International Journal of Multiphase Flow 115 (2019) 137–143

Contents lists available at ScienceDirect

International Journal of Multiphase Flow

journal homepage: www.elsevier.com/locate/ijmulflow

A quantitative comparison of phase-averaged models for bubbly,

cavitating flows

Spencer H. Bryngelson

∗, Kevin Schmidmayer , Tim Colonius

Division of Engineering and Applied Science, California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, USA

a r t i c l e i n f o

Article history:

Received 4 January 2019

Revised 5 March 2019

Accepted 31 March 2019

Available online 3 April 2019

a b s t r a c t

We compare the computational performance of two modeling approaches for the flow of dilute cavitation

bubbles in a liquid. The first approach is a deterministic model, for which bubbles are represented in

a Lagrangian framework as advected features, each sampled from a distribution of equilibrium bubble

sizes. The dynamic coupling to the liquid phase is modeled through local volume averaging. The second

approach is stochastic; ensemble-phase averaging is used to derive mixture-averaged equations and

field equations for the associated bubble properties are evolved in an Eulerian reference frame. For

polydisperse mixtures, the probability density function of the equilibrium bubble radii is discretized

and bubble properties are solved for each representative bin. In both cases, the equations are closed

by solving Rayleigh–Plesset-like equations for the bubble dynamics as forced by the local or mixture-

averaged pressure, respectively. An acoustically excited dilute bubble screen is used as a case study

for comparisons. We show that observables of ensemble- and volume-averaged simulations match

closely and that their convergence is first order under grid refinement. Guidelines are established for

phase-averaged simulations by comparing the computational costs of methods. The primary costs are

shown to be associated with stochastic closure; polydisperse ensemble-averaging requires many samples

of the underlying PDF and volume-averaging requires repeated, randomized simulations to accurately

represent a homogeneous bubble population. The relative sensitivities of these costs to spatial resolution

and bubble void fraction are presented.

© 2019 Elsevier Ltd. All rights reserved.

1

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. Introduction

We consider the modeling of a flowing mixture of dilute cavita-

ion bubbles. The associated phenomenologies are often complex:

ubbles can oscillate, expand significantly (cavitate), and collapse

iolently. Notably, the multiphase bulk flow is sensitive to indi-

idual bubble motion; the shock-waves emitting from a cavitation

vent are often comparable to those in the bulk flow ( Reisman

t al., 1998; Brennen, 1995 ), and even just a few bubbles are

ufficient to modify larger-scale pressure waves ( Mettin and

auterborn, 2003 ). While the flow of dilute, cavitating bubbles

s only one possible bubbly flow scenario in a host of others,

uch as buoyancy-driven bubble motion ( Risso, 2018 ), the broad

atural occurrence and application of this subset motivates further

tudy of their behavior. Such bubbles emerge naturally via, e.g.,

avitation nucleation in mantis ( Patek et al., 2004; Patek and Cald-

ell, 2005 ) and pistol shrimp strikes ( Bauer, 2004; Koukouvinis

t al., 2017 ), bubble-net feeding of humpback whales ( Leighton

∗ Corresponding author.

E-mail address: spencer@caltech.edu (S.H. Bryngelson).

B

a

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ttps://doi.org/10.1016/j.ijmultiphaseflow.2019.03.028

301-9322/© 2019 Elsevier Ltd. All rights reserved.

t al., 20 04; 20 07 ), and vascular plant tissues ( Pickard, 1981 ).

ilute bubbly suspensions are also generated in engineering flow

pplications; purposefully, bubbles are nucleated in biomedical

ettings, including shock wave lithotripsy ( Coleman et al., 1987;

ishchalnikov et al., 2003; Ikeda et al., 2006 ), as shielding via

ubble screens ( Surov, 1999 ), and during underwater explo-

ions ( Etter, 2013; Kedrinskii, 1976 ). Unfortunately, cavitation is

lso an undesirable consequence of the flow dynamics in other

pplications. For example, cavitation causes erosion, noise, and

erformance loss of pipe systems ( Weyler et al., 1971; Streeter,

983 ), hydraulic machinery ( Brennen, 1995; Naudé and Ellis, 1961 ),

nd propellers ( Sharma et al., 1990; Ji et al., 2012 ).

