Accepted Manuscript

Free surface wave interaction with an oscillating cylinder

Canan Bozkaya, Serpil Kocabiyik

PII: S0893-9659(13)00229-2DOI: http://dx.doi.org/10.1016/j.aml.2013.07.009Reference: AML 4420

To appear in: Applied Mathematics Letters

Received date: 24 June 2013Accepted date: 30 July 2013

Please cite this article as: C. Bozkaya, S. Kocabiyik, Free surface wave interaction with anoscillating cylinder, Appl. Math. Lett. (2013), http://dx.doi.org/10.1016/j.aml.2013.07.009

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Applied Mathematics Letters 00 (2013) 1–6

AppliedMathsLetters

Free surface wave interaction with an oscillating cylinder

Canan Bozkaya1, Serpil Kocabiyik2∗

1Department of Mathematics, Middle East Technical University, Ankara, Turkey2Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S7, Canada

Abstract

The numerical solution of the special integral form of two-dimensional continuity and unsteady Navier-Stokes equations is usedto investigate vortex states of a horizontal cylinder undergoing forced oscillations in free surface water wave. This study aimsto examine the consequence of degree of submergence of the cylinder beneath free surface at Froude number 0.4. Calculationsare carried out for a single set of oscillation parameters ata Reynolds number ofR = 200. Two new locked-on states of vortexformation are observed in the near wake region. The emphasisis on the transition between these states, which is characterized interms of the lift force on the cylinder and the instantaneouspatterns of vortex structures and pressure contours in the near wake.

c© 2011 Published by Elsevier Ltd.

Keywords: Free surface, Cylinder, Forced oscillation, Wake dynamics, Lift force, Computation.

1. Introduction

The interaction of a free surface wave motion with moving cylindrical bodies has been principally the subjectof experimental studies. Computations of nonlinear viscous free surface problems including cylindrical bodies arerelatively few [1, 2]. In these studies the free surface effects on the near wake development and vortex formationmodes are investigated. Neither fluid force descriptions nor the link between fluid forces and the changes in wakestructure is discussed. In this paper, a viscous incompressible two-fluid model with a circular cylinder is investigatednumerically. The present two-fluid model involve the fluids in the regionsΩ1 andΩ2 with densities,ρ1, ρ2, anddynamic viscosities,µ1, µ2, entering into the domain with uniform velocityU at the inlet and leaving through theoutlet boundary. The circular cylinder of radius,d, is submerged in the fluid regionΩ2 at the distanceh∗ below theundisturbed free surface. Initially, an infinitely long circular cylinder whose axis coincides with thez-axis is at rest,and then, at timet = 0, the cylinder starts to perform streamwise oscillations about thex-axis. The imposed oscillatorycylinder displacement is assigned byx(t) = Acos(2π f t). The aims of present investigation are (i) to characterizethepossible states of the wake and (ii) to establish link between the changes in wake dynamics of the cylinder and liftforce at three different submergence depths.

The relevant dimensionless parameters are the Reynolds numberR2 = Ud/ν2 (R1 = Ud/ν1); the forcing amplitudeof the cylinder oscillations,A = A∗/d; the frequency ratio,f / f0, with f = d f∗/U and f0 = d f∗0 /U being the dimen-sionless forcing frequency of the cylinder oscillation andthe natural vortex shedding frequency for correspondingstationary cylinder case in an unbounded medium; the cylinder submergence depth,h = h∗/d, and the Froude number,

∗Corresponding author.Email addresses:bcanan@metu.edu.tr (Canan Bozkaya1), serpil@mun.ca (Serpil Kocabiyik2)

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C. Bozkaya et al./ Applied Mathematics Letters 00 (2013) 1–6 2

Fr = U/√

dg∗. Here,ν1 = µ1/ρ1, ν2 = µ2/ρ2 are the kinematic viscosities of the fluids inΩ1 andΩ1, respectively,f ∗

is the dimensional forcing frequency of cylinder oscillation, f ∗0 is the dimensional natural vortex shedding frequencyof a stationary cylinder,g∗ is the acceleration due to gravity,~g∗ = (0, g∗, 0), t∗ = td/U is the dimensional time, andt being the dimensionless time. The dimensionless fluid pressure, p, is defined byp/ε = p∗/ρ2U2, whereε = ρ1/ρ2

when~x ∈ Ω1, andε = 1 when~x ∈ Ω2.

