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Swing and reverse swing of a cricket ball: laminar separation bubble, secondary vortex and wing-tip-like vortices
- Aman Parekh, Daksh Chaplot, Sanjay Mittal
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- Journal:
- Journal of Fluid Mechanics / Volume 983 / 25 March 2024
- Published online by Cambridge University Press:
- 18 March 2024, A23
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Large eddy simulation of flow past a cricket ball with its seam at $30^\circ$ to the free stream is carried out for $5 \times 10^4 \le Re \le 4.5 \times 10^5$. Three regimes of flow are identified on the basis of the time-averaged swing force coefficient ($\bar {C}_Z$) – no swing (NS), conventional swing (CS, $\bar {C}_Z>0$) and reverse swing (RS, $\bar {C}_Z<0$). The effect of seam on the boundary layer is investigated. Contrary to the popular belief, the boundary layer does not transition to a turbulent state in the initial stages of CS. The seam energizes the laminar boundary layer and delays its separation. The delay is significantly larger in a region near the poles, whose extent increases with an increase in $Re$ causing $\bar {C}_Z$ to increase. Here $\bar {C}_Z$ assumes a near constant value in the later stage of CS. The boundary layer transitions to a turbulent state via formation of a laminar separation bubble (LSB) in the equatorial region and directly, without a LSB, in the polar region. The extent of the LSB shrinks while the region of direct transition near the poles increases with an increase in $Re$. A LSB forms on the non-seam side of the ball in the RS regime. A secondary vortex is observed in the wake bubble. While it exists on the non-seam side for the entire range of $Re$ considered, the mixing in the flow introduced by the seam causes it to disappear beyond a certain $Re$ on the seam side. The pressure difference between the seam and non-seam sides sets up wing-tip-like vortices. Their polarity reverses with the switch from the CS to RS regime.
Is a baseball like knuckleball possible in cricket?
- Kunjal Shah, Sanjay Mittal
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- Journal:
- Flow: Applications of Fluid Mechanics / Volume 3 / 2023
- Published online by Cambridge University Press:
- 15 June 2023, E16
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The knuckleball is considered to be one of the hardest pitches to hit in baseball due to its seemingly unpredictable motion. It has gained popularity in cricket in recent times. It is shown that the delivery referred to as knuckleball in cricket, at present, does not exhibit a zigzag motion and is, therefore, a misnomer. We propose a delivery in cricket that is associated with an erratic trajectory similar to the knuckleball pitch in baseball. Force measurement experiments in a wind tunnel on a new cricket ball in various orientations of the seam to the incoming flow and at different Reynolds number are carried out. The results are utilized to estimate the trajectory of knuckleball deliveries. The key parameters are the seam angle, speed and spin rate of the ball at the time of its release. Their effect on the trajectory is studied in detail. The optimal combination of these parameters that result in a knuckleball, which is likely hard for the batter to play, is identified.
Streamwise vortices, cellular shedding and force coefficients on finite wing at low Reynolds number
- Jawahar Sivabharathy Samuthira Pandi, Sanjay Mittal
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- Journal:
- Journal of Fluid Mechanics / Volume 958 / 10 March 2023
- Published online by Cambridge University Press:
- 28 February 2023, A10
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Numerical simulations for flow past a finite rectangular wing with a NACA 0012 section at $Re=1000$ for various semi-aspect ratios ($0.25\le sAR \le 7.5$) over a range of angles of attack ($0^{\circ }\le \alpha \le 14^{\circ }$) reveal streamwise vortices, which increase in strength and number to occupy an increasing spanwise extent with increase in $\alpha$. They result in non-monotonic spanwise variation of local force coefficients and increased strength of wing-tip vortex for $\alpha >8^{\circ }$. Viscous and pressure drag dominate for low and high sAR, respectively. The time-averaged drag coefficient first decreases and then increases with increase in $sAR$. Vortex shedding for $\alpha =14^{\circ }$ is single cell and parallel for $sAR<3$. Shedding is in two cells with an oblique angle that varies with time, leading to large spanwise variation in the root mean square of local force coefficients for higher $sAR$. Various types of dislocations, reported earlier in wakes of bluff bodies, are seen for different $\alpha$ and $sAR$. Dislocations for $\alpha =14^{\circ }$ appear at the same spanwise location for $sAR=3$ and at different spanwise locations for $sAR\ge 4$. Vortex shedding for $\alpha =12^{\circ }$ and $sAR=5$ exhibits one cell structure in the near wake and two cells in the far wake due to splitting and reconnection of vortices near the mid-span in the moderate wake. Linkages form between counter-rotating spanwise vortices for $sAR\ge 1$. Additional linkages between shed- and wing-tip vortices are observed for lower $sAR$. At each $\alpha$, the strength of the wing-tip vortex and radius of its core, estimated using Rankine and Lamb–Oseen models, increases up to a certain $sAR$ beyond which it is approximately constant.
