Dynamics of gap flow interference in a vibrating side-by-side arrangement of two circular cylinders at moderate Reynolds number
TThis draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics Dynamics of gap flow interference in avibrating side-by-side arrangement of twocircular cylinders at moderate Reynoldsnumber
B. Liu and R. K. Jaiman † Department of Mechanical Engineering, National University of Singapore, Singapore.(Received xx; revised xx; accepted xx)
In this work, the coupled dynamics of the gap flow and the vortex-induced vibration(VIV) on a side-by-side (SBS) arrangement of two circular cylinders is numericallyinvestigated at moderate Reynolds number 100 (cid:54) Re (cid:54) Key words: vortex-induced vibration, side-by-side arrangement, proximity interference,gap flow, near-wake instability † Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] J a n B. Liu and R. K. Jaiman
1. Introduction
The canonical side-by-side arrangements of circular cylinders are common and havea wide range of applications in various fields such as the offshore, the wind and theaerospace engineering. In addition to their great practical relevance in the engineeringapplications, a side-by-side system has a fundamental value due to the richness ofnonlinear flow physics associated with the near-wake dynamics and the vortex-to-vortexinteractions. There is a considerable difference between the flow dynamics of an isolatedcylinder and the multiple-cylinder arrangements. Many comprehensive investigations,e.g., Zdravkovich (1987); Sumner et al. (1999, 2000); Lin et al. (2002); Sumner (2010),were performed to understand and describe the mutual flow interference in the basiccanonical multi-body systems, in which the importance of the wake and proximityinterference was discussed. Among them, the flip-flop of gap flow in SBS arrangementshas attracted the attention among researchers. Different from the other fundamentalflow regimes in a two-dimensional laminar flow, the bi-stable character and chaos-likefluctuation of the flip flop have intrigued the research community over the past fewdecades. The flip flop was reported in many experimental works, e.g., Ishigai et al. (1972);Bearman & Wadcock (1973); Williamson (1985); Kim (1988). The flip flop was interpretedby Kim (1988) as a dynamical system with a bi-stable state, a deflected gap flow regime.Kim (1988) reported that the gap flow intermittently switched its direction at a timescale which was few orders of the magnitude greater than the shedding frequency ofthe primary vortices. Besides the aforementioned experimental works in literature, theflip flop was also observed within a narrow gap ratio range from 0.3 to 1.25 in a two-dimensional laminar flow from various numerical investigations in literature, e.g., Kang(2003); Agrawal et al. (2006). In the deflected gap-flow regime, the narrow near-wakeregion incorporates an enhanced vortex-wake interaction, which results in a higher vortexshedding frequency and mean drag coefficient value. As a result, the vortex sheddingfrequency of each cylinder dynamically changes with the gap-flow kinematics in timedomain.The origin of the flip flop was discussed by many authors. Alam & Sakamoto (2005)reported that a perfect symmetric structure geometry was a critical condition whichoriginated intermittent switching of the gap flow. However, the gap-flow flip flop wasalso observed in the asymmetric VSBS arrangements from Liu & Jaiman (2016). Ishigai et al. (1972) also considered the Coanda effect as the origin of the gap-flow flip flop.Nonetheless the flip flop was found in the near-wake region behind a pair of side-by-sideflat plates by Bearman & Wadcock (1973); Williamson (1985). Peschard & Le Gal (1996)modeled the dynamics of the deflected gap-flow regime through a system of two coupledLandau oscillators. The study illustrated that the stable deflected gap-flow regime andthe flip flop were formed by different mechanisms. Following the earlier studies, Carini et al. (2014) reported that the flip flop could be explained as a secondary instabilitythrough the coupling between Hopf bifurcation (the in-phase vortex synchronization) andthe pitchfork bifurcation (the deflected gap flow regime). This finding was subsequentlyvisualized by Liu & Jaiman (2016) in which the evolution of the flip flop from theinteraction of these two bifurcations were shown in a series of streamline plots as theReynolds number increased in the laminar flow regime. The exact instants of the flipflop and the instantaneous vortex shedding frequencies were visualized via the Hilbert-Huang Transform (HHT) technique by Huang (2014). Liu & Jaiman (2016) also reportedthat the flip flop was suppressed at the lock-in stage of VIV in VSBS arrangements, inwhich the time-averaged streamwise velocity profile of the gap flow became asymmetric.On the other hand, the lock-in range with respect to the reduced velocity became hree-dimensional gap-flow and VIV interference b ) reported that there weretwo types of instabilities in the flow transition, the mode-A and the mode-B. Themode-A instability was believed to be associated with the waviness of the primaryK´arm´an vortices induced by the elliptic instability. The counter-rotating streamwisevortices were formed in the high-strain region between the main spanwise vortices andsomewhat irregular with varying size, shape and spatial organizations. The conversionof the spanwise vorticity from the K´arm´an vortex cores into the streamwise vorticesis a result from the elliptic instability and was the central for the mode-A instabilityof bluff body wakes. Williamson (1996 b ) further mentioned that the onset of mode-Ainstability exhibited a hysteretic discontinuity of Strouhal number St - Re relationshipwith a spanwise correlation wavelength about 3 ∼ D . Mode A naturally triggereda vortex dislocation in the wake of a stationary isolated circular cylinder during thewake transition. Whereas the mode-B with a spanwise wavelength about unity diameterexperienced a non-hysteretic transition from the mode A. A relatively high sheddingfrequency occurred with more organized three-dimensional state of the mode-B and viceversa.In the context of three-dimensionality associated with the elliptic instability, thehyperbolic critical points were investigated by Kerswell (2002), Le Dizes & Laporte(2002) and Meunier et al. (2005). From the topological theory of separated flows, thetwo-dimensional streamline orbitals resemble hyperbolas around a hyperbolic criticalpoint, where its central velocity magnitude is zero and all eigenvalues of velocity gradienthave the nonzero real parts. The hyperbolic critical points in the fluid domain hadbeen previously reported as an unstable factor by Lifschitz & Hameiri (1991) andLeblanc (1997), where the maximal perturbation growth was found precisely aroundthese hyperbolic points near the vortex wake. One of the primary focus in the presentstudy is to interlink the characteristics between the near-wake instability, the vortex wakeinteraction and the fluid momentum. In spite of the above investigations, many aspectsof the proximity interference and the wake interference from the gap flow remain largelyunexplored.Here, we will present unsteady results from well-resolved numerical simulations of twocircular cylinders of SBS arrangements in 3D flow at moderate Reynolds numbers. Asanother step further, the primary focus is to explore the spanwise characteristics of thegap-flow and VIV kinematics at 3D flow through a systematic numerical analysis usingthe recently developed variational finite element solver for fluid-structure interaction(Jaiman et al. a , b ). Of particular interest is to answer the following questions:How do the VIV kinematics, the gap flow instability and the hydrodynamic responses B. Liu and R. K. Jaiman
Parameter Value Description l ∗ = L/D g ∗ = g/D . m ∗ = mρπD l
10 Mass ratio ζ = C πmf n .
01 Damping ratio U r = Uf n D Re = ρUDµ Table 1: Dimensionless parameters. Here l , g , D , m , ρ , f n , U and µ are respectively thespanwise distance, the gap distance, the cylinder diameter, the cylinder mass, thestructural frequency, the free-stream velocity and the dynamic viscosity.accommodate themselves in a 3D flow? How does the gap-flow kinematics influence the3D flow features? How does the spanwise correlation response to the cylinder’s kinematicsand the gap flow instability? In most engineering applications, multiple-body structuressubjected to the subtle proximity and flow interference are much more common. Thein-depth analysis of such complex nonlinear coupling is essential in various engineeringdesign and operations in which the SBS submerged arrangements are prevailing andsubjected to the influence from a 3D flow. In particular, the incorporation of VIV inthe investigation is crucial to reflect the practical aspects of structural motion and theinterference on the hydrodynamic forces.The manuscript is organized as follows. The numerical formulation, the problem setupand the verification are briefly presented in Section 2. Following that, the regulation effectof VIV kinematics on the three-dimensional feature is discussed in Section 3. We nextinvestigate the mutual interference between the 3D flow and the gap-flow kinematicsin SSBS arrangements in Section 4. The triple-coupling flow regime characteristicsof the VSBS arrangements are presented in Section 5. The primary focus is on theVIV lock-in phenomenon in VSBS arrangements for a range of reduced velocity atrepresentative gap ratios in the flip-flopping regime. We investigate the flow physicsof the flip-flop phenomenon and the VIV kinematics in terms of the wake topology, theresponse characteristics, the force components, the phase and frequency characteristics.The spanwise correlation and the three-dimensional modal analysis are discussed inSection 6 and Section 7, respectively. Finally, we present the concluding remarks inSection 8.
