Modeling Steady-State and Transient Forces on a Cylinder∗

Osama Marzouk1, Ali H. Nayfeh1†, Haider N. Arafat2, Imran Akhtar1

1Department of Engineering Science and Mechanics, MC 02192Department of Electrical and Computer Engineering, MC 0111

Virginia Tech, Blacksburg, VA 24061

{omarzouk, anayfeh, harafat, akhtar}@vt.edu

April 26, 2006

Abstract

Numerical simulations of flow past a stationary circular cylinder at different Reynolds numbers have

been performed using a computational fluid dynamics (CFD) solver that is based on the Reynolds-

averaged Navier-Stokes equations (RANS). The results obtained are used to develop reduced-order mod-

els for the lift and drag coefficients. The models do not only match the numerical simulation results in

the time domain, but also in the spectral domain. They capture the steady-state region with excellent

accuracy. Further, the models are verified by comparing their results in the transient region with their

counterparts from the CFD simulations and a very good agreement is found. The work performed here

is a step towards building models for vortex-induced vibrations (VIV) encountered in risers, spars, and

other offshore structures.

Keywords: Vortex-induced vibrations, reduced-order modeling, circular cylinders, risers, transient re-

sponse.

1 Introduction

1.1 Literature Summary

When flow passes over a bluff body, a street of shedding vortices (known as the von Karman vortex street)develops in the wake and exerts oscillatory forces on the body, which are often decomposed into drag andlift components along the free-stream and the cross-flow directions, respectively. If the body is capable offlexing or moving rigidly, these forces can cause it to oscillate and the classical vortex-induced vibration(VIV) problem takes place. If the frequency of vortex shedding is close to a natural frequency of the body,the resulting resonance can generate large-amplitude oscillations, which may ultimately cause structuralfailure. Understanding this problem is of great interest in the design and maintenance of offshore structures,like mooring cables, rises, and spars, and of high aspect-ratio structures subject to air streams, like chimneys,high-rise buildings, bridges, and cable suspensions systems.

In order to model and predict the VIV problem, full simulation of the coupled flow and structure equationsis needed. This involves accounting for three-dimensional effects, turbulence structures, and elasticity, among

∗Submitted for publication to the Journal of Vibration and Control†Corresponding Author

1

other considerations. However, in general, many of these structures are quite long, some exceeding 7, 000feet, such as risers and spars, and flows around them may reach very high Reynolds numbers (Re). Underthese conditions, full simulation of the VIV problem is overwhelming, even with todays computing power.So, simplifying assumptions about the flow and structure are often made and efforts to produce simple,accurate representations of the problem via reduced-order modeling are still on-going.

Circular cylinders are extensively used in the study of bluff-body fluid dynamics due to their geometricsimplicity and common use in engineering applications. We give a brief introduction of the studies that havedealt with VIV of fixed cylinders, cylinders with prescribed motions or forces, elastically mounted cylinders,and round cross-section elastic structures. A thorough review on the subject of VIV is given by Williamsonand Govardhan (2004).

Bishop and Hassan (1963) were one of the earliest to suggest using a self-excited oscillator to representthe forces over a cylinder due to vortex shedding. Hartlen and Currie (1970) formulated a remarkable modelfor elastically restrained cylinders that are restricted to cross-flow motions. They used a Rayleigh oscillatorto describe the lift force and coupled it to the cylinder motion by a linear velocity term. Currie and Turnball(1987) proposed a similar model for the fluctuations in the drag. Skop and Griffin (1973) pointed out that theparameters in the model of Hartlen and Currie (1970) lacked clear connections to physical parameters of theproblem. They proposed a modified van der Pol oscillator to represent the lift coupled to a linear equationof motion for the structure. They also introduced the Skop-Griffin parameter, which is now commonly usedin studying VIV problems.

Iwan and Blevins (1974) considered the fluid mechanics of the vortex street and developed a model interms of a fluid variable that captures the fluid dynamics effects of the problem. Landl (1975) added anonlinear aerodynamic damping term of fifth order to the van der Pol oscillator in his two-equation model,suggesting that this enables better capturing of some physical characteristics. However, the model involvedmany constants to be determined.

Several attempts have been made to extend the wake-oscillator models to elastic structures, such asbeams and cables. Iwan (1975) extended the model of Iwan and Blevins (1974) to predict the maximumVIV amplitude of taut strings and circular cylindrical beams with different end conditions. Iwan (1981) alsoderived an analytical model for the VIV of elastic structures under non-uniform flows.

Skop and Griffin (1975) extended their (1973) rigid cylinder model to elastic cylinders. With the goalof accurately capturing the asymptotic, self-limiting structural response near zero damping, Skop and Bal-asubramanian (1997) modified the model of Skop and Griffin (1975) by separating the fluctuating lift intotwo components: one satisfying a van der Pol equation oscillator that is driven by the transverse motion ofthe cylinder and the other linearly proportional to the transverse velocity of the cylinder (called the stallterm). Kim and Perkins (2002) modeled the lift and drag by two nonlinearly coupled van der Pol equationsin their study of resonant responses of suspended elastic cables. The coupling terms were introduced basedon the fact that the main frequency of the drag component is twice the main frequency of the lift component.Krenk and Nielsen (1999) proposed a two-oscillator model in which the mutual forcing terms are developedbased on the premise that energy flows directly between the fluid and structure. This means that the forcingterms correspond to the same flow of energy at all times.

