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The 12th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-12) Log Number: KN4 Sheraton Station Square, Pittsburgh, Pennsylvania, U.S.A. September 30-October 4, 2007.

ON THE COMPUTATION OF MULTIPHASE FLOWS

Richard T. Lahey, Jr. Center for Multiphase Research Rensselaer Polytechnic Institute

Troy, NY, USA 12180-3590 [email protected]

ABSTRACT This paper presents an assessment of various models which can be used for the multidimensional analysis of single and multiphase flows in nuclear reactor systems. In particular, a model appropriate for the direct numerical simulation (DNS) of single and multiphase flows and a, three-dimensional, turbulent, four-field, two-fluid computational multiphase fluid dynamic (CMFD) model are discussed.

KEYWORDS Simulation, Multiphase Flow, DNS, CMFD

1. INTRODUCTION Turbulent and single and multiphase flows are very complicated and, as a consequence, in the past, models for these flows have always been empirically-based. The models used for the analysis of nuclear reactor thermal-hydraulics and safety have evolved from lumped parameter and one-dimensional models (e.g., homogeneous equilibrium models, phasic slip models, drift-flux models and two-fluid models), to quasi-multidimensional subchannel models, fully three-dimensional single-phase computational fluid dynamic (CFD) and computational multiphase fluid dynamic (CMFD) models, and most recently, to the direct numerical simulation (DNS) of multidimensional single and multiphase flows. The focus of this paper will be on the latest models for the multidimensional analysis of single and multiphase flow and heat transfer. This paper presents the state-of-the-art in our ability to predict multidimensional single and multiphase flows using DNS and CMFD models. It should be noted that CMFD models are needed for the design and analysis of multiphase systems and processes, such as nuclear reactors, since DNS is currently impractical for these purposes. Nevertheless, DNS gives one important insights into multiphase flow phenomena and these results can be used as “data” to help develop closure laws for CMFD models. 2. DISCUSSION – CMFD Let us begin with the discussion of a state-of-the-art turbulent, four-field, two-fluid model. A mechanistically-based, four-field, two-fluid model of two-phase flow and heat transfer is, shown

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Richard T. Lahey, Jr. NURETH-12 Log: KN4

schematically in Figure-1. This model can accurately predict the phasic velocity and pressure fields and the volumetric concentrations of the continuous vapor (cv), continuous liquid ( c ), dispersed vapor (dv), and dispersed liquid ( d ) fields. Indeed it is inherently capable of describing the essential features of vapor/liquid flows in many flow regimes of interest. It will be shown that computational multiphase fluid dynamic (CMFD) predictions using this model agree with a wide range of bubbly flow experimental data [1], [2], [3] and that this type of model may be extended to other flow regimes.

Figure 1. Typical Four-Field, Two-Fluid Model Results

A 3-D turbulent, four-field, two-fluid model is comprised of the following conservation equations:

Mass Conservation (field-j, phase-k)

( ) ( )vjk kjkjk k jk jkm

tα ρ

α ρ∂

′′′+∇• = Γ +∂

(1)

where, αjk is the volume fraction of field-j of phase-k, Γjk is the mass transfer rate density due to phase change in field-j of phase-k, and jkm′′′ is the mass source density of field-j from other fields of phase-k.

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Momentum Conservation (field-j, phase-k)

( ) ( ) (

vv vjkjk k

jk jkjk k jk jk )pt

α ρα ρ α

∂+∇• +∇

∂

v v

T wjkjk k jkjk jkjk

ijk jk jk

g M M

m

α τ τ α ρ⎛ ⎞⎡ ⎤⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠−∇• + − − −

′′′= Γ + (2)

'''i jk jkjk v m vΓ= +

where T

jkτ is the Reynolds stress for field-j of phase-k, jk jk≠ , iv is the velocity of the phasic

interface and jkM , wjkM are the interfacial and wall force densities for field-j of phase-k,

respectively.

Energy Conservation (field-j, phase-k)

( ) ( ) " ''vTjk k jk

jk jkjk k jk jk jk

hh q

tα ρ

α ρ α ⎛⎜⎝ ⎠

∂+∇• +∇• +

∂q ⎞

⎟

''' '' '''jki ijk jk jk jk jk jkjk i

Dpq q A u m u

Dtα ′′′− − − − = Γ +D (3)

where, is the internal energy at the phasic interface, and uiu jk is the internal energy, hjk is the enthalpy,

jkiq′′ is the interfacial heat flux, jkq′′′ is a volumetric heat source, and Djk is the viscous

dissipation for field-j of phase-k. It has been shown [4] that single-phase k-ε turbulence models can be extended to bubbly two-phase flows. In particular, we may use:

