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Insight into singular vortex flows
Fluid Dynamics North-Holland
Research
FLUIDDVMMKS RESEARCH
10 (1992) 101-115
Insight into singular vortex flows Gianni
Pedrizzetti
Dipartimento di Ingegneria Cirile, Unicersitri di Firenze, via di Santa Marta, 3, 50139 Florence, Italy Received
10 May 1991
Abstract. The method of three-dimensional vortex singularities is analyzed. The fact that it does not represent a weak solution of the Euler equations has little bearing on its validity as an operative model. Other evolution inconsistencies are a major problem. The non-solenoidality of the vortex field is analyzed and a linear filtering feedback procedure, requiring no additional computations, is introduced to allow alignment with the reconstructed solenoidal vorticity field. A rough estimation of the dissipative effect during vortex reconnection has shown a finite viscous effect that is implicitly present in the model, growing with stretching, representing the mechanism of shifting to reconnected topologies without observing strong velocity gradients. The dividing vorton method is rearranged in order to allow the quantized reproduction of a predefined core evolution law, an error estimation, due to the corpuscularity of the field, is given. Two different numerical computations are performed corresponding to a weak and a strong vortex interaction. The improvements are tested, and confrontations with existing numerical and experimental results are performed showing good agreement. The possibility of the singular vortex flow serving as a rough, but simple, model for very high Reynolds number comglex flows is discussed.
1. Introduction
to the vorton method
Construction of a basic computational vortex element for the representation of three-dimensional vertical flow has been a widely investigated field for a long time. The classic vortex filament method, with cut-off, was constructed for modeling vortex filament evolution in inviscid flows. The method is the only one to use when a perfectly ideal flow is to be investigated, and has been proven to converge in the limit of the ratio between transversal section radius and the curvature radius approaching zero (Moore and Saffman, 1972; Greengard, 1986). Another possibility is the use of a vortex particle of finite size (blob vortons). The computational elements in this case are spherical blobs of vorticity. An interesting way of using blob vortons is given by approximating thick vortex structures using clouds of overlapping elements. In this case, particle independence makes possible the correct introduction of viscous effects (Winckelmans and Leonard, 1988). This method has proven to be an actual Lagrangian alternative to direct grid or spectral simulations. The vortex singularities (vortons) model is not an alternative to the methods outlined above, being in fact less accurate; it is rather a “model” for very high Reynolds number flows, in which the phenomenon of vortex reconnection is easily reproduced. A vorton (Novikov, 1983) is a vortex &singularity which, if located at point x,, gives rise to a vorticity field
where
y is the vorton
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a-function. via di Santa
The Marta,
3,
singularity is representative Intensity y is given by
of a vortex volume,
V, characterized
by a value, LC),of the vorticity.
y=wv. A series becomes
(2a)
of singularities y=T
can be used for representing
filament,
in which case (2a)
dx,
(2bl
where I‘ is the filament circulation and dx the represented by the singularity. The velocity field given by the vortex field (1) is u(x)
a vortex
= -G
vector
length
of the
filament
portion
1 (x-X”)XY (3)
Ix-XVI3 The vortex singularities
move with the local velocity
and the intensities
evolves as fluid lines
x, =u(x\)>
(4al
+=r*
(4b)
Cu(x,),
where the self-interaction term is neglected. The method here outlined is a very clean and simple model of vortex dynamics, but it has a number of inconsistencies which must be investigated. The vortex field (1) is not divergence-free. If we start with a vorton configuration which well represents a divergence-free vortex structure, the discretization errors can drive the system toward a structure that is not divergence-free, without any feedback opposing it. In section 2 a filtering procedure helping the system to readjust in a divergence-free configuration is proposed. A vortex line has an infinite self-induced velocity, which can be bounded only if we consider a vortex tube with a finite core. In the vorton method the discretization implicitly introduces a cut-off, and consequently a core size, proportional to the neighboring vortons’ interspace. During stretching of the vortex tube, the distance between particles grows, leading to the incorrect result of core growth. In section 4 a dividing vorton procedure which takes into account a predefined core evaluation law is introduced. These are, to our knowledge and experience, basic points, from an operative viewpoint, which make applications of the vorton method difficult. From a theoretical point of view, system (4) is not equivalent to the Euler equations, as shown by Saffman and Meiron (1986) and Greengard and Thomann (1988); even when the number of vortons tends to infinity, it does not converge to it. Nevertheless it has been proven by Hou and Lowengrub (1990) that the convergence proof given for regularized vortex models is applicable to singular elements, due to the smoothness of the velocity field in the points of application. Vortex reconnection is observed in vorton simulation; this is in contrast with ideal flow evolution, so a clarification and estimation of what we are simulating must be given. In section 3 we try to give an estimation of the finite viscosity effect implicitly present in the method. In section 5 we present two different simulations of vortex evolution, one a smooth reconnection, and the other a strong interaction ending with reconnection, of vortex filaments. The results are then compared with numerical and experimental results, and the improvements here introduced are tested. In conclusion, we consider the singular r,ortex flow as an economic, even though not accurate, model for a first insight into the complex problem of vortex interaction at high Reynolds number.
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2. Divergence
filtering
103
in the vorton method
The vorticity field given by (1) is not divergence-free, however, a divergence-free vorticity field can be reconstructed as the curl of the velocity field (3) (Novikov, 1983). When the y vectors are well aligned with the reconstructed vorticity field, a vorton system is a good, discrete approximation of a divergence-free vortex structure, and subsequent evolution closely agrees with a divergence-free vortex evolution. This fact has been stressed by Greengard and Thomann (1988), Aksman and Novikov (1988>, and Pedrizzetti (1991). A parallel study of vortex-body interaction with classical cut-off method and vortex singularity method has clearly revealed the different characteristics of the two models (Pedrizzetti, 1991a). Discretization errors and numerical approximations can drive the y vectors away from a divergence-free configuration. Actually, even after a short time calculation, they usually depart from it slowly, because there is no effect helping them to readjust in a divergence-free configuration. This is a serious obstacle to long computations, and to simulations of complex interactions (Pedrizzetti and Becchi, 1990). For this reason we introduce a filtering procedure which gives y vectors a tendency to realign with VX u vector field. Let us define the aligned intensity
We introduce
an equation
dr,‘dt=
for evolution
-v(Y-?),
of the vorton
intensities, (5)
to be added to eq. (4b). The parameter 77 is the characteristic frequency of the linear filter operation (5). The filter time scale, T, = l/q, must not be longer than the timescale of the physical phenomena otherwise it has no effect; on the other hand, it must not be shorter to avoid artificial forcing, because a discrete vortex representation can pass through a not perfectly aligned configuration during the evolution. Nevertheless, the results are not very sensitive to the value of 7, once the order of magnitude is right. Typically for a curved vortex filament 77 N T/R’, R being the radius of curvature. The effect of this procedure on the evolution of vortex particle structures is very encouraging, giving the system a natural self-consistency which permits the computation of complex interactions and the study of long-time evolutions, as will be seen from the results of section 5. We want to point out that this operation requires no additional calculation in a numerical program. At each time step the velocity partial derivatives, &L,/&K, with i, j = 1, 2, 3, are computed for the evaluation of Vu, at each vorton location, from these we can compute V x u and q. Then we update the intensities by yne, = (1 - cu)y + a+, with cy = 77 At, and At the computational time step. This dirlergence filtering procedure has no clear physical meaning yet (a study is in progress with an analogy with the self-consistent field method in a system with weak interaction; Landau and Lifshitz, 1967; Novikov, 1991), but its significance, from an operative point of view, is clear as a methodology for making the system (4a, b) consistent with its physical basis.
