scholarly journals Three-dimensional stability of an elliptical vortex in a straining field

1984 ◽  
Vol 142 ◽  
pp. 451-466 ◽  
Author(s):  
A. C. Robinson ◽  
P. G. Saffman
Keyword(s):  
Steady State ◽  
Linear Stability ◽  
Cross Section ◽  
Dimensional Stability ◽  
Growth Rates ◽  
Finite Strain ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Elliptical Cross ◽  
Dimensional Instability

The three-dimensional linear stability of a rectilinear vortex of elliptical cross-section existing as a steady state in an irrotational straining field is studied numerically in the case of finite strain. It is shown that the instability predicted analytically for weak strain persists for finite strain and that the weak-strain results continue to be quantitatively valid for finite strain. The dependence of the growth rates of the unstable modes on the strain and the axial-disturbance wavelength is discussed. It is also shown that a three-dimensional instability is always more unstable than a two-dimensional instability in the range of parameters of most interest.

scholarly journals Computational Study of Zigzag Spacer Design with Elliptical Cross-Section Filaments

2020 ◽  
Vol 307 ◽  
pp. 01047
Author(s):  
Gohar Shoukat ◽  
Farhan Ellahi ◽  
Muhammad Sajid ◽  
Emad Uddin
Keyword(s):  
Mass Transfer ◽  
Cross Section ◽  
Cross Sections ◽  
Computational Study ◽  
Flow Direction ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Two Dimensional ◽  
Permeate Flux ◽  
Elliptical Cross

The large energy consumption of membrane desalination process has encouraged researchers to explore different spacer designs using Computational Fluid Dynamics (CFD) for maximizing permeate per unit of energy consumed. In previous studies of zigzag spacer designs, the filaments are modeled as circular cross sections in a two-dimensional geometry under the assumption that the flow is oriented normal to the filaments. In this work, we consider the 45° orientation of the flow towards the three-dimensional zigzag spacer unit, which projects the circular cross section of the filament as elliptical in a simplified two-dimensional domain. OpenFOAM was used to simulate the mass transfer enhancement in a reverse-osmosis desalination unit employing spiral wound membranes lined with zigzag spacer filaments. Properties that impact the concentration polarization and hence permeate flux were analyzed in the domain with elliptical filaments as well as a domain with circular filaments to draw suitable comparisons. The range of variation in characteristic parameters across the domain between the two different configurations is determined. It was concluded that ignoring the elliptical projection of circular filaments to the flow direction, can introduce significant margin of error in the estimation of mass transfer coefficient.

A Method to Approximate the Steady-State Creep Response of Three-Dimensional Pipe Bend Finite Element Models Under Internal Pressure Loading Using Two-Dimensional Axisymmetric Models

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
J. P. Rouse ◽  
W. Sun ◽  
T. H. Hyde ◽  
A. Morris ◽  
W. Montgomery
Keyword(s):  
Finite Element ◽  
Steady State ◽  
Cross Section ◽  
Three Dimensional ◽  
Pressure Vessels ◽  
Steady State Creep ◽  
3D Analysis ◽  
Two Dimensional ◽  
Rupture Stress ◽  
Pipe Bend

Pipe bends are regions of geometric discontinuities in the pipe systems used in power plants and most industry recorded failures have been located around similar regions. Understanding these potential locations of weakness is therefore of great interest for the safe and economic operation of piping components. Increased predictive accuracy would assist in component design, condition monitoring, and retirement strategy decisions. Modeling of piping components for finite element analysis (FEA) is complicated by the variation of the cross section dimensions (changes in wall thicknesses or cross section ovality) around the pipe bend due to the manufacturing procedure implemented. Quantities such as peak rupture stress and creep rupture life can be greatly affected by these geometric variation (Rouse, J. P., Leom, M. Z., Sun, W., Hyde, T. H., Morris, A., “Steady-state Creep Peak Rupture Stresses in 90 Pipe Bends with Manufacture Induced Cross Section Dimension Variations”International Journal of Pressure Vessels and Piping, Volumes 105–106, May–June 2013, pp. 1–11). Three dimensional (3D) models can be used to approximate to the realistic level of detail found in pipe bends. These simulations may however be computationally expensive and could take a considerable amount of time to complete. Two dimensional (2D) axisymmetric models are relatively straight forward to produce and quick to run, but of course cannot represent the full geometric complexity around the pipe bend. A method is proposed that utilises multiple 2D axisymmetric pipe bend models to approximate the result of a 3D analysis through interpolation, thus exploiting the greatly reduced computing time observed for the 2D models. The prediction of peak rupture stress (both magnitude and location) is assessed using a simple power law material model. Comments are made on the applicability of the proposed procedure to a range of bends angles (90 deg, 60 deg, and 30 deg), as well as the effect of the stress exponent (n) and tri-axial (α) material constants. Provided that peak stresses do not occur at the bend/straight interface, the magnitude and location of the peak rupture stress can be predicted by the 2D axisymmetric interpolation method with a typical percentage difference of less than 1%.

