::;=;.....

NASA Contractor Report i87586

ICASE Report No. 91-49

ICASE

OPTIMAL CONTROL OF _/I)RAG RATIOON

A ROTATING CYLINDER

_--i,n,J :K:,pC,'Ft (TCA_) 1_* i-" C_F..L ?Oia

C:31 34

Yuh-Roung Ou

John A. Burns

Contract No.

June 1991

NAS 1-18605

ii/

z

xiiiii:_

Institute for Computer Applications _ience and Engineering

NASA Langley Research Center ...... :_

Hampton, Virginia 23665-5225

Operated by the Universities Space Research Association

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Virginia 23665-5225

i

-2.....

!

q '-__.- 7 _ "-".._

Optimal Control of Lift/Drag Ratios on a Rotating Cylinder

Yuh-Roung Ou 1

Institute for Computer Applications in Science and Engineering

NASA-Langley Research Center

Hampton, VA 23665

and

John A. Burns 2

Interdisciplinary Center for Applied Mathematics

Department of Mathematics

Virginia Polytechnic Institute and State University

Blacksburg, VA 24061

ABSTRACT

We present the numerical solution to a problem of maximizing the lift to drag ratio by

rotating a circular cylinder in a two-dimensional viscous incompressible flow. This problem

is viewed as a test case for the newly developing theoretical and computational methods

for control of fluid dynamic systems. We show that the time averaged lift to drag ratio

for a fixed finite-time interval achieves its maximum value at an optimal rotation rate that

depends on the time interval.

1This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NASl-18605 and the Air Force Office of Scientific Research under AFOSR Grant No. 89-0079

while the author was in residence at the Institute for Computer Application in Science and Engineering

(ICASE), NASA Langley Research Center, Hampton, VA 23665.

=This work was supported in part by AFOSR under grant AFOSR-89-0001 and by the National Aeronautics and Space Administration under Contract No. NAS1-18605 while the author was in residence at the

Institute for Computer Application in Science and Engineering (ICASE), NASA Langley Research Center,

Hampton, VA 23665.

1. INTRODUCTION

Active control of fluid dynamic systemsis an area of researchthat has been considerablegrowth during the past ten years. This activity is motivated by severalpotential

applications in engineeringsciencesand optimal design. The developmentof theoretical

frameworks and the construction of practical computational algorithms for control design

are highly complex problems. In this paper we considerthe problem of maximizing lift to

drag ratio by rotating a circular cylinder. This problem is simple enough to be solvedby

direct numerical calculation (as we do here) and complex enough to serveas an excellent

test problem for future theoretical and computational approaches. The main objective of

this short note is to provide the solution to two optimization problems that can be usedas

test examplesby researchersin this area.

2. MATHEMATICAL FORMULATION AND NUMERICAL METHOD

In this paper we consideran optimal control problem for a two-dimensional viscous

incompressible flow generated by a circular cylinder started impulsively into a combined

steady rotatory and rectilinear motion. This problem is investigated numerically by solving a velocity/vorticity formulation of the Navier-Stokes equations [2, 4]. A nonrotating

reference frame translating with the cylinder is employed, and the cylinder rotates in the

counterclockwise direction with angular velocity f}. The Reynolds number Re = 2Ua/u is

based on the cylinder diameter 2a and the magnitude U of the rectilinear velocity. The

angular/rectilinear speed ratio a = f}a/U is the primary control parameter in this paper.

The speed ratio a can impose significant influences on the characteristics of resulting flow

field as well as the temporal evolution of forces on the cylinder surface.

In this formulation, the dimensionless governing equations consist of the vorticity transport equation

0_ 2 2

O--t--I-ft.Va_ = _eeV a3 (i)
and the vector Poisson equation for the velocity

V2,Z= -V × @e-;). (2)

The cylinder radius a is used as the length scale while a/U is used as the time scale. The

dimensionless boundary conditions for the problem of a rotating cylinder are

ff = -a sin 0_ + a cos 0e_ on the cylinder surface ff = e_ at infinity. (3)

This velocity/vorticity formulation is especially well suited to treating initial development

of the flow generated by impulsively started bodies, in which the flow field is composed of

a relatively small vortical viscous region embedded in a much larger inviscid potential flow.

