Effects of Oblique Inﬂow in Vortex-Induced Vibrations · EFFECTS OF OBLIQUE INFLOW IN...

Flow, Turbulence and Combustion 71: 375–389, 2003.© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

375

Effects of Oblique Inflow in Vortex-InducedVibrations

DIDIER LUCOR and GEORGE EM KARNIADAKIS�

Center for Fluid Mechanics, Division of Applied Mathematics, Brown University, U.S.A.;E-mail: {didi, gk}@dam.brown.edu

Received 3 October 2002; accepted in revised form 12 September 2003

Abstract. We investigate the validity of the independence principle for fixed yawed circular cylin-ders and free yawed circular rigid cylinders subject to vortex-induced vibrations (VIV) at subcriticalReynolds number using direct numerical simulation (DNS). We compare forces on the cylinder andcylinder responses for different angles of yaw and reduced velocities, and investigate the value of thecritical angle of yaw. We also present flow visualizations and examine flow structures correspondingto different angles of yaw and reduced velocities.

Key words: flow visualizations, independence principle, vortex induced vibrations, yawed circularrigid cylinders.

1. Introduction

When a cylinder is placed at an angle with the respect to the main flow, the hydro-dynamic forces and correspondingly the cylinder response may change comparedto the normal-incidence case. Despite several theoretical and numerical studies ofthe stability of three dimensional boundary layer on a yawed circular cylinder, thereis very little published on the vortex shedding of yawed cylinder placed in a steadycurrent or the effect of the angle of yaw on the force distributions and sheddingfrequency. Conflicting reported results are related to such basic characteristics asthe drag coefficient, the base pressure and the shedding frequency.

In particular, to our knowledge, there are no direct numerical simulations offree rigid yawed cylinders subject to VIV with large angle of yaw. Both theoreticaland most of experimental works have shown, at least in the subcritical range, thatthe yawed cylinder is similar to the normal-incidence case through the use of thecomponent of the free-stream velocity normal to the cylinder axis [5, 13]. This isknown as the Crossflow or Independence Principle (IP) and is also referred as theCosine Law. Several investigators reported deviations from the predictions basedon that principle [1, 2, 4, 12] and in particular at large yaw angles [14, 16]. Surryand Surry [15] found the Strouhal number based on the normal velocity compo-nent remains approximately constant in the wake of stationary yawed cylinders for

� Author for correspondence.

376 D. LUCOR AND G.E. KARNIADAKIS

Reynolds numbers in the range 4,000 ≤ Re ≤ 63,000 up to angles θ of 60◦ to70◦. Here (and in the following), θ = 0◦ corresponds to cross flow and θ = 90◦corresponds to axial flow. They found that the energy in the Strouhal peak dispersesand decreases significantly with increasing inclination; it is virtually submergedin the general wake turbulence spectrum for θ ≥ 60◦. Snarski and Jordan [14]measure the wall pressure spectra on a stationary cylinder for various angles ofincidence at sub-critical Reynolds number. They show that the variation in narrow-band (corresponding to periodic vortex shedding) and broadband (correspondingto laminar/transitional boundary layer) spectra levels with angle of incidence is notmonotonic. They also conclude that there is a fundamental shift in the separationmechanism for θ ≈ 55◦ between Re = 7,250 and Re = 14,500. Kozakiewicz etal. [9] show that the IP can be applied to stationary cylinders in the vicinity of aplane wall in the subcritical range for an angle of yaw 0◦ ≤ θ ≤ 45◦.