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.

Two-dimensional bubbles in slow viscous flows

1968 ◽  
Vol 33 (3) ◽  
pp. 475-493 ◽  
Author(s):  
S. Richardson
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Cross Section ◽  
Stokes Flow ◽  
Function Theory ◽  
Three Dimensional ◽  
Viscous Flows ◽  
Bubble Surface ◽  
Two Dimensional ◽  
Elliptical Cross

The representation of a biharmonic function in terms of analytic functions is used to transform a problem of two-dimensional Stokes flow into a boundary-value problem in analytic function theory. The relevant conditions to be satisfied at a free surface, where there is a given surface tension, are derived.A method for dealing with the difficulties of such a free surface is demonstrated by obtaining solutions for a two-dimensional, in viscid bubble in (a) a shear flow, and (b) a pure straining motion. In both cases the bubble is found to have an elliptical cross-section.The solutions obtained can be shown to be unique only if certain restrictive assumptions are made, and if these are relaxed the same methods may give further solutions. Experiments on three-dimensional inviscid bubbles (Rumscheidt & Mason 1961; Taylor 1934) demonstrate that angular points appear in the bubble surface, and an analysis is presented to show that such a discontinuity in a two-dimensional free surface is necessarily a genuine cusp and the nature of the flow about such a point is examined.

Microstructure Evolution of Epitaxial (Ba, Sr) TiO3/ (001) HgO Thin Films

1994 ◽  
Vol 361 ◽  
Author(s):  
V.A. Alyoshin ◽  
E.V. Sviridov ◽  
V.I.M. Hukhortov ◽  
I.H. Zakharchenko ◽  
V.P. Dudkevich
Keyword(s):  
Thin Films ◽  
Cross Section ◽  
Microstructure Evolution ◽  
Three Dimensional ◽  
Gas Pressure ◽  
Smooth Surfaces ◽  
Two Dimensional ◽  
Surface Versus ◽  
Deposition Conditions

ABSTRACTSurface and cross-section relief evolution of ferroelectric epitaxial (Ba,Sr)TiO3 films rf-sputtered on (001) HgO crystal cle-avage surface versus the oxygen worKing gas pressure P and subst-rate temperature T were studied. Specific features of both three-dimensional and two-dimensional epitaxy mechanisms corresponding to various deposition conditions were revealed. Difference between low and high P-T-value 3D epitaxy was established. The deposition of films with mirror-smooth surfaces and perfect interfaces is shown to be possible.

Plane turbulent jets in a bounded fluid layer

1992 ◽  
Vol 241 ◽  
pp. 587-614 ◽  
Author(s):  
T. Dracos ◽  
M. Giger ◽  
G. H. Jirka
Keyword(s):  
Experimental Investigation ◽  
Cross Section ◽  
Fluid Layer ◽  
Three Dimensional ◽  
Turbulent Jets ◽  
Two Dimensional ◽  
Vortical Structures ◽  
Secondary Currents ◽  
Fluid Layers ◽  
Bounding Surfaces

An experimental investigation of plane turbulent jets in bounded fluid layers is presented. The development of the jet is regular up to a distance from the orifice of approximately twice the depth of the fluid layer. From there on to a distance of about ten times the depth, the flow is dominated by secondary currents. The velocity distribution over a cross-section of the jet becomes three-dimensional and the jet undergoes a constriction in the midplane and a widening near the bounding surfaces. Beyond a distance of approximately ten times the depth of the bounded fluid layer the secondary currents disappear and the jet starts to meander around its centreplane. Large vortical structures develop with axes perpendicular to the bounding surfaces of the fluid layer. With increasing distance the size of these structures increases by pairing. These features of the jet are associated with the development of quasi two-dimensional turbulence. It is shown that the secondary currents and the meandering do not significantly affect the spreading of the jet. The quasi-two-dimensional turbulence, however, developing in the meandering jet, significantly influences the mixing of entrained fluid.

scholarly journals Derivation of Equations for a Size Distribution of Spherical Particles in Non-Transparent Materials

2021 ◽  
Vol 5 (4) ◽  
pp. 53-60
Author(s):  
Daniel Gurgul ◽  
Andriy Burbelko ◽  
Tomasz Wiktor
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Size Distribution ◽  
Density Function ◽  
Cross Section ◽  
Three Dimensional ◽  
Spherical Particles ◽  
Two Dimensional ◽  
Mathematical Formulas ◽  
Transparent Materials

This paper presents a new proposition on how to derive mathematical formulas that describe an unknown Probability Density Function (PDF3) of the spherical radii (r3) of particles randomly placed in non-transparent materials. We have presented two attempts here, both of which are based on data collected from a random planar cross-section passed through space containing three-dimensional nodules. The first attempt uses a Probability Density Function (PDF2) the form of which is experimentally obtained on the basis of a set containing two-dimensional radii (r2). These radii are produced by an intersection of the space by a random plane. In turn, the second solution also uses an experimentally obtained Probability Density Function (PDF1). But the form of PDF1 has been created on the basis of a set containing chord lengths collected from a cross-section.The most important finding presented in this paper is the conclusion that if the PDF1 has proportional scopes, the PDF3 must have a constant value in these scopes. This fact allows stating that there are no nodules in the sample space that have particular radii belonging to the proportional ranges the PDF1.