Turbulent Kinetic Energy

[ ]T

c cc c c c c c

k

Dk k PDt

υα α α εσ

⎡ ⎤= ∇• ∇ + − + Φ⎢ ⎥

⎣ ⎦c kα (4a)

where, 1 v v2c ck c′ ′= •

Turbulent Dissipation

1 2

2Tc c c c c

c c c c cc ck

D PC CDt k k cε ε ε (4b) ε υ ε εα α ε α α α

σ⎡ ⎤⎢ ⎥⎣ ⎦

=∇• ∇ + − + Φ

Sato et. al. [5] gives the two-phase turbulent viscosity as:

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2

v0.6 vT cc d

c

kC Dμυε

= + v d rα (5)

where the relative velocity is: vv v vr d c= − Note that for the special case of bubbly flows, only two fields are required. That is, the continuous liquid and dispersed vapor (dv) fields. ( )c For bubbly flows the source term for turbulent kinetic energy, Φk, is given by [6], [7], [3]:

( )3

43

vv

v1

D2

rcp Dk

c d

k C CC εε

αε

Φ = Φ = + d (6)

where Cp=0.25 for potential flow around a sphere [8]. Equation (6) accounts for the velocity fluctuations induced by liquid displacement due to the relative velocity (vr) and the liquid eddies which may be formed behind the bubbles due to separation. Similar turbulent transport equations can be written for the dispersed phase [9]. Alternatively, the Reynolds stresses, T

jkτ , of the dispersed vapor (dv) phase, and thus the associated turbulent

kinetic energies, kjk, which are just the trace of the respective Reynolds stresses, are [10], [11]:

( )3 20 2 5T Tc cdvdv c

k Iτ τ ρ ρ ρ= Ω + − Ωc (7a)

where the turbulence correlation parameter, Ω, is:

( )( )

21 exp

1 exp 2c dv

c dv

θ θ

θ θ

⎡ ⎤⎣ ⎦⎡ ⎤⎣ ⎦

−Ω =

− (7b)

and the time scales of the liquid eddies and the bubbles are, respectively: c c ckθ ε= (7c)

21

3 18dv dv

dv dv dvc cdv

V DD

ρθπμ μ

= = ρ (7d)

It should be noted that for polydispersed bubbly flow (dv), where there are different size bubbles, Eqs. (1) – (7) must be evaluated for each interacting bubble size group [12]. Let us next consider how to constitute the interfacial and wall transfer laws in a two-fluid model of bubbly two-phase flow to obtain closure. In order to achieve closure we must be able to express all the parameters in the two-fluid model and the associated jump conditions in terms of

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the state variables (i.e., the dependent variables) or the independent variables of the two-fluid model. It is customary to partition the interfacial force density into an aerodynamic drag (D) law and non-drag (ND) terms. That is, for the continuous liquid ( ) phase [3]: c

( ) ( )

i i

D NDc c c c c c

M M M p cα τ α= + + ∇ − ⋅∇ (8)

For bubbly flows, we can use the drag law for spheres and potential flow theory can be used to derive many of the non-drag interfacial transfer laws [8], [10]. Thus we have,

(D)(D)c r rc D iv

1 '''C v v A8dM M ρ= − = − (9a)

and [3],

( ) vvv

v v

v v

v v ...

ND d cc mcd

r cc L TD c cd d

M D DCDt Dt

C C k

α ρ

α ρ ρ α

⎡ ⎤⎢ ⎥⎣ ⎦

= −

+ ×∇× + ∇ + (9b)

The wall force density on the dispersed vapor field, w

dvM , has been given by Antal et al., [13] as:

( )

w wdv 1 2

*

dv20,

2

v v

5v v v

B

cdv cw dv c w

dv

axial axialcdvww w w

dvc

DM M Max C C n

D

CD

α ρδ

ρ α δ

•= − = +

⎧ ⎫⎡ ⎛ ⎞⎤⎨ ⎬⎜ ⎟⎢ ⎥⎣ ⎝ ⎠⎦⎩ ⎭

− −

y (9c)

where wn is a unit vector which is normal to the wall of the conduit, and, ( ) wwy x x n= − •

1

2

*v v v v

0.104 0.06 v 1.5

1.0,0.147

0.0, otherwise

B

axial r w r w wc c

rw w

dvw w

n n

C C

y DC

τ ρ

δ

⎡ ⎤⎣ ⎦

⎧⎨⎩

= − • =

= − − =

≤= =

(9d)

The first term on the right hand side of Eq. (9b) is the virtual mass force density [14], the second term is often called the lift force density [14] and the next term after that is the turbulent dispersion force density [15]. It is significant to note that there are no arbitrary (i.e., tunable) constants in these models.

