Eulerian Structure Functions Research Articles

Saturation and Multifractality of Lagrangian and Eulerian Scaling Exponents in Three-Dimensional Turbulence.

Inertial-range scaling exponents for both Lagrangian and Eulerian structure functions are obtained from direct numerical simulations of isotropic turbulence in triply periodic domains at Taylor-scale Reynolds number up to 1300. We reaffirm that transverse Eulerian scaling exponents saturate at ≈2.1 for moment orders p≥10, significantly differing from the longitudinal exponents (which are predicted to saturate at ≈7.3 for p≥30 from a recent theory). The Lagrangian scaling exponents likewise saturate at ≈2 for p≥8. The saturation of Lagrangian exponents and transverse Eulerian exponents is related by the same multifractal spectrum by utilizing the well-known frozen hypothesis to relate spatial and temporal scales. Furthermore, this spectrum is different from the known spectra for Eulerian longitudinal exponents, suggesting that Lagrangian intermittency is characterized solely by transverse Eulerian intermittency. We discuss possible implications of this outlook when extending multifractal predictions to the dissipation range, especially for Lagrangian acceleration.

Biases in Structure Functions from Observations of Submesoscale Flows

AbstractSurface drifter observations from the LAgrangian Submesoscale ExpeRiment (LASER) campaign in the Gulf of Mexico are paired with Eulerian (ship‐borne X‐band radar) data to demonstrate that velocity structure functions from drifters differ systematically from Eulerian structure functions over scales from 0.4 to 7 km. These differences result from drifters oversampling surface convergences and regions of intense vorticity. The first‐, second‐, and third‐order structure functions are calculated using quasi‐Lagrangian (drifter) and Eulerian data from approximately the same location and time. Differences between quasi‐Lagrangian and Eulerian structure functions are attributed to two forms of bias. The first bias results from the mean divergence or vorticity of the background flow creating nonzero first‐order structure functions. This background bias affects both quasi‐Lagrangian and Eulerian data when insufficiently time‐averaged. It severely biases the drifter third‐order structure functions but is smaller in Eulerian structure functions at both second and third order. This bias can be corrected for using lower‐order structure functions. The second form of bias results from drifters accumulating in regions with flow statistics that differ from undersampled regions. This accumulation bias is diagnosed by identifying the dependence of the Eulerian structure functions on divergence and vorticity as well as scale. Together, both biases suggest that caution is needed when interpreting second‐order drifter statistics and that linking raw third‐order drifter statistics to energy fluxes is often erroneous in ocean data: Even with background correction and sufficient time‐averaging, drifters overestimate the Eulerian estimate of the third‐order structure function by up to a factor of 5 when signs are consistent.

High-resolution measurement of cloud microphysics and turbulence at a mountaintop station

Abstract. Mountain research stations are advantageous not only for long-term sampling of cloud properties but also for measurements that are prohibitively difficult to perform on airborne platforms due to the large true air speed or adverse factors such as weight and complexity of the equipment necessary. Some cloud–turbulence measurements, especially Lagrangian in nature, fall into this category. We report results from simultaneous, high-resolution and collocated measurements of cloud microphysical and turbulence properties during several warm cloud events at the Umweltforschungsstation Schneefernerhaus (UFS) on Zugspitze in the German Alps. The data gathered were found to be representative of observations made with similar instrumentation in free clouds. The observed turbulence shared all features known for high-Reynolds-number flows: it exhibited approximately Gaussian fluctuations for all three velocity components, a clearly defined inertial subrange following Kolmogorov scaling (power spectrum, and second- and third-order Eulerian structure functions), and highly intermittent velocity gradients, as well as approximately lognormal kinetic energy dissipation rates. The clouds were observed to have liquid water contents on the order of 1 g m−3 and size distributions typical of continental clouds, sometimes exhibiting long positive tails indicative of large drop production through turbulent mixing or coalescence growth. Dimensionless parameters relevant to cloud–turbulence interactions, the Stokes number and settling parameter are in the range typically observed in atmospheric clouds. Observed fluctuations in droplet number concentration and diameter suggest a preference for inhomogeneous mixing. Finally, enhanced variance in liquid water content fluctuations is observed at high frequencies, and the scale break occurs at a value consistent with the independently estimated phase relaxation time from microphysical measurements.

