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Applied Mathematical Modelling xxx (2014) xxx–xxx

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Applied Mathematical Modelling
journal homepage: www.elsevier.com/locate/apm

A meshless numerical approach based on Integrated Radial
Basis Functions and level set method for interfacial flows q
L. Mai-Cao a, T. Tran-Cong b,⇑
a

Faculty of Geology and Petroleum Engineering, Ho Chi Minh City University of Technology, Viet Nam
Computational Engineering and Science Research Centre, Faculty of Health, Engineering and Sciences, The University of Southern Queensland, Toowoomba,
QLD 4350, Australia
b

a r t i c l e

i n f o

Article history:
Received 6 May 2013
Received in revised form 13 June 2014
Accepted 16 June 2014
Available online xxxx
Keywords:
IRBFN method
Level set method
Interfacial flows
Incompressible Navier–Stokes equations

a b s t r a c t
This paper reports a new meshless Integrated Radial Basis Function Network (IRBFN)
approach to the numerical simulation of interfacial flows in which the two-way interaction
between a moving interface and the ambient viscous flow is fully investigated. When an
interface between two immiscible fluids moves, not only its position and shape but also
the flow variables (i.e. velocity field and pressure) change due to the presence of surface
tension along the moving interface. The velocity field of the ambient flow, on the other
hand, causes the interface to move and deform as a result of momentum transport between
the two immiscible fluids on both sides of the interface. Numerical investigations of such a
two-way interaction is reported in this paper where the level set method is used in combination with high-order projection schemes in the meshless framework of the IRBFN
method. Numerical investigations on the meshless projection schemes are performed with
typical benchmark incompressible viscous flow problems for verification purposes. The
approach is then demonstrated with the numerical simulation of two bubbles moving,
stretching and merging in an incompressible ambient fluid under the action of buoyancy
force.
Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction
Fluid flows studied in this paper with a moving interface between two immiscible fluids can be classified as interfacial
flows. In general, a numerical approach to the simulation of such flows consists of (a) a flow modelling method, (b) an interface modelling algorithm, and (c) a flow-interface coupling technique. These three components should be coupled together
in a consistent framework in order to properly model complicated phenomena associated with the interfacial flows.
Regarding the flow modelling, there are two main approaches to formulating the governing equations for an interfacial
flow: one-fluid and two-fluid models. In the one-fluid model, a single flow equation is formulated to describe both fluid
flows, and a characteristic function is used to specify a particular fluid [1]. In the two-fluid model, on the other hand, each
fluid has its own governing equations and therefore the characteristics of each phase can be separately captured [2]. For
incompressible interfacial flows, the governing equations in either one-fluid or two-fluid model are formulated from the

q
This article belongs to the Special Issue: ICCM2012 – Topical Issues on computational methods, numerical modelling & simulation in Applied

Mathematical Modelling.
⇑ Corresponding author. Tel.: +61 7 4631 1332; fax: +61 7 4631 2110.
E-mail addresses: (L. Mai-Cao), (T. Tran-Cong).

/>0307-904X/Ó 2014 Elsevier Inc. All rights reserved.

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Navier–Stokes equations (NSEs). Among others, the projection/pressure correction method can be used to solve NSEs. First
proposed in [3], the projection method consists of a predictor–corrector procedure in which the momentum equation is first
solved using an initial approximation of the pressure to obtain an intermediate velocity field. A pressure correction is then
obtained by solving a Poisson equation. Finally, the new velocity field is updated using the intermediate velocity and the
pressure correction. Several improvements to the original projection method have been made by (a) improving intermediate
velocity boundary conditions [4]; or (b) improving accuracy order in time via pressure correction procedure [5]; or (c)
improving pressure boundary conditions [6]. In this work, a class of new meshless projection schemes is developed based
on the improved projection methods mentioned above to solve flow equations in the one-fluid model.
Numerical approaches to interface modelling can be classified in two groups: moving-grid and fixed-grid methods. For
the moving-grid methods, the interface is treated as the boundary of a moving surface-fitted grid [7]. This approach allows
a precise representation of the interface whereas its main drawback is the severe deformation of the mesh as the interface
moves. The second approach which is based on fixed grids includes tracking and capturing methods. The tracking methods
explicitly represent the moving interface by means of predefined markers [8]. In capturing methods, on the other hand, the
moving interface is not explicitly tracked, but rather captured via a characteristic function. Examples of the capturing methods are phase field method [9], volume-of-fluid method [10] and level set method [11]. The characteristic function used to
implicitly describe the moving interface is the order parameter in the phase field method, volume fraction in the volume-offluid method and level set function in the level set method. For these capturing methods, no rezoning/remeshing is needed to
maintain the overall accuracy even when the interface undergoes large deformation.
Regarding flow-interface coupling in the numerical simulation of interfacial flows, the surface tension is normally taken

into account in the computation of force balance at the interface where the difference in stresses of the two fluids in the
direction normal to the interface is balanced by the surface tension on the interface [7]. A simple and effective model that
alleviates the interface topology constraints was presented in [12] where the proposed model, known as the continuum surface force (CSF) model, interprets the surface tension as a continuous, three-dimensional effect across an interface rather
than as a boundary value condition on the interface. The advantage of the CSF model is that the moving interface needs
not be explicitly described for the interfacial boundary condition.
In this work, all of the aforementioned modelling techniques are implemented within the meshless framework of the
IRBFN method for interfacial flows. The idea of using radial basis functions for solving partial differential equations (PDEs)
was first proposed in [13,14] to solve parabolic, hyperbolic and elliptic PDEs. Since its introduction, various methods based
on radial basis functions have been developed and applied in different areas. A local radial point interpolation method
(LRPIM) was proposed in [15] for free vibration analyses of 2-D solids. The Linearly Conforming Radial Point Interpolation
Method (Lc-Rpim) was studied in [16] for solid mechanics. The IRBFN method has been reported to be a highly accurate tool
for approximating functions, their derivatives, and solving differential equations [17,18]. The method was then successfully
applied to transient problems [19] including those governed by parabolic as well as hyperbolic PDEs where comparisons of
performance of the IRBFN-based methods and others, including finite difference, boundary element and finite element methods, were made. Additionally, high-order meshless schemes have been implemented for passive transport problems in [20]
where the motion and deformation of moving interfaces in an external flow are fully captured by a unified numerical procedure combining the IRBFN method and the level set method together with the semi-Lagrangian method or Taylor series
expansions. Furthermore, two numerical meshless schemes were proposed in [21] for the numerical solution of Navier–
Stokes equations. Based on the projection method and coupled with high-order time integration techniques in the meshless
framework of the IRBFN method, the two schemes showed their good stability and accuracy when applied to the numerical
simulation of incompressible fluid flows and interfacial flows in [21]. In the present work, the two schemes are further
numerically investigated, specifically for Navier–Stokes equations with time-dependent boundary conditions as well as
for the numerical simulation of buoyancy-driven bubble flow. In comparison with finite difference or finite element methods, the unique advantages of the proposed IRBFN-based methods include (i) the meshless nature and implicit interface capturing technique of the present approach; and (ii) the ability to effectively ensure the conservation of mass during the
evolution of the interfaces between fluids.
The remaining of this paper is organized as follows. Firstly, the one-fluid model is formulated for interfacial flows with an
introduction to the CSF model and level set method. The new meshless projection schemes for the one-fluid continuum
model are then presented followed by the step-by-step procedure of the proposed meshless approach to interfacial flows.
Numerical investigations on the new projection schemes with typical viscous flows as well as the application of the proposed
meshless IRBFN-based approach to the numerical simulation of two bubbles moving, stretching and merging in an ambient
viscous flow are then performed for verification purposes.
2. Mathematical formulation
Consider a domain X and its boundary @ X containing two immiscible Newtonian fluids, both being incompressible. Let X1

be the region containing fluid 1 at time t. Similarly, let X2 be the region containing fluid 2 and bounded by the fluid interface
C at time t. The governing equations describing the motion of the two fluids in their own regions are given by the Navier–
Stokes equations,

