CHAPTER CONCLUSIONS AND RECOMMENDATIONS CHAPTER CONCLUSIONS AND RECOMMENDATIONS 8.1 Conclusions Combined experimental and numerical investigations have been carried out to study the vortex breakdown structure and its stability in a confined cylindrical container driven by one rotating endplate. The experiments included flow visualization and hot-film measurements, and the numerical simulations included the solution by solving axisymmetric Navier-Stokes equations (by the author) or the three-dimensional Navier-Stokes equations (by Lopez). The main conclusions are drawn as follows: 1. The experiment with flow visualization confirms the existence of an S-shape vortex structure and a spiral-type vortex breakdown, not only for H/R = 4.0 as was first observed in the numerical studies of Serre and Bontoux (2002), but also for the aspect ratio as low as 3.65. The results further show that a bubble-type vortex breakdown in a low aspect ratio container is extremely robust and introducing flow asymmetry merely distorts the bubble geometry without it transforming into an S-shape vortex structure or a spiral-type vortex breakdown. 152 CHAPTER 2. CONCLUSIONS AND RECOMMENDATIONS For the study on mode competition between two axisymmetric limit cycles in the neighborhood of a double Hopf point (Λ ≈ 1.72 and Re ≈ 2665), our experiments provide, for the first time 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 (LC1) and 22 (LC2). The experiment clearly shows that the flow state can be either LC1 or LC2 even at the same aspect ratio and Reynolds number depending on the initial condition. The dynamics are also captured in the 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. Furthermore, the study also found that these dynamics are robust to the low-level imperfections and noise which are inherently present in physical laboratory experiments. 3. The study on vortex breakdown oscillations control through predetermined harmonic modulation was attempted for the first time. At supercritical conditions where the flow supported self-sustained oscillations in the absence of modulations, the experimental and numerical investigations revealed rich dynamics under modulation. The study shows that for very small modulation amplitudes, the resultant flow is quasi-periodic, possessing both the natural 153 CHAPTER CONCLUSIONS AND RECOMMENDATIONS frequency of the unforced bubble and the forcing (modulation) frequency. As the amplitude is increased to between 2% and 5% (depending irregularly on the forcing frequency), the resultant flow locks onto the forcing frequency and the natural frequency is completely suppressed. This is a common result in periodically forced flows (Chiffaudel and Fauve, 1987). Windows of limit cycles locked to half the forcing frequency were also found. Both the experiments and the numerical simulations indicate that all these flow phenomena remain axisymmetric, at least for Reynolds numbers less than about 3000. What is particularly interesting in this study is how the spatial nature of the forced limit cycle (locked to the forcing frequency) changes with the forcing frequency. For low forcing frequencies (less than about twice the natural frequency), the forced limit cycle consists of an enhanced vortex breakdown recirculation bubble on the axis oscillating with larger amplitude than in the unforced case, whereas for larger forcing frequencies, the locked limit cycle has a (nearly) stationary vortex breakdown bubble on the axis, and its oscillations are most pronounced near the cylinder sidewall. 4. For the mean rotation rates below the critical level for self-sustained oscillations, the experimental and numerical investigations have revealed three distinct regimes in the response to harmonic modulations, characterized by the modulation frequency. 154 CHAPTER CONCLUSIONS AND RECOMMENDATIONS For low modulation frequencies, we have a regime of quasistatic adjustment, where the swirling flow adjusts to the steady unmodulated solution at the instantaneous value of the rotation rate. In this regime, the boundary layers on the cylinder sidewall have sufficient time to fully develop during the long modulation period. At the other extreme, for high modulation frequencies, the sidewall layer does not have sufficient time to develop. As the rotating disk quickly accelerates and decelerates during the short modulation period, junction vortices form at the junction between the rotating disk and the stationary cylinder sidewall. As a junction vortex propagates up the sidewall it establishes the boundary layer. When the next junction vortex is generated, it is of opposite sense and the boundary layer development process is stopped and another layer of opposite signed vorticity is initiated. The result is a sequence of junction vortices of alternating sign propagating up the sidewall. Their short wavelength and high frequency tends to inhibit the natural (Hopf) instability of the steady axisymmetric basic state, accounting for the quenching of the oscillations. The third regime is characterized by modulation frequencies close to the Hopf frequencies of the basic state. By comparing the spatio-temporal structure of the sequence of junction vortices produced by the modulations in this range of frequencies with the vorticity eigenfunctions responsible for the self-sustained oscillations in the unmodulated problem, we have clearly identified the mechanism responsible for the large amplitude pulsations of the vortex 155 CHAPTER CONCLUSIONS AND RECOMMENDATIONS breakdown recirculations on the axis at mean rotation rates well below critical for the self-sustained vortex breakdown oscillations. An important consequence of this study is that to achieve a strong resonant effect, it is not sufficient to only consider the temporal characteristics of the flow state, but that the imposed forcing must also match the spatial characteristics. This may have wide-ranging implications for flow control issues in general. 