6 results
Intermittency in free vibration of a cylinder beyond the laminar regime
- Navrose, Sanjay Mittal
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- Journal:
- Journal of Fluid Mechanics / Volume 870 / 10 July 2019
- Published online by Cambridge University Press:
- 15 May 2019, R2
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Vortex-induced vibration of a circular cylinder that is free to move in the transverse ($Y$) and streamwise ($X$) directions is investigated at subcritical Reynolds numbers ($1500\lesssim Re\lesssim 9000$) via three-dimensional (3-D) numerical simulations. The mass ratio of the system for all the simulations is $m^{\ast }=10$. It is observed that while some of the characteristics associated with the $XY$-oscillation are similar to those of the $Y$-only oscillation (in line with the observations made by Jauvtis & Williamson (J. Fluid Mech., vol. 509, 2004, pp. 23–62)), notable differences exist between the two systems with respect to the transition between the branches of the cylinder response in the lock-in regime. The flow regime between the initial and lower branch is characterized by intermittent switching in the cylinder response, aerodynamic coefficients and modes of vortex shedding. Similar to the regime of laminar flow, the system exhibits a hysteretic response near the lower- and higher-$Re$ end of the lock-in regime. The frequency spectrum of time history of the cylinder response shows that the most dominant frequency in the streamwise oscillation on the initial branch is the same as that of the transverse oscillation.
Transient growth in the near wake region of the flow past a finite span wing
- Navrose, V. Brion, L. Jacquin
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- Journal:
- Journal of Fluid Mechanics / Volume 866 / 10 May 2019
- Published online by Cambridge University Press:
- 13 March 2019, pp. 399-430
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We investigate optimal perturbation in the flow past a finite aspect ratio ($AR$) wing. The optimization is carried out in the regime where the fully developed flow is steady. Parametric study over time horizon ($T$), Reynolds number ($Re$), $AR$, angle of attack and geometry of the wing cross-section (flat plate and NACA0012 airfoil) shows that the general shape of linear optimal perturbation remains the same over the explored parameter space. Optimal perturbation is located near the surface of the wing in the form of chord-wise periodic structures whose strength decreases from the root towards the tip. Direct time integration of the disturbance equations, with and without nonlinear terms, is carried out with linear optimal perturbation as initial condition. In both cases, the optimal perturbation evolves as a downstream travelling wavepacket whose speed is nearly the same as that of the free stream. The energy of the wavepacket increases in the near wake region, and is found to remain nearly constant beyond the vortex roll-up distance in nonlinear simulations. The nonlinear wavepacket results in displacement of the tip vortex. In this situation, the motion of the tip vortex resembles that observed during vortex meandering/wandering in wind tunnel experiments. Results from computation carried out at higher $Re$ suggest that, even beyond the steady flow regime, a perturbation wavepacket originating near the wing might cause meandering of tip vortices.
Optimal perturbation for two-dimensional vortex systems: route to non-axisymmetric state
- Navrose, H. G. Johnson, V. Brion, L. Jacquin, J. C. Robinet
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- Journal:
- Journal of Fluid Mechanics / Volume 855 / 25 November 2018
- Published online by Cambridge University Press:
- 21 September 2018, pp. 922-952
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We investigate perturbations that maximize the gain of disturbance energy in a two-dimensional isolated vortex and a counter-rotating vortex pair. The optimization is carried out using the method of Lagrange multipliers. For low initial energy of the perturbation ($E(0)$), the nonlinear optimal perturbation/gain is found to be the same as the linear optimal perturbation/gain. Beyond a certain threshold $E(0)$, the optimal perturbation/gain obtained from linear and nonlinear computations are different. There exists a range of $E(0)$ for which the nonlinear optimal gain is higher than the linear optimal gain. For an isolated vortex, the higher value of nonlinear optimal gain is attributed to interaction among different azimuthal components, which is otherwise absent in a linearized system. Spiral dislocations are found in the nonlinear optimal perturbation at the radial location where the most dominant wavenumber changes. Long-time nonlinear evolution of linear and nonlinear optimal perturbations is studied. The evolution shows that, after the initial increment of perturbation energy, the vortex attains a quasi-steady state where the mean perturbation energy decreases on a slow time scale. The quasi-steady vortex state is non-axisymmetric and its shape depends on the initial perturbation. It is observed that the lifetime of a quasi-steady vortex state obtained using the nonlinear optimal perturbation is longer than that obtained using the linear optimal perturbation. For a counter-rotating vortex pair, the mechanism that maximizes the energy gain is found to be similar to that of the isolated vortex. Within the linear framework, the optimal perturbation for a vortex pair can be either symmetric or antisymmetric, whereas the structure of the nonlinear optimal perturbation, beyond the threshold $E(0)$, is always asymmetric. No quasi-steady state for a counter-rotating vortex pair is observed.
