Singularity formation in three-dimensional motion of a vortex sheet

1995 ◽  
Vol 300 ◽  
pp. 339-366 ◽  
Author(s):  
Takashi Ishihara ◽  
Yukio Kaneda
Keyword(s):  
Asymptotic Analysis ◽  
Numerical Integration ◽  
Finite Time ◽  
Fourier Coefficients ◽  
Evolution Equations ◽  
Three Dimensional ◽  
Vortex Sheet ◽  
Three Dimensions ◽  
Leading Order ◽  
Singularity Formation

The evolution of a small but finite three-dimensional disturbance on a flat uniform vortex sheet is analysed on the basis of a Lagrangian representation of the motion. The sheet at time t is expanded in a double periodic Fourier series: R(λ1, λ2, t) = (λ1, λ2, 0) + Σn,mAn,m exp[i(nλ1 + δmλ2)], where λ1 and λ2 are Lagrangian parameters in the streamwise and spanwise directions, respectively, and δ is the aspect ratio of the periodic domain of the disturbance. By generalizing Moore's analysis for two-dimensional motion to three dimensions, we derive evolution equations for the Fourier coefficients An,m. The behaviour of An,m is investigated by both numerical integration of a set of truncated equations and a leading-order asymptotic analysis valid at large t. Both the numerical integration and the asymptotic analysis show that a singularity appears at a finite time tc = O(lnε−1) where ε is the amplitude of the initial disturbance. The singularity is such that An,0 = O(tc−1) behaves like n−5/2, while An,±1 = O(εtc) behaves like n−3/2 for large n. The evolution of A0,m(spanwise mode) is also studied by an asymptotic analysis valid at large t. The analysis shows that a singularity appears at a finite time t = O(ε−1) and the singularity is characterized by A0,2k ∝ k−5/2 for large k.

Propeller Wake Sheet Roll-up Modeling in Three Dimensions

1997 ◽  
Vol 41 (01) ◽  
pp. 81-92
Author(s):  
Sangwoo Pyo ◽  
Spyros A. Kinnas
Keyword(s):  
Three Dimensional ◽  
Vortex Sheet ◽  
Experimental Information ◽  
Higher Order ◽  
Three Dimensions ◽  
Tip Vortex ◽  
Propeller Wake ◽  
Wake Interaction ◽  
Radial Circulation ◽  
Wake Sheet

An algorithm for predicting the complete three-dimensional vortex sheet roll-up is developed. A higher order panel method, which combines a hyperboloidal panel geometry with a bi-quadratic dipole distribution, is used in order to accurately model the highly rolled-up regions. For given radial circulation distributions, the predicted wake shapes are shown to be convergent and consistent with those predicted from other methods. Then, a previously developed flow-adapted grid and the three-dimensional wake sheet roll-up algorithm are combined in order to estimate the propeller loading/trailing wake interaction. The complete wake geometry is determined by the method without the need of any experimental information on the shape of the wake. Predicted forces and tip vortex trajectories are shown to agree well with those measured in experiments.

On singularity formation in three-dimensional vortex sheet evolution

1999 ◽  
Vol 11 (11) ◽  
pp. 3198-3200 ◽  
Author(s):  
M. Brady ◽  
D. I. Pullin
Keyword(s):  
Three Dimensional ◽  
Vortex Sheet ◽  
Singularity Formation

Linearized oscillations of a vortex column: the singular eigenfunctions

2014 ◽  
Vol 741 ◽  
pp. 404-460 ◽  
Author(s):  
Anubhab Roy ◽  
Ganesh Subramanian
Keyword(s):  
Critical Radius ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Vortex Sheet ◽  
Initial Distribution ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Rankine Vortex ◽  
Parallel Flows ◽  
Lord Kelvin

