We survey attempts to construct vortex models of the inertial-range and fine-scale range of high Reynolds number turbulence. An emphasis is placed on models capable of quantitative predictions or postdictions.

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Vortex dynamics in turbulence

Fully developed rotating turbulent channel flow has been studied, through direct numerical simulations, for the complete range of rotation numbers for which the flow is turbulent. The present inves ...

Direct numerical simulations of rotating turbulent channel flow

The linear stability of Burgers and Lamb–Oseen vortices is addressed when the vortex of circulation Γ and radius δ is subjected to an additional strain field of rate s perpendicular to the vorticity axis. The resulting non-axisymmetric vortex is analysed in the limit of large Reynolds number RΓ=Γ/v and small strain s[Lt ]Γ/δ2 by considering the approximations obtained by Moffatt et al. (1994) and Jimenez et al. (1996) for each case respectively. For both vortices, the TWMS instability (Tsai & Widnall 1976; Moore & Saffman 1975) is shown to be active, i.e. stationary helical Kelvin waves of azimuthal wavenumbers m=1 and m=−1 resonate and are amplified by the external strain in the neighbourhood of critical axial wavenumbers which are computed. The additional effects of diffusion for the Lamb–Oseen vortex and stretching for the Burgers vortex are proved to limit in time the resonance. The transient growth of the helical waves is analysed in detail for the distinguished scaling s∼Γ/
 (δ2R1/2Γ). An amplitude equation describing the resonance is obtained and the maximum gain of the wave amplitudes is calculated. The effect of the vorticity profile on the instability characteristic as well as of a time-varying stretching rate are analysed. In particular the stretching rate maximizing the instability is calculated. The results are also discussed in the light of recent observations in experiments and numerical simulations. It is argued that the Kelvin waves resonance mechanism could explain various dynamical behaviours of vortex filaments in turbulence.

Three-dimensional instability of Burgers and Lamb–Oseen vortices in a strain field

We consider the dynamics of axial velocity and of scalar transport in the stretched-spiral vortex model of turbulent fine scales. A large-time asymptotic solution to the scalar advection-diffusion equation, with an azimuthal swirling velocity field provided by the stretched spiral vortex, is used together with appropriate stretching transformations to determine the evolution of both the axial velocity and a passive scalar. This allows calculation of the shell-integrated three-dimensional spectra of these quantities for the spiral-vortex flow. The dominant term in the velocity (energy) spectrum contributed by the axial velocity is found to be produced by the stirring of the initial distribution of axial velocity by the axisymmetric component of the azimuthal velocity. This gives a k(-7/3) spectrum at large wave numbers, compared to the k(-5/3) component for the azimuthal velocity itself. The spectrum of a passive scalar being mixed by the vortex velocity field is the sum of two power laws. The first is a k(-1) Batchelor spectrum for wave numbers up to the inverse Batchelor scale. This is produced by the axisymmetric component of the axial vorticity but is independent of the detailed radial velocity profile. The second is a k(-5/3) Obukov-Corrsin spectrum for wave numbers less than the inverse Kolmogorov scale. This is generated by the nonaxisymmetric axial vorticity and depends on initial correlations between this vorticity and the initial scalar field. The one-dimensional scalar spectrum for the composite model is in satisfactory agreement with experimental measurement.

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Axial motion and scalar transport in stretched spiral vortices

A new classification method for structures in turbulent flow is proposed and applied to the analysis of homogeneous isotropic turbulence. The criteria for the classification of the structures into three groups, namely, the group of structures similar to the core region of the Burgers’ vortex tube in which vorticity is predominant, that of the structures similar to the curved sheet in the circumference of the tube core in which strain is predominant, and that of the flat sheets similar to the Burgers’ vortex layer in which vorticity and strain are comparably large, were considered. This method was developed based on the eigenvalue solutions of the λ2 method [Jeong and Hussain, J. Fluid Mech. 285, 69 (1995)] on the basis of the principal strain eigenvectors, which were reordered according to the degrees of alignment with the vorticity vector. Assessment of the proposed method was carried out in fully developed homogeneous isotropic turbulence and in the process of rolling up of the vortex layer in ABC flow....

