Skip to main content Accessibility help

Longitudinal–transverse aerodynamic force in viscous compressible complex flow

  • L. Q. Liu (a1), Y. P. Shi (a1), J. Y. Zhu (a1), W. D. Su (a1), S. F. Zou (a1) and J. Z. Wu (a1)...


We report our systematic development of a general and exact theory for diagnosis of total force and moment exerted on a generic body moving and deforming in a calorically perfect gas. The total force and moment consist of a longitudinal part (L-force) due to compressibility and irreversible thermodynamics, and a transverse part (T-force) due to shearing. The latter exists in incompressible flow but is now modulated by the former. The theory represents a full extension of a unified incompressible diagnosis theory of the same type developed by J. Z. Wu and coworkers to compressible flow, with Mach number ranging from low-subsonic to moderate-supersonic flows. Combined with computational fluid dynamics (CFD) simulation, the theory permits quantitative identification of various complex flow structures and processes responsible for the forces, and thereby enables rational optimal configuration design and flow control. The theory is confirmed by a numerical simulation of circular-cylinder flow in the range of free-stream Mach number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}M_{\infty }$ between 0.2 and 2.0. The L-drag and T-drag of the cylinder vary with $M_{\infty }$ in different ways, the underlying physical mechanisms of which are analysed. Moreover, each L-force and T-force integrand contains a universal factor of local Mach number $M$ . Our preliminary tests suggest that the possibility of finding new similarity rules for each force constituent could be quite promising.


Corresponding author

Email address for correspondence:


Hide All
Burgers, J. M. 1921 On the resistance of fluids and vortex motion. Koninklijke Nederlandse Akademie van Wetenschappen Proceedings Series B Physical Sciences 23, 774782.
Chang, C. C. & Lei, S. Y. 1996 On the sources of aerodynamic forces: steady flow around a cylinder or a sphere. Proc. R. Soc. Lond. A 452, 23692395.
Chang, C. C., Su, J. Y. & Lei, S. Y. 1998 On aerodynamic forces for viscous compressible flow. Theor. Comput. Fluid Dyn. 10, 7190.
Chu, B. T. & Kovásznay, L. S. G. 1958 Non-linear interactions in a viscous heat conducting compressible gas. J. Fluid Mech. 3, 494514.
Fiabane, L., Gohlke, M. M. & Cadot, O. 2011 Characterization of flow contributions to drag and lift of a circular cylinder using a volume expression of the fluid force. Eur. J. Mech. (B/Fluids) 30, 311315.
Gilbarg, D. & Paolucci, D. 1953 The structure of shock waves in the continuum theory of fluids. J. Rat. Mech. Anal. 2, 617642.
Huang, G. C. 1994 Unsteady Vortical Aerodynamics: Theory and Applications. Shanghai Jiaotong University Press, (in Chinese).
von Kármán, Th. 1941 Compressibility effects in aerodynamics. J. Aero. Sci. 8, 337356.
von Kármán, Th. & Sears, W. R. 1938 Airfoil theory for non-uniform motion. J. Aeronaut. Sci. 5, 379390.
Kovásznay, L. S. G. 1953 Turbulence in supersonic flow. J. Aero. Sci. 20, 657674.
Lagerstrom, P. A. 1964 Laminar Flow Theory. Princeton University Press.
Lagerstrom, P. A., Cole, J. D. & Trilling, L.1948 Problems in the theory of viscous compressible fluids. GALCIT Tech Rep. 6.
Li, G. J. & Lu, X. Y. 2012 Force and power of flapping plates in a fluid. J. Fluid Mech. 712, 598613.
Liu, L. Q., Wu, J. Z., Shi, Y. P. & Zhu, J. Y. 2014 A dynamic counterpart of Lamb vector in viscous compressible aerodynamics. Fluid Dyn. Res. (in press).
Luo, Y. B.2004 Boundary vorticity dynamics, derivative moment theory, and their applications in flow diagnosis and control. Master thesis, Peking University.
Mao, F., Shi, Y. P. & Wu, J. Z. 2010 On a general theory for compressing process and aeroacoustics: linear analysis. Acta Mechanica Sin. 26, 355364.
Mao, F., Shi, Y. P., Xuan, L. J., Su, W. D. & Wu, J. Z. 2011 On the governing equations for the compressing process and its coupling with other processes. Sci. China. Phys. Mech. Astron. 54, 11541167.
Marongiu, C. & Tognaccini, R. 2010 Far-field analysis of the aerodynamic force by Lamb vector integrals. AIAA J. 48, 25432555.
Marongiu, C., Tognaccini, R. & Ueno, M. 2013 Lift and lift-induced drag computation by Lamb vector integration. AIAA J. 51, 14201430.
McCune, J. E. & Tavares, T. S. 1993 Perspective: unsteady wing theory—the Kármán/Sears legacy. J. Fluids Engng 115, 548560.
Mele, B. & Tognaccini, R. 2014 Aerodynamic force by Lamb vector integrals in compressible flow. Phys. Fluids 26, 056104.
Oswatitsch, K. 1956 Gas Dynamics. Academic.
Pierce, A. D. 1989 Acoustics: An Introduction to its Physical Principles and Applications. Acoustical Society of America.
Prandtl, L.1918 Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse. Tragflügeltheorie. I. Mitteilung, pp. 451–477.
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.
Sherman, F. S.1955 A low-density wind-tunnel study of shock structure and relaxation phenomena in gases. NACA Tech Rep. 3298.
Wu, J. C. 1981 Theory for aerodynamic force and moment in viscous flows. AIAA J. 19, 432441.
Wu, J. Z., Lu, X. Y., Yang, Y. T. & Zhang, R. K. 2010 Vorticity dynamics in complex flow diagnosis and management. Pei-Yuan Chou memorial lecture. In Proceedings 13th Asian Congr. Fluid Mech. Bangladesh Society of Mechanical Engineering, Dhaka (ed. Sadrul Islam, A. K. M.), pp. 1938.
Wu, J. Z., Lu, X. Y. & Zhuang, L. X. 2007 Integral force acting on a body due to local flow structures. J. Fluid Mech. 576, 265286.
Wu, J. Z., Ma, H. Y. & Zhou, M. D. 2006 Vorticity and Vortex Dynamics. Springer.
Wu, J. Z. & Wu, J. M. 1993 Interactions between a solid surface and a viscous compressible flow field. J. Fluid Mech. 254, 183211.
Wu, J. Z. & Wu, J. M. 1996 Vorticity dynamics on boundaries. Adv. Appl. Mech. 32, 19275.
Wu, J. Z., Wu, H. & Li, Q. S. 2009 Boundary vorticity flux and engineering flow management. Adv. Appl. Math. Mech. 1, 353366.
Wu, J. Z., Wu, J. M. & Wu, C. J. 1988 A viscous compressible flow theory on the interaction between moving bodies and flow field in the $(\boldsymbol{\omega}, \vartheta )$ framework. Fluid Dyn. Res. 3, 203208.
Wu, T. Y. 1956 Small perturbations in the unsteady flow of a compressible, viscous and heat-conducting fluid. J. Math. Phys. 35, 1327.
Wu, T. Y. 2007 A nonlinear theory for a flexible unsteady wing. J. Engng Maths 58, 279287.
Xu, C. Y., Chen, L. W. & Lu, X. Y. 2009 Effect of mach number on transonic flow past a circular cylinder. Chin. Sci. Bull. 54, 18861893.
Xu, C. Y., Chen, L. W. & Lu, X. Y. 2010 Large-eddy simulation of the compressible flow past a wavy cylinder. J. Fluid Mech. 665, 238273.
Yang, Y. T., Zhang, R. K., An, Y. R. & Wu, J. Z. 2007 Steady vortex force theory and slender-wing flow diagnosis. Acta Mechanica Sin. 23, 609619.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed