compressible navier-stokes formulation for a perfect gas

Compressible Navier-Stokes formulation for a perfect gas of 5

1. M

Here we derive the mathematical model describing the behavior of a compressible, perfect gas. Specialattention is paid to the origins of all conservation laws and constitutive relations employed. Direct notationis employed to ease conversion to an arbitrary coordinate system. The model will nondimensionalized afterderivation is complete.

1.1. Conservation laws.

1.1.1. Reynolds transport theorem. Consider a time-varying control volume Ω with surface ∂Ω and unitoutward normal n. For any scalar, vector, or tensor field quantity T , Leibniz’ theorem states

ddt

∫Ω(t)

T (x, t) dV =

∫Ω

∂

∂tT dV +

∫∂Ω

n · wT dA =

∫Ω

∂

∂tT + ∇ · wT dV

where w is the velocity of ∂Ω. When Ω follows a fixed set of fluid particles, w becomes the fluid velocity u.

1.1.2. Mass continuity. Since mass M =∫Ωρ dV and mass conservation requires d

dt M = 0,

0 =ddt

M =ddt

∫Ω

ρ dV =

∫Ω

∂

∂tρ + ∇ · uρ dV.

Because the result must hold for any control volume, we obtain∂

∂tρ + ∇ · ρu = 0.

1.1.3. Momentum equation. Separating total force into surface forces and a body force density∑F =

∫∂Ω

fs dA +

∫Ω

ρ fb dV =

∫∂Ω

σn dA +

∫Ω

ρ fb dV =

∫Ω

∇ · σ + ρ fb dV

where σ is the Cauchy stress tensor. Examining momentum I =∫Ωρu dV and its conservation d

dt I =∑

F,∫Ω

∂

∂tρu + ∇ · (u ⊗ ρu) dV =

∫Ω

∇ · σ + ρ fb dV.

Because the control volume may be arbitrary,∂

∂tρu + ∇ · (u ⊗ ρu) = ∇ · σ + ρ fb.

We further separate pressure p and viscous contributions τ to the Cauchy stress tensor so that σ = −pI + τ,∂

∂tρu + ∇ · (u ⊗ ρu) = −∇p + ∇ · τ + ρ fb.

Lastly, observing that u ⊗ ρu = 1ρρu ⊗ ρu is symmetric,

∂

∂tρu +

12∇ · (u ⊗ ρu + ρu ⊗ u) = −∇p + ∇ · τ + ρ fb.

1.1.4. Energy equation. Lumping internal and kinetic energy into an intrinsic density e, the energy E is

E =

∫Ω

ρe dV.


Treating heat input Q as both a surface phenomenon described by an outward heat flux qs and as a volumetricphenomenon governed by a body heating density qb,

Q =

∫Ω

ρqb dV −∫∂Ω

n · qs dA =

∫Ω

ρqb − ∇ · qs dV.

Power input P = F · v accounts for surface stress work and body force work to give

P =

∫∂Ω

σn · u dA +

∫Ω

ρ fb · u dV =

∫Ω

∇ · σu + ρ fb · u dV.

Demanding energy conservation ddt E = Q + P,∫

Ω

∂

∂tρe + ∇ · uρe dV =

∫Ω

ρqb − ∇ · qs dV +

∫Ω

∇ · σu + ρ fb · u dV.

Again, since the control volume was arbitrary,∂

∂tρe + ∇ · ρeu = −∇ · qs + ∇ · σu + ρ fb · u + ρqb.

After splitting σ’s pressure and viscous stress contributions we have∂

∂tρe + ∇ · ρeu = −∇ · qs − ∇ · pu + ∇ · τu + ρ fb · u + ρqb.

