scholarly journals Comparison of three-dimensional and two-dimensional statistical mechanics of shear layers for flow between two parallel plates

2012 ◽  
Vol 25 (2-4) ◽  
pp. 269-272
Author(s):  
L. Shirkov ◽  
V. Berdichevsky
Keyword(s):  
Statistical Mechanics ◽  
Three Dimensional ◽  
Shear Layers ◽  
Two Dimensional ◽  
Parallel Plates

The anatomy of the mixing transition in homogeneous and stratified free shear layers

2000 ◽  
Vol 413 ◽  
pp. 1-47 ◽  
Author(s):  
C. P. CAULFIELD ◽  
W. R. PELTIER
Keyword(s):  
Shear Layer ◽  
Three Dimensional ◽  
Streamwise Vortex ◽  
Shear Layers ◽  
Finite Amplitude ◽  
Small Scale ◽  
Two Dimensional ◽  
Streamwise Vortices ◽  
Free Shear Layers ◽  
Nonlinear Amplification

We investigate the detailed nature of the ‘mixing transition’ through which turbulence may develop in both homogeneous and stratified free shear layers. Our focus is upon the fundamental role in transition, and in particular the associated ‘mixing’ (i.e. small-scale motions which lead to an irreversible increase in the total potential energy of the flow) that is played by streamwise vortex streaks, which develop once the primary and typically two-dimensional Kelvin–Helmholtz (KH) billow saturates at finite amplitude.Saturated KH billows are susceptible to a family of three-dimensional secondary instabilities. In homogeneous fluid, secondary stability analyses predict that the stream-wise vortex streaks originate through a ‘hyperbolic’ instability that is localized in the vorticity braids that develop between billow cores. In sufficiently strongly stratified fluid, the secondary instability mechanism is fundamentally different, and is associated with convective destabilization of the statically unstable sublayers that are created as the KH billows roll up.We test the validity of these theoretical predictions by performing a sequence of three-dimensional direct numerical simulations of shear layer evolution, with the flow Reynolds number (defined on the basis of shear layer half-depth and half the velocity difference) Re = 750, the Prandtl number of the fluid Pr = 1, and the minimum gradient Richardson number Ri(0) varying between 0 and 0.1. These simulations quantitatively verify the predictions of our stability analysis, both as to the spanwise wavelength and the spatial localization of the streamwise vortex streaks. We track the nonlinear amplification of these secondary coherent structures, and investigate the nature of the process which actually triggers mixing. Both in stratified and unstratified shear layers, the subsequent nonlinear amplification of the initially localized streamwise vortex streaks is driven by the vertical shear in the evolving mean flow. The two-dimensional flow associated with the primary KH billow plays an essentially catalytic role. Vortex stretching causes the streamwise vortices to extend beyond their initially localized regions, and leads eventually to a streamwise-aligned collision between the streamwise vortices that are initially associated with adjacent cores.It is through this collision of neighbouring streamwise vortex streaks that a final and violent finite-amplitude subcritical transition occurs in both stratified and unstratified shear layers, which drives the mixing process. In a stratified flow with appropriate initial characteristics, the irreversible small-scale mixing of the density which is triggered by this transition leads to the development of a third layer within the flow of relatively well-mixed fluid that is of an intermediate density, bounded by narrow regions of strong density gradient.

scholarly journals Nonlinear evolution of interacting oblique waves on two-dimensional shear layers

1989 ◽  
Vol 207 ◽  
pp. 97-120 ◽  
Author(s):  
M. E. Goldstein ◽  
S.-W. Choi
Keyword(s):  
Wave Amplitude ◽  
Numerical Solutions ◽  
Nonlinear Evolution ◽  
Three Dimensional ◽  
Shear Layers ◽  
Instability Wave ◽  
Two Dimensional ◽  
Instability Waves ◽  
Downstream Distance ◽  
Parallel Streams

We consider the effects of critical-layer nonlinearity on spatially growing oblique instability waves on nominally two-dimensional shear layers between parallel streams. The analysis shows that three-dimensional effects cause nonlinearity to occur at much smaller amplitudes than it does in two-dimensional flows. The nonlinear instability wave amplitude is determined by an integro-differential equation with cubic-type nonlinearity. The numerical solutions to this equation are worked out and discussed in some detail. We show that they always end in a singularity at a finite downstream distance.

