1. II. Theory for numerical analysis of multi-phase flow (Practice)
MN74465: Computational multi-phase flow
II. Theory for numerical analysis of multi-phase flow (Practice)
Autumn, 2018
June Kee Min

2. 1/18
I. One-fluid approach

3. (Example#01) Capillary rise (1)
2/18
Stationary capillary height
Dynamics of capillary rise
GE (Newton’s law)
BCs
Singularity at t = 0
A remedy
Szekely et al.
Model for λ is necessary

4. (Example#01) Capillary rise (2)
3/18
Lucas-Washburn equation
Numerical method
Quasi-steady stage : capillary force is compensated by gravity and viscous drag
Asymptotic solution for short-time limit (t  0)
Asymptotic solution for long-time limit (t  ∞ , z  z∞)

5. (Example#01) Capillary rise (3)
Practice - One-fluid model (VOF)
VOF, Axisymmetric, Laminar model
Liquid fraction
Pressure distribution
Height vs time

6. : Film thickness varies temporally and spatially
(Example#02) Falling wavy film (1)
Classical: Nusselt film theory
Wavy film flow g
δ(x)
: Film thickness varies temporally and spatially

7. No wavy laminar film flow
(Example#02) Falling wavy film (2)
VOF, Youngs interface reconstruction scheme
Re < 30
30< Re<1,600
1,600<Re<3,200
Re > 3,200
No wavy laminar film flow
Wavy laminar film flow
Transitional flow
Turbulent flow

8. Average film thickness
(Example#02) Falling wavy film (3)
Film thickness and wave celerity
Average film thickness
Wave celeity

9. 8/18
II. Two-fluid approach

10. (Example#01) Fluidized bed (1)
9/18
Fluidized beds : Problem definition (Gidaspow*, 1994, see also Trygvasson’s book)
Drag modeling
Collision modeling
Assume constant
αparticle, t=0 = 0.55
For ideal gas model
Gas property
Grid

11. Solid pressure, Distribution, Packing etc.
(Example#01) Fluidized bed (2)
Numerical methods
ANSYS/Fluent
Eulerian multiphase model (Two fluid approach : Eulerian-Eulerian model)
Granular flow model in Fluent (Disperse flow: Fluid - Solid)
Consider flow field only (no heat transfer)
Models for corresponding Viscosity (Granular, Bulk, Frictional viscosities)
Solid pressure, Distribution, Packing etc.
Modeling of the interfacial force
+ fcollision
Drag
Added mass
Memory
Lift
Drag force : Wen and Yu (1996) model
Collision force : Syamlal (1987)
Boundary condition
Slip wall (no-slip is also possible)
Jet inlet

12. (Example#01) Fluidized bed (3)
UDFs
Initial condition (particles)
Boundary conditions
#include "udf.h"
real ymax = 0.20, xmin = 0.02, xmax = 0.03;
DEFINE_INIT(bed_init,d) {
int phase_domain_index;
cell_t c;
Thread *t;
Domain *sd;
real xc[ND_ND], fracx, fracy;
sub_domain_loop(sd,d, phase_domain_index)
/* loop over all cell threads in the domain */
thread_loop_c(t,sd)
/* loop over all cells */
begin_c_loop_all(c,t)
C_CENTROID(xc,c,t);
if(xc[1]<=ymax){
/* secondary phase = 3, primary phase = 2 */
if(DOMAIN_ID(sd) == 3) C_VOF(c,t) = 0.55;
if(DOMAIN_ID(sd) == 2) C_VOF(c,t) = 0.45; }
else{
if(DOMAIN_ID(sd) == 3) C_VOF(c,t) = 0.0;
if(DOMAIN_ID(sd) == 2) C_VOF(c,t) = 1.0;
end_c_loop_all(c,t)
real tstart = 0.05, tend = 0.15;
real bvel0 = 0.05, bvelmax = 0.25;
DEFINE_PROFILE(bed_prof,t,i) {
face_t f;
real xc[ND_ND], ctime;
ctime = CURRENT_TIME;
begin_f_loop(f,t)
F_PROFILE(f,t,i)= bvel0;
F_CENTROID(xc,f,t);
if((ctime>=tstart && ctime<=tend)
&& (xc[0]>=xmin && xc[0]<=xmax)){
F_PROFILE(f,t,i)= bvelmax; }
end_f_loop(f,t)

13. (Example#01) Fluidized bed (4)
Results
t = 0.25 s
t = 0.47 s
t = 0.66 s
t = 0.25 s
t = 0.47 s
t = 0.66 s
αAir = 0.5
Velocity (Air)
Volume fraction (air)

14. (Example#01) Fluidized bed (5)
13/18
Results by MFIX* (See Tryggvason’s book)
* MFIX (Multiphase Flow with Interface eXchange)
: Open source available from

15. (Example#03) Simplified models (1)
Mixture model with slip velocity model
Eulerian model

16. (Example#03) Simplified models (2)
(Cf) Mixture model with slip velocity model
Momentum equation (neglecting surface tension)
where J : the generalized drift flux
Homogeneous model
Assumption
The two phases move with the same velocity
Correlation terms are usually ignored

17. (Example#03) Simplified models (3)
Drift flux model (Wallis, 1969; Ishii, 1975; Toumi, 1996)
Correlation terms are usually ignored so that
Slip velocity
Slip velocity model : Manninen et al. (1996)
Slip velocity
Particle relaxation time
Re < 1000
Schiller and Naumann (1935)
Re > 1000

18. (Example#03) Simplified models (4)
(a) Mixture model with slip velocity model
Slip velocity model
Volume fraction(Air)
Velocity (Water)
Velocity (Air)
(b) Eulerian model
Velocity (Water)
Velocity (Air)
Volume fraction(Air)

19. 18/18
Thank you!