TRANSITION TO CHAOS OF RAYLEIGH-BÉNARD CELLS IN A CONFINED RECTANGULAR CONTAINER HEATED LOCALLY FROM BELOW

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1 TRANSITION TO CAOS OF RAYEIG-BÉNAR CES IN A CONFINE RECTANGUAR CONTAINER EATE OCAY FROM BEO iroyoshi Koizumi epartment of Mechanical Engineering & Intelligent Systems, The University of Electro-Communications, Chofu, Tokyo , Japan koizumi@mce.uec.ac.jp (phone & fax, direct) Keywords: Rayleigh-Bénard Cell, Transition, Chaos, Rectangular Container, eated ocally Bottom all Abstract Experimental and numerical studies concerning the effects of the Grashof number Gr, Prandtl number Pr, and the container height on the flow pattern and the bifurcation process to chaos of Rayleigh-Bénard cells in a confined rectangular container heated locally from below are presented. The dimensionless container size : : is 7:3:arbitrary height, where is the height of the container and is the width of the heated bottom wall. The flow pattern map for air flow (Pr=0.71) which depends on Gr and / was obtained mainly by experimentation. Two-dimensional (2-) rolls with their axes parallel to the long sidewalls are produced above both adiabatic bottom walls next to the heated bottom wall. In addition to these two rolls, three types of convective flow patterns above the heated bottom wall are observed: one pair of steady 2- rolls with their axes parallel to the long sidewalls or one pair of steady 3- roll-cells in which cell structures exist within the rolls, unsteady 3- cells, and 3- oscillatory flow. For Gr= and /=0.20, the 3- roll-cell pattern changes directly from time-dependent sinuous oscillatory motion (limit cycle) to chaotic flow. This non-cascade-type transition is due to the sudden flow pattern change from the 3- roll-cell to the chaotic cell structure. For water flow (Pr=7.1), one pair of steady 2- rolls and unsteady but almost 2- rolls with their axes parallel to the long sidewalls, which ascend from the center of the heated bottom wall and descend along both long sidewalls, always appear for Grashof numbers ranging from to and />0.1 by numerical simulation. These one pair of rolls also exhibit the non-cascade-type transition to chaos at Gr= and /=0.60, but their basic flow patterns do not change during the transition. 1 Introduction Recently, there is an idea to submerge a large amount of CO 2, which is a major cause of the global warming, into the deep sea [1]. This idea is that clustered CO 2 like sherbet is transported into the bottom of the deep sea by a pipe line and it is pooled in the bottom of a ravine. It is necessary to keep the clustered CO 2 for a long period in the deep sea, but there is an anxiety that unsteady strong vortices around a transport pipe enhance convective diffusion of clustered CO 2 in the region near the pipe outlet. The flow and heat transfer performance

2 around a cold horizontal cylinder placed near the bottom of the sea can be understood by the natural convective flow around a hot horizontal cylinder (outer diameter ) placed near a flat ceiling. The former experimental research for air flow [2] reveals that three types of flow patterns are observed depending on Grashof number Gr, and the dimensionless distance / between the ceiling and the cylinder: low dimensional chaotic flow in which the flow along the cylinder separates on its upper surface, 2- steady flow, and 3- oscillatory flow. owever, the effect of the Prandtl number Pr on the flow pattern and the bifurcation process to chaos could not be revealed. The aim of these experimental and numerical studies is to investigate the bifurcation process to chaos of Rayleigh-Bénard cells in a confined rectangular container heated locally from bottom wall, which roughly simulates the hot horizontal cylinder placed near a flat ceiling for air and water flows as shown in Fig. 1. Nomenclature width of the heated bottom wall in the short sidewall direction (Fig. 2), 76 mm yapunov dimension d m embedding dimension 2 Gr Grashof number, gβ ( Tbh Ttc ) 3 / ν height of the container which is able to set an arbitrary one length of the container, 7= 532 mm (length of the long sidewall) : : dimensionless container size ( 7:3: arbitrary height ) P pressure Pr Prandtl number, ν / α t time T temperature u, v, w x, y, z components of the velocity width of the container, 3= 228 mm (length of the short sidewall) x, y, z coordinates (Fig. 2), x=y=z=0; bottom