SYMMETRIZATION OF CONSERVATION LAWS WITH ENTROPY FOR HIGH-TEMPERATURE HYPERSONIC COMPUTATIONS(1) by Fr´ed´eric Chalot(2) and Thomas J. R. Hughes(3) Division of Applied Mechanics Durand Building Stanford University Stanford, California 94305

Farzin Shakib(4) NASA Ames Research Center Moffett Field, California 94035-4000

Appeared in: Computing Systems in Engineering, 1, 495–521 (1990)

(1)

This research was supported by NASA Langley Research Center under Grant NASANAG-1-361, the Avions Marcel Dassault-Br´eguet Aviation, St. Cloud, France, and the National Research Council. (2) Graduate Research Assistant. (3) Professor of Mechanical Engineering. (4) NRC Associate. Present Affiliation: CENTRIC Engineering Systems, 3801 East Bayshore, Palo Alto, California 94303.

ABSTRACT Results of Hughes, Franca, and Mallet are generalized to conservation law systems taking into account high-temperature effects. Symmetric forms of different equation sets are derived in terms of entropy variables. First, the case of a general divariant gas is studied; it can be specialized to the usual Navier-Stokes equations, as well as to situations where the gas is vibrationally excited, and undergoes equilibrium chemical reactions. The case of a gas in thermochemical nonequilibrium is considered next. Transport phenomena, and in particular mass diffusion are examined in the framework of symmetric advective-diffusive systems. Suitably defined finite element methods are shown to satisfy automatically the second law of thermodynamics, which a priori guarantees the stability of the discrete solution.

Citius, Altius, Fortius

ii

CONTENTS

Abstract

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Nomenclature

. . . . . . . . . . . . . . . . . . . . . . . . . . . . v

1. INTRODUCTION

. . . . . . . . . . . . . . . . . . . . . . . . . .

2. SYMMETRIC ADVECTIVE-DIFFUSIVE SYSTEMS

1

. . . . . . . . . .

1

3. GENERAL DIVARIANT GAS . . . . . . . . . . . . . . . . . . . . .

4

3.1 Systems of Conservation Laws 3.2 Constitutive Relations

. . . . . . . . . . . . . . . . . .

4

. . . . . . . . . . . . . . . . . . . . .

5

3.3 Entropy Variables and Symmetrization 3.4 Equations of State

. . . . . . . . . . . . . .

6

. . . . . . . . . . . . . . . . . . . . . . .

8

3.4.1 Calorically Perfect Gas

. . . . . . . . . . . . . . . . . .

3.4.2 Thermally Perfect Gas

. . . . . . . . . . . . . . . . . . 10

3.4.3 Mixture of Thermally Perfect Gases in Chemical Equilibrium 4. MIXTURE OF GASES IN THERMOCHEMICAL NONEQUILIBRIUM 4.1 Systems of Conservation Laws 4.2 Constitutive Relations 4.3 Equation of State

9

. 13 . . 15

. . . . . . . . . . . . . . . . . . 16

. . . . . . . . . . . . . . . . . . . . . 18

. . . . . . . . . . . . . . . . . . . . . . . 19

4.4 Computation of Entropy and of the Chemical Potentials 4.5 Entropy Variables and Symmetrization

. . . . . . 20

. . . . . . . . . . . . . . 23

5. ADVANTAGES OF SYMMETRIC CONSERVATION LAWS

. . . . . . . 26

f . . . . . . . . . . . . . . . . . . 26 5.1 Positive Semidefiniteness of K fmass 5.1.1 The Mass Diffusion Diffusivity Matrix, K

. . . . . . . . 26

fvisc . . . . . . . . . 26 5.1.2 The Viscous Stress Diffusivity Matrix, K iii

fheat 5.1.3 The Heat Conduction Diffusivity Matrix, K

5.2 Clausius-Duhem Inequality

. . . . . . . 27

. . . . . . . . . . . . . . . . . . . 27

5.3 Mathematical Foundations and Elements of Convergence Proof 6. CONCLUSIONS Acknowledgements

. . . 28

. . . . . . . . . . . . . . . . . . . . . . . . . . . 29 . . . . . . . . . . . . . . . . . . . . . . . . . . 31

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Appendix A – COEFFICIENT MATRICES FOR A GENERAL DIVARIANT GAS 35 Appendix B – A SIMPLE EQUILIBRIUM CHEMISTRY MODEL FOR AIR

40

Appendix C – COEFFICIENT MATRICES FOR A MIXTURE OF THERMALLY PERFECT GASES IN THERMOCHEMICAL NONEQUILIBRIUM

. . . . 43

Appendix D – COUPLING OF MASS DIFFUSION AND HEAT CONDUCTION: THE SORET AND DUFOUR EFFECTS

iv

. . . . . . . . . . . . . . . . 54

NOMENCLATURE

Roman Symbols Ai

advective Jacobian matrices with respect to conservative variables

ei A

advective Jacobian matrices with respect to entropy variables

e0 A

Riemannian metric tensor

a

speed of sound (m/s)

c

total number of moles per unit volume (mol/m3 )

cp

specific heat at constant pressure (J/kg·K=m2 /s2 ·K)

cps

specific heat at constant pressure of species s (J/kg·K=m2 /s2 ·K)

cs

mole concentration of species s (mol/m3 )

cv

specific heat at constant volume (J/kg·K=m2 /s2 ·K)

cvib v

vibrational specific heat at constant volume (J/kg·K=m2 /s2 ·K)

cˆvib v

vibrational specific heat at constant volume per mole (J/mol·K=kg·m2 /mol·s2 ·K)

cvs

specific heat at constant volume of species s (J/kg·K=m2 /s2 ·K)

cˆvs

specific heat at constant volume per mole of species s (J/mol·K=kg·m2 /mol·s2 ·K)

cvib vs

vibrational specific heat at constant volume of species s (J/kg·K=m2 /s2 ·K)

cˆvib vs

vibrational specific heat at constant volume per mole of species s (J/mol·K=kg·m2 /mol·s2 ·K)

DT

multicomponent thermal diffusion coefficient vector

DsT

multicomponent thermal diffusion coefficient of species s (kg/m·s)

Dsr

multicomponent diffusion coefficient of species s into species r (m2 /s)

dsi

gradient of species s in direction i used for the computation of diffusion velocities (m−1 )

E tot

total energy per unit volume (J/m3 =kg/m·s2 ) v

E vib

vibrational energy per unit volume (J/m3 =kg/m·s2 )

E

internal energy (J=kg·m2 /s2 )

E vib

vibrational energy (J=kg·m2 /s2 )

e

specific internal energy (J/kg=m2 /s2 )

etot

specific total energy (J/kg=m2 /s2 )

evib

specific vibrational energy (J/kg=m2 /s2 )

evib

specific vibrational energy vector

es

specific internal energy of species s (J/kg=m2 /s2 )

es 0

reference specific internal energy of species s (J/kg=m2 /s2 )

eˆs

specific internal energy per mole of species s (J/mol=kg·m2/mol·s2 )

erot s

specific rotational energy of species s (J/kg=m2 /s2 )

etot s

specific total energy of species s (J/kg=m2 /s2 )

etrans s

specific translational energy of species s (J/kg=m2 /s2 )

evib s

specific vibrational energy of species s (J/kg=m2 /s2 )

eˆvib s

specific vibrational energy per mole of species s (J/mol=kg·m2 /mol·s2 )

evib s0

reference specific vibrational energy of species s (J/kg=m2 /s2 )

e0

reference specific internal energy (J/kg=m2 /s2 )

evib 0

reference specific vibrational energy (J/kg=m2 /s2 )

Fiadv

advective flux vector in direction i

Fidiff

diffusive flux vector in direction i

Fiheat

heat flux vector in direction i

Fimass

mass diffusion flux vector in direction i

Fivisc

viscous stress flux vector in direction i

F

source vector in terms of conservative variables

e F

source vector in terms of entropy variables

G

Gibbs free energy (J=kg·m2 /s2 )

g

specific Gibbs free energy (J/kg=m2 /s2 ) vi

gs

specific Gibbs free energy of species s (J/kg=m2 /s2 )

gˆs

specific Gibbs free energy per mole of species s (J/mol=kg·m2/mol·s2 )

H

generalized entropy function (J/m3 ·K=kg/m·s2 ·K)

h

specific enthalpy (J/kg=m2 /s2 )

h

specific enthalpy vector

htot

specific total enthalpy (J/kg=m2 /s2 )

hs

specific enthalpy of species s (J/kg=m2 /s2 )

htot s

specific total enthalpy of species s (J/kg=m2 /s2 )

h0s

specific heat of formation of species s (J/kg=m2 /s2 )

ˆ0 h s

specific heat of formation per mole of species s (J/mol=kg·m2/mol·s2 )

h0

reference specific enthalpy (J/kg=m2 /s2 )

Ji

mass diffusion flux in direction i

Jsi

mass diffusion flux of species s in direction i

K

diffusivity matrix with respect to conservative variables

f K

diffusivity matrix with respect to entropy variables

fmass K

mass-diffusion diffusivity matrix with respect to entropy variables

fheat K

heat conduction diffusivity matrix with respect to entropy variables

fmass−heat K

mass-diffusion/heat conduction diffusivity matrix with respect to entropy

ftherm K

thermal diffusion matrix with respect to entropy variables

variables

fvisc K

viscous stress diffusivity matrix with respect to entropy variables

KcR

equilibrium constant of reaction R

Kij

diffusivity coefficient matrix with respect to conservative variables

fij K

diffusivity coefficient matrix with respect to entropy variables

fheat K ij

fmass K ij

fmass−heat K ij

heat conduction diffusivity coefficient-matrix with respect to entropy variables mass-diffusion diffusivity coefficient-matrix with respect to entropy variables mass-diffusion/heat conduction diffusivity coefficient-matrix with respect to vii

entropy variables ftherm K ij

fvisc K ij

viscous stress diffusivity coefficient-matrix with respect to entropy variables

ˆ M

molar mass (kg/mol)

ˆs M

molar mass of species s (kg/mol)

N

total number of moles (mol)

Ns

number of moles of species s (mol)

n

number of species

p

pressure (Pa=kg/m·s2 )

ps

partial pressure of species s (Pa=kg/m·s2 )

ps 0

reference partial pressure of species s (Pa=kg/m·s2 )

p0

reference pressure (Pa=kg/m·s2 )

QT −T

vib

thermal diffusion coefficient-matrix with respect to entropy variables

translation-vibration energy transfer rate (W/m3 =kg/m·s3 )

Q

heat received by the system (J=kg·m2 /s2 )

q

heat flux

qi

heat flux in direction i (W/m2 =kg/s3 )

qivib

vibrational heat flux in direction i (W/m2 =kg/s3 )

R

specific gas constant (J/kg·K=m2 /s2 ·K)

ˆ R

universal gas constant (= 8.3143 J/mol·K)

Rs

specific gas constant of species s (J/kg·K=m2 /s2 ·K)

S

entropy (J/K=kg·m2 /s2 ·K)

S ext

entropy due to interactions with exterior (J/K=kg·m2 /s2 ·K)

S int

entropy due to internal irreversibilities (J/K=kg·m2 /s2 ·K)

s

specific entropy (J/kg·K=m2 /s2 ·K)

ss

specific entropy of species s (J/kg·K=m2 /s2 ·K)

sˆs

specific entropy per mole of species s (J/kg·K=m2 /s2 ·K)

s0

reference specific entropy (J/kg·K=m2 /s2 ·K) viii

ss 0

reference specific entropy of species s (J/kg·K=m2 /s2 ·K)

T

(translational-rotational) temperature (K)

T vib

vibrational temperature (K)

T0

reference temperature (K)

T0vib

reference vibrational temperature (K)

t

time (s)

U

conservative variables vector

u

velocity vector

ui

velocity in direction i (m/s)

V

entropy variable vector

V

volume (m3 )

v

specific volume (m3 /kg)

vˆs

specific volume per mole of species s (m3 /kg)

vsi

diffusion velocity of species s in direction i (m/s)

xs

mole fraction of species s

y

mass fraction vector

ys

mass fraction of species s

Greek Symbols α

Onsager’s phenomenological coefficient matrix for mass diffusion

¯ α

Onsager’s phenomenological coefficient matrix for mass diffusion and heat conduction

αp

expansivity (K−1 )

