Modeling Steady-State and Transient Forces on a Cylinder

Transcription

1 Modeling Steady-State and Transient Forces on a ylinder Osama Marzouk, Ali H. Nayfeh, Haider N. Arafat 2, Imran Akhtar Departmenf Engineering Science and Mechanics, M 29 2 Departmenf Electrical and omputer Engineering, M Virginia Tech, Blacksburg, VA 246 {omarzouk, anayfeh, harafat, akhtar}@vt.edu April 26, 26 Abstract Numerical simulations of flow past a stationary circular cylinder at different Reynolds numbers have been performed using a computational fluid dynamics () solver that is based on the Reynoldsaveraged Navier-Stokes equations (RANS). The results obtained are used to develop reduced-order models for the lift and drag coefficients. The models do nonly 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 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 response. Introduction. Literature Summary When flow passes over a bluff body, a streef shedding vortices (known as the von Kármán vortex street) develops in the wake and exerts oscillatory forces on the body, which are often decomposed into drag and lift components along the free-stream and the cross-flow directions, respectively. If the body is capable of flexing 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 structural failure. 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 equations is needed. This involves accounting for three-dimensional effects, turbulence structures, and elasticity, among Submitted for publication to the Journal of Vibration and ontrol orresponding Author

2 other considerations. However, in general, many of these structures are quite long, some exceeding 7, feet, such as risers and spars, and flows around them may reach very high Reynolds numbers (Re). Under these 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. ircular cylinders are extensively used in the study of bluff-body fluid dynamics due to their geometric simplicity and common use in engineering applications. We give a brief introduction of the studies that have dealt 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 subjecf VIV is given by Williamson and Govardhan (24). Bishop and Hassan (963) were one of the earliest to suggest using a self-excited oscillator to represent the forces over a cylinder due to vortex shedding. Hartlen and urrie (97) formulated a remarkable model for elastically restrained cylinders that are restricted to cross-flow motions. They used a Rayleigh oscillator to describe the lift force and coupled it to the cylinder motion by a linear velocity term. urrie and Turnball (987) proposed a similar model for the fluctuations in the drag. Skop and Griffin (973) pointed out that the parameters in the model of Hartlen and urrie (97) lacked clear connections to physical parameters of the problem. They proposed a modified van der Pol oscillator to represent the lift coupled to a linear equation of motion for the structure. They also introduced the Skop-Griffin parameter, which is now commonly used in studying VIV problems. Iwan and Blevins (974) considered the fluid mechanics of the vortex street and developed a model in terms of a fluid variable that captures the fluid dynamics effects of the problem. Landl (975) added a nonlinear 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 involved many constants to be determined. Several attempts have been made to extend the wake-oscillator models to elastic structures, such as beams and cables. Iwan (975) extended the model of Iwan and Blevins (974) to predict the maximum VIV amplitude of taut strings and circular cylindrical beams with different end conditions. Iwan (98) also derived an analytical model for the VIV of elastic structures under non-uniform flows. Skop and Griffin (975) extended their (973) rigid cylinder model to elastic cylinders. With the goal of accurately capturing the asymptotic, self-limiting structural response near zero damping, Skop and Balasubramanian (997) modified the model of Skop and Griffin (975) by separating the fluctuating lift into two components: one satisfying a van der Pol equation oscillator that is driven by the transverse motion of the cylinder and the other linearly proportional to the transverse velocity of the cylinder (called the stall term). Kim and Perkins (22) modeled the lift and drag by two nonlinearly coupled van der Pol equations in their study of resonant responses of suspended elastic cables. The coupling terms were introduced based on the fact that the main frequency of the drag component is twice the main frequency of the lift component. Krenk and Nielsen (999) proposed a two-oscillator model in which the mutual forcing terms are developed based on the premise that energy flows directly between the fluid and structure. This means that the forcing terms correspond to the same flow of energy at all times..2 Motivation Mosf the aforementioned analytical works are based on modeling the lift by either the Rayleigh or the van der Pol oscillator. Nayfeh et al. (23) numerically simulated the two-dimensional flow past a stationary cylinder for a wide range of Reynolds numbers and calculated the lift and drag. They employed higher- 2