Where do small, weakly inertial particles go in a turbulent flow?

AbstractWe report experimental results on the dynamics of heavy particles of the size of the Kolmogorov scale in a fully developed turbulent flow. The mixed Eulerian structure function of two-particle velocity and acceleration difference vectors $\langle {\delta }_{r} \mathbi{v}\boldsymbol{\cdot} {\delta }_{r} {\mathbi{a}}_{\mathbi{p}} \rangle $ was observed to increase significantly with particle inertia for identical flow conditions. We show that this increase is related to a preferential alignment between these dynamical quantities. With increasing particle density the probability for those two vectors to be collinear was observed to grow. We show that these results are consistent with the preferential sampling of strain-dominated regions by inertial particles.

Statistical properties of supersonic turbulence in the Lagrangian and Eulerian frameworks

AbstractWe present a systematic study of the influence of different forcing types on the statistical properties of supersonic, isothermal turbulence in both the Lagrangian and Eulerian frameworks. We analyse a series of high-resolution, hydrodynamical grid simulations with Lagrangian tracer particles and examine the effects of solenoidal (divergence-free) and compressive (curl-free) forcing on structure functions, their scaling exponents, and the probability density functions of the gas density and velocity increments. Compressively driven simulations show significantly larger density contrast, more intermittent behaviour, and larger fractal dimension of the most dissipative structures at the same root mean square Mach number. We show that the absolute values of Lagrangian and Eulerian structure functions of all orders in the integral range are only a function of the root mean square Mach number, but independent of the forcing. With the assumption of a Gaussian distribution for the probability density function of the velocity increments for large scales, we derive a model that describes this behaviour.

Inertial range Eulerian and Lagrangian statistics from numerical simulations of isotropic turbulence

We present a study of Eulerian and Lagrangian statistics from a high-resolution numerical simulation of isotropic and homogeneous turbulence using the FLASH code, with an estimated Taylor microscale Reynolds number of around 600. Statistics are evaluated over a data set with 18563 spatial grid points and with 2563 = 16.8 million particles, followed for about one large-scale eddy turnover time. We present data for the Eulerian and Lagrangian structure functions up to the tenth order. We analyze the local scaling properties in the inertial range. The Eulerian velocity field results show good agreement with previous data and confirm the puzzling differences previously found between the scaling of the transverse and the longitudinal structure functions. On the other hand, accurate measurements of sixth-and-higher-order Lagrangian structure functions allow us to highlight some discrepancies from earlier experimental and numerical results. We interpret this result in terms of a possible contamination from the viscous scale, which may have affected estimates of the scaling properties in previous studies. We show that a simple bridge relation based on a multifractal theory is able to connect scaling properties of both Eulerian and Lagrangian observables, provided that the small differences between intermittency of transverse and longitudinal Eulerian structure functions are properly considered.

Bulk turbulence in dilute polymer solutions

By tracking small particles in the bulk of an intensely turbulent laboratory flow, we study the effect of long-chain polymers on the Eulerian structure functions. We find that the structure functions are modified over a wide range of length scales even for very small polymer concentrations. Their behaviour can be captured by defining a length scale that depends on the solvent viscosity, the polymer relaxation time and the Weissenberg number. This result is not captured by current models. Additionally, the effects we observe depend strongly on the concentration. While the dissipation-range statistics change smoothly as a function of polymer concentration, we find that the inertial-range values of the structure functions are modified only when the concentration exceeds a threshold of approximately 5 parts per million (p.p.m.) by weight for the 18 × 106 atomic mass unit (a.m.u.) molecular weight polyacrylamide used in the experiment.

