Approximate Riemann Solvers for Multi-component flows

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Approximate Riemann Solvers for Multi-component
flows

• Ben Thornber
• Academic Supervisor D.Drikakis
• Industrial Mentor D. Youngs (AWE)
• Aerospace Sciences
• Fluid Mechanics Computational Science

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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.

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Outline

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

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Governing equations

• Begin with the Euler equations in primitive
  variables

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Governing Equations

• Augment them with two multicomponent models
• Mass Fraction
• See, for example, Abgrall (1988) or Larrouturou
  (1989)

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Governing Equations

• 2) Total Enthalpy Conservation of the Mixture
  (ThCM)
• See Wang, S.P. et al (2004)

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Method of Solution

• Godunov finite volume method
• Dual time stepping method

Godunov (1959)
Jameson (1991)
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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)

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Characteristics Based Approximate Riemann Solver

• An extension of Eberles 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

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Non-Conservative Invariants

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

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Converting to conservative form

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

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Converting to Conservative Form

• For pressure this is a little more complex
• Giving
• Where

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Compact form

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

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Compact form

• Where

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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

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Test A Weak Post-shock Contact Discontinuity
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Test A

• 2nd order accuracy with Minbee characteristic
  bump in the MF density profile

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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

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Test B Shock-Contact surface interaction
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Test B

• Oscillation free results for all limiters
• Mass fraction model captures the contact surface
  over fewer points

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Test C Modified Sod shock tube
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Test C

• All profiles are captured reasonably well

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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

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Test D Shock interaction with a Helium Slab
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Test D

• Very complex problem oscillatory results for
  the Mass Fraction model
• Dissipative solution for the ThCM model

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Test D - Convergence

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

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Test E Convection of an SF6 slab
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Test E Results after 1 time step

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

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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

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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.

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References
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References
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References
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References