A classification method for vortex sheet and tube structures in turbulent flows

the mechanism of wrap, tilt and stretch of vorticity lines around a strong thin straight vortex tube of circulation Γ starting with a vortex filament in a simple shear flow (U=SX2x^1, S being a shear rate) is investigated analytically. an asymptotic expression for the vorticity field is obtained at a large reynolds number Γ/ν » 1, ν being the kinematic viscosity of fluid, and during the initial time St « 1 of evolution as well as St « (Γ/ν)1/2. the vortex tube, which is inclined from the streamwise (X1) direction both in the vertical (X2) and spanwise (X3) directions, is tilted, stretched and diffused under the action of the uniform shear and viscosity. the simple shear vorticity is on the other hand, wrapped and stretched around the vortex tube by a swirling motion, induced by it to form double spiral vortex layers of high azimuthal vorticity of alternating sign. the magnitude of the azimuthal vorticity increases up to O((Γ/ν)1/3S) at distance r=O((Γ/ν)1/3 (νt)1/2) from the vortex tube. the spirals induce axial flows of the same spiral shape with alternate sign in adjacent spirals which in turn tilt the simple shear vorticity toward the axial direction. as a result, the vorticity lines wind helically around the vortex tube accompanied by conversion of vorticity of the simple shear to the axial direction. the axial vorticity increases in time as s2t, the direction of which is opposite to that of the vortex tube at r=O((Γ/ν)1/2 (νt)1/2) where the vorticity magnitude is strongest. in the near region r « (Γ/ν)1/3 (νt)1/2, on the other hand, a viscous cancellation takes place in tightly wrapped vorticity of alternate sign, which leads to the disappearance of the vorticity normal to the vortex tube. only the axial component of the simple shear vorticity is left there, which is stretched by the simple shear flow itself. as a consequence, the vortex tube inclined toward the direction of the simple shear vorticity (a cyclonic vortex) is intensified, while the one oriented in the opposite direction (an anticyclonic vortex) is weakened. the growth rate of vorticity due to this effect attains a maximum (or minimum) value of ±S2/33/2 when the vortex tube is oriented in the direction of X^1+X^2[minus-or-plus sign] X^3. the present asymptotic solutions are expected to be closely related to the flow structures around intense vortex tubes observed in various kinds of turbulence such as helical winding of vorticity lines around a vortex tube, the dominance of cyclonic vortex tubes, the appearance of opposite-signed vorticity around streamwise vortices and a zig-zag arrangement of streamwise vortices in homogeneous isotropic turbulence, homogeneous shear turbulence and near-wall turbulence.

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Wrap, tilt and stretch of vorticity lines around a strong thin straight vortex tube in a simple shear flow

The mechanism of wrap, tilt and stretch of vorticity lines around
 a strong thin straight vortex tube of circulation Γ starting with a
 vortex filament in a simple shear flow ( U = SX 2 Xˆ 1 , S being a shear rate) is investigated analytically. An asymptotic
 expression for the vorticity field is obtained at a large Reynolds number
 Γ/ ν [Gt ]1, ν being the kinematic viscosity of
 fluid, and during the initial time St [Lt ]1 of evolution as well as St [Lt ](Γ/ ν ) 1/2 . The vortex tube,
 which is inclined from the streamwise ( X 1 ) direction both
 in the vertical ( X 2 ) and spanwise ( X 3 )
 directions, is tilted, stretched and diffused under the action of the uniform shear
 and viscosity. The simple shear vorticity is on the other hand, wrapped and stretched around the vortex
 tube by a swirling motion, induced by it to form double spiral vortex layers of high
 azimuthal vorticity of alternating sign. The magnitude of the azimuthal vorticity
 increases up to O ((Γ/ ν ) 1/3 S ) at
 distance r = O ((Γ/ ν ) 1/3 ( νt ) 1/2 ) from the vortex tube. The spirals induce
 axial flows of the same spiral shape with alternate sign in adjacent spirals which
 in turn tilt the simple shear vorticity toward the axial direction. As a result, the vorticity lines wind helically around the vortex tube accompanied
 by conversion of vorticity of the simple shear to the axial direction. The axial vorticity increases in time as S 2 t ,
 the direction of which is opposite to that of the vortex tube at r = O ((Γ/ ν ) 1/2 ( νt ) 1/2 ) where the vorticity magnitude is strongest.
 In the near region r [Lt ](Γ/ ν ) 1/3 ( νt ) 1/2 , on the other hand, a viscous cancellation
 takes place in tightly wrapped vorticity of alternate sign, which leads to the disappearance of the vorticity normal to the vortex tube. Only the axial component of the
 simple shear vorticity is left there, which is stretched by the simple shear flow itself. As a consequence, the vortex tube inclined toward the direction of the simple
 shear vorticity (a cyclonic vortex) is intensified, while the one oriented
 in the opposite direction (an anticyclonic vortex) is weakened. The growth rate of vorticity due to this
 effect attains a maximum (or minimum) value of ± S 2 /3 3/2 when the vortex
 tube is oriented in the direction of Xˆ 1 + Xˆ 2 ∓
 Xˆ 3 . The present asymptotic solutions are expected
 to be closely related to the flow structures around intense vortex tubes
 observed in various kinds of turbulence such as helical winding of vorticity lines
 around a vortex tube, the dominance of cyclonic vortex tubes, the appearance of
 opposite-signed vorticity around streamwise vortices and a zig-zag arrangement of streamwise vortices in homogeneous isotropic turbulence,
 homogeneous shear turbulence and near-wall turbulence.

