Computers and Fluids 179 (2019) 301–308

Contents lists available at ScienceDirect

Computers and Fluids

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

Benchmark solutions

Simulations of gas-liquid compressible-incompressible systems using

SPH

Prapanch Nair a , ∗, Gaurav Tomar b

a Institute for Multiscale Simulation, Friedrich-Alexander Universität Erlangen-Nürnberg, Erlangen, Germany b Department of Mechanical Engineering, Indian Insitute of Science, Bengaluru, India

a r t i c l e i n f o

Article history:

Received 29 March 2018

Revised 12 November 2018

Accepted 13 November 2018

Available online 14 November 2018

Keywords:

Compressible-Incompressible flow

Rayleigh–Plesset equation

Incompressible SPH

Effervescent atomization

Gas bubble

a b s t r a c t

Gas bubbles immersed in a liquid and flowing through a large pressure gradient undergo volumetric de-

formation in addition to possible deviatoric deformation. While the high density liquid phase can be as-

sumed to be an incompressible fluid, the gas phase needs to be modeled as a compressible fluid for such

bubble flow problems. The Rayleigh–Plesset (RP) equation describes such a bubble undergoing volumet-

ric deformation due to changes in pressure in the ambient incompressible fluid in the presence of capil-

lary force at its boundary, assuming axisymmetric dynamics. We propose a compressible-incompressible

coupling of Smoothed Particle Hydrodynamics (SPH) and validate this coupling against the RP model

in two dimensions. This study complements the SPH simulations of a different class of compressible-

incompressible systems where an outer compressible phase affects the dynamics of an inner incom-

pressible phase. For different density ratios, a sinusoidal pressure variation is applied to the ambient

incompressible liquid and the response of the bubble in terms of volumetric deformation is observed and

compared with the solutions of the axisymmetric RP equation.

© 2018 Elsevier Ltd. All rights reserved.

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. Introduction

The response of compressible gas bubbles flowing across a pres-

ure gradient has a wide range of industrial applications. For ex-

mple, effervescent flow atomizers (EFA) atomize liquids more ef-

ciently than conventional atomizers by introducing gas bubbles

nto liquid jets. The gas bubbles expand and explode due to the

ressure gradient across the nozzle and eventually cause fragmen-

ation of the surrounding liquid phase. Due to their insensitivity to

iquid viscosity [17] and reduced maintenance cost [30] , EFAs have

ecome a commonplace device to achieve efficient drop distribu-

ion and larger spray cone angles [33] . So far, experimental stud-

es have been helpful in only understanding the bulk properties

f effervescent flows [2,9,13] . A spatially and temporally resolved

nderstanding of the compressible bubbles deforming due to pres-

ure changes is generally lacking in experimental studies.

Numerical simulations where two phases with different com-

ressibility treatments are required, on the other hand, face

ther difficulties. The following are the three most important is-

ues in numerical modeling of such two phase compressible-

ncompressible (CI) systems [3] :

∗ Corresponding author.

E-mail address: prapanch.nair@fau.de (P. Nair).

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ttps://doi.org/10.1016/j.compfluid.2018.11.015

045-7930/© 2018 Elsevier Ltd. All rights reserved.

1. Density is constant in the incompressible phase and is depen-

dent on pressure and temperature in the compressible phase.

2. A zero divergence constraint has to be applied in the incom-

pressible phase, and modeling this constraint near the interface

involves numerical challenges.

3. The normal stress balance at the interface can become compli-

cated due to pressure waves in the compressible phase.

