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Interaction of inviscid vortices having orthogonal orientations.

Introduction

The efficient transport of scalars such as heat and species concentration in fluids can be brought about by methods which produce better mixing. Even though the generation of turbulence is a very effective cause of mixing, it becomes really difficult to generate it at low Reynolds numbers. A different but complementary mechanism is that of laminar, chaotic advection (Aref 1984, Ottino 1989) in which chaotic particle paths can be used to enhance heat transfer (Acharya et al. 1992, Mokrani et al. 1997). Ryrie (1992) studied mixing by chaotic advection in a class of spatially periodic flows. Lim (1989) studied the quasi-periodic dynamics of desingularized vortex models.

Chaotic fluid motion has been explored from Lagrangian perspective by Aref et al. (1989) caused mainly due to non-integrable vortex motions. Low-dimensional, nonlinear, conservative or dissipative dynamical systems can exhibit, under suitable conditions, sensitive dependence on initial conditions and produce complicated phase-space trajectories. An analysis of the literature of the last few decades shows that many researchers have studied chaotic dynamical systems. One of these systems, in which the chaotic phenomenon takes place, is large-scale vortex structures in which even laminar flows may lead to chaotic particle paths. It is simpler to consider fixed vortices rather than those that are free to move with the flow. The existence of chaotic path lines can be studied by using the Lagrangian approach of tracking individual fluid particles. Designs of industrial devices can be made using this principle, such as stirred tanks (Yoon et al. 2001, Gerson and Kole 2001), micro-biological flows (Orme et al. 2001), microchannels and micromixers (Liu et al. 1995, Stroock et al. 2002, Engler et al. 2004, Wong et al. 2004) and combustors (Fritz et al. 2004).

The idea behind most heat transfer enhancement techniques is to promote mixing and hence the rate of transport of the scalar. Previous work has also dealt with specific aspects of flow due to interacting vortices (e.g. Luton et al. 1995, Hwu et al. 1997, Kuznetsov and Zaslavsky 2000). Romero-Mendez et al. (1998) have studied the influence of Rankine vortices in an inviscid thermal boundary layer flow for different vortex orientations. Moffatt (2000) studied the interaction of two propagating vortex pairs, each pair being aligned along the principal axis of strain associated with the other that is skewed in space. Tabeling et al. (1990) studied chaos in a linear array of forced vortices experimentally. Pumir and Kerr (1987) carried out numerical simulation of interacting vortex tubes using spectral method in 3-D incompressible hydrodynamics at Reynolds number of 1000. A merging criterion for two-dimensional co-rotating vortices was studied by Meunier et al. (2002). In the study, vortices were generated experimentally by roll-up of a vortex sheet and phenomenon was followed up to study the transition for which critical parameters were measured.

The objective of the present work is to understand how the presence and orientation of vortices inserted into the flow affects particle paths and the mixing which ultimately decides the mechanism towards enhancement of heat transfer. For simplicity inviscid vortices have been considered, and the interaction between two or more of these is determined. Visualization of a trajectory and dependence of a velocity component of a particle on time, and its Fourier spectrum are used in the investigation as tools for analysis. For some cases, the behaviors of Poincare sections of particle path on an arbitrarily chosen plane are also studied. The transition in the flow is revealed using time series corresponding to Lagrangian velocity of the particle.

Fluid particle's pathline equations

Pathlines are described by particle positions, [??](t), which are governed by ordinary differential equations of the form,

d[??]/dt = [??]([??], t), (1)

with initial conditions, [??](0) = [[??].sub.0], where [??]([??], t)is the velocity field, and t is time. The fixed points or stationary points (Jordan and Smith, 1987) for the above system can be obtained by solving for the zeros of right hand side in Eqn. (1) above. The eigen values (Jordan and Smith, 1987) correspond to the linearized form of the general nonlinear system.

For a single inviscid vortex of strength [GAMMA], the velocity vector of a fluid particle at a perpendicular distance r from the axis of the vortex is

[??] = [GAMMA]/2[pi]r [[??].sub.[theta]], (2)

where [[??].sub.[theta]] is the tangential unit vector at the location of the fluid particle.

