Chaotic advection in open flows

Transcription

1 Chaotic advection in open flows J.J.B. Biemond DCT Traineeship report Coaches: dr. Gy. Károlyi dr. A. demoura Supervisor: prof. dr. H. Nijmeijer prof. dr. dr. h.c. C. Grebogi University of Aberdeen King s College College of Physical Sciences Eindhoven University of Technology Department Mechanical Engineering Dynamics and Control Group Eindhoven, February, 2008

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3 Contents Abstract v 1 Introduction Research goal and approach Outline Paper Physical Review Letters 5 3 Residence time Fractals in residence time Box-counting dimension Bibliography 23 iii

4 iv CONTENTS

5 Abstract In this report we investigate the transition to chaos in the motion of particles advected by open flows with obstacles. By means of a topological argument, we show that the separation points on the surface of the obstacle imply the existence of a saddle point downstream from the obstacle, with an associated heteroclinic orbit. We argue that as soon as the flow becomes time-periodic, these orbits give rise to heteroclinic tangles, causing passively advected particles to experience transient chaos. The transition to chaos thus coincides with the onset of time-dependence in flows with stagnant points. This is in contrast with open flows with no stagnant points, in which first the transition to time-dependence takes place, followed only later by the chaotic regime. We also show that the nonhyperbolic nature of the dynamics near the walls causes anomalous scalings in the vicinity of the transition parameter. These results are confirmed by numerical simulations of the two-dimensional flow around a cylinder. v

6 vi ABSTRACT

7 Chapter 1 Introduction In literature, many studies are available that study the mixing properties of specific fluid flows. However, the available theoretical framework is not sufficient to provide general conditions under which good mixing occurs. Extension of this framework could benefit different research fields, such as oceanography, biophysics and the process industry. Attempts to extend this framework are made both in theorical and in experimental studies, see e.g. [2, 10, 11]. When discussing mixing behaviour of a flow, turbulent and laminar flows have to be treated separately. In turbulent flows, the velocity field of the fluid behaves chaotic and mixing occurs in a chaotic manner. However, also in laminar flows, where the velocity field behaves in a non-chaotic manner, advected particles can experience chaotic motion. In literature, passively advected particles are studied, which means that the particles do not change the velocity field, because they are very small, and they move with the same velocity as the surrounding fluid, which implies that the particle density equals the density of the fluid. The passively advected particles will show the same dynamics as the fluid particles and chaotic motion of the advected particles implies good mixing of the fluid. Specific laminar flows, experiencing chaotic motion of advected particles, are studied in literature. For computational reasons and the availability of theoretical results, nearly all these flows are twodimensional. As we will show in the end of this section, stationary, incompressible flows in two dimensions can not show chaotic motion. Therefore, most literature concerns the dynamics of timedependent, two-dimensional flow models with incompressible fluids. In most of these studies, a modelofthevelocityfield intheflowisconsidered tobeknown,suchthatnonavier-stokesequations haveto besolved. Examples of the models used are theblinking vortexflow, journal bearing flow and cavity flow [1,10]. In these flows, all fluid particles remain in a confined region, such that they are called closed flows. In the opposite case of open flows, fluid particles can enter the region of flow under investigation and all those particles will exit this region after a finite time, which we will call the residence time. Examples of open flows experiencing chaotic motion are the meandering jet,[3, 8], obstructed oceanic jet,[5], and the von Karman flow [6]. In this research, the von Karman flow is studied in more detail. This type of flow may occur when a fluid has to pass an obstacle, for example a cylinder. For very small Reynolds numbers, the flow around a cylinder behaves stationary and laminar, see Figure 1.1a. When the Reynolds number increases, separation points will occur on the boundary of the cylinder, such that two stationary vortices are created in the wake of the cylinder, see Figure 1.1b. By increasing the Reynolds number even further, these vortices become unstable and will be shed of the surface of the cylinder and leave the flow downstream. In that case, vortices will emerge on the upper and lower side of the cylinder and leave the region close to the cylinder in an alternating manner, creating a von Karman vortex street, as depicted in Figure 1.1c. As mentioned before, two-dimensional incompressible flows cannot show chaotic motion. To show this, Hamiltonian dynamical theory is used. All two-dimensional incompressible flows can be described by a streamfunction ψ(x, y, t), such that: u = ψ(x, y, t), y v 1 = ψ(x, y, t), (1.1) x

