Flow control using bifrequency motion

Theor. Comput. Fluid Dyn. (2011) 25:381–405 DOI 10.1007/s00162-010-0206-6 O R I G I NA L A RT I C L E Osama A. Marzouk Flow control using bifrequency motion Received: 12 June 2009 / Accepted: 30 August 2010 / Published online: 16 September 2010 © Springer-Verlag 2010 Abstract We used a second-order approximation for the periodic lift coefficient of a circular cylinder under monofrequency and bifrequency cross-flow motions. Two lock-in modes exist under monofrequency fundamental (i.e., near the Strouhal number) motion. In the first mode, the work is done by the flow on the cylinder, whereas in the second mode the work is done by the cylinder on the flow. Under monofrequency superharmonic (i.e., near three times the Strouhal number) motion, the work is always done on the flow. We then replaced the monofrequency motions by a bifrequency one, consisting of a fundamental term combined with a small -magnitude superharmonic term. We examined the effect of the magnitude and phase of the superharmonic motion term on the two modes of lock-in which we obtained when only the fundamental motion term is applied, considering two different frequencies that belonged to the two lock-in modes. Under the bifrequency motion, the work can be done on the flow or on the cylinder. This can be controlled using the superharmonic motion term, even when its magnitude is 5% of magnitude of the fundamental motion term. Other flow variables, such as the magnification of the lift, can be remarkably altered through the added superharmonic motion term. The phase of the third superharmonic lift-coefficient component relative to the fundamental one is the most responsive variable to the phase of the superharmonic motion component relative to the fundamental one. Keywords Cylinder · Lift · Lock-in · Bifrequency · Multifrequency · Superharmonic List of symbols C L (t) fY fo L 1 (t) L 3 (t) Stdo (C L ) Std(C L ) Lift coefficient Nondimensional angular frequency of the fundamental motion term Nondimensional angular shedding frequency when no motion is applied Cosine-wave component of C L (t) at f L Cosine-wave component of C L (t) at 3 f L Standard deviation of C L when no motion is applied Standard deviation of C L when motion is applied Communicated by M. Y. Hussaini O. A. Marzouk (B) Department of Engineering Science and Mechanics (Mail Code 0219), Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA E-mail: omarzouk@vt.edu 382 t W Y (t) Y1 (t) Y3 (t) φL φY λk ηk O. A. Marzouk Nondimensional time Nondimensional work done by the flow on the cylinder per one motion period Nondimensional cross-flow displacement of the cylinder Cosine-wave component of Y (t) at f Y Cosine-wave component of Y (t) at 3 f Y Phase angle of L 3 w.r.t. L 1 Phase angle of Y 3 w.r.t. Y 1 Magnitude of L k (k = 1, 3) Magnitude of Yk (k = 1, 3) 1 Introduction The problem of a circular cylinder forced to oscillate in the cross-flow direction with a prescribed sinusoidal motion when subject to a uniform incoming flow has been studied experimentally [2,3,8,13–15,22– 24,34,35,37,43,45,49,52] and numerically [1,6,11,19,21,25,26,28,30,39,54] at different Reynolds numbers, motion magnitudes, and motion frequencies. In these studies, a simple monofrequency oscillation was applied, which is completely described by two parameters, namely the magnitude and frequency. It is known that the shedding and lift force on the cylinder will be locked onto the applied oscillation frequency over a certain range of frequencies. Moreover, two different modes of the lock-in may exist. The first mode corresponds to lower frequencies within the lock-in range and the second one corresponds to higher frequencies within the lock-in range. In the first mode, the lift is small and the work is being done on the cylinder; whereas in the second mode, the lift is large and the work is being done on the flow. This work refers to a mechanical energy that is transferred from the fluid to the cylinder motion due to the co-linearity of the exerted lift on the cylinder and its motion. In the case of a spring-mounted cylinder with damping, sustained free vibration occurs under the balance between this work and the dissipation due to structural damping [6,34,35]. Therefore, the work (on the average) must be done on the cylinder [45]. On the other hand, when the motion is forced (as in the present study), the work can be done either on the flow or on the cylinder. Work done on the cylinder indicates that the vortex shedding tends to accentuate the motion, whereas work done on the flow indicates that the vortex shedding resists the motion [14]. Flow visualizations and measurements [8,9,13,18,24,37,53] show that the increase in the lift is related to shorter vortex-formation region (in the streamwise direction). In particular, Krishnamoorthy et al. [24] found a sharp decrease in this length as the wake mode changes from the first to the second. In the first mode, the vortex located on the opposite side to the displacement of the cylinder is shed when the cylinder is near the maximum amplitude. In the second mode, it is the vortex adjacent to the cylinder displacement which is shed when the maximum amplitude is nearly reached. Both observations were reproduced in numerical simulations [6,19,26,30]. In references, [24,34,35,52] it is observed that in the first mode, the vortex shedding pattern is 2P (two vortex pairs per shedding cycle), whereas it is 2S (two single vortices per shedding cycle) in the second mode. However, this depends on the Reynolds number and the amplitude of oscillation. For example, at large motion amplitudes (e.g., above one cylinder diameter), the 2S pattern might not exist at all; and at very low Reynolds number (e.g., below 300), the 2P pattern can be replaced by an asymmetric P+S pattern [46,52]. We note that the direct numerical simulations, using the spectral element method, of Blackburn and Henderson [6] do not show a 2P pattern although they correspond to a Reynolds number of 500. Therefore, the suggested limit of 300 does not seem to be robust. At a fixed magnitude of monofrequency forced oscillation, they aimed at examining the effect of traversing the frequency on lock-in. They also found two modes of symmetric shedding. When plotting the nondimensional work versus the frequency, each mode corresponded to a branch (a similar plot is presented here in Sect. 4.1). The nondimensional work was viewed as a single variable to characterize each periodic mode of shedding. They also examined the bifurcation of these branches and the hysteresis transfer, which required traversing the frequency in small increments (known as a sine dwell or stationary sweep) during the same simulation. We do not focus on this aspect, and the frequency is kept fixed over each simulation. There have been some attempts to apply results from the problem of forced monofrequency oscillation (when only the fundamental motion term is applied) to the problem of free vibration (e.g., to predict the lock-in range or the fluid force in the latter problem). Our study suggests that this approach should be considered cautiously. The cylinder motion in the former problem does not include the superharmonic motion term, whereas the latter problem does. Our results, as we will show later, indicate that this ignored term (which contributes Flow control using bifrequency motion 383 to the work or energy transfer between the cylinder motion and the fluid) can have considerable effects even when it is small. This can explain some disagreements reported in earlier attempts to project results of monofrequency forced oscillation on the problem of free vibration or to relate the two problems. We will describe some of these attempts here. Staubli [45] used a forced linear damped system to model the nondimensional cross-flow displacement of a spring-mounted cylinder with damping, which is undergoing free vibration. The forcing term was taken to