nonlinear wave theory for transport phenomena · unified hyperbolic model for fluids and solids...
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Nonlinear Wave Theory for Transport Phenomena
ILYA PESHKOV CHLOE, University of Pau, France
EVGENIY ROMENSKI Sobolev Institute of Mathematics, Novosibirsk, Russia
MICHAEL DUMBSER University of Trento, Trento, Italy
OLINDO ZANOTTI University of Trento, Trento, Italy
JOSO 2016 March 9-11 2015
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Motivation for a New Fluid Dynamics
micro macro
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Motivation for a New Fluid Dynamics
micro macro
Classical parabolic transport theories (Navier-Stokes, Fourier, Fick) are not “wave” theories in a rigorous sense.
Classical Kirchhoff equation (dispersion relation for NS) says that at high frequencies
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Unified hyperbolic model for fluids and solids
First order Hyperbolic model (genuinely wave theory) Can describe fluids and solids in a one system of PDEs Free of empirical steady-state transport relations (Newton’s law of viscosity, Fourier heat conduction law etc.) Applicable to non-Newtonian, non-Fourier, non-Fickian transport Has less numerical issues than parabolic theory (mesh quality, discontinuities, singularities)
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Gas Solid Liquid
Continuum mechanics
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Continuum Gas
Continuum Liquid
Continuum Solid
Continuum mechanics
Gas Liquid Solid
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Continuum mechanics
Continuum Gas
Continuum Liquid
Continuum Solid
Gas Liquid Solid
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Continuum mechanics
Flow is the Particle Rearrangement process
Continuum Gas
Continuum Solid
Gas Solid
Continuum Liquid
Liquid
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Frenkel’s idea to describe fluidity of liquids is to introduce time
Deformed particle
Undeformed particle
Distortion (non-symmetric)
Now, molecules = fluid particles or fluid parcels
Particle rearrangements is a way to the Unified Flow Theory
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Main ingredients
Dissipation Time
Distortion field
Energy potential (equation of state):
Equation of State
micro meso macro
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Main ingredients
Dissipation Time
Distortion field
Energy potential (equation of state):
Equation of State
micro meso macro
In classical theory
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Governing equations
Momentum:
Equation for the distortion:
Visc. stresses
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Governing equations
Momentum:
Navier-Stokes stress tensor:
Visc. stresses
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Wave theory
Meso scale Macro scale
Waves Fluid particles
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Wave theory
Meso scale Macro scale
Waves Fluid particles
t
x
Longitudinal pressure wave
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Wave theory
Meso scale Macro scale
Waves Fluid particles
t
x
Longitudinal pressure wave
She
ar s
tre
ss
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Wave theory
Meso scale Macro scale
Waves Fluid particles
t
x
Longitudinal pressure wave
Viscosity coefficient
She
ar s
tre
ss
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Equation of State
micro meso macro
Hyperbolic Heat Conduction
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Chloromethane
Ref: Data from Sette, Busala, Hubbard, The Journal of Chem. Phys., 23 (5), 1955
Ph
ase
velo
city
, [m
/s]
f/p, [MHz/atm]
But how to get the parameters? High frequency measurements
Viscosity coefficient
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Chloromethane
Ref: Data from Sette, Busala, Hubbard, The Journal of Chem. Phys., 23 (5), 1955
Ph
ase
velo
city
, [m
/s]
f/p, [MHz/atm]
Viscosity coefficient
But how to get the parameters? High frequency measurements
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ADER-WENO-FVM-DG framework, (also PNPM methods)
t
x
Generalized Riemann Problem GRP
(smoothed initial data)
1. WENO reconstruction (degree N) 2. Solve GRP coupled with the source terms (degree M>N) (Cauchy-Kovalevski or DG) 3. Update at n+1
See papers by E. Toro, V. Titarev, M. Dumbser since 2000
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ADER-WENO-FVM-DG framework, (also PNPM methods)
• Explicit globally (implicit locally)
• Massively Parallel
• Arbitrary order (up to 10 implemented)
• Equally High Order in both, space and time
• One step in time
• Robust WENO FV or ultra compact DG
• Unstructured grids (complex geometries)
• Stiff source terms (asymptotic preserving)
Code characteristics:
See papers by Michael Dumbser since ~2008
Unified PNPM family of methods
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Blasius boundary layer, Re=1000
x=0.5 cut
Velocity contours
Inflow V=1
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Lid driven cavity flow at Re=100
Velocity, u
A11 A12
Symbols are NS model, velocities
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Double shear layer, visc=2∙10-4
time=0.8
time=1.2
time=1.8
Vorticity
Right: Navier-Stokes model staggered Semi- implicit DG P3
Tavelli , Dumbser 2014
Left: Hyperbolic model
ADER-WENO 4th order scheme from
Dumbser, Enaux, Toro, 2008
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Double shear layer,
time=0.8
time=1.2
time=1.8
Vorticity A12
Right: Navier-Stokes model staggered Semi- implicit DG P3
Tavelli , Dumbser 2014
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Compressible mixing layer, Re=250
Velocity Top: 6th order PNPM scheme
Navier-Stokes model Dumbser, Zanotti, 2009
Hyperbolic model 3rd order ADER-WENO
Dumbser, Enaux, Toro, 2008
Vorticity
A12
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Flow around a circular cylinder, Re=150
A12
Pressure field.