However, the validity of the IP remains questionable when the angle of yawbecomes very large and the flow direction is almost parallel to the cylinder axis.Hanson [4] studied vortex shedding from vibrating yawed hot wires in an airstream for low Reynolds number. He validated the IP for θ ≤ 68◦ but he founddiscrepancies in shedding frequencies for an angle of yaw θ = 72◦. Van Atta [1]investigated the region of an apparent discontinuity for angle of yaw in the range50◦ ≤ θ ≤ 75◦. He concluded that the discontinuity observed by Hanson was notdue to the large angle of yaw but was due to the existence of locked-in modes de-pending on the value of the wire tension. He also reported that for a given tension,the wire does not necessarily vibrate with the frequency of the harmonic that isnearest to the natural shedding frequency, but always locks-in to the frequencythat is lower than the natural shedding frequency. In the case of non-vibratingyawed cylinders Van Atta showed that for θ ≤ 35◦ the vortex shedding frequencydecreases nearly like the Cosine Law, whereas for larger angles the decrease withincreasing angle of yaw θ is slower than the proposed Cosine Law.

Koopman [8] carried out tests specifically designed to study flow-induced os-cillations of yawed cylinders. He noticed a drastic decrease in the cylinder periodicresponse as well as a drop in the correlation of the lift forces along the span forangles of yaw larger than 15◦. King [6] investigated VIV of yawed circular cylin-ders for Reynolds number in the range 2,000 ≤ Re ≤ 20,000 and for yaw angles−45◦ ≤ θ ≤ 45◦. He measured drag forces and fluctuating in-line and cross-flowcylinder responses. He observed that the maximum cross-flow amplitude corre-sponds to reduced velocity Vrn (based on the normal component of velocity ofthe inflow) in the range 5.8 ≤ Vrn ≤ 7.0. The fact that the profiles of instabilityregions collapse was presented as experimental justification of the generalisationof the Cosine Law to the case of the oscillating yawed cylinders. He attributed theincrease in crossflow response with yaw angle to a corresponding decrease in thereduced damping. He also recorded sustained oscillations at yaw angles θ = ±65◦and showed that the cylinder response is virtually identical for positive and negativeangles of yaw.

EFFECTS OF OBLIQUE INFLOW IN VORTEX-INDUCED VIBRATIONS 377

Figure 1. Simulation setup. Left: view perpendicular to the cross-flow direction (y-direction).Right: 3D view.

Finally, Ramberg [12] studied the effect of yaw angle (θ = 0–60◦) and end-conditions for stationary and forced vibrating circular cylinders (with aspect ratio20–100) in the Reynolds-number range 150–1,000. He determined that the resultswere very sensitive to end-conditions especially at the lower Reynolds numbers.He showed that slantwise shedding at angles other than the cylinder yaw angle isintrinsic to stationary inclined cylinders in the absence of end-effects. He reportedthat the IP fails in the case of stationary yawed cylinders because the sheddingfrequency is always greater than expected from the IP, while the shedding angle,the vortex-formation length, the base pressure and the wake width are all less thanexpected. However, he concluded that locked-in vortex wakes of vibrating yawedcylinders can be described successfully by the IP. In this case, frequency lock-in between the vortex wake and the cylinder motion was accompanied by vortexshedding parallel to the cylinder axis.

In the current paper we investigate the validity of the IP for fixed yawed circularcylinders and free yawed circular rigid cylinders subject to vortex-induced vibra-tions (VIV) at subcritical Reynolds number. Our goal is to verify if large cylinderresponses are possible for large angles of yaw (i.e. ≥ 60◦). We also examinetime- and span-averaged forces on the cylinder and compare them with the valuespredicted by the IP.

2. Simulation Parameters and Formulation

We report here simulation results at constant Reynolds number Re = Ud/ν =1,000 based on the free-stream velocity for stationary and vibrating rigid yawedcylinders. The angle of yaw θ (see Figure 1), defined by the direction normal tothe free-stream velocity direction and the cylinder axis, is such that a zero valuecorresponds to a cylinder normal to the free-stream velocity. We consider mainlytwo angles of yaw θ = −60◦ and θ = −70◦. The negative signs come from


the orientation of the frame of reference that is commonly used in the litterature(see Figure 1). By changing θ we modify the normal Reynolds number Ren =(Ucosθ)d/ν based on the normal component of the inflow.