3. Vorton structure
reconnection
In computational vortex models, when core size is evolved, the energy is not an invariant of motion (e.g. Agishtein and Migdal, 1986; Pumir and Siggia, 1987). This is obvious because swirl velocity and axial flow are not taken into account in energy computation, and during
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stretching of vortex lines there is a transfer between such “hidden” energy and the modeled macroscopic structure. In the vorton method the total energy of the system is given by the interaction energy between pairs of singularities, 1 y”‘. p _ 16rr i 1x(i) _p
E’12’= I”t
y’l’.(x 1
+
(1) _ p9
p
. (#)
1x(l) _ @) 13
_ x(29 (6) I>
where superscripts 1 and 2 indicate two different vortons, and the self-interaction energy of each singular element is infinite. During stretching, interaction energy can be transformed into self-energy and vice versa (Aksman et al., 1985). This implies an energy dissipation during stretching, which can be partially counterbalanced by a remeshing method, as indicated in the next section. In very high Reynolds number vertical flows, the moment of reconnection between vortex structures is connected with the formation of a singularity or very high velocity gradients. In vorton simulations reconnection is usually observed, in contrast with the ideal flow Kelvin theorem, and even without observing very high velocity gradients. This is due to the presence, for construction, of a spurious dissipative mechanism, which permits the reproduction of these phenomena. Consider a vorton as representative of a portion of vortex filament of length L and core radius c. During stretching of the vorton filament the length L of the portion increases, along with the vorton intensity as given by eqs. (4b) and (2b). The vorton method implicitly assumes a value of the core c proportional to the neighboring vorton distance c = Q L (Aksman and Novikov, 1988; Pedrizzetti, 1991), and implies a core evolution law dc/dt
= (Y dL/dt.
(7)
This evolution law implies that the vortex core grows during stretching of vortex lines instead of decreasing as required by the volume conservation law, as described in the next section. As a first rough but simple way to estimate the presence of a viscous effect, the phenomena of core growth in the vorton method can be compared with the core spread due to a viscous diffusion of vorticity. The core evolution law relative to a viscous diffusion, with kinematic viscosity V, is given by ;=;(_&zj,
(8)
where the additional term for local volume conservation is written Equaling the right-hand sides of eqs. (7) and (8), we obtain u -c
dL/dt.
(9)
This is a very rough estimation, but it shows effect exists implicitly in the vorton method. In the mechanics which permits jumping over the reconnection which, otherwise, could hardly be
4. Volume preserving
in parentheses,
dividing/fusing
that, in first approximation, a finite viscosity particular, it grows with stretching, and it is moment of local intense stretching as vortex followed numerically.
vortons
A vortex line of zero cross-sectional area moves with an infinite self-induced velocity due to a non-zero curvature. A vortex filament moves with a finite velocity only if a finite transversal radius, or core size, is taken into account. When a vortex filament is computed with a single series of computational points along the filament axis, the core size can be taken into account as proportional to a cut-off length which eliminates the singular character of the integral in
G. Pedrizzetti / Insight inlo singular clortex flows
105
computing the velocity (e.g., Chorin, 1982; Leonard, 1985; Winckelmans and Leonard, 1988). With singular vortons a cut-off is implicitly introduced due to the discretization, and a core size c, proportional to the vortons distance Ax, is assumed; in particular, considering the velocity of a vortex ring of circular cross section of radius c with uniform distribution of vorticity, it is found that Ax/c = 1.143.. . , with great accuracy (error < 10%) already with 4 vortons. During the filament motion the vortex cross section also must evolve. In a vortex model the motion within the core is not resolved, and some additional consideration must be given in order to introduce an evolution law for the radius c of the core, which is assumed to be circular. From a physical point of view it is required that the vortex volume be maintained, even locally. This implies that for a portion of filament of length L(t) and core c(t) the product L(t)c*(t) is constant. This consideration leads to variation of c along the filament. Moore and Saffman (1972) showed that a variation in the internal structure along the filament produces an axial flow that redistributes the volume along the filament, so as to eliminate inhomogeneities, in a time that is short compared with the time needed by the filament to move significantly (see Pumir and Siggia (1987) for different process timescales). The actual core evolution law is thus confined between the limiting situations of local volume conservation, in which axial flow smoothing is not taken into account, and global volume conservation, in which axial flow is assumed to be instantaneous with respect to the filament geometry evolution. Another extreme possibility is given by maintaining the core c, i.e. the cut-off, constant. In this way we have, for continuous filament, an Hamiltonian structure of the equation of motion (Anderson and Greengard, 1984; Agishtein and Migdal, 1986). A major problem in the vorton method is that, during stretching, vortons move apart and Ax and so c grow, instead of remaining constant or decreasing. The opposite happens during compression, even if stretching is much more frequent than compression. A dividing vorton model was introduced by Kuwabara (1986) in order to keep good discretization during vortex evolution. We propose an analogous method of vorton division arranged in such a way that a core evolution law, even if in a quantized way, can be introduced. We consider here the evolution laws for the two extreme cases of volume conservation and constant cut-off. Consider a vorton representing a portion of vortex tube of length L and core c. Call c, the core represented by the vorton model, and c, the core which should keep the volume of the portion constant. Assume that at a certain initial time the vorton discretization represents the correct volume, During or core size, c, = c, = cg = aL,, where the suffix 0 means at the initial instant. stretching of the vortex filament the core represented by the vorton discretization evolves as c, = aL (being the constant a = l/1.143.. . >, giving the incorrect result of core growth. If the local volume conservation is imposed, the core should evolve as c, = cc,L)/2/L”2. During stretching c, will always be greater than cr, until after some time the value of c, will come to be the double of c,. At this time, when c,/c, 2 2, we can divide the vorton into two parts so that, signing with a prime the variables after division, c: = c,/2 = c,. As CC= aL’ = c~L/2, imposing c:/cr = 1, c:/c, we obtain
= a( L/2)(
that division
L”*/(uL;‘*)
= ;( L/L(p2
= 1,
must take place when
L 2 22/‘L,. In exactly the same obtaining the condition, L 2 2L”.
(10) manner we can introduce corresponding to (101,
the core law, c, = co, of constant
cut-off,
(10’)
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l’ortex flows
ratio R/c Fig. 1. Relative
error ranges
in computing the velocity of a vortex ring, when relative error in computing the core radius from 0 to 1, as a function of the ratio between curvature radius and core radius.
An implementation of the algorithm requires memorization of the initial value L, of the length of each vorton, updated after each division, and to evolve the values L(t) by the vortex intensity values using eq. (2b). With this operation the ratio c,/c, between the core assumed by the vorton method and the desired vortex core is not constantly equal to 1 but ranges from 1 to 2 (or from 0.5 to 1.5). An estimation of the error on the self-induced velocity has been performed by computing the velocity of a vortex ring, and the results are shown in fig. 1. We note that the average error falls below 10% for ratio of radius of curvature and core radius, R/c,over 10, and is less than 20% even far R/c --+ 1. In analogy with the division routine, we briefly outline a technique for coupling vortons during compression. In the same way as before, we conclude that the coupling must occur when
L I 222/3L I1
(11)
or
L I 2PL,,
(11’)
for volume conservation and constant core respectively. The only difference with division operation lies in the fact that, when a vorton satisfies eq. (11) or (11’1, it must find the nearest vorton and fuse with it, adding the intensities and positioning in their center of vorticity. An interesting fusion procedure, removing tightly folded hairpins, is presented by Chorin (1990). Such technique leads to a regularization procedure for strong vortex interaction acting as a sort of sub-grid scale model, and permits a reasonable computational effort to be maintained. The remeshing methods outlined here are still local and do not require any numbering of the vortex element. The implicit dissipative effect, described in section 3, is able to act between two successive vorton divisions. The division process transforms a part of the infinite self-energy of the vorton in interaction energy (6) between the two newly created vortons,
G. Pedrizzetti / Insight into singular uortex flows
107
partially counterbalancing the dissipative effect. During reconnection, stretching grows and division is faster, so that in a strong vortex interaction there may be not enough time, during division interlaps, for viscous effect to permit reconnection before strong velocity gradients appear. This is particularly true when the local volume conservation law is imposed, as will be shown in section 6. We have used the dividing/fusing method to introduce a local vortex volume conservation, or a constant cut-off law; nevertheless a different one can be introduced, as an alternative. The main concept we want to note here is that the evolution law of a secondarily important quantity has been introduced in a particle method, in a quantized way without dramatic errors, retaining and using the corpuscularity of the field.
