Drag Optimization For A Circular Cylinder At HighReynolds Number By Rotary Oscillation Using

Genetic Algorithms

Tapan K. Sengupta, Kalyanmoy Deb

, Srikanth B. Talla

,

Professor, Department of Aerospace Engineering, Indian Institute of Technology Kanpur,Kanpur 208016, India, tksen@iitk.ac.in

Professor, Department of Mechanical Engineering, Indian Institute of TechnologyKanpur, Kanpur 208016, India, deb@iitk.ac.in

Research Associate, Department of Aerospace Engineering, Indian Institute ofTechnology Kanpur, Kanpur 208016, India, tsbabu@iitk.ac.in

KanGAL Report Number 2004018

Abstract

We propose here a new approach to optimally control incompressible viscous flow pasta circular cylinder for drag minimization by rotary oscillation. The flow at is simulated by solving 2D Navier-Stokes equations in stream function-vorticity formula-tion. High accuracy compact scheme for space discretization and four stage Runge-Kuttascheme for time integration makes such simulation possible. While numerical solution forthis flow field has been reported using a fast viscous-vortex method, to our knowledge,this has not been done at such a high Reynolds number by computing the Navier-Stokesequation before. The importance of scale resolution, aliasing problem and preservation ofphysical dispersion relation for such vortical flows of the used high accuracy schemes [1]is highlighted.

For the dynamic problem, a novel genetic algorithm based optimization technique hasbeen adopted, where solutions of Navier-Stokes equations are obtained using small time-horizons at every step of the optimization process, called a GA generation. Then the objec-tive functions is evaluated that is followed by GA determined improvement of the decisionvariables. This procedure of time advancement can also be adopted to control such flows ex-perimentally, as one obtains time-accurate solution of the Navier-Stokes equation subject todiscrete changes of decision variables. The objective function – the time-averaged drag – isoptimized using a real-coded genetic algorithm [2] for the two decision variables, the max-imum rotation rate and the forcing frequency of the rotary oscillation. Various approachesto optimal decision variables have been explored for the purpose of drag reduction and thecollection of results are self-consistent and furthermore match well with the experimentalvalues reported in [3].

Key words: Bluff body flows; Flow control; Drag optimization; High accuracy schemes;

Preprint submitted to Elsevier Science 14th December 2004

Compact schemes; DNS; Optimal dynamic control strategy; Genetic algorithms

1 Introduction

The problem of bluff-body flow control is of significant interest to research com-munity due to its importance in many practical application areas. It is also of con-siderable theoretical interest to flow control and optimization. Flow past a circu-lar cylinder is a canonical problem often used to understand the bluff-body flows.Modifications of the wake of cylinder by means of mechanical oscillatory excita-tions that alters vortex shedding pattern have been studied and reported in literature.However, study of flow over oscillating circular cylinder has mainly focused on in-line or transverse linear oscillations as reviewed in by [4,5,6] and in [7] A largenumber of studies also exist for flow past steadily rotating circular cylinders-as re-ported in [8,9,10,11,12] and other references contained therein. For high rotationrates and Reynolds numbers, vortex shedding was shown to have been suppressedin [13] While steady state rotation can effectively reduce drag and unsteadiness forflow past a circular cylinder, it does not work for all geometries. It is in this context,rotary oscillation becomes interesting- as it can produce reduced vortex sheddingfor bodies with non-circular cross section. Flow control by rotary oscillation hasbeen the theme of far fewer studies. For this flow field, there are three major pa-rameters. The first parameter is the Reynolds number, defined as ,where is the translational speed of the cylinder of diameter and is the kine-matic viscosity. The other two parameters are the maximum rotation rate ( !#" ) andthe forcing frequency ( $&% ) or the Strouhal number of the rotary oscillation definedby, !'(!)"+*-,/.&021435$6%87-9: (1)

If ;)<>=@? and A are the dimensional physical peak rotation rate and the forcing fre-quency respectively, then !B"CD;)<>=@?41E and $F%GHAIJK . In [14], flow vi-sualization results for LMON PN LMM have been reported to demonstrate thequalitative alteration of shedding pattern of circular cylinder by rotary oscillation.It was noted that for ( QEM and RR:TS43VU $F%OU 1XW43 , the vortex sheddingwas completely eliminated. In [15], the forces acting on a cylinder, in the rangeQEMYNZ[NVR\XM and LMESXMYN][N^\+RMM , were measured for M+:_1YN]!`"aNVR:bMand M+:bME1S43cNd$6%eNfM+:gRhS43 that also displayed vortex lock-on state. Similar resultswere reported in [16] for `fLXMM and SXMXM . Subsequent experimental results werereported in [3] and [17]. In [3], the important finding of drag reduction by a factorof six for the flow at BiRhSjkMMM was reported. Alteration of primary Karman vor-tex shedding by rotational oscillation of the cylinder in the Reynolds number rangeof 250 and 1200 was reported in [17], when the peripheral speed due to rotationaloscillation was between 0.5 and LEl of the free stream speed. For such forced ex-citation of the flow, apart from the modification of primary shedding, the separatedshear layer displayed unstable mode that is dictated by the absolute instability of

2

the near-wake.

In comparison to experimental investigations, very few researches have reportednumerical solutions, specifically at high Reynolds numbers. Finite difference sim-ulation of Navier-Stokes equation has been undertaken in [18] by a fractional stepmethod for m1XMM , RMMXM and LMMXM for the other two parameters varying in therange of M+:gRnNf!)"oN(L+:TM and M+:TS43[Nf$6%pNqQX3 . In [19], similar numerical methodswere used for eHRXRM with very low peak rotation rates and Strouhal numbers.[20] solved two-dimensional Navier-Stokes equation in stream function-vorticityformulation for rHSXMM and RMMM by spectral- finite difference method. An at-tribute of the formulation is the use of a grid that becomes coarse with the growingshear layer. The major observations are related to the presence of co-rotating vor-tex pair and a time variation of drag coefficient that switches frequency abruptlyat a discrete time. The authors also report a strong dependence of the flow fieldon Reynolds numbers- as opposed to that reported in [18]. There are other lowReynolds number flow simulations by solving Navier-Stokes equation, as reportedin [21,22].

In [23], the two-dimensional flow for sVRSXMMM have only been studied usinghigh resolution viscous vortex method, to verify the experimental results of [3].The authors noted multi-pole vorticity structures during particular cases of rotaryoscillation revealing bursting phenomenon in the boundary layer that delays sepa-ration and very large drag reduction- a phenomenon not present at low Reynoldsnumbers studied in other numerical simulations. In the present work, we report forthe first time the solution of Navier-Stokes equation at tuRhSvMMM with the helpof very high accuracy compact schemes developed in [12,24,25]. Details of thesehigh accuracy methods can be found in [1].

There have been some studies that investigated computational methods for dragminimization by rotary oscillation for flow past circular cylinder using two-dimensionalNavier-Stokes equation as the flow model. In [26], this problem is studied by per-forming gradient-based classical optimization for 1XMMwNxCNdRjkMXMM and reported30 to \MEl drag reduction. The numerical simulation was performed using finiteelement discretization and the evaluation of cost function gradient by adjoint equa-tion approach. In [27], rotary control of cylinder wake at tDWS and RhSvM usingoptimal control approach using adjoint equations over a time interval is reported.This time interval- called the time-horizon- is used to specify an optimum for theunsteady problem. One important aspect of the study is the use of velocity-vorticityformulation. The authors noted the advantage of this formulation as the usage ofthe more localized and compact variable, the vorticity, as opposed to velocity andpressure. In [28], a real-coded GA was used to study drag optimization for flow pastcircular cylinder by using two types of actuators- the belt type that provides mo-mentum transfer at the cylinder surface by providing slip velocity and another onethat depends on mass transfer through apertures on the cylinder surface by blowingor sucking fluids. In [28], two-dimensional flow was simulated at ySvMM by

3

using a Navier-Stokes solver on a staggered grid, using second-order central dif-ference method in generalized coordinates, following the methodology of [29]. ASXMEl drag reduction was noted for this low Reynolds number flow.

