PHYSICS OF FLUIDS
VOLUME 14, NUMBER 12
DECEMBER 2002
Axial core-variations of axisymmetric shape on a curved slender vortex filament with a similar, Rankine, or bubble core Daniel Margerit Institut de Me´canique des Fluides de Toulouse, UMR 5502 CNRS, Alle´e du Professeur Camille Soula, 31400 cedex Toulouse, France
共Received 31 July 2001; accepted 28 August 2002; published 7 November 2002兲 The dynamics of axial core-variations of axisymmetric shape on a vortex filament is derived from the Navier–Stokes equations in the slenderness limit. The core of the vortex is of similar, Rankine, or bubble type with a centerline of any shape. In this limit, a two-time-scale asymptotic approach is used to study the dynamics of the axial core-variations and of the centerline. The short-time dynamical equations of the axial core-variations are given and are inviscid at leading and first orders. The induced short-time and normal-time dynamics of the centerline is obtained. The full two-time-scale dynamics of the axial variations and of the filament motion is discussed qualitatively. The normal-time dynamics of vortex filaments without axial core-variations is given in a short form. Within the two-time-scale framework, the dynamics of axial core-variations around this one-time base flow is then studied in the small amplitude limit. The normal-time equations of a vortex bubble are given. The bubble has no axial variations, a centerline of any shape and can have a nonpotential core. The equation for the ultra-short-time dynamics of axial variations on this bubble is given. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1516210兴
circulation of the vortex filament, and is the kinematic viscosity. In dimensionless variables with the wavelength L ⫽11␦ as the characteristic length, these parameters of simulation correspond to ⫽ ␦ /L⫽1/11⬇0.09, and to a viscous parameter 共see Sec. II兲 ␣ ⬅Re⫺1/2/⬇0.43. Arendt et al.8 give the evolution of any initial axial variations of small amplitude in the dynamics of the linearized axisymmetric equations. In this paper I derive and study the leading-order set of equations of motion of curved slender vortex filaments of any shape with axial core-variations. From the study of the eigen-oscillations of a circular vortex ring, Kopiev and Chernyshev9 give the dynamics of the bending modes that oscillates on the time t of the motion of the vortex ring. The dynamics of these bending modes can also be found from a linear stability study of the asymptotic equation of motion of vortex filaments.10,11 For a circular vortex ring Kopiev and Chernyshev9 also give the dynamics of bulging modes that can be shown to oscillate on the short time ⫽t/. It shows that a two-time-scale analysis can be used to study them for a general curved filament. This would extend the one-time analysis of the previous asymptotic theory1 without axial core-variations. It also extends to a curved filament the oneand short-time analysis by Souza12 of a straight filament with core-variations. In this paper, I carry out this two-time-scale analysis. The paper is organized as follows. In Sec. II, I give the geometrical description of the centerline and of the vortex core in local coordinates near the centerline of the filament. In these coordinates, I give the dynamical equations of the velocity field, those of the interfaces 共if there are any: i.e., vortex sheet when there is a jump of axial velocity or bubble free-boundary兲, and all the two-time-scale asymptotic expan-
I. INTRODUCTION
Slender vortex filaments are high concentration of vorticity along a geometrical curve of the fluid flow; they have been studied since more than a century, as their dynamics gives the one of the flow. Almost all the vorticity is inside a tube of small thickness ␦ compared to the characteristic radius of curvature R of its centerline; the ratio ␦ /R defines a small parameter . In the small thickness limit, this tube is a ‘‘boundary layer’’ 共in the singular perturbation method point of view兲; and, except for a straight filament, it is moving with a velocity that depends on its core structure, as can be found from the Biot–Savart law of induction.1,2 This small moving region of the flow is difficult to track in experimental measurements and induces stiffness in direct numerical simulations of vortex filaments. Asymptotic methods have been used to get ride of this stiffness1,2 and to extract, at leading order, a nonstiff equation for the filament motion. This asymptotic theory assumes that the leading-order core is axisymmetric and without axial core-variations. Generalized systems of equations for filaments with axial core-variations have been proposed.3– 6 In these studies, the characteristic length of the axial variations is of the order of the radius of curvature which is bigger than the thickness ␦; short wavelengths are not taken into account. These systems are ad hoc equations because they are not asymptotically derived from the Navier–Stokes 共N–S兲 equations. For a straight vortex filament with axial core-variations 共bulging waves兲 Melander and Hussain7 give a spectral computation of the axisymmetric equations. They compute a bulging wave of wavelength ⬇11␦ and amplitude ⬇0.5␦ at Re⫽⌫/⬇665, where ␦ is the mean core radius, ⌫ is the 1070-6631/2002/14(12)/4406/23/$19.00
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© 2002 American Institute of Physics
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Phys. Fluids, Vol. 14, No. 12, December 2002
sions. In Sec. III the cascade of asymptotic equations is given in the two-time-scale framework. From the axisymmetric part of the first-order asymptotic equations, I obtain short-time-scale dynamical equations for the axial corevariations of axisymmetric shape. These equations are the same as those obtained by Souza12 for a straight filament. For a curved filament they were first given in Margerit and Brancher.13 Souza pointed out 共private communication兲 that the equations for a straight filament might be relevant to a curved filament. Here, I prove that curvature does not give adding terms in these equations and I also give the induced short-time dynamics of the curved vortex centerline. In Sec. IV the one-time 共normal-time兲 dynamics of vortex filaments without axial core-variations of the Callegari and Ting theory1 is given in a short form. This gives the one-time base flow that is used in the linear stability study of Sec. V. This analysis is carried out in the two-time-scale framework to study the dynamics of small axial core-variations around this one-time base flow. In Sec. VI the one-time 共normal-time兲 continuous vortex-core of the Callegari and Ting theory1 is extended for a vortex bubble with a centerline of any shape and with a nonpotential core. An equation for the ultra-fast dynamics of axial variations of the bubble free-boundary is given. This generalizes the theory of Genoux14 –16 of vortex bubbles of circular centerline and potential core. Finally a conclusion is given in Sec. VII. Several steps of the derivations are given in the appendices. Appendix A gives the asymmetric part of the equations at first order and Appendix B gives the axisymmetric part of the equations at second order. In Appendix C the core without axial variations appears to be the unique stationary solution of the short-time-scale equations given in Sec. III. Finally in Appendix D the axisymmetric part of the stationary solution of the short-time-scale equations at next order is proven to be the sum of a part without axial variations and of a part with axial variations. The structure of this second part is unique and is induced from the local stretching of the centerline. Fortunately this structure was that introduced in Margerit17 to generalize the Callegari and Ting theory1 at next order. II. NOTATIONS AND TWO-TIME EXPANSIONS
Here, I give the geometrical description of the flow field and of the filament, and the local coordinates that are used. A discussion of the characteristic scales of the asymptotic slender filament regime and the basic assumptions of the asymptotic study are then given. Finally, the equations for the flow are written on the local coordinates and the twotime-scale expansions are given. The closed centerline of the slender vortex of circulation ⌫ and length S is described by the vector function X ⫽X(s,t) where s stands for the arc-length at t⫽0. At each point of this curve the Frenet vector basis 共t, n, b兲 exists with, respectively, the tangent, normal, and binormal vectors. I introduce a local curvilinear coordinate system M(r, ,s) and the curvilinear vector basis (er ,e ,t) valid near this line. This system is defined in the following manner; if P(s) is the projection on the centerline of a point M then PM is in the
Axial core-variations of axisymmetric shape
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plane 共n, b兲, and thus polar coordinates 共r, 兲 can be used in this plane with the associated polar vectors (er ,e ) and with the angle between n and PM. The relative velocity V is defined by ˙ 共 s,t 兲 ⫹V共 r, ,s,t 兲 , v⫽X
共1兲
˙ the local filament where v is the velocity of the fluid and X velocity. The radial, circumferential, and axial components (u, v ,w) of the relative velocity are defined by V⫽uer ⫹ v e ⫹wt. The vorticity field is given by ⫽“ÃV ˙ s /h 3 , where h 3 ⫽ (1⫺rK(s)cos()), K is the local ⫹tÃX curvature, and ⫽ 兩 Xs 兩 . In the asymptotic theory of vortex motion, the thickness ␦ of the ring is of order l and the other length scales, for example the local radius of curvature 1/K or the length S of the closed filament are of the same order L. Since the vortex is slender the small parameter Ⰶ1 is defined as the ratio l/L. I nondimensionalize the velocity field with ⌫/L, all lengths with L, and the time with L 2 /⌫. The outer problem is defined by the outer limit: →0 with r fixed, which describes the flow far from the centerline; and the inner problem by the inner limit: →0 with ¯⫽r/ r fixed, which describes the flow near the centerline. The Reynolds number R e ⫽⌫/ , where is the kine⫽ ␣ . Here, the vismatic viscosity, is related to by R ⫺1/2 e cous number ␣ ⫽O(1) is defined by ␣ 2 ⫽¯ /⌫, where ¯ ⫽ / 2 . The inviscid vortex ring is obtained in the limit ␣ ⫽0. The asymptotic ansatz based on the small slenderness ratio allows to unify the related analyses for the Navier– Stokes and Euler equations. In this study I assume that the vorticity field is centered near the centerline and rapidly decays at large distance. I will assume that the vorticity distribution is of bounded support or decays exponentially. The same standard assumption is also taken for the axial velocity field. A. General equations
The continuity equation in these curvilinear coordinates (r, ,s) is1 ˙ s •t, 共 urh 3 兲 r ⫹ 共 h 3 v 兲 ⫹rw s ⫺Tr w ⫽⫺r X
共2兲
where T is the local torsion of the filament. The Navier– Stokes equation becomes1 a⫽⫺ⵜ p⫹ ⌬V⫹
冉 冊
1 ˙ X h3 h3 s
共3兲
, s
where is the kinematic viscosity, and the acceleration a is a⫽
冉 冊 V t
r, ,s
⫹ 共 V⫺re˙r 兲 •ⵜV⫹
˙s X ¨, 共 w⫺re˙r •t兲 ⫹X h3
with
冉 冊 V t
⫽ r, ,s
u e ⫹ue˙r ⫹... . t r
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Here and in the whole paper, the pressure is in fact the pressure divided by the constant density of the incompressible fluid. The boundary conditions of these equations are u⫽ v ⫽0 at r⫽0. B. Discontinuous vorticity field
In an inviscid fluid, if the vorticity is inside a tube of thickness ␦ t , the location of this interface r⫽ ␦ t ( ,s,t) is an unknown function. At this interface one has to satisfy both the continuity of pressure 关关 p 兴兴 ⬅p 共 ␦ t⫹ 兲 ⫺p 共 ␦ t⫺ 兲 ⫽0,
共4兲
and the continuity of the normal velocity 关关 v"N兴兴 ⫽0,
共5兲
where N is the normal vector to the interface and is given by N⫽⫺ⵜF/ 兩 ⵜF 兩 with F⫽ ␦ t ( ,s,t)⫺r and
冉
ⵜF⫽ ⫺er ⫹
冉
冊冊
␦t 1 ␦t 1 ␦t ⫺T e ⫹ t . t ␦ h3 s
The dynamical equation for the interface is given by the kinematic boundary condition
␦t ⫹ 共 V共 ␦ t⫾ 兲 ⫺ ␦ t e˙r 兲 •ⵜF⫽0, t
共6兲
where ␦ t⫾ is used to allow the possibility of the interface to be a vortex sheet of strength ␥ ⫽N⫻ 关关 v兴兴 . Following Wu,18 a dynamical equation for this strength ␥ can be written but will not be used here. For this inviscid fluid the thin shear layer is assumed to be without thickness and there is no viscosity ( ⫽0) in Eq. 共3兲 which becomes the Euler equation. In local coordinates, Eqs. 共5兲 and 共6兲 become ⫺ 关关 u 兴兴 ⫹
冉
冊
1 ␦t 1 ␦t ␦t ⫹ T ⫺ 关关v兴兴 关关 w 兴兴 ⫽0, ␦ t h3 s
冉
冊
共7兲
␦t ␦t 1 ␦t 1 ␦t ⫺u⫹ t ⫺T w⫺ ␦ t e˙r •ⵜF⫽0, v⫹ t ␦ h3 s 共8兲 where all velocity components are taken on the interface r ⫽ ␦ t⫾ . C. Vortex ring bubble
In the case of an inviscid vortex ring bubble, the location of the free-boundary r⫽ ␦ b ( ,s,t) is another unknown function. The pressure jump at the free-boundary of the bubble is 关关 p 兴兴 b ⫽p 共 ␦ b⫹ 兲 ⫺p 共 ␦ b⫺ 兲 ⫽2⌼ ,
共9兲
where ⌼ is the surface tension divided by the constant density of the incompressible fluid and ( ,s,t) the mean curvature of the free-boundary. The bubble14 contains liquid vapor of pressure P v and noncondensables of partial pressure P g ⫽ P g0 (V b0 /V b ) k , where V b is the volume of the bubble and k is the polytropic constant of the ideal gas in the bubble. Here, V b0 and P g0 are their initial values. The pressure p( ␦ b⫺ )⫽ P v ⫹ P g inside the
bubble is uniform. As pointed out by one of the referees, polytropic variations are merely an approximate simplification of the full thermodynamics as they are based on an ad hoc coupling between two otherwise independent thermodynamics variables. I will not remove this assumption of polytropic variations by including an energy balance equation in the analysis as was suggested by this referee and I postpone this work for the future. The dynamical equation of the free-boundary is
␦b ⫹ 共 V共 ␦ b⫹ 兲 ⫺ ␦ b e˙r 兲 •ⵜF⫽0, t
共10兲
where ⵜF is as before but with ␦ t replaced by ␦ b . For this inviscid fluid the thin diffusion layer along the interface is assumed to have no thickness and there is no viscosity ( ⫽0) in Eq. 共3兲 which becomes the Euler equation. As the fluid is outside of such a bubble, singularities can exist inside the bubble, which is not possible in a homogeneous fluid, and the condition u⫽ v ⫽0 at r⫽0 is no longer valid. Vortex ring bubbles without axial variations and with a circular centerline have been studied by Genoux14 for a potential flow. In fact, vortex ring bubble can be embedding in a vortical flow: for example, one can easily consider a vortex ring bubble with a vortex core of Rankine type. The circulation ⌫ of the ring is then ⌫⫽⌫ 1 ⫹⌫ 2 , where ⌫ 1 is the circulation induced by the vortex sheet on the free-boundary and ⌫ 2 is the added circulation due to the vortical core. ¯ The Weber number is W e ⫽L⌼/(⌫ 2 ) and I define ⌼ 2 2 ⫽⌼, ¯P v ⫽ P v , ¯P g0 ⫽ P g0 . I am interested by the regime W e ⫽O( ⫺2 ) because the effect of the surface tension will come at leading order in the pressure jump 共9兲. D. Two-time analysis and expansions
In the two-time analysis, the expansion of the velocity ˙ X⫽ t X⫹ ⫺1 X of the centerline is ˙ ⫽ t X共 0 兲 ⫹ X共 1 兲 ⫹O 共 log 兲 , X with the following expansion of the centerline: X⫽X共 0 兲 共 s,t 兲 ⫹X共 1 兲 共 s,t, ⫽t/ 兲 ⫹... . Here, fast oscillations of the centerline of amplitude can exist. The inner expansions of the relative velocity components and of the pressure are r ,s,t, 兲 ⫹..., u inn ⫽u 共 1 兲 共¯, r 兲 ⫹ v 共 1 兲 共¯, r ,s,t, 兲 ⫹..., v inn ⫽ ⫺1 v 共 0 兲 共¯,s,t, w inn ⫽ ⫺1 w 共 0 兲 共¯,s,t, r 兲 ⫹w 共 1 兲 共¯, r ,s,t, 兲 ⫹..., p inn ⫽ ⫺2 p 共 0 兲 共¯,s,t, r 兲 ⫹ ⫺1 p 共 1 兲 共¯, r ,s,t, 兲 ⫹..., where ¯⫽r/ r is the stretched radial coordinate in the core. Here, the leading-order velocity field is axisymmetric, as in the previous asymptotic theories, but can change along the filament. There is no radial velocity at leading order. This is consistent with an axisymmetric leading order: e.g., an ellip-
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Phys. Fluids, Vol. 14, No. 12, December 2002
Axial core-variations of axisymmetric shape
tic core at leading order will behave like a Kirchhoff ellipse which would rotate on the ultra-short time t/ 2 and have a nonzero radial velocity at leading order. Short wavelengths of order are out of the scope of this theory: their dynamics would be on the time ¯t ⫽t/ 2 , would use a stretched axial coordinate ¯⫽s/ s and was studied by Widnall and Tsai19 in the linear regime around a circular vortex ring. If the vorticity is inside a vortex tube, the interface has the following expansion: ¯␦ t ⫽ ␦ t /⫽¯␦ t 共 0 兲 共 ,s,t, 兲 ⫹¯␦ t 共 1 兲 共 ,s,t, 兲 ⫹... and its time-derivative is
␦˙ t ⫽ ¯␦ t ⫹ t¯␦ t ⫽ ¯␦ t 共 0 兲 ⫹ 共 ¯␦ t 共 1 兲 ⫹ t¯␦ t 共 0 兲 兲 ⫹... . For a vortex bubble, one have the same expansion for ␦ b , and the expansions of the volume of the bubble and of the curvature of the free-boundary are Vb ⫽V 2
b共 0 兲
⫹V
b共 1 兲
⫹...,
⫽ ⫺1 共 0 兲 ⫹ 共 1 兲 ⫹... . When the leading order is axisymmetric, it comes V ⫽ 2 b
冕
2
0
共 0 兲 关 ¯␦ b 共 0 兲 兴 2 ds⫹
冕
2
0
共 共 1 兲 关 ¯␦ b 共 0 兲 兴 2
⫹2 共 0 兲¯␦ b 共 0 兲¯␦ cb 共 1 兲 兲 ds⫹O 共 2 兲 ,
⫽⫺
1
1
2¯␦ b 共 0 兲 共 s,t, 兲
⫹
共11兲
1 ¯␦ b 共 1 兲 共 ,s,t, 兲 2 关 ¯␦ b 共 0 兲 共 s,t, 兲兴 2
1 1 关 ¯␦ b 共 1 兲 共 ,s,t, 兲兴 ⫹ K 共 0 兲 cos共 兲 ⫹ ⫹O 共 兲 , 2 2 关 ¯␦ b 共 0 兲 共 s,t, 兲兴 2 共12兲 where ¯␦ cb(1) is the axisymmetric part of ¯␦ b(1) ( ,s,t, ). III. TWO-TIME-SCALE DYNAMICS OF AXIAL VARIATIONS
The substitution of the previous expansions into Eqs. 共2兲–共3兲 leads to a cascade of asymptotic equations as in the one-time analysis.1 In this section I give the two-time-scale equations for the dynamics of axial variations at leading order. It consists in the leading-order short-time axisymmetric equations in the filament 关Eqs. 共13兲–共16兲兴 and the leadingorder short-time 关Eq. 共33兲兴 and normal-time 关Eq. 共32兲兴 equations of motion of the centerline. In the first subsection the leading-order equations of the short-time axisymmetric dynamics in the filament are given. The leading-order shorttime asymmetric dynamics in the filament is slaved by the axisymmetric dynamics and is given in Appendix A. The motion of the centerline is slaved by this core dynamics and its induced velocity is given in the next subsection. Finally a qualitative description of the two-time dynamics in the filament is given.