Effect of free stream turbulence on the topology of laminar separation bubble on a sphere
- Aditya Desai, Sanjay Mittal
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- Journal:
- Journal of Fluid Mechanics / Volume 948 / 10 October 2022
- Published online by Cambridge University Press:
- 09 September 2022, A28
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The topology of a laminar separation bubble (LSB) on a sphere in the critical regime is investigated via experiments at five turbulent intensities: $T_u=0.06\,\%$, $0.42\,\%$, $0.71\,\%$, $1.00\,\%$ and $1.36\,\%$. The drag crisis occurs at a lower $Re$ and becomes gradual with increasing $T_u$. The flow is devoid of the LSB at the onset of the critical regime. It forms on a small part of the sphere and not at all azimuthal locations, early in the critical regime. The LSB forms at more azimuthal locations with increasing $Re$. This azimuthal expansion of the LSB is accompanied by intermittency for a small range of $Re$. Towards the end of the critical regime, an axisymmetric LSB forms on the sphere at all time instants. A model is proposed to estimate the azimuthal extent and distribution of the LSB from the mean force coefficients of a flow state. The model predicts that the LSB forms as multiple segments for a large part of the critical regime. During the spatial growth of the LSB with $Re$ in the critical regime, some of its fragments relocate to alternate locations. Moderate increase in $T_u$ ($0.42\,\% \leq T_u \leq 0.71\,\%$) leads to rich dynamics with several intermittent flow states. However, fewer intermittent states are observed beyond a certain $T_u$ (${\geq }1.00\,\%$).
Secondary vortex, laminar separation bubble and vortex shedding in flow past a low aspect ratio circular cylinder
- Gaurav Chopra, Sanjay Mittal
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- Journal:
- Journal of Fluid Mechanics / Volume 930 / 10 January 2022
- Published online by Cambridge University Press:
- 08 November 2021, A12
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Large eddy simulation of flow past a circular cylinder of low aspect ratio ($AR=1$ and $3$), spanning subcritical, critical and supercritical regimes, is carried out for $2\times 10^3 \le Re \le 4\times 10^5$. The end walls restrict three-dimensionality of the flow. The critical $Re$ for the onset of the critical regime is significantly lower for small aspect ratio cylinders. The evolution of secondary vortex (SV), laminar separation bubble (LSB) and the related transition of boundary layer with $Re$ is investigated. The plateau in the surface pressure due to LSB is modified by the presence of SV. Proper orthogonal decomposition of surface pressure reveals that although the vortex shedding mode is most dominant throughout the $Re$ regime studied, significant energy of the flow lies in a symmetric mode that corresponds to expansion–contraction of the vortex formation region and is responsible for bursts of weak vortex shedding. A triple decomposition of the time signals comprising of contributions from shear layer vortices, von Kármán vortex shedding and low frequency modulation due to the symmetric mode of flow is proposed. A moving average, with appropriate size of window, is utilized to estimate the component due to vortex shedding. It is used to assess the variation, with $Re$, of strength of vortex shedding as well as its coherence along the span. Weakening of vortex shedding in the high subcritical and critical regime is followed by its rejuvenation in the supercritical regime. Its spanwise correlation is high in the subcritical regime, decreases in the critical regime and improves again in the supercritical regime.