Consider a system of three inviscid and fixed vortex filaments having strengths [[GAMMA].sub.i], i = 1,2,3 and axes parallel to the coordinate directions [x.sub.1], [x.sub.2] and [x.sub.3] respectively, as shown in Fig. 1. In general, the axes of vortices 1 and 2 do not intersect, and the shortest normal distance between them is 2d. The axes are assumed to be stationary with respect to time. Moreover, the vortices 1 and 2 have equal strength and relative strength of the third vortex is the only a parameter. The origin of coordinates is assumed to lie midway between the first two fixed vortices and the axis of vortex filament 3 is along the common normal of 1 and 2. The space in which the motion of a fluid particle is being considered is unbounded and time evolution of motion could be presented even for much longer time intervals. However, at far distances from the vortex filaments, the contributed velocity field due to each vortex would be very small. The combined velocity field for a given fluid particle can be written as a vector superposition of individual velocities contributed by each of the fixed vortex filaments. Assuming [[GAMMA].sub.1] = [[GAMMA].sub.2] = [GAMMA] and [[GAMMA].sub.3] = [gamma] [GAMMA] (where [gamma] is a positive or negative real constant), the velocity components of the fluid particle along respective coordinate directions are given as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where [??] = [([X.sub.1], [X.sub.2], [X.sub.3])T. Here, [u.sub.1] is the uniform external velocity field (called translational velocity) in [X.sub.1]-direction.

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Non-dimensionalization

The above system of equations can be non-dimensionalized with respect to the length scale [r.sub.0] = [[X.sup.2.sub.10] + [X.sup.2.sub.20] + [X.sup.2.sub.30]].sup.1/2], where ([X.sub.10], [X.sub.20], [X.sub.30]) is the initial position of the particle, and time scale [tau] = 2[pi][r.sup.2.sub.0]/[GAMMA]. Using the same symbols for the non-dimensional space variables and non-dimensional time, we have

d[X.sub.1]/dt = u + ([X.sub.3] + [epsilon])/[X.sup.2.sub.1] + [([X.sub.3] + [epsilon]).sup.2] - [gamma][X.sub.2]/[X.sup.2.sub.1] + [X.sup.2.sub.2], (6)

d[X.sub.2]/dt = [gamma][X.sub.1]/[X.sup.2.sub.1] + [X.sup.2.sub.2] - ([X.sub.3] - [epsilon])/[X.sup.2.sub.2] + [([X.sub.3] - [epsilon]).sup.2], (7)

d[X.sub.3]/dt = [X.sub.2]/[X.sup.2.sub.2] + [([X.sub.3] - [epsilon]).sup.2] - [X.sub.1]/[X.sup.2.sub.1] + [([X.sub.3] + [epsilon]).sup.2], (8)

where 2[epsilon] = 2d / [r.sub.0] is the non-dimensional separation between the axes of first two vortex filaments, and u = [u.sub.1] /([r.sub.0]/[tau]) is the non-dimensional velocity superimposed in the [X.sub.1]-direction.

Numerical solutions

The position and velocity of the fluid particle are numerically calculated as functions of time, and from the velocity signals, Fourier spectra are obtained. The separation between axes of vortices 1 and 2 and the strengths of vortex 3 are varied systematically, keeping the other parameters fixed. Three special cases have been considered: two vortices, three vortices and two vortices with uniform flow (translational superposition). The system of equations are solved numerically (here we have used a fourth-order Runge-Kutta scheme).

Two-vortex system

In the absence of third vortex and of uniform external velocity, i.e. for [gamma] = 0 and U = 0 ([gamma] is the relative strength of the third vortex and U is the non-dimensionalized translational velocity superimposed on the vortex field), Eqns. (6)-(8) become

d[X.sub.1]/dt = ([X.sub.3] + [epsilon])/[X.sup.2.sub.1] + [([X.sub.3] + [epsilon]).sup.2], (9)

d[X.sub.2]/dt = ([X.sub.3] - [epsilon])/[X.sup.2.sub.2] + [([X.sub.3] - [epsilon]).sup.2], (10)

d[X.sub.3]/dt = [X.sub.2]/[X.sup.2.sub.2] + [([X.sub.3] - [epsilon]).sup.2] - [X.sub.1]/[([X.sup.3]).sup.2] + [X.sup.2.sub.1], (11)

