Vortex sheet roll-up revisited

2018 ◽  
Vol 855 ◽  
pp. 299-321 ◽  
Author(s):  
A. C. DeVoria ◽  
K. Mohseni
Keyword(s):  
Classical Problem ◽  
Critical Time ◽  
Vortex Sheet ◽  
The Self ◽  
Physical Consideration ◽  
Considerable Difficulty ◽  
Circulation Distribution ◽  
Continuous Object ◽  
Induced Velocity ◽  
Vorticity Transport

The classical problem of roll-up of a two-dimensional free inviscid vortex sheet is reconsidered. The singular governing equation brings with it considerable difficulty in terms of actual calculation of the sheet dynamics. Here, the sheet is discretized into segments that maintain it as a continuous object with curvature. A model for the self-induced velocity of a finite segment is derived based on the physical consideration that the velocity remain bounded. This allows direct integration through the singularity of the Birkhoff–Rott equation. The self-induced velocity of the segments represents the explicit inclusion of stretching of the sheet and thus vorticity transport. The method is applied to two benchmark cases. The first is a finite vortex sheet with an elliptical circulation distribution. It is found that the self-induced velocity is most relevant in regions where the curvature and the sheet strength or its gradient are large. The second is the Kelvin–Helmholtz instability of an infinite vortex sheet. The critical time at which the sheet forms a singularity in curvature is accurately predicted. As observed by others, the vortex sheet strength forms a finite-valued cusp at this time. Here, it is shown that the cusp value rapidly increases after the critical time and is the impetus that initiates the roll-up process.

The self-induced motion of vortex sheets

1985 ◽  
Vol 150 ◽  
pp. 203-231 ◽  
Author(s):  
J. J. L. Higdon ◽  
C. Pozrikidis
Keyword(s):  
Vortex Sheet ◽  
The Self ◽  
Trigonometric Polynomials ◽  
Infinite Plane ◽  
Vortex Sheets ◽  
Periodic Disturbances ◽  
Circular Arcs ◽  
Special Cases ◽  
Induced Motion ◽  
Circulation Distribution

A method is presented for following the self-induced motion of vortex sheets. In this method, we use a piecewise analytic representation of the sheet consisting of circular arcs with trigonometric polynomials for the circulation. The procedure is used to study the evolution of the motion in two special cases: a circular vortex sheet with sinusoidal circulation distribution and an infinite plane vortex sheet subject to periodic disturbances. The first problem was studied by Baker (1980) as a test of the method of Fink & Soh (1978), while the second has been studied by a number of authors, notably Meiron, Baker & Orszag (1982). In each case, we follow the motion of the sheet up to the appearance of a singularity at a finite time. The singularity takes the form of an exponential spiral with the simultaneous development of singularities in the curvature and in the circulation distribution. In the final stages of the calculations, up to 155 marker points are used to specify the position of the sheet. If it were possible to execute a stable calculation with equally spaced point vortices, approximately 106 points would be required to achieve the same resolution. Problems with instabilities have been reduced, but not entirely eliminated, and prevent a rigorous verification of the results obtained.

scholarly journals Geometry and dynamics of vortex sheets in 3 dimension

2002 ◽  
pp. 55-76 ◽  
Author(s):  
D.A. Burton ◽  
R.W. Tucker
Keyword(s):  
Fluid Velocity ◽  
Normal Component ◽  
Vortex Sheet ◽  
The Self ◽  
Vortex Filament ◽  
Vortex Sheets ◽  
Filament Dynamics ◽  
De Rham ◽  
Induced Velocity

We consider the properties and dynamics of vortex sheets from a geometrical, coordinate-free, perspective. Distribution-valued forms (de Rham currents) are used to represent the fluid velocity and vorticity due to the vortex sheets. The smooth velocities on either side of the sheets are solved in terms of the sheet strengths using the language of double forms. The classical results regarding the continuity of the sheet normal component of the velocity and the conservation of vorticity are exposed in this setting. The formalism is then applied to the case of the self-induced velocity of an isolated vortex sheet. We develop a simplified expression for the sheet velocity in terms of representative curves. Its relevance to the classical Localized Induction Approximation (LIA) to vortex filament dynamics is discussed. .

