sino-german symposium on modern numerical...

Sino-German Symposium on

Modern Numerical Methods for Compressible Fluid Flows

and Related Problems

Beijing, May 21-27, 2014

Hyperbolic Techniques for Diffuse Interfaces in Multiphase Flow

Christian RohdeUniversitat Stuttgart

May 23, 2014 1

Plan of the Talk

1) Compressible Navier-Stokes-Korteweg Equations

2) A Hyperbolic Relaxation Approximation

3) The Korteweg/Relaxation Limit

4) Summary and Outlook

Joint work withA. Corli (Ferrara), J. Neusser, V. Schleper (Stuttgart), A. Viorel (Cluj)

May 23, 2014 2

Compressible Liquid-Vapour Dynamics:

Bubble rise and evaporation in vesselunder decrease of pressure

(Source: Peters/Binninger, Institut fur Technische Verbrennung, RWTH Aachen

May 23, 2014 3

1) Compressible Navier-Stokes-Korteweg Equations

(NSK Equations)

May 23, 2014 4

The NSK Equations

The Pressure Function (at Constant Temperature):

ρ

α2 b

α1

β1

β2

p

Graph of the Van-der-Waals pressure p = p(ρ).

Definition: (phases)The fluid is in the vapour (liquid) phase iff we have

ρ ∈ (0, α1)(ρ ∈ (α2, b)

).

Note:The fluid with density ρ ∈ (0, b) is in the vapour or liquid phase iffp′(ρ) > 0 holds.

May 23, 2014 5

The NSK Equations

The Free Energy (at Constant Temperature):

Graph of the free energy W = W (ρ):

p′(ρ) = ρW ′′(ρ).

May 23, 2014 6

The NSK Equations

Van-der-Waals Functional: (D ⊂ Rd open, bounded)

For ε > 0 find a minimizer ρε ∈ H1(D) of

F εvdW [ρ] :=

∫DW (ρ(x)) +

ε2

2|∇ρ(x)|2 dx,

∫Dρ(x) dx = m.

Theorem: [Modica ’87]The functional ε−1F εvdW Γ-converges in L1(D) to F : L1(D)→ [0,∞] with

F [ρ] :=

{σ∞Hd−1(S(ρ)) : ρ ∈ BV (D), ρ ∈ {β1, β2} a.e.

∞ : otherwise,

where the surface tension σ∞ is given by

σ∞ =

∫ β2

β1

√2W (s) ds.

May 23, 2014 7

The NSK Equations

Navier-Stokes-Korteweg Model:(Dunn&Serrin ’85, Anderson&McFadden&Wheeler ’98)

ρt + div(ρv) = 0

(ρv)t + div(ρv ⊗ v + p(ρ)I) = ε div(T) + ρ∇D[ρ]

D[ρ] = ε2∆ρ

in D × R>0,

v = 0, ∇ρ · n = 0 in ∂D × R>0

Energy inequality:

d

dt

(F εvdW [ρ(·, t)] +

∫D

1

2ρ(x, t)|v(x, t)|2 dx

)≤ 0

Wellposedness: classical solution → Hattori&Li ’94, Kotschote ’08Lions-Feireisl solution→ Bresch&Desjardins&Lin ’03

May 23, 2014 8

The NSK Equations

Direct Numerical Simulation (Diehl&Haink&Kroner&R. ’07-’10)


Local DG-code using

dynamical meshrefinement/coarsening in2D/3D

polynomial degree 3

semi-implicit RK timediscretization

currently ε = 0.001 possibleon unit cube.

