Refined Similarity Hypothesis Research Articles

Forward and Inverse Energy Cascade in Fluid Turbulence Adhere to Kolmogorov's Refined Similarity Hypothesis.

We study fluctuations of the local energy cascade rate Φ_{ℓ} in turbulent flows at scales (ℓ) in the inertial range. According to the Kolmogorov refined similarity hypothesis (KRSH), relevant statistical properties of Φ_{ℓ} should depend on ε_{ℓ}, the viscous dissipation rate locally averaged over a sphere of size ℓ, rather than on the global average dissipation. However, the validity of KRSH applied to Φ_{ℓ} has not yet been tested from data. Conditional averages such as ⟨Φ_{ℓ}|ε_{ℓ}⟩ as well as of higher-order moments are measured from direct numerical simulations data, and results clearly adhere to the predictions from KRSH. Remarkably, the same is true when considering forward (Φ_{ℓ}>0) and inverse (Φ_{ℓ}<0) cascade events separately. Measured ratios of forward and inverse cascade probability densities conditioned on ε_{ℓ} also confirm the applicability of the KRSH to analysis of the fluctuation relation from nonequilibrium thermodynamics.

Intermittency in the Expanding Solar Wind: Observations from Parker Solar Probe (0.16 au), Helios 1 (0.3–1 au), and Voyager 1 (1–10 au)

Abstract We examine statistics of magnetic-field vector components to explore how intermittency evolves from near-Sun plasma to radial distances as large as 10 au. Statistics entering the analysis include autocorrelation, magnetic structure functions of the order of n (SF n ), and scale-dependent kurtosis (SDK), each grouped in ranges of heliocentric distance. The Goddard Space Flight Center Space Physics Data Facility provides magnetic-field measurements for resolutions of 6.8 ms for Parker Solar Probe, 6 s for Helios, and 1.92 s for Voyager 1. We compute SF2 to determine the scales encompassing the inertial range and examine SDK to investigate the degree of non-Gaussianity. Autocorrelations are used to resolve correlation scales. Correlation lengths and ion inertial lengths provide an estimate of effective Reynolds number (Re). Variation in Re allows us to examine for the first time the relationship between SDK and Re in an interplanetary plasma. A conclusion from this observed relationship is that regions with lower Re at a fixed physical scale have on average lower kurtosis, implying less intermittent behavior. Kolmogorov refined similarity hypothesis is applied to magnetic SF n and kurtosis to calculate intermittency parameters and fractal scaling in the inertial range. A refined Voyager 1 magnetic-field data set is generated.

Finite Reynolds number effect and the 4/5 law

Kolmogorov [A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 30, 299 (1941)] formulated a theory of small-scale turbulence (K41), valid at extremely large Reynolds numbers, based on two similarity hypotheses and on an exact result derived from the transport equation for the second-order structure function, known as the 4/5 law. Although K41 was praised for its simplicity and elegance, Kolmogorov [A. N. Kolmogorov, J. Fluid Mech. 13, 82 (1962).] proposed a new refined similarity hypothesis (K62) mainly to account for the effect of the large scales on the small scales. It has been widely interpreted in the literature as a correction to K41 arising from the intermittency of the instantaneous energy dissipation rate ε. In this paper we argue that since K62 retains the 4/5 law, it must satisfy the same constraints as K41, viz., extremely large Reynolds number and flow stationarity. The retention of the 4/5 law is not however consistent with the presence of nonstationarity due to the effect of the large scales, as postulated by K62. A relatively extensive survey of published data shows that, indeed, the 4/5 law has not yet been observed in either experiments or simulations due to the Reynolds number not being sufficiently large. The use of the transport equation for the second-order structure function, together with an empirical model for the Kolmogorov-normalized second-order velocity structure function, confirms that the 4/5 law is established only after this structure function becomes independent of the Reynolds number.Received 25 April 2019DOI:https://doi.org/10.1103/PhysRevFluids.4.084602©2019 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasTurbulenceFluid Dynamics

Direct assessment of Kolmogorov's first refined similarity hypothesis

Using volumetric velocity data from a turbulent laboratory water flow and numerical simulations of homogeneous, isotropic turbulence, we present a direct experimental and numerical assessment of Kolmogorov's first refined similarity hypothesis based on three-dimensional measurements of the local energy dissipation rate $\epsilon_r$ measured at dissipative scales $r$. We focus on the properties of the stochastic variables $V_L = \Delta u(r)/(r \epsilon_r)^{1/3}$ and $V_T = \Delta v(r)/(r\epsilon_r)^{1/3}$, where $\Delta u(r)$ and $\Delta v(r)$ are longitudinal and transverse velocity increments. Over one order of magnitude of scales $r$ within the dissipative range, the distributions of $V_L$ and $V_T$ from both experiment and simulation collapse when parameterised by a suitably defined local Reynolds number, providing the first conclusive experimental evidence in support of the first refined similarity hypothesis and its universality.

