SOLVING CHEMICALLY REACTIVE KINETIC EQUATIONS

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SOLVING CHEMICALLY REACTIVE KINETIC EQUATIONS

• Maria Groppi and Giampiero Spiga
• Dipartimento di Matematica
• UNIVERSITÀ DI PARMA
• 19th ICTT Budapest July 2005

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(dedicated to Francesco Premuda)
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INTRODUCTION

• 1 - Kinetic models for chemical reactions
• Prigogine and Xhrouet (1949), Xystris and Dahler
  (SRS model, 1978), Giovangigli (book, 1999),
  Rossani and Spiga (1999), Desvillettes et al.
  (2004), ...

2 Approximate solutions

• Closure strategies for moment equations
• (Euler, Chapman-Enskog, Grads 13
  Moments...)
• Approximation at kinetic level
• BGK models

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STARTING POINTS

• A recent consistent BGK-type model for inert gas
  mixtures ( Andries, Aoki, Perthame (AAP), JSP
  106, 2002)
• only one global collision operator for each
  species
• all basic physical properties are fulfilled
  positivity, correct momentum and energy exchange,
  entropy inequality, indifferentiability principle.

• An extended kinetic model for bimolecular
  reversible chemical reactions (Rossani, Spiga,
  Phys. A 272, 1999), reproducing the main physical
  properties macroscopic balance equations,
  collision equilibria and mass action law,
  H-theorem,...
• (analysis of the mathematical properties in
  Groppi, Polewczak, JSP 117, 2004)

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BGK OPERATOR
?s (Ms f s)

• f s distribution function
• Ms suitable local Maxwellian ensuring the
  exact
• exchange rates for mass, momentum and energy
• ?s proper collision frequency (relaxation
  parameter)
• Remark transfer of mass and of energy of
  chemical bond must be taken
• into account in the reactive case, contrary to
  inert mixtures

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REACTIVE KINETIC MODEL
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EXTENDED BOLTZMANN EQUATIONS
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COLLISION EQUILIBRIA
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Exchange Rates (I)

• Elastic contributions for Maxwellian molecules

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Exchange Rates (II)

• Chemical contributions (slow chemical reactions)

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Reactive BGK equations

• where

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BGK exchange rates
The parameters ns , us , Ts of Ms are
determined by imposing that reactive Boltzmann
and BGK equations prescribe the same exchange
rates
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• By equating the exchange rates coming from the
  two models we obtain, for any given ns , a
  non-singular algebraic system of 20 equations for
  the 20 parameters ns, us, Ts (functions of x and
  t)

Consistency properties

• Conservation laws are exactly reproduced by the
  BGK model
• Reactive Boltzmann and BGK equations share the
  correct collision equilibria, including mass
  action law

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H - Functional

• For reactive Boltzmann equations H is a strict
  Lyapunov functional
• For reactive BGK model we have numerical
  evidence that H is monotonically decreasing

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Hydro-dynamic Equations

• At Navier-Stokes level, seven PDEs for the seven
  conserved quantities (or eight differential-algebr
  aic equations for the classical fields)
• Newtonian constitutive equations for diffusion
  velocities, viscous stress and heat flux, plus
  additional reactive correction for scalar pressure

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Consistency

• Same equations as from the Boltzmann kinetic
  equations (Bisi, Groppi, Spiga 2005), apart from
  different definition of viscosity coefficient and
  thermal conductivity
• If reaction is switched off, same constitutive
  equations as for the non-reactive case (AAP,2002)

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Relaxation parameters

• ns ns mech ns chem
• First step
• Estimation of the actual number of collisions
  undergone by each species of the mixture
• Second step
• Validation by comparison with exact solutions
  (analytical BKW-modes for inert mixtures,
  accurate kinetic computations)

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Estimation of number of collisions
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Remarks

• Such a choice ensures the positivity of auxiliary
  temperature Ts for inert mixtures (AAP 2002) and
  yields the positivity condition for the auxiliary
  number densities ns

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Numerical Test Reactive Mixture

• Assumptions
• spatially homogenous reacting gas mixture
• isotropic distribution functions f s(v,t )f
  s(v,t )

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Deviation of f 2 from a local Maxwellian shape
for Problem A
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Differences between actual (n s) and auxiliary (n
s) number densities
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Differences between actual (T s) and auxiliary (T
s) temperatures
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Initial and final distribution functions for
Problem B
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Distribution function f 3, initial data given by
linear splines
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Deviation of f 2 from a local Maxwellian shape
for Problem B
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Trend of the H-functional versus time starting
from Maxwellian initial data (solid line) or from
linear splines (dotted line)
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Differences ns-ns when relaxation parameters are
doubled (C1) or halved (C2)
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H functional in the initial transient, when
relaxation parameters are doubled (C1) or halved
(C2)
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Exact solutions of Boltzmann equations for inert
mixtures

• Assumption spatially homogeneous, isotropic
  Maxwellian molecules.
• BKW-modes solution (Grigoriev, Meleshko 1997)

as, ?0 , ? determined by certain conditions
from collision frequencies
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BGK-BKW Comparison
Differences at kinetic level only, since our BGK
model exactly reproduces macroscopic quantities
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Optimization of ?s

• Estimation of an optimal global reduction factor
  ? such that
• ns opt ns / ?
• by systematic comparison of the H-functional
  computed along the solutions coming from the two
  approaches

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Computed H-functional
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BGKopt-BKW comparison
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Work in progress

• Space-dependent problems evaporation-condensation
  , ...
• Comparison with kinetic calculations discrete
  ordinate approach, semi-continuous
  approximation,...