1.2. Constitutive relations and other assumptions.

1.2.1. Perfect gas. We assume our fluid is a thermally and calorically perfect gas governed by

p = ρRT

where R is the gas constant. The constant volume Cv specific heat, constant pressure specific heat Cp, andacoustic velocity a relationships follow:

γ =Cp

CvCv =

Rγ − 1

Cp =γRγ − 1

R = Cp −Cv a2 = γRT

We assume γ and therefore Cv and Cp are constant. The total (internal and kinetic) energy density is

e = CvT +u · u

2=

RTγ − 1

+u · u

2.

See a gas dynamics reference, e.g. Liepmann & Roshko 1957, for more details.

1.2.2. Newtonian fluid. If we seek a constitutive law for the viscous stress tensor τ using only velocity in-formation, the principle of material frame indifference implies that uniform translation (given by velocityu) and solid-body rotation (given by the skew-symmetric rotation tensor ω = 1

2

(∇u − ∇uT

)) may not in-

fluence τ. Considering contributions only up to the gradient of velocity, extensional strain (dilatation) andshear strain effects may depend on only the symmetric rate-of-deformation tensor ε = 1

2

(∇u + ∇uT

)and its

principal invariants.

Assuming τ is isotropic and depends linearly upon only ε, we can express it as

τi j = ci jmnεmn

=(Aδi jδmn + Bδimδ jn + Cδinδ jm

)εmn for some A, B,C ∈ R

= Aδi jεmm + Bεi j + Cε ji

= Aδi jεmm + (B + C) ε ji

= 2µεi j + λδi j∇ · u


where µ = 12 (B + C) is the viscosity and λ = A is the second viscosity. Reverting to direct notation we have

τ = 2µε + λ (∇ · u) I

= µ(∇u + ∇uT

)+ λ (∇ · u) I

1.2.3. Stokes hypothesis. We generally assume the second viscosity λ = − 23µ. However, because we antic-

ipate separately maintaining λ being useful, we will not combine µ and λ terms in the model.

1.2.4. Power law viscosity. We assume that viscosity varies only with temperature according to

µ

µ0=

(TT0

)βwhere µ0 and T0 are suitable reference values. This relationship models air well for temperatures up toseveral thousand degrees K. See Svehla’s 1962 NASA technical report R-132.

1.2.5. Fourier’s equation. We neglect the transport of energy by molecular diffusion and radiative heattransfer. We seek a relation between the surface heat flux qs and the temperature T . The principle of frameindifference implies we may only use the temperature gradient so that

qs = κ · ∇T

where κ is a thermal conductivity tensor. Consistent with our assumption that τ is isotropic, we assume κ isisotropic to obtain

qs = −κ∇T

where κ is the scalar thermal conductivity. We introduce the negative sign so that heat flows from hot to coldwhen κ > 0.

1.2.6. Constant Prandtl number. We assume the Prandtl number Pr =µCpκ is constant. Because Cp is

constant the ratio µκ must be constant. The viscosity and thermal conductivity must either grow at identical

rates or they must grow according to an inverse relationship. The latter is not observed in practice for ourclass of fluids, and so we assume

µ

µ0=κ

κ0.

1.2.7. Body force density. We generally assume fb = 0. However, in the formulation, we allow the fbto vary in all spatial directions and across time. Retaining body force will simplify using the method ofmanufactured solutions for implementation verification.

1.2.8. Body heating density. We assume a space- and time-varying body heating density qb.

1.3. Nondimensionalization.


1.3.1. Dimensional equations. By combining the conservation laws with our constitutive relations and as-sumptions, we arrive at the dimensional equations

∂

∂tρ = −∇ · ρu

∂

∂tρu = −

12∇ · (u ⊗ ρu + ρu ⊗ u) − ∇p + ∇ · τ + ρ fb

∂

∂tρe = −∇ · ρeu + ∇ ·

κ0

µ0µ∇T − ∇ · pu + ∇ · τu + ρ fb · u + ρqb

where terms in the right hand side make use of

T =

(γ − 1

R

) (e −

u · u2

)p = ρRT

µ = µ0

(TT0

)βτ = µ

(∇u + ∇uT

)+ λ (∇ · u) I.

