Universal aspects of small-scale motions in turbulence

2010 ◽  
Vol 662 ◽  
pp. 514-539 ◽  
Author(s):  
G. E. ELSINGA ◽  
I. MARUSIC
Keyword(s):  
Strain Rate ◽  
Shear Layer ◽  
Turbulent Flows ◽  
Local Coordinate System ◽  
Flow Patterns ◽  
Small Scale ◽  
Strain Rate Tensor ◽  
Average Flow ◽  
Third Invariant ◽  
Vorticity Vector

Two aspects of small-scale turbulence are currently regarded universal, as they have been reported for a wide variety of turbulent flows. Firstly, the vorticity vector has been found to display a preferential alignment with the eigenvector corresponding to the intermediate eigenvalue of the strain rate tensor; and secondly, the joint probability density function (p.d.f.) of the second and third invariant of the velocity gradient tensor, Q and R, has a characteristic teardrop shape. This paper provides an explanation for these universal aspects in terms of a spatial organization of coherent structures, which is based on an evaluation of the average flow pattern in the local coordinate system defined by the eigenvectors of the strain rate tensor. The approach contrasts with previous investigations, which have relied on assumed model flows. The present average flow patterns have been calculated for existing experimental (particle image velocimetry) or numerical (direct numerical simulation) datasets of a turbulent boundary layer (TBL), a turbulent channel flow and for homogeneous isotropic turbulence. All results show a shear-layer structure consisting of aligned vortical motions, separating two larger-scale regions of relatively uniform flow. Because the directions of maximum and minimum strain in a shear layer are in the plane normal to the vorticity vector, this vector aligns with the remaining strain direction, i.e. the intermediate eigenvector of the strain rate tensor. Further, the QR joint p.d.f. for these average flow patterns reveals a shape reminiscent of the teardrop, as seen in many turbulent flows. The above-mentioned organization of the small-scale motions is not only found in the average patterns, but is also frequently observed in the instantaneous velocity fields of the different turbulent flows. It may, therefore, be considered relevant and universal.

Vorticity, strain-rate and dissipation characteristics in the near-wall region of turbulent boundary layers

1997 ◽  
Vol 350 ◽  
pp. 29-96 ◽  
Author(s):  
ANANT HONKAN ◽  
YIANNIS ANDREOPOULOS
Keyword(s):  
Strain Rate ◽  
Joint Probability ◽  
Rate Of Change ◽  
Small Scale ◽  
Wall Region ◽  
Strain Rate Tensor ◽  
Viscous Effects ◽  
Vorticity Vector ◽  
Vorticity Flux ◽  
Near Wall

Experimental results are presented that reveal the structure of a two-dimensional turbulent boundary layer which has been investigated by measuring the time-dependent vorticity flux at the wall, vorticity vector, strain-rate tensor and dissipation-rate tensor in the near-wall region with spatial resolution of the order of 7 Kolmogorov viscous length scales. Considerations of the structure function of velocity and pressure, which constitute vorticity flux and vorticity, indicated that, in the limit of vanishing distance, the maximum attainable content of these quantities which corresponds to unrestricted resolution, is determined by Taylor's microscale. They also indicated that most of the contributions to vorticity or vorticity flux come from the uncorrelated part of the two signals involved. The measurements allowed the computation of all components of the vorticity stretching vector, which indicates the rate of change of vorticity on a Lagrangian reference frame if viscous effects are negligible, and several matrix invariants of the velocity gradient or strain-rate tensor and terms appearing in the transport equations of vorticity, strain rate and their squared fluctuations. The orientation of vorticity revealed several preferential directions. During bursts or sweeps vorticity is inclined at 35° to the longitudinal direction. It was also found that there is high probability of the vorticity vector aligning with the direction of the intermediate extensive strain corresponding to the middle eigenvector of the strain-rate matrix. The results of the joint probability distributions of the vorticity vector orientation angles showed that these angles may be related to those of hairpin vortex structures. All invariants considered exhibit a very strong intermittent behaviour which is characterized by large-amplitude bursts which may be of the order of 10 r.m.s. values. Small-scale motions dominated by high rates of turbulent kinetic energy dissipation and high enstrophy density are of particular interest. It appears that the fluctuating strain field dominates the fluctuations of pressure more than enstrophy. Local high values of the invariants are also often associated with peaks in the shear stress.

scholarly journals Measurements and Characterization of Turbulence in the Tip Region of an Axial Compressor Rotor

