Compressible Navier-stokes Formulation For A Perfect Gas

Compressible Navier-Stokes formulation for a perfect gas

Page 1 of 5

1. M  Here we derive the mathematical model describing the behavior of a compressible, perfect gas. Special attention is paid to the origins of all conservation laws and constitutive relations employed. Direct notation is employed to ease conversion to an arbitrary coordinate system. The model will nondimensionalized after derivation is complete. 1.1. Conservation laws. 1.1.1. Reynolds transport theorem. Consider a time-varying control volume Ω with surface ∂Ω and unit outward normal nˆ . For any scalar, vector, or tensor field quantity T , Leibniz’ theorem states Z Z Z Z ∂ ∂ d T (x, t) dV = T dV + nˆ · wT dA = T + ∇ · wT dV dt Ω(t) ∂t ∂t Ω ∂Ω Ω where w is the velocity of ∂Ω. When Ω follows a fixed set of fluid particles, w becomes the fluid velocity u. ρ dV and mass conservation requires Z Z d d ∂ ρ + ∇ · uρ dV. 0= M= ρ dV = dt dt Ω Ω ∂t Because the result must hold for any control volume, we obtain ∂ ρ + ∇ · ρu = 0. ∂t 1.1.2. Mass continuity. Since mass M =

R

Ω

d dt M

= 0,

1.1.3. Momentum equation. Separating total force into surface forces and a body force density Z Z Z Z Z X F= f s dA + ρ fb dV = σˆn dA + ρ fb dV = ∇ · σ + ρ fb dV ∂Ω Ω ∂Ω Ω Ω R P where σ is the Cauchy stress tensor. Examining momentum I = Ω ρu dV and its conservation dtd I = F, Z Z ∂ ρu + ∇ · (u ⊗ ρu) dV = ∇ · σ + ρ fb dV. Ω ∂t Ω Because the control volume may be arbitrary, ∂ ρu + ∇ · (u ⊗ ρu) = ∇ · σ + ρ fb . ∂t We further separate pressure p and viscous contributions τ to the Cauchy stress tensor so that σ = −pI + τ, ∂ ρu + ∇ · (u ⊗ ρu) = −∇p + ∇ · τ + ρ fb . ∂t Lastly, observing that u ⊗ ρu = ρ1 ρu ⊗ ρu is symmetric, ∂ 1 ρu + ∇ · (u ⊗ ρu + ρu ⊗ u) = −∇p + ∇ · τ + ρ fb . ∂t 2 1.1.4. Energy equation. Lumping internal and kinetic energy into an intrinsic density e, the energy E is Z E= ρe dV. Ω

Compressible Navier-Stokes formulation for a perfect gas

Page 2 of 5

Treating heat input Q as both a surface phenomenon described by an outward heat flux q s and as a volumetric phenomenon governed by a body heating density qb , Z Z Z Q= ρqb dV − nˆ · q s dA = ρqb − ∇ · q s dV. Ω

∂Ω

Ω

Power input P = F · v accounts for surface stress work and body force work to give Z Z Z P= σˆn · u dA + ρ fb · u dV = ∇ · σu + ρ fb · u dV. ∂Ω

Ω

Ω

= Q + P, Demanding energy conservation Z Z Z ∂ ρe + ∇ · uρe dV = ρqb − ∇ · q s dV + ∇ · σu + ρ fb · u dV. Ω ∂t Ω Ω Again, since the control volume was arbitrary, ∂ ρe + ∇ · ρeu = −∇ · q s + ∇ · σu + ρ fb · u + ρqb . ∂t After splitting σ’s pressure and viscous stress contributions we have ∂ ρe + ∇ · ρeu = −∇ · q s − ∇ · pu + ∇ · τu + ρ fb · u + ρqb . ∂t d dt E

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 C p , and acoustic velocity a relationships follow: Cp R γR γ= Cv = Cp = R = C p − Cv a2 = γRT Cv γ−1 γ−1 We assume γ and therefore Cv and C p are constant. The total (internal and kinetic) energy density is u·u RT u·u e = Cv T + = + . 2 γ−1 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 information, the principle of material frame indifference implies that uniform translation (given   by velocity T 1 u) and solid-body rotation (given by the skew-symmetric rotation tensor ω = 2 ∇u − ∇u ) may not influence τ. Considering contributions only up to the gradient of velocity, extensional strain  (dilatation)  and T 1 shear strain effects may depend on only the symmetric rate-of-deformation tensor ε = 2 ∇u + ∇u and its principal invariants. Assuming τ is isotropic and depends linearly upon only ε, we can express it as τi j = ci jmn εmn   = Aδi j δmn + Bδim δ jn + Cδin δ jm εmn = Aδi j εmm + Bεi j + Cε ji = Aδi j εmm + (B + C) ε ji = 2µεi j + λδi j ∇ · u

for some A, B, C ∈ R

Compressible Navier-Stokes formulation for a perfect gas where µ =

1 2

Page 3 of 5

(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 = − 32 µ. However, because we anticipate 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 !β µ T = µ0 T0 where µ0 and T 0 are suitable reference values. This relationship models air well for temperatures up to several 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 heat transfer. We seek a relation between the surface heat flux q s and the temperature T . The principle of frame indifference implies we may only use the temperature gradient so that q s = κ · ∇T where κ is a thermal conductivity tensor. Consistent with our assumption that τ is isotropic, we assume κ is isotropic to obtain q s = −κ∇T where κ is the scalar thermal conductivity. We introduce the negative sign so that heat flows from hot to cold when κ > 0. µC

1.2.6. Constant Prandtl number. We assume the Prandtl number Pr = κ p is constant. Because C p 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 our class of fluids, and so we assume µ κ = . µ0 κ0 1.2.7. Body force density. We generally assume fb = 0. However, in the formulation, we allow the fb to vary in all spatial directions and across time. Retaining body force will simplify using the method of manufactured solutions for implementation verification.

