Numerical Investigation of Viscous Laminar Supersonic Flow Past an Asymmetric Biconvex Circular-Arc Aerofoil

Saif Akram1, a, Nadeem Hasan2, b, Aqib Khan3, c 1,2,3Department of Mechanical Engineering, Zakir Hussain College of Engineering & Technology,

Aligarh Muslim University, Aligarh, India-202002

asaifakram.akram@gmail.com, bnadhasan@gmail.com, caqib.amu.khan@gmail.com

Keywords: Viscous, Laminar, Compressible Flow, Asymmetric biconvex circular-arc aerofoil

Abstract. A numerical investigation of two-dimensional unsteady, viscous and laminar

compressible flow past an asymmetric biconvex circular-arc aerofoil in supersonic regime is carried

out. The focus of the present work is to investigate the effects of variation of Mach number, at two

different angles of attack, on the flow and force characteristics on NACA 2S-(50)(04)-(50)(20)

aerofoil. The value of Reynolds number is taken as 5x105. The computations are carried out at

Mach numbers of 1.25, 1.5 and 2.0 at an angle of attack of α=00 and α=10

0. It is found that the

aerofoil works well in the supersonic flow and, unlike the conventional symmetric biconvex

aerofoil, generates finite lift at α=00 due to stronger shock waves at the lower surface. Moreover,

the L/D ratio at α=100 is always found to be more than 2.5.

Introduction

A tremendous amount of work has been done on biconvex circular-arc aerofoil and aerodynamic

characteristics including lift, drag, moment, wake surveys, etc. at different Mach numbers have

been investigated. Mayer [1] estimated the forces on symmetric biconvex circular-arc aerofoils at

near sonic speeds and in the presence of detached shock waves. Chapman [2] made a preliminary

theoretical and experimental investigation of the supersonic aerodynamic characteristics of

symmetric blunt-trailing-edge aerofoils. Katzen et al. [3] experimentally investigated the effects of

profile shape on aerodynamic and structural characteristics of thin, two-dimensional symmetric

airfoils at supersonic speeds. Magi et al. [4] investigated the near-wake flow of castellated blunt-

trailing-edge aerofoils at a Mach number of 2. Kuzmin et al. [5] studied the inviscid steady flow

over a symmetric biconvex circular-arc aerofoil with flat midpart and found an unexpected

bifurcation and intricate instability of its entire structure at certain freestream conditions. Kuzmin

further extended the work and examined the viscous transonic flow over such surfaces. Roohani et

al. [6] examined the influence of acceleration and deceleration on shock wave dynamics on and

around a symmetric biconvex aerofoil in transonic flight. Chen et al. [7] numerically investigated

compressible flow past an 18% thick symmetric circular arc aerofoil using DES for a free stream

Mach number M =0.76 and a Reynold’s number Re=1.1*107.

The above literatures show that all the investigations, till date, have been performed on

symmetric biconvex circular-arc aerofoil. Since, the thickness of upper and lower surfaces are

same, there is no lift generation with a finite value of drag at α=00. It is anticipated that if we

provide some asymmetry to the biconvex circular-arc aerofoil, a finite lift will be generated at α=00.

The characteristics of compressible flow over an asymmetric biconvex circular-arc aerofoil remain

completely unexplored. Since, the supersonic flow dynamics mainly depend on shock waves, the

thickness of the lower surface is made much larger as compared to the upper surface so that a much

stronger shock wave is formed at the lower surface, producing a finite lift at 00 angle of attack.

According to NACA, the aerofoil used for the present study can be designated as NACA 2S-

(50)(04)-(50)(20). The coordinates of the aerofoil are generated by writing a short programme in

Fortran and utilizing the equations given by Kuzmin [5]. Since practical flight does not occur only

at 00 angle of attack, the spatio-temporal dynamics of the aerofoil are also investigated at an angle

of attack of 100.

Applied Mechanics and Materials Vol. 390 (2013) pp 147-151Online available since 2013/Aug/30 at www.scientific.net© (2013) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMM.390.147

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Computational Overview and Validation

A two-dimensional viscous laminar supersonic flow past NACA 2S-(50)(04)-(50)(20) aerofoil(Fig.

1) at free stream Mach numbers of M∞ = 1.25, 1.5 & 2.0 and angle of attacks of α=00

& α=100 is

considered. The Reynolds number based on chord length and free stream condition is kept constant

at 5x105, which is the transition value, so that assuming laminar flow would be reasonable.

Moreover, since the flow parameters in supersonic regime are mainly governed by shock waves, the

use of laminar model does not lead to much error as can be observed from the validation studies.

ANSYS Fluent is used as the CFD software with density-based implicit formulation for this study.