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To derive an expression for the interfacial area density, , let us consider the Boltzmann transport equation for the dispersed bubbles (dv). In particular, the Boltzmann transport equation for bubbles of volume v is given by Valenti et al., [16] as:

'''iA

vc b phc

f df df dff f vt v dt dt dt

⎡ ⎤⎣ ⎦∂ ∂+∇• + = + +∂ ∂

(10)

where the third term on the left-hand side of Eq. (10) quantifies the effect of bubble expansion/contraction, and the terms on the right-hand side of Eq. (10) account for bubble coalescence (c), breakup (b) and phase change (phc), respectively. If fdv is the probability of having a bubble with a volume between v and v + dv, then the coalescence (c) and breakup (b) terms are given by [10]:

( ) ( ) ( )

( ) ( ) ( )

0

0

, , , , ,

1 , , , , ,2

c

v

df c v v f v x t f v x t dvdt

c v v v f v x t f v v x t dv

∞′ ′ ′= −

′ ′ ′ ′+ − −

∫

∫ ′

(11a)

and,

( ) ( )

( ) ( )

0

0

, , ,

, , ,bv

df b v v f v x t dvdt

b v v v f v x t dv

∞′ ′= −

′ ′ ′ ′+ −

∫

∫ (11b)

where, c and b are the coalescence and breakup rate kernels, respectively.

If we multiply Eq. (10) by the surface area of a spherical bubble of volume v, 2

3344v

vA ππ

⎛ ⎞= ⎜ ⎟⎝ ⎠

,

and integrate over all volumes, v, we obtain [10] for dispersed field–jk (e.g., jk=dv, spherical bubbles; jk=cv, large cap/Taylor bubbles):

( )jk

jkjk jk

'''''' vi

ii A

AA

tφ

∂+∇• =

∂ (12)

where

jkAφ is the generalized interfacial area density source/sink term, which for field-jk of the

bubbly flow is:

jkjk jk jkexpA b c phcφ φ φ φ= + + (13a)

and,

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jk

2 / 3jk

jk

jk

34

4jkb c phc co bphc

vdf dfdv

dt dtdfdt

φ φ ππ

∞+ = +

⎡ ⎤ ⎛ ⎞+⎢ ⎥ ⎜ ⎟⎝ ⎠⎣ ⎦

∫ (13b)

Phenomenological formulations for

jkAφ have been proposed by Wu et. al [17] for a two-field,

two-fluid model, and Ishii& Hibiki [18] for a three-field, two-fluid model. For polydispersed bubbly/slug flows having Mcrit discrete size bins and a minimum diameter Ddv,o for the bubbles (dv):

jkexpφ =( ) ( )

( )

crit

jk,m

jk

Mv

i jk jk,m-1 jk jk,mm=1v

v vi jk crit

v

D p2A H 1- H - ; jk = dv

3ρ Dt

d D p2A H ; jk = cv

3 dp Dt

v v v v v

v v

ddpρψ

ρ

ρ ′′′− −

′′′− −

⎧ ⎡ ⎤⎪ ⎣ ⎦⎪⎨⎪⎪⎩

∑ (14a)

(where ) critcrit jk,Mv v≡

( ) ( )[ ]

( )

critjk,m

jk

Mi jk,m

jk jk,m 1 jk jk,mm=1v jk,m

phc

jki jk jk critv jk

A ΓH 1 H ;

ρ α=

ψA

ρΓ H ; jk = cv

α

jk

jk

v v v v jk d

v v

χ

φχ

−

′′′v− − − =

′′′

⎧⎪⎪⎨⎪ −⎪⎩

∑ (14b)

(Normally, for m, 0.0dv mΓ = >2 and cv 0.0Γ = )

( ) ( )

( )

crit

jkb c

Mjk,m

jk jk,m-1 jk jk,mm=1v

jk

i jk,m

,

i jk,mjk crit

v jk

=

H - 1- H - ; jk = dρ

A m

A mψH - ; jk = cv

ρ α

jk

jk m

jk

jk

v v v v

v vφ

χ

α

χα

⎧⎡ ⎤⎪ ⎣ ⎦

⎪⎨⎪⎪⎩

v′′′ ′′′

′′′ ′′′

∑ (14c)

where jkχ is a geometric factor (~ 1.0 ), jkχ ψ ~ 1.5 includes a shape factor for the cap/Taylor bubbles (jk=cv), and the critical bubble volume for the transition from a spherical (dv) to a cap or bullet shaped Taylor bubble (cv) is given by,

crit

3crit v6 dv Dπ= (15a)

and the critical bubble diameter is [19],

( )1 3

critdv,crit

dv

64c

vDg

σπρ ρ

⎛ ⎞⎜ ⎟⎝ ⎠

= =−

(15b)

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We note in Eq. (14c) that there is a unique relationship between the field-to-field mass transfer term ( ) in Eq. (1) and the interfacial area density source term m′′′ ( )jkb cφ due to breakup and

coalescence.