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q1



@v 1
ỵ v 1 rv 1 ẳ rp1 ỵ r 2l1 D1 ị ỵ q1 g;
@t

x 2 X1 ;

1ị

q2



@v 2
ỵ v 2 rv 2 ẳ rp2 ỵ r 2l2 D2 ị ỵ q2 g;
@t

x 2 X2 ;

2ị

with incompressibility constraints

r v 1 ẳ 0;

x 2 X1 ;

3ị

r v 2 ẳ 0;

x 2 X2 ;

4ị

where v i is the velocity field, qi is the density, g is the gravity, pi is the pressure and
tensor Di is dened as

Di ẳ


1
rv i ỵ rv Ti ;
2

li is the viscosity. The rate of strain

i ẳ 1; 2:

5ị

The subscript i in the above equation denotes the ith fluid under consideration. Assuming that the moving interface C is
impermeable and that there is no mass transfer between the two fluids, velocity continuity condition at the interface yields

v 1 ¼ v 2;

x 2 C:

ð6Þ

The jump in normal stresses along the fluid interface is balanced by the surface tension as follows.

À

Á
2l1 D1 À 2l2 D2 n ẳ p1 p2 ỵ rjịn;

x 2 C;

ð7Þ

where j is the curvature of the interface, r the surface tension coefficient, and n is the unit vector normal to the fluid interface C, pointing outwards from fluid 2 (bounded by the interface) into fluid 1 (the surrounding fluid). The continuum surface
force (CSF) model is used in this work to embed the surface tension into the momentum equation rather than imposing surface tension as a boundary condition on the interface [12].
Let the fluid interface be the zero level of the level set function /,

C ẳ fxj/x; tị ẳ 0g;

8ị

8
>
< > 0; x 2 X1
/x; tị ẳ 0; x 2 C
>
:
< 0; x 2 X2

ð9Þ

where

The unit vector normal to the interface and the curvature of the interface can be expressed in terms of /x; tị as follows.



nẳ





r/ 
r/ 
and j ¼ r Á
jr/j/¼0
jr/j /¼0

ð10Þ

Let

v¼

&

v 1;
v 2;

x 2 X1
x 2 X2

ð11Þ

be the fluid velocity continuous across the interface, since the interface moves with the fluid particles, the evolution of / is
then given by Osher and Fedkiw [22].

@/
ỵ v r/ ẳ 0:
@t

12ị

By dening the Heaviside function H/ị

8
if / < 0;

>
<0
H/ị ẳ 1=2 if / ẳ 0;
>
:
1
if / > 0

13ị

and the uid properties

q/ị ẳ q2 ỵ q1 q2 ịH/ị;

14ị

l/ị ẳ l2 þ ðl1 À l2 ÞHð/Þ;

ð15Þ

together with the CSF model [12], one obtains the one-fluid continuum formulation [1] for the two-phase incompressible
viscous flow
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@v

1
ỵ v rịv ẳ
rp ỵ r 2l/ịDị ỵ rj/ịddịnị ỵ g;
@t
q/ị

16ị

r v ẳ 0:

17ị

where f c ẳ rjð/ÞdðdÞn is the surface tension, dðdÞ is the Dirac delta function, d is the normal distance to the interface. For the
numerical simulation of two bubbles rising in a viscous fluid studied in this work, Eq. (16) can be written in dimensionless
form as follows.



@v
1
1
1
ỵ v rịv ẳ
rp ỵ r 2l/ịDị ỵ j/ịd/ịr/ ỵ g u
Re
Bo
@t
q/ị

18ị

where the scaling factors are

p ¼

p
;
pref

tà ¼

t
;
tref

v ref

¼ ðgRÞ1=2 ;

và ¼
qà ¼

v
;
v ref

xà ¼

x
;

R

q
l
; là ¼ ;
qc
lc
pref ¼ qc v 2ref ;

tref ¼

v ref
R

:

ð19Þ

Fluid 1 is hereafter referred to as the fluid surrounding the bubbles with density qc and viscosity lc . Similarly, fluid 2 is
referred to as the fluid inside the bubbles, of initial radius R, that has the corresponding density qb and viscosity lb . g u is the
unit gravitational vector pointing downward. The dimensionless groups in the above equation are the Reynolds number

Re ẳ

p
2Rị3=2 g qc

lc

20ị

and the Bond number

Bo ẳ

4qc gR2

r

21ị

The dimensionless density and viscosity in Eq. (18) are dened as

q/ị ẳ k ỵ 1 kịH/ị; and l/ị ẳ g ỵ 1 gịH/ị;

22ị

where k ẳ qb =qc is the density ratio, g ¼ lb =lc is the viscosity ratio.
Eq. (18) is rewritten in the form similar to the Navier–Stokes equation where the gravity force, the surface tension and
f
nonsymmetric part of the rate of strain tensor are treated as the forcing term 

where

@v
1
1
ỵ v rịv ẳ
rp ỵ
r ẵl/ịrv ỵ f in X

@t
q/ị
q/ịRe

23ị

r v ẳ 0 on X;

24ị

v ẳ vb

25ị

vb

on @ X:

f is given by
is the Dirichlet boundary condition for velocity, and 

f ẳ g ỵ
u

h
i
1
1
r l/ịrv ịT ỵ
j/ịd/ịr/:

q/ịRe
q/ịBo

26ị

3. Meshless projection schemes for unsteady incompressible NavierStokes equations (NSEs)
This section presents the formulation of the new meshless IRBFN-based projection schemes for unsteady incompressible
Navier–Stokes equations proposed in [21]. As can be seen in the next section, with a straightforward adaptation, these proposed schemes can be applied to solve the flow Eqs. (23)–(26).
Four different projection schemes implemented within the meshless framework of the IRBFN method and coupled with
the high-order multistep time integration are presented in this section. They include: (a) Standard IPC-IRBFN, a meshless
incremental pressure correction scheme in the standard form inspired by Van Kan [5]; (b) Rotational IPC-IRBFN, a meshless
incremental pressure correction scheme in the rotational form motivated by Timmermans et al. [23]; (c) Standard IPCPPIRBFN, a meshless incremental pressure correction scheme in the standard form with pressure prediction based on [23];
and (d) Rotational IPCPP-IRBFN, a meshless incremental pressure correction scheme in the rotational form with pressure prediction motivated by Timmermans et al. [23].
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5

3.1. The Navier–Stokes equations
Consider a domain X & R2 with boundary @ X. The Navier–Stokes equations that govern incompressible viscous flows are
comprised of the momentum and continuity equations and written in dimensionless form as follows.

