Bulletin of the American Physical Society
60th Annual Meeting of the Divison of Fluid Dynamics
Volume 52, Number 12
Sunday–Tuesday, November 18–20, 2007; Salt Lake City, Utah
Session ND: Turbulence: Theory II |
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Chair: R. Rubinstein, NASA Langley Research Center Room: Salt Palace Convention Center 151 A-C |
Tuesday, November 20, 2007 11:35AM - 11:48AM |
ND.00001: Scale sensitivity of velocity and pressure correlations in channel flow Olof Grundestam, Sharath S. Girimaji The scale sensitivity of the GCM -- generalized central moments (Germano, M. 1992 Turbulence: the filtering approach. J. Fluid Mech. 238 325-336) -- of velocity, pressure-strain and dissipation correlations is investigated in channel flow. Pressure and velocity correlations are computed from direct numerical simulation data sets of nonrotating plane channel flow at Reynolds numbers 180 and 250 based on the wall-shear velocity and channel half-width ($\delta )$. The GCMs are evaluated using a top-hat filter in the physical periodic stream- and spanwise directions. It is found that, depending on wall-normal position, 70 - 90{\%} of energy of the velocity fluctuations is carried by scales smaller than $\delta $. This fraction is higher closer to the wall than in the center of the channel. The pressure strain and dissipation correlations are also to a major part carried by scales smaller than $\delta $. The rapid pressure strain part is, however, more sensitive to large filter widths than its slow counterpart and the dissipation. The implications for multi-resolution turbulence modeling (e.g. DES, hybrid RANS=LES) will be discussed. [Preview Abstract] |
Tuesday, November 20, 2007 11:48AM - 12:01PM |
ND.00002: Scaling of the velocity-scalar cross-correlation spectrum in two-dimensional turbulence Wouter J.T. Bos, Benjamin Kadoch, Kai Schneider, Jean-Pierre Bertoglio Two-dimensional isotropic turbulence with an imposed uniform scalar gradient is investigated. Dimensional arguments are presented to predict the inertial range scaling of the turbulent scalar flux spectrum both in the inverse cascade range and in the enstrophy cascade range. The predictions are checked by Direct Numerical Simulations. In the inverse cascade range perfect agreement with predictions is observed, while for the forward enstrophy cascade range the results differ from the predictions due to the emergence of coherent vortices. A link between the scalar flux and the Lagrangian particle displacement-velocity correlation, allows to evaluate the Lagrangian timescale needed in Markovian two-point closures of turbulence. [Preview Abstract] |
Tuesday, November 20, 2007 12:01PM - 12:14PM |
ND.00003: Modeling The Divergence Of The Two-Point, Third-Order Velocity Correlation Amitabh Bhattacharya, Robert Moser Knowledge of $\Gamma_{ij}(\mathbf{x},\mathbf{r})=\partial \langle u'_i(\mathbf{x})u'_k(\mathbf{x})u'_j(\mathbf{x}+\mathbf{r})\rangle/\partial r_k$ is crucial for modeling the subgrid force using the Optimal LES approach (Langford and Moser, (1999)). Here, a method is developed to obtain an approximation to $\Gamma_{ij}$ in statistically stationary turbulence, given a finite-dimensional representation for $R_{ij}(\mathbf{x},\mathbf{r})=\langle u'_i(\mathbf{x}) u'_j(\mathbf{x}+\mathbf{r})\rangle$. The rotationally invariant representation of $R_{ij}$ is in terms of Structure Tensors (Kassinos, 2001), and accounts for componental and directional anisotropy. Our method is based on the fact that the evolution equation for $R_{ij}$ is $D R_{ij}/D t=F_{ij}(\Gamma_{ij},\Pi_{ij},R_{ij},G_{ij})$, where $\Pi_{ij}(\mathbf{x},\mathbf{r})$ is the two-point pressure-strain correlation and $G_{ij}$ is the mean velocity gradient. For $D R_{ij}/D t=0$, a symmetrized form of $\Gamma_{ij}$ can be obtained in terms of a production term involving $R_{ij}$ and $G_{ij}$ (after projecting out $\Pi_{ij}$ using continuity). The resulting model and it's underlying assumptions are validated by comparison with DNS. [Preview Abstract] |
