Second-order Velocity Structure Function Research Articles

Finite Reynolds Number Effect on Small-Scale Statistics in Decaying Grid Turbulence

Since about 1997, the realisation that the finite Reynolds number (FRN) effect needs to be carefully taken into account when assessing the behaviour of small-scale statistics came to the fore. The FRN effect can be analysed either in the real domain or in the spectral domain via the scale-by-scale energy budget equation or the transport equation for the energy spectrum. This analysis indicates that the inertial range (IR) is established only when the Taylor microscale Reynolds number Reλ is infinitely large, thus raising doubts about published power-law exponents at finite values of Reλ, for either the second-order velocity structure function (δu)2¯ or the energy spectrum. Here, we focus on the transport equation of (δu)2¯ in decaying grid turbulence, which represents a close approximation to homogeneous isotropic turbulence. The effect on the small-scales of the large-scale forcing term associated with the streamwise advection decreases as Reλ increases and finally disappears when Reλ is sufficiently large. An approach based on the dual scaling of (δu)2¯, i.e., a scaling based on the Kolmogorov scales (when the separation r is small) and another based on the integral scales (when r is large), yields (δu)2¯∼r2/3 when Reλ is infinitely large. This approach also yields (δu)n¯∼rn/3 when Reλ is infinitely large. These results seem to be supported by the trend, as Reλ increases, of available experimental data. Overall, the results for decaying grid turbulence strongly suggest that a tendency towards the predictions of K41 cannot be dismissed at least at Reynolds numbers which are currently beyond the reach of experiments and direct numerical simulations.

Effect of Surface Roughness on Aerodynamic Loads of Bluff Body in Vicinity of Smoothed Moving Wall

This paper contributes to a new Lagrangian vortex method for the statistical control of turbulence in two-dimensional flow configurations around a rough circular cylinder in ground effect when considering higher subcritical Reynolds numbers, namely 3 × 104 ≤ Re ≤ 2 × 105. A smoothed moving wall (active control technique) is used to include the blockage effect in association with the variation in cylinder surface roughness (passive control technique), characterizing a hybrid approach. In contrast with the previous approaches of our research group, the rough cylinder surface is here geometrically constructed, and a new momentum source term is introduced and calculated for the investigated problem. The methodology is structured by coupling the random Discrete Vortex Method, the Lagrangian Dynamic Roughness Model, and the Large Eddy Simulation with turbulence closure using the truncated Second-Order Velocity Structure Function model. This methodological option has the advantage of dispensing with the use of both a refined near-wall mesh and wall functions. The disadvantage of costly processing is readily solved with Open Multi-Processing. The results reveal that intermediate and high roughness values are most efficient for Reynolds numbers on the orders of 105 and 104, respectively. In employing a moving wall, the transition from the large-gap to the intermediate-gap regime is satisfactorily characterized. For the conditions studied with the hybrid technique, it was concluded that the effect of roughness is preponderant and acts to anticipate the characteristics of a lower gap-to-diameter ratio regime, especially with regard to intermittency.

Adopting molecular tagging velocimetry for high-resolution measurements of oscillating grid turbulence