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A. The leading-order short-time axisymmetric dynamics of axial variations in the filament
The leading-order equations of the short-time axisymmetric dynamics in the filament come from the axisymmetric part at leading and first orders. I will give these equations in this subsection. 1. Continuous vorticity field
The leading-order compatibility conditions of the onetime analysis1 become the dynamical equations of the axisymmetric part of the leading-order relative velocity field, i.e., of the axial core-variations. a. The velocity form of the equations. At leading order p
共0兲
⫽⫺
冕
⬁
¯r
2
v共0兲 dr ¯⫹ p 共 0 兲 共 ⬁ 兲 , ¯r
共13兲
and at first order r c 共 1 兲 兲¯r ⫹r ¯w 共z0 兲 ⫽0, 共¯u
共14兲
v共0兲 ⫹ 共 0 兲 u c 共 1 兲 ⫹w 共 0 兲 v 共z0 兲 ⫽0,
共15兲
w共0兲 ⫹w¯共r0 兲 u c 共 1 兲 ⫹ p 共z0 兲 ⫹w 共 0 兲 w 共z0 兲 ⫽0,
共16兲
where (0) ⫽(r ¯ v (0) )¯r /r ¯ is the leading-order axial vorticity, c(1) is the axisymmetric part of the radial velocity at u first order, p (0) is the leading-order pressure, and z ⫽ 兰 s0 (0) (s ⬘ ,t)ds ⬘ . This system for p (0) , v (0) , w (0) , and u c(1) is closed. It gives the short-time-scale dynamics of the axial corevariations of axisymmetric shape. These equations are the same as the ones obtained by Souza12 for a straight filament. They are the ‘‘long wave scaling’’ shallow water equations derived from studies of vortex breakdown of a straight filament.20,12 Let us point out that in the studies of vortex breakdown and swirling-jets, the velocity field is often nondimensionalized using ⌫/l, all lengths using l, and the time using l 2 /⌫, where l is the small characteristic length and is of the thickness size. From this point of view, the O(1) wavelength of the asymptotic theory of vortex motion is a long wavelength and the short wavelength of the Tsai and Widnall19 study is a usual O(1) wavelength. At this order and on this short time the previous derivation shows that the curvature of the filament has no effect on the dynamics of axial variations. This proves the intuition of Souza who pointed out 共private communication兲 that the equations for a straight filament might be relevant to a curved filament. For a curved filament they were first given in Margerit and Brancher.13 I will now give other useful forms of these equations. b. The stream-function form of the equations. Let us define the meridional stream function c(1) , with 1 u c 共 1 兲 ⫽⫺ zc 共 1 兲 , ¯r
1 w 共 0 兲 ⫽ ¯rc 共 1 兲 , ¯r
and introduce the following transformation: K共 0 兲 ⫽r ¯ v 共 0 兲,
y⫽r ¯ 2,
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Phys. Fluids, Vol. 14, No. 12, December 2002
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usually used to study these equations.20 In these new variables the previous system becomes 共0兲
K ⫺2 zc 共 1 兲 K共y0 兲 ⫹2 cy 共 1 兲 K共z0 兲 ⫽0, D2
共17兲
c共 1 兲 2 ⫹2 cy 共 1 兲 D 2 zc 共 1 兲 ⫹ K共 0 兲 K共z0 兲 ⫺2y zc 共 1 兲 y ⫻ 关 y ⫺1 D 2 c 共 1 兲 兴 y ⫽0,
共0兲
1 K ⫹ ⵜ⬜ c 共 1 兲 •ⵜK共 0 兲 ⫽0, ¯r
共19兲
L c 共 1 兲 ⫽r ¯ 2 共 0 兲,
共20兲
共0兲
1 2 ⫹ ⵜ⬜ c 共 1 兲 •ⵜ 共 0 兲 ⫹ 4 K共 0 兲 K共z0 兲 ⫽0, ¯r ¯r
共21兲
¯ , and where ⵜ⫽( ¯r , z ), ⵜ⬜ ⫽(⫺ z , ¯r ), L⫽ ¯r2 ⫺ ¯r /r ⫺r ¯ (0) ⫽⫺w¯r(0) is the leading-order circumferential 共azimuthal兲 vorticity. The boundary conditions to these equations are the periodicity between z⫽0 and z⫽S (0) , where S (0) is the length of ¯→0 and K(0) →⌫/2 at infinity, the closed vortex, ¯rc(1) /r c(1) and / ¯⫽0 r and K(0) / ¯⫽0 r at ¯⫽0. r d. The equations in the Von Mises variables. One can ¯,s) by ( , c(1) ,s) at replace the independent variables ( ,r (0) all point where w ⫽0. In these Von Mises variables, the previous system becomes the nonstationary Bragg– Hawthorne 共or Squire–Long兲 equation
where H ⫽p (0)
H共 0 兲 K共 0 兲 共0兲 c 共 1 兲 ⫺K c共 1 兲 (0)
⫹( v
(0) 2
⫹w
(0) 2
共22兲
)/2.
2. Discontinuous vorticity field For a vortex with the vorticity inside a vortex tube the interface dynamics has to be found and the previous equations have to be completed. The leading order of the condition of continuity 共4兲 of the pressure on the interface yields
关关 p
共0兲
共26兲
where all velocity components are taken on the interface ¯r ⫽¯␦ t(0)⫾ . This system for p (0) , v (0) , w (0) , u c(1) , and ¯␦ t(0) is closed.
共18兲
where D 2 c(1) ⫽4y c(1) yy . c. The Souza form of the equations. The previous equations can also be written in the form12
L c 共 1 兲 ⫽r ¯2
¯␦ t 共 0 兲 ⫺u c 共 1 兲 ⫹w 共 0 兲¯␦ zt共 0 兲 ⫽0,
兴兴 ⫽0.
共23兲
Here and in the following, I use the notation 关关 f 兴兴 ⬅ f 共 ¯␦ t 共 0 兲 ⫹ 兲 ⫺ f 共 ¯␦ t 共 0 兲 ⫺ 兲
for the jump on the interface. This continuity of the leadingorder pressure means that the expression 共13兲 for p (0) is correct even through the interface. The first order of the continuity of the normal velocity 共7兲 yields 关关 u c 共 1 兲 兴兴 ⫽¯␦ zt共 0 兲 关关 w 共 0 兲 兴兴 ,
共24兲
which can be written for c(1) as 关关 zc 共 1 兲 兴兴 ⫽⫺¯␦ t 共 0 兲¯␦ zt共 0 兲 关关 w 共 0 兲 兴兴 .
共25兲
The axisymmetric part of the kinematic boundary condition 共6兲 at first order gives
3. Vortex ring bubble
For a vortex bubble the interface dynamics has to be found and the previous equations have to be completed. The leading-order of the axisymmetric part of the pressure jump 共9兲 is p
共0兲
共 ¯␦ b 共 0 兲 ⫹ 兲 ⫽
冉 冊 ⌫
2
2
C p 共 s, ,t 兲 共 ¯␦ b 共 0 兲 兲 2
⫽ ¯P v ⫹ ¯P g0
冉 冊 V
b共 0 兲 0
V
b共 0 兲
⫹ p 共 0 兲共 ⬁ 兲 k
⫺
¯ ⌼ ¯␦ b 共 0 兲
,
共27兲
where
冉
2 ¯␦ b 共 0 兲 C p 共 s, ,t 兲 ⫽⫺ ⌫
冊冕 2
2
r ,t 兲 v 共 0 兲 共¯,s, dr ¯. ¯␦ b 共 0 兲 ¯r ⬁
Equation 共27兲 is the equation of the thickness ¯␦ b(0) of the bubble. The leading-order for axisymmetric part of the dynamical equation of the free-boundary 共10兲 is
¯␦ b 共 0 兲 ⫺u c 共 1 兲 ⫹w 共 0 兲¯␦ zb 共 0 兲 ⫽0,
共28兲
where all velocity components are taken on the freeboundary ¯⫽ r ¯␦ b(0)⫹ . The bubble allows to have a solution of the equation of continuity 共14兲 in the form u c共 1 兲⫽
Dc 共 1 兲 共 s, ,t 兲 ⫹u c 共 1 兲 , ¯r
where u c(1) is regular at ¯⫽0. r As the thickness ¯␦ b(0) is given by Eq. 共27兲, Eq. 共28兲 is indeed the equation for Dc(1) . This system for p (0) , v (0) , w (0) , u c(1) , Dc(1) , and ¯␦ b(0) is closed. B. The two-time-scale dynamics of the centerline
In the previous subsection the short-time-scale dynamical equations of the axial core-variations were given for the velocity field in the core. In this subsection the dynamical equation of the induced velocity of the centerline is given. It comes from the matching law of the inner and outer velocity fields. This motion of the centerline is slaved by the leadingorder core dynamics.
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Phys. Fluids, Vol. 14, No. 12, December 2002
Axial core-variations of axisymmetric shape
1. Combined form of the equation
At leading order the outer velocity is v共 x兲 ⫽
⌫ 4
冕
共 0 兲 共 s,t 兲 t共 s,t 兲共 x⫺X共 0 兲 共 s,t 兲兲 ds. 兩 x⫺X共 0 兲 共 s,t 兲 兩 3
共29兲
This is the velocity induced by the vorticity ⫽ ␦ (x ⫺X(0) )t concentrated on the leading-order centerline X(0) , where ␦ (x⫺X(0) ) is the delta-function on X(0) (s,t). The axial core-variations have no leading-order effect in the outer region. This results of the assumption that fast oscillations of the centerline are of amplitude as stated in the form of the expansion of X in Sec. II. At first order the matching law between this outer solution and the inner solution 共found in Sec. III A, and Appendix A兲 yields
t X共 0 兲 共 s,t 兲 ⫹ X共 1 兲 共 s, ,t 兲
冋 冉 冊
册
S共0兲 K 共 0 兲 共 s,t 兲 ⫽A共 s,t 兲 ⫹⌫ log ⫺1⫹C v ⫹C w b共 s,t 兲 , 4
The regime studied in this section is not the same as the one considered by Ting and Klein,23 who studied axial corevariations on an open vortex filament by means of a singletime-scale t and double-axial-scale (s,˜ ⫽s) analysis. For an open filament, the double-time-scale analysis (t, ⫽t/) induces that the normal-time t behavior of a core-variation perturbation, which evolves at short-time-scale ⫽t/, is to reach the far-distance ˜ ⫽s of the Ting and Klein23 regime. Thus, except if the open filament is periodical, a doubletime-scale analysis (t, ⫽t/) coupled to a double-axialscale (s,˜ ⫽s) analysis would be needed to describe the dynamics of the open filament. For a vortex bubble, the matching law between the outer solution and the inner solution yields the equation
t X共 0 兲 共 s,t 兲 ⫹ X共 1 兲 共 s, ,t 兲 ⫽A共 s,t 兲 ⫹⌫
A共 s,t 兲 ⫽
⌫ 4
冕
⫹
⫺
a⫽ 共 0 兲 共 s⫹s ⬘ ,t 兲 共0兲
冋
a ds ⬘ ,
册
t共 s⫹s ⬘ ,t 兲 ⫻g K 共 0 兲 共 s,t 兲 b共 s,t 兲 ⫺ , 兩 g兩 3 2 兩 共 s,s ⬘ ,t 兲 兩
共0兲
g⫽X 共 s,t 兲 ⫺X 共 s⫹s ⬘ ,t 兲 , and (s,s ⬘ ,t)⫽ 兰 ss⫹s ⬘ (0) (s * ,t)ds * . In this Eq. 共30兲, C v (s, ,t) and C w (s, ,t) are known functions, which describe the circumferential and axial evolution of the inner velocity in the core:
冉 冕
42 1 C v 共 s, ,t 兲 ⫽ ⫹ lim 2 ¯r →⫹⬁ ⌫ 2
冊
¯r
0
¯r ⬘ v
共 0 兲2
¯⬘ 共¯r ⬘ ,s, ,t 兲 dr
⫺log ¯r , C w 共 s, ,t 兲 ⫽⫺
2
⬁
0
2. Time averaging and splitting form of the equation
The -average of a function f ( ,t), is denoted Mf , and is defined by Mf ⫽ lim ˜T →⫹⬁
1 T
冕
˜ ⫺1 t⫹T
f 共 ,t 兲 d ,
⫺1 t
˜ Ⰶt. The -average where ˜T is an intermediate variable: ⰆT of Eq. 共30兲 yields the leading-order equation of motion of the filament in the normal-time scale
冋 冉 冊
S共0兲 K 共 0 兲 共 s,t 兲 log ⫺1 4
册
2
¯w r 共 0 兲 共¯,s, r ,t 兲 dr ¯.
