Third-order Velocity Structure Function Research Articles

Empirical constraints on the turbulence in QSO host nebulae from velocity structure function measurements

ABSTRACT We present the first empirical constraints on the turbulent velocity field of the diffuse circumgalactic medium around four luminous quasi-stellar objects (QSOs) at z ≈ 0.5–1.1. Spatially extended nebulae of ≈50–100 physical kpc in diameter centred on the QSOs are revealed in [O ii] $\lambda \lambda \, 3727,3729$ and/or [O iii] $\lambda \, 5008$ emission lines in integral field spectroscopic observations obtained using Multi-Unit Spectroscopic Explorer on the Very Large Telescope. We measure the second- and third-order velocity structure functions (VSFs) over a range of scales, from ≲5 kpc to ≈20–50 kpc, to quantify the turbulent energy transfer between different scales in these nebulae. While no constraints on the energy injection and dissipation scales can be obtained from the current data, we show that robust constraints on the power-law slope of the VSFs can be determined after accounting for the effects of atmospheric seeing, spatial smoothing, and large-scale bulk flows. Out of the four QSO nebulae studied, one exhibits VSFs in spectacular agreement with the Kolmogorov law, expected for isotropic, homogeneous, and incompressible turbulent flows. The other three fields exhibit a shallower decline in the VSFs from large to small scales. However, with a limited dynamic range in the spatial scales in seeing-limited data, no constraints can be obtained for the VSF slopes of these three nebulae. For the QSO nebula consistent with the Kolmogorov law, we determine a turbulence energy cascade rate of ≈0.2 cm2 s−3. We discuss the implication of the observed VSFs in the context of QSO feeding and feedback in the circumgalactic medium.

Hydrodynamics of flow through a degraded channel bed

This article presents experimental results of turbulent flow measured in a bimodal degraded channel bed consisting of sand-gravel mixture. Sand and gravel of uniform sizes 0.25 and 3.5 mm were mixed in the same proportions (by weight) to create a bimodal sedimentary bed. A three-dimensional Vectrino velocimeter was employed to collect three-dimensional velocities over bimodal degraded bed under equilibrium condition. The streamwise velocity, Reynolds stresses, turbulent kinetic energy (TKE), and TKE fluxes profiles were compared with the literature. However, the advancement of the existing knowledge was done by exploring the laws of turbulence. To this end, the velocity structure function method was applied. Second and third-order streamwise velocity structure functions followed by mixed third-order velocity structure functions revealed the existence of inertial subrange. The TKE dissipation rate was estimated using Kolmogorov’s and Monin–Yaglom’s scaling laws of turbulence. The anisotropy analysis indicated anisotropic turbulence in the near-bed, whereas above the initial bed-level, the anisotropy tends to follow three-dimensional isotropy. The present study notably enhances the understanding of turbulent flow through a degraded bed by demonstrating the legitimacy of laws of turbulence at different locations over the bed and providing a comprehensible acquaintance in TKE budget and Reynolds stress anisotropy.

Approach to the 4/3 law for turbulent pipe and channel flows examined through a reformulated scale-by-scale energy budget

In this study, we propose a scale-by-scale (SBS) energy budget equation for flows with homogeneity in at least one direction. This SBS budget represents a modified form of the equation first proposed by Danaila et al. (J. Fluid Mech., vol. 430, 2001, pp. 87–109) for the channel centreline – the primary difference is that, here, we consider the role of pressure along with the errors associated with the isotropic approximations of the interscale divergence and Laplacian of the squared velocity increment. The term encompassing the effects of mean shear is also characterised such that the present analysis can be extended straightforwardly to locations away from the centreline. We show, based on a detailed analysis of previously published channel flow direct numerical simulations and pipe flow experiments near the centreline, how several terms in the present SBS budget equation (including the third-order velocity structure function) behave with increasing Reynolds number. The behaviour of these terms is shown to imply a rate of emergence and subsequent growth of the 4/3 law scale subrange at the channel centreline and pipe axis. The analysis also suggests that the peak magnitude of the third-order velocity structure function occurs at a scale that is fixed in proportion to the Taylor microscale at sufficiently high Reynolds number.