A statistical theory of turbulent relative dispersion

The laws governing the spread of a cluster of particles in homogeneous isotropic turbulence are derived using a theoretical approach based on inertial subrange scaling and statistical diffusion theory. The equations for the mean square dispersion of a puff admit an analytical solution in the inertial subrange and at large scales. The solution is consistent with Taylor's theory of absolute dispersion. An analytical derivation of the Richardson–Obukhov constant of relative dispersion is presented. A time scale for relative dispersion is identified, as well as relations between Lagrangian and Eulerian structure functions. The results are extended to turbulence at finite Reynolds number. A closure assumption for the relative kinetic energy, based on Taylor's theory, is presented. Comparisons with direct numerical simulations and laboratory experiments are reported.

Eulerian and Lagrangian Structure Function’s Scaling Exponents in Turbulent Channel Flow

The relation between Eulerian structure function’s scaling exponents and Lagrangian ones in turbulent channel flows is explored both theoretically and numerically. A nonlinear parametric transformation between Eulerian structure function’s scaling exponents and Lagrangian ones is derived, following Landau and Novikov’s frame work. This relation is then compared to some known experimental and numerical results, but mainly to our DNS (direct numerical simulation) results of a fully developed channel flow with Reτ = 100. The scaling exponents are evaluated in terms of the ESS (extended self-similarity) method, since the Reynolds number is too low to make the standard scaling laws applicable. The agreement between theory and simulation is satisfactory

Lagrangian statistics and temporal intermittency in a shell model of turbulence.

We study the statistics of single-particle Lagrangian velocity in a shell model of turbulence. We show that the small-scale velocity fluctuations are intermittent, with scaling exponents connected to the Eulerian structure function scaling exponents. The observed reduced scaling range is interpreted as a manifestation of the intermediate dissipative range, as it disappears in a Gaussian model of turbulence.

Relative dispersion in fully developed turbulence: the Richardson's law and intermittency corrections.

Relative dispersion in fully developed turbulence is investigated by means of direct numerical simulations. Lagrangian statistics is found to be compatible with Richardson description although small systematic deviations are found. The value of the Richardson constant is estimated as C2 approximately equal to 0.55, in a close agreement with recent experimental findings [S. Ott and J. Mann, J. Fluid Mech. 422, 207 (2000)]. By means of exit-time statistics it is shown that the deviations from Richardson's law are a consequence of Eulerian intermittency. The measured Lagrangian scaling exponents require a set of Eulerian structure function exponents zeta(p) which are remarkably close to standard ones known for fully developed turbulence.

Estimation Of The Lagrangian Structure Function Constant C0 From Surface-Layer Wind Data

Eulerian turbulence observations, madein the surface layer under unstable conditions (z/L > 0),by a sonic anemometer were used to estimatethe Lagrangian structure function constant O. Twomethods were considered. The first one makes use of arelationship, widely used in the Lagrangian stochasticdispersion models, relating O to the turbulent kineticenergy dissipation rate e, wind velocity variance andLagrangian decorrelation time. The second one employsa novel equation, connecting O to the constant of thesecond-order Eulerian structure function. Beforeestimating O, the measurements were processed in orderto discard non-stationary cases at least to a firstapproximation and cases in which local isotropy couldnot be assumed. The dissipation e was estimated eitherfrom the best fit of the energy spectrum in theinertial subrange or from the best fit of the third-orderlongitudinal Eulerian structure function. Thefirst method was preferred and applied to the subsequentpart of the analysis. Both methods predict thepartitioning of O in different spatial components as aconsequence of the directional dependence of theEulerian correlation functions due to the isotropy.The O values computed by both methods are presented anddiscussed. In conclusion, both methods providerealistic estimates of O that compare well withprevious estimations reported in the literature, evenif a preference is to be attributed to the second method.