kinetic methods for cfd - ossanworld.com

Kinetic Methods for CFD

Li-Shi Luo

Department of Mathematics and StatisticsCenter for Computational Sciences

Old Dominion University, Norfolk, Virginia 23529, USAEmail: [email protected]

URL: http://www.lions.odu.edu/~lluo

National Institute of Aerospace, Hampton, USATuesday, January 17, 2012

Collaborators: Wei Liao, Yan Peng, Manfred Krafczyk, et al.Supports from: NSF, DoD/AFOSR, NASA, ODU-RF

Luo (Math Dept, ODU) Kinetic Methods for CFD NIA, 01/17/’12 1 / 53

[email protected]

http://www.lions.odu.edu/~lluo

Outline

1 MotivationsThe Role of Kinetic TheoryScales and Related Methods

2 Mathematicsal BackgroundsKinetic Theory – A Multiscale FormalismLBE: Finite Difference Scheme for Incompressible FlowsD3Q19 LBE ModelGKS: Finite Volume Scheme for Compressible Flows

3 Numerical ResultsDNS of Homogeneous Isotropic TurbulenceCompressible and Nonequilibrium FlowsSome Animations

4 Conclusions


Outline




4 Conclusions


Navier-Stokes Equations: Continuum Theory

Conservation laws of mass, momentum, and energy:

∂tρ+ ∇·ρu = 0 (1a)

ρ∂u+ ρu·∇u = −∇·P (1b)

ρ∂te+ ρu·∇e = −P : ∇u−∇·q (1c)

Pαβ := pδαβ − µ(∂αuβ + ∂βuα −

2

3δαβ∇·u

)− ζδαβ∇·u

q = −κ∇T , e = e(T ), p = p(ρ, T )

Dimensionless Navier-Stokes equations (similarity law):

ρ∂u+ ρu·∇u = − 1

γMa2∇p+

1

Re∇·S (2a)

ρ∂te+ ρu·∇e = − 1

γMa2p∇·u+

1

α∇2T +

1

ReS : ∇u (2b)

α = (γ − 1)PrMaRe


Hierarchy of Scales and PDEs

Microscopic Scale

qk =∂H

∂pk, pk = −∂H

∂qk

H =∑(D+K)N

k=1 p2k + V

i~ψ = Hψ

H = − ~2

2m

∑Nj=1∇

2j+V

h ≈ 6.62 · 10−34(J · s)c ≈ 2.99 · 108(m/s)

a ≈ 5·10−11(m)

ta ≈ 2.41·10−17(s)

m ≈ 10−27(kg)

N = 1, 2, . . . , N0

Mesoscopic Scale

∂tf + ξ ·∇f =1

εQ(f, f)

f = f(x, ξ, t)

ε = Kn =`

L, Ma =

U

cskB ≈ 1.38·10−23(J/K)

` ≈ 102 – 103 (A)

≈ 10 – 100 (nm)

τ ≈ 10−10(s)

cs ≈ 300 (m/s)

N 1

Macroscopic Scale

ρDtu = −∇p+1

Re

∇·σ

Dt = ∂t + u·∇σ =

ρν

2[(∇u)+(∇u)†]

+2ρζ

DI(∇·u)

Reδ =UL

ν∼ Ma

Knν ≈ 10−6 – 10−4 (m2/s)

Kn ≈ 0 Ma < 103

L ≥ 10−5(m) = 10(µm)

T ≥ 10−4(s)

N ≥ NA ≈ 6.02 · 1023


Micro-, Meso-, Macro-Descriptions of Fluids

Knudsen Number Kn :=`

L=

Mean Free Path

Characteristic Hydrodynamic Length

Microscopic

Theory

DeterministicNewton’s Law

MolecularDynamics

Mesoscopic

Theory

StatisticalMechanics

LiouvilleEquation

BBGKYHierachy

BoltzmannEquation

xi, tn ci

Direct SimulationMonte Carlo

DiscreteVelocity Models

DiscreteOrdinance Method

MomentEquations

Macroscopic (Continuum) Theory (Kn ≈ 0)