8 2 CHAPTER 1. INTRODUCTION (a) Re (b) (c) Figure 1.1: Particle trajectories for a flow around a cylinder for increasing Reynolds number Re. (a) Stationary regime without separation. (b) Stationary regime with separation, resulting in stationary vortices. (c) Von Karman vortex street which is time-periodic. where x and y are the spatial coordinates, u and v the fluid velocities in x- and y-direction and t represents the time. Note, that the streamfunction ψ acts like a Hamiltonian. When small particles are embedded in the fluid, the velocity of these particles will be the same as the surrounding fluid if the particle density equals the fluid density. Therefore, the motion of a passively advected particle is given by: ẋ(t) = u = ψ(x, y, t), y ẏ(t) = v = ψ(x, y, t), (1.2) x where ẋ(t) and ẏ(t) represent the particle velocities in x- and y-directions. For stationary flows, where the streamfunction and velocity field are independent of time, all particles will move along lines of constant ψ. Due to this restriction, the particles cannot show chaotic behaviour. 1.1 Research goal and approach In several time-dependent, two-dimensional, incompressible flow models, advected particles experience chaotic motion. However, stationary two-dimensional incompressible flows show regular motion. Therefore, a transition has to take place between regular and chaotic motion, when a timeperiodic velocity-component is added to a stationary flow. In this research, the following question is investigated: Does the motion of passively advected particles become chaotic as soon as a two-dimensional incompressible flow becomes time-dependent? To answer this question, an analytical model of the flow behind a cylinder is studied. In this model, the particle velocity, given by x(t) and y(t), are known a priori, i.e. no Navier-Stokes equations will be solved. A discrete-time sytem is constructed by taking a Poincaré section. The dynamics of this system is then analysed. Hereto, the stable and unstable manifolds of fixed points in the discrete-time system, corresponding to periodic points in the real flow, are visualized with a numerical method.

9 1.2. OUTLINE Outline This report is organised as follows. In Chapter 2, a model describing the flow behind a cylinder is introduced. By studying the manifolds of the equilibria in the discrete-time system, general conditions will be formulated for the appearance of chaotic motion in open flows. Furthermore, consequences of theboundaryofthecylinderarediscussed. InChapter3,theresidencetimeofparticlesinthefloware analysed. With this analysis, the fractal dimension of the flow is calculated for some flow parameters.

10 4 CHAPTER 1. INTRODUCTION

11 Chapter 2 Paper Physical Review Letters 5

12 Stagnation and time-dependence implies chaotic advection in open flows J.J.Benjamin Biemond, 1 Alessandro P.S. de Moura, 2 György Károlyi, 2 Celso Grebogi, 2 and Henk Nijmeijer 1 1 Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands 2 College of Physical Sciences, King s College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom (Received Februari 21, 2008) Abstract In this Letter we investigate the transition to chaos in the motion of particles advected by open flows with obstacles. By means of a topological argument, we show that the separation points on the surface of the obstacle imply the existence of a saddle point downstream from the obstacle, with an associated heteroclinic orbit. We argue that as soon as the flow becomes time-periodic, these orbits give rise to heteroclinic tangles, causing passively advected particles to experience transient chaos. The transition to chaos thus coincides with the onset of time-dependence in open flows with stagnant points, in contrast with flows with no stagnant points. We also show that the nonhyperbolic nature of the dynamics near the walls causes anomalous scalings in the vicinity of the transition. These results are confirmed by numerical simulations of the two-dimensional flow around a cylinder. PACS numbers: a, Ky, Cn 1