Most well-known classical optimization techniques suffer from the difficulties as-sociated with the requirement of local gradient information and the problem ofusing deterministic functions alone. Genetic algorithm (GA), as a stochastic opti-mization technique, circumvents both these problems as the cost functions can bedirectly obtained in terms of the control parameter(s), without the need to obtaingradient information. Additionally, a GA usually does not get stuck at local op-timum and instead converge to the global optimum. It is to be noted that in [3]local basins were detected experimentally. Another attractive feature of GA is thesimultaneous parallel estimation of the cost functions for different control param-eter combinations. Such parallelism can be used to speed up the optimal searchtremendously, as opposed to gradient-based search methods.

The use of GA to actively control unsteady flows at high would be prohibitive ifconventional method is used to search for the optimum, due to high computationalcosts. Capability of evolutionary algorithms (EAs) to find solutions for dynamicproblems is discussed in Morrison (2004). A survey of methods, as applied to vari-ous benchmark problems can be found in [30]. A practical problem for greenhousecontrol is given in [31], where the authors discuss about the role of control-horizonsfrom direct online control point of view. The performance of GAs, in particular, indynamic environment requires the methodology to detect changes in the problemenvironment and respond to these changes. Quantification of attributes to improvedetection and response has been provided in [32]. Instead of following the abovemethodology in [32], we propose here a new optimization strategy for unsteadyproblems.

The GA is applied over a running time interval 0zj|z~z&9 , with z6 termed as thetime-horizon, and the control parameters are chosen based on the performance inthe previous time-horizon. At the beginning of the optimization process, we com-pute cases with a number ( ) of random parameter combinations. Rather than find-ing the optimal solution over a single large time interval, we look for the best popu-lation member of the current GA time-horizon. The decision variables are changedbased on some GA procedures used on the solution over the time-horizon to comeup for the next set of decision variables for the subsequent time-horizon. This GAprocedure on successive time-horizons will eventually converge to optimal deci-sion variables after a finite number of time- horizons. One can perform multipleGA operations involving iteration within each time-horizon.

The paper is formatted in the following manner. In the next section the governingfluid dynamic equations are provided along with the numerical methods to solvethem. In section 3, the elements of the GA used for this study is provided. Resultsand discussion is given in section 4. The paper is concluded with a summary and

4

recommendation for future work in section 5.

2 Governing Equations and Numerical Method

Two-dimensional Navier-Stokes equations, in stream function-vorticity formula-tion, are given in non-dimensional form as, j (2) 7 [: G R e j (3)

where is the out-of-plane component of vorticity and the velocity is related tothe stream function by

, where 02M+jM+j 9 . 0 9 formulation

is preferred here due to its inherent accuracy and computational efficiency in sat-isfying mass conservation exactly every where. The flow is computed using thetransformed orthogonal grid 0#[9 , where x09+*021436E9j q09+*-,/.601436E9jso that MNOjN R . The grid is stretched smoothly in the radial direction by thetransformation 09¡x¢> £ ¤ £ ¢K E¢1 0 £ ¥R9§¦2j (4)

where ¢ is the radius of the cylinder and£ h¢ is the spacing of the first radial line; ¢

is the increment of successive radial grid line spacing. This type of grid is preferredas it removes alteration of convection and lowest order numerical dissipation [1].

The pressure field is obtained numerically by solving the governing Poisson equa-tion for the total pressure (ª© ), given in the orthogonal coordinate system by, «&¬ ¬ " ¨+© B­ «&¬ "¬ ¨© ­ 0 ¬ k® 9s 0 ¬ "@¯9°j (5)

where¬ " and

¬ are the scale factors of the transformation given by,¬ "±²0 ³ ´ ³ 9 "2µ j and¬ ²0 ¶ Y ¶ 9 "2µ :

No-slip boundary condition on the cylinder wall is satisfied:« ­¸·¹º§» ¬ ¢!`j (6)

where ! is as defined in (1). Additional condition arising out of the no-slip condi-

5

tion is given by, (8X.*-¼½X.E¼: (7)

This condition is used to solve the stream function equation (2), while both (6) and(7) are used to evaluate the wall vorticity ( · ) that provides the boundary conditionfor the vorticity transport equation (VTE) given by (3).

At the outer boundary, uniform flow conditions are applied at the inflow (Dirichletcondition on

) and convective boundary conditions on radial velocity at the out-

flow, as shown in Figure 1. The actual radiative condition applied at the outflow isgiven in [33,34], ¯ª¾ 7 Y¯À¿07-9 ¯ª¾ fM+j (8)

where ¯ª¾ is the radial component of velocity and ¯I¿07-9 is the convection velocityat the outflow at time 7 , which is obtained from the radial component of velocity atthe previous time step i.e. ¯Á¿07-9¸O¯ª¾07¡ £ 7-9 . The initial condition is given by animpulsive start of the cylinder in a fluid at rest. The vorticity on the outer boundaryis obtained by using the kinematic definition of vorticity, as given in (2).

The equations (2) and (5) are solved using Bi-CGSTAB variant of the conjugategradient method given in [35]. The Bi-CGSTAB method is made to converge fasterby using ILUT pre-conditioners as described in [36]. To solve the PPE, the requiredNeumann boundary conditions on the physical surface and in the far-field are ob-tained from the normal ( ) momentum equation given by,¬ "¬ ¨ i ¬ "@¯ª R ¬ " ® 7 : (9)

This boundary condition is exact and conforms to the governing PPE, as comparedto the Poisson equation for stream function. Thus the PPE can be solved in anysmall part of the computational domain with boundary conditions obtained from(9), thereby providing an efficient and faster means of calculating loads and mo-ments acting on the cylinder.

The VTE is solved by discretizing the diffusion term by second order central dif-ference scheme and the time derivative by four stage Runge-Kutta scheme. Theconvection terms are evaluated using spectrally accurate compact schemes. Theuse of very high accuracy compact scheme allows solving Navier- Stokes equationfor rotary oscillation case, which were otherwise not possible for high Reynoldsnumber flows by low accuracy methods. The particular compact scheme used herewas introduced in [12], a brief description and comparison of it with the methodused in [28] follows.

In solving the VTE, one distinguishes between two types of discretization methods.In the first class, the functions and their derivatives are periodic in one direction thatdoes not require any boundary conditions- as the azimuthal direction in the presentcase. In the other class, the unknowns are non-periodic and one requires specific

6

boundary closure schemes. In choosing compact schemes, this distinction is keptin mind while evaluating first derivatives (as indicated by primes below). A generalrecursion relation, to evaluate first derivatives, of the following form is used incompact schemes,Â@ÃkÄ "§¯ªÅÃÄ " §à ¯ªÅà §Ã-Æ "§¯ªÅÃ|Æ " R¬ ÇÈkÉ Ä §Ê Ã|Æ È ¯ Ã|Æ È j (10)

where¬

is the constant spacing in the transformed plane. In the periodic azimuthaldirection, this implicit equation is adequate to evaluate the convective derivativeterms. The error in numerical evaluation of this can be minimized (as reportedin [37]) if one chooses the following coefficients in the above equation:

ÂkÃ-Ë "aM+:bLEWXÌLÍXÌXQEÌ+Rh1 ; §à R ; Ê Ã|Ë "GÎ#M+:TWXÍEWWXÍX\Í ; Ê Ã-Ë ÎnMÏ:TMXQESXÍM+Rh1 and Ê Ã ÐM .One solves the associated periodic tridiagonal system to evaluate the required firstderivatives in this direction.