In all vibrating cases we only allow vertical motions in the crossflow y-direction,i.e. we do not allow any motion in the streamwise x-direction. The cylinder is rigidand thus its motion has no spanwise z-dependence. The mass ratio of the system(cylinder mass over displaced fluid mass) is taken to be m = 2 and the structuraldamping coefficient to be ξ = 0.003.

The governing equations are the incompressible Navier–Stokes equations cou-pled with the structural dynamical equation. The cylinder is represented by a singledegree of freedom viscously damped second-order oscillator subject to the externalhydrodynamic forcing, i.e.

η̈(t) + 4πξ

Vr

η̇(t) + 4π2

V 2r

η(t) = CL(t)

2m, η(0) = η0 and η̇(0) = η̇0, (1)

where η represents the crossflow cylinder response, Vr = U/f d is the reduced ve-locity based on the free-stream velocity U and the natural frequency f of the struc-ture, ξ is the structural damping coefficient, and CL(t) is the spanwise-averagedlocal lift coefficient. The reduced velocity Vrn = (U cos θ)/(f d) is based on thenormal component of the inflow. All of the variables are non-dimensionalized withthe cylinder diameter d and the free-stream inflow velocity U .

The coupled Navier–Stokes/structure dynamics equations are discretized in spaceusing the spectral/hp element method that employs a hybrid grid in the (x, y)-planeand Fourier expansions in the z-direction (cylinder axis) with a dealiasing 3/2 rule.The parallel code N εκT αr is employed in all simulations [7]. A boundary-fittedcoordinate system is employed similar to the laminar flow simulations in [11],which has been validated against an Arbitrary Lagrangian Euler (ALE) formulation[3] that was also developed for moving domains [17]. The computational domainfor the (x, y)-plane extends 80d (cylinder diameters) downstream and 20d in frontof the cylinder; it extends 30d above and below it. The aspect ratio is L/d = 22. Ahybrid mesh with 2,340 elements is used. It is refined around the cylinder, and weuse a fourth order polynomial expansion per element. Also, 64 z-planes (32 Fouriermodes) are used along the spanwise direction. Periodic boundary conditions areimposed at the two ends along the cylinder axis. This is equivalent to treat thestructure as infinitely long, and then employ (free) periodic boundary conditionson a piece of finite length. By doing so, strong three-dimensional end effects thatare usually associated with oblique inflows past cylinders are avoided. In the caseof very large angles of incidence, or the extreme case of an axial flow, the flowcan be thought as a fully developed, three-dimensional boundary layer flow alongthe spanwise direction of the cylinder surface. A Newmark integration scheme wasused to solve for the structure [10].


Figure 2. Mean (left plot) and rms (right plot) drag coefficients versus reduced velocity Vr

based on free-stream inflow velocity.

Figure 3. Rms lift coefficients (left plot) and rms and maximum cross-flow cylinder responses(right plot) versus reduced velocity Vr based on free-stream inflow velocity.

3. Force Distributions

We consider stationary and rigidly moving yawed cylinders (SYC and RYC) andinvestigate the distribution of hydrodynamic forces acting on the body both in timeand along the span. We normalize these forces based on the component of theinflow normal to the cylinder axis to obtain drag and lift coefficients. In order topresent more concise information, we will mainly present here time- and span-averaged quantities for the different cases, instead of instantaneous quantities.

In Figures 2 and 3, we plot mean, maximum and rms (root mean square) val-ues of drag coefficient CD, lift coefficient CL, and cross-flow cylinder responseη against the reduced velocity Vr used in our simulations. All these statisticalquantities correspond to averaged values over both the time and space domains.


Here is the definition we use to compute our rms results (for the drag coefficientfor instance):

CDrms =

∑ni=1

(CD(z, ti) − (∑n

j=1 CD(z,tj )

n

))2

n

1/2

, (2)

where the bar denotes averaging over space and n is the number of terms in our timeseries for each spanwise location z. Another definition, where we first compute therms values along the space direction and then average in time, for the lift coefficientwill be used as well and described later on.

