optimized drag reduction and wake dynamics associated with … · 2018. 11. 19. · 4826 a. mehmood...

Contemporary Engineering Sciences, Vol. 11, 2018, no. 97, 4825 - 4843HIKARI Ltd, www.m-hikari.com

https://doi.org/10.12988/ces.2018.89519

Optimized Drag Reduction and Wake Dynamics

Associated with Rotational Oscillations of

a Circular Cylinder

A. Mehmood, M. R. Hajj, I. Akhtar, M. Ghommem

Department of Engineering Science and MechanicsVirginia Polytechnic Institute and State University

Blacksburg, VA 24061, USA

L. T. Watson

Departments of Mathematics, Computer Science, andAerospace and Ocean Engineering

Virginia Polytechnic Institute and State UniversityBlacksburg, VA 24061, USA

T. C. H. Lux

Department of Computer ScienceVirginia Polytechnic Institute and State University

Blacksburg, VA 24061, USA

Copyright c© 2018 A. Mehmood et al. This article is distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduc-

tion in any medium, provided the original work is properly cited.

Abstract

The drag reduction on a circular cylinder through rotational oscil-lations is globally maximized by coupling a CFD solver with a novelparallel global deterministic optimization algorithm. The simulationsare performed at a Reynolds number of 150 and the amplitude and fre-quency of the rotational oscillations are respectively constrained to theranges 0.1 ≤ Ω ≤ 1.0 and 0.1 ≤ fΩ ≤ 3.0. The novelty of this work liesin the use of a massively parallel derivative free optimization algorithmto find the globally (not locally, as is typically done) optimal values of

4826 A. Mehmood et al.

the rotational amplitude and frequency for minimum drag. Rotationaloscillations significantly affect the pattern of shedding vortices and themean drag is reduced at high forcing frequencies in relation to the shed-ding frequency. The results also show that there is a threshold of therotational oscillation amplitude below which the mean drag is not re-duced for any excitation frequency. The wake dynamics associated withthe globally optimized and nonoptimized configurations are presented.The contributions of the viscous and pressure components to the totaldrag in these configurations are also determined and discussed.

Keywords: Rotational oscillations, VTDIRECT95, Global optimization,Synchronization, Critical amplitude/frequency, Drag reduction

1 Introduction

The flow behind a circular cylinder exhibits an organized and periodic mo-tion of a regular array of concentrated vorticity that sheds from the cylinder.For a stationary cylinder, these vortices are shed with a specific frequencyrepresented by a nondimensional Strouhal number St. This vortex sheddingresults in oscillatory forces on the body, which can be decomposed into liftand drag components along the cross-flow and in-line directions, respectively([24, 33, 6]). Modifying the shedding mechanism or controlling the motion ofthe cylinder in an appropriate manner is thus a way of controlling the fluidforces.

Many active and passive control mechanisms for the vortex shedding havebeen explored for the purpose of reducing the forcing components. For in-stance, splitter plates have been used to modify the shedding patterns ([16,31]). Active controls such as suboptimal blowing and suction ([19, 3]), acousticfeedback ([7, 14]), inline and transverse oscillations ([28, 18, 27]), and rota-tional oscillations ([30, 8, 26, 23, 29]) have also been proposed. [3] observed asuppression of the fluctuating forces and obtained more than 40% reduction inthe mean drag coefficient using a couple of suction actuators on a stationarycylinder. [7] experimentally observed that the shedding frequency and forcescan be altered by introducing sound waves with frequencies close to the naturalvortex shedding frequency. [14] introduced local sound waves into the flow toinfluence the shear layer on one side of the cylinder. Suppression of the vortexshedding was observed by the destructive interference of the two shear layers.[27] observed a reduction in the drag force for a moderately spaced staggeredarrangement of circular cylinders.

