CHAPTER COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES CHAPTER 5* COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES 5.1 Introduction The competition organized by double Hopf bifurcations between an axisymmetric mode (m1 = 0) and a non-axisymmetric mode (m2 ≠ 0) has been studied numerically by Marques et al. (2002) and Lopez and Marques (2003). In both the m1 ≠ m2 ≠ and m1 = 0, m2 ≠ the double Hopf bifurcations, the competition has been between modes that reside in orthogonal subspaces and the quasi-periodic mixed mode (which may be stable or unstable) evolved in the product space. In this chapter, the mode competition between two axisymmetric limit cycles in the neighborhood of a double Hopf point was investigated experimentally and numerically, so that the mode competition takes place wholly in the axisymmetric subspace. This work is motivated partly by the linear stability analysis of Gelgat et al. (1996, 2001),which showed the existence of an axisymmetric double Hopf bifurcation, and the purpose of this experiment is to see if the dynamics associated with this double Hopf bifurcation can be captured in laboratory conditions. A close examination of the linear stability results of Gelfaget et al (2001) shows that in the range of the aspect ratio between about 1.6 to 2.8, where the steady basic state loses stability to axisymmetric time-periodic flow, there are two distinct Hopf bifurcations leading to axisymmetric states with different frequencies. The crossover point between these two Hopf bifurcations (the double Hopf bifurcation point) is at Λ ≈ 1.72 and Re ≈ 2665. * Part of this work has also appeared in Phys. Fluids. 18, 2006. 68 CHAPTER COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES However, the linear stability analysis itself says nothing about the nonlinear behavior of the flow following the bifurcation. In particular, it says nothing about the competition between the two limit cycles. In order to say anything in this regard, one must some nonlinear analysis, as done by Guckenheimer and Holmes (1983), Marsden and McCracken (1976) to perform a center manifold reduction and a normal form analysis in the neighborhood of the double Hopf bifurcation. Doing such an analysis gives a multitude of possible nonlinear scenarios describing all possible competition dynamics. In order to pin-point which scenario corresponds to a particular flow problem requires detailed spatio-temporal information about the bifurcating limit cycles, which may be obtained from quantitative experimental measurements or fully nonlinear computations. Thus, in this study, a combined experimental and numerical study was performed. The experimental measurements provide, for the first time, the laboratory evidence of the existence of an axisymmetric double Hopf bifurcation, involving the competition between two stable coexisting axisymmetric limit cycles with periods (non-dimensionalized by the rotation rate of the endwall) of approximately 31 and 22. The dynamics is also captured in our nonlinear computations, which clearly identify the double Hopf bifurcation as “type I simple” with the characteristic signatures that the two Hopf bifurcations are supercritical and that there is a wedge-shaped region in [Λ, Re] parameter space where both limit cycles are stable, delimited by Neimark-Sacker bifurcation curves. 5.2 Experimental Method The experiments presented here were carried out in the test rig described in Chapter 2, and only the essential features are presented here. The working fluid was a 69 CHAPTER COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES mixture of glycerin and water (roughly 76% glycerin by weight) with kinematic viscosity ν = 0.404 ± 0.002 cm2/s at a room temperature of 23.0°C. The aspect ratio Λ tested was between 1.67 and 1.81, and Reynolds number between 2600 and 2800. To capture the oscillatory behavior of the flow, flow visualization and hot-film measurements were used. Note that all flow visualization photos were inverted for ease of comparison with numerical results. 5.3 Numerical Method The numerical simulation results presented here were provided by Lopez J.M. using the spectral code described in Lopez et al. (2002) and previously used to explore the nonlinear dynamics of confined vortex breakdown flows (Marques and Lopez 2001, Marques et al. 2002, Lopez 2006). The solutions presented here have 48 Legendre modes in the radial and axial directions, and up to N = 16 (resolving up to azimuthal wavenumber m = 16; these were used to test the stability of the axisymmetric solutions to three-dimensional perturbations); the time-step used is t = × 10−3, which is much smaller than needed for stability of the code in the parameter regime investigated. 