Multiple lock-ins in vortex-induced vibration of a filament
- Mohd Furquan, Sanjay Mittal
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- Journal:
- Journal of Fluid Mechanics / Volume 916 / 10 June 2021
- Published online by Cambridge University Press:
- 06 April 2021, R1
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The vortex-induced vibration of a flexible filament attached behind a stationary cylinder is studied in the two-dimensional, laminar flow regime. We explore the response of the filament for a wide range of flexibility and inertia. Lock-in with a large number of normal modes of the filament, each in a different regime of reduced speed, is observed. Reduced speed is the free-stream speed of the incoming flow non-dimensionalized with the first natural frequency of the structure and the diameter of the cylinder. Several branches, based on response of the filament, are identified and the contributions of various structural modes along these branches are quantified. Contribution from a particular structural mode increases significantly during lock-in, accompanied by a large amplitude of vibration. The transition between different branches is found to be hysteretic and intermittent. The flow exhibits a variety of vortex-shedding patterns, including the $\mathsf {2P+2S}$ mode. The modes of shedding show a systematic variation with amplitude and frequency. The map of vortex-shedding patterns in the amplitude–frequency plane resembles the corresponding map for forced vibration of a rigid cylinder. The transformation of wake from one mode of shedding to another is explained phenomenologically. Variation of rate of energy transfer between the fluid and filament with space and time is analysed to determine optimal placement of transducers for harvesting energy.
Cellular vortex shedding from a cylinder at low Reynolds number
- Sanjay Mittal, Jawahar Sivabharathy Samuthira Pandi, Mainak Hore
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- Journal:
- Journal of Fluid Mechanics / Volume 915 / 25 May 2021
- Published online by Cambridge University Press:
- 18 March 2021, A74
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Flow past a cylinder in the presence of side walls is investigated numerically for various combinations of Reynolds numbers ($50 \leqslant \textit {Re} \leqslant 100$) and aspect ratio ($5 \leqslant AR \leqslant 90$). Various attributes such as cellular shedding, dislocations, oblique angle of vortices and their structure near the end wall are studied. The complexity of the transitions in the wake, with $\textit {Re}$, increases with increase in $AR$. The flow is associated with a single cell wake for $AR=20$, at all the $\textit {Re}$ considered. For $AR$$=90$ the flow transitions from a one-cell to two-cell wake at $\textit {Re} \sim 50$, to a four-cell structure at $\textit {Re} \sim 55$ to a three-cell wake at $\textit {Re} \sim 58$, and then to a two-cell wake at $\textit {Re} \sim 61$. The vortices near the end wall diffuse for $Re\leqslant 58$ whereas linkages form between adjacent vortices of opposite polarity at larger $Re$. The frequency of dislocations increases with increase in $\textit {Re}$. Two types of dislocations have been identified: the fork type at relatively low Re and connected fork type at larger Re. Linkages between vortices in connected fork-type dislocations may lead to ring-like structures. The end conditions and nonlinear mechanisms play a significant role in the evolution of cellular shedding. A single cell across the span is the dominant mode of shedding in the linear regime. Compared with the linear analysis, nonlinearities result in smaller oblique angles, higher shedding frequency and larger streamwise speed of the convection of vortices.