Ishii & Hibiki [18] have presented detailed phenomenological models for ( jkb cφ ) which account

for the bubble coalescence and breakup mechanisms associated with: turbulence-induced bubble collisions, the breakup due to the impact of liquid eddies on the bubbles, wake-induced entrainment and coalescence effects, breakup due to surface instabilities, etc. A similar modeling approach has also been successfully applied to bubble clusters for the analysis of the bubbly/slug flow regime transition [20]. Moreover, it has been shown [21] that a properly formulated two-fluid model will be well-posed (i.e., the void wave eigenvalues are real for all αdv, until a flow regime transition occurs). 3. MODEL ASSESSMENT This four-field, two-fluid model has been evaluated using a state-of-the-art CFMD solver (i.e., NPHASE), and good agreement with available experimental data was found without “tuning” the parameters in the model [3]. Figure 2a shows good agreement with the bubbly air/water upflow pipe data of Serizawa [22]. Note that the model properly predicts the peaking of the dispersed vapor volume fraction ( )dvα near the wall of the conduit. Similarly, figure 2b shows that the same model predicts the bubbly air/water downflow pipe data of Wang et al. [23] in which the dispersed vapor volume fraction peaks at the center of the vertical pipe. This remarkable change in phase distribution is a direct consequence of a change in the sign of the relative velocity in the lateral lift force. Figure 2c shows that the same model is also able to predict bubbly flows in more complex geometry conduits [24], and, as seen in figure 2d, for free bubbly jets [25].

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Figure 2a. Comparison with air/water bubbly upflow data

[Serizawa, 1974]: void fraction distribution (αG), [3].

αG

Figure 2b. Comparison with air/waterbubbly downflow data [Wang et al., 1987]: void fraction distribution (αG), [3].

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αG

Figure 2c. Comparison with air/water bubbly upflow data in an isosceles triangle [Lopez de Bertodano et al., 1994]: void fraction distribution (αG) , [3].

r (m)

0.00 0.01 0.02 0.03 0.04 0.05 0.060

1Model at L/D = 24Model at L/D = 40Model at L/D = 60Data at L/D = 24Data at L/D = 40Data at L/D = 60

αG/αo

Figure 2d. Comparison with bubbly jet flow data [Lopez de Bertodano et al., 2003]:

normalized void fraction distribution (αG/αo), [3].

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Figures-3 show that the same two-fluid model is also able to predict the near-wall void fraction distribution in a developing boundary layer [26], and figure-4 shows that, when using an appropriate interfacial heat transfer coefficient and a suitable model for the near-wall ebullition process [1], it can also predict subcooled boiling.

Figure 3a. Bubbly air/water two-phase flow over a flat plate [Moursali et al., 1995], [3].

Figure 3b. The prediction of the near-wall

void distribution at x=1.0m , [3].

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Figure 4. Predictions for subcooled boiling R-113 in an internally heated annulus

[Velindandla et al., 1995], [3]. Finally, figures-5 are examples of the model’s predictions [11] for various buoyancy dispersed particle data [27], [28], and for the air/water data of Kamp et al. [29] which was taken at microgravity.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

0.025

Radial Position (r/R)

Voi

d F

ract

ion

(αdv

)

predictiondata

Figure-5a. The prediction of positive buoyant spherical particles (2 mm φ) for water flowing upward in a pipe [Alajbegovic et al, 1994], [3].

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.01

0.02

0.03

0.04

0.05


Voi

d F

ract

ion

(αdv

)

predictiondata

Figure-5b. The prediction of negative buoyant spherical particles (2 mm φ) for water flowing upward in a pipe [Alajbegovic et al, 1994], [3].

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035


Voi

d F

ract

ion

(αdv

)

predictiondata

Figure-5c. The prediction of neutral buoyant spherical particles (2 mm φ) in a pipe [Assad et al, 2000], [3].

0.05

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045


Voi

d F

ract

ion

(αdv

)

predictiondata

Figure-5d. The prediction of air/water bubbly flow at microgravity [Kamp et al, 1993], [3].

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It can be seen that a properly formulated CMFD model is capable of predicting a wide range of conduit and external bubbly two-phase flows for both adiabatic and diabatic conditions and for various gravity levels.

4. OTHER FLOW REGIMES A properly formulated two-fluid model can also predict a wide variety of phenomena in other flow regimes. Unfortunately, the approach that proved to be successful for achieving closure in the bubbly flow regime (i.e., exact analytical solutions of the flow field using potential flow theory) is normally not possible for other flow regimes. Rather, it appears that, while simplified analysis can be done to help identify the functional form of the closure laws for some other flow regimes, such as slug [30], [31] and annular flows [32], [33]. Detailed separate effect experiments, and/or the ensemble-averaged results of multidimensional direct numerical simulations (DNS), will likely be required to develop realistic closure laws over a wide range of flow regimes [34]. Moreover, these models must be formulated to be well-posed [10], satisfy objectivity [35], the second law of thermodynamics [36], and so that flow regime transitions are a natural consequence of physical instabilities [37]. If the instantaneous conditions at, and around, the various interfaces are computed, using a suitable adaptive grid direct numerical simulation (DNS) technique, then the two-fluid closure laws can be deduced. That is, as noted previously, we may partition the interfacial force density in Eq. (2) into drag and non-drag force densities as:

