Computation of Stiffness and Damping Derivatives of an Ogive in a Limiting Case of Mach Number and Specific Heat Ratio

1  Introduction

The analysis of supersonic/hypersonic of simple shapes like wedges, cones, and Ogives are of great interest when oscillating at high Mach numbers, and significant incidences are of great interest. Researchers have shown great interest owing to the advent of the space program and high-speed fighter aircraft because of the cost involved in conducting experiments at high Mach numbers. Therefore, simple but reasonably accurate techniques are needed to compute the aerodynamic load and stability derivatives during re-entry. And more so, such parametric computations are very valuable in the initial design process when various geometrical and inertia parameters are investigated.

The present work aims to evaluate pitch aerodynamic stiffness and damping derivatives for limiting cases. In this situation, flow parameters are not dependent on the inertia levels. However, flow parameters are a function of geometry. In the Newtonian limit, Mach numbers will tend to infinity. The specific heat ratio (γ) at constant pressure and volume, with a constant value of 1.4 (i.e., γ = 1.4) at a standard ballistic atmosphere, becomes unity. The knowledge of aerodynamic derivatives at the Newtonian limit can be handy for the space program. For launch vehicles, the flow Mach number becomes very high. In aerodynamic vehicles like hypersonic missiles and the space shuttle, the focus of study shifts from optimizing the aerodynamic shapes to reduce the drag force to focusing on aerodynamic heating. At supersonic flow, the primary concern is to minimize the drag of the projectiles and missiles. The easiest option is to have a blunt nose instead of a sharp nose, which will give immediate relief from the high-temperature build-up at the nose. Unique material is used for the nose portion of the hypersonic missiles and the aircraft to address the issue of high temperature at the design stage. For space shuttles, tiles are used to protect from aerodynamic heating.

Appleton [1] evaluated stiffness and damping derivatives in pitch for a wedge in hypersonic flow. Brong [2] and Ericsson [3] studied the flow field of the right circular cone in unsteady flight. The large deflection hypersonic similitude of Ghosh [4] was applied by Ghosh et al. [5]. They devised another hypersonic similarity for the attached shock case. The Mach after the shock must be more than 2.5 (i.e., M2 > 2.5). He ignored the impact of the Lee surface as the contribution from the Lee surface was negligible. He mainly focussed on the windward side accompanied by an oblique shock wave at the plate. A unified supersonic/hypersonic theory for delta wings and cones was developed by Ghosh [6,7]. Hui [8,9] studied oscillating wedges and caret wings to evaluate supersonic/hypersonic flow stability derivatives. Hui et al. [10,11] used the unsteady Newton-Buseman theory. The role of dynamic during re-entry or maneuver was discussed by Orlik-Ruckman [12–14]. Shabana et al. [15,16] computed the stability derivatives for supersonic/hypersonic flow cones. They used these theories, resulting in the Newtonian limit and a fixed specific heat ratio. Monis et al. [17] studied the effect of secondary wave reflections on supersonic/hypersonic flow wings. Crasta et al. [18] estimated the surface pressure distribution on a delta wing with curved leading edges in hypersonic/supersonic flow using mathematical modelling and results shows impact on finding the pressure distribution over an delta wing. Later on, softcomputing approach was found via computational fluid dynamics (CFD) method and Khan et al. [19–21] computed the flow field around the 2-D wedge. Interestingly, the study of high-speed flow was limited to a slim body, and small angles of incidence are extended for high angles of attack.

2  Mathematical Modelling

An axisymmetric Ogive is obtained by the revolution of the plane Ogive of semi-nose angle δ = (θc+λ), round the streamwise axis (x-axis) as shown in Fig. 1. The Ogive has a slope, dy′dx′=λ(1−2x′L) , where x ′ and y ′ are aligned, as shown in Fig. 1, and λ is the slope at the apex of the Ogive.

Figure 1: Geometry of cone and Ogive

As the Ogive move with a velocity of U∞ in to the slab, the velocity of the equivalent piston is Up . The pressure ratio can be expressed in the instantaneous local piston Mach number.

Pbp∞=1+γMpB2[1+142+(γ−1)MpB22+(γ+1)MpB2] (1)

where MpB is the local piston Mach number at any arbitrary point B, in the cross-sectional plane located at an axial distance of x from the apex (Fig. 1), and the local density ratio, is

∈ = 2+(γ−1)MpB22+(γ+1)MpB2 (2)

The projection of point A (Fig. 1) on the axis of symmetry has a distance of x, and the pivot point 01 has a length of xo from the apex.

