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Numerical simulations of isotropic, homogeneous, forced and dissipative two-dimensional (2D) turbulence in the energy transfer subrange are complicated by the inverse cascade that continuously propagates energy to the large scale modes.... more
Numerical simulations of isotropic, homogeneous, forced and dissipative two-dimensional (2D) turbulence in the energy transfer subrange are complicated by the inverse cascade that continuously propagates energy to the large scale modes. To avoid energy condensation in the lowest modes, an energy sink, or a large scale drag is usually introduced. With a few exceptions, simulations with different formulations of the large scale drag reveal the development of strong coherent vortices and steepening of energy and enstrophy spectra that lead to erosion and eventual destruction of Kolmogorov-Batchelor-Kraichnan (KBK) statistical laws. Being attributed to the intrinsic anomalous fluctuations independent of the large scale drag formulation, these coherent vortices have prompted conjectures that KBK 2D turbulence in the energy subrange is irreproducible in long term simulations. Here, we advance a different point of view, according to which the emergence of coherent vortices is triggered by the inverse energy cascade distortion directly attributable to the choice of a large scale drag formulation. We subdivide the computational modes into explicit and implicit, or supergrid scale (SPGS), which are the few lowest wave numbers modes that adhere to KBK statistics. Then, we introduce a new concept of the large scale drag—rather than being an energy sink, it accounts for the energy and enstrophy exchange between the explicit and SPGS modes. The new SPGS parameterization was used in both direct numerical simulations (DNS) and large eddy simulations (LES) in a doubly periodic box setting. It was found that the new technique enables both DNS and LES to reach a steady state preserved for many large scale eddy turnover times. For the entire time of integration, the flow field remained structureless and in good agreement with the KBK statistical laws. We conclude that homogeneous, isotropic, forced, dissipative 2D turbulence in the energy subrange is statistically stable, does not produce coherent structures, and obeys the KBK statistical laws for as long as its inverse energy cascade remains undisturbed. The proposed new technique of computing the intermediate modes while the statistics of the largest scales is known may find a wide range of applications.
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The dynamics of velocity fluctuations governed by the Burgers equation, driven by the white-in-time random forcing function with [ f\(x+r,t\)-f\(x,t'\)]2¯~rxidelta\(t-t'\) is considered on the interval... more
The dynamics of velocity fluctuations governed by the Burgers equation, driven by the white-in-time random forcing function with [ f\(x+r,t\)-f\(x,t'\)]2¯~rxidelta\(t-t'\) is considered on the interval 0<x<L. The properties of the probability density function of velocity differences P\(Deltau,r\) are investigated for the three cases xi = \{0;1/2;2\}. It is shown that the tail of the probability density function in the interval
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Large eddy simulation (LES) of forced, homogeneous, isotropic two-dimensional (2D) turbulence in the energy transfer subrange is the subject of this paper. A difficulty specific to this LES and its subgrid scale (SGS) representation is in... more
Large eddy simulation (LES) of forced, homogeneous, isotropic two-dimensional (2D) turbulence in the energy transfer subrange is the subject of this paper. A difficulty specific to this LES and its subgrid scale (SGS) representation is in that the energy source resides in high wave number modes excluded in simulations. Therefore, the SGS scheme in this case should assume the function of the energy source. In addition, the controversial requirements to ensure direct enstrophy transfer and inverse energy transfer make the conventional scheme of positive and dissipative eddy viscosity inapplicable to 2D turbulence. It is shown that these requirements can be reconciled by utilizing a two-parametric viscosity introduced by Kraichnan (1976) that accounts for the energy and enstrophy exchange between the resolved and subgrid scale modes in a way consistent with the dynamics of 2D turbulence; it is negative on large scales, positive on small scales and complies with the basic conservation laws for energy and enstrophy. Different implementations of the two-parametric viscosity for LES of 2D turbulence were considered. It was found that if kept constant, this viscosity results in unstable numerical scheme. Therefore, another scheme was advanced in which the two-parametric viscosity depends on the flow field. In addition, to extend simulations beyond the limits imposed by the finiteness of computational domain, a large scale drag was introduced. The resulting LES exhibited remarkable and fast convergence to the solution obtained in the preceding direct numerical simulations (DNS) by Chekhlovet al. (1994) while the flow parameters were in good agreement with their DNS counterparts. Also, good agreement with the Kolmogorov theory was found. This LES could be continued virtually indefinitely. Then, a simplified SGS representation was designed, referred to as the stabilized negative viscosity (SNV) representation, which was based on two algebraic terms only, negative Laplacian and positive biharmonic ones. It was found that the SNV scheme performed in a fashion very similar to the full equation and it was argued that this scheme and its derivatives should be applied for SGS representation in LES of quasi-2D flows.