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Low-Dimensional Modeling of Flow-Induced Vibrations via Proper Orthogonal Decomposition

D.Newman, G.Karniadakis (Brown University, USA)

ABSTRACT: We investigate the use of proper orthogonal decomposition analysis on direct numerical simulations of flow over vibrating cylinders and cables. We first simulate the 2-d and 3-d wakes behind vibrating cylinders and cables, and then use the method of snapshots to compute the most energetic eigenmodes of these wakes. We examine the eigenmode energy decay versus mode number, and discuss the possibility of constructing low-dimensional dynamic models to simulate and predict the behavior of these systems.

1
Introduction

Fluid flows over flexible cables arise in many engineering situations, such as marine cables towing instruments, flexible risers used in petroleum production and mooring lines [1], [2], [3], [4], [5]. It is therefore important to understand and be able to predict the hydrodynamic forces and motion of cables caused by flow-induced vibration. Direct numerical simulations provide a means of computing these flows; however, as the problem size increases and as the Reynolds number increases, the computational requirements grow rapidly. One alternative method for simulating these flows is to use databases of existing computations to construct low-dimensional dynamical models to efficiently simulate the system at other operating conditions. In this paper we examine the use of proper orthogonal decomposition analysis to model the complex dynamics in the wake of vibrating cylinders and cables.

This paper is organized as follows: First, we describe the fluid/structure interaction problem, outline the solution method and describe the proper orthogonal decomposition procedure of computing the most energetic eigenmodes. Next we present results of 2-d simulations of flow over a vibrating cylinder, and compute the 2-d eigenmodes. Finally we consider 3-d simulations of flow-induced vibrations of flexible cables, and we compute the 3-d eigenmodes for these wakes, and discuss the construction of low-dimensional dynamic models.

2
Formulation
2.1
Governing Equations

We consider the interaction of an incompressible fluid flowing past a long flexible cable under tension. The equations that describe this problem are the coupled system of fluid equations and cable equations. The fluid equations are given by the Navier-Stokes equations and the continuity equation. In a stationary, Cartesian coordinate system (x′,y′,z′) these equations are:

For forced cable vibration, the motion of the cable is prescribed, usually in the form of a standing wave (for simplicity, constrained to move only in the crossflow direction) with an amplitude A, wavelength L, and frequency ωf:

ζ(z,t) = A cos ωft cos2πz/L.



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Twenty-First Symposium on NAVAL HYDRODYNAMICS Low-Dimensional Modeling of Flow-Induced Vibrations via Proper Orthogonal Decomposition D.Newman, G.Karniadakis (Brown University, USA) ABSTRACT: We investigate the use of proper orthogonal decomposition analysis on direct numerical simulations of flow over vibrating cylinders and cables. We first simulate the 2-d and 3-d wakes behind vibrating cylinders and cables, and then use the method of snapshots to compute the most energetic eigenmodes of these wakes. We examine the eigenmode energy decay versus mode number, and discuss the possibility of constructing low-dimensional dynamic models to simulate and predict the behavior of these systems. 1 Introduction Fluid flows over flexible cables arise in many engineering situations, such as marine cables towing instruments, flexible risers used in petroleum production and mooring lines [1], [2], [3], [4], [5]. It is therefore important to understand and be able to predict the hydrodynamic forces and motion of cables caused by flow-induced vibration. Direct numerical simulations provide a means of computing these flows; however, as the problem size increases and as the Reynolds number increases, the computational requirements grow rapidly. One alternative method for simulating these flows is to use databases of existing computations to construct low-dimensional dynamical models to efficiently simulate the system at other operating conditions. In this paper we examine the use of proper orthogonal decomposition analysis to model the complex dynamics in the wake of vibrating cylinders and cables. This paper is organized as follows: First, we describe the fluid/structure interaction problem, outline the solution method and describe the proper orthogonal decomposition procedure of computing the most energetic eigenmodes. Next we present results of 2-d simulations of flow over a vibrating cylinder, and compute the 2-d eigenmodes. Finally we consider 3-d simulations of flow-induced vibrations of flexible cables, and we compute the 3-d eigenmodes for these wakes, and discuss the construction of low-dimensional dynamic models. 2 Formulation 2.1 Governing Equations We consider the interaction of an incompressible fluid flowing past a long flexible cable under tension. The equations that describe this problem are the coupled system of fluid equations and cable equations. The fluid equations are given by the Navier-Stokes equations and the continuity equation. In a stationary, Cartesian coordinate system (x′,y′,z′) these equations are: For forced cable vibration, the motion of the cable is prescribed, usually in the form of a standing wave (for simplicity, constrained to move only in the crossflow direction) with an amplitude A, wavelength L, and frequency ωf: ζ(z,t) = A cos ωft cos2πz/L.

