1D Riemann Problem (Shock Tube) with JWL EOS
 

Dear CFD fellows!

I am wondering if anybody knows the analytic solution to 1D Riemann Problem (Shock Tube) if the Equation of State is Johns-Wilkins-Lee (JWL)? What I know is that an isentropic condition is imposed across a rarefaction while the Rankine-Hugoniot jump condition is imposed across a shock. But I have no idea what the equations are, neither how they are derived. Any type of help is much appreciated!

WB

Hello,

Did you have a look to the book of Toro? He derives the exact solution for the Riemann problem associated to the Euler equations and closed by ideal gas eos. Once you understand how it is developed for such a simple eos you should see the additional difficulties with more sophisticated eos. Keep in mind that even with ideal gas the solution is not analytic, it involves a non linear scalar equation for the star pressure.

I suggest this report from which you could be able to compute the solution using your equation for the pressure
https://books.google.it/books/about/...AJ&redir_esc=y

The exact Riemann solution for general EOS is a non-trivial task. If you can exclude non-convex and non-smooth cases you at least need two informations from your thermodynamics.

1. The Caloric EOS
2. A Model for speed of sound

Then you have to iterate your solution for

• Shock via calculation of Rankine Hugeniot condition and for
• Rarefaction Wave via path integration of Generalized Riemann Invariants

This has to be done with two nested iteration loops.

The non-nonvex and non-smooth case is much more challenging!

Regards

Here is a reference of Kamm:

https://cstools.asme.org/csconnect/F...w=yes&ID=45379

The JWL EOS is mentioned there.

Exact soln for Convex Equation of state
 

There is a paper by James R.Kamm:
'An exact compressible one dimensional Riemann solver for general, convex equation of state'.

This paper outlines the numerical procedure for solving the exact solution for Sod's problem with JWL-EoS. As it was mentioned in previous reply, analytical solution is not available and can only be solved using numerical techniques such as Secant method or Newton-Rhapson method.

Also refer to the paper by Collela and Glaz,1985 ( http://seesar.lbl.gov/anag/publicati...ella/A_1_7.pdf ) on this topic.

May I know why is the Generalised Riemann Invariant used for deriving rarefaction waves property? I understand why Rankine-Hugoniot condition is applied for shock but not expansion fan as I don't understand what Generalised Riemann Invariant means physically? Is Generalised Riemann Invariant the same as Riemann invariant? As far as I know, Riemann Invariant is the property transported along the characteristics.

I am reading the book from E.F. Toro but he stop elaborating on Generalised Riemann Invariant and mentioned: "For a detailed discussion see the book by Jeffrey [259]". Can someone explain what the Generalised Riemann Invariant means?

You may also derive all thermodynamic quantities in the rarefaction fan with the isentropic condition, which is equivalent to some particular Riemann invariants. Generalized means that the Riemann invariants are written in abstract mathematical notation, e.g. ratio of dw to the respective right eigenvectors. Note that the velocity can not be calculated with the isentropic relation, because it is no thermodynamic quantity.

Another reference for general EOS with NIST and multi-components is given here

https://link.springer.com/article/10...93-019-00896-1

Further informations for non-classical Riemann theory e.g. split and composite waves, can be found in papers of Menikoff and Plohr (1989) or Smith (1979).

The GRI can be deduced by introducing the integral curves in the eigenvectors space as described in the Leveque's textbook.


As far as the Riemann invariant in the physical space are concerned, they exist only in the case of omo-entropic flows, different is the case of iso-entropic flows.