1.2 Motivation

Most of the aforementioned analytical works are based on modeling the lift by either the Rayleigh or thevan der Pol oscillator. Nayfeh et al. (2003) numerically simulated the two-dimensional flow past a stationarycylinder for a wide range of Reynolds numbers and calculated the lift and drag. They employed higher-

2

order spectral analyses to determine the phase relations among the different spectral components of thelift and compared them with the phase information one obtains from closed-form approximate solutionsof the van der Pol and Rayleigh oscillators. They concluded that the van der Pol oscillator is the moreaccurate representation of the lift. They also proposed to model the drag as the sum of a mean term and atime-varying term proportional to the product of the lift and its time derivative.

1.3 Background of Work Presented

In all of the previous models, the emphasis has been on reproducing the steady-state lift and drag. However,to investigate this fluid-structure interaction problem more thoroughly, modeling the transient as well as thesteady-state lift and drag is crucial. The goals of this work are two-fold:

• First, we extend the results of Nayfeh et al. (2003) on the flow over a stationary cylinder by investigatingthe validity of their models in the transient region. In other words, are the models identified basedon the steady-state lift and drag CFD results capable of predicting the transient lift and drag resultsobtained with the CFD code?

• Second, we propose modifications and improvements to both of the lift and drag models that takefull and explicit advantage of the phase relations between the different components. These improvedmodels are validated against the CFD results for both the steady-state and transient behaviors.

The order of this paper is as follows:

• First, we discuss the CFD code used to generate the numerical solutions of lift and drag on a stationarycylinder.

• Second, we present some of the lift and drag results of Nayfeh et al. (2003) for steady-state responsesand investigate the applicability of their models within the transient response.

• Third, we discuss the improved lift and drag models and compare their performances to the CFDresults in the steady-state as well as transient regions.

• Fourth, we conclude with some remarks and general notes.

2 Numerical Simulation

Direct numerical simulation (DNS) of flows with engineering relevance remains a challenging task, even withcurrent advances in hardware and software capabilities. Hence, some modeling assumptions or approxi-mations are often needed to simplify the flow and render their computations feasible. Usually, this entailsrepresenting the turbulent scales with turbulence models, which significantly reduce the spatial and temporalresolution requirements. One alternative to DNS is the large-eddy simulation (LES) approach. In LES, theenergy-containing turbulent scales are resolved, whereas the subgrid scales (SGS) are modeled. LES providesinformation about a wide range of spatial and temporal scales in the flow at a cost that is significantly lowerthan DNS. Nonetheless, for high Reynolds-number flows over complex geometries, LES computations stillremain to be a formidable task.

In the conventional Reynolds-averaged Navier-Stokes (RANS) modeling, the effects of all of the turbulentfluctuations are modeled. Extensively used within the fluid engineering community, the RANS approach andits variants are efficient and much cheaper than DNS and LES in computation cost, yet it yields results thatare accurate enough to capture most of the important physical characteristics of the problem.

3

Therefore, we compute the time-dependent, incompressible flow past a cylinder by solving the Reynolds-averaged Navier-Stokes (RANS) equations. We employ the artificial compressibility method to speed upthe solution convergence. This method introduces an artificial time-derivative of pressure term into thecontinuity equation, thereby explicitly coupling the pressure and the velocity field (Rogers et al., 1987;Chorin, 1997).

The nondimensional equations governing the two-dimensional flow have the form

∂q∂t

+∂F∂x

+∂G∂y

− 1Re

H∇2q = 0 (1)

where Re ≡ UD/ν is the Reynolds number and

q =

p

u

v

,F =

βu

u2 + p

uv

,G =

βv

uv

v2 + p

, H =

0 0 00 1 00 0 1

(2)

Here, U is the free-stream velocity, D is the cylinder diameter, ν is the kinematic viscosity, p is the pressure,u and v are the velocity components in the horizontal (streamwise) and vertical (cross-stream) directions,respectively, and β is an artificial compressibility parameter. The original continuity equation correspondsto having β −→ ∞. The problem is solved on a curvilinear O-grid with a 25D radius in order to avoid anyblocking effects without using buffer domains. The convective terms are discretized using a second-orderupwinding difference scheme. The physical time terms, which represent flow unsteadiness, are switched tothe right-hand side and used as source terms; they are discretized using a second-order three-level backward-difference formula.