A numerical investigation of the rolling-up of vortex sheets

1980 ◽  
Vol 373 (1752) ◽  
pp. 67-91 ◽  
Keyword(s):  
Numerical Integration ◽  
Numerical Investigation ◽  
Equations Of Motion ◽  
Additional Term ◽  
Vortex Sheet ◽  
Time Step ◽  
Vortex Sheets ◽  
Numerical Instabilities ◽  
Sinusoidal Disturbance ◽  
Induced Velocity

In this paper the development of a vortex sheet due to an initially sinusoidal disturbance is calculated. When determining the induced velocity in points of the vortex sheet, it can be represented by concentrated vortices but it is shown that it is analytically more correct to add an additional term that represents the effect of the immediate neighbourhood of the point considered. The equations of motion were integrated by a Runge-Kutta technique to exclude numerical instabilities. The time step was determined by the requirement that a quantity (Hamiltonian) that remains invariant as a result of the equations of motion, should not change more than a certain amount in the numerical integration of the equations of motion. One difficulty is that if a greater number of concentrated vortices are introduced to represent the vortex sheet, the effect of round-off errors becomes more important. The number of figures retained in the computations limits the number of concentrated vortices. Where the round-off errors have been kept sufficiently small, a process of rolling-up of vorticity clearly occurs. There is no point in pursuing the calculations much beyond this point, first because the representation of the vortex sheet by concentrated vortices becomes more and more inaccurate and secondly because viscosity will have the effect of transforming the rolled-up vortex sheet into a region of vorticity.

scholarly journals Instantaneous energy and enstrophy variations in Euler-alpha point vortices via triple collapse

2012 ◽  
Vol 702 ◽  
pp. 188-214 ◽  
Author(s):  
Takashi Sakajo
Keyword(s):  
Weak Solution ◽  
Euler Equations ◽  
Weak Solutions ◽  
Point Vortex ◽  
Critical Time ◽  
Point Vortices ◽  
The Self ◽  
Two Dimensional ◽  
Graphic System ◽  
Self Similar

AbstractIt has been pointed out that the anomalous enstrophy dissipation in non-smooth weak solutions of the two-dimensional Euler equations has a clue to the emergence of the inertial range in the energy density spectrum of two-dimensional turbulence corresponding to the enstrophy cascade as the viscosity coefficient tends to zero. However, it is uncertain how non-smooth weak solutions can dissipate the enstrophy. In the present paper, we construct a weak solution of the two-dimensional Euler equations from that of the Euler-$\ensuremath{\alpha} $ equations proposed by Holm, Marsden & Ratiu (Phys. Rev. Lett., vol. 80, 1998, pp. 4173–4176) by taking the limit of $\ensuremath{\alpha} \ensuremath{\rightarrow} 0$. To accomplish this task, we introduce the $\ensuremath{\alpha} $-point-vortex ($\ensuremath{\alpha} \mathrm{PV} $) system, whose evolution corresponds to a unique global weak solution of the two-dimensional Euler-$\ensuremath{\alpha} $ equations in the sense of distributions (Oliver & Shkoller, Commun. Part. Diff. Equ., vol. 26, 2001, pp. 295–314). Since the $\ensuremath{\alpha} \mathrm{PV} $ system is a formal regularization of the point-vortex system and it is known that, under a certain special condition, three point vortices collapse self-similarly in finite time (Kimura, J. Phys. Soc. Japan, vol. 56, 1987, pp. 2024–2030), we expect that the evolution of three $\ensuremath{\alpha} $-point vortices for the same condition converges to a singular weak solution of the Euler-$\ensuremath{\alpha} $ equations that is close to the triple collapse as $\ensuremath{\alpha} \ensuremath{\rightarrow} 0$, which is examined in the paper. As a result, we find that the three $\ensuremath{\alpha} $-point vortices collapse to a point and then expand to infinity self-similarly beyond the critical time in the limit. We also show that the Hamiltonian energy and a kinematic energy acquire a finite jump discontinuity at the critical time, but the energy dissipation rate converges to zero in the sense of distributions. On the other hand, an enstrophy variation converges to the $\delta $ measure with a negative mass, which indicates that the enstrophy dissipates in the distributional sense via the self-similar triple collapse. Moreover, even if the special condition is perturbed, we can confirm numerically the convergence to the singular self-similar evolution with the enstrophy dissipation. This indicates that the self-similar triple collapse is a robust mechanism of the anomalous enstrophy dissipation in the sense that it is observed for a certain range of the parameter region.

