JFM classification

Vortex Flows: Vortex shedding Instability: Parametric instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 588 , 10 October 2007 , pp. 279 - 308
Copyright
Copyright © Cambridge University Press 2007

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References

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Wernz and Fasel supplementary movie

Movie 1. Ejection of a vortex dipole from a harmonically forced wall jet in response to additional forcing with a small-amplitude, broad-spectrum pulse disturbance (Case 3). Time-evolution of vorticity (represented by colour contours) from time t=tp to tp+40. The pulse disturbance leads to the development of a downstream travelling wave packet that is strongly amplified due to resonant interaction with the the primary harmonic disturbance wave (subharmonic resonance). The wave packet thus disrupts the double vortex row of the primary disturbance, resulting in repeated vortex mergings and, eventually, in the ejection of a dipolar vortex pair from the wall jet.

Download Wernz and Fasel supplementary movie(Video)
Video 4.5 MB

Wernz and Fasel supplementary movie

Movie 1. Ejection of a vortex dipole from a harmonically forced wall jet in response to additional forcing with a small-amplitude, broad-spectrum pulse disturbance (Case 3). Time-evolution of vorticity (represented by colour contours) from time t=tp to tp+40. The pulse disturbance leads to the development of a downstream travelling wave packet that is strongly amplified due to resonant interaction with the the primary harmonic disturbance wave (subharmonic resonance). The wave packet thus disrupts the double vortex row of the primary disturbance, resulting in repeated vortex mergings and, eventually, in the ejection of a dipolar vortex pair from the wall jet.

Download Wernz and Fasel supplementary movie(Video)
Video 6.6 MB

Wernz and Fasel supplementary movie

Movie 2. Dependence of wave-packet development on the amplitude of the pulse. Top: smaller-amplitude broad-spectrum pulse (Case 4); bottom: larger-amplitude broad-spectrum pulse (Case 5). Time-evolution of vorticity (represented by colour contours) from time t=tp to tp+40. Note that only one frame per fundamental forcing period is shown. Due to its larger initial amplitude, the wave packet in Case 5 reaches the nonlinear saturation level and leads to vortex ejection much farther upstream than the wave packet in Case 4.

Download Wernz and Fasel supplementary movie(Video)
Video 925.7 KB

Wernz and Fasel supplementary movie

Movie 2. Dependence of wave-packet development on the amplitude of the pulse. Top: smaller-amplitude broad-spectrum pulse (Case 4); bottom: larger-amplitude broad-spectrum pulse (Case 5). Time-evolution of vorticity (represented by colour contours) from time t=tp to tp+40. Note that only one frame per fundamental forcing period is shown. Due to its larger initial amplitude, the wave packet in Case 5 reaches the nonlinear saturation level and leads to vortex ejection much farther upstream than the wave packet in Case 4.

Download Wernz and Fasel supplementary movie(Video)
Video 1.4 MB

Wernz and Fasel supplementary movie

Movie 3. Dependence of wave-packet development on pulse frequency. Top: a pulse with spectral amplitude peak at 1/2 the frequency of the harmonic disturbance (Case 6); bottom: a pulse with amplitude peak at 3/2 the frequency of the harmonic disturbance (Case 7). Time-evolution of vorticity (represented by colour contours) from time t=tp to tp+40. Note that only one frame per fundamental forcing period is shown. While in both cases subharmonic resonance leads to rapid disturbance growth, receptivity and initial linear growth near the forcing location is far greater for the high-frequency pulse in Case 7 than for the low-frequency pulse in Case 6. As a result, vortex ejection in Case 7 occurs much farther upstream than in Case 6.

