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Analytical and numerical study of the nonlinear interaction between a point vortex and a wall-bounded shear layer
- OLIVER V. ATASSI
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- Journal:
- Journal of Fluid Mechanics / Volume 373 / 25 October 1998
- Published online by Cambridge University Press:
- 25 October 1998, pp. 155-192
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The unsteady interaction between a vortex and a wall-bounded vorticity layer is studied as a model for transport and mixing between rotational and irrotational flows. The problem is formulated in terms of contour integrals and a kinematic condition along the interface which demarcates the vortical and potential regions. Asymptotic solutions are derived for linear, weakly nonlinear and nonlinear long-wave approximations. The solutions show that the initial process of ejection of vorticity into the irrotational flow occurs at a stationary point along the interface. A nonlinear model is derived and shows that such a stationary point is more likely to exist when the circulation of the vortex is counter to the vorticity in the layer. A Lagrangian numerical method based on contour dynamics is then developed for the general nonlinear problem. Two sets of results are presented where for every initial height of the vortex its magnitude and sign are varied. In both sets, it is observed that when the magnitude of the vortex is held constant a much stronger interaction occurs when the sign of the vortex circulation is opposite to that of the vorticity in the layer. Moreover, when the horizontal velocity of the vortex is close to the velocity of the interfacial waves a strong nonlinear interaction between the vortex and the layer ensues and results in the ejection of thin filaments of vorticity into the irrotational flow. In order to study the dynamical consequences of strong unsteady interaction, the wall pressure distribution is computed. The results indicate that a significant rise in the magnitude of the wall pressure is associated with ejection of vorticity from the wall. The present analysis confirms that coherent vortical structures in the outer layer of a turbulent boundary layer can cause ejection of concentrated wall-layer vorticity and explains how and when this process occurs.
The interaction of a point vortex with a wall-bounded vortex layer
- OLIVER V. ATASSI, ANDREW J. BERNOFF, SETH LICHTER
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- Journal:
- Journal of Fluid Mechanics / Volume 343 / 25 July 1997
- Published online by Cambridge University Press:
- 25 July 1997, pp. 169-195
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The interaction of a point vortex with a layer of constant vorticity, bounded below by a wall and above by an irrotational flow, is investigated as a model of vortex–boundary layer interaction. This model calculates both the evolution of the interface which separates the vortex layer from the irrotational flow and the trajectory of the vortex. In order to determine the conditions which lead to sustained unsteady interaction, three cases are investigated where the mutual interaction between the vortex and interface is initially assumed to be weak. (i) When a weak point vortex lies outside the layer, the vortex moves with a horizontal speed that is small relative to the long-wave phase speed of interfacial waves. A uniformly valid solution is found for the interface evolution. This solution shows that for long times the interface and the vortex approach an equilibrium state. (ii) When a weak vortex lies inside the layer, the vortex is convected by the mean flow and moves with a horizontal speed which matches the phase speed of an interfacial wave. This results in a strong interaction between the vortex and the interfacial wave. On the interface, a monochromatic wavetrain forms upstream of the vortex and acts to attract or repel the point vortex. The displacement of the vortex due to the wavetrain results in the modulation of the amplitude and wavelength of the wavetrain. If the point vortex is attracted toward the interface the horizontal speed of the vortex slows and disturbances directly above the vortex focus and grow leading to the ejection of vorticity. (iii) When the point vortex lies close to the wall and it is sufficiently strong it propagates downstream with a large horizontal velocity. In this case, the amplitude of the interfacial disturbance is independent of the vortex strength. Again, the vortex and the interface approach an equilibrium state. The results of this paper indicate that when the horizontal speed of the vortex matches the phase speed of the interfacial disturbance, it is necessary to account for the vertical displacement of the vortex in order to predict the behaviour of vortex–boundary layer interactions.