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Steady streaming confined between three-dimensional wavy surfaces

Published online by Cambridge University Press:  03 August 2010

ROMAIN GUIBERT ,
FRANCK PLOURABOUÉ  and
ALAIN BERGEON
ROMAIN GUIBERT
Affiliation:
Université de Toulouse; INPT, UPS; IMFT (Institut de Mécanique des Fluides de Toulouse); 1 Allée du Professeur Camille Soula, F-31400 Toulouse, FranceCNRS; IMFT; F-31400 Toulouse, France
FRANCK PLOURABOUÉ*
Affiliation:
Université de Toulouse; INPT, UPS; IMFT (Institut de Mécanique des Fluides de Toulouse); 1 Allée du Professeur Camille Soula, F-31400 Toulouse, FranceCNRS; IMFT; F-31400 Toulouse, France
ALAIN BERGEON
Affiliation:
Université de Toulouse; INPT, UPS; IMFT (Institut de Mécanique des Fluides de Toulouse); 1 Allée du Professeur Camille Soula, F-31400 Toulouse, FranceCNRS; IMFT; F-31400 Toulouse, France
*
†Email address for correspondence: [email protected]
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Abstract

We present a theoretical and numerical study of three-dimensional pulsatile confined flow between two rigid horizontal surfaces separated by an average gap h, and having three-dimensional wavy shapes with arbitrary amplitude σh where σ ~ O(1), but long-wavelength variations λ, with h/λ ≪ 1. We are interested in pulsating flows with moderate inertial effect arising from the Reynolds stress due to the cavity non-parallelism. We analyse the inertial steady-streaming and the second harmonic flows in a lubrication approximation. The dependence of the three-dimensional velocity field in the transverse direction is analytically obtained for arbitrary Womersley numbers and possibly overlapping Stokes layers. The horizontal dependence of the flow is solved numerically by computing the first two pressure fields of an asymptotic expansion in the small inertial limit. We study the variations of the flow structure with the amplitude, the channel's wavelength and the Womersley number for various families of three-dimensional channels. The steady-streaming flow field in the horizontal plane exhibits a quadrupolar vortex, the size of which is adjusted to the cavity wavelength. When increasing the wall amplitude, the wavelengths characterizing the channel or the Womersley number, we find higher-order harmonic flow structures, the origin of which can either be inertially driven or geometrically induced. When some of the channel symmetries are broken, a steady-streaming current appears which has a quadratic dependence on the pressure drop, the amplitude of which is linked to the Womersley number.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 657 , 25 August 2010 , pp. 430 - 455
Copyright
Copyright © Cambridge University Press 2010

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