5.4 Results and Discussions 5.4.1 Basic state Before we present the results on mode competition between two axisymmetric limit cycles in the neighborhood of a double Hopf point, it is useful to briefly review some aspects of the issues regarding the basic steady state and its bifurcation to unsteady state. 70 CHAPTER COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES For the range of aspect ratio of interest (Λ from 1.64 to 1.80 and 2600 ≤ Re ≤ 2800), the basic steady state from which the limit cycles bifurcate does not have recirculation zones on the axis. There is however a very pronounced radial deflection of the flow from the axis which may still be considered a vortex breakdown region. Figure 5.1 shows contours of the streamfunction ψ and the three components of velocity for the basic state at Re = 2600, Λ = 1.75. For comparison, the contours of Re = 1850 are also attached. Fig. 5.1 Contours of ψ, u, v, and w for the axisymmetric steady-state solution at Λ = 1.75 and (a) Re = 1850, and (b) Re = 2600. There are 20 positive and 20 negative contours quadratically spaced, i.e. contour levels are [min|max] (i/20)2 with i = 1→ 20, and ψ∈[-0.0078, 0.000045], u∈[-0.16, 0.16], v∈[0, 1], and w∈[-0.16, 0.16]. The solid (broken) contours are positive (negative). The left boundary is the axis and the bottom is the rotating endwall. 71 CHAPTER COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES The Re = 1850 case has a very distinct recirculation zone (vortex breakdown bubble) on the axis near the top stationary endwall, together with the corresponding reversed axial flow (w component of velocity). The Re = 2600 case does not have such a recirculation zone, but it does have a comparable radial deflection of the flow in the same region. The Re = 1850, Λ = 1.75 case has a notorious history stemming from the experimental results of Spohn et al. (1998) in which they reported that the flow at this point in parameter space was not axisymmetric. Linear stability analysis of Gelfgat et al. (2001) however, showed theoretically that the steady axisymmetric state is stable to general three-dimensional unsteady perturbations, and numerous direct numerical simulations using the three-dimensional unsteady Navier-Stokes equations with small random perturbations in the initial conditions also show evolution to the steady axisymmetric basic state. Subsequently, Sotiropoulos et al. (2002) have conducted experiments at this point in parameter space and have shown that the degree to which the observed flow is three-dimensional can be reduced by reducing the level of imperfection in the apparatus. They obtained flow visualization results very similar to those Spohn et al. (1998) by tilting the stationary endwall by approximately 0.4º from horizontal (see their figure 3). Thompson and Hourigan (2003) also investigated the same flow numerically and were also able to produce streak-lines in very close agreement with the dye visualization results of Spohn et al. (1998) by numerically imposing a small misalignment of the rotating endwall (misalignment angle of about 0.1º). They also re-did the linear stability analysis of the basic state using a different numerical method to that of Gelfgat et al. (2001) and reached the same conclusion: the axisymmetric state is stable to all three-dimensional perturbation at Λ = 1.75 for Re < 2800 and unsteady axisymmetric flow appears at about Re = 2650. It would thus seem 72 CHAPTER COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES that the three-dimensional nature of the flow visualization at low Re = 1850 is not intrinsic to the flow but is due to extrinsic imperfections. That the flow visualizations at Re = 1850 are extremely sensitive to small imperfections is not surprising. The basic state has stagnation points on the axis; in the language of dynamical systems these hyperbolic fixed points are structurally unstable (Guckenheimer and Holmes 1983), and as pointed out by Holmes (1984) “…certain degenerate invariant manifolds of homoclinic or heteroclinic orbits connecting the stagnation points can be expected to break up under arbitrary small perturbations.” The question, of course, is whether these arbitrarily small imperfections have any effect on the dynamics of the flow (rather than on the kinematics, such as the dye steaks). (a) (b) Fig. 5.2 Flow visualization at Re = 1853, Λ = 1.75, using (a) fluorescent dye illuminated by a laser sheet and (b) food dye with ambient lighting. 