Analytical theory of homogeneous mean shear turbulence

2013 ◽  
Vol 727 ◽  
pp. 256-281 ◽  
Author(s):  
Jerome Weinstock
Keyword(s):  
Shear Flow ◽  
Exact Expression ◽  
Turbulence Energy ◽  
Velocity Spectrum ◽  
Nonlinear Behaviour ◽  
Local Isotropy ◽  
Characteristic Energy ◽  
Nonlinear Growth ◽  
Homogeneous Mean ◽  
Dominant Characteristic

AbstractA compact nonlinear expression for the velocity spectra of homogeneous mean shear flow is derived by means of a simplified two-point closure. It applies to all scales and times. The derived equation can be viewed as a nonlinear extension of the linear, rapid-distortion-theory (RDT) equation. The principal simplification is to model the nonlinear pressure–strain rate as first-order in the spectral anisotropy: a spectral Rotta-equation. This simplified equation and its solution are expressed in terms of the RDT solution. That solution helps reveal the role of nonlinearity. An equation for the velocity spectrum is then obtained at all scales and times. A dominant characteristic predicted for nonlinear behaviour is that the turbulence energy grows exponentially with time, with the spectrum simultaneously moving to smaller and smaller wavenumbers. The nonlinear growth rate is determined. Other analytical predictions of the derived equation include: the conditions for self-similarity; local isotropy; various properties of mean shear flow, including characteristic energy, length and temporal growth scales; and a critique of perturbation theory. Comparisons are made with laboratory experiments and direct numerical simulations. Although the theory applies to all scales and times, including an exact expression for RDT, the calculations are focused on nonlinear behaviour at large times. Several approximations used in this work are examined.

scholarly journals Characteristic energy losses of electrons scattered from incandescent solids

1930 ◽  
Vol 127 (804) ◽  
pp. 111-140 ◽  
Keyword(s):  
High Vacuum ◽  
Adsorbed Layer ◽  
Resolving Power ◽  
Velocity Spectrum ◽  
Deep Minimum ◽  
Characteristic Energy ◽  
The Subject ◽  
Solid Conductors ◽  
Theoretical Considerations ◽  
Primary Electrons

The electron emission produced when solid conductors are bombarded with electrons of controlled speed has formed the subject of a great number of investigations. It is now generally recognized that this emission consists of three different parts: (1) Primary electrons, truly reflected without loss of energy; (2) electrons scattered back with reduced energy; and (3) secondary electrons proper, with very low velocities, which would seem to be produced from the atoms of the target by the same collision processes that give rise to the second group. In recent years considerable attention has been paid to the reflected electrons, the angular distribution of which reveals their wave character, if the target is a definitely orientated crystal of the substance in question. Some time ago I made some measurements on the velocity spectrum of the emission produced by electron bombardment, using a magnetic deflection apparatus of fairly high resolving power. The principal object of this investigation was to look for evidence of groups of electrons with character­istic velocities related to the soft X-ray levels of the substance. From certain theoretical considerations such electrons might be expected to be present in the emission. Targets of lithium, beryllium, boron, carbon and aluminium were tried, but in no case was there any evidence of electrons of the kind in question. These results are discussed in the paper mentioned. The distribu­tion curves obtained for different targets and bombarding voltages ranging from 40 to 900 volts were all similar in shape. The reflected electrons pro­duced a sharp and narrow peak, separated from the rest of the curve by a very deep minimum. The curve then rapidly rose to a maximum, corresponding to scattered electrons which had lost an energy equivalent to 25 volts in the collision. In addition to these, some experiments were made with targets of platinum and carbon, which could be kept at incandescence also when readings were taken. It was found that new maxima appear at high temperature, nearer to the reflected peak, and that the 25 volt maximum becomes very faint but reappears after some time on cooling. These changes were repeated several times. It was concluded that the 25 volt maximum was produced by an adsorbed layer formed on the cold target in the high vacuum, whilst the new maxima with hot targets should probably be regarded as characteristic of the target substance itself. Somewhat similar effects have been observed by Brown and Whiddington using a photographic method.

