Simple Eulerian Methods for Compressible Fluids in Domains with Moving Boundaries

Alina Chertock

North Carolina State University chertock@math.ncsu.edu

joint work with Armando Coco, Yuanzhen Cheng, Smadar Karni, Alexander Kurganov and Giovanni Russo

supported in part by

Compressible Euler Equations

 

ρ ρu ρv E

  t

+

 

ρu ρu2 + p ρuv

u(E + p)

  x

+

 

ρv ρuv

ρv2 + p v(E + p)

  y

= 0

E = p

γ − 1 + ρ

2 (u2 + v2)

• ρ is the fluid density;

• u, v are the velocities;

• E is the total energy;

• p is the pressure;

• c = √ γp

ρ is the speed of sound.

1

Two Types of Problems

• Multicomponent Flow: We assume that γ is a piecewise constant function propagation with the fluid velocity according to

γt + uγx + vγy = 0.

• Problems with Changing Geometries:

– The right boundary is a moving solid wall with a prescribed equation of motion x = xB(t)

B

Moving Boundary

x

– Flow around an (oscillating) solid circle.

I II

2

Conservative Methods

wt + F (w)x = 0 ⇔

 

ρ ρu E ργ

  t

+

 

ρu ρu2 + p u(E + p) uργ

  x

= 0

Godunov-type discretization:

w̄n+1j = w̄ n j −

∆t

∆x

( Hj+1/2 −Hj−1/2

) ,

Hj+1/2 = H(. . . , w̄ n j , w̄

n j+1, . . .), H(. . . ,w,w, . . .) = F (w).

The major difficulty: to ensure that the pressure equilibrium between fluid components is not disturbed by the numerical scheme.

3

What is going wrong? (S. Karni)

4

Multicomponent Flow

wt + f(w)x = 0 ⇐⇒

 

ρ ρu E

  t

+

 

ρu ρu2 + p u(E + p)

  x

= 0

EOS: p = (γ − 1) ( E − 1

2 ρu2 ) − γp∞

w = (ρ, ρu,E)T

We assume that γ is a piecewise constant function propagation with the fluid velocity according to

γt + uγx = 0.

5

Multi-Fluid Example

0 0.2 0.4 0.6 0.8 1 0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

pressure

CONSERVATIVE METHOD EXACT SOLUTION

(ρ, u, p, γ)T =

{ (1.0, 1.0, 1.0, 1.6)T , if x < 0.25, (0.1, 1.0, 1.0, 1.4)T , if x > 0.25.

Different EOS...

6

What is going wrong?

J-1 J J+1

!

t n

!

t n+1

!

xJ"1/2

!

xJ+1/2

material interface

• Fluxes are computed using the information in the mixed cell.

• No valid EOS in mixed cells.

7

Some Models/Schemes

• Conserve not only total mass and momentum but also total energy.

– Volume-of-Fluid Method (Noh & Woodward; Collela, Glaz & Ferguson).

– Internal Energy Correction Algorithm (Jenny, Mueller & Thomann). – A Fluid-Mixture Type Algorithm (Shyue).

• Conserve only total mass and momentum but not total energy.

– Pressure Evolution Method (Karni). – Ghost-Fluid Method for the Poor (Abgrall & Karni).

• Nonconservative

– Ghost-Fluid Method (Fedkiw, Aslam, Merriman & Osher).

• – Conservative locally moving mesh method for multifluid flows (A.C. & A. Kurganov)

– A second-order finite-difference method for compressible fluids in domains with moving boundaries, (A. C., A. Coco, A. Kurganov and G. Russo)

– Interface tracking method for compressible multifluids, (A.C., S. Karni & A. Kurganov)

8

Semi-Discrete Central-Upwind Scheme – 1-D Case

wt + f(w)x = 0 ⇐⇒

 

ρ ρu E

  t

+

 

ρu ρu2 + p u(E + p)

  x

= 0

w := (ρ, ρu,E)T

Computational cells: Cj := [xj−12 , xj+12

]

The cell averages: wj(t) := 1

∆x

∫

Cj

w(x, t) dx

Time evolution:

d

dt wj(t) = −

Hj+12 (t)−Hj−12(t)

∆x

{ Hj+12

} : numerical fluxes

9

{wj(t)} → w̃(·, t)→ { w± j+12

(t) } → { Hj+12

(t) } → {wj(t+ ∆t)}

(Discontinuous) piecewise-linear reconstruction:

w̃(x, t) := wj(t) + (wx)j(x− xj), x ∈ Cj

It is conservative, second-order accurate, and non-oscillatory provided the slopes, {(wx)j}, are computed by a nonlinear limiter