@v
ỵ v rịv ẳ rp ỵ mr2 v ỵ f ;
@t

in X;

27ị

;
r v ẳ 0 in X

28ị

 ẳ X [ @ X; v x; tị ẳ u; v ịT is the velocity eld, px; tÞ is the kinematic pressure, f ðx; tÞ is the body force vector, and m
where X
is the kinematic viscosity. The velocity field is subject to boundary and initial conditions as follows.

v x; tị ẳ v b x; tị
v x; 0ị ¼ v 0
where v 0 ðxÞ and
posed.

on @ X;

ð29Þ

at t ¼ 0;

ð30Þ

v b ðx; tÞ are given functions satisfying the following constraints for the Navier–Stokes equations to be well

;
r Á v 0 ẳ 0 in X

31ị

n v 0 ẳ n Á v b ðx; 0Þ on @ X:

ð32Þ

Since neither initial nor boundary conditions are prescribed for the pressure in the Navier–Stokes equations, p is determined
up to an additive constant corresponding to the level of hydrostatic pressure. In addition, global mass conservation must be
imposed through the boundary conditions, leading to the constraint [24]

Z

n v b dC ẳ 0:

33ị

@X

3.2. Meshless incremental pressure correction IPC-IRBFN schemes
Consider the original projection method [3] in which Eq. (27) is first solved for the intermediate velocity field by using the
backward Euler time stepping with the linearized convective term and without the pressure gradient


1 nỵ1
v~ v n ỵ ẵv rịv n ẳ mr2 v~ nỵ1 ỵ f nỵ1
Dt

v~ nỵ1 ẳ v nỵ1
b

in X;

34ị

on @ X:

35ị

The new time-level (end-of-step) velocity

v

nỵ1

nỵ1

and pressure p

are then obtained by solving


1 nỵ1
v v~ nỵ1 ỵ rpnỵ1 ẳ 0 in X;
Dt

36ị

r v nỵ1 ẳ 0 in X;

37ị

v nỵ1 n ẳ v nỵ1
n
b

on @ X:

38ị

Rather than simultaneously solving for the velocity and pressure, a Poisson pressure equation (PPE) is formulated from the
above equations to solve for the new pressure separately. This is done by taking the divergence of Eq. (36) and using the
incompressibility constraint described in Eq. (37)

r2 pnỵ1 ẳ

1
r v~ nỵ1
Dt

in X;

39ị

The boundary condition for the PPE is obtained by taking the normal component of Eq. (36) and taking into account the
boundary conditions described in Eqs. (35) and (38)

@pnỵ1
ẳ 0 on @ X:
@n

40ị

The end-of-step velocity eld can be then updated using Eq. (36)

v nỵ1 ẳ v~ nỵ1

1
rpnỵ1 :
Dt

41ị

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In the incremental pressure correction methods [5], the pressure gradient from the previous step is taken into account rather
than ignored as in the original projection method. More specifically, the intermediate velocity field in this case can be found
by solving the following equations


1 nỵ1
v~ v n ỵ ẵv rịv n ẳ rpn ỵ mr2 v~ nỵ1 ỵ f nỵ1
Dt

v~ nỵ1 ẳ v nỵ1
b

in X;

on @ X:

42ị
43ị

The end-of-step velocity and pressure can be then obtained by solving the following equations




1 nỵ1
v v~ nỵ1 ỵ r pnỵ1 pn ẳ 0 in X;
Dt

44ị

r v nỵ1 ẳ 0 in X;

45ị

v nỵ1 n ẳ v nỵ1
n
b

on @ X:

46ị

Let qnỵ1 ẳ pnỵ1 pn be the pressure increment. By taking the divergence of Eq. (44), using Eq. (45) and taking into account
the boundary conditions in Eqs. (43) and (46), one has the Poisson equation for the pressure increment qnỵ1 along with the
boundary condition as follows.

r2 qnỵ1 ẳ

1 nỵ1
v~
Dt

in X;

@qnỵ1
ẳ 0 on @ X:
@n

ð47Þ

ð48Þ

The end-of-step velocity and pressure are then given by

pnỵ1 ẳ qnỵ1 ỵ pn ;

49ị

v nỵ1 ẳ v~ nỵ1 Dtrqnỵ1 :

50ị

3.2.1. The Standard IPC-IRBFN scheme
On the basis of the incremental pressure correction method previously presented, the Standard IPC-IRBFN scheme can be
formulated with the following modifications motivated by Karniadakis et al. [6]:
1. High-order Backward Differentiation Formula (BDF) integration method is used for time stepping rather than the firstorder backward Euler method. In particular, the temporal derivative is discretized in time as follows.

Z

t nỵ1

tn

!
 
Jv
X
@v
1
nỵ1
nỵ1k
dt %
b v

bk v
@t
Dt 0
kẳ1

51ị

The values of coefficients b’s corresponding to J v are given in the next section.

2. High-order Adam–Bashforth extrapolation method is used to linearized the convective term. For this method,
the convective term at a time level is calculated from multiple previous steps instead of just relying on the last
value.

Z

t nỵ1

tn

ẵv rịv dt %

JX
v 1

ak ẵv rịv nk

52ị

kẳ0

The values of coefcients a’s corresponding to J v are given in the next section.
3. Instead of just taking into account the value of the pressure from the last time step in solving for the intermediate
velocity field, the IPC-IRBFN scheme uses a pressure predictor which is extrapolated from multiple previous steps as
follows.

jtẳtnỵ1 ẳ
p

JX

v 1

ak pnk

53ị

kẳ0

By taking the above modications, the Standard IPC-IRBFN scheme consists of the following steps for each time level
tnỵ1 ẳ n ỵ 1ịDt; n ẳ 0; 1; 2; . . ..

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nỵ1
1. Calculate a predictor for the pressure, p

nỵ1 ẳ
p

Jp
X

ak pnk

kẳ0

8
Jp ¼ 0
>
< 0;
n
Jp ¼ 1
¼ p ;
>
: n
2p À pnÀ1 ; J p ẳ 2

54ị

~ nỵ1 , by solving
2. Compute a predictor for the velocity eld, v
J

v
X
1
~ nỵ1
b0 v
bk v nỵ1k
Dt
kẳ1

!
ỵ

JX
v 1

ak ẵv rịv nk ẳ rpnỵ1 ỵ mr2 v~ nỵ1 ỵ f nỵ1 in X;

55ị

kẳ0

with the Dirichlet boundary condition

v~ nỵ1 ẳ v nỵ1
b
where

on @ X:

56ị

8 n
Jv ẳ 1
>
Jv
X
nỵ1k
n
1 n1
Jv ẳ 2
bk v

ẳ 2v 2 v ;
>
: n 3 n1 1 n2
kẳ1
3v 2 v
ỵ 3 v ; Jv ẳ 3

b ẳ 1ị
0 3
b ẳ
0 211
b0 ẳ 6

57ị

and
JX
v 1

ak ẵv rịv

nk

kẳ0

8
n
J v ẳ 1;
>
< ẵv rịv ;

n
n1
ẳ 2ẵv rịv ẵv rịv ;
J v ẳ 2;
>
:
3ẵv rịv n 3ẵv rịv n1 ỵ ẵv rịv n2 ; J v ẳ 3:

