2014 Numerical Simulation of Vortex-Induced Vibration of a Circular Cylinder at Low Mass and Damping...

8/19/2019 2014 Numerical Simulation of Vortex-Induced Vibration of a Circular Cylinder at Low Mass and Damping With Diffe…

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Numerical simulation of vortex-induced vibration of a

circular cylinder at low mass and damping with

different turbulent models

Wei Lia, b, Jun Lia, Shengyu Liua a School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China b Hubei Key Laboratory of Naval Architecture and Ocean Engineering Hydrodynamics, Huazhong University of Science and

Technology, Wuhan 430074, China

E-mail address: [email protected] (Wei Li), [email protected] (Jun Li)

Abstract —Due to the great damage to widely utilized flexible

structures in ocean engineering, vortex-induced vibration (VIV)

of such long flexible marine structures is still a hot issue that

needs more theoretical research, and CFD techniques become

gradually indispensable to study the VIV problem. In this paper,two-dimensional Reynolds-averaged Navier-Stokes (RANS)

equations are adopted to investigate transverse VIV of elastically

mounted rigid cylinder with low mass-damping, and two typical

turbulent models are applied to solve the RANS equations: RNG

k-ε model and SST k-ω model. By comparing the cylinder

displacement response and vortex shedding modes of three

different response branches, analysis of differences between two

turbulence models are presented. The numerical results indicate

that SST k-ω model is more appropriate for VIV of the elastically

mounted rigid cylinder. Subsequently, other hydrodynamic

coefficients obtained by SST k-ω model are discussed and

compared with previous research in detail. This investigation

provides theoretical evidence for the numerical simulation of

VIV of marine riser in engineering application.

Keywords — vortex-induced vibration (VIV); vortex shedding; RANS; turbulent model; riser

I.

I NTRODUCTION

Vortex-induced vibration (VIV) involved in a great manyfields of engineering, especially in subsea pipelines, flexiblerisers and other marine structures. As flexible materials areincreasingly utilized in the deep-sea, VIV of such crucialstructures with low mass-damping renews much attention toresearch recently, which produced a large number offundamental studies [1,2,3,4,5].

The case of an elastically mounted rigid cylinder

constrained to oscillate transversely to a free stream is one ofthe most foundational research in the field of VIVs. AndWilliamson’s group have made a significant contribution inthis respect with a series of classical experimental studies[6,7,8,9], where the relevant conclusion of the experimentscould be summarized as follows: for an elastically mountedrigid cylinder with low mass-damping, three distinct branchesof amplitude response for the transverse oscillation wereobserved, which were denoted “the initial branch”, “the upper branch” and “the lower branch”, respectively. The transition

between different branches were associated with two phaseangles: ‘total phase’ and ‘vortex phase’, which were defined asthe phase angle between relevant hydrodynamic coefficient andthe cylinder response. Additionally, the modes of vortex

formation were also well present in their papers. Thoseconclusions presented in Williamson’s research have asignificant value for the future studies.

Recently, with the fast development of computer,numerical calculation become one of most important methodsto solve the VIV problems and many researchers come toutilize the computational fluid dynamics (CFD) techniques todo the modeling, which primarily include three differentcomputational approaches to solve the turbulent properties ofthe flow: Reynolds-averaged Navier-Stokes (RANS), Direct Numerical Simulation (DNS) and Large Eddy Simulation(LES). Since the RANS is robust enough and can acquire asimulated result with relative accuracy under much less time-consuming compared with other two approaches, it has a widerange of application into this field. Pan and Cui [10] usedRANS code to simulate a two-dimensional numerical model based on the experiment by Khalak and Williamson [6]. Thevortex modes and transition between different branches wereconsistent with the experimental results, but the responseamplitudes obtained in the simulation were much lower thanthe experimental results, especially the maximum responseamplitude occurred in only one timestamp, which seemsirregular compared with the experimental results. Meanwhile itwas pointed out that the loss of the spanwise correlation andthe low value of mass-damping may bring about the randomdisturbance and characteristics of the vortex-shedding processwith the vortex shedding.

It is well-known that the SST k-ω model and RNG k-ε model, basing on different hypothesis about the flow field, arethe most common turbulent models in solving RANSequations. The main objective of this paper is to assess arelatively accuracy turbulent model for an elastically mountedrigid cylinder with low mass-damping constrained to oscillatetransversely. Then the numerical simulations based on theturbulent model chosen are presented for our case. The physical parameters are referred to the experiment of Khalak

978-1-4799-3646-5/14/$31.00 ©2014 IEEE


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and Williamson[6]: * 2.4=m and * 0.013ξ =m ; Meanwhile

the reduced velocity ( ( )* /inlet nU U f D= ) is changed from 2.0 to

13.9 with increasing velocity, and the corresponding Reynoldsnumber is from 1700 to 11600. During the whole numerical process, the results are always compared with the experimentaldata provided by Khalak and Williamson [6].

