Second-order Structure Function Research Articles

Structure Functions and Structure Parameters of Velocity Fluctuations in Numerically Simulated Atmospheric Convective Boundary Layer Flows

AbstractWe extend our previous study, which dealt with structure functions of potential temperature fluctuations, and focus on the characteristics of second-order velocity structure functions and corresponding structure parameters in the atmospheric convective boundary layer. We consider the three previously reported methods to compute the structure parameters of turbulent velocity fields: the direct method, the true spectral method, and the approximate spectral method. The methods are evaluated using high-resolution gridded numerical data from large-eddy simulations of shear-free and shear-driven convective boundary layers. Results indicate that the direct and true spectral methods are more suitable than the approximate spectral method, which overestimates the structure parameters of velocity as a result of assuming the inertial-subrange shape of the velocity spectrum for all turbulence scales. Results also suggest that structure parameters of vertical velocity fluctuations are of limited utility because of violations of local isotropy, especially in shear-free convective boundary layers.

Observational Quantification of Three-dimensional Anisotropies and Scalings of Space Plasma Turbulence at Kinetic Scales

Abstract A statistical survey of spectral anisotropy of space plasma turbulence is performed using five years of measurements from the Magnetospheric Multiscale mission in the magnetosheath. By measuring the five-point second-order structure functions of the magnetic field, we have for the first time quantified the three-dimensional anisotropies and scalings at sub-ion scales (<100 km). In the local reference frame defined with respect to local mean magnetic field , the “statistical eddies” are found to be mostly elongated along and shortened in the direction perpendicular to both and local field fluctuations. From several d i (ion inertial length) toward ∼0.05 d i , the ratio between eddies’ parallel and perpendicular lengths features a trend of rise then fall, whereas the anisotropy in the perpendicular plane appears scale-invariant. Specifically, the anisotropy relations for the total magnetic field at 0.1–1.0 d i are obtained as , and , respectively. Our results provide new observational evidence to compare with phenomenological models and numerical simulations, which may help to better understand the nature of kinetic-scale turbulence.

Structural Properties of the Static Armor during Formation and Reestablishment in Gravel-Bed Rivers

The formation and reestablishment of bed structural properties in the static armor layer is an important research subject. To address this issue, we conducted a series of static armor layer experiments in a laboratory flume that focused on formation and reestablishment. Through an automatic measurement system, we obtained a real-time bed load transport rate. The bed surface elevation at different flow intensities was obtained using a PTS (Photo Terrain Scanning) system. The results show that the formation and reestablishment of the bed load transport rate in the static armor layer increased from zero to its peak before decaying. The bed structure response was found to be highly dependent on the unevenness of the bed surface elevation. The bed surface elevation of a static armor layer in a laboratory flume is considered as a two-dimensional random field. In a two-dimensional random field, the changes in bed elevation are characterized using statistical parameters. Statistical parameters are evaluated from precise digital elevation models (DEMs) of bed surfaces. Experimental results provide the change of probability distribution functions (PDFs) and second-order structure functions of bed elevations between formation and reestablishment after breaking the static armor layer. By quantitatively analyzing the changes in these statistical parameters, we quantified the difference between the bed structure in the static armor layer formation and the new static armor layer formed after being broken. Thus, this finding reveals that the bed structure of the static armor layer formed by different flow intensities is different, and this difference can be quantified using statistical methods.

Structure function tensor equations in inhomogeneous turbulence

Exact budget equations for the second-order structure function tensor ⟨𝛿u$_{i}$𝛿u$_{j}$⟩, where 𝛿u is the difference of the i th fluctuating velocity component between two points, are used to study the two-point statistics of velocity fluctuations in inhomogeneous turbulence. The anisotropic generalised Kolmogorov equations (AGKE) describe the production, transport, redistribution and dissipation of every Reynolds stress component occurring simultaneously among different scales and in space, i.e. along directions of statistical inhomogeneity. The AGKE are effective to study the inter-component and multi-scale processes of turbulence. In contrast to more classic approaches, such as those based on the spectral decomposition of the velocity field, the AGKE provide a natural definition of scales in the inhomogeneous directions, and describe fluxes across such scales too. Compared to the generalised Kolmogorov equation, which is recovered as their half-trace, the AGKE can describe inter-component energy transfers occurring via the pressure–strain term and contain also budget equations for the off-diagonal components of ⟨𝛿u$_{i}$𝛿u$_{j}$⟩. The non-trivial physical interpretation of the AGKE terms is demonstrated with three examples. First, the near-wall cycle of a turbulent channel flow at a friction Reynolds number of Re$_{𝜏}$ = 200 is considered. The off-diagonal component ⟨-𝛿u𝛿υ⟩, which cannot be interpreted in terms of scale energy, is discussed in detail. Wall-normal scales in the outer turbulence cycle are then discussed by applying the AGKE to channel flows at Re$_{𝜏}$ = 500 and 1000. In a third example, the AGKE are computed for a separating and reattaching flow. The process of spanwise-vortex formation in the reverse boundary layer within the separation bubble is discussed for the first time.