Due to the cost of DNS, we have chosen a few reduced velocities for eachangle. The reduced velocities for which we find maximum cylinder response anda “capture” of the natural vortex shedding by the cylinder motion are in the lock-in range given by Ramberg, i.e. Vrn ≈ 4.5 to 6.0 [12] and King, i.e. Vrn = 5–6[6]. We also include for reference some of the corresponding values for rigid withnormal (inflow: θ = 0◦) cylinders (RNC) subject to VIV at lock-in (numericalresults from [3]). It is worth mentionning that these reference results were obtainedwith a different aspect ratio of L/d = 4π . Each one of these reference casescorresponds to a single reduced velocity (mentionned in the legend). Nevertheless,for convenience reason, it has been represented by a line in the plots. Otherwise,circle symbols refer to cases with an angle of yaw θ = −60◦, and triangle symbolsrefer to θ = −70◦.

Figure 2 displays the drag coefficient (CD) results. We first examine the caseof the stationary yawed cylinder with θ = −70◦ and Ren = 342. By comparingwith the case of a stationary cylinder with normal inflow at the same Reynoldsnumber, we can check the validity of the IP. The numerical study of this case wasnot available to us for comparison. Instead, we considered one of our studies whereRe = 300 and we also compared with the base pressure coefficient results forRen = 342 by Ramberg [12]. It appears that the ratio between the mean basepressure coefficient of the yawed cylinder and the base pressure coefficient of thenormal cylinder is greater than unity (in fact equal to 1.2). This is consistent withRamberg’s findings [12]; it confirms that the base pressure on the yawed cylinder islower than the value predicted by the IP. Therefore, it produces a drag significantlygreater (CD ≈ 1.34) than predicted by the IP (CD ≈ 1.2).

If we now look at the moving yawed cylinders with θ = −70◦, we see thatthe cases outside the lock-in region give very similar results to the stationary case.In particular, mean and rms values of the drag coefficient for reduced velocitiesVr = 7.55 and 9.55 are almost identical to the stationary case for the same angleof yaw. Reduced velocity Vr = 14.5 or Vrn = 4.9593 (referred as case I) leads tolock-in. We see that the drag quantities are larger than predicted by the IP. Indeed,they are quite larger than the reference case of the free cylinder at normal incidenceat Re = 300 (again, an exact value of Re = 342 was not available for comparison).


They are closer to the results for the free cylinder at normal incidence at Re =1,000.

Results for free yawed cylinders with θ = −60◦ exhibit a very clear trend.As the reduced velocity increases toward the lock-in region, around Vr = 9.55 orVrn = 4.775 (referred as case II), mean and rms values of the drag force increaseand approach the case of the free cylinder at normal incidence at Re = 1,000.

Figure 3 displays the lift coefficient CL and cross-flow cylinder response η

results. Different rms values of the lift coefficient are presented. Filled symbolsand reference lines follow the previous definition. Hollow symbols (with an addedstar sign in the legend) follow a different definition, i.e.:

CLrms =(∑p

i=1

(CL(zi, t) − (∑p

j=1 CL(zj ,t)

p

))2

p

)1/2

, (3)

where, this time, the bar denotes averaging over time and p is the number of z-planes along the cylinder axis. With the latest definition, low values of rms clearlyindicate a good spatial correlation of the lift forces.

For both θ = −60◦ and θ = −70◦, the maximum cylinder response increasesas the cylinder gets closer to lock-in. For the case I, it reaches the value given bythe IP (η = 0.52). For case II, no comparison was available for the IP, but the value(η = 0.63) remains lower than the value for the free cylinder at normal incidenceat Re = 1,000 (η = 0.75). We notice that the maximum amplitude at lock-indecreases with a yaw angle increase. This is in disagreement with King’s work [6].Nevertheless, King attributed the increase in crossflow response with yaw angle toa corresponding decrease in the reduced damping (due to a change in the immersionlength). This explanation is not obvious as pointed out by Ramberg in his reviewof the paper. Moreover, King found differences (up to 50%) between his main andsubsidiary test responses that he attributed to some three-dimensional effects in themain test cylinder configuration. This bottom end effect was not quantified by theauthor. It would appear that there are important Reynolds number, yaw, aspect ratioand end effects that cannot be accounted for through the reduced velocity.