αsr

Onsager’s phenomenological coefficients for mass diffusion (kg s/m3 )

α ¯ sr

Onsager’s phenomenological coefficients for mass diffusion and heat conduction

βT

isothermal compressibility (Pa−1 =m·s2 /kg)

γs

ratio of specific heats of species s

δi

Kronecker delta vector ix

δij

Kronecker delta

ǫ

strain tensor

ǫ′

deviatoric part of the strain tensor

ǫ′′

dilatational part of the strain tensor

Θvib s

characteristic vibrational temperature of species s (K)

κ, κ′

coefficient of thermal conductivity (W/m·K=kg·m/s3 ·K)

κvib

coefficient of thermal conductivity of the vibrational energy (W/m·K=kg·m/s3 ·K)

λvisc

second viscosity coefficient (Pl=kg/m·s)

µvisc

first viscosity coefficient (Pl=kg/m·s)

µvisc B

bulk viscosity coefficient (Pl=kg/m·s)

µ

specific chemical potential (J/kg=m2 /s2 )

µ

specific chemical potential vector

µs

specific chemical potential of species s (J/kg=m2 /s2 )

µ ˆs

specific chemical potential per mole of species s (J/mol=kg·m2 /mol·s2 )

µ0s

specific chemical potential of species s in the pure state, at unit pressure (J/kg=m2 /s2 )

µ ˆ0s

specific chemical potential per mole of species s in the pure state, at unit pressure (J/mol=kg·m2 /mol·s2 )

µs 0

reference specific chemical potential of species s (J/kg=m2 /s2 )

νsR

stoichiometric coefficient of species s in reaction R

ρ

density (kg/m3 )

ρs

density of species s (kg/m3)

ρs 0

reference density of species s (kg/m3 )

σi

entropy flux in direction i (W/m2 ·K=kg/s3 ·K)

τ

viscous stress tensor

τij

viscous stress in direction i on a plane normal to direction j (Pa=kg/m·s2 ) x

Ω

production rate vector

Ωs

production rate of species s (kg/m3·s)

Subscripts ( )i;j;k

direction i, j, or k

( )R

reaction R

( )s;r;t

species s, r, or t

( )0

reference state

Superscripts ( )T

transpose

( )′s

all species, except species s

Overlines (ˆ)

quantity per mole

(f)

coefficient matrix with respect to entropy variables

xi

F. Chalot, T. J. R. Hughes, and F. Shakib

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I . INTRODUCTION. From the beginning, aeronautics and space flight have been driven by the urge to fly faster and higher. With current intense activity in the area of hypersonic transatmospheric vehicles, computational fluid dynamics has entered a new era. Until recently, the compressible Navier-Stokes equations were viewed by many as representing the state of the art. But the combination of high speed and altitude results in very strong shock waves and the departure of air from a calorically perfect gas, due to the excitation of internal energy modes and chemical reactions. Over some of the flight regimes the assumption of thermochemical equilibrium is valid. However, regions of thermal and chemical nonequilibrium need to be accounted for in order to better predict aerothermal loads on the vehicle. Although there is still much improvement needed in accuracy and reliability of existing Navier-Stokes codes, the extension of numerical techniques to systems taking account of the mentioned phenomena is required. In particular, the symmetric form of the NavierStokes equations, a basis of certain methods, must be reconsidered in order to accommodate more general forms of the conservation equations. An outline of this paper follows: In Section 2, we review the theory of symmetric advective-diffusive and hyperbolic systems. In Section 3 we consider the case of a general divariant gas, which we specialize to two kinds of frozen gases (the calorically and thermally perfect gases) and to a mixture of perfect gases in thermochemical equilibrium. In Section 4 we describe the more complex example of a multicomponent gas in thermochemical nonequilibrium. In Section 5 we summarize the advantages of the symmetric form. The detailed arrays for the so-called entropy-variables form of the equations and a simple equilibrium chemistry model are presented in the appendices. II . SYMMETRIC ADVECTIVE-DIFFUSIVE SYSTEMS. We consider systems of conservation laws in the form adv diff U,t + Fi,i = Fi,i +F

(1)

F. Chalot, T. J. R. Hughes, and F. Shakib

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where U is the vector of conservative variables; Fiadv and Fidiff are, respectively, the advective and the diffusive fluxes in the ith -direction; and F is the source vector. In subsequent sections, we will present different systems and provide expressions for the corresponding vectors. Inferior commas denote partial differentiation and repeated indices indicate summation. With reference to (1), we assume: i ) The diffusive fluxes can be written in the form Fidiff = Kij U,j

(2)

ii ) The purely advective form of (1), namely, adv U,t + Fi,i =0

(3)

is hyperbolic, that is, any linear combination of the Jacobian matrices Ai = Fi,adv U possesses real eigenvalues and a complete set of eigenvectors [36]. It is useful to rewrite (1) in so-called quasi-linear form: U,t + Ai U,i = (Kij U,j ),i + F

(4)

where K = [Kij ] is the diffusivity matrix. The Ai ’s and K do not necessarily possess any particular properties of symmetry or positiveness and, in general, are functions of U . Under the change of variables U = U (V ), (1) becomes:

where

e0 V,t + A ei V,i = (K fij V,j ),i + F e A e0 = U,V A

e0 e i = Ai A A

e0 fij = Kij A K

e = F(U (V )) F

(5)

(6) (7) (8) (9)

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We seek a particular change of variables satisfying the following conditions: e0 is symmetric, positive-definite; i) A

ei ’s are symmetric; ii ) The A

f = [K fij ] is symmetric, positive-semidefinite. iii ) K

When a change of variables U = U (V ) exists that engenders these properties, (5) is referred to as a symmetric advective-diffusive system. As we will see, symmetric systems and the notion of a generalized entropy function are intimately linked. A generalized entropy function is a scalar-valued function H = H(U ) that possesses the following properties: i ) H is convex; ii ) There exist scalar-valued functions σi = σi (U ), called entropy fluxes, such that H,U Ai = σi,U

(10)

Harten [8] and Tadmor [33] discuss the symmetrization of hyperbolic conservation laws and the satisfaction of generalized entropy inequalities. Harten presents two theorems first proved by Mock [22] and Godunov [5]: Theorem (Mock): A hyperbolic system of conservation laws possessing a generalized entropy function H becomes symmetric hyperbolic under the change of variables VT =

∂H ∂U

(11)

Theorem (Godunov): If a hyperbolic system can be symmetrized by introducing a change of variables, then a generalized entropy function and corresponding entropy fluxes exist for this system. Harten also considered the compressible Navier-Stokes equations, neglecting heat conduction, and proposed a family of admissible generalized entropy functions. Hughes et

F. Chalot, T. J. R. Hughes, and F. Shakib

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al. [12] extended Harten’s work to the full Navier-Stokes equations and showed that the only suitable members of Harten’s family of generalized entropy functions are the ones that are at most trivially different from the physical entropy (up to an affine transformation). Following along these lines, we derive herein symmetrized forms of conservation laws which apply to high-temperature hypersonic computations, beginning with the case of a general divariant gas. III . GENERAL DIVARIANT GAS. The thermodynamic properties of a divariant gas, including its chemical composition in the case of a mixture of reacting gases, are completely defined in terms of two thermodynamic quantities, such as pressure p and temperature T . A hypersonic flow can be described as a divariant gas under the following assumptions: i ) The gas is in thermal equilibrium, i.e., the translational and internal energy modes are characterized by the same temperature T ; ii ) The gas mixture is either frozen, i.e., nonreacting, or in chemical equilibrium. In addition, we assume temperature levels low enough to preclude any ionization or radiative heat transfer. In the next section, we give explicit forms for the different vectors of equation (1), in the framework of the above assumptions. 3.1. Systems of Conservation Laws. In three dimensions, the vectors of (1) read:         1 U 1                    u U 1 2     U = U3 = ρ u2             u3    U4           etot  U  5

(12)

F. Chalot, T. J. R. Hughes, and F. Shakib

Fiadv

final (corrected)

   0           δ 1i    = ui U + p δ2i       δ   3i     u 

5

(13)

i

Fidiff

     0  0               0    τ1i  τ2i 0 + =         τ3i      0      −q τ u   ij j

F =0

i

              

(14)

(15)

where ρ is the density; u = {u1 , u2 , u3 }T is the velocity vector; etot is the total energy per unit mass, which is the sum of the internal energy per unit mass, e, and of the kinetic energy per unit mass, |u|2 /2; p is the thermodynamic pressure; δij is the Kronecker delta (viz., δii = 1, and δij = 0 for i 6= j); τ = [τij ] is the viscous-stress tensor; q = {q1 , q2 , q3 }T is the heat-flux vector. 3.2. Constitutive Relations. The system of partial differential equations of the previous section is supplemented with the following constitutive relations: i ) The viscous stress tensor τ is given by τij = λviscuk,k δij + µvisc(ui,j + uj,i)

(16)

where λvisc and µvisc are the viscosity coefficients. λvisc may be defined in terms of µvisc and the bulk viscosity coefficient µvisc by B 2 visc λvisc = µvisc B − µ 3

(17)

For perfect monatomic gases, kinetic theory predicts that µvisc B = 0. Stokes’ hypothesis states that µvisc B can be taken equal to zero in the general case. However, as shown

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in Vincenti and Kruger [35], behaviors such as small departures from rotational equilibrium can be represented by means of bulk viscosity. In the present discussion, where thermal equilibrium is assumed, Stokes’ hypothesis is valid. ii ) The heat flux is given by the usual Fourier law, qi = −κT,i

(18)

where κ is the coefficient of thermal conductivity. All the parameters (λvisc, µvisc, µvisc B , κ) appearing in the preceding constitutive relations are assumed to be functions of the thermodynamic state of the gas. In the present model, we have not considered any effect due to mass diffusion. Mass diffusion is indeed a phenomenon which by nature is not divariant, but depends on the actual chemical composition of the gas. In order to incorporate its effects into a model, one continuity equation is needed per species present in the mixture, which precludes a divariant description of the system. A few authors include the contribution of mass diffusion to the heat flux [11, 1, 7, and 9]. Since the densities of the different species are only functions of the thermodynamic state of the system (e.g., p and T ), the heat flux due to mass diffusion can be written in terms of a pressure and a temperature gradient. However, one can show that this yields a nonsymmetric contribution to the diffusivity f Moreover, correct entropy production is not guaranteed. Suitable modelization matrix K. of mass diffusion will be presented in the next section, in which we deal with mixtures in chemical nonequilibrium. 3.3. Entropy Variables and Symmetrization. We consider the generalized entropy function H, H = H(U ) = −ρs

(19)

where s is the thermodynamic entropy per unit mass. We will verify a posteriori that H meets the requirements of a generalized entropy function and, in particular, we will check

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its convexity. We introduce a change of variables U 7→ V defined by VT =

∂H ∂U

(20)

V is referred to as the vector of (physical ) entropy variables. For a divariant gas, (20) yields

  2   µ − |u| /2           u1   1 V = u2  T      u3         −1

(21)

where µ = e + pv − T s is the chemical potential per unit mass; v = 1/ρ is the specific volume. To derive (21) we used Gibbs’ relation, ds =

1 (de + pdv) T

(22)

The computation was made more manageable by the introduction of the auxiliary variables ( ) v Y = u (23) e and the application of the chain rule: H,U = H,Y (U,Y )−1 e0 reads The Riemannian metric tensor A 

u1 u2 1    v  u1 u2 u21 +  βT  e0 = βT T  A  v v2  u22 +  βT    symm.  

u3

u1 u3 u2 u3 u23 +

v βT

 |u|2 vαpT h+ −   2 βT  |u|2 v(αp T − 1)  u1 (h + ) −  2 βT   2 |u| v(αp T − 1)  u2 (h + − )  2 βT  2 |u| v(αp T − 1)  u3 (h + − )  2 βT  a55

(24)

(25)

F. Chalot, T. J. R. Hughes, and F. Shakib where

and

final (corrected)


|u|2 2 v a55 = h + + cp T − 2hαp T − |u|2 (αpT − 1) 2 βT

h = e + pv = µ + T s   1 ∂v αp = v ∂T p   1 ∂v βT = − v ∂p T   ∂h cp = ∂T p

8

(26)

(specific enthalpy)

(27)

(expansivity)

(28)

(isothermal compressibility)

(29)

(specific heat at constant pressure)

(30)

These coefficients are related to the specific heat at constant volume, namely, cv =



∂e ∂T



(31) v

by Mayer’s relation: cp − cv =

α2p vT βT

(32)

e0 are positive provided that T > 0, v > 0, cv > 0 The leading principal minors of A

and βT > 0; these conditions are met for any thermally and mechanically stable fluid away e0 is positive definite from T = 0 [37]. The positiveness of these determinants imply that A

e−1 = V,U = [6], which in turn implies that H is a strictly convex function of U (viz., A 0 H,U U ).

