Estimation of stability derivatives of an oscillating hypersonic delta wings with curved

INTERNATIONALMechanical Engineering and Technology (IJMET), ISSN 0976 –
International Journal of JOURNAL OF MECHANICAL ENGINEERING
6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME
AND TECHNOLOGY (IJMET)

ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)
IJMET
Volume 3, Issue 3, September - December (2012), pp. 483-492
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ESTIMATION OF STABILITY DERIVATIVES OF AN OSCILLATING
HYPERSONIC DELTA WINGS WITH CURVED LEADING EDGES

Asha Crasta 1, S. A. Khan2
1. Research Scholar, Department of Mathematics, Jain University, Bangalore, India
Email: excelasha1@rediffmail.com
2. Principal, Department of Mechanical Engineering, P.A. College of Engineering,
Mangalore, India Email: sakhan06@gmail.com
ABSTRACT

In the present study hypersonic similitude has been used to obtain stability derivatives in
pitch and roll of a delta wing with curved leading edges for the attached shock case. A strip
theory is used in which strips at different span-wise locations are independent. This combines
with the similitude to give a piston theory. The present theory is valid only for attached shock
case. Effects of wave reflection and viscosity have not been taken into account. Some of the
results have been compared with those of Hui et al, Ghosh and Liu & Hui. Results have been
obtained for hypersonic flow of perfect gas over a wide range of mach numbers, incidences
and sweep angles.

Keywords: Attached shock wave, Curved leading edges, delta wings, Hypersonic, Pitch ,Roll

1. INTRODUCTION
The analysis of hypersonic flow over flat deltas (with straight leading edge
and curved leading edge) over a considerable incidence range is of current interest with the
advent of space shuttle and high performance military aircrafts. The knowledge of
aerodynamic load and stability for such types is a need for simple but reasonably accurate
methods for parametric calculations facilitating the design process. The dynamic stability
computation for these shapes at high incidence (which is likely to occur during the course of
reentry or maneuver) is of current interest. When descending shock waves which are usually
strong and can be either detached or attached.
The idea of hypersonic similitude is due to Tsien [1], who investigated the 2-D and
axi-symmetric irrotational equations of motion. Sychev’s [2] large incidence hypersonic
similitude is applicable for wings of extremely small span. Cole and Brainard [3] have given
a solution for a delta wing of very small span on large incidence. Messiter [4] has found a
solution, in the realm of thin shock layer theory, for steady delta wing with detached shock
case at small incidence based on hypersonic small disturbance theory. Pike [5] and Hui [6]

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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –

have given theories for steady delta wings in supersonic/hypersonic flow with attached
shocks. For 2-D flow exact solutions were given by Carrier [7] and Hui [6] for the case of an
oscillating wedge and by Hui[8] for an oscillating flat plate, which is valid uniformly for all
supersonic Mach numbers and wedge angles or angles of attack with attached shock wave.
Hui [9] calculated pressure on the compression side of a flat delta. The numerical solutions
for the attached shock case have been given by Babaev [10], Voskresenkii [11]. The role of
dynamic stability at high incidence during re-entry or maneuver has been pointed out by
Orlik-Ruckemann [12]. The shock attached relatively high aspect ratio delta is often preferred
for its high lift to drag ratio.
Hui and Hemdan [13] have studied the unsteady shock detached case in the context of
thin shock layer theory. Liu and Hui [14] have extended Hui’s [9] theory to a shock attached
pitching delta. Light hill [15] has developed a “Piston Theory” for oscillating airfoils at high
Mach numbers. A parameter δ is introduced, which is a measure of maximum inclination
angle of Mach wave in the flow field. It is assumed that M∞ δ is less than or equal to unity
(i.e. M∞ δ ≤ 1) and is of the order of maximum deflection of a streamline. Light hill [15]
likened the 2-D unsteady problem to that of a gas flow in a tube driven by a piston and
termed it “Piston Analogy”.
Ghosh [16] has developed a large incidence 2-D hypersonic similitude and piston
theory. It includes Light hill’s [15] and Mile’s [17] piston theories. Ghosh and Mistry [18]
have applied this theory of order of ¢2 where ¢ is the angle between the attached shock and
the plane approximating the windward surface. For a plane surface, ¢ is the angle between the
shock and the body. The only additional restriction compared to small disturbance theory is
that the Mach number downstream of the bow shock is not less than 2.5.
Ghosh[19] has obtained a similitude and two similarity parameters for shock attached
oscillating delta wings at large incidence. Crasta & Khan [20] have extended the Ghosh
similitude to supersonic flows past a planar wedge. Further, Khan & Crasta [21] have
obtained stability derivatives in pitch and roll of a delta wing with curved leading edges for
supersonic flows .In the present analysis this similitude has been extended to shock attached
delta wings with curved leading edges at large incidence for Hypersonic flows. The pressure
on the lee surface is assumed zero.