2. Numerical methodology
Formulation of coupled fluid-structure system
A Petrov-Galerkin finite element formulation is employed to investigate the fluid-structure interaction problem where the body interface is tracked accurately by thearbitrary Lagrangian-Eulerian technique. The traction and the velocity continuity con-ditions are imposed on the body-conforming fluid-solid interface via the non-linear hree-dimensional gap-flow and VIV interference Parameter Description St = f vs DU Strouhal number A maxy = √ A rmsy Dimensionless transverse displacement ∆φ = φ A y − φ C l Phase angle difference λ ∗ = λ/D Dimensionless spanwise wavelength λ = n n (cid:80) t i =1 λ m ( t i ) Averaged spanwise wavelength C l = F x ρU Dl Lift coefficient C d = F y ρU Dl Drag coefficient C z = F z ρU Dl Spanwise hydrodynamic coefficient C e = (cid:82) T C l v ∗ dτ Energy transfer coefficient
Table 2: Derived dimensionless quantities for post-processing. Here f vs , A rmsy , φ A y , φ C l , F x , F y , F z and v ∗ = v/U are respectively the vortex shedding frequency, theroot-mean-square transverse vibration amplitude, the instantaneous phase angle of A y ,the instantaneous phase angle of C l , the streamwise hydrodynamic force, the transversehydrodynamic force, the spanwise hydrodynamic force and dimensionless transversevelocity of a vibrating cylinder. λ m ( t i ) is the mean value of instantaneous spanwisewavelength along cylinder span at time t i and n is the number of sample locations in ashedding cycle.iterative force correction procedure (Jaiman et al. a , b ). The coupling scheme relieson a dynamic interface force sequence parameter to stabilize the coupled fluid-structuredynamics with strong inertial effects of incompressible flow on immersed solid bodies.The temporal discretizations of both the fluid and structural equations are formulated inthe variational generalized- α framework and the systems of linear equations are solvedvia the Generalized Minimal Residual (GMRES) solver (Jaiman et al. b ).The validation results of the two- and three-dimensional simulations are reported inLiu & Jaiman (2016), Mysa et al. (2016) and Li et al. (2016). These validation resultsand convergence investigations provide the validity and reliability of the fluid-structuresolver to simulate the gap flow and VIV interaction. The corresponding dimensionlesssimulation parameters and the post-processing quantities are listed in table 1 and table 2,respectively. 2.2. Dynamic mode decomposition
While the proper orthogonal decomposition (POD) modes may not necessarily providea description of a dynamically evolving flow driven by a momentum input, the dynamicmodal decomposition allows to extract the dominant spatial and temporal informationabout the flow (Schmid 2010). Therefore, we employ the dynamic modal decompositionto fit a discrete-time linear system to a set of snapshots from three-dimensional wake
B. Liu and R. K. Jaiman (a) (b)
Figure 1: Three-dimensional computational setup of SBS arrangements: (a) schematicdiagram of the fluid domain and the boundary conditions; (b) representativeunstructured mesh distribution in ( x, y )-plane at g ∗ = 0 .
8. Here
Cylinder1 is a freelytransverse vibrating cylinder and
Cylinder2 is stationary in the VSBS arrangements.
Spanwise resolution C meand C rmsl St∆z = 0 . ∆z = 0 .
15 1.196(0.8%) 0.358(2.5%) 0.2051(0.0%) ∆z = 0 .
075 1.195 0.349 0.2051
Table 3: Convergence of the global flow quantities at different spanwise meshresolutions for a stationary isolated circular cylinder at Re = 500 and l ∗ = 10 x - y plane element Number C meand C rmsl St × × × Table 4: Convergence of the global flow quantities at different x - y plane mesh for astationary isolated circular cylinder at Re = 500, ∆z = 0 .
15 and l ∗ = 10 hree-dimensional gap-flow and VIV interference α ) is computed through the best-fit betweenthe linearized modes and original snapshot data in a least-squares sense (Jovanovic et al. et al. et al. (2014), to analyze the near-wake stability and to decomposethe complex flow dynamics in the near-wake region behind the SBS system. Details ofthe SP-DMD formulation can be found in Jovanovic et al. (2014).2.3. Problem setup and verification
The problem setup for the 3D flow analysis is a spanwise extension of the two-dimensional setup implemented in Liu & Jaiman (2016), where the upstream distance,the downstream distance and the overall height of the fluid domain are respectively 50 D ,50 D and 100 D . A schematic diagram of the three-dimensional SBS arrangement is shownin figure 1a. The traction-free boundary conditions are respectively implemented alongthe domain boundaries Γ t , Γ b and Γ o . The top cylinder, Cylinder1 , is elastically-mountedin the transverse direction for the VSBS arrangements. The blockage ratio is taken as2 %. A pair of equal diameter D cylinders is placed in a three-dimensional hexahedrondomain, where a uniform free-stream flow with velocity U is along the streamwise x -axis,while the axis of the cylinder is along the spanwise z -axis. A representative ( x, y )-planesectional mesh configuration is exhibited in figure 1b. Based on the mesh convergenceanalysis in Liu & Jaiman (2016), the spatial discretization error is less than 1 % inthe ( x, y )-plane mesh. For the 3D flow at Re = 500, the x - y sectional mesh is furtherrefined, particularly the mesh within the boundary layer and the near-wake regions. Thedimensionless wall distance y + is kept less than one (within the viscous sublayer) for thefirst layer of the structural mesh around bluff bodies. The incremental ratio of elementsize from the boundary layer to the near-wake region and far field is less than 1 . × elementsand 120 × elements on each x - y section for the isolated cylinder cases and SBSarrangement cases, respectively.The spanwise length is taken as l ∗ = 10 D , based on the aspect ratio analysis inthe numerical simulations from Lei et al. (2001) and the experiments from Szepessy &Bearman (1992). A periodic boundary condition is employed at the ends of the cylinderspan to eliminate the end-plate effect. The mesh convergence study along the z -axis isshown in table 3. The spanwise resolution ∆z = 0 .
15 is chosen such that the spanwisespatial discretization error is controlled within 2 .
5% while maintaining the computationalefficiency for our parametric study. Furthermore, the x - y plane mesh convergence analysisin table 4 shows that the spatial discretization error is within 1% at chosen x - y planemesh resolution 81 × . Since the gap flow instability is the key concern of the present B. Liu and R. K. Jaiman C meand C rmsl S t Simulation Zhang et al. (1995) 1.44 0.68 0.216Persillon & Braza (1998) 1.366 0.477 0.206Behara & Mittal (2010) 1.390 0.594 0.210Present 1.26 0.5 0.205Experiment Wieselsberger (1921) 1.208 - -Williamson (1996 a ) - - 0.203 Table 5: Comparison of numerical and experimental results for a stationary isolatedcircular cylinder at Re = 300, where C meand is the mean drag coefficient, C rmsl is theroot-mean-square of lift coefficient fluctuation and St is the Strouhal number. Re U r Simulation(Blackburn et al. et al.
Table 6: Validation of transverse amplitude A maxy for a freely transverse vibratingcylinder in three-dimensional flow at m ∗ = 5 .
08 and ζ = 0 . g ∗ = 0 . g ∗ = 1 .
0. However, the investigations on the boundary circumstances,e.g., around g ∗ ≈ . g ∗ ≈ . et al. (2016 b ). The L norm error was reported at about 1% at a constant time step ∆t =0 .