58ị

3. Calculate the pressure increment qnỵ1

r2 qnỵ1 ẳ

b0
r v~ nỵ1
Dt

in X;

59ị

@qnỵ1
ẳ 0 on @ X:
@n

60ị

4. Perform the correction step for pressure pnỵ1

nỵ1
pnỵ1 ẳ qnỵ1 ỵ p

61ị

5. Perform the correction step for velocity eld

v nỵ1

v

nỵ1

Dt
~ nỵ1 rqnỵ1 :
ẳv
b0

62ị

3.2.2. The Rotational IPC-IRBFN scheme
In this scheme, a consistency requirement is explicitly imposed on the numerical solutions stating that the end-of-step
velocity and pressure,v nỵ1 and pnỵ1 , must numerically satisfy the momentum and continuity equations regardless of how the
~ nỵ1 , is calculated. More specically, the momentum Eq. (27) and the continuity Eq. (28) must hold for
velocity predictor, v
v nỵ1 and pnỵ1 in the semi-discrete form in time as follows.
J

v
X

1
b v nỵ1
bk v nỵ1k
Dt 0
kẳ1

!
ỵ

JX
v 1

ak ẵv rịv nk ẳ rpnỵ1 ỵ mr2 v nỵ1 ỵ f nỵ1 in X;

63ị

kẳ0

r v nỵ1 ¼ 0 on @ X:

ð64Þ

The above equations are now used to derive the corresponding steps for the new Rotational IPC-IRBFN scheme as follows.
First, subtracting Eq. (63) from Eq. (55) yields

Á
b0 nỵ1
v v~ nỵ1 ẳ rpnỵ1 pnỵ1 ị þ mr2 ðv nþ1 À v~ nþ1 Þ;
Dt

ð65Þ

By taking the divergence of Eq. (65), one has





b0
r v nỵ1 v~ nỵ1 ẳ r rpnỵ1 pnỵ1 ị ỵ mr ẵr2 v nỵ1 v~ nỵ1 ị
Dt

66ị

Simplifying and rearranging terms in the above equation yields





r2 pnỵ1 pnỵ1 ỵ mr v~ nỵ1 ẳ

b0
r v~ nỵ1
Dt

67ị

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nỵ1 ỵ mr v
~ nỵ1 be the pressure increment in this case, one obtains the Poisson equation for qnỵ1 as
By letting qnỵ1 ẳ pnỵ1 p
follows.

r2 qnỵ1 ẳ

b0
r v~ nỵ1
Dt

68ị

A homogeneous Neumann boundary condition for qnỵ1 is suggested in [23] as follows.

@qnỵ1
ẳ 0 on @ X:
@n

69ị

The solving procedure in the Rotational
tnỵ1 ẳ n ỵ 1ịDt; n ẳ 0; 1; 2; . . . as follows.
1.
2.

3.
4.

IPC-IRBFN

scheme

is

now

summarized

for

each

time

level

nỵ1 , using Eq. (54).
Calculate a predictor for the pressure, p
~ nỵ1 , by solving Eqs. (55) and (56).
Compute a predictor for the velocity eld, v
Calculate the pressure increment, qnỵ1 , by solving Eqs. (68) and (69) as in the Standard IPC-IRBFN scheme.
Perform the correction step for the new pressure pnỵ1

nỵ1 mr v
~ nỵ1 :

pnỵ1 ẳ qnỵ1 ỵ p

70ị

5. Perform the correction step for velocity eld,v

nỵ1

, using Eq. (62) as in the Standard IPC-IRBFN scheme.

As can be seen from the solving procedure of the IPC-IRBFN schemes in both standard and rotational forms, the two forms of
the IPC-IRBFN schemes differ in the manner that the pressure increment, qnỵ1 , is dened, and thus in the pressure correction
step.
3.3. The IPCPP-IRBFN schemes
Instead of extrapolating the pressure at the beginning of each time step as in the IPC-IRBFN schemes, the IPCPP-IRBFN
schemes solve a Poisson equation with Neumann boundary condition [25] for the pressure predictor at each time step.
By taking divergence of Eq. (27) and making use of Eq. (28), the Poisson equation for pressure is derived as

r2 p ẳ r ẵv rịv f Š in X;

ð71Þ

with the Neumann boundary condition being derived by taking the normal component of the momentum Eq. (27) as

@p
@v
¼nÁ
v rịv ỵ mr2 v ỵ f
@n
@t

!
on @ X:

ð72Þ

Therefore, the pressure predictor in the IPCPP-IRBFN schemes is calculated by solving the above equations in which the
implicit BDF method is used to discretize the temporal derivative and the forcing term with respect to time whereas the
explicit AB method is used for the nonlinear convective term and the viscous term in Eqs. (71) and (72) as follows.
2 nỵ1


r p

(J 1
v
X

)

ak ẵv rịv

ẳr

nk

ỵf

nỵ1

in X;

73ị

kẳ0

nỵ1
@p
ẳn
@n

(


b0 v nỵ1 ỵ

PJv 1
kẳ1

bk v nỵ1k

Dt



JX
v 1

ak ẵv rịv

kẳ0

nk

JX
v 1

ỵm

)
nk

ak ẵr r v ị

ỵf

nỵ1

on @ X:

74ị

kẳ0

where Dirichlet boundary condition on velocity, v nỵ1 ẳ v nỵ1
, is applied to v nỵ1 in Eq. (74). It is noted that in the Neumann
b
boundary condition for the pressure prediction, the viscous term is decomposed into

r2 v ẳ rr v ị r ðr  v Þ

ð75Þ

and the incompressibility constraint is used accordingly [25].
Like the IPC-IRBFN schemes, the IPCPP-IRBFN schemes are implemented in both standard and rotational forms. The solving procedure in the IPCPP-IRBFN schemes is summarized for each time level t nỵ1 ẳ n ỵ 1ịDt; n ẳ 0; 1; 2; . . . as follows.
 Step 1: Calculate the predictor for the pressure by solving Eqs. (73) and (74);
 Steps 2–5: The same as in the IPC-IRBFN schemes.
4. IRBF approximation of functions and their derivatives
All numerical schemes presented in this paper are based on the Integrated Radial Basis Function (RBF) method which is
briefly captured here. Interested readers are referred to [19,20] for further details.
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Let uðx; tÞ be an unknown function continuously defined on Q T :ẳ 0; Tị X, where X & Rd ; d ¼ 1; 2; 3 is a bounded
domain. For convenience, the coordinates of a typical point are denoted by x ẳ x; y; zị and typically derivatives with respect
to x are used to illustrate the derivation of the method. Let fxk gM
k¼1 be a set of discrete data points in X, and
utị ẳ ẵu1 tị; u2 tị; . . . ; uM ðtފT , the corresponding nodal values of the function at a certain point in time t. Let
xj ; j ¼ 1; . . . ; N be the centres of N RBFs. The IRBF formulation for the approximation of the function and its derivatives
(e.g. with respect to x), pertinent to second order systems, is written as follows.