Compressible magnetoconvection in three dimensions: planforms and nonlinear behaviour

1995 ◽  
Vol 305 ◽  
pp. 281-305 ◽  
Author(s):  
P. C. Matthews ◽  
M. R. E. Proctor ◽  
N. O. Weiss
Keyword(s):  
Magnetic Field ◽  
Shear Flow ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Nonlinear Behaviour ◽  
Two Dimensional ◽  
Horizontal Shear ◽  
Mean Flow ◽  
The Magnetic Field

Convection in a compressible fiuid with an imposed vertical magnetic field is studied numerically in a three-dimensional Cartesian geometry with periodic lateral boundary conditions. Attention is restricted to the mildly nonlinear regime, with parameters chosen first so that convection at onset is steady, and then so that it is oscillatory.Steady convection occurs in the form of two-dimensional rolls when the magnetic field is weak. These rolls can become unstable to a mean horizontal shear flow, which in two dimensions leads to a pulsating wave in which the direction of the mean flow reverses. In three dimensions a new pattern is found in which the alignment of the rolls and the shear flow alternates.If the magnetic field is sufficiently strong, squares or hexagons are stable at the onset of convection. Both the squares and the hexagons have an asymmetrical topology, with upflow in plumes and downflow in sheets. For the squares this involves a resonance between rolls aligned with the box and rolls aligned digonally to the box. The preference for three-dimensional flow when the field is strong is a consequence of the compressibility of the layer- for Boussinesq magnetoconvection rolls are always preferred over squares at onset.In the regime where convection is oscillatory, the preferred planform for moderate fields is found to be alternating rolls - standing waves in both horizontal directions which are out of phase. For stronger fields, both alternating rolls and two-dimensional travelling rolls are stable. As the amplitude of convection is increased, either by dcereasing the magnetic field strength or by increasing the temperature contrast, the regular planform structure seen at onset is soon destroyed by secondary instabilities.

scholarly journals The kinematics of the reduced velocity gradient tensor in a fully developed turbulent free shear flow

2015 ◽  
Vol 767 ◽  
pp. 627-658 ◽  
Author(s):  
P. K. Rabey ◽  
A. Wynn ◽  
O. R. H. Buxton
Keyword(s):  
Shear Flow ◽  
Strain Rate ◽  
Velocity Gradient ◽  
Mixing Layer ◽  
Joint Probability ◽  
Strain Rate Tensor ◽  
Gradient Tensor ◽  
Local Isotropy ◽  
Velocity Gradient Tensor ◽  
Kinematic Behaviour

AbstractThis paper examines the kinematic behaviour of the reduced velocity gradient tensor (VGT),$\tilde{\unicode[STIX]{x1D608}}_{ij}$, which is defined as a$2\times 2$block, from a single interrogation plane, of the full VGT$\unicode[STIX]{x1D608}_{ij}=\partial u_{i}/\partial x_{j}$. Direct numerical simulation data from the fully developed turbulent region of a nominally two-dimensional mixing layer are used in order to examine the extent to which information on the full VGT can be derived from the reduced VGT. It is shown that the reduced VGT is able to reveal significantly more information about regions of the flow in which strain rate is dominant over rotation. It is thus possible to use the assumptions of homogeneity and isotropy to place bounds on the first two statistical moments (and their covariance) of the eigenvalues of the reduced strain-rate tensor (the symmetric part of the reduced VGT) which in turn relate to the turbulent strain rates. These bounds are shown to be dependent upon the kurtosis of$\partial u_{1}/\partial x_{1}$and another variable defined from the constituents of the reduced VGT. The kurtosis is observed to be minimised on the centreline of the mixing layer and thus tighter bounds are possible at the centre of the mixing layer than at the periphery. Nevertheless, these bounds are observed to hold for the entirety of the mixing layer, despite departures from local isotropy. The interrogation plane from which the reduced VGT is formed is observed not to affect the joint probability density functions (p.d.f.s) between the strain-rate eigenvalues and the reduced strain-rate eigenvalues despite the fact that this shear flow has a significant mean shear in the cross-stream direction. Further, it is found that the projection of the eigenframe of the strain-rate tensor onto the interrogation plane of the reduced VGT is also independent of the plane that is chosen, validating the approach of bounding the full VGT using the assumption of local isotropy.

scholarly journals Nonlinear saturation of Rayleigh-Taylor instability and generation of shear flow in equatorial spread-F plasma

2001 ◽  
Vol 8 (3) ◽  
pp. 181-190
Author(s):  
N. Chakrabarti ◽  
G. S. Lakhina
Keyword(s):  
Shear Flow ◽  
Mode Coupling ◽  
Nonlinear Behaviour ◽  
Coupling Equations ◽  
Spread F ◽  
Rayleigh Taylor Instability ◽  
Secondary Instabilities ◽  
Rt Instability ◽  
Collisional Regime

Abstract. An analysis of low order mode coupling equations is used to describe the nonlinear behaviour of the Rayleigh-Taylor (RT) instability in the equatorial ionosphere. The nonlinear evolution of RT instability leads to the development of shear flow. It is found that there is an interplay between the nonlinearity and the shear flow which compete with each other and saturate the RT mode, both in the collisionless and collisional regime. However, the nonlinearly saturated state, normally known as vortices or bubbles, may not be stable. Under certain condition these bubbles are shown to be unstable to short scale secondary instabilities that are driven by the large gradients which develop within these structures. Some understanding of the role of collisional nonlinearity in the  shear flow generations is also discussed.