II.

THEORETICAL FORMULATION The unsteady incompressible RANS equations can be

written as follow:

0∂

=∂

i

i

u

x (1)

( ) ( ) ( )2 ' ' ρ ρ µ ρ ∂ ∂ ∂ ∂

+ = − + −∂ ∂ ∂ ∂

i i j ij j i

j i i

pu u u S u u

t x x x (2)

Where u, p represents the time-average value of the velocityand pressure, respectively; μ is the molecular viscosity; S ij is

the mean stress tensor; and ' ' j iu u is the Reynolds stress tensor,

which all can be solved by means of an Newtonian model asfollow:

2' '

3 ρ µ ρ µ δ

⎛ ⎞∂ ⎛ ⎞∂ ∂− = + − +⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠⎝ ⎠

ji i

i t t ij

j i i

uu uu u k

x x x (3)

Here, the eddy viscosity μt is given by the turbulence model;δij is the Kronecker delta; and the turbulent kinetic energyk can be represented as

( )'2 '2' ' 1

2 2= = +i i

u uk u v (4)

Meanwhile, considering a riser oscillating transversely to a

free stream, to simplify the flow into two dimensions, strategyknown as the “strip theory approach” is utilized to combinewith the CFD method, in which 2-D CFD models first arecarried out for a number of strips along the riser and loads areapplied to the structure later for the dynamic analysis.Furthermore, each strip can be regarded as an elasticallymounted rigid cylinder and analyzed as a linear spring-mass-damper vibration system of single freedom. Fig.1 illustrates thesimplification process.

Fig. 1.

the simplification process of a 3D riser

Thus the nondimensional motion equations generally usedto express the VIV of the cylinder in the transverse y-directioncould be

2

* * *

4 2 2 LC y y yU U m

πζ π

π

⎛ ⎞+ + =⎜ ⎟

⎝ ⎠

(5)

Where U * is the reduced velocity; ζ is the system structural

damping ratio; m* is the mass ratio; y is the nondimensionaltransverse cylinder displacement; and C L is the lift coefficient.

Once the vortex shedding frequency was fully locked ontothe oscillation frequency of cylinder, which referred to thelock-in region. Following formulas prevailed in the relevantresearch as [11, 12]

( )*sin ω = ex y A t (6)

( )0sin ω ϕ = + L L exC C t (7)

( ) ( )cos sinω ω = + −⎡ ⎤⎣ ⎦ L Lv ex La exC C t C t (8)

0 0sin , cosϕ ϕ = = − Lv L La LC C C C (9)

( )φ arctan /= − Lv LaC C (10)

Where C L0 C Lv C La are the amplitude of the lift coefficient and

its components in phase with velocity and acceleration,respectively; ωex is the oscillation frequency and φ representedas the phase angle between the lift coefficient and responsedisplacement.

By Parkinson [13], the response amplitude ratio

( * /= max A Y D ) and the frequency ratio (* /= ex n f f ) can be

derived as

2

*

3 *

1

4 π ζ

⎛ ⎞= ⎜ ⎟

⎝ ⎠

Lv n

ex n

C f U A

f f Dm (11)

12 2

*

3 * *

11 2 π

⎡ ⎤⎛ ⎞

⎢ ⎥= + ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

La

n

C U f f Dm A (12)

These equations originated from the linearization with energy balance between the structure and the surrounding fluid. Oncethe response amplitude ratio and frequency ratio were obtained,C Lv and C La could be calculated by the above two equations,as well as the phase angle φ.

Furthermore, by Lighthill [14], the total fluid force F actingon the cylinder can be separated into a ‘potential force’component F potential , given by the potential added-mass force,and a ‘vortex force’ component F vortex that is due to thedynamics of what is called the ‘additional vorticity’. Similarlyfollowing equations were set up after nondimensional disposal:

0= −vortex L potential C C C (13)

C potential could be derived as:

( )

*3

2* *

2/

potential

AC

U f π = (14)

Besides, by using ‘vortex coefficient’ the nondimensionalequation of motion can be written as follow:


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( )2

* * *

4 2 2sin

πζ π ω ϕ

π

⎛ ⎞+ + = +⎜ ⎟

+⎝

′

⎠

vortex

ex vortex

a

C y

U U m y

m y (15)

Where ma is the potential added-mass corresponding to the‘potential coefficient’; ζ ′ is the system structural damping

ratio that containing the potential added-mass; and φvortex is the phase angle between the ‘vortex coefficient’ and responsedisplacement, which could be solved by the similar method

given above.