AbstractIn 1880 Lord Kelvin analysed the linearized inviscid oscillations of a Rankine vortex as part of a theory of vortex atoms. These eponymously named neutrally stable modes are, however, exceptional regular oscillations that make up the discrete spectrum of the Rankine vortex. In this paper, we examine the singular oscillations that make up the continuous spectrum (CS) and span the entire base state range of frequencies. In two dimensions, the CS eigenfunctions have a twin-vortex-sheet structure similar to that known from earlier investigations of parallel flows with piecewise linear velocity profiles. The vortex sheets are cylindrical, being threaded by axial lines, with one sheet at the edge of the core and the other at the critical radius in the irrotational exterior; the latter refers to the radial location at which the fluid co-rotates with the eigenmode. In three dimensions, the CS eigenfunctions have core vorticity and may be classified into two families based on the singularity at the critical radius. For the first family, the singularity is a cylindrical vortex sheet threaded by helical vortex lines, while for the second family it has a localized dipole structure with radial vorticity. The presence of perturbation vorticity in the otherwise irrotational exterior implies that the CS modes, unlike the Kelvin modes, offer a modal interpretation for the (linearized) interaction of the Rankine vortex with an external vortical disturbance. It is shown that an arbitrary initial distribution of perturbation vorticity, both in two and three dimensions, may be evolved as a superposition over the discrete and CS modes; this modal representation being equivalent to a solution of the corresponding initial value problem. For the restricted case of an initial axial vorticity distribution in two dimensions, the modal representation may be generalized to a smooth vortex. Finally, for the three-dimensional case, the analogy between rotational flows and stratified shear flows, and the known analytical solution for stratified Couette flow, are used to clarify the singular manner in which the modal superposition for a smooth vortex approaches the Rankine limit.

Computational Methods With Vortices—The 1988 Freeman Scholar Lecture

1989 ◽  
Vol 111 (1) ◽  
pp. 5-52 ◽  
Author(s):  
Turgut Sarpkaya
Keyword(s):  
Computational Methods ◽  
Evolution Equations ◽  
Operator Splitting ◽  
Three Dimensional ◽  
Separated Flow ◽  
Body Representation ◽  
Vortex Sheet ◽  
Personal View ◽  
Discrete Vortex ◽  
Practical Applications

A comprehensive review is presented of the computational methods based upon Helmholtz’s powerful concepts of vortex dynamics, making use of Lagrangian or mixed Lagrangian-Eulerian schemes, the Biot-Savart law or the Vortex-in-Cell methods. The ingenious approximations and smoothing schemes developed in search of predictive models, qualitative solutions, new insights, or just some inspiration in the simulation of often two-dimensional, occasionally three-dimensional, and almost always incompressible fluids are described in detail. One is forewarned at the onset that chaos awaits at the end of the road. The challenge is to produce results in the face of ever accumulating errors within a time scale appropriate for the investigation. The review is organized around two major sections: Theoretical foundations and practical applications of vortex methods. The first covers topics such as vorticity and laws of transportation, evolution equations for a vortex sheet, real vortices and instabilities, Biot-Savart law, smoothing techniques (cutoff schemes, amalgamation of vortices, subvortex methods), cloud-in-cell or vortex-in-cell methods, body representation (Routh’s rule, surface singularity distributions), operator splitting and the random walk method (description and convergence), and asymmetry introduction. The next section covers contra flowing streams, vortical flows in aerodynamics (vortex sheet roll-up; slender-body, two-vortex, multi-discrete vortex, and segment or panel methods; three-dimensional flow models, and vortex-lattice methods), separated flow about cylindrical bodies (circular cylinder, sharp-edged bodies, arbitrarily-shaped bodies), general three-dimensional flows (vortex rings, turbulent spots, temporally and spatially-growing shear layers, and other applications (vortex-blade interactions, combustion phenomena, acoustics, contour dynamics, interaction of line vortices, chaos, and turbulence). The review is concluded with a brief comparison of these methods with others used in computational fluid dynamics and a personal view of their future prospects.

scholarly journals CANCELLATION OF UV DIVERGENCES IN THE ${\mathcal N} = 4$ SUSY NONLINEAR SIGMA MODEL IN THREE DIMENSIONS

2001 ◽  
Vol 16 (25) ◽  
pp. 1643-1650 ◽  
Author(s):  
TAKEO INAMI ◽  
YORINORI SAITO ◽  
MASAYOSHI YAMAMOTO
Keyword(s):  
Cotangent Bundle ◽  
Sigma Model ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Target Space ◽  
Quantum Corrections ◽  
Nonlinear Sigma Model ◽  
Leading Order ◽  
Β Function

