Moving Interface Problems: Methods & Applications Tutorial ...€¦ · Moving Interface Problems: Methods & Applications Tutorial Lecture IV ... Part I-Two-dimensional Flow. J. Comput

Moving Interface Problems—Complex Flows

Moving Interface Problems:Methods & Applications

Tutorial Lecture IV

Grétar TryggvasonWorcester Polytechnic Institute

Moving Interface Problems and Applications in Fluid Dynamics

Singapore National University, 2007


Outline

Flows with phase changeSolidificationBoiling

Electrohydrodynamics

Flows with topology changesRegime changes in bubbly flowsAtomization and sprays

Outlook

Phase Change


The phase change between liquid and solid or betweenliquid and vapor is the critical step in the processing ofmost material as well as in energy generation.Computations will make it possible to predict the smallscale evolution of systems undergoing phase changefrom first principles.

To simulate such flows, it is necessary to solve theenergy equation for the temperature distribution and toaccount for the change of phase at the phase boundary.

Phase Change


In addition to solving the energy equation and includingthe phase change, we must

• Account for volume expansion at the interface for boiling

• Accommodate a zero velocity field in the solid, for the solidification problem.

In reality there is a slight volume change for thesolidification as well, but this can usually be neglected.

Phase Change


Formation ofMicrostructure

duringSolidification

Moving Interface Problems—Complex FlowsPhase Change

Early papers on dendritic growth inthe presence of flow:

Two-dimensional systemsTonhardt and Amberg (1998)Beckermann, et al (1999)

Juric (1998),Shin and Juric (2000)Al-Rawhai and Tryggvason (2001)

Three-dimensional system:Danzig et al (2001)Al-Rawhai and Tryggvason (2002)


!

"cT

"t+# $ucT =# $ k#T + q%& (x ' x f )dA

Tf = Tm (1+(

L+L)

q = LVn

dx f

dt=Vnn

Pure material

• D. Juric and G. Tryggvason, "A Front Tracking Method for Dentritic Solidification."J. Comput. Phys. 123, 127-148, (1996).

• N. Al-Rawahi and G. Tryggvason. Computations of the growth of dendrites in thepresence of flow. Part I-Two-dimensional Flow. J. Comput. Phys. 180, 471–496(2002)

• N. Al-Rawahi and G. Tryggvason. “Numerical simulation of dendritic solidificationwith convection: Three-dimensional flow.” Journal of Computational Physics. 194(2004) 677–696

Solidification


(C, D) =(c1/ k, kD

1) in the solid

(c2,D

2) in the liquid

! " #

k = c1/ c

2

$C

$t=% &D%T + s'( (x ) x f )dA

s = C(1 ) k)Vn

Tf = Tm (1+*

L) Cm)

m: slope of liquidus line

Alloy

Solidification

In addition to the energy equation, wemust solve a species concentrationequation


Compute the heat source at the interface

T+

T-

Ts

(x0,y0)

(x1,y1)

!

˙ q = k"T

"n

#

$ %

l

& k"T

"n

#

$ %

s

Solidification

Originally we found theheat source iterativelysuch that the interfacetemperature matchedthe target value.Currently we use“normal probes,”following Udaykumar etal.


u* = un + !tA(un)

u * * = u* "!t#P

un +1

= $u * *

#%un +1

= 0

!"un +1

= ! "#u * * = #!"u* * + u * * "!# = 0

!"u * * = 0

Including the solid:

Simplified Procedure

Enforcing incompressibility ~0

Solidification


Dendrite growing in auniform flow

401 by 401grid

Anisotropy=0.4

St =c(T! " Tm )

L= "0.3

Re =#UZ

µ= 600

Solidification


Dendrite growing in auniform flow

401 by 401grid

Anisotropy=0.4

St =c(T! " Tm )

L= "0.3

Re =#UZ

µ= 600

Solidification


Dendrites growing ina uniform flow

401 by 401 grid

Anisotropy=0.4

St =c(T! " Tm )