( ) ( )D Njk

Djk jkjk jk jkjk

jk jk jkjki i

M p X X M M

p

τ

α τ α

≡ ∇ − ⋅∇ = +

+ ∇ − ⋅∇ (16)

where, the overbars denote ensemble averaging, and,

( ) ( )( ) ( )( )

i i

DDDjk jk jk jk jjk jk kM p p X Xτ τ≡ − ∇ − − ⋅∇ (17a)

( )( )( )( )ND

i jki

NDNDjk jk jk jk jjk kXM p p X τ τ ⋅∇⎛ ⎞⎜ ⎟

⎝ ⎠≡ − ∇ − − (17b)

i jkjk jk jk jkp p n X n X= ⋅∇ ⋅∇ jk (17c)

/jki

jk jk jkjk jkn X n Xτ τ= ⋅∇ ⋅∇ (17d)

We note that there are many non-drag (ND) forces (e.g, virtual mass, lift, etc.) which may include both pressure and shear induced forces. Nevertheless, once the two-fluid model’s drag

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(D) and non-drag (ND) components are constituted (e.g., using DNS) in terms of the various relevant spatial and temporal derivatives and algebraic terms (e.g., for aerodynamic type drag laws), then closure can be achieved for the various flow regimes of interest. These phenomenological models can then be assessed against suitable analytical/numerical results and data to verify their validity [34]. An example of this approach for wavy stratified flow was presented by Galimov et al. [33]. 5. DISCUSSION - DNS Let us now consider the direct numerical simulation (DNS) of both single and multiphase flows. There have been a number of interesting applications of the DNS technique to problems in multiphase flow [38], [39] [40]. In the latter paper, [40], a Galerkin least-squares finite element method (FEM) was used for the differencing of the various conservation equations, and the associated interfacial jump conditions. The interface was resolved using a level set algorithm [41]. The resultant code is called PHASTA [42], [43], [44]. PHASTA is a 3-D FEM code which is based on a fully unstructured, adaptive grid. The time step is dynamically readjusted to yield a specified accuracy and the computational grid is refined in regions where steep gradients occur, an coarsened in other regions. This enhances accuracy and resolution while significantly reducing computational time. In addition, hierarchical basis functions [45], [46] may be used to achieve higher order accuracy (e.g., third order resolution) in space as desired.

The PHASTA code can be spatially nodalized down to at least the local Kolmogorov scale and used to evaluate the Navier-Stokes equation and the corresponding phasic continuity and energy equations, which, when written in matrix form, are:

adv diff, , ,t i i i i+ − =U F F S (18a)

where, for each phase, we have:

1

2 1

3 2

4 3

5

1

tot

UU uU uU uU e

ρ

⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎩ ⎭⎩ ⎭

= =U , 1

adv2

3

0i

i i i

i

i

u p

u

δδδ

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

= +F U , 1

diff2

3

0

i

ii

i

ij j iu q

τττ

τ

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

=

′′−

F (18b)

and, S is a body force (or source) vector, such as gravity and surface tension, and:

( ) ( )123ij ij ijkkS u S uτ μ δ⎛ ⎞

⎜ ⎟⎝ ⎠

= − , ( ) ,

2i j j i

iju u

S u ,+=

(18c)

, "iq ,iκT=− tot 2

i iu ue e= + , ve c= Τ

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The state variables are the velocity components, ui, pressures, p, densities, ρ, temperatures, T, and the total specific energy, . Constitutive laws relate the stress tensor,tote ijτ , to the deviatoric

part of the strain rate tensor, dij ij kk ij

1S S S3

= − δ , through a molecular viscosity, μ. Similarly, the

heat flux, , is proportional to the gradient of temperature with the proportionality constant given by the molecular thermal conductivity, κ. It should be noted that because the spatial grid resolves the local Kolmogorov scales, turbulence modeling is not required. This yields true DNS predictions for each interacting phase and, using a level set model for multiphase flows, accounts for the various (continuous/dispersed) deformable interfaces.

"iq

Closure of the DNS model is achieved by the use of appropriate equations of state for the liquid and vapor phases, and the interfacial jump conditions, which, for multiphase flows, include surface tension, σ, (e.g., Marangoni) effects which are modeled as an interfacial force density vector,

( ) ( )1

Ni i i i is s ji j j

d dM x x n T C x x tdT dCσ σκσδ δ

⎡⎢⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

== − + ∇ + ∇ −∑

⎤⎥ (19)

where κ is the local instantaneous interfacial curvature, s∇ is the gradient operator in the interface, Cj is component concentration (if we have a multi-component mixture), ( )ixx −δ is a Dirac delta function (which is only non-zero at the position of the interface, xi), and ni and ti are the unit interfacial normal and tangential vectors, respectively. It is convenient to define the quasi-linear operator associated with Eq. (18a) as:

0

⎛ ⎞⎜⎜⎝ ⎠

⎟⎟∂ ∂ ∂ ∂≡ + −∂ ∂ ∂ ∂i ij

i i jL A A K

t x x x (20a)

which can be decomposed into time, advective, and diffusive terms:

adv diff= + +L L L Lt (20b) Using this notation, we can rewrite Eq. (18a) as,

=LY S (21) To derive the finite element form of Eq. (21), the entire equation is dotted with a vector of weight functions, W , and integrated over the spatial domain. Integration by parts is then performed to move the spatial derivatives onto the weight functions. This process leads to the following integral equation:

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( ) ( )adv diff adv, , ,0

difft i i i i i i id n dΓ ( )

10

e

e

n

edΤ

= ΩΩ Γ

⋅ − ⋅ + ⋅ + ⋅ Ω − ⋅ − +∫ ∫W A Y W F W F W S W F F + ⋅ − Ω =∑ ∫ L W LY Sτ (22)

where,

1 2 3( , , , , )Tp u u u T=Y (23)

The first two terms of Eq. (22) contain the Galerkin nodalization (interior and boundary) and the last term contains a least-squares stabilization. It is interesting to note that the well known SUPG (Streamline Upwind Petrov Galerkin) stabilization can be obtained by replacing with

TLTadvL . The stabilization matrix, τ, is an important part of these methods and it has been well

documented [47]. The integrals in Eq. (22) are evaluated using Gauss quadratures resulting in a system of non-linear ordinary differential equations which can be written as,

( )=1 1M Y N Y (24)

where the under bar is added to make clear that Y is the vector of solution values at discrete points (spatially interpolated using the finite element basis functions). Typical DNS results for turbulent single-phase channel flow are shown in figure 6 for

/178w h

Reττ ρν

≡ = (where h is the half-width of a rectangular channel;

4 , 2, 4 / 3x y zL L Lπ π= = = ). It can be seen in this figure that there is good agreement between DNS results of PHASTA and the spectral method of Moser et al. [48], and both results yield the classical “law of the wall”. Moreover, the non-isotropy of the single-phase turbulence field is clearly seen in figure 7, while the corresponding Reynolds and viscous stress profiles are shown in figure 8. The various one-dimensional energy spectra, which are Fourier transforms of the spatial correlation functions, for this conduit flow are given in figure 9, where we note the expected dependence versus wave number ( 2k π

λ= ).

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Figure 6. The mean streamwise velocity in wall coordinates, . 178Re =τ

Figure 7. RMS velocity fluctuations in wall coordinates, 178Re =τ

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Figure 8. Nondimensional Reynolds and viscous stress components, . 178Re =τ

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Figure 9. One-dimensional energy spectra of velocity fluctuations for

178Re =τ at the channel centerline

Finally, as can be seen in figure 10, typical instantaneous velocity contours show a “slow speed streak lift off” of liquid from the low velocity (wall) region which subsequently results in bursts of turbulence production. Similar, but more complicated phenomena occurs for turbulent multiphase flows.

Figure 10. Instantaneous velocity contours in the x-direction of the (x,y) cross-section for 178Re =τ ; PHASTA Mesh = 295x129x129.

For multiphase flows, the vapor/liquid interfaces are resolved using the level set method. The level set approach of Sussman et al. [49] and Sethian et al [41], represents the interface as the zero level set of a smooth function, φ , which is the signed distance from the interface. Hence instead of explicitly tracking the interface, we implicitly capture the interface within a field which is interpolated between the nodes like any other state variable. This enables one to predict the shape of the interface between the two phases accurately and to track its position (including any breakup and coalescence).

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Since the interface moves with the fluid, the evolution of φ is governed by the following transport equation:

0D uDt tφ φ φ∂≡ + ⋅∇ =

∂ (25)

This additional transport equation for the level set scalar is solved in a manner similar to the flow equations, however, appropriate redistancing must be done at each time step [49] to account for the nonuniform convection of φ and to assure mass conservation. The resulting FEM formulation of Eq. (25) is:

( ) (, , , ,1

0e

e

n

t i i t i ie

w wu wS d L w u S dφ φ τ φ φΤ

Ω Ω=+ + Ω+ + − Ω =∑∫ ∫ ) (26)

These integrals are also evaluated using Gauss quadratures, resulting in a system of non-linear ordinary differential equations which can be written as,

( )φ φ=2 2M N (27) Finally, the system of coupled non-linear ordinary differential equations given by Eqs. (24) and (27), are discretized in time via a generalized-α time integrator, resulting in a non-linear system of algebraic equations. This system of equations is then linearized, which yields a set of linear algebraic system of equations which are solved using Newton’s method. Newton iterations continue until the local residuals are sufficiently small at each time step, after which the method proceeds to the next time step, where the process is started all over again. The PHASTA code has been applied to the 3-D DNS analysis of bubbly flows [40] and typical results for initially spherical single and multiple interacting bubbles in an incompressible liquid (ICL) are show in Figs. 11 and 12, respectively. We can see in Fig. 11 the well known fact that rapidly rising large bubbles can deform and break up. Also Figs. 12 show a typical 3-D bubble coalescence simulation; note that bubble coalescence is complete for non-dimensional time τ > 0.15.

