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International Journal of Advanced Research ADVANCED RESEARCH IN ENGINEERING
INTERNATIONAL JOURNAL OF in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 7, November – December (2013), © IAEME

AND TECHNOLOGY (IJARET)

ISSN 0976 - 6480 (Print)
ISSN 0976 - 6499 (Online)
Volume 4, Issue 7, November - December 2013, pp. 276-289
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IJARET
©IAEME

STABILITY DERIVATIVES IN THE NEWTONIAN LIMIT
Asha Crasta1,

S. A. Khan2

1

2

Research Scholar, Department of Mathematics, Jain University, Bangalore, Karnataka, India
Principal, Mechanical Engineering Department, Bearys Institute of Technology, Innoli Mangalore,
Karnataka, India

ABSTRACT
This paper presents an analytical method to predict the aerodynamic stability derivatives of
oscillating delta wings with curved leading edge. It uses the Ghosh similitude and the strip theory to
obtain the expressions for stability derivatives in pitch and roll in the Newtonian limit. The present
theory gives a quick and approximate method to estimate the stability derivatives which is very
handy at the design stage. They are applicable for wings of arbitrary plan form shape at high angles
of attack provided the shock wave is attached to the leading edge of the wing. The expressions
derived for stability derivatives become exact in the Newtonian limit. The stiffness derivative and
damping derivative in pitch and roll are dependent on the geometric parameter of the wing. It is
found that stiffness derivative linearly varies with the amplitude. Whenever, the plan form area is
increased the stiffness derivative is also increased and vice versa. There is a shift of the center of
pressure towards the trailing edge whenever wing plan form is changed from concave to convex plan
form. In the case of damping derivative since expressions for these derivatives are non-linear and the
same is reflected in all the results. Good agreement is found with existing theories in some special
cases.
Keywords: Curved Leading Edges, Newtonian Limit, Strip Theory.
1. INTRODUCTION
Unsteady supersonic/hypersonic aerodynamics has been studied extensively for moderate
supersonic Mach number and hypersonic Mach number for small angles of attack only and hence
there is evidently a need for a unified supersonic/hypersonic flow theory that is applicable for large
as well as small angles of attack.
For two-dimensional flow, exact solutions were given by Carrier [1] and Hui [2] for the case
of an oscillating wedge and by Hui [3] for an oscillating flat plate. They are valid uniformly for all

276

International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –

supersonic Mach numbers and for arbitrary angles of attack or wedge angles, provided that the shock
waves are attached to the leading edge of the body.
For an oscillating triangular wing in supersonic/hypersonic flow, the shock wave may be
attached or detached from the leading edges, depending on the combination of flight Mach number,
the angle of attack, the ratio of specific heats of the gas, and the swept-back angle of the wing. The
attached shock case was studied by Liu and Hui [4] where as the detached shock case in hypersonic
flow was studied by Hui and Hemdan both are valid for moderate angles of attack. Hui et.al [5]
applied the strip theory to study the problem of stability of an oscillating flat plate wing of arbitrary
plan form shape placed at a certain mean angle of attack in a supersonic/hypersonic stream. For a
given wing plan form at a given angle of attack, the accuracy of the strip theory in approximating the
actual three-dimensional flow around the wing is expected to increase with increasing flight Mach
Number. The strip theory becomes exact in the Newtonian limit since the Newtonian flow, in which
fluid particles do not interact with each other is truly two-dimensional locally. In this paper the
Ghosh theory [6] is been extended to oscillating delta wings with curved leading edges and the
relations have been obtained for the stability derivatives in pitch and roll in the Newtonian limit.
2. ANALYSIS
Consider a wing whose leading edge is given by a sine wave superimposed on a straight
leading edge.
The Stiffness Derivative is given by
− Cmα =

sin α 0 cos α 0 f ( S 1) 2
1
[( − h) cot ε + {AF + AH .2.(2h − 1)}
4 AH
3
π
(cot ∈ −
)

π
Damping derivative in pitch is given by
− C mq =

sin α 0 f ( S 1)
4
1
1
( h 2 − h + ) cot ε −
4 AH )
3
2
π
(cot ∈ −

π

(1)