2017 ◽  
Vol 139 (12) ◽  
Author(s):  
Yuanchao Li ◽  
Huang Chen ◽  
Joseph Katz
Keyword(s):  
Strain Rate ◽  
Shear Layer ◽  
Turbulent Flows ◽  
Reynolds Stress ◽  
Suction Side ◽  
Stress And Strain ◽  
Spatial And Temporal Variability ◽  
Mean Flow ◽  
The Mean ◽  
Stress And Strain Rate

Modeling of turbulent flows in axial turbomachines is challenging due to the high spatial and temporal variability in the distribution of the strain rate components, especially in the tip region of rotor blades. High-resolution stereo-particle image velocimetry (SPIV) measurements performed in a refractive index-matched facility in a series of closely spaced planes provide a comprehensive database for determining all the terms in the Reynolds stress and strain rate tensors. Results are also used for calculating the turbulent kinetic energy (TKE) production rate and transport terms by mean flow and turbulence. They elucidate some but not all of the observed phenomena, such as the high anisotropy, high turbulence levels in the vicinity of the tip leakage vortex (TLV) center, and in the shear layer connecting it to the blade suction side (SS) tip corner. The applicability of popular Reynolds stress models based on eddy viscosity is also evaluated by calculating it from the ratio between stress and strain rate components. Results vary substantially, depending on which components are involved, ranging from very large positive to negative values. In some areas, e.g., in the tip gap and around the TLV, the local stresses and strain rates do not appear to be correlated at all. In terms of effect on the mean flow, for most of the tip region, the mean advection terms are much higher than the Reynolds stress spatial gradients, i.e., the flow dynamics is dominated by pressure-driven transport. However, they are of similar magnitude in the shear layer, where modeling would be particularly challenging.

Measurements and Characterization of Turbulence in the Tip Region of an Axial Compressor Rotor

2017 ◽  
Author(s):  
Yuanchao Li ◽  
Huang Chen ◽  
Joseph Katz
Keyword(s):  
Strain Rate ◽  
Shear Layer ◽  
Turbulent Flows ◽  
Reynolds Stress ◽  
Axial Compressor ◽  
Suction Side ◽  
Stress And Strain ◽  
Spatial And Temporal Variability ◽  
Mean Flow ◽  
The Mean

Modeling of turbulent flows in axial turbomachines is challenging due to the high spatial and temporal variability in the distribution of the strain rate components, especially in the tip region of rotor blades. High-resolution stereo PIV measurements performed in a refractive index matched facility in a series of closely-spaced planes provide a comprehensive database for determining all the terms in the Reynolds stress and strain rate tensors. Results are also used for calculating the turbulent kinetic energy production rate and transport terms by mean flow and turbulence. They elucidate some but not all of the observed phenomena, such as the high anisotropy, high turbulence levels in the vicinity of the tip leakage vortex (TLV) center, and in the shear layer connecting it to the blade suction side (SS) tip corner. The applicability of popular Reynolds stress models based on eddy-viscosity is also evaluated by calculating it from the ratio between stress and strain components. Results vary substantially, depending on which components are involved, ranging from very large positive to negative values. In some areas, e.g., in the tip gap and around the TLV, the local stresses and strains do not appear to be correlated at all. In terms of effect on the mean flow, for most of the tip region, the mean advection terms are much higher than the Reynolds stress spatial gradients, i.e., the flow dynamics is dominated by pressure-driven transport. However, they are of similar magnitude in the shear layer, where modeling would be particularly challenging.

scholarly journals Differential Sea-Ice Drift. I. Spatial and Temporal Variations in Sea-Ice Deformation

1974 ◽  
Vol 13 (69) ◽  
pp. 437-455 ◽  
Author(s):  
W. D. Hibler ◽  
W. F. Weeks ◽  
A. Kovacs ◽  
S. F. Ackley
Keyword(s):  
Sea Ice ◽  
Strain Rate ◽  
Low Frequency ◽  
Temporal Variations ◽  
Small Scale ◽  
Strain Rate Tensor ◽  
Pack Ice ◽  
Average Strain Rate ◽  
Average Residual ◽  
Sea Ice Drift

Measurements of mesoscale sea-ice deformation over a region approximately 20 km in diameter were carried out over a five-week period in the spring of 1972 at the main AIDJEX camp in the Beaufort Sea. They have been analyzed to determine non-linearities in the ice velocity field (due to the discrete small-scale nature of the ice pack), as well as a continuum mode of deformation represented by a least-squares strain-rate tensor and vorticity. The deformation-rate time series between Julian day 88 and 112 exhibited net areal changes as large as 3% and deformation rates up to 0.16% per hour. In the principal axis co-ordinate system, the strain-rate typically exhibited a much larger compression (or extension) along one axis than along the other. Persistent cycles at ≈ 12 h wavelengths were observed in the divergence rate.A comparison of the average residual error with the average strain-rate magnitude indicated that strains measured on a scale of 10 km or greater can serve as a valid measure of the continuum motion of the sea ice. This conclusion is also substantiated by a comparison between the mesoscale deformation, and macroscale deformation measured over a ≈ 100 km diameter region.Regarding pack-ice rotation, vorticity calculations indicate that at low temporal frequencies (0.02 h−1), the whole mesoscale array rotates essentially as an entity and consequently the low-frequency vorticity can be estimated accurately from the rotation of a single floe.

