Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 

 

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

$\displaystyle O\left( {\frac{1}{{AR}},\frac{1}{{AR}}\ln AR} \right)$

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.

References [Enhancements On Off] (What's this?)

  • [1] N. Bleistein and R. Handelsman, Asymptotic expansions of integrals, Holt, Rinehart and Winston, New York, 1975
  • [2] 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
  • [3] H. K. Cheng and M. M. Hafez, Transonic equivalence rule: a nonlinear problem involving lift, J. Fluid Mech. 72, 161-187 (1975)
  • [4] Julian D. Cole, Modern developments in transonic flow, SIAM J. Appl. Math. 29 (1975), no. 4, 763–787. MR 0386435, https://doi.org/10.1137/0129065
  • [5] L. Pamela Cook, A uniqueness proof for a transonic flow problem, Indiana Univ. Math. J. 27 (1978), no. 1, 51–71. MR 0483965, https://doi.org/10.1512/iumj.1978.27.27005
  • [6] L. Pamela Cook and Julian D. Cole, Lifting line theory for transonic flow, SIAM J. Appl. Math. 35 (1978), no. 2, 209–228. MR 0483999, https://doi.org/10.1137/0135016
  • [7] 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
  • [8] P. R. Garabedian and D. G. Korn, Analysis of transonic airfoils, Comm. Pure Appl. Math. 24 (1971), 841–851. MR 0297209, https://doi.org/10.1002/cpa.3160240608
  • [9] 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
  • [10] R. D. Small, Transonic lifting line theory: numerical procedure for shock-free flows, to appear, AIAA J., June 1978
  • [11] 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
  • [12] 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 0183199, https://doi.org/10.1002/cpa.3160180410
  • [13] Milton Van Dyke, Perturbation methods in fluid mechanics, Annotated edition, The Parabolic Press, Stanford, Calif., 1975. MR 0416240

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 76H05

Retrieve articles in all journals with MSC: 76H05


Additional Information

DOI: https://doi.org/10.1090/qam/542990
Article copyright: © Copyright 1979 American Mathematical Society


Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2017 Brown University
Comments: qam-query@ams.org
AMS Website