Kolmogorov turbulence in a random-force-driven Burgers equation: anomalous scaling and probability density functions

High-resolution numerical experiments, described in this work, show that velocity fluctuations governed by the one-dimensional Burgers equation driven by a white-in-time random noise with the spectrum $\overline{|f(k)|^2}\propto k^{-1}$ exhibit a biscaling behavior: All moments of velocity differences $S_{n\le 3}(r)=\overline{|u(x+r)-u(x)|^n}\equiv\overline{|\Delta u|^n}\propto r^{n/3}$, while $S_{n>3}\propto r^{\zeta_n}$ with $\zeta_n\approx 1$ for real $n>0$ (Chekhlov and Yakhot, Phys. Rev. E {\bf 51}, R2739, 1995). The probability density function, which is dominated by coherent shocks in the interval $\Delta u<0$, is ${\cal P}(\Delta u,r)\propto (\Delta u)^{-q}$ with $q\approx 4$.Comment: 12 pages, psfig macro, 4 figs in Postscript, accepted to Phys. Rev. E as a Brief Communicatio

Kolmogorov turbulence in a random-force-driven Burgers equation

The dynamics of velocity fluctuations, governed by the one-dimensional Burgers equation, driven by a white-in-time random force with the spatial spectrum \overline{|f(k)|^2}\proptok^{-1}, is considered. High-resolution numerical experiments conducted in this work give the energy spectrum $E(k)\propto k^{-\beta}$ with $\beta =5/3\pm 0.02$. The observed two-point correlation function $C(k,\omega)$ reveals $\omega\propto k^z$ with the "dynamical exponent" $z\approx 2/3$. High-order moments of velocity differences show strong intermittency and are dominated by powerful large-scale shocks. The results are compared with predictions of the one-loop renormalized perturbation expansion.Comment: 13 LaTeX pages, psfig.sty macros, Phys. Rev. E 51, R2739 (1995)

Large eddy simulation of two-dimensional isotropic turbulence

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 Chekhlov et 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 simplifiedComment: 34 pages plain tex + 18 postscript figures separately, uses auxilary djnlx.tex fil

Large Scale Drag Representation in Simulations of Two-dimensional Turbulence

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

The Effect of Small-scale Forcing on Large-scale Structures in Two-dimensional Flows

The effect of small-scale forcing on large-scale structures in β-plane two-dimensional (2D) turbulence is studied using long-term direct numerical simulations (DNS). We find that nonlinear effects remain strong at all times and for all scales and establish an inverse energy cascade that extends to the largest scales available in the system. The large-scale flow develops strong spectral anisotropy: k−53 Kolmogorov scaling holds for almost all φ, φ =arctan(kykx) except in the small vicinity of kx = 0, where Rhines\u27s k−5 scaling prevails. Due to the k−5 scaling, the spectral evolution of β-plane turbulence becomes extremely slow which, perhaps, explains why this scaling law has never before been observed in DNS. Simulations with different values of β indicate that the β-effect diminishes at small scales where the flow is nearly isotropic. Thus, for simulations of β-plane turbulence forced at small scales sufficiently removed from the scales where β-effect is strong, large eddy simulation (LES) can be used. A subgrid scale (SGS) parameterization for such LES must account for the small-scale forcing that is not explicitly resolved and correctly accommodate two inviscid conservation laws, viz. energy and enstrophy. This requirement gives rise to a new anisotropic stabilized negative viscosity (SNV) SGS representation which is discussed in the context of LES of isotropic 2D turbulence