Self-Induced Velocity of a Vortex Ring Using Straight-Line Segmentation

2014 ◽  
Vol 59 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Mahendra J. Bhagwat ◽  
J. Gordon Leishman
Keyword(s):  
Vortex Ring ◽  
Logarithmic Singularity ◽  
The Self ◽  
Numerical Calculations ◽  
Core Radius ◽  
Line Vortex ◽  
The Core ◽  
Analytical Approximations ◽  
Straight Line ◽  
Induced Velocity

The accuracy of discretized induced velocity calculations that can be obtained using straight-line vortex elements has been reexamined, primarily using the velocity field induced by a vortex ring as a reference. The induced velocity of a potential (inviscid) vortex ring is singular at the vortex ring itself. Analytical results were found by using a small azimuthal cutoff in the Biot–Savart integral over the vortex ring and showed that the singularity is logarithmic in the cutoff. Discrete numerical calculations showed the same behavior, with the self-induced velocity exhibiting a logarithmic singularity with respect to the discretization, which introduces an inherent cutoff in the Biot–Savart integral. Core regularization or desingularization can also eliminate the singularity by using an assumed “viscous” core model. Analytical approximations to the self-induced velocity of a thin-cored vortex ring have shown that the self-induced velocity has a logarithmic singularity in the core radius. It is further shown that the numerical calculations require special treatment of the self-induced velocity caused by curvature, which is lost by the inherent cutoff in the straight-line discretization, to correctly recover this logarithmic singularity in the core radius. Numerical solution using straight-line vortex segmentation, augmented with curved vortex elements only for the self-induced velocity calculation, is shown to be second-order accurate in the discretization.

Evaluation by Mental Stress of Paced and Self-Paced Conditions on Simple Repetitive Tasks in Design of Optimum Working Speed

1980 ◽  
Vol 24 (1) ◽  
pp. 655-659
Author(s):  
Masaharu Kumashiro ◽  
Tetsuya Hasegawa ◽  
Kageyu Noro
Keyword(s):  
The Self ◽  
The Other ◽  
Maximum Output ◽  
Considerable Difficulty ◽  
Time Value ◽  
Repetitive Tasks ◽  
Hand Coordination ◽  
Psychological Functions ◽  
Qualitative Performance ◽  
Better Than

Eight working speeds, conveyor-paced and self-paced, were established for a repetitive task with considerable difficulty in eye-hand coordination. The procedure for calculating pace allowances in this type of repetitive task was experimentally studied from the two angles of physiological and psychological functions of the subjects and quantative and qualitative variations of the task. The results obtained are as follows: (1) The maximum output speed under the self-paced condition was 125% of that when the subjects performed the task at their free pace. The quantative and qualitative performance of the subjects under the self-paced condition was better than that attained under the other conditions, but the physiological and psychological functions of the subjects lowered greatly 90 minutes after the start of the task. (2) When establishing the working speed for conveyor-paced operations, the basic time value calculated by the MTM procedure was satisfactorily used to set the standard speed per cycle. The time value obtained by the WF procedure, on the other hand, was not preferred in terms of both the performance and the physiological and psychological functions of the subjects.

A minimal flow-elements model for the generation of packets of hairpin vortices in shear flows

2014 ◽  
Vol 747 ◽  
pp. 30-43 ◽  
Author(s):  
Jacob Cohen ◽  
Michael Karp ◽  
Vyomesh Mehta
Keyword(s):  
Transition Process ◽  
Vortex Sheet ◽  
Base Flow ◽  
Minimal Flow ◽  
Streamwise Vorticity ◽  
Hairpin Vortices ◽  
Robust Model ◽  
Flow Velocity Profile ◽  
Spanwise Vortex ◽  
Induced Velocity

AbstractPackets of hairpin-shaped vortices and streamwise counter-rotating vortex pairs (CVPs) appear to be key structures during the late stages of the transition process as well as in low-Reynolds-number turbulence in wall-bounded flows. In this work we propose a robust model consisting of minimal flow elements that can produce packets of hairpins. Its three components are: simple shear, a CVP having finite streamwise vorticity magnitude and a two-dimensional (2D) wavy (in the streamwise direction) spanwise vortex sheet. This combination is inherently unstable: the CVP modifies the base flow due to the induced velocity forming an inflection point in the base-flow velocity profile. Consequently, the 2D wavy vortex sheet is amplified, causing undulation of the CVP. The undulation is further enhanced as the wave continues to be amplified and eventually the CVP breaks down into several segments. The induced velocity generates highly localized patches of spanwise vorticity above the regions connecting two consecutive streamwise elements of the CVP. These patches widen with time and join with the streamwise vortical elements situated beneath them forming a packet of hairpins. The results of the unbounded (having no walls) model are compared with pipe and channel flow experiments and with a direct numerical simulation of a transition process in Couette flow. The good agreement in all cases demonstrates the universality and robustness of the model.