A theoretical understanding of complex bubbly flows is often

rohibitive without significant simplifications. Further, a vast

ange of scales is usually present. The radius of single bubbles

an be as small as microns and can grow to as large as mil-

imeters ( Brennen, 1995 ), whereas bubble clouds and turbulent

eatures are often on the order of meters or larger ( d’Agostino and

rennen, 1983 ); the natural frequency and nominal collapse times

re usually on the order of microseconds, and the flow observation

ime scale is on the order of seconds ( Brennen, 1995 ). This makes

138 S.H. Bryngelson, K. Schmidmayer and T. Colonius / International Journal of Multiphase Flow 115 (2019) 137–143

Fig. 1. Schematic of (a) volume- and (b) ensemble-averaging models.

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computer simulations of the fully-resolved flow dynamics pro-

hibitive. Instead, modeling techniques are required to accurately

represent the flow.

The first models for dilute bubbly flows include theories

for linear scattering ( Foldy, 1945 ) and nonlinear oscillatory

systems ( Iordanskii, 1960; Kogarko, 1964; Wijngaarden, 1964;

1968 ). Since then, most models can be broadly classified as

either ensemble- ( Zhang and Prosperetti, 1994 ) or volume-

averaging ( Commander and Prosperetti, 1989 ). Herein, we focus on

two specific examples, one of each model, and assess their relative

computational cost and convergence.

We first discuss the technical differences between ensemble-

and volume-averaged models in Section 2 . The mathematical

formulation of each method is presented in Section 3 and the

numerical methods used to solve the associated equations are

outlined in Section 4 . In Section 5 , we consider an acoustically

excited bubble screen and compare the computational costs and

convergence of the methods. Key points and conclusions are

discussed in Section 6 .

2. Outset model comparison

The mixture-averaged flow equations associated with both

ensemble- and volume-averaged techniques represent the bubbles

as features that interact with the flow. However, the bubbles

are tracked and coupled to the liquid phase differently. Volume-

averaged models are formulated in an Euler–Lagrange framework,

where individual bubbles are Lagrangian particles, each sampled

from an underlying spatial distribution (see Fig. 1 (a)). The vol-

ume of gas per-unit-volume of the mixture is obtained locally

for each computational cell by projecting the volume of bubbles

onto the grid. The disturbances induced by the bubbles on the

flow is computed by decomposing the potential generated inside

each cell into background and bubble parts: the background

flow is constant inside a cell, whereas the potential generated

by each bubble decays with the distance from the bubble cen-

ter ( Fuster and Colonius, 2011 ). The ensemble-averaged approach

is an Euler– Euler method and is depicted in Fig. 1 (b); instead

of solving for the dynamics of individual bubbles, it evaluates

the statistically-averaged mixture dynamics by assuming a large

number of stochastically scattered bubbles dispersed within each

computational grid cell ( Ando et al., 2011 ).

Besides algorithmic differences, there are also differences

in assumptions that lead to their respective closures. In the

volume-averaged case, for the mixture dynamics to be considered

homogeneous, the length scale of the averaging volume (shown

in Fig. 1 (a)) must be much larger than mean bubble spacing and

uch smaller than the mixture length scale ( Nigmatulin, 1979;

rosperetti, 2001 ). Ensemble-averaged models are not beholden to

his assumption from the outset, though ultimately the separation

f scales is still invoked for model closure. In these theoretical

imits, ensemble- and volume-averaging are statistically equivalent

rocedures ( Batchelor, 1970; Biesheuvel and Wijngaarden, 1984 );

owever, neither the sensitivity of their respective closures to

nder-resolution nor their computational costs have been com-

ared for practical simulations. Herein, we formally contrast these

bservables.