The secondary flow and its stability for natural convection in a tall vertical enclosure

1989 ◽  
Vol 200 ◽  
pp. 189-216 ◽  
Author(s):  
Arnon Chait ◽  
Seppo A. Korpela
Keyword(s):  
Three Dimensional ◽  
Pseudospectral Method ◽  
Base Flow ◽  
Temperature Fields ◽  
Two Dimensional ◽  
Parallel Plates ◽  
Grashof Number ◽  
Practical Applications ◽  
Prandtl Numbers ◽  
Dependent Flow

The multicellular flow between two vertical parallel plates is numerically simulated using a time-splitting pseudospectral method. The steady flow of air, and the time-periodic flow of oil (Prandtl numbers of 0.71 and 1000, respectively) are investigated and descriptions of these flows using both physical and spectral approaches are presented. The details of the time dependency of the flow and temperature fields of oil are shown, and the dynamics of the process is discussed. The spectral transfer of energy among the axial modes comprising the flow is explored. The spectra of kinetic energy and thermal variance for air are found to be smooth and viscously dominated. Similar spectra for oil are bumpier, and the dynamics of the time-dependent flow are determined to be confined to the lower end of the spectrum alone.The three-dimensional linear stability of the multicellular flow of air is parametrically studied. The domain of stable two-dimensional cellular motion was found to be constrained by the Eckhaus instability and by two types of monotone instability. The two-dimensional multicellular flow is unstable above a Grashof number of about 8550 (with the critical Grashof number for the base flow being 8037). Therefore the flow of air in a sufficiently tall vertical enclosure should be considered to be three-dimensional for most practical applications.

scholarly journals Maximal heat transfer between two parallel plates

2018 ◽  
Vol 851 ◽  
Author(s):  
Shingo Motoki ◽  
Genta Kawahara ◽  
Masaki Shimizu
Keyword(s):  
Heat Transfer ◽  
Velocity Field ◽  
Large Scale ◽  
Three Dimensional ◽  
Temperature Fields ◽  
Two Dimensional ◽  
Parallel Plates ◽  
Solution Branch ◽  
Search Range ◽  
Dimensional Solution

The divergence-free time-independent velocity field has been determined so as to maximise heat transfer between two parallel plates with a constant temperature difference under the constraint of fixed total enstrophy. The present variational problem is the same as that first formulated by Hassanzadeh et al. (J. Fluid Mech., vol. 751, 2014, pp. 627–662); however, the search range for optimal states has been extended to a three-dimensional velocity field. A scaling of the Nusselt number $Nu$ with the Péclet number $Pe$ (i.e., the square root of the non-dimensionalised enstrophy with thermal diffusion time scale), $Nu\sim Pe^{2/3}$, has been found in the three-dimensional optimal states, corresponding to the asymptotic scaling with the Rayleigh number $Ra$, $Nu\sim Ra^{1/2}$, expected to appear in an ultimate state, and thus to the Taylor energy dissipation law in high-Reynolds-number turbulence. At $Pe\sim 10^{0}$, a two-dimensional array of large-scale convection rolls provides maximal heat transfer. A three-dimensional optimal solution emerges from bifurcation on the two-dimensional solution branch at $Pe\sim 10^{1}$, and the three-dimensional solution branch has been tracked up to $Pe\sim 10^{4}$ (corresponding to $Ra\approx 2.7\times 10^{6}$). At $Pe\gtrsim 10^{3}$, the optimised velocity fields consist of convection cells with hierarchical self-similar vortical structures, and the temperature fields exhibit a logarithmic-like mean profile near the walls.

Enhancement of three-dimensional instability of free shear layers

1999 ◽  
Vol 379 ◽  
pp. 23-38 ◽  
Author(s):  
VIVEK SAXENA ◽  
SIDNEY LEIBOVICH ◽  
GAL BERKOOZ
Keyword(s):  
Growth Rate ◽  
Growth Rates ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Maximum Growth ◽  
Shear Layers ◽  
Base Flow ◽  
Two Dimensional ◽  
Free Shear Layers ◽  
Dimensional Instability

Enhancement of the temporal growth rate of inviscid three-dimensional instability waves in free shear layers by deformation of the basic flow is studied. The deformation of a two-dimensional mixing layer is assumed to yield a base flow that remains unidirectional, but has a steady spanwise speed variation in addition to the two- dimensional shear. The computed growth rates for hyperbolic tangent base flow, perturbed this way, show enhanced instability in the sense that the neutral waves of the unperturbed flow exhibit positive growth rates. For each imposed spanwise periodicity, an oblique mode is selected that shows maximum growth rate. The results are consistent with related theoretical studies and with qualitative observations in experiments.

scholarly journals Damping of quasi-two-dimensional internal wave attractors by rigid-wall friction

2018 ◽  
Vol 841 ◽  
pp. 614-635 ◽  
Author(s):  
F. Beckebanze ◽  
C. Brouzet ◽  
I. N. Sibgatullin ◽  
L. R. M. Maas
Keyword(s):  
Viscous Dissipation ◽  
Gravity Waves ◽  
Three Dimensional ◽  
Rigid Wall ◽  
Internal Gravity Waves ◽  
Shear Layers ◽  
Wall Friction ◽  
Extended Model ◽  
Two Dimensional ◽  
Lateral Walls

The reflection of internal gravity waves at sloping boundaries leads to focusing or defocusing. In closed domains, focusing typically dominates and projects the wave energy onto ‘wave attractors’. For small-amplitude internal waves, the projection of energy onto higher wavenumbers by geometric focusing can be balanced by viscous dissipation at high wavenumbers. Contrary to what was previously suggested, viscous dissipation in interior shear layers may not be sufficient to explain the experiments on wave attractors in the classical quasi-two-dimensional trapezoidal laboratory set-ups. Applying standard boundary layer theory, we provide an elaborate description of the viscous dissipation in the interior shear layer, as well as at the rigid boundaries. Our analysis shows that even if the thin lateral Stokes boundary layers consist of no more than 1 % of the wall-to-wall distance, dissipation by lateral walls dominates at intermediate wave numbers. Our extended model for the spectrum of three-dimensional wave attractors in equilibrium closes the gap between observations and theory by Hazewinkel et al. (J. Fluid Mech., vol. 598, 2008, pp. 373–382).

