Freely vibrating circular cylinder in the vicinity of a stationary wall

Journal of Fluids and Structures 59 (2015) 103–128

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Freely vibrating circular cylinder in the vicinity of a stationary wall Daniel Mun Yew Tham a, Pardha S. Gurugubelli a, Zhong Li b, Rajeev K. Jaiman a,n a b

Department of Mechanical Engineering, National University of Singapore, 21 Lower Kent Ridge Road, Singapore 119077, Singapore NUS Graduate School for Integrative Sciences and Engineering (NGS), 28 Medical Drive, Singapore 117456, Singapore

a r t i c l e i n f o

abstract

Article history: Received 12 April 2015 Accepted 7 September 2015

We present a numerical study on vortex-induced vibration (VIV) of a freely vibrating two degree-of-freedom circular cylinder in close proximity to a stationary plane wall. Fully implicit combined field scheme based on Petrov–Galerkin formulation has been employed to analyze the nonlinear effects of wall proximity on the vibrational amplitudes and hydrodynamic forces. Two-dimensional simulations are performed as a function of decreasing gap to cylinder diameter ratio e=D A ½0:5; 10 for reduced velocities U  A ½2; 10 at ReD ¼ 100 and ReL ¼ 2900, where ReD and ReL denote the Reynolds numbers based on the cylinder diameter and the upstream distance, respectively. We investigate the origin of enhanced streamwise oscillation of freely vibrating near-wall cylinder as compared to the isolated cylinder counterpart. For that purpose, detailed analysis of the amplitudes, frequency characteristics and the phase relations has been performed for the isolated and near-wall configurations. Initial and lower branches in the amplitude response are found from the gap ratios of 0.75 to 10, similar in nature to the isolated cylinder laminar VIV. A third response branch has been found between the initial and the lower branch at the gap ratio of e=D r 0:60. For near-wall cases, phase relation between drag force and streamwise displacement varies from close to 0° to 180°. Between e=D A ½5; 7:5, the effect of wall proximity on the frequency response tends to disappear. The effect of mass-ratio is further investigated. Finally, we introduce new correlations for characterizing peak amplitudes and forces as a function of the gap ratio for a cylinder vibrating in the vicinity of a stationary plane wall. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Fully implicit combined field Near-wall vortex-induced vibration Streamwise vibration Gap ratio Amplitude branches Mass ratio

1. Introduction The offshore industry is increasingly considering designs of deepwater pipeline system traversing across escarpment, trough or depressions over the seafloor. The uneven nature of seafloor topography and randomness of scouring result in the formation of free spans that usually would exceed the allowable stress and fatigue limits required for the safety of pipelines (Tsahalis, 1983; Sumer and Fredsøe, 2006). In the presence of steady current close to the seafloor, free span pipelines can undergo severe flow-induced vibrations. In the short run, small amplitude vibrations with high frequencies may not be detrimental but they can lead to serious fatigue damage in the long run. The challenge is then to understand the coupled dynamics of free span pipelines undergoing vibrations in the vicinity of seabed. The problem of a free span vibration along n

Corresponding author. E-mail address: [email protected] (R.K. Jaiman).

http://dx.doi.org/10.1016/j.jfluidstructs.2015.09.003 0889-9746/& 2015 Elsevier Ltd. All rights reserved.

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Nomenclature αf ; αfm αs ; αsm δ Γ fh Γ sh Γ Γf μf ν Ωf Ωs ρf ρs σf τm ; τ c f b s b c Fs k m uf

fluid generalized-α parameters solid generalized-α parameters boundary layer thickness non-interface fluid Neumann boundary non-interface solid Neumann boundary interface between fluid and rigid body non-interface fluid domain boundary fluid dynamic viscosity fluid kinematic viscosity fluid domain rigid body fluid density solid density fluid Cauchy stress tensor Galerkin least square stabilization parameters body force acting on fluid body force acting on rigid body damping vector per unit length fluid traction acting on rigid body stiffness vector per unit length mass vector per unit length fluid velocity

w Arms x =D

mesh velocity nondimensional root-mean-squared streamwise displacement of rigid body Amax =D maximum nondimensional transverse disy placement of rigid body C rms root-mean-square drag coefficient D C rms root-mean-square lift coefficient L CD drag coefficient CL lift coefficient CP pressure coefficient Cp0 forward stagnation pressure coefficient D cylinder diameter e gap between cylinder and wall fn natural frequency of the cylinder in vacuum k spring stiffness Lu ; Ld ; H computational domain parameters M mass per unit length of cylinder mn structure to fluid mass ratio p fluid pressure p1 freestream fluid pressure ReD Reynolds number based on cylinder diameter ReL Reynolds number based on upstream distance U free-stream velocity Un reduced velocity

the pipeline length can be modeled as a spring-mounted cylinder near a plane wall. The presence of wall will alter the oncoming flow profile and vortex patterns during oscillations, which in turn will change the hydrodynamic loads and coupled fluid–structure responses. Most of the existing literature concerning VIV of a circular cylinder have focused on the case of an isolated circular cylinder placed in uniform flow without the wall-proximity effects. Comprehensive reviews have been done by Sarpkaya (2004), Williamson and Govardhan (2004, 2008) and Bearman (2011) for the current state of this area. For the isolated cylinder, Govardhan and Williamson (2000) found that VIV of a one degree-of-freedom (1-DoF) cylinder with high mass damping exhibits two response branches, namely the initial and lower branches. The vortex wake exhibits in 2S mode on the initial branch and 2P mode on the lower branch. The proximity of a wall introduces complex interactions between the wall boundary-layer and the shear layer over the circular cylinder. One of the earliest experiments studying the wall effects on a circular cylinder was reported in Taneda's experiment (1998), where a circular cylinder was towed through stagnant waters close to a fixed ground. In this experimental study, since both water and ground moved relative to the cylinder, the effects of wall boundary-layer formed on the bottom wall were neglected. Regular alternate vortex shedding occurred at a gap ratio, i.e. ratio of height of gap between cylinder and wall to the diameter of the cylinder, e/D¼0.6, while only a weak single row of vortices was shed at e/D¼0.1. Bearman and Zdravkovich (1978) investigated the effect of gap ratio on the vortex shedding in the Reynolds number regime of between 2.5  104 and 4.8  104, showing that vortex shedding is suppressed if e=D o0:3. Studies by Zdravkovich (1985b) and Lin et al. (2005), carried out at Reynolds number ReD of 3550 and 780 respectively, showed the cessation of regular vortex shedding for a stationary cylinder near a wall, where ReD ¼ UD=ν with U being the freestream velocity and ν the kinematic viscosity. Other studies on a stationary cylinder near a fixed wall include Lei et al. (1999, 2000), Wang and Tan (2008a) and Ong et al. (2010). In Lei et al. (1999), experiments were conducted in different boundary layer thicknesses, δ=D ¼ 0:14–2:89, at ReD from 1.31  104 to 1.45  105, and they found that both lift and drag are largely affected by e/D and influenced by δ=D as well. They also reported that vortex shedding is suppressed when gap ratio is between 0.2 and 0.3. Numerical studies were preformed by Lei et al. (2000) to study the vortex shedding for different Reynolds numbers ranging from 80 to 1000 at various gap ratios, where the vortex shedding mechanism was analyzed and critical gap ratios were identified at different ReD . Wang and Tan (2008a) looked into the near-wake flow characteristics of a circular cylinder proximity to a flat bed at ReD ¼ 1:2  104 and δ/D¼0.4, observing that the instantaneous flow field highly depends on e/D. These three works concluded that ReD, e/D and boundary layer thickness, δ=D, are three governing parameters affecting the flow over a cylinder near a fixed wall. In Wang and Tan (2008b), the authors investigated experimentally the flow characteristics in the near wake of a cylinder located close to a fully developed turbulent boundary layer. For small and intermediate gap ratios, the wake flow develops a distinct asymmetry about the cylinder centerline. Similarly, Ong et al. (2010) conducted numerical investigations by k–ϵ model on stationary near-wall cylinder in the turbulent regime. It was found that the drag coefficient increases as e/D increases for small e/D, reaching a maximum value before decreasing to approach a