PDEs of Conservation Laws

Navier-StokesEquations

EulerEquations

Turbulence ModelsRANS, LES, · · ·

Super-Burnett

Equation

BurnettEquation

Kinetic Theory

Hilbert and Chapman-Enskog analysis

Lattice GasAutomata

Lattice BoltzmannEquation

Gas-KineticScheme

Linear Relaxation ModelsSystem of Finite Moments


Outline




4 Conclusions


Kinetic Equation

The kinetic equation for the single particle distribution function f inphase space Γ := (x, ξ):

∂tf + ∇ · (ξf) = Q(f, f), f := f(x, ξ, t), (3)

The Uehling-Uhlenbeck collision model:

Q =

∫dξ2

∫dΩK

[(1 + ηf1) (1 + ηf2) f ′1f

′2 −

(1 + ηf ′1

) (1 + ηf ′2

)f1f2

],

(4)where K := K(ξ1, ξ2,Ω) is the collision kernel, depending on the solidangle Ω, and

η =

+1, Bose-Einstein,

0, Maxwell-Boltzmann,−1, Fermi-Dirac.

(5)


From KE to Hydrodynamics Equations

Expansion of f in terms of the Knudsen number ε = Kn:

f = f (0) + εf (1) + ε2f (2) + · · ·, (6)

f (0) =ρ

(2πRT )D/2e−(ξ−u)2/2RT , (7)

ρ =

∫fdξ =

∫f (0)dξ, (8a)

ρu =

∫fξdξ =

∫f (0)ξdξ, (8b)

ρe =D

2ρkBT =

∫1

2c2cf dξ =

∫1

2c2cf (0) dξ, c := ξ − u, (8c)

P =

∫ξξf dξ, q =

∫1

2c2cf dξ, (8d)


Multiscale Formalism

∂tρ+ ∇ · ρu = 0 (9a)

∂tρu+ ∇ (ρuu+ P) = 0 (9b)

∂tρE + ∇ · (ρuE + u · P + q) = 0, E := ρe+1

2ρu2 (9c)

P = P(0) + P(1) + P(2) + · · ·, (9d)

q = q(0) + q(1) + q(2) + · · ·, (9e)

f = f (0) =⇒

P(0) = 32ρkBTδij

q(0) = 0

=⇒ Euler Eqns

f = f (0) + f (1) =⇒

P(1) = −1

2µ[(∇u) + (∇u)†]

q(1) = κ∇T

=⇒ NS Eqns

f = f (0) + f (1) + f (2) + · · ·+ f (n) =⇒ ?


Linearize Collision Operator

The collision operator for the classical (Maxwell-Boltzmann) particle:

Q(f, f)=

∫dµ2

(f ′1f′2 − f1f2

)=

∫dµ2f

′1f′2 − f1

∫dµ2f2

≈ f (0)

1

∫dµ2f

(0)

2 − f1

∫dµ2f2 = − 1

λ

[f1 − f (0)

1

](10)

The above renowned Welander-Bhatnagar-Gross-Krook model reducesthe Boltzmann equation to:

∂tf + ∇ · (ξf) = − 1

λ

[f − f (0)

], f (0) =

ρ

(2πRT )3/2e−(ξ−u)2/2RT (11)

and its moment equations forms a hierarchy of first-order PDEs:

∂tm(n) + ∇ ·m(n+1) = − 1

λ

[m(n) −m

(n)0

](12a)

m(n) := 〈ξ · · · ξ︸︷︷︸n

〉 :=

∫dξf ξ · · · ξ︸︷︷︸

n

, m(n)0 :=

∫dξf (0) ξ · · · ξ︸︷︷︸

n

(12b)


A Priori Derivation of Lattice Boltzmann Equation

The Boltzmann Equation for f := f(x, ξ, t) with BGK approximation:

∂tf + ξ ·∇f =

∫[f ′1f

′2 − f1f2]dµ ≈ L(f, f) ≈ − 1

λ[f − f (0)] (13)

The Boltzmann-Maxwellian equilibrium distribution function:

f (0) = ρ (2πθ)−D/2 exp

[−(ξ − u)2

2θ

], θ := RT (14)

The macroscopic variables are the first few moments of f and f (0):

ρ =

∫fdξ =

∫f (0)dξ , (15a)

ρu =

∫ξfdξ =

∫ξf (0)dξ , (15b)

ρε =1

2

∫(ξ − u)2fdξ =

1

2

∫(ξ − u)2f (0)dξ . (15c)


Integral Solution of Continuous Boltzmann Equation

Rewrite the Boltzmann BGK Equation in the form of ODE:

Dtf +1

λf =

1

λf (0) , Dt := ∂t + ξ ·∇ . (16)

Integrate Eq. (35) over a time step δt along characteristics:

f(x+ ξδt, ξ, t+ δt) = e−δt/λ f(x, ξ, t) (17)

+1

λe−δt/λ

∫ δt

0et′/λ f (0)(x+ ξt′, ξ, t+ t′) dt′ .