Study of the interaction of two decaying grid-generated turbulent flows

Hot-wire measurements are carried out in a decaying turbulence downstream of a grid made up of two juxtaposed perforated plates with different mesh sizes but same solidity. The two perforated plates generate two interacting (quasi–) homogeneous and isotropic decaying turbulent flows with distinct turbulence intensities and integral length scales. The interaction between these two flows leads to the development of a shearless turbulent mixing layer (STML). The main focus is on the decay of the turbulence centerline of the STML. Along the downstream distance x, the Taylor microscale Reynolds number, Reλ, remains constant, the streamwise velocity variance behaves like x−1, and the Taylor microscale (λ) varies as x1/2. This indicates that the turbulence on the centerline of the STML decays in a perfectly self-preserving manner at all scales of motion. This is further supported by the very good collapse of the velocity spectra, second-, and third-order velocity structure functions.

Modeling the third-order velocity structure function in the scaling range at finite Reynolds numbers

A model for the third-order velocity structure function, S3, is proposed for closing the transport equation of the second-order velocity structure function, S2, in the scaling range of finite Reynolds number homogeneous and isotropic turbulence (HIT). The model is based on a gradient type hypothesis with an eddy-viscosity formulation. The present model differs from previous ones in that no assumptions are made with regard to the behaviors of S2 and S3 in the scaling range. This allows S3 to be modeled whether the intermittency of the energy dissipation ɛ (as modeled by the intermittency phenomenology) is considered or not. In both cases, the model predicts the same (infinite Reynolds number) asymptotic behavior for S2. This corresponds to the K41 prediction, i.e., S2 ∼ r2/3. The model yields good agreement against direct numerical simulation data for forced HIT at all scales of motion except in the transition region between the dissipative and scaling ranges. The introduction of a simple bridging function in the model improves significantly the agreement in this region. Furthermore, the model illustrates the effect of the finite Reynolds number on the scaling range and shows that this effect is responsible for the deviation from a power-law behavior.

Sensitivity analysis of the second and third-order velocity structure functions to the Reynolds number in decaying and forced isotropic turbulence using the EDQNM model

Numerical calculations based on a recent version of the eddy-damped quasi-normal model (EDQNM-LMFA) are carried out for homogeneous isotropic turbulence (HIT) with the aim of investigating the dependency on the Reynolds number of second and third order velocity structure functions. The quantities investigated include the energy spectrum E, the non-linear energy transfer T as well as the second (S2) and third (S3) order moments of the longitudinal velocity increment. Both free decaying HIT and (steady state) forced HIT are considered. The analysis of the structure functions for Reλ∈[50,106] indicates that, regardless of whether one considers decaying or forced HIT, the large scales affect S2 and S3 in the scaling range. In that range, forcing affects S2 and S3 differently. For forced HIT, S2∕(ϵ¯r)2∕3 exhibits a distinct “bump” near the upper end of the scaling range while no such bump is seen for S3∕(ϵ¯r). The latter quantity remains approximately constant for values of r which extend to the scale corresponding to the forcing. For decaying HIT, there is no discernible bump in either S2∕(ϵ¯r)2∕3 or S3∕(ϵ¯r). The slope of S3 in the scaling range approaches the theoretical value of 1, when Reλ is sufficiently large. However, at similar Reλ, the slope of S2 has not yet reached a constant for either decaying or forced HIT.

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.

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.

