Session BC: Turbulence Theory I

Dynamical Origins for Non-Gaussian Vorticity Distributions in Turbulent Flows

The problem of turbulence can be attacked either from a statistical or a dynamical point of view. Although a statistical theory of turbulent flows from first principles is still lacking, there have been promising attempts by Monin, Lundgren and Novikov to establish such theories. Furthermore it is known that fully developed three-dimensional turbulence is dominated by the complex interaction of filamentary vortices. These structures have a severe impact on statistical properties and are known to cause intermittency both in the Eulerian and Lagrangian frames. With the power of modern supercomputers direct numerical simulations help to gain insights into both of these questions. We will present results on the connection between non-Gaussian vorticity statistics and coherent structures. The underlying theory is cast in form of conditional averages, which allow to separate different dynamical influences like vortex stretching and vorticity diffusion.   

Averaging-invariance of compressible Navier-Stokes equation

While the averaging-invariance property of incompressible Navier-Stokes (iNS) is well documented; there is a need to formally establish the property for the compressible Navier-Stokes (cNS) equations. We put forth two new weighted-moment definitions and derive the averaging-invariant form of the continuity, momentum and energy equations for a general compressible flow. The averaging-invariant equations have the form of the Favre-averaged Navier-Stokes (FANS) equations and reduce to it in the appropriate limit. Furthermore, we derive the average-invariant forms of the evolution equations of various turbulent fluxes encountered in compressible turbulence. This formalization of the averaging invariance property is expected to contribute towards developing mathematically rigorous RANS-LES hybrid and/or bridging models.   

Representing the inhomogenous two-point second-order velocity correlation

Knowledge of a finite-dimensional representation for the velocity correlation tensor $R_{ij}(\mathbf{x},\mathbf{r})=\langle u'_i(\mathbf{x}-\mathbf{r}/2)u'_j(\mathbf{x}+\mathbf{r}/2)\rangle$ can be useful for extracting the correlation from finite-dimensional filtered LES correlation. Previously (Bhattacharya \emph{et al} 2008), a structure-tensor based model for $R_{ij}(\mathbf{x},\mathbf{r})$, homogeneous in $\mathbf{x}$, has been shown to agree well with data from channel flow for the normal $(i=j)$ components. However, this homogeneous model does not represent the shear ($i\neq j$) components well, due to the antisymmetric-in-$\mathbf{r}$ part of the correlation arising because of inhomogeneity. In this work, we formulate an inhomogeneous representation that is also based on structure tensors, satisfies the inhomogeneous continuity equation, and satisfies some additional consistency conditions. The inhomogeneity is expressed in terms of variation in $\mathbf{x}$ of both the integral length scale of the correlation and the velocity scale. We fit our model to correlations from DNS of turbulent channel flow and report these variations, as well as the improvements seen in the quality of the fit when compared to the homogeneous model.   

Vorticity Alignment with Local and Nonlocal Strain Rate Eigenvectors in Turbulent Flows

The anomalous alignment of vorticity $\omega_i(\textbf{x})$ with the strain rate eigenvectors in turbulent flows is examined using a decomposition of the strain rate $S_{ij}(\textbf{x})$ into the sum of local $S^R_{ij}(\textbf{x})$ and nonlocal (background) $S^B_{ij}(\textbf{x})$ contributions. Unlike previous work where alignment properties have been examined using the coupled differential transport equations for the vorticity and strain rate, we here consider instead the integro-differential equation for the vorticity that results when the strain rate is represented by a Biot-Savart integral over all vorticity in the flow. The decomposition of the strain rate as $S_{ij}(\textbf{x}) = S^R_{ij}(\textbf{x}) + S^B_{ij}(\textbf{x})$, which is achieved by splitting the integration domain into local $(r \leq R)$ and nonlocal $(r>R)$ domains, clearly distinguishes the linear (nonlocal) and nonlinear (local) contributions to the vorticity dynamics. The calculation of $S^R_{ij}(\textbf{x})$ and $S^B_{ij}(\textbf{x})$, including an operator method involving laplacians of $S_{ij}(\textbf{x})$, is demonstrated. Using data from highly-resolved direct numerical simulations of statistically-stationary homogeneous, isotropic turbulence, we show that while vorticity tends towards anomalous alignment with the intermediate eigenvector of the \textit{combined} strain rate $S_{ij}(\textbf{x})$, it aligns with the most extensional eigenvector of the \textit{background} strain rate $S^B_{ij}(\textbf{x})$, resulting in a significant linear contribution to the vorticity dynamics.   