Consequently, the computational domain may be restricted to a smaller region where the

vorticity contributions are contained.

The vorticity transport equation (1) is first discretized by a second order central difference

in the radial direction and a pseudospectral transform method in the circumferential direction

for all spatial derivatives. This semi-discretization form of equation (1), consisting of a system

of ordinary differential equations in time can be written as

d_5

dt - F(_), _2 = (_a22,2,''' ,(a2)_I_l,N_l) T, (4)

for all the interior grid points. Therefore, the calculation procedure consists of the following

steps to advance the solution for any given time increment:

Step i: Internal vorticity over the fluid region at each interior field point is calculated b q

solving the discretized vorticity transport equation. An explicit second-order rational RungeKutta marching scheme based on the work of [8] is used to advance in time for (4).

The discretization in time of (4) thus can be written as

= + (5)

with

tlx = F(_'_) At,
t_2 = F( &'_ + 0.5t_:)At, (6)

g3 = 2_:-_,

where (tla,g3) denotes the scalar product of if: and tTz. This step consists of the kinetic part

of the computational loop.

Step 2: Using known internal vorticity values at all the interior grid points from step I,

the generalized Biot-Savart law of induced velocity

:(r_,t) --

1

1

_(e,t) × (_- r_)dA

I_'- _1_

2_(?',t) × (e- r_)dA + (7)

:(r_, t) =

1

1

_(7, t) × (7- r_O)dA

I:- ,_l _

2fi(_',t) × (_'- r_)dA + U.

I_'- _l '

Here _o denotes the points located on the outer perimeter of the computational domain.

In fact, this integral representation allows us to determine the velocity point-by-point

explicitly if all vorticity values are known everywhere in the domain of interest. Moreover,

(s)

is used to update the boundary vorticity values at all the surface nodes. Here 7_ represent all

grid points located on the solid boundary.

One of difficulties encountered in the simulation of viscous flow is to prescribe the appropriate nonvelocity boundary conditions at the solid surface. The boundary vorticity is

required for the formulation based on the velocity/vorticity (or stream-function/vorticity)

variables. In order to overcome this particular difficulty, we pose the kinematic relationship between velocity and vorticity fields on the infinite domain in the integral form of (7).

This boundary integral method proposed by Wu and Thompson [9] provides the basic link

between the velocity and vorticity fields throughout the numerical computations.

Step 3: At this stage, all the vorticity values in the computational domain are known at

the new time level. Then, the velocity at points on the outer perimeter of the computational

domain is calculated by this integral kinematic constraint
it often exhibits more realistic behavior at the outer perimeter of the computational domain

than asymptotic techniques used in other formulations [1]. This indicates that the difficulty

resulting from the imposed far-field condition is removed by the application of this integral

constraint.

Step 4: The new velocity field can be established by solving the Poisson equations (2) with

prescribed solid boundary conditions and outer boundary conditions that have been determined

fl'om step 2.

The final form of the discretized Poisson equations can be written

Au Av =fl = f2, (9)

where u and v are the vectors of unknown interior nodal values. The resulting 11-banded

matrix equations are then solved by a preconditioned biconjugate gradient routine [3]. This

step completes the computational loop for each time level.

One further important point to be noted in the boundary integral approach is the determination of the initial flow field. In contrast to the special technique used by Badr & Dennis

[1], this integral approach enables the numerical code to generate the initial velocity field by

executing one cycle of the solution procedure (from step 2 to step 4) rather than employing

any additional treatments.

3. RESULTS AND DISCUSSIONS

To assess the numerical algorithm, calculations were performed over a wide range of

angular/rectilinear speed ratio o up to 3.25 at a Reynolds number of 200. In this model,

the rectilinear velocity is fixed as a constant value while the angular velocity is treated as a

control variable. Although the angular velocity can be time-dependent, in this paper angular

velocity has been restricted to the constant values. For the works of a rotating cylinder undergoing various kind of time-dependent rotation rate (which include time-harmonic rotatory

oscillations and time-periodic rotations with different angular amplitudes and frequencies),

the reader is referred to Ou [6] and in the forthcoming paper [7].
We havetested the model against the experimental work of CoutanceauL: M6nard [5]

with excellent agreement. Selected instantaneous streamline plots for constant value of speed

ratio at a = 2.07 are presented in figures l(a,b). We note that for higher values of a vortex

shedding continues to occur as a result of cylinder rotation [4]. However, the formation

of the vortex street behind the rotating cylinder at high a (a = 3.25) is not observed in

the experiments of Coutanceau & M6nard. The experiments were unable to detect this

vortex shedding because of the length limitation of the water tank and the flow visualization

techniques used in their work [5].