@ 2 uðx; tÞ ^ T À1
% hðxÞ H uðtÞ;
@x2

ð76Þ

@uðx; tÞ ~ T À1
% hðxÞ H uðtÞ;
@x

ð77Þ

uðx; tÞ % hðxÞT H À1 uðtÞ;

ð78Þ

^
~
^
is a set of basis functions, hðxÞ
and hðxÞ are obtained by symbolically integrating hðxÞ
in the x direction once and
where hðxÞ
th
^
is defined as follows.
twice, respectively. Constants of integrations appear as a result. The j component of hxị

^ xị ẳ ujjx x jjị; j ẳ 1; . . . ; N;
h
j
j
^j xị ẳ 0; j ẳ N ỵ 1; . . . ; N;
h

ð79Þ

in which N will be defined shortly and uðjjx À xj jjÞ are radial basis functions such as Hardys multiquadrics

ujjx xj jjị ẳ

q
r 2j ỵ s2j ;

j ẳ 1; . . . ; N;

80ị

or Duchons thin plate splines (TPS)

ujjx xj jjị ẳ r2m
j ẳ 1; . . . ; N;
j log r j ;

ð81Þ

where m is the TPS order, r j ¼ jjx À xj jj is the Euclidian norm, and sj is the RBF shape parameter given by Moody and Darken [26]
min

sj ¼ b dj

ð82Þ

;

in which b is a user-defined parameter and
matrix H is dened as

2

h1 x1 ị
6
6 h1 x2 ị
6
H ẳ 6.
6 ..
4
h1 ðxM Þ

h2 ðx1 Þ

. . . hN ðx1 Þ

h2 ðx2 Þ
..
.

. . . hN ðx2 Þ
. . ..
. .

h2 ðxM Þ . . . hN ðxM Þ

min
dj

is the distance from the jth data point to its nearest neighbouring point.The

3
7
7
7
7;
7
5

ð83Þ

where hj xị; j ẳ 1; . . . ; N is the jth component of hxị, and N ẳ N þ P in which P is the number of discrete points needed to
approximate the constants of integration. Details on the derivation of the IRBFN formulation and numerical investigations
of the IRBFN method can be found in [19].
In this work we choose N ¼ M and the RBF centres to be the same as the data points.
For a more compact form, the IRBFN formulation can be written as follows.

uxx ðx; tÞ 
ux ðx; tÞ 

@ 2 uðx; tÞ
% w@ xx ðxÞT uðtÞ;
@x2

ð84Þ

@uðx; tÞ
% w@x ðxÞT uðtÞ;

@x

ð85Þ

uðx; tÞ % wðxÞT uðtÞ;

ð86Þ

^ T H 1 ;
w@xx xị ẳ hxị

87ị

~ T H 1 ;
w@x xị ¼ hðxÞ

ð88Þ

where

wðxÞ ¼ hðxÞT H À1 :

ð89Þ

Let S be a certain differential operator in space that operates on the scalar function ux; tị in X 2 R ; d ẳ 1; 2; 3, the IRBFN
formulation above can be then rewritten in a generic form for approximating function uðx; tÞ and/or its derivatives as follows.
d

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L. Mai-Cao, T. Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx

Suðx; tÞ % wTS ðxÞuðtÞ;

ð90Þ

where wS ðxÞ is the vector whose components are the results of the application of operator S on the corresponding components of wxị,

wS xị ẳ ẵSw1 ðxÞ; Sw2 ðxÞ; . . . ; SwM ðxފT :

ð91Þ

For a special case where S is the identity operator, S ¼ I , one gets the approximation of function uðx; tÞ. Otherwise, one
@
obtains the corresponding derivative of the function. For example, if S ¼ @y
 @ y , one has the approximation of the first order
derivative of uðx; y; tÞ in the y direction as follows.

Sux; y; tị ẳ

@
ux; y; tÞ % wT@ y ðx; yÞuðtÞ:
@y

ð92Þ

0

10

−1

Velocity Error Norm

10

−2

10

−3

10

Standard IPC−IRBFN
Rotational IPC−IRBFN
Standard IPCPP−IRBFN
Rotational IPCPP−IRBFN

−4

10

10

20

30

40

50

60

70

80

90

100

70

80

90

100

Number of time steps
2

10

1

Pressure Error Norm

10

0

10

−1

10

−2

10

−3

10

−4

10

Standard IPC−IRBFN
Rotational IPC−IRBFN
Standard IPCPP−IRBFN
Rotational IPCPP−IRBFN
10

20

30

40

50

60

Number of time steps
Fig. 1. Stability analysis of the IPC-IRBFN and IPCPP-IRBFN schemes in terms of velocity field (top) and pressure (bottom) with Dt ¼ 0:01 in Test 1.

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5. Meshless numerical approach to interfacial flows
The solving procedure consists of the following steps.
Step 0: Initialize the level set function /ðxÞ to be the signed distance to the interface; For each nth time step, n ¼ 1; 2; . . .
Step 1: Compute the interface normal, curvature, and the density and viscosity of the fluids based on the level set function
value at the previous step.
Step 2: Solve the one-fluid continuum equations for the flow variables taking into account the interface dependency of density and viscosity as well as the surface tension.
Step 3: Advance the level set function from the previous step to the current one with the most updated velocity field calculated from Step 2.
Step 4: Re-initialize the level set function to a signed distance function at the current time step.
Step 5: Adjust the level set function by using the mass correction algorithm to ensure the mass conservation.

Step 6: The interface as the zero contour of the level set function has now been advanced one time step. Go back to step 1 for
further evolution of the moving interface until the predefined time is reached.
−1

10

−2

10

−3

Velocity Error Norm

10

−4

10

−5

10

−6

10

−7

Standard IPC−IRBFN
Rotational IPC−IRBFN
Standard IPCPP−IRBFN
Rotational IPCPP−IRBFN

10

−8

10

20

40

60

80

100

120

140

160

180

200

140

160

180

200

Number of time steps
2

10

1

10

0

Pressure Error Norm

10

−1

10

−2

10

−3

10

−4

10

−5

10

Standard IPC−IRBFN
Rotational IPC−IRBFN
Standard IPCPP−IRBFN
Rotational IPCPP−IRBFN
20

40

60

80

100

120

Number of time steps
Fig. 2. Stability analysis of the IPC-IRBFN and IPCPP-IRBFN schemes in terms of velocity field (top) and pressure (bottom) with Dt ¼ 0:005 in Test 1.

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5.1. Compute interface properties (normal and curvature) and fluid properties (density and viscosity)
The normal and curvature of the interface can be calculated by Eq. (10) whereas the density and viscosity are given
by Eq. (22). For the computation of the above fluid properties, the Heaviside function is used. A simple implementation of
Eq. (13) poses numerical difficulty since large jumps in q and l across the interface might cause numerical instabilities.
In order to avoid this issue, it is common to introduce an interface thickness to smooth the density and viscosity at the
interface. This can be done by replacing the Heaviside function in Eq. (13) with a smoothed Heaviside function H ð/Þ defined
as [1]

8
if / < 
>
<0
H /ị ẳ / ỵ ị=2ị ỵ sinðp/=Þ=ð2pÞ if j/j 6 
>
:
1
if / > 

ð93Þ

Fig. 3. Convergence of the u-component velocity along the mid-vertical line (top) and the v-component velocity along the mid-horizontal line (bottom) in
the lid-driven cavity flow.

