Lock-on to quasi-periodic flow transformation for a rotationally oscillating cylinder due to gust impulse Open Access

Flow around bluff bodies, particularly circular cylinders, has been a focus of investigation due to its importance in a wide range of applications.^1–8 For a bluff body immersed in uniform flow, alternating vortices shed in the wake rearward of the body, which induces vibrations in the structure that are called vortex-induced vibrations (VIVs). Analysis of such flows is particularly critical for the design of tall buildings, cooling towers, high-tension transmission lines, submarine periscopes, tubes of heat exchangers, fuel rods in nuclear plants, micro hot wire anemometers, pipelines, and cables on the ocean bed as VIVs may induce structural damage, modified heat transfer, etc. The flow dynamics involve the formation of the shear layer, separation, and further roll-over into vortices, which shed downstream alternately, making the study of these flows challenging.⁹ The shed vortices result in oscillating forces of lift and drag on the cylinder,¹⁰ and hence, the dependence of shedding frequency with the Reynolds number is critical¹¹—the Reynolds number is characterized as $Re= ρ U ∞ d μ$⁠, where fluid density, dynamic viscosity, cylinder diameter, and free-stream velocity are represented by $ρ$⁠, $μ$⁠, d, and $U ∞$⁠, respectively.

Various studies focused on altering these oscillating forces through diverse mechanisms of inline, crossflow, and rotational oscillations of the cylinder have been previously conducted. Bishop et al.¹² studied the transversely oscillating circular cylinder in free-stream flow and reported lock-on flow for this configuration. For lock-on flow, a synchronization occurs between vortex shedding and cylinder oscillation frequencies. As a result, large amplitudes of VIVs are observed. Among these external excitation mechanisms, rotational oscillation of the cylinder provides a pragmatic approach for controlling VIVs in a range of oscillation parameters. Regions of non-lock-on, quasi-periodic, and lock-on flows were defined by varying oscillation amplitude $θ$ measured in radians and Strouhal forcing frequency $S t f = f f d U ∞.$¹³ Here, $f f$ is the dimensional forcing frequency. The flow categories are based on the frequency response of VIVs. Non-lock on and lock-on flows are characterized by prominent peaks of frequencies in the Fourier transform of lift, which correspond to natural shedding and oscillation forcing frequencies, respectively. The frequency response of the lift signal of quasi-periodic flow results in two prominent frequency peaks, one of which follows the natural shedding frequency and the other oscillation forcing frequency. Hence, these flow categories have been commonly identified by analyzing the frequency responses of lift signals. Initially, lock-on for a circular cylinder rotationally oscillating was reported by Okajima et al.¹⁴ Furthermore, lift and drag were reported to increase in locked-on, Wu et al.¹⁵ and Fujisawa et al.¹⁶ Similar studies have been conducted to further provide insights into the lock-on flow at various Reynolds numbers. For instance, Baek and Sung¹⁷ studied the vortex dynamics of a cylinder performing rotational oscillations at Re = 110 in the lock-on range for varied oscillation amplitudes. A change of vortex shedding mode across the natural Strouhal frequency was reported. The natural Strouhal frequency is defined as $S t n= f n d U ∞$⁠, where $f n$ represents the dimensional natural shedding frequency. With decreasing $θ$⁠, a decrease in the lock-on range and an increase in the rate of change of vortex formation phase change were also reported. Fujisawa et al.¹⁸ found the occurrence of lock-on near $S t n$ for small $θ$⁠. At greater values of $θ$⁠, lock-on in a range of $S t f$ was reported; however, the lock-on frequency for the peak value of flow-induced forces was observed to shift to a lower value compared to $S t n$⁠. Lock-on near $S t n$⁠, i.e., primary lock-on, was as well reported by Cheng et al.^19,20 at Reynolds numbers of 1000 and 200, respectively. Lu²¹ studied lock-on flow and the effects of $S t f$ and $θ$ on vorticity dynamics and identified some basic patterns of vortex shedding.

