DSpace at VNU: A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows

DSpace at VNU: A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfa...

A meshless numerical approach based on Integrated Radial

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1 Introduction

Fluid flows studied in this paper with a moving interface between two immiscible fluids can be classified as interfacialflows In general, a numerical approach to the simulation of such flows consists of (a) a flow modelling method, (b) an inter-face 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 interfacialflow: one-fluid and two-fluid models In the one-fluid model, a single flow equation is formulated to describe both fluidflows, and a characteristic function is used to specify a particular fluid[1] In the two-fluid model, on the other hand, eachfluid has its own governing equations and therefore the characteristics of each phase can be separately captured[2] Forincompressible interfacial flows, the governing equations in either one-fluid or two-fluid model are formulated from the

⇑Corresponding author Tel.: +61 7 4631 1332; fax: +61 7 4631 2110.

E-mail addresses: maicaolan@hcmut.edu.vn (L Mai-Cao), thanh.tran-cong@usq.edu.au (T Tran-Cong).

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Applied Mathematical Modelling

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / a p m

Navier–Stokes equations (NSEs) Among others, the projection/pressure correction method can be used to solve NSEs Firstproposed in[3], the projection method consists of a predictor–corrector procedure in which the momentum equation is firstsolved using an initial approximation of the pressure to obtain an intermediate velocity field A pressure correction is thenobtained by solving a Poisson equation Finally, the new velocity field is updated using the intermediate velocity and thepressure correction Several improvements to the original projection method have been made by (a) improving intermediatevelocity 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 Forthe 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 interfacemoves The second approach which is based on fixed grids includes tracking and capturing methods The tracking methodsexplicitly represent the moving interface by means of predefined markers[8] In capturing methods, on the other hand, themoving interface is not explicitly tracked, but rather captured via a characteristic function Examples of the capturing meth-ods are phase field method[9], volume-of-fluid method[10]and level set method[11] The characteristic function used toimplicitly describe the moving interface is the order parameter in the phase field method, volume fraction in the volume-of-fluid method and level set function in the level set method For these capturing methods, no rezoning/remeshing is needed tomaintain 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 takeninto account in the computation of force balance at the interface where the difference in stresses of the two fluids in thedirection normal to the interface is balanced by the surface tension on the interface[7] A simple and effective model thatalleviates the interface topology constraints was presented in[12]where the proposed model, known as the continuum sur-face force (CSF) model, interprets the surface tension as a continuous, three-dimensional effect across an interface ratherthan as a boundary value condition on the interface The advantage of the CSF model is that the moving interface needsnot be explicitly described for the interfacial boundary condition

In this work, all of the aforementioned modelling techniques are implemented within the meshless framework of theIRBFN 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 InterpolationMethod (Lc-Rpim) was studied in[16]for solid mechanics The IRBFN method has been reported to be a highly accurate toolfor approximating functions, their derivatives, and solving differential equations[17,18] The method was then successfullyapplied to transient problems[19]including those governed by parabolic as well as hyperbolic PDEs where comparisons ofperformance of the IRBFN-based methods and others, including finite difference, boundary element and finite element meth-ods, 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 cedure combining the IRBFN method and the level set method together with the semi-Lagrangian method or Taylor seriesexpansions 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 meshlessframework of the IRBFN method, the two schemes showed their good stability and accuracy when applied to the numericalsimulation of incompressible fluid flows and interfacial flows in[21] In the present work, the two schemes are furthernumerically investigated, specifically for Navier–Stokes equations with time-dependent boundary conditions as well asfor the numerical simulation of buoyancy-driven bubble flow In comparison with finite difference or finite element meth-ods, the unique advantages of the proposed IRBFN-based methods include (i) the meshless nature and implicit interface cap-turing technique of the present approach; and (ii) the ability to effectively ensure the conservation of mass during theevolution of the interfaces between fluids

pro-The remaining of this paper is organized as follows Firstly, the one-fluid model is formulated for interfacial flows with anintroduction to the CSF model and level set method The new meshless projection schemes for the one-fluid continuummodel 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 proposedmeshless IRBFN-based approach to the numerical simulation of two bubbles moving, stretching and merging in an ambientviscous flow are then performed for verification purposes

2 Mathematical formulation

Consider a domainXand its boundary @Xcontaining two immiscible Newtonian fluids, both being incompressible LetX1

be the region containing fluid 1 at time t Similarly, letX2be the region containing fluid 2 and bounded by the fluid interface

Cat time t The governing equations describing the motion of the two fluids in their own regions are given by the Navier–Stokes equations,

inter-Let the fluid interface be the zero level of the level set function /,

where k ¼qb=qcis the density ratio,g¼lb=lcis the viscosity ratio.

