Numerical solution of the unsteady navier-stokes equations in curvilinear coordinates: The hypersonic blunt body merged layer problem

  • Keith J. Victoria
  • George F. Widhopf
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 19)


The Navier-Stokes solution of this study agrees with the Monte Carlo result in the shock layer region within the statistical scatter of the Monte Carlo calculation. Thus, it appears that the origin of the disagreement between the thin layer Navier-Stokes solution and the molecular simulation solution in the shock layer region is in the thin layer approximations and not in the Navier-Stokes stress-strain model and Fourier heat conduction law. It should be pointed out, however, that at this transitional flow condition (Kn = 0.10) the Monte Carlo solution shows that the temperature is not in equilibrium. Thus, for this particular flow condition, the molecular simulation technique is more appropriate.

The study also revealed interesting results regarding the numerical computation procedure in addition to the physical results. In particular, it was found that application of the leap-frog/Dufort-Frankel finite difference approximation to the continuity equation in curvilinear coordinates can lead to numerical instabilities. A simple time-averaging of the density in the inhomogeneous term in the continuity equation eliminated these instabilities. With this modification, it was possible to converge the numerical solution (cycle to cycle) for all flow variables to six significant figures.

Global conservation of mass, momentum and total energy can be accurately maintained with use of the non-conservative form of the Navier-Stokes equations. These quantities were found to be conserved to within one percent of their respective freestream inflow values.


Shock Layer Hypersonic Flow Finite Difference Approximation Finite Difference Equation Stagnation Line 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bird, G. A., “Shock Wave Structure in a Rigid Sphere Gas,” Rarefied Gas Dynamics, edited by J. H. deLeeuw, Supplement 3, Vol. I, 1965, p. 216.ADSGoogle Scholar
  2. Cheng, H. K., “The Blunt-Body Problem in Hypersonic Flow at Low Reynolds Number,” Cornell Aero Lab Report No. AF-1285-A-10, June 1963.Google Scholar
  3. Dellinger, T. C., “Computation of Nonequilibrium Merged Stagnation Shock Layers by Successive Accelerated Replacement,” AIAA Preprint No. 69-655, June 16–18, 1969.Google Scholar
  4. Hayes, W. D. and Probstein, R. F., Hypersonic Flow Theory, Academic Press, New York, 1959, p.p. 375–395.zbMATHGoogle Scholar
  5. Levinsky, E. S. and Yoshihara, H., Hypersonic Flow Research edited by F. R. Riddell, Academic Press, New York, 1962, p. 81.Google Scholar
  6. Liepmann, H. W., Narasimha, R. and Chahine, M. T., “Structure of a Plane Shock Layer,” Physics of Fluids, Vol. 5, No. 11, November 1962.Google Scholar
  7. Liepmann, H. W., Narasimha, R. and Chahine, M. T., “Theoretical and Experimental Aspects of the Shock Structure Problem,” Proc. of the 11th International Congress of Applied Mechanics, Munich, Germany, 1964, Springer-Verlag, edited by Henry Görtler.Google Scholar
  8. Vogenitz, F. W. and Takata, G. Y., “Monte Carlo Study of Blunt Body Hypersonic Viscous Shock Layers,” TRW Systems Group Report No. 06488-6470-RO-00, September 1970.Google Scholar

Copyright information

© Springer-Verlag 1973

Authors and Affiliations

  • Keith J. Victoria
    • 1
  • George F. Widhopf
    • 1
  1. 1.The Aerospace CorporationLos Angeles

Personalised recommendations