L= "0.3

Re =#UZ

µ= 600

Solidification

Moving Interface Problems—Complex FlowsSolidification

Moving Interface Problems—Complex FlowsSolidification


Velocity of the tip of the arms

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time

Tip velocity

Upstream tip velocity with Pef=0.002

Downstream tip velocity with Pef=0.002

Perpendicular tip velocity with Pef=0.002

Upstream tip velocity with Pef=0.006

Downstream tip velocity with Pef=0.006

Perpendicular tip velocity with Pef=0.006

Tip velocity with Pef=0

x106

x10-3

Solidification


Key challenges include:•The extension of the numerical methods to alloys•Inclusion of more complex interfacial effects•The use of simulations to predict microstructure offully solidified materials and the bulk properties ofthe material•More complex processes, such as solidification ofstirred melts

Solidification

Moving Interface Problems—Complex FlowsDroplet Impingement and Solidification

Solid

Experimental picture from anindustrial laboratory


Simulations ofBoiling Flows


Early papers on boiling

Juric and Tryggvason (1998)Son and Dhir (1998)Son, Ramanujapu, and Dhir (2002)Welch and Wilson (2000)Song and Juric (2002)Esmaeeli and Tryggvason (2002)Kunugi et al., (2001,2002)

Boiling Flows


!

"cT

"t+# $ u T =# $ k#T + q%& (x ' x f )dA

Tf :

q = L(V ' u ) $n

dx f

dt= Vnn+ u

# $ u =1

(

D(

Dt

Mass conservation

Energy equation

Thermodynamic

Velocity of bdryHeat source

Modified Clausius-Clapeyron eq.

D. Juric and G. Tryggvason. Computations of Boiling Flows. Int’l. J.Multiphase Flow. 24 (1998), 387-410.

A. Esmaeeli and G. Tryggvason. Computations of Explosive Boiling inMicrogravity. J. Scient. Comput. 19 (2003), 163-182

Boiling Flows


Vn

ug

ul

Computing the volume source

!

˙ m = "l

ul#V

n( ) = "v

uv#V

n( )

Volume expansion:

!

uv" u

l= ˙ m

1

#v

"1

#l

$

% &

'

( )

!

Vn

=1

2u

v+ u

l( ) "˙ m

2

1

#v

+1

#l

$

% &

'

( )

Normal velocitySource term

!

" #u =˙ q

L

1

$v

%1

$l

&

' (

)

* + , x % x f( )ds-

Boiling Flows

Moving Interface Problems—Complex FlowsBoiling Flows


Nusselt number versus time

Experimental correlationBerentson, 1961

!

NuB = 0.425 " (Gr "Pr/Ja)1/ 4

Boiling Flows


The effect ofthe Jacobinumber onthe boiling fornear criticalfilm boiling

Ja=0.035

Ja=0.117

Ja=0.234

Ja=0.468

Ja=1.167

Boiling Flows

Moving Interface Problems—Complex FlowsBoiling Flows


Film boiling from an embedded solid object. A hot solidcylinder is represented by an indicator function on arectangular structured grid. As the vapor region around thecylinder grows, bubbles periodically break off and rise to thefree surface

Time

Boiling Flows


Nucleate Flow Boiling Assumption : Surfacenucleation characteristicsdetermined by sizedistribution of potentiallyactive sites• Random spatial sitedistribution• Random conical cavitysize (mouth radius, r)distribution• Assume vapor embryoradius = r• Assume near wall liquidfilm is stationary

Nucleation site is active if rmin > r*

[ ]satllv

lvsat

TTh

Tr

!=

"#2* Carey (1992)

Boiling Flows


Microlayer Modeling

Model of: Son, Dhir, Ramanujapu (1999)

!

˙ q =kl Twall "Tint( )

#

!

˙ q = 2(M /2" R_

Tv )1/ 2 #vhlv

2

Tv

Tint $Tv + (Pl $ Pv )Tv /#lhlv[ ]

!

pl " pv = "#$ "A

% 3+

˙ q

&vhlv

2

Heat Conduction across Liquid Film

Modified Clausius-Clapeyron equation

Modified Laplace-Young equation

!

Tint = Tsat(P)

Combine to find:

!

˙ q = ˙ q (")

Vaporbubble

Liquidmicrolayer

!

Vapor Tint

Twall

!

