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E. Romenski , D. Drikakis Fluid Mechanics & Computational Science Group,

Cranfield University, UK

CONSERVATIVE FORMULATION AND NUMERICAL METHODS FOR MULTIPHASE COMPRESSIBLE MEDIA

The financial support from the EU Marie Curie Incoming International Fellowship Programme(contract MIF1-CT-2005-021368) is acknowledged.

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The problem of multiphase flow modeling lies in the mathematical and numerical formulation of the problem -

there is not yet a widely accepted formulation for the governing equations of multiphase flows

The challenge is associated with the development of a mathematical model that satisfies three important properties:

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. hyperbolicity (symmetric hyperbolic system in particular)

. fully conservative form of the governing equations

consistency of the mathematical model with thermodynamic laws.-------------------------------------------------------------------------

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Two-phase compressible flow models with different velocities, pressures and temperatures

1. Bayer-Nunziato-type model (Baer&Nunziato,1986; Saurel&Abgrall,1999). Governing equations are based on the mass, momentum, and energy balance laws for each phase in which interfacial exchange terms (differential and algebraic) included. Equations are hyperbolic (non-symmetric), but non-conservative

2. We propose extended thermodynamics approach and thermodynamically compatible systems formalism (Godunov-Romenski) to develop multiphase model. Governing equations are written in terms of parameters of state for the mixture and are taking into account a two-phase character of a flow.

Equations are hyperbolic (symmetric) and conservative

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Class of thermodynamically compatible systems of hyperbolic conservation laws

Thermodynamically compatible system is formulated in terms of generating potential and variables

All equations of the system are written in a conservative form and the system can be transformed to a symmetric hyperbolic form

Many well-posed systems of mathematical physics and continuum mechanics can be written in the form of thermodynamically compatible system.

Examples: gas dynamics, magneto-hydrodynamics, nonlinear and linear elasticity, electrodynamics of moving media, etc.

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Thermodynamically compatible system in Lagrangian coordinates

is determined by the generating potential M and variables:

The system is symmetric and hyperbolic if

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Energy conservation law:

is a convex function.

Flux terms are formed by invariant operatorsgrad, div, curl

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Thermodynamically compatible system in Eulerian coordinates can be obtained by passing to the new coordinates and corresponding transformation of generating potential and variables

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-- energy conservation law

The system can be transformed to the symmetric oneand hyperbolic if L is a convex function

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Development of the two-phase flow model

1. Introduction a new physical variables in addition to the classical variables (velocity, density, entropy) characterizing two-phase flow.

2. Formulation of new conservation laws for these new variables

in addition to the mass, momentum and energy conservation laws.

3. Introduction a source terms modelling phase interaction and dissipation.

4. Formulation of closing relationships, such as Equation of State for the mixture and functional dependence of source terms on the parameters of state

consists of several closely interrelated steps:

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Physical variables characterizing two-phase flow

- the phase number,

- volume fraction of i-th phase,

- mass density of i-th phase

- velocity vector of i-th phase

- specific entropy of i-th phase

- thermal impuls of i-th phase

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Physical variable is connected with the heat flux vector by the relation

- heat flux relaxation time, temperature and thermal conductivity coefficient

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Governing equations for compressible two-phase flow with different pressures and temperatures of phases

-- total mass conservation law,

-- volume fraction balance,

-- 1st phase mass balance law

-- total momentum conservation law

-- relative velocity balance law

-- phase heat flux balance laws

-- phase entropy balance laws

17 equations, 13 algebraic source terms

Derivation is based on the extended irreversible thermodynamics laws and thermodynamically compatible system formalism (Godunov-Romenski).

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SOURCE TERMSare responsible for phase interaction and dissipation.

The total energy conservation laws for the mixture must be fulfilled

The total mixture entropy production must be non-negative

The partial phase entropy production must be non-negative

Onsager’s principle of dissipative coefficients symmetry is held

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The following source terms are introduced in the governing equations:

- Phase pressure relaxation to the common value through the process of pressure waves propagation

- Phase to phase transition

- Interfacial friction force (the Stokes drag force)

- Heat flux relaxation to the steady Fourier heat transfer process

- Phase temperatures relaxation to the common value through the heat transfer between phases

- Phase entropies production caused by phase interaction

REQUIREMENTS:

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SOURCE TERMS definition

The total entropy production is positive

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- Drag coefficient

- thermal conductivity coefficients

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- mass fraction of the 1st phase----------------------------------------------

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1D equations for single temperature modela consequence of the general model under assumption

-- volume fraction balance law

-- mass conservation for the 1st phase

-- total mass conservation

-- relative velocity balance law

-- total energy conservation

Simplification of the model applicable in the case of small dispersed phase particles, if the phase temperatures equalizing process is fast

-- total momentum conservation

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- pressure relaxation

- interfacial friction

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1D conservative equations for isentropic model

-- total mass conservation

-- total momentum conservation

-- volume fraction balance law with pressure relaxation source term

-- mass conservation for the 1st phase

-- relative velocity balance law with drag force source term

Further simplification applicable in the case of negligible thermal variations

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-- pressure relaxation

-- interfacial friction (drag)

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Comparison of Conservative Model with Baer-Nunziato-Type Model

Isentropic one-dimensional case

B-N -type model (Saurel&Abgrall) Conservative model

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Systems are similar if to denote

Definition of interfacial pressure is different:

Definition of interfacial velocity is the same:

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Comparison of Conservative Model with Baer-Nunziato-Type Model

Isentropic multidimensional case

The difference between two models becomes more significant -

The momentum equations in conservative model can be written as follows:

Extra terms appear which are not presented in the B-N-type model:

These are forces arising for the flow with nonzero relative velocity, caused by the phase vorticities and are called as lift forces

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NUMERICAL METHOD

Standard finite-volume method is employed for solving the system of conservation laws

we apply recently proposed GFORCE method for flux evaluation (E.F. Toro, V.A. Titarev, 2006).