Instead of enhancing VIVs as is the case of locked-on flows, decreasing VIVs through both active and passive methods has been deeply investigated. Drag reduction by rotational oscillation is one of the active methods, and it has been previously investigated,^{19,20,22–31} while passive methods of drag reduction include helical wires on cylinder, longitudinal grooves on cylinder, passive jet flow control, upstream introduction of I-type bluff body and using a wavy cylinder instead of a regular one to name a few and these are discussed by Ishihara and Li,³² Zhou et al.,³³ Chen et al.,³⁴ Triyogi et al.,³⁵ and Karthik et al.³⁶ More recently, a combined effect of active and passive methods was presented by Mao et al.³⁷ In addition, Choi et al.³⁸ and Baek and Sung³⁹ presented flow characteristics in lock-on to quasi-periodic flow boundary for Re = 100 and 110.

More recent work on the topic investigates a variety of avenues. For instance, low Reynolds number flow over rotationally oscillating square cylinder, circular cylinders, controlled generation of periodic vortical gusts via rotationally oscillating cylinder with an attached plate, and control of flow over a rotationally oscillating cylinder using deep reinforcement learning are reported by Singh,⁴⁰ Mikhailov et al.,⁴¹ Ping et al.,⁴² Rockwood and Medina,⁴³ and Tokarev et al.,⁴⁴ respectively. However, a lack of understanding about the effects of an upstream disturbance in the lock-on to quasi-periodic flow boundary. Hence, this study aims to further analyze such flows. It enhances the previous study⁴⁵ focused on the effects of aperiodic/sporadic upstream disturbance. The key objective of this work is to quantify the effects of gust impulse on the lock-on flow of a cylinder with rotational oscillations near the lock-on to quasi-periodic flow boundary. This would shed some light on structural response in this condition, enhancing the design process of cylindrical objects which are relevant to a multitude of applications like tall buildings, cooling towers, high-tension transmission lines, submarine periscopes, etc.

An upstream gust impulse's effects on the flow development are investigated for an oscillating circular cylinder. The flow of Prandtl number 7 is considered. Three flow configurations are chosen based on oscillation amplitude: $θ=15°,30°$⁠, and $45°$⁠. Inlet flow velocity is varied such that Reynolds numbers are 100, 110, and 120. Rotational oscillation forcing frequency is chosen to be in the regions where flow transitions between quasi-periodicity to lock-on and vice versa. A one-time disturbance in the form of an upstream gust impulse is introduced in each flow configuration. The gust impulse is modeled by relating gust Strouhal frequency, $St G$ (reciprocal of time-period of disturbance), to the natural shedding Strouhal frequency, i.e., $S t G=0 St n,0.5S t n,1S t n,1.5S t n$⁠. Here, $S t G= f G d U ∞$⁠, where $f G$ is the dimensional gust frequency or reciprocal of time-period of disturbance. A two-dimensional (2D) gust profile is overlaid onto inlet velocity once the initial transient flow has passed. The gust profiles induce less than $8°$ deflection, which is little enough to expect non-reversing flow.

Dimensionless 2D Navier–Stokes equations for an incompressible flow are as follows:

• Continuity equation

  $∂ U ∂ X+ ∂ V ∂ Y=0.$

  (1)
• X-momentum equation

  $∂ U ∂ τ+ ∂ U U ∂ X+ ∂ V U ∂ Y=− ∂ P ∂ X+ 1 R e ∂ 2 U ∂ X 2 + ∂ 2 U ∂ Y 2.$

  (2)
• Y-momentum equation

  $∂ V ∂ τ+ ∂ U V ∂ X+ ∂ V V ∂ Y=− ∂ P ∂ Y+ 1 R e ∂ 2 V ∂ X 2 + ∂ 2 V ∂ Y 2.$

  (3)

Here, the non-dimensional stream-wise and cross-stream direction coordinates are $X= x d$ and $Y= y d$⁠, the non-dimensional stream-wise and cross-stream velocities are $U= u U ∞$ and $V= v U ∞$⁠, the non-dimensional time is $τ= tU ∞ d$⁠, and the dimensional time is $t$⁠, respectively. The orthogonal direction coordinates are noted as x and y, whereas the orthogonal components of flow velocity are u and v. Dimensionless pressure is $P= p * ρ U ∞ 2$⁠, where $p *$ is the dimensional pressure and $ρ$ the fluid density.

A no-slip boundary condition and an angular velocity about z-axis as $ω t=Acos 2 π f f t$ where angular velocity amplitude is $A=2π f fθ$ are applied to the cylinder's surface. A 2D, transient, pressure-based segregated solver is used for the solution of incompressible flow in ANSYS Fluent. Pressure-linked equations in Navier–Stokes equations for no-slip boundary condition are solved via a semi-implicit method. Discretization of convective and diffusive terms of momentum equations is done with a second-order upwind and central difference methods, respectively.