Equation 共30兲 extends the Callegari and Ting1 equation of vortex filament motion to axial core-variations of the leading-order velocity field. It holds both for a continuous or a discontinuous vorticity field. Any initial condition that does not satisfy the induced asymmetric flow field 关Eq. 共A8兲 in Appendix A兴, and Eq. 共30兲 will need a three-time-scale analysis (¯t ⫽t/ 2 , ,t). These small-amplitude oscillations of order 2 have already been introduced by Ting and Tung21 and Gunzburger22 to study a straight vortex filament with an initial velocity that is different from the potential background velocity on the filament. An example of such a curved filament, that does not satisfy the induced asymmetric flow field 关Eq. 共A8兲 in Appendix A兴, is given in Margerit and Brancher.13
共31兲
¯ e (s, ,t)⫽4 2¯␦ b(0) (s, ,t)⌼ ¯ /⌫ 2 . Here, the inner vewhere W locity field of Sec. III A and Appendix A was used. The adding term in Eq. 共31兲 as regard of Eq. 共30兲 is due to the difference between the inner velocity fields.
t X共 0 兲 共 s,t 兲 ⫽A共 s,t 兲 ⫹⌫
冉 冊冕
1 4 2 ⌫
冋 冉 冊 册
S共0兲 K 共 0 兲 共 s,t 兲 log 4
¯ e b共 s,t 兲 , ⫺1⫹C v ⫹C w ⫹W
共30兲 where
4411
⫹MC v ⫹MC w b共 s,t 兲 , where
冉 冕
42 1 MC v ⫽ ⫹ lim 2 ¯r →⫹⬁ ⌫ 2 MC w ⫽⫺
冉 冊冕
1 4 2 ⌫
2
⬁
0
¯r
0
2
共32兲
冊
¯r ⬘ M共 v 共 0 兲 兲 dr ¯ ⬘ ⫺log ¯r , 2
¯M r 共 w 共 0 兲 兲 dr ¯.
The subtraction of Eq. 共32兲 from 共30兲 leads to the equation for X(1) in the short-time scale:
X共 1 兲 共 s, ,t 兲 ⫽
⌫K 共 0 兲 共 s,t 兲 共 ⌬C v ⫹⌬C w 兲 b共 s,t 兲 , 4
共33兲
where
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4412
Phys. Fluids, Vol. 14, No. 12, December 2002
⌬C v ⫽
42 ⌫2
⌬C w ⫽
冕
⬁
0
2
冉 冊冕
1 4 2 ⌫
2
¯r 关v 共 0 兲 ⫺M共 v 共 0 兲 兲兴 dr ¯, 2
⬁
0
2
2
¯r 关 w 共 0 兲 ⫺M共 w 共 0 兲 兲兴 dr ¯.
The first-order equation of motion of the filament in the normal-time scale, i.e., the equation for t X(1) , would come from the matching at next order. An important result stated by Eq. 共33兲 is that the fast oscillations of small -amplitude of the centerline are only in the binormal direction and are proportional to the local curvature. C. The two-time-scale dynamics of axial variations in the filament
The previous subsection shows that we need to know the 2 2 normal-time average M( v (0) ) and M(w (0) ) of the square of the velocity components to find the normal-time evolution of the centerline. I will now give qualitative ideas of the full two-time-scale dynamics. 2 2 Let us first assume that M( v (0) )⫽M( v (0) ) and that the axial velocity also satisfies this property. Let us have the conjecture that core equations 共13兲–共16兲 give the dynamics of the short-time variations around this averaged state M( v (0) ) but do not give the dynamics of this averaged state and that these variations are bounded. We need to look at the axisymmetric part of the equations at second order to extract the needed dynamical equations of the averaged state. The second-order equations 关Eqs. 共B1兲–共B4兲 in Appendix B兴 is a linear system of equations for p c(1) , v c(1) , w c(1) , and u c(2) . This system gives the dynamics of the first-order axisymmetric axial variations. It has inhomogeneous terms and nonconstant coefficients, which depend only on the leading-order velocity field is satisfied. Let us also have the second conjecture that this linear operator is not uniquely invertible. The Fredholm alternative implies that this inhomogeneous linear system of equations has bounded solutions only if a compatibility condition for the leading-order velocity field. This compatibility condition is the dynamical equation of the leading-order time-averaged state. More theoretical and numerical works have to be done to prove these two conjectures. I will not do this work in this paper. Nevertheless, as a first step, Sec. IV gives one-time solutions of the equations and Sec. V studies the two-timescale dynamics of small axial variations around these onetime-scale solutions. The study of this linearized leadingorder operator may also help to study the linear system of equations for p c(1) , v c(1) , w c(1) , and u c(2) and to carry out its Fredholm alternative.
Daniel Margerit
solution at leading order, which may be called a quasistationary solution of these equations. Leading-order velocity fields without axial variations and u c(1) ⫽0 are solutions of these leading-order compatibility conditions. Appendix C considers the uniqueness of these compatibility conditions. The study of small perturbations around the solutions without axial variations seems to indicate that they are the unique solutions to these compatibility conditions. Appendix D considers the uniqueness of the compatibility conditions at second order. Assuming that the compatibility conditions at first order have the only solution without axial variations it is found that these compatibility conditions at second order also have a unique solution, which is given. This solution is the one introduced by Margerit17 to generalize the Callegari and Ting theory1 at next order. A. The one-time equations
I now consider the leading-order velocity fields without axial variations. For a closed filament, the s-average of quasi-stationary 共one-time兲 solutions of axisymmetric equations at second order 关Eqs. 共B1兲–共B4兲 in Appendix B兴 satisfies1 ˙S 共 0 兲 1 v共0兲 ⫺¯ 共r0 兲 ⫽ ¯r 共 0 兲 共 0 兲 , t 2 S
冉 冊
w共0兲 1 w共0兲 1 r ¯共r0 兲 兴¯r ⫽ ¯r 3 ⫺¯ 关¯w t ¯r 2 ¯r 2
˙S 共 0 兲 共0兲 , ¯r S
共35兲
where the leading-order quasi-stationary velocity is without axial variations and u c(1) ⫽0 as previously stated. Equations 共32兲, 共34兲, and 共35兲 derived by Callegari and Ting1 are a complete set of equations for the one-time solution, which is without axial variations. This one-time solution is in some sense the generalization to vortex filaments with centerline of any shape of the stationary circular vortex ring solution in a translative frame. B. The one-time solutions in dimensionless form
Callegari and Ting1 used a special transformation to find the solutions of Eqs. 共32兲, 共34兲, and 共35兲. In the remaining of this section the core-function C v (t) and C w (t) are given and displayed in a simple way. These expressions of the core functions and Eq. 共32兲 are a complete set of equations for the one-time motion of the centerline of the filament. Let us define the following similarity functions v * 共 0 兲 ⫽ v 共 0 兲¯␦ /⌫,
* 共 0 兲 ⫽ 共 0 兲¯␦ 2 /⌫,
IV. THE ONE-TIME FILAMENT SOLUTIONS
In this section I give solutions to the one-time equations. These solutions will be used in Sec. V to study the two-timescale dynamics of small axial variations around this base flow. If the short-time scale derivative is removed from Eqs. 共13兲–共16兲 or from the equivalent equations 共17兲–共18兲, these equations become compatibility equations for the one-time
共34兲
w*
共0兲
⫽w
共0兲
¯␦ 2
*
⫽
⫺2
共0兲 ⌫¯␦ 0 S 共 t 兲
K* 共 0 兲 ⫽K共 0 兲 /⌫, c共 1 兲
冉 冊 S 共00 兲
c共 1 兲
1
冉 冊 S 共00 兲
共0兲 ⌫¯␦ 0 S 共 t 兲
,
⫺2
,
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Phys. Fluids, Vol. 14, No. 12, December 2002
Axial core-variations of axisymmetric shape
冕 * 冕 *
where ¯␦ 0 , S (0) are the initial thickness and length of the 0 (0) ¯ vortex and ␦ , S their values at time t.
C* n⫽
1. Inviscid fluid
D* n⫽
If the fluid is inviscid (¯ ⫽0) the solutions are in the form24
C 0* ⫽S w 共 0 兲 /2 ,
⬁
0
w * 共 0 兲 ⫽w * 0 共 兲, ¯/¯␦ , ( v * where ⫽r 0 ( ),w * 0 ( )) are the initial velocity ¯ fields, and ␦ (t) is the -stretched thickness of the core ¯␦ 2 共 t 兲 ⫽¯␦ 20 共 S 共00 兲 /S 共 0 兲 共 t 兲兲 , S (0) 0
where is the initial length of the filament. The associated core functions are
⬁
C w 共 t 兲 ⫽C w 共 0 兲共 S 共00 兲 /S 共 0 兲 共 t 兲兲 3 , where C v (0) and C w (0) are the associated initial core constant:
冉 冕
1 C v 共 0 兲 ⫽ ⫹ lim 4 2 2 →⫹⬁
0
⬘ v 0* 2 共 ⬘ 兲 d ⬘ ⫺log
1 C v 共 t 兲 ⫽⫺log ¯␦ ⫹ 共 1⫹ ␥ ⫺log 2 兲 2 ⫹4 2
冕 ⬁
0
2 w* 0 共 兲d.
冋
⬁
册
1 2 2 2 ⫺n ⫽ D* , 共 1⫺e ⫺ 兲 ⫹e ⫺ n P n 共 兲 1 ¯ 2 n⫽1
冋
兺
⬁
册
S w共 0 兲 ⫺2 2 e ⫹2e ⫺ C n* L n 共 2 兲 1 ¯⫺n , n⫽1
兺
¯/¯␦ , and ¯␦ (t) is the -stretched thickness of the where ⫽r core ¯␦ 2 共 t 兲 ⫽¯␦ 20
1 ¯ ⫽1⫹ ¯␦ ¯2 ⫽4¯
冉 冊 S 共00 兲
S 共 0 兲共 t 兲
¯␦ ¯2
¯␦ 20
冕
1 ¯ ,
冕
⬁
S 共 0 兲共 t ⬘ 兲
0
S 共00 兲
冉 冊冉 冊冋 2
S 共 0 兲共 t 兲
¯␦
兺
共 n,m 兲 苸N2 \ 共 0,0兲
1 ¯⫺ 共 n⫹m 兲 ,
4
S 共00 兲
S w2 共 0 兲
册
* A nm 1 ¯⫺ 共 n⫹m 兲 , C n* C m
0
e ⫺2x L n 共 x 兲 L m 共 x 兲 dx⫽
dt ⬘ .
Here, L n are the Laguerre polynomials, P n ( 2 )⫽L n⫺1 ( 2 ) ⫺L n ( 2 ), ␥ is the Euler’s constant, ¯␦ ¯ is the diffusion-added -stretched thickness of the core, and (C n* ,D n* ) are the Fourier components of the initial axial velocity and tangential vorticity
共 n⫹m 兲 ! . n!m!2 m⫹n⫹1
In the inviscid limit ¯ →0, we recover the inviscid velocity field previously given. This clearly shows the continuity of the analyses for the Navier–Stokes and Euler equations in the asymptotic ansatz based on the small slenderness ratio. 3. Similar vortex core
For a viscous similar vortex core1 v *共 0 兲⫽
1 2 共 1⫺e 兲 , 2
w *共 0 兲⫽
S w共 0 兲 ⫺2 e ,
1 C v 共 t 兲 ⫽⫺log ¯␦ 共 t 兲 ⫹ 共 1⫹ ␥ ⫺log 2 兲 , 2 C w 共 t 兲 ⫽⫺2
,
t
n⫹m
2 \ 0,0兲 共
¯␦ 0
⫹8 2
A nm ⫽
If the flow is viscous (¯ ⫽0) the solutions of Eqs. 共34兲– 共35兲 and 共30兲 are in the form25
w *共 0 兲⫽
C w 共 t 兲 ⫽⫺2
* A nm D* n Dm
兺 共 n,m 兲 苸N
where
2. Viscous fluid
v*
共 兲 L n共 2 兲 d ,
S w (0)⫽m 0 /(⌫¯␦ 0 ) is the initial swirl number, where m 0 is the initial axial flux. The swirl number S w (t) at time t is defined by S w (t)⫽m(t)/(⌫¯␦ ), where m(t) is the axial flux at time t. The associated core functions are
冊
⫺log ¯␦ 0 ,
1
0
0
C v 共 t 兲 ⫽C v 共 0 兲 ⫺log共 ¯␦ 共 t 兲 /¯␦ 0 兲 ,
共0兲
w 0 共 兲 L n共 2 兲 d ,
D 0* ⫽1/2 .
v * 共 0 兲 ⫽ v 0* 共 兲 ,
C w 共 0 兲 ⫽⫺2 2
4413
冉 冊冉 冊 ¯␦ 0 ¯␦
2
S 20
S 共 0 兲共 t 兲
4
S w2 共 0 兲 .
The relative velocity field of this similar vortex also depends only on one parameter: the initial swirl number S w (0) and is independent of any parameter if the axial velocity w * (0) is divided by this parameter. 4. Discontinuous vorticity field
For a vortex with the vorticity inside a vortex tube, the s-average of quasi-stationary 共one-time兲 solutions of the axi-
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4414
Phys. Fluids, Vol. 14, No. 12, December 2002
Daniel Margerit
symmetric part of the kinematic boundary condition at second order 关Eq. 共B6兲 in Appendix B兴 satisfies 1 2
␦ថ t 共 0 兲 ⫽⫺ ¯␦ t 共 0 兲
˙S 共 0 兲 . S共0兲
共36兲
¯S˙ (0) /2, where 具 典 denotes the Here, I used 具 (0) u c(2) 典 ⫽⫺r s-average, as can be found from the s-average of the axisymmetric part of the continuity equation at second order 关Eq. 共B2兲 in Appendix B兴. It comes 共 ¯␦ t 共 0 兲 共 t 兲兲 2 ⫽ 共 ¯␦ t0共 0 兲 兲 2 共 S 共00 兲 /S 共 0 兲 共 t 兲兲 .