Hence numerical simulation approaches for solving CI two

hase flow problems are rather limited. Traditional grid based CFD

ethods attempt such problems using either a (1) Unified ap-

roach or a (2) Non-unified approach. Unified approach uses the

ame solver in the entire domain with localization parameters that

odify the terms in the Governing Equations based on the phase

f the node (discretized element) in question. Such methods tra-

itionally assume weak compressibility (see [11,12] ) in the incom-

ressible phase and hence compressible Navier–Stokes equation is

olved in the entire domain. Such methods address the incom-

ressibility as a limiting case. A major disadvantage of such an ap-

roach is the huge constraint on the time steps ( �t ) used in time

ntegration, since the wave speeds used in the time step criteria

re high in the near-incompressible region. Several methods use

imilar approach together with interface tracking schemes (for e.g.,

ow Mach Schemes, Generalized Projection Method, Stabilized Fi-

ite Element method, Marker and Cell scheme generalized to Euler

quations and the Discontinuous Galerkin scheme [25] ). The non-

302 P. Nair and G. Tomar / Computers and Fluids 179 (2019) 301–308

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unified schemes, on the other hand, solve both the fluid phases

differently in a spatially decomposed domain. Explicit or implicit

coupling is used to achieve stress balance. Such approaches are

relatively less explored in literature [12] . One difficulty with such

methods is the requirement to use conservative formulation in the

compressible phases, and non-conservative formulations based on

primitive variables in the incompressible phases. Separate set of

governing equations require different numerical approaches mak-

ing the solution complex to implement.

Meshless methods, such as the Smoothed Particle Hydrodynam-

ics (SPH), have advantages that could redress some of these dif-

ficulties. Primarily, mass conservation is naturally satisfied across

an interface in SPH. Due to its Lagrangian nature, different com-

ponents in the domain are naturally advected without the need

for detailed advection schemes. Also compressibility is achieved by

the corresponding variation in number density of particles while

an adaptive discretization length (smoothing length) ensures the

order of accuracy is not deteriorated.

We introduce a meshless method based on the Smoothed Parti-

cle Hydrodynamics (SPH) method, that couples a truly incompress-

ible phase [6] with a compressible phase. In most cases in litera-

ture (for example, [8,16] ), a stiff equation of state is used to obtain

pressure in the nearly incompressible phase. Numerically, such ap-

proaches result in severe time-step restrictions [31] . A more strict

incompressibility condition could be achieved by solving the pres-

sure Poisson equation (PPE) [6] or a condition for isochoricity [23] .

Owing to the Lagrangian nature of SPH, a coupling of phases with

different com pressibility treatments is feasible. In spite of exten-

sive research on both the compressible and incompressible ver-

sions of SPH, independently, a coupled approach has not been at-

tempted until recently. Coupled simulations where the presence of

air affects the flow of the incompressible fluid is validated and pre-

sented recently in Lind et al. [15] . However the response of the

gas bubble confined within an incompressible liquid, as is preva-

lent in effervescent atomization and cavitation phenomena, have

not been simulated using a coupled approach before to the best of

our knowledge. The compressible bubble responding to the pres-

sure fluctuations at the boundary of the outer fluid through vol-

umetric deformation and boundary expansion (or contraction) in

the presence of surface tension is a significantly stiff system. This

paper describes the coupling of compressible SPH with truly in-

compressible SPH where a compressible gas is immersed in an in-

compressible fluid.

This paper is organized as follows. In the following section we

present the SPH implementation of the CI two phase flow sim-

ulation (CI–SPH). In Section 3.1 we derive the Rayleigh–Plesset

(RP) problem applied to two dimensional finite domains of cir-

cular and rectangular shapes and proceed to validate the CI–SPH

method against the theory in Sections 3.2 and 3.3 . We compare the

time response of a compressible bubble against solutions of the RP

equation and present the motion of the interface.

2. SPH for compressible–incompressible flows

In what follows, we describe the SPH method used to couple

the compressible and incompressible phases and the method of

application of time varying ambient pressure, P ∞

( t ). The momen-

tum conservation equation for the entire domain is given by

d � u dt

∣∣a

= −∑

b m b p a + p b ρa ρb

∇ a W ab

+

∑

b m b μa + μb

ρa ρb F ab � u ab +

� f int ab

+

� f B a ,

(1)

where m is the mass, ρ is the density, p is the hydrodynamic pres-

sure, μ is the coefficient of viscosity and

� u is the velocity at an

SPH particle a or SPH particle b in the neighborhood of a and the

relative velocity, � u a − � u b is denoted by � u ab . Any body force act-

ing on the system is represented by � f B a , which assumes a value

f 0 for the simulations presented in this paper. The function W

s the SPH smoothing function (also known as the smoothing ker-

el) with a finite cut-off radius defined for an SPH particle pair

s W ab = W (r ab , h ) , where h is the smoothing length of the kernel.