By Taylor expansion, and with τ := λ/δt, we obtain:

f(x+ ξδt, ξ, t+ δt)− f(x, ξ, t) = −1

τ[f(x, ξ, t)− f (0)(x, ξ, t)] +O(δ2

t ) .

(18)Note that a finite-volume scheme or higher-order schemes can also beformulated based upon the integral solution.


Passage to Lattice Boltzmann EquationThree necessary steps to derive LBE:1,2,3

1 Low Mach number expansion of the distribution functions;

2 Discretize ξ-space with necessary and min. number of ξα;

3 Discretization of x space according to ξα.Low Mach Number (u ≈ 0) Expansion of the distribution functions f (0)

and f up to O(u2) is sufficient to derive the Navier-Stokes equations:

f (eq) =ρ

(2πθ)D/2exp

[−ξ

2

2θ

]1 +

ξ · uθ

+(ξ · u)2

2θ2− u

2

2θ

+O(u3) .(19a)

f =ρ

(2πθ)D/2exp

[−ξ

2

2θ

] 2∑n=0

1

n!a(n)(x, t) : H(n)(ξ) , (19b)

where a(0) = 1, a(1) = u, a(2) = uu− (θ − 1)I, and H(n)(ξ) aregeneralized Hermite polynomials.

1X. He and L.-S. Luo, Phys. Rev. E 55:R6333 (1997); ibid 56:6811 (1997).

2T. Abe, J. Comput. Phys. 131(1):241–246(1997).

3X. Shan and X. He, Phys. Rev. Lett. 80:65 (1998).


Discretization and Conservation Laws

To compute conserved moments (ρ, ρu, and ρε) and their fluxesexactly, one must evalute:

I =

∫ξmf (eq)dξ =

∫exp(−ξ2/2θ)ψ(ξ)dξ, (20)

where 0 ≤ m ≤ 3, and ψ(ξ) is a polynomial in ξ. The above integralcan be evaluated by quadrature:

I =

∫exp(−ξ2/2θ)ψ(ξ)dξ=

∑j

Wj exp(−ξ2j /2θ)ψ(ξj) (21)

where ξj and Wj are the abscissas and the weights. Then

ρ=∑α

f (eq)α =

∑α

fα, ρu=∑α

ξαf(eq)α =

∑α

ξαfα, (22)

where fα := fα(x, t) := Wαf(x, ξα, t), and f (eq)α := Wαf

(eq)(x, ξα, t).

The quadrature must preserve the conservation laws exactly!


Example: 9-bit LBE Model with Square Lattice

In two-dimensional Cartesian (momentum) space, set

ψ(ξ) = ξmx ξny ,

the integral of the moments can be given by

I = (√

2θ)(m+n+2)ImIn, Im =

∫ +∞

−∞e−ζ

2ζmdζ, (23)

where ζ = ξx/√

2θ or ξy/√

2θ.The second-order Hermite formula (k = 2) is the optimal choice toevaluate Im for the purpose of deriving the 9-bit model, i.e.,

Im =

3∑j=1

ωjζmj .

Note that the above quadrature is exact up to m = 5 = (2k + 1).Luo (Math Dept, ODU) Kinetic Methods for CFD NIA, 01/17/’12 16 / 53

Discretization of Velocity ξ-Space (9-bit Model)

The three abscissas in momentum space (ζj) and the correspondingweights (ωj) are:

ζ1 = −√

3/2 , ζ2 = 0 , ζ3 =√

3/2 ,ω1 =

√π/6 , ω2 = 2

√π/3 , ω3 =

√π/6 .

(24)

Then, the integral of moments becomes:

I = 2θ

[ω2

2ψ(0) +4∑

α=1

ω1ω2ψ(ξα) +8∑

α=5

ω21ψ(ξα)

], (25)

where

ξα =

(0, 0) α = 0,

(±1, 0)√

3θ, (0, ±1)√

3θ, α = 1 – 4,

(±1, ±1)√

3θ, α = 5 – 8.