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Twenty-First Symposium on NAVAL HYDRODYNAMICS Figure 1: The coordinate system is attached to the moving cable, producing an undeformed, stationary computational domain. In the case of flow-induced vibration, the equation of motion of the cable for its two directions of motion (i.e. in the x and y-directions) is given by a slightly modified forced wave equation: where ξ(z,t) = (ζ(z,t), η(z,t)) gives the cable displacement in the streamwise and crossflow directions and gives the phase speed of waves in the cable. The cable has mass per unit length m and tension T. To maintain a mean displacement, the cable is lightly elastically supported by linear springs with spring constant k, giving a natural frequency of The spring constant is selected sufficiently small to have negligible effect on the cable response. The fluid force on the cable is denoted by F(z, t). The components of F(z, t) in the streamwise and crossflow directions are the drag and lift force on the cable. Internal damping is neglected here as it does not significantly influence the response. To simplify the solution of the fluid equations we use a coordinate system attached to the cable. This maps the time-dependent and deforming problem domain to a stationary and non-deforming one as shown in Figure 1. This mapping is described by the following transformation: x=x′–ζ(x′,t′), y=y′–η(z′,t′), z=z′. Accordingly, the velocity components and pressure are transformed as follows: The Navier-Stokes equation and continuity equation are transformed to where the forcing term A(u,p,ξ) is the extra acceleration introduced by the transformation, consisting of both inviscid and viscous contributions. 2.2 Numerical Method Figure 2: View of cable showing “slices” of spanwise vorticity and spectral element mesh. To solve the three-dimensional Navier-Stokes equations, we use a parallel spectral element/Fourier method [6]. Spectral elements are used to discretize the x-y planes, while a Fourier expansion is used in the z-direction (i.e. along the cable). Consequently, all flow and cable variables are assumed to be periodic in the spanwise direction. The computational domain extends 35d (cable diameters) downstream, and 15d above and below the cable. The “cable span”, i.e. wavelength of vibrations in the cable was L/d=12.6 for the 3-d simulations. Each x-y plane is discretized by a 110 element mesh, with each element

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Twenty-First Symposium on NAVAL HYDRODYNAMICS having 81 collocation points. Typically 32 z-planes (16 Fourier modes) are used giving a total of 225,000 mesh points (see Figure 2). A non-dimensional time step of ΔtU/d=0.002 is used giving over 3,000 time steps per shedding cycle. For each simulation, the cable initial position and velocity are set and the simulations are run for at least 20 shedding cycles, or until the statistics are relatively stationary, 2.3 POD Analysis To get a more general indication of the structure of a particular wake, we use proper orthogonal decomposition. Proper orthogonal decomposition (POD) is a methodology that first identifies the most energetic “modes” in an evolving system, and second provides a means of obtaining a low dimensional description of the system's dynamics. POD was applied to turbulence problems by Sirovich [7], while Dean et al. [8] generated low-order models for incompressible flows in complex geometries. Once the POD model has been generated, we can use it to predict behavior outside the parameter range used to compute the modes, e.g. predict the dynamics of a particular incompressible flow at different Reynolds numbers. Here we apply POD to examine the structure of the 2-d and 3-d wakes behind vibrating cylinders and cables. The POD method starts by expressing the evolution of a field u(x,t), by a sum of functions of time, aj(t), multiplied by spatial eigenmodes, j(x), In this paper we will use POD to just compute eigenmodes j(x) of the wakes in order to study their structure. The procedure to compute the eigenmodes j can be described in a general form. Consider the time varying field u(x,t). Let us define the instantaneous energy as and the discrete approximation of u(x,t) be given by vector u(t), where the dimension of u(t) is n. This dimension n would correspond to the total number of mesh points used to discretize u(x,t). Let us consider u(t) at discrete time intervals t=iΔt, and denote ui=u(iΔt). Then the discrete version of the energy integral becomes where W would be a positive definite matrix (typically diagonal) of discrete integration weights from quadrature. Now we introduce the method of snapshots, where we examine a series of snapshots of u(x,t) taken at m time intervals, t=iΔt, i=1,…, m. Let us construct a matrix U of these snapshots ui, so U={u1,…,um} and the dimension of U is n×m. We compute the covariance matrix E by taking the weighted inner product of each snapshot ui with every other snapshot uj: Note that the ith diagonal entry of E is eii, the energy of snapshot ui. The dimension of E is m×m, and typically m n, i.e. the number of snapshots is much lower than the number of spatial degrees of freedom (e.g. for our applications O(m)=100 and O(n)=104–106). The eigenmodes j are computed by first computing the E's eigenvalues and eigenvectors: Eq=λq or EQ=ΛQ, where λ is the eigenvalue, q is the eigenvector, Λ is a diagonal matrix of all the eigenvalues, and Q is the corresponding matrix of eigenvectors. Note that E is symmetric positive definite (if U>0), and consequently (after normalization) the eigenvectors are orthonormal, with Q–1=QT. Proceeding, we now define our set of eigenmodes Φ as Φ=UQ Orthogonality of the eigenmodes is automatically guaranteed by their construction. Furthermore, the eigenvalues measure the energy of each eigenmode. Writing out the components of Φ={1,…, m} and the diagonal entries of Λ={λ1,…,λm}, we have the energy of eigenmode i is λi. One important quantity we will measure is how quickly the eigenmode energy λi decreases (i.e. convergence rate), as this will indicate an appropriate point to truncate the collection of eigenmodes. At this point we have the m eigenmodes i and corresponding mode energies λi. If we wish to reconstruct a snapshot u using J out of the m eigenmodes, we simply project the snapshot u onto the eigenmode basis to get the J coefficients. Let the projected (or reconstructed) snapshot be û. Then