hroughout this paper we have use the quintic Wendland kernel

32] function for SPH approximations owing to its superior stabil-

ty [7] . The gradient of the smoothing function appears as ∇ a W ab

or an SPH particle a with respect to its neighbor b . The pressure

radient term, first term on the right hand side, is based on the

pproximation given by [5] , suited for multiphase flows with large

ensity ratio. The viscous force term, second term on the right

and side, is given in Morris et al. [20] The radial derivative of the

ernel, given by F ab [18] is computed from the gradient of W as

ab =

� r ab · ∇ a W ab

r 2 ab

+ (εh ) 2

(2)

here ε is a small number ( ε ∼ 0.01) introduced to avoid division

y zero in the rare event of SPH particles overlapping in position.

ere, � r ab =

� r a − �

r b , where � r a and

� r b are the position vectors of par-

icles a and b , respectively, and r ab = ‖ � r ab ‖ . The incompressible phase is bounded by the interface with the

ompressible phase on one side and the free surface on the other

see Fig. 1 a and b). The PPE that ensures incompressibility and ac-

ounts for a Dirichlet boundary condition (BC) applied through the

runcated domain at the free surface [22] is given by

(p a − P ∞

) κ −∑

b

m b

ρb

4 p b ρa + ρb

F ab =

∑

b

m b

ρb

(� u ab · ∇ a W ab

�t − 4 P ∞

ρa + ρb

F ab

). (3)

his equation results in a linear system for unknown pressure val-

es p a . The linear system is then solved using the Biconjugate gra-

ient stabilized method (BiCGSTAB) [29] , for the unknown pres-

ures p a for the incompressible domain. Here P ∞

represents the

ime varying ambient pressure that is applied at the free surface.

he term κ in the above equation, given by

=

∑

b full

m b

ρb

4

ρa + ρb

� r ab · ∇ a W ab

r 2 ab

+ ε2 , (4)

here b full represents the full neighborhood of a particle (in the

ulk of the fluid) far from the free surface. This factor remains con-

tant for given smoothing parameters and constant density [22] .

he particles in the compressible region form the Dirichlet bound-

ry condition for the solver on the inner boundary of the incom-

ressible domain.

For the compressible phase, we consider a linear isothermal

ressure-density relation given by [24] :

p i = ρi c 2 . (5)

he velocities encountered are subsonic in all the cases considered

ere. The speed of sound in the compressible phase is denoted by

. In order to correspond to the RP equations we switch off the

iscous forces in the compressible phase. For successful application

f the time varying pressure at the ambience, the pressure gradi-

nt term also needs to account for deficiency near the free surface.

ince pressure at the free surface (Dirichlet BC for pressure) varies

n time, the pressure gradient computation should account for it

s a time varying boundary condition. We implement a penalty

pproach similar to that used in the solution for PPE to set this

irichlet BC. We assume pseudo particles across the free surface

uch that a particle at (or near) the free surface has a full neigh-

orhood that includes the pseudo particles. The pseudo particles

re assigned the time varying pressure P ∞

to serve as the Dirich-

et BC for pressure in computing the pressure gradient. Then the

ressure gradient approximation term for a particle a near the free

urface with a full neighborhood can be split as follows:

−∑

b full

m b

p a + p b ρa ρb

∇ a W ab

P. Nair and G. Tomar / Computers and Fluids 179 (2019) 301–308 303

Fig. 1. Schematic of the Rayleigh-Plesset problem in 2D.