Explicit definitions of the coefficient matrices are presented in Appendix A. 3.4. Equations of State. The system of equations presented in the previous sections must be closed by the addition of an equation of state, such as a relation giving the chemical potential in terms of the thermodynamical state, i.e., µ = µ(p, T )

(33)

F. Chalot, T. J. R. Hughes, and F. Shakib

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In general, there exists no explicit version of (33). However, for most aerodynamic applications the perfect gas assumption applies. Fortunately, the “real gas effects” of the aerodynamicist, such as vibrational excitation and dissociation, are unrelated to the “real gas” of the thermodynamicist. A true real gas is a gas for which intermolecular forces are important; this occurs either at very high pressures (≈ 1,000 atm), or at low temperatures (< 30 K). At room temperature air is essentially a calorically perfect gas. It remains so until the temperature reaches approximately 600 K. Then, as the temperature increases further, vibrational excitation becomes important, and air behaves as a thermally perfect gas. Above 2,000 K, chemical reactions occur and air becomes a chemically reacting mixture of perfect gases [1]. After 10,000 K ionization becomes significant. The description of the accompanying phenomena exceeds the scope of this paper. All relevant thermodynamic quantities can be obtained from (33):     ∂µ ∂µ , v= s=− ∂T p ∂p T h = µ + T s,     1 ∂ 2µ 1 ∂v = , αp = v ∂T p v ∂p∂T  2    ∂ µ ∂h = −T , cp = ∂T p ∂T 2 p

(34)

e = h − pv     1 ∂v 1 ∂ 2µ βT = − =− v ∂p T v ∂p2 T

(35)

cv = cp −

(37)

α2p vT βT

(36)

Relations (34)–(37) are sufficient to construct the flux vectors and the coefficient matrices. In the next sections, we review in greater detail the three categories of perfect gases mentioned above. 3.4.1. Calorically Perfect Gas. A calorically perfect gas satisfies the perfect gas law, viz., pv = RT

(38)

where R is the specific gas constant, R=

ˆ R ˆ M

(39)

F. Chalot, T. J. R. Hughes, and F. Shakib

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ˆ = 8.3143 J/mol.K is the universal gas constant and M ˆ is the molar mass of in which R the gas. In addition, the specific heats cp and cv are constant. Such a gas is often simply referred to as a perfect gas. A simple explicit expression is available for the chemical potential, µ = cp T − cp T ln

p T + RT ln − T s0 T0 p0

(40)

where (p0 , T0 ) defines the reference state of reference entropy s0 . Then, (34)–(37) yield, s = cp ln

T p − R ln + s0 T0 p0

(41)

h = cp T

(42)

e = cv T

(43)

1 T 1 βT = p cp − cv = R

(44)

αp =

(45) (46)

3.4.2. Thermally Perfect Gas. A thermally perfect gas is only assumed to follow the perfect gas law (38). In general, the differential of internal energy per unit mass (with temperature and specific volume viewed as arguments) is given by de = cv dT + (ℓ − p)dv with ℓ=T



∂p ∂T



(47)

(48) v

Therefore, in the case of a perfect gas,   ∂e =ℓ−p=0 ∂v T

(49)

which implies that e = e(T, v) does not depend on v. Consequently, cv is a function of T only. In the same way, it can be shown that cp depends solely on T : cv = cv (T ),

cp = cp (T )

(50)

F. Chalot, T. J. R. Hughes, and F. Shakib

final (corrected)

The chemical potential can be expressed as Z T Z µ= cp (θ) dθ + h0 − T T0

and

s= h=

Z

Z

T T0

T T0

cp (θ) p dθ + RT ln − T s0 θ p0

11

(51)

p cp (θ) dθ − R ln + s0 θ p0

(52)

cp (θ) dθ + h0

(53)

T T0

e = h − RT =

Z

T

cv (θ) dθ + e0

(54)

T0

1 T 1 βT = p cp (T ) − cv (T ) = R αp =

(55) (56) (57)

where e0 and h0 are respectively the values of e and h at the reference temperature T0 . The last piece of information that we need to complete the picture is an expression for the specific heat at constant pressure cp . In fact, statistical and quantum mechanics usually provide direct information concerning the internal energy. In terms of e(T ), the chemical potential reads: e

e0  µ = e(T ) + RT − T −T − T T0

Z

T

T0

e(θ) T p dθ − RT ln + RT ln − T s0 2 θ T0 p0

(58)

In deriving (58), we have used the fact that h0 = e0 − RT0

(59)

e(T ) is commonly obtained experimentally or theoretically through extensive computations. Results are available in the form of curve fits or look-up tables for fast computer access [19]. However, under reasonable assumptions, a simple closed form expression can be derived for e(T ). At room temperature, air is essentially a mixture of diatomic oxygen and nitrogen (79% N2 and 21% O2 by volume). These molecules, as well as NO, can be assimilated to

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rigid rotators, at least until 10,000 K [18]. In these conditions, vibrational and rotational partition functions are separable, and vibrational modes can be considered as independent from rotational modes. In addition, in the same temperature range, molecular potential wells can be approximated by parabolas; consequently, vibrational excitation obeys the harmonic oscillator model. If the vibrational states are assumed to extend to infinity, the summation over them can be performed explicitly, leading to the following expression for the specific vibrational energy of a diatomic species s [26]: evib s =

Rs Θvib s exp(Θvib /T )−1 s

(60)

where Rs is the specific gas constant for species s, and Θvib s is its characteristic vibrational temperature. For the molecules of interest, Θvib s is of the order of 2,500–3,500 K. Electronic excitation can be written in a form similar to (60). However, we will neglect its contribution in the expression of internal energy. This is legitimate, since the lowest significant excited electronic state, one for O2 , has a characteristic temperature of nearly 11,400 K. Therefore, the specific internal energy of species s can be written as the sum of a translational, rotational, and vibrational contribution: vib es = etrans + erot s s + es

(61)

1 etrans = 3 × Rs T s 2

(62)

1 erot s = 2 × Rs T 2

(63)

where

For diatomic molecules,

and evib is given by (60). Atoms do not have rotational or vibrational internal energy s modes. In the temperature range where air behaves as a thermally perfect gas (600–2,000 K roughly), it is chemically inert and its composition remains that of room temperature. Thus, e(T ) = yN2 eN2 (T ) + yO2 eO2 (T )

(64)

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where yN2 and yO2 are the constant mass fractions of N2 and O2 , given by: ˆN M ρ N2 2 = .79 = .7670 ˆ ρ M ˆO M ρO2 2 = .21 = = .2330 ˆ ρ M

yN2 =

(65)

yO2

(66)

ˆ is the molar mass of the mixture: M ˆ = .79M ˆ N + .21M ˆ O = 28.84 × 10−3 kg/mol M 2 2

(67)

with ˆ N = 28.0 × 10−3 kg/mol M 2

(68)

ˆ O = 32.0 × 10−3 kg/mol M 2

(69)

Density is still related to pressure and temperature by the perfect gas law (38) where R is obtained using (39) and (67). Equation (58) can be integrated exactly, yielding the following expression for the chemical potential:  evib evib  X T p 7 − 0 + µ = e+RT − RT ln +RT ln −T 2 T0 p0 T T0

s=N2 ,O2



 1 − exp(−Θvib s /T ) ys Rs T ln −T s0 1 − exp(−Θvib s /T0 ) (70)

where e is given by (60)–(66); the summation concerns molecules of N2 and O2 ; the term (evib /T − evib 0 /T0 ) in (70) comes from the term (e/T − e0 /T0 ) in (58): # " 5 5 vib vib X R T + e R T + e e0 e s s 0 s s0 − = ys 2 − 2 T T0 T T 0 s   X evib evib = ys s − s 0 T T0 s =

evib evib − 0 T T0

(71)

In turn, entropy can be written as   X p evib evib 1 − exp(−Θvib T 7 s /T ) 0 − R ln + − ys Rs ln + s0 − s = R ln 2 T0 p0 T T0 1 − exp(−Θvib s /T0 ) s

(72)

(72) differs from the calorically perfect case (41), only by the terms induced by vibrational excitation.

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3.4.3. Mixture of Thermally Perfect Gases in Chemical Equilibrium. We now release the restriction imposed on the gas to be chemically inert. In order to be able to continue describing the mixture as a divariant gas, we must assume that it is in thermal and chemical equilibrium. In other words, a fluid element is supposed to respond instantaneously to any local change in pressure and temperature; this requires infinite chemical and vibrational rates. We specialize our description to air. Before significant ionization, it can be considered as a mixture of N2 , O2 , NO, N, and O. Each of these species has a chemical potential given by es 0  −T − µs = es + Rs T − T T T0 e

s

=

µ0s (T )

+ Rs T ln ps =

µ0s (T )

Z

T

T0

es (θ) ps T + Rs T ln − T ss 0 dθ − Rs T ln 2 θ T0 ps 0

+ Rs T (ln p + ln xs )

(73)

where the xs ’s are the mole fractions of the different species; they are related to the ˆ s , and to the mass fractions by, concentrations, cs = ρs /M xs =

ˆ cs M y = ˆs s c M

(74)

c is the total number of moles per unit volume, c=

X

cs

(75)

s

ˆ is the molar mass of the mixture: M ˆ = M

X s

ˆs = xs M

X ys ˆs M s

!−1

(76)

According to the number of atoms in the particle, es takes the form (61) to which the heat of formation has been added: vib 0 es = etrans + erot s s + es + hs

(77)

F. Chalot, T. J. R. Hughes, and F. Shakib

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h0s is zero for the species present in the mixture at room temperature, i.e., N2 and O2 , and nonzero for NO, N, and O. ps is the partial pressure of species s in the mixture: ps = ρs Rs T

(78)

The pressure p is obtained using Dalton’s law, p=

X

ps

(79)

s

The chemical potential of the system can be written as µ(p, T ) =

X

ys (p, T )µs (p, T )

(80)

s

where the mass fractions are no longer constants, but functions of the thermodynamic state. Unfortunately, there exist no explicit expressions for the ys ’s, which thus must be evaluated numerically. Many computer programs are available for this purpose. They are essentially based on two equivalent methods: the equilibrium constant method and the free energy minimization method. The objective of this paper is not to describe equilibrium solvers in detail. However, we will stress the importance of using the Newton-Raphson scheme to solve the resulting nonlinear systems: once convergence has been achieved, any thermodynamic derivative can be obtained directly from the linear system of the last iteration at essentially no extra cost (see [19] and Appendix B). Data can then be tabulated and interpolation formula designed to facilitate computer access. In Appendix B, we present a simple example of equilibrium chemistry calculations, where the strategy mentioned above is made clearer. IV . MIXTURE OF GASES IN THERMOCHEMICAL NONEQUILIBRIUM. Energy exchanges between translational and internal degrees of freedom, and chemical processes take place through molecular collisions. Equilibration between translational and rotational modes is fast and requires only a few collisions, say 20. Therefore, the

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assumption of equilibrium of the rotation with the translation is often justified, and both modes can be represented by the same temperature T . On the contrary, in response to drastic changes in the flow, such as through a shock wave, vibrational degrees of freedom demand many more collisions, typically on the order of 20,000, to reach their equilibrium level. Thus, if we assume that the oscillators have a Boltzmann distribution over their energy states, it will be expressed in terms of a second temperature T vib , different from the translational-rotational temperature T . An even larger number of collisions, on the order of 200,000, is needed to break apart molecules and trigger dissociation.

These collision processes take time. Consequently, before a fluid element reaches equilibrium, it has moved a certain distance, in a region where nonequilibrium prevails. The size of this nonequilibrium region depends on the dimensions and velocity of the vehicle. It is clear that, when the characteristic time for readjustment by collisions is of the same order as the characteristic time of the fluid flow, the equilibrium assumption breaks down, and nonequilibrium effects must be accounted for. According to Park [26], a multi-temperature model is crucial to compute accurately the aerothermal loads on the vehicle; models assuming thermal equilibrium tend indeed to predict that the flow is closer to equilibrium than it actually is, leading to incorrect values of pitching moment and trim angle.