Numerical study of flow-induced vibration of a circular cylinder with attached flexible splitter plate at low $Re$
- Tulsi Ram Sahu, Mohd Furquan, Sanjay Mittal
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- Journal:
- Journal of Fluid Mechanics / Volume 880 / 10 December 2019
- Published online by Cambridge University Press:
- 10 October 2019, pp. 551-593
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Flow-induced vibration (FIV) of an elastically mounted circular cylinder with an attached splitter plate in uniform flow is studied numerically via a stabilized space–time finite element method. The Reynolds number based on the cylinder diameter $D$ and the free-stream speed is restricted to 150. The ratio of the density of the body to that of the fluid, for the major part of the study, is 10. Two different reduced speeds are defined to quantify the compliance of the elastic support and flexibility of the splitter plate, respectively: $U_{s}^{\ast }$ based on the natural frequency of the spring–mass system and $U_{p}^{\ast }$ based on the fundamental natural frequency of the plate. Flow past a stationary cylinder ($U_{s}^{\ast }=0$) with a flexible splitter plate of length $3.5D$ is studied at different values of $U_{p}^{\ast }$. The vibration response of the plate exhibits lock-in with various eigenmodes of the plate in different ranges of $U_{p}^{\ast }$. The onsets of these lock-in regions are abrupt and hysteretic. The elastically mounted cylinder, without the splitter plate, undergoes large-amplitude vortex-induced vibration (VIV) for $4<U_{s}^{\ast }<7$. These large-amplitude oscillations are a consequence of synchronization, wherein the vortex shedding frequency locks in to the cylinder oscillation frequency. A rigid splitter plate attached to the cylinder reduces significantly the peak amplitude during VIV. Increasing the length of the plate from $1.5D$ to $3.5D$ only marginally affects the peak amplitude. It, however, leads to a wider range of lock-in. Unlike the case of an isolated cylinder, the lock-in and desynchronization regimes are not well demarcated in the presence of the splitter plate. Further, galloping is observed beyond a critical value of $U_{s}^{\ast }$; the amplitude of vibration increases with an increase in $U_{s}^{\ast }$ while the vibration frequency is relatively low and remains nearly constant. Increase in plate length delays, in terms of $U_{s}^{\ast }$, the onset of galloping. It is also found that the flexibility of the plate affects the maximum oscillation amplitude in the VIV regime. It also dictates the presence/absence of galloping. The system behaves similar to an isolated cylinder for a very flexible plate. The response is devoid of galloping, but relatively large amplitude of oscillation is observed during lock-in. The behaviour of the cylinder with a stiff plate is similar to that with the rigid one. The galloping instability sets in when the flexibility of the plate is less than a certain value ($U_{p}^{\ast }<4.7$, approximately for $U_{s}^{\ast }=22$). The VIV and galloping are separated by a range of $U_{s}^{\ast }$ in which the flow is either steady, for longer plates, or exhibits very weak vortex shedding. In the VIV regime, the plate tip and cylinder vibrate in phase for low $U_{p}^{\ast }$; their motion is out of phase for larger $U_{p}^{\ast }$. The change in phase is also associated with change in the frequency of vibration. At low $U_{p}^{\ast }$, the frequency of vibration is close to the first natural frequency of the system, while at high $U_{p}^{\ast }$ it becomes closer to the second natural frequency. The vibration amplitude of the cylinder is close to maximum in the VIV regime for $U_{s}^{\ast }=6$. Computations for various $U_{p}^{\ast }$, for $U_{s}^{\ast }=6$ and $22$, are utilized to determine optimal flexibility that leads to minimal FIV. The effect of the length of the flexible splitter plate, mass ratio and damping ratio is studied. A strategy is proposed to utilize the computations from various combinations of $U_{s}^{\ast }$ and $U_{p}^{\ast }$ to choose the appropriate flexibility of the attached splitter plate to minimize FIV.
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.
