A Numerical Model for
Multiphase Flow, I:
The Interface Tracking
Algorithm
Frank 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-Liquid
Impact
Jet-Solid Impact
Jet Breakup
Melting and Mixing
Fluid Mixing
Phase Change:
melted glass
Droplets and Bubbles
Droplet Pinch Off
Sessile drops
Bubbles
Droplet Collisions and Shock
Impact
Two Droplets Colliding
Shock Wave/Droplet Impact
Droplet/Liquid Impact
Splash Corona & Rayleigh Jet Formation
Droplet Splash
Capillary Waves
Secondary Droplet Expulsion
Collision of Two Droplets
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 socalled 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 = 0.45
C2 = 0.00
C3 = 0.00
C1 = 0.00
C2 = 0.00
C3 = 0.23
C1 = 0.00
C2 = 0.32
C3 = 0.00
The phase indicator function is often the volume fraction
occupied by the fluid (m) in the volume V: Cm = Vm/V so that
C1 + C2 + C3 = 1 in V
Example: Grid Volume Fraction
C1 = 0.2
C2 = 0.8
C1 = 0
C2 = 1
C1 = 0.7
C2 = 0.3
C1 = 0
C2 = 1
C1 = 0.95
C2 = 0.05
C1 = 0.3
C2 = 0.7
C1 = 1
C2 = 0
C1 = 0.7
C2 = 0.3
Volume fraction information within grid cells
C1 – blue fluid, C2 – yellow fluid
One-Field Formulation
• For example, for 3 phase flow the density and viscosity
fields are:
• density: r(x,y,t) =r1C1(x,y,t) + r2C2(x,y,t) + r3C3(x,y,t)
• viscosity: m(x,y,t) =m1C1(x,y,t) + m2C2(x,y,t) + m3C3(x,y,t)
• so that
r(x,y,t) =r1C1(x,y,t) + r2C2(x,y,t) + r3[1-C1(x,y,t)-C2(x,y,t)]
m(x,y,t) =m1C1(x,y,t) + m2C2(x,y,t) + m3[1-C1(x,y,t)-C2(x,y,t)]
• Where the mi and ri are the constant viscosities and densities
within each fluid phase
• In general for M fluid phases we have
M
M
r ( x, y, t ) 

m 1
r m C ( x , y , t ),
m
m ( x, y, t ) 

m 1
m mC ( x, y, t )
m
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, MarkerParticle 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
x i  x1 / 2
y j  y1 / 2
1

  i  Dx
2

1

  j  Dy
2

• In Xmin < x < Xmax, Ymin< y < Ymax, x1/2 = Xmin, y1/2 =
Ymin, Dx = (Xmax-Xmin)/I, Dy = (Ymax-Ymin)/J
• Within each computational cell assign a set of
particles with positions (xp, yp)
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 C pm  C m  x p , y p  such that
•
C
m
p
1

0
if particle
if particle
p is located in fluid m
p is not located in fluid m
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
  x  xi
u ( x , y )  1  
  Δx
 x  xi

 Δx
    y-y j  
 x  xi


1

u

  
 i, j 

    Δy  
 Δx
  x  xi
   y-y j  
  u i  1, j  1  
 1  

   Δy  
  Δx
  y-y j 
 u i 1, j 1
 

  Δy 
   y-y j 
u i, j
  

   Δy 
Interpolation Function
or
J
u (x,y) 
I
  S(x
 x i ,y  y j )u i,j
j 1 i 1
Where the interpolation function S is given by

x  xi
1 
S   
Dx


y  yj

  1 

Dy

0




if
otherwise
0
x  xi
Dx
,
y  yj
Dy
1
Particle Kinematics
• Lagrangian particle advection: solve u = dx/dt
which moves fluid particles along characteristics
with velocity u
• Predictor
Dt
n 1 / 2
xp
y
• Corrector
n 1 / 2
p
 xp 
n
 y 
n
p
2
Dt