May 23, 2014 9

The NSK Equations

Direct Numerical Simulation


Problems of the NSK equations

Third-order terms lead torestrictive time steps

Implicit coupling of interfacialwidth and surface tension

Hyperbolic-elliptic structureprevents use of modernnumerics used forcompressible fluid flow

May 23, 2014 9

The NSK Equations

Elliptic-Hyperbolic Structure: (d=2)

ρt + div(ρv) = 0

(ρv)t + div(ρv ⊗ v + p(ρ)I) = ε div(T) + ρ∇D[ρ]

D[ρ] = ε2∆ρ ρ

α2 b

α1

β1

β2

p

Characteristic structure of the first-order part:The eigenvalues are given for some ξ ∈ Sd−1 by

λ1(ρ, ρv) = v · ξ −√p′(ρ), λ2(ρ, ρv) = v · ξ, λ3(ρ, ρv) = v · ξ +

√p′(ρ)

May 23, 2014 10

2) A Relaxation Approximation

May 23, 2014 11

Relaxation Approximation

A Relaxed Functional: (Brandon et al.’95, Rogers&Truskinovsky ’97 )

For ε, α > 0 find a minimizer (ρα, cα) ∈ L2(D)× H1(D) of

F ε,αRelax [ρ, c] :=

∫D

(W (ρ) +

α

2(ρ− c)2 +

ε2

2|∇c |2

)dx,

∫Dρ dx = m.

Theorem: [Solci&Vitali ’03]

(i) There is a function σ : (0,∞)→ (0,∞) such that for each α > 0 thefunctional ε−1F ε,αRelax Γ-converges in (L1(D))2 toF : (L1(D))2 → [0,∞] with

F [ρ, c] :=

{σ(α)Hd−1(S(ρ)) : ρ = c ∈ {β1, β2} a.e., ρ, c ∈ BV (D)

∞ : otherwise

(ii) The (surface tension) function σ is monotone increasing and satisfies

limα→∞

σ(α) = σ∞.

May 23, 2014 12


Low Order Relaxation Approximation:

ραt + div(ραvα) = 0

(ραvα)t + div(ραvα ⊗ vα + p(ρα)I) = εdiv(Tα)

+αρα∇(cα − ρα)

−ε2∆cα = α(ρα − cα)

in D × R>0,

vα = 0, ∇cα · n = 0 in ∂D × R>0

Energy inequality:

d

dt

(F ε,αRelax [ρα(·, t), cα(·, t)] +

∫D

1

2ρα(x, t)|vα(x, t)|2 dx

)≤ 0

May 23, 2014 13


Low Order Relaxation Approximation:

ραt + div(ραvα) = 0

(ραvα)t + div(ραvα ⊗ vα + p(ρα)I) = εdiv(Tα)

+αρα∇(cα − ρα)

−ε2∆cα = α(ρα − cα)

in D × R>0,

vα = 0, ∇cα · n = 0 in ∂D × R>0

Asymptotic Limits:

(i) Korteweg limit α→∞ towards NSK system

(ii) Sharp-interface limit ε→ 0 towards Euler system

May 23, 2014 13


Relaxation Approximation and Hyperbolicity: (d=2)

ρt + div(ρv) = 0

(ρv)t + div(ρv ⊗ v + p(ρ)I) = div(T) + αρ∇(c − ρ)

−ε2∆c = α(ρ− c)ρ

α2 b

α1

β1

β2

p

May 23, 2014 14


Relaxation Approximation and Hyperbolicity: (d=2)

ρt + div(ρv) = 0

(ρv)t + div(ρv ⊗ v +

(p(ρ) +

α

2ρ2︸︷︷︸

=:pα(ρ)

))= εdiv(T) + αρ∇c

−ε2∆c = α(ρ− c)

The new first-order part is strictly hyperbolic for α > max{−W ′′(ρ)}:

λ1(ρ, ρv) = v · ξ −√

p′α(ρ), λ2(ρ, ρv) = v · ξ, λ3(ρ, ρv) = v · ξ +√pα(ρ)

Note: Relaxation approximation has the structure of the shallow-watersystem with bottom topography c .