Finite Reynolds number effect on the scaling range behaviour of turbulent longitudinal velocity structure functions

The effect of large-scale forcing on the second- and third-order longitudinal velocity structure functions, evaluated at the Taylor microscale $r=\unicode[STIX]{x1D706}$, is assessed in various turbulent flows at small to moderate values of the Taylor microscale Reynolds number $R_{\unicode[STIX]{x1D706}}$. It is found that the contribution of the large-scale terms to the scale by scale energy budget differs from flow to flow. For a fixed $R_{\unicode[STIX]{x1D706}}$, this contribution is largest on the centreline of a fully developed channel flow but smallest for stationary forced periodic box turbulence. For decaying-type flows, the contribution lies between the previous two cases. Because of the difference in the large-scale term between flows, the third-order longitudinal velocity structure function at $r=\unicode[STIX]{x1D706}$ differs from flow to flow at small to moderate $R_{\unicode[STIX]{x1D706}}$. The effect on the second-order velocity structure functions appears to be negligible. More importantly, the effect of $R_{\unicode[STIX]{x1D706}}$ on the scaling range exponent of the longitudinal velocity structure function is assessed using measurements of the streamwise velocity fluctuation $u$, with $R_{\unicode[STIX]{x1D706}}$ in the range 500–1100, on the axis of a plane jet. It is found that the magnitude of the exponent increases as $R_{\unicode[STIX]{x1D706}}$ increases and the rate of increase depends on the order $n$. The trend of published structure function data on the axes of an axisymmetric jet and a two-dimensional wake confirms this dependence. For a fixed $R_{\unicode[STIX]{x1D706}}$, the exponent can vary from flow to flow and for a given flow, the larger $R_{\unicode[STIX]{x1D706}}$ is, the closer the exponent is to the value predicted by Kolmogorov (Dokl. Akad. Nauk SSSR, vol. 30, 1941a, pp. 299–303) (hereafter K41). The major conclusion is that the finite Reynolds number effect, which depends on the flow, needs to be properly accounted for before determining whether corrections to K41, arising from the intermittency of the energy dissipation rate, are needed. We further point out that it is imprudent, if not incorrect, to associate the finite Reynolds number effect with a consequence of the modified similarity hypothesis introduced by Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85) (K62); we contend that this association has misled the vast majority of post K62 investigations of the consequences of K62.

Anisotropic characteristics of the kraichnan direct cascade in two-dimensional hydrodynamic turbulence

Statistical characteristics of the Kraichnan direct cascade for two-dimensional hydrodynamic turbulence are numerically studied (with spatial resolution $8192\times 8192$) in the presence of pumping and viscous-like damping. It is shown that quasi-shocks of vorticity and their Fourier partnerships in the form of jets introduce an essential influence in turbulence leading to strong angular dependencies for correlation functions. The energy distribution as a function of modulus $k$ for each angle in the inertial interval has the Kraichnan behavior, $\sim k^{-4}$, and simultaneously a strong dependence on angles. However, angle average provides with a high accuracy the Kraichnan turbulence spectrum $E_k=C_K\eta^{2/3} k^{-3}$ where $\eta$ is enstrophy flux and the Kraichnan constant $C_K\simeq 1.3$, in correspondence with the previous simulations. Familiar situation takes place for third-order velocity structure function $S_3^L$ which, as for the isotropic turbulence, gives the same scaling with respect to separation length $R$ and $\eta$, $S_3^L=C_3\eta R^3$, but the mean over angles and time $\bar {C_3}$ differs from its isotropic value.

Quasi-two-dimensional and three-dimensional turbulence in rotational spherical liquid layers

Turbulent flows of a viscous incompressible fluid in a layer between rotating concentric spheres under the action of the modulation of the velocity of one of the spheres have been studied experimentally and numerically. The possibility of the formation of turbulence with spectra qualitatively similar to spectra obtained in measurements in the upper atmosphere is established: with the slope close to −3 at low frequencies and close to −5/3 at high frequencies and with the negative third-order longitudinal velocity structure function. It has been shown that such spectra are formed in the regions of a flow that are strongly synchronized under the action of the modulation of the rotation velocity.