Advances in the Fundamental Aspects of Turbulence: Energy Transfer, Interacting Scales, and Self-Preservation in Isotropic Decay

1998 ◽  
Vol 51 (4) ◽  
pp. 267-301 ◽  
Author(s):  
Ye Zhou ◽  
Charles G. Speziale
Keyword(s):  
Energy Transfer ◽  
Large Scale ◽  
Isotropic Turbulence ◽  
Future Research ◽  
Turbulence Energy ◽  
Physical Processes ◽  
Local Isotropy ◽  
Scale Modeling ◽  
Large Eddy ◽  
Subgrid Scale Modeling

The fundamental aspects of isotropic turbulence are reviewed in order to gain a better insight into the physical processes of turbulence. After first reviewing the Kolmogorov energy spectrum and the energy cascade, the Kolmogorov hypothesis of local isotropy is discussed in depth. Then, the detailed physical processes involving energy transfer and interacting scales in isotropic turbulence, including triad interactions, are reviewed. The inertial range and self-similarity are also discussed along with the response of the small scales to large-scale anisotropy and the final stages of the decay process. Results from direct and large-eddy simulations of isotropic turbulence—including a discussion of subgrid scale modeling—are then discussed in detail to illustrate these points. The article closes with a review of self-preservation in isotropic turbulence and a discussion of the prospects for future research. It contains 155 references.

Local Anisotropy in Strained Turbulence at High Reynolds Numbers

1991 ◽  
Vol 113 (4) ◽  
pp. 707-709 ◽  
Author(s):  
P. A. Durbin ◽  
C. G. Speziale
Keyword(s):  
Reynolds Number ◽  
Shear Flow ◽  
Reynolds Numbers ◽  
Local Anisotropy ◽  
Local Isotropy ◽  
Uniform Shear ◽  
Uniform Shear Flow ◽  
High Reynolds Numbers ◽  
High Reynolds ◽  
Mean Strain

It is shown that the hypothesis of local isotropy is implausible in the presence of significant mean rates of strain. In fact, it appears that in uniform shear flow near equilibrium, local isotropy can never constitute a systematic approximation, even in the limit of infinite Reynolds number. An estimate of the level of mean strain rate for which local isotropy is formally a good approximation is provided.

Local Isotropy in the Turbulent Wake of a Cylinder

1948 ◽  
Vol 1 (2) ◽  
pp. 161 ◽  
Author(s):  
AA Townnsend
Keyword(s):  
Shear Flow ◽  
Turbulent Flow ◽  
Turbulent Shear ◽  
Turbulent Shear Flow ◽  
General Applicability ◽  
Local Isotropy ◽  
Air Stream ◽  
Fully Developed Turbulent Flow ◽  
The Mean ◽  
High Reynolds

To investigate the validity of Kolmogoroff's theory of local isotropy, a series of turbulence measurements have been made in the wake of a circular cylinder, and the results have been compared with the predictions of the theory. Using a cylinder of 0.953 cm. diameter in an air-stream of velocity 1,280 cm. sec.-1, measurements have been made of the mean squares of the spatial derivatives in the mean-stream direction of the three components of the turbulent velocity fluctuation, and also of the skewness and flattening factors of the statistical distributions of these derivatives. Observations were taken at three traverses across the wake, respectively at 80, 120, and 160 cylinder diameters down-stream from the cylinder. Except in the immediate neighbourhood of the wake centre, the turbulent flow is observed to be intermittent, consisting of regions of fully developed turbulent flow separated by comparatively sharp boundaries from regions of almost completely laminar motion. If an "intermittency factor " is introduced to describe this phenomenon, and if inside each turbulent region local isotropy exists, then all the experimental results are consistent with the theory of local isotropy, and in agreement with previous measurements in flows possessing ordinary isotropy. It is concluded that, within the boundaries of the turbulent regions, local isotropy in the sense used by Kolmogoroff exists, and that the theory is applicable to this example of shear flow. The general applicability of the theory to turbulent shear flow at high Reynolds numbers must be considered very probable.

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