Computers & Fluids 39 (2010) 1261–1266

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Computers & Fluids

journal homepage: www.elsevier .com/ locate /compfluid

Computations of optimal cylinder flow control in weakly conductive fluids

Hui Zhang, Bao-chun Fan *, Zhi-hua ChenState Key Laboratory of Transient Physics, Nanjing University of Science and Technology, Nanjing 210094, China

a r t i c l e i n f o

Article history:Received 12 March 2009Accepted 23 March 2010Available online 27 March 2010

Keywords:Flow controlAdjoint optimal controlCylinder wakeNonlinear optimal controlElectromagnetic flow

0045-7930/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.compfluid.2010.03.007

* Corresponding author.E-mail address: bcfan@mail.njust.edu.cn (B.-c. Fan

a b s t r a c t

A nonlinear adjoint-based optimal control approach of cylinder wake by electromagnetic force has beeninvestigated numerically in the paper. A cost functional representing the balance of the regulated quan-tities with different weights and interaction parameter N (Lorentz force) has been constituted, where theregulated quantities related with flow and force are taken as targets of regulation and the Lorentz force,(as interaction parameter N), is taken as a control input. Based on the cost functional and Navier–Stokesequations, the corresponding adjoint equations have been derived and the sensitivity of the cost func-tional is found to be a simple function of the adjoint stream function in the adjoint field. For the differentregulations, the forms of optimal control rules are similar while the adjoint equations are different. Thereceding-horizon predictive control setting is employed to discuss the optimal control problems. Underthe action of optimal N(t), the flow separation is suppressed fully, so that the oscillations of drag and liftare suppressed and the total drag coefficient decreases dramatically. For the different regulations, thecontrol effects have some differences due to the different values of optimal inputs corresponding tothe different adjoint flow fields.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The viscous flow past a free bluff body may induce undesirablevibration of the body, accompanied by a large fluctuation of dragand lift forces, and acoustics noise. The above phenomena maybe suppressed through modern flow control methods and technol-ogies. Usually, flow control involves passive and active devices,where passive control modifies the flow without external energyinput [7,14]. On the contrary, in active control, energy is intro-duced into the flow. Active control methods have attracted moreattention in recent years, e.g. rotary oscillation control of a cylinderwake [15], sound wave disturbing [13], suction and blowing [9],thermal effect [8] and so on. A travelling wave (TW) technique isto trap a row of controllable vortices over the body surface, whichplays a role of ‘‘fluid roller bearing” (FBR) between the externalflows and hence suppresses the flow separation and vortex shed-ding [17,18]. Lorentz force was employed to control cylinder wakessuccessfully in the 60s of last century, and attracted new attentionagain due to its potential engineering application background, sofar, the suppression of its vortex shedding, reduction of the dragforce, absorption of vibration and suppression of noise have beeninvestigated widely with the use of electromagnetic actuators cov-ered on the object surface [3,4,6,12,16,19,20].

One of the important motivations for control is to reduce the in-put energy as small as possible to achieve the goal of the control,

ll rights reserved.

).

such as suppressions of flow separation and body vibration etc.,which concerns the optimal control theory. The optimal flow con-trol can be used to find a control input to make the cost functionminimal under the constraint of Navier–Stokes (N–S) equations,and it can be classified into two main types, the linear and nonlin-ear optimal control. For linear optimal control relied on linearizedN–S equations, the classical control theory can be applied straight-forwardly [10]. However, nonlinear optimal is applied to the fullynonlinear N–S equations which may provide perhaps the most rig-orous theoretical framework for flow control.

A suboptimal feedback control is developed and applied by Minand Choi. Three different cost functionals to be minimized or max-imized are investigated with this approach. The results show thatthe vortex shedding becomes weak or disappears, and the meandrag and drag/lift fluctuations significantly decrease [11]. One suchmethod is adjoint-based optimization [1,5], which requires defin-ing an adjoint field properly and getting the sensitivity of the costfunctional by solving the N–S equations and their adjoint equa-tions iteratively. Bewley et al. [2] makes use of adjoint equationsto fix adjoint flow, but the forms of adjoint equations vary withflow patterns, control methods and control purposes, and there isno general way to deduce the adjoint equations so farunfortunately.