To integrate the RANS equations with inflow, outflow, and no-slip boundary conditions, radial andcircumferential stretching is employed and a second-order finite-difference scheme is used. At the inflowboundary, the velocity components are specified and the pressure is extrapolated from the interior points.At the outflow boundary, the pressure is specified and the velocity components are extrapolated from theinterior points. For turbulent closure, a one-equation model, such as the Spalart-Allamras (SA) method(Spalart and Allamras, 1992), is used to represent the unresolved scales. Equivalent boundary conditionsare also set for the turbulence quantities.

The outcome of the simulation is the pressure distribution over the cylinder surface, which ultimately isintegrated to calculate the lift and drag coefficients. The time histories of the lift and drag coefficients arethen converted to the spectral domain using Fourier analysis. The time and frequency are nondimensionalizedusing the free-stream velocity and the cylinder diameter. The nondimensional frequency (Strouhal number,St) is defined as St ≡ fU/D, where f is the dimensional cyclic frequency in Hz.

Figure 1 shows typical time histories and corresponding power spectra for the case when Re = 4, 000.The lift coefficient CL is presented in the top two subfigures and the drag coefficient CD is presented inthe bottom two subfigures. The lift power spectrum shows a large peak with amplitude a1 at the sheddingfrequency fs ≈ 0.2 and smaller peaks (a3 and a5) at the odd harmonics 3fs and 5fs. The corresponding timehistory shows the lift coefficient fluctuating periodically about the origin. Therefore, one infers that the liftcoefficient oscillates at the shedding frequency and that its behavior is influenced by cubic, and, to lowerextent, higher-order odd nonlinearities.

By contrast, the drag power spectrum shows a large peak a2 at twice the shedding frequency (2fs) andsmaller peaks (a4 and a6) at the even harmonics 4fs and 6fs. The corresponding time history shows the dragfluctuating periodically about a non-zero mean that reaches a constant value at steady state. This impliesthat the drag coefficient consists of a time-varying mean term and a fluctuating term. The fluctuating termmainly varies quadratically with the lift coefficient, as the influence of the higher-order even nonlinearities is

4

quite small. The corresponding drag-polar plot in Figure 2 clearly illustrates this quadratic coupling betweenthe drag and lift coefficients.

3 Modeling Background

3.1 Lift

Nayfeh et al. (2003) investigated two wake-oscillator models to represent the lift, namely, the van der Poland Rayleigh oscillators. Using higher-order spectral moments analysis, they calculated, for a few cases, thephase angle φ13 between the lift components at fs and 3fs to fall around 90◦. Based on their findings, theyconcluded that the van der Pol oscillator

CL + ω2CL = µCL − αC2LCL (3)

is the more suitable choice as an efficient and simple model for the lift coefficient in steady state. Theangular frequency ω in equation (3) is related (but not equal) to the angular shedding frequency ωs = 2πfs

and the parameters µ and α represent the linear and nonlinear damping coefficients, respectively. The valuesof µ and α are taken positive, so that the linear damping is destabilizing while the nonlinear damping isstabilizing. As a consequence, small disturbances grow and large ones decay, both eventually approaching astable limit cycle. The values of the parameters in equation (3) depend on the Reynolds number and theirvalues are based on matching the steady-state CFD lift data.

Recently, the authors (Nayfeh et al., 2005) examined the possibility of using equation (3) to model thelift coefficient in the transient region as well. Using the method of multiple scales (Nayfeh, 1973, 1981)and assuming that the oscillator is weakly damped (i.e., µ = O(ε) and α = O(ε) where ε � 1 is a smallbookkeeping parameter), the following second-order approximate solution was obtained:

CL(t) = a(t)

√1 +

116ω2

[µ − 1

4αa(t)2

]2

sin[ωt + θ(t) + η(t)]

0 0.3 0.6 0.9 1.2 1.5Strouhal Number (St)

1x10-4

1x10-3

1x10-2

1x10-1

1x100

1x101

CL

0 0.3 0.6 0.9 1.2 1.5Strouhal Number (St)

1x10-4

1x10-3

1x10-2

1x10-1

1x100

1x101

CD

a1

a3

a5

a2

a4

a6

fs

<CD>ss

2fs

30 40 50 60 70 80 90 100Time

-1.6

-0.8

0

0.8

1.6

CL

30 40 50 60 70 80 90 100Time

1

1.2

1.4

1.6

CD

Figure 1: Time histories and corresponding power spectra of the lift and drag coefficients obtained from theCFD simulation at Re = 4, 000.

