Bulletin of the American Physical Society
68th Annual Meeting of the APS Division of Fluid Dynamics
Volume 60, Number 21
Sunday–Tuesday, November 22–24, 2015; Boston, Massachusetts
Session R19: Turbulence: Theory II |
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Chair: Bhimsen Shivamoggi, University Central Florida Room: 207 |
Tuesday, November 24, 2015 12:50PM - 1:03PM |
R19.00001: A Generalized Brownian Motion Model for Turbulent Relative Particle Dispersion Bhimsen Shivamoggi A generalized Brownian motion model has been applied to the turbulent relative particle dispersion problem (Shivamoggi [1]). The fluctuating pressure forces acting on a fluid particle are taken to follow an Uhlenbeck-Ornstein process while it appears plausible to take their correlation time to have a power-law dependence on the flow Reynolds number R$_{e}$. This ansatz provides an insight into the result that the Richardson-Obukhov scaling holds only in the infinite-R$_{e}$ limit and disappears otherwise. It provides a determination of the Richardson-Obukhov constant g as a function of R$_{e}$, with an asymptotic constant value in the infinite-R$_{e}$ limit. This ansatz is further shown to be in quantitative agreement, in the small-R$_{e}$ limit, with the Batchelor-Townsend ansatz for the rate of change of the mean square interparticle separation in 3D FDT. [1] B.K. Shivamoggi: arXiv: 1208.5786 (2014). [Preview Abstract] |
Tuesday, November 24, 2015 1:03PM - 1:16PM |
R19.00002: Turbulent Damping without Eddy Viscosity Simon Thalabard The intrinsic Non-Gaussianity of turbulence may explain why the standard Quasi-Normal cumulant discard closures can fail dramatically, an example being the development of negative energy spectra in Millionshtchikov's 1941 Quasi-Normal (QN) theory. While Orszag's 1977 EDQNM provides an ingenious patch to the issue, the reason why QN fails so badly is not so clear. Is it because of the Gaussian Ansatz itself? Or rather its inconsistent use? The purpose of the talk is to argue in favor of the latter option, using the lights of a new ``optimal closure'' recently exposed by [Turkington,2013], which allows Gaussians to be used consistently with an intrinsic damping. The key to this apparent paradox lies in a clear distinction between the ensemble averages and their proxies, most easily grasped provided one uses the Liouville equation rather than the cumulant hierarchy as a starting point. Schematically said, closure is achieved by minimizing a lack-of-fit residual, that retains the intrinsic features of the dynamics. For the sake of clarity, I will discuss the optimal closure on a problem where it can be entirely implemented and compared to DNS: the relaxation of an arbitrarily far from equilibrium energy shell towards the Gibbs equilibrium for truncated Euler dynamics. [Preview Abstract] |
Tuesday, November 24, 2015 1:16PM - 1:29PM |
R19.00003: Two-point Spectral Modeling of Anisotropic Rapid Distortion Timothy Clark, Susan Kurien, Charles Zemach We perform simulations of a two-point spectral model for the evolution of the energy tensor as function of wave-vector, for arbitrarily anisotropic turbulence in the limit of rapid distortion. The resulting Reynolds stress tensor for such flow is analysed for the effects of anisotropy during evolution. According to the SO(3) rotation group decomposition of the energy tensor, the leading order isotropic contribution is labelled by rotational mode index $j=0$, while higher order anisotropic contributions in statistically homogeneous flows contain a potentially very large array of rotational modes $j=2, 3, 4, \dots$. We compare our results to those of the classical Launder, Reece and Rodi class of models in the rapid distortion limit. These models only retain anisotropy in a nominal manner up to $j=2$, due to an a priori angle-averaging procedure on the energy tensor, reducing it to a function of wave-number alone. Although the Reynolds stress itself has maximum $j=2$ in the SO(3) representation, the terms that contribute to its evolution generate higher order rotational modes. The contributions from the higher order modes are shown to be responsible for the deviation of the LRR solution from the true solution over time. [Preview Abstract] |
Tuesday, November 24, 2015 1:29PM - 1:42PM |
R19.00004: Irreversibility and small-scale generation in 3-dimensional turbulent flows Alain Pumir, Haitao Xu, Rainer Grauer, Eberhard Bodenschatz In 3-dimensional turbulent flows, the irreversibility of turbulence manifests itself by an asymmetry of the probability distribution of the instantaneous power p of the forces acting on fluid elements: the third moment of p was found to be negative. Here, I will discuss the relation between this negative third moment and vortex stretching, which is traditionally related to the generation of small scales. The construction is based on a decomposition of the power p as a sum of a local contribution, due to the variation of velocity at a fixed point in space, plus a convective part, due to the displacement of particles in the flow. The third moment of the latter term, which dominates the statistics, is explicitly expressed in terms of vortex stretching, [Preview Abstract] |
Tuesday, November 24, 2015 1:42PM - 1:55PM |