共37兲
This equation is coherent with the function ¯␦ 2 (t)⫽¯␦ 20 (0) (S (0) (t)) previously introduced to describe the thickness 0 /S of the continuous vorticity field in an inviscid fluid (¯ ⫽0). In this case I choose the thickness ¯␦ of the vortex to be the interface thickness ¯␦ t . For a Rankine vortex core with a uniform axial jet:
v* 0 共 兲⫽
再
2 1 2
,
w 0* 共 兲 ⫽
再
S w共 0 兲 /
if ⬍1
0
if ⬎1.
the base flow, is split into the relative velocity field and the filament motion. In the previous analysis the perturbation that is used is the one of the absolute velocity field and is the superposition of axisymmetric modes e i ns e i and bending modes e ⫾i e i ns e i , where the coordinates are local to the base flow. In the case of a circular vortex ring, the results of my analysis can be derived from the one of Kopiev and Chernyshev9 by: 共i兲 deriving the velocity field from their displacement field; 共ii兲 writing this field in the usual coordi¯, ,s) of the moving frame; and 共iii兲 splitting the nates (r velocity perturbation into the relative velocity and the filament velocity. I prefer to derive it in a simpler and straightforward way as follows. A. Base flow and small perturbations
I introduce a small axisymmetric perturbation ˜ 共 0 兲 ,X ˜ 共 0 兲 ,X ˜ 共1兲兲 ˜ 共 0 兲 ,p ˜ 共 0 兲 , ˜ c 共 1 兲 ,K 共 ˜u c 共 1 兲 ,˜v 共 0 兲 ,w of a one-time flow without axial variations, denoted by គ (0) ,X(0) ,X(1) ⫽0): (uគ c(1) ⫽0,vគ (0) ,wគ (0) ,pគ (0) , គ c(1) ,K
共38兲
It comes C v (t)⫽3/4⫺log ¯␦ and C w (0)⫽⫺4S w (0) 2 . The relative velocity field of this Rankine vortex core with a uniform axial jet depends only on one parameter: the initial swirl number S w (0) and is independent of any parameter if the axial velocity w * (0) is divided by this parameter. V. TWO-TIME-SCALE DYNAMICS OF AXIAL VARIATIONS IN THE SMALL AMPLITUDE LIMIT
In this section, I consider small axisymmetric axial variations around the one-time scale solutions 共32兲, 共34兲, and 共35兲, which is the base flow. The leading-order equations of these perturbations will be found as a linearization near this base flow of the double-time-scale equations for the core 共17兲, 共18兲, and for the filament motion 共32兲–共33兲. This gives the equations 关Eqs. 共39兲–共41兲兴 of the dynamics of the small axial variations around the one-time base flow. From these equations the eigenvalue equations for linear Fourier modes 关Eqs. 共59兲–共61兲兴 are given for the stream function. This eigenvalue problem is then solved for both a Rankine and a similar core. This study is more general than previous ones because the vortex filament of the base flow is not restricted to be circular26,27,9 nor straight.28 –31 In the studies28 –30 of the stability of straight vortex filament, the characteristic lengthscale that is used is the one of the thickness of the filament and so the long-wavelength limit has to be carried out to obtain our O(1) wavelength regime, in which the characteristic length-scale that is used is the one of the radius of curvature of the filament. This study is a linear stability analysis in the small thickness limit and in the moving frame of the perturbed flow. The coordinates are local coordinates in this frame and are not local coordinates to the base flow. The perturbation, as
˜ c共 1 兲, c 共 1 兲 ⫽ គ c 共 1 兲 ⫹ ˜ 共 0 兲, គ 共 0 兲⫹ K K共 0 兲 ⫽K ˜ 共 1 兲. X共 1 兲 ⫽ X For a vortex with the vorticity inside a vortex tube, the interface function is also unknown. Its perturbation and the one of the pressure are given by ¯␦ t 共 0 兲 ⫽¯␦គ t 共 0 兲 ⫹ ␦ន t 共 0 兲 , p 共 0 兲 ⫽ pគ 共 0 兲 ⫹ ˜p 共 0 兲 . The base flow is the one-time scale solution given in Sec. IV and is without axial variations. Here, one has to restrict the form of the perturbations to the axisymmetric axial variations, for they are the perturbations we are interested by. In that sense I will not consider normal-time perturbations without axial variations of the relative velocity field or of the centerline. This induces the two following assumptions. The perturbations of the relative velocity field have axial core-variations and are assumed to have null axial average. With this assumption and with the uniqueness study of Appendix C we deduce that these perturbations have no normal-time-scale dynamics, i.e., M(˜v (0) )⫽0 and ˜ (0) )⫽0. Moreover as the motion of the centerline at M(w leading order X(0) is a normal-time-scale dynamics I assume ˜ (0) that the leading-order centerline is not perturbed, i.e., X ⫽0. These two assumptions are not restrictive; they only means that in the perturbation we do not have the bending modes of the one-time scale, which have already been studied elsewhere.10,11
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Phys. Fluids, Vol. 14, No. 12, December 2002
Axial core-variations of axisymmetric shape
B. Two-time-scale linear equations of small axial variations
for the jump on the interface. The continuity of the normal velocity 共25兲 becomes c共 1 兲
dd ˜ z
1. Continuous vorticity field
From Eqs. 共17兲–共18兲 we deduce that at first order in the small amplitude the perturbation of the relative velocity field satisfies ˜ 共0兲
˜ c共 1 兲
K ⫺2K គ 共y0 兲 z D2
⫹2 គ cy 共 1 兲
˜ 共0兲
K ⫽0, z
冉 冊
˜ 共0兲 ˜ c 共 1 兲 2 共 0 兲 K ˜ c 共 1 兲 គ ⫹G ⫹ K ⫽0, y z z
I define the following similarity functions for the perturbation y * ⫽ 2 ⫽y/¯␦ 2 ,
⌫K 共 0 兲 共 s,t 兲 共 ⌬C v ⫹⌬C w 兲 b共 s,t 兲 , 4
⬁
共41兲
* ⫽⌫ /¯␦ 20 , z * ⫽z¯␦ /¯␦ 20 , ˜ * 共 0 兲 ⫽K ˜ 共 0 兲 /⌫, K ˜ * c 共 1 兲 ⫽ ˜ c 共 1 兲 / 共 ⌫¯␦ 兲 , ˜v * 共 0 兲 ⫽˜v 共 0 兲¯␦ /⌫, ˜ * 共 0 兲 ⫽w ˜ 共 0 兲¯␦ /⌫, w ˜X * b 共 1 兲 ⫽X ˜ 共 1 兲 •b共 s,t 兲 / 共 ¯␦ 20 K 共 0 兲 共 s,t 兲兲 . For a vortex with the vorticity inside a vortex tube I also define
˜p * 共 0 兲 ⫽ 共 ¯␦គ t 共 0 兲 兲 2˜p 共 0 兲 /⌫ 2 .
2r ¯ vគ 共 0 兲˜v 共 0 兲 dr ¯,
冉 冊冕 2
⬁
0
These similarity functions are now used to simplify the system of linear equations for the small axial variations.
2r ¯w គ 共 0 兲w ˜ 共 0 兲 dr ¯.
This shows that for a curved vortex filament the perturbations with axial variations induce small oscillations of amplitude of the centerline and that these perturbations are in the binormal direction and proportional to the curvature. The system of Eqs. 共39兲–共41兲 gives the dynamics of the small axial variations around the one-time base flow.
1. Continuous vorticity field
With these functions the system 共39兲–共41兲 becomes ˜ *共 0 兲 ˜ *c共 1 兲 ˜ 共0兲 K c 共 1 兲 K* 共0兲 * ⫺2K គ* ⫹2R ⫽0, គ y* y* * z* z* D *2
2. Discontinuous vorticity field
For a vortex with the vorticity inside a vortex tube, the continuity of the leading-order pressure 共23兲 becomes dd ˜p 共 0 兲 cc ⫽⫺
共44兲
␦ន * t 共 0 兲 ⫽ ␦ន t 共 0 兲 /¯␦គ t 共 0 兲 ,
0
1 4 ⌬C w ⫽ 2 ⌫
␦ន t 共 0 兲 ⫺u ˜ c 共 1 兲 ⫹wគ 共 0 兲 ␦ន zt共 0 兲 ⫽0,
C. Dimensionless form of the linear equations
where
冕
The kinematic boundary conditions 共26兲 becomes
共40兲
These equations give the short-time dynamics of the small amplitude axial variations in the filament. 2 At first order in the -averages M( v (0) ) and 2 2 2 M(w (0) ) are given by M( v (0) )⫽M( vគ (0) ⫹2 vគ (0)˜v (0) ) 2 2 as M(˜v (0) )⫽0, and ⫽ vគ (0) ⫹2 vគ (0) M(˜v (0) )⫽ vគ (0) , (0) 2 (0) 2 គ . This means that the small perturbations of M(w )⫽w the relative velocity field have no normal-time-scale dynamics as it has previously been assumed. From Eqs. 共32兲 we can check that the leading-order centerline is not perturbed, i.e., ˜ (0) ⫽0, as it has previously been assumed. X From Eqs. 共33兲 we deduce that the perturbation of the filament velocity satisfies
42 ⌫2
共43兲
where all velocity components are taken on the free boundary ¯⫽ r ¯␦គ t(0)⫾ .
G⫽2 គ cy 共 1 兲 D 2 ⫺8y គ cy 共y1y兲 .
⌬C v ⫽
គ 共 0 兲 cc ␦ន zt共 0 兲 . cc ⫽⫺˜␦គ t 共 0 兲 dd w
共39兲
where
˜ 共 1 兲 共 s, ,t 兲 ⫽ X
4415
˜␦ t 共 0 兲 ¯␦គ t 共 0 兲
2
dd vគ 共 0 兲 cc ,
共42兲
where I used Eq. 共13兲 to have and in the following I use the notation
Here
冊
˜ *共 0 兲 ˜ * c 共 1 兲 K ˜ * c 共 1 兲 2 ⫹ K គ *共 0 兲 ⫹RG* ⫽0, * y* z* z* 共46兲
˜X * b 共 1 兲 1 ⫽ 共 ⌬C v ⫹⌬C w 兲 , * 4
共47兲
where c共 1 兲
2 (0) dd pគ ¯r cc ⫽ dd vគ (0) cc /¯␦ t(0) .
dd f cc ⬅ f 共 ¯␦គ t 共 0 兲 ⫹ 兲 ⫺ f 共 ˜␦គ t 共 0 兲 ⫺ 兲
冉
共45兲
G* ⫽2 គ * y* R⫽
¯␦ 0 ¯␦
c共 1 兲
D * 2 ⫺8y * គ * y*y*y* ,
冉 冊 S 共00 兲
S 共 0 兲共 t 兲
2
,
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4416
Phys. Fluids, Vol. 14, No. 12, December 2002
冕 冕
⬁
⌬C v ⫽4 2 ⌬C w ⫽16
0
2
K គ *共 0 兲 ⬁
0
Daniel Margerit
␦ន * t 共 0 兲 ⫽ ␦C t 共 S w 共 t 兲兲 e ⫺i *n * ⫹inzˆ ,
˜K* 共 0 兲 dy * , y*
Rw គ *共 0 兲
˜p * 共 0 兲 ⫽ pˆ 共 y * ,S w 共 t 兲兲 e ⫺i n* * ⫹inzˆ .
˜ * c 共 1 兲 dy * , y*
1. Continuous vorticity field
The linear system satisfied by the eigenfunctions is
and D * 2 ⫽4y * y * y * . 2. Discontinuous vorticity field
For a vortex with the vorticity inside a vortex tube, Eqs. 共42兲–共44兲 become 2 dd ˜p * 共 0 兲 cc ⫽⫺˜␦ * t 共 0 兲 dd vគ * 共 0 兲 cc ,
d
共48兲
c
˜ * c 共 1 兲 ␦ន * t 共 0 兲 ⫽ ⫺R dd w គ * 共 0 兲 cc , z* z*
共49兲
␦ន * t 共 0 兲 ˜ * c 共 1 兲 ␦ន * t 共 0 兲 ⫹ ⫹Rw គ *共 0 兲 ⫽0. * z* z*
共50兲
冕
˜ *共 0 兲 K គ *共 0 兲K dy * , y *2 冑y *
c 1兲
2
共 ˜ * y*
⫹
⬁
冉
c共 1 兲
⫹4R ⫺ គ * y*y*
2˜ c共 1 兲 ˜ * c 共 1 兲 c共 1 兲 * ⫹ គ * y* z* y* z*
共51兲
冊
˜p * 共 0 兲 ⫽0. z*
共52兲
m共 t 兲⫽
冉 冊 S 共 0 兲共 t 兲
2S w d 2 ˆ ˆ⫹ គ *共 0 兲K * c 共 1 兲 ˆ ⫽0, 2⫺ 2K dy * 2gy * g គ y*y*y*
共57兲
Xˆ b ⫽
* n
冉冕
⬁
0
K គ *共 0 兲
ˆ K dy * ⫹ y*
冕
⬁
0
4S w wគ * 共 0 兲
冊
d ˆ dy * , dy * 共58兲
where ⫽2 n¯␦ 20 /(S (0)¯␦ n* ) and c共 1 兲
g⫽1⫺2S w គ * y*
.
d 2 ˆ ⫹G 共 y * ,,S w 兲 ˆ ⫽0, dy * 2
共59兲
d ˆ 共 y * →⬁ 兲 ⫽0, dy *
共60兲
ˆ 共 y * ⫽0 兲 ⫽0,
共61兲
where
As the axial flux satisfies S 共00 兲
共56兲
ˆ from Eq. 共56兲 into Eq. 共57兲 gives The substitution of K the following eigenvalue problem for ˆ and :
Equations 共13兲 and 共16兲 become ˜p * 共 0 兲 ⫽⫺
共0兲 ˆ ˆ ⫽⫺2K K គ* y * /g,
共0兲
2
G 共 y * ,,S w 兲 ⫽
m0 ,
the swirl number satisfies S w (t)⫽RS w (0), which means that R is the ratio S w (t)/S w (0) if the axial flux is not zero: S w (0)⫽0. If the base flow is a Rankine vortex core with a uniform axial jet or a similar vortex, I divide both the axial velocity w គ * (0) and គ * c(1) by the initial swirl number S w (0): the base flow is then independent of any parameter and in Eqs. 共45兲– 共47兲 R becomes the swirl number S w (t) at t. For these vortices the stability analysis only depends on this oneparameter S w (t) and I carry out this study in the following.