left-side corner of the container Greek symbols α β thermal diffusivity coefficient of thermal expansion λ i yapunov exponent (i=1,, d m ) ν kinematic viscosity Θ nondimensional temperature, ( T Tm ) /( Tbh Ttc), where T m = ( Tbh + Ttc)/ 2 Physical properties are estimated at the mean temperature T. m Subscripts bh bottom, heated wall ba bottom, adiabatic wall f fluid lw long sidewall m mean temperature between the top cooled and bottom heated walls sw short sidewall tc top, cooled wall Former flow system [2] Rough simulation of the former flow system Present flow system eated cylinder suspended in an infinite flow medium near a flat ceiling Ceiling iameter of the heated cylinder Confined rectangular container heated locally from below Cold top wall idth of the heated bottom wall Fig. 1. Schematic drawing of the present flow system which simulates roughly the former flow system of a hot horizontal cylinder placed near a flat ceiling. g 2

3 2 Experimental method and numerical analysis 2.1 escription of experiment Experimental apparatus: Figure 2 shows a schematic drawing of the experimental apparatus and the coordinate system. The dimensions of the rectangular container are =532 mm (long sidewall), =228 mm (short sidewall), and mm (height can be set as an arbitrary height). The center part of one-third of the bottom wall in the short sidewall direction, of which width is 76 mm and length is 532 mm, was heated at constant temperature T bh. Each one-third of the bottom wall next to the heated bottom wall was adiabatic, which was made of foam glass. The top wall and the four sidewalls were cooled at constant temperature T tc. A dimensionless temperature is defined as Θ = ( T Tm ) /( Tbh Tct ), where T m = ( Tbh + Ttc ) / 2. imensionless container size /:/:/ is 7:3:arbitrary height, and the height / is able to be set below 0.4. Experiments were carried out with air (Pr=0.71) as the working medium at Grashof numbers ranging from to The Grashof number was set by changing the temperature difference between the cooled top wall T tc and the heated bottom wall T bh. z Cold top wall Θ tc = Adiabatic bottom wall ( Θ/ z' ) = 0 /:/:/= 7:3:(arbitrary height) ba y x idth of the heated bottom wall =76 mm, Θ bh =0.5 Fig. 2. Schematic drawing of the experimental apparatus and coordinate system. g Sensor and signal processing: The sensor used to measure the air temperature is a Cu-Co thermocouple, whose response time is approximately 0.2 s. Phase trajectory and yapunov exponents were obtained using the time series of a temperature at the center part of the container by using a microcomputer. The sampling time τ was set to s by using an analog to digital converter with 16 bit resolution. The power spectrum was obtained by using an FFT analyzer. Incense smoke was used as a tracer to enable instantaneous flow visualization. yapunov exponents: Using 131,072 data points from a time-series of a thermocouple at the center part of the container, an attractor was reconstructed in a d m -dimensional phase space by using a microcomputer. Then λ i was obtained from the orbits of points evolving in the time interval of length τ dev. The yapunov exponents λ i represent the time-development of the displacement vector between two very adjacent points in the phase space. Also the yapunov dimension, which shows the complexity of unsteady flow, was obtained from λ i. The exponents can be calculated by the method proposed by Sano and Sawada [3]. A detailed description of the analyzing method is given in a previous published paper [2]. 2.2 Numerical analysis Basic equations: The nondimensionalized governing equations for unsteady 3- natural convection flow can be written in Cartesian coordinates using the Boussinesq approximation as follows: u' v' w' + + = 0 x' y' z' (1) u' P' = + Pr 2 u' t' x (2) v' P' = + Pr 2 v' t' y' (3) w' P' 2 = + Pr w' + Gr Pr t' z' Θ 2 = Θ t' 2 Θ (4) (5) 3