2. NUMERICAL SOLUTION AND VALIDATION SUMMARY

The special integral form of two-dimensional continuity and unsteady Navier-Stokes equations are solved in theirprimitive variables formulation using an existing finite volume scheme. Detailed features of numerical method andsystematic validations have been outlined in [2], and only abrief description of points of direct relevance to the compu-tations will be provided here, further details of the implementation and validations can be found in [2]. The governingequations are the two-dimensional continuity and the Navier-Stokes equations (when a solid body is present) given by

dVdt+

∫

A(~u · ~n) dS = 0, (1)

ddt

∫

V~u dV+

∫

A(~n · ~u)~u dS= −1

ε

∫

A∪Ip~n dS+

1R

∫

A∪I~n · ∇~u dS+

∫

V~F dV (2)

whereV andA are the fractional volume and area, respectively, open to flow within the computational cell,V; I isthe length of the fluid-body interface open to flow;~u is the dimensionless velocity vector, where~u = (u, v, 0); ~nis the outward unit normal vector;S is the control volume boundary. These dimensionless quantities are defined interms of their dimensional counterparts:x = x∗/d, y = y∗/d, u = u∗/U, v = v∗/U; V = V∗/d2, S = S∗/d, V =V∗/d2, A = A∗/d, I = I∗/d. The external force,~F = (−a1, 1/Fr2 − a2, 0), is due to the dimensionless gravityforce,~g = (0, 1/Fr2, 0), and the dimensionless acceleration of the non-inertialframe of reference, (−a1, −a2, 0). Theboundary conditions are no-slip of the fluid on the cylinder surface,u = 0, v = 0; the uniform stream at the inflow,u = U − v1, v = −v2; and the free slip conditions at the top and bottom boundaries of the computational domain,∂u/∂x = 0, v = −v2. The well-posed open boundary conditions,1

R∂u/∂x+ h/Fr2 = p, ∂v/∂x = 0, are enforced at theoutflow boundary. Here,h is the height of the fluid at the outflow boundary. The uniform flow is used as the initialcondition. It is assumed that at timet = 0, the free surface is undisturbed.

Finite volume discretization of the governing equations are performed based on the aggregated-fluid approach,using single set of equations (1)-(2) in the computational domain,Ω = Ω1 ∪ Ω2. The values of fluid properties areset toρ1/ρ2=1/100 andµ1/µ2=1/100 (orν1/ν2=1) following the work of Reichlet. al. [1]. The Reynolds numbers inthe fluid regionsΩ1 andΩ2 are the same (R ≡ R1 = R2) which is varied by altering the viscosityµ (or ν). The maincomputational difficulty is solving the governing equations in an inertial frame of reference which results in pressurespikes. Professor Arthur E.P. Veldman’s group at University of Groningen attempted to overcome this difficultyunsuccessfully more than a decade (see e.g., [3]). In the present study, the computational difficulty is eliminated byemploying a non-inertial frame of reference. Free surface interface is discretized with the volume-of-fluid method[4]. Its advection in time is performed based on the strictlymass conserving volume-of-fluid advection method fortwo dimensional incompressible flows [5]. For the moving fluid-body interface the fractional area/volume obstaclerepresentation method [6], and the cut cell method [7] are employed. A second-order accurate central-differencescheme is used to discretize the governing equations in space in conjunction with first-order explicit forward Eulerscheme to advance the numerical solution in time.

The computational grid geometry is defined with respect to the meanposition of the cylinder and by specifyingthe locations of inflow and outflow boundaries,L1 andL2, along thex-axis and the location of the top and bottomboundaries,L3, along they-axis. The numerical simulations are carried out using the computational code which wasdeveloped by S. Kocabiyik’s research group at Memorial University. Code verification and validation testing wereconducted by L.A. Mironova and C. Bozkaya during their doctoral and postdoctoral studies at Memorial University,respectively. The numerical grid,L1 = 20, L2 = 30, L3 = 40, with 60 cells per cylinder diameter and∆t = 0.005are found to be satisfactory and are checked carefully. Tests are conducted in the absence of a free surface for astationary cylinder case atR = 200. The calculated value of natural shedding frequency,f0 = 0.198, is within 0.1%of the accepted value of 0.197 [8]. The predicted values of the mean drag coefficient,CD = 1.3399 and the maximum

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C. Bozkaya et al./ Applied Mathematics Letters 00 (2013) 1–6 3

Table 1. The effect of the submergence depth,h, on the local Froude number,Fr |L; the frequency ratio,f0/ f0; the averageu-velocity in the regiondirectly above the cylinder,u, for uniform flow past a stationary cylinder in the presence of a free surface atR = 180: Fr = 0.4, h = 0.22, 0.55,0.70. f0, f0 are the vortex shedding frequencies in the presence and absence of a free surface, respectively, andf0 = 0.191. Comparison with thenumerical results of Reichlet. al. [1].