[30] performed experiments on a rotationally oscillating circular cylinderin a steady uniform flow at Reynolds number Re = 15, 000. They observed areduction in the oscillatory component of the drag up to 80% for optimal am-plitude and frequency of oscillation. [29] experimentally investigated the effect

Optimized drag reduction and wake dynamics 4827

of rotatory oscillations at low Reynolds number Re = 150. They observed thatthe forcing parameters (oscillation amplitude and frequency) affect the patternof vortex shedding. Numerical studies on the flow past an oscillating rotat-ing cylinder have been performed by [26] over a range of 150 ≤ Re ≤ 15000.Their results are in qualitative agreement with those obtained experimentallyby [30]. They observed that drag reduction is mainly caused by the generationof multipole vorticity structures. These structures are generated as a result ofbursting of the boundary layer, triggered by appropriate rotational oscillations.They observed that this phenomenon leads to an overall separation delay andthus drag reduction. They also found that the impact of rotational oscilla-tions depends strongly on the Reynolds number and would be effective only ifRe ≥ 3000 because the viscosity suppresses the generation of multipole vor-ticity in the case of low Reynolds number. The Reynolds number dependencefor mean drag reduction was also observed by [8]. They studied the effectsof rotational oscillations in an unsteady laminar flow past a circular cylinderat Re = 100. They varied the rotational speed and frequency in the ranges0.2 ≤ Ω ≤ 2.5 and 0.02 ≤ fΩ ≤ 0.8, respectively. They found a reduction inthe mean drag of 12% for Reynolds number 100 and also performed numericalsimulation at Reynolds number 1000 and found a reduction in the mean dragof 60%. They observed that the lock-on frequency range becomes wider asrotational speed increases. They also determined that local minimum pointsfor the mean drag are near the boundaries of the lock-on regions. [23] numeri-cally investigated the flow over a cylinder undergoing rotational oscillations atRe = 150. They observed a reduction in the mean drag coefficient of 25% atthe higher forcing frequencies. They also suggested that the mean drag coef-ficient does not monotonically decrease as the forcing frequency is increased,but there is an optimal forcing frequency beyond which the mean drag againincreases. The value of this optimal frequency depends on Reynolds numberand rotational amplitude. More recently, [25] numerically and experimentallystudied the suppression of aerodynamic forces through rotational oscillations.They observed a complete suppression of vortex shedding phenomena at a rel-atively high rotation rate. [5] and [21] showed the existence of multiple lock-inregions in the flow past a circular cylinder subjected to a circular motion.[20] showed that the flow topology can be significantly affected by sinusoidalrotations of a cylinder placed in a flow.

The present study aims to globally optimize the drag reduction on a circu-lar cylinder through rotational oscillations, and understand the wake dynamicsthat are associated with maximum drag reduction. Beyond merely samplingin the parameter space, or following a descent direction, a parallel global op-timization code VTDIRECT95 is coupled with a CFD solver to find the rota-tional amplitude and frequency that globally minimize the mean value of thedrag. Section 2 discusses and validates the numerical methodology used to sim-


ulate the flow past a rotating cylinder in an unsteady uniform flow. Section 3presents the optimization algorithm VTDIRECT95 and its interface with theCFD solver. A discussion of numerical results and the frequency/amplituderesponse data is in Section 4, and Section 5 concludes.

2 Numerical Simulations

2.1 Numerical Scheme

In the CFD solver, the time dependent incompressible continuity and Navier-Stokes equations are solved in the generalized coordinates (ξ, η). These equa-tions are written in a strong conservative form as

∂Um∂ξm

= 0, (1)

∂(J−1ui)

∂t+∂Fim∂ξm

= 0, (2)

where the flux

Fim = Umui + J−1∂ξm∂xi

p− 1

ReDGmn ∂ui

∂ξn, (3)

J−1 = det(∂xi∂ξj

) is the Jacobian or the volume of the cell, Um = J−1 ∂ξm∂xj

uj is

the volume flux normal to the surface of constant ξm, and Gmn = J−1 ∂ξm∂xj

∂ξn∂xj

is the “mesh skewness tensor”. In the flow solver, the governing equations arenondimensionalized using the cylinder diameter D as a reference length andthe incoming free-stream velocity U∞ as a reference velocity. The Reynoldsnumber is defined as Re = U∞D/ν.

The equations are solved on a nonstaggered grid topology ([34]). TheCartesian velocity components u, v, w and pressure p are defined at the centerof the control volume in the computational space. A second order central dif-ference scheme is used for all spatial derivatives except for convective terms,which are discretized using QUICK ([17]). The temporal advancement is doneusing a fractional step method where a predictor step calculates an interme-diate velocity field, and a corrector step updates the velocity by satisfying thepressure Poisson equation at the new time step. A semi-implicit scheme withthe Adams-Bashforth method is used for the convection terms and the Crank-Nicolson scheme for the diffusion terms. In this study, an “O”-type grid isemployed to simulate the flow over a circular cylinder as shown in Figure 1.Further details of the numerical discretization and the parallel implementationcan be found in Refs. ([4, 2, 22]).


