Comp. Part. Mech. (2015) 2:233–246DOI 10.1007/s40571-015-0046-7

Hybrid grid-particle method for fluid mixing simulation

Takuya Matsunaga1 · Kazuya Shibata1 · Kohei Murotani1 · Seiichi Koshizuka1

Received: 14 January 2015 / Revised: 18 April 2015 / Accepted: 13 May 2015 / Published online: 19 May 2015© OWZ 2015

Abstract In this paper, we propose a hybrid grid-particlemethod for the numerical simulation of fluid mixing prob-lem. The method solves the incompressible Navier–Stokesequations using a grid method and the advection–diffusionequation of chemical species concentration using a particlemethod. With this framework, the fluid mixing problem atmoderate Reynolds number (∼102) and high Péclet num-ber (∼105), which is typical for liquid–liquid mixing inmicrosystem, can efficiently be solved utilizing the charac-teristics of each approach; the flow field is stably solvedeven with long time step, and the concentration field isaccurately solvedwithminimal numerical diffusion. The pro-posed hybrid method is examined through three test cases,and the results show that the hybrid method provides sub-stantial accuracy for simulating the fluid mixing problem athigh Péclet number.

Keywords Numerical simulation · Hybrid method ·Fluid mixing · Numerical diffusion

1 Introduction

Fluidmixing is ubiquitous in nature but an important physicalphenomenon encountered in many industrial applications. Itis associated with various physics like chemical reaction,synthesis, substance generation, separation, etc. In chemi-cal engineering field, predicting and controlling fluid mixing

B Takuya Matsunagamatsunaga@mps.q.t.u-tokyo.ac.jp

Seiichi Koshizukakoshizuka@sys.t.u-tokyo.ac.jp

1 Department of Systems Innovation, The University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

process in reactors is one of the intensive research subjects.That is because many types of chemical productions involveliquid blending process to initiate chemical reactions, andmoreover, properties of the resulting products can signifi-cantly be associated with involved mixing procedure. Forexample, mixing process in the nanoparticle precipitation isknown to influence the particle size distribution of the out-put [50,54]. In this context, the numerical simulation of fluidmixing attracts continuous attention from industry and acad-emia as a powerful tool to analyze and optimize the chemicalproduction process.

On the background of recent considerable progress inmicrofabrication technology, mechanical devices involvingmicrofluidic system, such as MEMS (micro-electro mechan-ical system) [14,16] and lab-on-a-chip device [17,48,51],have been prevalent as a promising technology for chem-ical, biological, and pharmaceutical industries [15,24,45].In those devices, microscale mixing, mixing in characteristiclength scale frommillimeter to submicrometer order, is oftenan essential and crucial unit and has an important role to playin [8]. That is because, in such small dimensions, viscousforce dominates, in other words Reynolds number is small,and the flow tends to be strongly stratified; mixing in suchlaminar regime most likely becomes inefficient and requireslong residence time to make mixture sufficiently homoge-neous. In pursuit of efficient rapid mixing in microsystems,a variety of geometrical designs is developed as a passivemicromixer [19]. For instance, herringbonemicromixer [52],three-dimensional serpentine micromixer [34], and T-shapedmicromixer [4,21,58] with modifications [2,39] are the pas-sive micromixers that achieve effective mixing enhancementby incorporating so-called chaotic advection [43] at lowReynolds number.

A number of numerical simulations have been carriedout for analysis of fluid mixing behavior in micromixer and

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234 Comp. Part. Mech. (2015) 2:233–246

assessment of its mixing performance [1,3,6,9,12,22,35],wherein the motion of incompressible Newtonian fluidsand the molar concentration of chemical species are solvedbased on the continuity and Navier–Stokes equations andthe advection–diffusion equation, respectively.Most of thosesimulations are performed using the finite volume method(FVM) [56]. FVM has several strong points such as con-servativeness of physical quantity, geometric adaptability ofcomputational mesh, and high numerical stability, while atthe same time, the numerical diffusion takes place, like manyother numerical methods based on the Eulerian description.The numerical diffusion leads to excessive diffusion rateand can be a crucial limiting factor for accurate solution.Influence of the numerical diffusion is known to be moresignificant at higher Péclet number for the solution of concen-tration (or Reynolds number for velocity), and in general, itdemands considerably high spatial resolution tomake numer-ical diffusion negligible at very high Péclet number. In manycases of the fluid mixing problems in micromixer, althoughthe Reynolds number is moderate, e.g. only in the order of102 atmost [42], since the characteristic length scale is small,the Péclet number can be very high because the Schmidtnumber is in the order of 103 if working fluid is liquid [3].Therefore, the fluid flow can accurately be calculated with nodifficulty, but solution of the chemical species concentrationfield requires considerable computational expense for satis-factory accuracy due to the unavoidable numerical diffusionin FVM [5,7].

