Explicit Equations to Transform from Cartesian to Elliptic Coordinates

Sun, Che

doi:10.11648/j.mma.20170204.12

Cited by 9 publications

(11 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We again note that for large µ the solution for F in (4.9) resembles that of F + A 2 F = 0 and therefore exhibits near periodicity in µ. Consequently, the contour plots of the corresponding stream function ψ consist of repeating blocks along some of the curves shown in the middle panel of Fig. 4 In order to obtain u and v from r and z, we can again use the conversion formulas from [19], which now have the form…”

Section: Paraboloidal Coordinates the Paraboloidal Coordinates Arementioning

confidence: 99%

See 1 more Smart Citation

On the axisymmetric steady incompressible Beltrami flows

Bělík¹,

Su²,

Dokken³

et al. 2019

Preprint

View full text Add to dashboard Cite

In this paper, Beltrami vector fields in several orthogonal coordinate systems are obtained analytically and numerically. Specifically, axisymmetric incompressible inviscid steady state Beltrami (Trkalian) fluid flows are obtained with the motivation to model flows that have been hypothesized to occur in tornadic flows. The studied coordinate systems include those that appear amenable to modeling such flows: the cylindrical, spherical, paraboloidal, and prolate and oblate spheroidal systems. The usual Euler equations are reformulated using the Bragg-Hawthorne equation for the stream function of the flow, which is solved analytically or numerically in each coordinate system under the assumption of separability of variables. Many of the obtained flows are visualized via contour plots of their stream functions in the rz-plane. Finally, the results are combined to provide a qualitative quasi-static model for a progression of flows that one might see in the process of a vortex breakdown. The results in this paper are equally applicable in electromagnetics, where the equivalent concept is that of a force-free magnetic field.

show abstract

Section: Paraboloidal Coordinates the Paraboloidal Coordinates Arementioning

confidence: 99%

“…In order to obtain u and v from r and z, we can use the conversion formulas from [19], which, in this case, result in…”

Section: Paraboloidal Coordinates the Paraboloidal Coordinates Arementioning

confidence: 99%

On the axisymmetric steady incompressible Beltrami flows

Bělík¹,

Su²,

Dokken³

et al. 2019

Preprint

View full text Add to dashboard Cite

show abstract

“…The curves with constant u are the ellipses and the curves with constant v are the hyperbolas. In order to obtain u and v from r and z, we can again use the conversion formulas from [32], which now have the form ( ) Therefore, necessary boundary conditions are ( ) ( )…”

Section: Oblate Spheroidal Coordinatesmentioning

confidence: 99%

“…In order to obtain u and v from r and z, we can use the conversion formulas from [32], which, in this case, result in ( )…”

Section: Prolate Spheroidal Coordinatesmentioning

confidence: 99%

On the Axisymmetric Steady Incompressible Beltrami Flows

Bělík¹,

Su²,

Dokken³

et al. 2020

OJFD

View full text Add to dashboard Cite

In this paper, Beltrami vector fields in several orthogonal coordinate systems are obtained analytically and numerically. Specifically, axisymmetric incompressible inviscid steady state Beltrami (Trkalian) fluid flows are obtained with the motivation to model flows that have been hypothesized to occur in tornadic flows. The studied coordinate systems include those that appear amenable to modeling such flows: the cylindrical, spherical, paraboloidal, and prolate and oblate spheroidal systems. The usual Euler equations are reformulated using the Bragg-Hawthorne equation for the stream function of the flow, which is solved analytically or numerically in each coordinate system under the assumption of separability of variables. Many of the obtained flows are visualized via contour plots of their stream functions in the rz-plane. Finally, the results are combined to provide a qualitative quasi-static model for a progression of tornado-like flows that develop as swirl increases. The results in this paper are equally applicable in electromagnetics, where the equivalent concept is that of a force-free magnetic field.

show abstract

“…By defining p = sin 2 (η) and q = − sinh 2 (ξ), we have p ≥ q since 0 ≤ p ≤ 1, q ≤ 0. According to [11], we have…”

Section: B Formation Constraints In Elliptic Coordinatesmentioning

confidence: 99%

Collaborative Target Tracking in Elliptic Coordinates: a Binocular Coordination Approach

Chang

Sun

Zhou

et al. 2020

Preprint

View full text Add to dashboard Cite

This paper concentrates on the collaborative target tracking control of a pair of tracking vehicles with formation constraints. The proposed controller requires only distance measurements between tracking vehicles and the target. Its novelty lies in two aspects: 1) the elliptic coordinates are used to represent an arbitrary tracking formation without singularity, which can be deduced from inter-agent distances, and 2) the regulation of the tracking vehicle system obeys a binocular coordination principle, which simplifies the design of the control law by leveraging rich physical meanings of elliptic coordinates. The tracking system with the proposed controller is proven to be exponentially convergent when the target is stationary. When the target drifts with a small velocity, the desired tracking formation is achieved within a small margin proportional to the magnitude of the target's drift velocity. Simulation examples are provided to demonstrate the tracking performance of the proposed controller.

show abstract

Explicit Equations to Transform from Cartesian to Elliptic Coordinates

Cited by 9 publications

References 9 publications

On the axisymmetric steady incompressible Beltrami flows

On the axisymmetric steady incompressible Beltrami flows

On the Axisymmetric Steady Incompressible Beltrami Flows

Collaborative Target Tracking in Elliptic Coordinates: a Binocular Coordination Approach

Contact Info

Product

Resources

About