The role of vegetation and large wood on the topographic characteristics of braided river systems

Riparian vegetation and large wood play a crucial role in shaping rivers. On the one hand, vegetation tends to increase bank stability, reduce channel width and reduce the braiding index in gravel-bed rivers. Conversely, large wood tends to increase channel dynamics and promote avulsions. The effects of vegetation and large wood have been rarely simulated together in flume experiments. In this paper we present a series of experimental runs conducted in a large flume facility, using cylindrical wooden dowels and alfalfa seedlings to represent logs and vegetation, respectively, in order to investigate their role on determining the topographical nature of braided river systems. A terrestrial laser scanner was used to measure in detail the topography of the channels and bars, and distribution of the elevations and second-order structure functions were used to explore the topography that large wood and vegetation imposed on the braided pattern in the flumes. Results show that vegetation reduced the braiding index and produced wider and deeper channels. The standard deviation of the bed elevations was higher than in flumes without vegetation. Large wood alone was not effective in changing the braiding index or the standard deviation of the elevations. When vegetation and large wood acted together in the flume, the braiding index reached a minimum value, and the skewness of the distribution of elevation increased, revealing shallower pools if compared with runs conducted with only vegetation.

Dynamics and invariants of the perceived velocity gradient tensor in homogeneous and isotropic turbulence

The perceived velocity gradient tensor (PVGT), constructed from four fluid tracers forming a tetrahedron, provides a natural way to study the structure of velocity fluctuations and its dependence on spatial scales. It generalizes and shares qualitatively many properties with the true velocity gradient tensor. Here, we establish the evolution equation for the PVGT, and, for homogeneous and isotropic incompressible turbulent flows, we analyse the dynamics of the PVGT in particular using its second- and third-order invariants. We show that, for PVGT based on regular tetrads with lateral size , the second-order invariants can be expressed solely in terms of the usual second-order velocity structure functions, while the third-order invariants involve the usual third-order longitudinal velocity structure function and a less well known three-point velocity correlation function. For homogeneous and isotropic turbulence, exact relations between the second moments of strain and vorticity, as well as enstrophy production and the third moments of the strain, are derived. These generalized relations are valid for all ranges of , and reduce to classical results for the velocity gradient tensor when is in the dissipative range. With the help of these relations, we quantify the importance of the various terms, such as vortex stretching, as a function of the scale . Our analysis, which is supported by the results of direct numerical simulations of turbulent flows in the Reynolds-number range , allows us to demonstrate that strain prevails over vorticity when is in the inertial range.

Scaling of Second-Order Structure Functions in Turbulent Premixed Flames in the Flamelet Combustion Regime

The second-order velocity structure function statistics have been analysed using a DNS database of statistically planar turbulent premixed flames subjected to unburned gas forcing. The flames considered here represent combustion for moderate values of Karlovitz number from the wrinkled flamelets to the thin reaction zones regimes of turbulent premixed combustion. It has been found that the second-order structure functions exhibit the theoretical asymptotic scalings in the dissipative and (relatively short) inertial ranges. However, the constant of proportionality for the theoretical asymptotic variation for the inertial range changes from one case to another, and this value also changes with structure function orientation. The variation of the structure functions for small length scale separation remains proportional to the square of the separation distance. However, the constant of proportionality for the limiting behaviour according to the separation distance square remains significantly different from the theoretical value obtained in isotropic turbulence. The disagreement increases with increasing turbulence intensity. It has been found that turbulent velocity fluctuations within the flame brush remain anisotropic for all cases considered here and this tendency strengthens towards the trailing edge of the flame brush. It indicates that the turbulence models derived based on the assumptions of homogeneous isotropic turbulence may not be fully valid for turbulent premixed flames.