Invariants of the reduced velocity gradient tensor in turbulent flows

2013 ◽  
Vol 716 ◽  
pp. 597-615 ◽  
Author(s):  
J. I. Cardesa ◽  
D. Mistry ◽  
L. Gan ◽  
J. R. Dawson
Keyword(s):  
Strain Rate ◽  
Turbulent Flows ◽  
Velocity Gradient ◽  
Three Dimensional ◽  
Joint Probability ◽  
Strain Rate Tensor ◽  
Gradient Tensor ◽  
Local Isotropy ◽  
Velocity Gradient Tensor ◽  
Asymmetric Shape

AbstractIn this paper we examine the invariants $p$ and $q$ of the reduced $2\times 2$ velocity gradient tensor (VGT) formed from a two-dimensional (2D) slice of an incompressible three-dimensional (3D) flow. Using data from both 2D particle image velocimetry (PIV) measurements and 3D direct numerical simulations of various turbulent flows, we show that the joint probability density functions (p.d.f.s) of $p$ and $q$ exhibit a common characteristic asymmetric shape consistent with $\langle pq\rangle \lt 0$. An explanation for this inequality is proposed. Assuming local homogeneity we derive $\langle p\rangle = 0$ and $\langle q\rangle = 0$. With the addition of local isotropy the sign of $\langle pq\rangle $ is proved to be the same as that of the skewness of $\partial {u}_{1} / \partial {x}_{1} $, hence negative. This suggests that the observed asymmetry in the joint p.d.f.s of $p{{\ndash}}q$ stems from the universal predominance of vortex stretching at the smallest scales. Some advantages of this joint p.d.f. compared with that of $Q{{\ndash}}R$ obtained from the full $3\times 3$ VGT are discussed. Analysing the eigenvalues of the reduced strain-rate matrix associated with the reduced VGT, we prove that in some cases the 2D data can unambiguously discriminate between the bi-axial (sheet-forming) and axial (tube-forming) strain-rate configurations of the full $3\times 3$ strain-rate tensor.

scholarly journals Simultaneous Invariants of Strain and Rotation Rate Tensors and Their Admitted Region

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Igor Vigdorovich ◽  
Holger Foysi
Keyword(s):  
Strain Rate ◽  
Stress Tensor ◽  
Turbulent Flows ◽  
Reynolds Stress ◽  
Building Block ◽  
Rotation Rate ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Strain Rate Tensor ◽  
Reynolds Stress Tensor

The purpose of this paper is to establish the admitted region for five simultaneous, functionally independent invariants of the strain rate tensorSand rotation rate tensorΩand calculate some simultaneous invariants of these tensors which are encountered in the theory of constitutive relations for turbulent flows. Such a problem, as far as we know, has not yet been considered, though it is obviously an integral part of any problem in which scalar functions of the tensorsSandΩare studied. The theory provided inside this paper is the building block for a derivation of new algebraic constitutive relations for three-dimensional turbulent flows in the form of expansions of the Reynolds-stress tensor in a tensorial basis formed by the tensorsSandΩ, in which the scalar coefficients depend on simultaneous invariants of these tensors.

scholarly journals Assessment of local and non–local turbulent flow components on turbulence–flame interaction

2021 ◽  
Vol 2116 (1) ◽  
pp. 012015
Author(s):  
Aimad Er-Raiy ◽  
Radouan Boukharfane ◽  
Linda Alzaben ◽  
Matteo Parsani
Keyword(s):  
Strain Rate ◽  
Small Scale ◽  
Strain Rate Tensor ◽  
Coupled Transport ◽  
Additive Decomposition ◽  
Local Contribution ◽  
Non Local ◽  
Flow Components ◽  
Scale Turbulence

