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Non-linear interactions in a viscous heat-conducting compressible gas

Published online by Cambridge University Press:  28 March 2006

Boa-Teh Chu  and
Leslie S. G. Kovásznay
Boa-Teh Chu
Affiliation:
The Johns Hopkins University, Baltimore, Maryland
Leslie S. G. Kovásznay
Affiliation:
The Johns Hopkins University, Baltimore, Maryland
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Abstract

The linearized equations of motion show that in a viscous heat-conducting compressible medium three modes of fluctuations exist, each one of which is a familiar type of disturbance. The vorticity mode occurs in an incompressible turbulent flow, the entropy mode is familiar as temperature fluctuations in low speed turbulent heat transfer problems, and the sound mode is the subject of conventional acoustics. A consistent higher order perturbation theory is presented with the only restrictions being that the Prandtl number is 3/4 and the viscosity and heat conductivity are monotinic functions of the temperature alone. The theory is based on expansion of the disturbance fields in powers of an amplitude parameter α. The non-linearity of the full Navier-Stokes equations can be interpreted as interaction between the three basic modes; in order to help physical insight the interactions are classed as ‘mass-like’, ‘force-like’, and ‘heat-like’ effects.

Besides the amplitude parameter α there is another subsidiary non-dimensional parameter ε which indicates the relative importance of viscosity and heat conduction effects as compared to the inertial effects, ε is proportional to the ratio of the molecular mean free path and the characteristic length of the flow pattern (Knudsen number). The main contribution of the paper is the outline of a consistent successive approximation for an arbitrary order in α and the presentation of explicit formulae for the second order (bilateral) interactions.

A special case of rather general significance is treated in more detail. This is when all three basic modes have intensities and length scales of the same orders of magnitude and in addition to α the parameter ε is also small; the second-order interactions are then relatively few and easily identifiable and are shown in table 1.

The present analysis also sheds some light on the ‘zero order’ approximation which treats the vorticity and entropy disturbances as a ‘frozen pattern’ and the sound field as propagating nondissipative waves. The interpretation of hot-wire measurements relies heavily on these simplified models and the present paper lends some support to these current hot-wire practices.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 3 , Issue 5 , February 1958 , pp. 494 - 514
Copyright
© 1958 Cambridge University Press

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References

Carrier, G. F. & Carlson, F. D. 1946 Quart. Appl. Math. 4, 1–12.
Chang, C. T. 1955 Thesis, Johns Hopkins University, ASTIA Document no. AD-95-263.
Cole, J. D. & Wu, T. Y. T. 1952 Proceedings, Heat Transfer and Fluid Mechanics Institute, 139–158.
Corrsin, S. 1952 J. Appl. Phys. 23, 113–118.
Favre, A. J., Gaviglio, J. J. & Dumas, R. 1954 La Rech. Aéro. 40, 7–14.
Favre, A. J., Gaviglio, J. J. & Dumas, R. 1957 J. Fluid Mech. 2, 313–342.
Kovásznay, L. S. G. 1953 J. Aero. Sci. 20, 657–674.
Kovásznay, L. S. G. 1955 Proceedings, Heat Transfer and Fluid Mechanics Institute.
Lagerstrom, P. A., Cole, J. D. & Trilling, L. 1949 California Institute of Technology, Contract Rep. ONR, Contract N6onr Task Order VIII.
Lighthill, M. J. 1952 Proc. Roy. Soc. A, 211, 564–587.
Lin, C. C. 1953 Quart. Appl. Math. 10, 295–306.
Morkovin, M. V. 1956 Johns Hopkins University Rep., AFORS TN-56-540, ASTIA Document no. AD-110-359.
Ribner, H. S. 1954 Nat. Adv. Comm. Aero., Wash., Tech. Note no. 3255.
Trilling, L. 1955 J. Acoust. Soc. Amer. 27, 425–431.
Westervelt, P. J. 1957 J. Acoust. Soc. Amer. 29, 199–203.
Wu, T. Y. T. 1952 Thesis, California Institute of Technology.