In this study, we perform two-dimensional numerical simulations of theflow passing over a circular cylinder undergoing rotary oscillations. For theinflow and outflow boundaries, Dirichlet and Neumann boundary conditionsare used, respectively. To simulate the flow field past a rotationally oscillatingcylinder in a uniform stream, we enforce the boundary conditions

(u, v) = (1, 0) (inlet),

∂u

∂n= 0 (outlet),

(u, v) = (−Ωy,Ωx) (surface),

where Ω is the angular velocity of the cylinder and is nondimensionalized as(ΩD)/(2U∞).

The fluid force on the cylinder is the manifestation of the pressure andshear stresses acting on the surface of the cylinder. The net force is decom-posed into two components, namely the lift and drag forces. These forces arenondimensionalized with respect to the dynamic pressure. The coefficients oflift and drag can thus be written in terms of the pressure and shear stresses as

CL = −∫ 2π

0(p sin θ − 1

ReDωz cos θ)dθ, (4)

CD = −∫ 2π

0(p cos θ +

1

ReDωz sin θ)dθ, (5)

where ωz is the spanwise vorticity component on the cylinder surface and θ isthe angle that the outer normal of the area component makes with the flowdirection. We employ curvilinear coordinates (ξ, η) in an Eulerian referenceframe. For the specific case of a circular cylinder, the generalized coordinates(ξ, η) can be represented by polar coordinates (r, θ).

2.2 Validation of CFD solver

We validate the rotational oscillation numerical results by comparing themwith other numerical simulations. Figure 2 shows a comparison of the dragcoefficients as obtained from the current simulations and those of [8] at Re =100 and Re = 1000 for three different oscillation frequencies. The comparisonshows good agreement. Note that the values are normalized by the respectivevalues of the stationary cylinder. We also compare our numerical results forthe drag reduction with those obtained by [23] as shown in Figure 3. In bothsimulations, it is shown that forcing the rotational oscillation at frequenciesnear or below the shedding frequency increases the drag coefficient. On theother hand, forcing it at a much higher frequency results in a reduction of thiscoefficient. The plot also shows good agreement between the results of the twosimulations.


Figure 1: A 2-D layout of an “O”-type grid in the (r, θ)-plane showing theinflow and outflow directions.

0 0.2 0.4 0.6 0.80.5

1

1.5

2

Re=1000

fΩ

=0.6

Ω=1.0

Re=100

fΩ

=0.6

Ω=1.0

Re=100

fΩ

=0.2

Ω=0.5Re=100

fΩ

=0.3

Ω=0.5

fΩ

CD

/CD

0

Current simulations

Choi et al. simulations

Figure 2: Comparison of the normalized mean drag coefficient on a rotationallyoscillating cylinder between current simulations and those of Choi et al. ([8]).The comparison encompasses simulations at Re = 100 and Re = 1000 fordifferent nondimensional forcing frequency fΩ and amplitudes Ω .

3 Global Optimization

For the global optimization, the parallel CFD code ([1]) that computes theflow past a circular cylinder undergoing rotational oscillation is coupled withthe parallel deterministic global optimization algorithm VTDIRECT95 ([11,10, 12]) through an interface function that computes the drag coefficient.


0 0.2 0.4 0.6 0.8 1 1.20.5

1

1.5

2

2.5

3

fΩ

CD

Strouhal number

Stationary case

Protas et al. simulations

Current simulations

Figure 3: Comparison of the mean drag coefficient as a function of the forcingfrequency of the rotational oscillations between current simulations and thoseof Protas et al. ([23]). The vertical dashed line is set at the Strouhal number ofthe stationary cylinder. The horizontal dashed line is set at the drag coefficientfor the stationary cylinder. The Reynolds number in both simulations is Re =150 and the amplitude of the oscillations is set at Ω = 2.0.