Direct and inverse energy cascades in a forced rotating turbulence experiment

We present experimental evidence for a double cascade of kinetic energy in a statistically stationary rotating turbulence experiment. Turbulence is generated by a set of vertical flaps, which continuously injects velocity fluctuations towards the center of a rotating water tank. The energy transfers are evaluated from two-point third-order three-component velocity structure functions, which we measure using stereoscopic particle image velocimetry in the rotating frame. Without global rotation, the energy is transferred from large to small scales, as in classical three-dimensional turbulence. For nonzero rotation rates, the horizontal kinetic energy presents a double cascade: a direct cascade at small horizontal scales and an inverse cascade at large horizontal scales. By contrast, the vertical kinetic energy is always transferred from large to small horizontal scales, a behavior reminiscent of the dynamics of a passive scalar in two-dimensional turbulence. At the largest rotation rate, the flow is nearly two-dimensional, and a pure inverse energy cascade is found for the horizontal energy. To describe the scale-by-scale energy budget, we consider a generalization of the Kármán-Howarth-Monin equation to inhomogeneous turbulent flows, in which the energy input is explicitly described as the advection of turbulent energy from the flaps through the surface of the control volume where the measurements are performed.

The Statistics of Supersonic Isothermal Turbulence

We present results of large-scale three-dimensional simulations of supersonic Euler turbulence with the piecewise parabolic method and multiple grid resolutions up to 20483 points. Our numerical experiments describe nonmagnetized driven turbulent flows with an isothermal equation of state and an rms Mach number of 6. We discuss numerical resolution issues and demonstrate convergence, in a statistical sense, of the inertial range dynamics in simulations on grids larger than 5123 points. The simulations allowed us to measure the absolute velocity scaling exponents for the first time. The inertial range velocity scaling in this strongly compressible regime deviates substantially from the incompressible Kolmogorov laws. The slope of the velocity power spectrum, for instance, is -1.95 compared to -5/3 in the incompressible case. The exponent of the third-order velocity structure function is 1.28, while in incompressible turbulence it is known to be unity. We propose a natural extension of Kolmogorov's phenomenology that takes into account compressibility by mixing the velocity and density statistics and preserves the Kolmogorov scaling of the power spectrum and structure functions of the density-weighted velocity υ ≡ ρ1/3u. The low-order statistics of υ appear to be invariant with respect to changes in the Mach number. For instance, at Mach 6 the slope of the power spectrum of υ is -1.69, and the exponent of the third-order structure function of υ is unity. We also directly measure the mass dimension of the "fractal" density distribution in the inertial subrange, Dm ≈ 2.4, which is similar to the observed fractal dimension of molecular clouds and agrees well with the cascade phenomenology.

Multi-fractal formulation of compressible fully developed turbulence: Parabolic-profile approximation for the singularity spectrum

A parabolic-profile approximation (PPA) for the singularity spectrum D(h) in the multi-fractal model for compressible fully developed turbulence (FDT) is considered and is then extrapolated to the Kolmogorov microscale regime. The generalization of Kolmogorov’s “4/5th law” relating the third-order velocity structure function to the mean energy dissipation rate ε to compressible FDT is considered. The PPA is also shown to afford, unlike the generic multi-fractal model, an analytical calculation of probability distribution function (PDF) of velocity gradients and to describe intermittency corrections for this PDF that complement those provided by homogeneous-fractal model.

Theories of convection and the spectrum of turbulence in the solar photosphere

AbstractClassical theories of turbulence do not describe accurately inertial range scaling laws in turbulent convection and notably fail to model the shape of the turbulent spectrum of solar photospheric convection. To understand these discrepancies, a detailed study of scale-by-scale budgets in turbulent Rayleigh-Bénard convection is presented, with particular emphasis placed on anisotropy and inhomogeneity. A generalized Kolmogorov equation applying to convection is derived and its various terms are computed using numerical simulations of turbulent Boussinesq convection. The analysis of the isotropic part of the equation shows that the third-order velocity structure function is significantly affected by buoyancy forcing and large-scale inhomogeneities. Anisotropic contributions to this equation are also shown to be comparable to their isotropic counterpart at moderate to large scales. Implications of these results for convection in the solar photosphere, mesogranulation and supergranulation are discussed.