5. Numerical
experiments
For testing and presenting the methodologies outlined above, we show here two different examples of vortex interactions for which experimental or numerical results are available. The first simulation presents the interaction, merging, and successive splitting of a two-vortex ring, as the characteristic merging process of two filament portions moving toward each other and fusing. The second simulation presents the motion of two infinite, periodic, orthogonally offset filaments, which interact reciprocally in a strong manner. 5. I. Incident
uortex rings
The fusion of two identical vortex rings has been investigated numerically. The initial configuration is given by two vortex rings of radius R = 1 and circulation r = 1, placed at a distance s = 3 center to center, and inclined at an angle 8 = 30” with respect to the horizontal, as shown in fig. 2. The filter parameter n has been chosen as equal to 2, but no great sensitivity to this parameter has been observed. The volume conservation dividing procedure has been used. The initial number of vortex singularities is 48, meaning an initial core c = 0.15, and an average of c = 1.225; at the end of simulation the number is 56. The time step is At = 0.01, with a fourth-order Runge-Kutta integration scheme. The two rings move with self-induced velocity toward each other, bringing two filament portions of opposite sign face to face (fig. 3). At this moment there is a localized stretching of the portions and a movement of the same portions toward each other in a fusion direction. When the portions are close enough with respect to the vortex core or, from a vorton point of
i r
-‘,
_z
..I
0
x
0.”
I
1
J
-5,
-2
-I
0
x
0.”
I
2
3
-‘,
-I
-1
0
f
*
,
Tad
Fig. 2. Initial configuration of two vortex rings. Vorton intensity vectors are plotted. The three orthogonal are reported, presenting top (left), front (center), and lateral (right) views.
projections
108
G. Pedrizzetii
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inro smgular
vortex flows
D t=14 -i. -2. -5.
r---‘~~_.,_ l- _-.p,
-.
Fig. 3. Axonometric projections of the stages of evolution, identical vortex rings of unit circulation and unit radius. unchanged at changing times. The initial number of vortons 1sq.
at different times. of two initially separate incident Vorton intensity vectors are plotted; the scales are per ring is 24. The projection angles are [e, []=[35”,
view, with respect to the discretization spacing, the vortex field recognizes the new vortex structure given by the two linked rings. The evolution continues passing through the reconnection instant, which occurs around t = 5, and a distorted vortex ring is formed. The ring
G. Pedrizzetti / Insight into singular r’ortex flows
10
109
15
20
l5
20
time
1.60
Fig. 4. Time evolution
of (a) Z-momentum
5
and (b) interaction
10
time
energy
in the interaction
of two incident
vortex rings.
oscillates and then splits again into two vortex rings between t = 14 and t = 15. The calculation is stopped at t = 20 when two nearly circular vortex rings are formed. The global vorticity, momentum, and angular momentum for the vorton system cannot be represented exactly as the sum of the vorton contributions (Aksman and Novikov, 1988); nevertheless we check these, corresponding to the non-solenoidal vortex field (1). All the invariants are constant below the numerical error O(lOPh), except for the momentum in the z-direction which after the first reconnection moves from - 11 to - 14. The helicity is also constant below the numerical error, while the interaction energy grows after the first reconnection from about 1.7 to 2.2. The time evolutions of the momentum I, and the interaction energy Eint are reported in fig. 4. A similar computation has been done by Winckelmans and Leonard (1988), with s = 2.7, 0 = 15”, c = 0.19, and Reynolds number R, = r/u = 400 with a viscous version of regularized vortex particle simulation for thick rings, using 6272 particles and double symmetry of the problem. If the evolution is very similar and the time of the first reconnection is in good agreement, then the simulation is stopped at t = 8; a decrease of the Z-momentum is observed there, too. A longer time confrontation can be done with a viscous low-resolution simulation made with the same technique as the precedent, with s = 3, 0 = 30”, c = 0.25, and R, = 200 (Winckelmans, 1989). The time of the first reconnection is in good agreement; thus the time needed to complete the reconnection is in our simulation much shorter, due to the highly viscous effect and a poor representation of the reconnection process itself. The agreement continues (with a shift on time equal to about 4 time units) until the second split, which again is faster. Another confrontation has been done by Schatzle (1987) experiments with s = 2.7,13 = 13.3”, c = 0.31, and R, = 1800. A very good agreement is observed, whereas the reconnection processes again are here reproduced faster than in the experiments. A last confrontation has been done with the viscous direct numerical simulation performed by Kida et al. (1989), with s = 3.73 and 8 = 30”. Qualitative agreement is observed until the
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/ Insight into singular cortex flows
i ?,
_I
-1
0 I xo,n
P
3 x0.0
10‘”
Fig. 5. Initial configuration of two rectilinear vortex filaments. Vorton intensity vectors are plotted. A thick dot represents the location of vortex singularities. The three orthogonal projections are reported, presenting top (left), front (center), and lateral (right) views.
second reconnection, even if the time evolution is slower due to the larger value of S. The second reconnection is accompanied by secondary vortex structures (bridges), which, of course, cannot be observed in a simulation like the one here presented. The evolution should be compared, also, with the original vorton simulations of roughly the same problem (Novikov, 1988; Aksman and Novikov, 1988; Pedrizzetti and Becchi, 19901, in order to be able to observe significant improvements due to filtering, and dividing, operations. The same simulation has been tried with a transpose (adjoint) vorton scheme (Winckelmans and Leonard, 1988); we noticed that such a scheme does not allow reconnection to take place, but at the moment of reconnection, when the structure is effectively a poor approximation of a divergence-free vortex, the rings loses connectivity, and vortons do not tend to realign but have the tendency to adjust orthogonally to the vortex field. 5.2. Orthogonally offset uortex filaments The interaction of two identical orthogonally offset vortex filaments is a challenging problem for vortex simulation, due to the fact that the tubes, before reconnecting, are subjected to a strong nonsymmetric, stretching. In this, it differs from the preceding experiment where two portions of rings move toward each other. The initial configuration (see fig. 5) is given by two filaments parallel to the x-axis and y-axis respectively; the filament circulation is r = 20 and the distance between the filament axes is 7r/3. The periodicity has been introduced in the X and Y directions, but not in the Z-direction because of the difficulty of inserting periodicity into a vortex method, so the simulation has been performed in a periodic square column of side 2~, rather than in a periodic box as in the numerical experiments discussed below, which are based on a spectral method; in particular, three periodic images per side have been considered and one image per side in the oblique direction, for a total of 16 images added to the original one. The initial number of vortons is 48. The time step is At = 0.001, with a fourth-order Runge-Kutta integration scheme. We first introduced the volume conservation dividing/fusing procedure, and parameter 77 = 60, observing a strong stretching of the vortex filaments as they approach. At time t = 0.95 the filaments are very close, and the stretching makes the number of vortons increase faster until the filaments connect before t = 1. The various stages of evolution are plotted in fig. 6. The number of vortons and the total length of filaments increase strongly before reconnection, and decrease once the reconnection instant has passed and the filaments have
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-
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/ Insight into singular cortex flows
111
-_
R -A
7r
Fig. 6. Axonometric projections of the stages of evolution, at different times, of two orthogonally offset identical vortex filaments, with circulation 20, filter parameter q = 60, and volume-preserving dividing routine. Vorton intensity vectors are plotted; the scales are unchanged at changing times. The initial number of vortons per filament is 24. The projection angles are [e, Cl = [25”, 207.
