We have performed direct numerical simulations of homogeneous and isotropic turbulence in a periodic box with 8,192 3 grid points. These are the largest simulations performed, to date, aimed at improving our understanding of turbulence small-scale structure. We present some basic statistical results and focus on "extreme" events (whose magnitudes are several tens of thousands the mean value). The structure of these extreme events is quite different from that of moderately large events (of the order of 10 times the mean value). In particular, intense vorticity occurs primarily in the form of tubes for moderately large events whereas it is much more "chunky" for extreme events (though probably overlaid on the traditional vortex tubes). We track the temporal evolution of extreme events and find that they are generally shortlived. Extreme magnitudes of energy dissipation rate and enstrophy occur simultaneously in space and remain nearly colocated during their evolution.turbulence | intermittency | extreme events | petascale computing | fluid dynamics F luid motions encountered in most circumstances are typically turbulent; therefore a good understanding of the subject is essential both for intrinsic scientific reasons and for advancing important technologies, e.g., improving jet engine performance. The difficulty of the subject (1, 2) has unfortunately consigned our present understanding to be partial at best. A milestone of turbulence theory consists of the similarity hypotheses of Kolmogorov (3, 4) and their various descendant scaling theories. In refs. 5 and 6, one can find a fair summary of the theoretical ideas as well as the considerable experimental work devoted to assessing their veracity. Rapid advances in computing power in recent decades have made computations increasingly important in advancing our understanding of the subject. Key quantities that cannot yet be measured in experiments can instead be computed by the socalled direct numerical simulation (DNS; e.g., see ref. 7), in which the exact equations of motion based on mass and momentum conservation are integrated numerically in time and space. The DNS data are capable of providing a wealth of quantitative detail (see, e.g., ref. 8) and improved qualitative understanding. In this paper, we present results from the largest DNS, to date, of isotropic turbulence aimed at the small-scale structure, rendered statistically stationary by large-scale forcing. We focus on the extreme events (to be made more precise momentarily).Turbulent flows consist of disorderly fluctuations in all measurable properties over a range of scales in both space and time. These fluctuations produce a combination of changes in shape and orientation of an infinitesimal fluid element and can affect quantities of practical interest, such as the tendency of tiny water vapor droplets to collide and grow to millimeter-size rain drops in atmospheric clouds (9). One of the key fluctuating quantities is the energy dissipation rate, e = 2νs ij s ij , where ν is the kinematic viscosity, s i...

The present paper is dedicated to the global large solutions and incompressible limit for the compressible Navier-Stokes system in R d with d ≥ 2. Motivated by the L 2 work of Danchin and Mucha [Adv. Math. 320, 904-925, 2017] in critical Besov spaces, we extend the solution space into an L p framework. The result implies the existence of global large solutions initially from large highly oscillating velocity fields.

The present paper is dedicated to the global well-posedness for the 3D inhomogeneous incompressible Navier-Stokes equations, in critical Besov spaces without smallness assumption on the variation of the density. We aim at extending the work by Abidi, Gui and Zhang (Arch. Ration.Mech. Anal. 204 (1):189-230, 2012, and J. Math. Pures Appl. 100 (1):166-203, 2013) to a more lower regularity index about the initial velocity. The key to that improvement is a new a priori estimate for an elliptic equation with nonconstant coefficients in Besov spaces which have the same degree as L 2 in R 3 . Finally, we also generalize our well-posedness result to the inhomogeneous incompressible MHD equations.

In this paper, we obtain the global solutions to the incompressible Oldroyd-B model without damping on the stress tensor in R n (n = 2, 3). This result allows to construct global solutions for a class of highly oscillating initial velocity. The proof uses the special structure of the system. Moreover, our theorem extends the previous result by Zhu [19] and covers the recent result by Chen and Hao [4].

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