73 CHAPTER COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES Great effort was made in the present experiments to reduce the level of imperfections in the apparatus, but of course, no apparatus is perfect. Figure 5.2 shows flow visualizations of the breakdown region at Λ = 1.75, Re = 1853, using both florescent dye illuminated by a laser sheet and food dye with ambient lighting. The level of imperfection is apparently lower than in the experiments of Spohn et al. (1998) and somewhere in between the “perfect” and the 0.4º tilted cases of Sotiropoulos et al. (2002), although the causes of the imperfections in the various apparatus are probably different. Note that the pictures in Fig. 5.2 have been reversed from the original blue dye with white background in order to improve their contrast. The dye sheet seen in the figure is steady, it does not precess, and there is a clear m = azimuthal wavenumber associated with it. That being said, it must be remembered that although the dye port is located at the center of the stationary endwall, there is no certainty that the dye will emerge axisymmetrically about the axis r = 0, and so even in a perfectly axisymmetric flow, a dye sheet that is released an arbitrarily small distance off-axis will have a non-axisymmetric appearance with a predominately m = azimuthal wavenumber (see Lopez and Perry1992). As Re is increased from 1850, the level of non-axisymmetry observed in the dye visualizations is gradually reduced. This is consistent with the fact that the basic state does not have stagnation points on the axis at higher Re (see Fig. 5.1 for Re = 2600), and hence the flow does not have the structural instability associated with the stagnation points commented on by Holmes (1984). Figure 5.3 shows dye visualizations of steady states for Λ = 1.75 with Re varying from Re = 2020 to Re = 2605, showing how the recirculation bubble disappears with increasing Re, and how dye released very close to the axis develops into a spiral kink in the vortex breakdown 74 CHAPTER COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES region when the recirculation bubble no longer exists (see the discussion in Hourigan et al. (1995), regarding spiral dye streaks in axisymmetric vortex flows). Re = 2020 2212 2306 2423 2452 2508 2605 Fig. 5.3 Dye visualization of steady states at Λ = 1.75 and Re as indicated. On increasing Re to 2688, the flow becomes time-periodic and the spiral dye filament undergoes an oscillatory excursion in the axial direction. The basic steady state (SS) bifurcates to axisymmetric limit cycle solutions LC1 and LC2 with different frequencies, and details will be presented in the following sections. 5.4.2 Hopf bifurcations of the basic state Over the parameter range reported here (Λ∈[1.64, 1.80] and Re ≤ 2800), the numerical solutions show that the basic steady state (SS) bifurcates to axisymmetric limit cycle solutions LC1 and LC2 with different frequencies, and that these are nonlinearly stable to three-dimensional perturbations. For several cases throughout the parameter regime in question, the three-dimensional governing equations resolving up to the m = 16 azimuthal wave number have been solved, using the axisymmetric solution together with small random perturbations in all m ≠ modes as initial conditions. In all cases, the m ≠ components of the flow decay toward machine zero. 75 CHAPTER COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES At larger Reynolds number (Re > 3000), some m ≠ modes grow and nonaxisymmetric solutions become stable (Lopez 2006). To characterize the axisymmetric time-periodic states LC1 and LC2, we use the oscillations in their kinetic energy as a global measure of their amplitudes, denoted as ΔE0, and the period in E0(t), where E0 = z =Γ r =1 u rdrdz ∫z =0 ∫r =0 (5.1) u0 is the 0-th Fourier mode of the velocity field. Figure 5.4 shows computed timeseries of E0 for LC1 (solid curve) and LC2 (dashed curve), both at Re = 2750 and Λ = 1.72. Fig. 5.4 Computed time-series of E0 for LC1 (solid curve) and LC2 (dashed curve), both at Re = 2750 and Λ = 1.72. Note that near the onset, the spatial characteristics of LC1 and LC2 are very similar to those of the steady state, SS, from which they bifurcate (see Fig. 5.1 for SS at Re = 