Elsevier

Journal of Computational Physics

Volume 227, Issue 21, 10 November 2008, Pages 9044-9062
Journal of Computational Physics

A contour dynamics algorithm for axisymmetric flow

https://doi.org/10.1016/j.jcp.2007.10.005Get rights and content

Abstract

The method of contour dynamics, developed for two-dimensional vortex patches by Zabusky et al. [N.J. Zabusky, M.H. Hughes, K.V. Roberts, Contour dynamics for the Euler equations in two-dimensions, J. Comp. Phys. 30 (1979) 96–106] is extended to vortex rings in which the vorticity distribution varies linearly with normal distance from the symmetry axis. The method tracks the motion of the boundaries of the vorticity regions and hence reduces the dimensionality of the problem by one. We discuss the formulation and implementation of the scheme, verify its accuracy and convergence, and present illustrative examples.

Introduction

In two-dimensions, the contour dynamics approach, initiated by Zabusky et al. [34], has made possible the study of the inviscid motion of vortex patches containing piecewise constant vorticity; see Pullin [28] for a review. Since vorticity follows the fluid, such a distribution remains unaltered in time within each region and only the boundaries between regions have to be tracked as they convect with the fluid velocity. The velocity can be expressed as a line integral along the contours, thus reducing the dimensionality of the problem by one. The approach was inspired by the so-called water-bag method in magnetohydrodynamics; for a recent work see [12]. In principle, arbitrary vorticity distributions may be approximated by piecewise constant ones, but to date most of the work has focussed on vortices containing single regions.

Contour dynamics has yielded mathematical insight into the nature of solutions of the Euler equations as well as increased understanding of physical processes in shear layers and two-dimensional turbulence. For instance Dritschel [7], [8] elucidated the role of energetics in the merger and fission of vortices and in more general topological changes that occur during their long time evolution. Specifically, perturbations of equilibrium solutions tend asymptotically to different equilibrium states which are energetically compatible with the original state. Dritschel [10] also studied nearly inviscid two-dimensional turbulence on a sphere using this approach and his results sharply contrast with behavior observed under the more viscous conditions of spectral simulations. Neu [21], motivated to explain the genesis of streamwise braid vortices in mixing layers, showed that highly flattened uniform vorticity cores “collapse” to a circular shape with concentrated vorticity when subjected to a three-dimensional strain which models the influence of spanwise rollers and neighboring streamwise vortices. Lin and Corcos [17], using finite-difference calculations of the two-dimensional Navier–Stokes equations with out of plane strain confirmed the mechanism for an array consisting of counter rotating pairs. Pullin and Jacobs [29] provided further evidence with contour dynamics simulations of vortex arrays employing multiple contours. As a final example, we mention [33] who studied some aspects of entrainment into a turbulent boundary layer by considering the behavior of disturbances on a uniform vorticity layer near a wall.

This work extends the method of contour dynamics to vortex rings in the hope that it may play a similar role in providing insight for axisymmetric flow that contour dynamics has for planar flows. The extension to axisymmetric flow offers the possibility of expanding the repertoire of possible vortex behavior by allowing an important effect lacking in planar flow, namely vortex stretching.

Section 2 gives a derivation of the contour dynamics formulation for the case in which ωϕ/σ (the ratio of azimuthal vorticity to cylindrical radius) is constant within each vorticity region. This form of the vorticity has been studied for over a century. Helmholtz [14] already started deriving the speed of translation of thin rings of this class in his 1858 paper and Kelvin put the finishing touches to the derivation in an appendix to the 1867 translation of Helmholtz’ paper. In 1894 Hill presented his famous spherical vortex, an exact steadily translating vortex in the thick core limit (see [1, p. 526]). Norbury [22] provided numerical solutions for the family of steadily propagating rings between the thin-core and Hill limits. Some dynamical aspects have also been studied. This includes Dyson’s [11] model from 1893 of thin interacting rings (for which core dynamics can be neglected) and the stability analysis of Hill’s vortex by Moffatt and Moore [20].