For both angles, ηrms follow the same trend and increase with the reduced veloc-ity. Maximum values are obtained at lock-in; value for case I matches the predictedvalue given by the IP (ηrms = 0.36). For θ = −60◦, the rms values of the liftcoefficient (see hollow symbols) peak at the reduced velocity corresponding tothe largest cylinder response (see case II). Therefore, it seems that there is a lossof spanwise coherence and a drop in the correlation of the lift forces when thecylinder response amplitude is maximum. For θ = −70◦, the situation is differentand the maximum rms value of the lift coefficient is reached at Vr = 7.55. Thecorrelation of the lift forces increase as we approach the lock-in region. However,this is of less importance in that case as the force amplitudes are small.

The study of spatio-temporal variation of CD and CL for case II (see Figure9) shows regions corresponding to moderate forces alternating with regions cor-


Figure 4. Freely vibrating yawed cylinder (θ = −60◦) at lock-in at Re = 1,000. Drag (top)and Lift (bottom) coefficients along the span versus non-dimensional time. (Original in colour)

responding to large forces. These regions look like inclined braids that follow atravelling wave pattern (see CL distribution). The angle of inclination does notdepend on time. At the present time, we can not explain the existence of thosebraids but we will relate it to near-wake flow visualizations in the next section.

4. Flow Visualizations

In two-dimensional bluff-body flows, irrotational fluid from outside the wake re-gion is swept into the vortex street. However, the yawed cylinder flow in the baseregion is inherently three-dimensional due to the additional spanwise vorticity andthe changing direction of the free-stream flow as it approaches the cylinder.

4.1. STATIONARY YAWED CYLINDERS

Flow visualizations for large values of yaw are useful in this case to prove theexistence of a vortex-shedding regime in the near-wake of the cylinder as opposedto a steady trailing vortex shedding regime. Moreover, it might also illustrate theslantwise shedding at angles other than the cylinder yaw angle. We present hereresults for the most extreme case of θ = −70◦ from our simulations. We plotin Figure 5 pressure isocontours at Re = 1,000. We do notice a vortex sheddingregime in the near-wake. This is consistent with the results from Ramberg [12]


Figure 5. Stationary yawed cylinder (θ = −70◦) at Re = 1,000. Pressure isocontour at value−0.025. View almost perpendicular to the plane of the inflow. The arrows represent the inflowcoming from left to right. (Original in colour)

for this range of Reynolds number and similar aspect ratio but a somewhat loweryaw angle (around 60◦) for flat end-shapes. Moreover, the shedding is parallel (2S

pattern) and slantwise (with a shedding angle smaller than the yaw angle) as foundby Ramberg [12]. The shedding angle α (defined with the same convention as theyaw angle) approaches a value of α = −58◦ for this case and it is stable in spaceand time. Measured shedding frequencies in the wake have shown that the ratioSt/Stn is greater than cos θ , which is consistent with α < θ , see [12]. Final resultsfor θ = −60◦ exhibit a similar trend with a shedding angle value that becomescloser still smaller to the yaw angle.