In present paper, the adjoint-based optimal control of cylinderwake by electromagnetic force has been investigated. Based onthe cost functional and N–S equations, the corresponding adjointequations have been derived and the sensitivity of the costfunctional is found to be a simple function of the adjoint stream

1262 H. Zhang et al. / Computers & Fluids 39 (2010) 1261–1266

function in the adjoint field. For the different regulations, the formsof optimal control rules are similar while the adjoint equations aredifferent. The receding-horizon predictive control setting is em-ployed to discuss the optimal control problems. The optimal inputcan be obtained by the forward and backward integrations and theiteration processes. Then, the flow is advanced some portion, andthe optimization process is obtained with this approach. Alterna-tive Direction Implicit algorithm and Fast Fourier Transformalgorithm are employed for numerical simulations. For the regula-tions with different weights, the control effect is discussed withthe application of the optimal input.

2. Lorentz force control of cylinder wake

For the control of a circular cylinder wake in weakly conductivefluid by Lorentz forces, the cylinder surface is mounted with anelectromagnetic actuator, which consists of two half cylinders,each obtained from the bend of the plate and consists of alternat-ing streamwise electrodes and magnets. In this way, the Lorentzforce is directed parallel to the cylinder surface and decays expo-nentially in the radial direction, which can be described by a dis-tributed function given by Weier [3,12,16]

jFhj ¼ e�aðr�1Þ ð1ÞFr ¼ 0

where r and h are polar coordinates, F is the distribution function ofdimensionless Lorentz force, subscripts r and h represent the com-ponents in r and h directions, respectively. a is a constant, repre-senting the effective depth of Lorentz force in the fluid.

Introducing the exponential-polar coordinates system (n, g),where r ¼ e2pn and h ¼ 2pg, the governing equations describingsuch fluid–structure problem can be written in the dimensionlessform

NðqÞq ¼�HX

NFðnÞ

� �ð2Þ

where the flow state q ¼ Xw

� �, NðqÞq is the N–S operator in the

exponential-polar coordinates,

NðqÞq ¼@2w@n2 þ @2w

@g2

H @X@t þ

@ðUrXÞ@n þ

@ðUhXÞ@g � 2

Re@2X@n2 þ @2X

@g2

h i0@

1A

and FðnÞ ¼ H12 @Fh

@n þ 2pFh

h iwhere stream function, w, is defined as

@w@g ¼ Ur ¼ H

12ur ;� @w

@n ¼ Uh ¼ H12uh, the vorticity, X, is defined as

X ¼ 1H

@Uh@n �

@Ur@g

� �, ur and uh are components of velocity in r and h

directions, respectively, H ¼ 4p2e4pn, Re ¼ 2u1am , u1 is the free-

stream velocity, m is the kinematic viscosity, a is the cylinder radius,the non-dimensional time t ¼ t�u1

a . N ¼ j0B0aqu21

is the interaction param-

eter, the current density j0 = rE0, r is the electric conductivity; E0 isthe electric field and B0 is the magnetic field.

The flow is considered to be potential initially, then

at t ¼ 0; w ¼ 0; X ¼ � 1H@2w

@n2 on n ¼ 0

w ¼ �2shð2pnÞ sinð2pgÞ; X ¼ 0 on n > 0

at t > 0; w ¼ 0; X ¼ � 1H@2w

@n2 on n ¼ 0

w ¼ �2shð2pnÞ sinð2pgÞ; X ¼ 0 on n ¼ n1

The dimensionless drag and lift coefficients are defined as

Cd ¼Fx

qu21a¼ 2

Re

Z 1

02pX� @X

@n

� �sinð2pgÞdg

� 2pNZ 1

0Fh sinð2pgÞdg ð3Þ

Cl ¼Fy

qu21a¼ 2

Re

Z 1

02pX� @X

@n

� �cosð2pgÞdg

� 2pNZ 1

0Fh cosð2pgÞdg ð4Þ

where Fx and Fy denote the total force component in streamwiseand normal direction, respectively. The total drag is composed ofthe two parts, i.e.

Fx ¼ Fpx þ Fsx ð5Þ

where Fpx and Fsx denote the pressure drag and the friction dragrespectively. Then, the pressure drag and friction drag coefficientsare given by

Cp ¼Fpx

qu21a¼ � 2

Re

Z 1

0

@X@n

sinð2pgÞdg� 2pNZ 1

0Fh sinð2pgÞdg ð6Þ

Cf ¼Fsx

qu21a¼ 2

Re

Z 1

02pX sinð2pgÞdg ð7Þ

3. Cost functional

The purpose of optimal flow control is to determine controlinput that effectively tailor the flow, and the key element of anoptimal flow control is the minimization of a cost functionalwhich provides a quantitative measure of the objective. The costfunctional contains three important physics variables i.e. thecontrol input, the state variable and the price of the control.