3 order spectral analyses to determine the phase relations among the different spectral components of the lift and compared them with the phase information one obtains from closed-form approximate solutions of the van der Pol and Rayleigh oscillators. They concluded that the van der Pol oscillator is the more accurate representation of the lift. They also proposed to model the drag as the sum of a mean term and a time-varying term proportional to the producf the lift and its time derivative..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 the steady-state lift and drag is crucial. The goals of this work are two-fold: First, we extend the results of Nayfeh et al. (23) on the flow over a stationary cylinder by investigating the validity of their models in the transient region. In other words, are the models identified based on the steady-state lift and drag results capable of predicting the transient lift and drag results obtained with the code? Second, we propose modifications and improvements to both of the lift and drag models that take full and explicit advantage of the phase relations between the different components. These improved models are validated against the results for both the steady-state and transient behaviors. The order of this paper is as follows: First, we discuss the code used to generate the numerical solutions of lift and drag on a stationary cylinder. Second, we present some of the lift and drag results of Nayfeh et al. (23) for steady-state responses and 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 results 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 with current advances in hardware and software capabilities. Hence, some modeling assumptions or approximations are often needed to simplify the flow and render their computations feasible. Usually, this entails representing the turbulent scales with turbulence models, which significantly reduce the spatial and temporal resolution requirements. One alternative to DNS is the large-eddy simulation (LES) approach. In LES, the energy-containing turbulent scales are resolved, whereas the subgrid scales (SGS) are modeled. LES provides information about a wide range of spatial and temporal scales in the flow at a cost that is significantly lower than DNS. Nonetheless, for high Reynolds-number flows over complex geometries, LES computations still remain to be a formidable task. In the conventional Reynolds-averaged Navier-Stokes (RANS) modeling, the effects of all of the turbulent fluctuations are modeled. Extensively used within the fluid engineering community, the RANS approach and its variants are efficient and much cheaper than DNS and LES in computation cost, yet it yields results that are accurate enough to capture mosf the important physical characteristics of the problem. 3

4 Therefore, we compute the time-dependent, incompressible flow past a cylinder by solving the Reynoldsaveraged Navier-Stokes (RANS) equations. We employ the artificial compressibility method to speed up the solution convergence. This method introduces an artificial time-derivative of pressure term into the continuity equation, thereby explicitly coupling the pressure and the velocity field (Rogers et al., 987; horin, 997). The nondimensional equations governing the two-dimensional flow have the form where Re UD/ν is the Reynolds number and p βu q = u, F = u 2 + p v uv q t + F x + G y Re H 2 q = (), G = βv uv v 2 + p,h = (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 corresponds to having β. The problem is solved on a curvilinear O-grid with a 25D radius in order to avoid any blocking effects without using buffer domains. The convective terms are discretized using a second-order upwinding difference scheme. The physical time terms, which represent flow unsteadiness, are switched to the right-hand side and used as source terms; they are discretized using a second-order three-level backwarddifference formula. To integrate the RANS equations with inflow, outflow, and no-slip boundary conditions, radial and circumferential stretching is employed and a second-order finite-difference scheme is used. At the inflow boundary, 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 the interior points. For turbulent closure, a one-equation model, such as the Spalart-Allamras (SA) method (Spalart and Allamras, 992), is used to represent the unresolved scales. Equivalent boundary conditions are also set for the turbulence quantities. The outcome of the simulation is the pressure distribution over the cylinder surface, which ultimately is integrated to calculate the lift and drag coefficients. The time histories of the lift and drag coefficients are then converted to the spectral domain using Fourier analysis. The time and frequency are nondimensionalized using 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 shows typical time histories and corresponding power spectra for the case when Re = 4,. The lift coefficient L is presented in the top two subfigures and the drag coefficient D is presented in the bottom two subfigures. The lift power spectrum shows a large peak with amplitude a at the shedding frequency f s.2 and smaller peaks (a 3 and a 5 ) at the odd harmonics 3f s and 5f s. The corresponding time history shows the lift coefficient fluctuating periodically about the origin. Therefore, one infers that the lift coefficienscillates at the shedding frequency and that its behavior is influenced by cubic, and, to lower extent, higher-order odd nonlinearities. By contrast, the drag power spectrum shows a large peak a 2 at twice the shedding frequency (2f s ) and smaller peaks (a 4 and a 6 ) at the even harmonics 4f s and 6f s. The corresponding time history shows the drag fluctuating periodically about a non-zero mean that reaches a constant value at steady state. This implies that the drag coefficient consists of a time-varying mean term and a fluctuating term. The fluctuating term mainly varies quadratically with the lift coefficient, as the influence of the higher-order even nonlinearities is 4