Eq.(18)is rewritten in the form similar to the Navier–Stokes equation where the gravity force, the surface tension andnonsymmetric part of the rate of strain tensor are treated as the forcing term f

3 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 incompressibleNavier–Stokes equations proposed in[21] As can be seen in the next section, with a straightforward adaptation, these pro-posed 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 withthe high-order multistep time integration are presented in this section They include: (a) Standard IPC-IRBFN, a meshlessincremental pressure correction scheme in the standard form inspired by Van Kan[5]; (b) Rotational IPC-IRBFN, a meshlessincremental pressure correction scheme in the rotational form motivated by Timmermans et al.[23]; (c) Standard IPCPP-IRBFN, 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 pre-diction motivated by Timmermans et al.[23]

3.1 The Navier–Stokes equations

Consider a domainX R2with boundary @X The Navier–Stokes equations that govern incompressible viscous flows arecomprised of the momentum and continuity equations and written in dimensionless form as follows

@v

@t þðvrÞv¼ rp þm r2vþ f ; inX; ð27Þ

where X¼X[ @X;vðx; tÞ ¼ ðu;vÞTis the velocity field, pðx; tÞ is the kinematic pressure, f ðx; tÞ is the body force vector, andm

is the kinematic viscosity The velocity field is subject to boundary and initial conditions as follows

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 beimposed through the boundary conditions, leading to the constraint[24]

Z

@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 thebackward Euler time stepping with the linearized convective term and without the pressure gradient

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

by solving the following equations

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 beformulated 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 order backward Euler method In particular, the temporal derivative is discretized in time as follows

The values of coefficients b’s corresponding to Jvare 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 lastvalue

Zt nþ1

t n ½ðvrÞvdt XJv 1

k¼0

The values of coefficientsa’s corresponding to Jvare 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 intermediatevelocity field, the IPC-IRBFN scheme uses a pressure predictor which is extrapolated from multiple previous steps asfollows

1 Calculate a predictor for the pressure, pnþ1

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-stepvelocity and pressure,vnþ1and pnþ1, must numerically satisfy the momentum and continuity equations regardless of how thevelocity predictor, ~vnþ1, is calculated More specifically, the momentum Eq.(27)and the continuity Eq.(28)must hold for

vnþ1and pnþ1in the semi-discrete form in time as follows

By letting qnþ1¼ pnþ1 pnþ1þmr ~vnþ1be the pressure increment in this case, one obtains the Poisson equation for qnþ1asfollows.

1 Calculate a predictor for the pressure, pnþ1, using Eq.(54)

2 Compute a predictor for the velocity field, ~vnþ1, by solving Eqs.(55) and (56)

3 Calculate the pressure increment, qnþ1, by solving Eqs.(68) and (69)as in the Standard IPC-IRBFN scheme

4 Perform the correction step for the new pressure pnþ1

pnþ1¼ qnþ1þ pnþ1m r ~vnþ1: ð70Þ

5 Perform the correction step for velocity field,vnþ1, 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 ofthe IPC-IRBFN schemes differ in the manner that the pressure increment, qnþ1, is defined, and thus in the pressure correctionstep

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-IRBFNschemes 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

where Dirichlet boundary condition on velocity,vnþ1¼vnþ1

b , is applied tovnþ1in Eq.(74) It is noted that in the Neumannboundary condition for the pressure prediction, the viscous term is decomposed into

r2

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 ing procedure in the IPCPP-IRBFN schemes is summarized for each time level tnþ1¼ ðn þ 1ÞDt; n ¼ 0; 1; 2; as follows

solv- 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 isbriefly captured here Interested readers are referred to[19,20]for further details

Let uðx; tÞ be an unknown function continuously defined on QT:¼ ð0; TÞ X, whereX Rd; d ¼ 1; 2; 3 is a boundeddomain 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 fxkgMk¼1 be a set of discrete data points in X, anduðtÞ ¼ ½u1ðtÞ; u2ðtÞ; ; uMðtÞT, 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

where hjðxÞ; j ¼ 1; ; N is the jth component of hðxÞ, and N ¼ N þ P in which P is the number of discrete points needed toapproximate 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

Number of time steps

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

Number of time steps

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

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.

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 functionvalue at the previous step

Step 2: Solve the one-fluid continuum equations for the flow variables taking into account the interface dependency of sity and viscosity as well as the surface tension

den-Step 3: Advance the level set function from the previous step to the current one with the most updated velocity field culated from Step 2

cal-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 forfurther evolution of the moving interface until the predefined time is reached

Number of time steps

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

Number of time steps

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

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.

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 inqandlacross 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 theinterface This can be done by replacing the Heaviside function in Eq.(13)with a smoothed Heaviside function Hð/Þ defined