The critical mass phenomenon in vortex-induced vibration at low $Re$
- Navrose, Sanjay Mittal
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- Journal:
- Journal of Fluid Mechanics / Volume 820 / 10 June 2017
- Published online by Cambridge University Press:
- 05 May 2017, pp. 159-186
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Vortex-induced vibration of a circular cylinder with low mass ratio ($0.05\leqslant m^{\ast }\leqslant 10.0$) is investigated, via a stabilized space–time finite element formulation, in the laminar flow regime where $m^{\ast }$ is defined as the ratio of the mass of the oscillating structure to the mass of the fluid displaced by it. Computations are carried out over a wide range of reduced speed, $U^{\ast }$, which is defined as $U/f_{N}D$, where $U$ is the free-stream speed, $f_{N}$ the natural frequency of the spring mass system in vacuum and $D$ the diameter of the cylinder. In particular, the situation where the lock-in regime extends up to infinite reduced speed is explored. Studies at large $Re$, in the past, have shown that the normalized amplitude of cylinder oscillation at infinite reduced speed, $A_{\infty }^{\ast }$, exhibits a sharp increase when $m^{\ast }$ is reduced below the critical mass ratio ($m_{crit}^{\ast }$). This jump signifies a shift from desynchronized response to lock-in state. In this work it is shown that in the laminar regime, a jump in $A_{\infty }^{\ast }$ occurs only beyond a certain $Re$ ($=Re_{j}\sim 108$). For $Re<Re_{j}$, the response increases smoothly with decrease in $m^{\ast }$ with no discernible jump. In this situation, therefore, the identification of $m_{crit}^{\ast }$ based on jump in response at $U^{\ast }=\infty$ is not possible. The difference in the $A^{\ast }-m^{\ast }$ variation on the two sides of $Re=Re_{j}$, is attributed to the difference in the transition between the lower branch of cylinder response and desynchronization regime. This transition is brought out more clearly by plotting $A^{\ast }$ with $f_{v_{o}}/f$, where $f_{v_{o}}$ is the vortex shedding frequency for the flow past a stationary cylinder and $f$ is the cylinder vibration frequency. In the $A^{\ast }-f_{v_{o}}/f$ plane, the response data as well as other quantities related to free vibrations, for different $m^{\ast }$, collapse on a curve. Unlike at high $Re$, the collapsed curves show a dependence on $Re$ in the laminar regime. The transition between the lock-in and desynchronized state, as seen from the collapsed curves, is qualitatively different for $Re$ on either side of $Re_{j}$. The collapsed curves, at a certain $Re$, are utilized to estimate $A^{\ast }$ for the limiting case of $(U^{\ast },m^{\ast })=(\infty ,0)$. Interestingly, unlike at large $Re$, this limit value is found to be lower than the peak amplitude of cylinder vibration at a given $Re$. Hysteresis in the cylinder response, near the higher-$U^{\ast }$ end of the lock-in regime, is explored. It is observed that the range of $U^{\ast }$ with hysteretic response increases with decrease in $m^{\ast }$. Interestingly, for a certain range of $m^{\ast }$, the response is hysteretic from a finite $U^{\ast }$ up to $U^{\ast }=\infty$. We refer to this phenomenon as hysteresis forever. It occurs because of the existence of multiple response states of the system at $U^{\ast }=\infty$, for a certain range of $m^{\ast }$. The study brings out the significant differences in the response of the fluid–structure system associated with the critical mass phenomenon between the low- and high-$Re$ regime.