Section 3 discusses the numerical implementation of the algorithm, including treatment of the local contribution to the velocity field resulting from those portions of the contour which neighbor the point at which the velocity is evaluated. Section 4 verifies the accuracy and convergence of the numerical scheme. Finally, §5 presents two examples: the behavior of an axisymmetric annular vortex layer and the head-on collision of two vortex rings. Appendix A provides details on the local contribution to the velocity field and Appendix B works out an expression for the Stokes’ streamfunction ψ in terms of a contour integral which might prove useful to those who wish to find steadily translating configurations; it also helps in computing the energy which is proportional to the integral of the product of ψ and the azimuthal vorticity ([1, p. 521]).

The present formulation and implementation was developed in the summer of 1984. Around the same time Pozrikidis independently also developed a contour dynamics formulation for axisymmetric flow and his work was reported in Pozrikidis [27] in which he studied the instability of Hill’s vortex in the non-linear regime. We will remark on the significant differences between the two formulations where appropriate. In 1990, Möhring sent us a 1978 diploma thesis by Poppe [26] containing a contour formulation for the streamfunction which the author uses to calculate some generalizations of Norbury’s rings [22] containing nested contours. Poppe’s contour formulation for the streamfunction appears to be different than the present one which is provided in Appendix B.

Section snippets

Axisymmetric contour dynamics formulation

In this section we derive the equations of motion for contours which bound regions in which the vorticity is a linear function of the cylindrical radius, σ. The reason for using this distribution will be given below.

Consider cylindrical polar coordinates (x, σ, ϕ) as shown in Fig. 1; x and σ measure distance along and normal to the axis of symmetry, respectively, and ϕ is the azimuthal angle. Let the vorticity ω be entirely azimuthal and independent of ϕ:ω=(0,0,ωϕ(x,σ)).The corresponding velocity

Numerical implementation

Eq. (31) is a non-linear integro-differential equation for motion of the boundaries of the vortex cores. For numerical purposes, the contours are represented by a discrete set of node points which are convected as material particles. The integrals are approximated by connecting the points with straight line segments. In the planar case, the segment integrals are carried out in closed form. However, quadrature is sometimes used to save computing time. This requires that the singularity be

Tests of accuracy and convergence

Fig. 5 shows the convergence in the discrete L2 norm of the axial and radial velocity as a function of the number of segments in the case of Hill’s vortex. The slope is close to −2, consistent with the second order accuracy of the segment discretization. There is a slight decrease in the slope as we approach machine round-off (single precision for this test). This represents a static test of the algorithm. A good dynamic test (suggested to us by Prof. Zabusky) is to ensure that for Norbury [22]

Examples

As a qualitative illustration of the method, we simulated a Hill’s spherical vortex with a region of vortical fluid removed. The removed region has as its initial boundary, one of the interior streamsurfaces of Hill’s vortex. The time evolution is shown in Fig. 7 where the shading indicates the vorticity containing region. Time tˆ has been normalized using the mean toroidal radius and speed of translation of a Hill’s vortex without the hole. A violent evolution occurs during the time that the

Summary

This work considered inviscid swirl-free axisymmetric flows consisting of regions in which the vorticity varies linearly with radius. The velocity field was expressed as a contour integral which reduces the problem to a one-dimensional integro-differential equation for the motion of the boundaries of the regions. The streamfunction was also formulated as a contour integral. The method was demonstrated for two vortex rings colliding head-on. A head-tail structure was formed which agrees with

Tribute to Tony Leonard for this Special Issue Dedicated to Him

I was lucky when in 1983 Joel Ferziger hooked me up with Tony to do my dissertation. Tony was then a research scientist at NASA Ames. A few thoughtful words spoken by him often turned out to provide the key way of looking at something. And, he had a knack for making one feel I one knew the answer all along: in discussions he might preface his insight with I think you are trying to say this . . . ” I am certain that others can recount similar phenomena. Tony has a warm, easy-going, and

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