ˆ 共 y * ,S w 共 t 兲兲 e ⫺i n* * ⫹inzˆ , ˜ * c 共 1 兲 ⫽⌿
共53兲
ˆ 共 y * ,S w 共 t 兲兲 e ⫺i *n * ⫹inzˆ , ˜ * 共 0 兲 ⫽K K
共54兲
˜X * b 共 1 兲 ⫽iXˆ b 共 S w 共 t 兲兲 e ⫺i *n * ⫹inzˆ ,
共55兲
where n* ⫽ n¯␦ 20 /⌫ and zˆ ⫽2 z *¯␦ 20 /(S (0)¯␦ ). For a vortex with the vorticity inside a vortex tube I also look for the perturbation of the interface function and of the pressure in the form
K គ *共 0 兲K គ* y* g 2y *2
c共 1 兲 គ* y*y*y* . ⫹2S w g
From Eq. 共59兲 it comes ˆ ⫽y * ⫹O(y * 2 ) near 0, where I used a normalization condition to select any eigenfunction of this homogeneous equation. 2. Discontinuous vorticity field
For a vortex with the vorticity inside a vortex tube, the linear system must be completed by 2 dd pˆ cc ⫽⫺ ␦C t dd vគ * 共 0 兲 cc ,
共62兲
គ * 共 0 兲 cc ␦C t , dd ˆ cc ⫽⫺S w dd w
共63兲
D. Eigenvalue equations for linear Fourier modes
I look for solution of the linear equations 共45兲–共47兲 in the form
2
pˆ ⫽⫺
冕
ˆ K គ *共 0 兲K dy * , 2 冑y * y * ⬁
冉
c共 1 兲
c共 1 兲
ˆ គ* pˆ ⫽4S w គ * y*y* ⫺ y*
共64兲
冊
d ˆ 2 d ˆ ⫺ , dy * dy *
共65兲
and by the kinematic boundary condition
␦C t ⫽ ˆ 共 1,S w 兲 /g,
共66兲
where all velocity components are taken on the interface ¯r ⫽¯␦គ t(0)⫾ . For a Rankine vortex core with a uniform axial jet, G is given by
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Phys. Fluids, Vol. 14, No. 12, December 2002
G⫽
再
⌳ 2 /y *
if y * ⬍1,
0
if y * ⬎1,
Axial core-variations of axisymmetric shape
4417
where ⌳ 2⫽
2 , 4 2 共 1⫺S w / 兲 2
and for a similar vortex, G is given by G⫽
2 共 1⫺e ⫺y * 兲 e ⫺y * S w e ⫺y * , ⫹ 42 y *2g 2 g
where g⫽1⫺S w e ⫺y * / .
FIG. 1. The first eigenmode 1 / ⫽2.39 of a perturbed Rankine vortex core without swirl (S w ⫽0). The solid line is ˆ (y * ,0), the dotted line is d ˆ /dy * , ˆ (y * ,0). and the dashed line is K
E. Examples of solutions
In this subsection I solve the eigenvalue equations for a Rankine and a similar vortex with a uniform axial jet. 1. Rankine vortex core with a uniform axial jet
2n¯␦ 20
⫽
1 /
⫽S w ⫾
1 j 0i
,
再
ˆ ⫽
⌳ J 1 共 2⌳ 兲 /⌳
再
J 0 共 2⌳ 兲 d ˆ ⫽ dy * 0 ˆ⫽ K
pˆ ⫽
再
再
if ⬍1, if ⬎1, if ⬍1, if ⬎1,
⫺2 J 1 共 2⌳ 兲
if ⬍1,
0
if ⬎1,
⫺
J 0 共 2⌳ 兲 ⌳ 1
⌳ n*
共70兲
c 共 1 兲 ⫽2 ˆ 共 y * ,0兲 sin共 nzˆ 兲 sin共 n 0i 兲 .
共71兲
if ⬍1, if ⬎1,
0
Xˆ b ⫽⫺
ˆ 共 y * ,0兲 cos共 nzˆ 兲 cos共 n 0i 兲 , K共 0 兲 ⫽K គ ⫹2 K
The selected first eigenmode of the velocity field is given in Fig. 1 and the associated core-thickness and centerline evolutions are given in Fig. 2. This result generalizes, to a vortex filament with a centerline of any shape, the bulging modes found by Kopiev and Chernyshev9 on a perturbed vortex of circular centerline. In the peculiar case of a perturbed vortex ring with a circular centerline, the modes found in this theory are the same as in their theory. For example, the interface disturbance of these bulging modes was given in Kopiev and Chernyshev9 in the absolute frame with help of the displacement field representation and one can show that it corresponds to the same
␦ˆ t ⫽2 J 1 共 2⌳ 兲 , J 1 共 2⌳ 兲
共69兲
with the associated velocity field
For a Rankine vortex core with a uniform axial jet, the solution of the linear system 共56兲–共61兲 is ¯␦ S 共 0 兲 * n
¯␦ t 共 0 兲 ⫽¯␦គ t 共 0 兲 ⫹2 ¯␦ 0 ␦C t 共 0 兲 cos共 nzˆ 兲 cos共 n 0i 兲 ,
共 J 2 共 2⌳ 兲 ⫺4S w J 1 共 2⌳ 兲兲 ,
where J 0 , J 1 are Bessel functions of the first kind, and j 0i is the ith zero of the Bessel function J 0 . The frequency n of these oscillations is
n ⫽n 共 Sw ⫾ 0i 兲 ,
共67兲
where
Sw ⫽2⌫S w / 共 ¯␦ S 共 0 兲 兲 , 0i ⫽2⌫/ 共 ¯␦ S 共 0 兲 j 0i 兲 . Without axial flux 关 S w (t)⫽0 兴 the shape of the filament and of the axial variations of the core are given by ᠪ 共 0 兲 ⫹2 共 ¯␦ 0 兲 2 Xˆ b 共 0 兲 cos共 nzˆ 兲 sin共 n 0i 兲 X共 0 兲 ⫽X ⫻K 共 0 兲 共 s,t 兲 b共 s,t 兲 ,
共68兲
FIG. 2. Fast axial-core oscillations 共mode n⫽4) of a vortex filament with a perturbed Rankine vortex core without swirl and the induced filament oscillations. 共a兲 Initial variation of the core without perturbation of the filament, evolution on 共b兲 one-fourth of the period, 共c兲 half of the period, 共d兲 threefourths of the period.
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4418
Phys. Fluids, Vol. 14, No. 12, December 2002
Daniel Margerit
binormal perturbation of the centerline and the same thickness perturbation as I found. In my approach it is easy to understand that the fast-oscillations of the centerline are in the binormal direction as the core of the vortex is known to be taken into account by core-functions coefficients C v and C w in the binormal term of the equation of motion of Callegari and Ting.1 It is also easy to see the consistence of my generalization with the theory of Callegari and Ting1 and of most studies of vortex dynamics as I use the same coordinates as most authors and I give the velocity field. In this sense this approach also completes the one of Kopiev and Chernyshev,9 who introduced their own interesting coordinates and give the components of the displacement field and some components of the velocity field on their special coordinates. The frequency of the bulging modes are found to be the same as the one they found for the perturbed vortex ring with a circular centerline, provided that ⌫/(¯␦ S (0) ) is used to have a dimensionless frequency. My approach is restricted to axisymmetric perturbations: the bulging modes on the shorttime and the bending modes on the normal-time t. Kopiev and Chernyshev9 considered all nonshort wavelength perturbations and so also have the ultra-short-time ¯t dynamics of nonaxisymmetric perturbations on the vortex ring with a circular centerline. This study also generalizes, to a vortex filament with a centerline of any shape, the bulging modes found on a perturbed straight filament28,30 in the long-wavelength limit. In the peculiar case of this straight filament the result is consistent with their long-wavelength limit for the axisymmetric and bending modes. 2. Similar vortex
For a similar vortex, Eq. 共59兲 gives ˆ ⫽c 0 ⫹c 1 y * ⫹O 关 exp(⫺y*)兴 at infinity, where c 0 and c 1 are two constants. For any value of , the solution of Eq. 共59兲 and ˆ ⬃y * near 0 asymptotically reaches a constant ˆ y * (y * →⬁), that is zero only for an infinity number of selected values i of . I use a shooting method and a Runge–Kutta solver to find these eigenvalues i (S w ). The frequency n of these oscillations is
n⫽
2⌫n
1
¯␦ S 共 0 兲 i 共 S w 兲 /
.
共72兲
FIG. 3. The first eigenmode of a similar vortex without swirl (S w ⫽0). The solid line is ˆ (y * ,0), the dotted line is d ˆ /dy * , and the dashed line is ˆ (y * ,0). The associated filament eigenmode is Xˆ b ⫽⫺0.096. K
vortex bubble and for the bubble thickness dynamics. These equations are used to study a circular vortex ring bubble and to compute the motion of a vortex ring bubble of elliptical shape. Finally the ultra-fast oscillations of a vortex ring bubble are studied on the ultra-short time scale. A. The one-time equations
The leading-order velocity fields without axial variations and u c(1) ⫽0 共and thus D (1) ⫽0) are also solutions of the leading-order compatibility conditions 共13兲–共16兲 and 共27兲– 共28兲 in the case of a vortex bubble. The non-axial variations of the thickness ¯␦ b(0) (s,t)⫽¯␦ b(0) (t) of the bubble gives the simplification Vb(0) ⫽S (0) (¯␦ b(0) ) 2 in Eq. 共11兲. For a closed vortex ring bubble, the s-average of the axisymmetric part of the continuity equation at second order 关Eq. 共B2兲 in Appendix B兴 gives
具 共 0 兲 u c 共 2 兲 典 ⫽⫺
S˙ 共 0 兲¯r D c 共 2 兲 共 t 兲 S 共 0 兲 ⫹ , 2 ¯r
where 具 典 denotes the s-average. Here, D c(2) (t)⫽0 is allowed and is required to because a singularity can exist at ¯⫽0 r satisfy the s-average of Eq. 共10兲: D c 共 2 兲 共 t 兲 ⫽¯␦ b 共 0 兲
冉
冊
¯␦ b 共 0 兲 1 S˙ 共 0 兲 b 共 0 兲 ¯␦ , ⫹ t 2 S共0兲
共73兲
where ¯␦ b(0) is given by Eq. 共27兲.
Without axial flux (S w ⫽0,) it gives 1 / ⫽⫾3.1, 2 / ⫽⫾6.0, and 3 / ⫽⫾9.1. The first three selected eigenmodes are given in Figs. 3, 4, and 5. With S w ⫽0.1, it gives 1 / ⫽(⫺3.3569,2.7520) and 2 / ⫽(⫺8.4073,4.4456). The selected first eigenmode 1 / ⫽2.7520 is given in Fig. 6. VI. ONE-TIME VORTEX RING BUBBLE
In this section I give the one-time equations of a vortex ring bubble without axial variations. A special transformation is then introduced to solve the core equations and the solutions to these equations are given. This gives coupled equations 关Eqs. 共79兲–共80兲兴 for the motion of the centerline of the
FIG. 4. The second eigenmode of a similar vortex without swirl (S w ⫽0). The solid line is ˆ (y * ,0), the dotted line is d ˆ /dy * , and the dashed line is ˆ (y * ,0). K
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Phys. Fluids, Vol. 14, No. 12, December 2002
Axial core-variations of axisymmetric shape
4419
Z 共 ,t 1 兲 ⫽ 共 0 兲 /S 共 0 兲 共 t 兲 . The previous equations become 2D c 共 2 兲 共 t 1 兲 ⫽
关共 ¯␦ b 共 0 兲 兲 2 S 共 0 兲 兴 V b 共 0 兲 ⫽ , t1 t1
1 ⫺¯ 共 兲 ⫽⫺D c 共 2 兲 共 t 兲 , t1
共76兲 共77兲
where stands for W and Z. I then use the following transformation FIG. 5. The third eigenmode of a similar vortex without swirl (S w ⫽0). The solid line is ˆ (y * ), the dotted line is d ˆ /dy * , and the dashed line is ˆ (y * ). K
The s-average of quasi-stationary 共one-time兲 solutions of axisymmetric equations at second order 关Eqs. 共B1兲–共B4兲 in Appendix B兴 satisfies
冋
册
1 v共0兲 S˙ 共 0 兲 2D c 共 2 兲 共 t 兲 , ⫺¯ ¯共r0 兲 ⫽ ¯r 共 0 兲 共 0 兲 ⫺ t 2 ¯r 2 S
冉 冊
w共0兲 1 w共0兲 1 ⫺¯ 关¯w r ¯共r0 兲 兴¯r ⫽ ¯r 3 t ¯r 2 ¯r 2
共74兲
S˙ 共 0 兲 D c 共 2 兲共 t 兲 ¯w¯共r0 兲 , 共 0 兲 ⫺r ¯r 2 ¯r S 共75兲
where the leading-order quasi-stationary velocity is without axial variations and u c(1) ⫽0 as previously stated. Equations 共31兲, 共27兲, 共73兲, 共74兲, and 共75兲 are a complete set of equations for the one-time solution. B. Transformation of the equations
I now solve Eqs. 共73兲–共75兲. I use the following transformation first introduced by Callegari and Ting1 共often referred as the transformation of Lundgren32兲: t 1⫽
冕
t
0
S 共 0 兲 共 t ⬘ 兲 dt ⬘ ,
⫽r¯ 冑S 共 0 兲 共 t 兲 , W 共 ,t 1 兲 ⫽S 共 0 兲 共 t 兲 w 共 0 兲 ,
1 ⫽ 2 ⫺ 共 ¯␦ b 共 0 兲 兲 2 S 共 0 兲 ⫽ 关¯r 2 ⫺ 共 ¯␦ b 共 0 兲 兲 2 兴 S 共 0 兲 , which yields
⫽4¯ 共关 1 ⫹ 共 ¯␦ b 共 0 兲 兲 2 S 共 0 兲 兴 1 兲 1 . t1
共78兲
The bubble allows to have a solution of the equation (0) ⫽(r ¯ v (0) )¯r /r ¯ in the form v 共 0 兲⫽
⌫1 ⫹ v 共 0 兲, 2 ¯r
r and to have the associated where v (0) is regular at ¯⫽0, circulation field K共 0 兲 ⫽
⌫1 ⫹K 共 0 兲 . 2
C. The one-time solutions in dimensionless form
I define the following similarity functions v * 共 0 兲 ⫽ v 共 0 兲¯␦ /⌫, v * 共 0 兲 ⫽ v 共 0 兲¯␦ /⌫,
*共 0 兲⫽ 共 0 兲
冉 冊 冉 冊 冉
¯␦ 2 V ⌫ V0
w * 共 0 兲 ⫽w 共 0 兲
¯␦ 2
⫺1
,
V V0
⌫¯␦ 0
S 共00 兲
⫺1
S 共 0 兲共 t 兲
冊
⫺2
共 1⫺V b0 /V0 兲 1/2,
K* 共 0 兲 ⫽K共 0 兲 /⌫, K* 共 0 兲 ⫽K 共 0 兲 /⌫,
*c共 1 兲⫽ c共 1 兲
1
冉 冊 S 共00 兲
⌫¯␦ 0 S
共0兲
共t兲
⫺2
共 1⫺V b0 /V0 兲 1/2,
(0) where ¯␦ 0 , V0 ⬅(¯␦ 0 ) 2 S (0) 0 , and S 0 are the initial thickness, volume, and length of the vortex and ¯␦ , V⬅(¯␦ ) 2 S (0) , and S (0) their values at time t. If the fluid is inviscid (¯ ⫽0), the solutions are in the form
*共 0 兲⫽ * 0 共 y 兲, FIG. 6. The first eigenmode of a similar vortex with swirl S w ⫽0.1. The solid line is ˆ (y * ,0.1), the dotted line is d ˆ /dy * , and the dashed line is ˆ (y * ,0.1). K
K* 共 0 兲 ⫽K0* 共 y 兲 ⫽
1 ⌫1 1 ⫹ 2 ⌫ 2
冕 * y
0
0
共 y ⬘ 兲 dy ⬘ ,
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4420
Phys. Fluids, Vol. 14, No. 12, December 2002
v *共 0 兲⫽
K* 0 共y兲 ¯/ r ¯␦
Daniel Margerit
¯ e 共 t 兲 ⫽W ¯ e共 0 兲 W
,
where
w * 共 0 兲 ⫽w * 0 共 y 兲, where y⫽
冉冊 ¯r
共 1⫹  共 t 兲兲 ⫺  共 t 兲 ,
with  (t)⫽V b(0) (t)/(V0 ⫺V thickness of the core V⬅ 共 ¯␦ 兲 2 S 共 0 兲 ⫽V0 ⫹V
b共 0 兲
b(0) 0 )
⫺V
and ¯␦ (t) is the -stretched
K
共0兲
共 s,t 兲 关 log共 S 共 0 兲 / 兲 ⫺1 4
¯ e 共 t 兲兴 b共 s,t 兲 , ⫹C v 共 t 兲 ⫹C w 共 t 兲 ⫹W
冉 冊 冉 冊 共 t 兲 共 t 兲 ⫺ ˜P g0 共 0 兲 共 0 兲
共 1⫺k 兲
⫹
S共0兲 S 共00 兲
共79兲
共 C p 关  共 t 兲兴
¯ e 关  共 t 兲兴 兲 ⫽0, ⫹W ˜P v ⫽4
¯␦ t 共 0 兲
1 S˙ 共 0 兲
⫹
2 S共0兲
V
t共 0 兲
⫽V
˜P g0 ⫽4 2 共 ¯␦ b0 共 0 兲 兲 2 ⫺ ¯P g0 /⌫ 2 . These coupled equations 共79兲–共80兲 for the motion of the centerline of the vortex bubble and for the bubble thickness dynamics generalize the equation of motion of Genoux14 to a non-potential vortex bubble with a filament of any shape and with axisymmetric time-variations of its thickness. The expressions of the core functions C v (t) and C w (t), ¯ e that appear in these equations are given in the C p and W following.