Assessment of large-scale forcing in isotropic turbulence using a closed Kármán–Howarth equation

We attempt to model the effects of large-scale forcing on the statistical behavior of small scales in isotropic turbulence. More specifically, the effect of large-scale forcing on the second-order velocity structure function, S2, in the region beyond the dissipative range, is analyzed via the transport equation for S2 where a closure model for S3, the third-order velocity structure function, is introduced. The model [L. Djenidi and R. A. Antonia, Fluid Turbulence Applications in Both Industrial and Environmental Topics, Marseille, 9–11 July, 2019, https://fab60.sciencesconf.org/] is based on a gradient type with an eddy-viscosity formulation and has the following expression: S3=−CS3(S2)2ϵ¯∂S2∂r, where ϵ¯ is the mean rate of the turbulent kinetic energy dissipation, r is the spatial increment, and CS3 is a constant. The closed S2-transport equation is further exploited to derive a model for S2 for scales beyond the dissipative range. The model for S2 takes the form S2=CK(ϵrr)23 with ϵr=ϵ¯1−Br/Reλ1/2, where CK is a constant, Reλ is the Taylor microscale Reynolds number, and the function Br accounts for the effect of the large scales. The numerical solutions of the S2 equation and the predictions based on the model for S2 agree very well with direct numerical simulation data for steady-state forced homogeneous and isotropic turbulence. The solutions of the S2-transport equation without large-scale forcing show that S2 behaves like (ϵ¯r)2/3. When forcing is applied, S2 deviates from this behavior. However, increasing the Reynolds number tends to restore this behavior over an increasing range of scales. This is also observed in the predictions of the model for S2.

Estimation of mean turbulent kinetic energy and temperature variance dissipation rates using a spectral chart method

A method aimed at estimating εk and εθ, respectively, the mean dissipation rates of turbulent kinetic energy k and half the temperature variance θ2/2, is developed for slightly heated turbulent flows of air. It is limited to a Prandtl number near unity and applicable to flows where temperature can be treated as a passive scalar. A significant advantage of the method is that εk and εθ can both be estimated from the measurement of a temperature frequency spectrum, Gθθ(f). The method relies on the collapse in the dissipative range of one-dimensional temperature spectra, ϕθ(k1η), when normalized with εθ, εk, and ν. This collapse ensues from a similarity analysis of scale-by-scale budgets of the second-order structure function for the temperature. A generic spectrum ϕθG(k1η), defined in the wavenumber range 0.07 ≤ k1η ≤ 0.7, is used to construct a spectral chart. The method has been tested in several flows and found to be reliable. In particular, it is tested on the axis of a slightly heated round jet, where εk and εθ can be estimated accurately via the budgets of k and θ2/2, and the agreement between these estimates and the spectral chart results is almost perfect.

Isotropic Scaling Features Measured Locally in the Solar Wind Turbulence with Stationary Background Field

Abstract The scaling anisotropy is crucial to interpret the nonlinear interactions in solar wind turbulence. Previous observations provide diverse results and the structure function analyses are also reported to be an approach to investigate the scaling anisotropy based on a local magnetic field. However, the determination of the sampling angle with respect to the local background magnetic field requires that the observed time series for the average are time stationary. Whether or not this required time stationarity is compatible with the measurements has not been investigated. Here we utilize the second-order structure function method to study the scaling anisotropy with a time-stationary background field. We analyze 88 fast solar wind intervals each with time durations ≥ 2 days measured by WIND spacecraft in the period 2005–2018. We calculate the local magnetic field as the average of the time series whose time stationarity is fulfilled by our criterion ϕ < 10° (ϕ is the angle between the two averaged magnetic fields after cutting into two halves). We find for the first time the isotropic scaling feature of the magnetic-trace structure functions with scaling indices and , respectively, with the local magnetic field parallel and perpendicular to the solar wind velocity directions. The scaling for the velocity-trace structure functions is also isotropic and the indices are and . We also find that with increasing ϕ threshold to 90°, the scaling index of the magnetic-trace structure function in the parallel direction decreases to −0.81, while the rms of the instantaneous angle between magnetic field and solar wind velocity increases up to 45° at the timescale 150 s, indicating a mix of perpendicular measurements into parallel ones at large scales.

