Sharp Interface Approach for Liquid-Vapour Flow with Phase … · 2014. 2. 28. · Micro-MacroModellingand SimulationofLiquid-VapourFlow DFG - CNRS Project RO 2222/4-1 Sharp Interface

Micro-Macro Modelling andSimulation of Liquid-Vapour FlowDFG - CNRS Project RO 2222/4-1

Sharp Interface Approach for Liquid-Vapour Flowwith Phase Transition

Christoph Zeiler

Institute for Applied Analysis and Numerical Simulation

February 2014

Free Boundary Value Problem in Rd

[Binninger et al.]

Bulk Phases%t + div(% v) = 0,

(% v)t + div (% v ⊗ v + p(%) I ) = 0.

Phase BoundaryJ%(v · n − σ)K = 0,J%(v · n − σ) v + p(%) nK = (d − 1)γκn,qg(%) + 0.5 (v · n − σ)2

y= −k j.

Local Well-Posedness Results_ for k = 0 see

[Benzoni-Gavage, Freistühler 2004]_ for k > 0 see

[Kabil, Rohde 2013]

Christoph Zeiler Sharp Interface Approach for Liquid-Vapour Flow with Phase Transition 1/21

Outline

Cut Cell Method

Exact Riemann Solver

Numerical Results


Cut Cell Method (joint work with C. Chalons)Basic Steps

Γ(t) Domain Ω ∈ Rd with fluid in two

phases at time t > 0_ liquid Ωliq(t),_ vapour Ωvap(t),_ curved phase boundary

Γ(t) = ∂Ωliq(t) ∩ ∂Ωvap(t).

We consider physical quantities inEuler coordinates (x, t)_ density %(x, t) > 0,_ velocity v(x, t) ∈ Rd .



Γ(t) Domain Ω ∈ Rd with fluid in two

phases at time t > 0_ liquid Ωliq(t),_ vapour Ωvap(t),_ curved phase boundary


Cut (and Merge) Cell Approach:_ approximate Γ(t)→ Γh(t),

_ cut cells_ and merge, when cells are to

small.



Domain Ω ∈ Rd with fluid in twophases at time t > 0_ liquid Ωliq(t),_ vapour Ωvap(t),_ curved phase boundary


Cut (and Merge) Cell Approach:_ approximate Γ(t)→ Γh(t),_ cut cells

_ and merge, when cells are tosmall.



Domain Ω ∈ Rd with fluid in twophases at time t > 0_ liquid Ωliq(t),_ vapour Ωvap(t),_ curved phase boundary


Cut (and Merge) Cell Approach:_ approximate Γ(t)→ Γh(t),_ cut cells_ and merge, when cells are to

small.


Cut Cell MethodMacroscale

MacroscaleDynamics of the bulk phases Ωliq and Ωvap.Model: Isothermal Euler equation

%t + div(% v) = 0,(% v)t + div (% v ⊗ v + p(%) I ) = 0,

with a pressure function p that covers liquidand vapour phase.

Van-der-Waals like Pressure p

%︸︷︷︸vapour

︸︷︷︸liquid


Cut Cell MethodMicroscale

n

MicroscaleDynamics at the phase boundary.

We assume that it is sufficient to consider1D Riemann type problems normal to Γh._ sharp interface models_ exact (Liu) solver [Jägle, Rohde, Z 2012]_ approximate (KinRel) solver

[Rohde, Z 2013]_ exact (KinRel) Riemannsolver [today]

_ diffuse interface models_ molecular dynamic models


Cut Cell MethodTransfer Operator

n

Macro- to microscale_ initial states for Riemann type

problems

Micro- to macroscale_ fluxes for phase boundary edges


Cut Cell Method

Bulk: Euler equation_ liquid phase_ vapour phase

Phase Boundary:_ Riemann type problem_ sharp/diffuse interface model

Bulk:_ curvature reconstruction_ interface tracking

Reconstruction: Compression:

initial states

mean curvature κ

flux atthe phaseboundary

speed ofthe phaseboundary

The cut cell method with Godunov type two phase flux is conservative.


Exact Riemann SolverSharp Interface Model

For each macro time step and any interface segmentthere is a Riemann problem to solve:_ Initial condition (

%, v)(x, 0) : =

{(%l, vl) for x ≤ 0,(%r, vr) for x > 0,

w.l.o.g. %l in the liquid phase, %r in the vapour phase._ x = x · n, v = v · n._ Curvature is fixed._ "Bubble case" with κ > 0

"Droplet case" with κ < 0

Γhx

liquid(%l, vl)

vapour(%r, vr)

0


Exact Riemann SolverSharp Interface Model(

%%v

)t

+(

%v%v2 + p(%)

)x

=(

00

)in R× (0,∞).