Reappraisal of the velocity derivative flatness factor in various turbulent flows

We first analytically show, starting with the Navier–Stokes equations, that the value of the derivative flatness is controlled by pressure diffusion of energy, viscous destructive effects and large-scale effects (decay and/or production). The latter two terms tend to zero when the Taylor-microscale Reynolds number$Re_{\unicode[STIX]{x1D706}}$is sufficiently large. We argue that the pressure-diffusion term should also tend to a constant at large$Re_{\unicode[STIX]{x1D706}}$. Available data for the velocity derivative flatness,$F$, in different turbulent flows are re-examined and interpreted in the light of the finite-Reynolds-number effect. It is found that$F$can differ from flow to flow at moderate$Re_{\unicode[STIX]{x1D706}}$; for a given flow,$F$may also depend on the initial conditions. The data for$F$in various flows, e.g. along the axis in the far field of plane and circular jets, and grid turbulence, show that it approaches a constant, with a value slightly larger than 10, when$Re_{\unicode[STIX]{x1D706}}$is sufficiently large. This behaviour for$F$is supported, at least qualitatively, by our analytical considerations. The constancy of$F$at large$Re_{\unicode[STIX]{x1D706}}$violates the refined similarity hypothesis introduced by Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85) to account for the intermittency of the energy dissipation rate. It is not, however, inconsistent with Kolmogorov’s original similarity hypothesis (Dokl. Akad. Nauk SSSR, vol. 30, 1941, pp. 299–303), although we contend that the power-law relation$F\sim Re_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FC}_{4}}$(Kolmogorov 1962), which is widely accepted in the literature, has in reality been almost invariably used to ‘model’ the finite-Reynolds-number effect for the laboratory data and has been strongly influenced by the weighting given to the atmospheric surface layer data. The inclusion of the latter data has misled previous investigations of how$F$varies with $Re_{\unicode[STIX]{x1D706}}$.

Dissipation, intermittency, and singularities in incompressible turbulent flows.

We examine the connection between the singularities or quasisingularities in the solutions of the incompressible Navier-Stokes equation (INSE) and the local energy transfer and dissipation, in order to explore in detail how the former contributes to the phenomenon of intermittency. We do so by analyzing the velocity fields (a) measured in the experiments on the turbulent von Kármán swirling flow at high Reynolds numbers and (b) obtained from the direct numerical simulations of the INSE at a moderate resolution. To compute the local interscale energy transfer and viscous dissipation in experimental and supporting numerical data, we use the weak solution formulation generalization of the Kármán-Howarth-Monin equation. In the presence of a singularity in the velocity field, this formulation yields a nonzero dissipation (inertial dissipation) in the limit of an infinite resolution. Moreover, at finite resolutions, it provides an expression for local interscale energy transfers down to the scale where the energy is dissipated by viscosity. In the presence of a quasisingularity that is regularized by viscosity, the formulation provides the contribution to the viscous dissipation due to the presence of the quasisingularity. Therefore, our formulation provides a concrete support to the general multifractal description of the intermittency. We present the maps and statistics of the interscale energy transfer and show that the extreme events of this transfer govern the intermittency corrections and are compatible with a refined similarity hypothesis based on this transfer. We characterize the probability distribution functions of these extreme events via generalized Pareto distribution analysis and find that the widths of the tails are compatible with a similarity of the second kind. Finally, we make a connection between the topological and the statistical properties of the extreme events of the interscale energy transfer field and its multifractal properties.