. Mathematical model formulation

We describe the flow of a dilute suspension of dynamically

volving bubbles in a compressible liquid. Ensemble- and volume-

veraged models are presented; in addition to the assumptions

f Section 2 , we assume that there is no-slip between the gas

nd liquid phases and that the gas density is much smaller than

he liquid density. While phase-slip is required to describe key

henomenologies for some bubbly flows, such as buoyancy-driven

ixing flows ( Risso, 2018 ), they are thought to play a lesser role

n the cavitating bubble dynamics we consider here ( Matsumoto

nd Kameda, 1996 ). The average mixture equations of motion

ake their usual quasi-conservative form ( Commander and Pros-

eretti, 1989 ):

∂q

∂t + ∇ · F = s (1)

here q = { ρ, ρu , E } ar e the conservativ e variables, F =

ρu , ρuu + pI , (E + p) u } are the fluxes, and s are the source

erms associated with bubble modeling. Here, ρ , u , p , and E are

he mixture density, velocity vector, pressure, and total energy, re-

pectively. Mixture variables obey (·) = (1 − α)(·) l + α(·) g , where

is the void fraction and l and g denote the liquid and gas states,

espectively.

.1. Ensemble-averaged flow equations

Our formulation of the continuum ensemble-averaged equa-

ions generally follows that of Zhang and Prosperetti (1994) .

he equilibrium radii of the bubble population are represented

iscretely as R o , which are N bin bins of an assumed log-normal

DF with standard deviation σ p ( Colonius et al., 2008 ). The bins

re distributed using a Gauss–Hermitian quadrature, though pre-

ious works have shown that similar results are obtained with

he same number of quadrature points when using Simpson’s

ule ( Ando, 2010 ). The instantaneous bubble radii are a function of

S.H. Bryngelson, K. Schmidmayer and T. Colonius / International Journal of Multiphase Flow 115 (2019) 137–143 139

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hese equilibrium states as R (R o ) = { R 1 , R 2 , . . . , R N bin } . In this case,

= 0 . The mixture pressure is

p = (1 − α) p l + α

(

R

3 p bw

R

3 − ρ

R

3 ˙ R

2

R

3

)

, (2)

here p l is the liquid pressure, which we model using the

tiffened-gas equation of state ( Menikoff and Plohr, 1989 )

l p l =

1

1 − α

(E − 1

2

ρv 2 )

− �∞ ,l , (3)

˙ are the bubble radial velocities, and p bw

are the bubble wall

ressures. The equation of state is parameterized by the specific

eat ratio γl = 1 + 1 / �l and �∞

is the liquid stiffness. Overbars

denote the usual moments with respect to the log-normal PDF.

he void fraction is transported as

t α = 3 αR

2 ˙ R

R

3 , (4)

here D t ≡ ∂ t + u · ∇ is the substantial derivative operator. The

ubble dynamics are evaluated as

∂n φ

∂t + ∇ · (n φu ) = n

φ, (5)

here φ ≡{

R , ˙ R , p b , m v }

is the vector of bubble dynamic variables

p b is the bubble pressure and m v is the vapor mass; model

etails are described in Section 3.3 ) and n is the conserved bubble

umber density per unit volume

=

3

4 π

α

R

3 . (6)

.2. Volume-averaged flow equations

Our volume-averaged approach follows that of Maeda and

olonius (2018) . There are N bub bubbles, each located at x i , i = , 2 , . . . , N bub and tracked as a Lagrangian point. The continuous

oid fraction field α( x ) is defined via volumetric bubble smearing

s

(x ) =

N bub ∑

i =1

V i δ(d i , h ) , (7)

here δ is the Gaussian regularization kernel,

(d i , h ) =

{

1 (2 π) 3 / 2 h 3

e − d 2

i 2 h 2 , 0 ≤ d i < 3 h,

0 , 3 h ≤ d i , (8)

i is the bubble volume, d i = | x − x i | , and h is the kernel support

idth. The void fraction advects as

∂α(x )

∂t =

∂

∂t

N bub ∑

i =1

V i δ =

N bub ∑

i =1

∂V i

∂t δ +

N bub ∑

i =1

V i

∂δ

∂t , (9)

here

∂V i

∂t = 4 πR

2 i

˙ R i and

∂δ

∂t = −u · ∇δ. (10)

hus, the source terms of (1) for the volume-averaged method

re

=

q

1 − αD t α, (11)

hich transport the void fraction on the mesh, and the mix-

ure pressure is simply p = p l (1 − α) , for which p l is recovered

rom (3) .