In the present work, control of weakly conductive fluids by Lor-entz forces is considered, therefore, the interaction parameter N, adimensionless magnitude of Lorentz force, is taken as a control in-put. The state variable, a quantity of physical interest regulated orminimized in control, can be chosen optionally such as enstrophyX2, skin-friction drag X sinð2pgÞ, X cosð2pgÞ and skin-pressuredrag @X

@n sinð2pgÞ etc. Several cases of physical interest may be rep-resented by a cost functional of the generic form

JðNÞ ¼ �Hd1

2

Z T

0

ZR

X2dv dt � 2d2

Re

Z T

0

Z 1

0XK1dgdt � 2d3

Re

�Z T

0

Z 1

0XK2dgdt � 2d4

Re

Z T

0

Z 1

0

@X@n

K1dgdt � 2d5

Re

�Z T

0

Z 1

0

@X@n

K2dgdt þ l2

2

Z T

0

Z 1

0N2dgdt ð8Þ

where K1 ¼ sinð2pgÞ; K2 ¼ cosð2pgÞ. l2 is a weighting factorwhich represents the price of the control, its quantity is small ifthe control is ‘cheap’ and large if it is ‘expensive’. Moreover, onwhich the final controlled flow field is dependent. The constantsdi are included to account for the relative weight of each individualterm and Ridi ¼ 1.

Considering a perturbation J0 to the cost function resulting froman arbitrary perturbation N0, then we have

H. Zhang et al. / Computers & Fluids 39 (2010) 1261–1266 1263

J0ðNÞ ¼Z T

0

DJ

DNN0dt

¼ �HZ T

0

ZR

d1XX0dv dt � 2Re

Z T

0

�Z 1

0d2X

0K1 þ d3X0K2 þ d4

@X0

@nK1 þ d5

@X0

@nK2

� �dgdt

þ l2Z T

0

Z 1

0NN0dgdt

ð9Þ

where the superscript ‘‘0” is defined as the Fréchet differential, T isthe period over which the control is optimized.

The purpose of optimal flow control is to determine the optimalcontrol input function N(t), which seeks the minimizing cost func-tion, thus

DJðNÞDN

¼ 0 ð10Þ

The desired optimal control input N, the solution of Eq. (10), isfound to be a simple function of the solution of the adjoint problemestablished appropriately.

4. Adjoint-based optimization approach

From Eq. (2), the linearized perturbation equations are given by

N0ðqÞq0 ¼ �HX0

N0F

� �ð11Þ

with initial and boundary conditions

at t ¼ 0; w0 ¼ 0; X0 ¼ 0;@w0

@n¼ 0;

@w0

@g¼ 0

at t > 0; w0 ¼ 0;@w0

@n¼ �U0h ¼ 0; U0r ¼ 0 on n ¼ 0;

w0 ¼ 0;@w0

@n¼ 0; X0 ¼ 0;

@X0

@n¼ 0 on n ¼ 1

where the flow perturbation state q0 ¼ X0

w0

� �, N0ðqÞq0 is given by

N0ðqÞq0 ¼@2w0

@n2 þ @2w0

@g2

H @X0

@t þ@ðUrX0þU0rXÞ

@n þ @ðUhX0þU0hXÞ@g � 2

Re@2X0

@n2 þ @2X0

@g2

h i0@

1A ð12Þ

Once the cost functional JðNÞ is defined mathematically, a properlydefined adjoint field has to be derived to determine the gradient ofthe cost functional [20].

