A Numerical Model A Numerical Model for Multiphase Flow, I:for Multiphase Flow, I:

The Interface Tracking The Interface Tracking AlgorithmAlgorithm

Frank BierbrauerFrank Bierbrauer

A Numerical Model for Multiphase Flow

• Part I, Kinematics: The Interface Tracking Algorithm (Marker-Particle Method)

• Part II, Dynamics: The Navier-Stokes Solver

Contents

• What should an Interface “Tracking” Algorithm be able to do ?

• Multiphase Flow, what does phase really mean ?• Interface “Tracking”• The Marker-Particle Method• Benchmark Tests of the MP Method• Conclusions• Associated Problems

Solid-Liquid Impact

Liquid Jets

Jet Breakup

Jet-Liquid Impact

Jet-Solid Impact

Melting and Mixing

Phase Change: melted glass

Fluid Mixing

Droplets and Bubbles

Sessile drops

Bubbles

Droplet Pinch Off

Droplet Collisions and Shock Impact

Two Droplets Colliding

Shock Wave/Droplet Impact

Droplet/Liquid Impact

Splash Corona & Rayleigh Jet Formation

Capillary Waves

Secondary Droplet Expulsion

Collision of Two Droplets

Droplet Splash

Example: Three-Phase Flow

Fluid phase 1 – droplet, fluid phase 2 – air, fluid phase 3 – other fluid

Fluid Phases

• Fluid Phases defined by individual densities and viscosities

• Can define physical properties such as density and viscosity as a single field varying in space, the so-called one-field formulation

• Then the interfaces between fluid phases represent a discontinuity in density or viscosity

• Can define these phases by a phase indicator function C

Volume Fraction

C1 = 0.45C2 = 0.00C3 = 0.00

C1 = 0.00C2 = 0.32C3 = 0.00

C1 = 0.00C2 = 0.00C3 = 0.23

The phase indicator function is often the volume fraction occupied by the fluid (m) in the volume V: Cm = Vm/V so thatC1 + C2 + C3 = 1 in V

Example: Grid Volume Fraction

Volume fraction information within grid cells

C1 – blue fluid, C2 – yellow fluid

C1 = 0C2 = 1

C1 = 0C2 = 1

C1 = 0.2C2 = 0.8

C1 = 0.7C2 = 0.3

C1 = 0.95C2 = 0.05

C1 = 1C2 = 0

C1 = 0.7C2 = 0.3

C1 = 0.3C2 = 0.7

One-Field Formulation• For example, for 3 phase flow the density and viscosity

fields are:• density: (x,y,t) =1C1(x,y,t) + 2C2(x,y,t) + 3C3(x,y,t)• viscosity: (x,y,t) =1C1(x,y,t) + 2C2(x,y,t) + 3C3(x,y,t)• so that

• Where the i and i are the constant viscosities and densities within each fluid phase

• In general for M fluid phases we have

M

m

mm tyxCtyx

1

),,,(),,(

(x,y,t) =1C1(x,y,t) + 2C2(x,y,t) + 3[1-C1(x,y,t)-C2(x,y,t)]

(x,y,t) =1C1(x,y,t) + 2C2(x,y,t) + 3[1-C1(x,y,t)-C2(x,y,t)]

M

m

mm tyxCtyx

1

),,(),,(

Interface Tracking

• Surface Tracking– The interface is explicitly tracked– The interface is represented as a series of

interpolated curves– A sequence of heights above a reference line, e.g.

level set method– A series of points parameterised along the curve,

e.g. front tracking

Surface Tracking

1. Points parameterised along a curve (x(s),y(s))2. Sequence of heights h above a reference line

Interface Capturing

• Volume Tracking– The interface is only implicitly “tracked”, it is

“captured”– The interface is the contrast created by the

difference in phase, e.g. MAC method, Marker-Particle method

– Or it can be geometrically re-constructed, e.g. VOF methods SLIC, PLIC

Interface Capturing

Volume Tracking

Eulerian Methods• Fixed Grid methods

– There is an underlying grid describing the domain, typically a rectangular mesh, e.g. FDM

– The interface does not generally coincide with a grid line or point

– Advantages: interface can undergo large deformation without loss of accuracy, allows multiple interfaces

– Disadvantage: the interface location is difficult to calculate accurately

Lagrangian Methods

• These methods are characterised by a coordinate system that moves with the fluid, e.g. fluid particles

• Advantages: accurately specifies material interfaces, interface boundary conditions easy to apply, can resolve fine structures in the flow

• Disadvantage: strong interfacial deformation can lead to tangled Lagrangian meshes and singularities

• Examples: SPH, LGM, PIC

Eulerian-Lagrangian Methods

• Makes use of aspects of both Eulerian and Lagrangian methods

• Particle-Mesh methods– use an Eulerian fixed grid to store velocity and

pressure information– Use Lagrangian particles to keep track of fluid

phase and thereby density and viscosity

The Marker-Particle Method• Define a fixed Eulerian mesh made up of computational

cells with centres

• In Xmin < x < Xmax, Ymin< y < Ymax, x1/2 = Xmin, y1/2 = Ymin, x = (Xmax-Xmin)/I, y = (Ymax-Ymin)/J

• Within each computational cell assign a set of particles with positions (xp, yp)

yjyy

xixx

j

i

2

1

2

1

2/1

2/1

Computational Cell & Initial Particle Configuration

Fluid Colour

• Each fluid phase (m) has a set of marker particles (p) located at position (xp, yp)

• Every marker particle of the mth set is assigned a colour such that

•

ppmm

p yxCC ,

mp

mpC m

p fluidin locatednot is particle if0

fluidin located is particle if1

Initial Particle Colours

• For example, for those particles of the 2nd phase:

Particle Velocities

• Particle velocities up = u(xp,yp) are interpolated from the nearest four grid velocities ui,j, ui+1,j, ui,j+1, ui+1,j+1

Grid-to-Particle Velocity Interpolation

jiji

jiji

jiji

jiji

Δy

y-y

Δx

xx

Δy

y-y

Δx

xx

Δy

y-y

Δx

xx

Δy

y-y

Δx

xxyx

,,1

1,1,

11

11),(

uu

uuu

Interpolation Function

or

Where the interpolation function S is given by

otherwise

y

yy

x

xxif

y

yy

x

xxS

jiji

0

1,011

J

ji,j

I

iji )y,yxS(x(x,y)

1 1

uu

Particle Kinematics

• Lagrangian particle advection: solve u = dx/dt which moves fluid particles along characteristics with velocity u

• Predictor

• Corrector

np

np

np

np

np

np

np

np

yxvt

yy

yxut

xx

,2

,,2

2/1

2/1

2/12/11

2/12/11

,

,,

np

np

np

np

np

np

np

np

yxtvyy

yxtuxx

Particle Boundary Conditions

• No-Slip: On approaching the boundary the fluid velocities there approach zero. The simplest way to impose this boundary condition is to reflect the particle back into the domain by the amount it has exceeded it

• Periodic: For periodic conditions the particle must exit the domain and appear out of the opposite face by the amount it exceeded the first boundary

Volume Fraction Update

• Require the updated grid volume fraction to update the grid densities and viscosities

• Use the same interpolation function, S, as defined previously

• Usually, particles-to-grid

interpolation involves many

irregularly placed particles,

in excess of four

1,,

nmjiC

Volume Fraction Interpolation• This requires a normalisation of the interpolation• Then, for each fluid m at the next time step n+1

N

pj

npi

np

N

p

mpj

npi

np

nmji

yyxxS

CyyxxS

C

1

11

1

11

1,,

,

,

Algorithm

1. Initialisation at t = 0

1. Assign a set number of particles per cell with a total number N in the domain

2. Assign an initial particle colour for each fluid

3. Construct initial grid cell volume fractions 0,

,m

jiC

mpC

Algorithm

2. For time steps t > 01. Given un and time centred grid velocities un+1/2 interpolate

velocities to all particles obtaining

2. Solve the equation of motion u = dx/dt using the predictor-corrector strategy already mentioned

3. Interpolate the new grid volume fractions from the advected particle colours

4. Update density and viscosity using the new volume fractions

5. Store old time particle positions as well as particle colour. Increment the time step n -> n+1 and go to step 1. above

2/1, np

np uu

1,,

nmjiC

mpC

Benchmark Tests• Two-Phase Flow test, droplet and

ambient fluid of different densities and viscosities in a unit domain. Let the droplet have volume fraction C = 1 and the ambient fluid have C= 0 (C = C1, C2 = 1 - C1).

• Apply various velocity fields up to time t = T/2 to the problem of a fluid cylinder, of radius R = 0.15, located at (0.50,0.75)

• Reverse velocity field at t = T/2 and measure difference between initial and final droplet configuration at t = T (T = total time)

Error Measures• Use a 642 grid (x = y = 1/64 = 0.016) with either 4 or 16

particles per cell (ppc)• At t = T measure droplet volume/mass given by

• Measure changes in transition width, the minimised, +ve, distance over which the volume fraction changes from C = 1 (droplet), in grid cell (x,y), to C = 0 (surrounding fluid), in grid cell (X,Y)

• Obtain relative percentage errors

1

0

1

0

),( dxdyyxC

0)()(min 22

1,;,0

yYxX

YyXx

Benchmark TestsTest Type Velocity Field Specified Field

Simple translation u(x,y) = (1,0)

Advection rotation u(x,y) = (y-1/2,-(x-1/2))

Topology shearing flow u(x,y) = (-sin2x sin2y, sin2x sin 2y)

Change vortex u(x,y) = (sin 4 (x+1/2) sin4 (y+1/2), cos 4 (x+1/2) cos4 (y+1/2))

Expected Shearing Flow Effect

Expected Vortex Field Effect

Translation: relative % errors

Rotation: relative % errors

Shearing Flow: relative % errors

Vortex: relative % errors

Translation: transition width

Rotation: transition width

Shearing Flow: transition width

Vortex: transition width

Relative Errors L1 norm

Conclusions

• Tests have shown the MP method can accurately “track” multiple fluid phases provided a sufficient number of marker particles are used

• The method performs well even for severely distorted flows

• The method maintains a constant interface width of about two grid cell lengths

• The method maintains particle colour permanently never losing this information

Local Mass Conservation

Local conservation of mass equation states, for incompressible fluids,

Or for M fluid phases

0

ut

01

mmM

mm C

t

Cu

0 u

Local Volume Conservation

So we could choose, for each fluid m:

1. Therefore also satisfies the discrete form of the equation:

0

mmm

Ct

C

Dt

DCu

0)( ,,

,1,

, nji

mnmji

nmji CtCC Gu

nmjiC ,

,

Total Mass

The total initial volume for M fluid phases is

With the corresponding total initial mass given by

M

m

Y

Y

X

X

m dxdyyxC1

max

min

max

min

)0,,(

M

m

Y

Y

X

X

mm dxdyyxC

1

max

min

max

min

)0,,(

Global Mass Conservation

This must be conserved for all time, i.e.

or

M

m

Y

Y

X

X

mm dxdytyxC

t 1

max

min

max

min

0),,(

M

m

Y

Y

X

X

mm dxdytyxC

t1

max

min

max

min

0),,(

Global Volume Conservation

max

min

max

min

0),,(Y

Y

X

X

m dxdytyxCt

max

min

max

min

max

min

max

min

),(),( ,1,Y

Y

X

X

Y

Y

X

X

nmnm dxdyyxCdxdyyxC

Can choose

Or

Discretised Volume Conservation

2. In discretised form

J

j

I

i

nmji

J

j

I

i

nmji CC

1 1

,,

1 1

1,,

Particle to Grid Volume Fraction Interpolation

3. Already know

4. And or

N

pj

npi

np

N

p

mpj

npi

np

nmji

yyxxS

CyyxxS

C

1

11

1

11

1,,

,

,

max

min

max

min

1),(1

Y

Y

X

X

ji dxdyyyxxSyx

J

j

I

iji yyxxS

1 1

1),(

Non-Solenoidal Particle Velocities

J

j

I

ijiji yyxxSyx

1 1, 0),(),( uu

5. Given a solenoidal velocity field ui,j the interpolated particle velocity field is not necessarily also solenoidal:

01 1

,

SJ

j

I

ijiu

Solutions ?

• How do you construct a modified interpolation function S which maintains solenoidality ?

• What equation does S have to satisfy when considering the previous points 1-5 ?