13 Two-dimensional flow around an obstacle is stationary for low Reynolds numbers, and the dynamics of passively advected particles is integrable. For higher Reynolds numbers, the flow displays a time-periodic shedding of vortices, leading to the formation of the von Kármán vortex street [1]. In this non-stationary regime, particles commonly experience transient chaotic motion [2, 3]. Chaotic advection is present in many important systems, especially in environmental flows such as the atmosphere and the oceans [4]. It is relevant for key environmental phenomena such as the depletion of the ozone layer [5] and plankton blooms [6 9]. A natural question to ask is whether the transition to chaos in open flows coincides with the transition from stationary to time-dependent flow; does chaotic advection appear as soon as the flow becomes time-dependent? In this Letter, the transition from the regular to the chaotic regime is investigated with a simplified analytical flow model, which incorporates the main features of the flow dynamics near the transition from stationarity to time dependence. The main goal is to determine if and under what conditions the transition from regular to chaotic advection is simultaneous with the transition from stationary to time dependent fluid motion, and to identify the dynamical mechanisms governing this transition. Our main result is that the nature of the transition to chaotic advection, and when it takes place, depend on whether or not there are stagnant points in the flow points where the flow velocity is zero, such as on the surface of walls and obstacles. In the absence of stagnant points, the advection dynamics remains regular immediately after the transition to non-stationarity. This means that there is a range of Reynolds numbers, above the critical value for the onset of non-stationarity, for which the advection remains regular, even though the flow is time-dependent. In this case, chaotic advection only sets in when the strength of timedependent perturbation of the velocity field exceeds a certain threshold. If, on the other hand, stagnant points are present in the flow, we show that the motion of particles becomes chaotic as soon as the time-dependence sets in. We argue in this work that this qualitative difference between the two cases is due to the presence of the stagnant points, resulting in the existence of heteroclinic orbits. These orbits break up and produce a heteroclinic tangle even for arbitrarily small time-dependent perturbations of the stationary flow, leading to chaos. This situation is very common in realistic flows, since stagnant points always exist in the presence of walls. We also study in this work the effects of the non-hyperbolic nature of the advection on 2

14 the dynamics, a result from the presence of walls in the flow. Due to the no-slip boundary conditions imposed on the Navier-Stokes equations, the surface of an obstacle consists of a set of fixed points with zero eigenvalues. Therefore, the advection dynamics is nonhyperbolic when obstacles or other boundaries are present. The heteroclinic orbits which exist before the time-dependence sets in break into heteroclinic tangles. Once this break-up occurs, particles coming from the outside can access the chaotic region present on the wake of the obstacle. In hyperbolic systems, the number of particles N v entering the region excluded by the previously existing separatrix is expected to depend on a bifurcation parameter p (such as the Reynolds number) as N v (p p c ) γ, with p c being the critical parameter value (it can be thought of as the Reynolds number for which the flow becomes time-dependent). Here, γ is a critical exponent, which depends on the eigenvalues of the fixed points [10, 11]. We find that due to the presence of the surface consisting of fixed points and the non-hyperbolic dynamics close to this surface, this relationship does not hold in our case. We have instead an anomalous law of the form N v exp[k(p p c ) η ], where k and η are constants. In order to analyze the transition from a stationary to a time-dependent flow, we use an analytic model for a two-dimensional incompressible flow with a cylindrical obstacle, adapted from a model introduced in [2]. This model can serve as a prototype for the flow of other bluff-body obstacles in a uniform background flow. For 2D incompressible flows, the velocity (ẋ, ẏ) of an advected particle is given by a time-dependent stream function ψ(x, y, t), such that ẋ(t) = ψ(x, y, t), y ẏ(t) = ψ(x, y, t). (1) x We use for ψ the form introduced in [2], which is an analytical approximation to the timedependent, periodic regime of the flow, with two vortices being shed from the back of the cylinder in an alternate manner: ψ(x, y, t) = f(x, y)g(x, y, t). (2) Here f(x, y) accounts for the no-slip boundary conditions on the surface of the cylinder of unit radius: f(x, y) = 1 exp [ a ( x2 + y 2 1 ) 2], (3) with a determining the width of the boundary layer. The factor g(x, y, t) describes the two 3

15 vortices and the background flow: g(x, y, t) = wh 1 (t)g 1 (x, y, t) + wh 2 (t)g 2 (x, y, t) +u 0 ys(x, y), (4) where h i (t), i = 1, 2 describe the growth and decay of the vortex amplitudes that are periodic with a period of T = 1 and vortex strength w, g i (x, y, t) determine the spatial structure of the vortices which are moving in the positive x-direction, as given in [2]. The term s(x, y) corresponds to the shielding effect of the cylinder on the uniform background flow of velocity u 0. The parameters of this flow model are set in [2] to fit the simulation results of the Navier-Stokes equations for a constant background velocity in a channel. We use the same parameter values except for the vortex strength w, which we choose as our bifurcation parameter. For w = 0, the velocity field is stationary; as w increases from 0, the flow becomes time-dependent. This flow model emulates the qualitative behavior of the real system. We note however, that in the limit of w 0, the model represents two moving vortices with infinitesimally small vorticity, whereas in real flows, two stationary vortices of finite strength exists before the transition to time-dependent fluid motion. However, by using this model, we uncover general features of the transition to chaotic advection in timedependent Hamiltonian flows. To confirm that our results are not dependent on the details of the model, we also consider later an alternative flow model, which has a limiting behavior (for w 0) closer to that of the realistic system; we find the same results. In order to investigate the dynamics, we continuously inject particles at a fixed position and plot a superposition of all their trajectories in a stroboscopic manner with the period T = 1 (i.e. we take snapshots of their positions at integer times). In this way we get a streakline. This is shown in Fig. 1 for w = 2. We clearly see that the motion near the cylinder s surface on the back of the obstacle is chaotic, with the presence of prominent KAM islands. Separation points S +, S and accumulation point S 0 on the cylinder s surface are also noticeable. In the region close to the KAM islands the particles whose trajectories get close to the surface of the cylinder exhibit transient chaotic motion. The particles that remain further away from the cylinder do not penetrate this chaotic region and are washed away rapidly by the background flow. To study the probability of particles getting in the chaotic region, a large number N of particles are injected into the flow upstream from the cylinder, and 4