be proportional to the lift coefficient, which was taken from forced-oscillation measurements (as a function of the oscillation frequency and its magnitude) and inserted into the forced linear system as an algebraic modeling of the lift. To assess their concept, they considered the free vibration experiments of Feng [12] in a wind tunnel and compared the amplitudes of the predicted and measured motion. The predicted results deviated from the measured ones, but the deviation was pronounced in the case of strong damping, where the predictions gave a stable large-amplitude attractor that was not observed experimentally. Gopalkrishnan [14] created a database of measured lift coefficient at different magnitudes and frequencies of monofrequency forced oscillation. Then, they located for each magnitude the frequency at which the maximum work done on the cylinder was recorded, and then evaluated the lift component at the oscillation frequency. Starting from a forced linear system to model the free vibration, and assuming the free vibration to be sinusoidal (as the forced oscillation), they derived a simple expression that predicts the magnitude of the nondimensional free vibration and allows the use of the lift-coefficient data obtained from the forced oscillation tests. To assess their concept, they predicted five magnitudes of the free vibration at different values of the nondimensional mass-damping, and plotted them on the Griffin plot [16,17], which is a compilation of maximum (peak-to-peak) cross-flow displacement of different circular cylindrical structures undergoing free vibrations over Reynolds numbers from 300 to 106 . Whereas their predicted points lied within the experimental data in the Griffin plot, this is a crude way to assess their concept because of the large scatter in the plot. Also, the experimental data in the plot represent a wide range of settings (e.g., spring-mounted rigid cylinders, cantilevered flexible cylinders, and pivoted rigid circular rods) and flow conditions. It also has a log–log scale which can mask large deviations. Hover et al. [20] experimentally found the work done (either on the flow or on the cylinder) when the monofrequency fundamental motion term is applied, over a range of magnitudes and frequencies, and generated contour lines for the work. When they overlaid measured results from free vibrations on their contours, many of these results lied in the region where the work (based on the forced motion) was done on the cylinder. This can establish a link between the two problems based on the work. However, there were some free vibrations in the region where the work (based on the forced motion) was done on the flow. Qin [40] tried to establish a link between the monofrequency forced oscillation and the free vibration in a different way. He used a forced linear damped system to model the nondimensional cross-flow displacement of a spring-mounted cylinder with damping as Staubli [45] did. Both Qin and Staubli used the lift coefficient to drive the linear damped system of the vibration. However, the lift coefficient in the case of Qin was described by an excited wake oscillator (the self-excited van der Pol equation with external excitations). The wake oscillator was excited by a linear combination of the displacement and the velocity of the cylinder. The connection between this reduced-order model for the free vibration and the forced oscillation comes through the model parameters for the excited wake oscillator (the linear and nonlinear damping coefficients, and the weights of the two excitation terms). The values of these parameters were tuned based on forced-oscillation results (from two-dimensional RANS simulations). When the predictive capabilities of the resulting model were assessed against free vibration experiments, these reduced-model parameters derived from the forced oscillation failed to predict the so-called upper and lower branches of the free vibration amplitude at a high mass ratio (cylinder mass to displaced fluid mass equal to 248). They also could not predict the correct characteristics of the free vibration at a low mass-damping (mass ratio times the damping ratio equal to 0.013). Dahl [10] extended the concept of using a database of force coefficients and nondimensional work corresponding to forced monofrequency oscillation to predict free vibrations to the case where both cross-flow and streamwise oscillations take place. They assumed a state of dual lock-in, where the drag frequency is twice the lift frequency, with latter being locked onto the oscillation frequency. Their procedure was based on the assumption that zero-work conditions correspond to free vibration with low damping. Szwalek and Larsen [47] also suggest that zero-work condition in forced-oscillation results correspond to free vibrations. The predicted amplitudes of cross-flow and streamwise free vibrations and the phase between them were not satisfactory. In the aforementioned attempts to take advantage of forced monofrequency oscillations to predict characteristics of free vibrations, where some of the outcomes were not promising, we identify two possible sources of errors, which can contribute to the deviations between the predicted free vibration and reference free vibration which is to be predicted. First, the Reynolds number (either a single value or a range) at which the forced oscillation results are collected is different from the one 384 O. A. Marzouk at which the reference vibration takes place. Physically, this means that the flow velocity, the fluid type, the cylinder diameter, or a combination of them is not the same for the forced oscillation and the reference free vibration. Second, the negligence of the superharmonic term in the forced oscillation puts forth a fact that the free vibration will never be predicted perfectly. This influence is mitigated for the free vibration cases where the cylinder displacement is nearly sinusoidal, such as the lower branch (low-amplitude) of free vibration. However, it can be a considerable source of error based on the results of the present study when a small superharmonic motion term exists. To check the influence of the different Reynolds number, we need to compare results from free vibrations and similar monofrequency forced oscillations at the same Reynolds number. This was done in the numerical study of Blackburn and Karniadakis [4] and the experimental one of Morse and Williams [33]. Blackburn and Karniadakis [4] compared the time histories of the lift and drag coefficients for a free-vibration simulation (laminar two-dimensional) at a Reynolds number of 200, with those corresponding to a monofrequency forced oscillation (with matching amplitude and frequency) at the same Reynolds number. The two sets of histories looked similar. In a similar manner, Morse and Williams [33] showed that the time histories of the fundamental component of the lift coefficient (denoted there by CY ) and its phase with respect to the fundamental term in the free vibration have comparable range of fluctuation (but the instantaneous profiles are of course different because the wake was turbulent in their case). One could infer from these two comparisons that the Reynolds number mismatch is solely responsible for any poor performance of freevibration prediction based on forced monofrequency oscillations. However, this is a misleading conclusion because the free vibrations in both comparisons just mentioned are almost sinusoidal (as commented by the authors themselves), thus the influence of the superharmonic motion term is not apparent. Also, we cannot rely on two discrete cases in making a judgment regarding the influence of the superharmonic motion term, since the parameters of the problem (such that the damping ratio and the mass ratio) can affect the degree of this influence, and need to be examined. Although Morse and Williams [33] showed that the root-mean-square of CY from free vibration