5 quite small. The corresponding drag-polar plot in Figure 2 clearly illustrates this quadratic coupling between the drag and lift coefficients. 3 Modeling Background 3. Lift Nayfeh et al. (23) investigated two wake-oscillator models to represent the lift, namely, the van der Pol and Rayleigh oscillators. Using higher-order spectral moments analysis, they calculated, for a few cases, the phase angle φ 3 between the lift components at f s and 3f s to fall around 9. Based on their findings, they concluded that the van der Pol oscillator L + ω 2 L = µċ L α 2 LĊL (3) is the more suitable choice as an efficient and simple model for the lift coefficient in steady state. The angular frequency ω in equation (3) is related (but not equal) to the angular shedding frequency ω s =2πf s and the parameters µ and α represent the linear and nonlinear damping coefficients, respectively. The values of µ and α are taken positive, so that the linear damping is destabilizing while the nonlinear damping is stabilizing. As a consequence, small disturbances grow and large ones decay, both eventually approaching a stable limit cycle. The values of the parameters in equation (3) depend on the Reynolds number and their values are based on matching the steady-state lift data. Recently, the authors (Nayfeh et al., 25) examined the possibility of using equation (3) to model the lift coefficient in the transient region as well. Using the method of multiple scales (Nayfeh, 973, 98) and assuming that the oscillator is weakly damped (i.e., µ = O(ɛ) and α = O(ɛ) where ɛ is a small bookkeeping parameter), the following second-order approximate solution was obtained: L (t) =a(t) + 6ω 2 [ µ 4 αa(t)2 ] 2 sin[ωt + θ(t)+η(t)].6.8 L D x x < D > ss x a x x - L x -2 a 3 a 5 x - D x -2 a 2 a 4 x -3 x -3 a 6 x -4 f s Strouhal Number (St) x f s Strouhal Number (St) Figure : histories and corresponding power spectra of the lift and drag coefficients obtained from the simulation at Re = 4,. 5

6 [ where η(t) = tan equations 6ω αa(t) 2 4µ α 32ω a(t)3 sin[3ωt +3θ(t)] + a (t) sin[ωt + θ(t) +η(t)] a 3 (t) sin[3ωt +3θ(t)] + (4) ] and the amplitude a(t) and phase θ(t) are governed by the modulation da dt = 8 (4µa αa3 ) (5) dθ dt = ( µ 2 3 8ω 2 αµa2 + ) 32 α2 a 4 (6) Setting ȧ = in Eqn. (5), the steady-state value of a is obtained from the solution of a(4µ αa 2 )=, which is either the trivial solution a = or the nontrivial solution a =2 µ/α. For the nontrivial solution, it follows from equation (4) that µ a =2 α and a 3 = µ µ 4ω α (7) Moreover, the angle η = π 2 and, from equation (6), the corresponding expression for θ = µ 2 /6. onsequently, the angular shedding frequency is given by ω s = ω + θ = ω µ2 (8) 6ω Equation (8) shows that the angular frequency ω of the van der Pol oscillator is not exactly equal to the angular shedding frequency ω s, as one would predict from a first-order expansion (Nayfeh et al., 23). Hence, an improved second-order approximate expression for the steady-state lift coefficient becomes L (t) a cos(ω s t)+a 3 cos(3ω s t + π 2 ) (9) The methodology used to identify the system parameters for a given Reynolds number is as follows:. The solver is used to calculate the time history of the lift coefficient. 2. Spectral analysis is performed on the steady-state parf the data to extract the values of a, a 3, and f s (or ω s )..5.5 L D Figure 2: A drag-polar plof L v.s. D at Re = 4,. 6