In the non-periodic direction, one needs stable boundary closure schemes and theone used here for the first and second nodes are (as introduced in [12]),¯ Å " BLX¯6"6´QX¯ ¯ÀÑ1 ¬ j (11)¯ Å 1vÒ'RL ¬ ¯F"> R\vÒG1\ ¬ ¯ QÒÓxR¬ ¯Àѱ R\XÒÓxR\ ¬ ¯ªÔK 14ÒL ¬ ¯°Õ8j (12)

with Ò as a parameter chosen to ensure accuracy and numerical stability of themethod for solving the VTE. Similar boundary closures are applied at the otherend of the non-periodic direction. This method of discretizing the derivative inthe non-periodic direction was termed the OUCS3 scheme in [12] and is usedhere. It requires the following coefficients: Ê Ã|Ë ÎnM+:bMXQJSvÍM+Rh1w ÖÑ×¢×¢ ; Ê Ã-Ë "~ÎnMÏ:_WXÍWWXÍ\ÍI ÖÑ×¢ ; Ê Ã Ø "×" Ö"ÙÕ×¢ ; §Ã|Ë " fM+:bLEWXÌLXÍÌXQEÌ+R1IÎ ÖÚ ¢ and

§à ØR and ÛpØ`1:TM .For the boundary closure, we have used ÒHBMÏ:TME1XS for Üs 1 and ÒmM+:bMÌ forÜw(m´R . To control aliasing and retain numerical stability an explicit fourth dis-sipation term Ò6" ¬ ÑÝÞàßÝ ? Þ is added at every point with ÒF"¡(MÏ:TML . The overall speed-upof the Navier-Stokes solver based on compact scheme is an order of magnitudehigher over the explicit higher order method based solvers, since this allows takingfar fewer points in all directions while the computed solutions have higher accuracy.

We compare the numerical property of this compact scheme vis- áÊ -vis the schemeused in [28] for discretizing the nonlinear convection terms in the spectral plane.Consider a function of single space dimension and time given in terms of its Fourier-Laplace transform by, ¯>0 j|7-9¡Pâ±âxã0äÁjkåX9-8æ/ç È ? Äè ©gé ä+JåÏ: (13)

7

One can evaluate its spatial derivative by spectral method from ¯ ââê äÀã02äÁjkå9@8æ/ç È ? Äè ©gé ä+Jå: (14)

This estimate of derivative by spectral method is the most accurate among all dis-crete numerical methods. For any general discrete method, the same derivative canbe expressed as, ¯ â±âê äëìw02äÁjkåX9- æ/ç È ? Äè ©gé ÏäÏå: (15)

In [12] and [1], the spectral accuracy of different numerical methods have beencompared by plotting äJë2ìä as a function of ä ¬ , whose limit is given by the NyquistCriterion, ä<>=§? ¬ P3 . For an ideal scheme, the above ratio should be equal to oneand any departure from it represents filtering or attenuation of the scheme. Thecompact scheme, given by (10)-(12), can be written notationally as

¤ ;)¦í¯ Å ¤ïî ¦/¯or ¯ Å ¤ïð ¦í¯ . It has been shown in [1] that the spectral accuracy at the Ü ©òñ node isobtained from, ê äEë2ì0 à 9± óÇ ô É " ð à ô 8æ/ç ô

Ävà é È ñ : (16)

In Figure 2, the spectral property of the OUCS3 scheme, is compared with theCD2 scheme used in [28] for an interior node. It is evident that the OUCS3 schemecan resolve more than 5.5 times the scales resolved by CD2 scheme. In additionto scale resolution, one must also ensure that the physical events depicted by thedifferential equation transfer energy at the correct speed. As the energy propagationspeed is given by the group velocity, one can ensure satisfaction of this property bycomparing the numerical dispersion relation with the physical dispersion relation.In [1], this has been done by comparing the numerically obtained group velocity for1D convection equation- that is a model of convection dominated phenomenon. Asthis equation is non-dispersive, the theoretical group velocity is equal to the phasespeed and hence any numerical scheme can be characterized by a comparison oftheoretical and numerical group velocity.

For the 1D convection equation the amplification factor of the numerical scheme([25]) is given by, õ ö õ ö÷ Ä æbøJj (17)

so that the unknown at any arbitrary time can be written as ,¯>0 <)j|7×ù9¡ â ¯À¢h0ä+9hö õ ö ùX æ/ç È ?ú Ä ù ø é äÁj (18)

where ¯°¢0ä9 is the Fourier-Laplace of the initial condition. Thus, the numericalphase speed is given by û ó øÈ|ü © and the ratio of the numerical and the physicalgroup velocities for the 1D convection equation is given in,ýªþ óû Ä "¿¬ ÿä : (19)

8

In Figure 3, the above expression is used to calculate the ratio, ¿ , for the methodsthat employ OUCS3 and CD2 spatial discretization schemes and four stage Runge-Kutta time integration scheme. It is clearly seen that the present OUCS3 scheme ismore accurate over larger space and time scales compared to the CD2 scheme usedin [28]. It is for these reasons, the method based on CD2 spatial scheme is restrictedto low Reynolds number application, while the present scheme is used successfullyat BiRhSjkMMM and compared with the experimental results of [3].

3 Optimization Problem Formulation and Genetic Algorithm

The objective functionð

, for a time interval¤ zj|zttz6±¦ is the time-averaged drag

co-efficient (ð

) of the cylinder, whereð 0Ù!)"j$6%h9± ½` 5í,/. Rh1+ : (20)

Hence, we obtain ð RzI â Æ

ð E7: (21)

where z6 is the time-horizon of one GA generation. The equations (2), (3) and (5),along with boundary conditions (6) - (9), define the system to be controlled withinput as 0Ù!B"jk$6%h9 and the output

ð, that is minimized. An efficient implementation

of a GA is suggested next. A population of input parameter vectors 02!#"jk$6%h9 isselected at each generation using three GA operators (details as given in [2]).

Selection is an operation through which members are selected for reproduction ac-cording to their fitness (objective function value). In this study, the tournament se-lection, with the participation size of two, is used. Crossover is an operation throughwhich new members are created from the members selected by the selection oper-ator. Simulated Binary Crossover (SBX) operator [2] has been used in this workand mutation is an operation which leads to random changes in the newly createdmembers. The polynomial mutation operator [2] is used in this study.