5

− α

32ωa(t)3 sin[3ωt + 3θ(t)] + · · ·

≡ a1(t) sin[ωt + θ(t) + η(t)] − a3(t) sin[3ωt + 3θ(t)] + · · · (4)

where η(t) = tan−1[

16ωαa(t)2−4µ

]and the amplitude a(t) and phase θ(t) are governed by the modulation

equations

da

dt=

18(4µa − αa3) (5)

dθ

dt= − 1

8ω

(µ2 − 3

2αµa2 +

1132

α2a4

)(6)

Setting a = 0 in Eqn. (5), the steady-state value of a is obtained from the solution of a(4µ − αa2) = 0,which is either the trivial solution a = 0 or the nontrivial solution a = 2

√µ/α. For the nontrivial solution,

it follows from equation (4) that

a1 = 2√

µ

αand a3 =

µ

4ω

õ

α(7)

Moreover, the angle η = π2 and, from equation (6), the corresponding expression for θ = −µ2/16. Conse-

quently, the angular shedding frequency is given by

ωs = ω + θ = ω − µ2

16ω(8)

Equation (8) shows that the angular frequency ω of the van der Pol oscillator is not exactly equal to theangular shedding frequency ωs, as one would predict from a first-order expansion (Nayfeh et al., 2003).Hence, an improved second-order approximate expression for the steady-state lift coefficient becomes

CL(t) ≈ a1 cos(ωst) + a3 cos(3ωst + π2 ) (9)

The methodology used to identify the system parameters for a given Reynolds number is as follows:

1. The CFD solver is used to calculate the time history of the lift coefficient.

2. Spectral analysis is performed on the steady-state part of the CFD data to extract the values of a1,a3, and fs (or ωs).

1 1.2 1.4 1.6

CD

-1.5

-1

-0.5

0

0.5

1

1.5

C

L

Figure 2: A drag-polar plot of CL v.s. CD at Re = 4, 000.

6

80 85 90 95 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

CL

Time

Re 200Lift modelCFD

(a) Re = 200

80 85 90 95 100−1.5

−1

−0.5

0

0.5

1

1.5

CL

Time

Re 100,000Lift modelCFD

(b) Re = 100,000

Figure 3: Comparisons between the simulated and modeled steady-state lift coefficients.

3. Equations (7) and (8) are then solved for the nonlinear and linear damping coefficients α and µ andthe angular frequency ω.

4. With all of the parameters identified, equation (3) is numerically integrated using a Runge-Kuttaroutine and the results are compared with the CFD results.

In Figure 3, we compare the time history of the steady-state lift obtained from the CFD simulation withthat obtained by integrating equation (3) for the Reynolds numbers Re = 200 and Re = 100, 000. We notethat the steady-state solution of the van der Pol equation is independent of the initial conditions. It is clearthat the van der Pol oscillator accurately models the CFD results for a wide range of Reynolds number flows.

Next, we check whether the van der Pol oscillator identified using the steady-state lift data can alsomodel the transient lift. This serves two purposes:

1. It gives confidence that the physics is modeled correctly.

2. It confirms that the model is capable of simulating the transient as well as the steady-state response.

Using the values of α, µ, and ω defined earlier and specifying the initial conditions CL(t0) and CL(t0),equation (3) is integrated for a long time. Because the CFD results only yield the values of CL, the initialvalues of CL are approximated using a five-point fourth-order central finite-difference formula. In Figures 4and 5, we compare the transient lift obtained from the model and the CFD analysis for Re = 200 andRe = 100, 000, respectively. For each case, results initiated at four different initial times are presented.

For the case Re = 200, we simulate the transient lift using the initial starting times t0 = 34, 39, 45, and50. We see from Figure 4 that, the larger the starting time is, the more accurate the model is in simulatingthe CFD results. For example, when t0 = 34, the van der Pol oscillator underestimates the amplitude ofthe lift markedly; also, there is a phase shift between its prediction and the CFD lift. These differences,however, diminish as the starting time t0 is increased, as shown in the results for t0 = 50, which are in verygood agreement with the CFD results.

For the case Re = 100, 000, we show in Figure 5 the predicted transient lift using the initial startingtimes t0 = 10, 12, 14, and 16. It follows from Figure 5 that the discrepancy in modeling the lift amplitudefor moderate Reynolds number flows is resolved. However, there is still a phase difference between the tworesults, which again is minimized when using larger starting times.

7

20 30 40 50 60 70 80−1

0

1

to=34

CL

Time

Lift modelCFD

20 30 40 50 60 70 80−1

0

1

to=39

CL

Time

20 30 40 50 60 70 80−1

0

1

to=45

CL

Time

20 30 40 50 60 70 80−1

0

1

to=50

CL

Time

Figure 4: Simulated and modeled transient lift coefficients for different starting times for Re = 200.

0 20 40 60−2

0

2

to=10

CL

Time

Lift modelCFD

0 20 40 60−2

0

2to=12

CL

Time

0 20 40 60−2

0

2

to=14

CL

Time

0 20 40 60−2

0

2to=16

CL

Time

Figure 5: Simulated and modeled transient lift coefficients for different starting times for Re = 100, 000.

One possible explanation for these deviations is the fact that the CFD results contain two types oftransients, those arising physically and those arising from numerical analysis effects (e.g., impulsive initialconditions), and it seems that the rate of decay of the latter depends on the Reynolds number. We notefrom Figures 4 and 5 that the larger the Reynolds number is, the faster the rate of convergence is to thephysical solution. For Re = 200, it follows from Figure 4 that the transients due to numerical effects in theCFD simulation might propagate for some time after the initial conditions and might take up to 50 timeunits to decay completely. On the other hand, for Re = 100, 000, it follows from Figure 5 that the transientsdue to numerical effects decay much faster, lasting for about 16 time units.