To obtain the nondimensional response displacement forthe transverse oscillation, the Newmark- β approach is utilizedto solve the above differential motion equation, and the forcecoefficient is obtained by solving the two-dimensional RANSequations.

III. GRID MODEL

The whole computational field in this investigation isdiscretized as shown in Fig.2(a), which is a rectangle with 20Din transverse and 30D in lengthways. The inflow boundary ofthe domain is located on the left hand side of the cylinder at adistance of 10D from the center of the cylinder, andconsequently the right side of the domain the outflow boundaryis defined. The symmetry boundary is located 10D away fromthe center of the cylinder in upward and downward directionsrespectively, and the no-slip condition is located on thecylinder surface. The hybrid meshing is used, shown inFig.2(b), and stretching of the mesh is performed to achieve afine resolution of the region closed to the cylinder surface.This mesh domain is large enough to enable obtainingoscillation of the wake down-stream of the cylinder. The first

points of the mesh away from the cylindrical face are located

where 0.5+ ≤ y for each reduced velocity considered.

10 D 20 D

1 0 D

1 0 D

0U x

y

10 D 20 D

1 0 D

1 0 D

0U x

y

(a)

(b)

(c)

Fig. 2.

Computational field and grid model

Grid independence tests are performed with three cases ofdifferent total grids, under the condition of reduced velocity U

*

can be 6.0, where the response amplitude ratio A* andfrequency ratio f

* can be obtained. Relevant testing results,

including comparing with the experimental ones, are listed inTable I.

TABLE I. NUMERICAL RESULTS OF RESPONSE AMPLITUDE AND

FREQUENCY WITH DIFFERENT GRID RESOLUTIONS

(* 2.4=m * 0.013ξ =m * 6.0U = )

Wall

grid

Near-

field

grid

Total

grid

A*

num A*

exp

f*

num

f*

exp

120 6000 8260 0.556 0.583 1.09 1.23

120 6000 10620 0.570 0.583 1.15 1.23

120 6000 11460 0.573 0.583 1.17 1.23

From Table I, it shows that there are good consistence between the results of response amplitude and frequency


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obtained by different grid resolutions, which means thesimulation in this research has good grid independence. Thus,latter investigation is based on the second grid scheme whichmakes the simulation more efficient.

IV. R ESULTS AND DISCUSSION

Before presenting results, some numerical parameters needto be specified. The nondimensional time step, /∞∆U t D , is set

to be 0.005 in the calculations. For each time step, a reductionof nonlinear residuals for the discrete momentum equations isrequired, and a two orders of magnitude of nonlinear residualsof discrete momentum equations is established for thereduction, which is exactly same with Guilmineau andQueutey [15]. The divergence of the velocity field is decreasedas 10-5.

A. Comparing between SST k-ω model and RNG k-ε model

Two typical turbulent models were adopted to solve theRANS equations: SST k-ω model and RNG k-ε model. Byusing of a k-ω formulation, SST k-ω model has an advantage innear wall treatment and it could be used as a Low-Re

turbulence model without any extra damping functions. On theother hand, based on the standard k-ε model, the eddy viscosityis modified and the rotating flows are taken into considerationin RNG k-ε model, which made it much more appropriate tosimulate the flow field around a circular cylinder. TheoreticallySST k-ω model seems much appropriate to the problem of flowaround circular cylinder. In order to figure out a better one,VIV of an elastically mounted rigid cylinder under designatedtypically reduced velocities are investigated. The responseamplitude ratio A*, obtained by different turbulent models, as afunction of reduced velocity U

* are given in Fig.3. Also the

results are compared with the experimental ones, which aredenoted by solid circular.

(a) SST k-ω model

(b) RNG k-ε model

Fig. 3.

Amplitude responses with different turbulent model ( * 2.4=m )

Fig.3 shows that both two turbulent models can get threeresponse branches. From Fig.3(a), it is obvious that transition

between the initial branch to upper branch happened aroundwhere reduced velocity U

* being 3.5, and when reduced

velocity U * reaches 5.2, the lower branch occurred. Also themaximum amplitude ratio reached 0.747 in the upper branch,while it was 0.642 in the lower branch. After the reducedvelocity U

* reach 11.0, the response amplitude of cylinder

becomes much lower than before, which corresponds to the‘lock-out’ region.