The role of the seam in the swing of a cricket ball
- Rahul Deshpande, Ravi Shakya, Sanjay Mittal
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- Journal:
- Journal of Fluid Mechanics / Volume 851 / 25 September 2018
- Published online by Cambridge University Press:
- 19 July 2018, pp. 50-82
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The role of the seam in the ‘swing’ of a cricket ball is investigated via unsteady force and surface-pressure measurements and oil-flow visualization in a low-turbulence wind tunnel. Various seam angles of the ball and flow speeds are considered. Static tests are carried out on a new ‘SG Test’ cricket ball as well as its idealized models: a smooth sphere with one and five trips. To study the effect of surface roughness of the ball as the game progresses, force measurements are also carried out on a cricket ball that is manually roughened, on one-half and completely, to model a ball that has been in play for approximately 40 overs (240 deliveries/balls). The Reynolds number ($Re$) is based on the free-stream speed and diameter of the respective model. A new cricket ball experiences three flow states with increase in $Re$: no swing (NS), conventional swing (CS) and reverse swing (RS). At relatively low $Re$, in the NS regime, the seam does not have any significant effect on the flow. The separation of the laminar boundary layer, with no subsequent reattachment, is almost axisymmetric with respect to the free-stream flow. Therefore, the ball does not experience any significant lateral force. Beyond a certain $Re$, the boundary layer on the seam side of the ball undergoes transition. The boundary layer on the non-seam side, however, continues to undergo a laminar separation with no reattachment, thereby creating a lateral force in the direction of the seam, leading to CS. The onset of the CS regime is marked by intermittent formation of a laminar separation bubble (LSB) on the surface of the ball in the region between the laminar separation of the boundary layer and its reattachment at a downstream location. Owing to the varying azimuthal location of the seam, with respect to the front stagnation point on the ball, the transition via LSB formation is localized to a specific region over the seam side. In other regions, the boundary layer either transitions directly without the formation of an LSB, or separates on encountering the seam with no further reattachment. The spatial extent of the region where the flow directly transitions to a turbulent state increases with increase in $Re$, while that of the LSB decreases. Interestingly, the flow dynamics is such that the magnitude of the swing force coefficient stays relatively constant with increase in $Re$. With further increase in $Re$, the boundary layer on the non-seam side undergoes a transition via formation of an LSB. This, along with an upstream shift of the separation point on the seam side, leads to a switch in the direction of the lateral force. It now acts away from the seam, and leads to RS. The transition from CS to RS occurs over a very narrow range of $Re$ wherein the flow intermittently switches between the two flow states. It is observed that the transition of the boundary layer on the seam side leads to an upstream shift of the separation point on the non-seam side at the onset of CS. A complementary effect is observed at the onset of RS. Experiments on a ball that is manually roughened bring out the relative effect of the seam and roughness on the transition of the boundary layer. Compared to a new ball, the magnitude of the maximum swing force coefficient for a rough ball is smaller during the CS regime, and larger during the RS regime. Unlike other models, the ball with roughened non-seam side and smooth seam side, for certain seam orientations, exhibits RS at relatively lower speeds and CS at higher speeds. The forces measured on the cricket ball are utilized to estimate the trajectory of the ball bowled at various initial speeds and seam angles. The lateral movement of the ball depends very significantly on the seam angle, surface roughness and speed of the ball at its delivery. The maximum lateral deviation of a new ball during RS is found to be less than half of that observed in CS. On the other hand, the lateral movement of a roughened ball during RS may significantly exceed its movement during CS. The range of the speed of the ball, for various seam orientations and surface roughnesses, are estimated wherein it undergoes CS, RS or one followed by the other. Optimal conditions are estimated for the desired lateral movement of the ball.
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.
Intermittency of laminar separation bubble on a sphere during drag crisis
- Rahul Deshpande, Vivek Kanti, Aditya Desai, Sanjay Mittal
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- Journal:
- Journal of Fluid Mechanics / Volume 812 / 10 February 2017
- Published online by Cambridge University Press:
- 05 January 2017, pp. 815-840