16 Chaotic Region 0.3 S y 0 S S x FIG. 1: Part of a numerically calculated streakline for w = 2 in the wake of the obstacle particles were injected per period at position (x, y) = ( 3, 0) into the flow, and their subsequent positions were plotted every period, at times given by t mod T = 0, until they leave the time dependent region and travel downstream. The solid black area on the left is the cylindrical obstacle. the number Nv of those which visit the region near the KAM surfaces is recorded. The ratio Nv /N is plotted for different values of w in Fig. 2a. We observed in our simulations that this ratio smoothly approaches zero as w is decreased to zero. Particles thus penetrate the chaotic region in the wake for any non-zero value of w. This means that the transition to chaotic scattering coincides with the transition from stationarity to time dependence: as soon as the flow ceases to be stationary, the advection becomes chaotic, and particles coming from the inflow region can access the chaotic region (albeit only in small numbers for w close to 0). For heteroclinic tangles connecting hyperbolic saddle points, one would expect the ratio to scale as Nv /N w γ, where γ = 1.5 for incompressible flows [10, 12]. However, this relationship does not hold for our simulations. We find that the relation Nv /N exp(kw η ) fits well with our numerical data, see Fig. 2a. This anomalous scaling is a direct consequence of the non-hyperbolic nature of the dynamics. We confirmed that the escape time of particles with initial conditions inside the chaotic regions (but outside the KAM islands) scales as a power law, which is characteristic of non-hyperbolic scattering. 5

17 log( log(nv/n)) log( log(nv/n)) (a) log(w) (b) log(β) FIG. 2: Ratio of the number N v of particles entering the wake to all N particles injected into the flow at time t = 0, as a function of the bifurcation parameter, with random initial positions distributed uniformly in the region 16.0 < x < 1.1, 0.05 < y < In figure (a) the original model with bifurcation parameter w is investigated by injecting N = particles, the fitted curve corresponds to N v /N exp(kw η ), where k = and η = In figure (b) the alternative model with bifurcation parameter β is investigated, the fitted curve corresponds to N v /N exp(kβ η ), where k = and η = In these computations, N = is used for parameters β > , whereas N = is used for smaller β. We verified that, even though the particular value of the ratio N v /N depends on where the particles are introduced in the flow, the scaling coefficient η is independent of this. To understand the transition between the regular and chaotic advection from a dynamical point of view, we focus on the stagnation points S +, S and accumulation point S 0 on the downstream surface of the cylinder, depicted in Fig. 3a. The lateral points S + and S have unstable manifolds emanating from them, which act as separatrices at the transition parameter w = 0, when the flow is stationary. Because of the incompressibility of the flow, there must be a similar point with a stable manifold, and that is the central point S 0. In this open flow the velocity has to be positive far downstream (x ). This implies that the stable manifold of S 0 cannot extend infinitely far in the downstream direction, otherwise the boundary condition would be violated, as there would be regions of negative flow velocity arbitrarily far downstream. The only way that incompressibility and the downstream boundary condition can be simultaneously satisfied is by the unstable manifolds of S + and S joining each other and forming a saddle point, as shown in Fig. 3a. For the stationary case w = 0, there are therefore heteroclinic orbits joining the saddle point with the separation points, as shown in Fig. 3a. These heteroclinic orbits act as separatrices, insulating the 6

