{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T12:30:20Z","timestamp":1774355420765,"version":"3.50.1"},"reference-count":38,"publisher":"Cambridge University Press (CUP)","license":[{"start":{"date-parts":[[2006,4,26]],"date-time":"2006-04-26T00:00:00Z","timestamp":1146009600000},"content-version":"unspecified","delay-in-days":4034,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Fluid Mech."],"published-print":{"date-parts":[[1995,4,10]]},"abstract":"<jats:p>Detailed theoretical and numerical results are presented for the eddy viscosity of three-dimensional forced spatially periodic incompressible flow.<\/jats:p><jats:p>As shown by Dubrulle &amp; Frisch (1991), the eddy viscosity, which is in general a fourth-order anisotropic tensor, is expressible in terms of the solution of auxiliary problems. These are, essentially, three-dimensional linearized Navier\u2013Stokes equations which must be solved numerically.<\/jats:p><jats:p>The dynamics of weak large-scale perturbations of wavevector <jats:italic>k<\/jats:italic> is determined by the eigenvalues \u2013 called here \u2018eddy viscosities\u2019 \u2013 of a two by two matrix, obtained by contracting the eddy viscosity tensor with two <jats:italic>k<\/jats:italic>-vectors and projecting onto the plane transverse to <jats:italic>k<\/jats:italic> to ensure incompressibility. As a consequence, eddy viscosities in three dimensions, but not in two, can become complex. It is shown that this is ruled out for flow with cubic symmetry, the eddy viscosities of which may, however, become negative.<\/jats:p><jats:p>An instance is the equilateral <jats:italic>ABC<\/jats:italic>-flow (<jats:italic>A<\/jats:italic> = <jats:italic>B<\/jats:italic> = <jats:italic>C<\/jats:italic> = 1). When the wavevector <jats:italic>k<\/jats:italic> is in any of the three coordinate planes, at least one of the eddy viscosities becomes negative for <jats:italic>R<\/jats:italic> = 1\/<jats:italic>v<\/jats:italic> &gt; <jats:italic>R<\/jats:italic><jats:sub><jats:italic>c<\/jats:italic><\/jats:sub> [bsime   ] 1.92. This leads to a large-scale instability occurring for a value of the Reynolds number about seven times smaller than instabilities having the same spatial periodicity as the basic flow.<\/jats:p>","DOI":"10.1017\/s0022112095001133","type":"journal-article","created":{"date-parts":[[2006,4,27]],"date-time":"2006-04-27T18:30:27Z","timestamp":1146162627000},"page":"249-264","source":"Crossref","is-referenced-by-count":24,"title":["Eddy viscosity of three-dimensional flow"],"prefix":"10.1017","volume":"288","author":[{"given":"A.","family":"Wirth","sequence":"first","affiliation":[]},{"given":"S.","family":"Gama","sequence":"additional","affiliation":[]},{"given":"U.","family":"Frisch","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2006,4,26]]},"reference":[{"key":"S0022112095001133_ref018","unstructured":"H\u00e9non, M. 1966 Sur la topologie des lignes de courant dans un cas particulier.C.R. Acad. Sci. 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