A Survey of Unsteady Hypersonic Flow Problems

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- 38 - where m is vehicle mass; p is air density; S and b are representative areas and lengths of the *vehicle. If p is very large, then the terms quartic (Duncan, Ref. 41, Chapter 5): C, D and E in the stability AX4 + Bh3 + Ch' + Dh + E = 0 . . . (3.2) are very much larger than the terms A and B and it is possible to give fairly simple expressions for the roots of the quartic in terms of the coefficients: which, if the dominant terms only are retained, become where CD is the drag coefficient; CL, is the rate of change of lift coefficient with angle of attaok: (q/ad ; 12% is the rate of dsnge of pitching moment coefficient with angle of attack: (acdad ; . . . (3.3) Cmq is the rate of change-of pitching moment coefficient with pitching 8b velocity parameter - : ; V iB is the non-dimensional form of the pitching moment of inertia about the oentre of gravity, IB : IB/pSb3+ Kz = ?+" and R is the radius of the flight path. P= The first pair of roots represent the usual phugoid type of oscillation and the seoond pair represent the usual predominantly pitching oscillation, provided the aircraft 1s statically stable, i.e., provded that
- 39 - The real and imaginary parts of the first patr of roots in equation (3.3) are both of order unity so that the period and the time constant of the rate of decay of the phugoid oscillation are both of the same order of magnitude as the natural unit of time . . . (3.6) which is large at high speeds ua , rided that KL is not very small, because of the size of the term - , the Froude number. In the second pair of ( & > roots, representing the pitching oscillation, the real part is again of order unity, but the imaginary part is, usually, large. The period of the pitching oscillation is even by: T = P T.+ q$(!$ 27c - ( A > [$.%(;)I+ = qb;-+Lf . . . (3.7) so that it depends on KL, and becomes large only at high altitudes where XL is large, and does not become very small beoause of the limit on the value of KL set by structural limitations on speed at low altitudes. The time constant of the rate of decay of the pitching oscillation becomes, then, very long in compdison with the period, And the osclllatlon is poorly damped. Assuming likely values of Cma and iB, Nonweiler estimates that, for KL of order O*Ol,whioh he consders a likely lower limit, the period of the pitching oscillation will be 2 or 3 seconds, and the time to half amplitude will occupy a few periods; for KL of order unity, the period will be 20 to 30 seconds, and the time to half amplxtude would be several minutes; and for KL large, which would correspond to conditions close to a high altitude orbit, the time to half amplitude could be an hour. The phugoid motion is very lightly damped and, in the time occupied by .a few cycles of the pitching motion, is equivalent to the motion of a system with two degrees of freedom in neutral equilibrium. This characteristic of the phugoid motion, coupled with the small damplng of the pitohing motion, could present novel control problems since pilot or automatic action to control the pitching motion could cause drift in the speed and altitude. The qua itstive cone usions of Nmweiler are confirmed by the detalled analysis of Etkint3 and. Rangi 4i for a vehicle with hypothetical characteristics flying at a constant altitude. The analysis takes account of the effects of using a rotating axis system, and of changes in the radius of the flight path, which are neglected by Nonweiler. Because of the additional terms which these factors introduce, the stability equation is a quintic, giving three normal modes, two oscillatory and one non-oscillatory. The characteristics of the oscillatory modes are shown in Figs. w(a), (b) and (c) with some particulars of the hypothetXa1 vehicle./
Page 1: MINISTRY OF AVlATlON AERONAUTICAL R
Page 4 and 5: -2- CONTENTS Int~~duction . . . . .
Page 6 and 7: General Surves and Conclusions -4-
Page 8 and 9: -6- and values of M6 < 1) and shock
Page 10 and 11: -a- have been ~0nce171ed with the f
Page 12 and 13: - to - APPENDICES Detailed Reviews
Page 14 and 15: - 12 - where W is the weight of the
Page 16 and 17: - 14 - In practice, equation (2.1)
Page 18 and 19: - 16 - nsteady Newtonian theory is
Page 20 and 21: - 18 - Then the transformed velocit
Page 22 and 23: - 20 - of a disturbance is of order
Page 24 and 25: - 22 - u cot cl ucota = . . . (2.35
Page 26 and 27: - 24 - minimum are that its variati
Page 28 and 29: - 26 - This simple form of the meth
Page 30 and 31: t - 28 - and r “” + u’ i + (1
Page 32 and 33: - 30 - The cut-of-phase component o
Page 34 and 35: - 32 - indication of their reliabil
Page 36 and 37: -34- between theory and. experiment
Page 38 and 39: - 36 - will no longer apply and a s
Page 42 and 43: - I+0 - vehicle. (The difference be
Page 44 and 45: -4z- where C . is the rate of chang
Page 46 and 47: -44- is to reduce further the range
Page 48 and 49: 46 - APPENDIX IV Review of Flutter
Page 50 and 51: where z.2 = ll& *.. (4.1) . . . (4.