(26)


Discretization of Velocity ξ-Space (9-bit Model)

IdentifyingWα = (2π θ) exp(ξ2

α/2θ)wα , (27)

with c := δx/δt =√

3θ, or c2s = θ = c2/3, δx is the lattice constant,

then:

f (eq)α (x, t) = Wα f

(eq)(x, ξα, t)

= wα ρ

1 +

3(cα · u)

c2+

9(cα · u)2

2c4− 3u2

2c2

, (28)

where weight coefficient wα and discrete velocity cα are:

wα =

4/9,1/9,1/36,

cα = ξα =

(0, 0), α = 0 ,(±1, 0) c, (0, ±1) c, α = 1 – 4,(±1, ±1) c, α = 5 – 8.

(29)

With cα|α = 0, 1, . . . , 8, a square lattice structure is constructed inthe physical space.


Discretized 2D Velocity Space (9-bit)

With Cartesian coordinates in ξ-space, a 2D square lattice is obtained:

cα =

(0, 0), α = 0 ,(±1, 0) c, (0, ±1) c, α = 1 – 4,(±1, ±1) c, α = 5 – 8.

If spherical coordinates are used, a 2D triangular lattice is obtained.

cα =

(0, 0), α = 0 ,(cos((α− 1)π/3), sin((α− 1)π/3)) c, α = 1 – 6.


D3Q19 MRT-LBE Model

f(xj + cδt, tn + δt) = f(xj , tn)−M−1 · S ·[m−m(eq)

], (30)

e(eq) = −11δρ+19

ρ0j · j, ε(eq) = 3δρ− 11

2ρ0j · j, (31a)

(q(eq)x , q(eq)

y , q(eq)z ) = −2

3(jx, jy, jz), (31b)

p(eq)xx =

1

3ρ0

[2j2x − (j2

y + j2z )], p(eq)

ww =1

ρ0

[j2y − j2

z

], (31c)

p(eq)xy =

1

ρ0jxjy, p(eq)

yz =1

ρ0jyjz, p(eq)

xz =1

ρ0jxjz, (31d)

π(eq)xx = −1

2p(eq)xx , π(eq)

ww = −1

2p(eq)ww , (31e)

m(eq)x = m(eq)

y = m(eq)z = 0, (31f)

where δρ is the density fluctuation, ρ = ρ0 + δρ and ρ0 = 1.Luo (Math Dept, ODU) Kinetic Methods for CFD NIA, 01/17/’12 20 / 53

D3Q19 MRT-LBE Model (cont.)

Conserved quantities:

δρ =

Q−1∑i=0

fi, j = ρ0u =

Q−1∑i=0

fici,

Transport coefficients and the speed of sound:

ν =1

3

(1

sν− 1

2

), ζ =

(5− 9c3s)

9

(1

se− 1

2

), c2

s =1

3c2,

where c := δx/δt.The transform between the discrete distribution functions f ∈ V = RQand the moments m ∈M = RQ:

m = M · f , f = M−1 ·m.

Note that M−1 is related M†.


The BGK Model Equation

The Boltzmann Equation with BGK approximation:

∂tf + ξ ·∇f =

∫[f ′1f

′2 − f1f2]dµ ≈ − 1

λ[f − f (0)] , f ≡ f(x, ξ, ζ, t)

(32)The Boltzmann-Maxwellian equilibrium distribution function:

f (0) = ρ (2π/β)−(D+K)/2 exp

[−β (c · c+ ζ · ζ)

2

], β := 1/kBT (33)

The hydrodynamic variables are moments of f and f (0):

ρ =

∫fdΞ =

∫f (0)dΞ , dΞ := dξdζ, (34a)

ρu =

∫ξfdΞ =

∫ξf (0)dΞ , (34b)

ρε =1

2

∫(c · c+ ζ · ζ)fdΞ =

1

2

∫(c · c+ ζ · ζ)f (0)dΞ . (34c)


Integral Solution of Continuous Boltzmann Equation

Rewrite the Boltzmann BGK Equation in the form of ODE:

Dtf +1

λf =

1

λf (0) , Dt ≡ ∂t + ξ ·∇ . (35)

Integrate Eq. (35) over a time step δt along characteristics:

f(x+ ξδt, ξ, t+ δt) = e−δt/λ f(x, ξ, t)

+1

λe−δt/λ

∫ δt

0et′/λ f (0)(x+ ξt′, ξ, t+ t′) dt′ . (36)

From Eq. (35), formally we have:

f = f (0) − λDtf

For the Navier-Stokes equations, f ≈ f (0) − λDtf(0).


Some Convenient Notations

We use the bold-face font for (D + 2)-tuple vectors:

Ψ := (1, ξ1 , ξ2, ξ3, (c2 + ζ2)/2)T, (37a)

W := (ρ, ρu1, ρu2, ux, ρε)T =

∫fΨdΞ =

∫f (0)ΨdΞ, (37b)

Fα :=

∫fΨξαdΞ, α ∈ x, y, z, (37c)

h := (ρ, u1, u2, u3, T )T, (37d)

h′ := (ρ−1, βu1, βu2, u3,1

2[β(c2 + ζ2)− (D +K)]T−1)T, (37e)

K = (5− 3γ)/(γ − 1) + 1, c := (ξ − u), Ξ := (ξ, ζ).


Initial and Equilibrium Values

The initial value f0 is approximated by its 1st-order CE expansion

f0 ≈ [1− λ(∂t + ξ ·∇)]f (0) =[1− λh′ · (∂t + ξ ·∇)h

]f (0) (38)

The equilibrium f (0) by its 1st-order Taylor expansion at x = 0:

f (0)(x, t) ≈ [1 + x·∇]f (0)(0, t) = f (0)(0, t)[1 + (x·∇h) · h′

](39)

f0(x, t) ≈ f (0)(0, t)[1 + (x·∇h) · h′

] [1− λh′ · (∂t + ξ ·∇)h

]≈ f (0)(0, t)[1 + a·x− λ(a·ξ +A)] (40)

a ≡∇ ln f (0) = ∇ ln ρ+

[(c2 + ζ2)

2RT− 3 + Z

2

]∇ lnT +

(∇u) · cRT

(41)

A ≡ ∂t ln f (0) = −a·ξ +

[(c2 + ζ2)

2RT− 5 + Z

2

]c·∇ lnT

+

[cc

RT− (c2 + ζ2)

5RTI

]: ∇u (42)


Initial and Equilibrium Values (2)

Coefficients a and A are related by the compatibility condition of f :∫f (n)ΨdΞ = 0, ∀n > 0 =⇒

∫Af (0)dΞ = −

∫a·ξ f (0)dΞ

because

∫f (1)dΞ = −λ

∫Dtf

(0)dΞ = −λ∫

(A+ a·ξ)f (0)dΞ = 0

Thus the equilibrium f (0)(x, t) in the solution (36) is approximated by:

f (0)(x, t) ≈ (1 + t∂t + x·∇)f (0)(0, 0)

=[1 + h′ · (t∂t + x·∇)h

]f (0)(0, 0)

= f (0)(0, 0)(1 + a · x+ At) (43)

where a and A are similar to a and A, respectively: in a and A, bothhydrodynamic varibles (ρ, u, T ) and their gradients may bediscontinuous, while in a and A, only the gradients may be so.


Finite Volume Formulation

Assuming the hydrodynamic variables h can be discontinuous at thecell boundary of xi+1/2 = 0, f (0) in both sides of xi+1/2 have to beevaluated differently:

On the left, h are interpolated to x−i+1/2 from three points xi−1, xi

and xi+1, and f (0)L is computed from h at x−i+1/2;

Similarly, f (0)R are evaluated from h, which are interpolated to the

x+i+1/2 from three points xi, xi+1 and xi+2.

Therefore:

f0(x, t) = [1 + aR(x− λξ1)− λAR]H(x)f (0)R (0, t)

+[1 + aL(x− λξ1)− λAL]H(−x)f (0)L (0, t), (44a)

f (0)(x, t) = [1 +H(−x)aLx+H(x)aRx+ At]f (0)(0, 0), (44b)

where H(x) is the Heaviside function.