4 with substantial derivative t' = + u' + v' + w' t' t' x' y' z' The x, y, and z coordinates were nondimensionalized by, the width of the heated bottom wall in the short sidewall direction, the velocities were scaled byα/, the time by 2 /α, and the pressure byρα 2 / 2, respectively, where α is the thermal diffusivity, and ρ is the density of fluid. The dimensionless temperature was defined by Θ = ( T Tm ) /( Tbh Tct ), where T m = ( Tbh + Ttc ) / 2. In the above equations, Gr is the Grashof number, and Pr is the Prandtl number. Solution method: Approximate forms of the Boussinesq equations were obtained using a control-volume based on a finite difference procedure. The convective terms were discretized using a hybrid scheme (a first-order accurate upwind difference scheme). The SIMPE algorithm [4] was used to solve these equations. The entire enclosure was treated as the full computational domain. A grid system with uniform spacing, whose size was , was typically used. A finer mesh size of was employed for unsteady 3- cells in region III of Fig. 3 at Gr= and /= Almost the same results were obtained for these two mesh sizes. Calculations were carried out both for air (Pr=0.71) and water (Pr=7.1) flows. Boundary condition: The boundary P' conditions were u ' = v' = w' = P' = = 0 at all n walls, and the wall temperature satisfying Θ Θ bh = 0.5 at the heated bottom wall, ( ) ba = 0 z' at the insulated bottom wall, and Θtc = Θsw = Θlw = 0.5 at the top wall and sidewalls. Sidewall temperature condition: In order to clarify the effects of the sidewall temperature condition on the flow pattern, both uniform and adiabatic temperature conditions were calculated. Both calculated flow patterns were almost the same except at only near the short sidewalls. Therefore, a uniform sidewall temperature condition was used in this study to facilitate the comparison with experiments. Initial condition and conversion criterion: The initial conditions as those of a stagnant fluid, and the temperature were set to Θ f = To ensure convergence, the maximum norm of the residue of each discretized equation was required to be less than The dimensionless time step Δt was for Gr , and Δt was at Gr= for the unsteady calculations. The time series of the temperature at the center grid point of the container were recorded and analyzed to obtain the phase trajectory and the power spectrum. All calculations were carried out with double precision. 3 Results and discussion 3.1 Flow pattern map for air flow Figure 3 shows the categorization of flow patterns for air flow obtained mainly by experimentation, and their dependence on the Grashof number Gr and the dimensionless container height /. Air flow Gr , Pr=0.71 Ⅱ(roll) Ⅰ Ⅳ Ⅱ(roll-cell) Ⅰ: Conduction Ⅱ: (roll-cell) Steady 4 rolls Ⅱ(roll): Steady 6 rolls { Steady 10 rolls Ⅲ: Unsteady 4 rolls-cells Ⅲ Enlargement Ⅱ(roll-cell) Ⅳ: Oscillatory flow Fig. 3. Flow pattern map for air flow. (Simulation) 4

5 2- rolls with their axes parallel to the long sidewalls are produced above both adiabatic bottom walls next to the heated bottom wall. In addition to these two rolls, three types of convective flow patterns above the heated bottom wall are observed: steady 2- rolls with their axes parallel to the long sidewalls at lower Gr flows or 3- roll-cells (cell structures exist within two rolls) at higher Gr flows in region Ⅱ, unsteady 3- cells in region Ⅲ, and 3- oscillatory flow in which an ascending flow begins to oscillate between top and bottom walls in region Ⅳ. Steady 2- rolls and 3- roll-cells in region II change their roll number from two to four above the heated bottom wall depending upon the /. These rolls have a wavelength close to about 2.016, which is equal to that of the unstable lowest mode in the critical flow with Rayleigh-Bénard convection in a layer of infinite horizontal extent with rigid-rigid boundaries [5]. 3.2 Transition to chaos of a steady 3- roll-cell pattern at Gr= for air flow One pair of steady 3- roll-cell pattern at /=0.200 in region II of Fig. 3 begins to exhibit time-dependent periodic flow (limit cycle) at about /=0.203 by numerical simulation. The roll-cell then begins to oscillate with a spectral peak frequency f 1 = z, and the resulting sinuous oscillation takes the form of a standing wave. Then, a successive bifurcation occurs from oscillatory motion directly into chaotic flow at /= That is, the roll-cell pattern does not show a cascade-type transition. Figure 4 shows chaotic flow characteristics of 3- cells obtained by experimentation at Gr= and /=0.20, which correspond to the cells just behind the transition to chaos at /=0.204 numerically as shown in the enlargement of the flow pattern map of Fig. 3. Figure 4(a) shows instantaneous flow visualization photos on the mid-plane of z/=0.10 above the heated bottom wall. One pair of 3- sinuous oscillatory roll-cells (cell structures exist within roll) with several necked parts which is shown by several arrows