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constant value. In Sarkar (2010), large-eddy simulations were used to investigate the modifications of wake dynamics and turbulence characteristics behind a near-wall cylinder for varying e/D. Recently, Rao et al. (2013b) investigated numerically the flow past a stationary cylinder at different heights above a no-slip plane from freestream case down to a small gap ratio e/D¼0.005. Critical Re were found where there is a transition from steady two-dimensional flow to three-dimensional flow. The authors (Rao et al., 2013a) further studied the dynamics and stability of the flow past two cylinders sliding along a wall in a tandem configuration for Re between 20 and 200 and streamwise separation distances between 0.1 and 10 diameters. Investigations for vibrating cylinder near a plane wall have been conducted in the moderate to high ReD regime. Tsahalis and Jones (1981) reported that the XY-trajectory of a near-wall cylinder is an oval-shape, instead of the common figure-ofeight shape for an isolated-cylinder. Fredsøe et al. (1985) found that the transverse vibration frequency is close to the vortex shedding frequency of a stationary cylinder when reduced velocity U  o3 and e=D 4 0:3. However, when 3 o U  o 8, the transverse vibrating frequency deviated considerably from the vortex shedding frequency of a stationary cylinder. Yang et al. (2008) studied experimentally 2-DoF VIV of near-wall cylinder in the turbulent regime, where they found a decrease in vibration amplitudes with decreasing e/D. Yang et al. (2011) compared the dynamic response of cylinder near a deformable wall in both 1-DoF and 2-DoF cases. Yang et al. (2009) showed that the dimensionless vibration frequency for smaller e/D of 0.06 and 0.30 are much larger than those for larger e/D of 0.66 and 0.88, respectively, at the same reduced velocity. For smaller mass ratio mn, the vibration range is larger. The frequency ratio, i.e. ratio of vibration frequency to the natural frequency of the system, is much larger for the smaller mn at the same Un. Zhao and Cheng (2011) numerically studied 2-DoF VIV of near-wall cylinder in the turbulent regime. Their study investigated low gap ratios of e/D¼0.002 and 0.3, where bounce-back from the plane boundary during VIV was considered. Wang and Tan (2013) conducted an experimental study for a 1-DoF vibrating cylinder near a plane wall at 3000 r ReD r 13 000 and a low mn of 1.0 for 1:53 rU  r 6:62. It was demonstrated that the wall in the proximity not only affects the amplitude and frequency of vibration, but also leads to nonlinearities in the cylinder response as evidenced by the presence of super-harmonics in the drag force spectrum. The vortices shed that would otherwise be in a double-sided vortex street pattern were arranged into a single-sided pattern, as a result of the wall. The review above indicates that there has been previous work on near-wall stationary and vibrating cylinder, in moderate to high ReD regime. However, numerical studies of 2-DoF VIV of an elastically supported circular cylinder near a wall in low ReD regime have not been conducted, to the best of our knowledge. It is known that the branching of cylinder response for VIV has its genesis in two-dimensional low ReD flow. In Leontini et al. (2006), the VIV regimes of cylinder response were captured via simulations at low ReD, similar in nature to the upper and lower branches seen at higher ReD. The initial developments of the upper and lower branch behavior were observed in low-Re VIV simulations at Re ¼200, with supporting results from the frequency, total phase, and instantaneous amplitude and phase behavior (Leontini et al., 2006). The main purpose of this study is to investigate the effects of gap ratio e/D and Un to characterize the hydrodynamic forces and vibration response of a vibrating circular cylinder near a plane wall in the low ReD regime. In this work, we consider the range of gap ratio e=D A ½0:5; 10. The interactions of boundary layer with freely vibrating cylinder result in a complex coupling of the wake and the oscillating cylinder. In the neighborhood of a stationary wall, the vortex dynamics and response characteristics are quite different from that for the isolated-cylinder vibrating in a freestream flow. The interaction dynamics between near-wall wake and oscillating structure depends strongly on the wall proximity and thereby altering the phenomenon of lock-in remarkably. Of particular interest here is to investigate the origin of enhanced streamwise vibrations of near-wall cylinder as compared to the freely vibrating isolated-cylinder configuration. For that purpose, detailed investigations of the fluid forces, the amplitude–frequency response characteristics and the phase relations have been performed for the isolated and nearwall configurations. We determine the gap ratios when the third branch appears in the amplitude response and the frequency response of the cylinder recovers to that of the isolated cylinder counterpart. We introduce new correlation functions for characterizing peak amplitudes and forces as a function of the gap ratio. This study represents a step towards an improved understanding of streamwise vibrations of subsea pipelines subject to ocean currents with varying gaps from the seabed. Such enhanced streamwise vibrations can lead to large bending stresses in the structure, which can in turn lead to premature fatigue failure. An understanding of the mechanisms of the large seabed-induced streamwise vibrations may improve future design of the subsea pipelines and may pave the way to design more effective suppression devices. The governing equations and the numerical method employed in this work are described in Section 2. The formulation of the problem and the key parameters are described in Section 3. This is followed by the verification of results for the isolatedcylinder in Appendix A. In Section 4, we present the characterization of the response dynamics of isolated and near-wall cylinders and discuss the basic differences between the two arrangements in terms of wake topology, response characteristics, force components, phase relations, frequency characteristics, and the effects of mass ratio and oncoming boundary layer development. In the end, we will present the concluding remarks.

2. Numerical methodology In the present study, the coupled fluid-rigid body simulations have been performed using the method proposed in Jaiman et al. (2015). For the sake of completeness, the methodology is briefly discussed in this section.

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2.1. Governing equations The Navier–Stokes equations governing an incompressible flow in an arbitrary Lagrangian–Eulerian (ALE) reference frame are ρf


 ∂uf f þρf uf  w  ∇uf ¼ ∇  σ f þ b ∂t

∇  uf ¼ 0

on Ωf ðt Þ;

ð1Þ

on Ωf ðtÞ;

ð2Þ

where u ¼ u ðx; tÞ and w ¼ wðx; tÞ represent the fluid and mesh velocities defined for each Eulerian coordinate x A Ω ðtÞ, f respectively. b is the body force applied on the fluid and σ f is the Cauchy stress tensor for a Newtonian fluid, written as 
T  σ f ¼ pI þ μf ∇uf þ ∇uf ; ð3Þ f

f

f

where p denotes the fluid pressure and μf is the dynamic viscosity of the fluid. An immersed rigid body experiences unsteady forces and when mounted elastically undergoes vortex-induced vibrations. The equation of motion of the body for the translational degrees of freedom is m

  ∂us s þc  us þk  φs ðz0 ; t Þ  z0 ¼ F s þ b ; ∂t

ð4Þ

where m; c and k denote the mass, damping and stiffness vectors per unit length for the translational degrees of freedom, s respectively, us ðt Þ represents the rigid-body velocity at time t, F s and b are the fluid traction and body forces acting on the s rigid body, respectively, φ is the function mapping the initial position z0 to its position at time t. The coupled system requires to satisfy the no-slip and traction continuity conditions at the fluid–body interface Γ as follows:   ð5Þ uf φs ðz0 ; tÞ; t ¼ us ðx; t Þ; Z

Z φðγ;tÞ

σ f ðx; t Þ  nf dΓ þ

γ

F s dΓ ¼ 0

8 γ A Γ;

ð6Þ

where nf and ns are respectively the outer normals to the fluid and the solid and ϕs is the function that maps each structural node from its initial position z to its deformed position at time t. Here, γ is any edge on the interface Γ. The weak form of the Navier–Stokes equations (1) and (2) can be written as Z Z Z

 f ρf ∂t uf þ uf  w  ∇uf  ϕf ðxÞ dΩ þ σ f : ∇ϕf ðxÞ dΩ ¼ b  ϕf ðxÞ dΩ þ Ωf ðtÞ Ωf ðtÞ Ωf ðtÞ Z Z
 f dh  ϕf ðxÞ dΓ þ σ f ðx; tÞ  nf  ϕf ðxÞ dΓ; ð7Þ Γ ðt Þ

Γ fh ðtÞ

Z Ωf ðtÞ

∇  uf qðxÞ dΩ ¼ 0:

ð8Þ

Here ∂t denotes the partial time derivative operator ∂ð U Þ=∂t, ϕf and q are test functions for the fluid velocity and pressure, f respectively. Γ fh ðtÞ represents the non-interface Neumann boundary along which σ f ðx; tÞ  nf ¼ h . The weak form for the rigid-body equation (4) is Z Z Z    s m  ∂t us þ c  us þ k  φs ðz0 ; t Þ  z0  ϕs dΩ ¼ F s  ϕs dΓ þ b  ϕs dΩ; ð9Þ Ωs

Γ

Ωs

s

where ϕ is the test function for the rigid-body velocity, which is a constant for any point on the body. Similarly, we can write the weak form for the traction boundary condition as Z Z
 σ f ðx; tÞ  nf  ϕf dΓ þ F s  ϕs dΓ ¼ 0 8 γ A Γ: ð10Þ φðγ;tÞ

γ

In the above equation, the condition ϕf ¼ ϕs can be enforced by considering a conforming mesh along the interface Γ. We next enforce the traction continuity condition (10) to combine the fluid and structural weak forms in (8), (7) and (9) to construct a single weak form for the combined system Z Z Z

 ρf ∂t uf þ uf  w  ∇uf  ϕf dΩþ σ f : ∇ϕf dΩ ∇  uf q dΩ Ωf ðtÞ Ωf ðtÞ Ωf ðtÞ Z Z Z Z    f f s þ m  ∂t us þ c  us þ k  φs ðz0 ; t Þ z0  ϕs dΩ ¼ b  ϕf dΩ þ h  ϕf dΓ þ b  ϕs dΩ; ð11Þ Ωs

Ωf ðtÞ

Γ fh ðtÞ

Ωs

and the velocity continuity condition is enforced in the function space. A solver employing a Petrov–Galerkin formulation and semi-discrete time stepping has been developed to investigate the incompressible flow characteristics of freely vibrating cylinders. For the sake of completeness, we briefly present the semi-discrete formulation employed in this study.

