A New Free Surface Capturing Finite Elements Methods

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.

[email protected]


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


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


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


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.


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


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


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


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


Fluid2

6. OVERLAPPING DOMAIN DECOMPOSITION TECHNIQUE

, 0t x

, 0t xFluid1 Ω3

Ω4

Ω5

Ω1

Ω2

Interface

, 0t x



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



• 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


7. Examples2-D Sloshing 2-D Sloshing


7. Examples2-D Dam Break2-D Dam Break

waterwater

AirAir

DamDam


7. Examples2-D Dam Break2-D Dam Break


7. Examples2-D Column Tank 2-D Column Tank

waterwater

AirAir

Solid WallsSolid Walls


7. Examples2-D Column Tank 2-D Column Tank


7. Examples2-D Filling of a glass mould2-D Filling of a glass mould


7. Examples2-D Loading of a tank 2-D Loading of a tank


7. Examples3-D tank sloshing 3-D tank sloshing

A New Free Surface Capturing Finite Elements Methods

Documents

Transcript of A New Free Surface Capturing Finite Elements Methods