On the Validity of Two-Dimensional Heat Transfer Simulation of Moving Droplets Between Parallel Plates

2011 ◽  
Author(s):  
V. Talimi ◽  
Y. S. Muzychka ◽  
S. Kocabiyik
Keyword(s):  
Heat Transfer ◽  
Three Dimensional ◽  
Computational Domain ◽  
Two Dimensional ◽  
Parallel Plates ◽  
Ansys Fluent ◽  
The Real ◽  
Dimensional Shape ◽  
Heat Transfer Simulation ◽  
Three Dimensional Shape

Use of moving droplets between two parallel plates has been investigated widely in recent years for cooling purposes. While the real shape of the droplets is a cylinder with curved side (convex or concave) i.e. a three dimensional shape, most of the researchers assumed a two-dimensional computational domain including vertical mid plane of the droplet, which is applicable for not realistic long droplets. In this paper, the differences between these two approaches are investigated numerically, using ANSYS Fluent package.

High Resolution Microscopy of Multi-Layered Biological Crystals

1982 ◽  
Vol 40 ◽  
pp. 80-83
Author(s):  
H.A. Cohen ◽  
T.W. Jeng ◽  
W. Chiu
Keyword(s):  
High Resolution ◽  
Low Dose ◽  
Three Dimensional ◽  
Data Interpretation ◽  
Two Dimensional ◽  
Biological Objects ◽  
Density Maps ◽  
Density Map ◽  
Complex Crystal ◽  
High Resolution Microscopy

This tutorial will discuss the methodology of low dose electron diffraction and imaging of crystalline biological objects, the problems of data interpretation for two-dimensional projected density maps of glucose embedded protein crystals, the factors to be considered in combining tilt data from three-dimensional crystals, and finally, the prospects of achieving a high resolution three-dimensional density map of a biological crystal. This methodology will be illustrated using two proteins under investigation in our laboratory, the T4 DNA helix destabilizing protein gp32*I and the crotoxin complex crystal.

Instrumentation For Quantitative Microscopy

1977 ◽  
Vol 35 ◽  
pp. 206-209
Author(s):  
B. Ralph ◽  
A.R. Jones
Keyword(s):  
Volume Fraction ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Automatic Data ◽  
Phase Volume Fraction ◽  
Statistical Confidence ◽  
Resultant Data ◽  
Automatic Data Collection ◽  
Third Dimension ◽  
Evolution Of Microstructure

In all fields of microscopy there is an increasing interest in the quantification of microstructure. This interest may stem from a desire to establish quality control parameters or may have a more fundamental requirement involving the derivation of parameters which partially or completely define the three dimensional nature of the microstructure. This latter categorey of study may arise from an interest in the evolution of microstructure or from a desire to generate detailed property/microstructure relationships. In the more fundamental studies some convolution of two-dimensional data into the third dimension (stereological analysis) will be necessary.In some cases the two-dimensional data may be acquired relatively easily without recourse to automatic data collection and further, it may prove possible to perform the data reduction and analysis relatively easily. In such cases the only recourse to machines may well be in establishing the statistical confidence of the resultant data. Such relatively straightforward studies tend to result from acquiring data on the whole assemblage of features making up the microstructure. In this field data mode, when parameters such as phase volume fraction, mean size etc. are sought, the main case for resorting to automation is in order to perform repetitive analyses since each analysis is relatively easily performed.

A linearity test for thick specimens imaged by HVEM over a 180° angular range

1991 ◽  
Vol 49 ◽  
pp. 552-553
Author(s):  
Yu Liu
Keyword(s):  
Phase Contrast ◽  
Three Dimensional ◽  
Angular Range ◽  
Two Dimensional ◽  
Back Projection ◽  
Line Integral ◽  
Exponential Law ◽  
Transmission Electron ◽  
Absorption Law ◽  
Linearity Test

The image obtained in a transmission electron microscope is the two-dimensional projection of a three-dimensional (3D) object. The 3D reconstruction of the object can be calculated from a series of projections by back-projection, but this algorithm assumes that the image is linearly related to a line integral of the object function. However, there are two kinds of contrast in electron microscopy, scattering and phase contrast, of which only the latter is linear with the optical density (OD) in the micrograph. Therefore the OD can be used as a measure of the projection only for thin specimens where phase contrast dominates the image. For thick specimens, where scattering contrast predominates, an exponential absorption law holds, and a logarithm of OD must be used. However, for large thicknesses, the simple exponential law might break down due to multiple and inelastic scattering.

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