05. Except stated otherwise, all positions and length scales are normalized by thecylinder diameter D , velocities with the free stream velocity U , and frequencies with U/D . To validate the numerical formulation in a three-dimensional flow, a comparison of astationary isolated circular cylinder at Re = 300 are presented in table 5. The comparisonof the overall VIV response with previous results from Blackburn et al. (2001) are shownin table 6. The results are in close agreement with the previous studies, and thus thecomputational setup is adequate for our investigation. A total of eighty-five simulationsis performed in the present investigation, comprising seven simulations for the validationof the three-dimensional FSI solvers, fifty-two cases for the principal investigation ofthe isolated, SSBS and VSBS arrangements; and twenty-six two-dimensional cases toinvestigate the relationship between the near-wake instability, the fluid shearing ratioand the fluid momentum. By taking into the consideration of large number of three-dimensional simulations and the involved computational resources, the selected timewindow is constrained at tU/D ∈ [250, 350], in which the fluid flow is already fully-developed for the extraction of flow statistics. In the selected time window, all fluid hree-dimensional gap-flow and VIV interference C z for an isolated cylinder at Re = 500, m ∗ = 10, ζ = 0 .
01 and U r ∈ [0, 7]. The VIV kinematics possesses a regulation effect tothe spanwise force fluctuation, whereby it is suppressed at the peak lock-in stage U r = 5 (a) (b)(c) (d) Figure 3: Instantaneous vortical structures using the Q-criterion for an isolated cylinderat Re = 500, Q = 0 . ω y = ± contours ) and tU/D = 300: (a) stationary; U r = (b) 3;(c) 5; and (d) 7 at m ∗ = 10 and ζ = 0 .
01 for a freely transverse vibrating cylinder. Thestreamwise vorticity clusters vanish at the peak lock-in stage U r = 5.features such as the hydrodynamic responses and the vibration amplitude, undergo atleast twenty cycles. In particular, we are interested in the behaviour of the flip flopsubjected to the influence of VIV and three-dimensionality within a short time window.0 B. Liu and R. K. Jaiman
3. Vortex-induced vibration in three-dimensional flow
Before we proceed to further investigation on the complex coupling between the 3Dflow, the VIV and the gap-flow kinematics in the SBS arrangements, we systematicallyexamine the interference of the VIV on the 3D flow dynamics. Similar to the work ofBorazjani & Sotiropoulos (2009), the spanwise hydrodynamic coefficient C z is consideredto quantify the overall spanwise fluctuation of the hydrodynamic forces induced fromthe 3D flow. Owing to the symmetrically-imposed periodic end-wall boundary conditionsalong the cylinder span, the magnitude of C z is nearly two orders of magnitudes smallerthan C d and C l . This small value of the spanwise force arises from the intrinsic 3D flowdynamics along the cylinder, and is not related to any numerical errors. In figure 2,the fluctuation of C z is found negligible at the lock-in stage U r = 5, in contrastto its counterpart at the off-peak stage. Such weakening effect of the spanwise forcesuggests that there exists a particular regulation mechanism which causes a recoveryof two-dimensional (2D) hydrodynamic responses along the cylinder. This regulation orstabilization effect is further visualized by the iso-surfaces of the vortical structures usinga vortex-identification based on Q -criterion (Hunt et al. x, y )-sectionalsnapshots at l ∗ = 5 D in the near-wake regions at Re = 500 and Re = 800. The intensestreamwise vortex rollers are formed along the interface of the counter-signed vortexseparating layers and across the saddle-point regions. This observation conforms with thetopological description of the turbulent flow pattern from Perry & Chong (1987). Basedon the definition of streamline saddle point (SSP), Appendix A shows that the inflectionvelocity profiles are indeed in the saddle-point region. These inflection velocity profiles aresubsequently visualized in figures 5 (a,b) and (c,d) at t = 3 / T when the SSP appears atthe investigated location, (1 . D , − . D , 5 D ). In particular, a high streamwise vorticityconcentration of the same sign appears on both sides of the saddle point along the z -axisin figure 5e, which conforms with the observation from Zhou & Antonia (1994), wherebythe streamwise vortical structures are inclined and crossed approximately at the saddle-point region on the ( x, y )-plane. Hence the presence of the high-strain rates reflects thesignificance of the two-dimensional SSP to the three-dimensionality of this flow.Following that, a stability analysis using the DMD technique is performed in a saddle-point region at low Reynolds number in Appendix B. The near-wake instability aroundthe saddle-point is found to be dependent upon the intensity of the fluid momentum andthe vorticity concentration difference of adjacent voritcity clusters. Recently, Huang et al. (2010) also reported that a planar shear flow could enhance the three-dimensionality inthe wake behind a circular cylinder. † In the present investigation, we further analyze therelationship between the near-wake instability and the resultant imbalanced vortex-to-vortex interaction from the planar shear flow. The fluid shearing is known to be critical tothe Kelvin-Helmholtz instability. As the Reynolds number increases, the fluid instability † Here a planar shear flow refers to an inflow with a constant velocity gradient along y -axisin the present problem configuration. hree-dimensional gap-flow and VIV interference (a) (b)(c) (d) Figure 4: Illustration of saddle-point regions along the interface of imbalancedcounter-signed vorticity clusters for a stationary cylinder at tU/D = 300 and l ∗ = 5 for Re = (a,b) 500; (c,d) 800 whereby ω x = ± . contours , ω z = ± . the solid and dashlines in (a, c) and sectional streamlines in (b, d)can develop around the inflection points along a velocity profile which have a directconnection with the fluid shear stresses. Since significant shear stresses are observedon the interface of the imbalanced counter-signed vorticity clusters, the interactionbetween different vorticity clusters is believed to be crucial to the near-wake instability.A supportive example could be the saddle-point at the tip of the formation region behinda stationary isolated circular cylinder at Re (cid:46)
48, where the symmetric counter-signedcirculations are interacting and no instability is observed. It is when the perturbationapproaches to the brink of a critical value, the perturbation becomes non-negligibleand induces the need for the extra dimension to quantify itself e.g., the introductionof new dimension in the Hopf bifurcation and the flow transition from the laminar flowto the chaotic turbulent flow. These two factors associated with the near-wake instabilityfacilitate the understanding of the proximity interference induced from the gap flowbehaviour in a three-dimensional flow in Section 4.To link the observations on the saddle-point regions to the recovery of two-dimensionalhydrodynamic responses, a series of streamline-contour plots of a locked-in cylinder are in-vestigated for one primary vortex shedding cycle in figure 6. Based on the discussion abouta two-dimensional streamline saddle point in Appendix A, the saddle-point generates alocal stagnant region which inhibits the transfer of kinetic energy from the mean flow. Thesaddle-point moves with the kinematics of the separating shear layer along the interface2
B. Liu and R. K. Jaiman (a) (b)(c) (d)(e) (f)
Figure 5: Instantaneous velocity profiles across a streamline saddle-point at(1.33D,-0.72D, 5D) for a stationary isolated circular cylinder at Re = 500: (a,b) u vs. y from (1.33D, -0.22D to -1.22D, 5D), u ∈ [ − .
5, 0 .
5] contour; (c,d) v vs. x from (0.83Dto 1.83D, -0.72D, 5D), v ∈ [ − .
5, 0 .
5] contour; (e,f) w vs z from (1.33D, -0.72D, 4.5D to5.5D), w ∈ [ − .
25, 0 .
25] contour. T is one period of primary vortex shedding cycle. Theinflection points are in the saddle-point region. hree-dimensional gap-flow and VIV interference (a) (b)(c) (d)(e) (f) Figure 6: Flow contours and sectional streamline topology at ( x, y )-plane for atransversely vibrating isolated cylinder at the peak lock-in stage in one shedding cycleof the primary vortex (anti-phase): Re = 500, m ∗ = 10, ζ = 0 . U r = 5, l ∗ = 5, ω z = ± . contours in (a,c,e) , and velocity amplitude | U | = √ U + V ∈ [0 , . contours in (b,d,f ) B. Liu and R. K. Jaiman (a) (b)
Figure 7: Hydrodynamic forces as a function of the reduced velocity for atransversely-vibrating isolated cylinder at m ∗ = 10 and ζ = 0 .
