Dissipation and enstrophy in isotropic turbulence: Resolution effects and scaling in direct numerical simulations

Abstract

Existing experimental and numerical data suggest that the turbulence energy dissipation and enstrophy ͑i.e., the square of vorticity͒ possess different scaling properties, while available theory suggests that there should be no differences at sufficiently high Reynolds numbers. We have performed a series of direct numerical simulations with up to 2048 3 grid points where advanced computational power is used to increase the Reynolds number ͑up to 650 on the Taylor scale͒ or to resolve the small scales better ͑down to 1 / 4 of a Kolmogorov scale͒. Our primary goal is to assess the differences and similarities between dissipation and enstrophy. Special attention is paid to the effects of small-scale resolution on the quality and reliability of the data, in view of recent theoretical work ͓V. Yakhot and K. R. Sreenivasan, "Anomalous scaling of structure functions and dynamic constraints on turbulence simulations," J. Stat. Phys. 121, 823 ͑2005͔͒ which stipulates the resolution needed to obtain a moment of a given order. We also provide error estimates as a function of small-scale resolution. Probability density functions of dissipation and enstrophy at high Reynolds number reveal the presence of extreme events several thousands times of the mean. The extreme events in dissipation and enstrophy fields appear to scale alike, substantially overlap in space, and are nearly statistically isotropic, while fluctuations of moderate amplitudes, at least for the present Reynolds numbers, show persistent differences. Conditional sampling shows that intense dissipation is likely to be accompanied by similarly intense enstrophy, but intense enstrophy is not always accompanied by intense dissipation.

On the accuracy of simulations of turbulence

“…In this way, the criteria and tests that we propose may well be useful in computational problems of this type. Finally we remark that the present study is complementary to the recent study by Donzis et al 5 who were concerned mainly with accuracy of higher-order statistics in hydrodynamic turbulence. The present work can be viewed as applying some of those ideas to MHD in two dimensions, and developing practical tests to be used in 2D MHD applications.…”

“…The present work can be viewed as applying some of those ideas to MHD in two dimensions, and developing practical tests to be used in 2D MHD applications. What remains unclear at present is how the details of either Donzis et al 5 or the present study extrapolate to 3D MHD, or other systems such as 2D hydrodynamics, Hall MHD, etc., including compressible cases. However, we expect that the basic picture discussed above will remain a requirement for accurately computing coherent structures in various fluid systems, and, in particular, that runs would likely need to satisfy k max Ͼ 3.…”

“…Therefore adequate resolution of moments of very high order may admit further challenges. 4,5 In Fig. 8, we plot the high-pass filtered kurtosis of the electric current density and the vorticity for the Table II simulation set ͑fixed viscosity͒.…”

“…[1][2][3] Recent works have discussed the need for stricter conditions on resolution, 4 and have examined specifically the resolution required to accurately compute high-order statistical quantities that are of central interest in turbulence theory. 5 Herein we show that there are alternative, easily calculable, quantities that provide a more discerning test of the simulations well-resolvedness. We also examine the performance of these simple tests in the context of a magnetohydrodynamics ͑MHD͒ problem that is very demanding in terms of spatial resolution, namely, the determination of rates of magnetic reconnection in turbulence.…”

Extreme events in computational turbulence

“…See the legend for Table 1 for specific definitions of the Reynolds number; it suffices here to say that it determines the range of excited scales. This quest to increase the Reynolds number and the need to better resolve the small scales is the crux of DNS methods today (13,14). As the flow Reynolds number increases (which is the main reason for increasing the size of the simulations), e and Ω fluctuate more and more wildly in space and time.…”