AbstractSingle-component molecular tagging velocimetry (1c-MTV) experiments are conducted in an oscillating grid turbulence facility to systematically clarify the trade-offs between maximizing measurement dynamic range and minimizing measurement uncertainty. The primary aim is to obtain reliable turbulence data with the maximum possible vector resolution ($$\varDelta {x}_{1}$$ Δ x 1 ). Four optical magnifications ($$M{_{0}}$$ M 0 =0.11, 0.22, 0.45, 1.11) with four interframe time delay ($$\varDelta {t}$$ Δ t ) values, for each optical magnification case, ranging from 3 to 10 ms, are investigated. The grid oscillates with a frequency (f) of 3.2 and 4.1 Hz, and a single fixed stroke length (S) of 55 mm. The measurement quality is quantified using three important turbulence descriptors — the second-order transverse velocity correlation function $$(\langle {g(r)}\rangle )$$ ( ⟨ g ( r ) ⟩ ) , the second-order transverse velocity structure function $$([\Delta {_{g}u}]^2)$$ ( [ Δ g u ] 2 ) , and the turbulence energy dissipation rate ($$\langle \epsilon \rangle$$ ⟨ ϵ ⟩ ). Other descriptors such as the longitudinal integral length scale (l) and the longitudinal Taylor microscale ($${\lambda }$$ λ ) are also used to compare with reference values reported in literature. The degree of agreement with the reference values from literature, and insensitivity of the descriptor estimates to different MTV design parameters are used to determine the most robust means of obtaining high-resolution turbulence data from 1c-MTV. The experimental design parameters are contextualized using the Kolmogorov length ($${\eta }$$ η ), time ($${\tau _{\eta }}$$ τ η ), and velocity ($${u_{\eta }}$$ u η ) scales that are based on the most reliable $$\langle {\epsilon }\rangle$$ ⟨ ϵ ⟩ estimates from the current experiments. The pixel-displacement corresponding to $${u_{\eta }}$$ u η $$({\varDelta {x}{_{2}}{_{\eta }}})$$ ( Δ x 2 η ) describes the interdependencies between $${M_{0}}$$ M 0 and $$\varDelta {t}$$ Δ t . The data set with $${M_{0}}=0.45$$ M 0 = 0.45 ($${\eta =}18{\varDelta {x}_{1}}$$ η = 18 Δ x 1 for f=3.2 Hz, and $${\eta =}13{\varDelta {x}_{1}}$$ η = 13 Δ x 1 for f=4.1 Hz) represents the most reliable and the highest resolution case. This conclusion was deduced by taking the performance of all the turbulence descriptors into account. As far as the interframe time delay is concerned, $$\varDelta {t} \ge$$ Δ t ≥ 4 ms ($${\varDelta {x}{_{2}}{_{\eta }}}\ge 0.5 \text {pixels}$$ Δ x 2 η ≥ 0.5 pixels ) works well for all $${{M}_{0}}$$ M 0 . To the authors’ knowledge, the present study documents the first instance of a series of 1c-MTV experiments conducted for a systematic clarification of the nature of balancing the available dynamic range and the measurement uncertainty. Additionally, since the study involves turbulent flow with negligible mean-flow, it serves as an adequate representation of 1c-MTV performance in a three-dimensional flow.

Characterization of physical properties of a coastal upwelling filament with evidence of enhanced submesoscale activity and transition from balanced to unbalanced motions in the Benguela upwelling region

Abstract. We combine high-resolution in situ data (acoustic Doppler current profiler (ADCP), Scanfish, and surface drifters) and remote sensing to investigate the physical characteristics of a major filament observed in the Benguela upwelling region. The 30–50 km wide and about 400 km long filament persisted for at least 40 d. Mixed-layer depths were less than 40 m in the filament and over 60 m outside of it. Observations of the Rossby number Ro from the various platforms provide the spatial distribution of Ro for different resolutions. Remote sensing focuses on geostrophic motions of the region related to the mesoscale eddies that drive the filament formation and thereby reveals |Ro|<0.1. Ship-based measurements in the surface mixed layer reveal 0.5<|Ro|<1, indicating the presence of unbalanced, ageostrophic motions. Time series of Ro from triplets of surface drifters trapped within the filament confirm these relatively large Ro values and show a high variability along the filament. A scale-dependent analysis of Ro, which relies on the second-order velocity structure function, was applied to the latter drifter group and to another drifter group released in the upwelling zone. The two releases explored the area nearly distinctly and simultaneously and reveal that at small scales (<15 km) Ro values are twice as large in the filament in comparison to its environment with Ro>1 for scales smaller than ∼500 m. This suggests that filaments are hotspots of ageostrophic dynamics, pointing to the presence of a forward energy cascade. The different dynamics indicated by our Ro analysis are confirmed by horizontal kinetic energy wavenumber spectra, which exhibit a power law k−α with α∼5/3 for wavelengths 2π/k smaller than a transition scale of 15 km, supporting significant submesoscale energy at scales smaller than the first baroclinic Rossby radius (Ro1∼30 km). The detected transition scale is smaller than those found in regions with less mesoscale eddy energy, consistent with previous studies. We found evidence for the processes which drive the energy transfer to turbulent scales. Positive Rossby numbers 𝒪(1) associated with cyclonic motion inhibit the occurrence of positive Ertel potential vorticity (EPV) and stabilize the water column. However, where the baroclinic component of EPV dominates, submesoscale instability analysis suggests that mostly gravitational instabilities occur and that symmetric instabilities may be important at the filament edges.