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The phenomenon of drag crisis for uniform flow past a smooth sphere is investigated via experiments in a low-turbulence wind tunnel for $1.5\times 10^{5}\leqslant Re\leqslant 5.0\times 10^{5}$. The Reynolds number, $Re$, is based on the free-stream speed and the diameter of the sphere. Based on the activity related to the laminar separation bubble (LSB), the critical regime for the occurrence of drag crisis ($3.4\times 10^{5}<Re<4.4\times 10^{5}$) is further divided into three subregimes. The gradual decrease of mean drag coefficient ($\overline{C}_{D}$) with $Re$, in subregime I, is due to the increase of base pressure and suction near the shoulder of the sphere. The flow is devoid of an LSB in this regime. The coefficient $\overline{C}_{D}$ decreases very rapidly with increase in $Re$ in subregime II primarily due to the increase in mean base pressure ($\overline{C}_{P,b}$). This subregime is characterized by intermittent switching of $C_{D}$ and $C_{P,b}$ between bistable states. Statistical analysis of the surface-pressure and force coefficients relates this behaviour to the intermittent appearance/disappearance of the LSB. The two states of the flow are referred to as the LSB and non-LSB states. The frequency of appearance of the LSB and the duration of its stay increase with increase in $Re$. An intermittency factor $I_{f}$, defined as the fraction of time during which the LSB exists in the flow, is estimated at each $Re$. The value of $I_{f}$ is zero in subregime I and increases from zero to one, with increase in $Re$, in subregime II. The variation of $\overline{C}_{D}$ with $Re$ is found to follow the variation of ($1-I_{f}$) with $Re$. This shows that the decrease of $\overline{C}_{D}$ with increase in $Re$, during drag crisis, is primarily due to the increased probability of the LSB state as opposed to the non-LSB state. A spatio-temporal analysis of the surface pressure measured at various polar locations on the surface of the sphere confirms the axisymmetric nature of the intermittent LSB. In subregime III of the critical regime, the LSB exists at all time instants ($I_{f}=1$). The $\overline{C}_{D}$ value continues to decrease with $Re$ in this subregime due to increase in $\overline{C}_{P,b}$. Unlike the general belief that the decrease in $\overline{C}_{D}$ with increase in $Re$ is due only to the increase in $\overline{C}_{P,b}$, it is found that the increase in suction upstream of the shoulder of the sphere, with increase in $Re$, also plays an important role. In particular, in the high subcritical regime ($2.5\times 10^{5}<Re<3.4\times 10^{5}$), the gradual decrease in $\overline{C}_{D}$ with increase in $Re$ is due solely to the increase in suction in a region upstream of the shoulder of the sphere.
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.
On the origin of the secondary vortex street
- Bhaskar Kumar, Sanjay Mittal
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- Journal:
- Journal of Fluid Mechanics / Volume 711 / 25 November 2012
- Published online by Cambridge University Press:
- 24 September 2012, pp. 641-666
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The origin of the secondary vortex street, observed in the far wake in the flow past a circular cylinder, is investigated. The Reynolds number, based on the diameter of the cylinder, is 150. The von Kármán vortex street, which originates in the near wake, decays exponentially downstream of the cylinder. Beyond the region of decay, a broad band of frequencies are selectively amplified, leading to the formation of a secondary vortex street consisting of packets of large-scale vortex structures. The streamwise location of the onset of the instability, frequency of the generation of packets and their convection speed are estimated via direct numerical simulation (DNS). Global linear stability analysis of the time-averaged flow reveals the presence of unstable convective modes that travel at almost the same speed and have a structure similar to the packet-like disturbances as observed in the DNS. Sensitivity analysis of the global convective modes to structural perturbations is carried out to locate the region of the wake that is most significant in generating the modes responsible for the appearance of the secondary vortex street. This information is utilized to control the flow. By placing a ‘slip’ splitter plate along the wake centre line, in the overlap region of the direct and the adjoint modes, the oscillations in the far wake are significantly reduced, though the oscillations related to the primary vortex shedding in the near wake are not. It is also found that suppression of the primary vortex shedding leads to annihilation of the secondary vortex street as well. Linear stability analysis of the steady-state flow does not yield any modes that can explain the appearance of the secondary vortex street. The steady and time-averaged wake profiles, for the flow, are compared to bring out the differences in the two. The effect of free-stream oscillations on the evolution of the secondary vortex street is investigated. By reducing the amplitude of inlet excitation, a gradual transition from ordered shedding in the far wake to the appearance of a broad-band spectrum of frequencies, as in the unforced wake, is observed. All the computations have been carried out using a stabilized finite element method.
