Probability density functions, skewness, and flatness in large Reynolds number turbulence

An Experimental Study of Turbulent Mixing in Channel Flow Past a Grid

Grid turbulence is considered to be a canonical case of turbulent flow. In the presented paper, the flow structure is analyzed from the point of view of mixing properties, where vortical structures and their properties play a significant role. That is why the effect of various length-scales in turbulence is studied separately. The experimental study uses the Particle Image Velocimetry (PIV) method. The original method for spatial spectrum evaluation is applied. Results on vortex spatial spectrum and isotropy are presented. The scaling of turbulent kinetic energy (TKE) is measured; furthermore, the TKE is decomposed according to the length-scales of the fluctuations. By this method, we found that the decay of TKE associated with the smallest length-scales is more sensitive to the Reynolds number than that at larger length-scales. The TKE at the largest investigated length-scales decays more slowly. The turbulence decay-law is studied for various Reynolds numbers. The second and fourth statistical moments of vorticity are evaluated at various Reynolds numbers and distances from the grid. The isotropy is investigated in the sense of ratio of fluctuations in stream-wise to span-wise directions as the used data are captured using the planar PIV method. The full 3D fluctuation invariants were investigated in a representative position by means of the Stereo-PIV method.

Large deviations, singularity, and lognormality of energy dissipation in turbulence

We study implications of the assumption of power-law dependence of moments of energy dissipation in turbulence on the Reynolds number Re, holding due to intermittency. We demonstrate that at Re→∞ the dissipation's logarithm divided by lnRe converges with probability one to a negative constant. This implies that the dissipation is singular in the limit, as is known phenomenologically. The proof uses a large deviations function, whose existence is implied by the power-law assumption, and which provides the general asymptotic form of the dissipation's distribution. A similar function exists for vorticity and for velocity differences where it proves the moments representation of the multifractal model (MF). Then we observe that derivative of the scaling exponents of the dissipation, considered as a function of the order of the moment, is small at the origin. Thus the variation with the order is slow and can be described by a quadratic function. Indeed, the quadratic function, which corresponds to log-normal statistics, fits the data. Moreover, combining the lognormal scaling with the MF we derive a formula for the anomalous scaling exponents of turbulence which also fits the data. Thus lognormality, not to be confused with the Kolmogorov (1962) assumption of lognormal dissipation in the inertial range, when used in conjunction with the MF provides a concise way to get all scaling exponents of turbulence available at present.

Statistical Lyapunov Theory Based on Bifurcation Analysis of Energy Cascade in Isotropic Homogeneous Turbulence: A Physical-Mathematical Review

This work presents a review of previous articles dealing with an original turbulence theory proposed by the author and provides new theoretical insights into some related issues. The new theoretical procedures and methodological approaches confirm and corroborate the previous results. These articles study the regime of homogeneous isotropic turbulence for incompressible fluids and propose theoretical approaches based on a specific Lyapunov theory for determining the closures of the von Kármán–Howarth and Corrsin equations and the statistics of velocity and temperature difference. While numerous works are present in the literature which concern the closures of the autocorrelation equations in the Fourier domain (i.e., Lin equation closure), few articles deal with the closures of the autocorrelation equations in the physical space. These latter, being based on the eddy–viscosity concept, describe diffusive closure models. On the other hand, the proposed Lyapunov theory leads to nondiffusive closures based on the property that, in turbulence, contiguous fluid particles trajectories continuously diverge. Therefore, the main motivation of this review is to present a theoretical formulation which does not adopt the eddy–viscosity paradigm and summarizes the results of the previous works. Next, this analysis assumes that the current fluid placements, together with velocity and temperature fields, are fluid state variables. This leads to the closures of the autocorrelation equations and helps to interpret the mechanism of energy cascade as due to the continuous divergence of the contiguous trajectories. Furthermore, novel theoretical issues are here presented among which we can mention the following ones. The bifurcation rate of the velocity gradient, calculated along fluid particles trajectories, is shown to be much larger than the corresponding maximal Lyapunov exponent. On that basis, an interpretation of the energy cascade phenomenon is given and the statistics of finite time Lyapunov exponent of the velocity gradient is shown to be represented by normal distribution functions. Next, the self–similarity produced by the proposed closures is analyzed and a proper bifurcation analysis of the closed von Kármán–Howarth equation is performed. This latter investigates the route from developed turbulence toward the non–chaotic regimes, leading to an estimate of the critical Taylor scale Reynolds number. A proper statistical decomposition based on extended distribution functions and on the Navier–Stokes equations is presented, which leads to the statistics of velocity and temperature difference.

