1. 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 , 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)

2. Introduction

3. 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

4. Polyatomic gases Many results for diatomic gases, such as N2
Not many results for CO 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)

5. 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

6. 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

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

8. Problem

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

10. 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)

11. Basic equations

12. 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

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

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

15. 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

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

17. BC

18. Numerical analysis

19. 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

20. Numerical results

21. 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

22. 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

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

24. Mean free path in equilibrium at

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

26. 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

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

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

32. Mean free path in equilibrium at
New Rankine-Hugoniot relations

33. Slowly varying solution with length scale
Contracted coordinate

34. Slowly varying solution

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

36. Integral equations for

37. Compatibility cond.
Macroscopic equations
One more equation!

38. Solution
Definitions of in terms of

39. 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

40. Equations for or

41. Shock-wave structure
Spatially 1D
Equations for
ODE for

42. ODE for : Solution Inverse function of :
Explicit integration

43. 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

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

45. 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

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

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

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

49. Thank you very much!