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Applied Mathematical Modelling xxx (2014) xxx–xxx Contents lists available at ScienceDirect 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 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 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 4631 1332; fax: +61 4631 2110 E-mail addresses: maicaolan@hcmut.edu.vn (L Mai-Cao), thanh.tran-cong@usq.edu.au (T Tran-Cong) http://dx.doi.org/10.1016/j.apm.2014.06.018 0307-904X/Ó 2014 Elsevier Inc All rights reserved Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 L Mai-Cao, T Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx 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 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 at time t Similarly, let X2 be the region containing fluid 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, Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 L Mai-Cao, T Tran-Cong / Applied Mathematical Modelling xxx (2014) xxxxxx q1 @v ỵ v rv ẳ rp1 ỵ r 2l1 D1 ị ỵ q1 g; @t x X1 ; 1ị q2 @v ỵ v rv ẳ rp2 ỵ r 2l2 D2 ị ỵ q2 g; @t x X2 ; 2ị with incompressibility constraints r Á v ¼ 0; x X1 ; 3ị r v ẳ 0; x X2 ; ð4Þ where v i is the velocity field, qi is the density, g is the gravity, pi is the pressure and tensor Di is defined as Di ẳ rv i ỵ rv Ti ; li is the viscosity The rate of strain i ¼ 1; 2: ð5Þ 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 ¼ v 2; x 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 Á n ¼ ðp1 p2 ỵ rjịn; x C; 7ị 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 (bounded by the interface) into fluid (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; ð8Þ > < > 0; x X1 /x; tị ẳ 0; x C > : < 0; x X2 ð9Þ where The unit vector normal to the interface and the curvature of the interface can be expressed in terms of /ðx; tÞ as follows n¼ r/ r/ and j ¼ r jr/j/ẳ0 jr/j /ẳ0 10ị Let vẳ & v 1; v 2; x X1 x 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 Hð/Þ if / < 0; > : if / > ð13Þ and the fluid properties qð/Þ ẳ q2 ỵ q1 q2 ịH/ị; 14ị l/ị ẳ l2 ỵ l1 l2 ịH/ị; 15ị together with the CSF model [12], one obtains the one-fluid continuum formulation [1] for the two-phase incompressible viscous flow Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 L Mai-Cao, T Tran-Cong / Applied Mathematical Modelling xxx (2014) xxxxxx @v 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 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 ẳ p ; pref t ẳ t ; tref v ref ẳ gRị1=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 is hereafter referred to as the fluid surrounding the bubbles with density qc and viscosity lc Similarly, fluid 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 defined as qð/Þ ẳ k ỵ kịH/ị; and l/ị ẳ g ỵ 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 ỵ v rịv ẳ rp ỵ r ẵl/ịrv ỵ f in X @t q/ị q/ịRe 23ị r v ẳ 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 r l/ịrv ịT ỵ j/ịd/ịr/: q/ịRe q/ịBo 26ị Meshless projection schemes for unsteady incompressible Navier–Stokes 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] Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 L Mai-Cao, T Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx 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 in X 28ị ẳ X [ @ X; v x; tị ẳ u; v ÞT is the velocity field, pðx; 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 where v ðxÞ and posed on @ X; 29ị at t ẳ 0; 30ị v b ðx; tÞ are given functions satisfying the following constraints for the Navier–Stokes equations to be well ; r Á v ẳ in X 31ị n v ¼ 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 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: ð35Þ The new time-level (end-of-step) velocity v nỵ1 nỵ1 and pressure p are then obtained by solving nỵ1 v v~ nỵ1 ỵ rpnỵ1 ẳ in X; Dt 36ị r v nỵ1 ẳ 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 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 on @ X: @n ð40Þ The end-of-step velocity field can be then updated using Eq (36) v nỵ1 ẳ v~ nỵ1 rpnỵ1 : Dt 41ị Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 L Mai-Cao, T Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx 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 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 nỵ1 v v~ nỵ1 ỵ r pnỵ1 pn ẳ in X; Dt 44ị r v nỵ1 ẳ 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 nỵ1 v~ Dt in X; @qnỵ1 ẳ 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]: 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 nỵ1 nỵ1k dt % b v bk v @t Dt kẳ1 51ị The values of coefficients b’s corresponding to J v are given in the next section 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 ak ẵv rịv nk 52ị kẳ0 The values of coefficients a’s corresponding to J v are given in the next section 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 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; Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 L Mai-Cao, T Tran-Cong / Applied Mathematical Modelling xxx (2014) xxxxxx nỵ1 Calculate a predictor for the pressure, p nỵ1 ẳ p Jp X ak pnÀk k¼0 Jp ¼ > < 0; n Jp ¼ ¼ p ; > : n 2p À pnÀ1 ; J p ¼ 54ị ~ nỵ1 , by solving Compute a predictor for the velocity field, v J v X ~ nỵ1 b0 v bk v nỵ1k Dt kẳ1 ! ỵ JX v 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 n Jv ẳ > Jv : n nÀ1 nÀ2 kẳ1 3v v ỵ v ; Jv ¼ ðb ¼ 1Þ À 3Á b ¼ 211 b0 ẳ 57ị and JX v ak ẵv rịv nk kẳ0 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 Calculate the pressure increment qnỵ1 r2 qnỵ1 ẳ b0 r v~ nỵ1 Dt in X; 59ị @qnỵ1 ẳ on @ X: @n ð60Þ Perform the correction step for pressure