Page 52 and 53: - 50 - The thcorctlcal deduction th
Page 54 and 55: - 52- frequcnoy. Theoretical calcul
Page 56 and 57: -%- wiierc h and au are the flutter
Page 58 and 59: - 56 - mean incidence case; but, fo
Page 60 and 61: - 58 - pressure both for divergence
Page 62 and 63: - 60 - .h = -. q/MA-1 w3 12(1-l?) E
Page 64 and 65: a a, A b c cD CL %l %I c% C Q% C mq
Page 66 and 67: AP 9 P r 'b F a B % R % Be Rex s 9
Page 68 and 69: Y Y yi 6 F A K - 66 - instantaneous
Page 70 and 71: Dcfmitions of derivatives "B L, = -
Page 72 and 73: No. Author(s) 12 Ashley, Holt and Z
Page 74 and 75: &. Author(sl 36 Moore, F. K. 37 Oli
Page 76 and 77: - 74 - No. Author(s) Title, etc. 52
Page 78 and 79: No 65 66 67 68 69 70 . 71 72 73 Aut
Page 81 and 82: FIG. I (a) Slender wing/body combin
Page 83 and 84: h ft x From Ref.5 and calculation b
Page 85 and 86: FIGS.5-7 FIG 5 Axis system for thin
Page 87 and 88: %‘8 Ro (-X/R; e- -0 \t I FIG. 9 R
Page 89 and 90: -0.02 0 ’ 0.02 cP 0.04 0.06 O-08
Page 91 and 92:
O- I- 2 - 3 - 4 - ,5- / , / / / /
Page 93 and 94:
FIG. 15 (a) I
Page 95 and 96:
CP (C) of5 = o rrom Flg.ZJof R 19.9
Page 97 and 98:
O-/ 0.12 . 0 OS08 1. CP 0.04 0 -0.0
Page 99 and 100:
O-16 CP o-12 From Ref.29 ;c) M = 6.
Page 101 and 102:
Inviscid theory - - - - Theory with
Page 103 and 104:
I ‘6 2 FIG. I 9 From Fig.3.8 of R
Page 105 and 106:
1.2 O-8 0.4 0 -0 *4 -0.8 -1.2 0.8 A
Page 107 and 108:
0.4 0 From Fig. 3.26 of Ref. 26 FIG
Page 109 and 110:
I.0 / / / / ,: kMM; k&I M; A 2-o 0
Page 111 and 112:
k M Mq I.4 O-8 0.6 0.4 0*2 0 -0.2 F
Page 113 and 114:
From Ref. 38 -0.16 (a) Details of m
Page 115 and 116:
IO'4- 5 x 101J- IO' J- 5 x10; 2- 2-
Page 117 and 118:
kit From Fig. 2 of Ref. 48 FIG. 31
Page 119 and 120:
. 2.4 2.0 l-6 K crit l-2 0.0 0.4 0
Page 121 and 122:
M 4 3 2 (,t”, 10-4) Tern OF I Fro
Page 123 and 124:
From Fig.8 of Ref. 56 FIG. 36 50 10
Page 125 and 126:
“f,o 5 I From Ref. 56 I I 0 IO FI
Page 127 and 128:
From Fig.2 in Ref. 57 FIG. 39 ----L
Page 129 and 130:
From Ref 57 0.7 6 0 - - - - - 3% --
Page 131 and 132:
From Ref. 57 o-7. 0.61 0 - - - - 3%
Page 133 and 134:
6 I i 1 nbol 6 = From Ref.61--/ Tip
Page 135 and 136:
. b 0 oao b FIG .44 c) 0 N 0 N . 0
Page 137 and 138:
FIG. 46 From Fig. 5.1 of Ref. 26 0
Page 139 and 140:
From Ref. 57 n.Al I 0.5 FIG. 48 I I
Page 141 and 142:
From Ref. 57 14 IZ- FIG.50 c - 3% t
Page 143 and 144:
(“f Iaxp w Ah (“f )axp (“f )t
Page 145 and 146:
I From Ref. 65 Modes as in FIG.53 P
Page 147 and 148:
q 5 0 7 6 5 4 From Fink 73 nf Rrf~
Page 149 and 150:
2.1 2 -0 Data taken from Ref. 63 o-
Page 151 and 152:
$3 i I a( I.2 (Ib/sq 1 in.) ’ ‘
Page 153 and 154:
I From Ref.71 FIG. 62 0 5 IO M 15 2
Page 155 and 156:
0.00 o-00 FIG. 64 FImm -... Fio~ =
Page 157 and 158:
A.R.C. C.P. No.901 Harch. 1%5 Wood,
Page 160:
0 Crown copyright 1966 Prmted and p
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A Survey of Unsteady Hypersonic Flow Problems

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