Bulletin of the American Physical Society
2006 59th Annual Meeting of the APS Division of Fluid Dynamics
Sunday–Tuesday, November 19–21, 2006; Tampa Bay, Florida
Session AP: Turbulence Theory I |
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Chair: Huidan Yu, Los Alamos National Laboratory Room: Tampa Marriott Waterside Hotel and Marina Meeting Room 12 |
Sunday, November 19, 2006 8:00AM - 8:13AM |
AP.00001: Asymptotic Exponents from Low-Reynolds-Number Flows Joerg Schumacher, Katepalli R. Sreenivasan, Victor Yakhot We present detailed studies of turbulence in the crossover region between the inertial and viscous ranges of turbulence. The work is based on two results: (1) the scaling exponents $\rho_n$ of the moments of velocity derivatives $\langle(\partial u/\partial x)^n\rangle$, with respect to the large scale Reynolds number $Re$, can be expressed in terms of the inertial-range scaling exponents $\xi_n$ of the longitudinal structure functions $S_n(r)=\langle(\delta_r u)^n\rangle$. (2) High-resolution direct numerical simulations of isotropic and homogeneous turbulence in a periodic box, in which sub-Kolmogorov grid has been employed to accurately resolve the analytic parts of the structure functions, are conducted. They show that the derivative moments for orders $0\le n\le 8$, obtained from relatively low Reynolds number flows with Taylor-microscale Reynolds number $10\leq R_{\lambda} \leq 63$, are represented well as powers of the Reynolds number. The exponents $\rho_{n}$ in these flows, though exhibiting no developed inertial range, agree closely with the exponents $\xi_n$ ($0\leq n\leq 17$) corresponding to the inertial range of high-Reynolds-number fully developed turbulence. The existence of a whole range of {\em local} dissipation scales, rather than a single Kolmogorov dissipation scale, is discussed. [Preview Abstract] |
Sunday, November 19, 2006 8:13AM - 8:26AM |
AP.00002: Testing a similarity theory for isotropic turbulence on DNS data. Mogens Melander, Bruce Fabijonas Using direct numerical simulations, we consider the issue of self-similarity in 3D incompressible isotropic turbulence. The starting point for our investigation is a similarity theory we have developed on the basis of high Reynolds number shell model calculations. Like Kolmogorov's 1941 theory, our theory calls for similarity across all scales in the inertial range. Unlike K41, our theory does not fail on account of intermittency, but is developed to blossom in that environment. To observe self-similarity, it is essential that the correct variables are used, otherwise one sees only intermittency. The correct variables are reasonably easy to spot for the shell model, but they are more difficult to identify for the full Navier-Stokes equations. Moreover, one has to overcome the fact that the DNS has lower Reynolds numbers than in the shell model simulations so that the inertial range is shorter. Using the technique ESS, we clear this obstacle with only a minor modification to the theory. The DNS data then collapse on the theoretical pdf at all scales. [Preview Abstract] |
Sunday, November 19, 2006 8:26AM - 8:39AM |
AP.00003: The effects of pressure, inter-scale interaction and viscosity on the Lagrangian evolution of velocity increments Yi Li, Charles Meneveau A simple system of two ordinary differential equations (the advected delta-vee system)has been derived from the Navier-Stokes equations to describe the short-time Lagrangian evolution of velocity increments (Li and Meneveau, PRL, 2005 and Li and Meneveau JFM, 2006). It was shown that many important intermittency trends ubiquitous in turbulent flows are reproduced from the system even when the effects of part of the isotropic pressure, the anisotropic pressure Hessian, inter-scale interaction and viscosity are neglected. The truncated system thus provides simple dynamical explanations to many intermittency trends in turbulence. In this talk, the effects of those neglected terms are investigated based on direct numerical simulations, using conditional statistics. The results show that the neglected terms tend to reduce the probability of large fluctuations in velocity increments, consistent with the fact that they are needed to regularize the truncated system. Different terms behave distinctly in different regions of the phase space of the advected delta-vee system. The results suggest that different models are required to model the pressure Hessian, interscale and viscous effects. [Preview Abstract] |
Sunday, November 19, 2006 8:39AM - 8:52AM |
AP.00004: Rapid distortion analysis of compressible turbulence in ideal gas: Part 1 -- Reynolds averaged moments Tucker Lavin, Huidan Yu, Sharath Girimaji We solve the inviscid, compressible linearized Reynolds-averaged Navier-Stokes equations, invoking the ideal gas law rather than employing the simpler but less practical isentropic assumption. The formulation involves 26 ordinary differential equations. At the zero-Mach number limit, the incompressible RDT solutions are recovered for various mean velocity gradients. At the high-Mach number limit, the Burgers solution is recovered for the homogeneous mean shear case. For intermediate Mach numbers the pressure dilatation term is found to be influential in transferring energy between the kinetic and internal modes. The exchange causes high frequency oscillations in Reynolds stresses which are absent in the incompressible limit. The effects of compressible initial conditions are also examined. Initially compressible velocity field is found to be more conducive to internal energy conversion. Nearly half of the total energy is in the form of internal energy at large times. In comparison about 25 percent of energy is in the form of internal energy for the case of the initially- incompressible field. [Preview Abstract] |