Clearly, the agreement between the van der Pol oscillator and CFD results is quite good once the transientsarising from numerical effects have died out. Therefore, the van der Pol oscillator seems to be capable ofmodeling both of the steady-state lift and the transient lift on a cylinder in uniform flow.

3.2 Drag

Referring to Figure 1, we note that the drag consists of two major components. The first is a mean compo-nent that monotonously approaches a constant value in the steady state; this component is assumed to beindependent of the lift. The second is an oscillatory component related to the lift and has a frequency equal

8

to twice the lift frequency of oscillation.Since the lift and drag have a common source, the pressure distribution of the cylinder surface, and in

view of this two-to-one frequency relationship, Nayfeh et al. (2003) reasoned that the drag is quadraticallyrelated to the lift in some fashion. They examined the phase relation between the periodic components ofthe drag and lift and found that it falls around 270◦. Hence, they inferred that the periodic component ofthe drag must be proportional to −CLCL and proposed the drag model

CD(t) = 〈CD〉 − 2a2

ωsa21

CL(t)CL(t) (10)

where a2 is the amplitude of the drag component at 2fs and 〈CD〉 is the mean component of drag. Forsteady-state behavior, 〈CD〉 is constant, while, for transient behavior, it increases monotonously with time.The constant value of 〈CD〉 is determined from the CFD steady-state time history of the drag and the valueof a2 is determined from its spectral analysis.

In Figure 6, the drag results obtained from equation (10) are compared with the CFD calculations forthe Reynolds numbers Re = 200 and 100, 000. For Re = 200, excellent agreement is found between the twosolutions. For Re = 100, 000, the agreement is also quite good, but there appears to be a slight deviation inphase between the two results.

We also examined the capability of equation (10) to predict the transient drag. However, because themean drag 〈CD〉 is time-dependent in the transient region, modeling it as a constant following the steady-state results brings about a mismatch with the CFD solution. To account for this discrepancy, we firstextracted the transient profile of the mean drag from the CFD simulation and fitted it into a polynomialfunction in time to replace the constant value of 〈CD〉. Then, substituting the transient lift results at differentinitial times into equation (10), we calculated the corresponding transient drag.

In Figures 7 and 8, we compare the transient results from the CFD code and the drag model for theReynolds numbers Re = 200 and 100, 000. Again, we find that, in general, the accuracy of the proposedmodel improves for later starting times as the transient effects due to numerical computations diminish. Forthe low Reynolds number flow in Figure 7, we find that the proposed model underestimates the amplitudeof CD when starting at t0 = 40, but it does an excellent job when starting at t0 ≥ 60, which is still inthe transient region. For the moderate Reynolds number flow in Figure 8, we find that the model does an

80 85 90 95 100

1.14

1.16

1.18

1.2

1.22

1.24

CD

Time

Drag modelCFDRe 200

(a) Re = 200

80 85 90 95 1000.78

0.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

CD

Time

Drag modelCFD

Re 100,000

(b) Re = 100,000

Figure 6: Comparisons between the simulated and modeled steady-state drag coefficients.

9

40 50 60 70 801

1.05

1.1

1.15

1.2

1.25to=40

Time

CD

Drag modeCFD

40 50 60 70 801

1.05

1.1

1.15

1.2

1.25to=50

Time

CD

40 50 60 70 801

1.05

1.1

1.15

1.2

1.25to=60

Time

CD

40 50 60 70 801

1.05

1.1

1.15

1.2

1.25to=65

Time

CD

Figure 7: Simulated and modeled transient drag coefficients for different starting times for Re = 200.

10 20 30 40 50 600.6

0.7

0.8

0.9

1to=10

Time

CD

Drag modeCFD

10 20 30 40 50 600.6

0.7

0.8

0.9

1to=12

Time

CD

10 20 30 40 50 600.6

0.7

0.8

0.9

1to=14

Time

CD

10 20 30 40 50 600.6

0.7

0.8

0.9

1to=16

Time

CD

Figure 8: Simulated and modeled transient part of the drag coefficients for different starting times for Re= 100, 000.

excellent job in predicting the amplitude of CD; however, there exists a small phase difference between thetwo solutions.

4 Improved Lift and Drag Models

The results presented in the previous section are solely based on matching the amplitudes and frequenciesof the CFD and model results. Even with the good agreement found, especially at steady state, we believethere is still room for improvement. Although the use of higher-order spectral analysis of the CFD resultsshowed that the phase φ13 between the lift components at fs and 3fs is nearly 90◦, it actually differs fromone case of Reynolds number to another. In fact, in some of our calculations, we have found separation from90◦ as large as ±25◦. Obviously, this wide variation in the phase is not fully accounted for in the van derPol model in equation (3). This fact does not produce a real problem as long as one is concerned with thetime histories because a3 is usually two orders of magnitude smaller than a1. However, because we intend toextend this model to cases of forced and freely vibrating cylinders, we would like to have it be very accuratein both the time domain and the spectral domain.