Intersecting pair

For [epsilon] = 0, we have

d[X.sub.1]/dt = [X.sub.3]/[X.sup.2.sub.1] + [X.sup.2.sub.3], (12)

d[X.sub.2]/dt = - [X.sub.3]/[X.sup.2.sub.2] + [X.sup.2.sub.3], (13)

d[X.sub.3]/dt = [X.sub.2]/[X.sup.2.sub.2] + [X.sup.2.sub.3] - [X.sub.1]/[X.sup.2.sub.1] + [X.sup.2.sub.3], (14)

The fixed points or stationary points for the above system lie along the line [X.sub.3] = 0 and [X.sub.1] = [X.sub.2]. If the stationary point is taken as (1, 1, 0) about which the system of equations (12) to (14) are linearized to obtain the characteristic equation as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

the three eigen values of the corresponding linearized system being 0 and [+ or -] [square root of 2], representing an unstable saddle (Jordan and Smith 1987).

If we multiply Eqns. (12)-(14) by [X.sub.1], [X.sub.2] and [X.sub.3] respectively and add, it can be shown that the system is integrable and one of the integrals is

[X.sup.2.sub.1] + [X.sup.2.sub.2] + [X.sup.2.sub.3] = 1 (15)

This means, a fluid particle initially at unit distance from the origin will remain confined on the surface of a sphere having center at the origin and radius equal to unity. The particle may follow a three-dimensional path during its motion, but its distance from origin always remains constant. There is thus no mixing of the fluid in the radial direction, and the component of advection flux in that direction is absent. Fig. 2(a) shows the path of such a fluid particle in the [X.sub.1]-[X.sub.2]-[X.sub.3] phase-space and it is observed that the three-dimensional path followed by a fluid particle for [epsilon] = 0 is closed.

Skewed vortex pair, [epsilon] [not equal to] 0

In such a case, no real fixed points or stationary points are found to exist. The effect of separation between the axes of the two vortices appears to cause a deviation from the integral in Eqn. (15), and the dynamical system becomes non-integrable. However, for small values of [epsilon] ([epsilon] [much less than] 1), the solution can be shown to remain confined in the neighborhood of the unit sphere. Multiplying Eqns. (9)-(11) by [X.sub.1], [X.sub.2] and [X.sub.3] respectively and adding, we can show that for [epsilon] [much less than] 1, the integral can be expressed in the form

d/dt ([X.sup.2.sub.1] + [X.sup.2.sub.2] + [X.sup.2.sub.3]) = [epsilon] ([X.sub.1], [X.sub.2], [X.sub.3]) (16)

Here, after multiplying Eqns. (9)-(10) and (11) by [X.sub.1], [X.sub.2] and [X.sub.3] and considering power series expansion about [epsilon] = 0, we have neglected the terms containing [[epsilon].sup.2] and higher order. Thus, it is apparent from the integral in Eqn. (16) that for [epsilon] [not equal to] 0, the three-dimensional path of the fluid particle evolves from the surface of the sphere having radius equal to initial distance of the fluid particle from the origin. The distance of the fluid particle would increase or decrease depends upon the sign of the function f([X.sub.1], [X.sub.2], [X.sub.3]) in the neighborhood of the starting point. However, for the cases of [epsilon] [not equal to] 0, no asymptotic bound can be predicted within the range of time-interval considered for different values of e, as can be observed from the nature of particle paths presented in Figs. 2(b)-(d). The particle paths presented in these figures originate from an initial position [X.sub.10] = [X.sub.20] = [X.sub.30] = 0.577 such that its initial distance from origin equals unity, and their trajectories are computed for a non-dimensional time equal to t = 100. From the behavior of the trajectories it is not possible to conclude whether the position of the Lagrangian particle remains confined in a bounded space or its distance from origin increases with time. With an increase in the value of [epsilon], the paths do not close during finite time of observation, as shown in Figs. 2(b), (c) and (d). These paths are not confined to a spherical surface, but appear to spread in the neighborhood of the spherical surface. For larger values of [epsilon], the distance of the fluid particle grows with time at much faster rate and hence we cannot predict the boundedness of its path. In either case, an important observation which could be made is that with increase in value of [epsilon], certain order of mixing gets caused in the radial direction as well.