To capture the discontinuity caused by shock waves, a second-order upwind scheme with the Roe’s

flux-difference splitting is applied. The pressure far-field boundary conditions are used for both

inflow and outflow. Multiblock structured mesh was generated using ICEM CFD with proper

clustering near the aerofoil to capture the flow physics appreciably. The total number of nodes was

133000 with a minimum size of 10-4

dimensionless units. Solutions were run with finer meshes in

order to ensure mesh independence. The finer meshes were 150000 with minimum size of 10-5

and

325000 with minimum size of 10-4

. The maximum percentage differences in lift, drag and moment

coefficient is observed to be less than 1% with much finer meshes. Therefore, the results are

considered to be mesh independent. The validation of the various numerical procedures employed

for computations is done by comparing the results obtained for NACA 2S-(50)(05)-(50)(05) with

the experimental data of Mayer[1] in terms of lift, drag and moment coefficients as shown in Table

1. Comparison of coefficient of pressure plots is also shown in figure 2, which demonstrates the

accuracy of the methodology used for the present study.

Fig. 1 Geometry of NACA 2S-(50)(04)-(50)(20) aerofoil

Table 1: Comparison of global flow parameters for NACA 2S-(50)(05)-(50)(05)

The non-dimensional parameters are defined as follows:

2 2 2 21 1 1 12 2 2 2

, , , p D L m

p p D L MC C C C

u u A u A u Alρ ρ ρ ρ∞

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

−= = = =

Where A is the reference area (1 m2), l is the reference length (chord length), D is the drag

force, L is the lift force and M is the pitching moment about leading edge of the aerofoil.

α

(degrees)

M∞ Present Ref.[1]

Cd Cl Cm,LE Cd Cl Cm,LE

0 1.85

0.036 0.000 0.000 0.038 0.000 0.000

2.13

0.029 0.000 0.000 0.030 0.000 0.000

10 1.85

0.118 0.445 0.174 0.121 0.445 0.169

2.13

0.096 0.362 0.144 0.099 0.348 0.143

15 1.85

0.225 0.675 0.275 0.215 0.614 0.256

2.13

0.189 0.569 0.233 0.188 0.548 0.228

148 Mechanical and Aerospace Engineering IV

Fig. 2 Comparison of Cp profile for NACA 2S-(50)(05)-(50)(05) at M=1.85 and an angle of attack

of (a) α=100 (b) α=15

0

Results and Discussion

The pressure contours, showing the formation of shock waves at different Mach numbers are shown

in figure 3. The variation of Cd and Cl with Mach number at different angle of attacks are shown in

figure 4. Figure 5 shows the pressure coefficient plots.

Fig.3 Pressure contours at different Mach numbers for α=00 and α=10

0

x/c

Cp

0 0.2 0.4 0.6 0.8 1

0

0.5

1

NACA RM(Lower)

NACA RM(Upper)

Present(Lower)

Present(Upper)

M=1.85

x/c

Cp

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.5

1

1.5

NACA RM(Lower)

Present(Lower)

NACA RM(Upper)

Present(Upper)

M=1.85

Pressure

45000040000035000030000025000020000015000014093510000081312.35000025576.62286015395.510839.1405.8530.00844756

-50000-55160.6

Pressure

220000

200000

180000

160000

140000

120000

100000

80000

60000

40000

20000

17306.7

10373.1

678.841

-3.19739E-05

-77.5454

-137.391

-8866.09

-20000

-40000

-60000

Pressure

120000800006000051485.846277.64000032693.624207.612841.56460.511.39859E-06

-515.559-690.284-799.184-1282.2-1812.74-9743.66-13034.9-20000-36179-60000

Pressure4500003500002500001500005000011330.95206.07340.624

-1567.47-31901.1-48498.1-50000

Pressure

2058441800001400001000008000042798.13251.140

-36276.2-42758.6-60000

Pressure

14000012000010685996701.877291.764925.153710.347068.84000014711

-8829.43-20000-45399-60000

(a)M=1.25,α=00 (b)M=1.5,α=0

0 (c)M=2.0,α=0

0

(d)M=1.25,α=100 (e)M=1.5,α=10

0 (f)M=2.0,α=10

0

Applied Mechanics and Materials Vol. 390 149

Fig. 4 Variation of lift and drag coefficients with Mach number at different α

Fig. 5 Coefficient of pressure plots at different Mach numbers

The aerodynamic characteristics in supersonic regime are mainly governed by the formation of

shock waves. Since, the aerofoil used for the present study is a thick aerofoil (24% maximum

thickness) with a high included angle at the leading edge, a detached shock wave is formed at lower

supersonic Mach numbers. At Mach 1.25, a bow shock is formed as seen in figure 3. As we

increase the Mach number to 1.5, the gap between the shock wave and the leading edge decreases

and eventually an oblique shock completely attached to the leading edge is formed at Mach 2.0.