18 inner region near the wall from the outer region. When the flow becomes time dependent for w > 0, we expect the breakup of the separatrices and the formation of a heteroclinic tangle, which is schematically depicted in Fig. 3b. From these considerations we thus expect that the scattering becomes chaotic as soon as the flow becomes time-dependent; this is indeed what we observe in our simulations. Our reasoning above depends purely on the existence of stagnation points, and our conclusion is a result of the interplay of the incompressible character of the fluid with the openness of the flow with its associated boundary condition. We thus expect the general conclusion to be valid for other flows with stagnant points (or regions). For example, we should have a similar behavior for stagnant or trapped fluid bodies as well, such as the structures depicted in Fig. 3c and d. S + S 0 S - (a) (b) (c) (d) FIG. 3: Schematic picture of the manifolds escaping from the separation points in the flow around an obstacle. In (a), the manifolds are shown for an autonomous system, such that they form separatrices. In (b), the manifolds for a time-periodic flow are depicted, showing a heteroclinic tangle. Images (c) and (d) show the manifolds and streamlines in case of persistent vortices. We calculated the heteroclinic tangle numerically. For the parameter value w = 2, the manifolds are depicted in Fig. 4. We found that the heteroclinic tangle is present for all w > 0, consistent with our conclusion described above, that transient chaotic advection appears as soon as the flow becomes time dependent. This is also confirmed in Fig. 2a, where the ratio of particles accessing the chaotic region behind the cylinder is positive for all w > 0. As argued above, the fact that the transitions to non-stationarity and to chaotic scattering coincide is a consequence of the presence of stagnant points on the surface of the obstacle. This behavior is qualitatively different from that of flows without stagnant points, which have no separatrices like those shown in Fig. 3. To verify that our findings are not an artifact of the particular flow model we use, we investigated a different model. In this modified model, there are two stationary vortices 7

19 FIG. 4: Visualization of the heteroclinic tangle for w = 2. Unstable manifolds are visualized by injecting particles close to the separation points and plotting their subsequent positions stroboscopically for 18 periods. The stable manifolds are visualized by injecting particles uniformly in the region 0.9 < x < 1.6, 0.5 < y < 0.5 and checking if they leave downstream (reach x = +10) or pass through the region 1.25 < x < 1.35, 0.15 < y < 0.1; the boundary between these two outcomes marks the stable manifold of the saddle point. present in the time independent case, as expected in real flows, in contrast to the previous model. This is achieved by replacing (4) with: g(x, y, t) = β[ wh 1 (t)g 1 (x, y, t) + wh 2 (t)g 2 (x, y, t)] +(1 β)[ w s g 1s (x, y) + w s g 2s (x, y)] +u 0 ys(x, y), (5) where the two new functions are g 1s (x, y) = exp{ R 0 [(x x s ) 2 + αs 2 (y y s) 2 ]}, (6) g 2s (x, y) = exp{ R 0 [(x x s ) 2 + αs 2 (y + y s) 2 ]}, (7) describing the contribution to the streamfunction of two stationary vortices. These vortices are positioned at x s = 1.0 and y = ±y s = ±0.3. The parameters are chosen to be α s = 2.0, w s = 6.0, such that the stationary vortices behave qualitatively as expected in real flows. The value w = 24 is used in accordance with [2]. The parameter β is used in this model as the bifurcation parameter. For β = 0, the flow is stationary, whereas for β = 1 the model is the same as the one used in [2]. The streamlines of this flow for β = 0 are shown in Fig. 5a. 8

20 y y (a) x (b) x FIG. 5: Results for modified model. (a) Streamlines of the modified model with β = 0, representing a time-independent flow with two stationary vortices. (b) Heteroclinic tangle for β = The stable manifolds of the saddle point and the unstable manifolds of the separation points on the surface of the cylinder were computed in the same way as in Fig. 4. For all values of β, we find again a heteroclinic tangle, as depicted in Fig. 5b for β = This region is accessible for particles injected upstream from the obstacle. By injecting many particles into the flow upstream, the number Nv of particles entering the chaotic region is shown in Fig. 2b for different values of β. We see that the ratio of chaotic to all particle trajectories, again, follow Nv /N exp(kβ η ). In conclusion, we established that fluid flows with stagnant points or regions contain separatrices that are barriers to transport in the stationary case. These separatrices break up as soon as the flow becomes time dependent, resulting in a chaotic sea that can trap for a transient time the particles coming from upstream. Hence there is an immediate transition to chaotic advection in these systems. This behavior is to be contrasted with systems without such stagnant regions: there the lack of separatrices imply that a small time dependent velocity component is not enough to compensate for the background velocity that washes out the particles, and hence time periodicity does not necessarily imply chaos. An example of a boundary-less flow is a flow consisting of a constant velocity field, on which a timedependent component is superimposed. If the time-dependent part has small amplitude, a particle will simply be washed away by the constant background flow, and there will be no chaotic advection. Transition to chaos only happens for sufficiently high amplitudes, in contrast to the case with boundaries. Due to the presence of the surface and the KAM islands the dynamics around an obstacle is non-hyperbolic, which yields a non-trivial scaling near the bifurcation. As a final remark, we note that, although the stagnant points with zero velocity can only be distinguished for an obstacle with fixed position, a system with a 9