Here, the adjoint flow field is governed by

N0ðqÞ�q� ¼�Hðd1 þ d2 þ d3 þ d4 þ d5ÞX� � d1X

0

� �ð13Þ

With initial and boundary conditions

at t ¼ T; w� ¼ 0; X� ¼ 0

at t < T;@w�

@n¼ d2K1 þ d3K2; w� ¼ d4K1 þ d5K2 on n ¼ 0

w� ¼ 0; X� ¼ 0 on n ¼ 1

where the adjoint state q� ¼ X�

w�

� �, the adjoint operation N0ðqÞ�q�

is given by

N0ðqÞ�q� ¼� @2w�

@n2 þ @2w�

@g2

� �

� H @X�

@t þ@ðUrX�þU�r XÞ

@n þ @ðUhX�þU�hXÞ@g � 2

Re@2X�

@n2 þ @2X�

@g2

� �h i0B@

1CA

From Eq. (12), we get

b ¼Z T

0

ZRN0ðqÞq

0q�dv dt

¼Z

RHðX0w�ÞT dv � 2

Re

Z T

0

Z 1

0A1dg�

Z 1

0A1dg

� �dt ð14Þ

where A1 ¼ X0@w�

@nþ w�

@X0

@n

� �n¼1

A0 ¼ X0@w�

@nþ w�

@X0

@n

� �n¼0

According to the integrations by parts and Eqs. (11) and (12), wealso get

b ¼ N0ðqÞq

0; q�D E

� q0; N0ðqÞ�q

�D E

ð15Þ

where

N0ðqÞq

0; q�D E

¼Z T

0

ZRð�HX0X� þ N0Fw�Þdv dt

q0; N0ðqÞ�q

�D E

¼Z T

0

ZRð�Hðd1 þ d2 þ d3 þ d4 þ d5ÞX0X�

þ Hd1XX0Þdv dt

a; bh i denotes an inner product of vector a and b over the domain inspace–time.

From Eqs. (14) and (15), we getZ T

0

ZR

N0Fw�dv dt ¼Z T

0

ZR

d1HXX0dv dt

þ 2Re

Z T

0

Z t

0

�ðd2K1 þ d3k2ÞX0

þ ðd4K1 þ d5K2Þ@X0

@n

�dgdt ð16Þ

Using Eq. (9) and (10), we find thatZR

Fw�dv ¼ l2N ð17Þ

The desired optimal control input N is thus found to be a simplefunction of the solution of the adjoint problem proposed in Eq.(13), which is independent of the chosen state variable.

5. Numerical method

The evolutions of the flowfield governed by Eq. (2) and the ad-joint flowfield governed by Eq. (13) can be solved numerically byusing the Alternative Direction Implicit (ADI) algorithm and theFast Fourier Transform (FFT) algorithm, which have the accuracyof second order in space and first order in time [3].

During the optimal control process, the control input N(t) isdetermined by the control law described by Eq. (17), in whichthe right side term is in proportion to N and the left side term isrelated with an integration

RR Fw�dv , which is dependent upon N.

When the control input N and the period of the control time Tare set, the developments of the flow field q from t = 0 to t = Tcan be obtained by integrations of Eq. (2) numerically. Then thedevelopments of the adjoint flow field, q�, with marching back-ward from t = T to t = 0 are also obtained by integration of Eq.(13) numerically, where the computation of the adjoint field re-quires storage of the flow field q. Finally,

RR Fw�dv is obtained by

integrating the w� over the adjoint flow field at t = 0. Therefore,the optimal N can be determined at the intersection point of thetwo curves shown in Fig. 1, where the solid line 1 denotes the

Fig. 1. N(t) determined by intersection of two curves.

1264 H. Zhang et al. / Computers & Fluids 39 (2010) 1261–1266

integral curve related with the adjoint flow, and the dashed line 2denotes the line in proportion to N. The iteration processes are per-formed to get the intersection point.

The receding-horizon predictive control setting [2] is employedto discuss the optimal control problems in the paper. When theflow field at t = ta is determined and the period of the control time

Fig. 2. Variations of optimal actuator N(t) during the optimal control process.

(a)

Fig. 3. Variations of vorticity contours (left) and steamline

T is set, then the optimal input N can be obtained as describedabove by the forward and backward integrations and the iterationprocesses. Once the optimal input N is determined, the flow is ad-vanced some portion, and the optimization process is begun anew.

The calculations have been performed at Re ¼ 150 and T = 0.15.The computational steps are Dn ¼ 0:004, Dg ¼ 0:002 andDt ¼ 0:005.

6. Results and discussion

6.1. Optimal control input

Based on calculated results, the variations of the optimal inputN(t) with time t are shown in Fig. 2. For d1 = 1 i.e. enstrophy regu-lation, the value of N increases rapidly at the triggered time t = 500and after a short time interval, it tends to remain constant, due tothe tendency for the flow field to steady. For d2 = 1, i.e. friction dragregulation, the control input N(t) increases dramatically at thebeginning of the control, then decreases and oscillates and finallysteadies on a constant level. For di ¼ 0:2ði ¼ 1—5Þ i.e. the regula-tions with several cases of physical interest, the variations of opti-mal input N(t) are similar with that of d1 = 1 during the optimalcontrol process. However, there is no significant difference forthe final input N(t) of different regulations.

