Aerodynamic Shape Optimization of Airfoils In

Aerodynamic Shape Optimization of Airfoils in

Ultra-Low Reynolds Number Flow using

Simultaneous Pseudo-Time Stepping

S. B. Hazra, A. Jameson

Aerospace Computing Lab (ACL) Report, 2007-4,

2007

1

Aerodynamic Shape Optimization of Airfoils in

Ultra-Low Reynolds Number Flow using

Simultaneous Pseudo-Time Stepping

S. B. Hazra1, A. Jameson2

1 Department of Mathematics, University of Trier, D-54286 Trier, Germany

2 Department of Aeronautics and Astronautics, Stanford University, Stanford,

CA 94305, USA

Abstract. The paper presents numerical results of optimized airfoils at ultra-low

Reynolds numbers. These investigations are carried out to understand the aerodynamic

issues related to the low speed and micro scale air vehicle design and performance. The

optimization method used is based on simultaneous pseudo-time stepping in which sta-

tionary states are obtained by solving the preconditioned pseudo-stationary system of

equations representing the state, costate and design equations. Design examples of airfoils

of different thicknesses at Mach numbers between 0.25 to 0.3 and at Reynolds numbers

below 15000 are presented.

1 Introduction

Aerodynamic studies of low Reynolds number flows have begun to take momentum. This

is due to the fact that technological advances have made it possible to consider Micro Air

Vehicles (MAVs) for various difficult missions of interest.

Aerodynamic studies are quite mature for flows with Reynolds number greater than

106. However, there are not many analytical or numerical or experimental studies available

for flows at very low Reynolds numbers. Consideration of 2d geometries, i.e., airfoils, for

such studies seems to be a good starting point to improve our understanding of low

Reynolds number flows. Some experimental studies are reported in [7, 8, 9]. Numerical

studies, for analysis as well as for design, and experimental validations of airfoils at ultra-

low Reynolds number are presented in [11, 12].

The viscous effects are dominant at low Reynolds numbers, and the flow physics

is quite complicated due to the predominance of separations. If the flow is laminar, a

2

separation bubble is formed in the boundary layer on the upper surface of the airfoil at

a quite low lift coefficient. The position of this bubble depends on the Reynolds number

and the angle of attack. These are also the parameters which determine weather the flow

reattaches behind the bubble or remains separated. If the flow remains separated, there

is a a sharp drop in the lift coefficient and an increase in the drag coefficient, leading to

a drastic reduction in the performance.

The paper is organized as follows. In the next section we give a brief description of

the numerical method used in this study. Section 3 presents the numerical results and

discussions. We draw our conclusions in Section 4.

2 Computational framework for simulation and op-

timization of the flow

The flow conditions that we considered are in the Mach numbers between 0.3 and 0.25 and

the Reynolds numbers less than 15000. The flow is assumed to be laminar. The SYN103

code of A. Jameson [5, 6, 4] has been used for the simulation and optimization which

is capable of running in laminar as well as in turbulent mode. The Reynolds averaged

Navier-Stokes equations together with the turbulence model of Baldwin and Lomax is

used for the flow modelling. The cell-centered finite volume method together with the

artificial dissipation of Jameson, Schmidt and Turkel is used in this code. For the time

stepping it uses 5 stage, 4th order Runge-Kutta type of scheme. The code additionally

has acceleration techniques such as multigrid, enthalpy damping and residual averaging.

It has been used for numerous applications in analysis and design, specially in transonic

flows.

The simultaneous pseudo-time stepping optimization method has been developed in

[1]. In this method the preconditioned pseudo-stationary system of equations resulting

from the necessary optimality conditions of the optimization problem are solved using time

stepping method. The preconditioner used in this method stems from the reduced SQP

methods. This method has been integrated into SYN103 code and used for optimization

in viscous transonic flow in [2]. In this work we use the same method and the code for

optimization at the very low Reynolds numbers. The code seems to work quite well in

these flow conditions.

3

3 Numerical results and discussion

The numerical method is applied to 2D test cases for drag reduction with constant lift

and constant thickness. The computational domain is discretized using 512 × 64 C-grid.

On this grid pseudo-unsteady state, costate and design equations are solved using the

SYN103 code. All the points on the airfoil are used as design parameters which are 257

in numbers.

Case 1: RAE 2822 airfoil at Reynolds number 15000

In this case the optimization method is applied to a RAE2822 airfoil at Mach number

0.25. The constraint of constant lift coefficient is fixed at 0.2. Figure 1 presents the

cordwise pressure distribution (left) and the Mach contours (right) in baseline. As one

can see there is a separation bubble near the upper surface trailing edge of the airfoil.

Figure 2 presents the vector plots of the velocity where, in the zoomed trailing edge, one

can clearly see the circulation zone forming the separation bubble. In Figures 3 and 4

the optimized surface pressure, Mach contours and the velocity vectors are presented. In

the optimized profiles the separation bubble has been disappeared making the flow purely

attached laminar one. The total drag, which is the sum of pressure drag and the viscous

drag, has been reduced by 82 counts. The values of the force coefficients are given in

Table 1.