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107

2.2. Semi-discrete Petrov–Galerkin formulation The Petrov–Galerkin finite-element formulation followed here employs the same order of interpolation for velocity and pressure. The spatial domain Ωf is decomposed into finite elements Ωe , e ¼ 1; 2; …; nel ; with nel being the number of elements. We adopt a generalized-α method to integrate in time between t A ½t n ; t n þ 1 , which can be unconditionally stable and second-order accurate for linear problems (Chung and Hulbert, 1993). With the aid of the generalized-α parameters ðαf ; αfm ; αs ; αsm Þ, the coupled variational fluid-rigid body formulation along with the stabilization terms can be written as follows: Z Z

  f f f f;n þ αfm þ αf  ϕf dΩ þ ρf ∂t uh þ ufh;n þ α  wnh þ α  ∇uf;n σ fh;n þ α : ∇ϕf dΩ  h Ωfh ðt n þ 1 Þ

Z

∇  ufh;n þ α q dΩþ f

Ωfh ðt n þ 1 Þ

nel X

Ωfh ðt n þ 1 Þ

Z Ωe

e¼1



 f f τm ρf ufh;n þ α  wnh þ α  ∇ϕf þ ∇q

nel

  X f f f;n þ αfm f þ αf þ αf þ αf þ ρf uf;n wnh þ α  ∇uf;n  ∇  σ f;n f ðt n þ α Þ dΩe þ  ρf ∂t uh h h h

Z f

Ω

Z

e

f

∇  ϕ τc ρ ∇ 

Ωfh ðt n þ 1 Þ

þ αf uf;n h

e

dΩ þ Z f f b ðt n þ α Þ  ϕf dΩ þ

Z h

Γ fh ðt

Ω

s

m f

nþ1

Þ

s;n þ αsm ∂t uh

h  ϕf dΓ þ

Z

þc 

Ωs

þ αs us;n h

þk 




e¼1

þ αs φs;n ðz0 Þ  z0 h

b ðt n þ α Þ  ϕs dΩ; s

s

i

 ϕs dΩ ¼ ð12Þ

where the terms with element level summations stand for stabilization terms applied on each element locally and the remaining terms along with the right-hand side represents the Galerkin statement of the variational fluid-body problem. The stabilization parameters τm and τc are the least-squares metrics added to the fully discretized formulation. The fullycoupled finite-element formulation and implementation are discussed in detail in Jaiman et al. (2015). The kinematic compatibility at the interface is achieved by construction, which leads to a reduction in the size of the linear system solved per nonlinear iteration. The incremental velocity and pressure are computed via the matrix-free implementation of the

Fig. 1. Schematic of (a) problem definition and (b) computational domain and boundary conditions for freely vibrating cylinder near stationary wall.

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restarted Generalized Minimal RESidual (GMRES) solver proposed in Saad and Schultz (1986). The GMRES uses a Krylov space of 30 orthonormal vectors. Through the implicit time integration and the quasi-monolithic formulation, the FICF scheme achieves stability for large time step sizes while maintaining second-order accuracy for very low mass ratios.

3. Problem definition and key parameters We consider a two dimensional incompressible flow over a rigid circular cylinder with diameter D placed close to a fixed wall (see Fig. 1a). The rigid cylinder is mounted on spring–damper systems to allow 2-DoF displacement in the transverse and streamwise directions. Fig. 1b depicts a schematic of the computational domain along with the boundary conditions considered. The upstream and downstream boundaries are located at Lu and Ld respectively, from the center of the cylinder. H is the depth of the cylinder from the top boundary. The gap between the rigid cylinder and the fixed wall is denoted by e and δ is the boundary layer thickness at the cylinder center in the upcoming flow. In addition to the interface boundary conditions (5) and (6), a uniform freestream velocity U has been assigned at the inlet boundary. Slip-wall boundary condition has been implemented along the upper boundary. The lower boundary consists of two parts, where the first part is considered as slip-wall boundary and no-slip condition is implemented for the second part. A traction-free Neumann boundary condition is prescribed at the downstream outlet. The two standard VIV parameters are mass ratio m ¼ 4M=ρf πD2 and reduced velocity U  ¼ U=f n D where M denotes the pffiffiffiffiffiffiffiffiffiffi cylinder mass per unit length, fn is the natural frequency of the structure in vacuum and is given by f n ¼ ð1=2πÞ k=M . The natural frequency of the cylinder and thus Un is changed by varying the stiffness k of the springs on which the cylinder is mounted. The springs in both transverse and streamwise directions are assumed linear and have the same stiffness coefficients. The structural damping coefficient is set to zero to encourage high amplitude oscillations. ReD is kept constant at 100. Un is varied from 2 to 10. For the isolated-cylinder cases, blockage ratio b¼0.05, where blockage ratio refers to the ratio pffiffiffiffiffiffiffiffi of the diameter of the cylinder to the height of the domain. Boundary layer thickness δ is given as δ ¼ ðLu DÞ5:0= ReL , where ðLu  DÞ refers to the upstream distance for which the no-slip boundary condition applies and ReL ¼ UðLu  DÞ=ν. The fluid loading is computed by integrating the surface traction considering the first layer of elements located on the cylinder

0.6

/D Amax y

0.5 0.4 0.3 0.2 0.1 0

e/D = 0.5 e/D = 0.6 e/D = 0.75 e/D = 0.9 e/D = 1.0 e/D = 1.25 e/D = 1.5 e/D = 2.5 e/D = 5 e/D = 7.5 e/D = 10 isolated (b = 0.05)

Third branch

Lower branch

Initial branch

2

4

6

8

10

8

10

U*

Axrms/D

0.15

0.1

e/D = 0.5 e/D = 0.6 e/D = 0.75 e/D = 0.9 e/D = 1.0 e/D = 1.25 e/D = 1.5 e/D = 2.5 e/D = 5 e/D = 7.5 e/D = 10 isolated (b = 0.05)

0.05

0

2

4

6

U* Fig. 2. Variation of the maximum displacement amplitudes with reduced velocity at m ¼ 10: (a) transverse and (b) streamwise.

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109

surface. The instantaneous force coefficients are defined as Z
 1 σ f :n :nx dΓ: CD ¼ 1 f 2 Γ ρU D 2 Z
 1 CL ¼ σ f :n :ny dΓ; 1 f 2 Γ ρU D 2

ð13Þ

ð14Þ

Here nx and ny are the Cartesian components of the unit normal, n. In the present study, drag coefficient CD and lift coefficient CL are calculated as derived quantities using direct evaluation of the Cauchy stress on the boundary. Pressure coefficient CP is defined as CP ¼

p  p1 ; 1 f 2 ρU 2

ð15Þ

where p represents the pressure at the point at which pressure coefficient is being calculated and p1 denotes the freestream pressure. The numerical methodology and computational details used are validated against the published results in the literature. For the sake of completeness, the validation of VIV and the convergence results have been summarized in Appendix A.

4. Results and discussion The complexity of the coupled physical phenomena involved in a vibrating cylinder close to a plane stationary wall is enhanced by wake/boundary layer and cylinder/wall interactions. The unsteady wake of the cylinder interacting with the boundary layer makes the coupled response of near-wall configuration different from the isolated-cylinder arrangement. To

e/D = 0.5 e/D = 0.6 e/D = 0.75 e/D = 0.9 e/D = 1.0 e/D = 1.25 e/D = 1.5 e/D = 2.5 e/D = 5 e/D = 7.5 e/D = 10 isolated (b = 0.05)

1.5

rms L

1

0.5

0

2

4

6 U*

8

0.6

e/D = 0.5 e/D = 0.6 e/D = 0.75 e/D = 0.9 e/D = 1.0 e/D = 1.25 e/D = 1.5 e/D = 2.5 e/D = 5 e/D = 7.5 e/D = 10 isolated (b = 0.05)

0.5 0.4 rms D

10

0.3 0.2 0.1 0

2

4

6 U*

8

10

Fig. 3. Variations of force coefficients with respect to reduced velocity at m ¼ 10: (a) lift coefficient and (b) drag coefficient.