Let α be the pitch angle at any instant, and q be the pitch rate. Therefore, the local piston Mach number at A is

MpA=Upa∞=1a∞[U∞sin(θc−α)−cos(Φ−θc)(x−xo)cos∅.q+U∞cosθc.dy′dx′](3)

If Ψ is the azimuthal angle (Fig. 1) subtended by a tiny area element at B, then

MpB=1a∞[U∞sin⁡(θc−αcos⁡Φ)−(x−xo).cos⁡(∅−θc)cos⁡∅.qcos⁡Ψ+U∞cosθc.dy′dx′] (4)

After simplification, we obtain

MpB=1a∞[U∞sinθc−αU∞cosθccos⁡Ψ−(x−xo).cos⁡θc . qcos⁡Ψ(1+tan⁡∅tan⁡θc)+U∞cos∅c . dy′dx′] (5)

In the limit α, q →0,

[MpB]α,q→0=1a∞[U∞sinθc+U∞cosθcdy′dx′] (6)

d[MpB]α,q→0=M∞(sinθc+cosθcdy′dx′) (7)

where

dy′dx′=λ(1−2x′L)=λ(1−2xc)=λf(x) (8)

Hence

[MpB]α,q→0=M∞(sinθc+λfcos⁡θc) (9)

Putting λtan⁡θc=λ′ , and K = M∞ sinθc we have

[MpB]α,q→0=K(1+λ′f) (10)

Differentiating Eq. (9), concerning MpB , we get

d​∈dMpB=−8MpB[2+(γ+1)MpB2]2 (11)

And from Eq. (1),

dpbdMpB=2γp∞MpB[1+14(∈+12MpBd∈dMpB)] (12)

[∈]α,q→0=2+(γ−1)K2(1+λ′f)22+(γ+1)K2(1+λ′f)2 (13)

Since λ << tan θc , λ1 << 1, hence the terms containing λ12 and other higher powers of λ′ can be neglected compared to the other terms.

[∈]α,q→0=2+(γ−1)K2(1+2λ′f)N[1+2(γ+1)K2λ′fN] (14)

where

N=[2+(γ+1)K2] (15)

[∈]α,q→0=2+(γ−1)K2(1+2λ′f)N(1+Qλ′f) (16)

where

Q=2+(γ−1)K2N (17)

[∈]α,q→0=2+(γ−1)K2(1+2λ′f)N[1+Qλ′f]−1 (18)

Expanding Binomially, and neglecting higher-order terms and simplifying, we obtain

[∈]α,q→0=[2+(γ−1)K2][2+(γ+1)K2]+λ′f(−8K2N2) (19)

[∈]α,q→0=∈+λ′f(−8K2N2) (20)

From Eqs. (10) and (11)

[d∈dMpB]α,q→0=−8K(1+λ′f)[2+(γ−1)K2(1+λ′f)2]2 (21)

On simplification, we get

[d∈dMpB]α,q→0=−8KN2[1+λ′f][1+Rλ′f]−1 (22)

where

R=4K2(γ+1)N (23)

Further simplifying, we obtain

[d∈dMpB]α,q→0=−8KN2+λ′f{8K(3(γ+1)K2−2)N3} (24)

and

[MpB]2α,q→0=K2(1+2λ′f) (25)

Now expanding Eq. (12), we get

[dpbdMpB]=2γp∞[MpB+14MpB∈+18MpB2d∈dMpB] (26)

This expression must be evaluated in the limit α,q→0 .

Hence, from Eqs. (10), (13), (24), and (25) we have

[MpB∈]α,q→0=K∈+λ′f(K∈+−8k3n2) (27)

And

[MpB2d∈dMpB]α,q→0=K2[1+2λ′f][−8kN2+λ′8K(3(γ+1)K2−2)N3] (28)

[MpB2d∈dMpB]α,q→0=−8k3N2+λ′f[−16k3N2+8k3(3(γ+1)K2−2)N3] (29)

Therefore

[dpbdMpB]α,q→0=2γp∞[K+λ′f+K∈4+K∈4λ′f−2k3N2λ′f−K3N2−2k3N2λ′f+K3(3(γ+1)K2−2)N3λ′f] (30)

After simplification, we obtain

[dpbdMpB]α,q→0=2γp∞[(a1+λ′a2f)]K (31)

where

a1=1+∈4−K2N2 (32)

a2=1+∈4−K2(N+8)N3 (33)

[dpbdMpB]α,q→0=2γp∞K[(a1+λ′a2(1−2xc)] (34)

From Fig. 1, we have

tanϕ=xtanθc(x−x0)andtanϕ0=ctanθ(c−x0) (35)

where ϕ is an angle delimited by A at the x-axis. For the various position of A, ϕ differs from π to ϕ0 , θc is the flow deflection angle of the Ogive. And the chord length is c.