Ref. Reichlet. al. [1] Present

h Fr|L f0/ f0 u Fr|L f0/ f0 u0.22 1.08 - 0.97 1.08 - 0.970.55 0.82 1.06 1.30 0.82 1.07 1.300.70 0.76 1.10 1.34 0.75 1.10 1.34

lift coefficient, CL ,max = 0.70, are in good agreement with the previous numerical results of De Palmaet al. [9](CD = 1.34,CL ,max = 0.70). Thex- andy- components of the dimensionless force,~F = 2~F∗/(ρdU2), exerted by thecylinder on the fluid are the dimensionless drag and lift force coefficients (CD, CL):

CD =

2π∫

0

pcosθ dθ +1R

2π∫

0

∂ur∂~n

dθ, CL =

2π∫

0

psinθ dθ +1R

2π∫

0

∂v∂~n

dθ (3)

where~n = (cos(θ), sin(θ), 0) is the outward unit normal to the cylinder boundary. In Table 1, the effect ofh on predictedvalues ofFr |L, f0/ f0, andu at R = 180: Fr = 0.4, h = 0.22, 0.55, 0.70 are compared with the numerical results ofReichlet. al. [1]. The u-velocity is averaged based on the free surface height,h|L, in the region directly above thecylinder. The local Froude number,Fr |L, is calculated usingFr |L = u/

√(gh|L). Hereu is the maximumu-velocity in

the region directly above the cylinder at the time when the lift coefficient reaches its maximum. The predicted resultsare in excellent agreement with the numerical results of Reichl et. al. [1].

3. RESULTS AND CONCLUSIONS

The numerical simulations are carried out in the presence ofa free surface ath = 0.25, 0.5, 0.75 andFr = 0.4for the case ofR = 200 : A = 0.13 and f / f0 = 3.75. Small amplitude excitation atf / f0 = 3.75 is chosen outsidethe fundamental lock-on range (f / f0 ≈ 2) for R = 200. This leads to the formation of new locked-on laminar vortexshedding modes,C(2S)+S andC(4S)+2S. The well known classical modes are the2S mode (two single vortices percycle),2P mode (two vortex pairs per cycle), andP+S (one single and a pair of vortices per cycle). The coalescenceof smaller vortices may result inC(2S), C(2P), C(P+S) modes. TheC(2S)+S mode is similar toC(2S) mode withonly one additional vortex shed from the free surface per cycle. TheC(4S)+2S mode is the combination of the twoC(2S)+S mode per cycle. A single vortex shedding cycle,Tv, is defined byTv = nT, wheren is an integer number.The numerical simulations are also conducted in the absencea free surface (h = ∞) under the same oscillationconditions to better understand what differences result from the inclusion of the free surface. The unsteady flowcalculations are conducted for the time up tot = 150 whenh = ∞, and up tot = 100 whenh = 0.25, 0.5, 0.75. In allequivorticity patterns that follow, black colours correspond to counterclockwise rotation (positive) and gray coloursindicate clockwise rotation (negative).

Figure 1 displays the lift coefficient,CL (22.5T ≤ t ≤ 28.5T), and the snapshots of the equivorticity and pressurecontours att = 25.5T: Fr = 0.4, h = 0.25, 0.5, 0.75. The lift trace progresses rapidly from a near sinusoidaltrace tothe one that contains three peaks per 2T, with an increase inh (0.25→ 0.5→ 0.75). The mechanism governing theshift in the formation of vortex shedding modes,C(2S)+S→ C(4S)+2S, is partly due to the fact that cylinder motioncauses changing acceleration of low pressure in the wake region. At the deeper submergence depths,h = 0.5, 0.75,the high pressure region partly offsets the low pressure created behind the cylinder. Localized free surface distortionsare due to the proximity of vorticity concentrations from the free surface. This secondary vortex interacts with thenegative concentrations of vorticity originally shed fromthe upper portion of the cylinder. These positive-negativevortices translates in the downstream direction along the free surface as shown in Figure 1. Thus, the presence ofthe free surface atFr = 0.4 results in locked-on vortex formation not only from the cylinder, but also from the free

