a numerical model for multiphase flow, i: the interface tracking algorithm frank bierbrauer
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/1.jpg)
A Numerical Model A Numerical Model for Multiphase Flow, I:for Multiphase Flow, I:
The Interface Tracking The Interface Tracking AlgorithmAlgorithm
Frank BierbrauerFrank Bierbrauer
![Page 2: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/2.jpg)
A Numerical Model for Multiphase Flow
• Part I, Kinematics: The Interface Tracking Algorithm (Marker-Particle Method)
• Part II, Dynamics: The Navier-Stokes Solver
![Page 3: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/3.jpg)
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
![Page 4: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/4.jpg)
Solid-Liquid Impact
![Page 5: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/5.jpg)
Liquid Jets
Jet Breakup
Jet-Liquid Impact
Jet-Solid Impact
![Page 6: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/6.jpg)
Melting and Mixing
Phase Change: melted glass
Fluid Mixing
![Page 7: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/7.jpg)
Droplets and Bubbles
Sessile drops
Bubbles
Droplet Pinch Off
![Page 8: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/8.jpg)
Droplet Collisions and Shock Impact
Two Droplets Colliding
Shock Wave/Droplet Impact
![Page 9: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/9.jpg)
Droplet/Liquid Impact
Splash Corona & Rayleigh Jet Formation
![Page 10: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/10.jpg)
Capillary Waves
Secondary Droplet Expulsion
Collision of Two Droplets
Droplet Splash
![Page 11: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/11.jpg)
Example: Three-Phase Flow
Fluid phase 1 – droplet, fluid phase 2 – air, fluid phase 3 – other fluid
![Page 12: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/12.jpg)
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
![Page 13: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/13.jpg)
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
![Page 14: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/14.jpg)
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
![Page 15: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/15.jpg)
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
),,(),,(
![Page 16: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/16.jpg)
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
![Page 17: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/17.jpg)
Surface Tracking
1. Points parameterised along a curve (x(s),y(s))2. Sequence of heights h above a reference line
![Page 18: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/18.jpg)
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
![Page 19: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/19.jpg)
Interface Capturing
Volume Tracking
![Page 20: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/20.jpg)
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
![Page 21: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/21.jpg)
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
![Page 22: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/22.jpg)
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
![Page 23: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/23.jpg)
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
![Page 24: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/24.jpg)
Computational Cell & Initial Particle Configuration
![Page 25: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/25.jpg)
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
![Page 26: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/26.jpg)
Initial Particle Colours
• For example, for those particles of the 2nd phase:
![Page 27: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/27.jpg)
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
![Page 28: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/28.jpg)
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
![Page 29: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/29.jpg)
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
![Page 30: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/30.jpg)
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
![Page 31: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/31.jpg)
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
![Page 32: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/32.jpg)
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
![Page 33: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/33.jpg)
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,,
,
,
![Page 34: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/34.jpg)
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
![Page 35: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/35.jpg)
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
![Page 36: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/36.jpg)
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)
![Page 37: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/37.jpg)
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
![Page 38: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/38.jpg)
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))
![Page 39: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/39.jpg)
Expected Shearing Flow Effect
![Page 40: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/40.jpg)
Expected Vortex Field Effect
![Page 41: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/41.jpg)
Translation: relative % errors
![Page 42: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/42.jpg)
Rotation: relative % errors
![Page 43: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/43.jpg)
Shearing Flow: relative % errors
![Page 44: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/44.jpg)
Vortex: relative % errors
![Page 45: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/45.jpg)
Translation: transition width
![Page 46: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/46.jpg)
Rotation: transition width
![Page 47: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/47.jpg)
Shearing Flow: transition width
![Page 48: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/48.jpg)
Vortex: transition width
![Page 49: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/49.jpg)
Relative Errors L1 norm
![Page 50: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/50.jpg)
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
![Page 51: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/51.jpg)
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
![Page 52: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/52.jpg)
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 ,
,
![Page 53: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/53.jpg)
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,,(
![Page 54: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/54.jpg)
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),,(
![Page 55: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/55.jpg)
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
![Page 56: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/56.jpg)
Discretised Volume Conservation
2. In discretised form
J
j
I
i
nmji
J
j
I
i
nmji CC
1 1
,,
1 1
1,,
![Page 57: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/57.jpg)
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),(
![Page 58: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/58.jpg)
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
![Page 59: A Numerical Model for Multiphase Flow, I: The Interface Tracking Algorithm Frank Bierbrauer](https://reader031.vdocuments.net/reader031/viewer/2022020208/56813980550346895da11334/html5/thumbnails/59.jpg)
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 ?