Session MA: Turbulence Theory III

Closure theories for inhomogeneous turbulence

Although Kraichnan formulated the Direct Interaction Approximation and the Test-Field model for general problems of inhomogeneous turbulence, the resulting equations, requiring repeated integrations over the flow domain, are both difficult to understand and difficult to apply in practice; in the homogeneous case, triad interactions provide the key to unraveling the physics of the approximation. The goal of this work is to formulate some special inhomogeneous problems with comparable simplicity. It is done by decomposing the inhomogeneous problem into a set of coupled quasi-homogeneous problems, each of which admits a simple formulation. The formalism will be applied to the problem of weakly inhomogeneous turbulence, where previous heuristic theories have proven incomplete. The same formalism applies to problems admitting scaling transformations; it will be applied to give a simple formulation of the problem of turbulence in a half-space.   

Energy injection into two-dimensional turbulence: a scaling regime controlled by drag

The energy injection rate $\varepsilon$ is the most important single statistical quantity characterizing two-dimensional turbulence, and it plays a central role in Kraichnan's theory of inverse energy cascade. In most experiments and meteorological applications, $\varepsilon$ is not known a priori, as the fluid is driven by a body force rather than by prescribing $\varepsilon$. It is therefore important to understand the dependence of $\varepsilon$ on the external control parameters of a system. Drag is an important physical effect in many quasi-two-dimensional systems. Hence, we consider two-dimensional turbulence driven by steady sinusoidal body force at small scale, with linear drag of damping time scale $\mu^{-1}$ as the main dissipative mechanism. We present numerical results that reveal a new scaling regime in which $\varepsilon \sim \mu^{1/3}$. A theoretical model in which the directly forced mode is randomly swept by the large scale motion across the stationary sinusoidal forcing pattern is used to explain the observations.   

An argument against non-universal scaling coefficients for inertial range structure functions

In the inertial range of homogeneous isotropic turbulence with large scale forcing, the structure functions are statistical moments that depend parametrically on a length scale l. For each moment the scaling dependence with respect to l takes the form of a power law characterized by an exponent, a coefficient, and a virtual origin. The scaling exponent is generally believed to be a universal function of the order p of the moment. In contrast, the coefficient is not believed to be a universal function of p. Only when p is three or zero are universal values of the coefficient accepted. For other values of p one finds arguments in the literature that the coefficients can't be universal. These arguments build on the fact that the large scales are governed by the forcing which is essentially arbitrary. Supposedly, universal coefficients are in conflict with arbitrariness at the large scales, or so the argument goes. By a counterexample we show that there need not be any such conflict. Hopefully, this will clear the way for the idea of universal scaling coefficients for the inertial range.   

Theoretical Aspects of Level Crossing Scales in Turbulence

We consider theoretical aspects of the statistics of level crossing scales in turbulence. Our focus is on two basic quantities of fundamental and practical interest. The first quantity is the probability density function of the level crossing scales. The second quantity is the correlation function of thresholded signals or fields corresponding to the level crossings. We explore a general mathematical approach aimed toward establishing relations between these two quantities. In addition to the general approach, we study specific classes of level crossings with power-law, exponential, and log-normal statistics. These three types of level crossings are believed to be appropriate for various turbulent flows, according to available observations, and thus are particularly relevant for turbulence studies. Because the correlation function is closely related to the power spectrum, this approach has the potential to illuminate the relation between the probability density function of level crossing scales and the spectrum of the corresponding thresholded signals or fields.   

Effects of Particle Size on Acceleration Measurements in Intense Turbulence

We present 3D Lagrangian particle tracking measurements of large neutrally buoyant particles (0.4 $<$ d/$\eta <$30) in intense turbulence (400$<$ R$_{\lambda }<$ 815). Stereoscopic high speed cameras image polystyrene tracer particles in a flow between counter-rotating disks at a frame rate of 20kHz. For large particles with diameter in the inertial range, the acceleration variance decreases with diameter as d$^{-2/3}$. This result is predicted by a model that identifies the particle acceleration with the fluid acceleration at the scale of the particle diameter. We show that this model can be extended to describe the transition from particle sizes in the inertial to the dissipation range. In addition, the measured particle acceleration PDF displays no significant dependence on particle size, extending results from previous work to a larger range of particle sizes.   