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understand the coupled dynamics of near-wall VIV, we investigate the effects of gap ratio e/D for the fixed upstream distance Lu ¼ 30D (where δ=D ¼ 2:7), downstream distance Ld ¼ 45D and cylinder depth H ¼10D and contrast the VIV behavior with the isolated-cylinder counterpart. 4.1. Transverse and streamwise amplitudes Fig. 2 summarizes the nondimensional transverse maximum Amax =D and streamwise root-mean-squared Arms y x =D amplitudes of a circular cylinder, which is placed in the proximity of a fixed-wall, in 2-DoF VIV as a function of reduced velocity Un and gap ratio e/D. The 2-DoF vibration response for laminar-VIV of both isolated cylinder and near-wall cylinder show namely the initial and lower branches for e=D Z 0:75. The range of U  o4 represents the pre-lock-in regime, where vibration amplitudes are negligibly small. As Un increases, the vortex shedding frequencies (and thus the periodic force frequencies) synchronize with the vibration frequencies, and the lock-in range is entered. In the initial branch, the vibration amplitudes increase as Un is increased before reaching a peak. At the peak streamwise amplitudes, the phase difference between the drag force and the streamwise displacement is around 40–90° (see Fig. 18(b)) where there is a net power transfer from the fluid to the cylinder motion as further explained in Section 4.6. As Un increases further, the phase difference becomes 180°, the vibration amplitudes decrease, and the lower branch of the response is entered. Fig. 2(a) depicts the relationship between the dimensionless transverse vibration amplitude Amax y =D and reduced velocity Un for a range of gap ratios e=D A ½0:5; 1 (where e=D ¼ 1 refers to an isolated cylinder configuration). As e/D decreases from =D decreases, corresponding to the fall in peak lift forces as presented in Fig. 3(a). In Tsahalis and Jones (1981) 1.5, peak Amax y the authors similarly reported a decrease in peak transverse amplitudes as e/D decreases. However at different ranges of Un, the effect of wall proximity is to promote transverse vibration rather than suppressing it as might be expected (e.g. at e=D ¼ 0:75, for 6:7 r U  r7:3, Amax y =D is higher than that for other higher e/D). In general, the effect of decreasing e/D is an n increase in Un at which peak Amax =D occurs. As such, Amax y y =D is highest at different e/D at different ranges of U (e.g. at max  e=D ¼ 0:90, for 5:5 rU r 6:2, Ay =D is higher than that for every other e/D considered). An increase in Un at which the peak transverse amplitudes occur as e/D decreases is similarly observed in Sumer and Fredsøe (2006). The effect of wall proximity is a change in the overall shape of the response curve. At a low value of reduced velocity, e.g. U  ¼ 4, the cylinder is in pre-lock-in where the vibration amplitudes are very small. As Un is increased in increments to 4.65, max Amax y =D for the isolated cylinder increases abruptly to reach a peak of 0.57. For the lower e/D of 0.75, Ay =D does not increase abruptly to reach a peak at U  ¼ 4:65, instead growing to reach a peak of 0.53 at U  ¼ 7. As Un is increased further above 7 in  increments, there is a steep decline in Amax y =D before vibration is completely suppressed at U ¼ 8:2. As such the vibration 2 1

CL

Y/D

0.5 0 −0.5 450

0 −1

455

460

465

470

−2 450

475

455

460

tU/D

475

465

470

475

465

470

475

2

L

0

1

C

Y/D

470

tU/D

0.5

0 −0.5 450

455

460

465

470

−1 450

475

455

460

tU/D

tU/D 1.5

0.5

L

1

0

C

Y/D

465

0

−0.5 450

0.5

455

460

465

tU/D

470

475

−0.5 450

455

460

tU/D

Fig. 4. Time histories for transverse vibration (left) and lift force (right). Legend: Red — e=D ¼ 0:75, Blue - - e=D ¼ 0:90, Green    e=D ¼ 1:0, Magenta   e=D ¼ 1:5, Black (bold line) — Isolated cylinder. (a) U  ¼ 5, (b) U  ¼ 6 and (c) U  ¼ 7. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

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111

response for the near-wall case of e=D ¼ 0:75 likens to a mirror-image of that for the isolated cylinder case. As e/D decreases from 10, the shape of the curve tends towards the shape of a mirror-image of the curve for the isolated cylinder. Decrease in e/D widens the lock-in region. At low gap ratio of e/D¼ 0.50, three branches instead of two are observed in the response, whereby a transverse amplitude peak is seen at both U  ¼ 6:9 and U  ¼ 8:6. In Zhao and Cheng (2011), three branches are similarly observed at e/D¼0.3 in turbulent ReD regime. n The relationship between the root-mean-square of the streamwise vibrations Arms x =D, U , and e/D is plotted in Fig. 2(b). rms max Ax =D for the isolated cylinder is very small relative to Ay =D. The effect of decreasing e/D is an increase in the peak Arms x =D up to a maximum at e=D ¼ 0:90. At e=D ¼ 0:90, the peak Arms x =D is increased by greater than 20 times that for the isolated =D. As e/D is decreased further from e=D ¼ 0:90 to 0.50, the peak Arms cylinder, to become of the same order as Amax y x =D decreases. e=D ¼ 0:90 corresponds to the gap ratio at which the alternate vortex shedding has just been completely sup=D, as e/D decreases, there is an increase in the Un at which the peak Arms pressed. Similar to the trend for Amax y x =D occurs. In rms Yang et al. (2011), peak Ax =D increases with a decrease in e/D, and a similar increase in Un at which the peak amplitudes occur is observed.

4.2. Fluid forces n The relationship between root-mean-square lift coefficient C rms L , U and e/D is illustrated in Fig. 3(a). The isolated cylinder lift force curve has a similar shape to that found in VIV studies for isolated cylinder e.g. Prasanth et al. (2006). As e/D decreases to 2.5, the peak rms lift forces decrease and there is a slight shift of the curve to the left so that the peak rms lift forces occur at a lower Un. As e/D decreases from 2.5 to 1.5, the peak rms lift force at e/D ¼1.5 becomes almost twice that of the value at e/D ¼2.5 and the curve shifts to the right so that the peak rms lift force occurs at a higher Un. As e/D decreases from 1.5, the peak rms lift forces decrease. Similar to the vibration response trend, the Un at which the peak rms lift forces occur increases as e/D decreases. Furthermore, the mean lift is not zero. The lift force is directed away from the wall. n The relationship between root-mean-square drag coefficient C rms D , U and e/D is illustrated in Fig. 3(b). As e/D increases to 2.5, the peak rms drag forces increase. As e/D increases above 2.5, the peak rms drag force decreases slightly to reach an approximate constant between e/D¼ 5 and e/D ¼7.5. As pointed out in Sumer and Fredsøe (2006), one characteristic point in variation of CD with respect to e/D is that CD increases with increasing e/D up to a certain value of e/D, then it remains reasonably constant for further increase in e/D. This behavior has been linked in Zdravkovich (1985a) to the thickness of the boundary layer of the approaching flow.

3

0.15

2.5

D

X/D

0.1

C

0.05 0 −0.05 450

2 1.5 1

455

460

465

470

0.5 450

475

455

460

tU/D

D

C

X/D

0.1 0

465

470

475

465

470

475

1.5 1

−0.1 455

460

465

470

0.5 450

475

455

460

tU/D

tU/D

0.4

1.8 1.6

D

0.2

C

X/D

475

2

0.2

0 −0.2 450

470

2.5

0.3

450

465

tU/D

1.4 1.2 1

455

460

465

tU/D

470

475

450

455

460

tU/D

Fig. 5. Time histories for streamwise vibration (left) and drag force (right). Legend: Red — e=D ¼ 0:75, Blue - - e=D ¼ 0:90, Green    e=D ¼ 1:0, Magenta    e=D ¼ 1:5, Black (bold line) — Isolated cylinder. (a) U  ¼ 5, (b) U  ¼ 6 and (c) U  ¼ 7. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

112

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Fig. 6. Evolution of vorticity contours (left) and pressure contours (right) for e=D ¼ 0:75, U  ¼ 6 at tU=D ¼ (a) 1452.25, (b) 1453.25, and (c) 1455.5.

Fig. 4 shows the time evolution of transverse oscillation and the corresponding time evolution of the lift coefficient, while Fig. 5 shows the time evolution of streamwise oscillation and the corresponding time evolution of the drag coefficient, in the range of Un where vibration amplitudes are significant. Through the time histories it can be clearly seen that the streamwise vibration amplitude is significantly larger for the near-wall cylinder compared to the isolated cylinder. The presence of multiple frequencies can be observed in the time histories, particularly clearly in the time evolution of drag force for the near-wall cylinder. Phase angles are calculated through a Hilbert transform of the displacement and force signals, showing a region of Un where there is non-180° phase angle for the near-wall cylinders and thus a net power transfer. This is further highlighted in Section 4.6. 4.3. Flow field The vorticity and pressure contours at various nondimensional timesteps for e/D ¼0.75, U  ¼ 6 are analyzed in Figs. 6 and 7, with Fig. 8 showing the corresponding time history. At this Un, the phase difference between lift and transverse

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113

Fig. 7. Evolution of vorticity contours (left) and pressure contours (right) for e=D ¼ 0:75, U  ¼ 6 at tU=D ¼ (a) 1456.25, (b) 1457.5 and (c) 1458.

displacement is close to zero, while the phase difference between drag and streamwise displacement is close to 60°. When the cylinder is at the extremity of the transverse oscillation (most negative transverse displacement) at tU/D ¼1452.25, the force on the cylinder increases due to the formation of vortex on the side closer to the wake centerline, as illustrated in Fig. 6 (a). A high pressure region on the upper part of the cylinder wall is generated by the negative vortex on the cylinder top. This corresponds to the valley in the lift force signal. As soon as the cylinder begins to move upwards, the front stagnation point shifts towards the cylinder top, and a low pressure on part of the upper surface of the cylinder acts with the movement. The streamwise displacement is at its most negative as illustrated in Fig. 6(b). Once the cylinder begins to decelerate, the stagnation point shifts slowly back to the cylinder bottom. The transverse displacement is at its most positive in Fig. 6(c), while the streamwise displacement is at its most positive in Fig. 7(b). The distribution of surface pressure, CP, averaged over time, is presented in Fig. 9. For the isolated and stationary cylinders, as expected for a symmetric body, the C P –θ curve for the upper and lower halves of the cylinder is symmetric. Furthermore the maximum pressure is attained at the forward stagnation point. Between the points of minimum pressure, a favorable pressure gradient persists along the front surface of the cylinder, and an adverse pressure gradient along the rear

D.M.Y. Tham et al. / Journal of Fluids and Structures 59 (2015) 103–128

0.6

Y/D C

0.4

L

Y/D

0.2 0 −0.2

1.5 1

tU/D = 1455.5

tU/D = 1452.25

2

0.5

CL

114

0 −0.5

−0.4 −0.6 1452

1453

1454

1455

1456

1457

1458

−1 1460

1459

tU/D

1.8

D

1.6

tU/D = 1453.25

0.1

1.4

tU/D = 1456.25

0.05

1.2

D

0.2 0.15

X/D

2

X/D C

C

0.25

1

0

0.8

−0.05 −0.1 1452

1453

1454

1455

1456

1457

1458

1459

0.6 1460

tU/D Fig. 8. Time history of (a) transverse vibration and lift force and (b) streamwise vibration and drag force at e=D ¼ 0:75, U  ¼ 6.