Riemann Problem_ Classical entropy solution in the bulk phases._ There exists one phase boundary that satisfies

−σ J%K + J% vK = 0−σ J% vK+J% v2 + p(%)K=(d − 1)γκ.

_ The phase boundary satisfies the kinetic relationqg(%) + 0.5 (v − σ)2

y= −k j.

σ speed of the phase boundary J%K := %vap − %liqj = %(v − σ) mass flux τ = 1/% specific volume

p̃(τ) := p(τ−1) pressure (d − 1)γκ (const.) surface tensiong̃(τ) := g(τ−1) chemical potential ψ Helmholz free energy

Exact solvers are available ([LeFloch et al. (≥ 1991)], [Hantke et a. 2013]. . . , [Jägle, Rohde, Z 2012]), but only for simplified models or kinetic re-lations.


Exact Riemann SolverSharp Interface Model

Exact Riemann Solver_ Classical entropy solution in the bulk phases._ There exists one phase boundary that satisfies

−σ J%K + J% vK = 0−σ J% vK+J% v2 + p(%)K=(d − 1)γκ.

_ The phase boundary satisfies the kinetic relationqg(%) + 0.5 (v − σ)2

y= −k j.

Then, for k > 0, follows for the total energy e = ψ + 12 v2:

−σ(J%eK + (d − 1)γκ

)+ J(%e + p) vK = −k j2 ≤ 0

Exact solvers are available ([LeFloch et al. (≥ 1991)], [Hantke et a. 2013]. . . , [Jägle, Rohde, Z 2012]), but only for simplified models or kinetic re-lations.


Exact Riemann SolverGibbs-Thomson based Kinetic Relation

Admissible Phase Boundary Wave_ connects liquid state (τliq, vliq) and vapour state (τvap, vvap),_ satisfies

J%(v − σ)K = 0,j JvK + Jp̃(τ)K=(d − 1)γκ

}⇒ j = ±

√(d − 1)γκ− Jp̃(τ)K

JτK

_ and the kinetic relationqg̃(τ) + 0.5 j2 τ2

y= −k j.




Exact Riemann SolverGibbs-Thomson based Kinetic Relation

Example: Stationary Phase Boundary_ connects liquid state (τmwliq , vmwliq ) and vapour state (τmwvap , vmwvap ),_ satisfies

j = 0,Jp̃(τ)K = (d − 1)γκ

_ and the kinetic relationJg̃(τ)K = 0.





Admissible phase boundaries for k = 1 (without surface tension)

6 8 10 12 14 16 180.4645

0.465

0.4655

0.466

0.4665

0.467

0.4675

0.468

0.4685

volume vapour phase

volu

me

liqui

d ph

ase

complex jcondensation (j < 0)evaporation (j > 0)maxwell states (j = 0)

0 5 10 15

pressure

volume



Admissible phase boundaries for k → 0 (without surface tension)

6 8 10 12 14 16 180.4645

0.465

0.4655

0.466

0.4665

0.467

0.4675

0.468

0.4685

volume vapour phase

volu

me

liqui

d ph

ase


0 5 10 15

pressure

volume




6 8 10 12 14 16 180.4645

0.465

0.4655

0.466

0.4665

0.467

0.4675

0.468

0.4685

volume vapour phase

volu

me

liqui

d ph

ase


0 5 10 15

pressure

volume



Admissible phase boundaries for k = 3.5 (without surface tension)

6 8 10 12 14 16 180.4645

0.465

0.4655

0.466

0.4665

0.467

0.4675

0.468

0.4685

volume vapour phase

volu

me

liqui

d ph

ase


0 5 10 15

pressure

volume



Admissible phase boundaries for k = 1 with surface tension

6 7 8 9 10 11 12 13 14 15 16 170.4645

0.465

0.4655

0.466

0.4665

0.467

0.4675

0.468

0.4685

volume vapour phase

volu

me

liqui

d ph

ase

0 5 10 15

pressure

complex jcondensation (j < 0)evaporation (j > 0)maxwell states (j = 0)planar


Exact Riemann SolverKinetic Relation based on the Generalized Gibbs-Thomson LawThere ex. kinetic functions fc (condensation), fe (evaporation) such that

fc(τvap) = τliq∨ fe(τliq) = τvap

}⇔

qg̃(τ) + 0.5 j2 τ2

y= −k j.

k = 0 Reversible process fc(fe(τliq)) = τliq.0 ≤ k < k∗ Monotone functions τvap 7→ fc and τliq 7→ fe.

k > k∗ Phase transition in metastable region.Loss of monotonicity → uniqueness?

Monotone Kinetic Functions with Maximal Entropy DissipationDefine f mc (τvap) := τmwliq and f me (τliq) = τmwvap .Observation:_ The Liu entropy criterion applied on a modified pressure based on

the Maxwell equal area rule leads to f mc , f me .



