Mode-based energy transfer analysis of flow-induced vibration of two rigidly coupled tandem cylinders
Graphical abstract
Introduction
Flow-induced vibration (FIV), a typical fluid–structure interaction phenomenon, shows significant consequences concerning structural safety and energy harvesting. Recently, flow past an elastic isolated cylinder has been comprehensively studied [1], [2], [3], [4], [5], [6], [7], [8], [9]. In comparison, studies on two or group cylinders are negligible, even though real structures are always arranged in groups or clusters. The flow-induced vibration and flow structures for group structures seem more complex than those for an isolated structure [10], [11], [12], [13], [14], [15], [16]. Currently, the two-cylinder structure has become the canonical model in the study of fundamental physics of FIV for multiple-cylinder structures [17,18].
Cylinders with square cross-sections are commonly seen in practical engineering, such as girders, pylons, high-rise buildings, and offshore structures. However, compared to two circular cylinders, research on the FIV of two square cylinders is critically insufficient [19]. It is worthwhile to introduce the flow around stationary tandem square cylinders, which can be beneficial in understanding the FIV behavior. For the two stationary square cylinders, the flow features are highly sensitive to the gap [[10], [11], [12],20,[24], [25], [26], [27], [28], [29]]. Tatsutani et al. [25] divided the flow regimes of two tandem square cylinders into three types: (i) the single slender-body regime (L/D < 1.5), (ii) the reattachment regime (1.5 < L/D < 4), and (iii) the co-shedding regime (L/D > 4). Sohankar and associates [26,27] confirmed the existence of these three different regimes for Re = 40–1000 by numerical simulation. Kumar et al. [30] numerically investigated the steady and separated flow past two tandem square cylinders at Re = 40 and L/D = 2–30. They identified four distinct flow regimes based on separation topology. Shui et al. [31] numerically investigated the flow around tandem square cylinders at Re = 100 and 1.5 ≤ L/D ≤ 9.0. They found six distinct flow regimes based on the flow structures, especially far downstream of the square cylinder. The two-layered vortex formation and secondary vortices formation modes were defined.
Presently, the study on the FIV for two tandem square cylinders has attracted much attention [15,[32], [33], [34], [35], [36], [37], [38]]. Kumar and Gowda [39] experimentally investigated the interference effects of a downstream square cylinder (DC) on the transverse vibrations of a spring-mounted upstream square cylinder (UC). They reported that there is a critical combination of relative size (b / D, b is the width of the downstream cylinder) and the gap that gives rise to the maximum amplitude of vibration. The tandem arrangement with L/D = 1.25 and b/D = 0.5 gives rise to the maximum amplitude of vibration with 0.57 D. More et al. [40] reported the flow around two square cylinders in tandem arrangements for L/D = 1.5–5 by applying wind tunnel test. The UC oscillated with amplitude A / D = 0.1 and frequency f/f0 = 0.5, 1.0, and 2.0, whereas the DC was fixed. Jaiman et al. [32] studied the free vibration of a downstream square cylinder for the fixed L/D = 4 and mass ratio m∗ = 5. They observed the initial and lower branches in the lock-in regime and the galloping branches beyond the lock-in regime. Bhatt and Alam [15] numerically studied a transversely vibrating square cylinder in the wake of a stationary cylinder for L/D = 2 or 6 and Re = 100 or 200. They reported that the gap flow can significantly influence the vibration response, leading to the absence of galloping at all values of Re and L/D. The initial branch was delayed by the gap flow effects, and an upper branch occurred only for Re = 200 and L/D = 6. Han et al. [34] performed simulations for Re = 40–200, a fixed gap L/D = 5, and m∗ = 2. It was observed that the resonance changed from single to dual and that the vortex-shedding modes within the lock-in regime changed from “2S” to “P + S” and “2T” when Re was increased to 160 or 200. They further evaluated two cylinders that were free to oscillate in the in-line and transverse directions [35] and observed that the effect of changes to the Reynolds number on the dynamic responses of the DC was amplified due to the upstream wake. Kumar and Sen [36] studied the flow-induced bidirectional vibrations of two tandem square cylinders with the same size and mass ratio (m* = 10) for Re = 100 and L/D = 5. They reported that the responses of the two cylinders included desynchronization regimes and a lower branch but no initial branch. Additionally, although the two cylinders had identical frequency characteristics, the vortex-induced vibration (VIV) of the DC was significantly different from its upstream counterpart. Qiu et al. [37] numerically studied the spacing effect on the VIV of two tandem square cylinders with two degrees of freedom at Re = 150 with L/D = 2–6 and m* = 10. They reported that for small spacing (L/D = 2, 2.5), UC had no traditional lock-in region, and DC had a narrow lock-in region. However, there was a wide soft-lock-in region at high reduced velocities (Ur ≈ 9.5–12) for both cylinders. For moderate spacings (L/D = 3, 4), the UC had no lock-in or soft-lock-in regions, while the DC exhibited a relatively traditional and wide lock-in region. For large spacings (L/D = 5, 6), UC had no significant soft-lock-in or lock-in regions, while DC exhibited only a highly narrow and traditional lock-in region. Later, they also investigated the effect of the mass ratio on vortex-induced vibration (VIV) of two tandem square cylinders [38]. They reported that the mass ratio also plays an important role in the VIV response of the two cylinders.