GFORCE is a convex average of the Lax-Friedrichs flux and Lax-Wendroff flux:

- Lax-Friedrichs flux

- Lax-Wendroff flux

In Toro&Titarev (2006) it is reported that the GFORCE flux is upwind and reproduces the Godunov upwind flux for linear advection equation.

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Since the Riemann problem for general equations can not be easily solved because the eigenstructure can not be obtained explicitly

Here Δt is the local time step chosen without any relation to the global time step

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WATER FAUCET PROBLEM: water column flow in air annulus in a tube under the effect of gravity (Ransom,1987)

initialstate

steady state

t = 0.5 s

gravity

Initial data:

Boundary conditions:

Exact solution:

- air volume fraction

- water velocity

- air velocity

- uniform pressure

Inlet:

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Outlet:_______________________________________

Tube length – 10 m

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Numerical solution of water faucet problem using isentropic model equations

GFORCE flux (Toro&Titarev,2005,2006 )(MUSCL-Hancock 2nd order method)

Linearized Riemann Solver(1st order Godunov method)

Blue - exact solutionBlack - 200 mesh cellsRed - 400 mesh cellsGreen - 800 mesh cellsPurple - 1600 mesh cells

Instantaneous pressure relaxation is assumed, drag force is neglected

GFORCE is comparable with the 1st order linearized solver

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WATER/AIR SHOCK TUBETwo-phase flow test case (Saurel&Abgrall, 1999)

Numerical solution with the use of single temperature model (GFORCE flux)

Riemann problem, initial discontinuity at x=0.7 m

Initial data:

Black – exact solutionRed -- 200 cellsBlue --- 800 cells, gives a verygood agreement with the exact solution

Modelling of the moving water/air interface – instantaneous pressure relaxation, infinite drag coefficient

left: right:

water +small amount of

air

air +small amount of

water

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Shock – bubble interaction (single temperature model) Interaction of shock wave propagating in air with a cylindrical bubble

Shock wave with the Mach number 1.23

Mesh: 300 x 100 cellsRegion: 225 x 44.5 (in millimeter), Bubble radius: 25 (in millimeter)

Light (Helium) and Heavy (Freon R22) bubbles have been considered

[experiments: Haas&Sturtevant (1987)]

2D Test case

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Shock –bubble interaction (single temperature model)

Light (Helium) bubble

Mixture density

Perfect gas EOS with γ =1.4 for air and γ =1.648 for Helium

Both gases are initially at atmospheric pressure

Shock wave with the Mach number 1.23. The pressure behind the wave is 1.68 atmosphere

Instantaneous pressure relaxation is assumed.

Drag coefficient is

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Shock –bubble interaction (single temperature model)

Heavy (Freon R22) bubble

Mixture density

Perfect gas EOS with γ =1.4 for air and γ =1.249 for R22Both gases are initially at atmospheric pressure

Shock wave with the Mach number 1.23. The pressure behind the wave is 1.68 atmosphere

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Instantaneous pressure relaxation is assumed.

Drag coefficient is

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Further Developments

1. Implement a phase to phase transition kinetics2. Include dispersed phase coalescence and breakdown3. Develop high-accuracy numerical methods for 3D flows

Conclusions

A new approach in multiphase flow modelling based on thermodynamicallycompatible systems theory is proposed. A hierarchy of conservative hyperbolic models for two-phase compressible flow is presented and robust numerical method for solving equations of the models is developed.

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REFERENCES1. Godunov S.K., Romenski E., Elements of continuum mechanics and conservation laws. Kluwer Academic/Plenum Publishers, New York (2003).

2. Godunov S.K., Romenski E., Thermodynamics, conservation laws and symmetric forms of differential equations in mechanics of continuous media, in Computational Fluid Dynamics Review 1995. John Wiley & Sons, New York, (1995), 19--31.

3. Godunov S.K. , Mikhailova T.Yu., Romenski E.I., Systems of thermodynamically coordinated laws of conservation invariant under rotations. Siberian Math. J., V. 37 (1996), 690--705.

4. Romensky E., Thermodynamics and hyperbolic systems of balance laws in continuum mechanics, in Godunov Methods: theory and applications, Kluwer Academic/Plenum publishers, (2001), 745--761.5. Romenski E., Toro E.F., Compressible two-phase flow models: two-pressure models and numerical methods, Computational Fluid Dynamics Journal, V.13 (2004), 403--416.6. Romenski E., Resnyansky A.D., Toro E.F., Conservative hyperbolic formulation for compressible two-phase flow with different phase pressures and temperatures, Quarterly of Applied Mathematics V. 65 (2007), 259-279.

7. Romenski E., Drikakis D., Compressible two-phase flow modelling based on thermodynamically compatible systems of hyperbolic conservation laws, submitted toInt. J. for Numerical Methods in Fluids.