R19.00005: On the power law of passive scalars in turbulence Toshiyuki Gotoh, Takeshi Watanabe It has long been considered that the moments of the scalar increment with separation distance $r$ obey power law with scaling exponents in the inertial convective range and the exponents are insensitive to variation of pumping of scalar fluctuations at large scales, thus the scaling exponents are universal. We examine the scaling behavior of the moments of increments of passive scalars 1 and 2 by using DNS up to the grid points of $4096^3$. They are simultaneously convected by the same isotropic steady turbulence at$R_\lambda=805$, but excited by two different methods. Scalar 1 is excited by the random scalar injection which is isotropic, Gaussian and white in time at law wavenumber band, while Scalar 2 is excited by the uniform mean scalar gradient. It is found that the local scaling exponents of the scalar 1 has a logarithmic correction, meaning that the moments of the scalar 1 do not obey simple power law. On the other hand, the moments of the scalar 2 is found to obey the well developed power law with exponents consistent with those in the literature. Physical reasons for the difference are explored. [Preview Abstract] |
Tuesday, November 24, 2015 1:55PM - 2:08PM |
R19.00006: The structure of energy transfer in homogeneous turbulence Jose I. Cardesa, Alberto Vela-Martin, Adrian Lozano-Duran, Javier Jimenez The filtering approach to scale decomposition in physical space leads to several energy evolution equations in incompressible flows. By comparing the statistics of the different inter-scale energy transfer terms that arise, we choose a novel optimal expression for the energy transfer which exhibits large-scale independence of the ratio of forward to backward energy flux, in addition to a milder filter-width dependence of this ratio than the classical subgrid-scale dissipation. We study the flow regions of intense energy transfer between resolved and subgrid scales from a geometrical point of view, to gain insight into the mechanism by which a predominant forward energy cascade is obtained in homogeneous 3D turbulence. The concept of length scale becomes a difficulty with this approach, because both a filter width $r$ and an object size $L$ are present. We show that with our energy transfer marker, $L$ depends on $r$ only to the extent that two regimes are observed: one above and one below $r\approx 30\eta$, where $\eta$ is the Kolmogorov length scale. Such a clear distinction is not observed with the usual expression for the subgrid-scale dissipation. [Preview Abstract] |
Tuesday, November 24, 2015 2:08PM - 2:21PM |
R19.00007: Large Deviation Statistics of Vorticity Stretching in Isotropic Turbulence Perry Johnson, Charles Meneveau A key feature of 3D fluid turbulence is the stretching/re-alignment of vorticity by the action of the strain-rate. It is shown using the cumulant-generating function that cumulative vorticity stretching along a Lagrangian path in isotropic turbulence behaves statistically like a sum of i.i.d. variables. The Cramer function for vorticity stretching is computed from the JHTDB isotropic DNS (Re\textunderscore $\backslash $lambda $=$ 430) and compared to those of the finite-time Lyapunov exponents (FTLE) for material deformation. As expected the mean cumulative vorticity stretching is slightly less than that of the most-stretched material line (largest FTLE), due to the vorticity's preferential alignment with the second-largest eigenvalue of strain-rate and the material line's preferential alignment with the largest eigenvalue. However, the vorticity stretching tends to be significantly larger than the second-largest FTLE, and the Cramer functions reveal that the statistics of vorticity stretching fluctuations are more similar to those of largest FTLE. A model Fokker-Planck equation is constructed by approximating the viscous destruction of vorticity with a deterministic non-linear relaxation law matching conditional statistics, while the fluctuations in vorticity stretching are modelled by stochastic noise matching the statistics encoded in the Cramer function. The model predicts a stretched-exponential tail for the vorticity magnitude PDF, with good agreement for the exponent but significant error (30-40{\%}) in the pre-factor. [Preview Abstract] |
Tuesday, November 24, 2015 2:21PM - 2:34PM |
R19.00008: Spatial-temporal spectra of velocity fluctuations in turbulent shear flows Guowei He, Ting Wu, Xin Zhao Space-time correlation or its Fourier form, spatial-temporal spectrum, is a minimal quantity to statistically characterize the temporal evolutions of spatial structures in turbulent flows. The Kraichnan-Tennekes random-sweeping model is well-known for spatial-temporal spectra in isotropic and homogeneous turbulence. Recently, Wilczek, Stevens and Meneveau (J. Fluid Mech. 2015 vol. 769, R1) have developed a simple model for spatial-temporal spectra in the Logarithmic layer of wall turbulence. In this study, we propose a model equation for turbulent shear flows. This model equation includes both sweeping and stretching effects and its solution gives an analytical expression for spatial-temporal spectra of stream-wise velocities. The results obtained are compared with the data from direct numerical simulation (DNS) of turbulent channel flows. It is found that this model is reasonably consistent with the DNS results for either small or large shear rates. This model is also discussed in comparison with the EA (elliptic approximation) model for space-time correlations in turbulent shear flows (Phys. Rev. E 79 046316 2009). [Preview Abstract] |
