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
where the 0 subsript denotes the upstream flow with Mach number equal to two. The domain is large enough and the final time short enough, to minimize the boundary errors.
The newly implemented CFL number evaluation for Clawpack in 1D is
where the maximum is take over all cell centers. This is derived by using Von Neumann analysis and ignoring the term.
[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 |