On the two-point correlation of potential vorticity in rotating and stratified turbulence

A framework is developed to describe the two-point statistics of potential vorticity in rotating and stratified turbulence as described by the Boussinesq equations. The Kármán–Howarth equation for the dynamics of the two-point correlation function of potential vorticity reveals the possibility of inertial-range dynamics in certain regimes in the Rossby, Froude, Prandtl and Reynolds number parameters. For the case of large Rossby and Froude numbers, and for the case of quasi-geostrophic dynamics, a linear scaling law with 2/3 prefactor is derived for the third-order mixed correlation between potential vorticity and velocity, a result that is analogous to the Kolmogorov 4/5-law for the third-order velocity structure function in turbulence theory.

Opportunities for use of exact statistical equations

Exact structure function equations are an efficient means of obtaining asymptotic laws such as inertial range laws, as well as all measurable effects of inhomogeneity and anisotropy that cause deviations from such laws. “Exact” means that the equations are obtained from the Navier–Stokes equation or other hydrodynamic equations without any approximation. A pragmatic definition of local homogeneity lies within the exact equations because terms that explicitly depend on the rate of change of measurement location appear within the exact equations; an analogous statement is true for local stationarity. An exact definition of averaging operations is required for the exact equations. Careful derivations of several inertial range laws have appeared in the literature recently in the form of theorems. These theorems give the relationships of the energy dissipation rate to the structure function of acceleration increment multiplied by velocity increment and to both the trace of and the components of the third-order velocity structure functions. These laws are efficiently derived from the exact velocity structure function equations. In some respects, the results obtained herein differ from the previous theorems. The acceleration–velocity structure function is useful for obtaining the energy dissipation rate in particle tracking experiments provided that the effects of inhomogeneity are estimated by means of displacing the measurement location.

Reynolds dependence of third-order velocity structure functions

We study the experimental dependence of the third-order velocity structure function on the Taylor based Reynolds number, obtained in different flow types over the range 72⩽Rλ⩽2260. As expected, when the Reynolds number is increasing, the third-order velocity structure functions (plotted in a compensated way) converge very slowly to a possible −4/5 plateau value according to the Kolmogorov 41 theory. Actually, each of these normalized third-order functions exhibits a maximum, at a scale close to the Taylor microscale λ. In this Brief Communication, we show that experimental data are in good agreement with the recent predictions of Qian and Lundgren. We also suggest that, from an experimental point of view, a log-similar plot suits very well to study carefully the behavior of the third-order velocity structure functions with the flow Reynolds number.

Scaling Range Exponents From X-Wire Measurements In The Atmospheric Surface Layer

This paper focuses on the behaviour of moments, up to order 6, of longitudinal and vertical velocity increments, measured in the atmospheric surface layer, at a height of 1.7 m. The local derivatives of these moments with respect to the spatial separation between two points indicate that inertial range power-law exponents cannot be determined unambiguously. This is supported by the local slopes of moments of the locally averaged energy dissipation rate but contrasts with the extensive power-law ranges indicated by spectra of longitudinal and vertical velocity fluctuations. The third-order longitudinal velocity structure function provides stronger evidence of anisotropy in the inertial range than either the second-order velocity structure functions or the velocity spectra.

Next-order structure-function equations

Kolmogorov’s equation [Dokl. Akad. Nauk SSSR 32, 16 (1941)] relates the two-point second- and third-order velocity structure functions and the energy dissipation rate. The analogous next higher-order two-point equation relates the third- and fourth-order velocity structure functions and the structure function of the product of pressure-gradient difference and two factors of velocity difference, denoted Tijk. The equation is simplified on the basis of local isotropy. Laboratory and numerical simulation data are used to evaluate and compare terms in the equation, examine the balance of the equation, and evaluate components of Tijk. Atmospheric surface-layer data are used to evaluate Tijk in the inertial range. Combined with the random sweeping hypothesis, the equation relates components of the fourth-order velocity structure function. Data show the resultant error of this application of random sweeping. The next-order equation constrains the relationships that have been suggested among components of the fourth-order velocity structure function. The pressure structure function, pressure-gradient correlation, and mean-squared pressure gradient are related to Tijk. Inertial range formulas are discussed.