Download Wernz and Fasel supplementary movie(Video)
Video 919 KB

Wernz and Fasel supplementary movie

Movie 3. Dependence of wave-packet development on pulse frequency. Top: a pulse with spectral amplitude peak at 1/2 the frequency of the harmonic disturbance (Case 6); bottom: a pulse with amplitude peak at 3/2 the frequency of the harmonic disturbance (Case 7). Time-evolution of vorticity (represented by colour contours) from time t=tp to tp+40. Note that only one frame per fundamental forcing period is shown. While in both cases subharmonic resonance leads to rapid disturbance growth, receptivity and initial linear growth near the forcing location is far greater for the high-frequency pulse in Case 7 than for the low-frequency pulse in Case 6. As a result, vortex ejection in Case 7 occurs much farther upstream than in Case 6.

Download Wernz and Fasel supplementary movie(Video)
Video 1.4 MB

Wernz and Fasel supplementary movie

Movie 4. Dependence of wave-packet development on the relative phase angle between pulse disturbance and harmonic forcing. Top: less favourable phase angle (Case 8); bottom: more favourable phase angle (Case 9). Time-evolution of vorticity (represented by colour contours) from time t=tp to tp+40. Note that only one frame per fundamental forcing period is shown. Deployment of the pulse disturbance at an unfavourable phase angle relative to the primary harmonic disturbance can delay the onset of subharmonic resonant growth and, as a consequence, shift the ejection location farther downstream. This is particularly true for Cases 8 and 9 where the amplitude of the harmonic fundamental disturbance is very high, four times higher than in Cases 4-7.

Download Wernz and Fasel supplementary movie(Video)
Video 1.1 MB

Wernz and Fasel supplementary movie

Movie 4. Dependence of wave-packet development on the relative phase angle between pulse disturbance and harmonic forcing. Top: less favourable phase angle (Case 8); bottom: more favourable phase angle (Case 9). Time-evolution of vorticity (represented by colour contours) from time t=tp to tp+40. Note that only one frame per fundamental forcing period is shown. Deployment of the pulse disturbance at an unfavourable phase angle relative to the primary harmonic disturbance can delay the onset of subharmonic resonant growth and, as a consequence, shift the ejection location farther downstream. This is particularly true for Cases 8 and 9 where the amplitude of the harmonic fundamental disturbance is very high, four times higher than in Cases 4-7.

Download Wernz and Fasel supplementary movie(Video)
Video 1.9 MB

Wernz and Fasel supplementary movie

Movie 5. Wave-packet development during wall-jet transition. A forced wall jet perturbed by small-amplitude three-dimensional random perturbations and by a small-amplitude (two-dimensional) high-frequency pulse (Case 7, 3-D). Vortical structures represented by the Q-criterion (iso-surfaces of Q=0.002) from time t=tp to tp+40. Note that only one frame for every five fundamental forcing periods is shown. During transiton to turbulence, the subharmonic resonance mechanism causing the rapid amplitude growth of the two-dimensional wave packet is in competition with three-dimensional breakdown mechanisms leading to the breakup of two-dimensional disturbances into three-dimensional turbulent motion. For Case 7, subharmonic resonance 'wins' this competition leading to vortex merging and vortex ejection ahead of the region of turbulent breakdown.

Download Wernz and Fasel supplementary movie(Video)
Video 505.3 KB

Wernz and Fasel supplementary movie

Movie 5. Wave-packet development during wall-jet transition. A forced wall jet perturbed by small-amplitude three-dimensional random perturbations and by a small-amplitude (two-dimensional) high-frequency pulse (Case 7, 3-D). Vortical structures represented by the Q-criterion (iso-surfaces of Q=0.002) from time t=tp to tp+40. Note that only one frame for every five fundamental forcing periods is shown. During transiton to turbulence, the subharmonic resonance mechanism causing the rapid amplitude growth of the two-dimensional wave packet is in competition with three-dimensional breakdown mechanisms leading to the breakup of two-dimensional disturbances into three-dimensional turbulent motion. For Case 7, subharmonic resonance 'wins' this competition leading to vortex merging and vortex ejection ahead of the region of turbulent breakdown.

Download Wernz and Fasel supplementary movie(Video)
Video 1 MB