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Twenty-First Symposium on NAVAL HYDRODYNAMICS 3 2-D DNS Results In this section we will examine the effect of cylinder oscillation on the 2-d wake. Experimental crossflow vibration studies were first performed by Koopmann [9]. Ongoren and Rockwell [10] studied vortex pairing in the wake of an oscillating cylinder also at low Reynolds numbers. Williamson and Roshko [11] did an extensive study classifying the wake modes as a function of the forcing frequency and amplitude. We will concentrate on vibration frequencies and amplitudes that are close to those observed in flow-induced vibration cases. The forced cylinder oscillation cases are parameterized by the non-dimensional vibration amplitude A and frequency ωf. We will refer to the forcing frequency ratio ωf/ω0, which is defined as the ratio of forcing frequency ωf to the frequency of vortex shedding of the fixed cylinder, ω0. Figure 3: Forced vibration frequency-amplitude plot showing lock-in region from Koopmann, and Re=100 simulation results. Koopmann [9] conducted a series of experiments at several Reynolds numbers measuring the flow behind cylinders oscillating in the crossflow direction. He considered crossflow vibration amplitudes from η/d=0.05 to η/d=1, and he varied the vibration frequency above and below the vortex shedding frequency of the fixed cylinder at that Reynolds number. We conducted 2-d simulations for several frequency-amplitude pairs, and the results are plotted in Figure 3. The points show the simulation results, while the shaded region is from Koopmann's data at Re=100. Here we illustrate the wake “lock-in” phenomenon, where the frequency of vortex shedding in the wake matches the frequency of cylinder vibration. The shaded region delineates the frequency-amplitude envelope where we observe a locked-in wake. We see that the simulation results agree with the lock-in region measured by Koopmann. Figure 4: Forced vibration wakes at Re=100: vorticity for frequency ratios ωf/ω0=0.6, 0.8 and 1.4 and amplitude η/d=0.5. To more closely examine this lock-in/no lock-in behavior, we concentrate on the Re=100 results for non-dimensional crossflow amplitude η/d=0.5, and frequency ratios ωf/ω0=0.6, 0.8, 1.0, 1.2 and 1.4 (i.e. synchronous, and 20% and 40% above and below synchronous). These simulations were run for 120 time units (20 shedding cycles for the fixed cylinder case). To illustrate the different wake patterns, we highlight three cases: sub-lock-in ωf/ω0=0.6, lock-in (but sub-synchronous) ωf/ω0=0.8, and super-lock-in ωf/ω0=1.4. Contour plots of vorticity for these frequency ratios of 0.6, 0.8 and 1.4 are shown in Figure 4. We immediately see from the picture of the wake vorticity that when ωf/ω0=0.8 (middle plot) a regular wake structure is displayed, and classed as locked-in, but when ωf/ω0=0.6 and

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Twenty-First Symposium on NAVAL HYDRODYNAMICS 1.4, the wakes are not classed as locked-in. Note that although the wake in case ωf/ω0=0.8 looks similar to that of the fixed cylinder, it is in fact different due to the lower frequency of vortex shedding, producing vortices with streamwise spacing of 6.8d compared to 5.4d for the fixed cylinder, commensurate with the ratio of 0.8. Although the wake patterns in cases ωf/ω0=0.6 and 1.4 do not resemble the usual Karman vortex street, we do observe some regularity and pattern to the wake, with vortex pairing observed when ωf/ω0=0.6 and vortex merging occurring when ωf/ω0=1.4. 3.1 2-D POD Eigenmodes Here we compute the eigenmodes for the 2-d simulations. Note that although the POD procedure was described for some scalar field u, and energy function we simply apply the procedure to the 2-d flow problem by concatenating the u and v velocities, giving the consistent energy function (and similarly for the 3-d flows). We computed the eigenmodes for the fixed cylinder flow as well as the five forced vibration cases ωf/ω0=0.6,0.8,1.0,1.2 and 1.4. In each case we took 300 snapshots (m=300) over tU/d=60 time units (on average, about 10 shedding cycles), giving the interval between snapshots of ΔtU/d=0.2. We examine the results for the fixed cylinder and the three forced vibration cases ωf/ω0=0.6, 0.8 and 1.4. We compute the vorticity eigenmode directly from the computed velocity eigenmodes, i.e. . Note that this gives the desired answer, rather than computing the eigenmodes from snapshots of vorticity, which would be based on enstrophy We also mention here two points: first, we are computing the eigenmodes based on the fluctuations from the mean flow, and second, the eigenmodes are computed in the reference frame of the cylinder, i.e. in the transformed coordinate system where the cylinder (or cable) appears stationary and straight. Figures 5, 6, 7 and 8 show the six most energetic eigenmodes for the fixed cylinder and forced vibration cases ωf/ω0=0.6, ωf/ω0=0.8, and ωf/ω0=1.4 respectively. First looking at the fixed cylinder case, where the wake is the most regular, we see that the eigenmodes occur in “conjugate pairs”, i.e. 2 is 90° out of phase with 1. This phase relationship is the same for the eigenmode pairs and . The size of the structures in the first pair match the vortex spacing in the snapshots, and these eigenmodes are symmetric about the x-axis. We classify the patterns in these 2 eigenmodes as (1,1), i.e. structures with a scale of 1 vortex in the x- Figure 5: Eigenmodes 1–6 for the fixed cylinder. Figure 6: Eigenmodes 1–6 for forced vibration ωf/ω0=0.6.

