Couette flow in channels with wavy walls

Summary Three–dimensional Couette flows enclosed by a plane and by a wavy wall are addressed; the wave amplitude is proportional to the mean clearance of the channel multiplied by a small dimensionless parameter ɛ. A perturbation expansion in terms of the powers of ɛ of the full steady Navier–Stokes...

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Veröffentlicht in:Acta mechanica 2008-05, Vol.197 (3-4), p.247-283
Hauptverfasser: Malevich, A. E., Mityushev, V. V., Adler, Pierre M.
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creator Malevich, A. E.
Mityushev, V. V.
Adler, Pierre M.
description Summary Three–dimensional Couette flows enclosed by a plane and by a wavy wall are addressed; the wave amplitude is proportional to the mean clearance of the channel multiplied by a small dimensionless parameter ɛ. A perturbation expansion in terms of the powers of ɛ of the full steady Navier–Stokes equations yields a cascade of boundary value problems which are solved at each step in closed form. The supremum value of ɛ for which the expansion converges, is determined as a function of the Reynolds number The analytical-numerical algorithm is applied to compute the velocity in the channel to O (ɛ 4 ). Even in the first order approximation O (ɛ), new results are obtained which complement the triple deck theory and its modifications. In particular, the incipient separation–detachment is discussed using the Prandtl-Schlichting criterion of starting eddies. The value ɛ e for which eddies start in the channel, is analytically deduced as a function of as well as analytical formulas for the coordinates of the separation points. These analytical formulas show that ɛ e in 3D channels is always less than ɛ e in 2D channels. For non-smooth channels, a criterion of infinitesimally small ɛ e is deduced. The critical value of ɛ up to which bifurcation of the solutions can occur is estimated.
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In particular, the incipient separation–detachment is discussed using the Prandtl-Schlichting criterion of starting eddies. The value ɛ e for which eddies start in the channel, is analytically deduced as a function of as well as analytical formulas for the coordinates of the separation points. These analytical formulas show that ɛ e in 3D channels is always less than ɛ e in 2D channels. For non-smooth channels, a criterion of infinitesimally small ɛ e is deduced. 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E.</au><au>Mityushev, V. V.</au><au>Adler, Pierre M.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Couette flow in channels with wavy walls</atitle><jtitle>Acta mechanica</jtitle><stitle>Acta Mech</stitle><date>2008-05-01</date><risdate>2008</risdate><volume>197</volume><issue>3-4</issue><spage>247</spage><epage>283</epage><pages>247-283</pages><issn>0001-5970</issn><eissn>1619-6937</eissn><coden>AMHCAP</coden><abstract>Summary Three–dimensional Couette flows enclosed by a plane and by a wavy wall are addressed; the wave amplitude is proportional to the mean clearance of the channel multiplied by a small dimensionless parameter ɛ. A perturbation expansion in terms of the powers of ɛ of the full steady Navier–Stokes equations yields a cascade of boundary value problems which are solved at each step in closed form. The supremum value of ɛ for which the expansion converges, is determined as a function of the Reynolds number The analytical-numerical algorithm is applied to compute the velocity in the channel to O (ɛ 4 ). Even in the first order approximation O (ɛ), new results are obtained which complement the triple deck theory and its modifications. In particular, the incipient separation–detachment is discussed using the Prandtl-Schlichting criterion of starting eddies. The value ɛ e for which eddies start in the channel, is analytically deduced as a function of as well as analytical formulas for the coordinates of the separation points. These analytical formulas show that ɛ e in 3D channels is always less than ɛ e in 2D channels. For non-smooth channels, a criterion of infinitesimally small ɛ e is deduced. The critical value of ɛ up to which bifurcation of the solutions can occur is estimated.</abstract><cop>Vienna</cop><pub>Springer-Verlag</pub><doi>10.1007/s00707-007-0507-z</doi><tpages>37</tpages></addata></record>
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subjects Algorithms
Channels
Classical and Continuum Physics
Control
Couette flow
Criteria
Dynamical Systems
Engineering
Engineering Thermodynamics
Exact sciences and technology
Flows in ducts, channels, nozzles, and conduits
Fluid dynamics
Fundamental areas of phenomenology (including applications)
General theory
Heat and Mass Transfer
Mathematical analysis
Mathematical models
Mathematics
Mechanical engineering
Navier-Stokes equations
Physics
Reynolds number
Rotational flow and vorticity
Solid Mechanics
Theoretical and Applied Mechanics
Theory
Three dimensional
Vibration
Walls
title Couette flow in channels with wavy walls
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