# TABULATION OF RESULTS—COMPRESSIBLE FLOW

In the case of supersonic flow, many authors give numerical tables of aerodynamic coefficients for oscillating airfoils. For flutter-calculation purpose the following reference, prepared by E. C. Kennedy, is the most comprehensive:

1. Handbook of Supersonic Aerodynamics. liM M 1.1 (0.1) 2.0 (0.2)

4.0 (0.5) 5.0 (1.0) 12. Q: 0.01 (0.01) 0.04 (0.02) 0.10 (0.05) 0.40 (0.10)

1.0 (0.20) 3.0 (0.50) 5.0, 7.5, 10, 15, 20. (8 fig.)

The coefficients CLh, CL(t, CMh, CMa listed in this Handbook are, respectively, the Lh, Le, Mh, Ma defined in Chapter 6. The independent entry in the Handbook is the frequency parameter Cl, which is related to the reduced frequency к and Mach number M by the relation

In the following reference, compiled by Y. L. Luke on the basis of tables published by Garrick and Rubinow and Jordan, the independent entries are the reduced velocity Ujbco = l/к, and the coefficients Lh,

Mh, Ma:

2. Tables of Coefficients for Compressible Flutter Calculations M: if, , if, I, 2, f, if, 5. 1/k: approx. 0.1 (irreg.) 100. (5 dec.)

The scope of these tables is indicated above in the ranges of M and Cl[40] Note that in the supersonic case it suffices to tabulate the four fundamental aerodynamic coefficients named above. The lift and moment coefficients

References |
Possio, Ref. 14.12 |
Possio, Ref. 14.38 |
von Borbely, Ref. 14.29 |
Kussner, Ref. 15.12 |
Schwarz, Ref. 14.16 |
Dietze, Ref. 14.2 |
Temple Jahn, Ref. 14.42 |

Chord |
/ |
1 |
t |
21 |
2 |
l |
c |

Free-stream velocity |
Vs |
K, |
u0 |
V |
V |
V |
V |

Density, undisturbed fluid |
P |
Pi |
Po |
p |
p |
P CO |
Po |

Circular frequency |
Q |
Q |
a |
V |
V |
CO |
2-іrf |

Reduced frequency |
CO |
(0 |
— |
CO |
cor |
COr |
X 2 |

Frequency parameter ^ ) |
— |
— |
O) |
— |
— |
— |
У |

Kernel of Possio’s equation |
V, + iVg aa) |
— |
— |
— |
ж |
ЗҐ |
— |

Mach number |
X |
X |
Mach angle § |
p |
X |
M |
M |

Downward displacement at reference point |
-n |
a |
– Y |
_ |
_ |
b |
z’o |

Pitching angle (+ nose up) |
a |
b |
E |
— |
• — |
— |
a’ |

Lift on wing (+ upward) |
p |
p |
P |
— |
— |
— |
L’ |

Pitching moment (+ nose up) |
– M |
– M |
– M |
— |
— |
— |
AT |

References |
Garrick Rubinow, Ref. 14.32 |
Karp, et al., Ref. 14.1 |
Luke, Ref. 14.8 |
Turner, Ref. 14.21 |
Timman, Ref. 14.17 |
Fettis, Ref. 14.4 |
Supersonic Handbook Ref. 14.44 |
This Book |

Chord |
2b |
21 |
2b |
/ |
21 |
2b |
2b |
2b |

Free-stream velocity |
V |
U |
V |
V |
V |
V |
V |
U |

Density, undisturbed fluid |
p |
Po |
p |
P |
Po |
p |
P |
P |

Circular frequency |
CO |
CO |
CO |
CO |
V |
V |
CO |
CO |

Reduced frequency – |
к |
COT |
к |
cor |
CO |
CO |
к |
к |

/ 2M2k Frequency parameter pp——— j-J |
CO |
X |
— |
— |
— |
/t |
Q. |
Q. |

Kernel of Possio’s equation |
— |
К |
К |
К |
— |
К |
— |
К |

Mach number |
M |
M |
м |
M |
/3 |
X |
M |
M |

Downward displacement at refer- |
||||||||

ence point |
К |
– Y |
h |
4s |
A |
h |
h’ |
h |

Pitching angle (+ nose up) |
«0 |
E |
a |
4d |
В |
a |
a |
a |

Lift on wing (+ upward) |
— p |
Lq |
-L |
&Ps |
– К |
– L |
-L |
L |

Pitching moment (+ nose up) |
Mx |
M0 |
M |
amd |
м |
M |
M |
Mx* |

439 |

,r0 is the point about which the moment is taken. Moment about the mid-chord is written as Mi/S, etc. |

involving the motion of a flap and a tab can be expressed in terms of these four fundamental coefficients. Explicit formulas expressing these relations involving a flap are given in Ref. 14.44. A complete list of formulas including both the flap and the tab can be found in Ref. 14.1.

For subsonic flow, the published data are meager. The principal sources are the papers by Possio, Frazer, Frazer and Skan, Dietze, Schade, Schwarz, Turner and Rabinowitz, Timman, Van de Vooren and Greidanus, and Fettis (see bibliography). The numerical results of the first eight authors named above have been compiled and converted into the Lh, Mh, • • ■ coefficients by Luke in Ref. 14.8. A summary of the published tables is given in Table 14.1. In Table 14.2, the notations used in some of the most important references, for both the supersonic and the subsonic cases, are listed.

In Table 14.1, the symbols Lh, Mh> etc., are defined in § 6.10. These coefficients are referred to the 1/4-chord axis for both the rotation a and the moment Mx. In Timman, van de Vooren and Greidanus’s papers, the rotation and moment are referred to the mid-chord axis; hence, a transformation is needed when comparison of the data is to be made. This is indicated in Table 14.1. Under each column the factors in parentheses are the tabulated quantity expressed in the author’s notation. The adjacent coefficients are the factors necessary to convert to the corresponding Lh, La, Mh, or Ma, which is listed at the left on the same horizontal

line. Thus, under Timman and opposite Lh, we find the entry ~ (ka).

This indicates that (ka) is given by Timman and that Lh = (ka)/k2. With the exception of the symbols for the quantities actually tabulated in the references, the notation of this book is used throughout. The notations of the original authors may be found in Table 14.2.

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