Post on 20-May-2015
TOPICAL PROBLEMS IN FLUID MECHANICS 2006Institute of thermomechanics AS CR, Prague
A NEW FREE SURFACE CAPTURING FINITE A NEW FREE SURFACE CAPTURING FINITE ELEMENTS METHOD FOR ANALYSIS OF SHIP ELEMENTS METHOD FOR ANALYSIS OF SHIP
HYDRODYNAMICSHYDRODYNAMICS
Aleix Valls Tomas
International Centre for Numerical Methods in Engineering
(CIMNE), Spain.
aleix@cimne.upc.edu
TOPICAL PROBLEMS IN FLUID MECHANICS 2006Institute of thermomechanics AS CR, Prague
1. INTRODUCTION• The innovation of this method is the application of domain
decomposition techniques to improve the accuracy of capturing algorithm for free surface (level set) as well as the resolution of governing equations in the interface between two fluids.
1 2p h g h h g
Fluid 1
Fluid 2
1 1 1 1 1
2 2 2 2 2
, , ,, , ,
, , ,
p xp
p x
uu
u
1p hg
0
h
h
TOPICAL PROBLEMS IN FLUID MECHANICS 2006Institute of thermomechanics AS CR, Prague
2. PROBLEM STATEMENT
, 1, 2,3
0 1,2,3
ijii j i
j i j
i
i
u pu u b i j
t x x x
ui
x
1 1 1 1 1
2 2 2 2 2
, , ,, , ,
, , ,
p xp
p x
uu
u 1 2int
1 2
,
, ,
u
ij ij p
nj j j ij i j ij i
on
p p and n t on
u n u n g t n s t on
u u
Navier-Stokes equationsTwo-fluids
Boundary Conditions
TOPICAL PROBLEMS IN FLUID MECHANICS 2006Institute of thermomechanics AS CR, Prague
3. LEVEL SET METHOD (LSM)
1
1 2
2
0
, 0
0
t
x
x x
x
| , 0t t x x
k
n
Interface localization (capturing technique)
Level Set Function
Some interface properties
TOPICAL PROBLEMS IN FLUID MECHANICS 2006Institute of thermomechanics AS CR, Prague
3. LEVEL SET METHOD (LSM)
0D
Dt t
u
Since the interface moves with the fluid, the evolution of is defined by the following hyperbolic equation:
,t x
To solve a problem involving an interface is necessary to integrate the Navier-Stokes equations coupled with transport equation for Level Set function.
TOPICAL PROBLEMS IN FLUID MECHANICS 2006Institute of thermomechanics AS CR, Prague
3. LEVEL SET METHOD (LSM)For numerical considerations is quite common to use a signed distance to the interface as Level Set function.
Cut at Y=2.5
-6
-5
-4
-3
-2
-1
0
1
2
3
0 2 4 6 8 10
X Coord.
Level Set Function
Signed Distance
Jump
Fluid2
Fluid1Cut Y=2.5
TOPICAL PROBLEMS IN FLUID MECHANICS 2006Institute of thermomechanics AS CR, Prague
4. FIC STABILIZED PROBLEM
_ 0A Bq q =
2
2 02
dq hd qdx dx
- = 1 2h d d= -
Taylor until second order
FIC stabilizationterm
Characteristic Length
Balance of fluxes
TOPICAL PROBLEMS IN FLUID MECHANICS 2006Institute of thermomechanics AS CR, Prague
4. FIC STABILIZED PROBLEM
, 1, 2,3i
ijim i j i
j i j
u pr u u b i j
t x x x
10 , 1,2,3
2
10 1,2,3
2
i
i i j
mm m
j
dd j
j
rr h on i j
x
rr h on j
x
The stabilized Finite Increment Calculus (FIC) form of governing differential equations
1,2,3id
i
ur i
x
1 2
,
1
2
1 1, ,
2 2
i
i i
u
ij ij j j m p
nj j j ij i j j m i j ij i j j m i
u u on
p p and n h n r t on
u n u n g h n r g t n s h n r s t on
TOPICAL PROBLEMS IN FLUID MECHANICS 2006Institute of thermomechanics AS CR, Prague