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

1.3. Nondimensionalization.

Compressible Navier-Stokes formulation for a perfect gas

Page 4 of 5

1.3.1. Dimensional equations. By combining the conservation laws with our constitutive relations and assumptions, we arrive at the dimensional equations ∂ ρ = −∇ · ρu ∂t ∂ 1 ρu = − ∇ · (u ⊗ ρu + ρu ⊗ u) − ∇p + ∇ · τ + ρ fb ∂t 2 κ0 ∂ ρe = −∇ · ρeu + ∇ · µ∇T − ∇ · pu + ∇ · τu + ρ fb · u + ρqb ∂t µ0 where terms in the right hand side make use of ! γ−1  u · u T= e− R 2 p = ρRT !β T µ = µ0 T0   τ = µ ∇u + ∇uT + λ (∇ · u) I. 1.3.2. Introduction of nondimensional variables. We rewrite the dimensional equations using nondimensional variables combined with arbitrary reference quantities. For each dimensional quantity in the dimensional model we introduce a nondimensional variable or operator denoted by a superscript star, e.g. ∇∗ . We introduce t∗ =

t x ∗ t0 and x = l0 ∂t∗ 1 ∂

∂ ∂ = = ∂t ∂t∗ ∂t t0 ∂t∗

for some reference t0 and l0 . This induces the following relationships: ∂ ∂ ∂x∗ 1 ∂ = ∗ = ∂x ∂x ∂x l0 ∂x∗

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

ρ ρ0 )

∇ = eˆ i

∂ 1 ∂ 1 = eˆ i = ∇∗ ∂xi l0 ∂xi∗ l0

and use them to reexpress the model

ρ0 ∂ ∗ ρ0 u 0 ∗ ∗ ∗ ρ =− ∇ ·ρ u ∗ t0 ∂t l0 1 ρ0 u20 ∗ p0 τ0 ρ0 u0 ∂ ∗ ∗ ρ u = − ∇ · (u∗ ⊗ ρ∗ u∗ + ρ∗ u∗ ⊗ u∗ ) − ∇∗ p∗ + ∇∗ · τ∗ + ρ0 f0 ρ∗ fb∗ ∗ t0 ∂t 2 l0 l0 l0 ρ0 e0 u0 ∗ ∗ ∗ ∗ κ0 T 0 ∗ ∗ ∗ ∗ p0 u0 ∗ ∗ ∗ ρ0 e 0 ∂ ∗ ∗ ρ e =− ∇ ·ρ e u + 2 ∇ ·µ ∇ T − ∇ ·p u t0 ∂t∗ l0 l0 l0 τ0 u 0 ∗ ∗ ∗ + ∇ · τ u + ρ0 f0 u0 ρ∗ fb∗ · u∗ + ρ0 q0 ρ∗ q∗b l0 where terms in the right hand side are computed using ! ! ∗ ∗ 1 γ−1 ∗ 2u · u ∗ T = e0 e − u0 T0 R 2 ρ RT 0 0 ∗ ∗ p∗ = ρT p0
µ∗ = T ∗ β 
i µ0 u0 h ∗  ∗ ∗ τ∗ = µ ∇ u + ∇∗ u∗ T + λ∗ ∇∗ · u∗ I . l0 τ0 Notice that λ has been nondimensionalized using µ0 . At this stage, we have many more reference quantities than the underlying dimensions warrant.

Compressible Navier-Stokes formulation for a perfect gas

Page 5 of 5

1.3.3. Reference quantity selections. We choose a reference length l0 , temperature T 0 , and density ρ0 . These selections fix all other dimensional reference quantities: a3 a2 l0 µ0 a0 p0 = ρ0 a20 τ0 = f0 = 0 q0 = 0 a0 l0 l0 l0 Because we assume viscosity varies only with temperature, µ0 = µ(T 0 ) is fixed by T 0 . Because we assume a constant Prandtl number, κ0 = κ(µ(T 0 )) is also fixed by T 0 . a0 =

p γRT 0

u0 = a0

e0 = a20

t0 =

1.3.4. Nondimensional equations. We employ the reference quantity relationships after multiplying the continuity, momentum, and energy equations by ρt00 , ρ l0a2 , and ρ0t0e0 respectively. Henceforth we suppress 0 0

the superscript star notation because all terms are dimensionless. We arrive at the following nondimensional equations: ∂ ρ = −∇ · ρu ∂t ∂ 1 1 ρu = − ∇ · (u ⊗ ρu + ρu ⊗ u) − ∇p + ∇ · τ + ρ fb ∂t 2 Re ∂ 1 1 ρe = −∇ · ρeu + ∇ · µ∇T − ∇ · pu + ∇ · τu + ρ fb · u + ρqb ∂t Re Pr (γ − 1) Re where Re =

ρ0 u 0 l 0 µ0

and Pr =

µ0 C p κ0 .

The nondimensional quantities appearing above are given by:  u · u (γ T = γ − 1) e − 2 1 p = ρT γ µ = Tβ   τ = µ ∇u + ∇uT + λ (∇ · u) I