1

U

0.5

CP

0

1.5

e/D = 0.50 e/D = 0.75 e/D = 0.90 e/D = 1.0 e/D = 1.25 e/D = 1.5 vibrating isolated

1

C

P

0.5

1.5

θ

0.5

0

0

−0.5

−0.5

−0.5

−1

−1

−1

−1.5

−1.5

−1.5

−2

0

100

200

θ

300

e/D = 0.50 e/D = 0.75 e/D = 0.90 e/D = 1.0 e/D = 1.25 e/D = 1.5 vibrating isolated

1

P

e/D = 0.50 e/D = 0.75 e/D = 0.90 e/D = 1.0 e/D = 1.25 e/D = 1.5 vibrating isolated stationary isolated

C

1.5

−2

0

100

200

θ

300

−2

0

100

200

300

θ

Fig. 9. Averaged distribution of CP with angular position θ over the circumference of cylinder due to pressure (a) U  ¼ 5, (b) U  ¼ 6 and (c) U  ¼ 7.

surface. For the near-wall cases, in the initial branch of the response, the C P –θ curve is asymmetric, such that the lower half of the cylinder experiences smaller favorable and adverse pressure gradients at the forward and rear surfaces respectively, compared to the upper half of the cylinder. In the lower branch of the response where vibration amplitudes are decreasing, the C P –θ curve tends towards symmetry, though the pressure coefficient at the cylinder bottom (θ¼90°) remains larger than the pressure coefficient at the cylinder top (θ¼270°). E.g. at e/D ¼1.25, at Un ¼5 (in the initial branch of the response), the curve is largely asymmetric, while at Un ¼6 and 7 (in the lower branch of the response), the curve tends towards symmetry. As the cylinder is brought towards the wall, the point at which maximum pressure is attained moves towards the upper half of the cylinder. At Un ¼5 and 6, the forward stagnation pressure coefficient, Cp0, increases for e/D¼1.5, then it monotonically decreases until e/D¼0.75. At Un ¼7, the forward stagnation pressure coefficient, Cp0, increases for e/D ¼1.5, then it decreases until e/D¼0.75, before increasing again for e/D ¼0.50. At Un ¼5, the base suction coefficient,  C pb , decreases monotonically as e/D goes down. The vortex shedding mode at e/D¼10 and for the isolated cylinder is 2S. In the 2S mode of vortex shedding, a single vortex is alternately shed from each side of the cylinder per vortex-shedding cycle, in a line parallel to the free stream, as

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115

Fig. 10. Instantaneous vorticity contours for isolated-cylinder (left) and for e=D ¼ 1:5 (right): (a) U  ¼ 5 and (b) U  ¼ 6.

Fig. 11. Instantaneous vorticity contours for e=D ¼ 1:25 (left) and for e=D ¼ 1 (right): (a) U  ¼ 5, (b) U  ¼ 6 and (c) U  ¼ 8.

seen at U  ¼ 6 in Fig. 10. The vortex shedding develops into two parallel rows and coalesces in the far wake to give rise to the Cð2SÞ mode of vortex shedding, for the values of Un where the cylinder executes high-amplitude transverse vibrations e.g. U  ¼ 5 in Fig. 10. The 2S and Cð2SÞ vortex shedding modes are described in Williamson and Roshko (1988).The 2S and Cð2SÞ vortex shedding mode is similarly reported in the literature for laminar-VIV of isolated cylinder such as Singh and Mittal (2005), Zhou et al. (2012), and Tang et al. (2013). When the cylinder is brought closer to the wall below a critical e/D (Huerre and Monkewitz, 1990), several changes occur in the flow around the cylinder, such as the break-up of symmetry in the flow and the suppression of vortex shedding. The

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Fig. 12. Instantaneous vorticity contours for e=D ¼ 0:9 (left) and for e=D ¼ 0:75 (right): (a) U  ¼ 5, (b) U  ¼ 6 and (c) U  ¼ 8.

Fig. 13. Instantaneous vorticity contours for e=D ¼ 0:60 (left) and for e=D ¼ 0:50 (right): (a) U  ¼ 6 and (b) U  ¼ 8:5.

suppression of vortex shedding is closely linked with the asymmetry in the development of the vortices on both sides of the cylinder. When the cylinder is placed close to a wall, the shear layer on the wall-side will not develop as strongly as the shear layer on the free-stream side. This supposedly leads to a weak interaction between the shear layers, or to practically no interaction, resulting in partial or complete suppression of the regular vortex shedding. The combined VIV and wall-induced oscillations are due to this asymmetric wake dynamics, which is also reported in Sumer and Fredsøe (2006). At e/D¼1.5, alternate vortex shedding continues to take place, although the positive vortices are weaker than the opposing negative vortices. For high-amplitude vibrations e.g. at U  ¼ 5 in Fig. 10, the vortex shedding mode is Cð2SÞ, as it is in the isolated

D.M.Y. Tham et al. / Journal of Fluids and Structures 59 (2015) 103–128

0.6

0.6

U* = 4.7

0.4

0

0

0.2

0.2

0.2

0.4

0.4

0.4

0

0.1

0.2

0.3

0.1

0

X/D

0.1

0.2

0.3

0.1

Y/D

Y/D

0.2

0

0

0.2

0.2

0.2

0.4

0.4

0.4

0

0.1 X/D

0.2

0.3

0.3

U* = 7.8

0.4

0.2

0

0.2

0.6

U* = 7.1

0.4

0.2

0.1 X/D

0.6

U* = 7.05

0.1

0

X/D

0.6 0.4

U* = 6

0.2 Y/D

0

0.1

Y/D

0.4

0.2 Y/D

Y/D

0.6

U* = 5

0.4

0.2

117

0.1

0

0.1

0.2

0.3

X/D

0.1

0

0.1

0.2

0.3

X/D

Fig. 14. XY-trajectory of freely vibrating near-wall cylinder at e/D¼ 0.75.

cylinder case. The vortices shed develops into two parallel rows and coalesces in the far wake. The effect of the wall proximity at e/D ¼1.5 is such that the near-wall vortices are considerably weaker than the vortices for the isolated cylinder case. As e/D decreases to 1.0, alternate vortex shedding is largely suppressed as seen in Fig. 11. A single discrete positive vortex is shed which dissipates further downstream, in the range of Un corresponding to the lower branch of the response e.g. U  ¼ 6. As e/D decreases to 0.90, vortex shedding is completely suppressed for U  o4:5 and U  4 7:4. In the synchronization range, alternate vortex shedding is suppressed as seen in Fig. 12. The upper shear layer emanates from the cylinder in a series of negative vortices. The lower shear layer is weak in magnitude and small in size, and no shedding of discrete positive vortices occurs. As e/D further decreases to 0.75, vortex shedding is completely suppressed for U  o 4:5 and above U  4 8:1. Similar to e=D ¼ 0:90, alternate vortex shedding is suppressed in the synchronization range due to the effect of the wall proximity, as seen in Fig. 13. The suppression of alternate vortex shedding becomes even more pronounced as e/D is further reduced from e=D ¼ 0:75 to e=D ¼ 0:50. Vortex shedding mode at e=D ¼ 0:60 and e=D ¼ 0:50 remains as the single vortex shedding for the entirety of the response. This shows that there are critical e/D below which there is a break-up of symmetry and suppression of regular vortex shedding. The suppression of alternate vortex shedding leads to changes in the vibration and hydrodynamic force response. Generally, the flow asymmetry gradually decreases as e/D increases. The flow behavior becomes increasingly similar to that of an isolated cylinder as e/D increases. 4.4. Trajectory The XY-trajectory of the vibration for the isolated cylinder takes the form of a figure of eight, as observed in many available literature for VIV of isolated cylinder (Sarpkaya, 2004; Zhou et al., 2012). As the cylinder is brought towards the wall, the XY-trajectory becomes an enclosed clockwise elliptic circle whose top leans towards downstream in the initial branch and upstream in the lower branch (corresponding to the change in phase difference between force and displacement). The XY-trajectory of the near-wall cylinder at e=D ¼ 0:75 is shown in Fig. 14. Tsahalis and Jones (1981) similarly showed experimentally that the XY-trajectory of a near-wall cylinder is an oval-shape. The change in trajectory from initial to lower branch corresponds to the shift in phase difference between force and displacement to 180°. 4.5. Frequency characteristics As it is known for the isolated-cylinder, the transverse vibration is characterized by single-mode response. The frequencies are computed for the forces and vibration using a Fast Fourier Transform. In the synchronization range, the lift force has the presence of the third harmonic, at a frequency thrice that of the transverse vibration, as similarly reported by Tang et al. (2013). The streamwise motion for the isolated-cylinder has a dominant mode which is at a frequency twice that of the transverse vibration frequency, and a smaller response frequency which is at the transverse vibration frequency. This