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The above Heaviside function defines the smoothed Dirac delta function d as follows.

&
d /ị ẳ

1=21 ỵ cosp/=ịị= if j/j < 
0

ð94Þ

otherwise

5.2. Solve the one-fluid continuum equations
The meshless IRBFN-based projection schemes introduced in Sections (3.2) and (3.3) can be simply modified to solve the
one-fluid continuum Eqs. (23)–(26). The adaptation for the Standard IPC-IRBFN projection scheme is presented here in
details from which the other meshless projection schemes can be modified in the same manner.
The Standard IPC-IRBFN scheme is modified to take into account the fact that fluid densities and viscosities in the onefluid continuum equations can vary from one location to another instead of being constant in the Navier–Stokes equations. In
particular, the numerical procedure of the IPC-IRBFN scheme for the one-fluid continuum equations consists of the following
steps for each time level tnỵ1 ẳ n ỵ 1ịDt; n ẳ 0; 1; 2; . . ..

Streamfunction value at the primary vortex centre

−0.03
Re=1000
Re=400
Re=100

−0.04
−0.05
−0.06
−0.07
−0.08
−0.09
−0.1
−0.11
−0.12

0

5

10

15

20

25

30

Time

Vorticity value at the primary vortex centre

−2

−3
Re=1000
Re=400
Re=100

−4

−5

−6

−7

−8

−9

0

5

10

15

20

25

30

Time
Fig. 4. The evolution of the streamfunction (top) and the vorticity (bottom) at the centre of the primary vortex at different points in time with different
Reynolds numbers. For any of the Reynolds numbers, the streamfunction and the vorticity values at the centre of the primary vortex change rapidly in the
beginning. The rate of change then slows down, and finally the streamfunction and the vorticity reach their steady state.

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Table 1
Streamfunction wc and vorticity xc at the centre of the primary vortex at different time steps corresponding to Re = 100, 400 and 1000.
t

Re ẳ 100
wc xc ị

Re ẳ 400
wc xc ị

Re ẳ 1000
wc xc ị

1
2.5
5
7.5
10
12.5
15
17.5
20
25
30

0.073624 (3.668184)
0.092665 (3.551824)
0.100557 (3.247018)
0.102477 (À3.181559)
À0.10298 (À3.147246)
À0.103111 (À3.147487)
À0.103145 (À3.147548)
À0.103154 (À3.147563)
À0.103156 (À3.147567)
À0.103157 (À3.147568)
À0.103157 (À3.147568)

À0.047149 (À5.825875)
À0.072358 (À4.363596)
À0.090033 (À3.441605)

À0.099643 (À2.819742)
À0.105461 (À2.55873)
À0.108959 (À2.427785)
À0.110951 (À2.366845)
À0.112037 (À2.327834)
À0.11262 (À2.305005)
À0.11309 (À2.296747)
À0.113224 (À2.291365)

À0.035008 (À8.523135)
À0.059933 (6.84297)
0.080071 (4.62247)
0.091677 (3.596885)
0.099589 (3.049279)
0.105255 (2.710798)
0.109306 (2.484563)
0.11212 (2.333954)
0.113977 (2.231622)
0.115874 (2.115058)
0.116652 (2.062449)

nỵ1
1. Calculate a predictor for the pressure, p

nỵ1 ẳ
p

Jp
X

ak pnk :

95ị

kẳ0

~ nỵ1 , by solving
2. Compute a predictor for the velocity eld, v
J

v
X
1
~ nỵ1
b0 v
bk v nỵ1k
Dt
kẳ1

!
ỵ

JX
v 1
kẳ0

ak ẵv rịv nk ẳ

1

q/n ị

rpnỵ1 ỵ

1

q/n ịRe





r l/n ịrv~ nỵ1 ỵ f n in X;

96ị

with the Dirichlet boundary condition

v~ nỵ1 ẳ v nỵ1
b

on @ X:

97ị

Fig. 5. Streamlines at t ¼ 1; 5; 10; 15 of the lid-driven cavity flow (Re ¼ 1000).

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15

3. Calculate the pressure increment qnỵ1

r

!
1
b
nỵ1
~ nỵ1
ẳ 0rv
r
q
Dt
q/n ị

in X;

98ị

@qnỵ1
ẳ 0 on @ X:
@n

ð99Þ

4. Perform the correction step for pressure pnỵ1

nỵ1 :
pnỵ1 ẳ qnỵ1 ỵ p
5. Update velocity eld

v nỵ1 ẳ v~ nỵ1

v

100ị
nỵ1

Dt
rqnỵ1 :
b0 q/n ị

In the above equations, the dimensionless density and viscosity qð/n Þ and

ð101Þ

lð/n Þ are given by

q/n ị ẳ k ỵ 1 kịH /n ị; k ẳ qb =qc ;

102ị

l/n ị ẳ g ỵ 1 gịH /n ị; g ẳ lb =lc :

103ị

where the smoothed Heaviside function H ð/n Þ is calculated from Eq. (93) with the corresponding value of the level set function /n .

Fig. 6. Streamlines at t ¼ 17; 20; 25; 30 of the lid-driven cavity flow (Re ¼ 1000).

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Fig. 7. Contours of vorticity at t ¼ 1; 5; 10; 15 of the lid-driven cavity flow (Re ¼ 1000).

5.3. Advance the level set function
The level set function is advanced by solving the convective transport equation

/t ỵ v r/ ¼ 0;

/ðx; t ¼ 0Þ ¼ /0 ðxÞ;

ð104Þ

for one time step using meshless semi-Lagrangian or Taylor series expansions schemes (SL-IRBFN or Taylor-IRBFN) reported
in [20].

5.4. Re-initialize the level set function
Due to numerical error, the level set function is not necessarily a distance function as desired even after one time step.
Reinitialization is therefore needed to make the function signed distance after certain time steps. This could be achieved by
solving the following PDE to steady state [27].


jr/jị;
/t ẳ S /ị1

 yị
/x; y; t ẳ 0ị ẳ /x;

105ị

where S denotes the smoothed sign function


/
 ẳ p
S /ị
2

/ ỵ 2

106ị

in which  can be chosen to be the minimum distance from any data point to the others. Eq. (105) is solved to steady-state
using the semi-discrete IRBFN scheme with the fourth-order Runge–Kutta method reported in [19].
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17

Fig. 8. Contours of vorticity at t ¼ 17; 20; 25; 30 of the lid-driven cavity flow (Re ¼ 1000).

5.5. Adjust the level set function with the mass correction algorithm
The reinitialization procedure might introduce some numerical diffusion which results in an inaccuracy of the interface
location and some loss of mass [28]. The mass correction is then performed to prevent any losses of mass. After advancing
the level set function at time step t ẳ tn ỵ 1, one gets the moving interface C that bounds the domain X2 ¼ x 2 X : / < 0. To
correct the area of X2 , one changes the zero level set to certain neighbouring isoline based on the fact that it has almost the
same shape since / is a distance function. This can be done by simply moving the level set function upward or downward by
an amount of c/ , where jc/ j is the distance between old and new zero-level sets

/new ¼ / À c/ ;
new

where /
is the new (raised or lowered) level set function, X
tion procedure was described in [20].