2

 D tv  x
 x p  D tu x p
n 1
 yp
yp
n
n
n
n
p
n
p
v x ,y
n 1
xp
n

u xp, yp ,
n 1 / 2
n 1 / 2
p

n 1 / 2
, yp
n 1 / 2
, yp
,

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
m , n 1
• Require the updated grid volume fraction C i , j
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
Volume Fraction Interpolation
• This requires a normalisation of the interpolation
• Then, for each fluid m at the next time step n+1
N
 S x
C
m , n 1
i, j

n 1
p
 xi , y
n 1
p

 yj C
p 1
N
 S x
p 1
n 1
p
 xi , y
n 1
p
 yj

m
p
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 C pm for
each fluid
3. Construct initial grid cell volume
m ,0
fractions C i , j
Algorithm
2. For time steps t > 0
Given un and time centred grid velocities un+1/2 interpolate
velocities to all particles obtaining u np , u np 1 / 2
2. Solve the equation of motion u = dx/dt using the
predictor-corrector strategy already mentioned
m , n 1
3. Interpolate the new grid volume fractions C i , j from the
m
advected particle colours C p
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
1.
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 (Dx = Dy = 1/64 = 0.016) with either 4 or 16
particles per cell (ppc)
• At t = T measure droplet volume/mass given by
1 1
  C ( x , y ) dxdy
0 0
• 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)
min
0  x , X ; y ,Y 1
( X  x )  (Y  y )  0
• Obtain relative percentage errors
2
2
Benchmark Tests
Test 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) = (-sin2px sin2py, sin2px
sin 2py)
Change
vortex
u(x,y) = (sin 4p (x+1/2) sin4p
(y+1/2), cos 4p (x+1/2) cos4p
(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,   u  0
r
t
 u r  0
Or for M fluid phases
 C m
m
 r m   t  u   C
m 1

M

0


Local Volume Conservation
So we could choose, for each fluid m:
DC
m

Dt
C
t
m
 u  C
m
0
1. Therefore C im, j, n also satisfies the discrete form
of the equation:
C
m , n 1
i, j
C
m ,n
i, j
 D t (u  G C )
m
n
i, j
0
Total Mass
The total initial volume for M fluid phases is
M
Y max X max
  C
m
( x , y , 0 ) dxdy
m 1 Y
min X min
With the corresponding total initial mass given
by
Y max X max
M
 r  C
m
m 1
Y min X min
m
( x , y , 0 ) dxdy
Global Mass Conservation
This must be conserved for all time, i.e.

Y max X max
M

t

rm
m 1
 C ( x , y , t ) dxdy  0
m
Y min X min
or

M
r
m 1
m
t
Y max X max
 C
Y min X min
m
( x , y , t ) dxdy  0
Global Volume Conservation
Can choose

t
Y max X max
 C
m
( x , y , t ) dxdy  0
Y min X min
Or
Y max X max
 
Y min X min
Y max X max
C
m , n 1
( x , y ) dxdy 

C
Y min X min
m ,n
( x , y ) dxdy
Discretised Volume Conservation
2. In discretised form
J
I
C
j 1 i 1
m , n 1
i, j
J

I
C
j 1 i 1
m ,n
i, j
Particle to Grid Volume Fraction
Interpolation
3. Already know
N
 S x
m , n 1
C i, j

n 1
p
n 1
 xi , y p

 yj Cp
p 1
N
 
n 1
S xp
n 1
 xi , y p
 yj
p 1
4. And
1
D xD y
Y max X max
  S (x  x , y  y
i
j
) dxdy  1
or
Y min X min
J
I
  S (x  x , y  y
i
j 1 i 1
m
j
) 1

Non-Solenoidal Particle Velocities
5. Given a solenoidal velocity field ui,j the
interpolated particle velocity field is not
necessarily also solenoidal:
J
  u ( x, y )    
I
 S (x  x , y  y
i
j 1 i 1
J

I
u
j 1 i 1
i, j
 S  0
j
)u i , j  0
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 ?