Lifting-line theory for a swept wing at transonic speeds
Author:
L. Pamela Cook
Journal:
Quart. Appl. Math. 37 (1979), 177-202
MSC:
Primary 76H05
DOI:
https://doi.org/10.1090/qam/542990
MathSciNet review:
542990
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The boundary-value problems describing the first-order corrections \[ O\left ( {\frac {1}{{AR}},\frac {1}{{AR}}\ln AR} \right )\] to two-dimensional flow about a lifting swept wing at transonic speeds $\left ( {{M_\infty } < 1} \right )$ are derived. The corrections are found by the use of the method of matched asymptotic expansions on the transonic small disturbance equations. The wing is at a sweep angle of $O\left ( {{{\left ( {1 - M_\infty ^2} \right )}^{1/2}}} \right )$ in the physical plane, hence of $O\left ( 1 \right )$ in the transonic small disturbance plane. As has been noted for subsonic flow, the finite sweep angle necessitates the introduction of terms $O\left ( {{{\left ( {AR} \right )}^{ - 1}}\ln AR} \right )$. These terms arise naturally in the matching process. Of particular interest is the derivation of the near field of a skewed lifting-line which is found by Mellin transform techniques. Also of interest is the fact that the influence of the nonzero sweep angle can be completely separated from the unswept solution.
N. Bleistein and R. Handelsman, Asymptotic expansions of integrals, Holt, Rinehart and Winston, New York, 1975
J. W. Boerstoel and G. H. Huizing, Transonic shock-free airfoil design by an analytic hodograph method, AIAA paper No. 74–539, presented AIAA 7th Fluid and Plasma Dynamic Conf. 1974
H. K. Cheng and M. M. Hafez, Transonic equivalence rule: a nonlinear problem involving lift, J. Fluid Mech. 72, 161–187 (1975)
- Julian D. Cole, Modern developments in transonic flow, SIAM J. Appl. Math. 29 (1975), no. 4, 763–787. MR 386435, DOI https://doi.org/10.1137/0129065
- L. Pamela Cook, A uniqueness proof for a transonic flow problem, Indiana Univ. Math. J. 27 (1978), no. 1, 51–71. MR 483965, DOI https://doi.org/10.1512/iumj.1978.27.27005
- L. Pamela Cook and Julian D. Cole, Lifting line theory for transonic flow, SIAM J. Appl. Math. 35 (1978), no. 2, 209–228. MR 483999, DOI https://doi.org/10.1137/0135016
William Fredrick Durand, ed., Aerodynamic theory: a general review of progress: Vol. II, Division E, General aerodynamic theory, perfect fluids, Th. von Karman and J. M. Burgers, 1943, Durand Reprinting Committee, Calif. Inst, of Technology, pp. 197–201
- P. R. Garabedian and D. G. Korn, Analysis of transonic airfoils, Comm. Pure Appl. Math. 24 (1971), 841–851. MR 297209, DOI https://doi.org/10.1002/cpa.3160240608
E. M. Murman and J. D. Cole, Inviscid drag at transonic speeds: studies in transonic flow III, UCLA School of Eng. Rpt. 7603, Dec. 1974; also AIAA paper No. 75–540, presented at AIAA 7th Fluid and Plasma Dynamic Conference, June 1974
R. D. Small, Transonic lifting line theory: numerical procedure for shock-free flows, to appear, AIAA J., June 1978
R. D. Small, Calculation of a transonic lifting line theory, Studies in Transonic Flow VI, UCLA School of Eng. and Applied Science Report, April 1978
- James K. Thurber, An asymptotic method for determining the lift distribution of a swept-back wing of finite span, Comm. Pure Appl. Math. 18 (1965), 733–756. MR 183199, DOI https://doi.org/10.1002/cpa.3160180410
- Milton Van Dyke, Perturbation methods in fluid mechanics, Annotated edition, The Parabolic Press, Stanford, Calif., 1975. MR 0416240
N. Bleistein and R. Handelsman, Asymptotic expansions of integrals, Holt, Rinehart and Winston, New York, 1975
J. W. Boerstoel and G. H. Huizing, Transonic shock-free airfoil design by an analytic hodograph method, AIAA paper No. 74–539, presented AIAA 7th Fluid and Plasma Dynamic Conf. 1974
H. K. Cheng and M. M. Hafez, Transonic equivalence rule: a nonlinear problem involving lift, J. Fluid Mech. 72, 161–187 (1975)
Julian D. Cole, Modern developments in transonic flow, S.I.A.M. J. Appl. Math. 29, 763–787 (1975)
L. Pamela Cook, A uniqueness proof for a transonic flow problem, Indiana Univ. Math. J. 27, 51–71 (1978)
L. Pamela Cook and Julian D. Cole, Lifting line theory for transonic flow, S.I.A.M. J. Appl. Math. 27, 1978
William Fredrick Durand, ed., Aerodynamic theory: a general review of progress: Vol. II, Division E, General aerodynamic theory, perfect fluids, Th. von Karman and J. M. Burgers, 1943, Durand Reprinting Committee, Calif. Inst, of Technology, pp. 197–201
P. Garabedian and D. G. Korn, Analysis of transonic airfoils, Comm. Pure Appl. Math. 24, 848–851 (1971)
E. M. Murman and J. D. Cole, Inviscid drag at transonic speeds: studies in transonic flow III, UCLA School of Eng. Rpt. 7603, Dec. 1974; also AIAA paper No. 75–540, presented at AIAA 7th Fluid and Plasma Dynamic Conference, June 1974
R. D. Small, Transonic lifting line theory: numerical procedure for shock-free flows, to appear, AIAA J., June 1978
R. D. Small, Calculation of a transonic lifting line theory, Studies in Transonic Flow VI, UCLA School of Eng. and Applied Science Report, April 1978
James K. Thurber, An asymptotic method for determining the lift distribution of a swept-back wing of finite span, Comm. Pure Applied Math. XVIII, 733–756 (1965)
M. D. Van Dyke, Perturbation methods in fluid mechanics, Parabolic Press, 1975
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
76H05
Retrieve articles in all journals
with MSC:
76H05
Additional Information
Article copyright:
© Copyright 1979
American Mathematical Society