.3. Single-bubble dynamics

We model the single-bubble dynamics under the assumption

hat the bubbles remain a spherical, ideal, and spatially uniform

as region, which does not interact with other bubbles, break-up,

r coalesce. The bubble dynamics are driven by pressure fluctua-

ions of the surrounding liquid; in our model, their radial acceler-

tions R are computed by the Keller–Miksis equation ( Keller and

iksis, 1980 ):

R

(1 −

˙ R

c

)+

3

2

˙ R

2

(1 −

˙ R

3 c

)=

p bw

− p ∞

ρ

(1 +

˙ R

c

)+

R

˙ p bw

ρc ,

(12)

here c is the speed of sound, p ∞

is the bubble forcing pressure,

nd

p bw

= p b −4 μ ˙ R

R

− 2 σ

R

(13)

s the bubble wall pressure, for which p b is the internal bubble

ressure, σ is the surface tension coefficient, and μ is the liq-

id viscosity. The evolution of p b is evaluated using the model

f Ando (2010) :

˙ p b =

3 γb

R

(− ˙ R p b + R v T bw

˙ m v +

γb − 1

γb

k bw

∂T

∂r

∣∣∣∣r= R

), (14)

here T is the temperature, k is the thermal conductivity, R v is

he gas constant, γ b is the specific heat ratio of the gas, and sub-

cript w indicates properties evaluated at the bubble wall r = R .

ass transfer of the bubble contents follows the reduced model

f Preston et al. (2007) :

˙ v =

Dρbw

1 − χv w

∂χv

∂r

∣∣∣∣r= R

, (15)

here χ v is the vapor mass fraction and D is the binary diffusion

oefficient. This single-bubble model includes thermal effects, vis-

ous and acoustic damping, and phase change, and its full formu-

ation and ability to represent actual bubble dynamics have been

resented elsewhere ( Preston et al., 2007; Ando, 2010 ).

. Numerical methods

Our numerical scheme generally follows that of Coralic and

olonius (2006) . For this, the spatial discretization of (1) in three-

imensional Cartesian coordinates is

∂q

∂t +

F x (q )

∂x +

F y (q )

∂y +

F z (q )

∂z = s (q ) , (16)

here F i are the i ∈ ( x, y, z ) flux vectors. We spatially integrate

16) within each cell-centered finite volume as

d q i, j,k

d t +

1

�x i

[F x i +1 / 2 , j,k − F x i −1 / 2 , j,k

]+

1

�y j

[F y

i, j+1 / 2 ,k − F y

i, j−1 / 2 ,k

]+

1

�z k

[F z i, j,k +1 / 2 − F z i, j,k −1 / 2

]= s (q i, j,k ) . (17)

e reconstruct the primitive variables at the finite-volume-cell

aces via a 5th-order WENO scheme ( Coralic and Colonius, 2006 )

nd use the HLLC approximate Riemann solver to compute the

uxes ( Toro et al., 1994 ). The time derivative is computed using the

rd-order TVD Runge–Kutta algorithm ( Gottlieb and Shu, 1998 ).

. Results

.1. Dilute bubble screen setup

We consider an acoustically excited dilute bubble screen as a

ase study of the differences and behaviors of the ensemble- and

140 S.H. Bryngelson, K. Schmidmayer and T. Colonius / International Journal of Multiphase Flow 115 (2019) 137–143

Fig. 2. The model flow system.

Fig. 4. Spatial convergence of bubble-screen-centered mixture pressure.

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of the stochastic closures discussed next.

volume-averaged flow models. Indeed, bubble screens are a prac-

tical configuration; they can serve as a reduced model for di-

lute bubble clouds and have been utilized to mediate structural

damage due to underwater explosions ( Langefors and Kihlström,

1967; Domenico, 1982 ), modify the open-channel flow topol-

ogy ( Blackaert et al., 2008 ), and even manipulate fish behav-

ior ( Patrick et al., 1985 ).

A schematic of the problem setup is shown in Fig. 2 . The do-

main is a square prism with x, y ∈ [ −H/ 2 , H/ 2] and z ∈ [ −L/ 2 , L/ 2] ,

where L = 25 mm and H = L/ 5 ; the boundaries are non-reflective.

A bubble screen occupies the cubic region x, y, z ∈ [ −H/ 2 , H/ 2] ,

wherein the initial void fraction is αo = 4 × 10 −5 . In the volume-

averaged case, the bubble positions are distributed uniformly in

the bubble-screen region via a pseudo-random number generator.

