Contents
Navier-Stokes
NS equations are a set of non-linear partial derivative equations of conservation of mass, momentum, and energy:
∂t∂ρ+∇⋅(ρv)∂t∂(ρv)+∇⋅(ρvv+pI)∂t∂(ρE)+∇⋅(ρvH)=0=∇⋅τ+fm=∇⋅(k∇T)+∇⋅(τ⋅v)+fm⋅v
Equations of state:
- p=ρRgT
- e=cvT
- Rg=cp−cv
- E=e+V2/2
- H=E+p/ρ
These equations are even more complicated in CFD as one needs to add artificial viscosity and equations for modeling turbulences.
Some notes:
- One should use conservative variables to solve discontinuities correctly
- Convective terms are more completed than viscous terms
Linearized Euler (1D)
Approximation of ideal gas, no viscosity and heat transfer (τ′=q=0, with τ=−pI+τ′):
∂t∂ρρuρE+∂x∂ρuρu2+pρuH=000
Defining conservative variables:
- u1=ρ
- u2=ρu
- u3=ρE
And with the vector of conservative variable: U=(u1,u2,u3)T
∂t∂U+∂x∂F=0
Where F is the vector of conservative flux:
f1f2f3=u2=u1u22+(γ−1)u3−21(γ−1)u1u22=u2(γu1u3−21(γ−1)u12u22)
In quasi-linear form:
∂t∂U+∂U∂F∂x∂U=0
And by defining A=∂F/∂U:
A=∂uj∂fi=021(γ−3)(u1u2)2−γu12u2u3+(γ−1)(u3u2)31(3−γ)u1u2γu1u3−23(γ−1)(u1u2)20γ−1γu1u2
Using conservative variables and a=γRgT:
A=021(γ−3)u221(γ−2)u3−γ−1a2u1(3−γ)u23−2γu2+γ−1a20γ−1γu
Perturbations
Supposing U(x,t)=U0+u′(x,t) with ∣∣u′∣∣≪∣∣U0∣∣ where:
- U0: Uniform and stationary solution
- u′(x,t): General perturbation
Using Taylor expansion with the quasi-linear form, retaining only 1st order components:
∂t∂u′+A(U0)∂x∂u′=0
This equation is a hyperbolic system if the eigenvalues for A(U0) are reals: A(U0)uiR=λiuiR
- λ1=u0
- λ2=u0+a0
- λ2=u0−a0
With a0=γRgT0=γp0/ρ0, and right eigenvectors:
u1R=1u02u02;u2R=1u0+a02u02+γ−1a02+a0u0;u3R=1u0−a02u02+γ−1a02−a0u0
Defining T0=(u1R,u2R,u3R) and w:
T0−1A0T0=Λ0=λ1000λ2000λ3;w=T0−1u′
Adding to the linear equation:
∂t∂w+Λ0∂x∂w=0
where:
- w1=p′−a02ρ′
- w2=p′+ρ0a0u′
- w3=p′−ρ0a0u′
Boundary conditions:
- M0<1:
- 2 conditions in x=0 for downstream waves (entropy and acoustic)
- 1 condition in x=L for upstream wave
- M0>1:
- 3 conditions in x=0 for all three waves
Linearized Navier-Stokes
∂t∂ρρuρE+∂x∂ρuρu2+pρuH=∂x∂0μxμuux+κTx
Compact form:
∂t∂U+∂x∂f(U)=∂x∂fv(U)
- f: Convective flux
- fv: Viscous flux
For analytical study, better user primitive variables: V=(ρ,u,p)T. Therefore:
∂t∂ρup+u00ρuγp01/ρu∂x∂ρup=0(μux)x(γ−1)((kTx)x+μux2)
By linearizing the system:
∂t∂ρ′u′p′+A0u000ρ0u0γp001/ρ0u0∂x∂ρ′u′p′=0μuxx′ρ0Rg(γ−1)k0(pxx′−ρ0p0ρxx′)
Eigenvalues for A0 are the same, but the eigenvectors are now the columns of Q0:
Q0=1002a0ρ0212ρ0a0−2a0ρ021−aρ0a0;Q0−1=100011−a021ρ0a01−ρ0a01
