Session D23: Turbulence: Theory II - General

Effects of anisotropy on the fluctuating dissipation scale

The invariance of the dissipation scale distribution, $Q(\eta)$, to local measures of anisotropy at high Reynolds number is a necessary condition to support the notion of a universal and isotropic small-scale structure of turbulence. We examine the effects of varying levels of anisotropy on $Q(\eta)$ using a monte-carlo approach to model correlated spatial gaussian velocity ensembles. Anisotropy was modeled as a linear variation in velocity rms in space as is the case locally in strongly anisotropic turbulence. $Q(\eta)$ calculated from isotropic simulations matched recent mathematical distributions from the equations of motion and the multifractal formalism. However $Q(\eta)$ in anisotropic cases, where spatially increasing rms was modeled, systematically deviated from the isotropic expectations. Peak locations of $Q(\eta/\eta_{0})$ shift left with increasing anisotropy, however a significant reduction in Reynolds number can induce an overall right shift. These results illustrate contrasting effects between local anisotropy and low Reynolds number with respect to the small-scale structure of the dissipative scales of motion.   

Local dissipation scales in turbulent shear flows

Recent studies of homogeneous isotropic turbulence and wall-bounded shear flows have indicated that dissipation of kinetic energy occurs over a broad range of scales, including scales significantly larger than the classical mean Kolmogorov scale. It is thus possible to construct a field of local dissipation scales by examining the local Reynolds number at every point in a flow. Distributions of the resulting scales have proven to be similar in the flows examined to date, although substantial variations are observed as the wall is approached in turbulent channel flow. These variations could be due to one or several effects in the near-wall region, including decreased Reynolds number, increased flow two-dimensionality, or increased mean shear. In this talk, the effect of mean shear on local dissipation scales is examined by analyzing direct numerical simulations of homogeneously sheared turbulence. The simulations are performed for a range of shear strengths and Reynolds numbers, and the resulting distributions of dissipation scales are compared. The implications of these results for wall-bounded flows are discussed, and the results are also used to test the validity of assumptions concerning local isotropy and scale separation in turbulent shear flows.   

Nonlocal pressure and viscous contributions to the velocity gradient statistics based on Gaussian random fields

The velocity gradient tensor characterizes the small scales of fully developed turbulence comprehensively. The challenge in understanding its statistical properties in terms of exact statistical evolution equations lies in specifying the nonlocal pressure and viscous effects. Based on the assumption of incompressible Gaussian velocity fields, these statistically unclosed terms are evaluated analytically, and the dynamics of this Gaussian closure and generalizations thereof are discussed and compared to data from direct numerical simulations. The results help to explain how nonlocal pressure Hessian contributions prevent the restricted Euler singularity, and yield insights into the origin of the velocity gradient skewness related to a breaking of the time-reversal symmetry.   

Exploring the link between intermittency in scalar dissipation ($\chi )$ and energy dissipation ($\varepsilon )$ rates

The occurrence of spatial and temporal intermittency in $\chi $, analogous to that seen in $\varepsilon $ for the velocity field, poses a formidable challenge in the formulation of subgrid scale models for $\chi $. As the scalar transport equation is known to be linear, intermittency in the scalar field must be inherited largely from non-linearity in the momentum equation. This occurrence may be explained physically as the coincidence of steepest gradients in the scalar field (which correspond to the largest magnitudes of $\chi )$ with those in the velocity field (largest magnitudes of $\varepsilon )$, caused by strong straining of material particles. To determine the extent of the inheritance, we attempt to establish a qualitative as well as quantitative correlation between intermittency in $\varepsilon $ and $\chi $. Any external role of the scalar forcing term in the intermittency of $\chi $ is also assessed by using two scalar forcing techniques in homogeneous isotropic turbulence, namely mean scalar gradient forcing and linear scalar forcing. A third, unforced configuration, the turbulent mixing layer is used as well, where scalar fluctuations are sustained naturally by a mean gradient present in the cross-stream direction. Appropriate conclusions are also drawn regarding the relevance of the Schmidt number to the extent of intermittency inheritance, in light of the spectral de-linking that happens at very high Schmidt numbers.   