The lift and drag coefficients can be calculated in r-0 coordinates as

- + cob cos OdO (10)

CL- Re -3-7r b

CD = _ _ b -- COb sin OdO,

and

(11)

where the subscript b denotes quantities evaluated on the cylinder surface. In particular,

we denote the positive values of CL in the -y direction. The effect of the speed ratio

(0 < o_ _< 3.25) on the temporal evolution of lift/drag coefficient is shown in Figures 2.

Note that the lift/drag increment is manifested timewise for the speed ratio up to o_ = 2.07.

However, it does not imply that as the speed ratio further increases, the flow field will result

in a further improvement of lift and reduction of drag on the cylinder. On the contrary, the

adverse effect of speed ratio on the lift/drag is quite evident if a comparison is made between

= 3.25 and c_ = 2.07. Notice that for all o_ less than 2, in the initial time interval, the

slope of the lift/drag curve seems to increase with increasing of a. When c_ becomes greater

than 2, the slopes decrease gradually as c_ increases. For all speed ratios considered here,

a significant increase in (CL/CD)max is obtained with increment of _. However, (CL/CD)

achieves its maximum value at much later time for a higher c_. We are interested in the

problem of maximizing the averaged (CL/CD) on a fixed finite time interval.

Figure 3 shows the effect of the speed ratio on time-average (for 0 _< t _< 24) lift,

drag and lift/drag coefficients in the range of 0 < c_ _< 3.25. It illustrates that the lift

5
average is almost linearly proportional to the speed ratio. On the other hand, the drag

average remains almost constant up to a = 2, then monotonically increases with speed ratio

thereafter. Consequently, the resulting lift/drag average is not linearly proportional to the

speed ratio a. Moreover, the maximum lift/drag average occurs approximately at (_ = 2.38,

and it represents a substantial increase of 440% with respect to a = 0.5.

It is also of interest to study effect of speed ratio on the variation of the (total lift)/(total

drag) force ratio. As shown in Figure 4 this ratio is similar to the lift/drag average in Figure

3 and achieves its maximum value between a = 2.0 and a = 2.38.

4. CONCLUSIONS

_Ve have presented numerical solutions of two "flow control problems". The first

problem is to find c_ that maximizes the time-averaged lift to drag functional

1 [24 "CL(t,o_)]

Yl(Ol) = N JO CD(t, og)j dt,

and the second problem is to find a_ that maximizes the total lift to total drag functional

fx24CL(t, , )dt

C19( t, a )dt "

Based on a velocity/vorticity formulation of the Navier-Stokes equations, a finite-difference

scheme was used to directly calculate ,71(a) and ,72(c_) for 0 < a _< 3.25. It is hoped that

these solutions will useful as base line values for comparison with other approaches to these

problems.

Acknowledgments

The authors gratefully acknowledge Dr. Yeng-Ming Chen for many valuable discussions

during initial stage of this work. Thanks are also due to Dr. M. Coutanceau for providing

the experimental results.

6
References

[1] H. M. Badr and S. C. R. Dennis, Time-dependent viscous flow past an impulsively started

rotating and translating circular cylinder, J. Fluid Mech. 158,447-488 (1985).

[2] Y.-M. Chen, Numerical simulation of the unsteady two-dimensional flow in a timedependent doubly-connected domain, phD thesis, University of Arizona, (1989).

[3] Y.-M. Chen, A. E. Koniges and D. V. Anderson, ILUBCG2-11: Solution of ll-banded

nonsymmetric linear equation systems by a preconditioned biconjugate gradient routine,

Comp. Phys. Comm. 55,359-365 (1989).

[4] Y.-M. Chen, Y.-R. Ou and A. J. Pearlstein, Development of the wake behind a circular

cylinder impulsively started into rotatory and rectilinear motion: intermediate rotation

rates, ICASE Report 91-10, submitted to J. Fluid Mech. (1991).