Finite Volume Formulation (2)

The value of f at a cell boundary can be obtained by using f0(x, t)and f (0)(x, t):

f(xi+1/2, t) =[

(1− Aλ)(1− e−t/λ) + At]

+[(t+ λ)e−t/λ − λ

][aLH(ξ1) + aRH(−ξ1)] ξ1

f (0)

0 (0, 0)

+e−t/λ

[(1− ξ1(t+ λ)aL)− λAL]H(ξ1)f (0)0L (0, 0)

+ [(1− ξ1(t+ λ)aR)− λAR]H(−ξ1)f (0)0R (0, 0)

(45)

where f (0)

0 , f (0)0L and f (0)

0R are initial values of f (0), f (0)L and f (0)

R evaluatedat cell boundary xi+1/2.


Calculation of Fluxes

With f given at cell boundaries, the time-dependent fluxes are

Fi+1/2,jα =

∫ξαΨf(xi+1/2, t)dΞ (46)

By integrating the above equation over each time step, we obtain thetotal fluxes:

Fi±1/2,jx =

∫ ∆t

0Fi±1/2,jx dt, Fi,j±1/2

y =

∫ ∆t

0Fi,j±1/2y dt. (47)

The FV formulation for the governing equations are:

Wn+1ij = Wn

ij−1

∆x(Fi+1/2,j

x − Fi−1/2,jx )− 1

∆y(Fi,j+1/2

y − Fi,j−1/2y ), (48)

which is used to update the flow field.


GKS for Near Continuum Flows

In the near-continuum or nonequilibrium flows, the viscosity is not aconstant (e.g., within a shock). Therefore, λ is no longer a constant.Assume:

f = f (0) − λDtf =⇒ f ≈ f (0) − λ∗Dtf(0)

where λ∗ is the modified relaxation time. Also

f ≈ f (0) − λDtf(1) = f (0) − λDtf

(0) + λλ∗D2t f

(0) (49)

thenλ∗Dtf

(0) ≈ λ(Dtf(0) − λ∗D2

t f(0)) (50)

λ∗ =λ

1 + λ⟨D2t g⟩/ 〈Dtg〉

(51)

where 〈D2t g〉 and 〈Dtg〉 can be computed as functions of hydrodynamic

variables and their derivatives up to second order.


Outline




4 Conclusions


In the Course of Hydrodynamic Events ...

Ma (Mach) and Kn (Knudsen) characterize nonequilibrium

Van Karmen relation based on Navier-Stokes equation (Kn=O(ε)): Ma=Re·Kn

Re1 Re≈1 Re1

Stokes Flows Incompressible Navier-Stokes FlowsMa1

Ma=O(ε2), Re=O(ε) Ma=O(ε), Re=O(1) Ma=O(ε1−α), Re=O(ε−α)

Sub/Transonic FlowsMa≈1

Ma=O(1), Re=O(ε−1)

Super/Hypersonic FlowsMa1

Ma=O(ε−1), Re=O(ε−2)

With the framework of kinetic theory (Boltzmann equation)

Hydrodynamics Slip Flow Transitional Free Molecular

Kn<10−3 10−3<Kn<10−1 10−1<Kn<10 10<Kn


Decaying Homogeneous Isotropic Turbulence

The decaying homogeneous isotropic turbulence is the solution of theincompressible Navier-Stokes equation

∂tu+ u ·∇u = −∇p+ ν∇2u, ∇ · u = 0, x ∈ [0, 2π]3, (52)

with periodic boundary conditions. The initial velocity satisfies a giveninitial energy spectrum E0(k)

E0(k) := E(k, t = 0) = Ak4e−0.14k4, k ∈ [ka, kb]. (53)

The initial velocity u0 can be given by Rogallo procedure:

u0(k) =αkk2 + βk1k3

k√k2

1 + k22

k1 +βk2k3 − αk1k

k√k2

1 + k22

k2 −β√k2

1 + k22

kk3, (54)

where α =√E0(k)/4πk2eıθ1 cosφ, β =

√E0(k)/4πk2eıθ2 sinφ,

ı :=√−1, and θ1, θ2, φ ∈ [0, 2π] are uniform random variables.