f

Lift

0.5

0.5

0.2

0.3

0.4

0.5

0.6

0.7

0 0.8

f

Drag

0.5

0

0.5

0

0.1

0.2

x

Strouhal Number

Lift

1

0.5 0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.8

Lift

1

0.5 0.5

0.2

0.3

0.4

0.5

0.6

0.7

0 0.8

Single−Sided Amplitude Spectrum of Ax

1

Single−Sided Amplitude Spectrum of CL

y

Single−Sided Amplitude Spectrum of A

Dispy

f

0.1

fDrag

1

1 0.5 0.5

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

f

0 0.8

Dispx

1

fDrag

1

0.5 0.5

0

0

0.1

0.2

Strouhal Number

0.3

0.4

0.5

0.6

0.7

0 0.8

fDispy f

1

Lift

0.5

0.5

0

1

0

0.1

0.2

0.3

0.4

0.5

Strouhal Number

0.6

0.7

0 0.8

D

Single−Sided Amplitude Spectrum of Ax

L

Strouhal Number

Single−Sided Amplitude Spectrum of C

Single−Sided Amplitude Spectrum of Ay

0 0.8

0.7

Strouhal Number

f

0

0.6

fDispx

Strouhal Number

0

0.5

x

Single−Sided Amplitude Spectrum of A

Dispy

f

Single−Sided Amplitude Spectrum of CL

y

Single−Sided Amplitude Spectrum of A

f

0

0.4

Strouhal Number

1

0

0.3

Single−Sided Amplitude Spectrum of CD

0.1

Dispx

Single−Sided Amplitude Spectrum of CD

0

1

f

fDispx f

Drag 1

1

0.5

0.5

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.8

Single−Sided Amplitude Spectrum of C

0

1

1

Single−Sided Amplitude Spectrum of CD

1

Single−Sided Amplitude Spectrum of A

Single−Sided Amplitude Spectrum of A

fDispy

Single−Sided Amplitude Spectrum of CL

D.M.Y. Tham et al. / Journal of Fluids and Structures 59 (2015) 103–128

y

118

Strouhal Number

Fig. 15. Power spectral analysis for transverse vibration and lift (left), streamwise vibration and drag (right) at e=D ¼ 0:75. (a) U  ¼ 5. (b) U  ¼ 7. (c) U  ¼ 7:32. (d) U  ¼ 8.

1 0.5 0.5

0.2

0.3

0.4

0.5

0.6

0.7

0 0.8

0.5 0.5

0 0

0.1

0.2

1 0.5 0.5

0 0.2

0.3

0.4

0.5

0.6

0.7

0 0.8

D

f

Dispx

fDrag

fLift 1 0.5 0.5

0 0.4

0.5

0.6

0.7

0 0.8

0.5

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.8

D

Single−Sided Amplitude Spectrum of Ax

Dispy

0.3

1

0.5

0

Single−Sided Amplitude Spectrum of CL

f

0.2

0 0.8

0.7

Strouhal Number

1

0.1

0.6

1

Strouhal Number

0

0.5

x

Single−Sided Amplitude Spectrum of A

1

Single−Sided Amplitude Spectrum of CL

Single−Sided Amplitude Spectrum of Ay

Dispy

fLift

0.1

0.4

Strouhal Number

f

0

0.3

fDispx

1

1

f

Drag

0.5

0.5

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.8

Strouhal Number

f

Dispy

1

f

Lift

0.5 0.5

0

0.1

0.2

0.3 0.4 0.5 Strouhal Number

0.6

0.7

0 0.8

Single−Sided Amplitude Spectrum of A

1

Single−Sided Amplitude Spectrum of CL

Single−Sided Amplitude Spectrum of Ay

x

Strouhal Number

0

Single−Sided Amplitude Spectrum of C

0.1

1

Single−Sided Amplitude Spectrum of C

0

Drag

1

fDispx fDrag

1

0.5 0.5

0

0.1

0.2

0.3 0.4 0.5 Strouhal Number

0.6

0.7

0 0.8

Single−Sided Amplitude Spectrum of CD

0

Dispx

f

Single−Sided Amplitude Spectrum of CD

Lift

Single−Sided Amplitude Spectrum of Ax

f

f 1

Strouhal Number

Single−Sided Amplitude Spectrum of Ay

119

L

fDispy 1

Single−Sided Amplitude Spectrum of C

Single−Sided Amplitude Spectrum of Ay

D.M.Y. Tham et al. / Journal of Fluids and Structures 59 (2015) 103–128

Fig. 16. Power spectral analysis for transverse vibration and lift (left), streamwise vibration and drag (right) at e=D ¼ 1:5. (a) U  ¼ 2. (b) U  ¼ 5. (c) U  ¼ 7. (d) U  ¼ 8.

has been observed for the isolated-cylinder in many available literature such as in Vandiver and Jong (1987) or Anagnostopoulos (1994). The drag force for the isolated cylinder has a dominant mode which is at a frequency twice that of the transverse vibration frequency, and a smaller response frequency which is at the transverse vibration frequency.

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4 1.05

1

3

y

f /f

n

0.95

2

0.9

0.85 4

4.5

5

5.5

6

6.5

7

7.5

8

1

0 2

4

f x / fn

6 U*

8

10

1.05

3.5

1

3

0.95

2.5

4

0.9 0.85 4.5

2

5

5.5

6

6.5

7

7.5

8

1.5 1 0.5 0 2

3

4

5

6 U*

7

8

9

Fig. 17. Vibration frequency responses at m ¼ 10: (a) transverse, (b) streamwise. ð    Þ Strouhal's law, ( (

) e/D ¼0.9, (

) e/D¼ 1.0, (

) e/D ¼ 1.25, (

) e/D ¼1.5), (

) e/D ¼ 2.5), (

) e/D¼ 5, (

10

) e/D¼ 0.5, (

) e/D ¼7.5, (

) e/D¼ 0.6, (

) e/D ¼10 and (

) e/D¼ 0.75,

) isolated cylinder.

As the cylinder is brought to the wall so that the alternate vortex shedding becomes more suppressed, the transverse vibration remains characterized by single-mode response. The spectral plots for e=D ¼ 0:75 are shown in Fig. 15. However the frequency characteristics for the hydrodynamic forces varies at different branches of the response. In the pre-lock-in range and initial branch of the response, the vibrations and forces have the same dominant frequencies. However in the lower-branch of the response, the frequency twice that of the transverse vibration dominant frequency grows to become the dominant frequency for the hydrodynamic forces. In post-lock-in range of the response, the dominant frequency for the forces once again becomes that of the transverse vibration dominant frequency. As the cylinder is brought further from the wall so that the alternate vortex shedding is less suppressed, the frequency characteristics tend towards those of the isolated-cylinder. The spectral plots for e=D ¼ 1:5 are shown in Fig. 16. At e=D ¼ 1:5, the transverse vibration is similarly characterized by single-mode response. However, the drag force dominant frequency which is twice that of the transverse vibration dominant frequency grows to become the dominant frequency earlier in the initial branch instead of later in the lower branch. Also, the lift force dominant frequency remains at that of the transverse vibration frequency for the entire response, following the isolated-cylinder case. In the synchronization range, the lift force consists of three modes, where the dominant mode instead remains at the frequency of the transverse vibration, similar to the isolated-cylinder case. The second and third modes for the lift force are at frequencies twice and thrice the dominant mode respectively. The frequency responses are plotted using the dominant frequencies. The relationship between vibration frequency ratio fn (ratio of vibration frequency, to the natural frequency of the cylinder), e/D and Un is illustrated in Fig. 17. A characteristic feature of VIV is that of the lock-in phenomenon, where the vortex-shedding frequency diverges from that predicted by Strouhal's Law (vortex shedding frequency of a stationary cylinder) and becomes equal or close to the cylinder natural frequency of vibration. The vibration frequency generally increases with Un at all e/D considered. The general effect of decreasing e/D is a decrease in vibration frequency, such that fn deviates further from unity. Experimental studies by Wang and Tan (2013) in the high ReD regime showed a decrease in frequency as e/D decreased from 1.0 to 0.5. For e=D r5, the

D.M.Y. Tham et al. / Journal of Fluids and Structures 59 (2015) 103–128

200

e/D = 0.5 e/D = 0.6 e/D = 0.75 e/D = 0.9 e/D = 1.0 e/D = 1.25 e/D = 1.5 e/D = 2.5 e/D = 5 e/D = 7.5 e/D = 10 isolated (b = 0.05)

L

Phase Difference between Y and C

121

150

100

50

0

2

4

6 U*

8

10

Phase Difference between X and CD

200

150

100 e/D = 0.5 e/D = 0.6 e/D = 0.75 e/D = 0.9 e/D = 1.0 e/D = 1.25 e/D = 1.5 e/D = 2.5 e/D = 5 e/D = 7.5 e/D = 10 isolated (b = 0.05)