Turbulence Dissipation Equation for Particle Laden Flow

A governing equation for turbulent dissipation is derived from fundamental principles for the carrier phase in particle-laden flow. The governing equation is valid for incompressible flows with no mass transfer between the dispersed and continuous phases. The equation is obtained by volume averaging the same equation used for single phase flows which includes the effects of the particle surfaces. The governing equation shows an additional production of dissipation term that is related to the instantaneous relative velocity gradients at the particle surface. The terms in the governing dissipation transport equation were simplified to produce a model that is similar to the time averaged dissipation model currently used for single phase flows. The model was then applied to experimental data for particles falling in an initially quiescent flow. The ratio of the new production of dissipation coefficient (due to the presence of particles) to the dissipation of dissipation coefficient was found to be related to the particle diameter and the Taylor length scale. The ratio of the coefficients was appeared to have a remarkable relationship with the relative Reynolds number of the particle. Future studies should address the validity of the equation under various flow condition and also be compared with DNS studies.   

Development of non-Gaussian statistics in rotating turbulence

The short-time evolution of non-Gaussian statistics in rotating turbulence is studied. Generalizing the idea in Y. Li and C. Meneveau, Phys. Rev. Lett. 95, 164502 (2005), dynamical equations for the velocity increments defined over an evolving material line element are obtained in a Lagrangian coordinate frame defined by the directions of the line element and the rotation axis. The equations provide simple representation for the effects of nonlinear advection and the Coriolis force. From these equations, we show that several observations in rotating turbulence, including the development of two-dimensional three-componential state and the positive skewness in cyclonic vorticity, can be qualitatively explained by the interaction between the Coriolis force and elementary nonlinear self-interaction processes.   

The decay of anistropic homogeneous turbulence

One of the curious aspects of grid tunnel turbulence has been its non-return-to-isotropy; e.g., in $[1]$ $\langle v^2 \rangle/\langle u^2 \rangle = 0.72$ and $\langle w^2 \rangle/\langle u^2 \rangle = 0.88$ throughout decay, while in $[2]$ $\langle v^2 \rangle / \langle u^2 \rangle \approx 0.77$. We extend the equilibrium similarity analysis of $[3]$ to the component spectral equations for anisotropic turbulence, and show the existence of permanently anisotropic solutions for which all components decay at the same rate. Moreover, the (physical) integral and Taylor microscales remain proportional during decay and the spectra and structure functions collapse using only Taylor variables. The theory is in excellent agreement with all of the available data, $[2]$ and $[3]$ in particular. \\ 1. Antonia, R.A. {\it et al.\ } {\bf JFM, 487}, 245-269, 2003. \\ 2. Kang, H.S. {\it et al.\ } {\bf JFM, 480}, 129,160, 2003. \\ 3. George, W.K. {\bf Phys Fluids A, 4}, 1493-1509, 1992.   

Measuring spectra using burst-mode LDA

The phrase ``burst-mode LDA'' refers to an LDA which operates with at most one particle present in the measuring volume at a time. For the signal to be interpreted correctly to avoid velocity bias, one must apply residence time-weighing to all statistical analysis. In addition, for time-series analysis, even though the randomly arriving particles eliminate aliasing, the self-noise from the random arrivals must be removed or it will dominate the spectra and correlations. A flaw in the earlier theory [1],[2], the goal of which was to provide an unbiased and unaliased spectral estimator from the random samples, is identified and corrected. The new methodology is illustrated using recent experiments in a round jet and a turbulent boundary layer. \\ 1. Buchhave, P. {\bf PhD Thesis}, SUNY/Buffalo, 1979. \\ 2. George, W.K. {\bf Proc. Marseille.-Balt. Dyn. Flow Conf. 1978},757-800.   

Magnetohydrodynamic Turbulence: Generalized Formulation and Extension to Compressible Cases

A general framework that incorporates the Iroshnikov-Kraichnan (IK) and Goldreich- Sridhar (GS) phenomenologies of magnetohydrodynamic (MHD) turbulence is developed [1]. This affords a clarification of the regimes of validity of IK and GS models of MHD turbulence. This formulation is generalized further to include compressibility effects. [1] B.K. Shivamoggi: Ann. Phys., vol. 323, p.1295, (2008).