multiphase flows Application to evaporation phenomena Olivier Le - PowerPoint PPT Presentation

Dynamic relaxation processes in compressible multiphase flows Application to evaporation phenomena Olivier Le Métayer Aix-Marseille University, Polytech Marseille UMR CNRS 7343

Some industrial applications where evaporation is crucial Fuel injectors at high pressure Cryogenic injection device in space launcher BLEVE phenomena (Security) : rapid depressurization of liquefied gas + combustion

Typical physical processes we are interested in Low pressure Liquid flow from a Valve/diaphragm chamber high pressure chamber  « Flashing » (rapid depressurization)  Liquid/vapor mixture (drops, bubbles, pockets, ,…) generated by the initial pressure ratio  Cavitation created by geometric  Drops evaporation and jet explosion singularities (strong rarefaction waves) Two-velocities flow (inter-penetration) Interface problems How can be solved the multiphase flow when the topology strongly varies ? A way is to consider relaxation effects

Plan  Description of the non-equilibrium multiphase flow model  Presentation of the relaxation effects  Some multidimensional numerical results

Non-equilibrium multiphase compressible flow model

Multiphase compressible flow model  Initially proposed by (Baer & Nunziato,IJMF,1986) for two phases  Each phase has its own set of equations and associated variables (velocity, pressure temperature, entropy,…) :   α     α k u . 0  I k t     ( )     k . ( u ) 0  k  t e ( P , ) + Equation of state : k k k            ( u )       ρ α k . u u P I P  k I k t           ( E )       ρE α k . P u P u .  k I I k t  Solved by the Discrete Equations Method (Abgrall & Saurel, JCP, 2003 ; Saurel & al., JFM, 2003 ; Chinnayya & al., JCP, 2004 ; Le Métayer & al., JCP, 2005 ; Saurel & al., IJNMF, 2007 ; Le Métayer & al., JCP, 2011)

Discrete Equations Method : Godunov’s concept applied to multiphase cells •Godunov’s method =averaging procedure by solving a single Riemann problem at each cells interface Cell (i+1) Cell (i-1) Cell i Fixed control volume = volume filled with the fluid • DEM=averaging procedure by solving Riemann problems between pure fluids at each cells interface Cell (i-1) Cell i Cell (i+1)       Fixed control volume = sum of the sub-volumes filled with the fluids Each phase has a time-dependent sub-volume (volume fraction)

Major contributions of DEM (1)  Discrete equations traducing the time evolution of phases and mixture variables are obtained = Numerical scheme  When dealing with a two-phases bubbly flow, a continuous model has been obtained :   α α All interface variables are determined !      μ 1 1 u P P   I 1 2 t x P P  1 2            ρu α 2 Z Z Z u Z u P ( u )         1 2 P λ 1 1 2 2 1 1 1 u P u u I  1 1    I I 2 1  Z Z t x x 1 2 Z Z           1 2 ρE α     ( E ) P u       λu μP 1 1 1 P u P P u u    I I I 1 2 I 2 1 t x x Interfacial area between phases A Z Z A     I 1 2 I   Z Z Z Z 1 2 1 2  µ and λ express the rates at which pressure and velocity equilibrium are reached respectively

Major contributions of DEM (2)  This non-equilibrium model is able to treat simultaneously : - mixtures with several velocities, temperatures,… - material interfaces (contact) - permeable interfaces (interfaces separating a cloud of drops and a gas for example,…)

Evaporation effects  This model is completed by heat and mass transfer source terms (evaporation, condensation) at infinite rates (relaxations)  Relaxation processes (at infinite rate) allow the obtention of solutions in limit cases corresponding to reduced models.  They are particularly relevant when : - the interfacial area and the flow topology are unknown, - the Direct Numerical Simulation of interface problems is considered.

Two-phase compressible flow model + relaxation terms Pressure Heat Velocity Mass Hyperbolic relaxation transfer relaxation transfer             m α μ 1 u . P P  I 1 1 2 ~  t I     ( )       m 1 .( u )  1 t               ( u )          ρ α  ~ 1 λ  . u u P I P  u u m u  I 1 1 2 1 I t        ~           ( E )                 ρE α μ λ 1 m H . P u P u . P P P u . u u H T T  I 1 I I 1 I 1 2 T 2 1 I 2 1 t     ( )      2 .( u )  m  2 t             ( u )           ρ α   2 ~  . u u P I P   λ m u u u  2 I 1 t I 2 1              ( E )       ~             λ    ρE α μ 2 u . u u H T T m H . P u P u . P P P  I I 1 I 2 1 T 2 1 I 2 I 1 2 t ~      m ( g g ) I 1 2

Relaxation coefficients  µ and λ express the rate at which pressure and velocity equilibria are reached respectively :         μ λ P P u u 1 2 2 1  Mechanical interactions (drags, compressibility ratio) between phases  H T expresses the rate at which temperature equilibrium is reached :     Heat exchange between phases H T T T 2 1  ν expresses the rate at which Gibbs free energy equilibrium is reached : ~       Mass transfer (evaporation/condensation process) m ( g g ) I 1 2 μ , H T , ν → ∞ Instantaneous exchanges : relaxation effects

Relaxation effects at infinite rates

Hierarchy of compressible flow models : reduced models Multi-velocity flow model Non-equilibrium flows Mechanical (u 1 ,u 2 ,p 1 ,p 2 ,T 1 ,T 2 ,g 1 ,g 2 ) relaxation (velocity and pressure) Single-velocity flow model  ,    Interface problems Thermal (u,p,T 1 ,T 2 ,g 1 ,g 2 ) relaxation (temperature)   H T Multi-species Euler mixture model Chemical Thermal equilibrium flows relaxation (Gibbs free (u,p,T, g1,g2 ) energy)    Homogeneous Euler model Thermodynamic equilibrium flows (u,p,T,g)

Velocity relaxation procedure : λ → ∞  α 1  0  Momentum and energy t  1  cst   conservation 1  ( ) 0   ( ) t   1 Y cst       1  ( u )    λ 1 u u  2 1  t ( )   2 Y cst    ( E )      2   1 λ u . u u  I 2 1 t      *   u Y u Y u 2  ( ) 1 1 2 2 0  t     1      * * * e e ( u u ).( u u )      ( u )   1 1 1 1  λ 2 2 - u u  2 1 t     1     * * *   e e ( u u ).( u u )  ( E )     2 2 2 2   2 2 λ - u . u u  I 2 1 t EOS of both phases are not used explicitly

Pressure relaxation procedure : μ → ∞  α 1     μ P P  1 2  t ( )   1 Y cst Mass and energy    1 1  ( ) 0 conservation   t ( )   2  Y cst    1  2 ( u ) Use of EOS 0   t u 1   cst * * * e ( p , )   ( E )   1 1    1 μ P P P   I 1 2 u 2  t  cst * * * e ( p , )   2  2 2 ( )   0 1 1       t * * e e p      1 1   *   2  ( u ) 1 1 0     t * * 1 1 ( p )      * * 1 e e p       ( E ) 2 2 *       2  μ * * P P P 2 2 ( p )  I 1 2 2 t 1 Y Y Function of the final   1 2    * * * * ( p ) ( p ) relaxed pressure only 1 2 When considering ideal gas or ‘Stiffened Gas’ EOS an analytical relation is available for the pressure