2600, Λ = 1.75). For both limit cycles, the oscillations consist of pulsations in the vortex breakdown region. Figures 5.5 and 5.6 show contours of the axial component of velocity w for LC1 and LC2, respectively, both at Re = 2700 and Λ = 1.72. A 76 CHAPTER COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES significant feature in these sequences is that for a short time interval during the period of each limit cycle, there exists a small region of reversed flow on the axis precisely in the vortex breakdown region. Fluid particles originating near the center of the top stationary disk (where dye is released in the experiment) form a kinked spiral in the steady flow and in these time-periodic flows the kink oscillates up and down. Note that if the Hopf bifurcation were to break axisymmetry, then the kink would process in the azimuthal direction without change of form. Fig. 5.5 Contours of w for the axisymmetric time-periodic state LC1 at Re = 2700, Λ = 1.72 at six phases over one oscillation period (T ≈ 31.89); there are 20 positive and 20 negative contours quadratically spaced, i.e. contour levels are ± 0.15(i/20)2 with i = 1→20. The solid (broken) contours are positive (negative). The left boundary is the axis and the bottom is the rotating endwall. Fig. 5.6 Contours of w for the axisymmetric time-periodic state LC2 at Re = 2700, Λ = 1.72 at six phases over one oscillation period (T ≈ 22.01); there are 20 positive and 20 negative contours quadratically spaced, i.e. contour levels are ± 0.15(i/20)2 with i = 1→20. The solid (broken) contours are positive (negative). The left boundary is the axis and the bottom is the rotating endwall. 77 CHAPTER COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES It can be seen from the figure that at early stages of the flow development (4 minutes ≈ 0.86 viscous time units), the hot-film signal is quasi-periodic with two dominant frequencies 0.31 Hz and 0.46 Hz, corresponding to the non-dimensional periods T1 of ≈ 31.9 (LC1) and T2 of ≈ 21.5 (LC2), respectively (time has been scaled by 1/Ω). With increasing time, the LC1 mode becomes progressively more dominant, until about 16 minutes after start-up (3.44 viscous time units) when LC1 prevailed. At another nearby point in parameter space, Λ = 1.76 and Re = 2806 (see Fig. 5.18), the initial evolution again has a quasi-periodic character typical of the competition between the two limit cycles, but in this case LC2 eventually wins. 0.4 0.08 Amplitude Hot-film output 0.2 0.0 -0.2 -0.4 20 40 0.00 60 (a) 0.04 0.29 0.42 t (s) f 0.4 0.08 Amplitude Hot-film output 0.2 0.0 -0.2 -0.4 20 40 0.00 60 (b) 0.04 0.29 0.42 t (s) f 0.08 0.4 Amplitude Hot-film output 0.2 0.0 -0.2 -0.4 20 40 t (s) 60 (c) 0.04 0.00 0.42 f Fig. 5.18 Hot-film data (time series over minute and corresponding the amplitude of FFT results) taken (a) minutes, (b) 12 minutes, and (c) 18 minutes after start-up from rest with Re = 2806 and Λ = 1.76, showing evolution to an LC2 state. 89 CHAPTER 5.4.3.2 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES Coexistence of the two limit cycles LC1 and LC2 In this part of the investigation, Λ was kept constant and measurements were taken at different Re. The results obtained in this manner are represented by open and filled squares in Fig. 5.16. For Λ >1.739 it is found that LC2 can be obtained by impulsively starting the rotating disk from rest, and likewise for Λ < 1.728 LC1 can be obtained in the same manner. Depending on the initial starting condition, it is found that both LC1 and LC2 can be obtained at the same Λ and Re. Figures 5.19 and 5.20 show this very clearly for Λ = 1.733, where the first figure shows the evolution to LC2 following an impulsive start of the bottom rotating disk from rest to a predetermined speed (corresponding to Re = 2750), and the second figures shows how LC1 was generated, also from rest, but with Re increasing gradually to Re = 2750 at the rate ∂Re/∂t ≈ 50/s. This is just one of many experimental observations we have made of the coexistence of the two limit cycles LC1 and LC2 at the same point in parameter space (Λ, Re); which one dominates depends on the initial starting condition. The coexistence region is in between the two Neimark-Sacker curves NS1 and NS2 (see Fig. 5.16). 