In all the cases for which we considered the path of a fluid particle above, the velocity in the [X.sub.3]-direction, being called as w-velocity, has been shown in Fig. 3. It is seen from Figs. 3(a)-(d) that the amplitude as well as frequency of the velocity signal decreases with increase in the axial separation ([epsilon]). It is interesting to note that for the case of intersecting vortices, the time period of oscillations remains same but for the case of non-intersecting vortices, with increase in e, the time period (T) increases with time. In Fig. 3(d) for large value of [epsilon], it may be seen that the number of cycles decreases compared with Figs. 3(a), (b) and (c). This, in other words, gives an indication about unbounded nature of the particle paths for large values of axial separations. The Fourier spectra of [X.sub.3]-component of velocity (w) are shown in Figs. 4(a)-(d) respectively. Fourier spectrum of a signal gives the contributions from the dominant harmonics present in the signal. For the present case of two intersecting vortices, the spectra consist of discrete peaks of well defined harmonics. It is apparent that lower frequencies have more dominant contribution compared with higher ones. An increase in axial separation (increasing value of skewness parameter e) results into spreading of Fourier spectra from well defined characteristic peaks into a band. In other words, the increase in axial separation leads to velocity fluctuations having contributions from more number of higher frequencies with their individual contributions becoming relatively weaker compared with those of smaller separation value. The appearance of wide band in the spectrum with increasing separation indicates more disordered motion of fluid particles and hence better fluid mixing Berge et al., 1984).

The Poincare sections (Berge et al. 1984) of the path of the fluid particle on the bisector plane of [X.sub.1]-[X.sub.3] plane and [X.sub.2]-[X.sub.3] plane and projected on [X.sub.1]-[X.sub.3] plane, are shown in Figs. 5(a)-(d). In order to demonstrate the role of e as a perturbation parameter, with respect to the integrable behavior of the intersecting vortex pair, small values of [epsilon] are considered in the range from 0.0 to 0.1. It is apparent that with an increase in the axial separation, the Poincare maps indicate a larger spread of the point of intersection of the trajectory of the fluid particle with the considered plane (Figs. 5(b) and 5(c)) and for certain value of [epsilon] it reaches a maximum. In other words, these Poincare maps on a particular plane show that a perturbation with respect to an integrable or closed form solution for the intersecting vortex system results into an increased degree of wandering of a fluid particle which reaches its maximum for a particular value of e. For the range of [epsilon] considered, it is seen that the interaction effect of the field increases with increase in separation from the original intersecting configuration, reaches a maximum for a particular separation and then begins to decrease. One of the aims of the present study could be to identify the separation of the orthogonal skewed pair in presence of the third vortex which would cause most effective particle mixing. With that objective, results below present the effect of presence of third vortex for various values of separation parameter ([epsilon]) and also to examine if any qualitative change occurs in the vortex field as result of increase in strength of the third vortex.

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Three-vortex system

For the system of Eqns. (6)-(8), for [gamma] [not equal to] 0 and also [gamma] [not equal to] 1, in absence of translation (U=0), it has been found that the existence offixed points is conditional, i.e. for [epsilon] = 0, no critical points exist. Even though [X.sub.1]=[X.sub.2]=[X.sub.3]=0 is a trivial solution of a fixed point, it is unacceptable due to being a point of singularity for the inviscid vortex filament (velocities shoot up near the axis of an inviscid vortex). However, for [epsilon] = 0 and [gamma] = 1, the fixed points of the dynamical system are clearly seen to be given as [X.sub.1] = [X.sub.2] = [X.sub.3]. For [epsilon] [not equal to] 0, no real zeros are found to exist for the set of Eqns. (6)-(8) and hence no fixed points are expected to exist for this case.

Once again, using Eqns. (6)-(8), we can show that for U=0, and [epsilon] = 0, the dynamical system is integrable and one of the integrals is the same as obtained in Eqn. (15) above. It is found that even in the presence of the third vortex, if the fixed axes of vortex filaments are concurrent and uniform external velocity field is absent, the path of a fluid particle remains confined on the surface of a sphere having radius equal to the initial distance of the particle from origin. Also, for [epsilon] [much less than] 1, the system of Eqns. (6)-(8) can be expressed in a form similar to Eqn. (16) above (the nature of function f will of course change). This indicates that even for the system consisting of all the three fixed vortex filaments, the paths of fluid particles evolve from the surface of the sphere having radius equal to initial distance of the particle from origin.