Equivalent Circuits of the Elastic Field

1944 ◽  
Vol 11 (3) ◽  
pp. A149-A161
Author(s):  
Gabriel Kron
Keyword(s):  
Steady State ◽  
Reference Frames ◽  
Three Dimensional ◽  
Elastic Field ◽  
Companion Paper ◽  
Theory Of Elasticity ◽  
Equivalent Circuits ◽  
Two Dimensional ◽  
Arbitrary Boundary ◽  
Nonlinear Stress

Abstract This paper presents equivalent circuits representing the partial differential equations of the theory of elasticity for bodies of arbitrary shapes. Transient, steady-state, or sinusoidally oscillating elastic-field phenomena may now be studied, within any desired degree of accuracy, either by a “network analyzer,” or by numerical- and analytical-circuit methods. Such problems are the propagation of elastic waves, determination of the natural frequencies of vibration of elastic bodies, or of stresses and strains in steady-stressed states. The elastic body may be non-homogeneous, may have arbitrary shape and arbitrary boundary conditions, it may rotate at a uniform angular velocity and may, for representation, be divided into blocks of uneven length in different directions. The circuits are developed to handle both two- and three-dimensional phenomena. They are expressed in all types of orthogonal curvilinear reference frames in order to simplify the boundary relations and to allow the solution of three-dimensional problems with axial and other symmetry by the use of only a two-dimensional network. Detailed circuits are given for the important cases of axial symmetry, cylindrical co-ordinates (two-dimensional) and rectangular co-ordinates (two- and three-dimensional). Nonlinear stress-strain relations in the plastic range may be handled by a step-by-step variation of the circuit constants. Nonisotropic bodies and nonorthogonal reference frames, however, require an extension of the circuits given. The circuits for steady-state stress and small oscillation phenomena require only inductances and capacitors, while the circuits for transients require also standard (not ideal) transformers. A companion paper deals in detail with numerical and experimental methods to solve the equivalent circuits.

scholarly journals Membrane analogy for multi-material bars under torsion

2019 ◽  
Vol 475 (2225) ◽  
pp. 20190124 ◽  
Author(s):  
Laura Galuppi ◽  
Gianni Royer-Carfagni
Keyword(s):  
Finite Element ◽  
Cross Section ◽  
Cross Sections ◽  
Three Dimensional ◽  
Element Model ◽  
Elastic Problem ◽  
Two Dimensional ◽  
Torsion Problem ◽  
Linear Elastic ◽  
Physical Apparatus

Prandtl's membrane analogy for the torsion problem of prismatic homogeneous bars is extended to multi-material cross sections. The linear elastic problem is governed by the same equations describing the deformation of an inflated membrane, differently tensioned in regions that correspond to the domains hosting different materials in the bar cross section, in a way proportional to the inverse of the material shear modulus. Multi-connected cross sections correspond to materials with vanishing stiffness inside the holes, implying infinite tension in the corresponding portions of the membrane. To define the interface constrains that allow to apply such a state of prestress to the membrane, a physical apparatus is proposed, which can be numerically modelled with a two-dimensional mesh implementable in commercial finite-element model codes. This approach presents noteworthy advantages with respect to the three-dimensional modelling of the twisted bar.

scholarly journals On inertial-range scaling laws

1996 ◽  
Vol 306 ◽  
pp. 167-181 ◽  
Author(s):  
John C. Bowman
Keyword(s):  
Steady State ◽  
High Resolution ◽  
Scaling Laws ◽  
State Energy ◽  
Three Dimensional ◽  
Inertial Range ◽  
Energy Spectra ◽  
Two Dimensional ◽  
Unified Framework ◽  
Asymptotic Nature

Inertial-range scaling laws for two- and three-dimensional turbulence are re-examined within a unified framework. A new correction to Kolmogorov's k−5/3 scaling is derived for the energy inertial range. A related modification is found to Kraichnan's logarithmically corrected two-dimensional enstrophy-range law that removes its unexpected divergence at the injection wavenumber. The significance of these corrections is illustrated with steady-state energy spectra from recent high-resolution closure computations. Implications for conventional numerical simulations are discussed. These results underscore the asymptotic nature of inertial-range scaling laws.

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