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W

fi

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T

−

w

l

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−

= −∑

b

m b

p a + p b ρa ρb

∇ a W ab −∑

b pseudo

m b

p a + P ∞

ρa ρb

∇ a W ab

= −∑

b

m b

p a + p b ρa ρb

∇ a W ab

− p a + P ∞

ρa

( ∑

b full

m b

ρb

∇ a W ab −∑

b

m b

ρb

∇ a W ab

)

≈−∑

b

m b

p a + p b ρa ρb

∇ a W ab −p a + P ∞

ρa

(

0 . 0 −∑

b

m b

ρb

∇ a W ab

)

. (6)

Here the summation �b is over all the actual particles b ; ∑

b full

s over the particles b in the entire neighborhood including pseu-

oparticles, if present; ∑

b pseudo is over all particle b which are

seudo particles outside the free surface and within the neighbor-

ood of the particle a . This modified pressure gradient term re-

laces the pressure gradient term of Eq. 1 for all particles a be-

onging to the incompressible phase. Thus the time varying ambi-

nt pressure BC can be applied to both the pressure solver as well

s the pressure gradient operator for the incompressible phase.

Different surface tension models are available in literature for

wo phase flows using SPH. These can be broadly classified into

ontinuum surface forced (CSF) based approaches [1,28] ), pairwise

orce based approach [21] or geometric reconstruction based ap-

roaches [35] . Pairwise forces have the issue of virial pressure, an

dditional pressure due to pairwise forces, that would affect the

ressure in the compressible phase of our setup. Geometric re-

onstruction based approaches apply forces on one layer of par-

icles and this leads to deterioration in the order of accuracy since

he forces are applied at interface width typically smaller than

he smoothing length. Therefore, we use a simple model based

n the continuum surface force (CSF) that is based on computa-

ion of gradients of a discontinuous color function across the two

hase interface [19] . For details on this CSF implementation please

ee [19] .

. Validation of the CI-SPH

The RP equation in 2D is derived in this section for a rectan-

ular and a circular geometry. We then preform the validation of

he above introduced CI-SPH method to solve the RP equation in

ectangular and circular domains as depicted in Fig. 1 .

.1. Rayleigh-Plesset equation in two dimensions

The RP equation governs the radial and axisymmetric deforma-

ion of a (compressible) spherical bubble in an infinite body of

ncompressible liquid [14,26,27] . The equation assumes no mass

ransfer within the bubble. For a given ambient time varying pres-

ure P ∞

( t ) and known initial pressure within the bubble, P B (t = 0) ,

he RP equation can be used to solve for the bubble radius R ( t ).

hile the original RP equation [27] is written for a 3D bubble con-

ned in an infinite incompressible fluid, its 2D version can be read-

ly derived for a drop within a finite incompressible fluid domain.

In the rest of this paper we use only a two dimensional ver-

ions of the RP equation after deriving them for a circular and a

ectangular geometry schematically represented in Fig. 1 .

From conservation of momentum we have,

∂u

∂t + u

∂u

∂r = − 1

ρ

∂ p

∂r (7)

here r is the radial coordinate with respect to the origin, u is the

elocity (radial) of the fluid at any position r . From the require-

ent for continuity, the velocity of the interface dR / dt and the ve-

ocity at any point in the fluid can be related as

=

R

r

dR

dt . (8)

herefore,

1

ρ

∂ p

∂r =

∂

∂t

(R

r

dR

dt

)+

R

r

dR

dt

∂

∂r

(R

r

dR

dt

)(9)

=

1

r

(

R

d 2 R

dt 2 +

(dR

dt

)2 )

− R

2

r 3

(dR

dt

)2

, (10)

here p is the pressure. Integrating this equation by applying the

imits from a point just outside the interface (excludes surface ten-

ion effects) to the outer boundary of the incompressible fluid,

p ∞ ∫ p ′

B

1

ρdp =

R ∞ ∫ R

(

1

r

(

R

d 2 R

dt 2 +

(dR

dt

)2 )

− R

2

r 3

(dR

dt

)2 )

dr (11)

304 P. Nair and G. Tomar / Computers and Fluids 179 (2019) 301–308

H

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b

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⇒

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a

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t

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o

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T

E

b

t

r

t

c

t

r

i

i

i

a

a

i

i

p

t

l

d

⇒ R̈ =

(p B − p ∞

ρ+

˙ R

2

(1

2

(1 −

(R

R ∞

)2 )