in Fig. 4(a-1), and it changes to just unsteady 3- cell structures as shown in Fig. 4(a-2) within a few seconds after being set from /=0.19 to Each cell size is almost the same as the width of the heated bottom wall and is aligned in a zigzag along the heated bottom wall. Gr= , Pr=0.71 and /=0.20 (a-1) (a-2) Flow pattern above the heated bottom wall changes from photo (a-1) to photo (a-2) within a few seconds after being set from /=0.19 to /= (a) Instantaneous flow visualization photos in the x-y plane at z/=0.10. (b) d m = 5 d m = 6 d m = 7 λ ± ± ±0.4 λ ± ± ±0.2 λ ± ± ±0.2 λ ± ± ±0.4 λ ± ± ±1.0 λ ± ±1.6 λ ± ± ± ±0.3 yapunov exponents and their dimension. Fig. 4. Chaotic flow characteristics of air flow which was obtained by experimentation for Gr= , Pr=0.71 and /=

6 Figure 4(b) is the yapunov exponents λi and yapunov dimension, when the embedding dimension takes from d m =5 to d m =7. The sign ± in the yapunov exponents λ i is calculated from several runs with different parameters τ dev, ε r, N within the following ranges. The developing time is 3 τ τ dev 6 τ for sampling time τ = s. hen the extent of the phase space is normalized as unity, the radius of a small ball is 0.03 ε r 0.06, and the number of the phase points included in it is 10 N 30. Two positive yapunov exponents appear and is about 3.7, therefore this flow may be strongly chaotic. This non-cascade-type transition is due to the sudden flow pattern change from one pair of 3- sinuous oscillatory roll-cells in which cell structures exist within two rolls shown by the flow visualization photo of Fig. 4(a-1) to unsteady 3- cells of Fig. 4(a-2). Figure 5 shows computed chaotic flow characteristics at /=0.204, which correspond to the characteristics just behind the transition to chaos at /=0.20 experimentally as shown in Fig. 4. Figure 5(a) shows instantaneous velocity vectors and isotherms at various cross-sections, but the container size ratio of length and breadth was not to scale. (a-3) Isotherms in the y-z cross-section at x/=1.5 (a-4) Isotherms on the mid-plane of z/=0.102 Gr= , Pr=0.71 and /=0.204 /=0.204 Isotherms /=0.20 [Fig. 4(a-2)] Flow visualization photo (a-1) Velocity vectors in the x-z cross-section at y/=3.5 (a-5) Comparison of 3- cell-pattern between calculation and flow visualization above the heated bottom wall. (a) Instantaneous velocity vectors and isotherms at various cross-sections. (a-2) Isotherms in the x-z cross-section at y/=3.5 Fig. 5. Chaotic flow characteristics of air flow which was obtained by numerical simulation for Gr= , Pr=0.71 and /=

7 Θ(t) t [z] (b-1) Time series and its power spectrum. Θ(t+m) Θ(t) (b-2) Phase trajectory (b) Flow characteristics. Fig. 5. continued. Figure 5(a-1) shows the velocity vectors, and Fig. 5(a-2) shows the isotherms in the x-z cross-section at y/=3.5, Fig. 5(a-3) is the isotherms in the y-z cross-section at x/=1.5, and Fig. 5(a-4) shows the isotherms on the mid-plane of z/=0.102, respectively. Figure 5(a-5) reveals the comparison of 3- cells between the calculated result in the region from 2.5<y/<5.5 to 1.0<x/<2.0 above the heated bottom wall and the corresponding flow visualization photo of Fig. 4(a-2). Figure 5(b) shows flow characteristics. Figure 5(b-1) is the time series of fluid temperature and its power spectrum, and Fig. 5(b-2) is the phase trajectory. From the velocity vectors of Fig. 5(a-1) and isotherms of Figs. 5(a-2), (a-3) and (a-4), several numbers of large 3- cells, which descend in the center and ascend around the outer part of each cell, are formed and these cells are aligned in a zigzag pattern with smaller cells between them along the center part of the long heated bottom wall. In addition to these cells, large 2- two rolls with their axes parallel to the long sidewalls, which differ in their circulation direction above both adiabatic bottom walls, are produced. hen the instantaneous flow visualization photo in the right-side of Fig. 5(a-5) is compared to the isotherms obtained numerically in the left-side of Fig. 5(a-5), almost the same cell-pattern was obtained, but the simulated cell size is smaller than that of the experimental observation. The difference between these two cell sizes is due to the lack of the spatial resolution, especially near the heated bottom wall in the simulation. These numerical results have qualitatively elucidated the nature of the bifurcation process to chaos observed in the experimentation as shown in Fig. 4(a). A considerable number of modes may be relevant in the transition due to the fact that the power spectrum is continuous and below about 1 z, and the trajectory in the phase space follows very complicated orbits as shown in Fig. 5(b). 