For the current work, a circular domain centered by a circular cylinder is chosen as displayed in Fig. 1. “D” and “d” represent the diameters of the polar domain and the cylinder, respectively. For the study, three domains of D/d equal to 100, 200, and 300 are examined. Table I presents the comparison between maximum lift coefficient, $C L − max$⁠, of rotationally oscillating cylinder using our solution scheme with those of Baek and Sung¹⁷ for Re of 110 and $θ$ of $15°$⁠. As the forcing frequency advances toward natural frequency, percent error decreases. A maximum error of 5.89% is observed at $St f$ of 0.18 and a minimum error of 1.71% at $St f$ of 0.17. The deviations from the literature slightly increase for increasing D/d at respective $S t f$⁠. The small increment in error can be attributed to being numerical in nature, and hence, the smallest of D/d = 100 is chosen for independence study of grids.

Multiple grid structures of increased finer mesh distribution near the cylinder are obtained by varying the number of radial node points while retaining the number of peripheral nodes constant. The grids G1, G2, G3, and G4 contain 200 peripheral grid points and 250, 300, 350, and 400 radial grid points resulting in 25 000, 30 000, 35 000, and 40 000 grid elements, respectively. Comparison of $C L − max$ between values reported by Baek and Sung¹⁷ and the current solution scheme is presented in Table II for Re of 110 and $θ$ of $15°$⁠. Evidently, increasing the mesh density reduces the error. The reduction of error is significant for the transition from G1 to G2; however, this reduction decreases as we further increase the mesh density. A maximum of 0.78% for a transition from G1 to G2 at $St f=0.15$ and minimum of 0.01% in the transition from G1 to G2, G2 to G3, and G3 to G4 at $St f=0.17$ occur in error reduction. Based on the numerical sensitivity of the analysis, a domain of D/d = 100 and a grid of 30 000 cells (G2) is chosen for the numerical analysis.

Initially, it is essential to determine the effectiveness of the current solution for a stationary cylinder and a cylinder with rotational oscillation. For the stationary cylinder, natural shedding Strouhal frequency $St n$⁠, spatiotemporally averaged drag force coefficient, $C D ¯$⁠, and root-mean-squared value of lift force coefficient, $C L − rms$⁠, are compared with those of Qu et al.⁴⁷ and Baranyi⁴⁸ in Table III. $St n$ shows a minimum deviation of 0.44% from Baranyi⁴⁸ at Re of 100, while it has a maximum deviation of 0.90% from Qu et al.⁴⁷ at Re of 120. Similarly, $C L − rms$ has a minimum and maximum deviation of 0.57% and 1.85% at Re of 100 and 120 from Qu et al.,⁴⁷ respectively. Moreover, $C D ¯$ showed deviations of 0.95% and 1.22% from Qu et al.⁴⁷ and Baranyi⁴⁸ at Re of 100, respectively. For the rotationally oscillating cylinders, the current solution scheme is validated against the work of Baek and Sung¹⁷ by comparing $C L − max$ at Re of 110 and $θ$ of $15°$⁠. As shown in Table IV, a minimum deviation of 1.71% at $St f$ of 0.17 and a maximum deviation of 3.67% at $St f$ of 0.18 are observed from the literature. The excellent agreement between current numerical data and previously published works confirmed the model is suitable for the analysis considered in this study.

Lock-on boundary identification is the first part of this study. As stated by Baek and Sung,³⁹ upon increasing $St f$⁠, flow configuration transitions from quasi-periodic flow to lock-on and then back to quasi-periodic flow. The lock-on flow region exists surrounding the natural frequency. Analysis conducted by Baek and Sung³⁹ was for Re = 110 and $0.15≤ St f≤0.19$ with increments in $S t f$ of 0.005. However, for the current study, the resolution of $St f$ is increased to 0.0001 for better identification of lock-on to quasi-periodic flow boundary at all combinations of Re = 100, 110, 120 and $θ=15°,30°,45°$⁠, which would further be introduced to an upstream gust impulse.