Low-Order Turbulence Statistics

The energy and the compensated spectra:

E(k, t) :=1

2u(k, t) · u†(k, t), Ψ(k) := ε(k)−2/3k5/3E(k), (55)

And the pressure spectrum P (k, t). Low-order moments of E(k, t):

K(t) :=

∫dkE(k, t), Ω(t) :=

∫dk k2E(k, t) (56a)

ε(t) := 2νΩ(t), η := 4√ν3/ε (56b)

Sui(t) =〈(∂iui)3〉〈(∂iui)2〉3/2

, Su(t) =1

3

∑i

Sui (56c)

Fui(t) =〈(∂iui)4〉〈(∂iui)2〉2

, Fu(t) =1

3

∑i

Fui (56d)

We will also compare instantaneous flows fields u(x, t) and ω(x, t).


Pseudo-Spectral Method

The pseudo-spectral (PS) method solve the Navier-Stokes equation inthe Fourier space k, i.e.,

u(x, t) =∑ku(k, t)eık·x, −N/2 + 1 ≤ kα ≤ N/2.

The nonlinear term u ·∇u computed in physical space x byinverse Fourier-transform u and ku to x for form the nonlinearterm; and it is transformed back to k space;

De-aliasing: u(k, t) = 0 ∀‖k‖ ≥ N/6;

Time matching: second-order Adams-Bashforth scheme:

u(t+ δt)− u(t)

δt= −3

2T (t) +

1

2T (t− δt)e−νk2δt,

where T := F [ω × u]− (F [ω × u] · k)k.


Parameters in DNS

Use N3 = 1283 and [ka, kb] = [3, 8].In LBE: ν = 1/600 (cδx), c := δx/δt = 1, Mamax = ‖u0‖max/cs ≤ 0.15,A = 1.4293 · 10−4 in E0(k), and K0 ≈ 1.0130 · 10−2, u′0 ≈ 8.2181 · 10−2.The time t is normalized by the turbulence turn-over time t0 = K0/ε0.In SP method, K0 = 1 and u′0 =

√2/3.

Method L δx u′0 δt ν δt′

LBE 2π 2π/N√

2K0/3 2π/N ν 2π/Nt0PS 2π 2π/N

√2/3 2π

√K0/N ν/

√K0 2π/Nt0

The Taylor microscale Reynolds number:

Reλ :=u′λ

ν, λ :=

√15

2Ωu′ :=

√15ν

εu′ (57)

The resolution criterion:

SP: N ∼ 0.4Re3/2λ , η/δx ≥ 1/2.1, N = 128→ Reλ = 46.78

LBE: ηkmax = η/δx ≥ 1, N = 128→ Reλ = 24.35.


Initial Conditions

For the psuedo-spectral method:

Generate u0(k) in k-space with a given E0(k) (Rogallo’sprocedure) with K0 = 1 and u′ =

√3/2;

The initial pressure p0 is obtained by solving the Poisson equationin k-space.

For the LBE method:4

Use the initial velocity u0 as in PS method except a scaling factorso that Mamax = 0.15;

The pressure p0 is obtained by an iterative procedure with a givenu0.

4R. Mei, L.-S. Luo, P. Lallemand, and D. d’Humieres. Computers & Fluids 35(8/9):855–862 (2006).


‖u(t′)/u′‖ and ‖ω(t′)/u′‖ at Reλ = 24.37, t′ = 4.048

LBE vs. PS1 (equal δt) and PS2 (δt/3), PS1 vs. PS2ve

loci

tyu/u′ 0

vort

icit

yω/u′ 0


‖u(t′)/u′‖ and ‖ω(t′)/u′‖ at Reλ = 24.37, t′ = 29.949

LBE vs. PS1 (equal δt) and PS2 (δt/3), PS1 vs. PS2ve

loci

tyu/u′ 0

vort

icit

yω/u′ 0


L2 ‖δu(t′)‖ and ‖δω(t′)‖

LBE vs. PS1 (equal δt) and PS2 (δt/3), PS1 vs. PS2.


Flow past 25-55 Double-Cone Ma = 9.59 (Run 28)

The free stream parameters for Run 28:Ma∞ = 9.59, p∞ = 36.06 (Pa), T∞ = 185.6 (K), Tw = 293.3 (K)Re∞ = 12, 838 (based on L = 3.625′′), Kn∞ ≈ 1.1× 10−3

The mesh size is 481× 201, minimum distant to wall ≈ 2.81 · 10−6 (m).Convergence residual: ≥ 10−7.