Liquid nitrogen in fluid dynamics: visualization and velocimetry using frozen particles

High-Reynolds-number flows are common both in nature and industrial applications, but are difficult to attain in laboratory settings using standard test fluids such as air and water. To extend the Reynolds number range, water and air have been replaced at times by low-viscosity fluids such as pressurized air, sulfur hexafluoride, and cryogenic nitrogen gas, as well as liquid and gaseous helium. With a few exceptions, liquid nitrogen has been neglected despite the fact that it has a kinematic viscosity of about a fifth of that of water at room temperature. We explore the use of liquid nitrogen here. In particular, we study the use of frozen particles for flow visualization and velocimetry in liquid nitrogen. We create particles in situ by injecting a gaseous mixture of room-temperature nitrogen and an additional seeding gas into the flow. We present a systematic study of potential seeding gases to determine which create particles with the best fidelity and optical properties. The technique has proven capable of producing sub-micrometer sized tracers that allow particle tracking and particle image velocimetry. We review possible high-Reynolds-number experiments using this technique, and discuss the merits and challenges of using liquid nitrogen as a test fluid.

Experimental evidence of a phase transition in a closed turbulent flow

We experimentally study the susceptibility to symmetry breaking of a closed turbulent von Kármán swirling flow from Re=150 to Re≃10⁶. We report a divergence of this susceptibility at an intermediate Reynolds number Re=Re(χ)≃90,000 which gives experimental evidence that such a highly space and time fluctuating system can undergo a "phase transition." This transition is furthermore associated with a peak in the amplitude of fluctuations of the instantaneous flow symmetry corresponding to intermittencies between spontaneously symmetry breaking metastable states.

Velocity difference statistics in turbulence

We unify two approaches that have been taken to explain the non-Gaussian probability distribution functions (PDFs) obtained in measurements of longitudinal velocity differences in turbulence, and we apply our approach to Couette-Taylor turbulence data. The first approach we consider was developed by Castaing and co-workers, who obtained the non-Gaussian velocity difference PDF from a superposition of Gaussian distributions for subsystems that have a particular energy dissipation rate at a fixed length scale [Castaing, Physica D 46, 177 (1990)]. Another approach was proposed by Beck and Cohen, who showed that the observed PDFs can be obtained from a superposition of Gaussian velocity difference PDFs in subsystems conditioned on the value of an intensive variable (inverse "effective temperature") in each subsystem [Beck and Cohen, Physica A 322, 267 (2003)]. The intensive variable was defined for subsystems assuming local thermodynamic equilibrium, but no method was proposed for determining the size of a subsystem. We show that the Castaing and Beck-Cohen methods are related, and we present a way to determine subsystem size in the Beck-Cohen method. The application of our approach to Couette-Taylor turbulence (Reynolds number 540,000) yields a log-normal distribution of the intensive parameter, and the resultant velocity difference PDF agrees well the observed non-Gaussian velocity difference PDFs.

Statistics of intense turbulent vorticity events

We investigate statistical properties of vorticity fluctuations in fully developed turbulence, which are known to exhibit a strong intermittent behavior. Taking as the starting point the Navier-Stokes equations with a random force term correlated at large scales, we obtain in the high Reynolds number regime a closed analytical expression for the probability distribution function of an arbitrary component of the vorticity field. The central idea underlying the analysis consists in the restriction of the velocity configurational phase-space to a particular sector where the rate of strain and the rotation tensors can be locally regarded as slow and fast degrees of freedom, respectively. This prescription is implemented along the Martin-Siggia-Rose functional framework, whereby instantons and perturbations around them are taken into account within a steepest-descent approach.