Full computational domain
Strouhal number
Flow direction
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Why would one use order Hyperbolic PDEs?
2nd order Parabolic
1st order Hyperbolic
Severe time step restriction in Parabolic problems (explicit scheme) critical for complex flows and HPC (# turbulence, viscoacoustics, 2-phase pore-scale modeling, etc.)
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Turbulence
Navier-
Stokes-
Fourier
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Turbulence
Navier-
Stokes-
Fourier
Proposed Hyperbolic
Theory
Now we have 2 fundamentally different models
Parabolic vs. Hyperbolic
Dumbser, Peshkov, Romenski, Zanotti “High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids” Journal of Computational Physics. 2016. (Open access)
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Turbulence
Navier-
Stokes-
Fourier
Proposed Hyperbolic
Theory
Tu
rbu
len
ce
Most of us believes that
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Turbulence
Navier-
Stokes-
Fourier
Proposed Hyperbolic
Theory
Tu
rbu
len
ce
But what if ???
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Turbulence
Navier-
Stokes-
Fourier
Proposed Hyperbolic
Theory
Blasius boundary layer problem
Distortion, A11
Velocity
Tu
rbu
len
ce
But what if ???
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Turbulence
Navier-
Stokes-
Fourier
Double shear layer problem
Vorticity
Distortion, A11
Proposed Hyperbolic
Theory
Tu
rbu
len
ce
But what if ???
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Turbulence
Navier-
Stokes-
Fourier
3D Taylor-Green vortex
Proposed Hyperbolic
Theory
Tu
rbu
len
ce
But what if ???
Distortion, A11
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Solid dynamics
Using the same(!) system of PDEs we can simulate dynamics of solids as well
Plastic Elastic
Bending of a plate
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Solid dynamics
Plastic Elastic
Using the same(!) system of PDEs we can simulate dynamics of solids as well
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Seismic wave propagation
HPR model Linear elasticity
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Poroelasticity
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Biot’s Theory
Drawbacks:
Established as a Linear theory from the very beginning • modification problems (viscoelastic media, fraction time derivative, etc.)
Composite elastic modulus Q of the whole media (phase coupling parameter)
• measurement problems • interpretation
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Nonlinear Mixture Theory: state parameters
Velocities of the solid and fluid phase
Volume fraction of the solid matrix
Mixture density
Mass fractions (concentrations)
Deformation gradient
The missing parameter in the Biot’s theory
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Nonlinear Mixture Theory: state parameters
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Solid-Fluid mixture model
momentum
deformation
Mass fraction
Relative velocity
Volume fraction
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Linearised model: single pressure model
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Linearised model: single pressure model
Volume fraction of the solid matrix
Mass fractions (concentrations)
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km
/s
porosity
Longitudinal sound speed in pure solid
Fluid sound speed
Two Longitudinal sound waves (P-waves)
fluid solid
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km
/s
porosity fluid solid
Transverse sound speed in pure solid
Transverse sound in
the saturated media
Transverse sound wave
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Conclusion
Hyperbolic (wave theory) for viscous, heat and mass transport
The unified model can describe fluids and solids in a single system of PDEs
The model was implemented in the ADER-FVM-DG code and tested on a large number of test cases
Nonlinear solid-fluid mixture model was presented. Application to poroelasticity is expected
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Thank you for your attention