b共 0 兲
冉
冊
冕
2 K* 0 共 y⬘兲
1 dy ⬘ ⫺ log共 y y ⫹  t 2 兲 共 0 ⬘ y
冉 冊
1 1 V ⫹  共 t 兲兲 ⫺ log 共 0 兲 ⫹ log共 1⫹  共 t 兲兲 , 2 2 S C w 共 t 兲 ⫽⫺4 2 共 S 共00 兲 /S 共 0 兲 共 t 兲兲 3 C p 共 t 兲 ⫽⫺2 2  共 t 兲
冕
⬁
0
冕
⬁
0
w 0* 2 共 y 兲 dy,
2 K* 0 共y兲
共 y⫹  共 t 兲兲
2
dy,
共81兲
,
共82兲
⫺V
b共 0 兲 0
w* 0 共 y 兲⫽
再
再
冉
1 ⌫1 共 1⫺y 兲 ⫹y 2 ⌫
冊
1 2
if 0⬍y⬍1, 共83兲 if y⬎1,
S w共 0 兲 /
if 0⬍y⬍1,
0
if y⬎1,
共84兲
where S w (0)⫽m 0 /(⌫¯␦ 0 冑1⫺V b0 /V0 ) is the initial swirl number, and m 0 is the initial axial flux. The swirl number S w (t) at time t is defined by S w (t)⫽m(t)/(⌫¯␦ 冑1⫺V b /V), where m(t) is the axial flux at time t. These two swirl numbers are related by S w共 t 兲 ⫽
¯␦ 0 ¯␦
冉 冊冉 冊 S 共00 兲
S 共 0 兲共 t 兲
It comes
1 C v 共 t 兲 ⫽ ⫹ lim 2 2 2 y→⫹⬁
¯␦ t 共 0 兲
previously introduced to describe the volume of the continuous vorticity field in an inviscid fluid (¯ ⫽0). In this case, I choose the thickness ¯␦ of the vortex to be ¯␦ t . For a vortex bubble with a core of Rankine type and a uniform axial jet:
K* 0 共 y 兲⫽
The associated core functions are
D c 共 2 兲共 t 兲
t共 0 兲 b共 0 兲 ⫺V b0 共 0 兲 , 0 ⫹V
V⫽V0 ⫹V
共 ¯␦ b0 共 0 兲 兲 2 共 p 共 0 兲 共 ⬁ 兲 ⫺ ¯P v 兲 /⌫ 2 ,
1. Inviscid fluid
¯␦ t 共 0 兲 ⫽
where I used the value of 具 (0) u c(2) 典 . From this equation and Eq. 共73兲, it comes
共80兲
with 2
共 t 兲 , 共 0 兲
where V t(0) ⫽(¯␦ t(0) ) 2 S (0) (t) is the volume of the vortex tube. This equation is coherent with the volume
where  (t) 关and so V b(0) (t) or ¯␦ b(0) ] is obtained from Eq. 共27兲 and is solution of ˜P v
S
共0兲
¯ e (0)⫽4 2¯␦ b(0) ¯ 2 W 0 ⌼ /⌫ .
t
b共 0 兲 . 0
Here, (K* 0 (y),w 0* (y)) are the initial circulation and axial velocity fields. The equation of the filament motion 共31兲 is
t X共 0 兲 共 x,t 兲 ⫽A共 s,t 兲 ⫹⌫
S 共00 兲
2. Discontinuous vorticity field For a vortex bubble with the vorticity inside a vortex tube, the s-average of quasi-stationary 共one-time兲 solutions of the first-order interface equation 关Eq. 共B6兲 in Appendix B兴 satisfies
2
¯␦
冑 冑
2
V
1/2
V0
S w共 0 兲 .
冉 冊 冉
V 1 1 1 C v 共 t 兲 ⫽⫺ log 共 0 兲 ⫹ 2 2 S ⫹
冕
1
0
冊
关共 ⌫ 1 /⌫ 兲共 1⫺y 兲 ⫹y 兴 2 dy , y⫹  共 t 兲
C w 共 t 兲 ⫽⫺4S w2 共 0 兲共 S 共00 兲 /S 共 0 兲 共 t 兲兲 3 ,
冉
1 1 C p 共 t 兲 ⫽⫺  共 t 兲 2 1⫹  共 t 兲 ⫹
冕
1
0
冊
关共 ⌫ 1 /⌫ 兲共 1⫺y 兲 ⫹y 兴 2 dy . 共 y⫹  共 t 兲兲 2
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Phys. Fluids, Vol. 14, No. 12, December 2002
Axial core-variations of axisymmetric shape
4421
The relative velocity field of this Rankine vortex core with a uniform axial jet depends only on two parameters: the ratio ⌫ 1 /⌫ and the initial swirl number S w (0). It depends on only one parameter if the axial velocity w * (0) is divided by S w (0). In the case of a potential vortex ring bubble (⌫ 2 ⫽0), the added condition ⫽0 implies the following leadingr and w (0) ⫽0. It comes order velocity field: v (0) ⫽⌫ 1 /(2 ¯)
冉 冊 冉
冊
V 1 1 1⫹  C v 共 t 兲 ⫽⫺ log 共 0 兲 ⫹ 1⫹log , 2 2  S 1 ⫽ ⫺log ¯␦ b 共 0 兲 , 2
FIG. 7. Thickness ¯␦ b(0) 共m兲 of a potential vortex ring bubble versus 0 the surface tension ⌼ 共Nm2/kg兲. The physical parameters are ⌫⫽1 m2 /s, P g0 ⫽0.2870(273⫹20) atm/m3 /kg, p(⬁)⫽101.3 atm/m3 /kg, P v ⫽2.026 atm/m3 /kg.
C w 共 t 兲 ⫽0, 1 C p 共 t 兲 ⫽⫺ . 2
In the case of a circular vortex the global integral A is A⫽⌫K log(8/2 )b/(4 ) and the velocity of this bubble is
D. Study of typical cases
1. Isothermal transformation
In the peculiar interesting case k⫽1 共isothermal transformation14兲, Eq. 共80兲 is the following polynomial of second degree in x: x 2 ⫹ax⫺1⫽0, where x⬅ 冑 共 t 兲 /  共 0 兲
¯ e共 0 兲 a⬅W
冒冑
冒 冉冑 冑 ˜P v
˜P g0 / ˜P v ⫹
1 S共0兲 1 2 S 共00 兲 ˜P v
,
冊
S 共00 兲 1 ⫹ ˜P g0 共 0 兲 . 2 S
The solution is 1 x⫽ 共 ⫺a⫹ 冑4⫹a 2 兲 . 2
共85兲
In order to find the thickness of the bubble ¯␦ b(0) (t) at time t we compute a, deduce x from Eq. 共85兲, obtain 冑 (t)/  (0) from the definition of x and use 冑 (t)/  (0) . This thickness exists for any values of the ⫽¯␦ b(0) (t)/¯␦ b(0) 0 parameters and at any time. It decreases with increasing val¯ e . Its initial value ues of the surface tension parameter W b(0) ¯␦ 0 is found by solving ¯ ¯␦ b0 共 0 兲 ⫽ 共 p 共 0 兲 共 ⬁ 兲 ⫺ ¯P v ⫺ ¯P g0 兲共 ¯␦ b0 共 0 兲 兲 2 ⫹⌼
⌫2 , 82
V⫽
⌫ 4
冋
K log
8 ¯ b共 0 兲
␦
册
1 ¯ ¯␦ b 共 0 兲 /⌫ 2 . ⫺ ⫹4 2 ⌼ 2
Figure 8 shows the velocity V as a function of the surface tension ⌼ of a circular vortex ring bubble. Figure 9 shows the evolution of a perturbed circular vortex bubble in the moving frame of the nonperturbed vortex. The perturbation is of elliptic shape 共mode 2 of the polar Fourier expansion11兲 and its amplitude is 0.15. The computation was performed with the EZ – vortex code 共see our submitted paper, Margerit et al., ‘‘Implementation and validation of a slender vortex filament code: Its application to the study of a four-vortex wake model’’兲 by implementing the bubble thickness equations 共85兲–共86兲 and the Weber number ¯ e computation. W 2. Almost adiabatical transformation
In the peculiar interesting case k⫽1.5 共close to the adiabatical transformation14 k⫽1.4), Eq. 共80兲 is the following polynom of degree three in x:
共86兲
which is Eq. 共80兲 at t⫽0. as a function In Fig. 7, I give the initial thickness ¯␦ b(0) 0 of the surface tension ⌼. The physical parameters are P g0 ⫽0.2870(273⫹20) atm/m3 /kg, p(⬁) ⌫⫽1 m2 /s, 3 ⫽101.3 atm/m /kg, P v ⫽2.026 atm/m3 /kg. Let us recall that this surface tension and these pressures are divided by the mass density ⫽1000 kg/m3 and that 1 atm⫽1.013 ⫻105 Pa.
FIG. 8. Velocity V 共m/s兲 of a circular vortex ring bubble versus the surface tension ⌼ 共Nm2/kg兲. The physical parameters are as in Fig. 7.
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4422
Phys. Fluids, Vol. 14, No. 12, December 2002
Daniel Margerit
E. Ultra-fast oscillations of a vortex ring bubble
The potential vortex cannot support axisymmetric axial variations on a short-time , because the only potential fields that are solutions of the two-time-scale equations are v (0) ⫽⌫ 1 /(2 ¯) r and w (0) ⫽0, which are without axial variation. However, axial variation are possible on a ultra-short time ¯t ⫽t/ 2 . In this subsection I give the system of equations 关Eqs. 共92兲–共95兲兴 for the ultra-short-time dynamics of axial variations on the vortex ring bubble. I solve this system and give a closed equation 关Eq. 共97兲兴 for the the ultra-short-time dynamics for the thickness of the bubble. 1. General equations
The bubble allows to have a solution of the leading¯u (0) )¯r ⫽0 in the form order equation of continuity (r D共 0 兲 共 s,¯t , ,t 兲 . ¯r
u 共 0 兲⫽
At leading order the Navier–Stokes equations are
u共0兲 FIG. 9. Numerical simulation of the motion of a potential vortex ring bubble of elliptical shape (mode⫽2, amplitude⫽0.15) in the isothermal k⫽1 case. The physical parameters are as in Fig. 7 and ⌼⫽0.7. The frame is moving with the unperturbed circular vortex ring bubble velocity.
¯t v共0兲 ¯t w共0兲
a x ⫹bx 冑a⫺ x⫺1⫽0, 2 3
2
¯t
where x⬅ 冑 共 t 兲 /  共 0 兲 / 共 ˜P g0 / ˜P v 兲 1/3, a⬅ 共 S
共0兲
/S 共00 兲 兲共 ˜P g0
/ ˜P v 兲
⫺2/3
/ ˜P v ,
¯ e / 冑˜P v . b⬅W The solution is
冉
冊
a ab 2 x⫽ d⫹6 ⫹4 ⫺2b 冑a /6 d d
⫹ p¯共r0 兲 ⫺
d ⬅⫺18a b⫹108⫺8a b 3
3/2
3/2 3
⫹6) 冑⫺2a 3 ⫺a 3 b 2 ⫺36a 3/2b⫹108⫺16a 3/2b. This shows that a thickness of the bubble ¯␦ b(0) (t) not always exists. The values allowed for a is between 0 and a maximum value, which decreases with increasing values of b and ¯ e . This maximum so of the surface tension parameter W value is almost a⫽4 when b⫽0. The thickness decreases with both increasing values of the surface tension parameter ¯ e , and of a. W
2
⫹u 共 0 兲 u¯共r0 兲 ⫽0,
¯r
共87兲
⫹ 共 0 兲 u 共 0 兲 ⫽0,
共88兲
⫹w¯共r0 兲 u 共 0 兲 ⫽0,
共89兲
¯ v (0) )¯r /r ¯ is the leading-order axial vorticity. where (0) ⫽(r Equation 共87兲 gives the pressure p
共0兲
⫽⫺
冕
⬁
¯r
v共0兲
2
¯r
dr ¯⫹ p 共 ⬁ 兲 ⫺
D共 0 兲 2r ¯2
2
⫺
D共 0 兲 ¯t
log ¯, r
共90兲
where the log ¯r term has to be matched with the outer velocity induced by a sink concentrated on the leading-order centerline X(0) . The leading order of the axisymmetric part of the dynamical equation of the free-boundary 共10兲 is
¯␦ b 共 0 兲
with
v共0兲
¯t
⫺u 共 0 兲 ⫽0,
共91兲
where all velocity components are taken on the freeboundary ¯⫽ r ¯␦ b(0)⫹ . As the thickness ¯␦ b(0) is given by Eq. 共27兲, this Eq. 共91兲 is indeed the equation for D(0) and gives D共 0 兲 ⫽
1 共 ¯␦ b 共 0 兲 兲 2 2
¯t
.