Characteristics of clustered particle relative velocity in homogeneous and isotropic turbulence

Particle collisions are mainly governed by the preferential concentration of inertia particles and the formation of fold caustics. By fold caustics, we mean that relative velocity does not go smoothly to zero when the particle separations decrease due to inertia. Despite the importance of the second-order relative velocity structure function, there has been relatively little experimental research on the formation of caustics due to the high accuracy requirements for the particle relative velocity measurements. In the dissipation range, an obvious departure between the second-order structure function of particles normalized by the square of the Kolmogorov velocity and the Kolmogorov turbulent scaling of r2 was observed for all four experimental conditions. In the inertial range, the second-order structure function normalized by the square of the Kolmogorov velocity was consistent with the scale form of r2/3 in all cases. The conditional second-order relative velocity structure function of particles in clusters differs from that of the arithmetically averaged particle statistics in both the dissipation range and the inertial subrange. In the dissipation range, caustics are present for all categories of particles. In the inertial range, the scaling of the second-order velocity structure function is almost identical for all categorized particle types.

Perturbative model for the second-order velocity structure function tensor in turbulent shear flows

A model for the second order velocity structure function tensor, based on Kolmogorov's theory and valid for the inertial range, is proposed, incorporating the effect of anisotropy as a linear perturbation to the standard isotropic form. Structure functions measured from direct numerical simulations of channel flow and from experimental measurements in turbulent boundary layers are compared with predicted behavior and reasonable agreement is found.

Velocity structure functions in multiphase turbulence: interpreting kinematics of Hα filaments in cool-core clusters

ABSTRACT The central regions of cool-core galaxy clusters harbour multiphase gas, with gas temperatures ranging from $10$ to $10^7\, \mathrm{K}$. Feedback from active galactic nuclei jets prevents the gas from undergoing a catastrophic cooling flow. However, the exact mechanism of this feedback energy input is unknown, mainly due to the lack of velocity measurements of the hot-phase gas. However, recent observations have measured the velocity structure functions (VSFs) of the cooler molecular (${\sim} 10\, \mathrm{K}$) and Hα filaments (${\sim} 10^4\, \mathrm{K}$) and used them to indirectly estimate the motions of the hot phase. In the first part of this study, we conduct high-resolution (3843–15363 resolution elements) simulations of homogeneous isotropic subsonic turbulence, without radiative cooling. We analyse the second-order velocity structure functions (VSF2) in these simulations and study the effects of varying spatial resolution, the introduction of magnetic fields, and the effect of projection along the line of sight (LOS) on it. In the second part of the study, we analyse high-resolution (7683 resolution elements) idealized simulations of multiphase turbulence in the intracluster medium from the companion study Mohapatra et al. We compare the VSF2 for both the hot ($T\sim 10^7\, \mathrm{K}$) and cold ($T\sim 10^4\, \mathrm{K}$) phases and find that their amplitude depends on the density contrast between the phases. They have similar scaling with separation, but introducing magnetic fields steepens the VSF2 of only the cold phase. We also find that projection along the LOS steepens the VSF2 for the hot phase and mostly flattens it for the cold phase.

The Finite Size Lyapunov Exponent and the Finite Amplitude Growth Rate

The finite size Lyapunov exponent (FSLE) has been used extensively since the late 1990s to diagnose turbulent regimes from Lagrangian experiments and to detect Lagrangian coherent structures in geophysical flows and two-dimensional turbulence. Historically, the FSLE was defined in terms of its computational method rather than via a mathematical formulation, and the behavior of the FSLE in the turbulent inertial ranges is based primarily on scaling arguments. Here, we propose an exact definition of the FSLE based on conditional averaging of the finite amplitude growth rate (FAGR) of the particle pair separation. With this new definition, we show that the FSLE is a close proxy for the inverse structural time, a concept introduced a decade before the FSLE. The (in)dependence of the FSLE on initial conditions is also discussed, as well as the links between the FAGR and other relevant Lagrangian metrics, such as the finite time Lyapunov exponent and the second-order velocity structure function.