However, for the result of RNG k-ε model, the transitiontimestamp between the initial branch to upper branch is a bitdelayed, and transition between the upper branch to lower

branch seems not quiet instinctive. Moreover, the maximumamplitude ratio is 0.713, which is lower than one obtained bySST k-ω model. Although the response amplitudes are all lessthan the experimental ones for both two turbulent models,which may due to the simplification of two-dimensional andother factor caused by turbulence model, the comparisonresults still indicate that SST k-ω model agreed better with theexperimental results than that of RNG k-ε model.

Besides, according to Govardhan and Williamson [16], themodes of vortex formation differ between each response

branch. A 2S mode (two single vortices shed per cycle) turnedup in the initial branch, while a 2P mode (two pair vorticesshed per cycle) is observed in both the upper and lower

branches, merely the intensity of the second vortex of each pair

in the upper branch differ from the other branch.Corresponding to some typical reduced velocity above,Contour maps of vortices in different timestamp during ashedding cycle are shown in Fig.4-5.


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x/D

y / D

0 2 4 6 8 10-3

-2

-1

0

1

2

3

x/D

y / D

0 2 4 6 8 10-3

-2

-1

0

1

2

3

x/D

y / D

0 2 4 6 8 10

-3

-2

-1

0

1

2

3

x/D

y / D

0 2 4 6 8 10

-3

-2

-1

0

1

2

3

(a) * 3.0U =

x/D

y / D

0 2 4 6 8 10

-3

-2

-1

0

1

2

3

x/D

y / D

0 2 4 6 8 10-3

-2

-1

0

1

2

3

x/D

y / D

0 2 4 6 8 10-3

-2

-1

0

1

2

3

x/D

y / D

0 2 4 6 8 10-3

-2

-1

0

1

2

3

(b) * 5.0U =

x/D

y / D

0 2 4 6 8 10-3

-2

-1

0

1

2

3

x/D

y / D

0 2 4 6 8 10-3

-2

-1

0

1

2

3

x/D

y / D

0 2 4 6 8 10

-3

-2

-1

0

1

2

3

x/D

y / D

0 2 4 6 8 10-3

-2

-1

0

1

2

3

(c) * 7.5U =

Fig. 4. Vorticity magnitude contours in a vortex shedding period (SST k-ω)

x/D

y / D

0 2 4 6 8 10

-2

-1

0

1

2

x/D

y / D

0 2 4 6 8 10

-2

-1

0

1

2

x/D

y / D

0 2 4 6 8 10-2

-1

0

1

2

x/D

y / D

0 2 4 6 8 10-2

-1

0

1

2

(a) * 3.0U =

x/D

y

0 2 4 6 8 10-2

-1

0

1

2

x/D

y / D

0 2 4 6 8 10-2

-1

0

1

2

x/D

y

0 2 4 6 8 10-2

-1

0

1

2

x/D

y / D

0 2 4 6 8 10-2

-1

0

1

2

(b) * 5.0U =

x/D

y / D

0 2 4 6 8 10-2

-1

0

1

2

x/D

y / D

0 2 4 6 8 10-2

-1

0

1

2

x/D

y / D

0 2 4 6 8 10-2

-1

0

1

2

x/D

y / D

0 2 4 6 8 10-2

-1

0

1

2

(c) * 7.5U =

Fig. 5.

Vorticity magnitude contours in a vortex shedding period (RNG k-ε)

In the case of reduced velocity U * being 3.0, which referred

to the initial branch. The mode of vortex formation acquired bySST k-ω model in Fig.4(a) seems like not a 2S mode but a P+Smode (two pair vortices shed on one side while one singlevortex shedding on the other during per cycle), which isconsidered as a transient, unsteady-state pattern. While a 2Svortex formation mode obtained by RNG k-ε model is shownin Fig.5(a), from where the vortex follows a pattern similar towhat is found in a classical Von Karman Street.

Then, when the reduced velocity U * comes to 5.0, whichrepresents the upper response branch, a typical 2P vortex

formation mode is successfully simulated by SST k-ω model inFig.4(b), which is completely in accord with the experimentalresult by Williamson’s. This is due to the good behavior ofSST k-ω model in adverse pressure gradients and separatingflow. However, the vortex formation mode acquired by RNGk-ε model is totally not consistent with the experimental one.

Finally, when the reduced velocity U * is 7.5, whichrepresents the lower response branch, it seems that both twoturbulent models do not simulate 2P mode successfully in therelevant figures.