A numerical method based on the Eulerian description,such as FVM, is called grid method, and in contrast, amethod based on the Lagrangian description is called particlemethod. Particle methods for the computational fluid dynam-ics, such as SPH [40] and MPS [28], have been attractinga great deal of attention for their capability to treat com-plex geometry and physics, and applied to various kindsof problems [31,33] including ones that are difficult to besolved using the grid method. Another important property ofthe particle method is that the numerical diffusion is muchless significant than in the grid method. In grid method, thenumerical diffusion occurs due to the discretization error ofthe convective term; on the other hand, in particle methodno convective term appears in the governing equations, andin this respect, occurrence of the numerical diffusion can beavoided.

Under these circumstances, this research has been doneaiming at developing a hybrid of grid and particle methodsthat provides substantial accuracy for simulating a fluid mix-ing problem even at high Péclet number, e.g. higher than 105.The current main target is a liquid–liquid mixing problem inmicrosystem that is solved under following assumptions; thedynamics of the fluid is described as continuum, incompress-ible, and Newtonian; the fluid flow is under laminar state; allfluids to be mixed have identical and homogeneous physi-

+

Grid system Flow field

Particle system Concentration field

Fig. 1 Concept of the hybrid grid-particle method

cal properties, by which the chemical species concentrationcan be treated as a passive scalar; no chemical reaction takesplace.

2 Hybrid grid-particle method

2.1 Overview

The calculation procedure of fluid mixing simulation can bedivided into two large parts: solution of fluid flow and solu-tion of chemical species concentration. As depicted in Fig. 1,the present hybridmethod combines grid-based approach andparticle-based approach in such a way that a grid method isused for the solution of fluid flow and a particle method isused for the solution of concentration field. With this frame-work, the fluid mixing problem of interest can efficiently besolved utilizing the characteristics of each approach.

In general, the numerical instability frequently arises assevere practical difficulties in computational fluid dynam-ics, and ensuring stability can be considered as primaryimportance. The grid method is more reliable to conduct thenumerical simulation stably than the particle method. In theparticle method, there exists a strict limitation of the timestep size for stable computation, whereas the grid methodallows much longer time step. Associated with this fact, thegrid method is suitable to find steady solution. This featurecan also be an important advantage in the current problembecause a majority of the fluid mixing processes in passivemicromixer deals with the steady flow in low Reynolds num-ber range [42]. Furthermore, at such a low Reynolds numberthe numerical diffusion can substantially be vanished withaffordable spatial resolution and hardly causes significanterror. For these reasons, the grid method is appropriate forthe solution of fluid flow. In the present hybrid method, thefluid flow is solved by means of the widely-used FVM, bywhich the simulation can be conducted quite stably evenwitha long time step.

On the other hand, the solution of chemical species con-centration requires a greater emphasis on the accuracy rather

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Comp. Part. Mech. (2015) 2:233–246 235

than the computational stability. For the accurate analysisof the mixing process, it is of importance to simulate faith-fully behavior of the contact surface between distinct fluidsand development of the boundary layer generated around thecontact surface. As stated in Sect. 1, the Péclet number cantake very high value in the fluid mixing problem with liq-uids. The nature of high Péclet number makes the boundarylayer thickness very thin and produces sharp discontinuitiesat the contact surface in the concentration field. Such a mov-ing sharp discontinuity is difficult to be dealt with in the gridmethod and requires sophisticated treatment as non-diffusiveinterface; otherwise, the thin boundary layer ismost probablydestroyed by the numerical diffusion. The multi-phase flowis a representative application of the non-diffusive movinginterface calculation. A number of numerical techniques hasbeen developed to address this problem; examples are thevolume of fluid (VOF) method [20], piecewise linear inter-face calculation (PLIC) method [47,60], coupled level setand volume-of-fluid (CLSVOF) method [53], etc [49]. How-ever, those interface calculation techniques assume absenceof diffusion and are basically inapplicable to the currentfluid mixing problem unless the Péclet number is infinite.In contract to the grid method, the particle method is ableto conserve the sharp discontinuity with faithful calculationof the diffusion. Moreover, the numerical stability in calcula-tion of the diffusion equation is guaranteed by employing theimplicit time integration, and even with an explicit schemethe stability condition is rarely violatedwhen the Péclet num-ber is very high. For these reasons, the particle method isappropriate for the solution of concentration.