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1
time
1
time
1.5
‘.5
Fig. 7. Time evolution of (a) the number of vortex singularities and (b) the total length of filaments, in the interaction of two orthogonally offset identical vortex filaments. Curve (1) is obtained with the filter parameter n = 40 and volume-preserving division operation; curve (2) with n = 60 and volume-preserving division operation; curve (3) with n = 40 and constant cut-off division operation.
shifted to the new configuration. The volume-preserving dividing procedure, in such a strong stretching phenomenon, leaves a short time for the viscous effect considered in section 3 to take place, making the simulation closer to an effective inviscid simulation. This has been observed in terms of the criticality of the reconnection moment, where a slight perturbation can render impossible the reconnection. In fig. 7 the number of vortons and total filaments length are plotted versus time for the case n = 60 (curve 21, discussed above, and n = 40 (curve 11, where the shift to the reconnection does not happen and a finite time collapse is observed. The same simulation has been performed with the dividing routine keeping a constant cut-off. The evolution is now much smoother; reconnection occurs for any value of the
I 0.5
1
time
1.5
2
Fig. 8. Time evolution of the interaction energy, in the interaction of two orthogonally offset identical vortex filaments. Curve (1) is obtained with the filter parameter n = 40 and volume-preserving division operation; curve (2) with TJ= 60 and volume-preserving division operation; curve (3) with n = 40 and constant cut-off division operation.
G. Pedrizzetti / Insight into singular Llortexjlows
113
parameter 77 (in the approximate range 10 to 1001, showing an absence of criticality. The number of vortons and total filament tength is plotted versus time in fig. 7 (curve 3). In fig. 8 the interaction energy, in the computational square column, for the three cases is plotted versus time. The dissipation due to stretching and the energy recovered due to division is here evident; in particular, strong dissipation occurs before reconnection, while the strong increase of energy due to recompression, just after reconnection, can lead to instabilities. An analogous computation has been performed by Zabusky and Melander (19891, with r= 20.944 and Gaussian core of size l/G, with a high-resolution (963> spectral simulation of viscous Navier-Stokes flow. The time at which the filaments lose their individuality appears close to t = 1, in agreement with our results, followed by the formation of secondary structures. The reconnection time is difficult to recognize, but appears to be between t = 2 and t = 3. In our simulation the reconnection is much faster, due to the presence of a high viscous effect at the moment of reconnection. Zabusky and Melander (1989) mention that at higher resolution or with hype~iscosity alone reconnection occurs earlier; they also observe that the major inffuence of the core size lies in the vorticity distribution into the secondary structures, but not in the global evolution.
6. Conclusion, singular flows, and perspective The vortex singularities method has been analyzed, and some inconsistencies have been studied. It has been clarified that this method is not intended as an alternative to the classic cut-off method, but rather a model for very high Reynolds number flows. A linear filter procedure, giving vortons a tendency to realign with the divergence-free vorticity field, has been introduced. This operation has been shown to give vorton simulation a natural self-consistency. The dynamics of reconnection have been analyzed. A crude estimation suggests the presence of a spurious viscous effect implicit in the method. The effect should grow with growing stretching, representing the regularization mechanism for passing through the moment of reconnection without exhibiting large velocity gradients. A cut-off, and thus the core size, is implicitly assumed by the discretization proportional to the vortons’ interspace. During stretching it grows, leading to an incorrect core evolution. The dividing (fusing) procedure has been rearranged in such a way as to permit the introduction of a core evolution law in a quantized model. An estimation of errors due to quantized evolution has been given. Two different kinds of vortex reconnection have been studied numericaily. The interaction, merging and successive splitting of two vortex rings represent the fusion process of two paired filament portions moving toward each other. The second simulation has analyzed the interaction of two infinite, periodic, orthogonaIly-offset identical fiIaments. The interaction is here much stronger, and the reconnection is preceded by an intense stretching to allow filament portions to adjust in a reconnectable configuration. The period before reconnection has proven to be a very critical moment; a tendency to vortex collapse has been observed when local volume conservation law is introduced. The results of simulations have been compared with existing numerical and experimental results and a good agreement has been observed. The very moment of reconnection is simulated in a spurious way, but the topological changes and successive evolution are reproduced with good agreement. The results presented here suggest the method of vortex singularities can be considered, rather than an approximation, as a “model” for computing vortex evolution in very high Reynolds flow. In many cases it permits evolution of vortex structures to be foIlowed with
114
(3. Pedrizzetti / Imight mto singular ivrtex fI0w.s
relatively little effort and with enough accuracy to define, if necessary, where higher precision methods must be used or further investigation must be performed. The vorton system is not a weak solution of the Euler equations, but such an argument has not much bearing on the validity of the model, as stressed by Greengard and Thomann (1988). It has been shown by Novikov (1988) that the energy of a vorton ring, for a large number of vortons (N > l), despite the singular character of the vortons, coincides with the total kinetic energy of the flow created by a continuous vortex ring. Hou and Lowengrub (1990) have shown that the point vortex approximation is representative of a smooth distribution of vorticity and is convergent, in a special integral norm, to the smooth solution of Euler equations if it exists. The choice of singular elements is due to the will of creating a model rather than an approximation. On the other hand, we do not have an existence theorem for solution of Navier-Stokes equations at very high Reynolds numbers, and we may expect as a real possibility the formation of singufarities and the branching of solutions (Novikov, 1990). Apart from perfect ideal vortex motion, which must be simulated by a different method, the observation of the main phenomena and tendencies of vortex metamorphoses in high Reynolds number flows can be described with a limited number of degrees of freedom. In this view the singular uortex ,flow is a corpuscular model, existing on its own as a dynamic system, which can help in undertanding complex vortex interactions. The calculations have been performed on a 80386/20 personal computer and on a SUN Spark 1 + workstation. The source codes of programs used, and the complete numerical results, are available from the author.
This work has been supported by MURST. I wish to thank Professor I. Becchi for many useful comments. I also want to thank Dr. E. Novikov for his scientific advice during the research work.
References Agishtein, M.E. and A.A. Migdal(1986) Computer simulation of three-dimensional vortex dynamics, Mod. Whys. Left. A I (33). 221. Aksman, M.J.. E.A. Novikov and S.A. Orszag, (lY85) Vorton method in three-dimensional hydrodynamics~ Phys fly:. Lett. 54 (221, 2410. Aksman, M.J. and EA. Novikov (1988) Reconnection of vortex filaments, Fluid Dyn. Res. 3, 239. Anderson, C. and C. Greengard (1984) On vortex method, SIAM J. Nume,: Anal. 22,413. Chorin, A.J. (1982) The evolution of a turbulent vortex, Comm. Math. Phys. 83, 517. Chorin, A.J. (1990) Hairpin removal in vortex interaction, J: Camp. fhys. 91, 1. Greengard, C. (1986) Convergence of the vortex filament method, M&I. Comp. 47 (12) 387. Greengard, C. and E. Thomann (1988) Singular vortex systems and weak solutions of the Euler equations, Phys. Fluids 31 (IO), 2810. Hou, T.Y. and J. Lowengrub (1990) Convergence of the point vortex method for the 3-D Euler equations, Comm. Pure A& Math. 43, 965. S. Kida, M. Takaoka and F. Hussain (1989) Reconnection of two vortex rings, Phys. Fluids A I (4), 630. Kuwabara, S. (1986) Studies on vortex motion based on the dividing vorton model, Proc. Symp. on h&c/z for Space Flight, Inst. of Space and Astronaut. Sci.. Rep. SP4, 1 I. Landau. L. and E. Lifshitz (1967) ~~~~~1~~~1~~ Quarlti~ue, T!&rie aon R~~ufi~~fe (Mir, Moscow) ch. 69. Leonard, A. (198S) Computing three-dimensional inc~~mpressible flows with vortex elements, Ann. Ret~. F&id Me& 17, 523. Moore, D.W. and P.G. Saffman (1972) The motion of a vortex with axial flow, Phil. Trans. R. Sac. A 272, 403. Novikov, E.A. (1983) Generalized dynamics of three-dimensional vertical singularities (vortons), SOL.. Phys. JETP 57 (31, 566.