pnỵ1 nỵ1 pnỵ1 ẳ qnỵ1 ỵ p ð61Þ Perform the correction step for velocity field v nỵ1 v nỵ1 Dt ~ nỵ1 rqnỵ1 : ẳv b0 ð62Þ 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 b v nỵ1 bk v nỵ1k Dt kẳ1 ! ỵ JX v ak ẵv rịv nk ẳ rpnỵ1 ỵ mr2 v nỵ1 ỵ f nỵ1 in X; 63ị kẳ0 r v nỵ1 ẳ 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 b0 nỵ1 v v~ nỵ1 ẳ rpnỵ1 pnỵ1 ị ỵ mr2 v nỵ1 v~ nỵ1 ị; Dt 65ị By taking the divergence of Eq (65), one has À Á  à b0 r Á 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 ð67Þ Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 L Mai-Cao, T Tran-Cong / Applied Mathematical Modelling xxx (2014) xxxxxx 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 on @ X: @n 69ị The solving procedure in the Rotational tnỵ1 ẳ n ỵ 1ịDt; n ¼ 0; 1; 2; as follows 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 field, 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 ð70Þ Perform the correction step for velocity field,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 ¼ Àr ẵv rịv f in X; 71ị with the Neumann boundary condition being derived by taking the normal component of the momentum Eq (27) as @p @v ẳn 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 nỵ1 r p (J À1 v X ) ak ½ðv Á rịv ẳr nk ỵf nỵ1 in X; 73ị kẳ0 nỵ1 @p ẳn @n ( b0 v nỵ1 ỵ PJv kẳ1 bk v nỵ1k Dt JX v ak ẵv rịv kẳ0 nk JX v 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 ¼ rðr Á v Þ À 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 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 Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 L Mai-Cao, T Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx Let uðx; tÞ be an unknown function continuously dened on Q T :ẳ 0; Tị X, where X & Rd ; d ¼ 1; 2; is a bounded domain For convenience, the coordinates of a typical point are denoted by x ¼ ðx; y; zÞ 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 ðtÞT , 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 @ 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 hðxÞ ^ 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; ð80Þ or Duchon’s thin plate splines (TPS) uðjjx 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] sj ẳ b dj 82ị ; in which b is a user-defined parameter and matrix H is defined as h1 ðx1 Þ 6 h1 ðx2 Þ H ¼ 6 h1 ðxM Þ h2 ðx1 Þ hN ðx1 Þ h2 ðx2 Þ hN ðx2 Þ h2 ðxM Þ hN ðxM Þ dj is the distance from the jth data point to its nearest neighbouring point.The 7 7; 83ị where hj xị; j ẳ 1; ; N is the jth component of hðxÞ, and N ẳ N ỵ 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 @ uðx; tÞ % w@ xx ðxÞT uðtÞ; @x2 ð84Þ @uðx; tÞ % w@x ðxÞT uðtÞ; @x ð85Þ uðx; tÞ % wðxÞT uðtÞ; ð86Þ ^ T H ; w@xx xị ẳ hxị 87ị ~ T H ; w@x xị ẳ hxị 88ị where wxị ẳ hðxÞT H À1 : ð89Þ Let S be a certain differential operator in space that operates on the scalar function ux; tị in X 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 Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 10 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 wðxÞ, wS xị ẳ ẵSw1 xị; Sw2 xị; ; SwM ðxÞT : 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 @ uðx; y; tÞ % wT@ y ðx; yÞuðtÞ: @y ð92Þ 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 10 Pressure Error Norm 10 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 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 Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 11 L Mai-Cao, T Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx 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 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 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 10 10 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 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 Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 12 L Mai-Cao, T Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx 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] if / < > : if / > ð93Þ Fig 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 Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 13 L Mai-Cao, T Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx The above Heaviside function defines the smoothed Dirac delta function d as follows & d /ị ẳ 1=21 ỵ cosp/=ịị= if j/j < 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 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 10 15 20 25 30 Time Fig 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 Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 14 L Mai-Cao, T Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx Table 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 ¼ 100 wc xc ị Re ẳ 400 wc xc ị Re ¼ 1000 wc ðxc Þ 2.5 7.5 10 12.5 15 17.5 20 25 30 À0.073624 (À3.668184) À0.092665 (À3.551824) À0.100557 (À3.247018) À0.102477 (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 (À6.84297) À0.080071 (À4.62247) À0.091677 (À3.596885) À0.099589 (À3.049279) À0.105255 (À2.710798) À0.109306 (À2.484563) À0.11212 (À2.333954) À0.113977 (À2.231622) À0.115874 (À2.115058) 0.116652 (2.062449) nỵ1 Calculate a predictor for the pressure, p nỵ1 ẳ p Jp X ak pnk : 95ị kẳ0 ~ nỵ1 , by solving Compute a predictor for the velocity field, v J v X ~ nỵ1 b0 v bk v nỵ1k Dt kẳ1 ! ỵ JX v kẳ0 ak ẵv rịv nk ẳ q/n ị rpnỵ1 ỵ 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 Streamlines at t ¼ 1; 5; 10; 15 of the lid-driven cavity flow (Re ¼ 1000) Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 L Mai-Cao, T Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx 15 Calculate the pressure increment qnỵ1 r ! b nỵ1 ~ nỵ1 ẳ 0rv r q Dt q/n ị in X; 98ị @qnỵ1 ẳ on @ X: @n 99ị