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Twenty-First Symposium on NAVAL HYDRODYNAMICS direction, and 1 vortex in the y-direction. The next pair would then be classified as (2,2), since the size of the structures is half in both the x and y-directions. These eigenmodes are anti-symmetric about the x-axis. The third pair is classified as (3,1), again symmetric about the x-axis. Each subsequent pair of eigenmodes introduces successively finer scales into the structure of the wake. Figure 7: Eigenmodes 1–6 for forced vibration ωf/ω0=0.8. Figure 8: Eigenmodes 1–6 for forced vibration ωf/ω0=1.4. Table 1: Modal energy percentages of eigenmodes 1–6 for 2-d cases: fixed cylinder, and forced vibration cases ωf/ω0=0.6,0.8 and 1.4, Re=100. Mode Fixed 0.6 0.8 1.4 1 49.1 18.9 48.4 19.6 2 48.1 17.8 42.8 18.0 3 0.9 10.8 3.1 12.1 4 0.9 9.4 2.3 11.6 5 0.4 7.9 1.4 10.9 6 0.4 7.2 1.2 6.0 How do the energies of these modes decrease with increasing mode? Since the same contour levels are used in all the plots, we get some indication of the amount of energy in each mode. The values are given in Table 1 for the four cases, showing the the percentage of the total energy that that eigenmode contains. The pairing of the eigenmodes is clearly indicated by the energies, for the fixed cylinder case. The eigenmode energy rapidly decreases with increasing mode number (the first 2 modes contain 97.2% of the energy, while the first 6 modes contain 99.8% of the energy). This is also an indication of how unimodal time periodic the flow is. Now looking at the sub-lock-in eigenmodes for ωf/ω0=0.6 (Figure 6), we see quite a different set of patterns compared to the fixed cylinder case. Again, the first pair of eigenmodes show some regularity, this time with anti-symmetry about the x-axis, but again, approximately 90° out of phase with one another. Modes 3–6 are more difficult to classify specifically, but we do see the finer structures emerging with increasing mode number. Looking at the Table 1 in this case shows that the first 2 eigenmodes only contain 36.7% of the total energy. The decay rate of eigenmode energy is relatively low—the first 6 eigenmodes contain just 72% of the total energy. Furthermore, the total energy in this forced case is 3.2 times larger than that in the fixed case. The locked-in case ωf/ω0=0.8 (Figure 7) eigenmodes closely resemble those for the fixed cylinder (taking into account the 1/0.8 expansion of the length scales). The classification of the eigenmode pairs follow that of the fixed cylinder, however in this forced case the energy of each eigenmode is substan-

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Twenty-First Symposium on NAVAL HYDRODYNAMICS tially larger (comparing the numbers in the table)—and the total energy in this forced case is more than four times larger than that in the fixed case. Finally in the super-lock-in case ωf/ω0=1.4 (Figure 8), we see a reasonably regular first pair of modes. Here we point out that the ωf/ω0=0.6 flow case was the one where the first pair of eigenmodes did not have approximate symmetry about the x-axis. Furthermore, it looks like modes 3 and 6 are a conjugate pair, and modes 4 and 5 are a conjugate pair. This splitting of the conjugate pairs is possibly due to time sampling resolution. We see a similar decay rate of mode energy to the sub-lock-in case—37.6% of the energy is contained in the first 2 modes, and just 78.2% of the energy is contained in the first 6 modes. The total energy in this case was the largest, at 4.4 times that of the fixed cylinder. We demonstrate how the eigenmodes are used to reconstruct a snapshot. This gives us a qualitative indication of how a low-order dynamical model would reproduce the dynamics of a particular wake. Let us review the reconstruction procedure. We seek an approximation û to a snapshot u using the first J of m eigenmodes, i.e. Of course when we use all the eigenmodes J=m, we get a perfect reproduction, û=u, so for J<m, û approximates u. The coefficients aj are simply computed by projection. The three forced vibration cases were used to demonstrate the reconstruction procedure. In each case we reconstruct using (the highest) 2, 4, 8, 16 and 32 eigenmodes (the total number of eigenmodes available is equal to the number of snapshots, m=300). The results are shown in Figures 9, 10 and 11. We see that when ωf/ω0=0.6, the coarse structure of the wake is captured using 2–4 modes, however it requires at least 32 modes to get a reasonable qualitative match with the original snapshot. Note that this is still just using approximately one tenth of all the eigenmodes (32 out of 300). The locked-in case, (ωf/ω0=0.8, Figure 10) requires far fewer modes to get a qualitative match—here just 8 eigenmodes capture the details of the wake structure reasonably well. Finally, the super-lock-in case (ωf/ω0=1.4, Figure 11) shows a similar trend as for ωf/ω0=0.6, i.e. 32 modes are required to obtain a reasonably accurate reconstruction. The point at which we choose to truncate the collection of eigenmodes can be selected by examining the eigenmode energy spectrum. This is plotted for our fixed cylinder and forced vibration cases in Figure 12. Here we plot the eigenmode energy (as a proportion of the total energy for that case), λj/λtot Figure 9: Reconstruction of vorticity field using 2 to 32 modes: forced vibration ωf/ω0=0.6 Figure 10: Reconstruction of vorticity field using 2 to 32 modes: forced vibration ωf/ω0=0.8