Transition at dissipative scales in large-Reynolds-number turbulence

Among the available diagnostics of turbulence, the flatness of the velocity derivatives is particularly interesting because it represents a straightforward test of Kolmogorov theory, and provides a quantitative estimate for intermittency effects. It is commonly considered that the flatness factor increases with the Reynolds number, following a power law at high Reynolds numbers. At variance with this picture, evidence for a transitional behavior, taking place around the Taylor microscale Reynolds number R(lambda)=700, has been recently obtained in several experiments. In the present paper we study this transition in detail, and show it has the characteristics of a second order phase transition. We propose a physical picture for this transition, based on worm vortex breakdown, which leads as to suggest that intense sub-Kolmogorov structures might develop above the transition point. These results indicate that the existence of an asymptotic state at infinite Reynolds number may become questionable and more generally, that our current views on dissipative range intermittency probably need to be revised

Scaling of structure functions in homogeneous shear-flow turbulence

We apply spectral dynamics and non-Gaussian statistical model of velocity difference to study the scaling of structure functions in homogeneous shear-flow turbulence. Let L(S) be the shear length scale and eta the viscous scale. It is found that, when L(S)/eta is finite, due to a combined effect of viscosity and mean shear, the scaling deviates from normal scaling, and the deviation increases as L(S)/eta decreases. In the presence of a strong shear (L(S)/eta<100), the deviation is substantially larger than the prediction of typical intermittency models, in agreement with recent experiments. As L(S)/eta-->infinity, the normal scaling is valid in the inertial range where viscous and shear effects are negligible.

Turbulence in nature and in the laboratory

Fluid turbulence has attracted the attention of physicists, mathematicians, and engineers for over 100 years, yet it remains an unsolved problem. The reasons for the difficulties are outlined and recent advances in describing its small-scale statistical structure are described. Contrary to traditional notions, new experimental evidence indicates that the small scales are anisotropic, reflecting the overall character of the flow. The consequences of this finding with regard to the long-held postulate of the universality of the small-scale turbulence structure are discussed.

Dissipation field asymmetry and intermittency in fully developed turbulence

Experimental study of high Reynolds number turbulence provides additional evidence that asymmetry of turbulence is related to the intermittency. The refined similarity hypothesis (RSH), on the other hand, connects the intermittency of the longitudinal velocity increments with that of the dissipation field, implying in particular that the dissipation field should be asymmetric as well. The asymmetry of the latter is indeed found in these experiments. In addition, the study of the dissipation field asymmetry provides us with quantitative estimations of the deviations from the RSH.

Probability density functions of the enstrophy flux in two dimensional grid turbulence

Probability density functions of the enstrophy flux in two dimensional grid turbulence are found to be strongly non-Gaussian and can be mimicked by stretched exponential functions. Evidence of this behavior is found in experiments using turbulent soap films and numerical simulations. The enstrophy flux itself is found to be constant for a range of scales corresponding to the enstrophy cascade.

Closure approach to high-order structure functions of turbulence

Let T(p) be the structure function of order p of turbulence. First T2 is determined, then T3 is derived from T2, and finally T(p) (p>3) are derived from T2 and T3. This closure scheme is realized by a non-Gaussian statistical model. We use it to study the scaling law of T(p), and we find that the available data on scaling exponents favor Kolmogorov's 1941 theory rather than his 1962 theory. We also predict the high-order universal constants of inertial-range scaling.

Velocity structure functions, scaling, and transitions in high-Reynolds-number Couette-Taylor flow

Flow between concentric cylinders with a rotating inner cylinder is studied for Reynolds numbers in the range 2x10(3)<R<10(6) (Taylor Reynolds numbers, 10 < R(lambda)< 290) for a system with radius ratio eta=0.724. Even at the highest Reynolds number studied, the energy spectra do not show power law scaling (i.e., there is no inertial range), and the dissipation length scale is surprisingly large. Nevertheless, the velocity structure functions calculated using extended self-similarity exhibit clear power-law scaling. The structure function exponents zeta(p) fit Kolmogorov's log-normal model within the experimental uncertainty, zeta(p)=(p/3) [1+(mu/6)(3-p)] (for p < or =10) with mu=0.27. These zeta(p) values are close to those found in other flows. Measurements of torque scaling are presented that are an order of magnitude more accurate than those previously reported [Lathrop et al., Phys Rev. A 46, 6390 (1992)]. Measurements of velocity in the fluid core reveal the presence of azimuthal traveling waves up to the highest Reynolds numbers examined. These waves show evidence of a transition at R(T)=1.3 x 10(4); this transition was observed previously in measurements of torque, but our wave velocity and wall shear stress measurements provide the first evidence from local quantities of the transition at R(T). Velocity measurements indicate that at R(T) there is a change in the coherent structures of the core flow; this is consistent with our analyses of the scaling of the torque. Our measurements were made at two aspect ratios, and no significant dependence on aspect ratio was observable for R > R(T).