7 L Re 2 Lift model L Re, Lift model (a) Re = 2 (b) Re =, Figure 3: omparisons 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 and the angular frequency ω. 4. With all of the parameters identified, equation (3) is numerically integrated using a Runge-Kutta routine and the results are compared with the results. In Figure 3, we compare the time history of the steady-state lifbtained from the simulation with thabtained by integrating equation (3) for the Reynolds numbers Re = 2 and Re =,. We note that the steady-state solution of the van der Pol equation is independenf the initial conditions. It is clear that the van der Pol oscillator accurately models the 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 also model the transient lift. This serves two purposes:. 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 L (t ) and Ċ L (t ), equation (3) is integrated for a long time. Because the results only yield the values of L, the initial values of Ċ L are approximated using a five-point fourth-order central finite-difference formula. In Figures 4 and 5, we compare the transient lifbtained from the model and the analysis for Re = 2 and Re =,, respectively. For each case, results initiated at four different initial times are presented. For the case Re = 2, we simulate the transient lift using the initial starting times t = 34, 39, 45, and 5. We see from Figure 4 that, the larger the starting time is, the more accurate the model is in simulating the results. For example, when t = 34, the van der Pol oscillator underestimates the amplitude of the lift markedly; also, there is a phase shift between its prediction and the lift. These differences, however, diminish as the starting time t is increased, as shown in the results for t = 5, which are in very good agreement with the results. For the case Re =,, we show in Figure 5 the predicted transient lift using the initial starting times t =, 2, 4, and 6. It follows from Figure 5 that the discrepancy in modeling the lift amplitude for moderate Reynolds number flows is resolved. However, there is still a phase difference between the two results, which again is minimized when using larger starting times. 7

8 L Lift model = L = L = L = Figure 4: Simulated and modeled transient lift coefficients for different starting times for Re = 2. 2 Lift model 2 L = L = =2 2 =6 L L Figure 5: Simulated and modeled transient lift coefficients for different starting times for Re =,. One possible explanation for these deviations is the fact that the results contain two types of transients, those arising physically and those arising from numerical analysis effects (e.g., impulsive initial conditions), and it seems that the rate of decay of the latter depends on the Reynolds number. We note from Figures 4 and 5 that the larger the Reynolds number is, the faster the rate of convergence is to the physical solution. For Re = 2, it follows from Figure 4 that the transients due to numerical effects in the simulation might propagate for some time after the initial conditions and might take up to 5 time units to decay completely. On the other hand, for Re =,, it follows from Figure 5 that the transients due to numerical effects decay much faster, lasting for about 6 time units. learly, the agreement between the van der Pol oscillator and results is quite good once the transients arising from numerical effects have died out. Therefore, the van der Pol oscillator seems to be capable of modeling both of the steady-state lift and the transient lifn a cylinder in uniform flow. 3.2 Drag Referring to Figure, we note that the drag consists of two major components. The first is a mean component that monotonously approaches a constant value in the steady state; this component is assumed to be independenf the lift. The second is an oscillatory component related to the lift and has a frequency equal 8