The population for the first generation is created randomly in the allowed deci-sion variable space. The cost function

ðof the members is calculated for a time-

horizon z6 (user-defined) measured from an initial time z6¢ . The GA with its threeoperators mentioned above is applied to this initial population for a

õnumber of

iterations and the best solution ¤ ¦ ]0Ù!)"j$6%h9 which minimizes the cost func-tion

ðover the time interval

¤ zj|z'qz6±¦ is recorded. During this time interval,the optimum control approach is assumed to be the corresponding control strategy ¤ ¦ . This is the end of one GA generation. To find the best control strategy forthe next time-horizon

¤ zY´z6j-zY'14zI¡¦ , the population after the previous gener-ation is used as the initial GA population. Another GA generation is started with

9

this population and the best control strategy ¤ 0 9 ¦ is found and fixed forthat time-horizon. This procedure is continued till the best control strategy of con-secutive generations are similar to each other or a maximum number of generations( z"! # ) is reached, thereby signifying the achievement of a steady control strategy.A pseudo-code of this procedure is presented below.

beginT = T_0 // Control time scaleInitialize population P(0) // randomly createdrepeat

t = 0 // GA iteration counterrepeat

Evaluate P(t) for calculating C_DSelection, Crossover and Mutation to obtain P(t+1)t = t+1

until (t <= G)x(T)* = Solution corresponding to best P(t)P(0) = P(t)T = T + dT

until (T <= T_max)end

4 Results and Discussion

All computations are performed on an orthogonal grid of size ( RhSXM QJSXM ), with150 points in the azimuthal direction and 450 points in the radial direction. Thepoints in the azimuthal direction are spaced at equal angular intervals. In the radialdirection, points are distributed as per the relation given in (4), with the first pointM+:bMM+R away from the surface of the cylinder and the value of +¢ fixed in such a waythat the outer boundary is located at 40 diameter from the center of the cylinder.The time integration of VTE is performed by the four-stage Runge-Kutta scheme.The Poisson equations (2) and (5) have been solved by the Bi-CGSTAB method,as discussed before. The efficiency and accuracy of the CFD techniques have beenwell established and some representative applications can be seen for different flowconfigurations in [1] and [24]. The solutions of (3) and (5) provide the vorticity andtotal pressure respectively in the flow field. The surface pressure on the cylinder isreadily obtained from the total pressure and one can calculate the drag experiencedby the cylinder at any instant performing the following contour integral,ð 1

+ â ¿ » ô 0 ¨%$Á?)'& æ ?($ æ 9-*)×: (22)

10

where ¨ is the surface pressure and & æ ? the viscous stress tensor on the surface ofthe cylinder with $ æ as the unit normal vector in the ê ©òñ direction.

The investigated control problem depends explicitly upon the two parameters of(1), when the Reynolds number is kept fixed at HRSÏjkMMM to match the exper-imental conditions of [3]. As mentioned before, corresponding CFD calculationsare very expensive and hence in the present investigation a population size of 11 isused for most of the investigated cases for locating the optimum parameter condi-tions using GA. In two cases, where GA search was conducted over a large decisionvariable space, we have used a population size of ØL+R . Experimental observa-tions in [3] revealed that there exist local and global minima for drag of this flowconfiguration. These lie in the space: MaNO!`"NdQ and MaNi$6%tNOL+:TS and to beginwith, we use these ranges to define the initial decision variable space. Investigationsare performed with different time-horizons ( zF ). As the time scale of the probleminvestigated is governed by the Strouhal number ( $5% ), it is natural to choose a zFthat is at least three times the cycle-time to avoid dealing with flow transients. Itis for this reason, we have chosen a time horizon given by z&H L for the firstcase studied. The flow is computed up to 7G\ without any control and subse-quently the GA procedures begin. This is also done to reduce the undue transienteffects following the impulsive start of the flow past the cylinder that was set into arotary oscillation impulsively. The initial solution for the population members forany time-horizon is chosen as the best solution obtained from the previous time-horizon. The solution of each member of the population is time averaged over thetime- horizon to obtain

ðthat is compared among all the members to grade the

fitness of the solution.

In Figure 4, variation of drag coefficient with time is plotted for the first 14 gener-ations of the proposed GA, showing the best and the worst members of the chosenpopulation at each time- horizon in Figure 4(a). In Figure 4(b), all the 11 solutionsare shown during the fourteenth generation, showing small differences among themembers at this stage of GA search. Time-averaged

ðis brought down from

about 0.6 (during 7ÓV\ to Ì ) to less than 0.4 (during the fourteenth time- hori-zon). To show an effective comparison, we ran our two-dimensional CFD simu-lation with the optimum value of !`"y 1Ï:_WXS and $6%' M+:bÌ obtained from theexperimental results of [3]. The time-averaged drag coefficient for this simulationturned out as

ð M+:TS1XÌEW . In [3], the experimental drag coefficients was pre-sented with a non- dimensionalization that gives half the values presented here.The difference between the present optimum and that in [3] could be due to the useof two-dimensional governing equation while the experiments have three- dimen-sional effects. Interestingly, despite the difference in the dimension of the model-ing, the optimal control strategy at the end of GA procedure ( !#"rÐ1Ï:bÌEWSXW and$6%`fMÏ:T\ÍXLÍ ) is close to that observed experimentally. However, the GA optimizedvalue of the drag coefficient over the entire duration ( 7CH\ to QEÍ ) is found to beð xM+: QJ1vQEL for zI´(L (as in Figure 4) and is about 25% better than that obtainedfor the two-dimensional simulation of the experimental solution reported in [3]. To

11

perform an effective comparison for these exercises, an uncontrolled (no rotation)case was run at sVRhSXMMXM . A time averaged drag co-efficient

ð+ VR:T1XL wasobtained for the time interval of 7#\ to ÍM . This constitutes a \\El reduction indrag co-efficient value for zFYfL case.

To investigate the effect of time-varying control strategy on the reduction ofð,

,we rerun the CFD model with the GA optimized solution ( !"q 1Ï:bÌEWSW and$6%]M+:b\ÍLÍ ) as obtained at the end of fourteenth generation and keep this con-trol strategy fixed throughout the time duration ( 7¸d\ to QÍ ). In this case, the timeaveraged drag coefficient was found to be M+: QJSXÌEW , about 8.3% worse than that foundwith the time-varying control strategy. This may show the better practical utility ofusing a time-varying control and optimization strategy.

The time variation of !B" and $6% for the best member solution in each generation areshown in Figure 5, for various time-horizon cases that have been investigated. Inthe same figure, optimal ranges of these two parameters are identified by a shadedregion that were experimentally obtained in [3] from their long time-averaged data.There appears to be very good agreement between the experimental and the presentresults with z6YxL , i.e. the GA generated optimum values of !" and $6% are in closeneighborhood of the experimental optimum data-band. Apart from the fact thatdifferent averaging time was used in the experiment, it is also to be noted that theexperiments at high Reynolds number ( iRSÏjkMMM ) may have three-dimensionaleffects that are not predicted by the present two-dimensional computations. Theprogressive change of parameter values towards an optimum cluster is seen to occurin Figure 6, that shows the cluster of the 11 population members chosen by the GAwith selection, crossover and mutation operators.

4.1 Effect of Time-Horizon

The above optimization exercise may depend on the choice of z5 , as the effects offlow-transients cannot be ruled out. The experimental results of [3] were obtainedtaking long-time average of the data that is different from the procedure followedin the present numerical exercise, where the optimum has been sought over singletime-horizons. Present investigation is undertaken specifically to device alternativemethod of obtaining optimum solution in a dynamical environment and once a timeaccurate sequence of parameters are obtained they can be replicated experimentally,as the solution of fluid dynamical equations are time-accurate.