共92兲
The leading-order of the axisymmetric part of the pressure jump 共9兲 is
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Phys. Fluids, Vol. 14, No. 12, December 2002
p
共0兲
共 ¯␦ b 共 0 兲 ⫹ 兲 ⫽
冉 冊 ⌫
2
C p 共 s,¯t , ,t 兲
2
⫺
1 2
共 ¯␦ b 共 0 兲 兲 2
冋冉 冊 ¯␦ b 共 0 兲
⫽ P v ⫹ P g0 where C p 共 s,¯t , ,t 兲 ⫽⫺
冉
2
冉 冊 V
b共 0 兲 0
V
b共 0 兲
2 ¯␦ b 共 0 兲 ⌫
冊冕 2
The dynamical equation of the bubble ¯␦ b(0) (s,¯t , ,t) on this ultra-short-time is
⫹p 共 ⬁ 兲
共 ¯␦ b 共 0 兲 兲 2 ¯t 2
k
⫺
册 冉 冊 ¯␦ b 共 0 兲
2
⫹
¯t
Axial core-variations of axisymmetric shape
¯ b共 0 兲
log ␦
¯t
¯␦ b 共 0 兲
,
⫹
2 r ¯t , ,t 兲 v 共 0 兲 共¯,s, dr ¯. ¯␦ b 共 0 兲 ¯r
⬁
冋冉 冊
2
¯t
⫹
w共0兲 ¯t
¯共r0 兲
⫹
¯r
册
2 共 ¯␦ b 共 0 兲 兲 2 ⫹ log ¯␦ b 共 0 兲 , ¯t 2
D共 0 兲 ⫽0,
w¯共r0 兲 ¯r
共95兲
The system of Eqs. 共92兲–共95兲 is a closed system for the ultra-short-time dynamics of axial variations on the vortex ring bubble.
2. Solution
I use the following transformation
1 ⫽r¯ 2 ⫺ 共 ¯␦ b 共 0 兲 兲 2 , which yields
¯t
⫽0,
共96兲
where stands for (0) and w (0) . The solutions are in the form
共 0 兲 ⫽ 共 0 兲 共 1 ,s, ,t 兲 , K共 0 兲 ⫽K共 0 兲 共 1 ,s, ,t 兲 ⫽ v 共 0 兲⫽
K共 0 兲 共 1 ,s, ,t 兲 , ¯r
w 共 0 兲 ⫽w 共 0 兲 共 1 ,s, ,t 兲 .
⌫1 1 ⫹ 2 2
冕
1
0
⌫ 2 C p 共 s,¯t , ,t 兲 2
2
共 ¯␦ b 共 0 兲 兲 2
冉 冊
⫹
V
b共 0 兲 0
V
b共 0 兲
¯ 2⌼ ¯␦ b 共 0 兲
k
共97兲
,
兰 20 共00 兲 共 s 兲关 ¯␦ b0 共 0 兲 共 s 兲兴 2 ds , ⫽ Vb 共 0 兲 兰 20 共 0 兲 共 s,t 兲关 ¯␦ b 共 0 兲 共 s, ,¯t ,t 兲兴 2 ds
Vb0 共 0 兲
冉 冊冕 ¯␦ b 共 0 兲 ⌫
2
⬁
0
2
K共 0 兲 共 1 ,s, ,t 兲 共 1 ⫹ 共 ¯␦ b 共 0 兲 兲兲 2
d1 ,
and
共94兲
D共 0 兲 ⫽0.
log ¯␦ b 共 0 兲
C p 共 s,¯t , ,t 兲 ⫽⫺2 2
which can easily be checked from Eq. 共92兲. Equation 共93兲 is the equation of the thickness ¯␦ b(0) of the bubble. It is coupled with the ultra-short-time dynamics of the core, given by Eqs. 共88兲–共89兲, which can be written
共 0 兲
¯t 2
thickness
where
2
¯␦ b 共 0 兲 ¯t
2 共 ¯␦ b 共 0 兲 兲 2
共93兲
Here, we used the relation
1 ⫽⫺ 2
⫹
⫽2 共 p 共 ⬁ 兲 ⫺ P v 兲 ⫺2 P g0
¯ ⌼
D共 0 兲 D共 0 兲 ⫺ log ¯␦ b 共 0 兲 ⫺ 2 共 ¯␦ b 共 0 兲 兲 2 ¯t
2
4423
共 0 兲 共 1⬘ ,s, ,t 兲 d 1⬘ ,
K共 0 兲 ⫽
⌫1 1 ⫹ 2 2
冕
1
0
共 0 兲 共 1⬘ ,s, ,t 兲 d ⬘1 .
Here, the leading-order vorticity function (0) 关 1 ⫽r ¯2 ⫺(¯␦ b(0) ) 2 ,s, ,t 兴 is given initially and its short-time and normal-time t evolution are given by the equations at next orders. This equation 共97兲 generalizes the one of Genoux14 to a nonpotential vortex bubble with a filament of any shape and with axisymmetric axial variations of its thickness. The stationary solution of this equation 共when it exists兲 is without axial variations and satisfies Eq. 共80兲. VII. CONCLUSION
This two-time-scale asymptotic approach allows us to derive, from the Navier–Stokes equations, the dynamics of the axial core-variations of axisymmetric shape on a vortex filament. This gives an extension of the one-time-scale asymptotic theory of Callegari and Ting1 of vortex filament motion. This asymptotic theory is also an alternative to different ad hoc models of vortex filament with axial corevariations proposed by Marshall,3,4 Leonard,5 and Lundgren.6 The dynamics of these axial variations is on a short-time scale and is inviscid at leading and first orders. These axial variations induce a small amplitude 共first-order兲 motion of the curved centerline on the short-time scale. This motion is in the binormal direction of the leading-order centerline. The solutions of the two-time-scale equations have been given for axial core variations of small amplitude. More theoretical and numerical work is required to study the finite amplitude regime. The theory of Genoux14 of vortex ring bubbles has been extended to a vortex filament bubble with a centerline of any shape and with a nonpotential core. The axisymmetric part of the velocity field at first order 共it is the next order to the leading order兲 was proved to be
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4424
Phys. Fluids, Vol. 14, No. 12, December 2002
Daniel Margerit
composed of two parts: a part without axial variations and a part with axial variations. The expression of this second part was proved to be unique and to be related to the local stretching of the centerline. This form of the velocity field at first order was chosen by Margerit,17 who gave the one-time dynamical equations satisfied by the part without axial variations of this first-order axisymmetric part of the velocity field: it is the generalization of the Callegari and Ting theory to the next order. The implementation of the first-order thickness correction in a numerical code of slender vortex filament motion is currently under investigation. The associated first-order correction to the leading-order corrected vortex filament methods of Klein and Knio2,33 for slender vortex filament is also under investigation. I hope to extend these computations to vortex filaments with thicker cores and to be able to perform quantitative comparisons between these numerical computations and direct numerical computation of the Navier–Stokes equations. ACKNOWLEDGMENTS
APPENDIX A: THE LEADING-ORDER SHORT-TIME ASYMMETRIC DYNAMICS IN THE FILAMENT
In this appendix I give the leading-order equations of the short-time asymmetric dynamics for axial variations in the filament. This dynamics is slaved by the axisymmetric one. These equations come from the asymmetric part of the equations at first order. They are used in Sec. III B to perform the matching between the outer and inner velocity fields and so to obtain the equation of motion of the centerline. A stream function is introduced and the whole asymmetric field is described with this function. The stream function solution is given for a continuous and discontinuous vorticity field 关Eq. 共A8兲兴 and for a vortex ring bubble 关Eq. 共A20兲兴. 1. General equations
The equations of the asymmetric components u a , v a , u a at first order are1 1 a共 1 兲 ⫹ 共¯u r a 共 1 兲 兲¯r 兴 ⫽⫺ v 共 0 兲 K 共 0 兲 sin , 关v ¯r
u
共A1兲
v 共 0 兲 a共 1 兲 1 a共 1 兲 2 ⫹ v ⫹ p ⫽w 共 0 兲 K 共 0 兲 sin , ¯r ¯r
w a 共 1 兲 ⫽Swa共 1 兲 , a共 1 兲
共A2兲
v
关 ⫺u
a共 1 兲
⬁
a 共 1 兲 ⫽ 兺 共n11 兲 cos n ⫹ 共n21 兲 sin n .
共A7兲
n⫽1
(1) and is given by The only non-zero Fourier component is 11
共111 兲 共¯,s, r ,t 兲 ⫽ 共0兲 K 共 s,t 兲v 共 0 兲 共¯,s, r ,t 兲
冕
¯r
兰 z0 xD共 x,s, ,t 兲 dx
0
z 关v 共 0 兲 共 z,s, ,t 兲兴 2
⫹
冕
¯r
0
z
w共0兲 v共0兲
2
2
dz⫹
¯r 2 , 2
dz
共A8兲
2
2
r ,t 兲 ⫽ v 共 0 兲 ⫺2w 共 0 兲 . D共¯,s,
共A9兲
(1) (r ¯⫽0)⫽0 and Here, I used the boundary conditions 11 (1) ¯⫽0). This expression 共A8兲 has been written in this 关 11 兴¯r (r form, without derivatives of the velocity field, so that it is (1) easy to find 11 if the velocity field is discontinuous. Any expression with derivatives of the velocity field would be more difficult to use because of the contributions of the delta-functions that would be in the integrals. Equation 共A8兲 extends the Callegari and Ting1 equation of the stream func(1) to axial core-variations of the leading-order veloction 11 ity field and to discontinuous vorticity field. The asymmetric velocity field (u a(1) , v a(1) ,w a(1) ) and the asymmetric pressure p a(1) depend only on the leadingorder velocity field. From Eqs. 共A6兲, 共A3兲, and 共A2兲, it comes
u a 共 1 兲 ⫽u 共111 兲 sin ,
共A10兲
v a 共 1 兲 ⫽ v 共111 兲 cos ,
共A11兲
w a 共 1 兲 ⫽w 共111 兲 cos ,
共A12兲
p a 共 1 兲 ⫽ p 共111 兲 cos ,
共A13兲
with u 共111 兲 ⫽⫺ 共111 兲 /r ¯,
w 共111 兲 ⫽⫺ 共111 兲 w¯共r0 兲 / v 共 0 兲 , 2
p 共111 兲 ⫽⫺ 共111 兲 共 0 兲 ⫺r ¯K 共 0 兲 w 共 0 兲 ⫺ v 共 0 兲 v 共111 兲 .
2
⫽⫺w 共 0 兲 K 共 0 兲¯r cos , 共A4兲
where ¯r
and expand it in a Fourier series
v 共111 兲 ⫽⫺ 共 共111 兲 兲¯r ⫹K 共 0 兲¯r v 共 0 兲 ,
共A3兲
v 共 0 兲 u a 共 1 兲 ⫺2 v 共 0 兲 v a 共 1 兲 ⫹p¯r
Swa共 1 兲 ⫽ 共 0 兲
1 ¯ v 共 0 兲 K 共 0 兲 cos , u a 共 1 兲 ⫽ a 共 1 兲 , and v a 共 1 兲 ⫽⫺ ¯ra 共 1 兲 ⫹r ¯r 共A6兲
where
I would like to thank Dr. Max O. Souza, Professor Stephen J. Cowley, and Dr. Pierre Brancher for their stimulating discussions. I also thank Lu. Ting for electronic correspondence at the beginning of this work: it was very encouraging for me and helped me to go ahead.
共 0 兲 a共 1 兲
As s and are parameters in these equations 共there are no s-derivative nor -derivative兲, their solution is the same as in the one-time analysis. Following this analysis1 I define a stream function a(1) by
w¯共r0 兲 ⫺w 共 0 兲 v 共 0 兲 K 共 0 兲
sin 兴 .
共A5兲
2. Discontinuous vortex field
For a vortex with the vorticity inside a vortex tube, the first-order asymmetric part of the condition of continuity of the pressure 共4兲 yields
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Phys. Fluids, Vol. 14, No. 12, December 2002
关关 p a 共 1 兲 兴兴 ⫽⫺
¯␦ ta 共 1 兲 ¯␦ t 共 0 兲
关关v 共
0 兲2
Axial core-variations of axisymmetric shape
共111 兲 共¯,s, r ,t 兲 K 共 0 兲 共 s,t 兲v 共 0 兲 共¯,s, r ,t 兲
兴兴 ,
where ¯␦ ta(1) is the asymmetric part of the first-order thickness of the tube. Here, I used Eq. 共13兲 to have 关关 p¯r(0) 兴兴 2 ⫽ 关关v (0) 兴兴 /¯␦ t(0) . The same jump can be found from Eqs. 共A13兲 and 共A8兲. The first-order asymmetric part of the condition of continuity of the normal velocity 共7兲 gives 关关 u a 共 1 兲 兴兴 ⫽
¯␦ ta 共 1 兲 ¯␦ t 共 0 兲
4425
⫽
冕
z
¯r
兰 ¯␦ b 共 0 兲 xD共 x,s, ,t 兲 dx
¯␦ b 共 0 兲
⫹
dz⫹
z 关v 共 0 兲 共 z,s, ,t 兲兴 2
¯r 2 ⫺ 共 ¯␦ b 共 0 兲 兲 2 ¯ ⫹¯␦ b 共 0 兲 ⌼ 2
冕
冕
¯r
¯␦ b 共 0 兲
z
w共0兲 v共0兲
2
2
dz
1
¯r
¯␦ b 共 0 兲 z 关v 共 0 兲 共 z,s, ,t 兲兴 2
dz. 共A20兲
关关v 共 0 兲 兴兴 ,
¯r 兰 ¯␦ b(0) 1/(z 关v (0) (z,s, ,t) 兴 2 )dz
at infinity is The behavior of ¯/⌫, r where I used Hoˆpital’s rule and the behavior of v (0) at infinity.
and so t共 1 兲 关关 共111 兲 兴兴 ⫽¯␦ 11 关关v 共 0 兲 兴兴 , t(1) where ¯␦ ta(1) ⫽¯␦ 11 cos . The same jump can be found from (1) Eq. 共A8兲. It means that Eq. 共A8兲 of 11 is correct even through the interface. The asymmetric part of the kinematic boundary condition at first order gives
¯␦ ta 共 1 兲 ⫽
¯␦ t 共 0 兲 v共0兲
u a共 1 兲.
共A14兲
Here, all velocity components are taken on the interface ¯r ⫽¯␦ t(0)⫾ . This equation yields t共 1 兲 ¯␦ 11 ⫽ 共111 兲 / v 共 0 兲 .
共A15兲
3. Vortex ring bubble
For a vortex bubble, the first order for the asymmetric part of the continuity of pressure 共9兲 is p 共111 兲 ⫹
b共 1 兲 ¯␦ 11
¯␦ b 共 0 兲
v
共 0 兲2
¯K ⫽⌼
共0兲
共A16兲
,
b(1) is the asymmetric part of the first order freewhere ¯␦ 11 boundary of the bubble. Here, all fields are taken on the free-boundary ¯␦ b(0)⫹ . This equation is the equation of the b(1) of the bubble. The first order of the asymmetthickness ¯␦ 11 ric part of the dynamical equation of the free-boundary 共10兲 is
¯␦ ba 共 1 兲 ⫽
¯␦ b 共 0 兲 v共0兲
u a共 1 兲.
共A17兲
Here, all velocity components are taken on the free-boundary ¯⫽ r ¯␦ b(0)⫹ . This equation yields b共 1 兲 ¯␦ 11 ⫽ 共111 兲 / v 共 0 兲 .