9 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. (23) reasoned that the drag is quadratically related to the lift in some fashion. They examined the phase relation between the periodic components of the drag and lift and found that it falls around 27. Hence, they inferred that the periodic componenf the drag must be proportional to L Ċ L and proposed the drag model D (t) = D 2 a 2 ω s a 2 L (t)ċl(t) () where a 2 is the amplitude of the drag component at 2f s and D is the mean componenf drag. For steady-state behavior, D is constant, while, for transient behavior, it increases monotonously with time. The constant value of D is determined from the steady-state time history of the drag and the value of a 2 is determined from its spectral analysis. In Figure 6, the drag results obtained from equation () are compared with the calculations for the Reynolds numbers Re = 2 and,. For Re = 2, excellent agreement is found between the two solutions. For Re =,, the agreement is also quite good, but there appears to be a slight deviation in phase between the two results. We also examined the capability of equation () to predict the transient drag. However, because the mean drag D is time-dependent in the transient region, modeling it as a constant following the steadystate results brings about a mismatch with the solution. To account for this discrepancy, we first extracted the transient profile of the mean drag from the simulation and fitted it into a polynomial function in time to replace the constant value of D. Then, substituting the transient lift results at different initial times into equation (), we calculated the corresponding transient drag. In Figures 7 and 8, we compare the transient results from the code and the drag model for the Reynolds numbers Re = 2 and,. Again, we find that, in general, the accuracy of the proposed model improves for later starting times as the transient effects due to numerical computations diminish. For the low Reynolds number flow in Figure 7, we find that the proposed model underestimates the amplitude of D when starting at t = 4, but it does an excellent job when starting at t 6, which is still in the transient region. For the moderate Reynolds number flow in Figure 8, we find that the model does an Re 2 Drag model Re, Drag model D.8 D (a) Re = 2 (b) Re =, Figure 6: omparisons between the simulated and modeled steady-state drag coefficients. 9

10 D D =4. Drag mode t =5 o D D = = Figure 7: Simulated and modeled transient drag coefficients for different starting times for Re = 2. D D.9.8 =.7 Drag mode = D D = t =6 o Figure 8: Simulated and modeled transient parf the drag coefficients for different starting times for Re =,. excellent job in predicting the amplitude of D ; however, there exists a small phase difference between the two solutions. 4 Improved Lift and Drag Models The results presented in the previous section are solely based on matching the amplitudes and frequencies of the and model results. Even with the good agreement found, especially at steady state, we believe there is still room for improvement. Although the use of higher-order spectral analysis of the results showed that the phase φ 3 between the lift components at f s and 3f s is nearly 9, it actually differs from one case of Reynolds number to another. In fact, in some of our calculations, we have found separation from 9 as large as ±25. Obviously, this wide variation in the phase is not fully accounted for in the van der Pol model in equation (3). This fact does not produce a real problem as long as one is concerned with the time histories because a 3 is usually two orders of magnitude smaller than a. However, because we intend to extend this model to cases of forced and freely vibrating cylinders, we would like to have it be very accurate in both the time domain and the spectral domain. Similarly, for the drag calculations, we have found that the phase φ 2 between the lift component

11 at f s and the drag component at 2f s varies from 27 by as much as 85. Qin (24) proposed that the quadratic term in the drag model should be of the form 2 L instead of LĊL. He also found linear coherence between the drag and lift components at f s and 3f s and introduced a linear lift term in the drag model to account for it. 4. Lift Here, we modify the van der Pol oscillator by introducing a Duffing-type nonlinearity to equation (3), resulting in the new lift model L + ω 2 L = µċl α 2 LĊL γ 3 L () The coefficient γ is determined based on matching precisely the phase φ 3 obtained from the data to the phase obtained from solving equation (). In this process, we use the method of harmonic balance to determine approximate solutions of the model. These solutions along with spectral analysis of the data are used to identify all of the parameters in equation (). We seek a solution for the lift coefficienf the form L (t) =c cos(ω s t)+c 2 sin(ω s t)+c 3 cos(3ω s t)+c 4 sin(3ω s t) (2) Then, upon substituting equation (2) into equation () and separating the terms multiplying the different sine and cosine functions, we obtain the linear system Ay = b (3) where y = {ω 2,µ,α,γ}, b = {c,c 2, 9c 3, 9c 4 }ω 2 s, and the entries of the matrix A are A = c, A 2 = c 2, A 3 = c 3, A 4 = c 4, A 2 = c 2 ω s, A 22 = c ω s, A 32 = 3c 4 ω s, A 42 =3c 3 ω s, A 3 = ( c c 4 c c2 c 2 +2c 2 3 c 2 +2c 2 4 c 2 2c c 3 c 2 + c 2 c ) 4 ωs, A 23 = ( c 3 4 c 3 c 2 c2 2 c 2c 2 3 c 2c 2 4 c 2c 2 c 4 c + c 2 2 c ) 3 ωs, A 33 = ( c c 4 c c 2 c 2 +3c c 2 c 4 +3c 2 ) 3c 4 ωs, A 43 = ( c 3 4 6c 3 c 2 +3c 2 2c 3c 3 3 3c 3 c 2 4 6c 2 ) 2c 3 ωs, A 4 = ( ) 3c c 3 c 2 +3c 2 2c +6c 2 3c +6c 2 4c +6c 2 c 4 c 3c 2 2c 3, A 24 = ( ) 3c c 4 c c 2 c 2 +6c 2 3c 2 +6c 2 4c 2 6c c 3 c 2 +3c 2 c 4, A 34 = ( ) c c 3 c 2 3c 2 2c +3c c 3 c c 2 2c 3, A 44 = ( ) c c 4 c c 2 c 2 +3c c 2 c 4 +3c 2 3c 4 The c j are determined by matching the solution in equation (2) with the steady-state simulation solution which is expressed as This yields c = a, c 2 =,c 3 = a 3 cos φ 3, and c 4 = a 3 sin φ 3. L (t) =a cos(ω s t)+a 3 cos(3ω s t + φ 3 ) (4)