4.2. FREELY VIBRATING YAWED CYLINDERS AT LOCK-IN

We present flow patterns for freely vibrating yawed cylinders (θ = −70◦ andθ = −60◦) at lock-in (see cases I and II in Section 3). In Figures 6 and 7 we plotinstantaneous pressure isocontours of the flow. In both cases, we notice a vortex-shedding regime in the near-wake. This time, the vortex shedding is parallel tothe cylinder axis. This extends the findings by Ramberg [12] for forced vibrating


Figure 6. Freely vibrating yawed cylinder (θ = −70◦) at lock-in at Re = 1,000. Pressureisocontour at value −0.025. View almost perpendicular to the plane of the inflow. Inflowcoming from left to right. (Original in colour)

yawed cylinders with similar Reynolds number, aspect ratio and somewhat loweryaw angle (θ about 50◦) to the case of freely vibrating yawed cylinders. Parallelshedding to the cylinder axis was encountered in most of our cases as long as thecylinder crossflow amplitude was not too small. In the case of reduced velocityvalue for which the cylinder response was almost zero (therefore mimicking astationary cylinder), the vortex shedding had the tendency to slant.

Figure 6 shows the flow past a vibrating cylinder at lock-in for an angle ofyaw θ = −70◦. It corresponds to the instant of time when the cross-flow cylin-der displacement reaches its minimum value (η = −0.506). In this plot, we seethat the flow structures are much more contorted than in the stationary case. Theshedding pattern of the von Kármán vortices is of type 2S but we notice strongstreamwise vortices winding up around the spanwise vortices. The latter seem to beconnected via “braids” consisting of streamwise-oriented counter-rotating vortexpairs. Downstream in the wake, those streamwise vortices significantly distort andpinch the von Kármán vortices. Streamwise vortex tubes are not perpendicular tothe cylinder axis but they seem to follow helical paths around the spanwise vorticalstructures. Streamwise vortices appearing on the front side of Figure 6 are slantedalong the direction of the inflow. However, streamwise vortices from the other side(not visible on Figure 6) are slanted along the direction perpendicular to the inflow.


Figure 7. Freely vibrating yawed cylinder (θ = −60◦) at lock-in at Re = 1,000. Pressureisocontours at value −0.05. Views almost perpendicular to the plane of the inflow. Left: Frontview with inflow coming from left to right. Right: Back view with inflow coming from rightto left. (Original in colour)

This suggests that the free-stream flow direction may have a strong influence whichis in disagreement with the IP. The steady trailing regime, as mentioned in severalpublications, would correspond to the extreme case where the spanwise vorticeswould bend and loop over to the point where they would eventually detach from theshedding vortex filaments and form a wavy line streaming out behind the cylinder.

Figure 7 presents two different views of the same instantaneous flow past avibrating cylinder at lock-in for an angle of yaw θ = −60◦. It shows the front sideand the back side of the pressure isocontours in the near-wake.

Figure 8 shows smoothed spanwise vorticity isocontours. Specifically, the span-wise vorticity has been filtered several times using a three dimensional Gaussianconvolution kernel [3 × 3 × 3] with standard deviation σ = 0.65 in order to isolatethe large structures from the noise. Both Figures 7 and 8 are taken at the sametime (tU/d = 748.75). The cross-flow cylinder displacement is η = 0.262 and thecylinder is travelling on its way down to its minimum position (η ≈ −0.6). It showsthe front side and the back side of the pressure isocontours in the near-wake. Figure9 presents another view of the pressure isocontours in the wake at a different time(tU/d = 753.75). The corresponding cross-flow displacement is η = −0.278 andthe structure is travelling on its way up to its maximum position (η ≈ 0.6). Theseinstants are chosen in order to relate to case II (see Section 3) and examine howthe flow pattern correlates with the force distribution. Spanwise vortices close (andaway) from the cylinder are distorted (see Figure (8). The last (see green arrows onFigure 8) and the next to last (see yellow arrows on Figure 8) vortices shed fromthe cylinder could be described as making a X or H shape where the central part


Figure 8. Freely vibrating yawed cylinder (θ = −60◦) at lock-in at Re = 1,000. Spanwisevorticity isocontours at value ±0.25. View almost perpendicular to the plane of the inflow.Inflow coming from left to right. (Original in colour)

of the vortical structures in the spanwise direction remains closer to the cylinderbody while the two ends of the structure (toward the cylinder ends) stand furtherdownstream (see yellow arrows on Figure 8). The dynamics of this vortical patterncan be opposed to those of the generic “chevron” pattern where vortex sheddingnear the cylinder ends is delayed. In our case, the gap left between those two legsand the cylinder body is filled by the spanwise vortex of opposite sign shed onthe other side of the body (see green arrows on Figure 8). Therefore, close to thecylinder and toward the ends of the body, a 2P vortex shedding mode takes placewithin the 2S mode.