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C. Bozkaya et al./ Applied Mathematics Letters 00 (2013) 1–6 4

CL (t)(black),x(t)(gray)

C(2S) + S, per 5T

C(2S) + S, per 4T

C(4S) + 2S, per 7T

Figure 1. Lift coefficient, CL/cylinder displacement,x(t) (left - black/gray) and the corresponding equivorticity patterns (middle) and pressurecontours (right) forR = 200 : A = 0.13, f / f0 = 3.75 atFr = 0.4 : h = 0.25, 0.5,0.75 from top to bottom. Dots indicate the instantt = 34.34(t = 25.5T, T ≈ 1.3468) at which the images are drawn when the cylinder reachesits maximum displacement,x(t) = A.

surface at locations both upstream and downstream of the cylinder. This observation is consistent with experimentalfindings of the study [10]. Calculations are also conducted for the low Froude number case, Fr=0.2, for comparisonpurposes but are not shown here for brevity. A switchover from locked-on state to the non-locked-on state occurs atcertainFr-h combinations as shown in Table 2. Similar phenomena has beenreported in [11] for the case of uniform

Table 2. The effects of the free surface inclusion and the Froude number on vortex shedding modes and their periods,Tv, for f / f0 = 3.75 atR= 200:A = 0.13. The superscript “∗” denotes quasi-locked-on modes.

h = 0.25 h = 0.5 h = 0.75 h = ∞Fr Mode Tv Mode Tv Mode Tv Mode Tv

0.2 non-locked -[C(2S)+S]∗

9T ≤ t ≤ 73T4T non-locked - C(2S)∗

60T ≤ t ≤ 111T4T

0.4[C(2S)+S]∗

15T ≤ t ≤ 30T;

non-locked 31T ≤ t ≤ 73T

5T[C(2S)+S]∗

14T ≤ t ≤ 36T;

non-locked 37T ≤ t ≤ 73T

4T[C(4S)+2S]∗

16T ≤ t ≤ 35T;

non-locked 36T ≤ t ≤ 73T

7T C(2S)∗

60T ≤ t ≤ 111T4T

flow past a cylinder near a free surface.Figure 2 displays a series of instantaneous equivorticity patterns over slightly more than one vortex shedding cy-

cle,Tv = 4T together with lift coefficient,CL, at Fr = 0.4: h = 0.5. The snapshots are taken at the instants indicatedby dots onCL. The lift coefficient shows almost a repetitive behavior per 4T, with three peaks over two periods,

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C. Bozkaya et al./ Applied Mathematics Letters 00 (2013) 1–6 5

A

B

C D

E F

GH

I

L

K

J

A

B

C

D

E

F

G

H

I

J

K

L

Figure 2. Comparison of vorticity patterns with temporal variation of lift coefficient, CL (solid line) and the cylinder displacement,x(t) (dashedline), for R= 200 : A = 0.13, f / f0 = 3.75 atFr = 0.4 : h = 0.5 Dots indicate the instant at which vorticity images are obtained.