12 Table : Lift parameters at different Reynolds numbers. Re = 2 Re = 4, Re =, f s a a φ ω s ω µ α γ (a).6.8 (b) L L (c) Model Figure 9: omparisons of the steady-state time histories of the lift coefficients obtained with the simulation and the improved model: (a) Re = 2, (b) Re = 4,, and (c) Re =,. Listed in Table are the results for the cases Re = 2, 4,, and,, covering a fairly wide range of Reynolds numbers. Typically, risers operate in the lower subsef that range while spars operate in the upper subset and beyond it. From the values of α and γ, it is clear that the influence of the Duffing-type cubic term should not be neglected in the modeling. In Figure 9, we plot the steady-state time histories of the lift coefficienbtained from the improved model and the 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. In Figure, we show the transient variations of the lift coefficient with time for Re = 2. The results of the improved model are compared with the results for the starting times t = 4, 5, and 6 time units. It is clear that the improved model for the lift matches the simulation results in amplitude and phase with excellent accuracy for t 5. In Figure, the transient variations of the lift coefficient with time for Re = 4, are shown. The results of the improved model are compared with the results for the starting times t = 38, 42, and 45 time units and the improved model matches very well the simulation for t 45. Lastly, we present in Figure 2 the transient variations of the lift coefficient with time for Re =,. The results of the improved model are compared with the results for the starting times t = 2, 5, and 8 time units. The improved lift model matches very well the simulation for t 8. 2

13 L L (a) t = (c) t = (b) t = Model Figure : histories of the lift coefficients in the transient region from the and improved van der Pol model for Re = 2. L L (a) t = (c) t = (b) t = Model Figure : histories of the lift coefficients in the transient region from the and improved van der Pol model for Re = 4,. Nonetheless, a small deviation in phase still remains in the lift transient results for t = Drag In the simulations we have conducted thus far, we have found that the phase φ 2 is around 27. However, the actual value of the phase may vary from one case of Reynolds number to another by almost 85, which is quite significant. Therefore, we propose a new model in which the drag is proportional to both L Ċ L and L 2 in the following manner: a 2 ( D (t) = D +2k 2 a 2 L 2 a 2 L ) +2k2 ω s a 2 L Ċ L (5) where the brackets denote the mean value of the term inside. The first expression in equation (5) stands for the mean componenf the drag, which reaches a constant value D ss in the steady-state region. The term L 2 in the second expression in equation (5) negates the D component introduced by L 2. The amounf contribution from both quadratic expressions to the 3