Following the plan of the preceding section, we will present the symmetrization of a two-temperature model for non-ionized hypersonic flows in thermochemical nonequilibrium, and the corresponding entropy variables. In addition, since the thermodynamics of the irreversible processes involved [27] is much less documented in the literature than Gibbs’ classical thermodynamics, we will outline the derivations of entropy and of the chemical potentials of the different species present in the mixture.

F. Chalot, T. J. R. Hughes, and F. Shakib

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4.1. Systems of Conservation Laws. In three dimensions, and for a mixture of n species, the vectors appearing in (1) read:    U 1         .   ..          U    n      y           u   Un+1 U= =ρ vib e    Un+2        etot      Un+3                U   n+4      Un+5     0    n  Fiadv = ui U + p δi  0    u 

(81)

(82)

i

Fidiff = Fimass + Fivisc + Fiheat     −J i       0 3 mass Fi =  −Ji · evib       −J · h  i    0   n      τ δ ij j visc Fi =  0   τ u    

(83)

(84)

(85)

ij j

Fiheat =

     

0n 03 −qivib

     

       vib  −(qi + qi )       Ω       03 F= vib  Ω · evib + QT −T        0

(86)

(87)

where y = {ys } is the vector of mass fractions; δi is a generalized Kronecker delta, δi = {δij }; 0n and 03 are respectively null vectors of length n and 3; the diffusive flux,

F. Chalot, T. J. R. Hughes, and F. Shakib

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due to translational nonequilibrium effects [35], splits up into three parts: a mass diffusion part, Fimass , a viscous stress part, Fivisc, and a heat conduction part, Fiheat ; Ji is the mass diffusion flux in direction i, Ji = {Jsi } = {ρs vsi }, where vsi is the diffusion velocity of species s in direction i; evib and h are respectively the vibrational energy and enthalpy vecvib tors, evib = {evib are the heat fluxes in direction i, respectively s } and h = {hs }; qi and qi

due to gradients in the translational-vibrational temperature T and the vibrational temperature T vib ; Ω is the vector of the production rates of the different species, Ω = {Ωs }; QT −T

vib

is the translation-vibration energy transfer rate; other notations are common with

the preceding sections. 4.2. Constitutive Relations. The following constitutive relations are needed to complete the definition of the diffusive flux: i ) Nonequilibrium kinetic theory provides the following expression for the diffusion velocities (neglecting the thermal diffusion effect or Soret effect): Jsi = ρs vsi =

X M ˆ sM ˆr Dsr dri ρ ˆ2 M r

(88)

where dsi = xs,i + (xs − ys )(ln p),i

(89)

The Dsr ’s are the multicomponent diffusion coefficients, which are intricate functions of temperature and of the chemical composition of the mixture. They are not symmetric, i.e., Dsr 6= Drs

,

s 6= r

(90)

but satisfy the relations [10]: Dss = 0 X s

ˆ sM ˆ r Dsr − M ˆ sM ˆ t Dst ) = 0 (M

(91) ,

∀r, t

(92)

F. Chalot, T. J. R. Hughes, and F. Shakib

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Equation (92), together with X

dsi = 0

(93)

ρs vsi = 0

(94)

s

implies that X s

As shown in Appendix C, the pressure term in (89) is critical for obtaining a complete symmetrization of the system. However, the diffusion model presented here is apparently not widely used due to its ostensible complexity. Simpler models, which would retain symmetry, are currently under investigation. ii ) The viscous stress tensor is given as before by τij = λviscuk,k δij + µvisc(ui,j + uj,i)

(95)

iii ) The heat fluxes are given by Fourier’s law: qi = −κT,i qivib = −κvib T,ivib

(96) (97)

In (96) we have neglected the diffusion thermo-effect or Dufour effect. For completeness, the coupling between mass diffusion and heat conduction resulting from the Soret and Dufour effects, is considered in Appendix D. λvisc , µvisc, κ, and κvib can be computed from kinetic theory, but the corresponding formulae which involve temperatures, species densities and collision integrals, are too complex for practical use. Reasonable approximations can be obtained using Eucken’s relations [35, 10] and Wilke’s mixing rule [26]. 4.3. Equation of State. We assume that the gas is a mixture of thermally perfect gases. In addition, we adopt the rigid-rotator and harmonic-oscillator model. In these conditions, simple closed

F. Chalot, T. J. R. Hughes, and F. Shakib

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form expressions exist for the different energies: |u|2 etot = e + 2 X e= ys es

(98) (99)

s

evib = es (T, T

vib

)=

X

ys evib s

s trans es

(100)

vib 0 + erot s + es + hs

1 etrans (T ) = 3 × Rs T s 2 ( 0, for atoms rot es (T ) = 1 2 × Rs T, for diatomic molecules 2  for atoms  0, vib vib vib es (T ) = Rs Θs  , for diatomic molecules vib ) − 1 exp(Θvib s /T

(101) (102) (103)

(104)

4.4. Computation of Entropy and Chemical Potentials.

We consider a closed system, whose extensive properties are designated by script capital letters. Its change in entropy can be split into two parts (see Prigogine [27]): dS = dS ext + dS int

(105)

where dS ext is the flow of entropy into the system from its surroundings, and dS int is the production of entropy by irreversible processes within the system itself. dS ext is given by dS ext =

dQ T

(106)

where dQ is the heat received by the system at the temperature T . Using the first law of thermodynamics, (106) can be rewritten as dS ext =

p dE + dV T T

(107)

The internal irreversibilities inherent to the chemical and the vibrational nonequilibria produce the following entropy [35]: dS

int

  1X 1 1 =− µ ˆs dNs + dE vib − vib T s T T

(108)

F. Chalot, T. J. R. Hughes, and F. Shakib

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where Ns is the number of moles of species s in the system, and µ ˆs the molar chemical potential of species s. In general, we indicate per mole quantities by the addition of a circumflex over the symbol. The second law of thermodynamics states that dS int = 0 when the system undergoes reversible changes, and that dS int > 0 if the system is subject to irreversible processes. From (108), it is easy to derive the condition for chemical equilibrium (B.10), which is related to De Donder’s notion of chemical affinity [27]. The production of entropy due to vibrational nonequilibrium can be seen as the result of an exchange of heat between two subsystems, one being in equilibrium at temperature T , the other at temperature T vib . If we assume chemical equilibrium, the second law of thermodynamics dictates that, if T > T vib , then the vibrational energy must increase; accordingly, if T vib > T , then E vib will decrease. Combining (107) and (108), we get   dE 1 p 1X 1 dS = µ ˆ s dNs + dE vib + dV − − T T T s T vib T

(109)

In order to compute the entropy of the system, we need an expression for the chemical potentials µ ˆs . Let us introduce the Gibbs free energy, an extensive thermodynamic state variable defined by G = E + pV − T S

(110)

dG = dE + pdV + Vdp − T dS − SdT

(111)

The differential of (110) reads,

Substituting (109) into (111), we get dG = Vdp − SdT +

X

µ ˆs dNs −

s



 T − 1 dE vib T vib

(112)

But, E vib =

X s

Ns eˆvib s

(113)

F. Chalot, T. J. R. Hughes, and F. Shakib

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Thus, (112) becomes dG = Vdp − SdT − where N =

P

s Ns



    X T T vib vib vib − 1 N cˆv dT + µ ˆs − − 1 eˆs dNs T vib T vib s

(114)

is the total number of moles in the system, N cˆvib v =

X

Ns cˆvib vs

(115)

s

and, cˆvib vs =

vib ∂ˆ evib ) s (T vib ∂T

(116)

From (114), we deduce, 

∂G ∂Ns



=µ ˆs − p,T,T vib ,Ns′



 T − 1 eˆvib s T vib

(117)

G is extensive, and as a function of p, T , T vib , and Ns , must therefore be homogeneous in Ns :

X  ∂G  X G= Ns = Ns gˆs ∂Ns p,T,T vib ,N ′ s s

(118)

s

From (117) and (118), we obtain:

gˆs = µ ˆs −



 T − 1 eˆvib s T vib

(119)

We see that the well-known identity gˆs = µ ˆs is satisfied only when vibrational degrees of freedom are at equilibrium with the translation, that is, when T vib = T . Going back to the definition of G, we can derive another expression for the gˆs ’s: G = E + pV − T S X V Ns (ˆ es + ps = − T sˆs ) N s s X X = Ns (ˆ es + ps vˆs − T sˆs ) = Ns gˆs s

(120)

s

Therefore, gˆs = eˆs + ps vˆs − T sˆs

(121)

F. Chalot, T. J. R. Hughes, and F. Shakib

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Finally, (119) and (121) lead to:

µ ˆ s = eˆs + ps vˆs − T sˆs + P



 T − 1 eˆvib s T vib

for S into the expression for dS yields    X X  dˆ es ps 1 1 vib Ns dˆ ss = dˆ es Ns + dˆ vs + − T T T vib T s s    X T dNs vib + eˆs + ps vˆs − µ ˆs + − 1 e ˆ − T s ˆ s s T vib T s    X  dˆ 1 1 es ps vib dˆ es Ns + dˆ vs + − = T T T vib T s

Then, substituting

(122)

ˆs s Ns s

(123)

Thus, dˆ es ps + dˆ vs + dˆ ss = T T



1 T vib

1 − T



dˆ evib s

(124)

writing eˆs in the form, vib ˆ0 eˆs = cˆvs T + eˆvib )+h s (T s

(125)

(124) becomes dˆ ss = cˆvs

vib dT vs vib dT ˆ dˆ + cˆvib (T ) +R vs T vˆs T vib

(126)

ˆ s , and integrating from Converting into quantities per unit mass by dividing (126) by M the state of reference entropy ss 0 , we get:   vib 1 − exp(−Θvib ) ps evib evib T s /T s s0 − Rs ln + ss 0 − Rs ln + − ss = cps ln vib T0 ps 0 T vib T0vib 1 − exp(−Θvib s /T0 )

(127)

where cps = cvs + Rs

(128)

The specific entropy of the system is obtained from (127), s=

X s

ys s s

(129)

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4.5. Entropy Variables and Symmetrization. As in Section 3, we define the generalized entropy function H as H = −ρs

(130)

where s is given by (129). The change of variables VT =

∂H ∂U

(131)

yields the following vector of entropy variables:   2   |u|     1 µ −   n     2     1 u V =  T  T    1 − vib        T     −1

(132)

where µ = {µs } and 1Tn = {1, . . . , 1}. | {z } n terms

The assumptions made in Section 4.3 concerning the form of the different energies

are not necessary to derive (132). Knowing that es is the sum of a function of T and of a function of T vib is sufficient. However, if (101)–(104) are satisfied, we can express explicitly the primitive variables Y T = {ρT , uT , T vib , T } in terms of the entropy variables: ui = −

Vn+i Vn+5

,

i = 1, 2, 3

1 Vn+4 + Vn+5 1 T =− Vn+5    γ 1−1  vib s 1 − exp(−Θvib T µs − h0s µs 0 − h0s s /T0 ) ρs = ρs 0 exp − vib ) T0 1 − exp(−Θvib Rs T Rs T0 s /T

T vib = −

where

µs = T Vs + and

Vs 1 |u|2 2 2 2 =− + 2 (Vn+1 + Vn+2 + Vn+3 ) 2 Vn+5 2Vn+5

(133) (134) (135) (136)

(137)

F. Chalot, T. J. R. Hughes, and F. Shakib γs =

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cps cvs

(138)

ρs 0 and µs 0 being respectively the density and the chemical potential of species s in the state of reference entropy. The production rates of the different species, Ωs , are given in terms of the ρs ’s, T , and T vib [26]. These quantities are readily available through (133)– (136). In particular, there is no need to numerically solve for T vib , as it would be the case with a strategy based on the conservative variables U , unless one vibrational temperature is defined for each diatomic species [2]. The characteristics of the different transformations between primitive, conservative, and entropy variables for the rigid-rotator/harmonicoscillator model, are summarized in Table 1. Figure 1. also depicts the available explicit relations among the three sets of variables. While conservative variables appear clearly as a bottleneck, entropy variables are seen to bring in the ease of use of primitive variables, retaining the advantage of a conservation law form.

to:

from:

Primitive

Conservative

Entropy

Primitive

=

explicit

explicit

Conservative

numerical

=

numerical

Entropy

explicit

explicit

=

Table 1. Variable transformations for rigid-rotator/harmonic-oscillator model.