and matching monofrequency forced oscillations are in good agreement for all three branches of the free vibration (namely the initial, upper, and lower branch), this does not mean that the lift coefficient will be in such agreement since they look only at one (the fundamental) component, and did not report results for the superharmonic component, especially for those cases when the superharmonic motion term in the free vibration is not negligible. Moreover, Morse and Williams [33] compared the magnitude of the forced oscillation, which is well-defined, to the nondimensional peak amplitude of the free vibration (denoted there by A∗). Only when the free vibration is almost sinusoidal, the two quantities can be reasonably compared. However, when the superharmonic motion term exists in the free vibration, the comparison becomes less meaningful, and spectral analysis needs to be applied to separate the fundamental and superharmonic components of the free vibration. Only the magnitude of the former term should be compared with the magnitude of the monofrequency forced oscillation. Based of the aforementioned discussion and review of previous studies, combined with the results to be presented here, we see that there is a need for systematic and detailed work to examine whether accounting for a superharmonic motion term in the forced oscillation tests improves the matching between the characteristics of the free vibration and the equivalent second-order forced oscillation, and consequently better predictive capabilities for the free vibration based on results from forced oscillation. We hope that the present study will motivate this area of research to address these questions and confirm these hypotheses. In this numerical study (using two-dimensional simulations), our main goal is to investigate bifrequency forced motions of a circular cylinder in the cross-flow direction, with a wide range of motion configurations. This bifrequency motion is composed of a modified monofrequency motion (with a single ‘fundamental’ motion term) by adding a small third superharmonic term. However, we investigate first the case of monofrequency motion where only the fundamental motion term is applied and we obtain two modes of lock-in, each of which occurs over a certain range of motion frequencies. Whereas this part is not the main goal of the study, it is a prerequisite in which we identify two different reference frequencies (leading to two different lock-in modes), which will be fixed afterwards when we add the superharmonic motion term to form the bifrequency motion. We also examine the influence of the superharmonic motion term when it is applied alone, which is also a useful prerequisite that provides reference results over a range of magnitudes of this motion term to compare with when we examine the case of bifrequency motion. In addition to the two parameters that describe the traditional monofrequency motion (i.e., the magnitude and frequency of the only-existing fundamental motion term), we have two additional parameters to describe the bifrequency motion, which are the magnitude and phase of the superharmonic motion term. We vary the magnitude of the superharmonic motion term from 0 to 25% of the magnitude of the fundamental motion term and vary the phase of the superharmonic motion term relative to the fundamental motion term from 0◦ to 360◦ . We investigate how these parameters alter the flow Flow control using bifrequency motion 385 character and their effect on the two lock-in modes mentioned earlier. We monitor four flow variables, namely the amplification of the standard deviation of the lift coefficient, the nondimensional work done (and whether it is being done on the flow or on the cylinder), the phase angle by which the superharmonic lift-coefficient component leads the fundamental one, and the magnitude ratio of the fundamental lift-coefficient component to the superharmonic one. 2 Analysis of motion and lift coefficient The nondimensional motion displacement (nondimensionalized using the cylinder diameter) has the following periodic form: Y (t) = η1 cos ( f Y t) + η3 cos (3 f Y t + φY ) (1) Y (t) = Y1 (t) + Y3 (t) with Y1 (t) = η1 cos ( f Y t) Y3 (t) = η3 cos (3 f Y t + φY ) (2) or (3) (4) The term Y1 (t) is the fundamental motion term, and Y3 (t) is the superharmonic motion term. The nondimensional angular motion frequency f Y is kept close to the nondimensional angular shedding frequency when no motion is applied (or f o ). We obtained f o = 1.3236 (for the Reynolds number considered here, which is 300). This frequency is in agreement with reported values in the literature. For example, Roshko [42] measured a value of 1.282 at a Reynolds number of 302. The two-dimensional and three-dimensional (the spanwise length was 2.25 cylinder diameters) simulations of Persillon and Braza [38] using the finite volume method gave 1.313 and 1.294, respectively. The empirical formula of Norberg [36] gives 1.2667. Because each shedding cycle corresponds to a complete period of the lift, the above frequency was found by analyzing the liftcoefficient data. A nondimensional time interval of 398.76 was used for the spectral analysis. This contains 84 shedding cycles. The corresponding nondimensional angular frequency increment in the spectral domain is 2π/398.76 = 0.01576. The choice of the length of the analyzed interval was made carefully to avoid spectral leakage or the need for window functions. Taking advantage of the periodicity in the lift coefficient, we aimed at having the length of the analyzed interval a multiple of the shedding period. Because the latter is an unknown that needs to be estimated, we performed the estimation iteratively with the aid of the method of Poincaré section, which is generated by sampling the C L − dC L /dt orbit with a fixed sampling period. If the sampling period is equal to the correct shedding frequency, the Poincaré section should be a single point. The procedure is described as follows: For a nondimensional interval of 400, the spectral analysis gives an initial estimate of shedding frequency and period. The time history is analyzed by generating the Poincaré section using this initial period for sampling. Because this is not precisely the shedding frequency, the expected single-point form of the Poincaré section is replaced by an open curve (corresponding to a portion of the C L −dC L /dt orbit). The sampling period is adjusted repeatedly until the curve reduces to a point, and the converged sampling period is taken as the shedding period. This gives the correct shedding frequency, which is used to adjust the length of the analyzed interval. Finally, the spectral analysis is applied and the precise spectral data (e.g., magnitudes of the lift-coefficient components and their phases) are obtained. From Eqs. 1–4, there are four parameters to completely describe the applied bifrequency motion of the cylinder, namely the nondimensional magnitudes η1 and η3 , the nondimensional frequency f Y , and the phase angle φY of Y3 (t) relative to Y1 (t). Because we are more interested in the effects of the superharmonic motion term, we fix η1 at 0.3. This value was selected because it ensures that the two lock-in modes can take place when only the fundamental motion term is applied, as we will show later. We will present f Y as a fraction of f o . We chose two values of f Y/f o , namely 0.98 and 1.00. The former value corresponds to a lock-in mode with work being done on the cylinder, whereas the latter value corresponds to a lock-in mode with work being done on the flow. The two remaining parameters (η3 and φY ) describe the superharmonic motion term. We will present the parameter η3 as a fraction of η1 . We consider a range of η3 /η1 up to 25% and the entire range of φY (from 0◦ to 360◦ ). 