Lock-on flows occur in a band of $St f$ around $St n$⁠. The current study focuses on the boundary for $Re=100, 110, 120 andθ=15°,30°,45°$⁠. Figure 2 presents lift histories for the cases of $Re=120$ and $θ=30°$⁠. Figures 2(a) and 2(d) correspond to the cases of quasi-periodic flows, whereas Figs. 2(b) and 2(c) correspond to the cases of lock-on flows. A similar pattern of lift variation as reported by Baek and Sung³⁹ for flow transitioning from quasi-periodicity to lock-on and back to quasi-periodicity is observed here. As shown in Fig. 2, it is important to highlight here that flow in the upper lock-on boundary (⁠ $S t f>S t n$⁠) reaches its limit cycle oscillations (LCOs) relatively quickly as compared to the lower lock-on boundary (⁠ $S t f<S t n$⁠). Limit cycle oscillations are a state of the system in which it exhibits repeating behavior over a time after overcoming the initial transient period. The quick attainment of LCOs is because of the higher $S t f$ in the upper lock-on boundary as compared to the lower lock-on boundary. As $S t f$ is decreased, the flow starts to resemble the case of stationary cylinder. Hence, in similar fashion, flows with lower $S t f$ takes more time to overcome the initial transients and reach LCOs. Lock-on boundary is presented in Fig. 3 for $Re=100, 110, 120$ and $θ=15°,30°,45°$⁠. It is observed that since, $S t n$ is directly proportional to $Re$⁠, increasing $Re$ results in shifting of the lock-on region to the right or higher $S t f$⁠. Also, increasing $θ$ results in increasing the range of lock-on region.

Bluff bodies subjected to free-stream flow exhibit alternate vortex shedding above a certain Re. This alternate shedding of vortices induces a fluctuating lift force on the bluff body. The flow condition is categorized based on the observations of the lift evolution. Three distinct flow conditions are attributed to rotationally oscillating circular cylinder, namely, non-lock-on, quasi-periodic, and lock-on.¹³ In the current study, the cases of lock-on flow near the transition boundary of lock-on and quasi-periodic flows are divided into two categories based on the Fourier transforms of their lift histories. These categories include the cases, which transform from the lock-on flow to quasi-periodic and those which do not, due to the introduction of an upstream gust impulse.

It is observed that transformation from lock-on to quasi-periodicity occurs only in the upper lock-on boundary (⁠ $S t f>S t n$⁠). The gust impulse introduces frequencies greater than $S t f$ in the lift signal, Fig. 4. In the case of lower lock-on boundary (⁠ $S t f<S t n$⁠), the higher frequencies introduced by the gust impulse lie in the lock-on flow regime. However, this is not the case for the upper lock-on boundary, i.e., higher frequencies lie in the quasi-periodic flow regime. Hence, in the lower lock-on boundary region, since all the frequencies are in the lock-on regime, the flow remains locked-on. In such cases, the frequencies introduced by the gust impulse die down and the prominent frequency, which is $S t f$⁠, dictates the state of the locked-on flow. However, the frequencies introduced by the gust impulse in the upper lock-on boundary are quasi-periodic in nature. Hence, they cause instability in the initially locked-on flow. Since the lock-on phenomenon is the weakest at the lock-on flow region boundary, therefore, it is proposed that these instabilities persist. Resulting in transforming initially locked-on flow in the upper lock-on boundary region to quasi-periodic flow even though the cylinder is rotationally oscillating with $S t f$ in the lock-on region. The transformation is observed to occur at four descending values of $S t f$ at the investigated resolution near the upper lock-on boundary; e.g., for $Re=120,θ=30°$⁠, transformation is observed for $S t f=0.2042, 0.2043, 0.2044, and 0.2045$⁠. Decreasing $S t f$ further results in strengthening lock-on phenomena; hence, no transformation occurs.

Yet, in the upper lock-on boundary region, apparently $θ and St G$ are the main contributors in determining the above-mentioned transformation. It is observed that cases with relatively low $θ$ or higher $St G$ return to nominal lock-on flow condition once the gust is past the flow domain. Figure 5 presents $C L$ against non-dimensional time $τ$ and power spectral densities (PSD) against Strouhal number (St) for $Re=100, St f=0.1820, St G=0.5 St n, and θ=15°$⁠.