To check the effect of time-horizon on the obtained optimal solution, another sim-ulation is performed with a lower value of zFD 1 . In Figure 7, time history ofthe computed instantaneous drag coefficients are shown for this z5 , showing thebest and worst members in each generation using the same window for the decisionvariables !)" and $6% as before. As expected, one sees significant transient effects

12

in this figure as compared to Figure 4. The frequency content is much higher forthe data in Figure 7, as compared to zF¥OL case. Here, 20 generations have beentraversed for the search, starting from 7^\ . The corresponding variation of thedecision variables, !B" and $F% display totally different trend in Figure 5. The opti-mal values of these variables are different from the experimental ranges shown inthe same figure. It is also noted that the optimal values of !" appears to gravitatetowards the maximum of the search window value. This is clearly visible in Fig-ures 5 and 8. From Figure 8, it is seen that the decision variables starting from arandom distribution in the beginning, forms a nice cluster by tenth generation it-self, that stays in the same neighborhood for the next ten generations. A calculationperformed with the optimal values of decision variables produced an time-averageddrag coefficient of

ð OM+: QE\XQJW . Interestingly, this time-averaged drag coefficientis somewhat worse than that obtained with the z& HL case. Moreover, the con-trol strategies are also quite different for zFOD1 and z6O L cases, as shown inFigure 5. Also, the unsteadiness for z&Y(1 case is more than for the zFY(L case.

Having noted the effects of transients for z&m 1 case, another case is run withz6m]\ and a population size of VRR , so that the transient effects are mini-mized. The simulation is preformed over 13 generations and the drag co-efficientvariation with time is shown in Figure 9(a) for the best and worst solutions among11 participating members. In Figure 9(b), all the member solutions are shown dur-ing the thirteenth generation. It is to be noted that the unsteadiness increases signifi-cantly beyond 7¡xQEM . The corresponding distribution of GA population of decisionvariables are shown in each time-horizons in Figure 10. Starting from the randominitial distribution of decision variables one notices clustering of these from fifthgeneration onwards. These optimal values of !" and $6% are found to be betweenz6u1 and z6u L cases, as shown in Figure 5. This exercise shows that theflow control (to minimize drag) by rotary oscillation is better achieved via an activestrategy with a proper choice of zF . The time-avareged

ð(for 7V\ to QEÍ ) is

0.4023, which is even better than that obtained in the z5YfL case.

4.2 Enlarging the Search Window

In the above cases, the optimum variable values were close to the boundary of thedecision variable space. This indicates that the true optimum may lie outside thechosen search window. Thus, another case was investigated with a larger windowof variable space MfN!B"yNÍ and M(N$6%'N Q with a larger population sizeof ^L+R for z6 ZL . Except for the first few generations, the optimum deci-sion variables converge to the neighborhood of the experimental optimum band, asshown in Figure 5. The optimized drag coefficient variation with time for the bestand worst member of the population, in each generation, is shown in Figure 11.The corresponding distribution of the decision variables of the population in eachgeneration are shown in Figure 12. However, unlike the previous cases here the

13

variables (randomly scattered on the entire search space in the initial generations)seem to gravitate towards two distinct basins from eleventh generation onwards.At sixteenth generation, one can clearly see the presence of two distinct basins ofattractors. The best solutions in two basins have the following

ð+values for the

16th generation: (i) solution ( !`" dL+:íR\E1 , $6%nOMÏ:TLÌÏR ) withð dM+:bLÍMÏR and (ii)

( !)"BDR: QEL+R , $6% M+:bÍEWX\ ) withð M+:TSXÍMvQ . However, one notices that the sec-

ond basin with smaller value of !`" progressively depletes (having a worse objectivevalue), as can be seen from the clustering at 27th generation. At this generation, thebest solution corresponds to !B"CHL+:_WvMÌ and $6%auM+:b\EWXM . It is important to notethat at some intermediate generations (such as in generations 20, 22, 24 and others),the best solution has a !B" value greater than 4.0, thereby indicating the importanceof this larger window study.

In the presence of multiple optimum with almost identical objective values, previ-ous theoretical and experimental studies on GAs have amply shown that the bestGA solution may oscillate between different optima [38,39]. In the absence of anyspecialized niching operator, the population moves to one of the optima by meansof a genetic drift, but with a drifting time which can be enormously large [39].Since we used a GA without any niching operator here, the effect of oscillationof the best solution between the two apparent optimal basins is nicely captured bythe GA runs from 11th to 20th generations. If the GA is run for a long enoughgenerations (some indications can be seen from 21st to 27th generations), eventu-ally the complete GA population will converge to one of the optima, with a largerprobability to the optima having the better objective value.

4.3 Multiple GA Iterations

From the previous cases’ results it is seen that the investigated problem shows dis-tinct dynamical behavior with the optimal solutions changing in each generation.According to Morrison [32], such problems would require sensing the correct so-lution with the help of sentinel. This can be also attempted by iterating the GAprocedure in each generation (i.e. by using

õ.- R (refer to the pseudo-code at theend of Section 3)), as has been attempted elsewhere [31] in looking for an optimalgreenhouse effect. Thus, to study this aspect another case is studied where multipleiterations have been performed for the case with z5PmL and looking for optimalcondition in the large domain constituting the decision variables. The populationsize for the GA search is chosen as 31. In this study, first five generations progressedas before with only one iteration (

õ R ). On the sixth generation six iterations(õ i\ ) are performed, followed by four iterations (

õ OQ ) on the seventh gener-ation and two iterations (

õ 1 ) on the eighth generation. From ninth generationonwards, once again only one iteration (

õ ]R ) is performed. The correspondingtime variation of the drag coefficient is shown in Figure 13a, showing the best andworst cases in each generation by discrete and continuous symbols. In Figure 13b

14

the drag coefficient of first eleven members showing the best behavior is depicted atthe fifteenth generation, up to which this case was investigated. The correspondingdistribution of the decision variables in each generation are shown in Figure 14,with the best solution marked by the filled symbols. As before, we start with trulyrandom distribution of 31 cases at the beginning that shows slow clustering of thecombination for the first five generations when no iterations are performed. At thesixth generation, with six GA iterations performed, one immediately sees cluster-ing of the whole population very near the optimal solution. On seventh and eighthgeneration, when four and two iterations were performed respectively, the cluster-ing further improved around the optimal solution. This is further reinforced in theninth generation. The best solution remains more or less constant from then on-wards, thereby meaning that an optimal control strategy has been found and thesame strategy can be continued for an optimal result.

It is also interesting to note that the time-averagedð

value for 7s 1ÏR till SR(within which multiple GA iterations are performed) is 0.3540, compared to 0.3713obtained earlier with

õ ØR throughout (in the previous subsection) for an identicaltime duration. This indicates that the use of multiple GA iterations was helpful inreducing the drag coefficient value. It is to be noted that Ursem et al. [31] alsoperformed multiple iterations (20) at each generation for arriving at the optimalsolution. In the current problem, the evaluation of a solution for a time step

£ zfR takes about one hour on a Pentium IV, 3 GHz processor. Although the abovesimulations were performed on a 36-node cluster having each node a Pentium IV, 3GHz processor, the sheer computational time needed for CFD simulation prohibitsthe use of a large number of GA iterations per generation. Nevertheless, the limitedsimulations with multiple GA iterations indicate that the overall GA procedure isquite capable of finding the the near-optimal region in a reasonably small numberof generations.

