We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Bistabilities of two equilibrium states discovered in the coupled side-by-side Kármán wakes are investigated through Floquet analysis and direct numerical simulation (DNS) with different initial conditions over a range of gap-to-diameter ratio (
$g^*= 0.2\text {--}3.5$) and Reynolds number (
$Re = 47\text {--}100$). Two bistabilities are found in the transitional 
$g^*-Re$ regions from in-phase (IP) to anti-phase (AP) vortex shedding states. By initialising the flow in DNS with zero initial conditions, the flow in the first bistable region (i.e. bistable IP/FF
$_C$ at 
$g^*= 1.4 \text {--} 2.0$, where FF
$_C$ denotes the conditional flip-flop flow) attains flip-flop (FF) flow, it settles into the IP state by initialising the flow with an IP flow. The second bistability is observed between cylinder-scale IP and AP states at large 
$g^*$ (
$=$ 2.0–3.5). The transition from the FF
$_C$ to IP is dependent on initial conditions and irreversible over the parameter space, meaning that the flow cannot revert back to the FF
$_C$ state once it jumps to the IP state irrespective of the direction of 
$Re$ variations. Its counterpart for the bistable IP/AP state is reversible. We also found that the FF
$_C$ flow in the first bistable region is primarily bifurcated from synchronised AP with cluster-scale features, possibly because the cluster-scale AP flow is inherently unstable to FF flow instabilities. It is demonstrated that the irreversible bistability exists in other interacting wakes around multiple cylinders. A good understanding of flow bistabilities is pivotal to flow control applications and the interpretation of desynchronised flow features observed at high 
$Re$ values.
We focus on the population dynamics driven by two classes of truncated 
$\alpha$-stable processes with Markovian switching. Almost necessary and sufficient conditions for the ergodicity of the proposed models are provided. Also, these results illustrate the impact on ergodicity and extinct conditions as the parameter 
$\alpha$ tends to 2.
Transition to chaos through a cascade of period doublings of the primary 
$1/2$ synchronization mode is discovered in steady approaching flow around a forced inline oscillating cylinder near a plane boundary at a Reynolds number 
$(Re)$ of 175. The transition occurs well within the otherwise synchronized region (known as the Arnold tongue) in the frequency and amplitude space of the oscillating cylinder, creating two parameter strips of desynchronized flows within the Arnold tongue. Five orders of period doublings from mode 
$1/2$ to mode 
$16/32$ are revealed by progressively increasing the frequency resolution in the simulation. The ratio of frequency intervals of two successive period-doubling modes asymptotes towards the first Feigenbaum constant, reaching a value of 4.52 at mode of 
$16/32$. Additional three-dimensional simulations demonstrate the existence of period doubling with a regular spanwise flow structure similar to regular mode B of steady flow around an isolated cylinder. Although transition to chaos through cascades of period doublings is primarily reported for the primary 
$1/2$ synchronization mode, it is also observed for other synchronization modes 
$(p/q)$ (Tang et al., J. Fluid Mech., vol. 832, 2017, pp. 146–169), where 
$p$ and 
$q$ are integers with a non-reducible 
$p/q$, such as 
$2/3$. The physical mechanisms responsible for the present period-doubling bifurcations and transition to chaos through cascades of period doublings are ascribed to the interaction of asymmetric vortex shedding from the cylinder (due to a geometric asymmetry) and the boundary layer developed on the plane boundary, through specifically designed numerical tests.