VTDIRECT95 is a Fortran 95 package for deterministic global optimizationthat contains the subroutine VTdirect (serial) and pVTdirect(parallel) imple-menting a version of the algorithm known as DIRECT ([15]). This algorithmis widely used in multidisciplinary engineering problems and physical scienceapplications. It is an efficient global optimization method that avoids beingtrapped at local optimum points and performs the search for global optimumpoints through three main operations:

1. selection of potentially optimal boxes that are the regions most likely tocontain the global optimum point;

2. sampling at points within those selected boxes;

3. division of those selected boxes into smaller boxes.

A detailed description and both serial and parallel implementations of thecode are provided in [11, 10, 12]. A distinctive characteristic of determinis-tic algorithms like VTDIRECT95 is their frugal use of function evaluations,compared to population based evolutionary algorithms.

We consider a cylinder undergoing rotational oscillations and investigatethe reduction in the mean value of the drag coefficient as the amplitude Ω ofthe rotational oscillations and forcing frequency fΩ are varied. As such, theoptimization problem is formulated as

minvθ∈D

CD(vθ),


where vθ = (Ω, fΩ), D = vθ ∈ R2 | lθ ≤ vθ ≤ uθ is a 2-dimensional box, andCD is the mean value of the drag coefficient.

In order to show the advantage of combining the optimization code VT-DIRECT95 with the CFD solver, we note that most of the experimental ornumerical works dealing with drag reduction through oscillatory motion ofcircular cylinders have considered only variations of one control parameter([8, 23, 29]). To identify the optimal point, a refined sweep must be performedin a specified range of the control parameter. In the example of reducing themean drag, the frequency is usually taken as the control parameter while keep-ing the amplitude constant. Adding more control parameters would require alarge number of experimental runs or numerical simulations to determine theoptimal point where the mean drag is minimized. To show the significanceof coupling the CFD code with an optimizer, we keep the rotational speedconstant (Ω = 1.5) and vary the frequency in the range 0.1 ≤ fΩ ≤ 1.0 witha constant increment of 0.1. As a brute force approach, we perform ten sim-ulations to obtain the frequency response curve as shown in Figure 4). Notethat the results are presented in terms of percentage reduction in the meandrag coefficient relative to that of the stationary cylinder. From this figure,we observe that the maximum drag reduction of approximately 20% occursat fΩ = 0.7. In order to further refine our results, we would require anotherset of simulations in the proximity of this locally optimal point, thus the totalnumber of simulations may be of the order of 100. In comparison we use aone-dimensional version of the original algorithm DIRECT for one control pa-rameter, i.e., frequency, and consider the same range of variations. DIRECTspecifies the value of the input frequency and the simulations are performedto compute the mean drag. The interface function feeds back the computedmean drag to DIRECT, which specifies the rotational frequency for the nextsimulation. In this way, DIRECT searches for the globally optimum pointwhere minimum mean drag is achieved. The frequency search for the mini-mum mean drag is also plotted in Figure 4 for the sake of comparison. Theresults show that DIRECT was able to converge to a drag reduction factorof 20% corresponding to fΩ = 0.65 within five simulations. The brute forcemethod would have required many more simulations to identify the optimalpoint. Thus, DIRECT provides an efficient algorithm to locate the optimalconfiguration for reducing the mean drag. Considering other parameters wouldadd significantly to the computational cost of parameter sweeps. The sameargument can be extended to the application of DIRECT for more than onecontrol parameter where each parameter has its own sweeping range. If werequire Ns simulations for one control parameter, then using a brute forcemethod, P control parameters would require NP

s simulations, many orders ofmagnitude more than VTDIRECT95 would need.


0.2 0.4 0.6 0.8 1−40

−30

−20

−10

0

10

20

fΩ

Perc

enta

ge r

eduction in C

D

Unforced case

Optimization technique

Frequency sweep

Figure 4: Percentage variation of the reduction in the mean drag coefficientwith the forcing frequency fΩ for a rotationally oscillating cylinder at Re = 150.The rotation amplitude is kept constant at Ω = 1.5.

4 Results and Discussion

To optimize the drag reduction using oscillatory rotation of the cylinder as acontrol mechanism and understand the wake dynamics associated with optimalreduction, we vary the amplitude Ω and frequency fΩ of rotation as controlparameters. The upper and lower bounds of the amplitude of the rotationaloscillations Ω and forcing frequency fΩ are presented in Table 1.