"o

Boiling Flows


•Front Tracking with LevelContour Reconstruction

•Staggered gridMAC/Projection solution oftwo-phaseincompressible Navier-Stokes Equations

•2nd order ENO advection•Wall refined grid•BiCGSTAB solution of pressurePoisson equation

• Liquid/Vapor:density ratio = 1605viscosity ratio = 23thermal conductivity ratio = 27specific heat ratio = 1

• Wall superheat = 5K

• 15mm3 horizontally periodic box• Initial bubble radius 2.5 mm• Cavity radius 1 mm

From Damir Juric

Boiling Flows

Rise of a Steam Bubblein Saturated Water


Water at 1atm, Tsat=373.15K; liquid/vapor density ratio=1605;viscosity ratio=23; thermal conductivity ratio=27; specific heat ratio=1;domain size: 10.5x10.5x15.75 mm; Wall superheat: 18K40x40x60 grid resolution


Flow directionNucleate Boiling

• Liquid/Vapor:density ratio = 1605viscosity ratio = 23thermal conductivity ratio = 27specific heat ratio = 1• Wall superheat = 5K

• 15mm3 horizontallyperiodic box• Initial bubble radius 2.5mm• Cavity radius 1 mm

•2nd order ENO advection•Wall refined grid•BiCGSTAB solution ofpressure Poisson equation

Boiling Flows


There appears to be no significant technicalobstacles for conducting large scale simulationsof nucleate flow boiling—however, somedevelopment works still needs to be done!

Such simulations should allow us to• Assess the accuracy of the assumptions madein the modeling of the microlayer• Use the simulations to make predictions aboutboiling under conditions where experiments aredifficult or do not yield the necessary data.

Boiling Flows


Electrohydrodynamicsof Droplet Suspensions


Electrostatic fields are known to have strong influence onmultiphase flows:

Breakup of jets and drops

Phase distribution in suspensions

Here, we examine the effect of electrostatic fields on asuspension of drops in channel flows by direct numericalsimulations.


For fluids with small but finite conductivity, Taylor and Melcher(1969) proposed the “leaky dielectric” model. This modelallows both normal and tangential electrostatic forces on a twofluid interface.


!"u

!t+#$"u u = %#p + f

+# $µ #u +#Tu ( ) + &F' ( n ) x % x f( )da

!" u = 0

Momentum (conservative form, variable density and viscosity)

Mass conservation (incompressible flows)

The fluid flow

Surface tension

Electric force



!

Dq

Dt=" #$E

f = qE ! 1

2(E "E)#$

q = !" #E

The electric field is obtained from the equation for theconservation of current:

the charge accumulation is found by:

The force on the fluid is then found by:

0 neglecting alsoconvection ofcharge



!

"3D

=#o

#i

$i

$o

%

& '

(

) *

2

+1

%

&

' '

(

)

* * + 2 +

3

5

$i

$o

#o

#i

+1%

& '

(

) *

2µo

µi

+ 3%

& '

(

) *

µo

µi

+1

%

&

' ' ' '

(

)

* * * *

!

"2D

=#i

#o

$

% &

'

( )

2

+#i

#o

+1* 3+i

+o

Boundary between prolate and oblate drops

Taylor(1966)

!

" =

>1 Prolate

= 0 Spherical

<1 Oblate

#

$ %

& %

Rhodeset al.(1988)



(b)

Electrostatic deformationof axisymmetric drops.The steady stateobtained after followingthe transient motion of aninitially spherical drop.For the oblate drop in (a)the ratio of the dielectricconstant of the drop tothe dielectric constant ofthe suspending fluid ismuch larger than theconductivity ratio, but forthe prolate drop in (b)both ratios arecomparable

(a)

Deformation of a Single Drop



Drop distribution and streamlinesfor the interaction between twoprolate drops. S-1 = 0.01, R=0.1;initial distance between the dropscentroids r0=3.5 times the radius

Drop distribution and streamlinesfor the interaction between twoprolate drops. S-1 = 0.01, R=1.0;initial distance between the dropscentroids r0=3.5 times the radius

!

R ="i

"o

; S#1

=$i

$o

;



The motion of two oblate drops in a quiescent flow.The drops align with the electric field and attracteach other. The drops are also attracted to the wall

Interaction of Two Drops


Moving Interface Problems—Complex FlowsElectrohydrodynamics


!