1.1 Nomenclature
A γ(γ+1)/4
AF, AH amplitude of full and half sine wave
AR aspect ratio
B [4/(γ+1)] 2
C chord length
Cmα, Cmq stiffness &damping derivative in pitch
Clp rolling moment derivative due to rate of roll
L rolling moment
M ∞, U ∞ Free stream Mach number and velocity
M2 Mach number behind the shock
Mp Piston Mach number
Ms Shock Mach number
S1 Similarity parameters in hypersonic flow
a∞ free stream sound velocity

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b Semi span
h Non dimensional pivot position, x0/c
k π/c
m Pitching momentum
n Power of X in the equation for leading edge
P Pressure on the wing surface
P∞ Free stream pressure
q Rate of pitch
r distance along a ray from apex
t time in second
U, V velocity components in X, Y direction
X, Y, Z body fixed reference system
X0 pivot position for pitching oscillation
α Incidence angle
β Shock wave angle
αo Mean incident for an oscillating wing
γ Specific heat ratio
δ Inclination of characteristic lines
ε Sweep angle
θ Half wedge angle
φ Angle between shock and wing in the strip
φw angle between shock and wedge
ρ∞ free stream density

2. Analysis
To get a curved leading edge we superpose a full sine wave and or half sine wave on a
straight leading edge. X-axis is taken along the chord of the wing and the Z-axis is
perpendicular to the chord in the plane of the wing.
Equation of x-axis is z = o
 2πx   πx 
Equation of full and half sine wave are Z = −a F sin   and Z = −a H sin  
 c   c  1

Equation of straight L.E Z = x cot ∈ 2
Where aF & aH are the amplitudes of the full & half sine waves and c is chord length of the
wing. Hence the equation of the curved leading edge is
 2πx   πx 
Z = x cot ∈ − a F sin   − a H sin  
 c   c  3

Area of the wing:
C
π 4A
Area ABD = ∫ Zdx , Let k = . Hence, the wing Area = C 2 (cot ∈ − H )
0 c π
2.1 Strip theory
A thin strip of the wing, parallel to the centerline, can be considered independent of
the z dimension when the velocity component along the z direction is small. This has been
discussed by Ghosh’s [19]. The strip theory combined with Ghosh’s large incidence
similitude leads to the “piston analogy” and pressure P on the surface can be directly related

485


to equivalent piston mach no. Mp. In this case both Mp and flow deflections are permitted to
be large. Hence light hill piston theory [15] or miles strong shock piston theory cannot be
used but Ghosh’s piston theory will be applicable.
1
P 2 2
= 1 + AM P + AM P ( B + M P ) 2 , Where P∞ is free stream pressure… 4
P∞

2.2 Pitching moment derivatives
Let the mean incidence be α 0 for the wing oscillating in pitch with small frequency
and amplitude about an axis X0. The piston velocity and hence pressure on the windward
surface remains constant on a span wise strip of length 2z at x, the pressure on the lee surface
is assumed zero. Therefore, the nose up moment is
c
m = −2 ∫ p.z.( x − x0 ) dx
0 5

2.3 Stiffness derivative

The stiffness derivative is non-dimensionalized by dividing with the product
of dynamic pressure, wing area and chord length.

2 ∂m
− Cm α = ( ) 6
4AH ∂q α=α
ρ∞U∞ 2C3 (cot ∈ − ) 0
π q =0

Where ρ ∞ and U∞ are density and velocity of the free stream, and q is the rate of pitch
(about x = x0) defined positive nose up.
c
− Cm =
α
2 sin α 0 cos α 0 f ( S 1)
3 (cot ∈ − 4 A H ) 0
∫ ( x cot ∈ −aF sin 2kx − aH sin kx ) (x-x0) dx 7
c
π

By solving the above equation, we get
sin α 0 cosα 0 f ( S 1) 2 1
− Cmα = [( − h) cot ε + {AF + AH .2.(2h − 1)} 8
4 AH 3 π
(cot ∈ − )
π
1
(γ + 1) 2 ) ( B + S 12 ) 2 ]
Where f ( S 1) = [ 2 S 1 + ( B + 2S 1 9`
2S 1
by using above expression for stiffness derivative calculations have been carried out and
some of the results have been shown.