− ln

R ∞

R

))1

R ln

(R ∞ R

)(12)

Here the pressure p B is accounted for right outside the bubble in-

terface. If P B is the pressure within the bubble, then considering

the Young–Laplace pressure jump due to surface tension P B − p B =S/R, where S is the surface tension coefficient, the above expres-

sion can be written as, [P B − P ∞

ρL

− S

ρL R

−(

R

2

R

2 ∞

− 1

)˙ R

2

2

− ˙ R

2 ln

R ∞

R

]1

R ln

R ∞ R

= R̈ , (13)

which is the 2D equivalent of the RP equation for a ‘circular’ bub-

ble (see Fig. 1 b). The subscript L is used to represent quantities

in the incompressible liquid domain. For a rectangular bubble con-

fined by a finite incompressible fluid domain and separated by a

plane interface (see Fig. 1 a), this equation may be derived similarly

and is given by (P B − P ∞

ρL

)1

R ∞

− R

= R̈ (14)

In this paper the circular and rectangular versions of RP equations

( Eqs. 13 and 14 ) are solved numerically to obtain the time response

of the compressible bubble. The equations are solved for radius,

and from the corresponding volume change, the pressure in the

bubble P B ( t ) may be computed using the assumed pressure density

relation, Eq. 5 , which is also used in the SPH compressible phase

model.

Throughout this paper we use the explicit 4th order Runge–

Kutta method to solve the RP equations.

3.2. Rectangular domain

Simulations of the RP problem for a rectangular bubble are pre-

sented here. The compressible phase is bounded by a wall at the

bottom and the incompressible phase at the top. The wall is mod-

eled as layers of static SPH particles. The width of the wall is cho-

sen to be larger than the support radius of the smoothing function.

While velocities of these wall particles are set to zero, they are

included in the pressure solver, similar to the wall set up in Ref.

[22] . Though the dynamics are confined to one dimension, we use

a 2 dimensional domain with a finite width and periodic boundary

conditions at the vertical edges of the domain. In the rectangular

case we have neglected the surface tension. The rectangular bubble

has an initial radius (distance of interface from the bottom wall) of

2.5 units and R ∞

is 6 units. The following three different density

ratios have been considered 10 0 0:1, 10 0:1 and 10:1 and the initial

pressure inside the bubble is set to be at P B (t = 0) = ρc 2 = 20 in

each case. The value of speed of sound was changed correspond-

ingly for each density ratio. A time varying pressure given by

P ∞

(t) = 20 + 1 . 5 sin

(2 πt

70

)(15)

is applied at the free surface of the incompressible fluid. Initially

the pressure within the bubble is the same as that in the outer

boundary. As the pressure increases, the volume of the bubble de-

creases and its pressure increases, while the CI interface moves to-

wards the bottom wall.

The natural frequency for the 1D bubble can be derived from a

linear stability analysis. Assuming the initial pressure in the bubble

P B 0 to be the same as the far-field pressure outside of the incom-

pressible region, we have

P B − P ∞

ρL H

= R̈ . (16)

ere ρL is the density of the incompressible fluid and H = R ∞

− R .

ubscript is added to distinguish it from the density in the bub-

le, ρB . Let P B 0 = ρB 0 c 2 at t = 0 , where ρB 0 is the initial density.

et P B be the pressure at time t due to a small perturbation in R .

or a small time after t = 0 , R = R 0 + r ′ e iωt , where r ′ is the small

erturbation given to R .

R̈ =

ρB c 2 − ρB 0 c

2

ρL H

=

ρB 0 c 2

ρL H

(R 0

R

− 1

)=

ρB 0 c 2

ρL H

(R 0

R 0 + r ′ e iωt − 1

)=

ρB 0 c 2

ρL H

(− r ′

R 0

e iωt

)(using binomial theorem) (17)

ω

2 =

ρB c 2

HρL R 0

, (18)

here ρL is the density of the incompressible fluid, H is the dis-

ance between the CI interface and the free interface and R 0 is the

nitial radius of the bubble.