May 23, 2014 15

Relaxation Approximation...it works

Qualitative Test/Bubble Merging:

t = 0 t = 0.25

t = 2.0 t = 2.3 t = 40.0

May 23, 2014 16

Relaxation Approximation...it is more robust I

Sharp Interface Limit ε→ 0:Initial configuration:

ρ0(x) =

{1.8, x ∈ (0.3, 0.6)0.3, else

v0(x) = 0.∆x = 0.005, α = 100.

Density for the NSK System at t = 1.72:

computation crashed

ε = 0.01 ε = 0.001 ε = 0.0001

May 23, 2014 17


Sharp Interface Limit ε→ 0:Initial configuration:

ρ0(x) =

{1.8, x ∈ (0.3, 0.6)0.3, else

v0(x) = 0.α = 100.

Energy evolution for the NSK system with ε = 0.01:

May 23, 2014 17


Density for the relaxation approximation at t = 1.72:

ε = 0.01 ε = 0.001 ε = 0.0001

Energy evolution for the relaxation approximation with ε = 0.01:

May 23, 2014 18

Relaxation Approximation...it is more robust II

Larger Density Ratio:Initial configuration:

ρ0(x) =

{36, x ∈ (0.3, 0.6)6.0, else

v0(x) = 0.∆x = 0.00025, α = 400, ε = 0.01.

Density evolution for the relaxation approximation:

t = 0.01 t = 0.2 t = 0.5

May 23, 2014 19


An Energy-Dissipative Discretization:The first-order-part of the relaxation approximation for d = 1:(

ρm := ρv

)t

+ fα

((ρm

))x

= 0⇔ ut + fα(u)x = 0

It is equipped with the entropy/entropy flux pair

ηα(ρ,m) = Wα(ρ) +m2

2ρ, qα(ρ,m) =

m

ρ

(ηα(ρ,m) + pα(ρ)

).

The convexity of ηα implies that the mapping

u 7→ w(u) := ∇ηα(u)

is one-to-one. The relaxation approximation can be rewritten in the form

u(w)t + gα(w)x = 0.

May 23, 2014 20


Theorem (Tadmor ’84, ’03)For the relaxation approximation there is a 2-parameter family ofnumerical fluxes g∗ = g∗(w, z) such that the scheme

u′j(t) = − 1

∆x

(g∗j+ 1

2(t)− g∗

j− 12(t)), g∗

j+ 12(t) = g∗(wj(t),wj+1(t)),

is entropy-conservative, i.e. there is a numerical entropy flux q∗ = q∗(w, z)that satisfies q∗(w,w) = qα(w) with

ηα(uj(t))′ = − 1

∆x

(q∗j+ 1

2(t)− q∗

j− 12(t))

for all t ∈ (0,T ) and j ∈ Z.

May 23, 2014 21


Theorem: (Neusser&R.’14)There exists at least one Tadmor flux g∗ = g∗(w, z) such that for

h∗(w, z) =

g∗

1 (w,z)w2

: w2 6= 0,

W ′−1α

(w1 + w2

2

): w2 = 0.

the solution of

u′ +1

∆x

(g∗j+ 1

2− g∗

j− 12

)=

α

∆x

(0

h∗(wj ,wj−1)(cj+1 − cj

)),

ε2

∆x2(cj+1 − 2cj + cj−1) = α(cj − ρj)

satisfies the energy equality

d

dt

∑j∈Z

(m2

j

2ρj+ W (ρj) +

α

2(ρj − cj)

2 +ε2

2∆x2(cj+1 − cj)

2

)= 0.