Tuesday, November 20, 2007 12:14PM - 12:27PM |
ND.00004: Unbalanced vortex stretching in nonstationary turbulence Robert Rubinstein An important implication of Kolmogorov's theory is that at high enough Reynolds number, the large scales of motion become independent of viscosity. Associated to this idea is the famous observation of Tennekes and Lumley: if the enstrophy balance in a turbulent flow is to be independent of viscosity, then enstrophy generation by vortex stretching and enstrophy destruction by vorticity, both of order $Re^{1/2}$, must cancel, leaving an $O(Re^0)$ remainder. Some issues associated with this balance have recently been reconsidered by Tsinober. The failure of any number of theoretical investigations to justify this balance led Speziale and Bernard to investigate the possibility of `unbalanced vortex stretching' in turbulence, leading to novel predictions for decaying turbulence and homogeneous shear flow. We will show that unbalanced vortex stretching occurs in transient turbulence dynamics, but that in self-similar turbulent flows, the Tennekes-Lumley balance is recovered. This analysis implies that whereas the Kolmogorov theory does indeed represent a statistical {\em attractor} for the small scales of motion, it is not a permanent feature. Implications of unbalanced vortex stretching for modeling transient flows will be considered. [Preview Abstract] |
Tuesday, November 20, 2007 12:27PM - 12:40PM |
ND.00005: Hierarchical energy spectra in quasi-steady turbulence Kiyosi Horiuti, Takeharu Fujisawa The Kolmogorov $-5/3$ law, $E_0(k)=C_K \varepsilon^{2/3} k^{- 3/5}$, forms a base state for the energy spectrum in the inertial subrange, which is applied only to a steady state. An expansion for the spectrum about this base state using the perturbation method (Yoshizawa 1998, Woodroff \& Rubinstein 2006) yields a nonequilibrium spectrum as $$E(k)= E_0(k)+C_N \dot {\varepsilon}\;\varepsilon^{-2/3} k^{-7/3}+C_3(\ddot {\varepsilon}\;\varepsilon^{-1}-2\dot{\varepsilon}^2\varepsilon^ {-2}/3)k^{-9/3}+\cdots ,$$ where $\varepsilon$ and $\dot{\varepsilon}$ denote the dissipation rate and its time derivative, respectively. This formula indicates that the spectrum contains the hierarchical scaling exponents, and the $-7/3$ and $-9/3$ scalings can be induced by the fluctuation of $\varepsilon$. Long term-temporal average yields $E(k)\approx E_0(k)$, but the $-7/3$ component can be extracted by conditionally sampling on $\dot{\varepsilon} $. We carried out this extraction using the DNS data for quasi- steady forced homogeneous isotropic turbulence and homogeneous sheared turbulence. It is shown that the $-7/3$ spectrum is indeed identified in both flows. The relationship between the each decomposed spectra and those induced by the three modes of vorticity configurations in the stretched spiral vortex model (Lundgren 1982, Horiuti \& Fujisawa 2007) will be discussed. [Preview Abstract] |
Tuesday, November 20, 2007 12:40PM - 12:53PM |
ND.00006: Velocity gradient dynamics in compressible and incompressible flows using Homogenized Euler Equation Sawan Suman, Sharath Girimaji Velocity gradient dynamics in compressible flows is expected to be influenced by variations in temperature and density. Along the lines of the restricted Euler equation meant for incompressible flows, we propose Homogenized Euler Equation (HEE) to study the evolution of compressible velocity gradient dynamics. The HEE is solved in the Lagrangian reference frame for a calorically perfect gas. Unlike the restricted Euler equation, the HEE allows for the inclusion of both the isotropic and the anisotropic parts of the pressure Hessian. Computations are performed for a large number of initial conditions. Conditional statistics of principal strain rates, vorticity vector alignment and the second and third invariants of the normalized velocity gradient tensor are presented with particle dilatation as a parameter. Also, the orientation tendencies of vorticity with the eigenvectors of the pressure Hessian tensor are investigated. The HEE results conditioned at appropriate dilatation values agree well with (1) two stable fixed points of Burgers dynamics and, (2) direct numerical simulation data for decaying isotropic, incompressible turbulence. [Preview Abstract] |