More on the asymptotic state of high Reynolds number, smooth-wall turbulent flows

This is an update of a hypothesis (Pullin, Inoue \& Saito, {\it Phys. Fluids}, 2013) concerning the asymptotic state of some canonical, smooth-wall turbulent flows. There it was argued, based on the extrapolation to arbitrarily large Reynolds numbers ($Re_\tau$) of both the log-wake law for the mean velocity profile, and also of Townsend-Perry scaling for stream-wise turbulent velocity fluctuations, that over almost all of the turbulent-flow layer, turbulent velocity fluctuations on outer scales asymptotically decline with increasing $Re_\tau$. Presently this is extended to include vorticity fluctuations using scaling proposed by Panton ({\it Phys. Fluids}, 2009). This suggests that, at least for turbulent channel flow, the asymptotic state consists of vanishingly-small turbulent velocity fluctuations but unbounded enstrophy ($\overline{\omega^2}$) fluctuations on outer scales, over almost the whole turbulent-flow domain.   

Why the ``K41''/Batcher hypothesis of ``local equilibrium'' is wrong

The foundation of modern turbulence theory since Kolmogorov's pioneering paper in 1941 has been the hypothesis that the small scales of the turbulence were in ``local'' statistical equilibrium relative to those containing most of the energy. This hypothesis is shown to be fundamentally incorrect and internally inconsistent with deductions based upon it, no matter the Reynolds number. In fact deductions from the local equilibrium hypothesis are valid only in flows in strict statistical equilibrium; i.e., flows that are either already statistically stationary at all scales, or in a convective frame, statistically homogeneous. In other words, only in flows where the ``local equilibrium'' is in fact exact. Hence experiments in such flows (of which there are many) provide no proof at all, contrary to popular belief that ``K41'' is ``proven.'' Moreover there are many experiments in non-stationary flows that ``disprove'' ``K41,'' consistent with the conclusions here. The implications of this for a new theory of turbulence are explored using the material derivative of the dissipation rate.   

What are the origins of -5/3 spectra and related dissipation scalings?

In this numerical work we present results concerning the spatial development of energy spectra and their associated integral and Taylor scales in conjunction with the spatial developments of vorticity, strain and production rates of vorticity and strain obtained from Direct Numerical Simulations of spatially developing grid-generated turbulence. We use a fractal square grid and a single mesh grid where the mesh is similar to the largest square on the fractal square grid. We find two adjacent but physically different regions in these flows relatively close to the grid: one where the Q-R diagram has not yet formed its well-known, presumed universal, tear-drop shape but where the energy spectra are not too far from a -5/3 shape over a decade of a frequency range which is set by inlet conditions rather than Kolmogorov scalings: and one where the Q-R diagram immediately adopts the well-known tear-drop shape but where the energy spectra are just about proportional to -5/3 over nearly a decade of frequencies. In the present fractal grid simulation, the first region gives rise, as one moves downstream, to the non-equilibrium behaviour C$\varepsilon \approx $ 1/Re$\lambda $ (see Valente {\&} Vassilicos, PRL, 2012 and Gomes-Fernandes et al., JFM, 2012) whilst the second region leads to C$\varepsilon \approx $ Const.   

Axisymmetric turbulent wakes with new non-equilibrium similarity scalings

The recently discovered non-equilibrium turbulence dissipation law (Seoud \& Vassilicos PoF 19, 2007, Mazellier \& Vassilicos PoF 22, 2010, Valente \& Vassilicos JFM 687, 2011, Valente \& Vassilicos PRL 108, 2012, Gomes-Fernandes et al. JFM 711, 2012) implies the existence of axisymmetric turbulent wake regions where the mean flow velocity deficit decays as the inverse of the distance from the wake-generating body and the wake width grows as the square root of that distance. This behaviour is different from any documented boundary-free turbulent shear flow to date. Its existence is confirmed in wind tunnel experiments of wakes generated by plates with irregular fractal-like edges placed normal to an incoming free stream.   

Power Fluctuations and Irreversibility in Turbulence

We show that for fluid turbulence irreversibility manifests itself in the evolution of the kinetic energy of individual fluid elements. We found in experiment and numerical simulations of two-dimensional and three-dimensional turbulence that fluid elements decelerate faster than accelerate. This asymmetry gives rise to negative third moments of energy changes of a fluid element, which we observed to remain constant for time delays in the range characteristic of turbulent eddies, independently of the flow details including space dimensionality. However, turbulence in two and three dimensions show striking differences in how energy is exchanged between fluid elements: pressure forces redistribute energy from fast to slow elements in two dimensions; conversely, in three dimensions, pressure transfers energy from slow to fast ones.