The analytical solution is formulated in non-dimensional form, with the upstream density, pressure, and velocity in addition to an equivalent perfect gas mean free path are used to non-dimensionalize the solution. The mean free path is quite abritrary, and for the solution used is
All the above simplifies the 3 PDE's of the 1D Navier-Stokes equations to one ODE, which is solved analytically. See Kramer et al[1] for the implicit solution, which is expressed as a function of the Mach number and specific heat ratio and relates the non-dimensional velocity and position. The density, pressure, and hence the total energy are found with the relations,
This derivation is carried out for for the monoatomic gas case, , with different arbitrary perfect gas equivalent mean free path in the textbook by Zel'dovich and Raizer[3].
This specific exact solution is available in the weno and clawpack: applications/euler/1D/ViscousShock directories. For the following tests, we use the computational domain, x = [-30,30], inflow/outflow boundary conditions and the parameters
The newly implemented CFL number evaluation for Clawpack in 1D is
[1] R.M.J. Kramer, C. Patano, D.I. Pullin. A class of energy stable, high-order finite-difference interface schemes suitable for adaptive mesh refinement of hyperbolic problems. J. Comput. Phys., 226:1458-1484, 2007.
[2] A.C. Hindmarsh, P.M. Gresho, and D.F. Griffiths. The stability of explicit Euler time-integration for certain finite difference approximations of the multi-dimensional advection-diffusion equation. Inter. J. Num. Methods in Fluids, 4:853-897, 1984.
[3] Y.B. Zel'dovich and Y.P. Raizer. Physics of shock waves and high-temperature hydrodynamic phenomena. Dover. 2002.
-- JackZiegler? - 29 Jul 2008
I | Attachment ![]() | Size | Date | Who | Comment |
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![]() | velocity.ps | 40.5 K | 29 Jul 2008 - 12:54 | JackZiegler? | velocity |
![]() | velocity.png | 4.2 K | 29 Jul 2008 - 13:12 | JackZiegler? | velocity |
![]() | entropy.gif | 4.1 K | 29 Jul 2008 - 13:19 | JackZiegler? | entropy |