Effects of Inhomogeneity and Large Scale Intermittency on Small Scale Turbulence

We report on the effects of inhomogeneity and temporal fluctuations of energy injection on small scale turbulence statistics. We study a 1mx1mx1.5m flow between oscillating grids which produces Taylor Reynolds number 280 while containing regions of nearly homogeneous and highly inhomogeneous turbulence. Large data sets of 3D tracer particle velocities have been collected using stereoscopic high speed cameras with real-time image compression technology. The second and third order Eulerian structure functions are measured in both homogeneous and inhomogeneous regions of the flow. We condition the structure functions on the instantaneous large scale velocity or on the grid phase. At all scales, the structure functions depend strongly on the large scale velocity, but are independent of the grid phase. We see clear signatures of inhomogeneity, but even in the homogeneous region the dependence on the large scale velocity remains at all scales. Previous work has shown that similar correlations extend even to very high Reynolds numbers. Comprehensive measurements of these effects in a laboratory flow allows the possibility of separating the contributions from shear, inhomogeneity, and large scale intermittency.   

High order statistics of turbulence Lagrangian acceleration

It is well known that the Lagrangian acceleration of a fluid element in turbulence is highly intermittent. Therefore it has been a challenge to accurately measure higher order statistics of acceleration. We report measurements of the scaling of acceleration moments with Reynolds number and compare our results with several available theories, including the standard K41, K62 with intermittency correction, and a recent theory advocated by Yakhot and co-workers. We achieved these measurements by optically tracking neutrally buoyant polystyrene microspheres at high spatial and temporal resolution in a nearly homogeneous and isotropic water flow driven by 12 independently controlled propellers in a newly constructed turbulence generator.   

Relative dispersion of inertial particles in fully developed turbulence: an experimental study

We report experimental results of the motion of tracer and non-tracer heavy particles obtained by 3-D Lagrangian particle tracking in a fully developed turbulent water flow between counter-rotating disks. The sizes of the non-tracer particles are of the order of the Kolmogorov length scale and their densities range from 2.5 to 7.8. In this study, we focus on the effect of inertia on the separation of two particles in turbulence. The inertia effect is characterized by the Stokes number, the ratio of a particle time scale ($\tau_{p}$) to a flow time scale ($\tau_{f}$). As the separation distance ($r(t)$) changes, the particle time scale remains the same, but the flow time scale increases with the length scale considered according to the classical K41 phenomenology as $\tau_{f} \propto r^{\frac{2}{3}}$. We relate the experimental observations to this scale-dependent Stokes number.   

Effect of polymer additives on high order Lagrangian structure functions of a turbulent flow

According to K41 predictions, Lagrangian high-order moments of the velocity differences should display a universal scaling in the inertial range. Lagrangian Particle Tracking (LPT) measurements and DNS in Newtonian turbulence revealed an anomalous scaling substantially different from that predicted by K41 (N. Mordant et al. 2001 PRL 87 214501, H. Xu et al. 2006 PRL 96 024503, L. Biferale et al. 2005 Phys. Fluids 17 021701). Here we report the high-order Lagrangian structure functions measured in turbulent dilute polymer solutions. We used the LPT technique to perform the measurements in an axisymmetric turbulent flow at moderate Reynolds numbers to avoid polymer degradation. The Weissenberg numbers were above unity and the polymers were stretched by the flow. Under these conditions the polymers were able to exchange energy with the flow and to modify the dynamics of turbulence. The scaling behavior of the dilute polymer solution structure functions is strikingly different from the one observed for Newtonian flows at the same Reynolds number.