386 O. A. Marzouk To second order, the periodic lift coefficient (C L ) can be approximated as C L (t) ≈ L 1 (t) + L 3 (t) with L 1 (t) = λ1 cos ( f Y t + ψ L ) L 3 (t) = λ3 cos (3[ f Y t + ψ L ] + φ L ) (5) (6) (7) The even and higher odd superharmonic terms are negligible. When bifrequency motion is applied, periodic C L can still exist. However, the characteristics of L 3 (t) are sensitive to the applied Y3 (t), which can terminate the C L periodicity or cause transition in the periodic C L pattern, even when the magnitude of Y3 (t) is very small, as we will show later. From Eqs. 5–7, the phase angle φ L of the third superharmonic lift-coefficient term L 3 (t) is relative to the fundamental lift-coefficient term L 1 (t). Whereas previous studies examined variation of the phase angle ψ L of L 1 (t) relative to Y1 (t) within the lock-in range [9,25,29,44,45], no previous study to our knowledge has studied variation of φ L because a first-order approximation was utilized for the locked C L and the superharmonic component was neglected. We use a second-order approximation and account for this component. The nondimensional work done by the flow on the cylinder over one motion period is  1 dY CL W = dt (8) n dt n TW where TW is the period of the motion and n is an arbitrary integer. When only the fundamental motion term is applied, or when bifrequency motion is applied, then TW = 2π/ f Y (which is the largest of the two existing periods in the case of bifrequency motion) and n is typically 82 for f Y / f o = 0.98 and 84 for f Y / f o = 1.00. When only the superharmonic motion term is applied, then TW = 2π/3 f Y and n is typically 247 for f Y / f o = 0.98 and 252 for f Y / f o = 1.00. These values were chosen such that the nondimensional time interval used for the calculation and averaging of W is approximately 400. The integration in Eq. 8 is performed numerically using interpolated midpoint values for the dependent variables as follows:     t  C L (i) + C L (i + 1) d Y /d t (i) + d Y /d t (i + 1) W ≈ (9) n 2 2 i where i is a time-step index and t is the nondimensional time step. The time and frequencies in Eqs. 1–9 are nondimensionalized using the cylinder diameter as a reference length and the velocity of the incoming uniform flow as a reference velocity. 3 Numerical simulations We performed direct numerical simulations and solved the two-dimensional laminar Navier–Stokes equations. We used a modified version of the open source Fortran code INS2D [41], which solves the nondimensional incompressible Navier–Stokes equations in two-dimensional generalized coordinates. The method of artificial compressibility is used to formulate the equations into a hyperbolic set of partial differential equations. The convective terms are differenced using a third-order upwind biased flux-difference splitting, whereas the viscous terms are differences using second-order central difference. The time integration is done using the second-order backward (three-level) scheme. Therefore, the overall accuracy is second-order in both time and space. The equations are solved using an implicit line-relaxation method. In the original distribution, movinggrid simulations require the user to provide a sequence of external grid files to be read during the simulations (one per time step). In this study, we perform a large number of simulations, each of which corresponds to a particular configuration of the cylinder and grid motions, thus requires its own sequence of grid files (1.56 Mega Bytes each). To alleviate the excessive need for disk space and to have more flexibility in changing the cylinder motion from one simulation to another, we modified the code such that the moving grid is calculated at each time step based on the displacement of the cylinder, which is the internal boundary of the grid. The outer boundary is kept fixed at a radius of 24 cylinder diameters, thus we use a deforming O-type grid. The structured grid has a total of 43,200 grid points, distributed nonuniformly over 240 radial lines. The nondimensional time step is 0.02. These spatial and temporal resolutions were checked at the same Reynolds number considered Flow control using bifrequency motion 387 Present study Zheng and Zhang (2008) CL 2 0 -2 260 280 300 320 t Fig. 1 Comparison of the lift coefficient from our simulations and from the simulations of [54]. The results are obtained under low-frequency monofrequency motion with f Y = 0.8π here and were found to be satisfactory. For the case when no motion is applied, the differences in the peak lift coefficient and mean drag coefficient when the number of grid points is increased to 76,800 are only 0.25 and −0.03%, respectively. The corresponding differences when the nondimensional time step is reduced to 0.01 are −0.05 and −0.05% (equal changes). At the cylinder surface, the flow velocity is set equal to the prescribed cylinder velocity at the same time step. At the inflow, a uniform flow field is specified and the pressure is extrapolated. At the outflow, a unity pressure is specified and the velocities are extrapolated. The Reynolds number (based on the diameter and velocity of the incoming uniform flow) is set to 300, which corresponds to the end of the laminar shedding when no motion is applied, thus turbulence effects are insignificant. It is known that the onset of three-dimensional effects is at a lower Reynolds number of approximately 190 [27,50]. However, these effects do not occur promptly, but increase with the Reynolds number [51]. At a low Reynolds number of 300, these effects are not strong with regard to its influence on the lift coefficient. This was confirmed by comparing two-dimensional and three-dimensional simulations by Persillon and Braza [38]. The changes in the shedding frequency and the base pressure, and consequently the drag coefficient, are stronger [38,51]. The present study is limited to the analysis of the lift coefficient, which makes the use of two-dimensional simulations reasonable. Moreover, the applied motion and lock-in enhance the two-dimensionality of the flow and increase the spanwise correlation length as indicated in the experimental work described in the references [7,13,48]; and in the computational studies of Blackburn and Karniadakis [4] and Blackburn and Henderson [5,6]. This justifies the use of two-dimensional simulations in our study. As a validation of our simulations, we consider the case of low-frequency monofrequency motion with a magnitude η1 = 0.15 and a frequency f Y = 0.8π at a Reynolds number of 200. This case was simulated by Zheng and Zhang [54], who solved the two-dimensional laminar Navier–Stokes equations using an immersed-boundary method with a direct compensation force. They solved the equations over a Cartesian grid but a forcing term was added to the equations as a velocity corrector for the grid points inside the immersed boundary. The time histories of the lift coefficient from both simulations are compared in Fig. 1 and the agreement is excellent. More validations are available in recent studies [31,32], where the same numerical method was applied to simulate the flow over a cylinder with and without motion. 4 Results Before we examine the bifrequency motion, it is very useful to examine the effects of each term (fundamental and superharmonic) of the motion in Eq. 1 individually on four flow variables to be examined in this study, namely the amplification of the standard deviation of the lift coefficient Std(C L )/Stdo (C L ), the work done on the cylinder W , the relative phase of Y3 (t)φ L , and the magnitude ratio of the C L components λ1 /λ3 . The results will be organized in the coming sections as follows: we start with the case when only the fundamental motion term Y1 (t) is applied (thus η3 = 0) and contrast the two lock-in modes that take place. We then apply the superharmonic motion term Y3 (t) only (thus η1 = 0) and study the effects of the magnitude η3 . We then present three representative cases of the wake and lift-coefficient modes that take place under the applied bifrequency motion. We then focus on the effects of the magnitude of the superharmonic motion term and finally on the effects of its phase on altering the wake mode and the mentioned four flow variables. 