5 Conclusions

In this paper, we employ a genetic algorithm based optimal control strategy forarriving at optimal variation of time varying rotation rate to achieve smallest time-averaged drag co-efficient for a very high Reynolds number scenario. Several time-horizons and search windows are considered and the GA optimized solution isfound to be close to an experimentally observed optimal control strategy. This studyemphasizes two important aspects: (i) the use of reliable and accurate numericalmethod for DNS and (ii) the efficacy of the GA-based optimization strategy capa-ble of arriving at near-optimal solutions for a dynamic problem by a novel tech-nique. The use of high accuracy CFD method is central as evidenced from the factthat no solutions at `iRhSXMMM were produced by solving Navier-Stokes equationbefore this exercise. Interestingly, various strategies lead to more or less similarregions of optimality, that indicates the consistency and reliability of the proposed

15

optimization procedure.

The proposed generation-wise GA procedure is generic and can be applied to otherdynamical optimal control problems as well. An extensive parametric study re-vealing more insights into the effect of multiple GA iterations within each gener-ation remains as an important future research. Nevertheless, these first and proof-of-principle results on a difficult optimal control problem indicates the usefulnessof the proposed approach and should find more applicability in the near future.Another criterion for the objective function in a practical engineering applicationwould be reduction of flow unsteadiness, that can be studied in a multi-objectiveframe work in future.

References

[1] T. K. Sengupta, Fundamentals of Computational Fluid Dynamics, Univ. Press,Hyderabad, India, 2004.

[2] K. Deb, Multi-objective optimization using evolutionary algorithms, Chichester, UK:Wiley, 2001.

[3] P. T. Tokumaru, P. Dimotakis, Rotary oscillation control of a cylinder wake, J.FluidMech. 224 (1991) 77.

[4] E. Berger, R. Willie, Periodic flow phenomenon, Ann. Rev. Fluid Mech. 4 (1972) 313.

[5] P. W. Bearman, Vortex shedding from oscillating bluff bodies, Ann. Rev. Fluid Mech.16 (1984) 195.

[6] O. M. Griffin, M. Hall, Vortex shedding lock-on and flow control in bluff body wakes,Trans. ASME: J. Fluids Engng. 113 (1991) 526.

[7] B. M. S /0 mer, J. Freds 1 e, Hydrodynamics around Cylindrical Structures, (WorldScientific, Singapore), 1997.

[8] H. M. Badr, M. Coutanceau, S. C. R. Dennis, C. Menard, Unsteady flow past a rotatingcircular cylinder at reynolds numbers k Ñ and k Ô , J. Fluid Mech 220 (1990) 459.

[9] C.-C. Chang, R.-L. Chern, Vortex shedding from an impulsively started rotating andtranslating circular cylinder, J. Fluid Mech. 233 (1991) 265.

[10] Y.-M. Chen, Y.-R. Ou, A. Pearlstein, Development of the wake behind a circularcylinder impulsively started into rotatory and rectilinear motion, J. Fluid Mech. 253(1993) 449.

[11] M. Nair, T. Sengupta, U. Chauhan, Flow past rotating cylinders at high reynoldsnumbers using higher order upwind scheme, Computers Fluids 27 (1998) 47.

[12] T. K. Sengupta, G. Ganeriwal, S. De, Analysis of central and upwind compactschemes, J. Comp. Phys. 192 (2003) 677.

16

[13] F. Diaz, J. Gavalda, J. Kawall, J. Keffer, F. Giralt, Vortex shedding from a spinningcylinder, Phys. Fluids 26 (1983) 3454.

[14] S. Taneda, Visual observations of the flow past a circular cylinder performing arotatory oscillation, J. Phys. Soc. Japan 45 (1978) 1038.

[15] A. Okajima, H. Takata, T. Asanuma, Visocus flow around a rotationally oscillatingcylinder, Tech. Rep. Rept. 532, Inst. Space and Aero. Sci. (U. Tokyo) (1981).

[16] J. Wu, J. Mo, A. Vakili, On the wake of a cylinder with rotational oscillations, AIAAPaper 89 (1989) 1024.

[17] J. R. Filler, P. L. Marston, W. Mih, Response of the shear layers separating from acircular cylinder to small-amplitude rotational oscillations, J. Fluid Mech. 231 (1991)481.

[18] X.-Y. Lu, J. Sato, A numerical study of flow past a rotationally oscillating circularcylinder, J. Fluids Struct. 10 (1996) 829.

[19] S.-J. Baek, H. Sung, Numerical simulations of the flow behind a rotary oscillatingcircular cylinder, Phys. Fluids 10 (1998) 869.

[20] S. C. R. Dennis, P. Nguyen, S. Kocabiyik, The flow induced by a rotationallyoscillating and translating circular cylinder, J. Fluid Mech. 407 (2000) 123.

[21] S. Choi, H. Choi, S. Kang, Characteristics of flow over a rotationally oscillatingcylinder at low reynolds number, Phys. Fluids 14 (8) (2002) 2767.

[22] M. Cheng, Y. Chew, S. Luo, Numerical investigation of a rotationally oscillatingcylinder in mean flow, J. Fluids Structures 15 (2001) 981.

[23] D. Shiels, A. Leonard, Investigation of a drag reduction on a circular cylinder in rotaryoscillation, J. Fluid Mech. 431 (2001) 297.

[24] T. K. Sengupta, A. Guntaka, S. De, Incompressible navier-stokes solution by newcompact schemes, J. Sci. Comp. 21 (3) (2004) 269.

[25] T. K. Sengupta, A. Dipankar, A comparative study of time advancement methods forsolving navier-stokes equations, J. Sci. Comp. 21 (2) (2004) 225.

[26] J.-W. He, R. Glowinski, R. Metcalfe, A. Nordlander, J. Periaux, Active control anddrag optimization for flow past s circular cylinder, J. Comp. Phys. 163 (2000) 83.

[27] B. Protas, A. Styczek, Optimal rotary control of the cylinder wake in the laminarregime, Phys. Fluids 14 (7) (2002) 2073.

[28] M. Milano, P. Koumoutsakos, A clustering genetic algorithm for cylinder dragoptmization, J. Comp. Phys. 175 (2002) 79.

[29] R. Mittal, S. Balachander, Effect of three-dimensionality on the lift and drag ofnominally two-dimensional cylinders, Phys. Fluids 7 (8) (1995) 1841.

[30] J. Branke, Evolutionary approaches to dynamic optimization problems: Updatedsurvey, in: Proceedings of the GECCO workshop on Evolutionary Algorithms forDynamic Optimization Problems, 2001, p. 27.

17

[31] R. K. Ursem, B. Filipic, T. Krink, Exploring the performance of an evolutionaryalgorithm for greenhouse control, J. Comput. Information Tech. 10 (3) (2002) 195.

[32] R. W. Morrison, Designing Evolutionary Algorithms for Dynamic Environment,Springer Verlag, Berlin, Heidelberg, 2004.

[33] I. Orlanski, A simple boundary condition for unbounded hyperbolic flows, J. Comp.Phys. 21 (1976) 251.

[34] P. G. Esposito, R. Verzicco, P. Orlandi, Boundary condition influence on the flowaround a circular cylinder, in: IUTAM Symp. Proc. on Bluff-body Wakes, Dynamicsand Instabilities, Springer Verlag, Berlin, Heidelberg, 1993.

[35] H. A. V. der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG forthe solution of non-symmetric linear systems, SIAM J. Sci. Stat. Comput. 12 (1992)631.

[36] Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd Edition, SIAM, USA, 2003.

[37] Z. Haras, S. Ta’asan, Finite difference scheme for long time integration, J. Comp.Phys. 114 (1994) 265.