Thus the natural frequency of the rectangular RP system is

iven by

f N =

ω

2 π=

1

2 π

√

ρB 0 c 2

ρL HR 0

. (19)

The pressure and radius of the compressible 1D bubble is

hown in Fig. 2 , for different density ratios. The simulation pro-

eeds well during the first cycle of oscillation in all the cases. As

he density ratio increases, the simulation approaches the limit of

massless compressible bubble governed by the RP equation, as

een in the pressure plots in Fig. 2 . However, towards the end of

he first cycle of oscillations, the interface fails to remain stable,

ue to the absence of interface forces. This waviness in the inter-

ace reflects in the radius response as a deviation from the theoret-

cal response curve. On the other hand the pressure is computed as

he mean of the pressure of all particles within the bubble. More-

ver, with increasing density ratio, the simulated system becomes

loser to the Rayleigh-Plesset model where the bubble is assumed

o be massless. Hence, with increasing density ratio the response

f pressure is captured with increasing accuracy. Fig. 3 shows the

requency domain of the response of pressure within the bubble.

he natural frequency of the compressible bubble, as computed in

q. 19 and the forced frequency of the applied pressure are marked

y blue lines. For all the three density ratios shown, we see that

he natural frequency and force frequency show up in the pressure

esponse. The forcing frequency corresponds to the larger ampli-

ude. The RP theory, SPH simulations and the estimated frequen-

ies compare well for low density ratios. For high density ratios

he results compare well during the initial stages of the bubble’s

esponse.

Fig. 4 shows the particle configuration of the compressible and

ncompressible phases for a density ratio of 10:1 at different times

n the simulation. The phase in the upper part is the incompress-

ble phase. At time t/T = 3 . 2 ( T is the period of oscillation of the

pplied pressure) the interface has become unstable and is seen

s a wavy line. However, the particle arrangement at earlier times

s stable and is also reflected in the pressure and radius response

n Fig. 2 . We would like to note here that these simulations were

erformed in the absence of any stabilizing effects due to surface

ension forces at the interface. The artifical mixing that occurs at

ater times may be avoided by artificial inter-phase forces as intro-

uced in Ref. [10] .

P. Nair and G. Tomar / Computers and Fluids 179 (2019) 301–308 305

Fig. 2. Pressure and radius variation with time for different density ratios. Comparison with numerical solution of Rayleigh–Plesset equation. Time is non-dimensionalized

by the period of oscillation of the applied ambient pressure, T .

306 P. Nair and G. Tomar / Computers and Fluids 179 (2019) 301–308

Fig. 3. Discrete Fourier transform of the response of the pressure of the bubble for different density ratios. Comparison with the numerical solution of the 2D rectangular

Rayleigh–Plesset equation, shown by continuous lines. The amplitude of the pressure response is given by A p , the forcing frequency and the theoretical natural frequency are

given by f F and f N , respectively.

Fig. 4. Particle configuration of the compressible (green) and incompressible (red) phases, corresponding to a density ratio of 10:1. (For interpretation of the references to

color in this figure legend, the reader is referred to the web version of this article.)

−

ω

T

F

d

q

t

3.3. Circular domain

For the circular bubble case considered here (see Fig. 1 ), we

solve the RP equation with an applied surface tension coefficient

of S = 1 . The surface tension model is based on the CSF model and

is implemented as described in [19] . The time varying pressure ap-

plied at the free surface boundary of the incompressible fluid is

P ∞

(t) = 20 + 1 . 0 sin

(2 πt

30

)(20)

The corresponding natural frequency can be derived using a similar

procedure as with the 1D case. Using similar assumptions as in the

1D case,

R̈ =

[ρB 0 c

2

ρL

(R

2 0

R

2 − 1

)+

˙ R

2

2

(1 − R

2

R

2 ∞

)− ˙ R

2 ln

R ∞

R

]1

R ln

(R ∞ R

)(21)

ω

2 r ′ e iωt ≈(

−ρB 0 c 2

ρL

2 r ′ R 0

e iωt

)1

R 0 ln V R

(using V R =

R ∞ 0

R 0

)