388 O. A. Marzouk η 1 = 0.3, η 3 / η 1 = 0%, 2 f Y / f o = 1.00 CL 1 0 -1 -2 400 410 420 t Fig. 2 Time history of the lift coefficient when only the fundamental motion term is applied. The shown results correspond to a selected lock-in case with work being done on the flow (the solid triangle in Figs. 4, 5, 6, and 7) η 1 = 0.3, 101 10 CL 10 η 3 / η 1 = 0%, f Y / f o = 1.00 0 -1 10-2 10-3 10-4 0 1 2 3 4 5 6 4 5 6 f / fY 10 Y 10 0 -1 10-2 10-3 10-4 0 1 2 3 f / fY Fig. 3 Magnitude power spectra of the lift coefficient and the nondimensional displacement of the motion with only the fundamental motion term being applied. The shown results correspond to the lock-in case in Fig. 2 4.1 Monofrequency motion: fundamental term only As a representative locked-in case under monofrequency fundamental motion, we present in Figs. 2 and 3, respectively, the time histories of C L , the magnitude power spectra of C L and Y , and the orbit in the C L -Y phase plane for f Y / f o = 1.00. The lift is periodic, consisting mainly of two components at f Y and 3 f Y , and there is a single loop in the C L -Y orbit due to the lock-in and the dominance of the component at f Y . Due to the applied monofrequency fundamental motion, the lift coefficient is not only locked onto it, but its magnitude also changes from its value when no motion is applied. We present this change through the amplification (or attenuation) of the standard deviation of the lift coefficient Std(C L )/Stdo (C L ) in Fig. 4. The curve in this figure corresponds to values of f Y / f o from 0.790 to 1.009, which is the lock-in range for the applied η1 = 0.3. It is clear that the curve has two branches, one corresponding to the lock-in mode where the work is being done on the cylinder and the lift coefficient is attenuated, and the other corresponding to the lock-in mode where the work is being done on the flow and the lift coefficient is amplified. Figure 5 shows variation of the nondimensional work W with f Y / f o . In the first lock-in mode at lower values of f Y / f o , the work is positive or being done on the cylinder, whereas in the second mode at higher values of f Y / f o , the work is negative or being done on the flow. Again, the curve is discontinuous and each lock-in mode has a branch. Figure 6 provides the phase φ L of the third superharmonic component in the lift coefficient relative to its fundamental component, and Fig. 7 provides the magnitude ratio λ1 /λ3 of these components. The phase φ L is sensitive to f Y / f o , varying between −82◦ and 182◦ over the lock-in range. It increases with f Y / f o in the first branch mode, but decreases as f Y / f o in the second branch. The reference value of φ L when no motion is Flow control using bifrequency motion 389 Std (CL ) / Stdo (CL ) η 1 = 0.3, η 3 = 0 2.0 1.5 work on flow If η 1 = 0 1.0 0.5 work on cylinder 0.0 0.78 0.82 0.86 0.90 0.94 0.98 1.02 fY / fo Fig. 4 Amplification of the standard deviation of the lift coefficient at different frequencies with only the fundamental motion term being applied. The open circle corresponds to f Y / f o = 0.98 and the solid triangle corresponds to f Y / f o = 1.00. The dash-dot line corresponds to the obtained value when no motion is applied η 1 = 0.3, η 3 = 0 0.6 W 0.3 on work er cylind 0.0 If η 1 = 0 -0.3 -0.6 0.78 0.82 0.86 0.90 0.94 work on flow 0.98 1.02 fY / fo Fig. 5 Work (nondimensional) done by the flow on the cylinder over one oscillation period at different frequencies with only the fundamental motion term being applied. The open circle corresponds to f Y / f o = 0.98 and the solid triangle corresponds to f Y / f o = 1.00. The dash-dot line corresponds to the obtained value when no motion is applied η 1 = 0.3, η 3 = 0 240 work on flow If η 1 = 0 180 φL 120 60 0 k wor der ylin on c -60 -120 0.78 0.82 0.86 0.90 0.94 0.98 1.02 fY / fo Fig. 6 Relative phase of the superharmonic lift-coefficient component at different frequencies with only the fundamental motion term being applied. The open circle corresponds to f Y / f o = 0.98 and the solid triangle corresponds to f Y / f o = 1.00. The dash-dot line corresponds to the obtained value when no motion is applied applied is 95◦ . The reference magnitude ratio λ1 /λ3 when no motion is applied is 45.1. When the work is done on the cylinder, this ratio is below the reference value and decreases as f Y / f o increases, reaching a value of 14.0. When the work is done on the flow, the ratio λ1 /λ3 becomes larger than the reference value and increases as f Y / f o increases, reaching a value of 72.8 at the high-frequency end of the lock-in range. In Figs. 4, 5, 6, and 7, we superimposed a horizontal line that indicates the value when no motion is applied, which helps demonstrate the effect of applying the monofrequency motion within the lock-in range. We also designated one point on each branch ( f Y / f o = 0.98 for the first lock-in mode and f Y / f o = 1.00 for the second lock-in mode). These points correspond to the two fundamental motion terms to which we will add the superharmonic motion term. When f Y / f o = 0.98, Std(C L )/Stdo (C L ) = 0.52, W = 0.419, φ L = 134◦ , and λ1 /λ3 = 14.2. When f Y / f o = 1.00, Std(C L )/Stdo (C L ) = 1.92, W = −0.487, φ L = 174◦ , and λ1 /λ3 = 57.1. 390 O. A. Marzouk η 1 = 0.3, η 3 = 0 80 λ1 / λ 3 work on flow If η 1 = 0 60 40 work o n cylin der 20 0 0.78 0.82 0.86 0.90 0.94 0.98 1.02 fY / fo Fig. 7 Fundamental-to-superharmonic magnitude ratio of the lift-coefficient components at different frequencies with only the fundamental motion term being applied. The open circle corresponds to f Y / f o = 0.98 and the solid triangle corresponds to f Y / f o = 1.00. The dash-dot line corresponds to the obtained value when no motion is applied η 1 = 0, 2 η 3 / 0.3 = 5%, f Y / f o = 1.00 CL 1 0 -1 -2 400 410 420 t Fig. 8 Time history of the lift coefficient for a selected case with only the superharmonic motion term being applied η 1 = 0, 101 η 3 / 0.3 = 5%, f Y / f o = 1.00 CL 100 10 -1 10 -2 10-3 10-4 0 1 2 3 4 5 6 4 5 6 f / fY 100 Y 10-1 10 -2 10-3 10-4 0 1 2 3 f / fY Fig. 9 Magnitude power spectra of the lift coefficient and the nondimensional displacement of the motion for a selected case with only the superharmonic motion term being applied 4.2 Monofrequency motion: superharmonic term only We then examine the effect of applying only the superharmonic motion term Y3 (t), thus we set η3 = 0 in Eq. 1. As we have done in the previous section, we start here with a representative case for the C L behavior Std (CL ) / Stdo (CL ) Flow control using bifrequency motion 391 η1 = 0 3.0 f Y / f o = 0.98 f Y / f o = 1.00 2.5 2.0 wor 1.5 1.0 0 k on 5 f low 10 15 20 25 (η 3 / 0.3)% Fig. 10 Amplification of the standard deviation of the lift coefficient with only the superharmonic motion term being applied, for two fundamental frequencies of the motion. For f Y / f o = 0.98 and η3 /0.3 < 10.3%, the lift coefficient is chaotic η1 = 0 0.00 wor W -0.02 k on flow -0.04 -0.06 -0.08 f Y / f o = 0.98 f Y / f o = 1.00 0 5 10 15 20 25 (η 3 / 0.3)% Fig. 11 Work (nondimensional) done by the flow on the cylinder over one oscillation period with only the superharmonic motion term being applied, for two fundamental frequencies of the motion. For f Y / f o = 0.98 and η3 /0.3 < 10.3%, the lift coefficient is chaotic η1 = 0 420 360 work on flow φL 300 240 180 f Y / f o = 0.98 f Y / f o = 1.00 120 60 0 5 10 15 20 25 (η 3 / 0.3)% Fig. 12 Relative phase of the superharmonic lift-coefficient component with only the superharmonic motion term being applied, for two fundamental frequencies of the motion. For f Y / f o = 0.98 and η3 /0.3 < 10.3%, the lift coefficient is chaotic η1 = 0 12 f Y / f o = 1.00 f Y / f o = 0.98 λ1 / λ 3 10 8 6 4 work on flow 2 0 0 5 10 15 20 25 (η 3 / 0.3)% Fig. 13 Fundamental-to-superharmonic