Example — Generalized Minmod Limiter

(wx)j = minmod

( θ wj −wj−1

∆x , wj+1 −wj−1

2∆x , θ

wj+1 −wj ∆x

)

where

minmod(z1, z2, ...) :=

  

minj{zj}, if zj > 0 ∀j, maxj{zj}, if zj < 0 ∀j, 0, otherwise,

and θ ∈ [1, 2] is a constant 10

{wj(t)} → w̃(·, t)→ { w± j+12

(t) } → { Hj+12

(t) } → {wj(t+ ∆t)}

w± j+12

(t) are the point values of

w̃(x, t) = wj + (wx)j(x− xj), x ∈ Cj

at xj+12 :

w+ j+12

:= w̃(xj+12 + 0, t) = wj+1 −

∆x

2 (wx)j+1

w− j+12

:= w̃(xj−12 − 0, t) = wj +

∆x

2 (wx)j

11

{wj(t)} → w̃(·, t)→ { w±j (t)

} → { Hj+12

(t) } → {wj(t+ ∆t)}

Hj+12 (t) = H(w−

j+12 (t),w+

j+12 (t))

Example — Central-Upwind Flux:

[Kurganov, Lin, Noelle, Petrova, Tadmor, et al.; 2000–2007]

Hj+12 = a+ j+12

f(w− j+12

)− a− j+12

f(w+ j+12

)

a+ j+12 − a−

j+12

+ a+ j+12

a− j+12

a+ j+12 − a−

j+12

[ w+ j+12 −w−

j+12

]

a± j+12

(t) = a± j+12

( w+ j+12

,w− j+12

) are one-sided local speeds, estimated using

the largest and the smallest eigenvalues of the Jacobian ∂f

∂w .

d

dt wj(t) = −

Hj+12 (t)−Hj−12(t)

∆x

12

Boundary Treatment in 1-D

• Assume that at some time level t the wall is located in cell J

• All computational cells are divided into 3 groups:

– Internal cells with reliable data – Boundary cells with unreliable data – External cells with no data

All computational cells are divided into 3 groups:

• Internal cells with reliable data

• Boundary cells with unreliable data (as multi-fluid “mixed” cells)

• External cells with no data

x J!5/2

x J!3/2

x J!1/2

x J+1/2

x J+3/2

Moving Boundary

xB

43

13

Boundary Treatment in 1-D

• The cell averages wJ(t) will not be evolved

• To evolve the cell averages wJ−1(t) we need:

– the numerical flux HJ−12 (t)

– the point values w+ J−12

(t)

Assume that at some time level t the wall is located in cell J

The cell averages wJ(t) will not be evolved

To evolve the cell averages wJ−1(t) we need the numerical flux HJ−12 (t).

Thus we need the point values w+ J−12

(t)

x J!5/2

x J!3/2

x J!1/2

x J+1/2

x J+3/2

Moving Boundary

w J!1/2

J!1

xB

w

+

44

14

Solid Wall Extrapolation

The solid wall extrapolation of the solution values in cell (J − 1),

ρJ−1, uJ−1 = (ρu)J−1 ρJ−1

, pJ−1 = (γ − 1) [ EJ−1 −

ρJ−1u 2 J−1

2

]

To obtain the ”missing” ghost values:

ρ d2xB(t)

dt2

∣∣∣ x=xB(t)

= −px ∣∣∣ x=xB(t)

, ∂p

∂ρ

∣∣∣ x=xB(t)

= γp

ρ

∣∣∣ x=xB(t)

Solid Wall Extrapolation

The solid wall extrapolation of the solution values in cell (J − 1),

ρJ−1, uJ−1 = (ρu)J−1 ρJ−1

, pJ−1 = (γ − 1)  EJ−1 −

ρJ−1u2J−1 2

  ,

results in the following ghost-cell values:

ρgh := ρJ−1, ugh := 2ẋB(t) − uJ−1, pgh := pJ−1 ẋB(t): velocity of the wall at time t

x J!5/2

x J!3/2

x J!1/2

x J+1/2

x J+3/2

w J!1

w J!1/2

xB

+

wgh

45

15

Solid Wall Extrapolation

• Second-Order Approximation:

ρJ−1 + ρgh 2

· d 2xB(t)

dt2 = −pgh − pJ−1

∆x

pgh − pJ−1 ρgh − ρJ−1

= γ · pgh + pJ−1 ρgh + ρJ−1

• First-Order Approximation:

ρgh := ρJ−1, ugh := 2ẋB(t)− uJ−1, pgh := pJ−1

ẋB(t): velocity of the wall at time t

16

Phase Space Interpolation

if u∗ > 0 then

w+ J−12

=

{ w∗, if u∗ − c∗ < 0 wJ−1, otherwise

else

w+ J−12

=

{ w∗, if u∗ + c∗ > 0

wgh, otherwise

gh

ρ