"i"o

= 0.005 Re = 20

#i#o

= 0.01 We = 0.0625

$ = 20% E*

= 0.0182

!

"i"o

= 0.01 Re = 20

#i#o

= 0.1 We = 0.0625

$ = 20% E*

= 0.04



The interaction of many drops in channels, with andwithout flow has been examined.

Oblate drops always fibrate as the electrohydro-dynamically induced fluid motion works with the electricinteractions to line up the drops

Fluid shear breaks up the fibers, depositing them on thewalls for intermediate flow rate and keeping them insuspension for high enough flow rates

Prolate drops exhibit more complex interaction and formadditional structures



The instability of a thin film:


The interface and thevelocity field at time zeroand three subsequenttimes for S=1 and R=100.


Coalescence induced flowregime transitions


High bubble concentration at the walls is likely to lead tobubble collisions and coalescence. The collision of small andnearly spherical bubbles-which hug the wall-to form largedeformable bubbles—that are repelled by the wall—is likely tobe one of the major mechanism responsible for changing thevoid fraction distribution from “wall-peak” to a maximum in thecore. The figure shows a simulation of the collision of twonearly spherical bubbles and the evolution of the resulting largebubble.

Coalescence

Moving Interface Problems—Complex FlowsFlow regime changes

Coalescence induced flow regime transitions in a laminar bubbly channelflow: The figure shows a preliminary two-dimensional simulation of thetransition from a wall peaked distribution of many bubbles to a single largeslug in the channels center.


A simulation of a coalescence induced regime transitionin a small three-dimensional system

Time


!

1

Volnnda

S"

Thecomponents ofthe interfacearea tensorversus time


Atomization anddroplet breakup


In general, the interface separating two fluids willundergo topology changes where two regions of onefluid coalesce, or one region breaks in two. Of those, thecoalescence problem appears to be the harder one.

In their simples implementation, explicit tracking methodnever allow coalescence and method based on a markerfunction always coalesce two interfaces that are close.

In reality, films between two fluid interfaces take a finitetime to drain and rupture only when the thickness issufficiently small so the film is unstable to non-continuumattractive forces. In general this draining can not beresolved and must be modeled.


References:M.R.H. Nobari, Y.-J. Jan and G. Tryggvason. "Head-on Collision of Drops--A

Numerical Investigation.”Phys. of Fluids 8, 29-42 (1996).M.R.H. Nobari, and G. Tryggvason, "Numerical Simulations of Three-Dimensional

Drop Collisions.”AIAA Journal 34 (1996), 750-755.J. Qian, G. Tryggvason, and C.K. Law. An Experimental and Computational Study

of Bouncing and Deforming Droplet Collision. Submitted to Phys. Fluids

Binary Collision of Drops


Three frames from a simulationof the three-dimensional breakupof a jet. The initial two-dimensional fold becomesunstable and generates fingersthat eventually break into drops.Here, Re=1000, We=5, and thedensity ratio is 10. Thesimulation is done using 72 by48 by 38 unevenly spaced gridpoints in the radial, axial, andazimuthal direction, respectively.

Primary breakup of jets

Moving Interface Problems—Complex FlowsSecondary breakup of drops

Moving Interface Problems—Complex FlowsSecondary breakup of drops


Splatting drops

Thermocapillarymigration

Mass transfer & chemical reactions

DNS of Multiphase Systems

ExplosiveBoiling

Shocks inbubbly flows


Multifluid simulations of relatively simple systemsare well under control and can be used tounderstand such systems.

Large scale three-dimensional simulations areemerging. The challenge is to use the results toproduce engineering/scientific knowledge.

Methods for multiphase flows are in their infancy.

Summary

System size:<1980: Mostly two-dimensional systems1980: early three-dimensional studies1990: less than 1003 grid points2006 > 10003 grid points + new computational techniques

Documents

Moving Interface Problems: Methods & Applications Tutorial ...€¦ · Moving Interface Problems: Methods & Applications Tutorial Lecture IV ... Part I-Two-dimensional Flow. J. Comput