Case 3: RAE 2822 airfoil at Reynolds number 10000

In this case the optimization method is applied to a RAE2822 airfoil at Mach number 0.25.

The constraint of constant lift coefficient is fixed at 0.3. The reduction of Reynolds number

results in stronger separation bubble and hence larger drag value. Figure 5 presents the

cordwise pressure distribution and Mach contours. Figure 6 presents the vector plots of

the velocity. One can see the stronger separation bubble near the upper surface trailing

edge. The same optimized quantities are presented in Figures 7 and 8. The optimization

again has resulted the airfoil with attached laminar flow. The total drag in this case has

been reduced by about 170 counts. In both of these cases the viscous drag have been

increased by 8 and 22 counts respectively. The baseline and optimized force coefficients

are presented in Table 1. The polars are presented in Figure 9. As we can see there, the

optimized airfoil will have better performance around the lift coefficient upto 0.4.

Case 3: RE6K airfoil at Reynolds number 6000

It is generally believed that thin airfoils perform better than thick airfoils at low Reynolds

numbers. In [12] the RE6K airfoil was optimized using a numerical method. The INS2D

code, where the incompressible fluid model is used, was used for that design. The airfoil

4

Geometry Re CD CL CM AL CLV CDV

Baseline 15000 0.0431 0.200 -0.0011 4.728 0.0003 0.0194

Optimized 15000 0.0349 0.230 -0.0151 2.867 0.0005 0.0202

Baseline 10000 0.0618 0.300 -0.0068 6.872 0.0005 0.0214

Optimized 10000 0.0448 0.327 -0.0081 4.518 0.0009 0.0236

Table 1: Comparison of number of force coefficients and angle of attack for the baseline

and the optimized RAE2822 airfoil at different Reynolds numbers

is 2% thick and at Reynolds number 6000 it has an lift to drag ratio of about 12.9. We

use this optimized airfoil for our next investigation. We use the fixed lift value of 0.57 and

Mach number 0.3 for the optimization. Figure 10 presents the surface pressure distribution

and Mach contours for this flow condition. As one can see in the Mach contours there

appears a separation bubble near the upper surface trailing edge. In Figure 11, the vector

plots of the velocity, one can clearly see the circulation zone in the upper surface trailing

edge. In Figures 12 and 13 the optimized surface pressure, Mach contours and the velocity

vectors are presented. The optimization again resulted the airfoil which produces attached

laminar flow. In this case the total drag has been reduced by about 28 counts. Which

means that the optimized airfoil has almost 1% higher lift to drag ratio. The optimized

force coefficients are presented in Table 2. The polars of the baseline and optimized airfoils

are presented in Figure 14. As we see there, the optimized airfoil has better performance

upto the lift coefficient 0.7.

Geometry Re CD CL CM AL CLV CDV

Baseline 6000 0.0530 0.5710 -0.0863 3.698 0.0010 0.0329

Optimized 6000 0.0502 0.5710 -0.0506 4.370 0.0016 0.0307

Table 2: Comparison of number of force coefficients and angle of attack for the baseline

and the optimized RE6K airfoil

4 Conclusions

Numerical optimization method is used to study and optimize the airfoils in low Reynolds

number flows. The results show that the method is capable of producing optimized airfoils

5

with better performance at the prescribed flow conditions. We plan to extend our study

to further 2D and 3D aerodynamics shapes.

References

[1] Hazra, S. B., Schulz, V., Simultaneous pseudo-timestepping for PDE-model based

optimization problems, Bit Numerical Mathematics, 44(3): 457-472, 2004.

[2] Hazra, S. B., Jameson, A.,One-shot pseudo-time method for aerodynamic shape

optimization using the Navier-Stokes equations, AIAA 2007-1470, 2007.

[3] Jameson, A., Aerodynamic design via control theory, J. Scientific Computing, Vol.3,

pp.233-260, 1988.

[4] Jameson, A., Automatic design of transonic airfoils to reduce shock induced pressure

drag, In Proceedings of the 31st Israel Annual Conference on Aviation and Aeronau-

tics, pages 5-17, Tel Aviv, February, 1990.

[5] Jameson, A., Pierce, N., Martinelli, L., Optimum aerodynamic design using

the Navier-Stokes equations, AIAA 97-0101, 35th Aerospace Science Meeting and

Exhibit, Reno, Nevada, 1997.

[6] Jameson, A., Martinelli, L., Pierce, N., Optimum aerodynamic design using

the Navier-Stokes equations, Journal of Theoretical Computational Fluid Dynamics,

10: 213-237, 1998.

[7] Schmitz, F. W.,Aerodynamics of the model airplane. Part I. Airfoil measurements,

NACA TM X-60976, 1967.

[8] Sunada, S., Sakaguchi, A., Kawachi, K.,Airfoil section characteristics at a low

Reynolds number, Journal of Fluids Engineering, Vol.119, pp.129-135, 1997.

[9] Sunada, S., Yasuda, T., Yasuda, K., Kawachi, K.,Comparison of wing char-

acteristics at an ultralow Reynolds number, Journal of Aircraft, Vol.39, pp.331-338,

2002.