G. Pedrizzetti
/ Insight into singular twtex flows
115
Novikov, E.A. (1988) ‘Three-dimensional vortex reconnection, Proc. 6th Symp. Energy Eng. Sci., Argonne Nat]. Lab., p. 59. Novikov, E.A. (1990) The internal dynamics of flows and formation of singularities, Fluid Dyn. Rrs. 6, 79. Novikov, E.A. (1991) private communi~tion. Pedrizzetti G. and 1. Becchi (1990) Interazione di strutture vorticose e lore modellazione, Pruc. 1st ANIV Congress (Pitagora Bologna), Florence 28-30 October. Pedrizzetti, G. (1991) Stretching of a vortex filament over an approaching sphere using the vortex singularities method, DIG I-3. Dip. Ing. Civile, Florence. Pedrizzetti, G. (1991a) About modelling the interaction of a vortex filament and a rigid sphere, Euromech, Zst European Fluid Me&. Conf. Cambridge 16-20 September. Pumir, A. and E.D. Siggia (1987) Vortex dynamics and the existence of sotutions to the Navier-Stokes equations. Phys. F1uid.s 30 (71, 1606. Saffman, G., B.T. Meiron (1986) Difficulties with three-dimensional weak solutions for inviscid incompressible flow, Phys. Fluids 29 (S), 2373. Schatzle, P.R. (1987) An experimental study of fusion of vortex rings, Ph.D. thesis, Cal. Inst. Techn., Pasadena, CA. Winckelmans, G. and A. Leonard (1988) Improved vortex methods for three-dimensional flows, Proc. SMN workshop on Vortex L&vzamics, April 25-27, Leesburg VA, p. 25. Winckelmans, G.S. (1989) Topics in vortex methods for the computation of three- and two-dimensional incompressible unsteady flows, Ph.D. Thesis, Cal. Inst. Techn., Pasadena CA. Zabusky, NJ., M.V. Melander (1989) Three-dimensional vortex tube reconnection: morphology for orthogonally offset tubes, Physiaa D 37, 555.
Research
FLUIDDVMMKS RESEARCH
10 (1992) 101-115
Insight into singular vortex flows Gianni
Pedrizzetti
Dipartimento di Ingegneria Cirile, Unicersitri di Firenze, via di Santa Marta, 3, 50139 Florence, Italy Received
10 May 1991
Abstract. The method of three-dimensional vortex singularities is analyzed. The fact that it does not represent a weak solution of the Euler equations has little bearing on its validity as an operative model. Other evolution inconsistencies are a major problem. The non-solenoidality of the vortex field is analyzed and a linear filtering feedback procedure, requiring no additional computations, is introduced to allow alignment with the reconstructed solenoidal vorticity field. A rough estimation of the dissipative effect during vortex reconnection has shown a finite viscous effect that is implicitly present in the model, growing with stretching, representing the mechanism of shifting to reconnected topologies without observing strong velocity gradients. The dividing vorton method is rearranged in order to allow the quantized reproduction of a predefined core evolution law, an error estimation, due to the corpuscularity of the field, is given. Two different numerical computations are performed corresponding to a weak and a strong vortex interaction. The improvements are tested, and confrontations with existing numerical and experimental results are performed showing good agreement. The possibility of the singular vortex flow serving as a rough, but simple, model for very high Reynolds number comglex flows is discussed.
1. Introduction
to the vorton method
Construction of a basic computational vortex element for the representation of three-dimensional vertical flow has been a widely investigated field for a long time. The classic vortex filament method, with cut-off, was constructed for modeling vortex filament evolution in inviscid flows. The method is the only one to use when a perfectly ideal flow is to be investigated, and has been proven to converge in the limit of the ratio between transversal section radius and the curvature radius approaching zero (Moore and Saffman, 1972; Greengard, 1986). Another possibility is the use of a vortex particle of finite size (blob vortons). The computational elements in this case are spherical blobs of vorticity. An interesting way of using blob vortons is given by approximating thick vortex structures using clouds of overlapping elements. In this case, particle independence makes possible the correct introduction of viscous effects (Winckelmans and Leonard, 1988). This method has proven to be an actual Lagrangian alternative to direct grid or spectral simulations. The vortex singularities (vortons) model is not an alternative to the methods outlined above, being in fact less accurate; it is rather a “model” for very high Reynolds number flows, in which the phenomenon of vortex reconnection is easily reproduced. A vorton (Novikov, 1983) is a vortex &singularity which, if located at point x,, gives rise to a vorticity field
where
y is the vorton
Correspondence 50139 Florence,
intensity
to: G. Pedrizzetti, Italy.
0169-5983/92/$04.75
and
Dipartimento
0 1992 - The Japan
6c.j
is the three-dimensional
di Ingegneria
Civile, Universiti
Society of Fluid Mechanics.
All rights
Dirac di Firenze,
reserved
a-function. via di Santa
The Marta,
3,
singularity is representative Intensity y is given by
of a vortex volume,
V, characterized
by a value, LC),of the vorticity.
y=wv. A series becomes
(2a)
of singularities y=T
can be used for representing
filament,
in which case (2a)
dx,
(2bl
where I‘ is the filament circulation and dx the represented by the singularity. The velocity field given by the vortex field (1) is u(x)
a vortex
= -G
vector
length
of the
filament
portion
1 (x-X”)XY (3)
Ix-XVI3 The vortex singularities
move with the local velocity
and the intensities
evolves as fluid lines
x, =u(x\)>
(4al
+=r*
(4b)
Cu(x,),
where the self-interaction term is neglected. The method here outlined is a very clean and simple model of vortex dynamics, but it has a number of inconsistencies which must be investigated. The vortex field (1) is not divergence-free. If we start with a vorton configuration which well represents a divergence-free vortex structure, the discretization errors can drive the system toward a structure that is not divergence-free, without any feedback opposing it. In section 2 a filtering procedure helping the system to readjust in a divergence-free configuration is proposed. A vortex line has an infinite self-induced velocity, which can be bounded only if we consider a vortex tube with a finite core. In the vorton method the discretization implicitly introduces a cut-off, and consequently a core size, proportional to the neighboring vortons’ interspace. During stretching of the vortex tube, the distance between particles grows, leading to the incorrect result of core growth. In section 4 a dividing vorton procedure which takes into account a predefined core evaluation law is introduced. These are, to our knowledge and experience, basic points, from an operative viewpoint, which make applications of the vorton method difficult. From a theoretical point of view, system (4) is not equivalent to the Euler equations, as shown by Saffman and Meiron (1986) and Greengard and Thomann (1988); even when the number of vortons tends to infinity, it does not converge to it. Nevertheless it has been proven by Hou and Lowengrub (1990) that the convergence proof given for regularized vortex models is applicable to singular elements, due to the smoothness of the velocity field in the points of application. Vortex reconnection is observed in vorton simulation; this is in contrast with ideal flow evolution, so a clarification and estimation of what we are simulating must be given. In section 3 we try to give an estimation of the finite viscosity effect implicitly present in the method. In section 5 we present two different simulations of vortex evolution, one a smooth reconnection, and the other a strong interaction ending with reconnection, of vortex filaments. The results are then compared with numerical and experimental results, and the improvements here introduced are tested. In conclusion, we consider the singular r,ortex flow as an economic, even though not accurate, model for a first insight into the complex problem of vortex interaction at high Reynolds number.