Energy dissipation statistics along the Lagrangian trajectories in three-dimensional turbulent flows

Energy dissipation rate is relevant in the turbulent phenomenology theory, such as the classical Kolmogorov 1941 and 1962 refined similarity hypothesis. However, it is extremely difficult to retrieve experimentally or numerically. In this paper, the full energy dissipation, its proxy and the pseudo-energy dissipation rate along the Lagrangian trajectories in the three-dimensional turbulent flows are examined by using a state-of-art high resolution direct numerical simulation database with a Reynolds number Re λ = 400. It is found that the energy dissipation proxy e P is more correlated with the full energy dissipation rate e. The corresponding correlation coefficient ρ between the velocity gradient and e shows a Gaussian distribution. Furthermore, the coarse-grained dissipation rate is considered. The cross correlation ρ is found to be increased with the increasing of the scale τ. Finally, the hierarchical structure is extracted for the full energy dissipation rate, its proxy and the pseudo one. The results show a power-law behavior in the inertial range 10 ≤τ/τ η ≤ 100. The experimental scaling exponent of the full energy dissipation rate is found to be h L =0.69, agrees very well with the one found for the Eulerian velocity. The experimental values for e P and e S are around h L = 0.78, implying a more intermittent Lagrangian turbulence. Therefore, the intermittency parameter provided by e P and e S will be biased.

Reynolds number scaling of velocity increments in isotropic turbulence.

Using the largest database of isotropic turbulence available to date, generated by the direct numerical simulation (DNS) of the Navier-Stokes equations on an 8192^{3} periodic box, we show that the longitudinal and transverse velocity increments scale identically in the inertial range. By examining the DNS data at several Reynolds numbers, we infer that the contradictory results of the past on the inertial-range universality are artifacts of low Reynolds number and residual anisotropy. We further show that both longitudinal and transverse velocity increments scale on locally averaged dissipation rate, just as postulated by Kolmogorov's refined similarity hypothesis, and that, in isotropic turbulence, a single independent scaling adequately describes fluid turbulence in the inertial range.

Refined similarity hypothesis using three-dimensional local averages.

The refined similarity hypotheses of Kolmogorov, regarded as an important ingredient of intermittent turbulence, has been tested in the past using one-dimensional data and plausible surrogates of energy dissipation. We employ data from direct numerical simulations, at the microscale Reynolds number R(λ)∼650, on a periodic box of 4096(3) grid points to test the hypotheses using three-dimensional averages. In particular, we study the small-scale properties of the stochastic variable V=Δu(r)/(rε(r))(1/3), where Δu(r) is the longitudinal velocity increment and ε(r) is the dissipation rate averaged over a three-dimensional volume of linear size r. We show that V is universal in the inertial subrange. In the dissipation range, the statistics of V are shown to depend solely on a local Reynolds number.

Direct numerical simulation studies of Lagrangian intermittency in turbulence

Lag-averaged Lagrangian statistics from direct numerical simulations over a range of Reynolds numbers are analyzed to test the predictions of the Lagrangian Refined Similarity Hypothesis (LRSH). The analysis uses the Lagrangian integral time scale to scale the lag since it is the natural time scale to reveal trends and scaling with Reynolds number. Both the velocity difference and the dissipation rate probability density functions (PDFs) collapse across inertial sub-range and diffusive scales for approximately the same values of the scaled lag, and in the zero lag limit are independent of the lag and depend only on the Reynolds number. These findings are consistent with the LRSH. The velocity difference PDFs are characterized by stretched exponential tails, while the dissipation rate PDFs for small lags have a log normal core with power law tails at both large and small values of the dissipation rate. The velocity structure functions show inertial sub-range similarity scaling with Reynolds number which extends to smaller scales with increasing Reynolds number. Estimates of the scaling exponents obtained are consistent with those from previous studies. They tend to saturate at a value of about two for high order moments. Non-dimensional acceleration moments show a striking power law dependence on Reynolds number from which novel estimates of the scaling exponents have been determined. Similarity scaling is much more elusive to demonstrate in the dissipation rate moments. The data are consistent with, but do not confirm, the Oboukhov relationship connecting velocity structure functions and dissipation rate moments on inertial sub-range scales.

The direct enstrophy cascade of two-dimensional soap film flows

We investigate the direct enstrophy cascade of two-dimensional decaying turbulence in a flowing soap film channel. We use a coarse-graining approach that allows us to resolve the nonlinear dynamics and scale-coupling simultaneously in space and in scale. From our data, we verify an exact relation due to Eyink [“Local energy flux and the refined similarity hypothesis,” J. Stat. Phys. 78, 335–351 (1995); Eyink “Exact results on scaling exponents in the 2D enstrophy cascade,” Phys. Rev. Lett. 74, 3800–3803 (1995)] between traditional 3rd-order structure function and the enstrophy flux obtained by coarse-graining. We also present experimental evidence that enstrophy cascades to smaller (larger) scales with a 60% (40%) probability, in support of theoretical predictions by Merilees and Warn [“On energy and enstrophy exchanges in two-dimensional non-divergent flow,” J. Fluid Mech. 69, 625–630 (1975)] which appear to be valid in our flow owing to the ergodic nature of turbulence. We conjecture that their kinematic arguments break down in quasi-laminar 2D flows. We find some support for these ideas by using an Eulerian coherent structure identification technique, which allows us to determine the effect of flow topology on the enstrophy cascade. A key finding is that “centers” are inefficient at transferring enstrophy between scales, in contrast to “saddle” regions which transfer enstrophy to small scales with high efficiency.