magnitude ratio of the lift-coefficient components with only the superharmonic motion term being applied, for two fundamental frequencies of the motion. For f Y / f o = 0.98 and η3 /0.3 < 10.3%, the lift coefficient is chaotic 392 O. A. Marzouk Y, dY / d t 0.8 η 3 / η 1 = 9%, f Y / f o = 1.00, φ Y = 0 0.4 o Y dY / d t 0.0 -0.4 -0.8 400 405 t 410 415 410 415 d2 Y / dt 2 1.2 0.6 0.0 -0.6 -1.2 400 405 t Fig. 14 Time histories of the nondimensional displacement, velocity, and acceleration of the motion with both the fundamental and superharmonic motion terms being applied. The shown results correspond to the periodic wake and lift-coefficient case in Fig. 16 where work is done on the flow in response to the applied monofrequency superharmonic motion, and present in Figs. 8 and 9, respectively, the time histories of C L , the magnitude power spectra of C L and Y , and the orbit in the C L -Y phase plane for f Y / f o = 1.00 and η3 /0.3 = 5%. Whereas both λ1 and λ3 are present in C L as in the case of monofrequency fundamental motion at same f Y / f o (see Figs. 2 and 3), the component L 3 (t) now is much stronger and the ratio λ1 /λ2 now is 2.0 instead of 57.1. This is reflected in the three peaks per cycle in Fig. 8. In Figs. 10, 11, 12, and 13, we examine variations in the four flow variables we focus on in this study (i.e., Std(C L )/Stdo (C L ), W, φ L , and λ1 /λ3 ) for f Y / f o = 0.98 and 1.00 as η3 increases. Although η1 = 0, we use its nominal value 0.3 as a scale for η3 . We consider a range of η3 /0.3 = 0 to 25%, which is the same range to be considered when bifrequency motion is applied in the next sections. In these figures, the curves for f Y / f o = 0.98 are shown only for η3 /0.3 ≥ 10.3% because below this value, irregular patterns of C L take place, which are not of interest in this study. The curves for f Y / f o = 0.98 and 1.00 are very close to each other; the difference is that for f Y / f o = 1.00, we obtained periodic C L even at small values of η3 . Therefore, a stronger motion is required to achieve periodic C L if f Y / f o  = 1.00. Figures 10 and 11 show that the Std(C L ) is always amplified and the work is always done on the flow under monofrequency superharmonic motion. Both the Std(C L ) and the modulus of W increase monotonically with η3 . Only φ L exhibits a discontinuous behavior as shown in Fig. 12 with two narrow ranges of φ L near 200◦ at smaller η3 and near 350◦ at larger η3 . The ratio λ1 /λ3 in Fig. 13 starts from a large value (corresponding to 45.1 when no motion is applied) and decreases rapidly with η3 reaching a minimum value of unity at η3 /0.3 = 10.3% before it increases slowly at larger η3 . This value of η3 corresponds to the discontinuity in φ L in Fig. 12. 4.3 Bifrequency motion: different modes We apply the bifrequency motion with η1 = 0.3 and for f Y / f o = 0.98 and 1.00. Depending on η3 /η1 and φY , we can obtain one of three wake and lift-coefficient modes: periodic with work being done on the flow, periodic with work being done on the cylinder, or irregular. In this section, we present three cases which are representative of each mode. In Figs. 14, 15, 16, and 17, we show typical behavior of a periodic wake and lift-coefficient mode with work being done on the flow. This case corresponds to f Y / f o = 1.00, η3 /η1 = 9%, and φY = 0◦ . We start with the time histories of the applied motion (displacement, velocity, and acceleration) in Fig. 14. Whereas in all plots there are two discrete components at f Y and 3 f Y , the magnitude ratios of these components are different. This ratio is 9% in the case of displacement Y , 27% in the case of velocity dY /dt, and 81% in the case of acceleration d2 Y /dt 2 . These differences are better demonstrated in the corresponding magnitude power Flow control using bifrequency motion 393 10 o -1 Y 10 φY = 0 η 3 / η 1 = 9%, f Y / f o = 1.00, 0 10-2 10-3 0 1 2 3 4 5 6 4 5 6 4 5 6 f / fY d Y / dt 10 0 10-1 10-2 10-3 0 1 2 3 f / fY 0 2 2 d Y / dt 10 10-1 10-2 10 -3 0 1 2 3 f / fY Fig. 15 Magnitude power spectra of the nondimensional displacement, velocity, and acceleration of the motion with both the fundamental and superharmonic motion terms being applied. The shown results correspond to the periodic wake and lift-coefficient case in Fig. 16 where work is done on the flow 3.0 η 3 / η 1 = 9%, f Y / f o = 1.00, o work on flow 1.5 CL φY = 0 0.0 -1.5 -3.0 400 405 t 410 415 Fig. 16 Time history of the lift coefficient with both the fundamental and superharmonic motion terms being applied. The figure corresponds to a periodic wake and lift-coefficient mode where work is done on the flow spectra in Fig. 15. The resulting time history and magnitude power spectrum of C L are shown in Figs. 16 and 17. The Std(C L ) is large and λ1 > λ3 . At a slightly larger η3 /η1 = 10%, a mode transition occurs and the above periodic wake and lift-coefficient mode is replaced with another periodic mode in which the work is done on the cylinder, which is illustrated in Figs. 18 and 19. The resulting time history and magnitude power spectrum of C L are shown in Figs. 18 and 19. The Std(C L ) is smaller than its value under the previous periodic mode and λ1 < λ3 . As a typical case of the third wake and lift-coefficient mode, we present the case of f Y / f o = 1.00, η3 /η1 = 5%, and φY = 90◦ , which corresponds to chaotic C L . The motion pattern is illustrated in the time histories in Fig. 20. The resulting time history and magnitude power spectrum of C L are shown in Figs. 21 and 22. The broadband noise in the C L spectrum is a character of chaos. O. A. Marzouk CL 394 10 1 10 0 φY = 0 η 3 / η 1 = 9%, f Y / f o = 1.00, o work on flow 10-1 10-2 10 -3 10-4 0 1 2 3 4 5 6 f / fY Fig. 17 Magnitude power spectrum of the lift coefficient with both the fundamental and superharmonic motion terms being applied. The shown results correspond to the periodic wake and lift-coefficient case in Fig. 16 where work is done on the flow η 3 / η 1 = 10%, f Y / f o = 1.00, 3.0 φY = 0 o work on cylinder CL 1.5 0.0 -1.5 -3.0 400 405 t 410 415 Fig. 18 Time history of the lift coefficient with both the fundamental and superharmonic motion terms being applied. The figure corresponds to a periodic wake and lift-coefficient mode where work is done on the cylinder η 3 / η 1 = 10%, 101 work on cylinder 100 CL φY = 0o f Y / f o = 1.00, 10-1 10-2 10-3 10-4 0 1 2 3 4 5 6 f / fY Fig. 19 Magnitude power spectrum of the lift coefficient with both the fundamental and superharmonic motion terms being applied. The shown results correspond to the periodic wake and lift-coefficient case in Fig. 18 where work is done on the cylinder To contrast the periodic changes in the wake and the shedding pattern for the above cases with periodic lift coefficient but different sign of the work done, we visualize the vorticity fields for both cases (corresponding to Figs. 16 and 18) in Fig. 23. For each case, four successive views of the vorticity field near the cylinder are shown, which span one oscillation period. For the case with work being done on the flow, the clockwise top vortex is shed when the cylinder is moving downward and the bottom counterclockwise vortex is shed when the cylinder is moving upward. This is reversed for the case with work being done on the cylinder. In addition, the shedding length is shorter for the case with work being done on the flow, resulting in stronger variation in the pressure at the cylinder wall when the cylinder is at its largest displacement. The latter influence is better illustrated through the sequence of wall pressure coefficient for the same cases and same instants of the vorticity fields. These wall pressure coefficients are visualized in Fig. 24. The variable θ in this figure is an angular coordinate that traverses the cylinder wall from the nose (furthest upstream point on the cylinder surface), in the clockwise direction. The stronger imbalance (in the cross-flow direction) in the wall