486


2.4 Damping derivative

The damping derivative is non-dimensionalzed by dividing with the dynamic
C
pressure, wing area, chord length and characteristic time factor ( ).
U∞
2  ∂m 
4AH  
−Cmq =
)  ∂q 
4
ρ ∞ U ∞ C (cot ∈ − α =α 0

π q=0 10

∂m
Since m is given by integration to find ( ) differentiation within the integration is
∂q
necessary.
1
 ∂p  P ( x − x0 ) 2) 2)2 ]
 ∂q  α =α = A ∞ [ 2S 1 + ( B + 2S 1 (B + S1 11
  q =0 0 a∞

Substituting the value of the integral in the above equation
sin α 0 f ( S 1) 4 1 1 4 
− Cm q = [(h 2 − h + ) cot ε − ( 2h − 1) AF + 2( 2h 2 − 2h − 2 + 1) AH ]
(cot ∈ − π4 AH ) 3 2 π π 
12

2.5 Rolling Moment Derivative due to Rate of Roll

Let the roll be p and rolling moment is L, defined according to the right hand system
of reference

c Z = f ( x)

∴ L = 2 ∫  ∫ pzdz dx
 
13
0 0 

The local piston Mach number normal to the wing surface is given by
z
(M P ) = (M ∞ ) sinα0 p 14
a∞

The roll-damping derivative is non-dimensionalised by dividing with the product of
dynamic pressure, wing area, and span and characteristic time factor
∴ − Cl p =
1
4AH
C3 15
∫U∞C³b(cot ∈ - )
∞
π
sin α 0 f ( S1 ) cot 3 ∈ AF AH 2
− Cl p = [ + cot 2 ∈ − (π − 4) +
4 AH 12 2π π 3
(Cot 2 ε − cot ε )
π
1 2 4 16 AFAH 16 16
cot ∈ ( AF 2 + AH ) − AH 3 − − AF 2 AH ]
4 9π 9 π 2 15π
.

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Where S1 = M ∞ sinα 0

(γ + 1) ( B + 2S 12 )
f ( S 1) = [ 2S1 + 1
] 17
2S 1 2 2
(B + S1 )
2.6 Results and Discussions

The wing geometry for a full and half sine wave (convex and concave) is shown in
Fig.1 and 2. Stiffness and damping derivatives in pitch have been calculated for a full sine
wave and half sine wave (Fig. 3, 4, 5, 6) using present theory and have been compared with
Ghosh’s theory (Fig. 7 & 8). Results of present theory are better when compared to Ghosh as
it uses straight leading edges where as in the present case wing has curved leading edges. The
present theory invokes Ghosh strip theory arguments. Hui et al also use strip theory
arguments where by flow at span wise station is considered equivalent to an oscillating flat
plate flow; this flow is calculated by perturbing the known steady plate flow (oblique shock
solution) which serves as the basic flow for the theory. Hui et al have obtained closed form
expressions for stiffness and damping in pitch but have not calculated the rolling moment
derivative. The present theory calculates rolling moment derivative also, since it makes use of
Ghosh quasi steady theory which is simpler than both liu and Hui et al and brings out the
explicit dependence of derivatives on the similarity parameter S1.

Concave shaped wing Convex shaped wing

Fig1: Wing geometry of half sine wave AH=0.1 with Sweep angle = 50

Fig2: Wing geometry of full sine wave AH=0.1 and AH=-0.1 with Sweep angle = 50

488


Fig3&4: Variation of Stiffness derivative with pivot position

Fig 5&6: Variation of Stiffness derivative with pivot position.

Fig. 7&8: Variation of Stiffness and damping with pivot position

489


Fig. 9 shows the variation of Stiffness and Damping derivative with angle of incidence.
Stiffness derivative increases linearly up to angle of 30-35 degrees, it is seen that the damping
derivative increases with increase of angle of incidence and holds good up to a value of about
35 degrees. Fig. 10 compares the damping of a wing with the results of Ghosh’s theory for
different incidences. Both theories show same trend. Close to shock- detachment the present
theory is not valid. Fig. 11 shows that rolling damping decreases with Mach number initially
and confirms the Mach number independence principle for large Mach numbers.
Fig. 12 & 13 show variation of rolling derivative with sweep angle for a full
and half sine wave. As the sweep angle increases the rolling derivative decreases up to 75
degrees for a full sine wave and up to 65 degrees for a half sine wave. The mathematical error
in Ghosh [19], due to this the factor Clp is decreased to one-forth. This has been identified and
rectified (Fig 14.).