−→ ←−

Y ց

V ւ

U

Figure 1. Explicit relations for rigid-rotator/harmonic-oscillator model.

F. Chalot, T. J. R. Hughes, and F. Shakib

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e0 and the corresponding coefficient matrices are The Riemannian metric tensor A

described in Appendix C.

V . ADVANTAGES OF SYMMETRIC CONSERVATION LAWS. The advantages of the symmetry of (5) are practical as well as aesthetic. It can be shown that (5) expresses intrinsically the mathematical and physical stability provided by the second law of thermodynamics. Moreover, strong mathematical foundations support the analysis of convergence of finite element methods applied to the numerical solution of (5). We will exemplify these properties with the system of Section 4, for which the sources of irreversibility are more numerous and richer. We will begin this section with the proof f of the positive semidefiniteness of the diffusivity matrix K.

f 5.1. Positive Semidefiniteness of K.

As will be seen in the next section, when the dot product of (5) with V is taken, the

f fij V,j appears after an integration by parts. We will examine term ∇V · K∇V = V,i · K f namely, K fmass , K fvisc, and K fheat . it for each of the three components of K, fmass . 5.1.1. The Mass-diffusion Diffusivity Matrix, K

fmass ∇V = V,i · K fmass V,j = V,i · F mass ∇V · K ij i      XX ρr,i T,i ρs,i T,i + (T αsr )Rr + = Rs ρ T ρ T s r i s,r XX = [Rs (ln ps ),i (T αsr )Rr (ln pr ),i ] ≥ 0 i

(139)

s,r

if and only if α = [αsr ] is positive semidefinite. The αsr ’s are Onsager’s phenomenological coefficients for mass diffusion, as defined in Appendix C. The second law of thermodynamics requires the positiveness of α (see Onsager [24]).

F. Chalot, T. J. R. Hughes, and F. Shakib

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fvisc. 5.1.2. The Viscous Stress Diffusivity Matrix, K fvisc∇V = V,i · K fvisc V,j = V,i · F visc ∇V · K ij i  1  visc = λ (∇ · u)2 + 2µvisc (ǫ : ǫ) T

(140)

1 ǫij = (ui,j + uj,i ) 2

(141)

where ǫ is the strain tensor:

ǫ can be split into a deviatoric part ǫ′ , and a dilatational part ǫ′′ :

Then, fvisc

∇V · K

1 1 ǫ′′ij = ( trǫ)δij = ǫkk δij 3 3

(142)

ǫ′ij = ǫij − ǫ′′ij

(143)

  2 visc 1 visc 2 visc ′ ′ (λ + µ )(trǫ) + 2µ (ǫ : ǫ ) ≥ 0 ∇V = T 3

(144)

if and only if λvisc + 32 µvisc ≥ 0 and µvisc ≥ 0. fheat . 5.1.3. The Heat Conduction Diffusivity Matrix, K fheat ∇V = V,i · K fheat V,j = V,i · F heat ∇V · K ij i

κvib κ 2 = 2 (∇T ) + (∇T vib )2 ≥ 0 2 vib T T

(145)

if and only if κ ≥ 0 and κvib ≥ 0. 5.2. Clausius-Duhem Inequality. The dot product of (5) with V reads h i e0 V,t + A ei V,i = (K fij V,j ),i + F e V · A

(146)

Integrating by parts the diffusivity term, and rearranging yield e0 V,t + V · F adv − V · F diff V ·A i i,i




,i

e = −V,i · K fij V,j −V ·F

(147)

F. Chalot, T. J. R. Hughes, and F. Shakib

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From the preceding section, we know that the right-hand side in (147) is nonpositive. Moreover, one can show that e0 V,t = H,t V ·A

(148)

adv V · Fi,i = (Hui ),i X V · Fimass = ρs ss vsi

(149) (150)

s

V · Fivisc = 0

(151)

qivib T vib

qi + T   vib 1 1 1X e g s Ωs − QT −T − V ·F = vib T s T T

V · Fiheat =

(152) (153)

Substituting (148)–(153) into (147), we get !     X vib qi 1 qivib 1X 1 H,t + (Hui ),i − ρs ss vsi − g s Ωs + QT −T ≤ 0 + vib − − vib T T T s T T ,i s ,i

(154)

An interpretation of (154) is that the rate of growth of a convex function of the solution, H(U ), is bounded from above. Appropriately defined finite element methods inherit the fundamental stability property possessed by solutions of (5) (see Hughes, Franca and Mallet [12], Mallet [21], and Shakib [29]). Setting H = −ρs in (154) gives the local form of the Clausius-Duhem inequality, which governs entropy production for the system: !



 qi qivib (ρs),t + (ρsui),i + ρs ss vsi + + vib T T ,i s ,i   1 1 1X T −T vib g s Ωs − Q ≥0 − + T s T vib T X

(155)

5.3. Mathematical Foundations and Elements of Convergence Proof. The finite element methods developed by Hughes et al., variously referred to as Galerkin/least-squares, SUPG, and streamline diffusion, are the only ones for which con-

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vergence proofs and error estimates have been given for linear steady and unsteady multidimensional scalar advection-diffusion equations, over the full spectrum of advective-diffusive phenomena [13, 14, 15]. Direct extensions of these results have also been established for linear symmetric advective-diffusive systems. Linear symmetric advective-diffusive systems are of interest in their own right, since they serve as model equations for many nonlinear systems, such as those described in this paper. Until fairly recently, there had been virtually no successful analyses for nonlinear problems. In a series of penetrating articles, Johnson et al. have proved a number of important mathematical convergence results for our techniques in the context of hyperbolic conservation laws: in [16, 17], they established a convergence proof for Burger’s equation in one dimension. Szepessy extended this result to a general scalar conservation law in two dimensions [32]. For hyperbolic systems of conservation laws which possess a strictly convex entropy function and thus can be symmetrized via entropy variables, Johnson, Szepessy, and Hansbo [17] showed that limits of the finite element solutions are weak solutions of the original system and satisfy the entropy condition. Complete convergence proof appears as an extremely difficult, or even impossible task, since it is linked to the still unsolved problem of the existence of solutions to such systems. Nevertheless, the mathematical richness of symmetric advective-diffusive systems and of Galerkin/least-squares type methods sheds some rigor into an area of numerical analysis where it does not usually prevail. VI . CONCLUSIONS. In this paper, we have presented different advective-diffusive systems to compute high-temperature hypersonic flows, and their symmetrization via entropy variables. In generalizing our techniques to higher Mach number flows, involving chemistry and hightemperature effects, entropy variables may have been expected to engender complications. In fact, not only no fundamental impediment was encountered, but also entropy variables appear as the “natural” variables to use. In addition, it is interesting to see how correct

F. Chalot, T. J. R. Hughes, and F. Shakib

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entropy production conditions the form of the constitutive relations, and in particular that of mass diffusion, while demonstrating the deficiencies of widely employed models such as Fick’s law.

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ACKNOWLEDGEMENTS The authors would like to express their appreciation for helpful comments to Zdenˇek Johan, Yen Liu, Robert MacCormack, Marshal Merriam, Chul Park, Thomas Pulliam, and Marcel Vinokur.

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REFERENCES [1] J.D. ANDERSON, JR., Hypersonic and High Temperature Gas Dynamics, McGraw-Hill, 1989. [2] G. CANDLER, “The Computation of Weakly Ionized Hypersonic Flows in Thermochemical Nonequilibrium”, Ph.D. Thesis, Department of Aeronautics and Astronautics, Stanford University, 1988. [3] G. CANDLER and R.W. MACCORMACK, “The Computation of Hypersonic Flows in Chemical and Thermal Nonequilibrium”, Third National Aerospace Plane Technology Symposium, June 2–4, 1987. Paper number 107. [4] S. CHAPMAN, and T.G. COWLING, The Mathematical Theory of Non-uniform Gases, Third Edition, Cambridge University Press, 1970. [5] S.K. GODUNOV, “The Problem of a Generalized Solution in the Theory of Quasilinear Equations and in Gas Dynamics”, Russ. Math. Surveys, Volume 17, pp 145–156, 1962. [6] G.H. GOLUB, and C.F. VAN LOAN, Matrix Computations, Second Edition, Johns Hopkins, 1989. [7] C.F. HANSEN, “Approximations for the Thermodynamic and Transport Properties of High-temperature Air”, NASA Technical Report TR-R-50, 1959. [8] A. HARTEN, “On the Symmetric Form of Systems of Conservation Laws with Entropy”, Journal of Computational Physics, Volume 49, pp 151–164, 1983. [9] J.O. HIRSCHFELDER, “Heat Transfer in Chemically Reacting Mixtures. I.”, Journal of Chemical Physics, Volume 26, Number 2, pp 274–281, 1957. [10] J.O. HIRSCHFELDER, C.F. CURTISS, and R.B. BIRD, Molecular Theory of Gases and Liquids, Second Printing, John Wiley, New York, 1954. [11] H. HOLLANDERS, L. MARRAFFA, J.-L. MONTAGNE´ , PH. MORICE, and H. VIVIAND, “Computational Methods for Hypersonic Flows: Special Techniques and Real Gas Effects”, Second Joint Europe/U.S. Short Course in Hypersonics, January 16–20, 1989, Colorado Springs, Colorado. [12] T.J.R. HUGHES, L.P. FRANCA and M. MALLET, “A New Finite Element Formulation for Computational Fluid Dynamics: I. Symmetric Forms of the Compressible Euler and Navier-Stokes Equations and the Second Law of Thermodynamics”, Computer

F. Chalot, T. J. R. Hughes, and F. Shakib final (corrected) 33 Methods in Applied Mechanics and Engineering, North-Holland, Volume 54, pp 223– 234, 1986. [13] T.J.R. HUGHES, L.P. FRANCA, and M. MALLET, “A New Finite Element Formulation for Computational for Computational Fluid Dynamics: VI. Convergence Analysis of the Generalized SUPG Formulation for Linear Time-dependent Multidimensional Advective-diffusive Systems”, Computer Methods in Applied Mechanics and Engineering, Volume 63, pp 97–112, 1987. [14] T.J.R. HUGHES, L.P. FRANCA, and G. HULBERT, “A New Finite Element Formulation for Computational for Computational Fluid Dynamics: VIII. The Galerkin/Leastsquares Method for Advective-diffusive Equations”, Computer Methods in Applied Mechanics and Engineering, Volume 73, pp 173–189, 1989. [15] T.J.R. HUGHES, F. CHALOT, and Z. JOHAN, Finite Element Methods in Fluid Mechanics, text in preparation. [16] C. JOHNSON, and A. SZEPESSY, “On the Convergence of a Finite Element Method for a Nonlinear Hyperbolic Conservation Law”, Mathematics of Computation, Volume 49, Number 180, pp 427–444, 1987. [17] C. JOHNSON, A. SZEPESSY, and P. HANSBO, “On the Convergence of Shock-capturing Streamline Diffusion Finite Element Methods for Hyperbolic Conservation Laws”, Preprint No 1987-21, Department of Mathematics, Chalmers University of Technology, G¨oteborg, Sweden. [18] Y. LIU, F. SHAKIB and M. VINOKUR, “A Comparison of Internal Energy Calculation Methods for Diatomic Molecules”, AIAA 28th Aerospace Sciences Meeting, January 8–11, 1990, Reno, Nevada. Paper AIAA-90-0351. [19] Y. LIU and M. VINOKUR, “ Equilibrium Gas Flow Computations. I. Accurate and Efficient Calculations of Equilibrium Gas Properties”, AIAA 24th Thermophysics Conference, Buffalo, New York, June 12–14, 1989. Paper AIAA-89-1736. [20] Y. LIU and M. VINOKUR, “ Equilibrium Gas Flow Computations. II. An Analysis of Numerical Formulations of Conservation Laws”, AIAA 26th Aerospace Sciences Meeting, January 11–14, 1988, Reno, Nevada. Paper AIAA-88-0127. [21] M. MALLET, “A Finite Element Method for Computational Fluid Dynamics”, Ph.D. Thesis, Division of Applied Mechanics, Stanford University, 1985. [22] MOCK, “Systems of Conservation Laws of Mixed Type”, Journal of Differential Equations, Volume 37, pp 70–88, 1980. [23] L. ONSAGER, “Reciprocal Relations in Irreversible Processes. I.”, Physical Review, Volume 37, pp 405–426, 1931.