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.

Anisotropic Statistics of Lagrangian Structure Functions and Helmholtz Decomposition

AbstractWe present a new method to estimate second-order horizontal velocity structure functions, as well as their Helmholtz decomposition into rotational and divergent components, from sparse data collected along Lagrangian observations. The novelty compared to existing methods is that we allow for anisotropic statistics in the velocity field and also in the collection of the Lagrangian data. Specifically, we assume only stationarity and spatial homogeneity of the data and that the cross covariance between the rotational and divergent flow components is either zero or a function of the separation distance only. No further assumptions are made and the anisotropy of the underlying flow components can be arbitrarily strong. We demonstrate our new method by testing it against synthetic data and applying it to the Lagrangian Submesoscale Experiment (LASER) dataset. We also identify an improved statistical angle-weighting technique that generally increases the accuracy of structure function estimations in the presence of anisotropy.

Eulerian and Lagrangian second-order statistics of superfluidHe4grid turbulence

We use particle tracking velocimetry to study Eulerian and Lagrangian second-order statistics of superfluid $^4$He grid turbulence. The Eulerian energy spectra at scales larger than the mean distance between quantum vortex lines behave classically with close to Kolmogorov-1941 scaling and are almost isotropic. The Lagrangian second-order structure functions and frequency power spectra, measured at scales comparable with the intervortex distance, demonstrate a sharp transition from nearly-classical behavior to a regime dominated by the motion of quantum vortex lines. Employing the homogeneity of the flow, we verify a set of relations that connect various second-order statistical objects that stress different aspects of turbulent behavior, allowing a multifaceted analysis. We use the two-way bridge relations between Eulerian energy spectra and second-order structure functions to reconstruct the energy spectrum from the known second-order velocity structure function and vice versa. The Lagrangian frequency spectrum reconstructed from the measured Eulerian spectrum using the Eulerian-Lagrangian bridge differs from the measured Lagrangian spectrum in the quasi-classical range which calls for further investigation.

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.

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.

Can We Detect Submesoscale Motions in Drifter Pair Dispersion?

AbstractA cluster of 45 drifters deployed in the Bay of Bengal is tracked for a period of four months. Pair dispersion statistics, from observed drifter trajectories and simulated trajectories based on surface geostrophic velocity, are analyzed as a function of drifter separation and time. Pair dispersion suggests nonlocal dynamics at submesoscales of 1–20 km, likely controlled by the energetic mesoscale eddies present during the observations. Second-order velocity structure functions and their Helmholtz decomposition, however, suggest local dispersion and divergent horizontal flow at scales below 20 km. This inconsistency cannot be explained by inertial oscillations alone, as has been reported in recent studies, and is likely related to other nondispersive processes that impact structure functions but do not enter pair dispersion statistics. At scales comparable to the deformation radius LD, which is approximately 60 km, we find dynamics in agreement with Richardson’s law and observe local dispersion in both pair dispersion statistics and second-order velocity structure functions.