21 moving obstacle will experience the same dynamics in a co-moving coordinate system. [1] M. V. Dyke, An Album of Fluid Motion (Parabolic Press, Stanford, 1982). [2] C. Jung, T. Tél, and E. Ziemniak, Chaos 3, 555 (1993). [3] A. Péntek, Z. Toroczkai, T. Tél, C. Grebogi, and J. A. Yorke, Phys. Rev. E 51, 4076 (1995). [4] T. Tél, A. de Moura, C. Grebogi, and G. Károlyi, Phys. Rep. 413, 91 (2005). [5] A. Wonhas and J.C. Vassilicos, Phys. Rev. E 65, (2002). [6] E. Hernández-García and C. López, Ecological Complexity 1, 253 (2004). [7] G. Károlyi, A. Péntek, I. Scheuring, T. Tél, and Z. Toroczkai, Proc. Nat. Academy of Sci. 97, (2000). [8] C. López, Z. Neufeld, E. Hernández-García, and P. Haynes, Phys. Chem. Earth B 26, 313 (2001). [9] M. Cencini, G. Lacorata, A. Vulpiani, and E. Zambianchi, Journal of Physical Oceanography 29, 2578 (1999). [10] C. Grebogi, E. Ott, and J. A. Yorke, Phys. Rev. Lett. 57, 1284 (1986). [11] Y.-C. Lai, C. Grebogi, R. Blümel, and M. Ding, Phys. Rev. A 45, 8284 (1992). [12] C. Grebogi, E. Ott, F. Romeiras, and J. A. Yorke, Phys. Rev. A 36, 5365 (1987). 10

22 16 CHAPTER 2. PAPER PHYSICAL REVIEW LETTERS

23 Chapter 3 Residence time An alternative approach to analyse the dynamics of an open flow is the study of residence times of particles in the chaotic region. Results of this approach are presented here for the original model, introduced in Chapter 1. However, approaching the bifurcation parameter is becoming more difficult, due to computational limitations. Therefore, the adapted model is not studied in this manner. 3.1 Fractals in residence time For the flow described in the article, the residence time is studied for different initial positions. Particles are injected in the flow at the line x = 3 at different y-positions and their trajectories are computed by numerical integration. The residence time of this initial position is defined as the time it takes for particles to cross the line x = 10. Note, that outside the interval 3 < x < 10, the flow is approximately uniform. By depicting the residence time as function of the y-position, clearly a fractal structure is found for theoriginal parametersoftheflow,i.e. w = 24,accordingto[6]. ThisisshowninFigure3.1a. Ineach figure, we computed the residence times for uniformly distributed y-positions. The residence time shows a fractal structure. According to[4, 7], this implies that some particles in the flow experience chaotic mixing. A fractal structure of the residence times indicates, that particles with initial conditions close to each other show a structurally different trajectory, which causes the jumps in the residence times as shown in Figure 3.1. This indicates a strong dependence of initial conditions, which means the dynamics is chaotic. Since the resolution of the y-grid is finite, no infinite residence times are detected, as could be expected when a fractal structure existed. However, comparison of Figures 3.1a with a zoom-in of this figure, shown in Figure 3.1b, gives an indication that these spikes in the residence times can be expected when reducing the resolution of the y-grid. In Figure 3.1b., more and higher spikes can be found as in Figure 3.1a. Byreducingtheparameter w,thefractalstructureremainsvisibleforlargervaluesof w,i.e. w 2, see Figure 3.2. This is in correspondence with limitations of the method used. Showing the fractal structure of the residence times is getting more difficult for smaller parameters w. For smaller parameters w,thechaoticmixing, ifavailable,hastotakeplaceinthewakeofthecylinder, whichisonly accessible for particles that pass the cylinder close to the wall. Particles not getting close to the cylinder will be flushed away by the background flow in a regular manner. Since particle velocity close to the cylinder wall is very low, residence times of these particles become larger. However, the computation time is related, among others, to the residence time. Therefore, with decreasing w, it becomes harder to prove the structure of the residence times is fractal. Since the region of y-positions showing fractal residence times may become very narrow for small values of w and the y-resolution of the figures is finite, withthis approachit is notpossibleto provethedynamics is regular. 17