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Twenty-First Symposium on NAVAL HYDRODYNAMICS Figure 11: Reconstruction of vorticity field using 2 to 32 modes: forced vibration ωf/ω0=1.4 versus mode number j. The curves show quantitatively what we see qualitatively in the eigenmode and reconstruction plots—in the fixed cylinder and locked-in cases (ωf/ω0=0.8 and 1.0), the curves have a steep gradient (note the log axis), while in the non-locked-in cases (ωf/ω0=0.6, 1.2 and 1.4), the mode energy decreases at a much slower rate. These curves provide a systematic means of deciding where to truncate the series for low-order dynamical models. For example, let us assume that we wish to retain 99.9% of the total energy, we would need to keep approximately 10 modes for the locked-in cases, and approximately 30 modes for the non-locked in cases (i.e. where the curves intersect e=0.001). 4 3-D DNS Results We first wish to simulate flow-induced vibrations of a flexible cable, then compute the POD eigenmodes for these flows. The authors have been conducting an ongoing investigation of direct numerical simulations of flow over flexible cables—results for flow-induced vibrations are reported in [12], and a comparison of forced and flow-induced vibrations is given in [13]. For this paper, we concentrate on three flow-induced vibration cases: standing wave Figure 12: Eigenmode energy fraction versus mode number for fixed cylinder and 2-d forced vibration cases, Re=100. and traveling wave responses at Re=100, and the flow-induced vibration response at Re=200. For each of these simulations, we assume spanwise periodicity with a length of L/d=12.6. The standing wave and traveling wave flow-induced vibration responses were generated by placing the initial cable position in the form of a standing wave and traveling wave, and then running the simulation for several shedding cycles, at which point a time-periodic state was reached. The cable tension was selected so that vibrations of wavelength L/d=12.6 would respond to the forcing frequency (estimated from the fixed cylinder Strouhal number, i.e. St=0.17 at Re=100). The Re=200 flow-induced vibration case was started from the Re=100 simulations. For this case, neither the cable nor the flow reached a strictly time-periodic state, indicating the turbulent nature of the wake at this Reynolds number (here, the simulation was run for more than forty shedding cycles). At Re=100, flow over a cylinder is 2-d, and we see parallel vortex shedding. If we take a slice of the flow field perpendicular to the cylinder, we see the well-known von Karman vortex street pattern of staggered vortices with alternating signs. A top view of this (looking in the negative y-direction) would show parallel rolls of alternating sign vorticity being shed and convecting downstream. Starting with the familiarity of the wake structure in this simple case, we now look at iso-contours of spanwise vorticity in the wake of the flow-induced standing wave, and