G. Pedrizzerti / Insight into singular [orten flows
2. Divergence
filtering
103
in the vorton method
The vorticity field given by (1) is not divergence-free, however, a divergence-free vorticity field can be reconstructed as the curl of the velocity field (3) (Novikov, 1983). When the y vectors are well aligned with the reconstructed vorticity field, a vorton system is a good, discrete approximation of a divergence-free vortex structure, and subsequent evolution closely agrees with a divergence-free vortex evolution. This fact has been stressed by Greengard and Thomann (1988), Aksman and Novikov (1988>, and Pedrizzetti (1991). A parallel study of vortex-body interaction with classical cut-off method and vortex singularity method has clearly revealed the different characteristics of the two models (Pedrizzetti, 1991a). Discretization errors and numerical approximations can drive the y vectors away from a divergence-free configuration. Actually, even after a short time calculation, they usually depart from it slowly, because there is no effect helping them to readjust in a divergence-free configuration. This is a serious obstacle to long computations, and to simulations of complex interactions (Pedrizzetti and Becchi, 1990). For this reason we introduce a filtering procedure which gives y vectors a tendency to realign with VX u vector field. Let us define the aligned intensity
We introduce
an equation
dr,‘dt=
for evolution
-v(Y-?),
of the vorton
intensities, (5)
to be added to eq. (4b). The parameter 77 is the characteristic frequency of the linear filter operation (5). The filter time scale, T, = l/q, must not be longer than the timescale of the physical phenomena otherwise it has no effect; on the other hand, it must not be shorter to avoid artificial forcing, because a discrete vortex representation can pass through a not perfectly aligned configuration during the evolution. Nevertheless, the results are not very sensitive to the value of 7, once the order of magnitude is right. Typically for a curved vortex filament 77 N T/R’, R being the radius of curvature. The effect of this procedure on the evolution of vortex particle structures is very encouraging, giving the system a natural self-consistency which permits the computation of complex interactions and the study of long-time evolutions, as will be seen from the results of section 5. We want to point out that this operation requires no additional calculation in a numerical program. At each time step the velocity partial derivatives, &L,/&K, with i, j = 1, 2, 3, are computed for the evaluation of Vu, at each vorton location, from these we can compute V x u and q. Then we update the intensities by yne, = (1 - cu)y + a+, with cy = 77 At, and At the computational time step. This dirlergence filtering procedure has no clear physical meaning yet (a study is in progress with an analogy with the self-consistent field method in a system with weak interaction; Landau and Lifshitz, 1967; Novikov, 1991), but its significance, from an operative point of view, is clear as a methodology for making the system (4a, b) consistent with its physical basis.
3. Vorton structure
reconnection
In computational vortex models, when core size is evolved, the energy is not an invariant of motion (e.g. Agishtein and Migdal, 1986; Pumir and Siggia, 1987). This is obvious because swirl velocity and axial flow are not taken into account in energy computation, and during
G. Pedrizzetti
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/ Insight into singular cortex flows
stretching of vortex lines there is a transfer between such “hidden” energy and the modeled macroscopic structure. In the vorton method the total energy of the system is given by the interaction energy between pairs of singularities, 1 y”‘. p _ 16rr i 1x(i) _p
E’12’= I”t
y’l’.(x 1
+
(1) _ p9
p
. (#)
1x(l) _ @) 13
_ x(29 (6) I>
where superscripts 1 and 2 indicate two different vortons, and the self-interaction energy of each singular element is infinite. During stretching, interaction energy can be transformed into self-energy and vice versa (Aksman et al., 1985). This implies an energy dissipation during stretching, which can be partially counterbalanced by a remeshing method, as indicated in the next section. In very high Reynolds number vertical flows, the moment of reconnection between vortex structures is connected with the formation of a singularity or very high velocity gradients. In vorton simulations reconnection is usually observed, in contrast with the ideal flow Kelvin theorem, and even without observing very high velocity gradients. This is due to the presence, for construction, of a spurious dissipative mechanism, which permits the reproduction of these phenomena. Consider a vorton as representative of a portion of vortex filament of length L and core radius c. During stretching of the vorton filament the length L of the portion increases, along with the vorton intensity as given by eqs. (4b) and (2b). The vorton method implicitly assumes a value of the core c proportional to the neighboring vorton distance c = Q L (Aksman and Novikov, 1988; Pedrizzetti, 1991), and implies a core evolution law dc/dt
= (Y dL/dt.
(7)
This evolution law implies that the vortex core grows during stretching of vortex lines instead of decreasing as required by the volume conservation law, as described in the next section. As a first rough but simple way to estimate the presence of a viscous effect, the phenomena of core growth in the vorton method can be compared with the core spread due to a viscous diffusion of vorticity. The core evolution law relative to a viscous diffusion, with kinematic viscosity V, is given by ;=;(_&zj,
(8)
where the additional term for local volume conservation is written Equaling the right-hand sides of eqs. (7) and (8), we obtain u -c
dL/dt.
(9)
This is a very rough estimation, but it shows effect exists implicitly in the vorton method. In the mechanics which permits jumping over the reconnection which, otherwise, could hardly be
4. Volume preserving
in parentheses,
dividing/fusing
that, in first approximation, a finite viscosity particular, it grows with stretching, and it is moment of local intense stretching as vortex followed numerically.
vortons
A vortex line of zero cross-sectional area moves with an infinite self-induced velocity due to a non-zero curvature. A vortex filament moves with a finite velocity only if a finite transversal radius, or core size, is taken into account. When a vortex filament is computed with a single series of computational points along the filament axis, the core size can be taken into account as proportional to a cut-off length which eliminates the singular character of the integral in
G. Pedrizzetti / Insight inlo singular clortex flows
105
computing the velocity (e.g., Chorin, 1982; Leonard, 1985; Winckelmans and Leonard, 1988). With singular vortons a cut-off is implicitly introduced due to the discretization, and a core size c, proportional to the vortons distance Ax, is assumed; in particular, considering the velocity of a vortex ring of circular cross section of radius c with uniform distribution of vorticity, it is found that Ax/c = 1.143.. . , with great accuracy (error < 10%) already with 4 vortons. During the filament motion the vortex cross section also must evolve. In a vortex model the motion within the core is not resolved, and some additional consideration must be given in order to introduce an evolution law for the radius c of the core, which is assumed to be circular. From a physical point of view it is required that the vortex volume be maintained, even locally. This implies that for a portion of filament of length L(t) and core c(t) the product L(t)c*(t) is constant. This consideration leads to variation of c along the filament. Moore and Saffman (1972) showed that a variation in the internal structure along the filament produces an axial flow that redistributes the volume along the filament, so as to eliminate inhomogeneities, in a time that is short compared with the time needed by the filament to move significantly (see Pumir and Siggia (1987) for different process timescales). The actual core evolution law is thus confined between the limiting situations of local volume conservation, in which axial flow smoothing is not taken into account, and global volume conservation, in which axial flow is assumed to be instantaneous with respect to the filament geometry evolution. Another extreme possibility is given by maintaining the core c, i.e. the cut-off, constant. In this way we have, for continuous filament, an Hamiltonian structure of the equation of motion (Anderson and Greengard, 1984; Agishtein and Migdal, 1986). A major problem in the vorton method is that, during stretching, vortons move apart and Ax and so c grow, instead of remaining constant or decreasing. The opposite happens during compression, even if stretching is much more frequent than compression. A dividing vorton model was introduced by Kuwabara (1986) in order to keep good discretization during vortex evolution. We propose an analogous method of vorton division arranged in such a way that a core evolution law, even if in a quantized way, can be introduced. We consider here the evolution laws for the two extreme cases of volume conservation and constant cut-off. Consider a vorton representing a portion of vortex tube of length L and core c. Call c, the core represented by the vorton model, and c, the core which should keep the volume of the portion constant. Assume that at a certain initial time the vorton discretization represents the correct volume, During or core size, c, = c, = cg = aL,, where the suffix 0 means at the initial instant. stretching of the vortex filament the core represented by the vorton discretization evolves as c, = aL (being the constant a = l/1.143.. . >, giving the incorrect result of core growth. If the local volume conservation is imposed, the core should evolve as c, = cc,L)/2/L”2. During stretching c, will always be greater than cr, until after some time the value of c, will come to be the double of c,. At this time, when c,/c, 2 2, we can divide the vorton into two parts so that, signing with a prime the variables after division, c: = c,/2 = c,. As CC= aL’ = c~L/2, imposing c:/cr = 1, c:/c, we obtain
= a( L/2)(
that division
L”*/(uL;‘*)
= ;( L/L(p2
= 1,
must take place when
L 2 22/‘L,. In exactly the same obtaining the condition, L 2 2L”.