Dissipation of Turbulent Kinetic Energy in the Oscillating Bottom Boundary Layer of a Large Shallow Lake

AbstractMixing rates and biogeochemical fluxes are commonly estimated from the rate of dissipation of turbulent kinetic energy ε as measured with a single instrument and processing method. However, differences in measurements of ε between instruments/methods often vary by one order of magnitude. In an effort to identify error in computing ε, we have applied four common methods to data from the bottom boundary layer of Lake Erie. We applied the second-order structure function method (SFM) to velocity measurements from an acoustic Doppler current profiler, using both canonical and anisotropy-adjusted Kolmogorov constants, and compared the results with those computed from the law of the wall, Batchelor fitting to temperature gradient microstructure, and inertial subrange fitting to acoustic Doppler velocimeter data. The ε from anisotropy-adjusted constants in SFM increased by a factor of 6 or more at 0.2 m above the bed and showed a better agreement with microstructure and inertial method estimations. The maximum difference between SFM ε, computed using adjusted and canonical constants, and microstructure values was 25% and 50%, respectively. This difference was 30% and 55%, respectively, for those from inertial subrange fitting at times of high-intensity turbulence (Reynolds number at 1 m above the bed of more than 2 × 104). Comparison of the SFM ε to those from law of the wall was often poor, with errors as large as one order of magnitude. From the considerable improvement in ε estimates near the bed, anisotropy-adjusted Kolmogorov constants should be applied to compute dissipation in geophysical boundary layers.

Comparison of Different Techniques to Calculate Properties of Atmospheric Turbulence from Low-Resolution Data

In this work we study different techniques to estimate basic properties of turbulence, that is its characteristic velocity and length scale from low-resolution data. The methods are based on statistics of the signals like the velocity spectra, second-order structure function, number of signal’s zero-crossings and the variance of velocity derivative. First, in depth analysis of estimates from artificial velocity time series is performed. Errors due to finite averaging window, finite cut-off frequencies and different fitting ranges are discussed. Next, real atmospheric measurement data are studied. It is demonstrated that differences between results of the methods can indicate deviations from the Kolmogorov’s theory or the presence of external intermittency, that is the existence of alternating laminar/turbulent flow patches.

A new interpolation method for Lagrangian statistics in non-isothermal gas-solid flow

In this article, a new interpolation method is developed for the Lagrangian statistics in the non-isothermal gas-solid flow. The Lagrangian statistics are inseparable from interpolation, especially the optimal interpolation. The new interpolation method in this article is developed based on the smooth Dirac delta function, which is also called delta function interpolation method (DFIM). The performance of DFIM is compared with that of the spectral interpolation method (the interpolation method with the highest precision). The comparison results indicate that the performance of DFIM, especially the DFIM, is superior. The Lagrangian statistics method is one of the most effective methods for heat transfer analysis in the gas-solid flow. For the heat transfer analysis of gas-solid flow in this article, the relevant Lagrangian statistics are evaluated based on the newly developed DFIM, which include the temperature autocorrelation, the velocity autocorrelation, the second-order Lagrangian structure function, and the temperature velocity correlation function. The effect of the particle inertia on the heat transfer of the gas-solid flow is considered based on these Lagrangian statistics. The results show that the particle inertia has a great effect on the heat transfer in the gas-solid flow. As the particle inertia increases, the heat transfer in the gas-solid flow increases obviously. These results are helpful to understand the thermal behavior of the gas-solid flow.HighlightsNew interpolation method is developed for the Lagrangian statistics in non-isothermal gas-solid flow.Based on the relevant Lagrangian statistics, the heat transfer in the gas-solid flow is analyzed.The DFIM is much less computational cost than the spectral interpolation method.

The Vertical Structure of Open-Ocean Submesoscale Variability during a Full Seasonal Cycle

AbstractSubmesoscale dynamics are typically intensified at boundaries and assumed to weaken below the mixed layer in the open ocean. Here, we assess both the seasonality and the vertical distribution of submesoscale motions in an open-ocean region of the northeast Atlantic. Second-order structure functions, or variance in properties separated by distance, are calculated from submesoscale-resolving ocean glider and mooring observations, as well as a 1/48° numerical ocean model. This dataset combines a temporal coverage that extends through a full seasonal cycle, a horizontal resolution that captures spatial scales as small as 1 km, and vertical sampling that provides near-continuous coverage over the upper 1000 m. While kinetic and potential energies undergo a seasonal cycle, being largest during the winter, structure function slopes, influenced by dynamical characteristics, do not exhibit a strong seasonality. Furthermore, structure function slopes show weak vertical variations; there is not a strong change in properties across the base of the mixed layer. Additionally, we compare the observations to output from a high-resolution numerical model. The model does not represent variability associated with superinertial motions and does not capture an observed reduction in submesoscale kinetic energy that occurs throughout the water column in spring. Overall, these results suggest that the transfer of mixed layer submesoscale variability down to depths below the traditionally defined mixed layer is important throughout the weakly stratified subpolar mode waters.

