Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-19T03:28:48.388Z Has data issue: false hasContentIssue false
  • English
  • Français

An exact solution of linearized flow of an emitting, absorbing and scattering grey gas

Published online by Cambridge University Press:  29 March 2006

Ping Cheng  and
A. Leonard
Ping Cheng
Affiliation:
NASA-Ames Research Center, Moffett Field, California Present address: Department of Mechanical Engineering, University of Hawaii, Honolulu, Hawaii 96822.
A. Leonard
Affiliation:
Stanford University, Stanford, California
Get access Rights & Permissions [Opens in a new window]

Abstract

The governing equations for the problem of linearized flow through a normal shock wave in an emitting, absorbing, and scattering grey gas are reduced to two linear coupled integro-differential equations. By separation of variables, these equations are further reduced to an integral equation similar to that which arises in neutron-transport theory. It is shown that this integral equation admits both regular (associated with discrete eigenfunctions) and singular (associated with continuum eigenfunctions) solutions to form a complete set. The exact closed-form solution is obtained by superposition of these eigen-functions. If the gas downstream of a strong shock is absorption–emission dominated, the discrete mode of the solution disappears downstream. The effects of isotropic scattering are discussed. Quantitative comparison between the numerical results based on the exact solution and on the differential approximation are presented.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 45 , Issue 4 , 26 February 1971 , pp. 673 - 699
Copyright
© 1971 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Carlson, D. J. 1966 Metalized solid propellants exhaust plumes. AIAA Paper 66-652, Pt. II.Google Scholar
Case, K. M. & Zweifel, P. F. 1967 Linearized Transport Theory. Addison-Wesley.
Cheng, P. 1965 Study of the flow of a radiating gas by a differential approximation. Ph.D. Dissertation, Stanford University.
Cheng, P. & Leonard, A. 1970 Application of singular eigenfunction expansions to the propagation of periodic disturbances in a radiating grey gas. Submitted to Phys. Fluids.Google Scholar
Clarke, J. F. 1962 Structure of radiating resisted shock waves Phys. Fluids, 5, 1347.Google Scholar
Copson, E. T. 1955 Theory of Functions of a Complex Variable. Oxford University Press.
Fontenot, J. E. 1965 Thermal radiation from solid rocket plumes at high altitude A.I.A.A. J. 3, 970.Google Scholar
Heaslet, M. A. & Baldwin, B. S. 1963 Predictions of the structure of radiation-resisted shock waves Phys. Fluids, 6, 781.Google Scholar
Lighthill, M. J. 1960 Introduction to Fourier Analysis and Generalized Functions. Cambridge University Press.
Mendelson, M. R. & Summerfield, G. C. 1964 One-speed neutron transport in two adjacent half-spaces J. Math. Phys. 5, 668.Google Scholar
Mitchner, M. & Vinokur, M. 1963 Radiation smoothing of shock with and without a magnetic field Phys. Fluids, 6, 1682.Google Scholar
Muskhelishvili, N. I. 1953 Singular Integral Equations. Groningen: Noordhoff.
Pearson, W. E. 1964 On the direct solution of the governing equation for radiation-resisted shock wave. NASA TN D-2128.Google Scholar
Rochelle, W. C. 1967 Review of thermal radiation from liquid and solid propellant rocket exhausts. NASA TMX 53579.Google Scholar
Vincenti, W. G. & Kruger, C. H. 1965 Introduction to Physical Gas Dynamics. Wiley.