Tuesday, November 20, 2007 12:53PM - 1:06PM |
ND.00007: Parabolic Profile Approximation for the Singularity Spectrum in Fully Developed Turbulence: Applications to Enstrophy Cascade in Two Dimensions Bhimsen Shivamoggi The parabolic-profile approximation (PPA) for the singularity spectrum f ($\alpha )$ of fully-developed turbulence (FDT) is extended to the Kolmogorov microscale regime. The PPA also affords, unlike the multi-fractal model, an analytical calculation of probability distribution function of velocity gradient, and describes intermittency corrections that complement those provided by the homogeneous-fractal model. This formulation is extended to the two-dimensional (2D) enstrophy cascade. Intermittency (externally induced) in the 2D enstrophy cascade is shown to be able to maintain a finite enstrophy along with a vorticity conservation anomaly. Intermittency mechanisms of 3D energy cascade and 2D enstrophy cascade in fully-developed turbulence seem at least theoretically to have some universal features. [Preview Abstract] |
Tuesday, November 20, 2007 1:06PM - 1:19PM |
ND.00008: Joint multifractal analysis of intermittent fields in high-resolution DNS of turbulence Takashi Ishihara, Hirotaka Higuchi In high-Reynolds number turbulence, several intermittent fields coexist, among which are the rate \textit{$\varepsilon $} of dissipation of turbulent energy, vorticity \textit{$\omega $} and pressure gradients grad$p$, etc. These intermittent fields display different degrees of correlation among them. To characterize such coexisting distributions of intermittent fields in high-Reynolds number turbulence, we apply joint multifractal analysis to the data obtained by high-resolution DNS of turbulence in a periodic box. The analysis shows that the degree of correlation between \textit{$\alpha $}$_{\varepsilon }$ and \textit{$\alpha $}$_{P}$ is considerably high, but lower than between \textit{$\alpha $}$_{\varepsilon }$ and \textit{$\alpha $}$_{\Omega }$, where $P=\vert $grad$p\vert ^{2}$ and \textit{$\Omega $ }=\textit{$\omega $}$^{2}$/2, and \textit{$\alpha $}$_{{\rm A}}$ is a local singularity strength of $A$. [Preview Abstract] |
Tuesday, November 20, 2007 1:19PM - 1:32PM |
ND.00009: Thermalization and Turbulence Bottleneck Jian-Zhou Zhu, Uriel Frisch, Walter Pauls, Susan Kurien It is conjectured that for many equations of hydrodynamical type, including the three-dimensional Navier-Stokes equations, the Burgers equation and various models of turbulence, the use of hyperviscous dissipation with a high power $\alpha$ (dissipativity) of the Laplacian and suitable rescaling of the hyperviscosity becomes asymptotically equivalent to using a Galerkin truncation with zero dissipation and suppression of all Fourier modes whose wavenumber exceeds a cutoff $k_d$. The Galerkin-truncated Euler system will develop a thermalized range at high wavenumbers as presented by Cichowlas et al [{\it Phys. Rev. Lett.} {\bf 95} (2005) 264502]. It is therefore proposed to interpret the phenomenon of bottleneck, which becomes stronger with increasing $\alpha$, as an aborted thermalization. Numerical verification of these ideas are discussed, along with various artefacts which can appear when using hyperviscosity. [Preview Abstract] |
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