(22)

2 ≈ 2 ρB 0 c 2

ρL R

2 0

ln V R

. (23)

Here, R ∞ 0 is the initial radius of the incompressible domain.

he natural frequency of the circular RP system is

f N =

ω

2 π=

1

2 π

√

2 ρB 0 c 2

ρL R

2 0

ln

(R 0

R ∞ 0

) (24)

ig. 5 shows the pressure and radius response of the bubble for a

ensity ratio of 10:1. The pressure response is shown in the fre-

uency domain in Fig. 6 . As seen in Fig. 5 , the pressure in the in-

erior of the bubble is not as accurately simulated (with respect

P. Nair and G. Tomar / Computers and Fluids 179 (2019) 301–308 307

Fig. 5. Radius and pressure response of the 2D circular drop. Time is non-dimensionalized by the period of oscillation of the applied ambient pressure, T .

Fig. 6. Frequency domain of the pressure of the 2D circular drop.

t

f

d

l

o

Fig. 7. Position of interface at different times. (For interpretation of the references

to color in this figure legend, the reader is referred to the web version of this arti-

cle.)

t

f

a

p

F

l

o the theory) as the radius of the bubble. Possibly, this is the ef-

ect of inaccuracy in the surface tension model applied. At higher

ensity ratios the interface undergoes much severe distortions and

eads to mixing of the two fluids. Fig. 8 shows such an instance

f mixing where particles from the compressible phase penetrates

ig. 8. Particle mixing at the bubble interface for density ratio of 100:1, shown at two d

egend, the reader is referred to the web version of this article.)

he incompressible phase making it difficult to define bubble inter-

ace.

However, the initial time response is accurately captured. More

ccurate surface tension implementations, for example, that pro-

osed in [1] or [28] for larger density ratio interfaces may be im-

ifferent time instances. (For interpretation of the references to color in this figure

308 P. Nair and G. Tomar / Computers and Fluids 179 (2019) 301–308

[

[

[

[

plemented to mitigate the artificial mixing of the phases at the in-

terface. In Fig. 7 , the location of the interface at different times can

be seen. At t/T = 4 . 33 we see that the particles begin to get dis-

turbed from their initial lattice. We note that these distortions can

be reduced by remeshing [4] or redistributing [34] the particles.

4. Summary

Several two phase flow problems of practical importance in-

volve compressibility effects in at least one of the phases. We cou-

pled compressible and the incompressible SPH methods such that

the compressible and incompressible phases mutually provide the

Dirichlet boundary condition for pressure to each other. Simulation

of a compressible phase confined within an incompressible phase

was previously not attempted using SPH methods. To motivate ap-

plication to gas bubbles moving through strong pressure gradients,

we derived the RP equation for a rectangular and a circular bub-

ble in two dimensions and compared the CI–SPH simulations with

these results. A basic CSF based surface tension model was used.

For Dirichlet BC at the free surface, a penalty term was used in

both the PPE and the pressure gradient approximation.

Artificial mixing of the two phases at the interface was ob-

served. This could be due to the inaccuracies in the surface ten-

sion model. Improved surface tension models and other correction

approaches in gradient treatments available in SPH literature could

rectify this problem. Notwithstanding, the initial time solutions for

high density ratio of up to 10 0 0:1 compared well with the theory.

For low density ratios, the solutions compared well with the the-

ory without interface distortions. In effervescent atomization, more

complex scenarios, especially those involving deviatoric deforma-

tions of the bubble would occur. The CI–SPH model presented and

validated here is a significant step towards simulation of such com-

plex scenarios.

Acknowledgments

The authors gratefully acknowledge the support of the Clus-

ter of Excellence Engineering of Advanced Materials, ZISC, FPS and

the Collaborative Research Center SFB814 funded by the German

Science Foundation (DFG), and the Indo-German Partnership in

Higher Education Program (IGP) 2016, funded by DAAD-UGC. The

authors would also like to acknowledge the effort of the reviwers

in considerably improving this manuscript from its first submis-

sion.

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