level set based finite volume discretization for two-phase ...dlogashenko/dload/bubbletalk.pdf ·...
TRANSCRIPT
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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
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1 Model
2 Discretization
3 Reinitialization
4 Numerical results
5 Conclusions
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 2 / 21
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Rising bubble
Fluid 1:density 1, viscosity 1
Fluid 2:density 2 < 1, viscosity 2
!g
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 3 / 21
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Rising bubble
Fluid 1:density 1, viscosity 1
Fluid 2:density 2 < 1, viscosity 2
Interface:surface tension factor
!g
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 3 / 21
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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
)
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 3 / 21
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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] =
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 4 / 21
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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] =
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 4 / 21
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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] =
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 4 / 21
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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] =
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 4 / 21
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Features
Moving inner boundary (the interface)
Shape-dependent factors in the BC at the interface(curvature)
Non-smooth solution (considered on the whole domain)
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 5 / 21
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Features
Moving inner boundary (the interface)
Shape-dependent factors in the BC at the interface(curvature)
Non-smooth solution (considered on the whole domain)
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 5 / 21
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Features
Moving inner boundary (the interface)
Shape-dependent factors in the BC at the interface(curvature)
Non-smooth solution (considered on the whole domain)
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 5 / 21
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Methods
By the representation of the interface:
Moving gridsVOF-methodsLevel-Set methods
By the discretization of the curvature and the discontinuities:
SmoothingWith no smoothing
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 6 / 21
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Methods
By the representation of the interface:
Moving grids
VOF-methodsLevel-Set methods
By the discretization of the curvature and the discontinuities:
SmoothingWith no smoothing
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 6 / 21
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Methods
By the representation of the interface:
Moving gridsVOF-methods
Level-Set methods
By the discretization of the curvature and the discontinuities:
SmoothingWith no smoothing
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 6 / 21
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Methods
By the representation of the interface:
Moving gridsVOF-methodsLevel-Set methods
By the discretization of the curvature and the discontinuities:
SmoothingWith no smoothing
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 6 / 21
-
Methods
By the representation of the interface:
Moving gridsVOF-methodsLevel-Set methods
By the discretization of the curvature and the discontinuities:
Smoothing
With no smoothing
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 6 / 21
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Methods
By the representation of the interface:
Moving gridsVOF-methodsLevel-Set methods
By the discretization of the curvature and the discontinuities:
SmoothingWith no smoothing
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 6 / 21
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Smoothing vs. no smoothing
Sufrace tension force: f (x) = (x) n(x) (x).
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 7 / 21
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Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 8 / 21
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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.
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 9 / 21
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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.
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 9 / 21
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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.
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 9 / 21
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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.
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 9 / 21
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Computation of the curvature
(x) = (x) H(x , ) ((x))T
(x)32,
where
H(x , ) =
(x2x2(x) x1x2(x)x1x2(x) x1x1(x)
)
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 10 / 21
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Computation of the curvature
Grid h. uh : h R2, ph : h R.
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 10 / 21
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Computation of the curvature
Grid h/2. h/2 : h/2 R.
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 10 / 21
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Computation of the curvature
Quadratic interpolation of h/2 on h.
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 10 / 21
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Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 11 / 21
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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
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 12 / 21
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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
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 12 / 21
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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
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 12 / 21
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NS equation: Ghost fluid method
Assume [] = 0.
1
2
int
x1
x2
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 13 / 21
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NS equation: Ghost fluid method
Assume [] = 0.
1
2
int
x1
x2
Stored: p1(x1) and p2(x2).
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 13 / 21
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NS equation: Ghost fluid method
Assume [] = 0.
1
2
int
x1
x2
p1(x2) p2(x2) = Green FV: p2(x2) is stored, p1(x2) is computed.
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 13 / 21
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NS equation: Ghost fluid method
Assume [] = 0.
1
2
int
x1
x2
p1(x1) p2(x1) = Blue FV: p1(x1) is stored, p2(x1) is computed.
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 13 / 21
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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
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 14 / 21
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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
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 14 / 21
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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
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 14 / 21
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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
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 14 / 21
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Simulation
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 15 / 21
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Simulation
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 15 / 21
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Simulation
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 15 / 21
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Simulation with the reinitialization
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 16 / 21
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Simulation with the reinitialization
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 16 / 21
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Simulation with the reinitialization
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 16 / 21
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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
2: Compute ukh from uk1h for the phase interface
given by k1h/2 using the discretization
of the Navier-Stokes equations;
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
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 17 / 21
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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
2: Compute ukh from uk1h for the phase interface
given by k1h/2 using the discretization
of the Navier-Stokes equations;
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
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 17 / 21
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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
2: Compute ukh from uk1h for the phase interface
given by k1h/2 using the discretization
of the Navier-Stokes equations;
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
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 17 / 21
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Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 18 / 21
dcqReusken_cm-reggrid-lev6.aviMedia File (video/avi)
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Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 19 / 21
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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.
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 20 / 21
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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.
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 20 / 21
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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.
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 20 / 21
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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.
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 20 / 21
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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.
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 20 / 21
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Thank youfor your attention!
Frolkovic, Logashenko, Wehner LS+FV for Two-Phase Flows 21 / 21
ModelDiscretizationReinitializationNumerical resultsConclusions