Approximate Riemann Solvers for Approximate Riemann Solvers for Multi-component flowsMulti-component flows

Ben ThornberAcademic Supervisor: D.Drikakis

Industrial Mentor: D. Youngs (AWE)

Aerospace SciencesFluid Mechanics & Computational Science

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.

Outline

Introduction

– Governing equations

– Godunov method

– Higher Order Extensions

Characteristics-Based Solver

Test Cases and Validation

Conclusions

Governing equations

Begin with the Euler equations in primitive variables:

Governing Equations

Augment them with two multicomponent models:

1) Mass Fraction*:

* See, for example, Abgrall (1988) or Larrouturou (1989)

Governing Equations

2) Total Enthalpy Conservation of the Mixture (ThCM)*:

* See Wang, S.P. et al (2004)

Method of Solution

Godunov finite volume method:

Dual time stepping method:

Jameson (1991)

Godunov (1959)

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)

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:

Non-Conservative Invariants

After some manipulation this gives six characteristic equations for six unknown flow variables:

Converting to conservative form

Now we convert the equations to conservative form using the chain rule of differentiation:

Converting to Conservative Form

For pressure this is a little more complex:

Giving:

Where:

Compact form

After considerable manipulation the characteristics based variables with which the Godunov fluxes are calculated are:

Compact form

Where:

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

Test A : Weak Post-shock Contact Discontinuity

0.25 0.5 1.0

Mach 3.352 shock

Argon Nitrogen

0.0

Test A

2nd order accuracy with Minbee – characteristic ‘bump’ in the MF density profile

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

Test B: Shock-Contact surface interaction

0.5 1.00.0

Test B

Oscillation – free results for all limiters

Mass fraction model captures the contact surface over fewer points

Test C : Modified Sod shock tube

0.5 1.00.0

Test C

All profiles are captured reasonably well

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

Test D : Shock interaction with a Helium Slab

0.25 0.4 1.0

Air Helium

0.0

Air

0.6

Mach 1.22 shock

Test D

Very complex problem – oscillatory results for the Mass Fraction model

Dissipative solution for the ThCM model

Test D - Convergence

Dissipative solution for the ThCM model, with 2000 points it is more dissipative than the mass fraction model with 400 points

Test E: Convection of an SF6 slab

0.4 1.0

Air SF6

0.0

Air

0.6

Constant velocity u = 0.1

Test E: Results after 1 time step

Pressure equilibrium is not maintained for the ThCM model or the Mass Fraction model

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

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 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.

References

References

References

References