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[24] L. ONSAGER, “Reciprocal Relations in Irreversible Processes. II.”, Physical Review, Volume 38, pp 2265–2279, 1931. [25] C. PARK, “On Convergence of Computation of Chemically Reacting Flows”, AIAA 23rd Aerospace Sciences Meeting, January 14–17, 1985, Reno, Nevada. Paper AIAA85-0247. [26] C. PARK, Nonequilibrium Hypersonic Aerothermodynamics, Wiley-Interscience, 1990. [27] I. PRIGOGINE, Introduction to Thermodynamics of Irreversible Processes, Third Edition, Interscience, 1967. [28] J.V. RAKICH, H.E. BAILEY and C. PARK, “Computation of Nonequilibrium Threedimensional Inviscid Flow over Blunt-nosed Bodies Flying at Supersonic Speeds”, AIAA 8th Fluid and Plasma Dynamics Conference, Hartford, Connecticut, June 16–18, 1975. Paper AIAA-75-835. [29] F. SHAKIB, “Finite Element Analysis of the Compressible Euler and Navier-Stokes Equations”, Ph.D. Thesis, Division of Applied Mechanics, Stanford University, 1989. [30] S. SRINIVASAN, J.C. TANNEHILL, and K.J. WEILMUENSTER, “Simplified Curve Fits for the Thermodynamic Properties of Equilibrium Air”, NASA Reference Publication RP1181, 1987. [31] J.L. STEGER and R.F. WARMING, “Flux Vector Splitting of the Inviscid Gas Dynamics Equations with Applications to Finite Difference Methods”, NASA Technical Memorandum TM-78605, July 1979. [32] A. SZEPESSY, “Convergence of a Shock-capturing Streamline Diffusion Finite Element for Scalar Conservation Laws in Two Space Dimensions”, Preprint No 1988-07, Department of Mathematics, Chalmers University of Technology, G¨oteborg, Sweden. [33] E. TADMOR, “Skew-selfadjoint Form for Systems of Conservation Laws”, Journal of Mathematical Analysis and Applications, Volume 103, pp 428–442, 1984. [34] J.C. TANNEHILL, and P.H. MUGGE, “Improved Curve Fits for the Thermodynamic Properties of Equilibrium Air Suitable for Numerical Computation Using Timedependent or Shock-capturing Methods”, NASA Contractor Report CR-2470, 1974. [35] W.G. VINCENTI and C.H. KRUGER, JR., Introduction to Physical Gas Dynamics, Krieger, 1965. [36] R.F. WARMING, R.M. BEAM and B.J. HYETT, “ Diagonalization and Simultaneous Symmetrization of the Gas-dynamic Matrices”, Mathematics of Computation, Volume 29, Number 132, pp 1037–1045, 1975. [37] L.C. WOODS, The Thermodynamics of Fluid Systems, Oxford University Press, 1975.

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Appendix A – COEFFICIENT MATRICES FOR A GENERAL DIVARIANT GAS. In this appendix, we present the flux vectors and the coefficient matrices of the conservation law system for a general divariant gas, expressed in terms of the (physical) entropy variables. For convenience, U and V are repeated here:     1          u  1   1 u2 U=  v     u 3        e + |u|2 /2 

(A.1)

  2   µ − |u| /2           u 1   1 V = u2  T      u   3       −1

(A.2)

Given p, T , and µ = µ(p, T ), we have the following relations:     ∂µ ∂µ , v= s=− ∂T p ∂p T h = µ + T s,     1 ∂v 1 ∂ 2µ αp = = , v ∂T p v ∂p∂T   2   ∂ µ ∂h = −T , cp = ∂T p ∂T 2 p

(A.3)

e = h − pv     1 ∂v 1 ∂ 2µ βT = − =− v ∂p T v ∂p2 T

(A.4)

cv = cp −

(A.6)

α2p vT βT

(A.5)

We express the flux vectors and coefficient matrices with the help of the following variables: |u|2 , 2 v , βT

vαp T , βT v c2 = u22 + , βT

c¯1 = u21 + cv T,

c¯2 = u22 + cv T,

k=

c1 = u21 +

d=

γ¯ =

vαp , βT cv

(A.7)

v , βT

(A.8)

c¯3 = u23 + cv T,

(A.9)

c3 = u23 +

F. Chalot, T. J. R. Hughes, and F. Shakib e2 = e1 − d,

e1 = h + k, e4 = e2 + 2

final (corrected) e3 = e2 +

v , βT

v , βT e¯2 = e¯1 − d,

u12 = u1 u2 ,

e¯3 = e¯2 − cv T,

u23 = u2 u3 ,

v(2k + cp T ) , βT

(A.12)

u31 = u3 u1 ,

(A.13)

vcp cv βT

(A.14)

a2 =

u123 = u1 u2 u3 ,

e5 = e21 − 2e1 d +

(A.10) (A.11)

e¯1 = h − k,

as

36

e¯5 = e¯21 − 2¯ e1 d + 2kcv T +

vcp T βT

(A.15)

e0 = U,V and its inverse A e−1 = V,U can be written The Riemannian metric tensor A 0 

   β T T e0 =  A 2 v    and

e−1 A 0



  v   = cv T 2   

1

e¯5

u1 c1

u2 u12 c2 symm.

u1 e¯3 c¯1

u3 u31 u23 c3

u2 e¯3 u12 c¯2 symm.

 e2  u1 e3   u2 e3   u3 e3   e5

u3 e¯3 u31 u23 c¯3

 −¯ e2  −u1   −u2   −u3   1

(A.16)

(A.17)

The advective fluxes are

Fiadv

     1 0               u1    δ1i   u2 + p δ2i = ρui            u3 δ3i             u    e + |u|2 /2  i        

(A.18)

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The advective Jacobians with respect to U , Ai = Fi,adv U , are given by   0 1 0 0 0     a2 − u2 − e¯1 γ¯ −u (¯ γ − 2) −u γ ¯ −u γ ¯ γ ¯ 1 2 3 1     A1 =  −u12 u2 u1 0 0      −u u 0 u 0 31 3 1   −u1 (e1 + e¯1 γ¯ − a2 ) e1 − u21 γ¯ −u12 γ¯ −u31 γ¯ u1 (¯ γ + 1) 

    A2 =    



    A3 =    

0 0 u12 u2 a2 − u22 − e¯1 γ¯ −u1 γ¯ −u23 0 2 −u2 (e1 + e¯1 γ¯ − a ) −u12 γ¯

0 0 −u31 u3 −u23 0 a2 − u23 − e¯1 γ¯ −u1 γ¯ 2 −u3 (e1 + e¯1 γ¯ − a ) −u31 γ¯

Note that

1 0 u1 0 −u2 (¯ γ − 2) −u3 γ¯ u3 u2 2 e1 − u2 γ¯ −u23 γ¯

0 0 u3 −u2 γ¯ −u23 γ¯

(A.19)



0   0   γ¯    0  u2 (¯ γ + 1)

(A.20)



1 0   u1 0   u2 0    −u3 (¯ γ − 2) γ¯  e1 − u23 γ¯ u3 (¯ γ + 1)

     1 0               u1   c − vα p     δ1i  p p u2 + AiU = ρui p δ2i     βT cv p        u3 δ3i              e + |u|2 /2   u  i        

(A.21)

(A.22)

where (cp − vαp p)/βT cv p = 1 only for a thermally perfect gas.

by

ei = F adv = Ai A e0 , are given The advective Jacobian matrices with respect to V , A i,V 

    β T T  e1 = A v2    

u1

c1 u12 u1 (u21 + 3 v ) u2 c1 βT u1 c2 symm.

u31 u3 c1 u123 u1 c3

 u1 e3  e1 v + u21 e4   βT   u12 e4   u31 e4   v u1 (e5 + 2e1 ) βT

(A.23)

F. Chalot, T. J. R. Hughes, and F. Shakib 

    β T T e2 =  A v2     

   βT T  e A3 = 2  v    

u2

u3

u12 u2 c1

c2 u1 c2 2 u2 (u2 + 3 v ) βT symm.

u31 u3 c1

u23 u123 u3 c2 symm.

u23 u123 u3 c2 u2 c3

c3 u1 c3 u2 c3 u3 (u23 + 3 v ) βT

final (corrected)

38

 u2 e3   u12 e4  v 2 e1 + u2 e4   βT   u23 e4   v u2 (e5 + 2e1 ) βT

(A.24)

 u3 e3   u31 e4   u23 e4   v 2 e1 + u3 e4  βT   v u3 (e5 + 2e1 ) βT

(A.25)

The diffusive fluxes are given as

Fidiff

     0  0               0    τ1i  τ2i 0 + =        0  τ3i          −q   τ u  ij j

where

i

              

τij = λviscuk,k δij + µvisc(ui,j + uj,i) qi = −κT,i

(A.26)

(A.27) (A.28)

The spatial gradients of the velocity components and temperature are ui,j = T Vi+1,j + T ui V5,j

for i = 1, 2, 3

T,i = T 2 V5,i

(A.29) (A.30)

Let χvisc = λvisc + 2µvisc

(A.31)

F. Chalot, T. J. R. Hughes, and F. Shakib

final (corrected)

fij , where K fij V,j = F diff , are The diffusivity coefficient-matrices K i  0 0 0 0 0  visc visc  χ 0 0 χ u1   f11 = T  K µvisc 0 µviscu2   symm. µvisc µviscu3  χvisc u21 + µvisc (u22 + u23 ) + κT 

    f K22 = T    



   f33 = T  K    

0

0 visc

µ

0 0

0 0

χvisc symm.

0

0 µvisc

f12 = K fT K 21

µ

0 0

µvisc symm.

u1

χvisc u2 µviscu3 χvisc u22 + µvisc (u21 + u23 ) + κT

visc

0 0

µviscu1 µviscu2 χvisc u3

0 χvisc



0  0   = T 0  0  0

0 0

0  0   = T 0  0  0 0  0   = T 0  0  0

0 λvisc

µvisc 0 visc µ u2 0 0 0

0 0 λvisc u1 0 0 0

visc

µ µviscu3

        

        

       

0 0

λviscu2

0 0 0

µviscu1 0 visc (λ + µvisc)u12

0

0

0

λvisc 0

λviscu3 0

0 0 0 λvisc u1

visc

µ u1 (λvisc + µvisc)u31

0 0

0 0

0 0

0 0

0 0

0 µ

λvisc 0

λviscu3 µviscu2

0 µvisc u3

λvisc u2

(λvisc + µvisc)u23

visc

(A.32)

(A.33)



0

χvisc u23 + µvisc (u21 + u22 ) + κT



f23 = K fT K 32

µ

0



f13 = K fT K 31

0 visc

39

(A.34)

        

(A.35)

        

(A.36)

        

(A.37)

F. Chalot, T. J. R. Hughes, and F. Shakib

final (corrected)

40

Appendix B – A SIMPLE EQUILIBRIUM CHEMISTRY MODEL FOR AIR. In this appendix, air is considered as a thermally perfect mixture of N2 , O2 , NO, N, and O molecules (we number the species from 1 to 5 in this order). Given the thermodynamic state of the system (p, T ), we propose to compute the equilibrium mass fraction of each component, and the quantities (33)–(37). In order to solve for the five ys ’s, we need five independent equations. By definition of the mass fractions, we have X

ys = 1

(B.1)

s

In an inviscid flow, chemical reactions cannot change the elemental ratio between nitrogen and oxygen. Thus, 79 2xN2 + xNO + xN = 2xO2 + xNO + xO 21

(B.2)

Even in the case of a viscous flow, (B.2) holds approximately, since, nitrogen and oxygen atoms having nearly the same mass, the net diffusion velocities of the two elements are expected to be nearly identical [26]. In addition to the mass conservation (B.1), and the element conservation (B.2), we need three more equations. These are provided by three independent chemical reactions, e.g., N2 ⇀ ↽ 2N

(B.3)

O2 ⇀ ↽ 2O

(B.4)

NO ⇀ ↽ N+O

(B.5)

For each of these reactions, we can write the law of mass action, viz., c2N = Kc1 (T ) cN2 c2O = Kc2 (T ) cO2 cN cO = Kc3 (T ) cNO

(B.6) (B.7) (B.8)

F. Chalot, T. J. R. Hughes, and F. Shakib

final (corrected)

41

where the KcR ’s, R = 1, 2, 3, are the equilibrium constants of the different reactions. Ideally, these functions of temperature should be computed from the molar chemical potentials [19]: ˆ ln ps ˆs = µ µ ˆs = µs M ˆ0s (T ) + RT

(B.9)

where µs is given by (73). In terms of the µ ˆ s ’s, the condition for a reaction R to be at equilibrium reads, X

µ ˆs νsR = 0

(B.10)

s

where νsR is the stoichiometric coefficient of component s in reaction R. Equations (B.10) can be rewritten as the more familiar law of mass action:  P 0  µ ˆs νsR exp − sRT ˆ Y P KcR (T ) = cνssR = ˆ (RT ) s νsR s

(B.11)

For the sake of simplicity, we will use here curve fits for the KcR ’s, similar to those proposed by Rakich or Park [28, 25, 26]. The resulting system of five nonlinear equations for the ys ’s in terms of p and T ((B.1), (B.2), (B.6)–(B.8)) can formally be expressed as f (y, p, T ) = 0

(B.12)

where y = {ys }. Given an initial guess y (0) for y, we define the (n + 1)st Newton iterate by y (n+1) = y (n) + ∆y (n)

(B.13)

∆y (n) = −J −1 (y (n) , p, T )f (y (n), p, T )

(B.14)

where

and J=



∂f ∂y



p,T

A good initial guess assures a quadratic convergence of the process.

