Session H27: Turbulence: Theory II

Turbulence at high resolution: intense events in dissipation, enstrophy and acceleration

Access to the {\em Blue Waters} supercomputer under the NSF Track 1 Petascale Resource Allocations program has allowed us to conduct an $8192^3$ simulation of forced isotropic turbulence, with Taylor-scale Reynolds number close to 1300, and grid spacing at about 1.5 Kolmogorov scales. Extreme fluctuations in dissipation and enstrophy (over 10,000 times the mean) are observed, and found to scale similarly and occur together. Conditional sampling based on both dissipation and enstrophy shows that such extreme events in these variables are directly associated with strong intermittency in the fluid particle acceleration, which reaches values well beyond 100 standard deviations. An attempt is made to characterize in detail the formation of events of intense dissipation and enstrophy, including the transport, production and dissipation terms in the dissipation and enstrophy transport equations, as well as the nature of local flow conditions in principal strain-rate axes. Statistics of dissipation and enstrophy averaged over 3D sub-domains of linear size in the inertial range are also available. Both high Reynolds number and good small-scale resolution are important factors in these results.   

Energy Spectra of Higher Reynolds Number Turbulence by the DNS with up to $12288^3$ Grid Points

Large-scale direct numerical simulations (DNS) of forced incompressible turbulence in a periodic box with up to $12288^3$ grid points have been performed using K computer. The maximum Taylor-microscale Reynolds number $R_\lambda$, and the maximum Reynolds number $Re$ based on the integral length scale are over 2000 and $10^5$, respectively. Our previous DNS with $R_\lambda$ up to 1100 showed that the energy spectrum has a slope steeper than $-5/3$ (the Kolmogorov scaling law) by factor $0.1$ at the wavenumber range ($k\eta<0.03$). Here $\eta$ is the Kolmogorov length scale. Our present DNS at higher resolutions show that the energy spectra with different Reynolds numbers ($R_\lambda>1000$) are well normalized not by the integral length-scale but by the Kolmogorov length scale, at the wavenumber range of the steeper slope. This result indicates that the steeper slope is not inherent character in the inertial subrange, and is affected by viscosity.   

The turbulent cascade of individual eddies

The merging and splitting processes of Reynolds-stress carrying structures in the inertial range of scales are studied through their time-resolved evolution in channels at $Re_\lambda=100-200$. Mergers and splits coexist during the whole life of the structures, and are responsible for a substantial part of their growth and decay. Each interaction involves two or more eddies and results in little overall volume loss or gain. Most of them involve a small eddy that merges with, or splits from, a significantly larger one. Accordingly, if merge and split indexes are respectively defined as the maximum number of times that a structure has merged from its birth or will split until its death, the mean eddy volume grows linearly with both indexes, suggesting an accretion process rather than a hierarchical fragmentation. However, a non-negligible number of interactions involve eddies of similar scale, with a second probability peak of the volume of the smaller parent or child at 0.3 times that of the resulting or preceding structure.   

The transfer of kinetic energy in turbulent flows

We study the statistics of the point-wise inter-scale energy transfer across a given filter width in direct numerical simulations of homogeneous isotropic turbulence, homogeneous shear flow and turbulent channels. This is first done for the classical subgrid-scale (SGS) dissipation found in the kinetic energy equation for the filtered velocity field. It is then compared with an analogous term $T$ arising in the equation for the residual (small-scale) velocity field. $T$ can take several expressions, and we report on the one which minimises its variance. For all flows, the SGS dissipation exhibits a negative skewness which increases with the filter width, while $T$ has a positive skewness which decreases with filter width. This is consistent with the SGS dissipation being an average energy sink for the large scales, while $T$ is an average energy source for the small ones. The different dependence on filter width of the mean and standard deviations of these two quantities is explored, and joint probability density functions based on the two quantities are investigated to understand the observed discrepancies between forward scatter and backscatter events.   

The reversible 3D turbulent cascade

It has been known for some time that the dynamic Smagorinsky LES model is reversible. If the sign of the velocities in an isotropic turbulence simulation is inverted after it has decayed for some time, it evolves back to its original state, recovering its energy and other turbulent quantities. We use this reverse evolution, during which the cascade transfers energy from the small to the large scales, to gain new insights into the behavior and reversible features of the inertial energy range. The dynamics in the plane of the Q-R topological invariants are studied for the forward and backward evolutions, as well as the structure of the Lyapunov exponents in both regimes. Considerable differences are found. In particular, the Q-R pdf of the inverse evolution is reversed, with a stable Vieillefosse tail along negative R, and a main lobe in which vortex compression predominates. The contribution of the different terms in the equation is computed for both cases, both with and without an LES model.   