4


2
( 2h − 1) AF + 2(2h − 2h − 2 + 1) AH 
π



(2)
Rolling moment due to rate of roll is given by
sin α 0 f (S1 )
− Cl p =
4A
(Cot 2ε − H cot ε )

π

[

2
cot3 ∈
AF AH
1
4
16 AFAH 16
+ cot2 ∈ − 3 (π 2 − 4) + cot ∈ ( AF 2 + AH ) −
AH 3 −
−
AF 2 AH ]
2 15π
12
2π π
4
9π
9 π

(3)
Where S 1 = M ∞ sin α 0
Where ( f ( S1 ) =

γ +1
2 S1

1
2

2

[2s1 + ( B + 2S1 ) /( B + S1 ) 2 ]

S1 = M ∞ sin α ο
4 2
B=(
)
γ + 1 in all above cases.
277


In the Newtonian limit M ∞ tends to infinity and γ tends to unity. In the above expression
(1), (2) and (3) only f ( S1 ) contains M ∞ and γ .


2
(γ + 1)
( B + 2 S 21 )
 2 + (4 + 2 S 1 )  = 4
lim f ( S1 ) = lim
{2 S1 +
} = lim
1
1 
M ∞ →∞
M ∞ →∞ 2 S
S1 →∞ 
2 2
2 2
1
(B + S 1)
S1 (4 + s1 ) 




(4)

Therefore in the Newtonian limit, stiffness derivative in pitch,

−Cmα =

4sin α 0 cos α 0 2
1
[( − h) cot ε + { AF + AH .2.(2h − 1)}
4 AH 3
π
(cot ∈ −
)

π

−Cmα
sin 2α 0

2

=

(cot ∈ −

4 AH

π

2
1
[( − h) cot ε + { AF + AH .2.(2h − 1)}
π
) 3
(5)

The Damping derivative in Newtonian limit for a full sine wave is given by
sin α 0 f ( S 1) 2 4
1
1
−Cmq =
(h − h + ) cot ε − {(2h − 1) AF }
π
(cot ∈)
3
2
4
1
1
We define g(h) = (h 2 − h + ) cot ε − {(2h − 1) AF } which is a quadratic in pivot position h and
π
3
2
hence has a minimum value
4sin αο g (h)
−Cm q =
(cot ε )
In Eq. (5), only g (h) depends on h and other terms are constant. To get minimum value of Cmq only
g (h) is to be differentiated and putting

∂
g (h) equal to zero
∂h

∂
4
1
1
[(h2 − h + ) cot ε − {(2h − 1) AF }] = 0
∂h
3
2
π
2 A
∴h = + F
3 π
Let the value for h corresponding to [C m q ] min be denoted hm .
2 A
∴ hm = + F
3 π

4
1
1
Hence g (h) min = (h2 m − hm + )cot ε − {(2hm − 1) AF }
π
3
2
4sin αο g (h)min
∴ −C m q =
(cot ε )
− Cm
4 g (h)min
q min
=
∴
sin α ο
(cot ε )
(6)
The Damping derivative for a half sine wave in a Newtonian limit is given by
278


− Cm q =

4 sin α ο
2 AH
4
1
4
[(h 2 − h + ) cot ε −
( 2h 2 − 2h − 2 + 1)]
4A
3
2
π
π
(cot ε − H )

(7)

π

2 AH
4
1
4
h + ) cot ε −
( 2h 2 − 2h − 2 + 1)]
3
2
π
π
position h and hence has a minimum value.

We define g (h) = [(h 2 −

− Cm q =

4 sin α ο g (h)
4A
(cot ε − H )

which is a quadratic in pivot

(8)

π

In Eq. (8), only g (h) depends on h and other terms are constant. To get minimum value of C mq only
∂
g (h) equal to zero.
∂h
2A
∂
4
1
4
Or
[(h 2 − h + ) cot ε − H (2h 2 − 2h − + 1)] = 0
∂h
3
2
π
2
2(π cot ε − 4 AH + AH )
∴h =
3(π (cot ε − 4 AH )

g (h) is to be differentiated and putting

Let the value for h corresponding to [C m q ] min be denoted hm .
AH
2
∴ hm = [1 +
]
3
π cot ε − 4 AH )

(9)

Hence g (h) min =

(10)