01: (a) the mean dragcoefficient C meand with respect to the reduced velocity U r ; (b) the r.m.s. lift coefficient C rmsl with respect to the reduced velocity U r .and represents a communication barrier. As the vortex wake reaches its maximum growth,it breaks up and sheds downstream. The strict periodic motion of the cylinder at the peaklock-in stage also generated a well segregated vortex wakes downstream with relativelybenign interactions from the vortex wakes † . This vortex shedding mode inhibits theformation of the fluid shearings along the interface of vortex wakes. Consequently, the SSPvanishes and the corresponding streamwise vorticity concentration weakens downstream.A direct consequence is a recovery of the two-dimensional hydrodynamic response alongthe cylinder. The intense turbulent flow at the off lock-in stage results into a smaller C meand value than its laminar flow counterpart in figure 7, where C meand is over-predictedby the relatively large two-dimensional vortex wakes at the lock-in. Overall, the responseof C l shows an increment in the transverse fluctuating lift force and an earlier onset of theVIV lock-in. As reported by Liu & Jaiman (2016) for the similar problem setups withtwo-dimensional laminar flow, the earlier onset of VIV is attributed to the enhancedvortex interaction which leads to higher vortex shedding frequency. Both C d and C l at Re = 500 show respectively about 6% and 22% amplification compared to theirlaminar counterparts from Liu & Jaiman (2016) at the peak lock-in. These results indicatethat the VIV regulation effect has a profound influence to the transverse hydrodynamicresponse, compared with its streamwise one. A similar phenomenon was observed by Zhao et al. (2014), in which the spanwise correlations were discussed at the VIV lock-in stageand a uniformity of C l was observed along the span of a rigid circular cylinder. Here,the primary focus is to understand the complex near-wake flow physics in the SBSarrangements.
4. Three-dimensional flow interference to the gap flow
In this section, the relationship between the gap-flow proximity and the near-wakeinstability is discussed for the cylinders with SBS arrangements. An incorporation of theinterference from the VIV kinematics is important to analyze the practical applications † The focus is to investigate the influence of a saddle-point region to the vortex sheddingprocess. Hence the vortex shedding modes are not discussed here. hree-dimensional gap-flow and VIV interference (a) (b)(c) (d) Figure 8: Time series of the hydrodynamic forces for SSBS arrangement at g ∗ = 0 . Re = 100, the flip-flop are marked at tU/D = 270, 300 and 330; (b,d) Re = 500Figure 9: Comparison of the spanwise hydrodynamic force C z between a stationarycylinder and SSBS arrangement at Re = 500 and g ∗ = 0 .
8. When the gap flow deflectsto
Cylinder2 for tU/D ∈ [250, 350], C z amplitude ( the solid-dot star ) is amplified.6 B. Liu and R. K. Jaiman (a) (b)
Figure 10: Instantaneous vortical structures using the Q-criterion at Re = 500, tU/D = 300, Q = 0 . ω y = ± contours ): (a) stationary cylinder; (b) SSBSarrangement at g ∗ = 0 .
8. The streamwise vorticity concentration is higher in a narrownear wake regionFigure 11: Horizontal velocity profile of the gap flow for the SSBS arrangements at Re = 100 and Re = 500. The velocity profiles are extracted at the fluid jet location(0 . D , 0 . D to − . D ) where (0 D , 0 D ) is the center between the cylinders. Thetime-averaging is performed from tU/D ∈ [250, 350].and operations in side-by-side systems. So far the three-dimensional numerical investiga-tion of the gap flow instability in the SBS arrangements is rarely documented. Hence, thepresent investigation on the VSBS arrangements in a 3D flow is deemed as another stepfurther to understand the gap flow and the VIV kinematics. From a systematic analysisviewpoint, it is desirable to first focus merely on the interaction between the gap-flowkinematics and the 3D flow by eliminating the motion of the structure.The flip flop was frequently described as an intermittent deflection of the gap flow. Asreported by Liu & Jaiman (2016), the switch-over of C meand from each cylinder in theSSBS arrangement could indicate the direction of gap-flow deflection. Since the vortex-to-vortex interaction is enhanced in the narrow near-wake region, the corresponding f vs value is higher than its wide near-wake-region counterpart. To investigate the character-istics related to gap flow features without and with the presence of 3D effects, figure 8is plotted. In the two-dimensional laminar flow, figures 8a and 8c, f vs of the cylinderwith the narrow near-wake region is observed possessing a larger value. However, this hree-dimensional gap-flow and VIV interference (a) (b) Figure 12: Spanwise vorticity ω z contours of cylinders in SSBS arrangement at Re = 500, g ∗ = 0 . l ∗ = 5 and ω z = ± . contours ): (a) tU/D = 303; (b) tU/D = 317.Large interfaces of different vorticity concentrations are observed in the narrownear-wake regions (a) (b)(c) (d) Figure 13: Streamline and contours of the streamwise vorticity ω x and the spanwisevorticity ω z of cylinders in SSBS arrangement at ( x, y )-plane at Re = 500, g ∗ = 1 . tU/D = 300, ω x = ± . (contours) , ω z = ± . (solid-dash lines) in (a,c) and sectionalstreamlines in (b,d): (a,b) l ∗ = 4; (c,d) l ∗ = 88 B. Liu and R. K. Jaiman (a)(b)
Figure 14: ω x contours in ( y, z )-plane for the cylinders in SSBS arrangement at Re = 500, g ∗ = 0 . ω x = ± . x/D = 1: (a) tU/D = 175, the gap flowmomentarily deflects to Cylinder1 (top section); (b) tU/D = 350 the gap flowmomentarily deflects to
Cylinder2 (bottom section). (a) (b)
Figure 15: Hydrodynamic forces as a function of the gap ratio: (a) mean drag coefficientwith respect to the gap ratio, (b) r.m.s. lift coefficient with respect to the gap ratio.The subscripts t , n and w denote respectively the total, the narrow and the widenear-wake regions. C meand and C rmsl are higher along the cylinder with a narrownear-wake region, where a force modulation is observed. hree-dimensional gap-flow and VIV interference (a) (b)(c) (d) Figure 16: Time series of the drag coefficient of cylinders in the VSBS arrangementswhere
Cylinder1 vibrates in the cross-flow direction at Re = 500, g ∗ = 0 . m ∗ = 10and ζ = 0 . U r = (a) 3; (b) 4; (c) 6 and (d) 8tendency is not confirmed in its three-dimensional counterpart, as shown in figure 8b andfigure 8d. The figures show that there is no significant difference among the mean vortex-shedding frequency of two cylinders for the 3D flow, although the gap flow deflects. Inaddition, the flip flop is not observed in the selected time window tU/D ∈ [250, 350]in figure 8d, since f flip is remarkably low for the 3D flow. A comparison of C z betweenthe SSBS arrangement at g ∗ = 0 . Re = 500is plotted in figure 9. The fluctuation of C z is evidently amplified by a factor of 1 . Cylinder2 at tU/D ∈ [250, 350],compared with its isolated and wide counterparts respectively. This amplification factoris practically constant in all other SSBS arrangements at g ∗ ∈ [0 .
6, 1 . B. Liu and R. K. Jaiman
Figure 17: Time-averaged maximum transverse vibration amplitude as a function of thereduced velocity in the VSBS arrangements at Re = 100 and 500, m ∗ = 10 and ζ = 0 . Cylinder1 vibrates in the transverse direction. The onset of the VIV lock-inoccurs earlier at a smaller U r value for cases at Re = 500flow promotes additional instability in the near-wake region. Considering the two criticalfactors identified in Section 3, we also observe another unstable factor in the narrownear-wake region, large adjoining interfaces of primary vorticity clusters, as shown infigure 12. Remarkable shear stresses are present along these adjoining interfaces andresult in a significant streamwise vorticity concentration formed in the narrow near-wakeregion in figure 13. The locations with intensified streamwise vorticity clusters follow wellalong these interfaces in the near-wake region, and confirms the observation about SSPin Section 3, where the SSP lies right at the point with significant streamwise vorticityconcentration along the interface of primary vorticity clusters. A more straightforwardvisualization is exhibited by ( y, z )-sectional contour plots of ω x in figure 14. Figures 14aand 14b refer to the ω x contour plots of the SSBS arrangement, where the gap flowdeflects to Cylinder1 and
Cylinder2 at tU/D = 175 and tU/D = 350 respectively. Twodistinct streamwise vorticity distributions were evidently shown in the narrow and widenear-wake regions.This asymmetric distribution of streamwise vorticity has a profound influence to thehydrodynamic response. Here figure 15a shows distinctively higher and lower C meand values for the cylinders with a narrow and wide near-wake regions, respectively. Fur-thermore the algebraic sum of C meand shows a base-bleeding type effect, which is well-documented for the SSBS arrangements in the literature, e.g., Bearman & Wadcock(1973). Hence the overall response of C d is diminished. However, this base-bleeding effectis weakened as the value of g ∗ increases beyond the deflected gap-flow regime. To analyzethe transverse hydrodynamic response, C rmsl is adopted to indicate the fluctuating extentof C l as a function of the gap ratio g ∗ in figure 15b. Here C rmsl represents the fluctuationintensity (absolute value) of transverse force and is measured from the C meanl valuebetween a time interval, when the gap flow stably deflects to one particular side of theSSBS arrangements. It should be noted that the in-phase and anti-phase of C l fromboth cylinders have to be taken into account, when taking the measurement of theresultant transverse force fluctuation for SBS arrangements. Similar to figure 15a, adrastic transverse fluctuation of the lift appears along the cylinder with the narrow near-wake region. The overall transverse fluctuation of C l is calculated as a sum of C l from hree-dimensional gap-flow and VIV interference (a) (b)(c) (d) Figure 18: Time traces of the spanwise hydrodynamic forces of the cylinders in theVSBS arrangements at Re = 500, g ∗ = 0 . m ∗ = 10 and ζ = 0 . U r = (a) 3.5, (b) 4,(c) 5, (d) 6. A recovery of two-dimensional hydrodynamic responses is observed alongthe locked-in cylinder.each cylinder, the solid circle in figure 15b. A force modulation is clearly shown at thedeflected gap flow regime, g ∗ ∈ [0 .