共A18兲
This last equation combined with Eqs. 共A16兲 and 共A13兲 gives the following boundary conditions for the equation of (1) 11 ¯␦ b 共 0 兲 共 v 共 0 兲 2 ⫹w 共 0 兲 2 兲 ⫹⌼ ¯⫽
共 共111 兲 兲¯r
K共0兲
v 共 0 兲,
共A19兲
(1) ¯ b(0) (␦ ) where I used the second boundary condition 11 (1) ⫽0. The only solution of the equation of 11 with these two boundary conditions is
APPENDIX B: THE FIRST-ORDER SHORT-TIME AXISYMMETRIC DYNAMICS IN THE FILAMENT
In this appendix I give the first-order equations of the short-time axisymmetric dynamics for axial variations in the filament. These equations come from the axisymmetric part at second order. In the two-time scale framework the timeaverage of these equations gives 共Sec. III C兲 the leadingorder normal-time equations in the filament. In the one-time 共normal-time兲 framework the compatibility conditions to these equations also give 共Secs. IV and VI兲 the leading-order one-time equations in the filament. The one-time solution to these equations and its uniqueness is studied in Appendix D. 1. Continuous vorticity field
The first-order compatibility equations of the one-time analysis become the dynamical equations of the axisymmetric part of the first-order relative velocity field. At first order p c 共 1 兲 ⫽⫺
冕
⬁
¯r
2 v 共 0 兲v c共 1 兲 dr ¯, ¯r
共B1兲
and at second order r c 共 2 兲 兲¯r ⫹r ¯w zc 共 1 兲 ⫽Sc 共 1 兲 , 共¯u
共B2兲
v c共 1 兲 ⫹ c 共 1 兲 u c 共 1 兲 ⫹ 共 0 兲 u c 共 2 兲 ⫹w c 共 1 兲 v 共z0 兲 ⫹w 共 0 兲 v zc 共 1 兲 ⫽Scv共 1 兲 ,
共B3兲
w c共 1 兲 ⫹w¯共r0 兲 u c 共 2 兲 ⫹w¯rc 共 1 兲 u c 共 1 兲 ⫹ p zc 共 1 兲 ⫹w c 共 1 兲 w 共z0 兲 ⫹w 共 0 兲 w zc 共 1 兲 ⫽Swc共 1 兲 ,
共B4兲
where c(1) ⫽(r ¯ v c(1) )¯r /r ¯ is the axisymmetric part of the first-order axial vorticity and Sc 共 1 兲 ⫽⫺
˙ 共 0 兲 共1兲 r ¯rc 共 1 兲 兲 , 共 0 兲 ¯r ⫺ 共 0 兲 共¯u
Scv共 1 兲 ⫽w 共 0 兲 v 共z0 兲
冉
冊
共0兲 共¯r v¯r 兲¯r v 共 0 兲 共1兲 v共0兲 ⫹ ⫺ ¯ ⫺ 2 , t ¯r ¯r 共0兲
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4426
Phys. Fluids, Vol. 14, No. 12, December 2002
Swc共 1 兲 ⫽w 共 0 兲 w 共z0 兲
Daniel Margerit
共1兲 共 1 兲 ˙ 共 0 兲 共 0 兲 w 共 0 兲 共0兲 ⫹p ⫺ w ⫺ z t 共0兲 共0兲 共0兲
r ¯共r 兲 兲¯r 共¯w
␦
0
⫹¯
¯r
This system, of unknown p c(1) , v c(1) , w c(1) , u c(2) , and , is closed.
¯ tc(1)
3. Vortex ring bubble
.
Here, I used the first-order asymmetric field to state that the -average of (1) u (1) is c(1) u c(1) , and that the one of w¯r(1) u (1) is w¯rc(1) u c(1) . This system, of unknown p c(1) , v c(1) , w c(1) , and u c(2) , is closed.
For a vortex bubble the interface dynamics has to be found and the previous equations have to be completed. The first-order of the axisymmetric part of the continuity of pressure 共9兲 is p
c共 1 兲
⫹¯␦ bc 共 1 兲 p¯共 0 兲 ⫽ ¯P g0 r
2. Discontinuous vorticity field
For a vortex with the vorticity inside a vortex tube the interface dynamics has to be found and the previous equations have to be completed. The second-order axisymmetric part of the condition of continuity of the pressure 共4兲 yields 关关 p c 共 1 兲 兴兴 ⫽⫺
¯␦ tc 共 1 兲 ¯ t共 0 兲
␦
2
关关v 共 0 兲 兴兴 ,
where ¯␦ tc(1) is the axisymmetric part of the first-order interface thickness. Here, I used Eq. 共13兲 to have 关关 p¯r(0) 兴兴 2 ⫽ 关关v (0) 兴兴 /¯␦ t(0) . This jump condition means that the expression 共B1兲 of p c(1) is not correct through the interface and has to be replaced by
冉 冊冋
¯ ⫹⌼
V
b共 0 兲 0
V
b共 0 兲
¯␦ bc 共 1 兲 共 ¯␦ b 共 0 兲 兲 2
k
V V
b共 1 兲 0 ⫺ b共 0 兲 0
册
V
b共 1 兲 k
V
b共 0 兲
,
共B7兲
where all fields are taken on the free-boundary ¯␦ b(0)⫹ . This equation gives the axisymmetric part of the thickness ¯␦ bc(1) of the bubble. The first order for the axisymmetric part of the dynamical equation 共10兲 of the free-boundary is
¯␦ bc 共 1 兲 ⫺u c 共 2 兲 ⫹w 共 0 兲¯␦ zbc 共 1 兲 ⫹¯␦ zb 共 0 兲 w c 共 1 兲 ⫹w¯共r0 兲¯␦ zb 共 0 兲¯␦ bc 共 1 兲 共1兲 ⫽Sbc , b
共B8兲
where p c共 1 兲
⫽
冦
⫺ ⫺
冕 冕
⬁
2 v 共 0 兲v c共 1 兲 ¯r
¯r ⬁
2 v 共 0 兲v c共 1 兲 ¯r
¯r
dr ¯⫹
dr ¯
¯␦ tc 共 1 兲 ¯␦ t 共 0 兲
共1兲 ⫽⫺ Sbc b 2
关关v 共 0 兲 兴兴
if ¯⬍ r ¯␦ t 共 0 兲 ,
if ¯⬎ r ¯␦ t 共 0 兲 .
The second-order axisymmetric part of the continuity of the normal velocity 共7兲 yields
冉
关关 u c 共 2 兲 兴兴 ⫽ ¯␦ zt共 1 兲 ⫺
共1兲
冊
¯␦ t 共 0 兲 关关 w 共 0 兲 兴兴 ⫹¯␦ zt共 0 兲 关关 w c 共 1 兲 兴兴 共0兲 z
⫹¯␦ zt共 0 兲¯␦ tc 共 1 兲 关关 w¯共r0 兲 兴兴 .
共B5兲
Here, I used the first-order asymmetric field to state that the -averages of v (1)¯␦ t(1) /¯␦ t(0) and of u¯r(1)¯␦ t(1) are zero. The axisymmetric part of the kinematic boundary condition at second order gives
¯␦ tc 共 1 兲 ⫺u c 共 2 兲 ⫹w 共 0 兲¯␦ ztc 共 1 兲 ⫹¯␦ zt共 0 兲 w c 共 1 兲 ⫹w¯共r0 兲¯␦ zt共 0 兲¯␦ tc 共 1 兲 ⫽Scb共 1 兲 ,
共B6兲
where Scb共 1 兲 ⫽⫺
¯␦ t 共 0 兲 共1兲 ⫹w 共 0 兲 共 0 兲 ¯␦ zt共 0 兲 . t
In this equation all velocity components are taken on the interface ¯⫽ r ¯␦ t(0)⫾ .
¯␦ b 共 0 兲 共1兲 ⫹w 共 0 兲 共 0 兲 ¯␦ zb 共 0 兲 . t
In this equation, all velocity components are taken on the free-boundary ¯⫽ r ¯␦ b(0)⫹ . The bubble allows to have a solution of the equation of continuity 共B2兲 in the form u c共 2 兲⫽
Dc 共 2 兲 共 s, ,t 兲 ⫹u c 共 2 兲 , ¯r
r As the thickness ¯␦ bc(1) is where u c(2) is regular at ¯⫽0. given by Eq. 共B7兲, Eq. 共B8兲 is indeed the equation for Dc(2) . This system, of unknown p c(1) , v c(1) , w c(1) , u c(2) , Dc(2) , and ¯␦ bc(1) , is closed.
APPENDIX C: UNIQUENESS PROBLEM AT LEADING ORDER
In this appendix I consider the uniqueness problem of the solutions to the leading-order one-time compatibility conditions. These equations are obtained if the short-timescale derivative is removed from the leading-order short-time axisymmetric equations 共13兲–共16兲 for axial variations. Leading-order velocity fields without axial variations and u c(1) ⫽0 are solutions of these leading-order compatibility conditions. As we will see, the study of small perturbations around the solutions without axial variations seems to indicate that they are the unique solutions to these compatibility conditions. I introduce a small stationary perturbation c(1) (0) ˜ (0) ) of a flow without axial ˜ ,˜v ,w ˜ (0) ,p ˜ (0) , ¯ c(1) ,K (u
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Phys. Fluids, Vol. 14, No. 12, December 2002
Axial core-variations of axisymmetric shape
variations, denoted by (uគ c(1) ⫽0,vគ (0) ,w គ (0) , pគ (0) , គ c(1) ,K គ (0) ). At first order in its amplitude, the perturbation satisfies ˜p 共 0 兲 ⫽⫺
冕
⬁
¯r
2 vគ 共 0 兲˜v 共 0 兲 dr ¯, ¯r
共C1兲
r˜ c 共 1 兲 兲¯r ⫹r ¯w ˜ 共z0 兲 ⫽0, 共¯u
共C2兲
គ 共 0 兲˜u c 共 1 兲 ⫹wគ 共 0 兲˜v 共z0 兲 ⫽0,
共C3兲
˜ 共z0 兲 ⫹w គ 共 0 兲w ˜ 共z0 兲 ⫽0, w គ ¯共r0 兲˜u c 共 1 兲 ⫹p
共C4兲
or the equivalent system ˜ 共z0 兲 ⫽K គ c 共 1 兲 K គ 共y0 兲 f ,
共C5兲
1 共0兲 共0兲 ˜ z ⫺4y គ cy 共y1y兲 f ⫽0, គ K 4y គ cy 共 1 兲 f y y ⫹ K y
共C6兲
˜ c(1) K គ (0) where f ⫽ ˜ zc(1) . If គ c(1) y ⫽0 then Eq. 共C5兲 yields z y c(1) ˜ ⫽0 and so z ⫽0 is the only physical solution. Equation ˜ z(0) ⫽0 as K គ (0) ⫽0. So the only possible 共C6兲 then yields K stationary perturbation is without axial variations. If គ c(1) y ⫽0, Eqs. 共C5兲–共C6兲 yield f ⬙ ⫹G 共 y 兲 f ⫽0,
共C7兲
lim f ⫽0,
APPENDIX D: UNIQUENESS PROBLEM AT FIRST ORDER
In this appendix I consider the uniqueness problem of the solutions to the one-time compatibility conditions at next order. These equations are obtained if the short-time scale derivative is removed from the first-order short-time axisymmetric equations 共B1兲–共B4兲 for the axial variations. Assuming that the compatibility conditions at first order have the only solution without axial variations 共as suggested in Appendix C兲 it is found in this appendix that these compatibility conditions at second order also have a unique solution. Here, this solution is given and proves to be the one introduced by Margerit17 to generalize the Callegari and Ting theory1 at next order. I define ⫽
˙ 共 0 兲 S˙ 共 0 兲 ⫺ , 共0兲 S共0兲
z ˜⫽
冕 s
0
共 2 兲⫽
ds ⬘ ,
冉
冊
1 c 共 2 兲 1 ˙S 共 0 兲 u ⫹ ¯r 共 0 兲 , 2 S
 共 1 兲 ⫽w c 共 1 兲 ⫹z˜. The subtraction of Eqs. 共34兲–共35兲 from Eqs. 共B1兲–共B4兲 yields
y→⫹⬁
p c 共 1 兲 ⫽⫺
f ⬘ 共 0 兲 ⫽0,
冕
⬁
¯r
2 v 共 0 兲v c共 1 兲 dr ¯, ¯r
with G⫽
冋
共0兲
K គ
K គ 共y0 兲 c共 1 兲 c共 1 兲 គ y គ y y y 2 ⫺
4y
册冒
c共 1 兲2
គ y
.
,K ), f ⫽0 seems to be the For most flows ( គ unique solution of the linear equation 共C7兲 and so the only possible stationary perturbation seems to be without axial variations. The velocity fields without axial variations seems to be isolated solutions of the leading-order compatibility conditions. Souza12 used a standard comparison principle for quasi-linear elliptic operators and proved that there are no other stationary solutions of the Bragg–Hawthorne equation 共22兲 than the solutions without axial variations. Klein and Ting34 assumed that these compatibility conditions have stationary solutions with axial variations and derived the equations of evolution of these fields in a one-time analysis on the normal-time scale. Unfortunately, as was pointed out by Souza 共private communication兲, no field with axial variations and without axial velocity at infinity is solution of the leading-order compatibility conditions. For a vortex with the vorticity inside a vortex tube, it also comes ¯␦ t(0) (s,t)⫽¯␦ t(0) (t), and for a vortex bubble it comes ¯␦ b(0) (s,t)⫽¯␦ b(0) (t). c(1)
(0)
共D1兲
¯ ˜共z 兲 ⫽0, 共¯r 共 2 兲 兲¯r ⫹r
共D2兲
共 0 兲 共 2 兲 ⫹w 共 0 兲 v˜zc 共 1 兲 ⫽0,
共D3兲
w¯共r0 兲 共 2 兲 ⫹ p˜zc 共 1 兲 ⫹w 共 0 兲 ˜共z1 兲 ⫽0,
共D4兲
1
˜ 共z0 兲 ⫽ f K គ 共y0 兲 / គ cy 共 1 兲 , K
4427
or the equivalent system
c 共 1 兲 K˜zc 共 1 兲 ⫽K共y0 兲 f ,
共D5兲
1 4y cy 共 1 兲 f y y ⫹ K共 0 兲 K˜zc 共 1 兲 ⫺4y cy 共y1y兲 f ⫽0, y
共D6兲
where f ⫽ ˜zc 共 2 兲 , 1 共 2 兲 ⫽⫺ ˜zc 共 2 兲 , ¯r 1  c 共 1 兲 ⫽ ¯rc 共 2 兲 , ¯r ¯ v c共 1 兲. Kc 共 1 兲 ⫽r As the linear operator of the systems 共D1兲–共D4兲 and 共D5兲– 共D6兲 is the same as the one of Eqs. 共C1兲–共C4兲 and 共C5兲– 共C6兲, the unique solution of these systems is f ⫽0 and K˜zc(1) ⫽0. The unique stationary solutions of the compatibility conditions for the first-order axisymmetric field are
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4428
Phys. Fluids, Vol. 14, No. 12, December 2002
r r 兲, 兲 ⫽ v u 共 1 兲 共¯,t v c 共 1 兲 共¯,s,t w c 共 1 兲 共¯,s,t r r 兲⫹ 兲 ⫽w u 共 1 兲 共¯,t
冕冉 s
0
⫺ ˙ 共 0 兲 ⫹ 共 0 兲
Daniel Margerit
冊
˙S 共 0 兲 ds ⬘ , S共0兲
1 ˙S 共 0 兲 r u c 共 2 兲 共¯,s,t 兲 ⫽⫺ ¯r 共 0 兲 , 2 S where the evolution of ( v u(1) ,w u(1) ) in the normal-time scale can be found from the axisymmetric equations at third-order.17 For a vortex with the vorticity inside a vortex tube, the subtraction of Eq. 共36兲 from Eq. 共B6兲 written without the short-time scale derivative yields ¯␦ ztc(1) ⫽0, i.e., ¯␦ tc(1) (s,t) ⫽¯␦ tc(1) (t). 1
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