Oscillatory flow around a cluster of four circular cylinders in a diamond arrangement is investigated using two-dimensional direct numerical simulation over Keulegan–Carpenter numbers (KC) ranging from 4 to 12 and Reynolds numbers (Re) from 40 to 230 at four gap-to-diameter ratios (G) of 0.5, 1, 2 and 4. Three types of flows, namely synchronous, quasi-periodic and desynchronized flows (along with 14 flow regimes) are mapped out in the (G, KC, Re)-parameter space. The observed flow characteristics around four cylinders in a diamond arrangement show a few unique features that are absent in the flow around four cylinders in a square arrangement reported by Tong et al. (J. Fluid Mech., vol. 769, 2015, pp. 298–336). These include (i) the dominance of flow around the cluster-scale structure at 
$G=0.5$ and 1, (ii) a substantial reduction of regime D flows in the regime maps, (iii) new quasi-periodic (phase trapping) 
$\text{D}^{\prime }$ (at 
$G=0.5$ and 1) and period-doubling 
$\text{A}^{\prime }$ flows (at 
$G=1$) and most noteworthily (iv) abnormal behaviours at (
$G\leqslant 2$) (referred to as holes hereafter) such as the appearance of spatio-temporal synchronized flows in an area surrounded by a single type of synchronized flow in the regime map (
$G=0.5$). The mode competition between the cluster-scale and cylinder-scale flows is identified as the key flow mechanism responsible for those unique flow features, with the support of evidence derived from quantitative analysis. Phase dynamics is introduced for the first time in bluff-body flows, to the best knowledge of the authors, to quantitatively interpret the flow response (e.g. quasi-periodic flow features) around the cluster. It is instrumental in revealing the nature of regime 
$\text{D}^{\prime }$ flows where the cluster-scale flow features are largely synchronized with the forcing of incoming oscillatory flow (phase trapping) but are modulated by localized flow features.
This paper presents a numerical investigation of oscillatory flow around a circular cylinder that is placed in proximity to a plane boundary that is parallel to the cylinder axis. The onset and development of the Honji instability are studied over a range of Stokes numbers (
$\unicode[STIX]{x1D6FD}$) and gap-to-diameter ratios (
$e/D$) at a fixed Keulegan–Carpenter number (
$KC=2$). Four flow regimes are identified in the (
$e/D,\unicode[STIX]{x1D6FD}$)-plane: (I) featureless two-dimensional flow, (II) stable Honji vortex, (III) unstable Honji vortex and (IV) chaotic flow. As 
$e/D$ increases from 
$-0.5$ (embedment) to 
$1$, the critical Stokes number 
$\unicode[STIX]{x1D6FD}_{cr}$ for the onset of the Honji instability follows two side-by-side convex functions, peaking at the connection point of 
$e/D=0.125$ and reaching troughs at 
$e/D=0$ and 0.375. The Honji instability is always initiated on the gap side of the cylinder surface for 
$0.375\leqslant e/D\leqslant 2$ and occurs only on the top side for 
$-0.5\leqslant e/D<0.125$. The location for the initiation of the Honji instability switches from the gap side to the top side of the cylinder surface for 
$0.125<e/D<0.375$. No Honji instability is observed at 
$e/D=0.125$, where the flow three-dimensionality is developed through a different flow mechanism. Consistently, the three-dimensional kinetic energy of the flow, which represents a measure of the strength of flow three-dimensionality, varies with 
$e/D$ in a trend opposite to that of 
$\unicode[STIX]{x1D6FD}_{cr}$. Three physical mechanisms are identified as being responsible for the observed variation trend of 
$\unicode[STIX]{x1D6FD}_{cr}$ with 
$e/D$ and for various flow phenomena, which are the blockage effect induced by the geometry setting, the existence of the Stokes layer on the plane boundary and the favourable pressure gradient in the flow direction over the gap between the cylinder and the plane surface.
