Session KD: Turbulence: Theory I

On the issue of monotonicity of structure function exponents

Structure functions in the inertial range of isotropic turbulence are generally believed to be power laws in the length scale. In the presence of intermittency, the exponent varies nonlinearly with the order p of the structure function. It is a known mathematical fact that the exponent must be a concave down function of p. The literature moreover suggests that the exponent should increase monotonically with p. It is well known that experimental and computational evidence support that suggestion in the sense that no violation of the monotonicity has been observed. In this talk, we point out that the theoretical support for monotonicity crumples upon close scrutiny. Moreover, we present a series of numerical experiments with given log-normal exponents (not monotone) showing that astronomical ensemble sizes are required to capture the decreasing exponents. For realistic ensemble sizes, one is misled to believe the exponents follow log- Poisson statistics.   

Lagrangian velocity correlations in homogeneous and isotropic turbulence

Lagrangian velocity correlations are the important quantity to describe turbulent diffusion processes and pose the challenges for modeling efforts that do not explicitly represent subgrid scale motions in large eddy simulation. Our previous study investigates the effects of subgrid scale modeling on Eulerian velocity correlations (Phys. Fluids.14 2186), where the Kraichnan model for Eulerian velocity correlations is crucial (Phys. Fluids 16 3859). The present study develops a model for Lagrangian velocity correlations in homogeneous and isotropic turbulence, which relates the Lagrangian velocity correlations to the Eulerian velocity correlations via a characteristic velocity. The characteristic velocity is analytically calculated from the Navier-Stokes equations and provides the decorrelation time scales. The model is based on the two assumptions on the iso-correlation contours: (1) they can be well approximated by the elliptic curves; (2) they are self-similar at sufficient large Reynolds numbers, that is to say, they share a preference directions and aspect ratios. The data from direct numerical simulation of homogeneous and isotropic turbulence verifies the the model. The model is further discussed on its relationship with Kraichnan's straining hypothesis and its potential applications to the large eddy simulation of particle-laden turbulence.   

On the non-local geometry of turbulence

A multi-scale methodology for the study of the non-local geometry of structures in turbulence is applied to a passive-scalar field and the square of the vorticity field of a $5123$ periodic cube DNS of homogeneous isotropic turbulence. Results of its application to the vorticity field of a set of DNS with identical initial conditions and increasing grid resolutions ($256,512,10243$, with $k_{max}\eta\approx 1$,$2$,$4$) are also discussed. The methodology consists of three main steps: extraction, characterization and classification, starting from a 3D scalar field. Extraction is done via the curvelet transform (allowing a multi-scale decomposition), followed by isosurfacing of the set of scalar fields obtained by filtering in curvelet space. Characterization is based on the area-based probability density function of two differential-geometry properties, shape index and curvedness, complemented with global invariants of the surface, thus defining its signature. Classification uses a feature space of parameters obtained from the signature of each structure,where clustering techniques are applied searching for groups of structures with common geometry.   

An analysis of the energy transfer and the locality of nonlinear interactions in turbulence

Using results of direct numerical simulations of isotropic turbulence we compute detailed energy exchanges among different scales of motion, defined by decomposing velocity fields using sharp spectral and smooth, tangent hyperbolic filters. The elementary energy exchange event involves two scales interacting nonlinearly and producing an effect in the third scale. In Fourier decomposition wavevectors representing all three interacting scales form a triangle. The analysis of such detailed interactions reveals that individual nonlocal contributions are large, in agreement with results of other investigators. The global energy quantities such as the energy transfer, the spectral energy flux, and the subgrid-scale dissipation reflect an integrated effect of many individual triad interactions. We investigate how the detailed, triadic energy exchanges contribute to the global, integrated quantities. We find that while individual nonlocal contributions are large, significant cancellations lead to the global quantities asymptotically dominated by the local interactions. The conclusions are expressed quantitatively in terms of infrared and ultraviolet Kraichnan's locality functions. Implications of these results for turbulence modeling are discussed.   

Asymptotic Sensitivity of Homogeneous Turbulent Shear Flow to the

Our recent numerical studies of homogeneous turbulent shear flow suggest the dynamics of the large and small scales are sensitive to the initial value of the shear parameter. In particular for initial values of $S^{*} = S k /\epsilon\ge 10$, we find that the asymptotic state of the turbulence depends upon this parameter. Rapid distortion theory (RDT) predicts the dependence of both large- and small-scale statistics on $S^*$ reasonably well, but the theory is applicable only for relatively short times ($S t < 2$). Direct numerical simulation (DNS) has a somewhat longer window, but it too eventually fails when the integral length scale becomes too large. Motivated by this earlier work, we performed experimental measurements of large- and small-scale velocity statistics in homogeneous turbulent shear flow in a wind tunnel. We are able to vary the initial shear parameter over the relevant range and observe the aforementioned asymptotic statistics. The experimental results will be presented, including detailed comparisons with earlier DNS and RDT.   