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Consequently, compared to RNG k-ε model, the maximumresponse amplitude obtained by SST k-ω model agreed betterwith the experimental result, and the 2P wake mode in theupper branch was successfully simulated by SST k-ω model,which indicated SST k-ω model is more appropriate for VIV ofthe elastically mounted rigid cylinder. Thus more discussionabout VIV will be based on SST k-ω model subsequently.

B.

Response frequency and force coefficientGenerally, besides the response amplitude ratio, response

frequency ratio is also a significant parameter to assess thenumerical results. Here, the response frequency ratio f

*,

obtained by SST k-ω model, as a function of reduced velocityU

* is given in Fig.6, which also compared with theexperimental ones.

Fig.6 shows that the response frequency in the lower branch agrees well with the experimental ones. In the lock-inregion, the oscillation frequency of cylinder f ex separates fromthe vortex shedding frequency f st , and the frequency ratio goessmoothly from 1.03 to 1.29. On the other hand, when thecylinder is out of lock-in region, the oscillation frequency f ex equals the vortex shedding frequency f st , which is similar tothe previous research.

Fig.7 presents the lift and drag coefficients for someselected cases, as well as the nondimensional displacement

response. At * 3.0=U (Fig.7(a)), time traces of force

coefficients are extremely irregular, and it is distinct that themulti-frequency vibrations occurred, which is consistent with

the experimental result in the initial branch region. At * 5.0=U ,it turns into the lock-in region, time traces of force coefficientsare regularly periodic and the amplitude of the lift coefficient

also becomes larger. At * 7.5=U , the force coefficient decrease

a bit due to the lower branch, and it has an obvious decreasewhen the reduced velocity U

* equals 11.0, which indicates that

the system have turned into the lock-out regime.

Fig. 6.

Response frequency ratio responses with U * ( * 2.4=m SST k-ω)

(a) (b)

(c) (d)

Fig. 7.

Time traces of cylinder displacement, lift and drag coefficient

( * 2.4=m , SST k-ω)

C. Vortex coefficient and vortex phase

As mentioned earlier, once the values of A* and f

* were

obtained, the lift coefficient C L, ‘vortex coefficient’ C vortex, total phase angle φ , and the ‘vortex phase’ angle φvortex can alldeduced through relevant equations. Here omitting the processof calculate, the corresponding force coefficients and phaseangles are directly presented at Fig.8.

From Fig.8 we can figure out that in the initial branch the

vortex phase φvortex is closed to 0 , which means that ‘vortex

coefficient’ C vortex and ‘potential coefficient’ C potential are in phase, so in the initial branch the total lift coefficient C L

achieves a large value. Then in the upper branch althoughC vortex and C potential are out of phase because of φvortex closing

to180 , the large response amplitude in this branch causes a

large potential force, which makes C potential much larger thanC vortex , and still leading to a large total lift coefficient C L .Finally, C vortex and C potential are quite comparable and still outof phase in the lower branch, so C L becomes quite diminished.Furthermore, from the former research, there is a distinct jumpto both two phase angle: the total phase φ jumps at thetransition between initial-upper branches, while the vortex

phase φvortex jumps at the transition between upper-lower branches. The results from Fig.8 quite coincide withGovardhan and Williamson [16].


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Fig. 8.

Relevant force coefficients and phase angles varing with U *

( * 2.4=m SST k-ω)

V. SUMMARY AND CONCLUSIONS

In this paper, two-dimensional RANS equations were usedto calculate VIV of an elastically mounted rigid cylinder at lowmass-damping constrained to oscillate transversely, and twotypical turbulent models were adopted and compared to solvethe RANS equations: RNG k-ε model and SST k-ω model. Thecomparison results between two turbulent models showsignificant differences between two turbulent models: themaximum response amplitude obtained by SST k-ω model

agreed better with the experimental result than that of RNG k-ε model, and the 2P wake mode in the upper branch wassuccessfully simulated by SST k-ω model.

Subsequently, transverse VIV for our case is extensivelyinvestigated based on SST k-ω model. The response frequency,hydrodynamic coefficients, and relevant phase angles are ingood agreement with the experimental results on the whole.These results suggest that SST k-ω model is valid and effectivefor the VIV of cylinder with low mass-damping.

With the fast development of computer and computationaltechnique, the CFD method becomes much indispensable tostudy in the ocean engineering. The present research providedtheoretical evidence for the computation of VIV of marine riser

in engineering application. And the 3-D RANS issues areessential to be discussed in the further research.

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2014 Numerical Simulation of Vortex-Induced Vibration of a Circular Cylinder at Low Mass and Damping...

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