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  • 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