Since, the strength of the shock wave decreases with an increase in Mach number, from a much

stronger bow shock at Mach 1.25 to a weaker oblique shock at Mach 2.0, the lift also decreases as it

depends on the strength of the shock wave. Furthermore, since a high pressure region is developed

behind the shock, a high value of drag is observed when the shock wave is detached at Mach 1.25.

Mach Number

Co

eff

icie

nt

of

Lif

t

1 1.2 1.4 1.6 1.8 2 2.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Cl,a=0

Cl,a=10

Mach Number

Co

eff

icie

nt

of

Dra

g

1 1.2 1.4 1.6 1.8 2 2.20

0.05

0.1

0.15

0.2

0.25

0.3

Cd,a=0

Cd,a=10

(a)M=1.25,α=00 (b)M=1.5, α=0

0 (c)M=2.0,α=0

0

(e)M=1.5,α=100 (f)M=2.0,α=10

0

(d)M=1.25,α=100

150 Mechanical and Aerospace Engineering IV

The drag decreases as the shock wave moves closer to the leading edge at Mach 1.5 and a much less

drag is produced at Mach 2.0 when an attached oblique shock is formed. A similar trend is also

observed at high angle of attack as can be seen from the results at α=100. The lift and drag both

decreases with an increase in Mach number due to the same reason as for α=00, but with a much

higher L/D ratio. It is observed from figure 4 that, unlike symmetric aerofoil, some finite lift is

generated at α=00 due to stronger shock waves at the lower surface. Moreover, as the curvature of

lower surface of the aerofoil is much larger than the upper surface, a greater pressure gradient is

expected at the lower surface as compared to the upper surface. This fact is clearly visible in the

pressure contour plots. Since, the upper surface is nearly flat, pressure is almost constant,

especially at α=100 as can be seen from figure 5.

Conclusion

A preliminary numerical investigation of the supersonic aerodynamic characteristics of asymmetric

biconvex circular-arc aerofoil has been carried out. It is found that if a large curvature is provided to

the lower surface keeping the upper surface almost flat, we can get finite lift at α=00. As found for

NACA 2S-(50)(04)-(50)(20) in the present study, the L/D ratio is nearly equal to 1 at Mach=1.25

and 1.5 at α=00. Whereas, at Mach 2.0 we get L/D ratio of 1.34 at α=0

0. At higher angle of attacks,

the aerofoil is quite suitable with an L/D ratio greater than 2.5 at α=100 at all supersonic Mach

numbers. Therefore, it can be concluded from the present investigation that this type of aerofoil can

be utilized for supersonic cruise conditions. For subsonic flight, the aerodynamic characteristics

need to be investigated as the present asymmetric geometry is likely to yield negative lift at α=00.

However, with finite angles of attack, the asymmetric aerofoil should provide positive lift. Further,

the effects of Re needs to be investigated in detail as increase in Re can cause transition in the

boundary layers leading to increase in the value of drag. Efforts in this direction shall be made in

further studies.

References

[1] Mayer, J. P., “Estimation of lift and drag of airfoils at near sonic speeds and in the presence of

detached shock waves,” NACA RM No. L8L07, February 1949.

[2] Chapman, D. R., “Reduction of profile drag at supersonic velocities by the use of airfoil

sections having a blunt trailing edge,” NACA TN 3503, September 1955.

[3] Katzen, E. D., Kuehn, D. M., Hill, W. A., “Investigation of the effects of profile shape on the

aerodynamic and structural characteristics of thin, two-dimensional airfoils at supersonic

speeds,” NACA TN 4039, September 1957.

[4] Magi, E. C. and Gai, S. L., “Flow behind castellated blunt-trailing-edge aerofoils at supersonic

speeds,” Journal of Fluid Mechanics (1998), 375, pp 85-111 doi:10.1017/

S0022112098002699.

[5] Kuzmin, A., “Bifurcations of Transonic Flow Past Flattened Airfoils,” Keynote Lecture at the

Fifth International Conference on Computational Fluid Dynamics, July 8, 2008, Seoul National

University, Korea.

[6] Roohani, H., Skews, B. W., “The influence of acceleration and deceleration on shock wave

movement on and around aerofoils in transonic flight,” Shock Waves (2009) 19:297–305 DOI

10.1007/s00193-009-0207-9.

[7] Li-Wei Chen, Chang-Yue Xu and Xi-Yun Lu, “Numerical investigation of the compressible

flow past an aerofoil,” J. Fluid Mech. (2010), vol. 643, pp. 97–126. Cambridge University

Press 2009, doi:10.1017/S0022112009991960.

Applied Mechanics and Materials Vol. 390 151

Mechanical and Aerospace Engineering IV 10.4028/www.scientific.net/AMM.390 Numerical Investigation of Viscous Laminar Supersonic Flow Past an Asymmetric Biconvex Circular-

Arc Aerofoil 10.4028/www.scientific.net/AMM.390.147