level set based finite volume discretization for two-phase ...dlogashenko/dload/bubbletalk.pdf ·...

Level Set Based Finite Volume Discretizationfor Two-Phase Flows

Dr. Peter Frolkovic, Dr. Dmitry Logashenko, Christian Wehner

Goethe Center for Scientific Computing (Prof. Dr. G. Wittum)

Sept. 2009

Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 1 / 21

1 Model

2 Discretization

3 Reinitialization

4 Numerical results

5 Conclusions

Rising bubble

Fluid 1:density 1, viscosity 1

Fluid 2:density 2 < 1, viscosity 2

!g

Rising bubble

Fluid 1:density 1, viscosity 1

Fluid 2:density 2 < 1, viscosity 2

Interface:surface tension factor

!g

Rising bubble

!g

2 (u2,t + u2u2) = T2 + "g u2 = 0

1 (u1,t + u1u1) = T1 + "g u1 = 0

u1 = u2(T2 T1)n = n

( : mean curvature of int)

int

Ti = piI + i(ui + (ui)T

)

Discontinuities

Let u = un + vt. Then

[u] := u1 u2 = 0

[u n] = [u t] = [v t] = 0

[p] = + 2[]u n[v n] = []u t

If [] = 0 then[u] = 0, [u] = 0

[p] =

Discontinuities

[u] := u1 u2 = 0

[u n] = [u t] = [v t] = 0

[p] = + 2[]u n[v n] = []u t

If [] = 0 then[u] = 0, [u] = 0

[p] =

Discontinuities

[u] := u1 u2 = 0

[u n] = [u t] = [v t] = 0

[p] = + 2[]u n[v n] = []u t

If [] = 0 then[u] = 0, [u] = 0

[p] =

Discontinuities

[u] := u1 u2 = 0

[u n] = [u t] = [v t] = 0

[p] = + 2[]u n[v n] = []u t

If [] = 0 then[u] = 0, [u] = 0

[p] =

Features

Moving inner boundary (the interface)

Shape-dependent factors in the BC at the interface(curvature)

Non-smooth solution (considered on the whole domain)

Features

Methods

By the representation of the interface:

Moving gridsVOF-methodsLevel-Set methods

By the discretization of the curvature and the discontinuities:

SmoothingWith no smoothing

Methods

Moving grids

VOF-methodsLevel-Set methods

Methods

Moving gridsVOF-methods

Level-Set methods

Methods

Smoothing

With no smoothing

Methods

Smoothing vs. no smoothing

Sufrace tension force: f (x) = (x) n(x) (x).

Level-set formulation

Level-set function: a smooth function

int = {x : (x) = 0}

((x) > 0 outside, (x) < 0 inside.)

Level-set equation:t = u

Initial condition: for ex. the distance function.Coupling with the fluid dynamics: u and int.

int = {x : (x) = 0}

((x) > 0 outside, (x) < 0 inside.)Level-set equation:

t = u

Initial condition: for ex. the distance function.Coupling with the fluid dynamics: u and int.

int = {x : (x) = 0}

t = u Initial condition: for ex. the distance function.

Coupling with the fluid dynamics: u and int.

int = {x : (x) = 0}

t = u Initial condition: for ex. the distance function.Coupling with the fluid dynamics: u and int.

Computation of the curvature

(x) = (x) H(x , ) ((x))T

(x)32,

where

H(x , ) =

(x2x2(x) x1x2(x)x1x2(x) x1x1(x)

)

Grid h. uh : h R2, ph : h R.

Grid h/2. h/2 : h/2 R.

Quadratic interpolation of h/2 on h.

Time stepping + Operator splitting

ukh , pkh and

kh/2: grid functions for the velocity, the pressure

and the level-set function in time step k .

u0h, p0h and

0h/2 are given.

1: for k = 1, . . . do begin

2: Compute ukh from uk1h for the phase interface

given by k1h/2 using the discretization

of the Navier-Stokes equations;

3: Compute kh/2 from k1h/2 and u

kh using

the discretization of the level-set equation;4: end

ukh , pkh and

u0h, p0h and

0h/2 are given.

1: for k = 1, . . . do begin

kh using

ukh , pkh and

u0h, p0h and

0h/2 are given.

1: for k = 1, . . . do begin

kh using

NS equation: Ghost fluid method

Assume [] = 0.

1

2

int

x1

x2

Assume [] = 0.

1

2

int

x1

x2

Stored: p1(x1) and p2(x2).

Assume [] = 0.

1

2

int

x1

x2

p1(x2) p2(x2) = Green FV: p2(x2) is stored, p1(x2) is computed.

Assume [] = 0.

1

2

int

x1

x2

p1(x1) p2(x1) = Blue FV: p1(x1) is stored, p2(x1) is computed.

Extended approximation spaces

([] 6= 0.)

In intersected element e:

Additional basis functions:

ph = ph,lin + Pe Ne(p),

uh = uh,linn + vht,

wherevh = vh,lin + V

e Ne(v)

Pe and V e are eliminated using the discretized interfaceconditions

[p] = + 2[]u n[v n] = []u t

([] 6= 0.)In intersected element e:

uh = uh,linn + vht,

e Ne(v)

[p] = + 2[]u n[v n] = []u t

uh = uh,linn + vht,

e Ne(v)

[p] = + 2[]u n[v n] = []u t

uh = uh,linn + vht,

e Ne(v)

[p] = + 2[]u n[v n] = []u t

Simulation

Simulation with the reinitialization

Time stepping + Reinitialization

The reinitialization should preserve the position and thecurvature of the interface.

u0h, p0h and

0h/2 are given.

1: for k = 1, . . . do begin

3a: Compute kh/2 from k1h/2 and u

kh using

the discretization of the level-set equation;3b: Reinitialize kh/2 (not in every time step);

4: end

u0h, p0h and

0h/2 are given.

1: for k = 1, . . . do begin

kh using

4: end

u0h, p0h and

0h/2 are given.

1: for k = 1, . . . do begin

kh using

4: end

dcqReusken_cm-reggrid-lev6.aviMedia File (video/avi)

Conclusions

Done: Extending the single-phase Navier-Stokes discretizationto the two-phase case.

Done: Extended approximation spaces to capture the jumps.

Done: Adaptively created fine grid for the level-set function.

Done: No explicit reconstruction of the interface.

ToDo: Better coupling.

Conclusions

Thank youfor your attention!

ModelDiscretizationReinitializationNumerical resultsConclusions

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