Sunday, November 19, 2006 8:52AM - 9:05AM |
AP.00005: Rapid distortion analysis of compressible turbulence in ideal gas: Part 2 - Density averaged moments Sawan Suman, Huidan Yu, Sharath Girimaji, Tucker Lavin We perform rapid distortion analysis of linearized, inviscid Favre-averaged Navier Stokes equation. This study - Favre-averaged Rapid Distortion Theory (F-RDT)- investigates the evolution of density weighted fluctuating moments. The fluid is assumed to be an ideal gas. The F-RDT formulation comprises of a closed set of 65 ordinary differential equations for the case of homogenous mean shear field. With $S \mathord{\left/ {\vphantom {S {R\widetilde{T}}}} \right. \kern-\nulldelimiterspace} {R\widetilde{T}}$ as compressibility parameter ($S=$magnitude of mean shear, $R=$gas constant, $\widetilde{\mbox{T}}=$density weighted mean temperature), the versatility of the formulation is demonstrated by recovering both the incompressible and Burgers limit behaviors. Results for several intermediate cases-between the above two extreme limits-are also obtained. Favre-averaged Reynolds stresses, temperature variance, density variance and various cross-correlations will be discussed. [Preview Abstract] |
Sunday, November 19, 2006 9:05AM - 9:18AM |
AP.00006: Small scale response in periodically forced turbulence Robert Rubinstein, Wouter Bos, Timothy Clark The response of the small scales of an isotropic homogeneous turbulence subject to periodic large scale forcing is studied using two-point closures and other simpler models. Closure results are validated by comparison with available numerical and experimental data. The phase and amplitude of the dissipation perturbations show nontrivial dependence on the forcing frequency. Perturbation methods are used to understand the basic features of this dependence. Simple finite dimensional models are found to be fundamentally incapable of reproducing the essential features of this problem. [Preview Abstract] |
Sunday, November 19, 2006 9:18AM - 9:31AM |
AP.00007: An Analytical Model for the Three-Point Third-Order Velocity Correlation in Isotropic Turbulence Henry Chang, Robert Moser In turbulent flows, the three-point third-order velocity correlation $T_{ijk}({\bf r},{\bf r}') = \langle v_i({\bf x}) v_j({\bf x}+{\bf r}) v_k({\bf x}+{\bf r'}) \rangle$ is an important quantity. In particular, when considering large eddy simulation, the contribution of the nonlinear terms to evolution of the two-point second-order correlation of filtered velocities can be written in terms of integrals of the three-point correlation. In contrast, the two-point third order correlation appears in the equation for the unfiltered two-point correlation, and under the Kolmogorov scaling assumptions, this is sufficient to determine it. An analytic model for the three-point third-order correlation, under the same assumptions, would be very useful in the analysis of LES. There are constraints imposed by continuity and symmetry, and in 1954, Proudman and Reid determined a general form for the Fourier transform of this correlation that satisfies the constraints. Inverse transforming to physical-space yields a form for $T_{ijk}({\bf r},{\bf r}')$ in terms of derivatives of a scalar function of the magnitudes of the separation vectors. Considering the simplest possible forms of the scalar function that are consistent with the known two-point third-order correlation in the Kolmogorov inertial range yields a six-dimensional space of representations. The coefficients of the representation for $T_{ijk}$ are then determined from DNS data to yield the proposed model. [Preview Abstract] |
Sunday, November 19, 2006 9:31AM - 9:44AM |
AP.00008: Representation of Two-Point Velocity Correlations in Terms of Structure Tensors A. Bhattacharya, U. Godse, R.D. Moser, S. Kassinos A general representation of the homogeneous, anisotropic two-point second-order correlation of turbulent velocity fluctuation of the form $R_{ij}=\langle u'_i(\mathbf{x}) u'_j(\mathbf{x+r}) \rangle=\Sigma_n f^{(n)}(r) T^{(n)}_{ij}$ is constructed, where $12$ basis tensors $T^{(n)}_{ij}$ are expressed in terms of the separation vector $\mathbf{r}$ and structure tensors introduced by Kassinos and Reynolds (1995). The structure tensors are one-point correlations of the derivatives of fluctuating streamfunctions and are given by componentality $b_{ij}$, dimensionality $y_{ij}$ and stropholysis $Q_{ijk}$. These tensors are shown to contain information about the anisotropy of $R_{ij}$ (thus motivating such a representation). Using continuity and an additional constraint, only four scalar functions $f^{(n)}$ are shown to remain linearly independent. A comparison of the representation with two-point correlation data from DNS of channel flow turbulence is made in order to assess the suitability of this representation. [Preview Abstract] |