107ị
new
2

new

ẳx2X:/

< 0. Details of the IRBFN-based mass correc-

6. Numerical results
6.1. Numerical investigations of the new IRBFN-based projection schemes
This section presents the numerical results obtained by applying the IPC-IRBFN and IPCPP-IRBFN schemes to some

unsteady problems in CFD. The first test problem is an unsteady Navier–Stokes problem with an analytic solution. The meshless IRBFN-based projection schemes are used to solve the problem where the nonlinear convective term is discretized in
time by the second-order Adams–Bashforth method. The proposed schemes are then demonstrated with the well-known
lid-driven cavity flow focusing on the unsteady behaviour of the flow. In the test problems with analytic solutions, the error
norm of function uxj ; tị; u ẳ u; v ; pị, is used to verify the accuracy of the numerical schemes at each time step and defined
as follows.

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L. Mai-Cao, T. Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx
t=0.1

t=0.2

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6

0.8

1

0

0

0.2

0.4

t=0.3
2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0.2

0.4

0.8

1

0.6

0.8

1

t=0.4

2

0

0.6

0.6

0.8

1

0

0

0.2

0.4

Fig. 9. Numerical simulation of two bubbles rising up in a buoyancy-driven flow at time t ¼ 0:1; 0:2; 0:3; 0:4.

v
u N
u1 X
RMSEuị ẳ t
ẵuxj ; tị ua xj ; tị2
N jẳ1

108ị

where N is the number of collocation points, and ua is the corresponding analytic solution. The stability analysis on the new
schemes is carried out by checking the boundedness of the error norm over the time interval of interest.
6.2. Test 1: unsteady Navier–Stokes equations with known analytic solution
Consider the Navier–Stokes equations governing an unsteady flow in a square domain X ẳ ẵ0; p2 with the analytic solution as follows ([4]).
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L. Mai-Cao, T. Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx
t=0.5

t=0.6

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6

0.8

1

0

0

0.2

0.4

t=0.7
2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0.2

0.4

0.8

1

0.6

0.8

1

t=0.8

2

0

0.6

0.6

0.8

1

0

0

0.2

0.4

Fig. 10. Numerical simulation of two bubbles rising up in a buoyancy-driven flow at time t ¼ 0:5; 0:6; 0:7; 0:8.

ux; y; tị ẳ cosxị sinyị exp2tị;

109ị

v x; y; tị ẳ sinxị cosyị exp2tị;

110ị

1

px; y; tị ẳ cos2xị þ cosð2yÞÞ expðÀ4tÞ:
4

ð111Þ

The initial and boundary conditions as well as the forcing term are defined according to the analytic solution.
For this test problem, a Cartesian grid of M ¼ N ¼ 31 Â 31 points is used. Since the purpose of the analysis focuses on the
temporal errors of the present schemes, the point density is chosen so that the error contributed from the spatial discretization does not affect the ultimate error of the numerical schemes. In this test problem, the multistep BDF and AB methods of
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L. Mai-Cao, T. Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx
t=0.9

t=1.0

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6

0.8

1

0

0

0.2

0.4

t=1.1
2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0.2

0.4

0.8

1

0.6

0.8

1

t=1.2

2

0

0.6

0.6

0.8

1

0

0

0.2

0.4

Fig. 11. Numerical simulation of two bubbles rising up in a buoyancy-driven flow at time t ¼ 0:9; 1:0; 1:1; 1:2.

order two are used from which the analysis of the accuracy and stability of the IPC-IRBFN and IPCPP-IRBFN schemes is
accomplished.
The stability analysis of the two schemes in this test problem is shown in Fig. 1. As can be seen from the figures, both IPCIRBFN and IPCPP-IRBFN exhibit good stability over the computational time domain.In particular, the numerical solutions of
the velocity (top) and pressure (bottom) are highly stable over time just with a mild value of the time-step size ðDt ¼ 10À2 Þ.
For this test problem which is the Navier–Stokes equations with time-dependent boundary conditions, the IPCPP-IRBFN
schemes in both standard and rotational forms exhibit good stability and accuracy. It can also be seen from Fig. 1, the errors
of pressure and velocity field obtained by the Standard IPC-IRBFN scheme seem not to vary much with time. This scheme,
however, shows large errors in Fig. 2. From the numerical stability point of view that the errors should be kept unchanged or
not increased much with time, the rotational schemes studied in this work still exhibit better behaviour with respect to stability. In fact, although showing oscillations, the Rotational IPC-IRBFN scheme has its error values in acceptable ranges in all
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L. Mai-Cao, T. Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx
t=1.3

t=1.4

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6

0.8

1

0

0

0.2

0.4

t=1.5
2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0.2

0.4

0.8

1

0.6

0.8

1

t=1.6

2

0

0.6

0.6

0.8

1

0

0

0.2

0.4

Fig. 12. Numerical simulation of two bubbles rising up in a buoyancy-driven flow at time t ¼ 1:3; 1:4; 1:5; 1:6.

of the cases of interest. The other rotational scheme, the Rotational IPCPP-IRBFN, shows good stability in all of the numerical
experiments in this study.
For Dt ¼ 0:005, as can be seen in Fig. 2, the error norm of the velocity is bounded within Oð10À4 Þ, and the pressure error,
with rather high value in a short interval of time, is quickly bounded within Oð10À3 Þ. In addition, the errors of Standard IPCIRBFN scheme increases markedly with time when compared to those of the other schemes. This can be explained as follows.
The Rotational IPC-IRBFN scheme takes into account the divergence of the intermediate velocity field in updating the end-ofstep pressure in Eq. (70) whereas the Standard IPC-IRBFN scheme assumes a divergence-free intermediate velocity field
which is not necessarily true. This error accumulates with time as shown in Fig. 2 for the Standard IPC-IRBFN. This phenomenon, however, does not happen with the Standard IPCPP-IRBFN. This is owing to the fact that the intermediate velocity field
in this scheme is computed from a pressure predictor obtained by solving Eqs. (73) and (74) rather than extrapolating from
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t=1.7

t=1.8

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0.2

0.4

0.6

0.8

1

0

0

0.2

0.4

t=1.9
2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0.2

0.4

0.8

1

0.6

0.8

1

t=2.0

2

0

0.6

0.6

0.8

1

0

0

0.2

0.4

Fig. 13. Velocity field in the numerical simulation of two bubbles rising up in a buoyancy-driven flow at time t ¼ 1:7; 1:8; 1:9; 2:0.