[38] M. D. Vose, Simple Genetic Algorithm: Foundation and Theory, Ann Arbor, MI: MITPress, 1999.

[39] D. E. Goldberg, Genetic Algorithms for Search, Optimization, and Machine Learning,Reading, MA: Addison-Wesley, 1989.

18

Ωoutflow

inflow

U∞

Figure 1 Schematic of the flow configuration. A (150x450) grid has been used for the presentedresults for Re = 15000. Convective boundary condition (eq. (8)) has been used for the indicatedoutflow.

kh

(keq

/k) re

al

0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CD2

OUCS3

θ∆t

kh

1 2 3

0.5

1

1.5

2

2.5

3

CD2

(a)

θ∆t

kh

1 2 3

0.5

1

1.5

2

2.5

3

OUCS3(b)

Figure2. A comparison of theability of thenumerical schemes used hereand in [28] to represent thenon-linearconvection terms. Hereh is thegrid spacing used and k is thewave number. Theordinate is thenormalizedequivalent wavenumber of discretization.

Figure3. Theregion marked in the (kh-θ∆t) planewhere thenumerical group velocity matches thephysical groupvelocity in solving linear waveequation within 5% tolerance. Shown in (a) themethod used in [28] and (b) thepresent method. In both thecases four-stageRunge-Kutta time integration method is used.

t

CD

45 46 47 48-1

-0.5

0

0.5

1

1.5

(b)

Figure 4. Variation of drag co-efficient with time for Re = 15000. (a) Shown are the best (discrete symbols) and theworst (solid line) cases over the first 14 generations; (b) Drag co-efficient variation during the fourteenth time-horizon is shown for all the eleven members of GA.

t

CD

6 9 12 15 18 21 24 27 30 33 36 39 42 45 48

-0.5

0

0.5

1

1.5

(a)

TH = 3

t

Ω1

10 20 30 40 50 60 70 801

1.5

2

2.5

3

3.5

4

4.5

5TH = 2TH = 3TH = 6TH = 3 LD

t

Sf

10 20 30 40 50 60 70 800

0.5

1

1.5

2

2.5

3 TH = 2TH = 3TH = 6TH = 3 LD

Figure 5. Variation of decision variables of the best member with time. (a) Peak rotation rate Ω1 (b) Forcingfrequency f. Here [T,T+TH] corresponds to one generation of GA, with the first generation starting at T = 6 inall the cases. The top three cases in the legend box correspond to search space Ω1 = [0,4] and Sf = [0,3.5] andpopulation size of N = 11, while the last case corresponds to a larger search space Ω1 = [0,8] and Sf = [0,4]and population size of N = 31. The shaded regions correspond to the experimentally found [3] optimal results.

01

23

40

0.51

1.52

2.53

3.54

GE

N=

01

01

23

40

0.51

1.52

2.53

3.54

GE

N=

02

01

23

40

0.51

1.52

2.53

3.54

GE

N=

03

01

23

40

0.51

1.52

2.53

3.54

GE

N=

04

01

23

40

0.51

1.52

2.53

3.54

GE

N=

05

01

23

40

0.51

1.52

2.53

3.54

GE

N=

08

01

23

40

0.51

1.52

2.53

3.54

GE

N=

07

01

23

40

0.51

1.52

2.53

3.54

GE

N=

09

01

23

40

0.51

1.52

2.53

3.54

GE

N=

10

01

23

40

0.51

1.52

2.53

3.54

GE

N=

06

Fig

ure

6.T

hedi

stri

butio

nof

GA

popu

latio

nof

deci

sion

vari

able

ssh

own

atea

chtim

e-ho

rizo

nw

ithth

ebe

stm

embe

rsh

own

with

afi

lled

circ

lefo

rth

ein

dica

ted

gene

ratio

nsfo

rT

H=

3ca

se.

(con

td.)

01

23

40

0.51

1.52

2.53

3.54

GE

N=

12

01

23

40

0.51

1.52

2.53

3.54

GE

N=

13

01

23

40

0.51

1.52

2.53

3.54

GE

N=

14

01

23

40

0.51

1.52

2.53

3.54

GE

N=

11

Fig

ure

6.T

hedi

stri

butio

nof

GA

popu

latio

nof

deci

sion

vari

able

ssh

own

atea

chtim

e-ho

rizo

nw

ithth

ebe

stm

embe

rsh

own

with

afi

lled

circ

lefo

rth

ein

dica

ted

gene

ratio

nsfo

rT

H=

3ca

se.

t

CD

44 44.5 45 45.5 46-0.5

0

0.5

1

1.5

2

(b)

Figure 7. Variation of drag co-efficient with time for Re = 15000. (a) Shown are the best (discrete symbols) and theworst (solid line) cases over the first 20 generations; (b) Drag co-efficient variation during the twentieth time-horizonis shown for all the eleven members of GA.

t

CD

6 9 12 15 18 21 24 27 30 33 36 39 42 45-0.5

0

0.5

1

1.5

(a)

TH = 2

01

23

40

0.51

1.52

2.53

3.54

GE

N=

01

01

23

40

0.51

1.52

2.53

3.54

GE

N=

02

01

23

40

0.51

1.52

2.53

3.54

GE

N=

03

01

23

40

0.51

1.52

2.53

3.54

GE

N=

04

01

23

40

0.51

1.52

2.53

3.54

GE

N=

05

01

23

40

0.51

1.52

2.53

3.54

GE

N=

08

01

23

40

0.51

1.52

2.53

3.54

GE

N=

06

01

23

40

0.51

1.52

2.53

3.54

GE

N=

07

01

23

40

0.51

1.52

2.53

3.54

GE

N=

09

01

23

40

0.51

1.52

2.53

3.54

GE

N=

10

Fig

ure

8.T

hedi

stri

butio

nof

GA

popu

latio

nof

deci

sion

vari

able

ssh

own

atea

chtim

e-ho

rizo

nw

ithth

ebe

stm

embe

rsh

own

with

afi

lled

circ

lefo

rth

ein

dica

ted

gene

ratio

nsfo

rT

H=

2ca

se.

(con

td.)

01

23

40

0.51

1.52

2.53

3.54

GE

N=

11

01

23

40

0.51

1.52

2.53

3.54

GE

N=

12

01

23

40

0.51

1.52

2.53

3.54

GE

N=

13

01

23

40

0.51

1.52

2.53

3.54

GE

N=

14

01

23

40

0.51

1.52

2.53

3.54

GE

N=

15

01

23

40

0.51

1.52

2.53

3.54

GE

N=

16

01

23

40

0.51

1.52

2.53

3.54

GE

N=

17

01

23

40

0.51

1.52

2.53

3.54

GE

N=

18

01

23

40

0.51

1.52

2.53

3.54

GE

N=

19

01

23

40

0.51

1.52

2.53

3.54

GE

N=

20

Fig

ure

8.T

hedi

stri

butio

nof

GA

popu

latio

nof

deci

sion

vari

able

ssh

own

atea

chtim

e-ho

rizo

nw

ithth

ebe

stm

embe

rsh

own

with

afi

lled

circ

lefo

rth

ein

dica

ted

gene

ratio

nsfo

rT

H=

2ca

se.