0.08 0.4 Amplitude Hot-film output 0.2 0.0 -0.2 -0.4 20 40 0.00 60 (a) 0.04 0.30 0.43 f t (s) 0.08 0.4 Amplitude Hot-film output 0.2 0.0 -0.2 -0.4 20 40 t (s) 60 (b) 0.04 0.00 0.30 0.43 f 90 CHAPTER COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES 0.08 0.4 Amplitude Hot-film output 0.2 0.0 -0.2 -0.4 20 40 0.00 60 (c) 0.04 0.43 f t (s) Fig. 5.19 Hot-film data (time series over minute and corresponding the amplitude of FFT results) showing the evolution to an LC2 state at Λ =1.733 taken (a) 17 minutes, (b) 23 minutes, and (c) 32 minutes after start-up impulsively from rest to Re = 2750. 0.04 0.4 0.0 -0.2 -0.4 (a) Amplitude Hot-film output 0.2 20 40 0.00 60 0.30 0.43 t (s) f 0.4 0.12 0.08 Amplitude Hot-film output 0.2 0.0 -0.2 -0.4 20 40 (b) 0.04 0.00 60 0.30 t (s) f 0.12 0.4 0.08 Amplitude Hot-film output 0.2 0.0 -0.2 -0.4 20 40 t (s) 60 (c) 0.04 0.00 0.30 f Fig. 5.20 Hot-film data (time series over minute and corresponding the amplitude of FFT results) showing the evolution to an LC1 state at Λ =1.733 taken (a) 10 minutes, (b) 20 minutes, and (c) 25 minutes after starting gradually from rest to Re = 2750 at a rate of ∂Re/∂t ≈ 50/s. 91 CHAPTER 5.4.3.3 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES Determination of critical Reynolds numbers for the Hopf bifurcations With Λ fixed, reducing Re toward the Hopf bifurcation point led to a reduction in the amplitude of the hot-film signal (or velocity variation) as the flow approached the steady state. This behavior can be clearly seen in Fig. 5.21 for an LC2 mode at Λ = 1.769. The critical Reynolds number is then determined by extrapolating to zero the peak to peak amplitude of the hot-film. Figure 5.22 shows such an extrapolation from the data in Fig. 5.21, giving an estimated Rec ≈ 2647 for LC2 at Λ = 1.769. Figures 5.23 and 5.24 present a similar results for LC1 with Λ = 1.728. Using this procedure, the critical Reynolds numbers for other aspect ratios were also obtained to produce the Hopf bifurcation curves H1 for the onset of LC1 and H2 for the onset of LC2; their loci in (Λ, Re) parameter space is presented in Fig. 5.25 as filled and hollow circles, respectively. The figure also shows the computed H1 and H2 curves. The experimental measurements of the critical Re for the Hopf bifurcations at the various Λ are within 2% of the computed values. 0.3 Re = 2804 Re = 2698 Hot-film output 0.2 Re = 2776 Re = 2660 Re = 2746 Re = 2650 Re = 2724 0.1 0.0 -0.1 -0.2 -0.3 10 t Fig. 5.21 Hot-film outputs over 10 seconds, taken once flow transients had died down, of the LC2 state at Λ = 1.769 for various Re as indicated. 92 CHAPTER COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES Peak-to-peak amplitude 0.4 0.3 0.2 0.1 2620 2660 2700 2740 2780 2820 Re Fig. 5.22 Variation with Re of the peak-to-peak amplitudes of the time series shown in Fig. 5.21. 0.4 Re = 2816 Re = 2710 Re = 2760 Re = 2680 Re = 2730 Re = 2670 Hot-film output 0.2 0.0 -0.2 -0.4 10 t Fig. 5.23 Hot-film outputs over 10 seconds, taken once flow transients had died down, of the LC1 state at Λ = 1.728 for various Re as indicated. 93 CHAPTER COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES Peak-to-peak amplitude 0.5 0.4 0.3 0.2 0.1 2640 2680 2720 2760 2800 Re Fig. 5.24 Variation with Re of the peak-to-peak amplitudes of the time series shown in Fig. 5.23. 2800 2760 NS2 NS1 Re 2720 2680 H1 2640 H2 2600 1.68 1.72 1.76 1.80 Λ Fig. 5.25 Bifurcation curves. The curves H1, H2, NS1, and NS2 are the numerically determined Hopf and Neimark-Sacker bifurcation curves. The filled and hollow circles are experimental estimates of the Hopf bifurcations H1 and H2, determined by fixing Λ, measuring the amplitude of the oscillation at various Res and extrapolating in Re to zero amplitude. The symbols + are experimentally observed LC1 states that evolved from an LC2 initial condition on crossing the Neimark-Sacker curve NS2 as Λ was quasi-statically reduced, and the symbols × are experimentally observed LC2 states that evolved from an LC1 initial condition on crossing the Neimark-Sacker curve NS1 as Λ was quasi-statically increased. 94 CHAPTER 5.4.3.4 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES Oscillation periods of LC1 and LC2 The primary characteristic distinguishing modes LC1 and LC2 is their period of oscillation. Figure 5.26 shows how the periods T (scaled by 1/Ω) of LC1 and LC2 vary with Re for various Λ. Although there is a noticeable scatter in the data due to the vertical scale of the figure, the percentage variation is small ([...]