[dpbdMpB]α,q→0=2γp∞K[(a1+λ′a2(1−2xotan⁡∅c(tan⁡∅−tan⁡θc)))] (36)

Hence

[dpbdMpB]α,q→0=2γp∞K[((a1+λ′a2)−2a2λ′htan⁡∅(tan⁡∅−tan⁡θc)] (37)

where

h=xocandλ′=λtan⁡θc (38)

Substituting Eq. (35) in Eq. (5) and dropping the subscript B; subsequently, we have the local piston Mach number as

Mp=1a∞[U∞sin⁡θc−αU∞cos⁡θccos⁡ψ−x0sin⁡θcqcos⁡ψ(tan⁡ϕ−tan⁡θc)+xosin⁡θctan⁡∅tan⁡θcqcosΨ(tan⁡ϕ−tan⁡θc)+U∞cos⁡θcλ(1−2htan⁡∅(tan⁡ϕ−tan⁡θc))] (39)

Hence

[∂Mp∂∝]α,q→0=1a∞[U∞cos⁡(θc−cos⁡ψ)(−cos⁡ψ)] (40)

[∂Mp∂∝]α,q→0=M∞cos⁡θccos⁡ψ (41)

And

[∂Mp∂q]α,q→0=−xosin⁡θccosψ(1+tan⁡∅tan⁡θc)a∞(tan⁡∅−tan⁡θc) (42)

The expression for the total pitching moment of the Ogive is given by

M¯=∫∅=π∅o∫ψ=02π−Pbxo3tan3θctan∅sec2∅(1+tan⁡∅tan⁡θc)(tan⁡∅−tan⁡θc)4.cosψdϕdψ (43)

Here Pb is given by Eq. (1).

The stiffness derivative, Cm∝ is given by

Cm∝=[∂M¯∂α]α,q→0=112ρ∞U∞2Sbc2 (44)

where

Sb = Ogive surface area = π(ctanθc)2

c = chord size of an Ogive

Since Pb is a function of Mp,

[∂M¯∂α]α,q→0=∫∅=π∅o∫Ψ=02π−[∂pb∂α]α,q→0xo3tan3θccos⁡Ψtan⁡∅sec2∅sec2∅(1+tan⁡∅tanθc)(tan⁡∅−tan⁡θc)4dϕ

=∫∅=π∅o∫Ψ=02π−[dpbdMp.dMpdα]α,q→0xo3tan3θccos⁡Ψtan⁡∅sec2∅sec2∅(1+tan⁡∅tanθc)(tan⁡∅−tan⁡θc)4dϕdψ

Substituting from Eq. (41)

[∂M¯∂α]α,q→0=∫∅=π∅o∫Ψ=02π−[dpbdMp]α,q→0(−M∞cos⁡θccosΨ)cos⁡Ψxo3tan3θctan∅sec2∅(1+tan⁡∅tan⁡θc)(tan⁡∅−tan⁡θc)4d∅dΨ (45)

where

∫Ψ=02πcos2ΨdΨ=π (46)

Now, substituting for [dpbdMp]α,q→0 from Eq. (36), we have

∂M¯∂α=∫∅=π∅o2γp∞K[(a1+λ′a2(1−2λ′a2htan⁡∅c(tan⁡∅−tan⁡θc)))]M∞cos⁡θcπ xo3tan3θctan⁡∅sec2∅(1+tan⁡∅tanθc)(tan⁡∅−tan⁡θc)4d∅ (47)

Putting tan⁡θ=t, and tanθc=n ,

∂M¯∂α=2γp∞KM∞cosθcxo3n3π[(a1+λ′a2)∫otan∅t+nt2(t−n)4dt−2λ′a2h∫otan∅ot2+nt3(t−n)4dt] (48)

[∂M¯∂α]α,q→0=2γp∞KM∞cosθcxo3n3π(a3I1−a4I2)

where

a3=a1+λ′a2

a4=2λ′a2h

Since tan⁡ϕ0=n1−h and (tan⁡ϕ0−n)=nh1−h and Solving integrals I1 and I2 , we obtain

I1=16n2[(1−2n2)−(1−h){H(2h3+1h2)+n2(1h2+2h)}] (49)

and

I2=[112n2−14]−[n2(1−h)2(1+n21−h)4(nh1−h)4+n1−h(2+3n21−h)12(nh1−h)3+(1+3n21−h)12(nh1−h)3+n4(nh1−h)] (50)

Putting H=(1−h+n2) and simplifying

I2=112n2[(1−3n2)−(1−h){Hh4(3+2h+h2)+n2h3(1+2h+3h2)]} (51)

Substituting I1 and I2 in Eq. (21)