The bubbles are initially quiescent with mean (and equilib-

rium) radii uniformly sampled from a log-normal distribution

centered at R ∗o = 10 mm with standard deviation σ p . For the

volume-averaged method, this distribution must be sampled

multiple times via independent simulations to represent the

homogeneous mean flow. Ensemble-averaging instead samples

the most-probable equilibrium bubble radii N bin times and solves

the corresponding bubble dynamic equations for the statistically-

homogeneous flow; the relative costs of these procedures are

examined in Section 5.4 . The Cartesian grid has uniform mesh

spacing with N z = 250 and N x = N y = 50 unless otherwise stated.

The initial condition is quiescent at ambient pressure. A plane

acoustic source at z = −3 H/ 2 excites one cycle of a 300 kHz,

p A = 100 kPa sinusoidal pressure wave p ∞

in the + z direction.

The liquid is water with specific heat ratio � = 0 . 16 and stiffness

�∞

= 356 mPa ( Maeda and Colonius, 2018 ).

5.2. Comparison of observables

We compare the mixture-averaged pressure at the bubble-

screen center, p o ≡ p({ x, y, z} = { 0 , 0 , 0 } ) , for both methods.

Fig. 3 shows p o for (a) monodisperse and (b) polydisperse bubble

screens; the pressure grows then decays with the passage of

Fig. 3. Pressure p o for (a) monodisperse ( σp = 0 ) and (b) polydisperse ( σp = . 3 ) bubble dis

randomized bubble population, and their mean are shown, as well as the ensemble-avera

he wave, with additional oscillatory features present due to the

rapping of the wave in the screen region. In the volume-averaged

ase, multiple simulations are averaged to determine the homoge-

eous statistics; in Fig. 3 these are labeled as “single” and “mean”,

espectively. We compute the volume-averaged mean from 50

uch simulations, which have standard deviation σ d . In both cases,

he volume-averaged mean and ensemble-averaged pressures

atch closely, with the difference within 2 σ d of the individual

olume-averaged pressures almost everywhere.

.3. Spatial convergence

We next evaluate the simulation response to spatial under-

esolution. From the outset, it is unclear if the response should

e regular, as sub-grid modeling can introduce mesh-dependent

eatures. Fig. 4 shows the L 2 difference between the bubble-screen

entered mixture pressure and a high-resolution simulation for

oth methods. In the volume-averaged case, the mean pressure is

sed for comparisons and is computed via 200 individual simu-

ations at each spatial resolution. This, coupled with the necessity

f 3D simulations in the volume-averaged case (due to spatial

eterogeneity), limits the largest N z we consider. We see that

oth methods monotonically and linearly converge. Indeed, only

inearity can be expected despite higher-order numerical methods,

wing to the material discontinuities present and the truncation

rrors accumulated in the model assumptions, and thus stochastic

losures. While the relative error we show here is smaller for the

nsemble-averaged simulations, we emphasize that the methods

e use to compute it preclude a direct comparison of their ac-

uracy. Furthermore, the cost of considering a small number of

dditional spatial mesh points is relatively small compared to that

tributions. Individual volume-averaged simulations, each initialized with a different

ged result.

S.H. Bryngelson, K. Schmidmayer and T. Colonius / International Journal of Multiphase Flow 115 (2019) 137–143 141

Fig. 5. Convergence of bubble-screen-centered statistics with additional simulations N sim ; (a) mean pressure and (b) standard deviation.

5

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.4. Achieving stochastic closure and convergence

Following the previous discussion, in the volume-averaged case

ultiple volume-averaged simulations, each initialized with a dif-

erent sample of the bubble size and position distributions, are re-

uired to converge to the homogeneous mean flow. Fig. 5 (a) shows

he mean bubble-screen-centered pressure p o for additional simu-

ations N sim

; we see that it converges as 1 /N sim

for all N sim

and

z . The value of the difference is important when considering how

any simulations are required for a given accuracy.

Fig. 5 (b) shows the standard deviation of the individual simu-

ations associated with p o , ‖ σ [ p o (t) N sim

] . Its value is transient for

sim

� 40 and all N z , and is relatively constant for N sim

� 40 . That

s, at least 40 simulations are required for a faithful estimation of

d . Generally, ‖ σ d [ p o ( t )] ‖ increases with increasing N z , though this

hange is relatively small.