Tuesday, November 24, 2015 2:34PM - 2:47PM |
R19.00009: Effects of the mean velocity field on the renormalized turbulent viscosity and correlation function Abhishek Kumar, Mahendra Verma We perform renormalization group analysis of the Navier Stokes equation in the Eulerian framework in the presence of mean velocity field $U_0$, and observe that that the renormalized viscosity $\nu(k)$ is independent of $U_0$, where $k$ is the wavenumber. Thus we show that $\nu(k)$ in the Eulerian field theory is Galilean invariant. We also compute $\nu(k)$ using numerical simulations and verify the above theoretical prediction. The velocity-velocity correlation function however exhibits damped oscillations whose time period of oscillation and damping time scales are given by $1/k U_0$ and $1/( \nu(k) k^2)$ respectively. In a modified form of Kraichnan's direct interaction approximation (DIA), the ``random mean velocity field'' of the large eddies sweeps the small-scale fluctuations. The DIA calculations also reveal that in the weak turbulence limit, the energy spectrum $E(k) \sim k^{-3/2}$, but for the strong turbulence limit, the random velocity field of the large-scale eddies is scale-dependent that leads to Kolmogorov's energy spectrum.\footnote{M. K. Verma and A. Kumar, arXiv:1411.2693 (2015).} [Preview Abstract] |
Tuesday, November 24, 2015 2:47PM - 3:00PM |
R19.00010: Scaling of Lyapunov Exponents in Homogeneous, Isotropic Turbulence Prakash Mohan, Nicholas Fitzsimmons, Robert Moser Lyapunov exponents measure the rate of separation of infinitesimally close trajectories in a chaotic system. Here we use Lyapunov exponents describing the phase divergence of perturbed velocity fields to characterize the chaotic nature of homogenous isotropic turbulence. To compute the Lyapunov exponents we perform a DNS of forced isotropic turbulence and evolve a linear disturbance field along with the turbulence simulation. The average exponential growth rate of the linear disturbance field is the largest Lyapunov exponent of the system. We will discuss the scaling of this exponent with: a) the Reynolds number and b) the ratio of the integral length scale to the computational domain size. [Preview Abstract] |
Tuesday, November 24, 2015 3:00PM - 3:13PM |
R19.00011: Intermittency in an ensemble of Gaussian velocity fields with fluctuating characteristic scales Laura J. Lukassen, Michael Wilczek Turbulent velocity fluctuations exhibit intermittent behavior in both, the Lagrangian and the Eulerian description. Probability density functions of Lagrangian velocity increments, e.g., show a transition from a nearly Gaussian shape for large time lags to highly non-Gaussian shapes for smaller time lags. This inherent non-Gaussianity poses a challenge for both, phenomenological descriptions of turbulence as well as statistical approaches suffering from the closure problem. Here, we discuss the properties of an ensemble of Gaussian velocity fields in which the characteristic time (or length) scales of the ensemble members are drawn from an underlying distribution of those scales. Such an ensemble naturally exhibits non-Gaussian statistics, as has been demonstrated, e.g., in the context of multifractal modeling or superstatistics. In order to provide a more general approach, our model is based on the characteristic functional. Consequently, it captures the complete statistical information of the ensemble and thus, additional information such as joint statistics of increments at various scales can be obtained. In this context, we will discuss the potential of formulating novel closures based on our method as well as its relation to existing phenomenological models of turbulence. [Preview Abstract] |
Tuesday, November 24, 2015 3:13PM - 3:26PM |
R19.00012: Dynamical similarities of the direct and inverse turbulent cascades Alberto Vela-Martin, Javier Jimenez A fully reversible homogeneous isotropic turbulent system is constructed using inviscid LES to model energy fluxes in the inertial range. It recovers energy and other turbulent quantities when reversed after being allowed to decay. During the first phase, a direct cascade transfers energy from large to small scales while, during the second, an inverse cascade does the opposite. Short-time Lyapunov (STL) analysis is used to study and compare the dynamics of both cascades. This allows us to identify a smallest length scale for the chaotic flow behavior, below which the system behaves as a unit dynamically enslaved to larger motions by the contracting effect of the model. Above it, the inertial forces become relevant and the system is fully chaotic. When the inertial scales are isolated, the leading STL exponent is similar for both cascades, suggesting that the dynamics of the inertial range is conservative and time-symmetric, and that the direct and inverse energy cascades share similar energy transfer mechanisms. The cascade would thus be a bi-directional reversible process with similar up and down mechanisms, although, because the $L_2$ norm used in the STL analysis respects the geometry of phase space, the entropy-driven cascade directionally is not precluded. [Preview Abstract] |
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