Similarly, for the drag CFD calculations, we have found that the phase φ12 between the lift component

10

at fs and the drag component at 2fs varies from 270◦ by as much as 85◦. Qin (2004) proposed that thequadratic term in the drag model should be of the form C2

L instead of −CLCL. He also found linear coherencebetween the drag and lift components at fs and 3fs and introduced a linear lift term in the drag model toaccount for it.

4.1 Lift

Here, we modify the van der Pol oscillator by introducing a Duffing-type nonlinearity to equation (3),resulting in the new lift model

CL + ω2CL = µCL − αC2LCL − γC3

L (11)

The coefficient γ is determined based on matching precisely the phase φ13 obtained from the CFD data tothe phase obtained from solving equation (11). In this process, we use the method of harmonic balance todetermine approximate solutions of the model. These solutions along with spectral analysis of the CFD dataare used to identify all of the parameters in equation (11).

We seek a solution for the lift coefficient of the form

CL(t) = c1 cos(ωst) + c2 sin(ωst) + c3 cos(3ωst) + c4 sin(3ωst) (12)

Then, upon substituting equation (12) into equation (11) and separating the terms multiplying the differentsine and cosine functions, we obtain the linear system

Ay = b (13)

where y = {ω2, µ, α, γ}, b = {c1, c2, 9c3, 9c4}ω2s, and the entries of the matrix A are

A11 = c1, A21 = c2, A31 = c3, A41 = c4,

A12 = −c2ωs, A22 = c1ωs, A32 = −3c4ωs, A42 = 3c3ωs,

A13 =14

(c32 − c4c

22 + c2

1c2 + 2c23c2 + 2c2

4c2 − 2c1c3c2 + c21c4

)ωs,

A23 =14

(−c3

1 − c3c21 − c2

2c1 − 2c23c1 − 2c2

4c1 − 2c2c4c1 + c22c3

)ωs,

A33 =14

(−c3

2 + 6c4c22 + 3c2

1c2 + 3c34 + 6c2

1c4 + 3c23c4

)ωs,

A43 =14

(−c3

1 − 6c3c21 + 3c2

2c1 − 3c33 − 3c3c

24 − 6c2

2c3

)ωs,

A14 =14

(3c3

1 + 3c3c21 + 3c2

2c1 + 6c23c1 + 6c2

4c1 + 6c2c4c1 − 3c22c3

),

A24 =14

(3c3

2 − 3c4c22 + 3c2

1c2 + 6c23c2 + 6c2

4c2 − 6c1c3c2 + 3c21c4

),

A34 =14

(c31 + 6c3c

21 − 3c2

2c1 + 3c33 + 3c3c

24 + 6c2

2c3

),

A44 =14

(−c3

2 + 6c4c22 + 3c2

1c2 + 3c34 + 6c2

1c4 + 3c23c4

)

The cj are determined by matching the solution in equation (12) with the steady-state CFD simulationsolution which is expressed as

CL(t) = a1 cos(ωst) + a3 cos(3ωst + φ13) (14)

This yields c1 = a1, c2 = 0, c3 = a3 cos φ13, and c4 = −a3 sin φ13.

11

Table 1: Lift parameters at different Reynolds numbers.

Re = 200 Re = 4, 000 Re = 100, 000

fs 0.192 0.23 0.254a1 0.621 1.23 1.06a3 0.0043 0.042 0.021

φ13 78◦ 107◦ 100◦

ωs 1.206 1.445 1.596

ω 1.185 1.562 1.641µ 0.066 0.386 0.251α 0.68 1.050 0.902γ 0.192 −0.297 −0.166

100 104 108 112 116 120Time

-0.8

-0.4

0

0.4

0.8

CL

CFD

Model

80 84 88 92 96 100Time

-1.6

-0.8

0

0.8

1.6

100 104 108 112 116 120Time

-1.6

-0.8

0

0.8

1.6

CL

(a) (b)

(c)

Figure 9: Comparisons of the steady-state time histories of the lift coefficients obtained with the CFDsimulation and the improved model: (a) Re = 200, (b) Re = 4, 000, and (c) Re = 100, 000.

Listed in Table 1 are the results for the cases Re = 200, 4, 000, and 100, 000, covering a fairly wide rangeof Reynolds numbers. Typically, risers operate in the lower subset of that range while spars operate in theupper subset and beyond it. From the values of α and γ, it is clear that the influence of the Duffing-typecubic term should not be neglected in the modeling. In Figure 9, we plot the steady-state time histories ofthe lift coefficient obtained from the improved model and the CFD simulations. For all three cases of Re,there is excellent agreement between the two solutions.

