Lars Onsager, a giant of twentieth-century science and the 1968 Nobel Laureate in Chemistry, made deep contributions to several areas of physics and chemistry. Perhaps less well known is his groundbreaking work and lifelong interest in the subject of hydrodynamic turbulence. He wrote two papers on the subject in the 1940s, one of them just a short abstract. Unbeknownst to Onsager, one of his major results was derived a few years earlier by A. N. Kolmogorov, but Onsager's work contains many gems and shows characteristic originality and deep understanding. His only full-length article on the subject in 1949 introduced two novel ideas-negative-temperature equilibria for two-dimensional ideal fluids and an energy-dissipation anomaly for singular Euler solutions-that stimulated much later work. However, a study of Onsager's letters to his peers around that time, as well as his private papers of that period and the early 1970s, shows that he had much more to say about the problem than he published. Remarkably, his private notes of the 1940s contain the essential elements of at least four major results that appeared decades later in the literature: ͑1͒ a mean-field Poisson-Boltzmann equation and other thermodynamic relations for point vortices; ͑2͒ a relation similar to Kolmogorov's 4/5 law connecting singularities and dissipation; ͑3͒ the modern physical picture of spatial intermittency of velocity increments, explaining anomalous scaling of the spectrum; and ͑4͒ a spectral turbulence closure quite similar to the modern eddy-damped quasinormal Markovian equations. This paper is a summary of Onsager's published and unpublished contributions to hydrodynamic turbulence and an account of their place in the field as the subject has evolved through the years. A discussion is also given of the historical context of the work, especially of Onsager's interactions with his contemporaries who were acknowledged experts in the subject at the time. Finally, a brief speculation is offered as to why Onsager may have chosen not to publish several of his significant results.

The JHU turbulence database [1] can be used with a state of the art visualisation tool [2] to generate high quality fluid dynamics videos. In this work we investigate the classical idea that smaller structures in turbulent flows, while engaged in their own internal dynamics, are advected by the larger structures. They are not advected undistorted, however. We see instead that the small scale structures are sheared and twisted by the larger scales. This illuminates the basic mechanisms of the turbulent cascade. THE JHU TURBULENCE DATABASEIn [1] a database containing a solution of the 3D incompressible Navier-Stokes (NS) equations is presented. The equations were solved numerically with a standard pseudo-spectral simulation in a periodic domain, using a real space grid of 1024 3 grid points. A large-scale body force drives a turbulent flow with a Taylor microscale based Reynolds number R λ = 433. Out of this solution, 1024 snapshots were stored, spread out evenly over a large eddy turnover time. More on the simulation and on accessing the data can be found at http://turbulence.pha.jhu.edu. In practical terms, we have easy access to the turbulent velocity field and pressure at every point in space and time. VORTICES WITHIN VORTICESOne usual way of visualising a turbulent velocity field is to plot vorticity isosurfaces -see for instance the plots from [3]. The resulting pictures are usually very "crowded", in the sense that there are many intertwined thin vortex tubes, generating an extremely complex structure. In fact, the picture of the entire dataset from [3] looks extremely noisy and it is arguably not very informative about the turbulent dynamics.In this work, we follow a different approach. First of all, we use the alternate quantityfirst introduced in [4]. Secondly, the tool being used has the option of displaying data only inside clearly defined domains of 3D space. We can exploit this facility to investigate the multiscale character of the turbulent cascade. Because vorticity is dominated by the smallest available scales in the velocity, we can visualize vorticity at scale ℓ by the curl of the velocity box-filtered at scale ℓ. We follow a simple procedure:• we filter the velocity field, using a box filter of size ℓ 1 , and we generate semitransparent surfaces delimitating the domains D 1 where Q > q 1 ;• we filter the velocity field, using a box filter of size ℓ 2 < ℓ 1 , and we generate surfaces delimitating the domains D 2 where Q ≥ q 2 , but only if these domains are contained in one of the domains from D 1 ;and this procedure can be used iteratively with several scales (we use at most 3 scales, since the images become too complex for more levels). Additionally, we wish sometimes to keep track of the relative orientation of the vorticity vectors at the different scales. For this purpose we employ a special coloring scheme for the Q isosurfaces: for each point of the surface, we compute the cosine of the angle α between the ℓ 2 filtered vorticity and the ℓ 1 filtered vorticity: cos α = (∇ × u 1 ) · (∇ × u ...

No abstract