5. The study on modulation shows that low-amplitude modulations can either enhance the oscillations of the vortex breakdown bubble (for low frequencies) or quench them (for high frequencies). Enhancing the oscillations can be beneficial in some applications where mixing is desired, such as micro-bioreactor, or swirl combustion chambers, while suppressing the oscillations can be useful in applications where unsteady vortex breakdown is prevalent, such as the tail buffeting problem. 156 CHAPTER CONCLUSIONS AND RECOMMENDATIONS 8.2 Recommendations A confined cylindrical container driven by one rotating endwall has been used extensively to study the phenomenon of vortex breakdown and its dynamic behaviour because this configuration forms a well posed problem controlled by only two parameters (the aspect ratio and Reynolds number), which is suitable for theoretical, numerical and experimental investigations. Numerous studies of this nature have been published in international journals and conferences. Though the present study has made some valuable contributions to our understanding of the flow physics in the confined cylindrical container with one rotating endwall, there are still some issues that need to be addressed and explored. The following are recommendation for future work: The flow transition in an enclosed cylinder driven by the constant rotation of an endwall is an interesting topic. In our study, the Reynolds number considered is usually below 3000, where the flow remains axisymmetric even where the flow is unsteady. As the Reynolds number is increased, a transition to aperiodic flow occurs, which is characterized by intermittent bursting dynamics. The numerical study by Lopez (2006) has revealed some interesting phenomena with various branches of modulated rotating waves associated with subsequent bifurcations from the rotating wave. It will be a challenging task to examine if the numerically observed phenomena can be captured in laboratories. 157 CHAPTER CONCLUSIONS AND RECOMMENDATIONS To explore the above-mentioned issue experimentally, the effect of noise becomes increasingly important. Since every experimental setup is not 100% perfect there will always be small unavoidable imperfections. This extraneous noise can drive the system from one stable state to another state. If the noise is sufficiently large, it is possible that the system dynamics can be altered completely, e.g. via resonances. Thus, the effects of noise level on the flow state will be an interesting research topic to pursue. For the flow under modulation, although our study has revealed some interesting dynamic behavior such as resonance, period doubling and locked state, it is limited to the axisymmetry mode near the first Hopf bifurcation. It will be interesting to explore the modulation in other regimes: such as the region near a double Hopf bifurcation (Λ = 1.72 where two axisymmetric modes coexist and compete, or Λ = 1.6 where one axisymmetric mode and mode coexist and compete), or the region far away from the first Hopf bifurcation to examine if the modulation (forcing) stabilize the flow or not. 158 PUBLICATIONS LIST OF RELATED PUBLICATIONS 1. T.T. Lim and Y.D. Cui, “On the generation of a spiral-type vortex breakdown in an enclosed cylindrical container”, Physics of Fluids, Vol. 17, No. 044105, April 2005. 2. J.M. Lopez, Y.D. Cui and T.T. Lim, “An experimental and numerical investigation of the competition between axisymmetric time-periodic modes in an enclosed swirling flow”, Physics of Fluids, Vol. 18, No. 104106, October 2006. 3. J.M. Lopez, Y.D. Cui, F. Marques and T. T. Lim, “Quenching of vortex breakdown oscillations via harmonic modulation”, Journal of Fluid Mechanics, Vol. 599, No. 3, pp441-464, 2008. 4. Y.D. Cui, T.T. Lim and H.M. Tsai, “Blowing and suction effects on vortex breakdown in an enclosed cylindrical container”, Paper AIAA 2007-4360, 37th AIAA Fluid Dynamics Conference and Exhibit, 25-28 Jun 2007, Hyatt Regency, Miami, Florida, United States. 5. Y.D. Cui, J.M. Lopez, T.T. Lim and F. Marques “Harmonically forced enclosed swirling flow”, submitted to Physics of Fluids, 2008. 159 PUBLICATIONS 160 ACHIEVEMENT LIST OF RELATED ACHIEVEMENT Y.D. Cui, Winner of Andrew Fraser Prize 2008, Awarded by the Institution of Mechanical Engineers, Singapore Branch for Excellence in Postgraduate Research. 161 . the vortex breakdown structure and its stability in a confined cylindrical container driven by one rotating endplate. The experiments included flow visualization and hot-film measurements, and. H.M. Tsai, “Blowing and suction effects on vortex breakdown in an enclosed cylindrical container , Paper AIAA 2007-4360, 37 th AIAA Fluid Dynamics Conference and Exhibit, 25- 28 Jun 2007, Hyatt. international journals and conferences. Though the present study has made some valuable contributions to our understanding of the flow physics in the confined cylindrical container with one rotating