25-55 Double-Cone: Ma, p, T

The contours of Ma/Ma∞ (L), p/p∞ (C), and T/T∞ (R).


25-55 Double-Cone: Interaction Zone

The contours of ρ/ρ∞ (recirculation) and p/p∞ (shock interactions).


25-55 Double-Cone: Density schliere

ϕ(x, y) = exp

(−3000

‖∇ρ(x, y)‖2

‖∇ρ‖2max

)Luo (Math Dept, ODU) Kinetic Methods for CFD NIA, 01/17/’12 44 / 53

25-55 Double-Cone: Cp and St

The pressure coefficient Cp and Stanton number St, experiment,5 NS(1024× 256),6 and GKS (481× 201).

5M. Holden and T. P. Wadhams. AIAA Paper 2001-1031 (2001).

6P. Gnoffo. AIAA Paper 2001-1025 (2001); G. Candler. AIAA Paper 2001-1024 (2001); I. Nompelis. PhD

Thesis (2004).


25-55 Double-Cone: Separation/Reattachment

Abscissa x/L LAURA NS GKS

Separation 0.5555 0.5594 0.5755

Reattachment 1.291 1.287 1.263

Length 0.7355 0.7276 0.6875

mesh size 1024× 256 1024× 256 481× 201


Shock Structures: ρ and T at Ma=8.0 and 9.0

The density and temperature profiles of monoatomic gas inside a shockof Ma= 8.0 and 9.0. GKS results vs. experimental7 and DSMC8 data.

δρ

∆ρ:=

(ρ− ρ1)

(ρ2 − ρ1)

δT

∆T:=

(T − T1)

(T2 − T1)λ1 :=

2µ(7− 2ω)(5− 2ω)

15ρ1

√2πRT1

7Schmidt, J. Fluid Mech. 39:361 (1969); Alsmeyer, J. Fluid Mech. 74:497 (1976).

8Bird, Phys. Fluids 13:1172 (1970).


Shock Structures: τxx and qx at Ma=8.0

The stress τxx and heat-flux qx profiles of monoatomic gas inside theshock of Ma= 8.0. GKS results vs. DSMC data.9

9Bird, Phys. Fluids 13:1172 (1970).


Other Applications (Animations)

Ladd, Univ. Florida: Cluster of 1812 particles, Re = 0.3

Flow through Porous Media (Krafczyk et al., TUB):

Air through fluids in a packed-sphere bio-filterMulti-component flow past porous media: imbibe, drain, re-imbibe

Free-Surface Flow (Krafczyk et al., TUB):

Flow past a bridgeWater-Tunnel: 384× 64× 48, δx = 10(cm), LES, Re = 3 · 106

Tsumani over S. Manhatten: 512× 512× 80, δx = 2.35(m)

Krafczyk et al.:Digital Wind Tunnel, 64× 64× 320, δx = 10(cm), Re = 4 · 103

Droplet collision (Frohn et al., Univ. Stuttgart):

Merge, Separate, and one extra; LBE vs. Experiment.

More (Thury et al., Univ. Erlangen):

Bubbles rising in water tankMetal foams


Outline




4 Conclusions


Why Kinetic Methods?

The Boltzmann equation (BE) is 1st-order PDE with a linearadvection term, the Navier-Stokes equation is 2nd-order and nonlinear.It is more costly to solve the BE in phase space, kinetic schemes10

Requires the smallest possible stencil for accurate discretization,thus least need for inter-nodal data communication;

Nonlinearity is in local collision term, stiffness of which can beovercome by local techniques.

Discretized 1st-order systems may be easier to converge thanequivalent higher-order systems.

1st-order PDE’s yield the highest potential discretization accuracyon non-smooth, adaptively refined grids.

The systems of 1st-order PDE’s are better suited for functionaldecomposition, thus easier to parallelize.

The Boltzmann equation is valid for nonequilibrium flows whichcannot be modeled by the Navier-Stokes equations.

10B. van Leer, AIAA Paper 2001-2520 (2001).


Questions?


Food for Thought

For an n-th order scheme, how to guarantee that the operators ∇and ∇2 are isotropic to the order consistent/comparable with n?

How to construct physics-based CFD schemes beyond upwind,(approximated) multi-dimensional Riemann solver, artificialdissipation, and limiters?


kinetic methods for cfd - ossanworld.com

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