Reynolds stress and eddy viscosity in direct numerical simulations of sheared two-dimensional turbulence

• Published in: Journal of fluid mechanics Vol. 657; pp. 394 - 412
• Main Authors: CUMMINS, PATRICK F., HOLLOWAY, GREG
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 25.08.2010
• Subjects: Eddies; Eddy viscosity; Exact sciences and technology; Fluid dynamics; Fluid flow; Fluid mechanics; Fundamental areas of phenomenology (including applications); Inviscid flows with vorticity; Isotropic turbulence; homogeneous turbulence; Isotropy; Laminar flows; Marine; Physics; Reynolds number; Reynolds stress; Shear stress; Turbulence; Turbulence simulation and modeling; Turbulent flow; Turbulent flows, convection, and heat transfer; Two dimensional; Viscosity; Wavenumber; Reynolds stress; Non viscous fluid; Energy spectra; Shear flow; Digital simulation; Vorticity; Modelling; Isotropic turbulence; Eddy viscosity; Turbulence structure; Homogeneous turbulence
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/S0022112010001424

The Reynolds stress associated with the adjustment of two-dimensional isotropic eddies subject to a large-scale shear flow is examined in a series of initial-value calculations in a periodic channel. Several stages in the temporal evolution of the stress can be identified. Initially, there is a brief period associated with quasi-passive straining of the eddy field in which the net Reynolds stress and the associated eddy viscosity remain essentially zero. In spectral space this is characterized by mutual cancellation of contributions to the Reynolds stress at high and low eddy wavenumbers. Subsequently, eddy–eddy interactions produce a tendency to restore isotropy at higher eddy wavenumbers, leading to an overall positive eddy viscosity associated with the dominant contribution to the Reynolds stress at low eddy wavenumbers. These results are consistent with theoretical predictions of positive eddy viscosity for initially isotropic homogeneous two-dimensional turbulence. Due to the inverse cascade, the accumulation with time of energy at the scale of the channel produces a competing tendency to negative eddy viscosity associated with linear shearing of the disturbances. This finite-domain effect may become dominant if the nonlinearity of the eddy field is relatively weak.