High-order methods for decaying two-dimensional homogeneous isotropic turbulence Omer San ⇑ , Anne E. Staples Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, VA 24061, USA article info Article history: Received 7 October 2011 Received in revised form 18 February 2012 Accepted 9 April 2012 Available online 21 April 2012 Keywords: Incompressible flows High-order accurate schemes Finite difference methods Fourier–Galerkin pseudospectral method Double shear layer problem Two-dimensional decaying turbulence abstract Numerical schemes used for the integration of complex flow simulations should provide accurate solu- tions for the long time integrations these flows require. To this end, the performance of various high- order accurate numerical schemes is investigated for direct numerical simulations (DNS) of homoge- neous isotropic two-dimensional decaying turbulent flows. The numerical accuracy of compact differ- ence, explicit central difference, Arakawa, and dispersion-relation-preserving schemes are analyzed and compared with the Fourier–Galerkin pseudospectral scheme. In addition, several explicit Runge–Kutta schemes for time integration are investigated. We demonstrate that the centered schemes suffer from spurious Nyquist signals that are generated almost instantaneously and propagate into much of the field when the numerical resolution is insufficient. We further show that the order of the scheme becomes increasingly important for increasing cell Reynolds number. Surprisingly, the sixth-order schemes are found to be in perfect agreement with the pseudospectral method. Considerable reduction in computational time compared to the pseudospectral method is also reported in favor of the finite difference schemes. Among the fourth-order schemes, the compact scheme provides better accuracy than the others for fully resolved computations. The fourth-order Arakawa scheme provides more accu- rate results for under-resolved computations, however, due to its conservation properties. Our results show that, contrary to conventional wisdom, difference methods demonstrate superior performance in terms of accuracy and efficiency for fully resolved DNS computations of the complex flows considered here. For under-resolved simulations, however, the choice of difference method should be made with care. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction The physics of two-dimensional turbulence have been eluci- dated substantially during the past decades by theoretical models, intensive numerical investigations, and dedicated soap film exper- iments [1]. Two-dimensional turbulence research efforts have applicability in geophysics, astronomy and plasma physics, in which numerical experiments play a large role. One of the most important reasons for studying two-dimensional turbulence is to improve our understanding of geophysical flows in the atmosphere and ocean [2–8]. We may also find two-dimensional flows in a wide variety of situations such as flows in rapidly rotating systems and flows in a fluid film on top of the surface of another fluid or a rigid object [9]. Two-dimensional turbulence behaves in a profoundly different way from three-dimensional turbulence due to different energy cascade behavior, and follows the Kraichnan–Batchelor–Leith (KBL) theory [10–12]. In three-dimensional turbulence, energy is transferred forward, from large scales to smaller scales, via vortex stretching. In two dimensions that mechanism is absent, and it turns out that under most forcing and dissipation conditions energy will be transferred from smaller scales to larger scales. This is largely because of another quadratic invariant, the potential enstrophy, defined as the integral of the square of the potential vorticity. Despite the apparent simplicity in dealing with two rather than three spatial dimensions, two-dimensional turbulence is possibly richer in its dynamics than three-dimensional turbulence due to its conservation properties, such as its inverse energy and forward enstrophy cascading mechanisms. Danilov and Gurarie [13] and Tabeling [14] reviewed both theoretical and experimental two-dimensional turbulence studies along with extensions into geophysical flow settings. More recent reviews on two-dimensional turbulence are also provided by Clercx and van Heijst [15] and Boffetta and Ecke [16]. Recent studies in two-dimensional turbu- lence, both forced (stationary) turbulence [17–20] and unforced (decaying) turbulence [21–23] provide high resolution computa- tional confirmation of the KBL theory. Simulation of turbulent and other convection-dominated unsteady flows using direct numerical simulation (DNS) requires a 0045-7930/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compfluid.2012.04.006 ⇑ Corresponding author. E-mail address: omersan@vt.edu (O. San). Computers & Fluids 63 (2012) 105–127 Contents lists available at SciVerse ScienceDirect Computers & Fluids journal homepage: www.elsevier.com/locate/compfluid