3.3 Effect of the Prandtl number on the flow pattern and the transition process to chaos at Gr= The Prandtl number governs mainly characteristics of non-linearity in the flow system. Therefore, the onset of the time dependence and the transition to chaos are strongly dependent on Pr [6], [7]. The non-linear momentum advection terms of the equations (2)-(4) play a dominant role in low Pr instabilities of convection, while the stability of high Pr convection depends on the non-linear term in the energy equation (5). In order to reveal the effect of Pr on the flow pattern and the transition process to chaos in this flow system, only numerical simulations were carried out for moderate Prandtl number fluids such as water (Pr=7.1) at Gr= This Grashof number is the same as the air flow which was described in section 3.2 above. One pair of steady 2- rolls and unsteady but almost 2- rolls with their axes parallel to the long sidewalls, which ascend from the center of the heated bottom wall and descend along both 7

8 long sidewalls, are produced at />0.1 for water flow. This then clarifies that Pr is low, as can be observed in the former case of air, 2- rolls, 3- roll-cell and 3- cells above the heated bottom wall. On the other hand, one pair of steady 2- rolls and unsteady but almost 2- rolls with their axes parallel to the long sidewalls, of which rolls extended over both adiabatic and heated bottom walls, are only possible for water. Flow patterns are almost the same as shown in Fig. 6 at one order magnitude higher Grashof number of Gr= in section 3.4 which follows. One pair of steady 2- rolls begins to exhibit time-dependent periodic flow (limit cycle) at /=0.35 with a spectral peak frequency of f 1 =0.965 z. Then a successive bifurcation occurs from oscillatory motion into chaotic flow at /=0.80. ith a further increase of /, this unsteady roll pattern returns to limit cycle at / 1.1, and the same unsteady sinuous motions remained up to /=2. That is, it is found that one pair of roll-patterns show a chaotic flow within a very narrow range of /. 3.4 Transition process to chaos at a higher Grashof number of Gr= for water flow Next, calculations at a higher Grashof number of Gr= were carried out for water flow (Pr=7.1). One pair of steady 2- rolls, which is the same flow pattern for water flow at the lower Grashof number of Gr= in section 3.3 above, begin to exhibit time-dependent sinuous oscillatory motion (limit cycle) with a spectral peak frequency of f 1 =2.15 z at /=0.45. Then a successive bifurcation occurs from oscillatory motion into chaotic flow at /=0.60. That is, one pair of roll-patterns shows a non-cascade-type transition. Furthermore, the occurrence of a chaotic flow is within a very narrow range of /. hen / is increasing more, this chaotic roll-pattern returns to its limit cycle at / 1.1, and the same unsteady sinuous motions remained up to /=2. The more detailed retransition process from chaotic flow to regular motion (limit cycle) is now proceeding. Figure 6 shows flow characteristics for water flow obtained numerically at Gr= and /=0.60 when the flow fields become chaotic. Figure 6(a) shows the instantaneous vectors in the x-z cross-section at y/=3.5, and Fig. 6(b) shows the corresponding isotherms. Figure 6(c) shows the instantaneous isotherms on the mid-plane of z/=0.30. Figure 6(d) shows the time series at the center grid point of the container, and its power spectrum and the phase trajectory. Gr= , Pr=7.1 and /=0.60 (a) Instantaneous velocity vectors in the x-z cross-section at y/= y x (b) Instantaneous isotherms in the x-z cross-section at y/=3.5. y (c) Instantaneous isotherms on the mid-plane of z/=0.30. Fig. 6. Chaotic flow characteristics which was obtained by the numerical simulation at Gr= and /=0.60 for water flow. x 8