Table 1: Lower and upper bounds of the frequency and amplitude of therotational oscillations.

Parameter Lower bound Upper bound

Ω 0.1 3.0fΩ 0.1 1.0

To perform the optimization, we specify the maximum number of iterationsand function evaluations, the minimal relative change in the objective functionand minimum box diameter. These limits serve as the stopping conditions forVTDIRECT95. In the current study, we specify the limit on the number offunction evaluations used by VTDIRECT95 as 41. By specifying the abovestopping conditions, the optimization algorithm yielded the four best resultsreported in Table 2.

Table 3 shows a summary of the percentage reduction of the mean valuesof the drag coefficient with the forcing amplitude Ω and frequency fΩ. The


Table 2: Summary of best points.

Case Ω fΩ CD

1 2.08 0.78 0.9752 2.18 0.78 0.97593 2.30 0.78 0.97794 2.08 0.75 0.9796

percentage reduction is calculated relative to its value on a stationary cylin-der. The first observation to be made here is that below a certain oscillationfrequency (fΩ < 0.41), the drag coefficient is not reduced for any rotationaloscillation amplitude, as shown in Table 3 by the vertical double dashed line.As a matter of fact, increasing the amplitude of the oscillations results in asignificant increase in the value of the drag coefficient in this region. As theforcing frequency is increased to values above 0.41, the drag coefficient becomessmaller than that of the stationary cylinder. The results also show that thereis a threshold of the rotational oscillation amplitude below which the meandrag coefficient does not decrease for any rotational oscillation frequency, asshown in Table 3 by the single horizontal dashed line. These observationsindicate that there is a minimum threshold for the rotational oscillation fre-quency/amplitude to obtain drag reduction. The highest reduction of about21% is obtained over the forcing frequency range between 0.75 and 0.85, i.e.,4–5 times that of the Strouhal frequency and for oscillation amplitudes in therange 2.1–2.3.

Experimental ([30, 29]) and numerical studies ([26, 8]) show that changesin the value of drag coefficient as a result of rotational oscillations are associ-ated with changes in the structure of the wake of the cylinder. Depending onthe oscillation amplitude Ω and forcing frequency fΩ, the wake of the cylin-der exhibits different flow patterns. For some specific values of these controlparameters, the vortices are shed with the external excitation frequency andthe phenomenon is then referred to as “lock-in” or “synchronization” wherebythe frequency of vortex shedding becomes identical to the cylinder’s forcedoscillation frequency.

To determine the wake effects of forcing the cylinder with rotary oscilla-tions, we use flow over a stationary cylinder at Re = 150 as the reference case.Three snapshots of contours of the instantaneous vorticity in the wake of thestationary cylinder are shown in Figure 5. The plots show an alternating pat-tern for the vortex shedding from the upper and lower halves of the cylinder.Snapshots of vorticity contours for the optimal case of drag reduction whereΩ = 2.08 and fΩ = 0.78 obtained from VTDIRECT95 are shown in Figure 6.


Table 3: Summary of percentage reductions in the mean drag coefficient asobtained from VTDIRECT95, the rotational amplitude Ω versus the forcingfrequency fΩ for a rotationally oscillating cylinder at Re = 150, thresholdpoints in italics, best points in boldface.

2.50 -41.0 17.2 17.8 18.9 19.7 20.1 19.62.45 13.22.37 5.342.30 20.3 21.02.20 14.4 20.0 20.5 20.8 20.32.08 20.7 21.01.97 7.651.87 3.28 11.5 15.8 19.6 19.71.76 9.061.55 -34.2 10.8 17.2 20.1 19.7 17.8 14.51.20 13.9 18.3 18.6 13.20.90 16.00.58 -1.38 6.71 3.72

0.26 -0.98

Ω/fΩ 0.25 0.41 0.45 0.48 0.55 0.65 0.70 0.71 0.75 0.78 0.85 0.95

The contours show that the oscillatory cylinder releases single vortices of op-posite signs during each half cycle. The time history of the lift coefficient,presented in Figure 7, shows periodic variations with a frequency that is equalto the forcing frequency. This is confirmed by the power spectrum, presentedin the same figure, which shows a high peak at the rotational oscillation forcingfrequency and a smaller peak at the shedding frequency of the stationary cylin-der. The time history of the drag coefficient shows variation at a frequencytwice that of the forcing frequency.