Small-Scale Dispersion in the Presence of Langmuir Circulation

AbstractWe present an analysis of ocean surface dispersion characteristics, on 1–100-m scales, obtained by optically tracking a release of bamboo plates for 2 h in the northern Gulf of Mexico. Under sustained 5–6 m s−1 winds, energetic Langmuir cells are clearly delineated in the spatially dense plate observations. Within 10 min of release, the plates collect in windrows with 15-m spacing aligned with the wind. Windrow spacing grows, through windrow merger, to 40 m after 20 min and then expands at a slower rate to 50 m. The presence of Langmuir cells produces strong horizontal anisotropy and scale dependence in all surface dispersion statistics computed from the plate observations. Relative dispersion in the crosswind direction initially dominates but eventually saturates, while downwind dispersion exhibits continual growth consistent with contributions from both turbulent fluctuations and organized mean shear. Longitudinal velocity differences in the crosswind direction indicate mean convergence at scales below the Langmuir cell diameter and mean divergence at larger scales. Although the second-order structure function measured by contemporaneous GPS-tracked surface drifters drogued at ~0.5 m shows persistent r2/3 power law scaling down to 100–200-m separation scales, the second-order structure function for the very near surface plates observations has considerably higher energy and significantly shallower slope at scales below 100 m. This is consistent with contemporaneous data from undrogued surface drifters and previously published model results indicating shallowing spectra in the presence of direct wind-wave forcing mechanisms.

Statistical behaviors of conditioned two-point second-order structure functions in turbulent premixed flames in different combustion regimes

The second-order structure functions and their components conditioned upon various events have been analyzed for unweighted and density-weighted velocities using a Direct Numerical Simulation database. The heat release due to combustion has been shown to have significant influences on the structure functions and their components conditioned on different mixture states. The use of density-weighted velocities changes the relative magnitudes of differently conditioned structure functions but does not reduce the scatter of these magnitudes. The structure functions conditioned to constant-density unburned reactants at both points and normalized using the root-mean-square velocity conditioned to the reactants are larger at higher values of mean reaction progress variables c¯ (deeper within the flame brush), with this trend being not weakened with increasing turbulence intensity u′/SL. These results indicate that, contrary to a common belief, combustion-induced thermal expansion can significantly affect the incoming constant-density turbulent flow of unburned reactants even at u′/SL and Karlovitz number Ka as large as 10 and 18, respectively. The statistical behaviors of the structure functions reveal that the magnitude of the flame normal gradient of the velocity component tangential to the local flame can be significant, and it increases with increasing turbulence intensity. Moreover, the structure functions conditioned on both points in the heat release zone bear the signature of the anisotropic effects induced by the baroclinic torque for the flames belonging to the wrinkled flamelet and corrugated flamelet regimes. These anisotropic effects weaken with increasing turbulence intensity in the thin reaction zone regime.

Applicability of a modified breakage and coalescence model based on the complete turbulence spectrum concept for CFD simulation of gas-liquid mass transfer in a stirred tank reactor

A generalized model for bubble breakage and coalescence is proposed using Computational Fluid Dynamics – CFD for considering the complete energy spectrum. An eulerian model and balance equations are simultaneously used to simulate the multiphase flow and bubble size distribution, respectively. The turbulent kinetic energy and its dissipation are calculated using the standard turbulence model k-ε. A semi-empirical model that solves the second-order longitudinal structure function based on an interpolation function is coupled to CFD via UDF (User Defined Functions) code. CFD results are compared with experimental data obtained from Sauter mean diameter measurements at different bioreactor positions and stirred by a Rushton turbine. A reasonable prediction is obtained in comparison with the original Coulaloglou and Tavlarides (Break up) and Prince and Blanch (Coalescence) model. Further, the generalized model was extended to other stirring and aeration geometries using the same 10 litter tank bioreactor. The latter for evaluating strategies for overcoming gas-liquid mass transfer problems commonly found in bioreactors and a significant effect of the energy spectrum is reached in the geometries studied. The above, explained by the kLa oxygen transfer rate and bubble size determinations. It is numerically demonstrated that flow patterns and bubble size significantly influence the average kLa mass transfer in a bioreactor.