Lock-in in vortex-induced vibration
- Navrose, Sanjay Mittal
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- Journal:
- Journal of Fluid Mechanics / Volume 794 / 10 May 2016
- Published online by Cambridge University Press:
- 05 April 2016, pp. 565-594
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The phenomenon of lock-in in vortex-induced vibration of a circular cylinder is investigated in the laminar flow regime ($20\leqslant Re\leqslant 100$). Direct time integration (DTI) and linear stability analysis (LSA) of the governing equations are carried out via a stabilized finite element method. Using the metrics that have been proposed in earlier studies, the lock-in regime is identified from the results of DTI. The LSA yields the eigenmodes of the coupled fluid–structure system, the associated frequencies ($F_{LSA}$) and the stability of the steady state. A linearly unstable system, in the absence of nonlinear effects, achieves large oscillation amplitude at sufficiently large times. However, the nonlinear terms saturate the response of the system to a limit cycle. For subcritical $Re$, the occurrence of lock-in coincides with the linear instability of the fluid–structure system. The critical $Re$ is the Reynolds number beyond which vortex shedding ensues for a stationary cylinder. For supercritical $Re$, even though the aeroelastic system is unstable for all reduced velocities ($U^{\ast }$) lock-in occurs only for a finite range of $U^{\ast }$. We present a method to estimate the time beyond which the nonlinear effects are expected to be significant. It is observed that much of the growth in the amplitude of cylinder oscillation takes place in the linear regime. The response of the cylinder at the end of the linear regime is found to depend on the energy ratio, $E_{r}$, of the unstable eigenmode. $E_{r}$ is defined as the fraction of the total energy of the eigenmode that is associated with the kinetic and potential energy of the structure. DTI initiated from eigenmodes that are linearly unstable and whose energy ratio is above a certain threshold value lead to lock-in. Interestingly, during lock-in, the oscillation frequency of the fluid–structure system drifts from $F_{LSA}$ towards a value that is closer to the natural frequency of the oscillator in vacuum ($F_{N}$). In the event of more than one eigenmode being linearly unstable, we investigate which one is responsible for lock-in. The concept of phase angle between the cylinder displacement and lift is extended for an eigenmode. The phase angle controls the direction of energy transfer between the fluid and the structure. For zero structural damping, if the phase angle of all unstable eigenmodes is less than 90°, the phase angle obtained via DTI evolves to a value that is close to 0°. If, on the other hand, the phase angle of any unstable eigenmode is more than 90°, it settles to 180°, approximately in the limit cycle. A new approach towards classification of modes is presented. The eigenvalues are tracked over a wide range of $U^{\ast }$ while keeping $Re$ and mass ratio ($m^{\ast }$) fixed. In general, for large values of $m^{\ast }$, the eigenmodes corresponding to the two leading eigenvalues exhibit a decoupled behaviour with respect to $U^{\ast }$. They are classified as the fluid and elastic modes. However, for relatively low $m^{\ast }$ such a classification is not possible. The two leading modes are coupled and are referred to as fluid–elastic modes. The regime of such occurrence is shown on the $Re{-}m^{\ast }$ parameter space.
Three-dimensional flow past a rotating cylinder
- Navrose, Jagmohan Meena, Sanjay Mittal
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- Journal:
- Journal of Fluid Mechanics / Volume 766 / 10 March 2015
- Published online by Cambridge University Press:
- 30 January 2015, pp. 28-53
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Three-dimensional computations are carried out for a spinning cylinder placed in a uniform flow. The non-dimensional rotation rate is varied in the range $0.0\leqslant {\it\alpha}\leqslant 5.0$ . A stabilized finite element method is utilized to solve the incompressible Navier–Stokes equations in primitive variables formulation. Linear stability analysis of the steady state shows the existence of several new unstable three-dimensional modes for $200\leqslant \mathit{Re}\leqslant 350$ and $4.0\leqslant {\it\alpha}\leqslant 5.0$ . The curves of neutral stability of these modes are presented in the $\mathit{Re}{-}{\it\alpha}$ parameter space. For the flow at $\mathit{Re}=200$ and rotation rate in the ranges $0.0\leqslant {\it\alpha}\leqslant 1.91$ and $4.34\leqslant {\it\alpha}\leqslant 4.7$ , the vortex shedding, earlier reported in two dimensions and commonly referred to as parallel shedding, can also exist as oblique shedding. In this mode of shedding, the vortices are inclined to the axis of the cylinder. In fact, parallel shedding is a special case of oblique shedding. It is found that the span of the cylinder plays a significant role in the time evolution of the flow. Of all the unstable eigenmodes, with varied spanwise wavenumber, only the ones whose integral number of wavelengths fit the span length of the cylinder are selected to grow. For the flow at $\mathit{Re}=200$ , two steady states exist for $4.8\leqslant {\it\alpha}\leqslant 5.0$ . While one of them is associated with unstable eigenmodes, the other is stable to all infinitesimal perturbations. In this regime, irrespective of the initial conditions, the fully developed flow is steady and devoid of any instabilities.