This case is a representative of the prior mentioned category of non-transforming cases, which is comprised of cases for $θ=15° at St G=0.5 St n$ and for all $θ and Re at St G=1 St n and 1.5 St n$⁠. Figure 5(b) presents spectral density against St of lift signal before gust shown in Fig. 5(a). Evidently, flow is exhibiting a lock-on condition before gust impulse introduction since there is a single peak against forcing frequency in Fig. 5(b). On entry of gust impulse, the frequency response is disturbed, Fig. 5(c). However, once the gust exits the flow domain, flow acquires its initial state of lock-on flow, Fig. 5(d). Only the cases of $θ=30°,45°$ and $S t G=0.5S t n$ show transformation from lock-on to quasi-periodic flow for Re = 100, 110, 120. This transformation consistently persists for four descending values of $S t f$ at the upper lock-on boundary with decrements of 0.0001 for all the combinations of the aforementioned parameters except for $Re=120,θ=45°and St G=0.5 St n$⁠, which exhibit transformation for five descending values of $S t f$⁠.

Sections IV A and IV B will focus on the cases, which exhibit transformation from lock-on to quasi-periodicity due to gust impulse to further shed light on the lift, drag, spectral densities, and vorticity contours.

For stationary cylinder, the flow develops in an initial transient period in which the flow is initially separated, and later vortices shed downstream of the cylinder. Once the initial transient period has passed, the flow reaches a steady state. However, for a cylinder with rotational oscillations, vortex shedding onsets as the flow starts and the initial transient period is relatively smaller compared to the stationary case. As reported by Filler et al.,⁴⁹ rotational oscillations of small amplitudes introduce asymmetry in the flow behind the cylinder. This asymmetry promotes the early shedding of vortices and hence reduces the initial transient period. It can be reasoned that rotational oscillation accelerates flow on either top or bottom semi-circular part of cylinder and retards it on the other side. The accelerating and retarding flow results in early vortex shedding and quick flow development.

Rotationally oscillating excitations result in distinct flow conditions of lock-on and quasi-periodic flows based on $St f$⁠, Baek and Sung.³⁹  Figure 6 presents $C L$ against $τ$⁠, while Fig. 7 presents the corresponding PSDs against St of the lift histories for $Re=120,θ=30°$⁠.

Figure 6(a) shows the lift history for quasi-periodic flow near the lock-on to quasi-periodic flow transition boundary. It is observed that flow development occurs at $τ ≈ 350$⁠. Once the flow is fully developed a beating lift signal is visible, which signifies the presence of multiple frequencies that are resulting in the lift evolution. This statement is corroborated by PSDs of lift evolution for the quasi-periodic flow shown in Fig. 7(a), which shows the PSD peaks corresponding to $St=0.205 and 0.165$⁠. As reported by Baek and Sung,³⁹ one frequency, i.e., $St=0.205$⁠, follows the forcing frequency of $St f=0.2047$ and the other, i.e., $St=0.165$⁠, follows the natural shedding frequency of $St n=0.172$⁠.

Furthermore, Figs. 6(b)–6(e) represent the cases of lock-on flow near the lock-on to quasi-periodic flow transition boundary, which are subjected to an upstream gust impulse. A comparison of the initial transient period of the flow development for the cases of quasi-periodic and lock-on flow shows that it is much smaller for the cases of lock-on flow than for the quasi-periodic flow; i.e., lock-on flows reach full flow development well before $τ=135$⁠, while, as mentioned before, the quasi-periodic flows reach full flow development at $τ ≈ 350$⁠. However, once the flow is developed, the cases of lock-on flow exhibit a repetitive lift evolution, which would result in a single prominent frequency peak in spectral analysis, which is described in the description of Fig. 5. As reported in our earlier study,⁴⁵  $C L$ and $C D$ are increased during the interaction of gust impulse with the cylinder. However, once gust exits the flow domain, cases exhibiting transformation from lock-on to quasi-periodic flow show lift evolutions similar to that of quasi-periodic flow depicted in Figs. 6(b)–6(e). This similarity is further confirmed by the spectral analysis of $C L$ in a period when the flow transforms from lock-on to quasi-periodic, i.e., period of beating $C L$ signal, in Figs. 7(b)–7(e). Like the case of quasi-periodic flow, $C L$ for the transforming cases is also beating signals depicting the presence of multiple frequencies constituting the lift signals. The PSDs, shown in Figs. 7(b)–7(e), uphold this statement by representing two prominent frequency peaks for each transforming case. These peaks are not exactly equal to the peaks for the quasi-periodic flow case, but a resemblance can be observed. The rest of the transforming cases exhibit similar behavior.