Critical strength of the third vortex ([[gamma].sub.c])

Figures 6(a)-(f) present the fluid particle paths corresponding to two intersecting vortices ([epsilon] = 0) of equal strength superimposed with the third vortex having variable strength. As expected, for different values of [gamma], the path remains closed. However, in the neighborhood of [gamma] = 1, the nature of path changes as seen by comparing Figs. 6(c) and 6(e). For [gamma] = 1, as seen from Fig. 6(d), there is no path of the particle because the initial position ([X.sub.10] = [X.sub.20] = [X.sub.30] = 0.577) is a fixed point. The fluid particle initially located at this position remains stationary. That is the reason we also find empty block in the velocity signal of Fig. 7(d) and FFT of the velocity signal shown in Fig. 8(d). The nature of path of the fluid particle changes for values of [gamma] > 1 (Figs. 6(e) and (f)) compared with those for [gamma] < 1 (Figs. 6(a), (b) and (c)). This characterizes a transition in the vortex field and the strength of the third vortex corresponding to this transition is being referred to as the critical strength ([[gamma].sub.c]). The velocity signals and Fourier spectra of the velocity signals for the values of [gamma] in Fig. 5, are shown in Figs. 7 and 8 respectively. Comparison of path, velocity and Fourier spectra in Figs. 6(c), 7(c) and 8(c) with those of Figs. 6(e), 7(e) and 8(e) confirms the transition about the fixed point corresponding to [gamma] = 1. As the value of [gamma] is increased, the velocity signal changes to composite harmonic nature with new harmonics appearing in the neighborhood of pre-existing harmonics in absence of the third vortex, as seen from the velocity signals in Figs. 7(b) and 7(c) and their spectra in Figs. 8(b) and 8(c). However, for [gamma] > [[gamma].sub.c], the change in nature of particle path as seen from 7(c) to 7(e) is associated with a change in the harmonic nature of the signal as well. For [gamma] = 1.01, the velocity signal in Fig. 7(e) has reduced amplitude as well as frequency as compared to Fig. 7(c). This gets explained by observing the Fourier spectra presented in Figs. 8(c) and 8(e) where in spite of both having wide bands, the lower harmonics are seen to be more dominant in Fig. 8(e).

For small separation between the axes of the skewed pair, viz. [epsilon] = 0.01, the particle path, the w-velocity signal and the Fourier spectra of the velocity signal are presented in the Figs. 9, 10 and 11 respectively. In a similar way, for a larger separation of [epsilon] = 0.2, the corresponding results are presented in Figs. 12-14. It is seen that the critical transition is associated for each separation value. Corresponding to various separations investigated in the present study, it has been found that there exists a critical value of strength of the third vortex ([[gamma].sub.c]) associated with a transition in the vortex field. The results presented consider only one small ([epsilon] = 0.01) and one large ([epsilon] = 0.2) separation magnitudes. Values of [[gamma].sub.c] for larger separations are not investigated because the effect of interaction seems to diminish appreciably with increase in separation. For [epsilon] = 0.01, a comparison of velocity signals in Figs. 10(b) and 10(c) and their spectra in Figs. 11(b) and (c) shows that this separation is more effective in causing particle mixing before onset of transition. This is observed from the nature of path of the fluid particle presented in Fig. 10(b) as well corresponding to [gamma] = 0.5(< [[gamma].sub.c]) which shows a more dense spreading as compared to that in Fig. 10(d). For [epsilon] = 0.2, the particle paths in Fig. 12 show that the vortex configuration just before transition results into wider spread paths than beyond transition. However, the frequency of the velocity signal in Fig. 13(c) is much lesser before transition which gets significantly increased beyond transition. It is also apparent from spectra of velocity signals presented in Figs. 14(c) and (d). This indicates that for larger separations, the particle motion becomes chaotic beyond transition of the vortex field. This is certainly the most sought configuration for enhancement of particle mixing in the vortex field.