Hysteresis in vortex-induced vibrations: critical blockage and effect of m*
- T. K. PRASANTH, V. PREMCHANDRAN, SANJAY MITTAL
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- Journal:
- Journal of Fluid Mechanics / Volume 671 / 25 March 2011
- Published online by Cambridge University Press:
- 03 February 2011, pp. 207-225
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The hysteretic behaviour of a freely vibrating cylinder, near the low-Reynolds-number end of synchronization/lock-in, in the laminar regime is investigated. Computations are carried out using a stabilized finite-element method. The flow remains two-dimensional in this Reynolds number regime. This is verified via comparison of two- and three-dimensional computations. The cylinder is free to undergo crossflow as well as in-line vibrations. The combined effect of mass ratio (1 ≤ m* ≤ 100) and blockage (0.25% ≤ B ≤ 12.5%) is studied in detail. The existence of a critical mass ratio (m*cr = 10.11), below which hysteresis disappears for an unbounded flow situation, is identified. For higher mass ratio the hysteretic behaviour is observed for all blockage. However, the hysteresis loop width is found to vary with B; its variation with m* and B is studied. The concept of critical blockage Bcr is introduced. For B ≤ Bcr the response of the cylinder is virtually the same as that in an unbounded flow domain. The variation of Bcr with m* is investigated. Furthermore, Bcr is found to vary non-monotonically with m* for m* ≤ m*cr and is almost constant for m* ≥ 20. The effect of damping, as well as restricting the cylinder to undergo transverse vibrations only, on the hysteresis behaviour is studied. The transverse-only motion leads to a larger hysteresis loop width compared with the transverse and the in-line motion of the cylinder. An attempt is made to explain this by comparing the results from forced vibrations.
Global stability of flow past a cylinder with centreline symmetry
- BHASKAR KUMAR, JACOB JOHN KOTTARAM, AMIT KUMAR SINGH, SANJAY MITTAL
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- Journal:
- Journal of Fluid Mechanics / Volume 632 / 10 August 2009
- Published online by Cambridge University Press:
- 27 July 2009, pp. 273-300
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Global absolute and convective stability analysis of flow past a circular cylinder with symmetry conditions imposed along the centreline of the flow field is carried out. A stabilized finite element formulation is used to solve the eigenvalue problem resulting from the linearized perturbation equation. All the computations carried out are in two dimensions. It is found that, compared to the unrestricted flow, the symmetry conditions lead to a significant delay in the onset of absolute as well as convective instability. In addition, the onset of absolute instability is greatly affected by the location of the lateral boundaries and shows a non-monotonic variation. Unlike the unrestricted flow, which is associated with von Kármán vortex shedding, the flow with centreline symmetry becomes unstable via modes that are associated with low-frequency large-scale structures. These lead to expansion and contraction of the wake bubble and are similar in characteristics to the low-frequency oscillations reported earlier in the literature. A global linear convective stability analysis is utilized to find the most unstable modes for different speeds of the disturbance. Three kinds of convectively unstable modes are identified. The ones travelling at very low streamwise speed are associated with large-scale structures and relatively low frequency. Shear layer instability, with relatively smaller scale flow structures and higher frequency, is encountered for disturbances travelling at relatively larger speed. For low blockage a new type of instability is found. It travels at relatively high speed and resembles a swirling flow structure. As opposed to the absolute instability, the convective instability appears at much lower Re and its onset is affected very little by the location of the lateral boundaries. Analysis is also carried out for determining the convective stability of disturbances that travel in directions other than along the free stream. It is found that the most unstable disturbances are not necessarily the purely streamwise travelling ones. Disturbances that move purely in the cross-stream direction can also be convectively unstable. The results from the linear stability analysis are confirmed by carrying out direct time integration of the linearized disturbance equations. The disturbance field shows transient growth by several orders of magnitude confirming that such flows act as amplifiers. Direct time integration of the Navier–Stokes equation is carried out to track the time evolution of both the large-scale low-frequency oscillations and small-scale shear layer instabilities. The critical Re for the onset of convective instability is compared with earlier results from local analysis. Good agreement is found.