"Wrap, Tilt and Stretch of Vorticity Lines around a Strong Straight Vortex Tube in a Simple Shear Flow"

Stability characteristics of a rotating simple shear flow modulated periodically in the shear direction is investigated by direct numerical simulation The temporal evolution and the final profile of the mean flow are examined, starting with a random perturbation imposed on it under the assumption that the velocity and pressure fields are uniform in the mean flow direction It is found that the mean velocity profile changes due to the shear-Coriolis instability If there is a locally unstable region in a stable rotating simple shear, the mean velocity profile there is deformed into a linear one with nearly-zero absolute vorticity An analogy holds between rotating uniformly sheared turbulence and thermally convective turbulence that the region of nearly-zero absolute vorticity in the former corresponds to that of nearly-uniform temperature in the latter

Zero-absolute-vorticity state in a rotating turbulent shear flow

The mechanism of wrap, tilt and stretch of vorticity lines around a strong thin straight tubular vortex in a simple shear flow is investigated analytically. An asymptotic expression of the vorticity field around a vortex which starts with a filament is derived at large vortex Reynolds numbers and at short times of evolution. An oblique vortex, which inclines from the streamwise direction both to the vertical and the spanwise, wraps and stretches the simple shear vorticity to form double spiral vortex layers of high azimuthal vorticity of alternating sign in adjacent spirals. These spirals induce axial shear flows of the same spiral shape and sign which in turn tilt the simple shear vorticity toward the axial direction. Vorticity lines wind helically around the vortex, which results in conversion of the simple shear vorticity to the axial direction. The effects of system rotation on this interaction are also discussed.

Vorticity Dynamics around a Straight Vortex Tube in a Simple Shear Flow

Stability characteristics of a rotating simple shear flow modulated periodically in the shear direction is investigated by direct numerical simulation. The temporal evolution and the final profile of the mean flow are examined, starting with a random perturbation imposed on it under the assumption that the velocity and pressure fields are uniform in the mean flow direction. It is found that the mean velocity profile changes due to the shear-Coriolis instability. If there is a locally unstable region in a stable rotating simple shear, the mean velocity profile there is deformed into a linear one with nearly-zero absolute vorticity. An analogy holds between rotating uniformly sheared turbulence and thermally convective turbulence that the region of nearly-zero absolute vorticity in the former corresponds to that of nearly-uniform temperature in the latter.

Shigeo Kida

Papers

Wrap, tilt and stretch of vorticity lines around a strong thin straight vortex tube in a simple shear flow

"Wrap, Tilt and Stretch of Vorticity Lines around a Strong Straight Vortex Tube in a Simple Shear Flow"

Zero-absolute-vorticity state in a rotating turbulent shear flow

Vorticity Dynamics around a Straight Vortex Tube in a Simple Shear Flow

Zero-absolute-vorticity State in a Rotating Turbulent Shear Flow