From (8) and (10)
∴ −C m q =

4 sin α ο g (h) min
4A
(cot ε − H )

π

∴

− Cm

q

min

sin α ο

=

4 g (h) min
4A
(cot ε − H )

(11)

π

Rolling Moment due to roll in Newtonian limit becomes

−Clp =

4sin α0
4A
(Cot 2ε − H cot ε )

π

[

2
cot3 ∈
AF AH
1
4
16 AFAH 16 2
+ cot 2 ∈ − 3 (π 2 − 4) + cot ∈ ( AF 2 + AH ) − AH 3 −
−
AF AH ]
12
2π π
4
9π
9 π 2 15π

279


∴

[

−Cl p
sin α0

=
(Cot 2ε −

4
4 AH

π

cot ε )

2
cot3 ∈
1
4
16 AFAH 16 2
AF AH
+ cot 2 ∈ − 3 (π 2 − 4) + cot ∈ ( AF 2 + AH ) − AH 3 −
−
AF AH ]
12
2π π
4
9π
9 π 2 15π

(12)

3. RESULTS AND DISCUSSION
Before we discuss the results obtained from the present theory and by Hui et al (5) in the
Newtonian limit, it is important to discuss the matching of wing geometry of the present theory with
Ref. [5] power law wings ( y = bx n ; Eq. for leading edge) for n = 0.5, 1, 2. The wing geometries
have been approximated (as far as possible) by present half sine wave (Fig. 1). It is to be noted that
for n = 1 the matching is exact (straight leading edge). For n = 2, the matching is good only in the
trailing edge but in the leading the matching is poor. However, for n = 0.5 the matching is poor
throughout. The values of [ −C mq ] min / sin α ο are compared with Hui et al [5] in Fig. 2. The
agreement is good when the geometrical plan form matching is good and matching is poor when the
wing plan form matching is poor. Another reason for the disagreement could be that the present
theory is quasi steady one where as the theory of Ref. (5) is fully unsteady. Further, the results are
presented only for three values of n = 0.5, 1, and 2 as matching of the wing leading edge was done
only for these three values and for higher values of n the matching is not possible for higher values
of n even though Ref. (5) presented the results up to n = 8

Fig. 1: Comparison of wing geometry with Hui et al [5]

280


Fig. 2: Variation of minimum damping in pitch derivative with power n of a delta wing
Figure 3 presents the results of stiffness derivative for power law wing of Ref. (5) and the
present work. As discussed earlier for n = 0.5 and AH = -0.16 the matching of the wing plan form
area is poor, and the wing plan form area of Ref. (5) is more where as the wing plan form area of
present study is less. In view of the above the magnitude of stiffness derivative is more for the wings
having convex leading edge where as for the wing the having concave leading edge the trend is
reversed.

Fig. 3: Variation of stiffness derivative ratio with pivot position for a half sine wave

281


Fig. 4: Variation of damping derivative ratio with amplitude of half sine wave

Figure 4 presents the value of

( −Cm ) / sin α ,
q

min

ο

for various amplitude of the sine

wave here again it is seen that there is a gradual decrease in the minimum damping with the variation
in convexity of the wing plan form and this trend continues however, for concave plan form this
results in maximum drop at the highest value of the amplitude AH = 0.3 of half sine wave. The
reasons for progressive decrease in the minimum derivative are due to the decrease in the plan form
area of the wing and this sudden drop may be due to the larger area of the wing being shifted towards
the leading edge. Results for roll damping derivatives are shown in figure 5. The roll damping
derivative decreases with the reduction in concavity once convexity is introduced for lower values of
amplitude there is decrease in the magnitude of the roll damping derivative but for amplitude more
than 0.1 the trend is reversed. As discussed earlier this trend is due to the change in the wing plan
form and shift of the wing area towards the leading edge or trailing edge.
Figure 6 presents result for stiffness derivative in pitch for full sin wave and Power law wing of Ref.
(5). For full sine wave due to the increased wing plan form area the values are on the higher side as
compare to the values of Ref. (5). Further, there is a backward shift of the center of pressure for the
wing with full sine wave plan form compare to that power law wing of Ref. (5). This trend may be
due to the pressure distribution on the wing for these wing plan form.