[5] M. Coutanceau and C. M_nard, Influence of rotation on the near-wake development

behind an impulsively started circular cylinder, J. Fluid Mech. 158, 399-446 (1985).

[6] Y.-R. Ou, Active control of flow around a rotating cylinder, In Meeting on Turbulence

Structure and Control, Cosponsored by AFOSR and Ohio State University, Columbus,

April 1-3, (1991).

[7] Y.-R. Ou and J. A. Burns, Effect of rotation rate on the forces of a rotating cylinder,

ICASE Report in preparation, (1991).

[8] A. Wambecq, Rational Runge-kutta methods for solving systems of ordinary differential

equations, Computing 20, 333-342 (1978).

[9] J. C. Wu and J. F. Thompson, Numerical solutions of time-dependent incompressible

Navier-Stokes equations using an integral-differential fornmlation, Computers & Fluids

1, 197-215 (1973).

7
(a)

::::--..--.::-..-.--:i .... :: ::-.-:.....

- --__-;-'-:" 'ii_i"i-:

.-:;-::.;--v,.t;' ., -Ill.

Figure 1. Comparison of calculated (left) and experimental (right) instantaneous streamlines

for Re = 200, o_ = 2.07. (a) t = 5.0, (b) t = 9.0.
10

8

2.82

2.07

2.38

3.25

CLIO D 4

0

-2 I I = I I I = I = ,1

0 4 8 12 16 20 24

t

Figure 2. Temporal evolution of the lift/drag coefficient at various values of speed ratio

(0 < cr < 3.25) up to t = 24.0.

9
10.0 -

8.0

6.0

4.0

2.0

0.0

0.0

I

/"

m

m'

.

,_.. " I , I , I ,

_ ,A_ _--- ---- _

I

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

(X

Figure 3. Effect of speed ratio on time-average lift, drag and lift/drag coefficients for

0<a <3.25.

10
10.0 -

8.0

6.0

4.0

2.0

0.0 r_, I J I I I I I i I i I I I i I

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

__"_"'O ( cL )l°'al/(CD)I°I&I

(_

Figure 4. Effect of speed ratio on variation of total lift/total drag force ratio.

11


Report Documentation Page

1. Report No i 2 Government Accessmn No,

NASA CR- 187586 !

!

ICASE Report No. 91-49

4 Title and Subtitle

OPTIMAL CONTROL OF LIFT/DRAG RATIO ON A ROTATING

CYLINDER

7 Author{s}

Yuh- Roung Ou

John A. Burns

9 Performing Organization Name and Address

Institute for Computer Applications in Science

and Engineering

Mail Stop 132C, NASA Langley Research Center

Hampton, VA 23665-5225

12. Sponsoring Agency NameandAddress

National Aeronautics and Space Administration

Langley Research Center

Hampton, VA 23665-5225

3. Recipient's Catalog No,

5 Report Date

June 1991

6 Performing Organization Code

8. Performing Orgamzation Report No

91-49

10. Work Unit No.

505-90-52-01

11. Contract or Grant No.

NASI-18605

13. Type of Report and Period Covered

Contractor Report

14. Sponsoring _.gency Code

t5. Supplementary Notes

Langley Technical Monitor:

Michael F. Card

Submitted to Applied Mathematics

Letters

Final Report

16 Abstract

We present the numerical solution to a problem of maximizing the lift to drag

ratio by rotating a circular cylinder in a two-dimensional viscous incompressible

flow. This problem is viewed as a test case for the newly developing theoretical

and computational methods for control of fluid dynamic systems. We show that the

time averaged lift to drag ratio for a fixed finite-time interval achieves its maximum value at an optimal rotation rate that depends on the time interval.

17. Key Words(SuggestedbyAuthorls))

lift-to-drag ratio, flow control

18. Distribution Statement

34 - Fluid Mechanics and Heat Transfer

63 - Cybernetics

Unclassified - Unlimited

-_" SecuriW Cla=if. Iof this report)

Unclassified

20. SecuriW Cla_if. (of this page}

Unclassified

21. No. of pages

13

22. Price

A0 3

NASA FORM 1626 OCT86

NASA-Langley, 1991

PRECEDING PAGE BLANK NOT FILMED