pressure for the case with work being done on the flow, at the instants when the cylinder is at its largest (upward and Flow control using bifrequency motion 395 o η 3 / η 1 = 5%, f Y / f o = 1.00, φ Y = 90 Y, dY / d t 0.8 Y dY / d t 0.4 0.0 -0.4 -0.8 400 405 t 410 415 410 415 0.6 0.0 2 d Y / dt 2 1.2 -0.6 -1.2 400 405 t Fig. 20 Time histories of the nondimensional displacement, velocity, and acceleration of the motion with both the fundamental and superharmonic motion terms being applied. The shown results correspond to the chaotic case in Fig. 21 3.0 η 3 / η 1 = 5%, f Y / f o = 1.00, φ Y = 90 o CL 1.5 0.0 -1.5 -3.0 200 300 400 500 600 t Fig. 21 Time history of the lift coefficient with both the fundamental and superharmonic motion terms being applied. The shown results correspond to a chaotic wake and lift-coefficient mode η 3 / η 1 = 5%, 101 f Y / f o = 1.00, φ Y = 90 o CL 100 10-1 10-2 10-3 10 -4 0 1 2 3 4 5 6 f / fY Fig. 22 Magnitude power spectrum of the lift coefficient with both the fundamental and superharmonic motion terms being applied. The shown results correspond to the chaotic case in Fig. 21 downward) displacement justifies the stronger lift coefficient for this case, in which the largest (positive and negative, respectively) values of the lift occur approximately at the same instants. 4.4 Bifrequency motion: effect of superharmonic-term magnitude In this section, we select four values of φY ; namely 0◦ , 90◦ , 180◦ , and 270◦ ; and vary η3 /η1 from 0 to 25%. As mentioned before, we fix η1 at 0.3 and consider f Y / f o = 0.98 and 1.00, but we do not present results for f Y / f o = 1.00 and φY = 90◦ because no periodic C L is obtained except for η3 /η1 up to 1%. Similarly we 396 O. A. Marzouk Fig. 23 Visualization of the vorticity fields over one oscillation period. The shown results correspond to the periodic wake and lift-coefficient cases in Fig. 16 (left) and Fig. 18 (right) Flow control using bifrequency motion 397 o η 3 / η 1 = 9%, f Y / f o = 1.00, φ Y = 0 η 3 / η 1 = 10%, f Y / f o = 1.00, φ Y = 0 Y ~ 0, increasing Y ~ Ymax 1 0 C P, W C P, W Y ~ 0, increasing Y ~ Ymax 1 0 -1 -2 -1 -2 work on flow -3 0 60 120 180 work on cylinder -3 240 300 360 0 60 120 θ o 240 300 η 3 / η 1 = 10%, f Y / f o = 1.00, φ Y = 0 Y ~ 0, decreasing Y ~ Ymin 1 180 360 θ η 3 / η 1 = 9%, f Y / f o = 1.00, φ Y = 0 o Y ~ 0, decreasing Y ~ Ymin 1 0 C P, W 0 C P, W o -1 -1 -2 -2 work on flow -3 0 60 120 180 work on cylinder -3 240 300 360 0 60 120 θ 180 240 300 360 θ Fig. 24 Visualization of the wall pressure coefficient over one oscillation period. The shown results correspond to the periodic wake and lift-coefficient cases in Fig. 16 (left) and Fig. 18 (right) f Y / f o = 0.98 3.5 φY = 0 90 180 270 3.0 Std (CL) / Stdo (CL) o 2.5 on work flow 2.0 1.5 r nde cyli n rk o wo 1.0 0.5 0.0 0 5 10 15 20 25 (η 3 / η 1)% f Y / f o = 1.00 3.5 φY = 0 180 270 Std (CL) / Stdo (CL) 3.0 o on work flow 2.5 2.0 der ylin c n rk o wo 1.5 1.0 0.5 0.0 0 5 10 15 20 25 (η 3 / η 1)% Fig. 25 Amplification of the standard deviation of the lift coefficient at different values of the magnitude and phase of the superharmonic motion term, and for two fundamental frequencies of the motion. Segments of the curves where the lift coefficient is chaotic are not plotted 398 O. A. Marzouk f Y / f o = 0.98 0.5 work on cylinder 0.0 W work o n flow φY = 0o 90 180 270 -0.5 -1.0 0 5 10 work on 15 flow 20 25 (η 3 / η 1)% f Y / f o = 1.00 0.5 φY = 0 180 270 work on c ylinder W 0.0 o work on flow -0.5 -1.0 0 5 10 15 20 25 (η 3 / η 1)% Fig. 26 Work (nondimensional) done by the flow on the cylinder over one oscillation period at different values of the magnitude and phase of the superharmonic motion term, and for two fundamental frequencies of the motion. Segments of the curves where the lift coefficient is chaotic are not plotted do not show results for f Y / f o = 0.98 and φY = 90◦ for the range η3 /η1 > 12%. For f Y / f o = 0.98 and φY = 0◦ , we only obtain periodic wake and lift-coefficient mode with work done on the cylinder. In contrast, for f Y / f o = 1.00 and φY = 180◦ and 270◦ , we only obtain periodic wake and lift-coefficient mode with work done on the flow. For f Y / f o = 0.98 and φY = 180◦ and 270◦ , and for f Y / f o = 1.00 and φY = 0◦ , a mode transition occurs at different values of η3 /η1 , which lie between 9 and 17%. Therefore, a phase φY near 90◦ acts towards the termination of periodicity in the wake and lift-coefficient. In Fig. 25, the curves of the amplification of Std(C L ) nearly collapse for each wake and lift-coefficient mode regardless of φY . They increase with η3 /η1 as in the case of monofrequency superharmonic motion. However, the corresponding W curves exhibit disparities, especially when the work is done on the flow at large values of η3 /η1 as shown in Fig. 26. For example, for η3 /η1 = 25% and f Y / f o = 0.98, the values of Std(C L )/Stdo (C L ) with φY = 180◦ and 270◦ differ by only 0.012, whereas the difference in W is 0.606. Figure 27 indicates that φ L is controlled by the applied φY when φY = 270◦ , where φ L remains close to this angle for both values of f Y / f o . The curves of λ1 /λ3 are shown in Fig. 28. They exhibit a monotonic decrease with η3 unlike the case of monofrequency superharmonic motion in Fig. 13. The curves of the magnitude ratio η1 /η3 for Y , the magnitude ratio η1 /3η3 for dY /dt, and the magnitude ratio η1 /9η3 for d2 Y /dt 2 are shown in Fig. 29. Comparing this figure to Fig. 28, which has same axes range, indicates that the magnitude ratio λ1 /λ3 is controlled by the corresponding magnitude ratio of the motion acceleration d2 Y /dt 2 for the wake and lift-coefficient mode when the work is done on the cylinder, whereas it is controlled by the magnitude ratio of the motion velocity dY /dt for the wake and lift-coefficient mode when the work is done on the flow. Flow control using bifrequency motion 399 φY = 0o 360 f Y / f o = 0.98 90 180 270 300 work on flow work on cylinder 240 φL work on flow 180 work on cylinder 120 der work on cylin 60 0 0 5 10 15 20 25 (η 3 / η 1)% f Y / f o = 1.00 360 300 work on flow φL 240 φY = 0 180 180 270 120 work on 60 0 o 0 5 r work on cylinde flow 10 15 20 25 (η 3 / η 1)% Fig. 27 Relative Phase of the superharmonic lift-coefficient component at different values of the magnitude and phase of the superharmonic motion term, and for two fundamental frequencies of the motion. Segments of the curves where the lift coefficient is chaotic are not plotted 4.5 Bifrequency motion: effect of superharmonic-term phase In this section, we consider variations of the same four flow variables of interest as the applied phase φY is varied from 0◦ to 360◦ at two values of η3 /η1 = 1% and 5%. Again, we consider f Y / f o = 0.98 and 1.00 and fix η1 at 0.3. With η3 /η1 = 1%, there is only one wake and lift-coefficient mode over the entire range of φY ; where for f Y / f o = 0.98, we obtain the periodic mode with work being done on the cylinder, whereas for f Y / f o = 1.00, we obtain the periodic mode with work being done on the flow. On the other hand, mode transition occurs with η3 /η1 = 5%; where for f Y / f o = 0.98, the work is done on the cylinder except for φY = 210◦ to 240◦ (work is done on the flow), and for f Y / f o = 1.00, the work is done on the flow except for φY = 14◦ to 31◦ (work is done on the cylinder) and φY = 32◦ to 135◦ (irregular mode). The curves of Std(C L )/Stdo (C L ) in Fig. 30, of W in Fig. 31, and of λ1 /λ3 in Figs. 33 and 34 exhibit nonmonotonic wavy variations with φY . Increasing η3 /η1 intensifies these variations but they remain qualitatively similar for each mode. On the other hand, variations of φ L is monotonic and weakly sensitive to η3 /η1 as shown in Fig. 32. Therefore, the phase φ L can be controlled by the applied φY even with very small η3 that is only 1% of η1 . 