The above-mentioned resemblance is further studied by comparing $C D ¯$ and $C L − rms$ for the quasi-periodic flow and the cases presenting transformation from lock-on to quasi-periodicity in Figs. 8 and 9, respectively. Figures 8(a)–8(c) show $C D ¯$ for $θ=30°$ and Figs. 8(d)–8(f) for $θ=45°$⁠, for all Re in current work. The solid line in each graph of Fig. 8 shows $C D ¯$ for the cases of quasi-periodic flows, while the diamond symbols represent $C D ¯$ for the transforming cases after the gust exits the flow domain and the lock-on flow has transformed into a quasi-periodic flow. It can be easily assessed that values for the transforming cases are almost equal to those of quasi-periodic flows.

$C D ¯$ has a maximum deviation of 0.91% for the case of $Re=110,θ=45°, and St f=0.2087$ and a minimum deviation of 0.03% for the case of $Re=120,θ=30°, and St f=0.2045$ from the cases of quasi-periodic flows at the respective $Re and θ$⁠. All the transforming cases exhibit values of $C D ¯$ within 1% of the values for the quasi-periodic flows at the respective $Re and θ$⁠. The values of $C L − rms$ for the transforming cases also show conformity with the values for the respective quasi-periodic flows. However, this agreement is not as strong as that found for $C D ¯$⁠. For instance, $C L − rms$ has peak difference of 4.09% for $Re=110,θ=30°, and St f=0.1998$ and a minimum deviation of 0.06% for $Re=120,θ=30°, and St f=0.2046$ from their respective quasi-periodic flows. For the rest of the transforming cases, deviations of $C L − rms$ are below 3.4%. The deviations for $C L − rms$ are relatively large compared to $C D ¯$⁠; however, these changes are not significant enough to describe the flow not being quasi-periodic. Furthermore, these deviations can be justified by considering the different values of $S t f$ for the transforming cases from those of the quasi-periodic cases. Hence, it is concluded that lock-on flow has transformed into quasi-periodic flow due to an upstream gust impulse at $Re=100, 110, 120; θ=30°,45°; and St G=0.5 St n$⁠.

To further insight into the transformation from lock-on to quasi-periodic flow, vorticity contours along with the lift evolutions are presented in Figs. 10 for Re = 100 and $θ=30°$ and 11 for Re = 100 and $θ=45°$⁠. Each lift history includes vertical dashed lines, which encapsulate a specific portion of the lift in a quasi-periodic cycle. In the encapsulating region, flow transitions from 2S and P + S to a more chaotic P + S vortex shedding mode for $θ=30° and 45°$⁠, respectively. Here, 2S and P + S correspond to shedding of two vortices and a pair with a single vortex downstream of the cylinder in a single period of lift evolution. This transition is highlighted by the adjoining vorticity contours. Three dashed concentric circles are placed around the cylinder in each contour such that their distances from cylinder's center are 1d, 2d, and 3d. Also, vorticity contours at four distinct instances are presented to highlight the aforementioned vortex-shedding mode transition. Instance $τ 1$ represents the time before transition, i.e., 2S and P + S vortex shedding modes for $θ=30° and 45°$⁠, respectively; $τ 2$ represents the initiation of transition of vortex shedding mode; $τ 3$ represents a chaotic P + S vortex shedding mode, i.e., instance in the encapsulated region of lift; and $τ 4$ represents a recovery of vortex shedding mode back to 2S and P + S for $θ=30° and 45°$⁠, respectively.

Figure 10(a) presents the quasi-periodic flow, while Figs. 10(b)–10(e) illustrate the transforming cases at Re = 100 and $θ=30°$⁠. As mentioned earlier, for $θ=30°$⁠, the shedding mode alternates between 2S and a chaotic P + S mode. In each lift history, the portion enveloped by the vertical dashed lines in a quasi-periodic cycle represents the time for the chaotic P + S shedding mode. In the vorticity contours of Fig. 10(a), the dominance of 2S shedding mode at instances $τ 1$ and $τ 2$ is evident. However, at $τ 2$⁠, the transition starts and the chaotic P + S shedding mode becomes dominant, which is represented in the contour at $τ 3$⁠. Finally, the transition from chaotic P + S to normal 2S shedding mode is evident at $τ 4$⁠. From Figs. 10(b)–10(e), it can be observed that similar transitions of vortex shedding modes are evident for all the transforming cases at Re = 100 and $θ=30°$⁠.