24 18 CHAPTER 3. RESIDENCE TIME residence time [periods] residence time [periods] y-position y-position (a) (b) Figure 3.1: Residence time for the flow with parameter W = 24. Particles have been injected at x = 3 and the the time it takes to cross the line x = 10 is recorded. Figure a displays a large interval for the y-position, Figure b shows a zoom-in at some of the spikes. 3.2 Box-counting dimension By depicting the residence times as a function of the initial position, as used before, only a qualitative statementcanbegivenaboutthemixingpropertiesofaflow. Theresidencetimeplotsarefractal,orno fractal structure is distinguishable, where the fractal case implies chaotic mixing. To give a quantitative statement for the mixing properties, the box-counting dimension D 0 is calculated as described in [9]. Thebox-counting dimension is an upperbound ofthehausdorff dimension D H, i.e. D 0 D H, (3.1) which can be calculated analytically for some fractals. To calculate the box-counting dimension, wetake a large number of random initial y-positions y 0 in theinterval < y 0 < Foreach value y 0, wecalculatethe residence time of theparticle injected in the flow at the position (x, y) = ( 3, y 0 ) on t = 0. Furthermore, the residence times of particles at slightly different positions, i.e. (x, y) = ( 3, y 0 ± ǫ) are calculated, where ǫ is a small number. If the differences in residence times of these three points remain below a certain threshold, the point is considered ǫ-certain, otherwise, the point is considered ǫ-uncertain. In this study, we used a threshold of 1 period. By determining for a large number of particles whether they are ǫ-uncertain or ǫ-certain and countingthenumberofuncertainpoints,wecalculateafraction f(ǫ). Thisfractioniscomputedforvarying values of ǫ as shown in Figure 3.3 for different parameters w. For each different parameter, a fit is made according to the relation f(ǫ) ǫ α, and the constant α is derived by plotting the f(ǫ) versus ǫ on log-log scale. With these value α wecomputethebox-counting dimension D 0 according to: D 0 = N α, (3.2) where N = 2 is the dimension of the phase space. The resulting box-counting dimension is given in Table 3.1. In this table, the box-counting dimension appears to be increasing towards D 0 = 2 when the time-dependent velocity component of the flow is reduced. This corresponds with existing theory, [9], predicting that for 2-dimensional autonomous systems, i.e. w = 0, the dynamics is regular, such that the box-counting dimension under consideration equals the dimensionality of the phase space. Since the study of residence times can not be used to analyse the dynamical behaviour of flows with small time-dependent contributions to the velocity field, and it is therefore not well suited to analyse the transition to chaos for open flows, this study is not performed for the alternative model.

25 3.2. BOX-COUNTING DIMENSION residence time [periods] residence time [periods] y-position y-position (a) w=6 (b) w=6,zoom residence time [periods] residence time [periods] y-position (c) w= y-position (d) w=3,zoom x 10 3 residence time [periods] y-position (e) w=2 residence time [periods] y-position (f) w=2,zoom x 10 4 Figure 3.2: Residence time for the flow with different parameters w. Notice, that different axis scales have been used.

26 20 CHAPTER 3. RESIDENCE TIME f(ǫ) f(ǫ) ǫ ǫ (a) w=24 (b) w= f(ǫ) 10 1 f(ǫ) ǫ ǫ (c) w=3 (d) w=2 Figure 3.3: Fraction of ǫ-uncertain points f(ǫ) for different values of the flow parameter w A fit is shown according to f(ǫ) ǫ α. The gradient of this fit results in the box-counting dimensions given in Table 3.1.

27 3.2. BOX-COUNTING DIMENSION 21 w D Table 3.1: Box-counting dimension D 0 for different flow parameters w.

28 22 CHAPTER 3. RESIDENCE TIME

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