For polydisperse ensemble-averaged simulations, the bubble

ize distribution, given by a log-normal PDF, must be sampled mul-

iple ( N bin ) times. Sampling this distribution is expensive, as each

ample adds four equations and variables for each grid cell. In

ig. 6 , we show the convergence of bubble-screen-centered pres-

ure with N bin , as compared with a well-resolved N bin = 10 3 sim-

lation. Convergence appears to be exponential, with generally

arger error for larger σ p and fixed N bin . This is expected, as larger

p represents a broader bubble size distribution and thus, more

amples are required for the same accuracy. Small N bin entails rel-

tively large error; for σp = . 3 , N bin = 11 gives an error of 8% of

ig. 6. Convergence of ensemble-averaged simulations with N bin for varying degrees

f polydispersity σ p . F

a

A , whereas N bin = 101 gives an error of only 10 −4 % . We thus an-

icipate that, in this case, greater than N bin = 11 , but less than

bin = 101 samples are sufficient for most purposes.

.5. Computation cost

We compute the computational cost of each method by con-

idering the time-step cost of a simulation configuration. For this

enchmarking, simulations were performed with the same three-

imensional grid and matching time-step sizes on a single core of

twelve-core Intel Xeon E5-2670 Haswell 2.3 GHz processor. Here,

s is the time-step costs in seconds, which is computed as the

verage cost of a time-step over 10 0 0 time steps of a single simu-

ation. We emphasize that both methods have the same simulation

latform, following the general implementation of Coralic and

olonius (2006) , which ensures that the relative costs computed

or each method are restricted to the computational bubble model

tself.

Fig. 7 shows the relative cost of polydisperse simulations.

olydisperse volume-averaged simulations are no more costly than

onodisperse simulations, so T s is independent of N bin . The dashed

ines show their cost for different initial void fractions, αo , vary

ecause larger αo entails more bubbles, each of which is evolved

ia the Keller–Miksis equation. For the ensemble-averaged simula-

ions, T s is independent of αo , since it is represented as an Eulerian

ariable on the mesh. Instead, following the previous subsection,

he cost is paid when considering polydispersity. Indeed, this cost

ig. 7. Time-step cost T s for simulations of varying polydispersity resolution N bin

nd initial void fraction αo .

142 S.H. Bryngelson, K. Schmidmayer and T. Colonius / International Journal of Multiphase Flow 115 (2019) 137–143

Fig. 8. Time-step cost T s for simulations of varying spatial resolution N = N x N y N z , polydispersity resolution N bin , and either (a) void fraction αo or (b) number of bubbles

N bub for the volume-averaged simulations.

6

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t

is significant with T s going as N bin for N bin � 10 . Another consid-

eration for the volume-averaged case is that multiple simulations

are required for stochastic closure; following Fig. 5 (a), at least 40

simulations are likely required for most purposes and the dashed

lines of Fig. 7 should be ascended by this factor, accordingly. Thus,

for monodisperse simulations, the ensemble-averaging method is

cheaper for all αo . For the αo = 4 × 10 −5 bubble screen of previous

sections, if the bubble population is considered polydisperse with

σp = . 3 and we accept 1% relative errors in the stochastic closures

of both methods, then the methods have nearly the same cost,

with ensemble-averaging costing 81% that of volume-averaging. Of

course, smaller αo will instead favor volume-averaging.

Our final consideration is the dependence of T s on the spatial

grid resolution, and how it couples to the relative costs of polydis-

persity and number of bubbles N bub , where

N bub =

3 αo

4 π

(H

R o

)3

. (18)

Fig. 8 (a) shows this computational cost for several example cases.

In the volume-averaged case, T s is linear with N for small αo ,

and plateaus for small N if αo is sufficiently large. This is because

increasing N decreases the relative cost of computing the bubble

dynamics of volume-averaged simulations, as most of the time is

spent reconstructing variables and computing fluxes on the grid.

For ensemble-averaged simulations, T s increases linearly with N ,

as the bubble variables are computed in an Eulerian framework.