2T. During the time interval, 29.86 < t < 30.36 (A-C), CL steadily decreases, and reaches its local minimum valueat t ≈ 30.36 (C) and the fully formed negative vortex in a previous cycle sheds into the upper vortex layer by thesweeping of a large-scale, positive concentration of vorticity. The shedding of this vortex seems to induce a local freesurface rising near the upper left side cylinder which creates sufficiently high pressure region near the free surfaceinterface as the cylinder moves towards its maximum negative displacement at(C). The positive vorticity concentra-tion is formed from the bottom surface of the cylinder duringa previous cycle and sheds into the downstream of thecylinder att ≈ 28.96 whenCL reaches its maximum-positive valueCL = 0.6364. This occurs slightly before theinstantA. However, it reattaches to the lower vortex shedding layer and coalesces with smaller positive vortices(D-H),becomes detached shortly aftert ≈ 32.18 (H). The aforementioned process occurs for negative vorticityconcentrationduring the time interval, 31.93≤ t ≤ 32.43(G-K). Tentative and final detachments occur, due to the dominant high/lowpressure concentrations, att ≈ 31.56 (F) andt ≈ 34.95 shortly after(L) whenCL reaches its minimum-negative andmaximum-positive values: (−1.177, 0.535), respectively. Thus, the shedding of the significant positive/negative vor-ticity concentrations contributes to the occurrence of maximum-positive/minimum-negativeCL. Similar observationswere reported in [10] for the streamwise oscillating cylinder in the presence of a free surface. The occurrence of theminimum-negative and maximum-positiveCL at Fr = 0.4 andh = 0.5 has general features in common with its fullysubmerged counterpart in the absence of free surface whenh = ∞ (not shown here). In the absence of free surface, thepositive/negative vortex detachments into the downstream of the cylinder occur when theCL reaches its minimum-negative (CL ≈ −1.0370)/maximum-positive (CL ≈ 1.078) values. In both cases, concentrations of vorticity areshedfrom the top surface of the cylinder and their movement up/away from the surface of the cylinder contributes to thepeak negativeCL. The proximity of the surface both alters the values of the mean lift coefficient,CL, the root-mean-square of lift coefficient,CL,rms relative to its counterpart for the fully submerged case when h = ∞ as shown in Table

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C. Bozkaya et al./ Applied Mathematics Letters 00 (2013) 1–6 6

Table 3. The effects of the free surface inclusion ath = 0.25, 0.5, 0.75 and the Froude number,Fr = 0.2, 0.4 on the lift coefficient values forf / f0 = 3.75.

Fr = 0.2 Fr = 0.4 −h = 0.25 h = 0.5 h = 0.75 h = 0.25 h = 0.5 h = 0.75 h = ∞

CL,min -5.8432 -3.3979 -2.7351 -3.3384 -1.9443 -2.0772 -1.0852CL,max 3.7279 2.6467 2.1510 -0.2902 1.0255 1.0158 1.0863CL -0.6714 -0.0369 0.1414 -0.9946 -0.4147 -0.2454 -4.0648×10−3

CL,rms 2.1545 1.3673 0.9815 1.1708 0.6667 0.6486 0.3647

3. For the case where a small gap exists between the free surface and the upper surface of the cylinder,h = 0.25, thistable shows substantial increase/decrease inCL/CL,rms values, relative to the fully submerged counterpart ath = ∞.For the smallest submergence depth case,h = 0.25, the negative spikes of lift coefficient,CL, are generally of largermagnitude and more consistent than the deeper submergence cases,h = 0.5, 0.75.

The physics of vortex formation and development in the presence of a free surface atFr = 0.2, 0.4: h =0.25, 0.5, 0.75 differs from that of a fully submerged cylinder case whenh = ∞. The positive layer of vorticityfrom the lower surface of the cylinder remains at approximately the same vertical elevation as the upper surface ofthe cylinder rather than moving upward from the cylinder. When the cylinder is relatively close to the free surface,the positive vorticity layer from the free surface and the neighbouring layer of negative vorticity generated from thetop surface of the cylinder are immediately adjacent to one another and the irrotational region between them is small.Thus, there is a close relationship between the unsteady development of each of two vorticity layers and the evolutionof small-scale concentrations of vorticity in each of them appears to occur in a coupled fashion (Figure 2): (i) a sec-ondary positive vortex emanating from the free surface envelopes the negative vortex and enforces its reattachmentto the upper vortex shedding layer; (ii) the reattached negative vortex rises up and interacts with the free surface bythe sweeping of a large positive vortex from the bottom surface of the cylinder and coalesces with smaller positivevortices (D-H); (iii) finally, it sheds from the upper vortexshedding layer remaining attached to the free surface. Thus,the presence of a free surface markedly influences the mean lift force and prolongs the duration of vortex sheddingprocess. The vorticity layers origination from the free surface and the upper surface of the cylinder are central de-termining the wake states. These observations suggests that free surface could be used to bring control on the wakestructure.

The financial support for this research is provided by the Natural Sciences and Engineering Research Council of Canada.S. Kocabiyik gratefully acknowledges use of the services and facilities of the Institute of Applied Mathematics at Middle EastTechnical University during the final stages of this research, while visiting IAM/METU (May/June 2012).

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