Figure 9. Variation of drag co-efficient with time for Re = 15000. (a) Shown are the best (discrete symbols) andthe worst (solid line) cases over the first 13 generations; (b) Drag co-efficient variation during the thirteenthtime-horizon is shown for all the eleven members of GA.

t

CD

78 79 80 81 82 83 84-1

-0.5

0

0.5

1

1.5

2

2.5

3

(b)

t

CD

6 12 18 24 30 36 42 48 54 60 66 72 78 84-0.5

0

0.5

1

1.5

(a)

TH = 6

01

23

40

0.51

1.52

2.53

3.54

GE

N=

01

01

23

40

0.51

1.52

2.53

3.54

GE

N=

02

01

23

40

0.51

1.52

2.53

3.54

GE

N=

03

01

23

40

0.51

1.52

2.53

3.54

GE

N=

04

01

23

40

0.51

1.52

2.53

3.54

GE

N=

05

01

23

40

0.51

1.52

2.53

3.54

GE

N=

08

01

23

40

0.51

1.52

2.53

3.54

GE

N=

06

01

23

40

0.51

1.52

2.53

3.54

GE

N=

07

01

23

40

0.51

1.52

2.53

3.54

GE

N=

09

01

23

40

0.51

1.52

2.53

3.54

GE

N=

10

Fig

ure

10.T

hedi

stri

butio

nof

GA

popu

latio

nof

deci

sion

vari

able

ssh

own

atea

chtim

e-ho

rizo

nw

ithth

ebe

stm

embe

rsh

own

with

afi

lled

circ

lefo

rth

ein

dica

ted

gene

ratio

nsfo

rT

H=

6ca

se.

(con

td.)

01

23

40

0.51

1.52

2.53

3.54

GE

N=

11

01

23

40

0.51

1.52

2.53

3.54

GE

N=

12

01

23

40

0.51

1.52

2.53

3.54

GE

N=

13

Fig

ure

10.T

hedi

stri

butio

nof

GA

popu

latio

nof

deci

sion

vari

able

ssh

own

atea

chtim

e-ho

rizo

nw

ithth

ebe

stm

embe

rsh

own

with

afi

lled

circ

lefo

rth

ein

dica

ted

gene

ratio

nsfo

rT

H=

6ca

se.

t

CD

6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84-0.5

0

0.5

1

1.5

(a)

TH = 3 LD

Figure 11. Variation of drag co-efficient with time for Re = 15000. (a) Shown are the best (discrete symbols) andthe worst (solid line) cases over the first 27 generations; (b) Drag co-efficient variation during the twenty-seventhtime-horizon is shown for the best eleven members when population size used N = 31 for the GA. The search spaceis Ω1 = [0,8] and Sf = [0,4].

t

CD

81 81.5 82 82.5 83 83.5 84-1

-0.5

0

0.5

1

1.5

2

2.5

3

(b)

01

23

4012345678

GE

N=

01

01

23

4012345678

GE

N=

02

01

23

4012345678

GE

N=

03

01

23

4012345678

GE

N=

04

01

23

4012345678

GE

N=

05

01

23

4012345678

GE

N=

07

01

23

4012345678

GE

N=

08

01

23

4012345678

GE

N=

09

01

23

4012345678

GE

N=

10

01

23

4012345678

GE

N=

06

Fig

ure

12.T

hedi

stri

butio

nof

GA

popu

latio

nof

deci

sion

vari

able

ssh

own

atea

chtim

e-ho

rizo

nw

ithth

ebe

stm

embe

rsh

own

with

afi

lled

circ

lefo

rth

ein

dica

ted

gene

ratio

nsfo

rT

H=

3L

Dca

se.

(con

td.)

01

23

4012345678

GE

N=

11

01

23

4012345678

GE

N=

12

01

23

4012345678

GE

N=

13

01

23

4012345678

GE

N=

14

01

23

4012345678

GE

N=

15

01

23

4012345678

GE

N=

16

01

23

4012345678

GE

N=

17

01

23

4012345678

GE

N=

18

01

23

4012345678

GE

N=

19

01

23

4012345678

GE

N=

20

Fig

ure

12.T

hedi

stri

butio

nof

GA

popu

latio

nof

deci

sion

vari

able

ssh

own

atea

chtim

e-ho

rizo

nw

ithth

ebe

stm

embe

rsh

own

with

afi

lled

circ

lefo

rth

ein

dica

ted

gene

ratio

nsfo

rT

H=

3L

Dca

se.

(con

td.)

01

23

4012345678

GE

N=

21

01

23

4012345678

GE

N=

22

01

23

4012345678

GE

N=

23

01

23

4012345678

GE

N=

24

01

23

4012345678

GE

N=

25

01

23

4012345678

GE

N=

26

01

23

4012345678

GE

N=

27

Fig

ure

12.T

hedi

stri

butio

nof

GA

popu

latio

nof

deci

sion

vari

able

ssh

own

atea

chtim

e-ho

rizo

nw

ithth

ebe

stm

embe

rsh

own

with

afi

lled

circ

lefo

rth

ein

dica

ted

gene

ratio

nsfo

rT

H=

3L

Dca

se.

Figure 13. Variation of drag co-efficient with time for Re = 15000. (a) Shown are the best (discrete symbols) andthe worst (solid line) cases from the sixth to the fifteenth time-horizon; (b) Drag co-efficient variation during thefifteenth time-horizon is shown for the best eleven members when population size used N = 31 for the GA. Thesearch space is Ω1 = [0,8] and Sf = [0,4].

t

CD

48 49 50 51-1

-0.5

0

0.5

1

1.5

2

2.5

3

(b)

t

CD

21 24 27 30 33 36 39 42 45 48 51-0.5

0

0.5

1

1.5

(a)

TH = 3 LD-N

01

23

4012345678

GE

N=

01

01

23

4012345678

GE

N=

02

01

23

4012345678

GE

N=

03

01

23

4012345678

GE

N=

04

01

23

4012345678

GE

N=

05

01

23

4012345678

GE

N=

06

01

23

4012345678

GE

N=

07

01

23

4012345678

GE

N=

08

01

23

4012345678

GE

N=

09

01

23

4012345678

GE

N=

10

Fig

ure

14.T

hedi

stri

butio

nof

GA

popu

latio

nof

deci

sion

vari

able

ssh

own

atea

chtim

e-ho

rizo

nw

ithth

ebe

stm

embe

rsh

own

with

afi

lled

circ

lefo

rth

ein

dica

ted

time-

hori

zons

for

TH

=3

LD

-Nca

se.

Her

e6

GA

itera

tions

duri

ngsi

xth;

4G

Aite

ratio

nsfo

rse

vent

h;2

GA

itera

tions

duri

ngei

ghth

and

noite

ratio

nsin

the

rest

oftim

e-ho

rizo

nsw

ere

perf

orm

ed.

(con

td.)

01

23

4012345678

GE

N=

12

01

23

4012345678

GE

N=

13

01

23

4012345678

GE

N=

14

01

23

4012345678

GE

N=

15

01

23

4012345678

GE

N=

11

Fig

ure

14.T

hedi

stri

butio

nof

GA

popu

latio

nof

deci

sion

vari

able

ssh

own

atea

chtim

e-ho

rizo

nw

ithth

ebe

stm

embe

rsh

own

with

afi

lled

circ

lefo

rth

ein

dica

ted

time-

hori

zons

for

TH

=3

LD

-Nca

se.

Her

e6

GA

itera

tions

duri

ngsi

xth;

4G

Aite

ratio

nsfo

rse

vent

h;2

GA

itera

tions

duri

ngei

ghth

and

noite

ratio

nsin

the

rest

oftim

e-ho

rizo

nsw

ere

perf

orm

ed.