(10) manner we can introduce corresponding to (101,
the core law, c, = co, of constant
cut-off,
(10’)
G. Pedrizzetti
106
/ Insight
into smgular
l’ortex flows
ratio R/c Fig. 1. Relative
error ranges
in computing the velocity of a vortex ring, when relative error in computing the core radius from 0 to 1, as a function of the ratio between curvature radius and core radius.
An implementation of the algorithm requires memorization of the initial value L, of the length of each vorton, updated after each division, and to evolve the values L(t) by the vortex intensity values using eq. (2b). With this operation the ratio c,/c, between the core assumed by the vorton method and the desired vortex core is not constantly equal to 1 but ranges from 1 to 2 (or from 0.5 to 1.5). An estimation of the error on the self-induced velocity has been performed by computing the velocity of a vortex ring, and the results are shown in fig. 1. We note that the average error falls below 10% for ratio of radius of curvature and core radius, R/c,over 10, and is less than 20% even far R/c --+ 1. In analogy with the division routine, we briefly outline a technique for coupling vortons during compression. In the same way as before, we conclude that the coupling must occur when
L I 222/3L I1
(11)
or
L I 2PL,,
(11’)
for volume conservation and constant core respectively. The only difference with division operation lies in the fact that, when a vorton satisfies eq. (11) or (11’1, it must find the nearest vorton and fuse with it, adding the intensities and positioning in their center of vorticity. An interesting fusion procedure, removing tightly folded hairpins, is presented by Chorin (1990). Such technique leads to a regularization procedure for strong vortex interaction acting as a sort of sub-grid scale model, and permits a reasonable computational effort to be maintained. The remeshing methods outlined here are still local and do not require any numbering of the vortex element. The implicit dissipative effect, described in section 3, is able to act between two successive vorton divisions. The division process transforms a part of the infinite self-energy of the vorton in interaction energy (6) between the two newly created vortons,
G. Pedrizzetti / Insight into singular uortex flows
107
partially counterbalancing the dissipative effect. During reconnection, stretching grows and division is faster, so that in a strong vortex interaction there may be not enough time, during division interlaps, for viscous effect to permit reconnection before strong velocity gradients appear. This is particularly true when the local volume conservation law is imposed, as will be shown in section 6. We have used the dividing/fusing method to introduce a local vortex volume conservation, or a constant cut-off law; nevertheless a different one can be introduced, as an alternative. The main concept we want to note here is that the evolution law of a secondarily important quantity has been introduced in a particle method, in a quantized way without dramatic errors, retaining and using the corpuscularity of the field.
5. Numerical
experiments
For testing and presenting the methodologies outlined above, we show here two different examples of vortex interactions for which experimental or numerical results are available. The first simulation presents the interaction, merging, and successive splitting of a two-vortex ring, as the characteristic merging process of two filament portions moving toward each other and fusing. The second simulation presents the motion of two infinite, periodic, orthogonally offset filaments, which interact reciprocally in a strong manner. 5. I. Incident
uortex rings
The fusion of two identical vortex rings has been investigated numerically. The initial configuration is given by two vortex rings of radius R = 1 and circulation r = 1, placed at a distance s = 3 center to center, and inclined at an angle 8 = 30” with respect to the horizontal, as shown in fig. 2. The filter parameter n has been chosen as equal to 2, but no great sensitivity to this parameter has been observed. The volume conservation dividing procedure has been used. The initial number of vortex singularities is 48, meaning an initial core c = 0.15, and an average of c = 1.225; at the end of simulation the number is 56. The time step is At = 0.01, with a fourth-order Runge-Kutta integration scheme. The two rings move with self-induced velocity toward each other, bringing two filament portions of opposite sign face to face (fig. 3). At this moment there is a localized stretching of the portions and a movement of the same portions toward each other in a fusion direction. When the portions are close enough with respect to the vortex core or, from a vorton point of
i r
-‘,
_z
..I
0
x
0.”
I
1
J
-5,
-2
-I
0
x
0.”
I
2
3
-‘,
-I
-1
0
f
*
,
Tad
Fig. 2. Initial configuration of two vortex rings. Vorton intensity vectors are plotted. The three orthogonal are reported, presenting top (left), front (center), and lateral (right) views.
projections
108
G. Pedrizzetii
/
Insight
inro smgular
vortex flows
D t=14 -i. -2. -5.
r---‘~~_.,_ l- _-.p,
-.
Fig. 3. Axonometric projections of the stages of evolution, identical vortex rings of unit circulation and unit radius. unchanged at changing times. The initial number of vortons 1sq.
at different times. of two initially separate incident Vorton intensity vectors are plotted; the scales are per ring is 24. The projection angles are [e, []=[35”,
view, with respect to the discretization spacing, the vortex field recognizes the new vortex structure given by the two linked rings. The evolution continues passing through the reconnection instant, which occurs around t = 5, and a distorted vortex ring is formed. The ring
G. Pedrizzetti / Insight into singular r’ortex flows
10
109
15
20
l5
20
time
1.60
Fig. 4. Time evolution
of (a) Z-momentum
5
and (b) interaction
10
time
energy
in the interaction
of two incident
vortex rings.
oscillates and then splits again into two vortex rings between t = 14 and t = 15. The calculation is stopped at t = 20 when two nearly circular vortex rings are formed. The global vorticity, momentum, and angular momentum for the vorton system cannot be represented exactly as the sum of the vorton contributions (Aksman and Novikov, 1988); nevertheless we check these, corresponding to the non-solenoidal vortex field (1). All the invariants are constant below the numerical error O(lOPh), except for the momentum in the z-direction which after the first reconnection moves from - 11 to - 14. The helicity is also constant below the numerical error, while the interaction energy grows after the first reconnection from about 1.7 to 2.2. The time evolutions of the momentum I, and the interaction energy Eint are reported in fig. 4. A similar computation has been done by Winckelmans and Leonard (1988), with s = 2.7, 0 = 15”, c = 0.19, and Reynolds number R, = r/u = 400 with a viscous version of regularized vortex particle simulation for thick rings, using 6272 particles and double symmetry of the problem. If the evolution is very similar and the time of the first reconnection is in good agreement, then the simulation is stopped at t = 8; a decrease of the Z-momentum is observed there, too. A longer time confrontation can be done with a viscous low-resolution simulation made with the same technique as the precedent, with s = 3, 0 = 30”, c = 0.25, and R, = 200 (Winckelmans, 1989). The time of the first reconnection is in good agreement; thus the time needed to complete the reconnection is in our simulation much shorter, due to the highly viscous effect and a poor representation of the reconnection process itself. The agreement continues (with a shift on time equal to about 4 time units) until the second split, which again is faster. Another confrontation has been done by Schatzle (1987) experiments with s = 2.7,13 = 13.3”, c = 0.31, and R, = 1800. A very good agreement is observed, whereas the reconnection processes again are here reproduced faster than in the experiments. A last confrontation has been done with the viscous direct numerical simulation performed by Kida et al. (1989), with s = 3.73 and 8 = 30”. Qualitative agreement is observed until the
G. Pedrizzetrl
110
/ Insight into singular cortex flows
i ?,
_I
-1
0 I xo,n
P
3 x0.0
10‘”
Fig. 5. Initial configuration of two rectilinear vortex filaments. Vorton intensity vectors are plotted. A thick dot represents the location of vortex singularities. The three orthogonal projections are reported, presenting top (left), front (center), and lateral (right) views.