9 (d-1) Time series [ z] (d-2) Power spectrum (d-3) Phase trajectory (d) Flow characteristics. Fig. 6. -continued. Compared to the results of air flow patterns, water flow patterns become simple due to the large viscosity and small thermal diffusivity of water. Furthermore, higher temperature regions are limited only near the heated bottom wall and the center part of the cold top wall. From Figs. 6(a) and (b), the maximum velocity point moves slightly in the rightward part of the center line of the heated bottom wall, and the velocity of the descending flow along the left sidewall is larger than that of the right sidewall at this instance. The corresponding temperature field becomes asymmetric, and the higher temperature contour elongates towards the rightward direction along the top cooled wall. That is, the center part of the ascending flow moves slightly left and right towards the short sidewall direction. From Fig. 6(c), these unsteady motions show a slight three dimensionality, because isolated slender higher temperature contour extends almost symmetrically along the center of the long bottom wall. A considerable number of modes may be relevant in the transition due to the fact that the power spectrum is continuous and below about 10 z as shown in Figs. 5(d-2). Furthermore, the flow becomes chaotic due to the fact that the trajectory in the phase space follows very complicated orbits as shown in Fig. 6(d-3). From the results of the numerical simulation for water flow at this higher Grashof number of Gr= , it can be said that chaotic vortices are produced within certain limited ranges of /. In this Grashof number flow, the outer diameter of a transport pipe of clustered CO 2 corresponds to about 0.20 m if the temperature difference between the cooled pipe and the surrounding seawater are 5 K. Therefore, there is concern that such kind of chaotic vortices around a real transport pipe enhance convective diffusion of clustered CO 2 in the region near the pipe outlet in the deep sea [1]. 4 Conclusions I have reported the results of experimental and numerical studies concerning the transition to chaos of Rayleigh-Bénard cells in a confined rectangular container heated locally from below for both air (Pr=0.71) and water (Pr=7.1) flows. Experiments were performed with air at Grashof numbers ranging from to The conclusions obtained are as follows: (1) The transition process to unsteady flow of certain buoyant container-flow problems for simple geometries is extremely influenced by the thermal boundary conditions of a bottom wall and the Prandtl number. (2) Three types of flow patterns above the 9

10 locally heated bottom wall are observed for air flow depending on Grashof number Gr and dimensionless container height / between the cooled top wall and the locally heated bottom wall: steady 2- rolls with their axes parallel to the long sidewalls or 3- roll-cells (cell structures exist within two rolls), unsteady 3- cells, and 3- oscillatory flow. (3) For air flow, the 3- roll-cell pattern changes directly from time-dependent sinuous oscillatory motion (limit cycle) to chaotic flow at Gr= and /=0.20 experimentally. This non-cascade-type transition is due to the sudden change from sinuous oscillatory 3- roll-cells to chaotic 3- cells. Two positive yapunov exponents appear and the yapunov dimension is about 3.7. These transitional characteristics were qualitatively obtained by the numerical simulation at /= (4) For water flow, one pair of 2- rolls with their axes parallel to the long sidewalls, which ascend from the heated bottom wall and descend along both long sidewalls, are produced for Grashof numbers ranging from Gr= to Gr= and />0.1. These rolls also exhibit a non-cascade-type transition to chaos at Gr= and /=0.80. hen / is increased more, this chaotic roll-pattern returns to its limit cycle at / 1.1, and the same unsteady sinuous motions remained up to /=2. [4] Patankar S V. Numerical eat and Fluid Flow. emisphere, ashington C, [5] Chandrasekhar S. ydrodynamic and ydromagnetic Stability, pp. 9-75, Clarendon Press, Oxford, [6] Busse F. Non-linear properties of thermal convection. Reports on Progress in Physics, Vol. 41, pp , [7] Gollub J P and Benson S V. Many routes to turbulent convection. J. Fluid Mech., Vol. 100, Part 3, pp , References [1] Steinberg M, et al. A system study for the removal, recovery and disposal of carbon dioxide from fossil fuel power plants in the U.S. Report of Brookhaven National aboratory, RN-35666, [2] Koizumi and osokawa I. Chaotic behavior and heat transfer performance of the natural convection around a hot horizontal cylinder affected by a flat ceiling. Int. J. eat Mass Transfer, Vol. 39, No. 5, pp , [3] Sano M and Sawada Y. Measurement of the yapunov spectrum from a chaotic time series. Phys. Rev. ett. 55, pp ,