(B.15)

F. Chalot, T. J. R. Hughes, and F. Shakib

final (corrected)

42

The derivatives of y with respect to p and T can easily be obtained from the Jacobian J of the last Newton iteration. The differential of (B.12) is:     ∂f ∂f J dy + dp + dT = 0 ∂p y ,T ∂T y ,p Thus, dy = −J

−1



   ∂f ∂f −1 dp − J dT ∂p y ,T ∂T y ,p

(B.16)

(B.17)

and 





 ∂f = −J ∂p y ,T T     ∂y ∂f −1 = −J ∂T p ∂T y ,p ∂y ∂p

−1

From these derivatives, one can for instance compute   X X  ∂ys  ∂h = ys cps + hs cp = ∂T p ∂T p s s     1 ∂v 1 1 X ∂ys αp = = + Rs v ∂T p T R s ∂T p     1 1 X ∂ys 1 ∂v = − Rs βT = − v ∂p T p R s ∂p T α2p vT cv = cp − βT vc p a2 = cv βT

where cps is the specific heat at constant pressure of species s,   ∂hs cps = ∂T p

(B.18) (B.19)

(B.20) (B.21) (B.22) (B.23) (B.24)

(B.25)

and a is the speed of sound. Nondimensionalized αp , βT , cp , and a2 are plotted respectively in Figures 2–5, for the rigid-rotator/harmonic-oscillator model. αp is nondimensionalized using R0 , the specific gas constant of the mixture at room temperature, computed with a molar mass given by Eq. (67). In the range of temperatures where ionization can be neglected, they compare very well with classical computations [7], or with more recent curve fits [34, 30].

F. Chalot, T. J. R. Hughes, and F. Shakib

final (corrected)

43

4.0

p = 100 p = 101 p = 102 p = 103 p = 104 p = 105 p = 106

3.5

Tp

3.0

2.5

Pa Pa Pa Pa Pa Pa Pa

2.0

1.5

1.0 0

2

4

6

8

10

8

10

T (K )

E3

Figure 2. Dimensionless expansivity. 1.10 1.09 1.08 1.07

p T

1.06 1.05 1.04 1.03 1.02 1.01 1.00 0

2 E3

4

6

T (K )

Figure 3. Dimensionless isothermal compressibility.

F. Chalot, T. J. R. Hughes, and F. Shakib

final (corrected)

44

120

p = 100 p = 101 p = 102 p = 103 p = 104 p = 105 p = 106

100

cp =R0

80 60

Pa Pa Pa Pa Pa Pa Pa

40 20 0

0

2

4

E3

T (K )

6

8

10

Figure 4. Dimensionless specific heat at constant pressure. 1.7

1.6

c2 =p

1.5

1.4

1.3

1.2

1.1

1.0 0

2 E3

4

6

T (K ) Figure 5. Speed of sound parameter.

8

10

F. Chalot, T. J. R. Hughes, and F. Shakib

final (corrected)

45

Appendix C – COEFFICIENT MATRICES FOR A MIXTURE OF THERMALLY PERFECT GASES IN THERMOCHEMICAL NONEQUILIBRIUM. In this appendix, we present the flux vectors and the coefficient matrices of the conservation law system for a mixture of thermally perfect gases in thermochemical nonequilibrium, expressed in terms of the entropy variables. We assume that the internal energy modes obey the rigid-rotator and harmonic-oscillator model (see Section 4.3). We first recall the expressions of U and V :     y           u   1 vib U= e  v    2    |u|     e +   2   2   |u|     1 µ −   n     2     1 u V =  T  T    1 − vib        T     −1 

              e A0 =              

(C.1)

(C.2)

e0 = U,V and its inverse A e−1 = V,U read: The Riemannian metric tensor A 0

ρ1 R1

ρ1 R1 u1

ρ1 R1 u2

ρ1 R1 u3

ρ1 vib R1 e1

ρ1 tot R1 e1

.. .

.. .

.. .

.. .

.. .

ρn u Rn 1 ρs u21 R + ρT s

ρn u Rn 2 ρs u 1 u2 R s ρs u22 R + ρT s

ρn u Rn 3 ρs u1 u3 R s ρs u 2 u3 R s 2 ρs u3 Rs + ρT

ρn vib e Rn n ρs vib u1 R e s s ρs vib u2 R e s s ρs vib u3 R e s s

ρn tot e Rn n ρs tot u1 R h s s ρs tot u2 R h s s ρs tot u3 R h s s

ρs vib 2 Rs es

ρs vib tot Rs es es

0

..

. ρn Rn

symm.

vib +ρcvib v T

2

vib +ρcvib v T ρs tot 2 Rs es



2

+ ρu2 T 2

vib +ρ(cv T 2 + cvib ) v T

(C.3)

                           

F. Chalot, T. J. R. Hughes, and F. Shakib 

       1   =  2 ρcv T        

e−1 A 0

a ¯11

··· .. .

a ¯1n .. . a ¯nn

¯b1 u1 .. . ¯bn u1 u21 + cv T

final (corrected)

¯b1 u2 .. . ¯bn u2 u 1 u2 2 u2 + cv T

symm.

¯b1 u3 .. . ¯bn u3 u1 u3 u 2 u3 2 u3 + cv T 1+

46

d¯1 .. . d¯n

c¯1 .. . c¯n u1 u2 u3 cv T 2 vib cvib v T

2



       −u1   −u2    −u3    −1    1 (C.4)

where a ¯sr = (cvs T +

h0s

2 u2 u2 0 vib vib cv T − )(cvr T + hr − ) + es er vib 2 2 2 cvib v T

+ cv T u2 + ρcv T 2

Rs δsr ρs

(C.5)

2 ¯bs = cvs T + h0 − u − cv T s 2 u2 cv T 2 c¯s = cvs T + h0s − − evib s vib 2 2 cvib v T

u2 d¯s = −(cvs T + h0s − ) 2

(C.6) (C.7) (C.8)

e0 is positive-definite, if ρs > 0, cv > 0, cvib > 0, T > 0, and T vib > 0, where A v cv =

X

ys cvs

(C.9)

ys cvib vs

(C.10)

s

cvib v

=

X s

These conditions are satisfied in practice. The advective fluxes are:

Fiadv

   0     n δ i = ui U + p  0    u  i

(C.11)

F. Chalot, T. J. R. Hughes, and F. Shakib

final (corrected)

47

ei = F adv , are given by The advective Jacobians with respect to V , A i,V



                    e A1 =                     

ρ1 R1 u1

0 ..

ρ1 2 R1 (u1

+ R1 T ) .. .

ρ1 R1 u1 u2

ρ1 R1 u1 u3

ρ1 vib R1 e1 u1

.. .

.. .

.. .

+ Rn T )

ρn Rn u1 u2

ρn Rn u1 u3

ρn vib Rn en u1

ρs u31 R s

ρs u21 u2 R s

ρs u21 u3 R s

ρs vib u21 R e s s

+3ρT u1

+ρT u2

+ρT u3

+ρT evib

ρs u1 u2 u3 R s

ρs vib u1 u2 R e s s

. ρn Rn u1

ρn 2 Rn (u1

ρs u1 u22 R s

+ρT u1

ρs u1 u23 R s

+ρT u1 symm.

ρ1 tot R1 h1 u1

ρs vib u1 R e s s

     ρn tot  h u 1 Rn n   ρ  u21 ( Rss htot s + ρT )    +ρT htot     ρs tot u1 u2 ( Rs hs + ρT )       ρs tot  u1 u3 ( R h + ρT )  s s     ρs vib tot  u1 Rs es hs   vib vib 2  +ρu1 cv T    ρs tot 2  u1 Rs hs   2 +ρu1 T (u + cp T )    vib vib 2 +ρu1 cv T .. .

ρs vib e u1 u3 R s s 2

vib +ρu1 cvib v T

2



(C.12)

F. Chalot, T. J. R. Hughes, and F. Shakib



                    e2 =  A                    

ρ1 R1 u2

0 ..

ρ1 R1 u1 u2

ρ1 2 R1 (u2

+ R1 T ) .. .

ρ1 R1 u2 u3

ρ1 vib R1 e1 u2

.. .

.. .

+ Rn T )

ρn u u Rn 2 3

ρn vib e u2 Rn n

ρs u1 u2 u3 R s

ρs vib u1 u2 R e s s

ρs u32 R s

ρs u22 u3 R s

ρs vib e u22 R s s

+3ρT u2

+ρT u3

+ρT evib

.. .

. ρn u Rn 2

ρn u u Rn 1 2

final (corrected)

ρn (u22 Rn

ρs u21 u2 R s

ρs u1 u22 R s

+ρT u2

+ρT u1

ρs u2 u23 R s

+ρT u2 symm.

48

ρ1 tot R1 h1 u2

ρs vib e u2 R s s

     ρn tot  h u2 Rn n    ρs tot u1 u2 ( Rs hs + ρT )       2 ρs tot u2 ( Rs hs + ρT )    +ρT htot     ρs tot  u2 u3 ( R h + ρT )  s s     ρs vib tot  u2 Rs es hs   vib vib 2  +ρu2 cv T    2 ρs tot  u2 R h  s s  2 +ρu2 T (u + cp T )    vib vib 2 +ρu2 cv T .. .

ρs vib u2 u3 R e s s 2

vib +ρu2 cvib v T

2



(C.13)

F. Chalot, T. J. R. Hughes, and F. Shakib 

                    e3 =  A                    

ρ1 R1 u3

0 ..

ρ1 R1 u1 u3

ρ1 R1 u2 u3

.. .

.. .

ρn u u Rn 1 3

ρn u u Rn 2 3

. ρn u Rn 3

ρs u21 u3 R s

+ρT u3

ρs u1 u2 u3 R s

final (corrected)

ρ1 2 R1 (u3

ρn (u23 Rn

+ R1 T ) .. .

ρ1 vib R1 e1 u3

+ Rn T )

ρn vib e u3 Rn n

ρs u1 u23 R s

+ρT u1

ρs u22 u3 R s

ρs u2 u23 R s

+ρT u3

+ρT u2

ρ1 tot R1 h1 u3

ρs vib u1 u3 R e s s

ρs vib u2 u3 R e s s ρs vib e u23 R s s

+3ρT u3

+ρT evib ρs vib e u3 R s s

2

vib +ρu3 cvib v T

2

(C.14)

The diffusive fluxes are the sums of three contributions: Fidiff = Fimass + Fivisc + Fiheat

(C.15)

where Fimass , Fivisc, and Fiheat are respectively the mass diffusion, the viscous stress, and the heat flux vectors. The mass diffusion flux vectors are given by:     −J i       0 3 mass Fi =  −Ji · evib       −J · h  i

(C.16)

where

Jsi = ρs vsi = and

X M ˆ sM ˆr ρ Dsr dri ˆ2 M r



     ρn tot  h u 3 Rn n    ρs tot u1 u3 ( Rs hs + ρT )        ρs tot u2 u3 ( Rs hs + ρT )      2 ρs tot u3 ( Rs hs + ρT )     +ρT htot    ρs vib tot  u3 Rs es hs   vib vib 2  +ρu3 cv T    2 ρs tot  u3 R h  s s  2 +ρu3 T (u + cp T )    vib vib 2 +ρu3 cv T .. .