50

0

2

4

6

8

10

12

U* Fig. 18. Phase difference at m ¼ 10 between (a) transverse displacement and lift force and (b) streamwise displacement and drag force.

vortex shedding locks to the inline oscillation frequency of the cylinder and the body responds favorably to the drag forcing. Between e=D A ½5; 7:5, the inline vibration frequency response recovers to that of the isolated cylinder and the vortex shedding cannot have an appreciable impact on the forces and amplitudes, and it cannot lead to a significant resonance in the streamwise direction. 4.6. Average phase difference between force and displacement The average phase difference between lift force and transverse displacement was obtained through the use of a Hilbert transform, and illustrated in Fig. 18(a). Hilbert transform of a real signal generates an imaginary signal, whose phase equals the phase of the real signal. Therefore, for every signal, using Hilbert transform, one can obtain variation of phase angle with time or the instantaneous phase. For the isolated cylinder, the average phase difference is found to be close to 0° (i.e. the force and vibration are in-phase) for low values of Un before the onset of lock-in. As Un is increased, the average phase difference increases in the synchronization region to reach a value close to 180° (i.e. the force and vibration are out-ofphase). This is not as sudden as the jump observed in high-Re experiments. A similar trend for the non-abrupt rise in phase angle between lift force and transverse vibration is observed in laminar-VIV numerical studies such as Tang et al. (2013) and Leontini et al. (2006). The change in phase angle from a value close to 0° to a value close to 180° marks the transition from the initial branch to the lower branch. The average phase difference between drag force and streamwise displacement was obtained similarly through the use of a Hilbert transform, and illustrated in Fig. 18(b). The phase difference between the drag force and streamwise vibration is 180° for the isolated cylinder case for all Un considered. There is no net power transfer from the flow to the cylinder, thus streamwise vibration amplitudes remain small.

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For the near-wall cases, a similar trend is observed for the phase difference between lift force and transverse vibration. The initial branch has a phase angle close to 0°, which grows in value as the transition to the lower branch occurs. It is in the initial branch that the amplitude reaches its highest value for the reduced velocities investigated. Where the maximum amplitudes occur, the vibration and forces are not completely out-of-phase i.e. not 180°. It is just before the occurrence of the rise in phase difference to 180°, that the peak amplitudes occur. For example at e=D ¼ 0:75, the maximum transverse displacements occur from U  ¼ 6:7 to 7.2, which corresponds to the reduced velocity range for which the average phase difference is increasing but is not 180°, and there is a net power transfer to the cylinder from the flow. As Un increases further, the phase difference grows to become 180° where there is no net power transfer, marking the transition to the lower branch of the response where Amax y =D decreases. As the cylinder is brought towards the wall, the phase difference between drag force and streamwise vibration becomes closer to zero in the pre-lock-in range (i.e. drag force and streamwise vibration are not completely out of phase). Upon entering lock-in range, in the initial branch, the phase difference then grows to reach approximately 60°, before decreasing slightly, corresponding to the peak streamwise vibrations. There is a net power transfer from the flow to the vibrating cylinder in the streamwise direction, giving rise to the higher streamwise vibration amplitudes for the near-wall cases. As Un is increased further, the phase difference abruptly increases to 180°. This corresponds to the range of Un for which the streamwise vibration amplitudes decline. As e/D decreases, there is an increase in the Un at which the phase angle between drag force and streamwise vibration becomes 180°, corresponding to the higher Un at which the peak streamwise amplitudes occur as e/D decreases. The sudden rise in phase difference to 180° occurs when the dominant frequency for drag becomes twice the dominant frequency of the streamwise vibration. The general trend is that as e/D increases so that the alternate vortex shedding occurs and becomes more pronounced, the phase difference curve tends towards that of the isolated cylinder, corresponding to the lower streamwise vibration amplitudes.

4.7. Correlations for vibration amplitudes and force coefficients With the uniform trends of vibration amplitudes and force coefficients with the variation of e/D in all cases in the low ReD regime, observed in Fig. 19, it is our interest to provide empirical equations as a function of e/D. Such correlations can have implications for Offshore Industry Guidelines for freespan subsea pipelines. We model peak vibration amplitude and 0.65

0.20 max /D y Arms/D x

Peak A Peak

0.18 0.16

x

0.55

Peak Arms/D

Peak A

max /D y

0.6

0.14 0.5 0.12 0.45

0.4 0.5

0.1

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0.08 1.5

e/D 3.5

3.5

Peak Cmax L max D

Peak C

3

2.5

2.5

2

2

1.5

1 0.5

Peak Cmax D

Peak C

max L

3

1.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1 1.5

e/D Fig. 19. (a) Peak vibration amplitudes (b) peak force coefficients with the variation of e/D at ReD ¼100, m ¼ 10. Points represent present results.

D.M.Y. Tham et al. / Journal of Fluids and Structures 59 (2015) 103–128

123

Table 1 Correlation of vibration amplitudes and force coefficients for e=D r 1:5. R2

r

α

β

Arms x D Amax y D max

 0.22845

0.43019

 0.04163

0.981

 0.29379

0.74559

0.11956

0.968

 0.53328  0.39864

2.5775 2.5363

 0.039408 0.17363

0.939 0.917

CL max CD

γ

Table 2 Correlation of vibration amplitudes and force coefficients for e=D 41:5. γ

R2

0.04952

1.224

0.99

 0.02962

3.566

0.95

0.002374  0.00127

0.557 0.024

0.99 0.99

r

α

β

Arms x D Amax y D CL max max

0 0 0 0

CD

peak force coefficients as a function of e/D by the following form: r ¼ αðe=DÞ2 þ βðe=DÞ þ γ where the coefficients α, β and γ are constants, and r is the parameters of interest such as determine the relations between these coefficients, as shown in Tables 1 and 2.

ð16Þ max =D; C max ; C max Arms L D . x =D; Ay

We then

4.8. Effect of mass ratio The vibration, force and frequency responses at m ¼ 5 are illustrated in Figs. 20, 21 and 22, respectively. The phase difference plots at m ¼ 5 are shown in Fig. 23. The general trends for the force and response at varying e/D for the lower m ¼ 5 are similar to those for m ¼ 10. For the lower mn, there is an increase in maximum lift forces, but a decrease in rootmean-square drag forces. There is an increase in the peak transverse and streamwise vibration amplitudes. There is a general shift to the left for the response and hydrodynamic force curves, and thus a decrease in the Un at which the peak hydrodynamic forces and vibration amplitudes occur. There is also a widening of the lock-in range for the lower mn. The force and vibration frequencies are increased with the lowering of mn. Govardhan and Williamson (2000) similarly reported increases in frequencies as mn was reduced for an isolated cylinder case.

5. Conclusions Two-dimensional flow over a vibrating two degree-of-freedom circular cylinder near a fixed wall has been studied numerically using the fully implicit combined field formulation based on ALE finite elements. The fixed wall is used to include the shear-layer effect from the bottom wall in considering the near-wall VIV characteristics. Parameters such as vibration and vortex shedding frequency, drag and lift forces, transverse and streamwise vibration amplitudes are characterized at different gap ratios. In laminar near-wall VIV, the vibration frequencies in the lock-in region decrease as e/D decreases. The two amplitude branches are identified in the response for e=D ¼ 0:75 and above, while a third branch develops at e=D r 0:60 between the initial and the lower-type branches. The origin of enhanced streamwise vibration amplitudes, due to wall proximity, has been investigated through the fluid forces, response characteristics and their phase relations for both the isolated and near-wall cylinder configurations. As gap ratio e/D is decreased from 1.5, the peak lift forces decrease, corresponding to the decrease in peak transverse vibration amplitudes. The peak drag forces similarly decrease, however the peak streamwise vibration amplitudes undergo a trend where they reach a maximum at e=D ¼ 0:9 but decrease as gap ratio decreases further. The phase difference curves show the range of Un for which there is a maximum net power transfer, i.e. where the phase difference is around 40–90°. For the isolated cylinder, the phase difference between drag force and streamwise displacement remains at 180° and there is thus no net power transfer in the streamwise direction. Although the peak drag forces at e=D ¼ 0:9 are lower than those at higher gap ratios, there is net power transfer in the streamwise direction, thus resulting in the highest peak streamwise vibration amplitudes. As gap ratio decreases further, the drag forces decrease (whilst the phase difference curve remains in the same shape), thus leading to a decrease in peak streamwise vibration amplitudes. The change in net power transfer occurs due to the change in flow field, where at e=D ¼ 0:9 the most significant change occurs, i.e. alternate vortex shedding has just been completely suppressed and only negative vortices which travel further downstream are shed. The increase in phase difference between drag and streamwise

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0.2

m* = 10, e/D = 0.75 m* = 5, e/D = 0.75 m* = 10, e/D = 0.9 m* = 5, e/D = 0.9

0.6

0.15

0.4

Axrms/D

A

max /D y

0.5

m* = 10, e/D = 0.75 m* = 5, e/D = 0.75 m* = 10, e/D = 0.9 m* = 5, e/D = 0.9

0.3

0.1

0.2 0.05 0.1 0

0 2

3

4

5

6

7

8

9

10

2

3

4

5

6

U*

7

8

9

10

U*

Fig. 20. Variation of the maximum amplitudes with reduced velocity at m ¼ 5 and 10 for e=D ¼ 0:75 and 0.9: (a) transverse and (b) streamwise.