By multiplying Q0−1 to the linearized equation:
∂t∂w1w2w3+u0000u0+a0000u0−a0∂x∂w1w2w3=B∂x2∂2w1w2w3
Stripping dimensions with the base field values:
- u=u′/a0
- p=p′/(ρ0a02)
- t=t/(Lc/a0)
- x=x/Lc
The equation is therefore:
∂t∂w1w2w3+u0000u0+1000u0−1∂x∂w1w2w3=B∂x2∂2w1′w2′w3′B=Re100001/2−1/20−1/21/2+RePr11112γ−12γ2γ−22γ−12γ−22γ
- Re=ρ0u0Lc/μ0
- Pr=cpμ0/κ0
- w1=p′−ρ′
- w1=p′+u′
- w1=p′−u′
Some conclusions:
- If κ0=0: Entropy equation decouples, and s′ is conserved along the path lines
- If Re≫1: Waves decouple. Perturbed characteristic variables are conserved along the path lines, slightly damped due to the viscosity
Entropy equation:
ρTDtDs=∇⋅(κ∇T)+Φv+Q
Linearized entropy equation:
ρ0T0(∂ts′+u0∂xs′)=κ0∂x2T′+Φv′
Vorticity equation (ω=∇×u) for isentropic and ideal flow, without forces from masses:
DtD(ω/ρ)=ρω⋅∇V
Acoustics Equations
Using the following decomposition in the Euler’s equation:
uρp=u′=ρ0+ρ′=p0+p′
We can obtain:
0=∂t(ρ0+ρ′)+∂x[(ρ0+ρ′)u′]0=∂t[(ρ0+ρ′)u′]+∂x[(ρ0+ρ′)u′u′+p0+p′]
As ρ0 and p0 (with u0=0) also verifies Euler’s equation:
∂tρ0=0;∂xp0=0
Using these values we can simplify the equation above:
0=∂tρ′+∂x[(ρ0+ρ′)u′]0=ρ0∂tu′+∂t(ρ′u′)+∂x[(ρ0+ρ′)u′u′]+∂xp′
Looking for linear solutions, neglecting quadratic terms of the perturbations:
00=∂tρ′+∂x(ρ0u′)=ρ0∂tu′+∂xρ′
The relationship between p′ and ρ′ is:
pp(ρ)=p0+p′≈p(ρ0)+∂ρ∂pρ0ρ′⎭⎬⎫p′=∂ρ∂ps0,ρ0ρ′=a02ρ′
Using this property results in:
00=a021∂tp′+ρ0∂xu′=ρ0∂tu′+∂xp′
Which is a system of two linear equations with two unknowns.
Deriving the first equation with respect to time and the second with respect to x:
a021∂ttp′+ρ0∂tx(u′)ρ0∂xtu′+∂xxp′=0=0⎭⎬⎫a021∂ttp′=∂xxp′
Which is the equation of the perturbation in the pressure field. In 2D:
a021∂ttp′=∇2p′;∇2=∂xx+∂yy
Generalized Equation
Considering the acoustic equation with base flux speed u0=0, the linearized Euler’s equations in 1D:
00=a021∂t∂p′+ρ0∂x∂u′+a02u0∂x∂p′=∂t∂u′+ρ01∂x∂p′+ux∂x∂u′
Combining both:
a021Dt2D02p′=∇2p′;DtD0=∂t∂+u0∇a021(∂t2∂2p′+2u0∂t∂x∂2p′+u02∂x2∂2p′)=∂x2∂2p′
which transforms into an equation with a 2nd temporal derivative with respect to time into two of 1st order:
∂t∂(p1p2)=(0(a02−u02)(∂2/∂x2)1−2u0(∂/∂x))(p1p2)
where p1=p′ and p2=p˙′
Other Equations
- Linear waves: ∂tu+a∂xu=0
- Viscous lineal waves: ∂tu+a∂xu=ν∂x2u
- Burgers: ∂tu+u∂xu=0
- Viscous burgers (Convection-Diffusion): ∂tu+u∂xu=ν∂x2u
- Heat: ∂tu=α∂x2u
- Stationary heat (Poisson): ∂x2u=0
- Linear wave with source: ∂tu+a∂xu=αu
- 2D linear waves: ∂tu+A∂xu+B∂yu=0