6.2. Evolution of flowfield in control

Fig. 3 shows the evolution process of the vorticity (left) andstreamline (right) contours with the optimal Lorentz force(di = 0.2) switched on at time t = 500. When the flow passesthrough the cylinder, the boundary layer appears on the cylindersurface due to the viscous effect. The effects of ‘‘adverse” pressuregradient make the flow separate from the cylinder surface, justbehind the separation points. The vortexes appear and shedperiodically from the cylinder, forming the vortex street. Withthe application of optimal Lorentz force, the flow in the boundarylayer is accelerated by the Lorentz force which causes the flow

T = 500

T = 508

T = 515

T = 550(b)

s (right) of cylinder wake in the flow control process.

Fig. 4. The variations of vorticity distribution on the cylinder surface.

H. Zhang et al. / Computers & Fluids 39 (2010) 1261–1266 1265

separation point to move downstream along the cylinder surfaceand be suppressed fully. Finally, the vortex street behind the cylin-der is mitigated and the flow field turns to stable.

The vorticity distributions on the cylinder surface before andafter the control are displayed in Fig. 4. Before the application ofLorentz force (di = 0.2), X = 0 occurs at h ffi 120o, which denotesthat the flow separates from the cylinder surface at this point.Due to the shedding of vortex, the distribution curves of vorticityfluctuate periodically with time in the region between the twodashed lines as shown in Fig. 4. With the application of Lorentzforce, the vorticity on the wall increases considerably due to the in-crease of the normal velocity gradient. And the point X ¼ 0 movesto h ¼ 180

�, and the flow separation is fully suppressed with the

optimal Lorentz force.

6.3. Variation of force coefficients in control

The time histories of the total drag and lift coefficients in con-trol are shown in Fig. 5a and b individually. It has been shown that

(a) drag coefficient dC

(c) friction drag coefficient fC

Fig. 5. Time history of forces exe

both the drag and lift oscillate due to the shedding of vortex on thecylinder surface before the control. After the control, as vortexshedding is suppressed, the oscillation of lift coefficient decreasesand disappears finally and at that time the lift is kept at zero level.The total drag coefficient decreases dramatically at the beginningof the control, and finally becomes steady at a negative level. Thetotal drag coefficient is made up of the pressure drag and frictiondrag coefficients, denoted as Cp and Cf respectively. The time histo-ries of the friction drag and pressure coefficients in control areshown in Fig. 5c and d, respectively. The increase of vorticity onthe wall due to the control leads to the raise of the friction dragconsequently. However, the pressure drag coefficient decreases incontrol, which is dominant in the total drag force and makes thetotal drag force decrease.

For d1 = 1, the total drag coefficient and pressure drag coeffi-cient are larger and the friction drag coefficient is smaller than thatfor d2 = 1 and di ¼ 0:2ði ¼ 1—5Þ due to the smallest value of controlinput N(t) in the three cases. The lift force coefficient tends to besteady more quickly because the control input N(t) variessmoothly. For d2 = 1, the total drag coefficient and pressure dragcoefficient are smallest and the friction drag coefficient is largestdue to the smallest value of control input N(t) in the three cases.The lift force coefficient tends to be steady after the damping fluc-tuation because the control input N(t) varies acutely.

7. Conclusion

In this paper, the nonlinear optimal control approach based onthe adjoint flow field for the cylinder wake control is discussednumerically. A cost functional representing the balance of the reg-ulations with different weights and interaction parameter N hasbeen constituted, where the regulated quantities related with flowand force are taken as targets of regulation and the Lorentz force,(referred to interaction parameter N), is taken as a control input.The adjoint equation has been derived in an exponential-polar

(b) lift coefficient lC

(d) pressure drag coefficient pC

rted on cylinder in control.

1266 H. Zhang et al. / Computers & Fluids 39 (2010) 1261–1266

coordinates system for control as mentioned above. Based on thecost functional and N–S equations, the corresponding adjoint equa-tions have been derived and the sensitivity of the cost functional isfound to be a simple function of the adjoint stream function in theadjoint field. For the different regulations, the forms of optimalcontrol rules are similar while the adjoint equations are different.

The receding-horizon predictive control setting is employed todiscuss the optimal control problems. The optimal input can be ob-tained by the forward and backward integrations and the iterationprocesses. Then, the flow is advanced some portion, and the opti-mization process is obtained with this approach. Alternative Direc-tion Implicit algorithm and Fast Fourier Transform algorithm areemployed for numerical simulations. The optimal control N(t)varying with the transient flow field is obtained numerically. Un-der the action of N(t), the oscillations of drag and lift are sup-pressed, the total drag coefficient decreases dramatically wherethe pressure drag decrease is a dominant effect, and the lift keepsat zero level. For the different regulations, the control effects havesome differences due to the different values of optimal inputs cor-responding to the different adjoint flow fields.

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