14 .4.7 (a) (b) t = 5 L t = 2 L (c) t = Model Figure 2: histories of the lift coefficients in the transient region from the and improved van der Pol model for Re =,. Table 2: Drag parameters at different Reynolds numbers. Re = 2 Re = 4, Re =, D ss a φ k k overall behavior of the drag is determined by matching precisely the amplitude a 2 and phase φ 2 obtained from the model with the results at steady state. To this end, we substitute equation (4) into equation (5), expand the result, and obtain D (t) = D ss + a 2 [k cos(2ω s t) k 2 sin(2ω s t)] + (6) Then, by comparing equation (6) with the solution D (t) = D + a 2 cos(2ω s t + φ 2 )+ (7) we obtain k = cos φ 2 and k 2 = sin φ 2. In Table 2, we present the drag model parameters for Re = 2, 4,, and,. It is clear from the values of k and k 2 that the term L 2 can play a role nearly as influential as the term L Ċ L on the behavior of the drag coefficient. In Figure 3, we plot the steady-state time histories of the drag coefficient for Re = 2, = 4,, and =,. Results obtained from the improved drag model are compared with the simulation results 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 D as a time-dependent function, which is extracted from the data and asymptotes D ss with time. We present in Figure 4 the transient variations of the drag coefficient with time for Re = 2. The results of the improved model are compared with the results for the starting times t = 4, 5, and 6 time units. It is clear that the improved drag model matches the simulation results in amplitude and phase 4

15 .24.2 D (c).88 D (a) (b) Model Figure 3: omparisons of the steady-state time histories of the drag coefficienbtained from the simulation and the improved model: (a) Re = 2, (b) Re = 4,, and (c) Re =,. with excellent accuracy for t 5. We present in Figure 5 the transient variations of the drag coefficient with time for Re = 4,. The results of the improved model are compared with the results for the starting times t = 39, 43, and 47. The improved model matches very well with the simulation for t = 47 time units. Lastly, we present in Figure 6 the transient variations of the drag coefficient with time for Re =,. The results of the improved model are compared with the results for the starting times t = 2, 2, and 3 time units. The improved model matches very well with the simulation for t = 3. 5 onclusions Lift and drag coefficients for the two-dimensional flow over a stationary circular cylinder were examined and modeled as a preliminary step to understanding the complex problem of vortex-induced vibrations in risers and other offshore slender structures. The models account for the coupling between these coefficients.3.2 D D..9 (a) Model (c) t = 4 t = > D < (b) (d) t = 5 < D > ss Figure 4: histories of the drag coefficient in the transient region from the and improved van der Pol model for Re = 2. 5

16 .6 (a) (b) D.4.2 t = 39 Model t = (c) (d) D.4.2 t = 47 > D < < D > ss Figure 5: histories of the drag coefficient in the transient region from the and improved van der Pol model for Re = 4,..9 D (a) t = 2 Model (b) t = D.8.7 (c) t = 3 > D < (d) < D > ss Figure 6: histories of the drag coefficient in the transient region from the and improved van der Pol model for Re =,. and cover the steady-state and transient behaviors. Initially, these models were based on representing the lift coefficient by the van der Pol equation and representing the drag coefficient by the sum of a mean drag term and a nonlinear term proportional to the lift coefficient times its time derivative. Values for the parameters in these models were obtained by matching a second-order approximate solution of the van der Pol equation with the steady-state lift. The latter was obtained by numerically integrating the unsteady Reynolds-averaged Navier-Stokes equations (RANS). These initial models assumed that the phase between the main and third harmonic components of the lift and the phase between the main components of the lift and the drag are fixed at φ 3 =9 and φ 2 = 27, respectively. However, the fact is that these phases vary from the aforementioned values, sometimes quite considerably, with the Reynolds number. Therefore, we seut to present improved models for the lift and drag coefficients that explicitly account for these phases. For the lift coefficient, the van der Pol oscillator was modified by adding a Duffing-type cubic nonlinearity term. Then, the values of the model parameters were computed by matching the model solution, obtained 6