[10] Sun, Q., Boyd, I. D.,Flat-plate aerodynamics at very low Reynolds number, J.

Fluid Mech., Vol.502, pp.199-206, 2004.

6

RAE 2822 AIRFOIL MACH 0.250 ALPHA 4.728 RE 0.150E+05CL 0.2064 CD 0.0431 CM -0.0011 CLV 0.0003 CDV 0.0194GRID 512X64 NDES 0 RES0.378E-02 GMAX 0.100E-05

0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++

++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+

+

+

+

+

+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++

+

Figure 1: Pressure distribution and Mach contours of the RAE2822 airfoil at Re = 15000

(Baseline)

[11] Kroo, I., Kunz, P. J.,Meso-scale flight and miniature rotorcraft development,

Proceedings of the Conference on Fised, Flapping and Rotary Vehicles at very Low

Reynolds Numbers, edited by T. J. Mueller, Univ. of Notre Dame, Notre Dame, IN,

pp.184-196, 2000.

[12] Kunz, P. J., Kroo, I.,Analysis, design and testing of airfoils for use at ultra-low

Reynolds numbers, Proceedings of the Conference on Fised, Flapping and Rotary

Vehicles at very Low Reynolds Numbers, edited by T. J. Mueller, Univ. of Notre

Dame, Notre Dame, IN, pp.349-372, 2000.

7

X

Y

0 2 4 6

0.4

0.6

0.8

1

1.2

1.4

X

Y

4.5 5 5.5

0.8

1

1.2

Figure 2: Vector plots of the velocity of the RAE2822 airfoil at Re = 15000: (left) around

the airfoil and (right) zoomed around the trailing edge (Baseline)


0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++

+++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++

+++++++++

+

+

+

+

+

+

+

+++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

(a)


(Optimized)

8

X

Y

0 2 4 6

0.4

0.6

0.8

1

1.2

1.4

X

Y

4.5 5 5.5

0.8

0.9

1

1.1


the airfoil and (right) zoomed around the trailing edge (Optimized)


0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++

+

+

+

+

+

+

+

+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++

+


(Baseline)

9

X

Y

0 2 4 6

0.4

0.6

0.8

1

1.2

1.4

X

Y

4 4.5 5 5.5

0.8

1

1.2




0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++

++

+

+

+

+

+

+

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++

+


(Optimized)

10

X

Y

0 2 4 6

0.4

0.6

0.8

1

1.2

1.4

X

Y

4 4.5 5 5.5

0.8

1



CD

CL

0.03 0.045 0.06 0.075 0.09

0.15

0.225

0.3

0.375

0.45

BaselineOptimized

Alpha

CD

1 2 3 4 5 6 7 8 90.03

0.04

0.05

0.06

0.07

0.08

0.09

BaselineOptimized

Alpha

CL

1 2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

BaselineOptimized

Figure 9: Polars of the RAE2822 airfoil at Re = 10000

11

RE6K OPTIMAL LIFT TO DRAG AIRFOIL MACH 0.300 ALPHA 3.698 RE 0.600E+04CL 0.5710 CD 0.0530 CM -0.0863 CLV 0.0010 CDV 0.0329GRID 512X64 NDES 0 RES0.150E-02 GMAX 0.100E-05

0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++

+

+

+

++

+

+

+

+

+

+++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Figure 10: Pressure distribution and Mach contours of the RE6K airfoil at Re = 6000

(Baseline)

X

Y

0 2 4 6 8

0.4

0.6

0.8

1

1.2

1.4

X

Y

4.5 5 5.5 6

0.5

0.55

0.6

0.65

0.7

0.75

0.8

Figure 11: Vector plots of the velocity of the RE6K airfoil at Re = 6000: (left) around


12

RE6K OPTIMAL LIFT TO DRAG AIRFOIL MACH 0.300 ALPHA 4.370 RE 0.600E+04CL 0.5710 CD 0.0502 CM -0.0506 CLV 0.0016 CDV 0.0307GRID 512X64 NDES 100 RES0.434E-04 GMAX 0.142E-02

0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+++

+

+

+

+

+

+

+

+++

+

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

Figure 12: Pressure distribution and Mach contours of the RE6K airfoil at Re = 6000

(Optimized)

X

Y

0 2 4 6 8

0.4

0.6

0.8

1

1.2

1.4

X

Y

4.5 5 5.5 6

0.5

0.55

0.6

0.65

0.7

0.75

Figure 13: Vector plots of the velocity of the RE6K airfoil at Re = 6000: (left) around


13

CD

CL

0.04 0.05 0.06

0.2

0.4

0.6

0.8BaselineOptimized

Alpha

CD

-2 -1 0 1 2 3 4 5 6 70.04

0.05

0.06


Alpha

CL

-2 -1 0 1 2 3 4 5 6 70

0.2

0.4

0.6


Figure 14: Polars of the RE6K airfoil at Re = 6000

14

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Aerodynamic Shape Optimization of Airfoils In