282


Fig. 5: Variation of Roll damping derivative ratio with amplitude of half sine wave

Fig. 6: Variation of stiffness derivative ratio with pivot position for a full sine wave
283


Fig. 7: Variation of ratio of Stiffness derivative with amplitude of full sine wave
Variation of stiffness and damping derivatives in roll and pitch with amplitude of full sine
wave are shown in Figs. 7 to 9. All the derivatives vary linearly with the amplitude of the full sine
wave. Since all the stability derivatives are considered in the Newtonian limit. Hence the trend will
be different compared to that they are considered at hypersonic and supersonic or at low Mach
numbers. From Fig. 7 it is seen that the stiffness derivative linearly increases with full sine wave.
This trend is due to the shift in the wing area from leading edge to trailing edge whereas the damping
derivative decreases with the amplitude of the full sine wave as shown in Fig.8. The Rolling
derivative also increases linearly with full sine wave (Fig.9).

Fig. 8: Variation of ratio of Minimum Damping derivative with amplitude of full sine wave

284


Fig. 9: Variation of ratio of rolling derivative with amplitude of full sine wave

Fig.10: Variation of ratio of stiffness derivative with pivot position of a Half sine wave with AF = 0.1

285


Figure 10 and Figure 11 show the variation of stiffness derivative with pivot position in pitch
for Half and Full sine wave with the amplitude variation. From figure 10 it is seen that when the
concavity is being decreased the centre of pressure shifts towards the trailing edge of the wing.
Further, it is seen that for convexity of the wing the center of pressure of the wing varies from 65 %
to 85 % from the wing leading edge. Due to the shift of the center of pressure towards the leading
edge this will result in high value of static margin which means that due to the variation in the
amplitude of the half sine wave the static stability will be very high which may not be desirable in
most of the cases it may be desirable in some special case. Hence, if the wing is used for fighter
aircraft this increase in the stiffness derivative may not be desirable from maneuvering point of view
rather pilot may like to have static margin either negative or almost zero which is found in case of F16 fighter aircraft.
Figure 11 presents the variation of stiffness derivatives with pivot position. As discussed
earlier in the case of full sin wave the trend is on the similar lines except the variation in the
magnitude. In case of half sine wave the wings are either concave or convex which results in increase
or decrease in the area of the wing which directly increases or decreases the value of the stiffness
derivative. Whereas, when we superimpose only full sine wave net plan form area of the wing
remains the same. When the magnitude is negative the wing plan farm area from the leading is being
removed and shifted towards the trailing edge and just opposite happens when the magnitude is
positive. Another observation in figure 11 is that the movement of the center of pressure is less as
compared to the case when half sine wave was super imposed of the same magnitude; it varies from
62 % to 78 % from the leading edge. When both the amplitudes of Half and full sine wave are varied
together the matching is expected to be good with the wing plan form area of that of Ref (5) and also,
it is expected to give interesting results.

Fig.11: Variation of ratio of stiffness derivative with pivot position for a full sine wave with AH = 0.1
Fig 12 and Fig 13 shows the variation of damping derivative with half and full sine wave by
super imposing a full and half sine wave to the leading edge of a delta wing. The Damping derivative
decreases with the amplitude of sine wave. The Trend is similar to that in Fig. 8 with the exception
that the magnitude has increased in Fig. 12. The net reduction in the value of the damping derivative
is about 25 % for the entire range of the parameters considered in the present study. The physical
286


reasons for this behavior are geometrical change in the wing plan form area. Similar results are
shown in fig 13. In this case amplitude of the full sine wave has been varied for a given value of the
half sin wave. This result indicates that when concavity is introduced in the leading edge and
convexity is super imposed in the trailing edge, this will give the best result. Whereas, if convexity is
super imposed in the leading edge and concavity at trailing edge there is a drastic decrease in the
damping derivative, and for this combination it will result instability. In the case of stiffness
derivative this reduction may be desirable as this gives static stability whereas, damping derivative
will result dynamic stability and if a system is statically stable need not be dynamically stable,
however, if a system is dynamically stable then it is automatically statically stable.