Abstract In the framework of turbulence-flame interaction, the flame is characterized by the gradient of a reactive scalar such as the progress variable, whereas the turbulence is represented by the vorticity and the strain rate. Quantitative assessment of this interaction is performed trough the study of the coupled transport between these quantities that are subject to the effects of heat release and chemical reactions. The present analysis aims at improving the understanding of the small scale turbulence – flame interaction properties, through the introduction of an additive decomposition of the strain rate and vorticity fields into their local and non-local components. The respective role of the local and non-local effects is studied for a broad range of Karlovitz numbers, by virtue of direct numerical simulations (DNS) of turbulent, premixed, lean, and statistically planar flames of methane-air. In the conditions of the present study, the alignment between flame front normals and the strain rate is found to be dominated by the local contribution from the strain rate tensor.

scholarly journals Small-scale dynamics of dense gas compressible homogeneous isotropic turbulence

2017 ◽  
Vol 825 ◽  
pp. 515-549 ◽  
Author(s):  
L. Sciacovelli ◽  
P. Cinnella ◽  
F. Grasso
Keyword(s):  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Equations Of State ◽  
Vapour Phase ◽  
Small Scale ◽  
Strain Rate Tensor ◽  
Dense Gas ◽  
Homogeneous Isotropic Turbulence ◽  
Third Invariant ◽  
Strong Expansion

The present paper investigates the influence of dense gases governed by complex equations of state on the dynamics of homogeneous isotropic turbulence. In particular, we investigate how differences due to the complex thermodynamic behaviour and transport properties affect the small-scale structures, viscous dissipation and enstrophy generation. To this end, we carry out direct numerical simulations of the compressible Navier–Stokes equations supplemented by advanced dense gas constitutive models. The dense gas considered in the study is a heavy fluorocarbon (PP11) that is shown to exhibit an inversion zone (i.e. a region where the fundamental derivative of gas dynamics $\unicode[STIX]{x1D6E4}$ is negative) in its vapour phase, for pressures and temperatures of the order of magnitude of the critical ones. Simulations are carried out at various initial turbulent Mach numbers and for two different initial thermodynamic states, one immediately outside and the other inside the inversion zone. After investigating the influence of dense gas effects on the time evolution of mean turbulence properties, we focus on the statistical properties of turbulent structures. For that purpose we carry out an analysis in the plane of the second and third invariant of the deviatoric strain-rate tensor. The analysis shows a weakening of compressive structures and an enhancement of expanding ones. Strong expansion regions are found to be mostly populated by non-focal convergence structures typical of strong compression regions, in contrast with the perfect gas that is dominated by eddy-like structures. Additionally, the contribution of non-focal expanding structures to the dilatational dissipation is comparable to that of compressed structures. This is due to the occurrence of steep expansion fronts and possibly of expansion shocklets which contribute to enstrophy generation in strong expansion regions and that counterbalance enstrophy destruction by means of the eddy-like structures.

scholarly journals Differential Sea-Ice Drift. I. Spatial and Temporal Variations in Sea-Ice Deformation

1974 ◽  
Vol 13 (69) ◽  
pp. 437-455 ◽  
Author(s):  
W. D. Hibler ◽  
W. F. Weeks ◽  
A. Kovacs ◽  
S. F. Ackley
Keyword(s):  
Sea Ice ◽  
Strain Rate ◽  
Low Frequency ◽  
Temporal Variations ◽  
Small Scale ◽  
Strain Rate Tensor ◽  
Pack Ice ◽  
Average Strain Rate ◽  
Average Residual ◽  
Sea Ice Drift

Measurements of mesoscale sea-ice deformation over a region approximately 20 km in diameter were carried out over a five-week period in the spring of 1972 at the main AIDJEX camp in the Beaufort Sea. They have been analyzed to determine non-linearities in the ice velocity field (due to the discrete small-scale nature of the ice pack), as well as a continuum mode of deformation represented by a least-squares strain-rate tensor and vorticity. The deformation-rate time series between Julian day 88 and 112 exhibited net areal changes as large as 3% and deformation rates up to 0.16% per hour. In the principal axis co-ordinate system, the strain-rate typically exhibited a much larger compression (or extension) along one axis than along the other. Persistent cycles at ≈ 12 h wavelengths were observed in the divergence rate.A comparison of the average residual error with the average strain-rate magnitude indicated that strains measured on a scale of 10 km or greater can serve as a valid measure of the continuum motion of the sea ice. This conclusion is also substantiated by a comparison between the mesoscale deformation, and macroscale deformation measured over a ≈ 100 km diameter region.Regarding pack-ice rotation, vorticity calculations indicate that at low temporal frequencies (0.02 h−1), the whole mesoscale array rotates essentially as an entity and consequently the low-frequency vorticity can be estimated accurately from the rotation of a single floe.

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