Fig9: Variation of Stiffness and damping derivative with angle of incidence

Fig10: Variation of damping derivative with angle of incidence

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6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep Dec (2012) © IAEME
Sep-

Fig 11: Variation of rolling derivative with Mach Number

Fig 12&13: Variation of Rolling derivative with Sweep angle

Fig 14: Variation of rolling derivative with Aspect ratio

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2.7 CONCLUSION
In the present theory the similitude and piston theory of Ghosh have been extended to
a flat wing with curved leading edges. There is a strong impetus for further research by taking
into account the effects of shock motion, viscosity, bluntness of the wing and the real gas
effects. The concepts of the present theory can be extended to axi-symmetric case, for which
analytical or experimental data are quite limited.
REFERENCES

[1] Tsien .H. S, Similarity laws of hypersonic flow, Journal of Mathematical Physics, Vol. 25, 1946,
pp. 247-251.
[2] Sychev V. V., Three Dimensional Hypersonic Gas Glow Past Slender Bodies at High Angles of
Attack, Journal of Applied Mathematics and Mechanics, Vol. 24, August 1960, pp. 296-306.
[3] Cole, J. D. and Brainerd, J.J. Slender wings at high angles of attack in hypersonic flow, ARS
Reprint 1980-1961.
[4] Messiter, A. F. Lift of slender delta wings according to Newtonian theory, AIAA Journal, 1963, 1,
pp. 794-802.
[5] Pike. J, The pressure on flat and anhydral delta wings with attached shock waves, The
Aeronautical Quarterly, November 1972, XXIII, Part 4, pp. 253-262.
[6] Hui, W.H., Stability of Oscillating Wedges and Caret Wings in Hypersonic and Supersonic Flows
, AIAA Journal, Vol. 7, Aug. 1969, pp.1524-1530.
[7] Carrier, G.F. 1. 1949, The oscillating Wedge in Supersonic stream , Jr.Areo.Sci. Vol. 16, No.3,
pp. 150-152, March.
[8] Hui, W. H., Supersonic/hypersonic flow past on oscillating flat plate at high angles of attack ,
ZAMP, Vol. 29, 1978, pp. 414-427.
[9] Hui, W. H. Supersonic and hypersonic flow with attached shock waves over delta wings, Proc of
Royal Society, London, 1971, A. 325, pp. 251-268.
[10] Babev, D. A. 1963, Numerical solution of the problems of Supersonic flow Past the lower
surface of a delta wing , AIAA Jr., Vol. 1, pp. 2224-2231.
[11] Vorkresenskii, G.P. 1968, Numerical solution of the problems of supersonic gas flow past an
arbitrary surface of a delta wing in compression region , IZV Akad Nauk SSSR, mekh, zkidk, gaza,
no.4, pp.134-142.
[12] Orlik-Ruckemann, K. J., Dynamic stability testing of aircraft needs versus capabilities, Progress
in the Aerospace Sciences, Academic press, N.Y., 1975, 16, pp. 431-447.
[13] Hui, W. H. and Hemdan, H. T. Unsteady hypersonic flow over delta wings with detached shock
waves, AIAA Journal, April 1976, 14, pp. 505-511.
[14] Lui, D. D. and Hui W. H., Oscillating delta wings with attached shock waves, AIAA Journal ,
June 1977,15, 6, pp. 804-812.
[15] Light Hill, M. J., Oscillating Aerofoil at High Mach Numbers, Journal of Aeronautical Sciences,
Vol. 20, June 1953, pp. 402-406.
[16] Ghosh K, A new similitude for aerofoil in hypersonic flow , Proc of the 6th Canadian congress of
applied mechanics , Vancouver, 29th may-3rd June, 1977, pp. 685-686.
[17] Miles, J. W., Unsteady flow at hypersonic speeds, Hypersonic flow, Butterworths Scientific
Publications, London, 1960, pp. 185-197.
[18] Ghosh, K. and Mistry B. K. Large incidence hypersonic similitude and oscillating non-planar
wedges, AIAA Journal, August 1980,18, 8, pp. 1004-1006.
[19] Ghosh K., Hypersonic large deflection similitude for oscillating delta wings, The Aeronautical
journal,Oct.1984, pp. 357-361.
[20] Asha Crasta and Khan S. A., High Incidence Supersonic similitude for Planar wedge,
International Journal of Engineering research and Applications, Vol. 2, Issue5, Sep-Oct 2012, pp.
468-471.
[21] Khan S. A. and Asha Crasta, Oscillating Supersonic delta wings with curved leading edges,
Advanced Studies in Contemporary mathematics, Vol. 20(2010) , No.3, pp.359-372.

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