Similarly, Fig. 11(a) presents the quasi-periodic flow, while Figs. 11(b)–11(e) provided the results for the transforming cases at Re = 100 and $θ=45°$⁠. However, for $θ=45°$⁠, the shedding mode transitions from P + S to a chaotic P + S mode for the quasi-periodic flow presented in Fig. 11(a). The dominant normal P + S shedding mode is presented at $τ 1$ and $τ 2$ in Fig. 11(a). The transition of shedding mode initiates at $τ 2$ and is observable at $τ 3$⁠. The chaotic P + S vortex shedding mode transitions back to normal P + S mode shown at $τ 4$⁠. This transitioning behavior of vortex shedding mode observed for the quasi-periodic flow is also observed for every transforming case.

Similar behavior is observed for the rest of the transforming cases at varied Re and $θ$ in this study. Based on the similarities found in the vortex shedding modes for the quasi-periodic flows and the respective transforming flows, the proposition of the lock-on to quasi-periodic flow transformation due to gust impulse is further reinforced here.

Effects of an upstream gust impulse on a circular cylinder with rotational oscillations in the lock-on to quasi-periodic flow boundary are investigated. The Reynolds number is varied as Re = 100, 110, 120. Forcing Strouhal frequency is selected in a range such that flow transitions either from quasi-periodicity to lock-on or lock-on to quasi-periodicity, i.e., lock-on flow boundary region. Rotational oscillation amplitudes and gust impulse frequencies are set as $θ=15°,30°,45°$ and $St G=0.5 St n,1 St n,1.5 St n$⁠. The gust impulse's effects on the lock-on flow are listed below:

• Flow is found to return back to its locked-on condition after the gust impulse is out of the flow domain in both the lower lock-on boundary (⁠ $S t f<S t n$⁠) and for upper lock-on boundary (⁠ $S t f>S t n$⁠) at $θ=15°$ and $St G=0.5 St n$ or all values of $θ$ at $St G=1 St n and 1.5 St n$⁠, for all Reynold numbers in this study.

• A transformation from lock-on to quasi-periodic flow is observed for all Re at different combinations of $St G=0.5 St n$ and $θ=30°,45°$⁠.

• The gust impulse of $S t G=0.5S t n$ is observed to introduce frequencies greater than $S t f$⁠. For lower lock-on boundary, these frequencies are in the lock-on region; hence, the flow remains locked-on and does not transform. However, for the upper lock-on boundary, these frequencies are in quasi-periodic flow region. It is proposed that these quasi-periodic frequencies persist in the upper lock-on boundary region since the locking-on phenomenon is the weakest here.

• Spectral analysis of all the transforming flows depicted similar behavior as those of their respective quasi-periodic flows. The frequency peaks of transforming cases differed slightly from each other and from those of quasi-periodic flows.

• Similarly, $C D ¯$ and $C L − rms$ agree for the transforming flows with their respective quasi-periodic flows. A maximum deviation of 4.09%, from quasi-periodic flow, is observed for $C L − rms$ at Re = 110, $θ=30°, and St f=0.1998$⁠, while deviations of $C D ¯$ from quasi-periodic flow remained below 1%.

• Vorticity contours for $θ=30°$ showed a change from 2S into a chaotic P+S shedding mode, while, for $θ=45°$⁠, shedding mode transitioned from a normal P+S to a chaotic P+S shedding mode in a quasi-periodic cycle. These transitions of vortex shedding modes were found both for the quasi-periodic flows and the transforming flows in a similar way.

• Slight variations in all the above-mentioned comparisons were observed, which are mainly caused by different values of $St f$ for the transforming flows and their respective quasi-periodic flows. However, the transforming cases were found to behave in a similar manner as their respective quasi-periodic flows.

The author(s) received no financial support for the research, authorship, and/or publication of this article.

The authors have no conflicts to disclose.

Arsalan Yawar: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Fatemeh Salehi: Supervision (equal); Writing – original draft (supporting); Writing – review & editing (supporting). Shehryar Manzoor: Supervision (equal); Writing – review & editing (supporting).

The data that support the findings of this study are available within the article.