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Twenty-First Symposium on NAVAL HYDRODYNAMICS Figure 13: Top and perspective view of spanwise vorticity for standing wave cable response, Re=100. Contour level set at ω=0.2. Figure 14: Top and perspective view of spanwise vorticity for traveling wave cable response, Re=100. Contour level set at ω=0.2. flow-induced traveling wave. Figures 13 and 14 show a top view and perspective view of equal and opposite levels of spanwise vorticity (ωx=±0.2) for the standing wave cable wake and traveling wave cable wake respectively (for L/d=12.6 wavelength vibration case). The darker shade shows negative spanwise vorticity, while the lighter shade shows positive spanwise vorticity. The flow is from left to right, and the cable is located at x=0. We see a remarkably different structure to the flow in the wake depending on the flow-induced cable response. The standing wave cable response produces an interlocked lace-like structure to the spanwise vorticity. In contrast, the traveling wave cable response produced oblique shedding of spanwise vorticity, i.e. much like the shedding in the fixed cylinder case, but at an angle to the spanwise direction. Oblique shedding has been observed in flows over fixed cylinders under certain experimental conditions [14] [15]. The nodes of the cable's standing wave are located at the two ends and middle of the cable in the figure. In the case of the traveling wave, the “nodes” (point of zero displacement) move in the negative z-direction with the phase velocity c. The streamwise spacing of vortices is about six diameters in the standing wave case, and about five diameters in the traveling wave case. One obvious difference between the wake in both the standing and traveling wave flow-induced vibration cases with the wake of a fixed cylinder (all at Re=100) is that the former wakes are intrinsically three-dimensional, while the wake behind the fixed cylinder is two-dimensional. Consequently, the streamwise and normal vorticities (ωx and ωy) will be non-zero in the 3-d case. Figure 15 shows a top view of the three vorticity components for the standing wave (top plots) and traveling wave (bottom plots) wakes. The two shades are again, equal and opposite levels of the vorticity, and the same contour levels are used in all the plots to allow easy comparison. Looking first at the standing wave wake, we see that the cable vibration introduces significant streamwise vorticity; in fact further downstream the streamwise vorticity is the largest vorticity component. This is in direct contrast to the 2-d flows where we only see spanwise vorticity. The magnitude of this streamwise vorticity will be somewhat related to the cable vibration amplitude to wavelength ratio (since this is a direct means of introducing streamwise rotation into the flow), which is relatively large in this case. Note that the streamwise and normal vorticities are zero in the planes of the anti-nodes. Again, we see the staggered pattern of spanwise vorticity. Now looking at the top view of the three vorticity components for the traveling wave wake, we

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Twenty-First Symposium on NAVAL HYDRODYNAMICS Figure 15: Top view of three vorticity components for wake behind standing wave (top) and traveling wave (bottom), Re=100. Contour level set at ω= 0.2. see a similar picture for the three components. Note that in this traveling wave case, further downstream, the largest vorticity component is spanwise vorticity. Approximately, in this case, given a vorticity magnitude of ω, the streamwise and spanwise vorticity components will be ωx=ωsinθ and ωx=ωcosθ respectively, where θ is the shedding angle. The normal vorticity ωy in this case decreases rapidly with downstream distance. All the Re=100 flow-induced vibration simulations resulted in time-periodic cable and flow responses after the transients died out. The question naturally arises as to what happens at higher Reynolds numbers. We conducted simulations examining the case of unconstrained flow-induced vibrations at Re=200 for the L/d=12.6 spanwise wavelength case. We started the Re=200 simulation from the Re=100 unconstrained flow-induced vibration simulation. The simulation was run for more than 100 shedding cycles at which point the transients due to the change in Reynolds-number had died out, and the cable and wake response had reached statistical stationarity. The top view and perspective view of equal and opposite levels of spanwise vorticity (ωz=±0.2) for the the Re=200 flow-induced vibration wake is shown in Figure 16. These plots can be directly compared with those in Figures 13 and 14. Furthermore, Figure 17 shows a top view of the three vorticity components for the wake behind the cable undergoing flow-induced vibrations at Re=200. Note the highly three-dimensional nature of this flow. 4.1 3-D POD Eigenmodes We computed the eigenmodes for the three 3-d simulations: standing wave at Re=100, traveling wave at Re=100 and the flow-induced vibration response at Re=200. In this case, we took 300 snapshots (m=300) over tU/d=15 time units (about 3 shedding cycles), giving the interval between snapshots of ΔtU/d=0.05. Let us first show the eigenmode energy fraction decay rate in Figure 18. Not surprisingly, we see that the modal energy for the two Re=100 cases decays at more than twice the (exponential) rate of the Re=200 decay. The modal energy fraction for the traveling wave case is only slightly lower than that for the standing wave case. It is interesting to compare these with the 2-d modal energy fraction plots (Figure 12). The curves suggest that the 3-d Re=200 case has similar complexity/dimensionality to the 2-d non-lock-in cases. Next we plot the three vorticity components computed from the eigenmodes from the 3-d simulations. In each case we plot a top view (looking in

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Twenty-First Symposium on NAVAL HYDRODYNAMICS Figure 16: Top and perspective view of spanwise vorticity for Re=200 flow-induced vibration response. Contour level set at ω=0.2. the negative-y direction) of equal and opposite signs of ωx, ωy and ωx (the same levels are used for all plots, all ω=0.2), with the flow going left to right, and the cable positioned at x=0. For the Re=100 simulations we plot just the odd eigenmodes (eigenmodes 1, 3, 5 and 7) because the eigenmodes occur in conjugate pairs, and consequently the set of even eigenmodes are very similar (i.e. just differing by approximately 90°). Because of the slower eigenmode energy decay rate at Re=200, we plot eigenmodes 1, 4, 8 and 16. Starting with the standing wave wake at Re=100, the vorticity for eigenmodes 1, 3, 5 and 7 are plotted in Figures 19, 20, 21 and 22. We see that for all the eigenmodes, the symmetry of each vorticity component matches that of the original standing wave vorticity snapshots shown in Figure 15 (top). Furthermore, we see an indication of the decay of eigenmode energy by the decreasing volume of iso-surface. We observe an increasing amount of detail as we increase mode number—in fact going from mode 1 to mode 3, we see structures with half the length scale in both the x and z-directions (in the z-direction, this is seen as increasing number of planes of symmetries going from mode 1 to mode 3). Note again, that the largest vorticity component for all the standing wave eigenmodes is the streamwise vorticity. Figure 17: Top view of three vorticity components for wake behind cable flow-induced vibrations, Re=200. Contour level set at ω=0.2.