We study the uv properties of the three-dimensional [Formula: see text] SUSY nonlinear sigma model whose target space is T*(CPN-1) (the cotangent bundle of CPN-1) to higher orders in the 1/N expansion. We calculate the β-function to next-to-leading order and verify that it has no quantum corrections at leading and next-to-leading orders.

scholarly journals On contact-line dynamics with mass transfer

2015 ◽  
Vol 26 (5) ◽  
pp. 671-719 ◽  
Author(s):  
J. M. OLIVER ◽  
J. P. WHITELEY ◽  
M. A. SAXTON ◽  
D. VELLA ◽  
V. S. ZUBKOV ◽  
...  
Keyword(s):  
Mass Transfer ◽  
Surface Tension ◽  
Asymptotic Analysis ◽  
Contact Line ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Leading Order ◽  
Vapour Transport ◽  
Gravity Surface ◽  
Matched Asymptotic Analysis

We investigate the effect of mass transfer on the evolution of a thin, two-dimensional, partially wetting drop. While the effects of viscous dissipation, capillarity, slip and uniform mass transfer are taken into account, other effects, such as gravity, surface tension gradients, vapour transport and heat transport, are neglected in favour of mathematical tractability. Our focus is on a matched-asymptotic analysis in the small-slip limit, which reveals that the leading-order outer formulation and contact-line law depend delicately on both the sign and the size of the mass transfer flux. This leads, in particular, to novel generalisations of Tanner's law. We analyse the resulting evolution of the drop on the timescale of mass transfer and validate the leading-order predictions by comparison with preliminary numerical simulations. Finally, we outline the generalisation of the leading-order formulations to prescribed non-uniform rates of mass transfer and to three dimensions.

scholarly journals Heat transport in an anharmonic crystal

2018 ◽  
Vol 32 (11) ◽  
pp. 1850126
Author(s):  
Shiladitya Acharya ◽  
Krishnendu Mukherjee
Keyword(s):  
Thermal Conductivity ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Leading Order ◽  
Slab Geometry ◽  
Anharmonic Crystal ◽  
One Dimensional ◽  
Environment Temperature ◽  
Anharmonic Coupling ◽  
Temperature Surface

We study transport of heat in an ordered, anharmonic crystal in the form of slab geometry in three dimensions. Apart from attaching baths of Langevin type to two extreme surfaces, we also attach baths of same type to the intermediate surfaces of the slab. Since the crystal is uninsulated, it exchanges energy with the intermediate heat baths. We find that both Fourier’s law of heat conduction and the Newton’s law of cooling hold to leading order in anharmonic coupling. The leading behavior of the temperature profile is exponentially falling from high to low temperature surface of the slab. As the anharmonicity increases, profiles fall more below the harmonic one in the log plot. In the thermodynamic limit thermal conductivity remains independent of the environment temperature and its leading order anharmonic contribution is linearly proportional to the temperature change between the two extreme surfaces of the slab. A fast crossover from one-dimensional (1D) to three-dimensional (3D) behavior of the thermal conductivity is observed in the system.

The spontaneous appearance of a singularity in the shape of an evolving vortex sheet

1979 ◽  
Vol 365 (1720) ◽  
pp. 105-119 ◽  
Keyword(s):  
Evolution Equation ◽  
Asymptotic Analysis ◽  
Fourier Coefficient ◽  
Small Amplitude ◽  
Fourier Coefficients ◽  
Critical Time ◽  
Vortex Sheet ◽  
Initial Amplitude ◽  
Vortex Layer ◽  
Approximate Equations

The evolution of a small amplitude initial disturbance to a straight uniform vortex sheet is described by the Fourier coefficients of the disturbance. An approximation to the exact evolution equation for these coefficients is proposed and it is shown, by an asymptotic analysis valid at large times, that the solution of the approximate equations develops a singularity at a critical time. The critical time is proportional to In (ε -1 ), where ε is the initial amplitude of the disturbance and the singularity itself is such that the n th. Fourier coefficient decays like n -2.5 instead of exponentially. Evidence - not conclusive, however - is present to show that the approximation used is adequate. It is concluded that the class of vortex layer motions correctly modelled by replacing the vortex layer by a vortex sheet is very restricted; the vortex sheet is an inadequate approximation unless it is everywhere undergoing rapid stretching.

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