8, 1 . C l is excited by a factor of 2 .
4. Since the gap flow is significantly suppressed at g ∗ (cid:46) . C l is much benign.On the other hand, albeit C rmsl along individual cylinder is drastically amplified beyond g ∗ (cid:38) .
5, the overall value of the entire structure system is diminished and canceled outinstead, due to the dominant anti-phase vortex shedding regime at these gap ratios. Theabove observations show that the gap-flow instability is critical to the overall stabilityof the SBS systems in engineering operations with a relatively small gap ratio, where astrong force modulation is observed.It is observed that the 3D flow not only modulates the hydrodynamic responses,but also the frequency f flip value. Here the f flip value appears to be small in three-dimensional flow, compared with its two-dimensional laminar flow counterparts. Liu& Jaiman (2016) visualized the flip-flopping instant as a zero phase angle differencebetween C l in the SSBS arrangements. Different from a two-dimensional laminar flow,the existence of the streamwise vorticity clusters in the formation region varies f vs along2 B. Liu and R. K. Jaiman (a) (b)
Figure 19: Recovery of two-dimensional hydrodynamic responses of a freely-vibratingcylinder at the peak lock-in: Re = 800, m ∗ = 10, ζ = 0 .
01 and U r = 4 .
8: (a) C z at tU/D ∈ [250, 350], where two dimensional hydrodynamic responses are observed alongthe cylinder span; (b) iso-surfaces using the Q-Criterion at tU/D = 300, Q = 0 . ω y = ± contours ).the cylinder span and results in a repetitive temporal modulation of the f vs values alongthe span. To completely flip over the gap-flow direction, approximately half or one in-phase shedding of a primary vortex is necessary. However, due to the complex non-linearnature of the flow, this modulation of f vs on each cylinder is chaotic, swift and highlyunstable. Consequently, f flip value is significantly influenced in a three-dimensional flow.
5. Coupling of VIV and gap-flow kinematics
In this section, the VIV kinematics and the gap flow instability are coupled in theVSBS arrangements, where
Cylinder1 is elastically-mounted in the transverse direction.Similar to the SSBS arrangements, the f flip is barely observed on the VSBS arrangementat the off-lock-in stage in the selected time window in figure 16. On the contrary, the f flip values at the lock-in stage in figure 16c surge remarkably. Based on the discussion fromLiu & Jaiman (2016), f flip is related to the Reynolds number and gap ratio. Here f flip isalso found susceptible to the influence of three-dimensionality. As the three-dimensionalvortical structure is subjected to the modulation from the VIV kinematics, f flip becomesVIV-dependent. Some phenomena reported by Liu & Jaiman (2016) from the VSBSarrangements are also observed in a three-dimensional flow. For instance, a quasi-stabledeflected gap flow regime occurs at the peak lock-in stage, where the gap flow permanentlydeflects toward the locked-in vibrating cylinder. The onset of VIV occurs at a lower U r value, as shown in figure 17, owing to the enhanced vortex-to-vortex interaction in thenarrow near-wake region. The aforementioned recovery of two-dimensional hydrodynamicresponses is also found along the locked-in vibrating cylinder in the VSBS arrangementsin figure 18. It is worth noting that the amplitude and the frequency of C z are relativelyinsensitive to the variation of U r values, except the C z suppression at the peak lock-instage. Figure 18b shows a trivial C z oscillation at the peak lock-in stage, compared withthe aforementioned perfect C z suppression along isolated cylinder at present order ofmagnitude ( × − ) in Section 3. It means that a perfect two-dimensional recovery maynot be feasible when the fluid instability is intensified, e.g., proximity interference from hree-dimensional gap-flow and VIV interference (a) (b)(c) (d) Figure 20: Streamline and contours of the streamwise vorticity ω x and the spanwisevorticity ω z of the cylinders in VSBS arrangement at ( x, y )-plane for Re = 500, g ∗ = 0 . m ∗ = 10, ζ = 0 . U r = 4 . (the peak lock-in stage) , ω x = ± . (colourcontours) , ω z = ± . (solid-dash) in (a, c), and streamline in (b, d): (a,b) l ∗ = 4; (c,d) l ∗ = 8a strong gap-flow jet or higher Reynolds number. To investigate the Reynolds numbereffect alone, the Reynolds number is increased slightly from 500 to 800 and the proximityinterference is eliminated by considering very large gap ratio, i.e. a stationary isolatedcylinder. Figure 19a shows that the C z is completely suppressed along an isolated cylinderat its peak lock-in stage ( U r = 4 . solid cross ) at Re = 800. Nonetheless, streamwisevorticity ribs are still prominently visible further downstream in figure 19b. This evidencenot only supports that an intensified fluid momentum is detrimental to the stability offluid, but also indicates that the streamwise vorticity clusters are originated from thevortex-to-vortex interaction, in which the fluid shearing is prominent along the vorticalinterfaces. This observation is supported by the discussion from Chantry et al. (2016), inwhich the fluid shearing alone was reported to be essential to the flow transition, insteadof the boundary layers from the walls. As a result, the corresponding sectional contourplots of aforementioned VSBS arrangement show a distinctive concentration differenceof streamwise vorticity from the narrow and wide near-wake regions in figure 20 on the( x, y )-plane and figure 21 on the ( y, z )-plane.The VSBS arrangements at two typical gap ratios g ∗ = 0 . . B. Liu and R. K. Jaiman (a)(b)
Figure 21: ω x contours in ( y, z )-plane for the cylinders in VSBS arrangement at Re = 500, g ∗ = 0 . m ∗ = 10 . ζ = 0 . U r = 4 (the peak lock-in sage) , ω x = ± . (contours) and tU/D = 350: (a) x ∗ = 1 .
25; (b) x ∗ = 1 . U r value in figure 17. Albeitthe onsets of the VIV lock-in at various g ∗ values are different, the ends of their VIVlock-in approximately occur at an identical U r value for the VSBS arrangements. Thecorresponding time-averaged phase angle difference ∆φ between A y and C l is plottedin figure 22b. The mean ∆φ is at about 110 ◦ from 6 . (cid:46) U r (cid:46) .
5, which represents arelative equilibrium state where the energy transfer between the fluid and the structureis balanced. In addition, a gradient discontinuity is observed at about U r ≈ .
0, whichstabilizes the VIV lock-in stage. A similar discontinuity of phase angle difference wasalso reported by Leontini et al. (2006) at VIV lock-in stage, which is correlated to theVIV kinematics and the vortex wakes. Following that, as the U r value exceeds 8 .
0, the ∆φ value becomes completely anti-phase. The profile of ∆φ for an isolated cylinder ina three-dimensional flow ( dash square ) shows a similar profile to its two-dimensionalcounterpart ( solid circle ). However its amplitude is excited at a smaller U r value.Generally speaking, in the VSBS arrangements, as long as two cylinders are sufficientlyclose, the gap-flow proximity interference becomes remarkable and the corresponding ∆φ values are more close to 90 ◦ ( the equilibrium state ). This can be seen from the relativepositions of the ∆φ profiles ( dash-square, solid-star and dash-cross ) in figure 22b. Thisgap-flow proximity interference becomes very prominent at the peak lock-in stage, wherethe two cylinders are very close, because of the extensive VIV motion. Besides ∆φ , theenergy transfer coefficient C e indicates not only the energy flow direction, but also thethe amount of work done during the energy transfer, which is defined in table 2. For hree-dimensional gap-flow and VIV interference (a) (b)(c) Figure 22: Frequency and energy transfer analysis at Re ∈ [100 , m ∗ = 10 and ζ = 0 .