For the sake of comparison, snapshots of the vorticity contours for thecase where the mean drag coefficient is significantly increased (by 42%) arepresented in Figure 8. In this case, the wake structure is also synchronized withthe forcing frequency, as evident from the time series of the lift coefficient andits power spectrum, which are presented in Figure 9. The lift coefficient alsoexhibits a third harmonic with a significant amplitude. The time series of thedrag coefficient shows variations at twice the frequency of the lift coefficient.What is different from the optimal case is the wake dimension. In this case,the wake exhibits large-scale vortices behind the cylinder that are shed in analternating manner. Furthermore, the shed vortices are further away from thecenter line, resulting in a larger wake in comparison to the wake in the optimalcase, which explains the significant increase in the drag coefficient. The aboveresults show that rotational oscillations at a large enough amplitude force the


X

Y

1 0 1 2 3 42

1

0

1

2

(a)

X

Y

1 0 1 2 3 42

1

0

1

2

(b)

X

Y

1 0 1 2 3 42

1

0

1

2

(c)

Figure 5: Three snapshots of the vorticity contours for the flow over a station-ary cylinder.

X

Y

1 0 1 2 3 42

1

0

1

2

(a)

X

Y

1 0 1 2 3 42

1

0

1

2

(b)

X

Y1 0 1 2 3 4

2

1

0

1

2

(c)

Figure 6: Three snapshots of the vorticity contours for the case where thehighest percentage reduction in the mean drag coefficient(≈ 21%) at Ω = 2.08and fΩ = 0.78 is attained.

shedding to take place at the imposed oscillation frequency no matter whatit is. However, relatively slow rotational frequencies, near or less than theshedding frequency of the stationary cylinder, result in larger wakes than thewake of the stationary cylinder. On the other hand, high frequency forcingresults in a smaller wake than that of the stationary cylinder and, thus, in asmaller drag coefficient.

The nature of the flow passing over a cylinder strongly affects the inducedfluid forces. When a fluid separates from the rear surface of the cylinder,it forms a separated region between the cylinder and the fluid stream. Thedifference between the high pressure in the vicinity of the stagnation pointand low pressure on the opposite side in the wake produces a net force onthe cylinder in the direction of flow. At low Reynolds numbers, the majorcontribution to the drag force is from the viscous component. At high Reynoldsnumbers (i.e., Re > 5000), pressure drag is the major contributor to the totaldrag. In the intermediate range Reynolds number, both effects are significant.Figure 10(a) shows time histories of the viscous drag, pressure drag, and total


180 185 190 195 200

−0.4

−0.2

0

0.2

0.4

0.6

Nondimensional time

CL

(a)

180 185 190 195 200

0.9

0.95

1

1.05

Nondimensional time

CD

(b)

0 0.5 1 1.5 210

−8

10−6

10−4

10−2

100

Nondimensional frequency

Sp

ectr

al d

en

sity

fΩ

St

(c)

Figure 7: Time histories of the (a) lift coefficient, (b) drag coefficient, and(c) power spectrum of the of the lift coefficient for the optimal case whereΩ = 2.08 and fΩ = 0.78.

X

Y

1 0 1 2 3 42

1

0

1

2

(a)

X

Y

1 0 1 2 3 42

1

0

1

2

(b)

X

Y

1 0 1 2 3 42

1

0

1

2

(c)

Figure 8: Three snapshots of the vorticity contours for the case where themean drag coefficient is increased by ≈ 42% at Ω = 1.55 and fΩ = 0.25.

drag for the case of the flow over a stationary cylinder at Re = 150. Atthis Reynolds number, the viscous drag and pressure drag contribute 22% and78%, respectively, to the total drag. The time histories of the drag and itscomponents for the optimal case are shown in Figure 10(b). The plots showthat the viscous drag contributes 22% and the pressure drag contributes 78% tothe total drag. However, for the nonoptimal cases, the viscous drag contributesonly 12% to the total drag and the pressure drag contribution increases to88%. This is a result of the fact that the wake is significantly larger (Figure 8)than the wakes of the stationary cylinder (Figure 5) and optimal configuration


180 185 190 195 200

−0.4

−0.2

0

0.2

0.4

0.6

Nondimensional time

CL

(a)

180 185 190 195 2001.3

1.4

1.5

1.6

1.7

1.8

Nondimensional time

CD

(b)

0 0.2 0.4 0.6 0.8 110

−8

10−6

10−4

10−2

100

Nondimensional frequency

Sp

ectr

al d

en

sity

fΩ

3 fΩ

(c)

Figure 9: Time histories of the (a) lift coefficient, (b) drag coefficient, and(c) power spectrum of the of the lift coefficient for the nonoptimal case whereΩ = 1.55 and fΩ = 0.28.