.. .

ρs u33 R s

symm.

49

(C.17)

F. Chalot, T. J. R. Hughes, and F. Shakib

final (corrected)

dsi = xs,i + (xs − ys )(ln p),i

50 (C.18)

In terms of gradients of primitive variables, (C.17) can be rewritten as: X Rr ρs vsi = − T αsr ρr,i − βs T,i ρr r

(C.19)

with

βs =

X

αsr Rr

(C.20)

r

The αsr ’s are Onsager’s phenomenological coefficients for mass diffusion [23, 24]. They are given in terms of the Dsr ’s by αsr

ˆs M = ˆT M

X yt ρr Dst − Dsr + ρr Rr R t t

!

(C.21)

The fundamental theorem of the thermodynamics of irreversible processes states that the αsr ’s satisfy the reciprocal relations, αsr = αrs Moreover, one can also write X

αst =

t

X

∀ s, r

,

αtr = 0

,

(C.22)

∀ s, r

(C.23)

t

(C.23) is equivalent to (92), and ensures that X ρs vsi = 0

(C.24)

s

The symmetry property (C.22) is checked for a mixture of three species in Hirschfelder, Curtiss, and Bird [10]. If the pressure term were omited in the definition of the dsi ’s, a nonsymmetric contribution would replace the αsr ’s in (C.19). In addition, the implicit pressure part of −Ji · h in Fimass would remain unbalanced in the diffusivity matrix.

are

fmass , where K fmass V,j = F mass , The mass-diffusion diffusivity coefficient matrices K ij ij i fmass K ii



α

  =T  symm. 

fmass = 0 K ij

αevib 03

0n×3 03×3

,

evib · αevib i 6= j

αh 03



   evib · αh   h · αh

,

i = 1, 2, 3

(C.25)

(C.26)

F. Chalot, T. J. R. Hughes, and F. Shakib

final (corrected)

51

fmass = [K fmass ] is positive semidefinite if and only The mass-diffusion diffusivity matrix K ij if α is positive semidefinite (see Section 5).

The viscous stress flux vectors are given by:     0   n     τij δj visc Fi =  0     τ u  

(C.27)

ij j

where

τij = λviscuk,k δij + µvisc(ui,j + uj,i)

(C.28)

χvisc = λvisc + 2µvisc

(C.29)

Let

Then, the viscous stress diffusivity coefficient  0n×n 0n 0n 0n  visc  χ 0 0   µvisc 0 fvisc = T  K  11 visc  symm. µ    

fvisc K 22

     =T     

fvisc K 33

     =T    

0n×n

0n µvisc

fvisc, where K fvisc V,j = F visc, are matrices K ij ij i  0n 0n  visc  0 χ u1   0 µviscu2  (C.30)  visc  0 µ u3   0 0  visc 2 visc 2 2 χ u1 + µ (u2 + u3 )

0n 0

0n 0

0n 0

0n visc µ u1

χvisc

0

0 0 0

χvisc u2 µviscu3 0

µvisc

symm.

χvisc u22 + µvisc (u21 + u23 ) 0n×n

0n µvisc

symm.

0n 0

0n 0

0n 0

0n µviscu1

µvisc

0

0 0 0

µviscu2 χvisc u3 0

χvisc

χvisc u23 + µvisc (u21 + u22 )

          

(C.31)

          

(C.32)

F. Chalot, T. J. R. Hughes, and F. Shakib 

fvisc = K fvisc K 12 21

T

     =T     

fvisc = K fvisc K 13 31

T

     =T     

fvisc = K fvisc K 23 32

T

     =T    

final (corrected)

0n×n

0n

0n

0n

0n

0n

0Tn

0

λvisc

0

0

λvisc u2

0Tn

µvisc

0

0

0

µvisc u1

0Tn

0

0

0

0

0

0

0

0Tn

0

0

0Tn

visc

visc

µ

u2

λ

u1

0

0

0 visc

(λ

+ µvisc )u1 u2

0n×n

0n

0n

0n

0n

0n

0Tn

0

0

λvisc

0

λvisc u3

0Tn

0

0

0

0

0

0Tn 0Tn 0Tn

visc

0

0

0

visc

0

0

0

0

µ

0

visc

0n×n

0n

0n

0n

0n

0n

0Tn

0

0

0

0

0

0Tn

0

0

λvisc

0

λvisc u3

0Tn

0

µvisc

0

0

µvisc u2

0Tn

0

0

0

0

0

visc

visc

0Tn

µ

visc

u3

µ

u3

λ

λ

u1

u2

0

0

µ

u1

0 visc

(λ

+ µvisc )u1 u3

0 visc

(λ

+ µvisc )u2 u3

52           

(C.33)

          

(C.34)

          

(C.35)

fvisc = [K fvisc] is positive, semidefinite if and only if The viscous stress diffusivity matrix K ij µvisc ≥ 0 and (λvisc + 32 µvisc ) ≥ 0 (see Section 5). The heat flux vectors are given by:

Fiheat =

where

     

0n 03 −qivib

     

        −(qi + qivib )  

qi = −κT,i qivib = −κvib T,ivib

(C.36)

(C.37) (C.38)

F. Chalot, T. J. R. Hughes, and F. Shakib

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fheat , where K fheat V,j = F heat, are The heat conduction diffusivity coefficient matrices K ij ij i fheat K ii



   =  

fheat = 0 K ij

0n×n

0n×3 03×3

0n 03

κvib T vib

symm.



0n 03

2

κvib T vib

2

κT 2 + κvib T vib ,

i 6= j

2

     

,

i = 1, 2, 3 (C.39)

(C.40)

fheat = [K fheat ] is positive, semidefinite if and The heat conduction diffusivity matrix K ij only if κ ≥ 0 and κvib ≥ 0 (see Section 5).

f=K fmass + K fvisc + K fheat is symmetric and positiveFinally, the diffusivity matrix K

semidefinite, provided that:

α≥0

(C.41)

µvisc ≥ 0

(C.42)

2 λvisc + µvisc ≥ 0 3

(C.43)

κ≥0

(C.44)

κvib ≥ 0

(C.45)

F. Chalot, T. J. R. Hughes, and F. Shakib

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Appendix D – COUPLING OF MASS DIFFUSION AND HEAT CONDUCTION: THE SORET AND DUFOUR EFFECTS. According to elementary kinetic theory, mass diffusion results from a concentration gradient, and heat flux from a temperature gradient. The finer analysis provided by nonequilibrium kinetic theory indicates that, in addition to these phenomena, there exists a transport of mass due to a temperature gradient, and a transport of energy due to a concentration gradient. The thermal-diffusion effect was not known theoretically, nor observed experimentally prior to the development of the Chapman-Enskog theory (see Chapman & Cowling [4]). Its existence was later to be confirmed by experiments conducted by Chapman and Dootson [10]. This effect is an interaction phenomenon, which arises from the processing by collisions of the heat flow, a direct consequence of the temperature gradient. The reciprocal process to thermal diffusion, known as diffusion thermo-effect, is responsible for the heat flux in a gas mixture initially at uniform temperature, where an inhomogeneity of composition has triggered mass diffusion. These two phenomena, also referred to as respectively the Soret and Dufour effects, are frequently considered as “second-order” corrections, since they require retaining at least two terms in the Sonine polynomial expansion of the distribution function, solution of the Boltzmann equation. In fact, in most aerospace applications, they can be neglected [26]. However, as we will see, it is interesting to study them in the light of Onsager’s reciprocal relations, and to examine their impact on entropy production. Appending the thermal diffusion effect to (88), the diffusion velocities are now given by: Jsi = ρs vsi =

X M ˆ sM ˆr ρ Dsr dri − DsT (ln T ),i ˆ2 M

(D.1)

r

where the DsT ’s are the multicomponent thermal diffusion coefficient; they satisfy X

DsT = 0

(D.2)

s

which is required together with (92) to ensure that the mass average of the diffusion

F. Chalot, T. J. R. Hughes, and F. Shakib

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55

velocities (94) vanishes. In turn, the heat flux of (96) is augmented by a term describing the diffusion thermo-effect: qi = −κ′ T,i − RT

X DT s

s

ys

dsi

(D.3)

κ′ is not the usual coefficient of thermal conductivity. It is conventional to eliminate the gradients dsi from the expression for qi by means of the equation for the diffusion velocities (D.1). The heat flux is then given in terms of the diffusion velocities and the temperature gradients. Because of the thermal diffusion term in the expression for the diffusion velocities, a small term adds to κ′ to result in the quantity κ, which is the usual coefficient of thermal conductivity. Details concerning the transformation, and the relation between κ and κ′ can be found in Hirschfelder et al. [10]. The Dufour effect part of (D.3) can be written in terms of the primitive variables as RT

X DT s

ys

s

dsi =

X Rs s

ρs

T DsT ρs,i +

X

Rs DsT T,i

(D.4)

s

In order to account for the thermal diffusion and the diffusion thermo-effect, the diffusivity fmass and K fheat of coefficient-matrices K ij ij  0 0n×3  n×n  03×3 ftherm = T  K ii   symm. Letting

ftherm = 0 K ij

,

i 6= j

Appendix C must be supplemented by  0n DT   03 03  , i = 1, 2, 3  0 evib · DT  2h · DT

(D.5)

(D.6)

fmass−heat = K fmass + K fheat + K ftherm K

(D.7)

fmass−heat V,j = F mass + F heat K i i ij

(D.8)

fheat , we have where κ′ replaces κ in K

From (D.8), one can see that mass diffusion and heat conduction are now linked together ftherm . The coupling appears even more strongly through the thermal diffusion matrix K

F. Chalot, T. J. R. Hughes, and F. Shakib

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from the standpoint of entropy production. As we will see shortly, the positive semidefinitefmass and K fheat are not enough to guarantee that of K fmass−heat, the contribution ness of K ftherm being nonpositive. from K

fmass−heat ∇V = V,i · K fmass−heat V,j = V,i · (F mass + F heat ) ∇V · K i i ij "     X X ρs,i T,i ρr,i T,i Rs = + (T αsr )Rr + ρs T ρr T s,r i   X T,i ρs,i T,i + Rs (T DsT ) 2 + ρs T T s   X T,i ρr,i T,i T + + (T Dr ) 2 T ρ T r r # T,i 2 ′ T,i + 2 (T κ ) 2 T T

(D.9)

We define the phenomenological coefficient-matrix for mass diffusion and heat conduction as follows: ¯= α

"

α symm.

T

D T κ′

#

(D.10)

(D.10) can be used to rewrite (D.9), yielding

fmass−heat

∇V · K

∇V =

" X X i

s,r

Rs





ρs,i T,i + ρs T



(T α ¯ sr )Rr



ρr,i T,i + ρr T

 ρs,i T,i T,i + Rs + (T α ¯ s,n+1 ) 2 ρs T T s   X T,i ρr,i T,i + (T α ¯ n+1,r ) + 2 T ρ T r r # T,i T,i ¯ n+1,n+1 ) 2 + 2 (T α T T X

=

i X n+1 X h Zsi (T α ¯ sr )Zri ≥ 0 i

s,r=1



(D.11)

F. Chalot, T. J. R. Hughes, and F. Shakib

final (corrected)

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where the vectors Zi are defined by     ρ T  1,i ,i  R1 +    ρ1 T     ..   .   Zi = ρn,i T,i   + Rn   ρn T      T,i    T2

                        

(D.12)

¯ is positive semidefinite, which is implied The inequality in (D.11) is satisfied if and only if α by the second law of thermodynamics. If the diffusion velocities are taken according to (D.1), and the heat flux to (D.3), the Clausius-Duhem inequality remains formally the same as (155): !



 qi qivib (ρs),t + (ρsui),i + ρs ss vsi + + vib T T ,i s ,i   vib 1 1 1X g s Ωs − QT −T ≥ 0 − + vib T s T T X

(D.13)