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

Resolved and subgrid dynamics of Rayleigh–Bénard convection

In this work we present and demonstrate the reliability of a theoretical framework for the study of thermally driven turbulence. It consists of scale-by-scale budget equations for the second-order velocity and temperature structure functions and their limiting cases, represented by the turbulent kinetic energy and temperature variance budgets. This framework represents an extension of the classical Kolmogorov and Yaglom equations to inhomogeneous and anisotropic flows, and allows for a novel assessment of the turbulent processes occurring at different scales and locations in the fluid domain. Two relevant characteristic scales, $\ell _{c}^{u}$ for the velocity field and $\ell _{c}^{\unicode[STIX]{x1D703}}$ for the temperature field, are identified. These variables separate the space of scales into a quasi-homogeneous range, characterized by turbulent kinetic energy and temperature variance cascades towards dissipation, and an inhomogeneity-dominated range, where the production and the transport in physical space are important. This theoretical framework is then extended to the context of large-eddy simulation to quantify the effect of a low-pass filtering operation on both resolved and subgrid dynamics of turbulent Rayleigh–Bénard convection. It consists of single-point and scale-by-scale budget equations for the filtered velocity and temperature fields. To evaluate the effect of the filter length $\ell _{F}$ on the resolved and subgrid dynamics, the velocity and temperature fields obtained from a direct numerical simulation are split into filtered and residual components using a spectral cutoff filter. It is found that when $\ell _{F}$ is smaller than the minimum values of the cross-over scales given by $\ell _{c,min}^{\unicode[STIX]{x1D703}\ast }=\ell _{c,min}^{\unicode[STIX]{x1D703}}Nu/H=0.8$, the resolved processes correspond to the exact ones, except for a depletion of viscous and thermal dissipations, and the only role of the subgrid scales is to drain turbulent kinetic energy and temperature variance to dissipate them. On the other hand, the resolved dynamics is much poorer in the near-wall region and the effects of the subgrid scales are more complex for filter lengths of the order of $\ell _{F}\approx 3\ell _{c,min}^{\unicode[STIX]{x1D703}}$ or larger. This study suggests that classic eddy-viscosity/diffusivity models employed in large-eddy simulation may suffer from some limitations for large filter lengths, and that alternative closures should be considered to account for the inhomogeneous processes at subgrid level. Moreover, the theoretical framework based on the filtered Kolmogorov and Yaglom equations may represent a valuable tool for future assessments of the subgrid-scale models.

An empirical expression for on the axis of a slightly heated turbulent round jet

Self-preservation analyses of the equations for the mean temperature and the second-order temperature structure function on the axis of a slightly heated turbulent round jet are exploited in an attempt to develop an analytical expression for$\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$, the mean dissipation rate of$\overline{\unicode[STIX]{x1D703}^{2}}/2$, where$\overline{\unicode[STIX]{x1D703}^{2}}$is the temperature variance. The analytical approach follows that of Thiessetet al.(J. Fluid Mech., vol. 748, 2014, R2) who developed an expression for$\unicode[STIX]{x1D716}_{k}$, the mean turbulent kinetic energy dissipation rate, using the transport equation for$\overline{(\unicode[STIX]{x1D6FF}u)^{2}}$, the second-order velocity structure function. Experimental data show that complete self-preservation for all scales of motion is very well satisfied along the jet axis for streamwise distances larger than approximately 30 times the nozzle diameter. This validation of the analytical results is of particular interest as it provides justification and confidence in the analytical derivation of power laws representing the streamwise evolution of different physical quantities along the axis, such as:$\unicode[STIX]{x1D702}$,$\unicode[STIX]{x1D706}$,$\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$,$R_{U}$,$R_{\unicode[STIX]{x1D6E9}}$(all representing characteristic length scales), the mean temperature excess$\unicode[STIX]{x1D6E9}_{0}$, the mixed velocity–temperature moments$\overline{u\unicode[STIX]{x1D703}^{2}}$,$\overline{v\unicode[STIX]{x1D703}^{2}}$and$\overline{\unicode[STIX]{x1D703}^{2}}$and$\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$. Simple models are proposed for$\overline{u\unicode[STIX]{x1D703}^{2}}$and$\overline{v\unicode[STIX]{x1D703}^{2}}$in order to derive an analytical expression for$A_{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}$, the prefactor of the power law describing the streamwise evolution of$\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$. Further, expressions are also derived for the turbulent Péclet number and the thermal-to-mechanical time scale ratio. These expressions involve global parameters that are most likely to be influenced by the initial and/or boundary conditions and are therefore expected to be flow dependent.

Exact result for mixed triple two-point correlations of velocity and velocity gradients in isotropic turbulence

ABSTRACTOn the grounds of Kolmogorov's 4/5 law analytical relations for triple two-point correlations of velocity and velocity gradients in homogeneous isotropic incompressible turbulence are derived. The corresponding correlation tensor can be expressed in terms of dissipation ε, the second-order longitudinal velocity structure function and one more scalar function of distance between the points. However, some components of the tensor do not depend on . The derived analytical results are in agreement with the data obtained from direct numerical simulations. The function can be well approximated in the inertial range by a constant value that depends on the dissipation ε only. The application of the obtained correlators in the turbulent transport theory is discussed.