For small αo = 10 −8 , the volume-average cost of the bubbles is

small, and the ensemble-averaged simulations are more expensive

for any N bin . For larger αo , the ensemble-average simulations are

generally cheaper, except for cases with sufficiently large N bin .

The relative cost of simulating individual bubbles in the

volume-averaged case is shown in Fig. 8 (b). Here, �t b is the addi-

tional time-step cost of simulating one additional bubble for fixed

N , and �t g is this cost for one additional grid point for fixed N bub .

We compute �t b = 3 . 15 × 10 −4 s /N bub and �t g = 3 . 53 × 10 −5 s /N,

and confirm that these values are independent within 1% for vary-

ing N and N bub , respectively. From these, the total time-step cost

of a volume-averaged simulation is simply �t = N bub �t b + N�t g ,

which we confirm for independently selected cases is within

2% of the actual cost. In Fig. 8 (b) we also label the intersection

of this curve with the cost of an ensemble-averaged simula-

tion with polydisperse resolution N bin (which is independent of

N bub ). Volume-averaged cases with larger N bub for constant N bin

are more expensive. For example, for N bin = 11 and N = 10 3 , a

single ensemble-averaged simulation is cheaper than a single

volume-averaged simulation when N > 916 .

bub

. Discussion and conclusions

We presented a computational analysis of ensemble- and

olume-averaged dilute bubbly flow models in the context of

n acoustically excited dilute bubble screen. Results showed, for

he first time, that the mixture pressure at the bubble screen

enter closely matched for both the mean volume-averaged and

nsemble-averaged methods.

As a step towards assessing the relative computational cost of

ach method, we focused on the cost of closing the stochastic part

f the models. The volume-averaged numerical model requires

ultiple, deterministic simulations of heterogeneous, random-

zed dilute bubble populations to converge to the homogeneous

veraged flow. We showed that the error in the mean flow ap-

roximation decreased as O(N

−1 sim

) , with the associated coefficient

etting the required number of simulations for stochastic clo-

ure within a given error bound. In the case of an acoustically

xcited bubble screen, error in the bubble-screen averaged pres-

ure was about 1% for N sim

= 40 , and independent of the spatial

esolution. Polydisperse ensemble-averaged simulations require

ultiple ( N bin ) samples of the log-normal PDF of most-probable

quilibrium bubble radii. We showed that the error associated

ith undersampling this PDF decreased approximately expo-

entially with increasing N bin , with slower decay for larger PDF

tandard deviations. Ultimately, N bin � 10 was required for faithful

epresentation of the polydisperse flow physics. Together, these

nalyses provided a framework for computing total computational

ffort.

In the polydisperse case, the cost of ensemble-averaged simu-

ations was dominated by its stochastic closure. That is, the addi-

ional reconstructed variables and computed fluxes on the Eulerian

rid associated with the underlying PDF were the primary time-

ost of simulation for N bin > 5 . In such cases, ensemble-averaged

imulations were generally more expensive than their volume-

veraged counterparts. However, monodisperse simulations were

enerally cheaper for the ensemble-averaged case, as only four

dditional equations were added to the governing system and no

ndividual bubble dynamics needed to be resolved. These relative

osts were complicated by the separate costs of computing single-

ubble dynamics and the Euler flow equations in the volume-

veraged case. For this, we linearly decomposed these costs such

hat the relative cost of each method could be assessed for any

ombination of N, αo (or N bub ), and N bin . For low void fraction

imulations on large spatial grids, the relative cost of computing

ingle-bubble dynamics was small and volume-averaged simula-

ions were preferable. For larger void fractions on relatively coarse

S.H. Bryngelson, K. Schmidmayer and T. Colonius / International Journal of Multiphase Flow 115 (2019) 137–143 143

m

t

A

g

R

A

A

B

B

B

B

BC

C

C

C

d

D

E

F

F

G

I

I

J

K

K

K

K

L

L

L

M

M

M

M

N

N

P

P

P

P

P

P

P

R

R

S

S

S

T

W

W

W

Z

eshes, the relative cost of computing the bubble-dynamic equa-

ions was large and ensemble-average simulations were preferable.

cknowledgements

This work was supported by the Office of Naval Research under

rant numbers N0014-17-1-2676 and N0014-18-1-2625 .

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