Turbulent Lagrangian Velocity Statistics Conditioned on Extreme Values of Dissipation

Direct numerical simulation data for the Lagrangian velocity structure functions conditioned on the local (in time) rate of dissi- pation of turbulence kinetic energy are analyzed over a wide range of Reynolds numbers to seek direct evidence for Lagrangian Refined Similarity Hypothesis (RSH) scaling. The main focus is on testing the Refined Similarity predictions for the dependence on the local dissipation rate. In the dissipation sub-range there is clear evidence for RSH scaling at large values of the dissipation rate, but strong departures from RSH scaling at very small values. However, a detailed analysis of the dependence on Reynolds number of the results at small values of the local dissipation rate suggests a trend towards RSH scaling with increasing Reynolds number, but this can only be realized, if at all, for Taylor scale Reynolds numbers greater than 105. Within the inertial sub-range, there is limited evidence for RSH scaling, but again there is a trend towards more substantial scaling over a wider range of dis- sipation rates with increasing Reynolds number. The main conclusion is that RSH scaling may apply as an important theoretical constraint on the Lagrangian structure of turbulence in the large Reynolds number limit, but significant corrections to the theory exist at practical values of the Reynolds number.

Generation of spatial atmospheric turbulence field in aircraft motion simulation based on refined similarity hypothesis

It is necessary to build turbulence model to study the response of aircraft to atmospheric turbulence for high resolution earth observation. The conventional method is on the basis of Dryden’s model with the assumption that individual patches are Gaussian. In this paper, based on Kraichnan’s refined similarity idea, a new 1D atmospheric turbulence model is set up by introducing the energy transfer rate as an intermittency disturbance to a Gaussian process. Our results show that the turbulent fields generated by our new method exhibit an anomalous scaling described by the She-Leveque (SL) formula, which is now well accepted for homogenous and isotropic turbulence.

Lagrangian Refined Kolmogorov Similarity Hypothesis for Gradient Time Evolution and Correlation in Turbulent Flows

We study time evolution of velocity and pressure gradients in isotropic turbulence by quantifying their autocorrelation functions and decorrelation time scales. The Lagrangian analysis uses data in a public database generated by direct numerical simulation at a Reynolds number Re{lambda} approximately 433. It is confirmed that when averaging over the entire domain, correlation functions decay on time scales on the order of the global Kolmogorov turnover time scale. However, when performing the analysis in different subregions of the flow, turbulence intermittency leads to large spatial variability in the decay time scales. Remarkably, excellent collapse of the autocorrelation functions is recovered when using a locally defined Kolmogorov time scale. This provides new evidence for the validity of Kolmogorov's refined similarity hypothesis, but from a Lagrangian viewpoint that provides a natural frame to describe the dynamics of turbulence.

Statistics of Small-Scale Velocity Fluctuations and Internal Intermittency in Marine Stratocumulus Clouds

Abstract Clouds are known to be turbulent, but the details of their internal turbulent structure have been largely unexplored. Measurements of turbulent velocities in stratocumulus clouds presented here reveal an intermittent structure consistent with that observed in classic homogeneous isotropic turbulence. The measurements were taken close to cloud top in a 200-m-thick cloud layer over a path of approximately 6 km, using a hot-wire anemometer below a helicopter as part of the Airborne Cloud Turbulence Observation System (ACTOS) measurement system. Hot-wire signal artifacts resulting from droplet impacts are removed without significantly degrading the signal, such that high-order velocity structure functions can be evaluated. The structure function analysis for orders 2–8 show statistically significant departures from the Kolmogorov’s 1941 scaling, yielding scaling exponents consistent with the Kolmogorov–Obukhov refined similarity hypothesis, with an intermittency exponent of 0.25. This is in agreement with the accepted value determined in single-phase flows under carefully controlled conditions, and no evidence is found of any departure from the large body of knowledge obtained from the laboratory on the finescale turbulence structure. This suggests that processes depending on the finescale structure of turbulence that cannot presently be measured in clouds can be explored in the laboratory setting. Since these findings pertain to clouds with relatively low liquid water content and weak turbulence, further work will be required to determine their applicability to other cloud types.