5. ALE FORMULATIONConvection derivative
( ) ( ) ( )mj j
j
Du u
Dt t x
Where uj m is the relative velocity between the local reference system and the real velocity of the particle
ALE formulation of the residuals
, 1, 2,3
1,2,3
1,2,3
i
j ijmi im j j i
j j i j
id
i
mj j
j
uu u pr u u u i j
t x x x x
ur i
x
r u u jt x
TOPICAL PROBLEMS IN FLUID MECHANICS 2006Institute of thermomechanics AS CR, Prague
Fluid2
6. OVERLAPPING DOMAIN DECOMPOSITION TECHNIQUE
, 0t x
, 0t xFluid1 Ω3
Ω4
Ω5
Ω1
Ω2
Interface
, 0t x
TOPICAL PROBLEMS IN FLUID MECHANICS 2006Institute of thermomechanics AS CR, Prague
6. OVERLAPPING DOMAIN DECOMPOSITION TECHNIQUE
Governing equations for Fluid 1 in domain Ω1 are
1 1 1 11
1
1 1
1 1
1 2 2
, , , , ,
, 0
,0 , ,
1
1
1
t 1 t
t
0
1 2 1
c a b p l v
b q q Q
p n n p σκ on
1 1 1
1
1 0
u v u u v u v v v V
u
u x v u x v v V
u u , τ τ
Governing equations for Fluid 2 in domain Ω2 are
2 2 2 22
2
2 2
2 , , , , ,
, 0
,0 , ,
2
2
2
t 2 2 2 2 2 t
t
2 0
c a b p l k
b q q Q
0
u v u u v u v v v v v V
v
u x v u x v v V
New Terms allow to define correct pressure
in the interface
TOPICAL PROBLEMS IN FLUID MECHANICS 2006Institute of thermomechanics AS CR, Prague
6. OVERLAPPING DOMAIN DECOMPOSITION TECHNIQUE
• Given two velocity fields defined in the overlapping sub domains Ω1 and Ω2 , respectively, and two pressure fields at time tn and a guess for the unknowns at an iteration i-1 and at time tn+1, find and
at time tn+1, by solving previous discrete variational problem.
General Idea of computational approach
1 1,1 , ,2 ,,n nh h u h h uu V u V
,1 ,1 ,2 ,2,n nh h h hp Q p Q
1 2
, ,,1 , ,2 ,,n i n ih h u h h uu V u V
, ,,1 ,1 ,2 ,2,n i n ih h h hp Q p Q
TOPICAL PROBLEMS IN FLUID MECHANICS 2006Institute of thermomechanics AS CR, Prague
7. Examples2-D Sloshing 2-D Sloshing
TOPICAL PROBLEMS IN FLUID MECHANICS 2006Institute of thermomechanics AS CR, Prague
7. Examples2-D Sloshing 2-D Sloshing
TOPICAL PROBLEMS IN FLUID MECHANICS 2006Institute of thermomechanics AS CR, Prague
7. Examples2-D Sloshing 2-D Sloshing
TOPICAL PROBLEMS IN FLUID MECHANICS 2006Institute of thermomechanics AS CR, Prague
7. Examples2-D Dam Break2-D Dam Break
waterwater
AirAir
DamDam
TOPICAL PROBLEMS IN FLUID MECHANICS 2006Institute of thermomechanics AS CR, Prague
7. Examples2-D Dam Break2-D Dam Break
TOPICAL PROBLEMS IN FLUID MECHANICS 2006Institute of thermomechanics AS CR, Prague
7. Examples2-D Column Tank 2-D Column Tank
waterwater
AirAir
Solid WallsSolid Walls
TOPICAL PROBLEMS IN FLUID MECHANICS 2006Institute of thermomechanics AS CR, Prague
7. Examples2-D Column Tank 2-D Column Tank
TOPICAL PROBLEMS IN FLUID MECHANICS 2006Institute of thermomechanics AS CR, Prague
7. Examples2-D Filling of a glass mould2-D Filling of a glass mould
TOPICAL PROBLEMS IN FLUID MECHANICS 2006Institute of thermomechanics AS CR, Prague
7. Examples2-D Loading of a tank 2-D Loading of a tank
TOPICAL PROBLEMS IN FLUID MECHANICS 2006Institute of thermomechanics AS CR, Prague
7. Examples3-D tank sloshing 3-D tank sloshing