Shock wave structure for polyatomic gas with large bulk viscosity 2018 International Workshop on Hyperbolic and Kinetic Problems: Theory and Applications (Institute of Mathematics, Academia Sinica July 10 - 14, 2018) Shock wave structure for polyatomic gas with large bulk viscosity Kazuo Aoki Department of Mathematics, National Cheng Kung University and NCTS, National Taiwan University Collaborator: Shingo Kosuge (Kyoto University)

Introduction

Structure of a plane shock wave Rapid change over a few mean free paths Problem for kinetic theory (Boltzmann equation) One of the most fundamental problems Many papers and books …… since 1950’s e.g., textbooks Kogan (1969), Cercignani (1976, 2000), Bird (1977, 1994), Shen (2005), Sone (2007), ….. Analysis, numerics, experiments Mathematical study Existence of (weak) shock profile Caflisch & Nicolaenko (1982), Commun. Math. Phys. Positivity of (weak) shock profile Liu & Yu (2004), Commun. Math. Phys. Numerical DSMC Bird (1965, 1967), RGD, JFM, …. Deterministic Ohwada (1993), Phys. Fluids

Polyatomic gases Many results for diatomic gases, such as N2 Not many results for CO2 Large bulk viscosity Large thickness (slow relaxation of internal modes) Asymmetric profile and double-layer structure Bethe & Teller (1941), Smiley et al. (1952, 1954) Griffith et al. (1954, 1956), Johannesen et al. (1962), … Zhdanov (1968), McCormack (1968, 1970), … Moment methods Taniguchi, Arima, Ruggeri, & Sugiyama (2014, 2016) Extended Thermodynamics (ET)

Profiles (schematic) from Taniguchi et al. (2014), Phys Rev. E gas flow CO2 : Density profile NS Experimant Johannesen et al. (1962) ET Type C: Double-layer structure (thin and thick layers) Obtained also for higher Mach numbers

Extended thermodynamics: Macroscopic theory from Taniguchi et al. (2016), Non-Linear Mech. CO2 Density profile (Type C) Extended thermodynamics: Macroscopic theory Curiosity – Can we observe Type C profile (and also Types A & B profiles) using a simple kinetic model? Ellipsoidal Statistical (ES) model for a polyatomic gas Numerical and asymptotic analysis for CO2 gas

Based on S. Kosuge and K.A., Shock-wave structure for a polyatomic Gas with large bulk viscosity, Phys. Rev. Fluids, 3, 023401 (2018).

Problem

Rankine-Hugoniot relations Shock wave structure Standing plane shock 1D problem Rankine-Hugoniot relations

Rankine-Hugoniot relations Upstream Mach number ( gas constant) Ratio of specific heats Internal degrees of freedom Structure of shock wave Kinetic theory Ellipsoidal Statistical (ES) model Holway (1963, 1966) for the Boltzmann eq. for a polyatomic gas Andries, Le Tallec, Perlat, & Perthame (2000) Brull & Schneider (2009)

Basic equations

ES model for a polyatomic gas Andries, Le Tallec, Perlat, & Perthame (2000), Eur. J. Mech. B Brull & Schneider (2009), Cont. Mech. Thermodyn. Prandtl number Conservation laws, H theorem, Velocity-energy distribution function time energy variable (internal modes, per unit mass) position molecular velocity Mass density in the space at Number of molecules contained in : molecular mass

ES model for a polyatomic gas Internal degs. of freedom 1D, steady Ratio of specific heats Collision frequency gas const.

ES model for a polyatomic gas Internal degs. of freedom 1D, steady Ratio of specific heats Collision frequency gas const.

ES model for a polyatomic gas Internal degs. of freedom 1D, steady Ratio of specific heats Collision frequency gas const. thermal cond. viscosity Prandtl number bulk viscosity

ES model for a polyatomic gas Local equilibrium Local equilibrium Conservation laws Mean free path equilibrium at rest

BC

Numerical analysis

Marginal distribution functions ( [BC] ) System of int.-diff. equations for with only two indep. variables Chu (1965) Andries et al. (2000) Numerical solution by finite-difference method

Numerical results

Shock structure in CO2 gas Parameter setting Emanuel (1990), Phys. Fluids A, 2 Uribe et al. (1990), J. Phys. Chem. Ref. Data, 19 Span & Wagner (1996), J. Phys. Chem. Ref. Data, 25 Kustova (2017), private communication …… pseudo-CO2 gas A model of a gas with large bulk viscosity

ES model for a polyatomic gas Internal degs. of freedom 1D, steady Ratio of specific heats Collision frequency gas const. thermal cond. viscosity Prandtl number bulk viscosity

Shock structure in pseudo-CO2 gas Profiles ( chosen appropriately) Type C Type B Type A Good agreement with Extended Thermodynamics!! Taniguchi et al. (2014)

Mean free path in equilibrium at

Mean free path in equilibrium at New Rankine-Hugoniot relations?

Comparison with ET (Taniuchi et al.) Taniguchi, Arima, Ruggeri, & Sugiyama (2014), Phys Rev. E Ratio of Specific heats Non-polytropic ES model Polytropic Complete comparison not possible A setting close to Taniuchi et al. : Internal degs. of freedom

 

Rankine-Hugoniot relations for Conservation laws New Rankine-Hugoniot relations

Rankine-Hugoniot relations for (Frozen) Same as R-H for monatomic gas if is regarded as upstream/ downstream Mach number

Mean free path in equilibrium at New Rankine-Hugoniot relations

Slowly varying solution with length scale Contracted coordinate

Slowly varying solution

Slowly varying solution with length scale Contracted coordinate ES model (dimensionless) 3D problems Hilbert-type expansion Usual procedure of H-expansion

Integral equations for

Compatibility cond. Macroscopic equations One more equation!

Solution Definitions of in terms of

One of the two equations serves as the additional equation Definitions of in terms of Not independent One of the two equations serves as the additional equation

Equations for or

Shock-wave structure Spatially 1D Equations for ODE for

ODE for : Solution Inverse function of : Explicit integration

No front shock: Type A (full profile) No front shock: Type B (full profile) Corner No corner Front shock: Type C (thick rear layer) Downstream condition of a shock with

Rankine-Hugoniot relations for (Frozen) Same as R-H for monatomic gas if is regarded as upstream/ downstream Mach number

Profiles (schematic) from Taniguchi et al. (2014), Phys Rev. E CO2 : Density profile NS Experimant Johannesen et al. (1962) ET Type C: Double-layer structure (thin and thick layers) Obtained also for higher Mach numbers

Slowly varying solution Type A pseudo-CO2 gas Slowly varying solution

Slowly varying solution Type B pseudo-CO2 gas Slowly varying solution

Slowly varying solution Type C pseudo-CO2 gas Slowly varying solution Numerical solution for

Thank you very much!