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Figure 11. Change in shape of the interface of a larger rapidly rising, initially spherical,

3-D bubble displayed with velocity vectors (PHASTA, 000,1g =ρρ ) [Nagrath et al., 2005], [3].

τ = 0.05

τ = 0.0

τ = 0.1 Speed contours at τ = 0.05

τ = 0.15 Pressure contours at τ = 0.10

Figure 12. Interacting 3-D bubbles (PHASTA, 000,1g =ρρ ) [Nagrath et al., 2005], [3].

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Figures 13 show a comparison of the ensemble averaged DNS results of PHASTA for a stratified wavy flow with the analytical solutions given by Galimov et al. [33]. This analysis demonstrates that appropriately averaged DNS results (i.e., using Eqs 16 and 17) can be used to derive suitable closure laws for multidimensional two-fluid models of typical separated flow regimes.

y/a

αcl

-0.5 0 0.5

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 13a. Comparison between analytical results (lines), and ensemble averaged DNS

results (squares) for cα , [3].

y/a-0.5 0 0.5

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

11000

12000

13000

14000

Figure 13b. Comparison between analytical results (lines), and ensemble averaged DNS

results (squares) for ( )ic cp p cα− ∇ , [3].

Next, figures-14 show typical DNS results using PHASTA for the analyses of a plunging liquid jet impacting a static pool of liquid. Figure-14a shows a 3-D subsurface view of the entrained air which is just starting to break off a bubble from the air-filled gas cavity surrounding the plunging

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liquid jet. Figure 14b – 14e are 2-D sections which show the air ingestion process and the dynamic grid adaptation which was required to obtain adequate spatial resolution. As can be seen, at a particular time, the liquid which surrounds the annular air-filled cavity collapses into the cavity thus creating a relatively large bubble which, as shown in Figs. 14f & 14g, subsequently breaks-up due to the relatively intense liquid shear. It is significant to note that this mechanism for air bubble entrainment is different from what others have proposed [50], [51].

Figure-14a. Plunging liquid jet: a three-dimensional subsurface view at the time of

incipient air bubble formation (t = 0.1330s) , [3].

Figure-14b. Bubble formation (t = 0.1330s).

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Figure-14c Bubble pinch off (t = 0.1338s)

Figure-14d Bubble ingestion (t = 0.13341s)

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Figure-14e Bubble distortion and air cavity rebound (t = 0.1592s)

Figure-14f Bubble break-up (t = 0.1613s)

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Figure-14g 3-D View of Bubble Break-up and Air Cavity Rebound (t = 0.1640s)

Finally let us consider compressible flow PHASTA predictions of a rapid transient. Figures-15 show a time sequence of the implosion process for an air/water bubble suddenly exposed to 100 atms over-pressure, where a perfect gas law equation of state was used for the air [52]. As can be seen in figures 15c & 15d the rapidly imploding air bubble experiences significant shape and interfacial instabilities, but these instabilities tend to wash out during the bubble’s re-expansion phase (i.e., see fig-15e).

Figure 15a. An imploding air bubble in room temperature water and subjected to 100 atms

overpressure-early stage of implosion process (t = 0.06 μs), [3].

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Figure 15b. Subsonic; ( )gR C< ) stage of bubble implosion (t = 0.085 μs), [3].

Figure 15c. High Mach number ( 1.0a gM R C= > ) stage of bubble implosion (t =0.093 μs),[3].

Figure 15d. Final stage of bubble implosion (t = 0.095 μs), [3].

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Figure 15e. Dynamic conditions during bubble rebound (t = 0.1 μs), [3].

Significantly the pressure field deep within the imploding bubble remains approximately spherical during the implosion process and this leads to high compression-induce-pressures and temperatures near the center of the bubble. Figure-16a shows the transient bubble radius in a vertical (y-axis) cut of the 3-D bubble simulations. It can be seen that the results for bubble radius agree with the well-known Rayleigh-Plesset bubble dynamics equation for an incompressible liquid (ICL), except during the final stages of the implosion process, where the bubble becomes distorted and compressibility in both phases may be important. The corresponding predicted pressure profiles are also shown in Figure-16b. A shock wave can be seen to be developing within the gas bubble and this steepening and intensifying pressure front bounces off itself at the center of the bubble resulting in , as can be seen in figures 16b and 16c, relatively high local pressures and temperatures near the center of the collapsed air bubble.

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Time (μs)

Figure 16. Air bubble implosion results along the vertical (y-axis) [Nagrath et al., 2006]

While these simulations were done for an air/water bubble with a very simple equation of state and no heat losses, they appear to indicate conditions suitable for sonoluminescence [53]. Indeed, even sonofusion [54], [55] might occur for sufficiently energetic implosions, since the shock wave within the bubble is spherical and, even though the bubble distorts during the implosion process, it remains intact.