Lagrangian statistics in forced two-dimensional turbulence

In recent years the Lagrangian description of turbulent flows has attracted much interest from the experimental point of view and as well is in the focus of numerical and analytical investigations. We present detailed numerical investigations of Lagrangian tracer particles in the inverse energy cascade of two-dimensional turbulence. In the first part we focus on the shape and scaling properties of the probability distribution functions for the velocity increments and compare them to the Eulerian case and the increment statistics in three dimensions. Motivated by our observations we address the important question of translating increment statistics from one frame of reference to the other [1]. To reveal the underlying physical mechanism we determine numerically the involved transition probabilities. In this way we shed light on the source of Lagrangian intermittency.\\[1ex] [1] R. Friedrich, R. Grauer, H. Hohmann, O. Kamps, A Corrsin type approximation for Lagrangian fluid Turbulence , arXiv:0705.3132   

Lagrangian evolution of velocity increments and development of non-Gaussian statistics in rotating turbulence

Recently, a system describing the Lagrangian evolution of velocity and passive scalar increments has been shown to reproduce many well-known intermittency trends in incompressible turbulence (Li and Meneveau, JFM vol 558, p. 133, 2006). Here we generalize the system to consider rotating turbulence. To take into account the effects of the Coriolis force, a system describing the Lagrangian evolution of three components of the velocity increment aligned with an advected material line turns out to be most convenient. After reviewing previous results, the derivation of the system is presented. Numerical time-integration of the system starting from Gaussian initial distributions is conducted. The effects of rotation on the development of non-Gaussain tails in the distributions are examined and comparison with non-rotating cases is made.   

Real-time Image Compression for Lagrangian Particle Tracking

Optical particle tracking is a powerful tool for fluid measurements; however it faces serious constraints due to the huge data rates produced by high-speed cameras with high spatial resolution. Since particle tracking typically produces images that contain simple spots on a uniform background, these images are prime candidates for real-time image compression. We have implemented a system that uses a programmable logic unit to achieve real-time image compression factors ranging from 100 to 1000 for 1024 x 1280 pixel images at 500Hz. We present details of our existing implementation and discuss future developments in this field.   

Effects of Large Scale Intermittency on Small Scale Turbulence

We report on the effect of temporal fluctuations of the large scales in a turbulent flow on small scale turbulence statistics. A stereoscopic high-speed imaging system (3D PTV) is used to obtain Lagrangian trajectories in a flow between oscillating grids in 1m x 1m x 1.5m tank. A novel real-time image compression system allows us to obtain very large data sets. We report measurements of structure functions and the energy dissipation rate conditioned on the phase of the driving grid or conditioned on the instantaneous velocity.   

The Molecular Origin of Turbulence in a Flowing Gas According to James Clerk Maxwell

James Clerk Maxwell was an eminent physicist who operated out of the University of Edinburgh in the early 1800's. He is internationally famous for his derivation of the laws governing the propagation of electro-magnetic waves. He also derived an equation for the Viscosity of a gas ($\mu )$ in terms of its \textbf{molecular} parameters. This derivation established clearly and unequivocably that a real (viscous) flowing gas was a \textbf{molecular fluid}, that is, a flow of molecules which obeys the Kinetic Theory of Gases. Maxwell's derivation of the Viscosity of a gas takes place in a zone of a flowing gas which (1) is remote from any solid surface, and (2) is in a region having a linear velocity-gradient dv$_{x}$/dy . The derivation which I will present today takes place in a zone of the flowing gas which is (1) immediately adjacent a solid surface, and (2) where the velocity gradient is unknown. My analytical approach, the parameters I use, and the theoretical concepts are all taken from Maxwell's derivation. I have simply re-arranged some of his equations in order to solve the 1-dimensional case of boundary-layer growth over an infinite flat plate, starting with a step-function of flow velocity, namely: v$_{x}$(y,t) for the initial condition v$_{x}$(y=0+,t=0+) = U$_{0}$ ,viz: rectilinear flow as an initial condition. Using Maxwell's approach, we write the equation for Net Stream-Momentum Flux flowing through an element of area, da$_{y}$ . This quantity is shown to be the difference between two Convolution integrals which Laplace transform readily into an equation in the s-plane which equation has the same form as a positive-feedback, single closed-loop amplifier gain equation, viz: Output = (input)x(transfer function). The solution in the Real plane shows v$_{x}$(y,t) equal to the sum of two exponentials. The coefficients of the two exponents, r$_{1}$ and r$_{2}$ . are found by using the binomial equation which contains a square-root radical. If the argument under the radical (the radicand) is positive, the two roots are real, and turbulence does not occur. If the radicand is negative, the two roots are complex conjugates and turbulence will develop. The physical reality of the transfer function's feedback-loop format may be clarified by tracing backwards through the derivation to the earliest occurrence of v$_{x}$(y,t). Maxwell's derivation of Viscosity, adapted to solve for the boundary-layer growth over an infinite flat plate, is shown to be a nice application of the Kinetic Theory of Gases, and is well suited to revealing the molecular mechanisms at work in such a flowing pattern.