Notes on the structure of viscous and numerically captured shocks.
by J. Pike.
The Aeronautical Journal of the Royal Aeronautical Society, Vol 89, No 889, pp.335-338, Nov. 1985.

Summary.
An exact expression for the flow variables through a viscous shock wave is obtained from the Navier-Stokes equations. The Prandtl number is taken to be 3/4, which is close to the value for air, and the viscosity is assumed to be given by Sutherland's formula.
By considering the limit as the visosity tends to zero, it is shown that the solution to the Euler equations has an entropy spike at the shock wave. This explains certain, hitherto considered spurious, features of shock waves captured by numerical solutions of the Euler equations.

Comment.
Shock waves are commonly treated as discontinuities in the flow variables. However when the effects of viscosity are included (as within the Navier-Stokes equations), the shock wave has a small but finite width. The analytic solution demonstrates the smooth variation of the flow variables through the shock wave, with the entropy showing a peak value within the shock wave. However it should be noted, that for the continuum flow assumption on which the Navier-Stokes equations are based to remain valid, the thickness of the shock wave must be large compared to the mean free path length of the gas molecules. This applies a restriction the minimum shock wave thickness and the minumum viscosity for which the solution is valid.

Availability.
Copies of the paper can be obtained from the Royal Aeronautical Society at www.aerosociety.com
or email j a c k @ j a c k p i k e . c o . u k

Return to Jack Pike Published Papers

Last amended: Dec 13.