previous step as in the Standard IPC-IRBFN. As a result, the error of the end-of-step velocity field is under control at each time
step in the Standard IPCPP-IRBFN.
In summary, as discussed above, the Standard IPC-IRBFN scheme would not be a method of choice. For the other schemes,

despite the oscillatory behaviour of the error norms, the schemes are more accurate at smaller time steps. In this sense the
schemes are stable. Nevertheless, further study is required for a rigorous explanation of the oscillatory behaviour of the error
norms.
6.3. Numerical analysis of the unsteady lid-driven cavity flow
Consider the lid-driven cavity flow in a unit square domain X ẳ ẵ0; 12 . The upper side of the cavity (i.e. the lid) moves in
its own plane at unit speed from left to right while the other sides are fixed. There is a discontinuity in the boundary
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23

conditions at the two upper corners of the cavity. There are two options in dealing with the discontinuity in numerical
schemes for this problem: (a) The two upper corners are either considered as belonging to the moving upper side (leaky cavity); (b) they are assumed to belong to the fixed vertical walls (non-leaky). Option (a) is adopted in this work.
The objective of this analysis is to investigate the transient behaviour of the lid-driven cavity flow, rather than its steadystate solution. In particular, the influence of Reynolds number on the numerical solutions are of primary interest. Different
values of the Reynolds number are used to study the effect of this dimensionless number on the numerical solution of such
flow. The IPC-IRBFN scheme in rotational form is used in this problem. A Cartesian grid of M ¼ N ¼ 61 Â 61 is used to well
capture the vortices appearing near the cavity corners as well as the primary vortex. For this numerical analysis, the timestep size is fixed at Dt ¼ 0:001.
Fig. 3(top) shows the evolution of the velocity field along the mid-vertical line for the lid-driven cavity flow with
Re ¼ 1000. It is observed that the rate of change in shape of the velocity profile along this line is rather fast in the beginning.
This change slows down with time, and finally the velocity field reaches its steady state profile. Similarly, Fig. 3(bottom)
shows the evolution of the velocity along the mid-horizontal line.
In this work, the evolution of the streamfunction and the vorticity at the centre of the primary vortex at different points in
time are analyzed for different Reynolds numbers. For any of the Reynolds numbers, the streamfunction values at the centre
of the primary vortex change rapidly in the beginning. The rate of change then slows down, and finally the streamfunction
reaches its steady state. It can be seen in Fig. 4 that the higher the Reynolds number is, the longer it takes for the two quantities to reach their steady state. The values of streamfunction and vorticity at the centre of the primary vortex at different
time step size captured by the Rotational IPC-IRBFN scheme are presented in Table 1. Figs. 5 and 6 show the streamfunction
contours at different points in time for the case Re ¼ 1000. The contours of vorticity are shown in Figs. 7 and 8. As can be seen
from the figures, at each time step, the new schemes well capture the primary vortex.

Numerical solutions of unsteady lid-driven cavity flows are presented with Reynolds numbers up to 1000 which is the
case widely reported in the literature. For larger Reynolds numbers, there are some notes to be mentioned as follows. Physically, from a transient analysis point of view, the larger the Reynolds number is, the longer time it takes for the streamfunction and vorticity to reach steady state. This can be predicted from the analysis shown in Fig. 4. Numerically, Reynolds
number directly involves in Step 2 and Step 4 of the proposed schemes. In Eq. (55) of Step 2, Re plays the role of a scaling
factor for the discrete viscous term. This, however, has an insignificant effect on the condition number of the system matrix
to be solved in this step. Indeed, a simple check shows that with Re ẳ 100; 1000ị, the condition number of the system matrix
only changes in a small range of Oð10À2 Þ. Regarding Step 4 of the proposed schemes, as can be seen from Eq. (70), the larger
the Reynolds number is, the less contribution the divergence of the velocity predictor has to the pressure correction.
6.4. Numerical simulation of two buoyancy-driven bubbles
This section reports the application of the new meshless approach presented in Section 5 to simulate the motion and deformation of the two bubbles in an interaction with the surrounding fluid flow. In this numerical experiment, a rectangular cavity is
filled up with two immiscible fluids where the heavier one settles at the bottom and the lighter one at the top. Two bubbles,
containing the same light fluid as in the top layer, are initially embedded in the heavier fluid at the bottom, one above the other.
The bubbles are then released from rest and allowed to rise by buoyancy force. The five primary parameters are chosen as follows: Reynolds number, Re ¼ 10; Bond number, Bo ¼ 5; density ratio, k ¼ 1=10; viscosity ratio, g ¼ 1. It is noted that the density
ratio k indicates that the fluid inside the bubbles and in the top layer is ten times lighter than the heavier fluid. For this numerical
simulation, a Cartesian grid is used with 21 points in the x-direction and 41 points in the y-direction (M ¼ N ¼ 21 Â 41).
As the bubbles are lighter than the surrounding fluid, they will rise with time. The two bubbles start moving upwards
from the bottom of the cavity due to the buoyancy force as can be seen in Fig. 9(top). During the motion, the bubbles merge
together and continuously affect the surrounding fluid flow indicated by the change in direction and magnitude of the velocity around the bubbles, as shown in Fig. 9 (bottom) and Fig. 10. The merging bubbles finally reach the free surface in the
upper part of the cavity and totally diffuse into the body of fluid of the top layer as shown in Figs. 11–13.
As can be observed from the figures, although having the same density and viscosity, the lower bubble moves faster than
the upper one. This can be explained by the wake formation below the upper bubble. As time evolves, the lower one becomes
entrapped into the wake region identified by the large magnitude of the velocity field below the upper bubble making the
lower one move faster.
When the bubbles get closer to the free surface as shown in Fig. 10(top), due to its surface tension, the free surface tends
to prevent the upward motion of the bubble making it flatten remarkably. In turn, the upward motion of bubbles makes the
free surface bend upwards significantly. The figures clearly show the effect of the surface tension in keeping the kinematic
equilibrium on the free surface.
In addition, the presence of the vortices in Fig. 12(top) indicates the effect of the surface tension along the free surface on
the velocity field even when the bubbles completely diffuse into the surrounding fluid.
7. Concluding remarks
A meshless IRBFN-based numerical approach to the simulation of interfacial flows is reported in this paper where the

motion and deformation of the interface as well as the interaction between the moving interface and the surrounding fluid
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are fully captured. The proposed approach consists of (a) the flow modelling scheme, (b) the interface modelling scheme, and
(c) the flow-interface coupling model. Bringing those ‘‘ingredients’’ together requires appropriate modifications as well as
adaptations so as to suit the numerical simulation of the interfacial flows.
Regarding the flow modelling scheme, the IRBFN-based projection schemes not only show their good capability to solve
the unsteady incompressible Navier–Stokes equations but also give stable results for interfacial flows with variable density
and viscosity. With regards to the interface modelling, the meshless approach to capturing moving interfaces reported in
[20] is used with the application of the level set formulation based on smoothed Heaviside and Dirac delta functions. This
helps avoid numerical instabilities in solving the one-fluid continuum model of the interfacial flow. Finally, the flow-interface coupling model based on the CSF model [12] makes it easier to implement the new approach thanks to the fact that no
explicit description of the moving interfaces is needed to impose the kinematic equilibrium conditions on the interface at
each time step.
The proposed meshless approach is applied to the numerical simulation of interfacial flows of two immiscible fluids. The
numerical results show that the new approach is capable of capturing primary phenomena of flows such as the deformation
and topological change of the moving interfaces as well as the interaction between the interface and the surrounding fluid.
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