A Survey of Unsteady Hypersonic Flow Problems

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where z.2 = ll& *.. (4.1) . . . (4.2) & = Mlcr, . . . (4.3) & = M& ana F = aerodynamic force on section = -4pbUaW[(k+iLz)$ + (J&+&)oL] & = aerodynamic moment about pitching axis = -4pb2U2ka~(~+i.&); + (Ivl,+i&)aj. For simple harmonic motion, and piston theory aerodynamics b = an, = 0, Mz = b b = (I/& , b = (l/k)b = (l/k)b 3 and, taking a double-wedge section as typical, (y+l) G4 = I+- EW,Y+ 04S)=l 4 G = (I-23 c (y+l) 1 + - 4 [(Ma3)2 + (Ms)2 1 3 (y+l) $ = 4+x(l+q 1 + - [(MaJ2 + (MS)"] c 3 (y+l) - - @as) 4 . . . (4.4) . . . (4.5) (y+l) - 2(1-9 - (~6) . . . (4.6) 4 4 where 6 is the thickness ratio of the section, and cz3 is the mean incidence. These results show three aerodynamic parameters: IJM, an altitude parameter; Mb, a thickness parameter, and Mar,, a mean lncldence parameter. This result is specifically for a double-wedge section, but Chawla shows that the results are the same for more general sections, though the thickness parameter applies only within a family of shapes, of course. The slgniflcance of Id6 and Ma, is in agreement with the results of hypersonic small-disturbance theory (Appendix II) and of experiment, which show that these are similarity parameters for hypersonic flows. Besides the parameters from Chawla's analysis, other factors have been found to be significant. These are profile shape, the effeot of strong leadingedge shock waves, leading-edge bluntness, and aercdynamlc nonlinearities. The effects of changes of @d, M6, Ma, and these other parameters on flutter of the section will now be discussed. (W
(i) Altitude parameter. IJM - 49 - Fzgs. 36 and 37, reproduced from Ref. 56, show the relatIonships between flutter speed, frequency ratio and altitude parameter (9) for a flat plate, twc-dimensional section. The dependence of flutter speed on altitude is obscured III some degree by the form of the parameter @, since M is dependent on Vf. The relationship is a little clearer if the fl is re-expressed *a . . . (4.7) Then, from the graphs, it is clear that, for a given altitude (P and p fixed) and frequency ratlo, there will be a unique flutter speed and, since @d increases with altitude independently of any change m the,flutter speed (because of the changes III P and p), the flutter speed itself must also increase with altitude. The relationsiiip of flutter speed and altitude can be expressed more explxitly,in an approximate form, by making use of tpe fact that, u Fig. 36, Vf can be,shown to vary approximately as (pap. Using this relatlonship I+ 1s possible to write, for the flutter speeds at two altitudes . . . (4.8) and, if equation (4.8) 1s squared, and the following substitutxons are made: whcrc suff'ix o refers to B reference altxtude, then vf POcb Id=--;p=- . . . (4.9) * P v 2 f = (4p1) Vf i M f (aapa) . . . (4.10) and >=($Pi) h . . . (4.11) Mf 1 (a: pz) = ii- * If, then, the changes of p and .e with altitude are known, ourves of flutter speed and Mach number against altitude can be plotted. Such curves are shorn in Fig. 38(a) and (b). In Fig. 38(b) use is made of the fact that, for 35 330 ft < h < 105 000 ft, .%/a, c 1, and VA Mf Pi = 2 = _. . . . (4.12) The/ ____________________---------------------- *This result can also be seen from equation (4.1) since - - @[2x, L - L(l-ga)] >> (-i&C2 +z) for fl large.
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 40 and 41: - 38 - where m is vehicle mass; p i
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 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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