A Phenomenological Theory of Rotating Turbulence

We present direct numerical simulations of statistically-homogeneous, freely-decaying, rotating turbulence in which the Rossby number, $\mbox{Ro}=u_{\bot } /2\Omega \ell_{\bot } $, is of order unity. The initial condition consists of fully-developed turbulence in which Ro is sufficiently high for rotational effects to be weak. However, as the kinetic energy falls, so also does Ro, and quite quickly we enter a regime in which the Coriolis force is relatively strong and anisotropy grows rapidly, with $\ell_{\bot } <<\ell_{//} $. This regime occurs when $\mbox{Ro}\sim \mbox{0.4}$ and is characterised by an almost constant perpendicular integral scale, $\ell_{\bot } \sim \mbox{constant}$, a rapid linear growth in the integral scale parallel to the rotation axis, $\ell_{//} \sim \ell _{\bot } \Omega t$, and a slow decline in the value of Ro. We observe that the rate of dissipation of energy scales as $\varepsilon \sim {u^{3}} \mathord{\left/ {\vphantom {{u^{3}} {\ell_{//} }}} \right. \kern-\nulldelimiterspace} {\ell_{//} }$ and that both the perpendicular and parallel energy spectra exhibit an $k^{-5/3}$ inertial range; $E(k_{\bot } )\sim \varepsilon^{2/3}k_{\bot }^{-5/3} $ and $E(k_{//} )\sim \varepsilon ^{2/3}k_{//}^{-5/3} $. We show that these power-law spectra have nothing to do with Kolmogorov's theory and are not a manifestation of traditional critical balance theory, as this requires $\varepsilon \sim u^{3}/\ell _{\bot } $ and $E(k_{//} )\sim (\varepsilon^{4/5}/\Omega ^{2/5})k_{//}^{-7/5} $. Finally, we develop a spectral theory of the inertial range that assumes that the observed linear growth in anisotropy, $\ell_{//} /\ell_{\bot } \sim \Omega t$, also occurs on a scale-by-scale basis all the way down to the Zeman scale.   

Filtered linear forcing: a technique for simulating high Reynolds number turbulence in physical space

Low waveshell spectral forcing has been proven to be a simple and effective manner to generate isotropic turbulence in a periodic domain. This simplicity is lost for flow problems with complex boundary conditions such as resolved particle flows, fluid-fluid flows with interfaces, and wall-bounded flows. Lundgren's linear forcing in physical space is a straightforward and easy-to-implement method to tackle these problems; however, the use of this method results in a halving of the large turbulence length scale. The technique that will be presented in this talk applies a low-pass filter to the source term used in linear forcing. It is shown to recover the scale resolution of low waveshell spectral forcing which translates to an approximately 60 percent increase in the attainable Reynolds number for a given computation domain. The characteristics of homogeneous isotropic turbulence generated using filtered linear forcing will be discussed. Finally, extension of this idea to scalar forcing will be presented.   

Grid generated turbulence in the near-field

Using a conventional bi-planar turbulence-generating grid, we confirm the recent findings (Valente {\&} Vassilicos, Phys. Rev. Lett., vol. 108, 2012, art. 214503) that show there is a turbulence decay region close to the generating grid that departs from the ``classical'' turbulence decay (Comte-Bellot {\&} Corrsin, J. Fluid Mech., vol. 25, 1966, pp. 657--682). In this ``near field'' region, the turbulence energy decays more rapidly than in the far field and it exhibits unusual scaling properties. Based on the velocity decay laws, we show that for our conventional grid, the near field extends from x/M $\sim$ 6 to x/M $\sim$ 12 where x is the downstream distance from the grid and M is the mesh size. However, other statistics (velocity derivatives and length scales ratios) indicate that the extent of the initial period depends on the grid mesh Reynolds number, R$_{\mathrm{M}}$, extending further for higher values of R$_{\mathrm{M}}$. In the near field the turbulence approaches isotropy both at the large and small scales but there still is inhomogeneity in the derivative statistics. The derivative skewness also departs from values observed at comparable Reynolds numbers in the far field decay region, and in other turbulent flows at comparable Reynolds numbers. We do not believe that the near field scaling violates Kolmogorov phenomenology, which applies to systems that are not affected by proximity to initial and boundary conditions. These conditions are not met close to the grid.   

Turbulent wakes of irregular objects

Recently, flow regions with non-equilibrium high Reynolds number turbulence at odds with usual Richardson-Kolmogorov phenomenology have been discovered in a number of turbulent flows, in particular axisymmetric and self-preserving turbulent wakes of plates with irregular edges. These regions are characterised by streamwise evolutions of the mean flow profiles which have only recently been documented, see PRL 111, 144503 (2013). One of the main differences between the equilibrium and the non-equilibrium predictions involves the momentum thickness. We therefore have carried out experiments with bluff bodies that have various different chord lengths in the direction of the flow. We performed wind tunnel anemometry measurements of wakes generated by bluff plates with simple square edge peripheries and by bluff plates with irregular edge peripheries which allow the formation of jet-wake flows. The wakes generated by the irregular plates become axisymmetric much earlier than the wakes generated by regular ones, irrespective of chord length. Furthermore, the non-equilibrium wake scalings are present in the case of the bluff plates with irregular edges, again irrespective of chord length.