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Twenty-First Symposium on NAVAL HYDRODYNAMICS Table 2: Modal energy percentages of eigenmodes 1–6 for 3-d cases: standing wave (Re=100), traveling wave (Re=100) and traveling wave (Re=200) flow-induced vibrations. Mode Stand. Trav. Re=200 1 44.4 45.1 21.2 2 39.7 44.9 20.9 3 6.4 3.6 12.0 4 5.7 3.6 10.7 5 1.4 1.0 6.9 6 1.3 1.0 4.3 Now moving on to the traveling wave wake at Re=100, the vorticity for eigenmodes 1, 3, 5 and 7 are plotted in Figures 23, 24, 25 and 26. Again, we can compare these plots with the original traveling wave vorticity snapshots shown in Figure 15 (bottom). Here we see spanwise wavelengths of L, L/2, L/3 and L/4 in the structure of the vorticities for eigenmodes 1, 3, 5 and 7 respectively. It is interesting to observe that in the standing wave case we only saw even fractions of the spanwise wavelength (in contrast to the L/3 scale observed in the traveling wave mode 5 plot). Comparing each mode of the standing wave plots and the traveling wave plots indicates that the traveling wave vorticity eigenmodes exhibit approximately the same energy levels as the standing wave vorticity eigenmodes. This is consistent with the corresponding eigenspectra shown in Figure 18. Finally the flow-induced vibration wake at Re=200 (which was not time-periodic) eigenmodes 1, 4, 8 and 16 are plotted in Figures 27, 28, 29 and 30. Compare these plots with the original traveling wave vorticity snapshots shown in Figure 17. The relatively large (absolute) energy of these eigenmodes is illustrated by the large volume enclosed by the iso-surfaces. At this Reynolds number, we do not observe much scale difference between the structures in eigenmodes 1 and 4. However, we do start to see some finer structure emerging in eigenmodes 8 and 16. The traveling wave nature of the wake response is somewhat observed in the ωx vorticity eigenmodes. Table 2 lists the modal energy percentages of eigenmodes 1–6 for the three 3-d cases: standing wave (Re=100), traveling wave (Re=100) and Re=200 flow-induced vibrations. Again we see the pairing of the eigenmodes, and the rapid decay for the Re=100 cases—here after six modes we have captured 98.9% and 99.2% of the flow's energy compared with only 76% for the Re=200 flow-induced vibration wake. Figure 18: Eigenmode energy fraction versus mode number for 3-d flow-induced vibrations, Re=100 and Re=200. 5 Discussion In this paper we conducted direct numerical simulations of 2-d and 3-d flows over oscillating cylinders and flexible cables, and then performed proper orthogonal decomposition analyses on the wakes. Table 3 gives a summary of the cases run. ln the 2-d forced vibration cases, we see a big difference in the eigenmode energy spectrum depending on whether the wake is locked-in or not. When the wake is locked-in, the eigenmodes look similar to the case of a fixed cylinder, except for a length Table 3: Summary of POD cases analyzed. The 2-d simulations were fixed and forced vibration cases, while all the 3-d simulations were flow-induced vibration cases. Dim. Re Case 2-d 100 Fixed   100 ωf/ω0=0.6   100 ωf/ω0=0.8   100 ωf/ω0=1.0   100 ωf/ω0=1.2   100 ωf/ω0=1.4 3-d 100 Standing   100 Traveling   200 Traveling

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Twenty-First Symposium on NAVAL HYDRODYNAMICS Figure 19: Eigenmode 1 for Re=100 standing wave. Figure 20: Eigenmode 3 for Re=100 standing wave. Figure 21: Eigenmode 5 for Re=100 standing wave. Figure 22: Eigenmode 7 for Re=100 standing wave.

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Twenty-First Symposium on NAVAL HYDRODYNAMICS Figure 23: Eigenmode 1 for Re=100 traveling wave. Figure 24: Eigenmode 3 for Re=100 traveling wave. Figure 25: Eigenmode 5 for Re=100 traveling wave. Figure 26: Eigenmode 7 for Re=100 traveling wave.

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Twenty-First Symposium on NAVAL HYDRODYNAMICS Figure 27: Eigenmode 1 for Re=200 flow induced vibration wake. Figure 28: Eigenmode 4 for Re=200 flow induced vibration wake. Figure 29: Eigenmode 8 for Re=200 flow induced vibration wake. Figure 30: Eigenmode 16 for Re=200 flow induced vibration wake.