Oscillatory flow around a circular cylinder close to a plane boundary is numerically investigated at low-to-intermediate Keulegan–Carpenter (
$KC$) and Stokes numbers (
$\unicode[STIX]{x1D6FD}$) for different gap-to-diameter ratios (
$e/D$). A set of unique flow regimes is observed and classified based on the established nomenclature in the (
$KC,\unicode[STIX]{x1D6FD}$)-space. It is found that the flow is not only influenced by 
$e/D$ but also by the ratio of the thickness of the Stokes boundary layer (
$\unicode[STIX]{x1D6FF}$) to the gap size (e). At relatively large 
$\unicode[STIX]{x1D6FF}/e$ values, vortex shedding through the gap is suppressed and vortices are only shed from the top of the cylinder. At intermediate values of 
$\unicode[STIX]{x1D6FF}/e$, flow through the gap is enhanced, resulting in horizontal gap vortex shedding. As 
$\unicode[STIX]{x1D6FF}/e$ is further reduced below a critical value, the influence of 
$\unicode[STIX]{x1D6FF}/e$ becomes negligible and the flow is largely dependent on 
$e/D$. A hysteresis phenomenon is observed for the transitions in the flow regime. The physical mechanisms responsible for the hysteresis and the variation of marginal stability curves with 
$e/D$ are explored at 
$KC=6$ through specifically designed numerical simulations. The Stokes boundary layer over the plane boundary is found to be responsible for the relatively large hysteresis range over 
$0.25<e/D<1.0$. Three mechanisms have been identified to the change of the marginal stability curve over 
$e/D$, which are the blockage effect due to the geometry setting, the favourable pressure gradient over the gap and the location of the leading eigenmode relative to the cylinder.
A numerical analysis of flow around a circular cylinder oscillating in-line with a steady flow is carried out over a range of driving frequencies 
$(f_{d})$ at relatively low amplitudes 
$(A)$ and a constant Reynolds number of 175 (based on the free-stream velocity). The vortex shedding is investigated, especially when the shedding frequency 
$(f_{s})$ synchronises with the driving frequency. A series of modes of synchronisation are presented, which are referred to as the 
$p/q$ modes, where 
$p$ and 
$q$ are natural numbers. When a 
$p/q$ mode occurs, 
$f_{s}$ is detuned to 
$(p/q)f_{d}$, representing the shedding of 
$p$ pairs of vortices over 
$q$ cycles of cylinder oscillation. The 
$p/q$ modes are further characterised by the periodicity of the transverse force over every 
$q$ cycles of oscillation and a spatial–temporal symmetry possessed by the global wake. The synchronisation modes 
$(p/q)$ with relatively small natural numbers are less sensitive to the change of external control parameters than those with large natural numbers, while the latter is featured with a narrow space of occurrence. Although the mode of synchronisation can be almost any rational ratio (as shown for 
$p$ and 
$q$ smaller than 10), the probability of occurrence of synchronisation modes with 
$q$ being an even number is much higher than 
$q$ being an odd number, which is believed to be influenced by the natural even distribution of vortices in the wake of a stationary cylinder.
Two-dimensional direct numerical simulation and Floquet stability analysis have been performed at moderate Keulegan–Carpenter number (
$KC$) and low Reynolds number (
$Re$) for a square cross-section cylinder with its face normal to the oscillatory flow. Based on the numerical simulations a map of flow regimes is formed and compared to the map of flow around an oscillating circular cylinder by Tatsuno & Bearman (J. Fluid Mech., vol. 211, 1990, pp. 157–182). Two new flow regimes have been observed, namely A
$^{\prime }$ and F
$^{\prime }$. The regime A
$^{\prime }$ found at low 
$KC$ is characterised by the transverse convection of fluid particles perpendicular to the motion; and the regime F
$^{\prime }$ found at high 
$KC$ shows a quasi-periodic feature with a well-defined secondary period, which is larger than the oscillation period. The Floquet analysis demonstrates that when the two-dimensional flow breaks the reflection symmetry about the axis of oscillation, the quasi-periodic instability and the synchronous instability with the imposed oscillation occur alternately for the square cylinder along the curve of marginal stability. This alternate pattern in instabilities leads to four distinct flow regimes. When compared to the vortex shedding in otherwise unidirectional flow, the two quasi-periodic flow regimes are observed when the oscillation frequency is close to the Strouhal frequency (or to half of it). Both the flow regimes and marginal stability curve shift in the 
$(Re,KC)$-space compared to the oscillatory flow around a circular cylinder and this shift appears to be consistent with the change in vortex formation time associated with the lower Strouhal frequency of the square cylinder.