0.45

m* = 10, e/D = 0.75 m* = 5, e/D = 0.75 m* = 10, e/D = 0.9 m* = 5, e/D = 0.9

1

m* = 10, e/D = 0.75 m* = 5, e/D = 0.75 m* = 10, e/D = 0.9 m* = 5, e/D = 0.9

0.4 0.35

0.8

Crms

0.6

0.25

D

CL

rms

0.3

0.4

0.2 0.15 0.1

0.2

0.05 0

0 2

3

4

5

6

7

8

9

10

2

3

4

5

U*

6

7

8

9

10

U*

Fig. 21. Variations of force coefficients with reduced velocity at m ¼ 5 and 10 for e=D ¼ 0:75 and 0.9: (a) lift coefficient and (b) drag coefficient.

m* = 10, e/D = 0.75 m* = 5, e/D = 0.75 m* = 10, e/D = 0.9 m* = 5, e/D = 0.9

1.6 1.5

1.5 1.4

fx / f n

fy / f n

1.4 1.3 1.2

1.3 1.2

1.1

1.1

1

1

0.9

0.9

0.8

m* = 10, e/D = 0.75 m* = 5, e/D = 0.75 m* = 10, e/D = 0.9 m* = 5, e/D = 0.9

1.6

4

5

6

U*

7

8

0.8

4

5

6

7

8

U*

Fig. 22. Influence of mass ratio at varying reduced velocities on vibrational frequency responses: (a) transverse and (b) streamwise.

displacement to 180° occurs when the drag dominant frequency becomes twice that of the streamwise vibration dominant frequency. Decrease in gap e/D widens the lock-in region and the wall proximity generally reduces the vibration frequency of the streamwise direction when e=D r5:0. Between e=D A ½5; 7:5, the trend of inline vibration frequency recovers to that of the isolated cylinder. The effect of decreasing mn is a shift in the response curves to the left, where the peak vibration amplitudes and hydrodynamic forces occur at a lower Un. Also, there is a general increase in peak vibration amplitudes and

200

Phase difference between X and CD

Phase difference between Y and CL

D.M.Y. Tham et al. / Journal of Fluids and Structures 59 (2015) 103–128

m* = 10, e/D = 0.75 m* = 5, e/D = 0.75 m* = 10, e/D = 0.9 m* = 5, e/D = 0.9

150

100

50

0

4

5

6

7

8

200

125

m* = 10, e/D = 0.75 m* = 5, e/D = 0.75 m* = 10, e/D = 0.9 m* = 5, e/D = 0.9

150

100

50

0

4

5

U*

6

7

8

U*

Fig. 23. Influence of mass ratio at varying reduced velocities and gap ratio on phase difference between (a) transverse displacement and lift force and (b) streamwise displacement and drag force.

hydrodynamic forces, though there is a slight decrease in the root-mean-square drag forces. Vibration frequencies in the lock-in region also increase. Correlations for variation of vibration amplitudes and forces with gap ratio have been proposed for VIV of near-wall cylinder, which can have implications for setting of offshore industry guidelines. In future, threedimensional studies will be considered for a freely vibrating cylinder in the vicinity of stationary wall. Due to the presence of near-wall effects, the onset of three-dimensionality can occur at much lower Reynolds in the steady flow regime (Rao et al., 2013b).

Acknowledgments The authors wish to acknowledge support from the Ministry of Education, Academic Research Fund (AcRF), Singapore with Grant number R-265-000-420-133 and Singapore Maritime Institute. Finally, the first author acknowledges Drs. Subhankar Sen, Ravi Mysa and Kun Yang for their suggestions.

Appendix A. Verification and convergence study Simulations for flow past a circular cylinder at ReD ¼ 100 are carried out in a computational domain where upstream distance Lu ¼ 30D, downstream distance Ld ¼ 45D and cylinder depth H ¼10D. The results serve to verify the validity and accuracy of the computational method used in this study. A.1. Verification for isolated cylinder Validation of the numerical method is conducted with two-dimensional simulations of an unconfined flow past a stationary circular cylinder at ReD ¼100 in the laminar flow regime. A comparison is performed between the present results obtained and those reported in previous literature (see Table A1). The results are in close agreement with those reported in studies that were conducted. Therefore the accuracy and validity of the computational method utilized is verified and proven suitable for this study. Table A1 Validation for isolated stationary cylinder. Study

St

CD

CL

Present Study Mittal and Raghuvanshi (2001) Cao and Wan (2010) Henderson (1995) Kravchenko et al. (1999) Experiments St ¼ 0.21(1–20/ReD)

0.168 0.168 0.168 0.166 0.167 0.168

1.389 1.402 1.3932 1.35 – –

0.352 0.355 – 0.33 0.35 –

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Table A2 Comparison of results for isolated vibrating cylinder. max

rms

Study

St

Amax y D

Arms x D

CL

CD

Present Study Prasanth et al. (2006) Anagnostopoulos and Bearman (1992)

0.198 0.200 –

0.57 0.57 0.55

0.01 0.01 –

0.91 0.93 –

0.43 0.42 –

Fig. A1. (a) A representative computational domain with finite element mesh for near-wall cylinder and (b) close-up view of the mesh in its initial (left) and deformed (right) states.

Simulations were carried out for an isolated 2-DoF vibrating cylinder at U  ¼ 4:92 with domain size set to be similar to the study in Prasanth et al. (2006), and the results are compared (see Table A2). The slight differences in the results can be attributed to the hysteresis effects which were present in their study but not in the current study. Vibration amplitude results for an isolated 1-DoF vibrating cylinder in the laminar regime, in Anagnostopoulos and Bearman (1992), are included for comparison. A.2. Convergence study The parameters used for the mesh convergence study are ReD ¼100, m ¼ 10 and U  ¼ 5, at the different gap ratios. Three different grid resolutions are used: (i) 24K mesh: 23 818 elements, (ii) 46K mesh: 46 414 elements, (iii) 97K mesh: 97 090 elements. Fig. A1 shows the computational domain and a close-up view of the mesh around the cylinder for 23 818 elements. The percentage differences were obtained with a comparison with the finest resolution tested (97K mesh). Tables A3 and A4 compare the results among the three grid systems. The percentage difference were all below 1% and thus is acceptable. Based on the mesh convergence study results, subsequent simulations are run where the computational domains are meshed with a resolution of 23 818 elements. Table A5 shows the temporal convergence results for e=D ¼ 0:75, U  ¼ 5 at ReD ¼ 100. This shows that timestep Δt ¼ 0:050 is small enough to compute the unsteady flows accurately. The Δt of 0.050 is used for all computations in this study presented hereafter. A.3. Effect of downstream distance The downstream distance, Ld, was increased from 45D to 90D for e=D ¼ 0:75, U  ¼ 5. Table A6 shows the results. There are negligible effects on the results with an increase in downstream distance. Ld ¼ 45D was thereby chosen for the subsequent simulations. The cylinder depth H is kept at 10D for subsequent simulations, to be consistent with the isolated-cylinder case (blockage ratio b¼0.05) where the cylinder is also placed 10D from the upper boundary. The wall is then brought towards the cylinder to study the effects of wall proximity.

D.M.Y. Tham et al. / Journal of Fluids and Structures 59 (2015) 103–128

127

Table A3 Mesh convergence study for e/D¼ 0.75. Gap ratio e/D ¼0.75

Parameter

24K

46K

97K

Amax y

0.3278 (0.64%)

0.3260 (0.09%)

0.3257

D Arms x D

0.0315 (0.32%)

0.0314 (0.01%)

0.0314

0.1856 (0.38%)

0.1854 (0.27%)

0.1849

0.8782 (0.52%) 1.215 (0.25%)

0.8761 (0.27%) 1.212 (0.01%)

0.8737 1.212

0.1613 (0.19%)

0.1615 (0.06%)

0.1616

CL rms CL CD rms CD

Table A4 Mesh convergence study for e/D ¼1.0. Parameter

Gap ratio e/D¼ 1.0 24K

46K

97K

Amax y

0.4251 (0.54%)

0.4234 (0.14%)

0.4228

D Arms x D

0.05267 (0.34%)

0.05260 (0.21%)

0.05249

0.2480 (0.56%)

0.2492 (0.08%)

0.2494

1.147 (0.44%) 1.456 (0.48%)

1.146 (0.35%) 1.455 (0.41%)

1.142 1.449

0.2744 (0.40%)

0.2749 (0.22%)

0.2755

CL rms CL CD rms CD

Table A5 Timestep independence study. max

rms

max

rms

Δt

St

Amax y D

Arms x D

CL

CL

CL

CD

CD

CD

0.050 0.025

0.183 0.183

0.328 0.328

0.0315 0.0317

0.186 0.185

1.44 1.44

0.878 0.880

1.22 1.21

1.49 1.49

0.161 0.162

Table A6 Effects of downstream distance at e=D ¼ 0:75, U  ¼ 5. max

rms

max

rms

Ld

St

Amax y D

Arms x D

CL

CL

CL

CD

CD

CD

45D 90D

0.183 0.183

0.328 0.328

0.0315 0.0317

0.186 0.187

1.44 1.44

0.878 0.879

1.22 1.22

1.49 1.50

0.161 0.162

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