01: (a) frequency ratio f /f n as a function of the reduced velocity at Re = 500;(b) phase angle ∆φ between A y and C l as a function of the reduced velocity; (c)averaged energy transfer in transverse direction for one primary vortex shedding cycle. C e , v , τ = tU/D and T respectively are the transverse velocity of a vibrating cylinder,dimensionless time scale and one (dimensionless) shedding cycle of primary vortex. Themagnitude of C e quantifies the energy transfer between the fluid flow and the structure.Its sign indicates the direction of energy transfer. For instance, a positive value means theenergy is transferred from the fluid flow to structure, which corresponds to 0 ◦ (cid:46) ∆φ (cid:46) ◦ (in-phase). The C e value, the work done between the fluid and the structure, is averagedand computed over the duration of one primary vortex shedding cycle, as shown infigure 22c. The aforementioned earlier onset of VIV lock in is also confirmed for higherReynolds number and VSBS arrangement cases, U r ≈
4. Over the entire U r range of lock-in, the direction or the sign change of energy transfer is observed for both isolated andthe VSBS arrangement cases. This sign change of energy transfer is associated with theaforementioned phase discontinuity observed by Leontini et al. (2006) and in figure 22bof the present investigation. Although this phase discontinuity is relatively small for theisolated cylinder at lower Reynolds number Re = 100, it is still evidently observed in thezoomed-in plot ( red line at U r = 7) of figure 22c. Furthermore, the amount of energy6 B. Liu and R. K. Jaiman transferred over off-lock-in U r ranges is trivial compared to their lock-in counterparts.To analyze the relationship among the near-wake instability, the gap flow and the VIVkinematics from another point of view, the spanwise wavelength λ ∗ is discussed in thenext section. The next section will show that the magnitudes of the λ ∗ and the ω x renderexcellent means to analyze the 3D vortical structures.
6. Interference to the spanwise correlation
To quantitatively describe the interference of the VIV and the gap-flow kinemat-ics on the three-dimensional vortical structure, the dimensionless spanwise correlationwavelength λ ∗ is discussed herein. The wavelength λ ∗ is a good representation of thenumber of the streamwise vortex pair formed along the cylinder span. Williamson (1996 b )documented that the mode-A was typically associated with λ ∗ ∈ (3, 4) and the mode-B which supersedes the mode-A after the flow transition possess one wavelength. In thepresent investigation, the measurement of λ ∗ is undertaken along the cylinder span, sincethe aforementioned two-dimensional recovery and the gap-flow proximity interferencecould significantly change λ ∗ value in the near-wake. Based on the proposed formulaof spanwise wavelength in table 2, the observed λ ∗ value of the mode-B in figure 23aconforms well with the results reported by Williamson (1996 b ).For the isolated cylinders in figure 23, the values of λ ∗ at the off-lock-in stage aresimilar, except the lock-in stage where the wavelength λ ∗ value gradually strides over theentire cylinder span until the peak lock-in stage, whereby the magnitude of ω x is reducedby three orders of magnitudes. The λ ∗ value at the peak lock-in stage quantitativelyexhibits the aforementioned recovery of two-dimensional hydrodynamic responses along acircular cylinder via the VIV kinematics in Section 3. On the other hand, the wavelength λ ∗ value is also subjected to the influence from the gap-flow kinematics in the SBSarrangements. To investigate further, a pair of cylinders in the SSBS arrangement at Re = 500 and g ∗ = 0 . tU/D = 175 when the gap flow deflects to Cylinder1 , and figures 24c and 24dshow that the gap flow deflects to
Cylinder2 at tU/D = 300. The wavelength λ ∗ valueis noticed to be relatively large λ ∗ ≈ .
33 along the cylinder with a narrow near-wakeregion in the SSBS arrangement at g ∗ = 0 .
8. A similar tendency of the λ ∗ augmentationis also observed for all other SSBS arrangements with a deflected gap flow. In the VSBSarrangements, the primary focus is on the quasi-stable gap flow regime. Hence the iso-surfaces of ω x for the VSBS arrangement at the peak lock-in stage are taken as an exampleand visualized in figure 25 for analysis. Similar to the SSBS arrangements, the λ ∗ valueincreases along the locked-in vibrating cylinder to which the gap flow permanently deflectsin figure 25. The scaling factor of λ ∗ , approximately 3.33, is similar to the aforementionedSSBS arrangement at an identical g ∗ value. Furthermore the wavelength λ ∗ value of itsstationary counterpart in figure 25b is very close to that of an isolated cylinder case λ ∗ ≈ g ∗ decreases, the gap-flow proximity interference becomes intense. As a result, the λ ∗ value increases along the cylinder with a narrow near-wake region. On the contrary,the λ ∗ value is relatively small in the wide near-wake region, because of a weakenedvortex-to-vortex interaction. hree-dimensional gap-flow and VIV interference (a) stationary , ω x = ± . U r = 3 . ω x = ± . U r = 4, ω x = ± . U r = 6, ω x = ± .
02 (e) U r = 8, ω x = ± . Figure 23: Contours of ω x for an isolated cylinder at tU/D = 200: (a) stationarycylinder at Re = 500 and ω x ∈ [ ± . , ± .
2] with λ ∗ ≈ Re = 500, m ∗ = 10, ζ = 0 .
01 and ω x ∈ [ ± . , ± . λ ∗ value significantly enlarges at the peak lock-in stage.8 B. Liu and R. K. Jaiman (a) tU/D = 175,
Cylinder1 (b) tU/D = 175,
Cylinder2 (c) tU/D = 300,
Cylinder1 (d) tU/D = 300,
Cylinder2
Figure 24: Contours of ω x in SSBS arrangement: the dimensionless spanwise wavelengthlength at Re = 500, g ∗ = 0 . ω x = ± .
2. The wavelength λ ∗ increases in the SSBSarrangements. (a) tU/D = 325, Cylinder1 (b) tU/D = 325,
Cylinder2
Figure 25: Contours of ω x and the dimensionless spanwise wavelength length in VSBSarrangement at Re = 500, g ∗ = 0 . U r = 4 . ω x = ± . hree-dimensional gap-flow and VIV interference α on frequency at Re = 500 for stationarycylinder, SSBS arrangement at g ∗ = 0 .
8, and VSBS arrangement at g ∗ = 0 . m ∗ = 10, ζ = 0 .
01 and U r = 4. Here α value is the optimal amplitude of each DMD mode andobtained from an optimization process in the sparsity-promoting DMD analysis. (a) fD/U = 0, α = 97 (b) fD/U = 0 . α = 39(c) fD/U = 0 . α = 31 (d) fD/U = 0 . α = 32 Figure 27: Iso-surface plots of the primary vortex modes of a stationary cylinder at Re = 500, tU/D ∈ [250 , ω z = ± .
01. A strong third-order harmonic vortexmode is decomposed at f D/U ≈ .
7. Three-dimensional modal analysis
The three-dimensional wakes behind the multi-body systems at even a moderateReynolds number can exhibit complex temporal and spatial flow features. A modalanalysis has become a common practice to decompose physical important features or0
B. Liu and R. K. Jaiman (a) fD/U = 0, α = 206 (b) fD/U = 0 . α = 56(c) fD/U = 0 . α = 57 (d) fD/U = 0 . α = 22 Figure 28: Iso-surface plots of the primary vortex modes of SSBS arrangement at Re = 500, g ∗ = 0 . tU/D ∈ [250 , ω z = ± .
01. A vortex discontinuity isobserved in a wide near-wake region behind the right-hand-side cylinder in figure 28b.A strong third-order harmonic vortex mode is decomposed in a narrow near-wakeregion behind the left-hand-side cylinder in figure 28d (a)
Cylinder1 (b)
Cylinder2
Figure 29: Contours of the transverse velocity v at ( z, x )-plane for SSBS arrangementat Re = 500, g ∗ = 0 . tU/D = 320 and v = ± .