Properties of Convectively Induced Turbulence over Developing Oceanic Convection

Abstract Convectively induced turbulence (CIT) is an aviation hazard that continues to be a forecasting challenge as operational forecast models are too coarse to resolve turbulence affecting aircraft. In particular, little is known about tropical maritime CIT. In this study, a numerical simulation of a tropical oceanic CIT case where severe turbulence was encountered by a commercial aircraft is performed. The Richardson number (Ri), subgrid-scale eddy dissipation rate (EDR), and second-order structure functions (SF) are used as diagnostics to determine which may be used for CIT related to developing and mature convection. Model-derived subgrid-scale EDR in past studies of midlatitude continental CIT was shown to be a good diagnostic of turbulence but underpredicted turbulence intensity and areal coverage in this tropical simulation. SF diagnosed turbulence with moderate to severe intensity near convection and agreed most with observations. Further, SF were used to diagnose turbulence for developing convection. Results show that the areal coverage of turbulence associated with developing convection is less than mature convection. However, the intensity of turbulence in the vicinity of developing convection is greater than the turbulence intensity in the vicinity of mature convection highlighting developing convection as an additional concern to aviation.

Towards mapping turbulence in the intra-cluster medium

X-ray observations of the hot gas filling the intra-cluster medium (ICM) provide a wealth of information on the dynamics of clusters of galaxies. The global equilibrium of the ICM is believed to be ensured by non-thermal and thermal pressure support sources, among which gas movements and the dissipation of energy through turbulent motions. Accurate mapping of turbulence using X-ray emission lines is challenging due to the lack of spatially resolved spectroscopy. Only future instruments such as the X-ray Integral Field Unit (X-IFU) on Athena will have the spatial and spectral resolution to quantitatively investigate the ICM turbulence over a broad range of spatial scales. Powerful diagnostics for these studies are line shift and the line broadening maps, and the second-order structure function. When estimating these quantities, instruments will be limited by uncertainties of their measurements, and by the sampling variance (also known as cosmic variance) of the observation. Here, we extend the formalism started in our companion Paper I to include the effect of statistical uncertainties of measurements in the estimation of these line diagnostics, in particular for structure functions. We demonstrate that statistics contribute to the total variance through different terms, which depend on the geometry of the detector, the spatial binning and the nature of the turbulent field. These terms are particularly important when probing the small scales of the turbulence. An application of these equations is performed for the X-IFU, using synthetic turbulent velocity maps of a Coma-like cluster. Results are in excellent agreement with the formulas both for the structure function estimation (≤3%) and its variance (≤10%). The expressions derived here and in Paper I are generic, and ensure an estimation of the total errors in any X-ray measurement of turbulent structure functions. They also open the way for optimisations in the upcoming instrumentation and in observational strategies.

Second order structure functions for higher powers of turbulent velocity

We experimentally study the temporal second-order structure functions for integer powers of turbulent fluid velocity fluctuations , in three dimensional (3D) and two dimensional (2D) turbulence. Here is a composite time-series constructed by averaging the concurrent time-series () sampled at N spatially distributed Eulerian points. The N = 1 case has been extensively studied for velocity fluctuations (m = 1) and to a lesser extent for m > 1. The averaging method in case of N > 1 diverges from the Kolmogorov framework and has not been studied because fluctuations in are expected to smooth with increasing N leaving behind uninteresting large-scale mean flow information, but we find this is not so. We report the evolution of scaling exponents for in going from a single (N = 1) to a spatial average over several Eulerian points (). Our 3D experiments in a tank with rotating jets at the floor show for all m-values in agreement with prior results and evolves to an asymptotic value of . The evolution of follows the functional form , where points is the only fit parameter representing the convergence rate constant. Results for the 2D experiments conducted in a gravity assisted soap film in the enstrophy cascade regime are in sharp contrast with their 3D counterparts. Firstly varies polynomially with m and asymptotes to a constant value at m = 5. Secondly, the evolution of is logarithmic , where A and B are fit parameters and eventually deviates at large N and asymptotes to for all m. The starkly different convergence forms (exponential in 3D versus logarithmic in 2D) may be interpreted as a signature of inter-scale couplings in the respective turbulent flows by decomposing the two-point correlator for into a self-correlation and cross-correlation term. In addition to aiding in the theoretical development, the results may also have implications for determination of resolution in 2D turbulence experiments and simulations, wind energy and atmospheric boundary layer turbulence.