1 / 34

Approximate Riemann Solvers for Multi-component flows

Approximate Riemann Solvers for Multi-component flows. Ben Thornber Academic Supervisor: D.Drikakis Industrial Mentor: D. Youngs (AWE) Aerospace Sciences Fluid Mechanics & Computational Science. Aims. Describe the derivation of a new approximate Riemann solver for multi-component flows

dawson
Download Presentation

Approximate Riemann Solvers for Multi-component flows

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Approximate Riemann Solvers for Multi-component flows Ben Thornber Academic Supervisor: D.Drikakis Industrial Mentor: D. Youngs (AWE) Aerospace Sciences Fluid Mechanics & Computational Science

  2. Aims • Describe the derivation of a new approximate Riemann solver for multi-component flows • Present a series of test cases illustrating the performance of the scheme for two different model equations • Compare and contrast the Mass Fraction and Total Enthalpy Conservation of the Mixture models.

  3. Outline • Introduction • Governing equations • Godunov method • Higher Order Extensions • Characteristics-Based Solver • Test Cases and Validation • Conclusions

  4. Governing equations • Begin with the Euler equations in primitive variables:

  5. Governing Equations • Augment them with two multicomponent models: • Mass Fraction*: * See, for example, Abgrall (1988) or Larrouturou (1989)

  6. Governing Equations 2) Total Enthalpy Conservation of the Mixture (ThCM)*: * See Wang, S.P. et al (2004)

  7. Method of Solution • Godunov finite volume method: • Dual time stepping method: Godunov (1959) Jameson (1991)

  8. Higher Order Accuracy • Utilise the MUSCL method (Van Leer, 1977): • With 2nd order Superbee, Minmod, Van Leer, Van Albada and 3rd order Van Albada limiters (See Toro, 1997)

  9. Characteristics Based Approximate Riemann Solver • An extension of Eberle’s scheme (Eberle, 1987) • As the governing equations are identical then the derivation holds for both models • Considering the Euler equations split directionally, thus solving: • The time derivative is replaced by the Characteristic Derivative:

  10. Non-Conservative Invariants • After some manipulation this gives six characteristic equations for six unknown flow variables:

  11. Converting to conservative form • Now we convert the equations to conservative form using the chain rule of differentiation:

  12. Converting to Conservative Form • For pressure this is a little more complex: Giving: Where:

  13. Compact form • After considerable manipulation the characteristics based variables with which the Godunov fluxes are calculated are:

  14. Compact form • Where:

  15. Numerical Tests • Used 5 test cases to examine the performance of the new scheme and the multi-component models employed: • A ) Weak Post-shock Contact Discontinuity • See Wang et al (2004) • B ) Shock-Contact surface interaction • See Karni (1994), Abgrall (1996), Shyue (2001), Wang et al (2004) • C ) Modified Sod shock tube • See Abgrall and Karni (2000), Chargy et al (1990), Karni (1996) and Larroururou (1989) • D ) Shock interaction with a Helium Slab • See Abgrall (1996), Wang et al (2004) • E ) Convection of an SF6 Slab • All cases are run with the 3-D code on a mesh 400x4x4

  16. Argon Nitrogen Mach 3.352 shock 0.0 0.25 0.5 1.0 Test A : Weak Post-shock Contact Discontinuity

  17. Test A • 2nd order accuracy with Minbee – characteristic ‘bump’ in the MF density profile

  18. Test A : Limiters • Density profile at the contact surface a) 1st order, b) Superbee, c) Van Albada, d) Van Leer, e) 3rd order Van Albada

  19. 1.0 0.0 0.5 Test B: Shock-Contact surface interaction

  20. Test B • Oscillation – free results for all limiters • Mass fraction model captures the contact surface over fewer points

  21. 1.0 0.0 0.5 Test C : Modified Sod shock tube

  22. Test C • All profiles are captured reasonably well

  23. Test C – Density and velocity profiles • Mass fraction model has a typical density undershoot and a velocity jump at the contact surface • Slight oscillations in the ThCM model

  24. Helium Air Air Mach 1.22 shock 0.0 0.25 0.4 1.0 0.6 Test D : Shock interaction with a Helium Slab

  25. Test D • Very complex problem – oscillatory results for the Mass Fraction model • Dissipative solution for the ThCM model

  26. Test D - Convergence • Dissipative solution for the ThCM model, with 2000 points it is more dissipative than the mass fraction model with 400 points

  27. SF6 Air Air 0.0 0.4 1.0 0.6 Constant velocity u = 0.1 Test E: Convection of an SF6 slab

  28. Test E: Results after 1 time step • Pressure equilibrium is not maintained for the ThCM model or the Mass Fraction model

  29. Test E: Results after 1 time step • Considering a convected contact surface computed using finite volume upwind method: • Where this fraction = 0.6 in the case of SF6 to air

  30. Conclusions • A new multi-component approximate Riemann solver has been developed and validated • The Total Enthalpy Conservation of the Mixture model is better for flows where g is not close to 1, and the difference in gas densities is low. • The Mass Fraction model captures discontinuities in fewer points • Neither model preserves pressure equilibrium exactly in the case of a convected contact surface, however the extent of the error depends on the gases simulated.

  31. References

  32. References

  33. References

  34. References

More Related