Perform the correction step for pressure pnỵ1 nỵ1 : pnỵ1 ẳ qnỵ1 ỵ p 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 ị are given by q/n ị ẳ k ỵ kịH /n ị; k ẳ qb =qc ; 102ị l/n ị ẳ g ỵ 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 Streamlines at t ¼ 17; 20; 25; 30 of the lid-driven cavity flow (Re ¼ 1000) Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 16 L Mai-Cao, T Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx Fig 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/ ẳ 0; /x; 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] À jr/jị; /t ẳ S /ị1 yị /x; y; t ¼ 0Þ ¼ /ðx; ð105Þ where S denotes the smoothed sign function / ẳ p S /ị / 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] Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 L Mai-Cao, T Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx 17 Fig 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 X : / < 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] ð107Þ new new ¼x2X:/ < Details of the IRBFN-based mass correc- 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 ; pÞ, is used to verify the accuracy of the numerical schemes at each time step and defined as follows Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 18 L Mai-Cao, T Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx t=0.1 t=0.2 2 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0.2 0.4 0.6 0.8 0 0.2 0.4 t=0.3 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.2 0.4 0.8 0.6 0.8 t=0.4 0.6 0.6 0.8 0 0.2 0.4 Fig Numerical simulation of two bubbles rising up in a buoyancy-driven flow at time t ¼ 0:1; 0:2; 0:3; 0:4 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 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 ow in a square domain X ẳ ẵ0; p2 with the analytic solution as follows ([4]) Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 19 L Mai-Cao, T Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx t=0.5 t=0.6 2 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0.2 0.4 0.6 0.8 0 0.2 0.4 t=0.7 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.2 0.4 0.8 0.6 0.8 t=0.8 0.6 0.6 0.8 0 0.2 0.4 Fig 10 Numerical simulation of two bubbles rising up in a buoyancy-driven flow at time t ẳ 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 px; y; tị ẳ cos2xị ỵ cos2yịị exp4tị: 111ị 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  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 Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 20 L Mai-Cao, T Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx t=0.9 t=1.0 2 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0.2 0.4 0.6 0.8 0 0.2 0.4 t=1.1 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.2 0.4 0.8 0.6 0.8 t=1.2 0.6 0.6 0.8 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 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 Dt ẳ 102 ị 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 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 Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 21 L Mai-Cao, T Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx t=1.3 t=1.4 2 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0.2 0.4 0.6 0.8 0 0.2 0.4 t=1.5 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.2 0.4 0.8 0.6 0.8 t=1.6 0.6 0.6 0.8 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 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 Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 22 L Mai-Cao, T Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx t=1.7 t=1.8 2 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0.2 0.4 0.6 0.8 0 0.2 0.4 t=1.9 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.2 0.4 0.8 0.6 0.8 t=2.0 0.6 0.6 0.8 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 Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 L Mai-Cao, T Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx 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  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 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 Figs and show the streamfunction contours at different points in time for the case Re ¼ 1000 The contours of vorticity are shown in Figs and 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 Numerically, Reynolds number directly involves in Step and Step 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 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 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 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  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 (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 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 Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 24 L Mai-Cao, T Tran-Cong / Applied Mathematical Modelling xxx (2014) xxx–xxx 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 References [1] Y Chang, T Hou, B Merriman, S Osher, A level set formulation of Eulerian interface capturing method for incompressible fluid flows, J Comput Phys 124 (1996) 449–464 [2] M Ishii, T Hibiki, Thermo-Fluid Dynamics of 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press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell (2014), http://dx.doi.org/10.1016/j.apm.2014.06.018 ... steady state Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl... (73) and (74) rather than extrapolating from Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method. .. 67ị Please cite this article in press as: L Mai-Cao, T Tran-Cong, A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows, Appl Math Modell