Fig. 12: Variation of ratio of Damping derivative with AH when AF = 0.1

Fig. 13: Variation of ratio of Damping derivative with AF when AH = 0.1

287


Fig. 14: Variation of Ratio of Rolling derivative with AH when AF = 0.1

Fig. 15: Variation of ratio of Rolling derivative with AF when AH = 0.1
Results for roll damping derivatives are presented in Figs. 14 to 15 for a fixed value of
amplitude for half sine wave as well as full sine wave. As seen in figure 14 that roll damping
derivative continuously decreases with the amplitude changing from -0.2 to +0.2 that means when
wing plan form changes from concave to convex that is decreasing or increasing the area. There is an
overall 30 % decrease in the value of roll damping derivatives for the parameters of the present
study. Similar results are shown in Fig. 15 for roll damping derivates for fixed value of half sine
wave with variations of amplitude of full sine wave. From the figure it is seen that initially with the
288


amplitude the roll damping derivatives decreases then tends to increases for the range of amplitude
of the present study and there is overall decrease in the roll damping derivative is around 20 %.
CONCLUSION
The present theory is valid when the shock wave is attached to the leading edge. The effect of
secondary wave reflections and viscous effects are neglected. The expressions derived for stability
derivatives become exact in the Newtonian limit. From the results it is found that the stability
derivatives are independent of Mach number as they are estimated in the Newtonian limit where
Mach numbers will tend to infinity and specific heat ratio gamma will tend to unity. The stiffness
derivative and damping derivative in pitch and roll are dependent on the geometric parameter of the
wing in the Newtonian limit. It is found that stiffness derivative linearly varies with the amplitude.
Whenever, the plan form area is increased the stiffness derivative is also increased and vice versa.
There is a shift of the center of pressure towards the trailing edge whenever wing plan form is
changed from concave to convex plan form. In the case of damping derivative since expressions for
these derivatives are non-linear and the same is reflected in all the results.
REFERENCES
1.

Carrier G. F., The Oscillating Wedge in a Supersonic Stream, Journal of the Aeronautical
Sciences, Vol.16, March 1949, pp.150-152.
2. Hui W. H., Stability of Oscillating Wedges and Caret Wings in Hypersonic and Supersonic
Flows, AIAA Journal, Vol.7, August 1969, pp. 1524-1530.
3. Hui. W. H , Supersonic/Hypersonic Flow past an Oscillating Flat plate at Large angles of
attack, Journal of Applied Mathematics and Physics, Vol. 29, 1978, pp. 414-427.
4. Liu D. D and Hui W. H., Oscillating Delta Wings with attached Shock waves, AIAA Journal,
Vol. 15, June 1977, pp. 804-812.
5. Hui W. H. et al, Oscillating Supersonic/Hypersonic wings at High Incidence, AIAA Journal,
Vol. 20, No.3, March1982, pp. 299-304.
6. Asha Crasta, S. A. Khan, Estimation of stability derivatives of an Oscillating Hypersonic delta
wings with curved leading edges, International Journal of Mechanical Engineering &
Technology, vol. 3, Issue 3, Dec 2012, pp. 483-492.
7. Asha Crasta and Khan S. A., High Incidence Supersonic similitude for Planar wedge,
International Journal of Engineering research and Applications, Vol. 2, Issue 5, SeptemberOctober 2012, pp. 468-471.
8. 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.
9. Asha Crasta and Khan S. A, Oscillating Supersonic delta wing with Straight Leading Edges,
International Journal of Computational Engineering Research, Vol. 2, Issue 5, September
2012, pp.1226-1233.
10. M N Raja Shekar and Shaik Magbul Hussain, “Effect of Viscous Dissipation an Mhd Flow
and Heat Transfer of a Non-Newtonian Power-Law Fluid Past a Stretching Sheet with
Suction/Injection”, International Journal of Advanced Research in Engineering & Technology
(IJARET), Volume 4, Issue 3, 2013, pp. 296 - 301, ISSN Print: 0976-6480, ISSN Online:
0976-6499.
11. T.Tirupati, J.Sanadeep and Dr.B.Subashchandran, “Computational Study of Film Cooling in
Hypersonic Flows”, International Journal of Mechanical Engineering & Technology (IJMET),
Volume 4, Issue 3, 2013, pp. 327 - 336, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359.

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