(Figure 6). In the nonoptimal case, the pressure drop from shoulder to tail isso sharp that the boundary layer falls near the shoulder (maximum thicknessof the cylinder), yielding a broad eddying wake. Consequently, a large pressuregradient exists between the front and rear parts of the cylinder, resulting in alarger pressure drag. Figure 11(a) shows the effect of the rotational oscillationson the pressure drag. In the nonoptimal case, the pressure drag is increased by52% when compared to the pressure drag of the stationary case. On the otherhad, this drag is reduced by 20% in the optimal configuration. This standsin contrast with the viscous drag that is decreased by 24% and 28% in theoptimal and nonoptimal cases, respectively, of rotational oscillations as shownin Figure 11(b). As such, the rotational oscillations of the cylinder reduce theviscous drag irrespective of the frequency of oscillation. On the other hand,the large difference is in the pressure drag that is significantly affected by therate of rotation.

5 Conclusions

For harmonic rotational oscillation of a circular cylinder global minimizationof the drag coefficient in a flow with Reynolds number 150 has been done. Op-timal drag reduction happens over a frequency range that is five times largerthan the stationary Strouhal frequency and with oscillation velocity ampli-


180 185 190 195 2000

0.5

1

1.5

2

2.5

Nondimensional Time

CD

CD

−Pressure

CD

−Viscous

CD

−Total

(a)

180 185 190 195 2000

0.5

1

1.5

2

2.5

Nondimensional Time

CD

CD

−Pressure

CD

−Viscous

CD

−Total

(b)

180 185 190 195 2000

0.5

1

1.5

2

2.5

Nondimensional Time

CD

C

D−Pressure

CD

−Viscous

CD

−Total

(c)

Figure 10: Time histories of drag coefficient for (a) stationary case, (b) optimalcase, and (c) nonoptimal case at Re = 150.

180 185 190 195 200

0.8

1

1.2

1.4

1.6

1.8

2

Nondimensional time

CD

−P

ressu

re

C

D−Pressure for Stationary

CD

−Pressure for Non−optimal

CD

−Pressure for Optimal

(a)

180 185 190 195 2000.1

0.15

0.2

0.25

0.3

0.35

Nondimensional time

CD

−V

isco

us

C

D−Viscous for Stationary

CD

−Viscous for Non−optimal

CD

−Viscous for Optimal

(b)

180 185 190 195 2000

0.5

1

1.5

2

2.5

Nondimensional time

CD

C

D−Total for Stationary

CD

−Total for Non−optimal

CD

−Total for Optimal

(c)

Figure 11: Time histories of (a) pressure drag, (b) viscous drag, and (c) totaldrag at Re = 150.

tudes that are near 4.5(2U∞)/(D). The maximum drag reduction is about21%. There are thresholds of the rotational oscillation amplitude and fre-quency below which the mean drag does not decrease for any forcing frequencyor amplitude. Arguably the wake characteristics in terms of relatively smalland large vortices are responsible for low and high pressure gradients, thusmodifying the drag force. These modifications in the wake are associated withthe injection of external vortices generated by the rotational oscillations of thecylinder. Irrespective of the oscillation frequency, the viscous drag is reducedwhen performing rotational oscillation.

Acknowledgments. This work was supported by AFRL Grant FA8650009-2-3938. A. Mehmood gratefully acknowledges the University of Engineering &Technology (UET) Peshawar, Pakistan for financial support during his gradu-ate studies. Numerical simulations were performed on the university AdvancedResearch Computing machines System X and HokieOne. The computer timeallocation grant and support provided by university staff is also gratefully ac-knowledged.


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Received: October 8, 2018; Published: November 18, 2018

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