[ ∂M¯∂α ]α,q→0= 2γp∞KM∞cosθc xo3 n3 π[a36n2((1-2n2)-(1-h){ H(2h3+1h2)+n2(1h2+2h })     -a412n2((1-3n2)-(1-h)Hh4(3+2h+h2)+n2h3(1+2h+3h2))](52)

Substituting Eq. (24) in (20) and putting back the values of a3 and a4

Cmα=[∂M¯∂α]α,q→0=2ρ∞U∞π2tan2θcc3=2γp∞M2∞sinθccosθcxo3n3π2p∞U∞2πn2c3[(a1+λ′a2)6n2.((1−2n2)−(1−h){H(2h3+1h2)+n2(1h2+2h)})−2λ′a2h12n2((1−3n2)−(1−h){Hh4(3+2h+h2)+n2h3(1+2h+3h2)})] (53)

Hence on simplification, we obtain

Cmα=23(1+n2)[(a1+λ′a2)(h3(1−2n2)−(1−h){H(2+h)+n2h(1+2h)})−λ′a2(h4(1−3n2)−(1−h){H(3+2h+h2)+n2h(1+2h+3h2)})] (54)

Cmα=D[h3(1−2n2)−(1−h){H(2+h)+n2h(1+2h)}]+2λ′a23(1+n2)[h3{1−2n2−h(1−3n2)}+(1−h){H(1+h+h2)+3n2h3}] (55)

where a1,a2 is given by Eqs. (32) and (33) and λ′=λn , where n=tanθc

The Damping derivative Cmq is given by

Cmq=[∂M¯∂q]α,q→0=112ρ∞U∞2Sbc2 (56)

where

[∂M¯∂q]α,q→0=∫∅=π∅o∫Ψ=02π[−∂pb∂q . xo3tan3θccosΨtan∅sec2∅(1+tan∅tanθc(tan∅−tanθc)4d∅dΨ

=∫∅=π∅o∫Ψ=02π[dpbdMp]α,q→0=[dMbdq]α,q→0.xo3tan3θccos tan∅sec2∅(1+tan∅tanθc)(tan∅−tanθc)4d∅dΨ (57)

On solving similarly like above, we obtain

Cmq=13(1+n2)[a1(h4(2n2−3n4−1)−(1−h){H(3H+h(3H+h(3H+2n2)+2n2h2)+n4h2(1+3h))+λ′a25(h4{5(2n2−3n4−1)−4h(3n2−6n4−1)}+(1−h){H(9H+(2n2−3H)h+2(2H+3n2)h2+12n2h3)−n4h2(1+3h−24h2)}]

Cmq=(D2)[(h4(2n2−3n4−1)−(1−h){H(3H+h(3H+h(3H+2n2)+2n2h2)+n4h2(1+3h))+λ′a215(1+n2)(h4{5(2n2−3n4−1)−4h(3n2−6n4−1)}+(1−h){H(9H+(2n2−3H)h+2(2H+3n2)h2+12n2h3)−n4h2(1+3h−24h2)}] (58)

where

D=23(1+n2)[1+14(ε+12KdεdMpo)]

D=23(1+n2)[1+14{(2+(γ−1)K22+(γ+1)K2)+12K(−8K[2+(γ+1)K2]2)}]

D=23(1+n2)[1+14{(2+(γ−1)M∞2sin2θc)(2+(γ+1)M∞2sin2θc)+12M∞sin⁡θc(−8M∞sin⁡θc[2+(γ+1)M∞2sin2θc]2)}]

D=23(1+n2)[1+14{(2+(γ−1)M∞2sin2θc)(2+(γ+1)M∞2sin2θc)−82(M∞2sin2θc[2+(γ+1)M∞2sin2θc]2)}]

and

a2=1+ε4−K2(N+8)N3

a2=1+14{2+(γ−1)M∞2sin2θc2+(γ+1)M∞2sin2θc}−M∞2sin2θc[{2+(γ+1)M∞2sin2θc}+8][2+(γ+1)M∞2sin2θc]3

For limiting case M∞ inclines to infinity and γ leans to unity

D=23(1+n2)limM∞→∞γ→1⁡[1+14{(2+(γ−1)M∞2sin2θc)(2+(γ+1)M∞2sin2θc)−82(M∞2sin2θc[2+(γ+1)M∞2sin2θc]2)}]

a2=1+14limM∞→∞γ→1  { 2+(γ−1)M∞2 sin2 θc2+(γ+1)M∞2 sin2 θc }−M∞2 sin2 θc[ { 2+(γ+1)M∞2 sin2 θc }+8 ][ 2+(γ+1)M∞2 sin2 θc ]3

By applying limit

D=23(1+n2) and a2=1