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 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 C 1 = 0.45 C 2 = 0.00 C 3 = 0.00 C 1 = 0.00 C 2 = 0.32 C 3 = 0.00 C 1 = 0.00 C 2 = 0.00 C 3 = 0.23 The phase indicator function is often the volume fraction occupied by the fluid (m) in the volume V: C m = V m /V so that C 1 + C 2 + C 3 = 1 in V

Example: Grid Volume Fraction Volume fraction information within grid cells C 1 – blue fluid, C 2 – yellow fluid C 1 = 0 C 2 = 1 C 1 = 0 C 2 = 1 C 1 = 0.2 C 2 = 0.8 C 1 = 0.7 C 2 = 0.3 C 1 = 0.95 C 2 = 0.05 C 1 = 1 C 2 = 0 C 1 = 0.7 C 2 = 0.3 C 1 = 0.3 C 2 = 0.7

One-Field Formulation For example, for 3 phase flow the density and viscosity fields are: density:  (x,y,t) =  1 C 1 (x,y,t) +  2 C 2 (x,y,t) +  3 C 3 (x,y,t) viscosity:  (x,y,t) =  1 C 1 (x,y,t) +  2 C 2 (x,y,t) +  3 C 3 (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  (x,y,t) =  1 C 1 (x,y,t) +  2 C 2 (x,y,t) +  3 [1-C 1 (x,y,t)-C 2 (x,y,t)]  (x,y,t) =  1 C 1 (x,y,t) +  2 C 2 (x,y,t) +  3 [1-C 1 (x,y,t)-C 2 (x,y,t)]

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 X min < x < X max, Y min < y < Y max, x 1/2 = X min, y 1/2 = Y min,  x = (X max -X min )/I,  y = (Y max -Y min )/J Within each computational cell assign a set of particles with positions (x p, y p )

Computational Cell & Initial Particle Configuration

Fluid Colour Each fluid phase (m) has a set of marker particles (p) located at position (x p, y p ) Every marker particle of the m th set is assigned a colour such that

Initial Particle Colours For example, for those particles of the 2 nd phase:

Particle Velocities Particle velocities u p = u(x p,y p ) are interpolated from the nearest four grid velocities u i,j, u i+1,j, u i,j+1, u i+1,j+1

Grid-to-Particle Velocity Interpolation

Interpolation Function or Where the interpolation function S is given by

Particle Kinematics Lagrangian particle advection: solve u = dx/dt which moves fluid particles along characteristics with velocity u Predictor Corrector

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

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

Algorithm 2.For time steps t > 0 1.Given u n and time centred grid velocities u n+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

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 = C 1, C 2 = 1 - C 1 ). 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 64 2 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

Benchmark Tests Test TypeVelocity FieldSpecified Field Simpletranslationu(x,y) = (1,0) Advectionrotationu(x,y) = (y-1/2,-(x-1/2)) Topologyshearing flow u(x,y) = (-sin2  x sin2  y, sin2  x sin 2  y) Changevortex 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 L 1 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

Local Volume Conservation So we could choose, for each fluid m: 1. Therefore also satisfies the discrete form of the equation:

Total Mass The total initial volume for M fluid phases is With the corresponding total initial mass given by

Global Mass Conservation This must be conserved for all time, i.e. or

Global Volume Conservation Can choose Or

Discretised Volume Conservation 2. In discretised form

Particle to Grid Volume Fraction Interpolation 3. Already know 4. And or

Non-Solenoidal Particle Velocities 5. Given a solenoidal velocity field u i,j the interpolated particle velocity field is not necessarily also solenoidal:

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 ?