Next, we examine the validity of this model in predicting the transient behavior of the lift coefficient. InFigure 10, we show the transient variations of the lift coefficient with time for Re = 200. The results of theimproved model are compared with the CFD results for the starting times t0 = 40, 50, and 60 time units.It is clear that the improved model for the lift matches the CFD simulation results in amplitude and phasewith excellent accuracy for t0 ≥ 50. In Figure 11, the transient variations of the lift coefficient with timefor Re = 4, 000 are shown. The results of the improved model are compared with the CFD results for thestarting times t0 = 38, 42, and 45 time units and the improved model matches very well the CFD simulationfor t0 ≥ 45. Lastly, we present in Figure 12 the transient variations of the lift coefficient with time forRe = 100, 000. The results of the improved model are compared with the CFD results for the starting timest0 = 12, 15, and 18 time units. The improved lift model matches very well the CFD simulation for t0 ≥ 18.

12

30 40 50 60 70 80 90 100

-0.8

-0.4

0

0.4

0.8

CL

CFD

Model

30 40 50 60 70 80 90 100Time

30 40 50 60 70 80 90 100Time

-0.8

-0.4

0

0.4

0.8

CL

(a) (b)

(c)

t0 = 40

t0 = 50

t0 = 60

Figure 10: Time histories of the lift coefficients in the transient region from the CFD and improved van derPol model for Re = 200.

30 40 50 60 70 80

-1.6

-0.8

0

0.8

1.6

CL

CFD

Model

30 40 50 60 70 80Time

30 40 50 60 70 80Time

-1.6

-0.8

0

0.8

1.6

CL

(a) (b)

(c)

t0 = 38

t0 = 42

t0 = 45

Figure 11: Time histories of the lift coefficients in the transient region from the CFD and improved van derPol model for Re = 4, 000.

Nonetheless, a small deviation in phase still remains in the lift transient results for t0 = 18.

4.2 Drag

In the simulations we have conducted thus far, we have found that the phase φ12 is around 270◦. However,the actual value of the phase may vary from one case of Reynolds number to another by almost 85◦, whichis quite significant. Therefore, we propose a new model in which the drag is proportional to both CLCL andC2

L in the following manner:

CD(t) = 〈CD〉 + 2k1a2

a21

(C2

L −⟨C2

L

⟩)+ 2k2

a2

ωsa21

CLCL (15)

where the brackets 〈 〉 denote the mean value of the term inside.The first expression in equation (15) stands for the mean component of the drag, which reaches a constant

value 〈CD〉ss in the steady-state region. The term −⟨C2

L

⟩in the second expression in equation (15) negates

the DC component introduced by C2L. The amount of contribution from both quadratic expressions to the

13

10 20 30 40 50 60

-1.4

-0.7

0

0.7

1.4

CL

CFD

Model

10 20 30 40 50 60Time

10 20 30 40 50 60Time

-1.4

-0.7

0

0.7

1.4

CL

(a) (b)

(c)

t0 = 12

t0 = 15

t0 = 18

Figure 12: Time histories of the lift coefficients in the transient region from the CFD and improved van derPol model for Re = 100, 000.

Table 2: Drag parameters at different Reynolds numbers.

Re = 200 Re = 4, 000 Re = 100, 000

〈CD〉ss 1.18 1.42 0.85a2 0.039 0.152 0.065

φ12 −35◦ −18◦ −7◦

k1 0.82 0.95 0.992k2 −0.57 −0.314 −0.127

overall behavior of the drag is determined by matching precisely the amplitude a2 and phase φ12 obtainedfrom the model with the CFD results at steady state.

To this end, we substitute equation (14) into equation (15), expand the result, and obtain

CD(t) = 〈CD〉ss + a2 [k1 cos(2ωst) − k2 sin(2ωst)] + · · · (16)

Then, by comparing equation (16) with the CFD solution

CD(t) = 〈CD〉 + a2 cos(2ωst + φ12) + · · · (17)

we obtain k1 = cos φ12 and k2 = sin φ12. In Table 2, we present the drag model parameters for Re =200, 4, 000, and 100, 000. It is clear from the values of k1 and k2 that the term C2

L can play a role nearly asinfluential as the term CLCL on the behavior of the drag coefficient.

In Figure 13, we plot the steady-state time histories of the drag coefficient for Re = 200, = 4, 000,and = 100, 000. Results obtained from the improved drag model are compared with the CFD simulationresults and excellent agreement is demonstrated for all of the cases presented.