1: (a) wide near-wake region of
Cylinder1 at y = 0 . D , where the gap flow deflects away momentarily, (b) narrownear-wake region of Cylinder2 at y = − . D with the gap flow deflection.modes as a first step for the subsequent analysis. Similar to the modal analysis inLiu & Jaiman (2016), SP-DMD is employed to investigate the primary vortices behindthe SBS arrangements in a three-dimensional flow. The primary focus is to investigatethe presentation of the complex coupling between the three-dimensional wakes in the hree-dimensional gap-flow and VIV interference (a) fD/U = 0, α = 111 (b) fD/U = 0 . α = 51(c) fD/U = 0 . α = 101 (d) fD/U = 0 . α = 41 Figure 30: Iso-surface plots of the primary vortex modes of VSBS arrangement at Re = 500, g ∗ = 0 . m ∗ = 10, ζ = 0 . tU/D ∈ [250 , ω z = ± .
01. The vortexmodes are well-distributed at the peak lock-in stage. A strong low frequency vortexmode is observed in the middle path of the gap flow.space and the frequency domains. For the computational efficiency viewpoint, l ∗ valueis reduced to five diameters. In accordance to the discussion in Section 2.3, the adopted l ∗ value should be sufficient to extract and manifest all aforementioned phenomena. Asummary of the DMD amplitude as a function of frequency for the representative casesis shown in figure 26.The decomposed DMD modes are selected via the sparsity-promoting process in thestandard DMD technique, as demonstrated in the examples from Jovanovic et al. (2014).Here all selected cases show a strong mean flow vortex mode at f D/U ≈
0. In particular,a stationary isolated circular cylinder case shows three clusters of modal frequencies atabout 0 .
2, 0 . .
6. Based on the discussion in Liu & Jaiman (2016) and comparisonwith the other spectral analyses in literature, the DMD modes at the modal frequency0 .
0, 0 . . f D/U refers to a dimensionless frequency for a particular DMD mode. f represents the vortex-wake frequency in each DMD mode with SI unit of Hz. The observedDMD mode with f D/U = 0 manifests the mean flow across the cylinder from where thekinetic energy is extracted. In addition, a DMD mode at f D/U = 0 .
61 is not observedby Liu & Jaiman (2016) in the corresponding two-dimensional flow. Taking the mode infigure 27b as a fundamental one, the mode at f D/U = 0 .
61 can be treated as a third-order harmonic mode. On the other hand, the SSBS arrangement case in figure 26 showsmore concentrated vortex modes at low frequency f D/U ∈ (0 .
05, 0 . B. Liu and R. K. Jaiman
For further discussion, their corresponding DMD vortex modes are visualized in fig-ure 28. In the SSBS arrangement, two particular vortex DMD modes are extracted,which possess different modal frequencies and associate with the lift forces over eachcylinder. One is at f D/U ≈ .
12 behind
Cylinder1 and the other one f D/U ≈ .
24 isfor
Cylinder2 with a narrow near-wake region † . The aforementioned f vs augmentationis simply visualized to be induced from the enhanced vortex-to-vortex interaction in anarrow near-wake region. Furthermore a similar third-order harmonic vortex mode isalso extracted in this SSBS arrangement, while taking the mean of 0 .
12 and 0 .
24 as anormalized modal frequency. This third-order harmonic vortex mode is represented byan isolated vortex mode at f D/U ≈ . crosses ) in figure 26 and is highly concentratedin the narrow near-wake region, as visualized in figure 28d.It is known that the third-order harmonic mode is strongly related to the instabilityof a dynamical system, since it breaks down the equilibrium of the fundamental basemode and distorts the wave forms. These third-order harmonic modes of the primaryvortex are only observed after the flow transition and highly concentrated in the regionwith enhanced three-dimensional flow structures. The footprints of these modal vortexpatterns are found forming further away from the cylinders in both the isolated and theSBS configurations. Consistent with the discussion on the recovery of 2D hydrodynamicresponses in Section 5, it confirms again that the three-dimensional vortical structureis originated from the vortex interactions. In addition, figure 28b shows a discontinuousvortex roller mode behind Cylinder1 with a wide near-wake region momentarily. Theshedding cell strength is about three diameters along the spanwise direction. On thecontrary,
Cylinder2 with the narrow near-wake region is followed by a continuous vortexroller mode in figure 28c in the same time window. The above observation is confirmedfrom the contour plots of the y -component velocity in figure 29, where a vortex discon-tinuity is observed behind Cylinder1 with a wide near-wake region in figure 29a. Theprimary vortex roller of the same sign was dislocated in the streamwise direction andrepresents a discontinuity in f vs . Since the primary vortex roller momentarily or partiallydeflects away from the wide near-wake region, f vs of Cylinder1 is dynamically varied bythe motion of the gap flow and entails a vortex discontinuity.In the VSBS arrangement, the f vs on both cylinders are synchronized with the naturalfrequency f n of the locked-in vibrating cylinder. The vortex modes at the same modalfrequencies ( triangles ) in figure 30 is well distributed behind both cylinders. Owing to therecovery of 2D hydrodynamic response, the primary vortex modes in figures 30c and 30dare rather similar to their two-dimensional counterparts in figures 12(b) and 12(d) fromLiu & Jaiman (2016) at relatively higher modal frequencies respectively. Nonetheless,different from its two-dimensional laminar flow counterparts, an influential vortex modeat f D/U ≈ .
16 is observed in the middle path of the gap flow in figure 30b. The three-dimensional effect is minimized to the most extent along the locked-in vibrating cylinder,thus the three-dimensional vortical structure is merely coupled between the gap flow andthe stationary cylinder. By Fast Fourier Transform (FFT) of C z from the stationarycylinder in this VSBS arrangement, we find a resemblance between the spanwise force C z frequency and the modal frequency of this particular vortex mode. Further analyzingthe position of its vortex pattern, this vortex mode is believed to be originated from thesecondary vortex-to-vortex interaction from the gap flow. The undulation of the spanwisehydrodynamic response along the cylinder is strongly influenced by this particular vortexmode induced by the gap flow. † The gap flow deflects to
Cylinder2 at tU/D ∈ (250 , hree-dimensional gap-flow and VIV interference
8. Conclusions
The dynamics of three-dimensional gap flow and VIV interaction is numerically inves-tigated in the side-by-side circular cylinder arrangements at moderate Reynolds numbersranging 100 (cid:54) Re (cid:54) C z response was amplified in both SSBS and VSBS arrangements alongthe cylinder with a narrow near-wake region. This asymmetry of hydrodynamic responsesinduced from the deflected gap flow caused a spatial modulation, about a factor of 2.4amplification of the C rmsl response. The flip flop was also subjected to the influence by thethree-dimensionality of flow, while it was remarkably suppressed in both SSBS and VSBSarrangements at the off-lock-in and restored together with two-dimensional response atthe lock-in stage for the VSBS arrangements. The gap flow was also found to promotethe three-dimensional flow feature through enhanced fluid shearing and mixing, whichexerts a strong proximity interference stabilizing the energy transfer between the fluidand structure. In addition to the three-dimensional interference, the VIV kinematics andthe gap flow are mutually influenced by each other. The onset of the VIV lock-in in theVSBS arrangement was observed at a smaller U r value. A quasi-stable deflected gap-flowregime was observed in the VSBS arrangements at the peak lock-in, in which the gap flowdeflected steadily behind the locked-in vibrating cylinder. The spanwise wavelength λ ∗ value was significantly affected by both the VIV and the gap-flow proximity interference.Due to the regulation of energy transfer between the fluid flow and the structure, thewavelength λ ∗ become more uniform along the cylinder and resulted into increased λ ∗ at the peak lock-in.Through the modal analysis, the third-order harmonic primary vortex modes weredynamically decomposed for the isolated cylinder and the SSBS arrangements. Owing totheir odd-order mathematical characteristics, the third-order vortex modes are crucialto the stability of the dynamic fluid-structure system and represents an unstable factor.A vortex discontinuity originated from the gap-flow kinematics was observed using theDMD technique in the wide near-wake region. An additional influential primary vortexmode along the middle path of the gap flow in the VSBS arrangement was observed,which was related to the periodic undulation of spanwise hydrodynamic response alongthe cylinder and represented the promoted gap-flow instability.Overall, it was found that the vortex-to-vortex interaction between the imbalancedcounter-signed vorticity clusters plays an important role in the near-wake stability, be-cause of the significant fluid shearing along the vortical interfaces. In general, the intensivefluid shearings along the vortical interfaces were associated with the in-determinant4 B. Liu and R. K. Jaiman streamline saddle-point regions. The saddle-point region is found in all range of Reynoldsnumber and is interlinked with various flow dynamic events, e.g. the vortex shedding,the flip flop, the streamwise vorticity clusters. Furthermore, the near-wake instability isfound to be closely interlinked with the gap flow and the VIV kinematics. In particular,as the VIV kinematics increases and stretches the vortices, the vorticity clusters aremore separated to weaken the vortex-to-vortex interaction in the near-wake region. Asa result, the two-dimensional hydrodynamic responses are significantly restored alongthe cylinder. On the contrary, the interaction dynamics between the gap-flow proximityinterference and the gap-flow instability enhances the vortex-to-vortex interaction. Theseobservations and findings are important in multi-body systems, from both operations anddesign viewpoints, found in offshore and aeronautical engineering.The authors would like to thank Singapore Maritime Institute Grant (SMI-2014-OF-04) for the financial support.