Radial Position (μm)

Radial Position (μm)

0 0.02 0.04 0.06 0.08 0.1

11

10

9

8

0.121

2

3

4

5

6

7(a) Radius (μm)

(b) Pressure (atms)

(c) Temperature (K)

Time (μs)

103

2 4 6 8 10 12

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6. SUMMARY AND CONCLUSIONS Remarkable progress has been made during the last several decades in our ability to reliably predict multiphase flow phenomena. This paper summarizes some of this progress and shows that DNS can be used to not only directly analyze local phenomena of interest but also to provide detailed numerical “data” for the development of the closure laws required by multidimensional, multi-field, two-fluid models [34]. The development and evaluation of these models using suitable CMFD solvers is necessary if we are to have accurate and numerically efficient analytical tools for the design, analysis, and scale-up of multiphase flow and heat transfer processes and systems such as nuclear reactors. It is hoped that this paper will help stimulate more nuclear reactor thermal-hydraulic researchers to work on the simulation of multidimensional multiphase flows for application in nuclear energy technology.

NOMENCLATURE

iA′′′= Interfacial area density (1/m)

αjk= Volume fraction of field-j of phase-k

Cg = Speed of sound in gas/vapor

Ddv = Bubble diameter

g = Gravity

hjk= Enthalpy of field-j of phase-k

kjk= Turbulent kinetic energy of field-j of phase-k

kn = Wave number in direction-n

jkM = Interfacial force density of field-j of phase-k

εjk = Turbulent dissipation of field-j of phase-k

ρk = Density of phase-k

v jk =Velocity of field-j of phase-k

pjk = Static pressure of field-j of phase-k

Pjk = Turbulence production in field-j of phase-k

R = Bubble radius T

jkτ = Reynolds stress of field-j of phase-k

jkq′′ =Heat flux in field-j of phase-k

jkq′′′ =Volumetric heat source in field-j of phase-k

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jkτ =Viscous shear stress tensor of field-j of phase-k

ujk= Internal energy of field-j of phase-k

ui = Internal energy at the phasic interface

υ = Bubble volume

iv = Velocity of the phasic interface

v jk′ = Velocity fluctuations in field-j of phase-k

vr =Relative velocity ( vv vd c− )

Xjk = Phase indicator function of field–j of phase-k (+/- 1.0)

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Models Using the Second Law of Thermodynamics,” Int. J. Multiphase Flow, Vol. 16, No. 3, 481-494,1990.

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[46] Szabo, B. and Sahrmann, G.J., “Hierarchic Plate and and Shell Models Based on p- Extension,” Int. J. for Numerical Methods in Eng., Vol 26, 1855-1881, 1988. [47] Franca, L.P. and S. Frey, “Stabilized Finite Element Methods – II: The Incompressible Navier-Stokes Equations,” Comp. Meth. Appl. Mech. Engng., Vol. 99, 209-233, 1992. [48] Moser, R., Kim, J., and Mansour, N.,“Direct Numerical Simulation of Turbulent Channel Flow up to Re 590τ = ,” Phys. of Fluids, Vol. 11 Issue 4, 943-945, 1999. [49] Sussman, M., Smereka, P., and Osher, S.J., “A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flows,” J. of Computational Physics, Vol-14, 146, 1994. [50] Zhu, Y., Oguz, H.N. and Prosperetti, A., “On the Mechanism of Air Entrainment by Liquid Jets at a Free Surface”, J. Fluid Mech., Vol. 404, 151-177, 2000. [51] Bonetto,F., Drew,D.A., and Lahey,R.T.,Jr.,”The Analysis of a Plunging Liquid Jet-The Air Entrainment Process”,Chem. Eng. Comm.,Vol.130,11-29,1994. [52] Nagrath, S., Jansen, K.E., Lahey.R.T., Jr. and Akhatov, I., “Hydrodynamic Simulation of Air Bubble Implosion Using a FEM based Level Set Approach,” J. Comp. Physics, Vol. 215, No. 1, 98-132, 2006. [53] Moss, W.C., Clarke, D.B., White, J.W. and Young, D.A., “Hydrodynamic Simulations of Bubble Collapse and Picosecond Sonoluminescence,” Phys. of Fluids, Vol. 6 (9), 2979, 1994. [54] Taleyarkhan, R.P., West, C.D., Cho, J.S., Lahey, R.T., Jr., Nigmatulin, R.I., and Block, R.C., “Evidence for Nuclear Emissions During Acoustic Cavitation,” Science, Vol. 295, 1868-1873, March 8, 2002. [55] Taleyarkhan, R.P., West, C.D., Cho, J.S., Lahey, R.T., Jr., Nigmatulin, R.I., and Block, R.C., “Additional Evidence of Nuclear Emissions During Acoustic Cavitation,” Phys. Rev.-E, Vol. 69, 036109, 2004.

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