Beyond the studies above, most publications focus on two separated uncoupled cylinders, while few studies concentrate on two tandem cylinders that are rigidly coupled with each other. Nevertheless, a question arises whether the dynamic motion of two rigidly-coupled tandem cylinders behaves similarly to that of two separately oscillating ones or a single cylinder. Zhao [41] numerically investigated the FIV of two rigidly coupled identical circular cylinders at Re = 150. He reported that the lock-in region, influenced by the gap flow, became narrower at L/D = 1.5 and 2, while it became wider at L/D = 4 and 6 as compared to that of a single cylinder. Huera-Huarte and Jiménez-González [42] experimentally investigated the effect of the diameter ratio on the FIV of two rigidly coupled tandem cylinders with L/D = 1.3. The results showed that the vibration amplitude is reduced as the diameter ratio increased. However, the oscillation frequencies did not change significantly when the diameter ratio d/D ≤ 0.4. Zhu et al. [43] reported the results of a numerical investigation on the FIV of two rigidly coupled tandem cylinders with unequal diameters at Re = 150. The flow regime transition was not only closely related to the gap but also the reduced velocity. Ping et al. [44] conducted a two-dimensional numerical computation to study the FIV of two rigidly coupled cylinders of unequal diameters. The results were examined for Re= 250 and a fixed diameter ratio of d/D = 0.2. It was observed that the structural dynamics and hydrodynamic forces were strongly dependent on the positioning arrangements.
In summary, all studies on FIV of two tandem square cylinders can be divided into three categories: free vibration of a DC with a fixed UC; free vibration of the two cylinders; and free vibration of a UC with a fixed DC. The studies on two rigidly coupled cylinders are only concerned with circular-cylinder counterparts. Nevertheless, no publication has been published on the FIV of two rigidly coupled tandem cylinders with a square cross-section, although they are commonly available in engineering practices, such as the twin box girders or rigidly linked neighboring buildings and bridge towers. In this study, we focus on the FIV of two rigidly-coupled square cylinders in a tandem arrangement. In the course of the study, several issues can be raised: can we extrapolate the knowledge of the single-cylinder response for two cylinders, since two-coupled cylinders act as a single body? How does the response for two tandem cylinders behave in the two ranges of Reynolds number? How does the vibration response feature in the reattachment and co-shedding flow regimes? Does the UC or DC dominate the energy transfer between the fluids and cylinders? What does the dominant coherence mode look like, and which modes make the most contribution to the FIV?
In this study, the FIV of two rigidly coupled square cylinders in a tandem arrangement is investigated at Reynolds numbers 100 and 200. The distance between the cylinders is chosen as L/D = 2.0 and 6.0, lying in the reattachment and co-shedding flow regimes, respectively. The dynamic response, fluid force characteristics, and instantaneous flow structures are first studied. Mode-based energy transfer analysis method was developed based on proper orthogonal decomposition (POD). The energy transfer between the cylinders and the coherence modes was then analyzed to uncover the underlying mechanism of the FIV.
Section snippets
Numerical methodology and data process
The section starts with the description of the FIV of two rigidly coupled cylinders and the corresponding computational strategy. Subsequently, the mode-based energy transfer analysis method based on the multi-variable POD is introduced.
Verification and validation
In this section, verification and validation of the present results are performed. The mesh applied in this study is the same as our previous study including the stationary cylinders and forced vibration cases. The verification and validation can be found in our previous published paper [46]. Here, we also conduct the mesh independence test and comparison with other publications to further validate present FIV results. Table 1 displays the results of the test for the mesh independence for Re =
Results and discussion
In this section, results for the rigidly coupled tandem cylinders are presented at L/D = 2 and 6 for Re = 100 and 200, where the dynamic response, fluid force, flow structures, and the energy transfer between coherence modes and cylinders are analyzed.
Conclusions
Flow-induced vibration of two rigidly coupled square cylinders in a tandem arrangement was investigated at Reynolds numbers 100 and 200 for the gaps L = 2.0 D and 6.0 D, lying in the reattachment and co-shedding flow regimes, respectively. The dynamic response, fluid force characteristics, and instantaneous flow structures were systemically studied. Mode-based energy transfer analysis method was developed based on POD. The energy transfer between the cylinders and the coherence modes was then
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
CRediT authorship contribution statement
Hongfu Zhang: Validation, Writing – original draft, Writing – review & editing, Funding acquisition. Lei Zhou: Conceptualization, Validation, Writing – review & editing. Tim K.T. Tse: Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The support for this work by the National Natural Science Foundation of China (Grant No. 51908107), the Natural Science Foundation of Heilongjiang Province China (Grant No. LH2020E010), and the China Postdoctoral Science Foundation (Grant No. 2018M641791) is gratefully acknowledged. We also acknowledge Beijng PARATERA Tech Co., Ltd. (https://www. paratera.com/) for providing HPC resources that have contributed to the research results reported within this paper.
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