17 by the method of harmonic balance, to the steady-state results. As for the drag coefficient, the initial model was modified by introducing a second quadratic term that is proportional to 2 L 2 L. The amounf contribution from each quadratic term was computed based on exactly matching the phase φ 2 to the phase computed from the data. These improved models were then compared with the solutions in both the steady-state and transient regimes for the three Reynolds numbers: Re = 2, Re = 4,, and Re =,. For the steady-state lift and drag coefficients, we found that the improved lift and drag models do an excellent job in matching the results. For the transient lift and drag coefficients, we found that the accuracy of the model depends on the starting point at which the initial conditions are taken. The reason is that parf the transient results arise from numerical inaccuracies due to the impulsive initial conditions imposed. Because these numerical effects diminish with time, it stands to reason that the agreement between the models and the results in the transient regimes improves with later starting times. Nonetheless, it is clear that these models predict the transient behavior quite well. This solidifies our conviction in these models as an initial step 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. References Bishop, R.E.D., Hassan, A.Y., 963, The lift and drag forces on a circular cylinder in a flowing fluid, Proceedings of the Royal Society Series A 277, horin, A. J., 997, A Numerical method for solving incompressible viscous flow problems, Journal of omputational Physics 35, urrie, I. G., Turnball, D. H., 987, Streamwise oscillations of cylinders near the critical Reynolds number, Journal of Fluids and Structures, Hartlen, R. T., urrie, I. G., 97, Lift-oscillator model of vortex-induced vibration, ASE Journal of Engineering Mechanics 96, Iwan, W.D., Blevins, R.D., 974, A model for vortex-induced oscillation of structures, Journal of Applied Mechanics 4 (3), Iwan, W.D., 975, The vortex induced oscillation of elastic structures, Journal of Engineering for Industry 97, Iwan, W.D., 98, The vortex-induced oscillation of non-uniform structural systems, Journal of Sound and Vibration 79 (2), Krenk, S., Nielsen, S.R.K., 999, Energy balanced double oscillator model for vortex-induced vibrations, Journal of Engineering Mechanics 25 (3), Kim, W. J., Perkins, N.., 22, Two-dimensional vortex-induced vibration of cable suspensions, Journal of Fluids and Structures 6 (2),

18 Landl, R., 975, A mathematical model for vortex-excited vibrations of bluff bodies, Journal of Sound and Vibration 42 (2), Nayfeh, A. H., 973, Perturbation Methods, Wiley, New York. Nayfeh, A. H., 98, Introduction to Perturbation Techniques, Wiley, New York. Nayfeh, A. H., Owis, F., Hajj, M. R., 23, A model for the coupled lift and drag on a circular cylinder, Proceedings of DET3, ASME 23 Design Engineering Technical onferences and omputers and Information in Engineering onference. Nayfeh, A. H., Marzouk, O. A., Arafat, H. N., Akhtar, I., 25, Modeling the Transient and Steady-State Flow over a Stationary ylinder, ASME 2th Biennial onference on Mechanical Vibration and Noise, 5th International onference on Multibody Systems, Nonlinear Dynamics and ontrol, DET , Long Beach, A, Sep 25. Qin, L., 24, Developmenf Reduced-Order Models for Lift and Drag on Oscillating ylinders with Higher-Order Spectral Moments, Ph.D. Dissertation, Virginia Tech, Blacksburg, VA. Rogers, S. E., Kwak, D., Kaul, U., 987, On the accuracy of the pseudocompressibility method in solving the incompressible Navier-Stokes equations, Applied Mathematics Modelling, Skop, R. A., Griffin, O. M., 973, A model for the vortex-excited resonant response of bluff cylinders, Journal of Sound and Vibration 27 (2), Skop, R. A., Griffin, O. M., 975, On a theory for the vortex-excited oscillations of flexible cylindrical structures, Journal of Sound and Vibration 4 (3), Skop, R. A., Balasubramanian, S., 997, A new twisn an old model for vortex-excited vibrations, Journal of Fluids and Structures (4), Spalart, P. R., and Allamras, S. R., 992, A one-equation turbulence model for aerodynamic flows, In Proceedings of the AIAA 3th Aerospace Sciences Meeting and Exhibit, AIAA paper Williamson,. H. K., Govardhan, R., 24, vortex-induced vibrations, Annual Review of Fluid Mechanics, 36 (4)