To adapt this model to capture the transient behavior of the drag as well, we only need to express 〈CD〉as a time-dependent function, which is extracted from the CFD data and asymptotes 〈CD〉ss with time. Wepresent in Figure 14 the transient variations of the drag coefficient with time for Re = 200. The resultsof the improved model are compared with the CFD results for the starting times t0 = 40, 50, and 60 timeunits. It is clear that the improved drag model matches the CFD simulation results in amplitude and phase

14

100 104 108 112 116 120Time

1.12

1.16

1.2

1.24

CD

CFD

Model

80 84 88 92 96 100Time

1.2

1.3

1.4

1.5

1.6

100 104 108 112 116 120Time

0.76

0.82

0.88

0.94

CD

(a) (b)

(c)

Figure 13: Comparisons of the steady-state time histories of the drag coefficient obtained from the CFDsimulation and the improved model: (a) Re = 200, (b) Re = 4, 000, and (c) Re = 100, 000.

with excellent accuracy for t0 ≥ 50. We present in Figure 15 the transient variations of the drag coefficientwith time for Re = 4, 000. The results of the improved model are compared with the CFD results for thestarting times t0 = 39, 43, and 47. The improved model matches very well with the CFD simulation fort0 = 47 time units. Lastly, we present in Figure 16 the transient variations of the drag coefficient with timefor Re = 100, 000. The results of the improved model are compared with the CFD results for the startingtimes t0 = 12, 20, and 30 time units. The improved model matches very well with the CFD simulation fort0 = 30.

5 Conclusions

Lift and drag coefficients for the two-dimensional flow over a stationary circular cylinder were examinedand modeled as a preliminary step to understanding the complex problem of vortex-induced vibrations inrisers and other offshore slender structures. The models account for the coupling between these coefficients

30 40 50 60 70 80 90 100

0.9

1

1.1

1.2

1.3

CD

CFD

Model

30 40 50 60 70 80 90 100

30 40 50 60 70 80 90 100Time

0.9

1

1.1

1.2

1.3

CD

(a) (b)

(c)

30 40 50 60 70 80 90 100Time

<CD>

(d)

<CD>ss

t0 = 40

t0 = 50

t0 = 60

Figure 14: Time histories of the drag coefficient in the transient region from the CFD and improved van derPol model for Re = 200.

15

30 40 50 60 70 80

1

1.2

1.4

1.6

CD

CFD

Model

30 40 50 60 70 80

30 40 50 60 70 80Time

1

1.2

1.4

1.6

CD

(a) (b)

(c)

30 40 50 60 70 80Time

<CD>

(d)

<CD>ss

t0 = 39

t0 = 43

t0 = 47

Figure 15: Time histories of the drag coefficient in the transient region from the CFD and improved van derPol model for Re = 4, 000.

10 20 30 40 50 60

0.6

0.7

0.8

0.9

1

CD

CFD

Model

10 20 30 40 50 60

10 20 30 40 50 60Time

0.6

0.7

0.8

0.9

1

CD

(a) (b)

(c)

10 20 30 40 50 60Time

<CD>

(d)

<CD>ss

t0 = 12

t0 = 20

t0 = 30

Figure 16: Time histories of the drag coefficient in the transient region from the CFD and improved van derPol model for Re = 100, 000.

and cover the steady-state and transient behaviors. Initially, these models were based on representing thelift coefficient by the van der Pol equation and representing the drag coefficient by the sum of a meandrag term and a nonlinear term proportional to the lift coefficient times its time derivative. Values forthe parameters in these models were obtained by matching a second-order approximate solution of the vander Pol equation with the CFD steady-state lift. The latter was obtained by numerically integrating theunsteady Reynolds-averaged Navier-Stokes equations (RANS).

These initial models assumed that the phase between the main and third harmonic components of the liftand the phase between the main components of the lift and the drag are fixed at φ13 = 90◦ and φ12 = 270◦,respectively. However, the fact is that these phases vary from the aforementioned values, sometimes quiteconsiderably, with the Reynolds number. Therefore, we set out to present improved models for the lift anddrag coefficients that explicitly account for these phases.

For the lift coefficient, the van der Pol oscillator was modified by adding a Duffing-type cubic nonlinearityterm. Then, the values of the model parameters were computed by matching the model solution, obtained

16

by the method of harmonic balance, to the steady-state CFD results.As for the drag coefficient, the initial model was modified by introducing a second quadratic term that

is proportional to C2L −

⟨C2

L

⟩. The amount of contribution from each quadratic term was computed based

on exactly matching the phase φ12 to the phase computed from the CFD data.These improved models were then compared with the CFD solutions in both the steady-state and transient

regimes for the three Reynolds numbers: Re = 200, Re = 4, 000, and Re = 100, 000. For the steady-state liftand drag coefficients, we found that the improved lift and drag models do an excellent job in matching theCFD results. For the transient lift and drag coefficients, we found that the accuracy of the model dependson the starting point at which the initial conditions are taken. The reason is that part of the CFD transientresults arise from numerical inaccuracies due to the impulsive initial conditions imposed. Because thesenumerical effects diminish with time, it stands to reason that the agreement between the models and theCFD results in the transient regimes improves with later starting times. Nonetheless, it is clear that thesemodels predict the transient behavior quite well. This solidifies our conviction in these models as an initialstep towards modeling the full fluid-structure problem.

6 Acknowledgment

This work was supported by Vetco Gray and Starmark Offshore Inc., David W. Hughes and Robert Sexton,Technical Monitors.

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