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Dependence of [[gamma].sub.c] on [epsilon]

Fig. 15 shows variation of [[gamma].sub.c] with [epsilon] in the range of 0 [less than or equal to] [epsilon] [less than or equal to] 0.2. It is seen that [[gamma].sub.c] increases with increase in value of [epsilon]. In other words, with an increase in axial separation, the transition gets delayed. However, much delayed transitions are less important to us because the strength of third vortex becomes quite dominant relative to other two; thereby the effect of interaction between the skewed pair is rendered almost of negligible significance. We are interested to find the most effective configuration of the vortex system which results into efficient particle mixing.

As seen from the results presented in Figs. 12-14 that for [epsilon] = 0.2, the value of critical strength of the third vortex corresponding to which sudden change in temporal characteristics of the vortex field take place increases to 3.372 in contrast to a value of 1.0 for the intersecting system. An interesting distinction results in the nature of particle path (Figs. 12(c) and (d)), the velocity signal (Figs. 13(c) and (d)) and Fourier spectra (Figs. 14(c) and (d)) when the value of [gamma] changes from 3.732 to 3.733. In fact, the velocity signal and spectra give a clear indication that the particle path of Fig. 12(d) beyond transition has chaotic nature.

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Effect of translational superposition

Bryden and Brenner (1999) have shown that even at low Reynolds numbers chaotic flows can arise in a droplet by superposition of translation and simple shear flows. The presence of chaotic streamlines increases the rate of transport of solutes in a bulk fluid. Keeping such an observation in mind, attempt has been made in the present work to investigate the effect of uniform external velocity field on the considered field due to inviscid and fixed vortex filaments. If a uniform external velocity is superimposed upon the vortex system in [X.sub.1]-direction, then we consider u [not equal to] 0 in the system of Eqns. (6)-(8). Here u = [u.sub.1]/([r.sub.0] / [tau]) is the non-dimensional uniform velocity along [X.sub.1]-direction. The effect of such a translational superposition has been discussed below separately for two and three vortex systems.

Figs. 16(a)-(d) present the particle paths due to effect of translational superposition on the two and three vortex filaments. For all the cases considered, a small translational velocity of u = 0.01 has been superimposed in the [X.sub.1]-direction on the vortex field. For the two and three vortex systems, effect of translational superposition on particle path is shown for e = 0.0 in Figs. 16(a) and (b) and for [epsilon] = 0.01 in Figs. 16(c) and (d). A comparison of paths in presence of translation and those in absence of translation presented before, shows that for two vortex system, the effect of translation for intersecting as well as for [epsilon] = 0.01 appears in the form of stretching and turning (Figs. 16(a) and (c)) of the particle trajectory. Moreover, it is apparent from velocity signals in Figs. 17(a) and (c) and their spectra in Figs. 18(a) and (c) that the particle motion in the two vortex field possesses clear signatures of chaotic behavior as effect of translational superposition. The spectra in Figs. 18(a) and (c) have broad bands with quite significant power density magnitudes. On the other hand, in the presence of third vortex, the path of particle in Figs. 16(b) and (d) has more ordered kind of periodic characteristics but still consists of composite harmonics. For [epsilon] = 0.0, the spectra of Fig. 18(b) show a dominant peak with significant spread bands of higher harmonics. This means, for the intersecting vortex field of the three vortices, the effect of translational superposition changes the fixed point behavior into an interesting chaotic motion.

Since the initial position of the fluid particle ([X.sub.10] = [X.sub.20] = [X.sub.30] = 0.577) happens to be a fixed point for u = 0.0 corresponding to parametric values in Fig. 16(b), the velocity signal in Fig. 17(b) shows that while the particle gets displaced slightly away from the stationary state by the superimposed translational velocity, its velocity continues to remain close to zero. Once the position of the particle gets changed with respect to the initial stationary state, it comes under influence of changed (non-zero) velocity field due to which an interesting velocity signal emerges thereafter which has a chaotic spectral band, well spread as seen from Fig. 18(b). Fore = 0.01, the translational superposition on the field of all the three vortices is seen to give rise to a multi-periodic path, as indicated by the spectra of velocity having discrete peaks in Fig. 18(d). The velocity signal in such a case is apparently a composite harmonic having large number of discrete peaks in its spectra.