Further downstream, it seems that there is a strong interaction between spanwiseand streamwise vortical structures. The streamwise vorticity sheets, shed from theupper and lower part of the body, wrap around the Von Kármán vortices (see redarrows on Figure 9). They also induce a local spanwise velocity component op-posed to the spanwise component of the free-stream flow. This could result in theinstability of the Kármán vortices that become constricted and distorted; they loopand take an helical shape along the z-direction (see arrows on left side of Figure 7).From this view, we see that the half bottom part of the last roll shed from the back ofthe cylinder describes a left-handed helical form that coils counter-clockwise (seered arrows). The top bottom part describes a right-handed form that coils clock-wise (see yellow arrows). This mismatch in the rotation orientation constrains the


Figure 9. Freely vibrating yawed cylinder (θ = −60◦) at lock-in at Re = 1,000. Pressureisocontours at value −0.05. View almost perpendicular to the plane of the inflow. Inflowcoming from left to right. (Original in colour)

bottom structure curvature to change and this explains the aforementioned vortexloops.

Other instants in time for this case were also investigated and flow visualizationsprovided the same type of vortical topology. Moreover, we observed that the part ofthe “chevron” pattern attached to the cylinder was moving in time. For increasingtime, it moves at a constant speed in the positive z-direction along the cylinder inthe main direction of the inflow. This phenomenon seems to be strongly correlatedto the travelling-like wave seen in the lift coefficient spatio-temporal distributionof case II described in Section 3.

5. Summary

We have investigated the validity of the independence principle (IP) for fixedyawed circular cylinders and free yawed circular rigid cylinders subject to vortex-induced vibrations (VIV) at subcritical Reynolds number (Re = 1,000) usingdirect numerical simulation (DNS). We considered here two angles of yaw θ =−60◦ and θ = −70◦.

For stationary cylinders, we have shown that the IP must not be valid for largeangle of attack. We confirmed that slantwise shedding at angles other than the


cylinder yaw angle is intrinsic to stationary inclined cylinders in the absence of end-effects. The shedding angle is smaller than the yaw angle in the case of θ = −70◦.The base pressure is lower than the value predicted by the IP, and hence, the dragcoefficient is higher than the value predicted by the IP.

For freely moving cylinders at lock-in, we showed that we get large cylindercross-flow amplitude even for large angle of yaw (θ = −60◦ and θ = −70◦). Themean and rms values of the cylinder cross-flow amplitude decrease for increasingangles of yaw. The values of the corresponding reduced velocities are in the rangegiven by King [6] and Ramberg [12]; the shedding is parallel to the cylinder axis.The mean and rms values of the drag coefficient are larger than the values predictedby the IP as they approache the VIV values for a flow at normal incidence (θ = 0◦)at Re = 1,000. For θ = −70◦, lift forces are more correlated for reduced velocitiesin the lock-in range but their magnitude is small. For θ = −60◦, different rmsvalues seem to indicate a loss of spanwise coherence and a drop in the correlationof the lift forces along the cylinder as the reduced velocity gets closer to the lock-in region where the cylinder response is maximum. Near-wake flow visualizationsfor θ = −60◦ at lock-in provided a vortical topology that we related to the spatio-temporal distribution of forces.

Acknowledgements

This work was supported by the Office of Naval Research. Computations were per-formed on the NAVO T3E, on the MHPCC SP2, and on the SP Power 3 computerof Brown’s Center for Scientific Computing & Visualization.

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