... Marques et al (2002) was adopted in this study, i.e., once a desired state was obtained at a particular point in (Λ, Re) parameter space, a quasi-static parameter sweep was carried out either by fixing Λ and varying Re or vice versa Fixing Λ and varying Re allow us to estimate the critical Re for the Hopf bifurcations at particular values of Λ, and by fixing Re and varying Λ, we estimate Neimark-Sacker... bifurcation curves Following this approach, the results obtained from the hot-film measurements are summarized in a state diagram Fig 5. 16 Although the figure may appear confusing, we find that this is still the best way of presenting the data (after trying several other methods) as it gives an overall picture of the states and their stability boundaries in the neigborhood of double-Hopf point as predicted... Within this small range of velocity variation, the temporal variation in the hot-film signal is approximately proportional to the temporal variation in velocity 85 CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES To experimentally establish the characteristics of the double Hopf bifurcation and the associated Neimark-Sacker bifurcations, the same strategy as in the numerical work of Marques... placed 180º apart on the stationary endplate Part (a) of the figure is the output for an LC1 state at Re = 2760 and Λ = 1.704, and part (b) is for an LC2 state at Re = 2 750 and Λ = 1.780 (each state is asymptotically stable, and reached from different conditions) Notice that in both cases, the hot-film outputs are synchronized (peaks match in time), providing further experimental evidence of the axisymmetric... H2, NS1, and NS2 are the numerically determined Hopf and Neimark-Sacker bifurcation curves The filled and hollow circles are experimental estimates of the Hopf bifurcations H1 and H2, determined by fixing Λ, measuring the amplitude of the oscillation at various Res and extrapolating in Re to zero amplitude The symbols + are experimentally observed LC1 states that evolved from an LC2 initial condition... are represented by open and filled squares in Fig 5. 16 For Λ >1.739 it is found that LC2 can be obtained by impulsively starting the rotating disk from rest, and likewise for Λ < 1.728 LC1 can be obtained in the same manner Depending on the initial starting condition, it is found that both LC1 and LC2 can be obtained at the same Λ and Re Figures 5. 19 and 5. 20 show this very clearly for Λ = 1.733, where... Fig 5. 7 Dye sequence of LC1 at Λ = 1. 75 and Re = 2688, at times as indicated in seconds (time for the first frame is arbitrarily set to zero) 78 CHAPTER 5 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES Fig 5. 8 (a) Time series of the cross-correlation coefficient Cr of dye sequences for steady state at Re = 23 95 (dash line) and Re = 2660 (dot-dash line), and for LC1 at Re = 2688 all at Λ = 1. 75 and. .. on crossing the Neimark-Sacker curve NS2 as Λ was quasi-statically reduced, and the symbols × are experimentally observed LC2 states that evolved from an LC1 initial condition on crossing the Neimark-Sacker curve NS1 as Λ was quasi-statically increased 94 CHAPTER 5 5.4.3.4 COMPETITION OF AXISYMMETRIC TIME-PERIODIC MODES Oscillation periods of LC1 and LC2 The primary characteristic distinguishing modes... computations We repeat this procedure at a lower Re of 2700, starting at Λ = 1.716 where the flow evolved to LC1 following an impulsive start from rest As Λ was varied in small discreet steps, LC1 was maintained up to 1. 757 , as shown in Fig 5. 29 However, when Λ was increased from 1. 757 to 1.78, LC1 lost stability and transited to LC2 after a long transient during which energy was transferred from oscillation... numerics as shown in Fig 5. 15 There are three distinct states that are observed: the steady axisymmetric basic state SS which is stable below both Hopf curves, H1 and H2; the two axisymmetric period states, LC1 (stable above Hopf curve H1 and to the left of Neimark-Sacker curve NS1) and LC2 (stable above Hopf curve H2 and to the right of Neimark-Sacker curve NS2), so that in between NS1 and NS2 both LC1 and . adopted in this study, i.e., once a desired state was obtained at a particular point in (, Re) parameter space, a quasi-static parameter sweep was carried out either by fixing  and varying Re. versa. Fixing  and varying Re allow us to estimate the critical Re for the Hopf bifurcations at particular values of , and by fixing Re and varying , we estimate Neimark-Sacker bifurcation. endplate. Part (a) of the figure is the output for an LC1 state at Re = 2760 and  = 1.704, and part (b) is for an LC2 state at Re = 2 750 and  = 1.780 (each state is asymptotically stable, and reached