Steady separated flow past a circular cylinder at low Reynolds numbers
- SUBHANKAR SEN, SANJAY MITTAL, GAUTAM BISWAS
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- Journal:
- Journal of Fluid Mechanics / Volume 620 / 10 February 2009
- Published online by Cambridge University Press:
- 10 February 2009, pp. 89-119
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The steady two-dimensional laminar flow around a stationary circular cylinder has been investigated via a stabilized finite-element method. The Reynolds number Re is based on the cylinder diameter and free-stream speed. The results have been presented for 6 ≤ Re ≤ 40 and the blockages between 0.000125 and 0.80. The blockage B is the ratio of the cylinder diameter to the domain width. There is large scatter in the value of Res, reported in the literature, marking the onset of the flow separation. From the present study the Res is found to be 6.29, approximately for B = 0.005. The effect of the blockage on the characteristic flow parameters is found to be insignificant for B ≤ 0.01. The bubble length, separation angle and Res exhibit non-monotonic variation with the blockage. It is for the first time that such a behaviour for the separation angle and Res is being reported. Two types of boundary conditions at the lateral walls have been studied: the slip wall and towing tank. In general for high blockage, the results from the slip boundary condition are closer to the ones for the unbounded flow. In that sense, the use of the slip boundary condition as opposed to the towing tank boundary condition on the lateral walls is advocated. The bubble length, separation angle, base suction, total drag, pressure drag, viscous drag and maximum vorticity on the cylinder surface for the steady flow are found to vary as Re, Re−0.5, Re−1, Re−0.5, Re−0.64, Re−0.60 and Re0.5, respectively. The extrapolated results for the steady flow, for higher Re, are found to match quite well with the other results from the literature.
Vortex-induced vibrations at subcritical Re
- SANJAY MITTAL, SAURAV SINGH
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- Journal:
- Journal of Fluid Mechanics / Volume 534 / 10 July 2005
- Published online by Cambridge University Press:
- 21 June 2005, pp. 185-194
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Flow past a stationary cylinder becomes unstable at Re$\,{\sim}\,47$. Flow-induced vibrations of an elastically mounted cylinder, of low non-dimensional mass, is investigated at subcritical Reynolds numbers. A stabilized finite-element formulation is used to solve the incompressible flow equations and the cylinder motion in two dimensions. The cylinder is free to vibrate in both the transverse and in-line directions. It is found that, for certain natural frequencies of the spring–mass system, vortex shedding and self-excited vibrations of the cylinder are possible for Re as low as 20. Lock-in is observed in all cases. However, the mass of the oscillator plays a major role in determining the proximity of the vortex-shedding frequency to the natural frequency of the oscillator. A global linear stability analysis (LSA) for the combined flow and oscillator is carried out. The results from the LSA are in good agreement with the two-dimensional direct numerical simulations.
Flow past a rotating cylinder
- SANJAY MITTAL, BHASKAR KUMAR
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- Journal:
- Journal of Fluid Mechanics / Volume 476 / 10 February 2003
- Published online by Cambridge University Press:
- 10 March 2003, pp. 303-334
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Flow past a spinning circular cylinder placed in a uniform stream is investigated via two-dimensional computations. A stabilized finite element method is utilized to solve the incompressible Navier–Stokes equations in the primitive variables formulation. The Reynolds number based on the cylinder diameter and free-stream speed of the flow is 200. The non-dimensional rotation rate, α (ratio of the surface speed and freestream speed), is varied between 0 and 5. The time integration of the flow equations is carried out for very large dimensionless time. Vortex shedding is observed for α < 1.91. For higher rotation rates the flow achieves a steady state except for 4.34 < α < 4:70 where the flow is unstable again. In the second region of instability, only one-sided vortex shedding takes place. To ascertain the instability of flow as a function of α a stabilized finite element formulation is proposed to carry out a global, non-parallel stability analysis of the two-dimensional steady-state flow for small disturbances. The formulation and its implementation are validated by predicting the Hopf bifurcation for flow past a non-rotating cylinder. The results from the stability analysis for the rotating cylinder are in very good agreement with those from direct numerical simulations. For large rotation rates, very large lift coefficients can be obtained via the Magnus effect. However, the power requirement for rotating the cylinder increases rapidly with rotation rate.