Sunday, November 19, 2006 9:44AM - 9:57AM |
AP.00009: Studying a stochastic turbulence model for Lagrangian dynamics of the velocity gradient tensor using direct numerical simulation Charles Meneveau, Laurent Chevillard, Luca Biferale, Federico Toschi A new stochastic Lagrangian model for the evolution of the velocity gradient tensor in turbulent flow has been recently developed (Chevillard \& Meneveau, 2006). It includes closures for the pressure and velocity gradient Hessians based on the local deformation field seen by a fluid particle along its trajectory. The model yields statistically stationary statistics and reproduces several well-known properties of turbulent flows, such as alignments of vorticity, joint pdfs in the R-Q plane, as well as anomalous relative scaling of moments of the velocity gradient at moderate Reynolds numbers. In this talk, model predictions for the various model terms are compared in detail with results from DNS at various Reynolds numbers. In order to quantify trends in relation to local topology, conditional averages at fixed values of R and Q are used. We conclude that the model reproduces the dynamics quite well in strain-dominated regions, but shows qualitative differences in regions dominated by rotation and vortex-stretching. We speculate on possible effects of coherent small-scale vortex structures. [Preview Abstract] |
Sunday, November 19, 2006 9:57AM - 10:10AM |
AP.00010: Energy Dissipation in Fractal-Forced Flows Alexey P. Cheskidov, Charles R. Doering, Nikola P. Petrov The rate of energy dissipation in solutions of the body-forced 3D incompressible Navier-Stokes equations is rigorously estimated with a focus on its dependence on the nature of the driving force. For square integrable body forces the high Reynolds number (low viscosity) upper bound on the dissipation is independent of the viscosity, consistent with the existence of a conventional turbulent energy cascade. On the other hand when the body force is not square integrable, i.e., when the Fourier spectrum of the force decays sufficiently slowly at high wavenumbers, there is significant direct driving at a broad range of spatial scales. Then the upper limit for the dissipation rate may diverge at high Reynolds numbers, consistent with recent experimental and computational studies of ``fractal-forced'' turbulence. [Preview Abstract] |
Sunday, November 19, 2006 10:10AM - 10:23AM |
AP.00011: Interscale energy transfer for band decompositions using spectral sharp and smooth filters Julian Domaradzki, Daniele Carati There is a consensus that a bulk of energy transfer in turbulence occurs between scales of similar size (local energy transfer) but a controversy persists concerning a role of nonlocal interactions, i.e., of much larger scales, in this process. In particular, a possibility exists that even qualitative conclusions may be affected by a choice of filters employed to define decompositions of velocity fields into bands. To address this question we analyze interscale interactions in DNS databases and compare results for band decompositions defined using classical sharp and smooth spectral filters, including Gaussian. In both cases, and for the range of scales available in DNS, the dependence of energy transfer on large scales in the energy containing range, i.e., the effect of nonlocal interactions, is always significant. However, for the Gaussian bands this dependence is observed to slightly diminish for increasing nonlocality. Many other features of interscale interactions are qualitatively similar in both cases. The main differences are mainly due to the exclusion of classes of interactions for sharp spectral bands because of the triangle inequality. [Preview Abstract] |
Sunday, November 19, 2006 10:23AM - 10:36AM |
AP.00012: Loitsiansky was correct in the infinite domain Jonathan Gustafsson, William K. George Decaying isotropic, homogeneous, incompressible turbulence in a infinite domain is examined. The Saffman integral\footnote{P. G. Saffman, J. Fluid Mech. 27, 581 (1967).}: $\int_0^{\infty}r^2B_{ii}(r)dr$ is found to be zero not as previously assumed $\pi^2 M$. Under Saffman assumption the integral doesn't converge in the infinite domain. Using the same method on the Loitsiansky equation\footnote{L. G. Loitsiansky, Cent. Aero. Hydrodyn. Inst. Moscow, Report. No. 440 (Trans. NACA Tech. Memo. 1079), 1939.}: \begin{equation} \frac{\partial}{\partial t} \int^{\infty}_0 r^4 B_{LL}(r)dr = -2[B_{NN,L}(r) r^4]^{\infty}_0 + +2\nu[\frac{\partial B_{LL}(r)}{\partial r} r^4]^{\infty}_0 \end{equation} shows that for an infinite domain $\lim_{r \rightarrow \infty}r^4B_{NN,L}(r) = 0$. Contradicting the findings of I. Proudman \& W. H. Reid\footnote{I. Proudman and W. H. Reid, Philos. Trans. R. Soc. London Ser. A 247, 163 (1954).} and G. K. Batchelor \& I. Proudman\footnote{G. K. Batchelor and I. Proudman, Philos. Trans. R. Soc. London Ser. A 248, 369 (1956).}. However in a finite domain the above result found in this research do not hold. [Preview Abstract] |
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