Monotone Kinetic Functions with Maximal Entropy Dissipation

6 8 10 12 14 16 18 20 220.4645

0.465

0.4655

0.466

0.4665

0.467

0.4675

0.468

0.4685

volume vapour phase

volu

me

liqui

d ph

ase


0 5 10 15

pressure

volume



Based on fc and fe we construct the full wave fan. Phase boundaries ofLax type are allowed but undercompresive waves are "preferred".

Theorem [Z 2014]The Riemann problem for |κ| < C and the kinetic relation based on thegeneralized Gibbs-Thomson law with 0 ≤ k < k∗ or has a unique entropysolution. The solution consist of elementary waves and exactly one phaseboundary satisfying

−σ(J%eK + (d − 1)γκ

)+ J(%e + p) vK ≤ 0

for e = ψ + 12 v2.

For the construction, we follow [LeFloch 2002].


Exact Riemann SolverExample

Density of examples with varying surface tension:

−0.2 0 0.2 0.4 0.6 0.8 1 1.20.12

0.13

0.14

0.15−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5

2.12

2.14

2.16

2.18

κ0


Numerical Results

Benchmark Problem for Surface Tension

Cut cell method for radially symmetric solutionson Ω =

{x ∈ Rd

∣∣ 0 < |x| = r < 1 } with phaseboundary Γ(t) =

{x ∈ Rd

∣∣ |x| = rΓ(t) }. x1x2

rΓ(t)

_ Bulk system:(%% v

)t

+(

% v% v2 + p(%)

)r

= 1−dr

(% v% v2

)for r ∈ (0, rΓ) ∪ (rΓ, 1),t > 0.

_Simple interface tracking (rn+1Γ = rnΓ + ∆t σn)and curvature reconstruction (κn = 1/rnΓ ).

d = 1: Fully conservative [Chalons, Wiebe]d > 1: Mass conservative (multidimensional sence) [Rohde, Z 2013]


Numerical ResultsGlobal Energy (cf. [Gurtin 1985])

E(%, v) =∫

Ω

12% |v|

2 + %ψ(%) dx + γ |Γ| ,

Estat = min{

E(%,0)∣∣ ∫

Ω % dx = const.}

Energy

0 5 10 15−22.95

−22.9

−22.85

−22.8

−22.75

−22.7

−22.65

Time

KinRel k=0Mon.Max.Ent.LiuE

statE

stat planar

(Initial conditions %L = %mwliq , %R = %mwvap, vR = 0.1, vL = −0.1, γ = 0.01, d = 2)


Numerical Results

Experimental order of convergence

Cells Error Order100 1.7e-03200 1.1e-03 0.68500 5.7e-04 0.701000 3.4e-04 0.741500 2.5e-04 0.762000 2.0e-04 0.762500 1.7e-04 0.753000 1.5e-04 0.75

Velocity (zoom)

0 0.1 0.2 0.3 0.4 0.5 0.6

0

0.05

0.1

v

h

v

(Initial conditions %l = 2.1368, %r = 0.0333, v = 0, k = 0.0001, reference solutionfrom exact Riemann solver, d = 1)


Numerical ResultsApplication in 2d (with P. Engel)

Pure Liu Riemannsolver,

%(x, 0) ={0.319 : inside,1.806 : outside,

v(x, 0) = 0,γ = 0.001.


Summary_ Conservative front tracking scheme._ Exact Riemann solver

_ based on the generalized Gibbs-Thomson law._ for arbitrary pressure laws

(Peng Robinson, external library (FPROPS [IAPWS])).

OutlookApproximative solver for the full Euler system.


Literature[Jägle, Rohde, Z 2012] F. Jaegle, C. Rohde, and C. Zeiler.

A multiscale method for compressible liquid-vapor flow with surface tension.ESAIM: Proc., 38:387–408, 2012.

[Rohde, Z 2013] C. Rohde and C. Zeiler.A relaxation riemann solver for compressible two-phase flow with phase transitionand surface tension.accepted for publication in Applied Numerical Mathematics, 2013.

[Wiebe 2014] M. Wiebe.A sharp-interface approach for phase transition problems.Master’s thesis, Universität Stuttgart, 2014.

[Kabil, Rohde 2013] B. Kabil and C. Rohde.The influence of surface tension and configurational forces on the stability ofliquid-vapor interfaces.

[Hantke et a. 2013] M. Hantke, W. Dreyer, and G. Warnecke.Exact solutions to the riemann problem for compressible isothermal euler equationsfor two-phase flows with and without phase transition.

[LeFloch 2002] P.G. LeFloch.Hyperbolic systems of conservation laws.


Documents

Sharp Interface Approach for Liquid-Vapour Flow with Phase … · 2014. 2. 28. · Micro-MacroModellingand SimulationofLiquid-VapourFlow DFG - CNRS Project RO 2222/4-1 Sharp Interface