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Twenty-First Symposium on NAVAL HYDRODYNAMICS scaling that is proportional to the frequency ratio ωf/ω0. When the wake is not locked-in, the eigenmodes for the sub-lock-in case are quite different to the eigenmodes for the super-lock-in case. In the non-lock in cases, we need to retain three to four times as many modes to get the same accuracy as the locked-in cases. For example, to retain 99.9% of the flow energy, we would need about 10 eigenmodes to model a locked-in wake, and about 30 eigenmodes to model a non-locked-in wake. We see a similar effect with Reynolds number in the 3-d cases—for the Re=200 case we need to retain two to three times as many modes to get the same accuracy as the Re=100 cases. Here, to retain 99.9% of the flow energy, we would again need about 10 eigenmodes to model the Re=100 cases, and about 30 eigenmodes to model the Re=200 flow-induced vibration case. 6 Acknowledgements This work was supported by the Office of Naval Research, under the supervision of Dr. T.F.Swean. Computations were performed on the IBM-SP2s at the Cornell Theory Center and the Center for Fluid Mechanics at Brown University. References [1] R.D.Blevins. Flow Induced Vibration. Van Nostrand Reinhold Company, New York, New York, 1977. [2] J.K.Vandiver. Dimensionless parameters important to the prediction of vortex-induced vibrations of long, flexible cylinders in ocean currents. MIT Sea Grant Report, MITSG 91–93, 1991. [3] S.E.Ramberg and O.M.Griffin. The effects of vortex coherence, spacing, and circulation on the flow-induced forces on vibrating cables and bluff structures . Naval Research Laboratory Report 7945, 1976. [4] F.S.Hover, M.A.Grosenbaugh, and M.S. Triantafyllou. Calculation of dynamic motions and tensions in towed underwater cables . IEEE J. Oceanic Engineering, 19:449, 1994. [5] D.R.Yoerger, M.A.Grosenbaugh, M.S.Triantafyllou, and J.J.Burgess. Drag forces and flow-induced vibrations of a long vertical tow cable —part 1: Steady-state towing conditions. J. Offshore Mechanics and Arctic Engineering, 113:117, 1991. [6] R.D.Henderson and G.E.Karniadakis. Unstructured spectral element methods for simulation of turbulent flows. J. Computational Physics, 122:191, 1995. [7] K.S.Ball, L.Sirovich, and L.R.Keefe. Dynamical eigenfunction decomposition of turbulent channel flow. Intl. J. Num. Meth. Fluids, 12:585, 1991. [8] A.E.Deane, I.G.Kevrekidis, G.E.Karniadakis, and S.A.Orszag. Low-dimensional models for complex geometry flows: Application to grooved channels and circular cylinders. Physics of Fluids, 3(10):2337, 1991. [9] G.H.Koopmann. The vortex wakes of vibrating cylinders at low reynolds numbers. J. Fluid Mechanics, 28:501–512, 1967. [10] A.Ongoren and D.Rockwell. Flow structure from an oscillating cylinder—part 1: Mechanisms of phase shift and recovery in the near wake. J. Fluid Mechanics, 191:197–223, 1988. [11] C.H.K.Williamson and A.Roshko. Vortex formation in the wake of an oscillating cylinder. J. Fluids and Structures, 2:355–381, 1988. [12] D.J.Newman and G.E.Karniadakis. Simulations of flow past a freely vibrating cable. J. Fluid Mechanics, 1996. Submitted. [13] D.J.Newman and G.E.Karniadakis. Simulations of flow over a flexible cable: A comparison of forced and flow-induced vibration. J. Fluids and Structures , 1996. Submitted. [14] M.Hammache and M.Gharib. An experimental study of the parallel and oblique vortex shedding from circular cylinders. J. Fluid Mechanics, 232:567–590, 1991. [15] C.H.K.Williamson. Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers. Journal of Fluid Mechanics, 206:579–627, 1989.

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Twenty-First Symposium on NAVAL HYDRODYNAMICS DISCUSSION J.M.R.Graham Imperial College, United Kingdom The three-dimensional simulations have produced some very interesting results. In the case of the traveling wave, locking-on of vortex shedding should be similar to locking of two-dimensional vortex shedding by oscillation of the cylinder, with constant amplitude and phase along the span. In the case of the standing waves there are two opposite effects. The oscillation tends to control and possibly strengthen non-oblique vortex shedding, when locked on, but the three-dimensional effect of the phase variation limits this to finite length spanwise cells of alternating sign which might be expected to weaken the vortex shedding. Can any observations be made from your three-dimensional results about the lock-in boundaries (oscillation amplitude vs. frequency ratio) for these two cases (traveling and standing wave), as a function of spanwise wavelength? AUTHORS' REPLY The current paper deals with the question of modeling the coupled motions of cable and wake using low-dimensional constructs. It does not deal directly with the lock-in boundaries that Professor Graham addresses. From ongoing work, however, we have concluded that the same dynamics encountered in simply dynamical systems and in the two-dimensional case is also present in the cable system. We have not yet determined the exact boundaries of the lock-in for the standing and traveling wave response on which we will report in a future publication.