Velocity-gradient statistics along particle trajectories in turbulent flows: The refined similarity hypothesis in the Lagrangian frame

We present an investigation of the statistics of velocity gradient related quantities, in particular energy dissipation rate and enstrophy, along the trajectories of fluid tracers and of heavy/light particles advected by a homogeneous and isotropic turbulent flow. The refined similarity hypothesis (RSH) proposed by Kolmogorov and Oboukhov in 1962 is rephrased in the Lagrangian context and then tested along the particle trajectories. The study is performed on state-of-the-art numerical data resulting from numerical simulations up to Re(lambda) approximately 400 with 2048(3) collocation points. When particles have small inertia, we show that the Lagrangian formulation of the RSH is well verified for time lags larger than the typical response time tau(p) of the particle. In contrast, in the large inertia limit when the particle response time approaches the integral time scale of the flow, particles behave nearly ballistic, and the Eulerian formulation of RSH holds in the inertial range.

Studying anomalous scaling and heat transport of turbulent thermal convection using a dynamical model

Turbulent thermal convection is a well-studied problem with various issues of interest. In this paper, we review our work which shows the nature and origin of anomalous scaling and heat transport in the limit of very strong thermal forcing, can be gained by studying a dynamical model, known as shell model, of homogeneous turbulent thermal convection in which buoyancy acts directly at most scales. Specifically, we have obtained two results. The first result is that when buoyancy acts directly at most scales such that the dynamics are governed by a cascade of entropy, the scaling behavior is described by Bolgiano and Obukhov scaling plus corrections that are due to the variations of the local entropy transfer rate. This result indicates the validity of the extension of refined similarity hypothesis to turbulent thermal convection. The second result is that when buoyancy is acting directly at most scales, a damping term acting on the largest scale, which has to be added for the system to achieve stationarity, plays a crucial role in heat transport, and that the heat transport depends on the strength of thermal forcing in the same manner as that predicted for the ultimate state of very strong thermal forcing. With our interpretation of the damping term representing the effect of the boundaries, this result indicates that in the ultimate state of turbulent thermal convection, when buoyancy is acting at most scales, boundaries would play a significant role in heat transport.

Is the Scaling of Supersonic Turbulence Universal?

The statistical properties of turbulence are considered to be universal at sufficiently small length scales, i.e., independent of boundary conditions and large-scale forces acting on the fluid. Analyzing data from numerical simulations of supersonic turbulent flow driven by external forcing, we demonstrate that this is not generally true for the two-point velocity statistics of compressible turbulence. However, a reformulation of the refined similarity hypothesis in terms of the mass-weighted velocity rho1/3v yields scaling laws that are almost insensitive to the forcing. The results imply that the most intermittent dissipative structures are shocks closely following the scaling of Burgers turbulence.

Refined similarity hypotheses in shell models of homogeneous turbulence and turbulent convection

A major challenge in turbulence research is to understand from first principles the origin of the anomalous scaling of velocity fluctuations in high-Reynolds-number turbulent flows. One important idea was proposed by Kolmogorov [J. Fluid Mech. 13, 82 (1962)], which attributes the anomaly to variations of the locally averaged energy dissipation rate. Kraichnan later pointed out [J. Fluid Mech. 62, 305 (1973)] that the locally averaged energy dissipation rate is not an inertial-range quantity and a proper inertial-range quantity would be the local energy transfer rate. As a result, Kraichnan's idea attributes the anomaly to variations of the local energy transfer rate. These ideas, generally known as refined similarity hypotheses, can also be extended to study the anomalous scaling of fluctuations of an active scalar, such as the temperature in turbulent convection. We examine the validity of these refined similarity hypotheses and their extensions to an active scalar in shell models of homogeneous turbulence and turbulent convection. We find that Kraichnan's refined similarity hypothesis and its extension are valid.

A Paradox concerning the Refined Similarity Hypothesis of Kolmogorov for Isotropic Turbulence

It is proved that the famous third-order structure function of the velocity in homogeneous isotropic turbulence derived by Kolmogorov implies the statistical interdependence of the difference and sum of the velocities at two points separated by a distance r. However, this is inconsistent with Kolmogorov's hypothesis regarding the difference between such velocities, made in 1962, in the sense that it implies that the statistics of the non-dimensional prefactor of the velocity difference are not universal but, instead, depend on the velocity sum, at least. The only resolution of this apparent paradox that can maintain the spirit of Kolmogorov's original hypothesis, which has been so important in the study of intermittency in turbulence, is given.