Figure 19 presents the Poincare maps of the velocity signal corresponding the four cases considered above. The mapping points are considered on the bisector plane of [X.sub.1]-[X.sub.3] plane and [X.sub.2]-[X.sub.3] plane and then projected on [X.sub.1]-[X.sub.3] plane. In other words, the set of points at which the particle path crosses the mentioned bisector plane is considered and then the projection of this set on [X.sub.2]-[X.sub.3] plane is mapped. Even though all the four maps in Figs. 19(a)-(d) show spreading, no clear indication appears about nature of chaotic motion and three-dimensional spreading of the particle path for the cases pointed out above. But the same can be resolved by considering a plane chosen such that the paths cross it in the directions associated with maximum twists and turns. Here, the choice of plane was made arbitrarily to observe the nature of maps for the cases having otherwise interestingly distinguishable particle paths.

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Sensitivity to initial conditions

From discussion in the previous section it is observed that the translational superposition on the vortex field results into change in behavior of particle motion. The path of the fluid particle in absence of third vortex possesses more twists and turns (Fig.16) than in presence of third vortex. From spectra of velocity signal in Fig. 18 this observation gets better confirmed because wide bands are present in the spectra for the case of two vortices which had relatively more sharp peaks when the third vortex was present. However, one of the important tests for fluid motion to become chaotic is response of particle path with respect to the initial position of the fluid particle, i.e. 'sensitivity to initial conditions' (SIC). The test for SIC is conducted for the three vortex system ([gamma] = 1) with translational superposition and [epsilon] = 0.01. Fig. 20 shows evolution of particle paths for six different initial conditions. It is seen that with change in initial condition the evolution of particle path is clearly distinguishable. Such a sensitivity of the particle motion on initial condition with its path remaining bound is the necessary condition for onset of chaotic motion. In fact, the three-dimensional paths shown in Figs. 20 (a), (b), (e) and (f) would be expected to be associated with chaotic particle motion due to presence of more twists and turns in them. On an appropriately chosen plane, the Poincare sections of such paths would be expected to show more spread distribution.

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Conclusions

The present study is an analytical attempt from Lagrangian perspective to investigate the characteristic transformation appearing in an inviscid vortex field due to interaction of a system of inviscid and fixed vortex filaments having orthogonal orientations. For two vortices, the existence of fixed points has been confirmed only when they have their axes intersecting and no such points exist when their axes are skewed. For [gamma] [not equal to] 0 and also [gamma] [not equal to] 1, in absence of translation (u=0), it has been found that the existence of fixed points is conditional, i.e. for [epsilon] = 0, no critical points exist. However, for [epsilon] = 0 and [gamma] =1, the fixed points of the dynamical system are clearly seen to be given by [X.sub.1] = [X.sub.2] = [X.sub.3]. For various parametric combinations, the particle motion is studied by considering fluid particle paths, Poincare maps of the trajectories on a plane, temporal evolution of the velocity signal and its Fourier spectra. It is observed that for an orthogonal pair of two inviscid vortices, the axial separation behaves as a perturbation parameter and it affects the extent of spreading of particle path and thereby enhances fluid mixing. In presence of a third inviscid and fixed vortex filament, oriented along the line of shortest separation of the skewed pair, the flow field undergoes transition when the strength of the third vortex ([[gamma].sub.c]) crosses a critical value ([[gamma].sub.c]). The value of critical strength of the third vortex ([[gamma].sub.c]) is found to depend upon the skewness parameter. Effect of translational superposition on the three vortex system can give rise to chaotic particle paths which would get more pronounced in presence of the third vortex. The sensitivity of particle paths to initial conditions has also been confirmed.

References

[1] Acharya, N., Sen, M., Chang, H. C., 1992, "Heat Transfer Enhancement in Coiled Tubes by Chaotic Mixing," Int. J. Heat Mass Transfer, 35 (10), pp. 2475-2489.

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Shaligram Tiwari (1), Gautam Biswas (2) and Mihir Sen (3)

(1) Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai-600036, INDIA, Corresponding Author Email: shaligt@iitm.ac.in

(2) Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur-208016, INDIA Email: gtm@iitk.ac.in

(3) Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556, U.S.A. Email: Mihir.Sen.1@nd.edu
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Author:Tiwari, Shaligram; Biswas, Gautam; Sen, Mihir
Publication:International Journal of Dynamics of Fluids
Article Type:Report
Geographic Code:9INDI
Date:Dec 1, 2009
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