The attraction between a flexible filament and a point vortex

2012 ◽  
Vol 697 ◽  
pp. 481-503 ◽  
Author(s):  
Silas Alben
Keyword(s):  
Point Vortex ◽  
Critical Time ◽  
Power Laws ◽  
Vortex Sheet ◽  
Separation Distance ◽  
Temporal Distance ◽  
Pressure Loading ◽  
Governing Equations ◽  
Flexible Filament ◽  
Wide Range

AbstractWe determine the inviscid dynamics of a point vortex in the vicinity of a flexible filament. For a wide range of filament bending rigidities, the filament is attracted to the point vortex, which generally moves tangentially to it. We find evidence that the point vortex collides with the filament at a critical time, with the separation distance tending to zero like a square root of temporal distance from the critical time. Concurrent with the collision, we find divergences of pressure loading on the filament, filament vortex sheet strength, filament curvature and velocity. We derive the corresponding power laws using the governing equations.

scholarly journals Self-thermophoresis of laser-heated spherical Janus particles

2021 ◽  
Vol 44 (11) ◽  
Author(s):  
E. J. Avital ◽  
T. Miloh
Keyword(s):  
Temperature Field ◽  
Continuous Light ◽  
The Self ◽  
Soret Coefficient ◽  
Physical Parameters ◽  
Two Phase ◽  
Rayleigh Approximation ◽  
Analytic Expressions ◽  
Induced Velocity ◽  
Laser Heated

Abstract An analytic framework is presented for calculating the self-induced thermophoretic velocity of a laser-heated Janus metamaterial micro-particle, consisting of two conducting hemispheres of different thermal and electric conductivities. The spherical Janus is embedded in a quiescent fluid of infinite expanse and is exposed to a continuous light irradiation by a defocused laser beam. The analysis is carried under the electrostatic (Rayleigh) approximation (radius small compared to wavelength). The linear scheme for evaluating the temperature field in the three phases is based on employing a Fourier–Legendre approach, which renders rather simple semi-analytic expressions in terms of the relevant physical parameters of the titled symmetry-breaking problem. In addition to an explicit solution for the self-thermophoretic mobility of the heated Janus, we also provide analytic expressions for the slip-induced Joule heating streamlines and vorticity field in the surrounding fluid, for a non-uniform (surface dependent) Soret coefficient. For a ‘symmetric’ (homogeneous) spherical particle, the surface temperature gradient vanishes and thus there is no self-induced thermophoretic velocity field. The ‘inner’ temperature field in this case reduces to the well-known solution for a laser-heated spherical conducting colloid. In the case of a constant Soret phoretic mobility, the analysis is compared against numerical simulations, based on a tailored collocation method for some selected values of the physical parameters. Also presented are some typical temperature field contours and heat flux vectors prevailing in the two-phase Janus as well as light-induced velocity and vorticity fields in the ambient solute and a new practical estimate for the self-propelling velocity. Graphic abstract

scholarly journals Assessment of a Three-Dimensional Potential Code for Axial and Radial Blade Row Flows

1986 ◽  
Author(s):  
H.-H. Frühauf ◽  
D. Krämer ◽  
U. Küster
Keyword(s):  
Three Dimensional ◽  
Cross Flow ◽  
Vortex Sheet ◽  
Full Potential ◽  
Aerodynamic Design ◽  
Blade Row ◽  
Circulation Distribution ◽  
Flow Effects ◽  
Downstream Flow ◽  
New Formulation

An assessment of a three-dimensional potential code is presented for axial and radial blade row flows, for which experimental data or computational results are available. The prediction of a flow case with severe cross flow is included. Numerous 3-D potential flow effects will be discussed. The conservative full potential equation for 3-D transonic flow is solved by a robust SLOR method. A new formulation of the downstream flow ensures the computability of blade row flows with arbitrary spanwise circulation distributions. Blade circulation distribution and downstream flow properties are obtained as a part of the solution. The surface representation in parameter form in the geometry definition program and the mesh generation program, as well as the tensor notation of the basic equation and a configuration-independent vortex sheet approximation prediction, ensure the computability of flows through blade rows with arbitrary complex geometries. The CRAY-1/M incore version of the code is inexpensive enough to be used in the aerodynamic design process of new blade rows with irrotational inflows.

Sign in / Sign up

Export Citation Format

Share Document