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Appendix A. Streamline saddle point and ( x, y ) -plane velocity profile A streamline saddle point is located in a two-dimensional incompressible flow far fromany boundaries, where the velocity components are defined from stream function ψ ( x, y )as u = ∂ψ∂y , v = − ∂ψ∂x (A.1)A fourth-order biquartic polynomial of ψ ( x, y ) surface patch † is employed to approximatethe local continuous ψ ( x, y ) field around an SSP. Assuming that the parametric surfaceof two-dimensional stream function is smooth, continuous and their spatial derivativesare everywhere well-defined up to the highest order of the approximating function, a localflow field can be represented by a general form as follows ψ ( x, y ) = a + a x + a y + a xy + a x + a y + a x + a x y + a xy + a y + a x + a x y + a x y + a xy + a y + O ( x , y )(A.2)Here a i ( i = 0 , ...,
14) are arbitrary scalar constants and O ( x , y ) is the truncation error.Based on the second derivative test for local extreme values , Eq. (A.2) has to satisfy acriterion Eq. (A.3) to approximate a (non-degenerated) two-dimensional SSP at (0,0). ∂ψ∂x = 0; ∂ψ∂y = 0; ∂ ψ∂x ∂ ψ∂y − (cid:20) ∂ ψ∂x∂y (cid:21) < at ( x, y ) = (0 ,
0) (A.3)By imposing Eq. (A.3) on Eq. (A.2), a = a = 0 and 4 a · a < ( a ) . The approximatingfunction reduces to ψ ( x, y ) = a + a xy + a x + a y + a x + a x y + a xy + a y + a x + a x y + a x y + a xy + a y + O ( x , y ) (A.4) † where the fluid shear stresses are approximated linearly in the field hree-dimensional gap-flow and VIV interference x = 0 and y = 0 aresubstituted into the first and second derivatives of Eq. (A.4) as follows u (0 ,
0) = 0; v (0 ,
0) = 0 ∂u∂y | (0 , = 2 a ; ∂v∂x | (0 , = − a ∂ u∂y | (0 ,y ) = 6 a + 24 a y ; ∂ v∂x | ( x, = − a − a x∂ u∂y | (0 , = 6 a ; ∂ v∂x | (0 , = − a (A.5)Analyzing Eq. (A.5), the velocity is zero at a saddle point and generate a local stagnantregion. The directional gradients of u and v are non-zero scalar constants at the saddlepoint. The second-order derivatives of u and v are also found having a linear relationshipwith respect to y and x variables respectively. Hence there is a point along y and x axesrespectively across the streamline saddle point, where ∂ u∂y and ∂ v∂x switch their signs.The zero values of ∂ u∂y and ∂ v∂x are the locations of the inflection points of u and v .As the values of a and a approach zero, the velocity profile inflection points translatetoward the streamline saddle point. Since the linear odd-order terms in the approximatingbiquartic polynomial tend to destroy the symmetry of ψ ( x, y ) about the saddle point,the amplification of their coefficient values is detrimental to the formation of the saddlepoint. Therefore, the locations of the velocity profile inflection points is expected to benot far from the streamline saddle point. Appendix B. Stability analysis of streamline saddle point
This appendix is concerned with the near-wake instability in a saddle-point regionbehind cylinders in the SSBS arrangements. The goal is to investigate the correlationbetween the near-wake stability and its stability parameters, e.g., Reynolds number Re ,fluid shear ratio S = U /U and gap ratio g ∗ , using DMD technique. Here U and U are respectively x-component inlet velocities in front of the gap flow and the free-side ofcylinders in the SSBS arrangement. The development of the well-known vortex shedding Hopf bifurcation at a low Reynolds number Re (cid:46)
100 is taken as an indication of the near-wake instability. Its relationship with Re and S values are used to analyze the importanceof the fluid momentum and the fluid shearing ratio induced from the imbalanced counter-signed vorticity concentration to the near-wake stability. Here a large g ∗ value, g ∗ > . Re (cid:46) g ∗ value. The rest of parameters is identical to the ( x, y )-section of the three-dimensional computational setup.To analyze the dependency of the near-wake instability on Reynolds number, the DMDmode which is account for the vortex shedding is identified in a saddle-point region infigure A.1a. At smaller Re and S values, the unstable modes are supposed to decaysas the fluid flow develops, since | µ | <
1. Here µ = µ r + jµ i is the eigenvalue of thedecomposed primary vortex mode using DMD technique, where µ r , µ i and j are itsreal parts, imaginary parts and imaginary unit respectively. The | µ | is its correspondingmagnitude of µ . As the values of Re and S increase, this primary vortex mode reachesan equilibrium state | µ | = 1. Similar to the stability analysis in Mizushima & Ino (2008),a bifurcation diagram of y-component velocity at (2 D , 0 D ) is plotted in figure A.1b.The manifested fluid instabilities from both Re and S variation are identified as Hopf B. Liu and R. K. Jaiman (a) (b)
Figure A.1: Stability analysis of wake around a saddle-point region behind in the SSBSarrangements: (a) | µ | vs. Re and | µ | vs. S ; (b) bifurcation diagram at Re = 40 for v vs. Re and v vs. S , where v is an averaged peak value ( tU/D ∈ [300 , y -componentvelocity at the location (2 D , 0 D ) behind a cylinder (0 D , 0 D ). bifurcation . This conclusion is based on the correlation ( v + ( P i ) − v − ( P i )) ∝ ( P i − P s ) . ,where P i and P s are the stability parameter, e.g., Re and S , and its critical valuerespectively. Here v + ( P i ) and v − ( P i ) are respectively the time-averaged maximum andminimum of y-component velocity v ( P i ). Owing to the imbalanced fluid shearing S (cid:54) = 1,the curve of original v value is asymmetric about v = 0 in the stability analysis of S .To present a symmetric curve, the time-averaged maximum and minimum v values aremodulated as v m and plotted in figure A.1b, where v m ( P i ) = 0 . v + ( P i ) − v − ( P i )) and v m ( P i ) = − . v + ( P i ) − v − ( P i )). Observing from figures A.1a and A.1b, the onset ofvortex shedding occurs exactly at Re ≈
48 as Re value increases. Its modal frequencyand vortex pattern at saturated state conform with those of vortex shedding documentedin numerical and experimental investigations of a stationary isolated circular cylinder.Furthermore it also shows a similar saturation of the velocity oscillation at S ≈ . g ∗ and the near-wake stability. Here figure A.2 shows that the Hopf bifurcation starts todevelop as g ∗ (cid:46) .
33. It means the gap-flow proximity interference is critical to the near-wake stability in the SSBS arrangements as well, as g ∗ value decreases. To summarize,the fluid shearing ratio S , the fluid momentum intensity Re and the gap-flow proximityinterference g ∗ are significant to the near-wake stability. In multi-body systems, e.g., SBSarrangements, either or all of these parameters could be influential at a particular fluiddomain. To exceed their corresponding critical values are destructive to the near-wakestability. hree-dimensional gap-flow and VIV interference v with respect to the gap ratio g ∗ at Re = 45, g ∗ ∈ [0 .