The two-dimensional oscillations of straight vortex filaments with a power-law distribution of vorticity

1982 ◽  
Vol 384 (1786) ◽  
pp. 1-29 ◽  
Keyword(s):  
Asymptotic Analysis ◽  
Cross Section ◽  
Power Law ◽  
Circular Cross Section ◽  
Azimuthal Velocity ◽  
Two Dimensional ◽  
Power Law Distribution ◽  
Vortex Filaments ◽  
Single Root ◽  
Complex Roots

Straight vortex filaments are considered with a circular cross section, with zero vorticity outside a given radius a , and with a power-law distribution of vorticity within the radius a such that the azimuthal velocity varies like r β as a function of the radius r . It is shown that the vortices are stable to a two-dimensional disturbance if –1 < β < 1 and unstable if β > 1. Numerical values are given for the eigenvalue and compared with results obtained from asymptotic analysis in some limiting cases. The behaviour near β = 1 is discussed in some detail. Results are found for different values of the mode number m , but for each m and β it was only possible to find a single root or pair of conjugate complex roots.

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.

STATIC STATISTICAL APPROACH TO THE BTW SANDPILE MODEL

Fractals ◽  
2006 ◽  
Vol 14 (01) ◽  
pp. 55-61
Author(s):  
DAHUI WANG ◽  
WEITING CHEN ◽  
QIANG YUAN ◽  
ZENGRU DI
Keyword(s):  
Power Law ◽  
Stable Distribution ◽  
Particle Number ◽  
Statistical Approach ◽  
Negative Temperature ◽  
Two Dimensional ◽  
Power Law Distribution ◽  
Sandpile Model ◽  
Total Particle ◽  
The One

A static statistical approach to the Bak, Tang and Wiesenfeld (BTW) sandpile model is proposed. With this approach, the exact avalanche distribution of the one-dimensional BTW sandpile is given concisely. Furthermore, we investigate the two-dimensional BTW sandpile and obtain some interesting results. First, the total particle number of the two-dimensional BTW sandpile obeys some kind of stable distribution. With the increase of the sandpile scale, the stable distribution transits from Gamma to Normal distribution. Second, when the total number of particles is fixed, the avalanche distribution is not power law. The system, however, shows a kind of "negative temperature" phenomenon when the particle number increases. Third, power law distribution of the avalanche could be viewed as the result of the superposition of a series of weighted distributions which do not yield power law.

Unsteady Laminar Duct Flow With a Given Volume Flow Rate Variation

1999 ◽  
Vol 67 (2) ◽  
pp. 274-281 ◽  
Author(s):  
D. Das ◽  
J. H. Arakeri
Keyword(s):  
Cross Section ◽  
Flow Rate ◽  
Volume Flow Rate ◽  
Circular Cross Section ◽  
Volume Flow ◽  
Rate Variation ◽  
Two Dimensional ◽  
Duct Flow ◽  
Fully Developed Flow ◽  
Gradient Variation

In this paper we give a procedure to obtain analytical solutions for unsteady laminar flow in an infinitely long pipe with circular cross section, and in an infinitely long two-dimensional channel, created by an arbitrary but given volume flow rate with time. In the literature, solutions have been reported when the pressure gradient variation with time is prescribed but not when the volume flow rate variation is. We present some examples: (a) the flow rate has a trapezoidal variation with time, (b) impulsively started flow, (c) fully developed flow in a pipe is impulsively blocked, and (d) starting from rest the volume flow rate oscillates sinusoidally. [S0021-8936(00)01702-5]

On the Viscous Flow Through a Long Pipe With Non-Constant Pressure Boundary Condition

2018 ◽  
Vol 73 (7) ◽  
pp. 639-644 ◽  
Author(s):  
Eduard Marušić-Paloka ◽  
Igor Pažanin
Keyword(s):  
Boundary Condition ◽  
Asymptotic Analysis ◽  
Small Parameter ◽  
Cross Section ◽  
Constant Pressure ◽  
Viscous Incompressible Fluid ◽  
Fluid Velocity ◽  
Circular Cross Section ◽  
Pressure Boundary Condition ◽  
Flow Through

AbstractWe investigate the flow of a viscous incompressible fluid through a straight long pipe with a circular cross section. The flow is driven by the prescribed pressures at the pipe’s ends, where pressure p0 on the pipe’s entry is assumed to be non-constant. Using asymptotic analysis with respect to the small parameter (being the ratio between the pipe’s radius and its length), we replace the non-constant pressure boundary condition with the effective one governing the macroscopic flow. We also derive the optimal boundary pressure p0 such that the fluid velocity through a pipe is maximal.

On the Critical Plunger Speed and Three-Dimensional Effects in High-Pressure Die Casting Injection Chambers

2003 ◽  
Vol 125 (3) ◽  
pp. 529-537 ◽  
Author(s):  
J. Lo´pez ◽  
F. Faura ◽  
J. Herna´ndez ◽  
P. Go´mez
Keyword(s):  
Molten Metal ◽  
Cross Section ◽  
Die Casting ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Water Model ◽  
Two Dimensional ◽  
Shallow Water Model ◽  
One Dimensional ◽  
Pressure Die Casting

During the initial slow stage of the injection process in high-pressure die casting machines with horizontal cold chamber, a plunger pushes the molten metal which partially fills the injection chamber, causing the formation of a gravity wave. The evolution of the wave surface profile, which depends on the plunger acceleration law, may trap air in the molten metal, causing porosity when the metal solidifies. In this work, a one-dimensional shallow-water model, which is solved numerically using the method of characteristics, and a three-dimensional numerical model, based on a finite element formulation and the volume of fluid (VOF) method for treating the free surface, are used to analyze the flow of molten metal in an injection chamber of circular cross section. The results for the evolution of the free surface obtained from both models for different plunger motion laws and initial filling fractions of the injection chamber were in good agreement for broad ranges of operating conditions. The existence of a critical plunger speed, above which the reflection of the wave of molten metal against the chamber ceiling might appreciably increase air entrapment effects, is investigated. The results for the wave profiles in chambers of circular cross section are compared with those obtained in an equivalent two-dimensional configuration of the injection chamber, for which the shallow-water model is solved analytically. It is shown how the results obtained by applying the one-dimensional model to a two-dimensional chamber configuration can be used to reproduce, with an acceptable degree of accuracy, the salient characteristics of the flow of molten metal in a real injection chamber of circular cross section.

scholarly journals Comparisons of Local Scouring for Submerged Square and Circular Cross-Section Piles in Steady Currents

Water ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1820
Author(s):  
Shengtao Du ◽  
Bingchen Liang
Keyword(s):  
Cross Section ◽  
Boundary Point ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Scour Depth ◽  
Two Dimensional ◽  
Exponential Equation ◽  
Equilibrium Scour Depth ◽  
Over Time

The local scouring that occurs around submerged vertical piles in steady currents was studied experimentally in this paper. Three experiments were carried out for square cross-section (SC) piles and a circular cross-section (CC) pile with the same width. The key point scour depths, including the center of the upstream boundary point (KC) and the two upstream corners (KM), were observed over time. The two-dimensional profiles and the three-dimensional topography around each pile were measured using a Seatek. The different scouring characteristics of the SC and CC piles were investigated. The experiment results show that the scour depth at KC is much smaller than that of KM. The equilibrium scour depth of the CC pile is far less than that of the SC piles. The scour and deposition distributions were different between the CC and SC piles. The maximum scour depth was found at the lateral rear of the CC pile, and the maximum deposition was observed in sections of the SC piles. The evolutions of the scour depths at KM are predicted using a developed exponential equation.

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