1.3.2. Introduction of nondimensional variables. We rewrite the dimensional equations using nondimen-sional variables combined with arbitrary reference quantities. For each dimensional quantity in the dimen-sional model we introduce a nondimensional variable or operator denoted by a superscript star, e.g. ∇∗.

We introduce t∗ = tt0

and x∗ = xl0

for some reference t0 and l0. This induces the following relationships:

∂

∂t=

∂

∂t∗∂t∗

∂t=

1t0

∂

∂t∗∂

∂x=

∂

∂x∗∂x∗

∂x=

1l0

∂

∂x∗∇ = ei

∂

∂xi= ei

1l0

∂

∂x∗i=

1l0∇∗

We introduce more nondimensional quantities (e.g. ρ∗ =ρρ0

) and use them to reexpress the model

ρ0

t0

∂

∂t∗ρ∗ = −

ρ0u0

l0∇∗ · ρ∗u∗

ρ0u0

t0

∂

∂t∗ρ∗u∗ = −

12ρ0u2

0

l0∇∗ · (u∗ ⊗ ρ∗u∗ + ρ∗u∗ ⊗ u∗) −

p0

l0∇∗p∗ +

τ0

l0∇∗ · τ∗ + ρ0 f0ρ∗ f ∗b

ρ0e0

t0

∂

∂t∗ρ∗e∗ = −

ρ0e0u0

l0∇∗ · ρ∗e∗u∗ +

κ0T0

l20∇∗ · µ∗∇∗T ∗ −

p0u0

l0∇∗ · p∗u∗

+τ0u0

l0∇∗ · τ∗u∗ + ρ0 f0u0ρ

∗ f ∗b · u∗ + ρ0q0ρ

∗q∗b

where terms in the right hand side are computed using

T ∗ =1

T0

(γ − 1

R

) (e0e∗ − u2

0u∗ · u∗

2

)p∗ =

ρ0RT0

p0ρ∗T ∗

µ∗ =(T ∗

)βτ∗ =

µ0u0

l0τ0

[µ∗

(∇∗u∗ + ∇∗u∗T

)+ λ∗

(∇∗ · u∗

)I].

Notice that λ has been nondimensionalized using µ0. At this stage, we have many more reference quantitiesthan the underlying dimensions warrant.


1.3.3. Reference quantity selections. We choose a reference length l0, temperature T0, and density ρ0. Theseselections fix all other dimensional reference quantities:

a0 =√γRT0 u0 = a0 e0 = a2

0 t0 =l0a0

p0 = ρ0a20 τ0 =

µ0a0

l0f0 =

a20

l0q0 =

a30

l0Because we assume viscosity varies only with temperature, µ0 = µ(T0) is fixed by T0. Because we assumea constant Prandtl number, κ0 = κ(µ(T0)) is also fixed by T0.

1.3.4. Nondimensional equations. We employ the reference quantity relationships after multiplying thecontinuity, momentum, and energy equations by t0

ρ0, l0ρ0a2

0, and t0

ρ0e0respectively. Henceforth we suppress

the superscript star notation because all terms are dimensionless. We arrive at the following nondimensionalequations:

∂

∂tρ = −∇ · ρu

∂

∂tρu = −

12∇ · (u ⊗ ρu + ρu ⊗ u) − ∇p +

1Re∇ · τ + ρ fb

∂

∂tρe = −∇ · ρeu +

1Re Pr (γ − 1)

∇ · µ∇T − ∇ · pu +1

Re∇ · τu + ρ fb · u + ρqb

where Re =ρ0u0l0µ0

and Pr =µ0Cpκ0

. The nondimensional quantities appearing above are given by:

T = γ (γ − 1)(e −

u · u2

)p =

1γρT

µ = T β

τ = µ(∇u + ∇uT

)+ λ (∇ · u) I

compressible navier-stokes formulation for a perfect gas

Documents