May 23, 2014 22

3) Analysis of the Korteweg Limit

May 23, 2014 23

Korteweg Limit

A Numerical Example: Let d = 1, ε = 0.01, and ∆x = 0.00125

ρ0(x) =

{1.8 : x ∈ (0.3, 0.6) ∪ (0.85, 1.05)0.3 : else

, v0(x) = 0

Evolution for relaxation approximation uα∆x

t = 0.02 t = 0.04 t = 4

Numerical convergence towards NSK solution u∆x

α 1 5 10 100 1000

‖uα∆x − u∆x‖L2 1.039e-1 3.333e-2 1.802e-2 1.909e-3 1.683e-4

EOC - 0.708 0.885 0.975 1.055

May 23, 2014 24

Korteweg Limit

Conservation law with screened Poisson equation: (Corli&R. ’12)Let f ∈ C 1(R) with |f ′(u)| ≤ L and consider the scalar problem

uαt + f (uα)x = εuαxx − α(uα − cα)x in R× (0,T )

−ε2cαxx = α(uα − cα) in R× (0,T )

uα(·, 0) = u0 in R

(Pα)

A-priori estimates for t ∈ (0,T ):

(i) ‖∂kx cα(·, t)‖L2(R) ≤ ‖∂kx uα(·, t)‖L2(R)

(ii) ‖uα(., t)‖2L2(R) + 2ε‖uαx ‖

2L2(R×(0,t)) = O(1)

(iii) ddt

∫R(F (uα(·, t)) + α

2 (uα(·, t)− cα(·, t))2 + ε2 (cαx (·, t))2

)≤ 0, F ′ = f

May 23, 2014 25

Korteweg Limit

Theorem

Let u0 ∈ H3(R) ∩W 3,∞(R) be given.Then we have

(i) For each ε, α > 0 there is a unique classical solution of (Pα).

(ii) There exists a subsequence of {(uα, cα)}α>0 and a functionu ∈ L2(R× (0,T )) such that

uα → u, λα → u in L2loc(R× (0,T )) for α→∞.

(iii) The function u is a weak solution of

ut + f (u)x = εuxx + ε2uxxx in R× (0,T )

u(·, 0) = u0 in R(P∞)

May 23, 2014 26

Korteweg Limit

Outline of Proof:

a. Use the (α-independent) a-priori estimate (ii) to show

‖uα‖L2(0,T ;H3) = O(1).

b. Straightforward calculations lead to

‖uαt ‖L2(R×(0,T )) ≤ C (‖uα‖L2(0,T ;H2)) + α‖uαx − cαx ‖L2(R×(0,T ))

= O(1) + ε2‖cαxxx‖L2(R×(0,T ))

A-priori estimate (i) and Step a. imply

‖uαt ‖L2(R×(0,T )) = O(1).

c. Apply Lions-Aubin lemma to deduce uα → u in L2loc(R× (0,T )).

d. Apply a-priori estimate (iii) to deduce uα − cα → 0 a.e.

May 23, 2014 27

Korteweg Limit

Extensions of the Analysis:

Conservation law with screened heat equation (Corli&R.&Schleper ’14)

uαt + f (uα)x = εuαxx − α(uα − cα)xεαc

αt − ε2cαxx = α(uα − cα)

NSK system in Lagrangean coordinates (R.&Viorel ’14)

ταt − vαx = 0vαt + p(τα)x = εvαxx − α(τα − cα)x

ε2cαxx = α(uα − cα)

Note: Korteweg limit for NSK systems in Charve ’13 and Giesselmann ’14.

May 23, 2014 28

Korteweg Limit

Note on Relaxation Approximation for other Phase SeparationModels:

Cahn-Hilliard equation

ut = ∆( 1

ε2W ′(u)−∆u

)

Low Order Relaxation Approximation

uαt = ∆(

1ε2W

′(uα)− α(cα − uα))

= ∆( 1

ε2W ′(uα) + αuα

)︸︷︷︸monotone increasing

−αcα

−∆cα = α(uα − cα)

May 23, 2014 29

Summary and Outlook

Introduction of a low-order NSK-type system for liquid-vapour flow

The relaxation approximation is numerically more efficient (time-steprestriction, use of hyperbolic solvers for convection-dominatedregimes)

Rigorous analysis for the Korteweg limit for scalar model problem

Thanks for your attention

May 23, 2014 30

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