5 Summary and conclusions We numerically simulated the flow over a circular cylinder under many varieties of forced cross-flow monofrequency and bifrequency motions (consisting of a small third superharmonic term added to a fundamental motion term) at a Reynolds number of 300. The monofrequency motions were first investigated as reference 400 O. A. Marzouk f Y / f o = 0.98 12 φY = 0o 90 180 270 10 λ1/ λ 3 8 work on cylinder 6 4 work on flow 2 0 0 5 10 15 20 25 (η 3 / η 1)% f Y / f o = 1.00 12 φY = 0o 180 270 10 λ1/ λ 3 8 work on flow 6 4 2 0 work on cylinder 0 5 10 15 20 25 (η 3 / η 1)% Fig. 28 Fundamental-to-superharmonic magnitude ratio of the lift-coefficient components at different values of the magnitude and phase of the superharmonic motion term, and for two fundamental frequencies of the motion. Segments of the curves where the lift coefficient is chaotic are not plotted η1 / η3, η1 / 3η3, η1 / 9η3 12 Y d Y / dt 2 2 d Y / dt 10 8 6 4 2 0 0 5 10 15 20 25 (η3 / η1)% Fig. 29 Fundamental-to-superharmonic magnitude ratio of the two terms of the displacement, velocity, and acceleration of the motion at different values of the magnitude of the superharmonic motion term cases for the more-important bifrequency ones. We used a second-order (two terms) approximation for the periodic lift coefficient to account for details that cannot be captured with the common first-order (one term) approximation. Flow control using bifrequency motion 401 f Y / f o = 0.98 work on flow 0.75 if work on cylinder Std (CL ) / Stdo (CL ) 0.80 2.12 2.10 0.70 if work on flow η 3 /η 1 = 1% η 3 /η 1 = 5% 0.65 work on cylinder 0.60 0.55 0.50 0.45 If η 3 = 0 0 60 120 180 240 300 360 φY η 3 /η 1 = 1% η 3 /η 1 = 5% if work on flow f Y / f o = 1.00 work on flow 2.0 1.9 If η 3 = 0 1.8 0.78 work on cylinder 0 60 120 180 240 300 0.76 360 φY if work on cylinder Std (CL ) / Stdo (CL ) 2.1 Fig. 30 Amplification of the standard deviation of the lift coefficient at different values of the phase and magnitude of the superharmonic motion term, and for two fundamental frequencies of the motion. The dash-dot line corresponds to the obtained value when only the fundamental motion term is applied η 3 /η 1 = 1% η 3 /η 1 = 5% W f Y / f o = 0.98 0.46 work on cylinder 0.44 If η 3 = 0 0.42 work on cylinder 0.40 0.38 0.36 work on flow 0 60 120 180 240 300 -0.2 -0.4 360 φY W -0.33 if work on flow f Y / f o = 1.00 work on cylinder 0.51 0.50 work on flow -0.41 if work on cylinder η 3 /η 1 = 1% η 3 /η 1 = 5% if work on flow if work on cylinder 0.48 -0.49 If η 3 = 0 -0.57 -0.65 0 60 120 180 240 300 360 φY Fig. 31 Work (nondimensional) done by the flow on the cylinder per oscillation period at different values of the phase and magnitude of the superharmonic motion term, and for two fundamental frequencies of the motion. The dash-dot line corresponds to the obtained value when only the fundamental motion term is applied 402 O. A. Marzouk η 3 /η 1 = 1% η 3 /η 1 = 5% 420 f Y / f o = 0.98 r de lin y c on rk o w 360 work on flow φL 300 240 360 300 240 180 180 120 120 r nde cyli n o k wor 60 0 420 0 60 120 If η 3 = 0 180 240 300 60 0 360 φY η 3 //η 1 = 1% η 3 /η 1 = 5% 420 f Y / f o = 1.00 flow k on wor φL 300 work on 240 cylinder 300 240 180 180 flow k on wor 120 60 0 420 360 360 0 60 120 If η 3 = 0 120 60 180 240 300 0 360 φY Fig. 32 Relative phase of the superharmonic lift-coefficient component at different values of the phase and magnitude of the superharmonic motion term, and for two fundamental frequencies of the motion. The dash-dot line corresponds to the obtained value when only the fundamental motion term is applied We first considered monofrequency motions with a constant magnitude and with frequencies near the shedding frequency that is obtained when no motion is applied. Lock-in takes place and it is characterized by two modes having different structures of the periodic wake and the vortex-shedding patterns leading to contrasting features; where in one mode the work (due to the existence of a co-linear lift force acting on the moving cylinder) is done on the cylinder and the lift coefficient is attenuated, whereas in the other mode the work is done on the flow and the lift coefficient is amplified. When monofrequency superharmonic motions are applied, the work is always done on the flow, monotonic variations in the lift amplification and the work take place as the motion magnitude is increased, and the relative magnitude of the superharmonic lift-coefficient component exhibits a maximum in response to the motion magnitude. We then considered forced bifrequency motions with two fundamental motion terms, each one corresponds to a different lock-in mode, and examined the effect of adding a small third superharmonic motion term with a relative (with respect to the fundamental motion term) magnitude up to 25% and a relative phase angle from 0◦ to 360◦ . The small-magnitude superharmonic motion term is viewed as a tool to control the wake and lift-coefficient behavior while keeping the fundamental motion term unchanged. The added superharmonic motion term can cause transition from a periodic wake and lift-coefficient mode to another periodic mode or to an irregular mode. As the magnitude of the superharmonic motion term increases, the work becomes very sensitive to the phase of the superharmonic motion compared to the standard deviation of the lift coefficient. The maximum of the relative (with respect to the fundamental lift-coefficient component) magnitude of the superharmonic lift-coefficient component that occurs under monofrequency superharmonic motion no longer exists under bifrequency motion. The phase of the superharmonic lift-coefficient component relative to the fundamental component can be controlled by the applied phase of the superharmonic motion term relative to the fundamental term, even with very small magnitudes of the superharmonic motion term that is two orders of magnitude smaller than the magnitude of the fundamental term. Flow control using bifrequency motion 403 η 3 /η 1 = 1% f Y / f o = 0.98 10 work on cylinder λ1 / λ 3 8 6 4 If η 3 = 0: λ 1 / λ 3 = 14.2 2 0 60 120 180 240 300 360 φY η 3 /η 1 = 1% f Y / f o = 1.00 32 work on flow λ1 / λ 3 28 24 20 16 If η 3 = 0: λ 1 / λ 3 = 57.1 12 0 60 120 180 240 300 360 φY Fig. 33 Fundamental-to-superharmonic magnitude ratio of the lift-coefficient components at different values of the phase of the superharmonic motion term, and for two fundamental frequencies of the motion λ1 / λ 3 if work on cylinder f Y / f o = 0.98 work on flow 4.05 3.95 1.15 1.10 if work on flow η 3 /η 1 = 5% 1.20 work on cylinder work on cylinder 1.05 If η 3 = 0: λ 1 / λ 3 = 14.2 1.00 0 60 120 180 240 300 360 φY η 3 /η 1 = 5% f Y / f o = 1.00 5.5 λ1 / λ 3 work on flow 4.5 4.0 3.5 3.0 2.5 work on cylinder 0 60 120 If η 3 = 0: λ 1 / λ 3 = 57.1 180 φY 240 300 1.19 1.17 360 if work on cylinder if work on flow 5.0 Fig. 34 Fundamental-to-superharmonic magnitude ratio of the lift-coefficient components at different values of the phase of the superharmonic motion term, and for two fundamental frequencies of the motion 404 O. A. Marzouk The important effects of a small superharmonic motion term on the wake mode and the lift coefficient can interpret some unsatisfactory results reported in previous studies that attempted to apply forced monofrequency oscillation to predict characteristics of free vibrations. The findings reported here suggest that utilizing bifrequency oscillations rather than monofrequency ones can improve the predictive capabilities of the concepts presented in these studies due to better equivalence between the two problems. Acknowledgments The author would like to thank NASA Ames Research Center and the NASA’s Advanced Supercomputing (NAS) Division (previously known as the Numerical Aerospace Simulation). The author acknowledges deeply the valuable advice of Professor Ali H. Nayfeh in the Department of Engineering Science and Mechanics at Virginia Polytechnic Institute and State University, which inspired this work. References 1. 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