Laboratoire Jacques-Louis Lions, Paris 27/10/2015 A unified flow theory for viscous fluids ILYA PESHKOV CHLOE, University of Pau, France joint work with EVGENIY ROMENSKI Sobolev Institute of Mathematics, Novosibirsk, Russia MICHAEL DUMBSER OLINDO ZANOTTI University of Trento, Trento, Italy University of Trento, Trento, Italy
Continuum physics model Mathematical model Numerical results
What am I going to present? A unified VISCOSITY-coefficient-free model for Solids & Fluids First order PDEs, Hyperbolic, Causal No explicit flow division on Equilibrium and non-equilibrium, Newtonian and non-newtonian Fully consistent with Thermodynamics Free of empirical relations (closed to what called from first principles ) 2/22
Two ways to derive a continuum model, Navier-Stokes example Postulates of Continuum Mechanics Flow is the fluid particle rearrangements Material particle or Fluid parcel, Fluid element This is the only common thing for any fluid, solid, granular material etc. flows 3/22
Two ways to derive a continuum model, Navier-Stokes example Empirical approach to describe particle interactions by means of Newton s law of viscosity Acausal theory (parabolic) Postulates of Continuum Mechanics Second order PDE Issues with computing, discontinuities, mesh quality With phenomenological viscosity coefficient Only equilibrium flows Material particle or Fluid parcel, Fluid element OR Try to describe particle rearrangements in the mathematical model, free of the above shortcomings 3/22
Particle rearrangements is a way to a unified flow theory In 1930 th, Yakov Frenkel developed a microscopic theory of liquids, and proposed a fundamental characteristic particle settled life time We apply Frenkel s idea to continuum modeling. Now, molecules = material particles or fluid parcels Simple Consequences: During the time all connections are conserved The longer the bigger resistance to shearing defines the smallest length scale (particle scale) No free volume particles have to deform 5/22
Particle rearrangements is a way to a unified flow theory In 1930 th, Yakov Frenkel developed a microscopic theory of liquids, and proposed a fundamental characteristic particle settled life time We apply Frenkel s idea to continuum modeling. Now, molecules = material particles or fluid parcels Distortion (non-symmetric) 5/22 Undeformed particle Deformed particle
Particle rearrangements and Strain dissipation Undeformed particle Particles in Equilibrium Deformed particle Shear fluctuations cannot propagate across the slip plane They dissipate here Distortions and are incompatible (local field) 6/22
Two ingredients to describe particle rearrangements Particle deformability Dissipation of transversal fluctuations (slips)
Continuum physics model Mathematical model Numerical results
Governing equations Visc. stresses Momentum Equation for the distortion Mass conservation Entropy equation Ref: Godunov, Elements of continuum mechanics, 1978 (in Russian) Ref: Godunov, Romenski, Elements of continuum mechanics., 2003 7/22
Closure: equation of state Total energy conservation Equation of State equilibrium viscous kinetic micro meso macro The simplest example of the Viscous energy transverse sound 08/22
Closure: equation of state Total energy conservation Equation of State equilibrium viscous kinetic micro meso macro The simplest example of the Viscous energy For example Classical (equilibrium) pressure Non-equilibrium pressure 09/22
Closure: dissipation Equation for the distortion The dissipative dynamics is also generated by the Energy potential: Shear fluctuations dissipate here 10/22
Two steps to close the model: Define the energy potential Define
Continuum physics model Mathematical model Numerical results 11/22
Newton s viscous law as a steady state limit Newton s law Viscosity coefficient 12/22 Ref: Peshkov, Romenski, Continuum Mech. and Therm., 2014
Newton s law of viscosity, arbitrary shear 13/22 Ref: Peshkov, Romenski, Continuum Mech. and Therm., 2014
Phase velocity, [m/s] Dispersion relations for viscous gas, longitudinal sound Carbon tetrachloride Phase velocity, [m/s] Chloroform Phase velocity, [m/s] f/p, [MHz/atm] f/p, [MHz/atm] Chloromethane f/p, [MHz/atm] Frequency up to 2 MHz 14/22 Ref: Data from Sette, Busala, Hubbard, The Journal of Chem. Phys., 23 (5), 1955
ADER-WENO-FVM framework t Generalized Riemann Problem (smoothed initial data) x 1. WENO reconstruction 2. Solve GRP coupled with the source terms (Cauchy-Kovalevskaya or DG) 3. Update at n+1 Dumbser, Enaux, Toro, 2008 See papers by E. Toro, V. Titarev, M. Dumbser since 2000 15/22
First stokes problem y V, suddenly imposed x time=1 time=1 time=1 16/22
Steady laminar Hagen-Poiseuille flow, Re=50 inflow 17/22
Blasius boundary layer, Re=1000 Velocity contours Inflow V=1 x=0.5 cut 18/22
Double shear layer, visc=2 10-4 Left: Hyperbolic model ADER-WENO 4 th order Dumbser, Enaux, Toro, 2008 Vorticity Right: Navier-Stokes model staggered Semiimplicit DG P3 Tavelli, Dumbser 2014 time=0.8 time=1.2 19/22 time=1.8
Double shear layer, A 12 Vorticity Right: Navier-Stokes model staggered Semiimplicit DG P3 Tavelli, Dumbser 2014 time=0.8 time=1.2 time=1.8
Compressible mixing layer, Re=250 Velocity Top: 6 th order P N P M scheme Navier-Stokes model Dumbser, Zanotti, 2009 Vorticity Hyperbolic model 3 rd order ADER-WENO Dumbser, Enaux, Toro, 2008 A 12 20/22
From gases to solids, a unified model Viscous gas Elasto-plastic solid Elastic solid 21/22
Unified approach is an advantage for multifluid applications Immiscible multiphase flows in porous media. Displacement of a blue fluid by a red fluid in the CT-image-based geometry Fluids can have very different rheology 22/22
Conclusion A unified approach has been presented It can deal with gases, liquids, solids Extensive comparison with the NS theory has been provided Hyperbolic heat conduction extension and other numerical examples will appear in our paper shortly
Conclusion A unified approach has been presented It can deal with gases, liquids, solids Extensive comparison with the NS theory has been provided Hyperbolic heat conduction extension and other numerical examples will appear in our paper shortly Possible next steps Combustion Boundary layer Non-Newtoinan flows Nonequilibrium flows, rarified effects Viscoacoustics
Thank you for your attention!