second reconnection, even if the time evolution is slower due to the larger value of S. The second reconnection is accompanied by secondary vortex structures (bridges), which, of course, cannot be observed in a simulation like the one here presented. The evolution should be compared, also, with the original vorton simulations of roughly the same problem (Novikov, 1988; Aksman and Novikov, 1988; Pedrizzetti and Becchi, 19901, in order to be able to observe significant improvements due to filtering, and dividing, operations. The same simulation has been tried with a transpose (adjoint) vorton scheme (Winckelmans and Leonard, 1988); we noticed that such a scheme does not allow reconnection to take place, but at the moment of reconnection, when the structure is effectively a poor approximation of a divergence-free vortex, the rings loses connectivity, and vortons do not tend to realign but have the tendency to adjust orthogonally to the vortex field. 5.2. Orthogonally offset uortex filaments The interaction of two identical orthogonally offset vortex filaments is a challenging problem for vortex simulation, due to the fact that the tubes, before reconnecting, are subjected to a strong nonsymmetric, stretching. In this, it differs from the preceding experiment where two portions of rings move toward each other. The initial configuration (see fig. 5) is given by two filaments parallel to the x-axis and y-axis respectively; the filament circulation is r = 20 and the distance between the filament axes is 7r/3. The periodicity has been introduced in the X and Y directions, but not in the Z-direction because of the difficulty of inserting periodicity into a vortex method, so the simulation has been performed in a periodic square column of side 2~, rather than in a periodic box as in the numerical experiments discussed below, which are based on a spectral method; in particular, three periodic images per side have been considered and one image per side in the oblique direction, for a total of 16 images added to the original one. The initial number of vortons is 48. The time step is At = 0.001, with a fourth-order Runge-Kutta integration scheme. We first introduced the volume conservation dividing/fusing procedure, and parameter 77 = 60, observing a strong stretching of the vortex filaments as they approach. At time t = 0.95 the filaments are very close, and the stretching makes the number of vortons increase faster until the filaments connect before t = 1. The various stages of evolution are plotted in fig. 6. The number of vortons and the total length of filaments increase strongly before reconnection, and decrease once the reconnection instant has passed and the filaments have
G. Pedrizzetti
Y
-
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/ Insight into singular cortex flows
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Fig. 6. Axonometric projections of the stages of evolution, at different times, of two orthogonally offset identical vortex filaments, with circulation 20, filter parameter q = 60, and volume-preserving dividing routine. Vorton intensity vectors are plotted; the scales are unchanged at changing times. The initial number of vortons per filament is 24. The projection angles are [e, Cl = [25”, 207.
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time
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Fig. 7. Time evolution of (a) the number of vortex singularities and (b) the total length of filaments, in the interaction of two orthogonally offset identical vortex filaments. Curve (1) is obtained with the filter parameter n = 40 and volume-preserving division operation; curve (2) with n = 60 and volume-preserving division operation; curve (3) with n = 40 and constant cut-off division operation.
shifted to the new configuration. The volume-preserving dividing procedure, in such a strong stretching phenomenon, leaves a short time for the viscous effect considered in section 3 to take place, making the simulation closer to an effective inviscid simulation. This has been observed in terms of the criticality of the reconnection moment, where a slight perturbation can render impossible the reconnection. In fig. 7 the number of vortons and total filaments length are plotted versus time for the case n = 60 (curve 21, discussed above, and n = 40 (curve 11, where the shift to the reconnection does not happen and a finite time collapse is observed. The same simulation has been performed with the dividing routine keeping a constant cut-off. The evolution is now much smoother; reconnection occurs for any value of the
I 0.5
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time
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Fig. 8. Time evolution of the interaction energy, in the interaction of two orthogonally offset identical vortex filaments. Curve (1) is obtained with the filter parameter n = 40 and volume-preserving division operation; curve (2) with TJ= 60 and volume-preserving division operation; curve (3) with n = 40 and constant cut-off division operation.
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parameter 77 (in the approximate range 10 to 1001, showing an absence of criticality. The number of vortons and total filament tength is plotted versus time in fig. 7 (curve 3). In fig. 8 the interaction energy, in the computational square column, for the three cases is plotted versus time. The dissipation due to stretching and the energy recovered due to division is here evident; in particular, strong dissipation occurs before reconnection, while the strong increase of energy due to recompression, just after reconnection, can lead to instabilities. An analogous computation has been performed by Zabusky and Melander (19891, with r= 20.944 and Gaussian core of size l/G, with a high-resolution (963> spectral simulation of viscous Navier-Stokes flow. The time at which the filaments lose their individuality appears close to t = 1, in agreement with our results, followed by the formation of secondary structures. The reconnection time is difficult to recognize, but appears to be between t = 2 and t = 3. In our simulation the reconnection is much faster, due to the presence of a high viscous effect at the moment of reconnection. Zabusky and Melander (1989) mention that at higher resolution or with hype~iscosity alone reconnection occurs earlier; they also observe that the major inffuence of the core size lies in the vorticity distribution into the secondary structures, but not in the global evolution.
6. Conclusion, singular flows, and perspective The vortex singularities method has been analyzed, and some inconsistencies have been studied. It has been clarified that this method is not intended as an alternative to the classic cut-off method, but rather a model for very high Reynolds number flows. A linear filter procedure, giving vortons a tendency to realign with the divergence-free vorticity field, has been introduced. This operation has been shown to give vorton simulation a natural self-consistency. The dynamics of reconnection have been analyzed. A crude estimation suggests the presence of a spurious viscous effect implicit in the method. The effect should grow with growing stretching, representing the regularization mechanism for passing through the moment of reconnection without exhibiting large velocity gradients. A cut-off, and thus the core size, is implicitly assumed by the discretization proportional to the vortons’ interspace. During stretching it grows, leading to an incorrect core evolution. The dividing (fusing) procedure has been rearranged in such a way as to permit the introduction of a core evolution law in a quantized model. An estimation of errors due to quantized evolution has been given. Two different kinds of vortex reconnection have been studied numericaily. The interaction, merging and successive splitting of two vortex rings represent the fusion process of two paired filament portions moving toward each other. The second simulation has analyzed the interaction of two infinite, periodic, orthogonaIly-offset identical fiIaments. The interaction is here much stronger, and the reconnection is preceded by an intense stretching to allow filament portions to adjust in a reconnectable configuration. The period before reconnection has proven to be a very critical moment; a tendency to vortex collapse has been observed when local volume conservation law is introduced. The results of simulations have been compared with existing numerical and experimental results and a good agreement has been observed. The very moment of reconnection is simulated in a spurious way, but the topological changes and successive evolution are reproduced with good agreement. The results presented here suggest the method of vortex singularities can be considered, rather than an approximation, as a “model” for computing vortex evolution in very high Reynolds flow. In many cases it permits evolution of vortex structures to be foIlowed with
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relatively little effort and with enough accuracy to define, if necessary, where higher precision methods must be used or further investigation must be performed. The vorton system is not a weak solution of the Euler equations, but such an argument has not much bearing on the validity of the model, as stressed by Greengard and Thomann (1988). It has been shown by Novikov (1988) that the energy of a vorton ring, for a large number of vortons (N > l), despite the singular character of the vortons, coincides with the total kinetic energy of the flow created by a continuous vortex ring. Hou and Lowengrub (1990) have shown that the point vortex approximation is representative of a smooth distribution of vorticity and is convergent, in a special integral norm, to the smooth solution of Euler equations if it exists. The choice of singular elements is due to the will of creating a model rather than an approximation. On the other hand, we do not have an existence theorem for solution of Navier-Stokes equations at very high Reynolds numbers, and we may expect as a real possibility the formation of singufarities and the branching of solutions (Novikov, 1990). Apart from perfect ideal vortex motion, which must be simulated by a different method, the observation of the main phenomena and tendencies of vortex metamorphoses in high Reynolds number flows can be described with a limited number of degrees of freedom. In this view the singular uortex ,flow is a corpuscular model, existing on its own as a dynamic system, which can help in undertanding complex vortex interactions. The calculations have been performed on a 80386/20 personal computer and on a SUN Spark 1 + workstation. The source codes of programs used, and the complete numerical results, are available from the author.
This work has been supported by MURST. I wish to thank Professor I. Becchi for many useful comments. I also want to thank Dr. E. Novikov for his scientific advice during the research work.
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