The present hybrid method, which combines grid and par-ticle methods, use both computational grid and particles. Thegrid is fixed in space and the particle moves according to theflow field calculated in the grid system. The variables neededfor the solution of fluid flow, i.e. velocity and pressure, aredefined on the grid system, and each particle has only theconcentration variable. The flow field is discretized and cal-culated in the grid system, and the concentration field is inthe particle system; in this respect, the present hybridmethoddiffers from the existing hybridmethods, such asmarker-and-cell (MAC) method [18], particle-in-cell (PIC) method [13],Liu’s hybrid method [32] and particle finite element method(P-FEM) [23]. To solve the fluid mixing problem with thehybrid grid-particle system, transfer of the velocity field fromthe grid system to the particle system is needed. The pro-posed method achieves this with interpolation by means ofthe moving least square approach. Transfer of the concen-tration field from the particle system to the grid system isneeded only if the chemical species concentration is an activescalar, while this process can be skipped in the present studysince passive scalar is assumed. There are numerical meth-ods developed for the accurate simulation of fluid mixingproblem. The backward random-walk Monte Carlo method

[57] is a numerical diffusion-free computational method forthe advection–diffusion problem, and it has provided a rig-orous solution to the mixing process even for large valuesof the Péclet number [36,37]. A semi-Lagrangian methoddeveloped by Matsunaga and Nishino [38], which utilizesfluid particle trajectory for discretization of the advection–diffusion equation, achieves an accuracy equivalent to that ofthe backward random-walk Monte Carlo method with lessercomputational cost. However, those computational methodsare computationally quite expensive andpractically restrictedto the time-independent analysis of the passive scalar trans-port.

2.2 Solution of fluid flow (grid phase)

With the continuum assumption of flow, the dynamics of theincompressible Newtonian fluid is described by the continu-ity and the Navier–Stokes equations:

∇ · u = 0, (1)∂u∂t

+ ∇ · (uu) = −∇ p + 1

Re∇2u, (2)

where velocity u, pressure p, and time t are nondimen-sionalized by characteristic velocity U, ρU 2 (ρ is fluiddensity), and L/U (L is characteristic length), respectively.The Reynolds number (Re) is defined as

Re = LU

ν, (3)

where ν denotes kinematic viscosity. For simplicity of theproblem, fluids to bemixed are assumed to have identical andhomogeneous physical properties. By this assumption, themotion of fluid is independent from the concentration field;in other words, the concentration can be treated as passivescalar.

The governing equations of fluid motion is discretizedbased on FVM. The variables velocity u and pressure p arelocated at the cell center, i.e. collocated grid system [26,46] is used. As the velocity-pressure coupling method, thePISOalgorithm [25] and theSIMPLEalgorithm [44] are usedrespectively for unsteady and steady conditions.

2.3 Solution of concentration (particle phase)

2.3.1 Governing equations

The molar concentration of chemical species is governedby the advection–diffusion equation (so-called species equa-tion):

Dc

Dt= 1

Pe∇2c, (4)

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236 Comp. Part. Mech. (2015) 2:233–246

where concentration c is scaled to take on a value from 0 to1. The Péclet number (Pe) is defined as

Pe = LU

D, (5)

where D denotes diffusion coefficient. To solve it withLagrangian particle system, the advection–diffusion equa-tion (Eq. 4) is split into the advection equation (Eq. 6) andthe diffusion equation (Eq. 7):

drdt

= u, (6)

∂c

∂t= 1

Pe∇2c, (7)

where r denotes the particle position. The variable of con-centration c is defined on each particle and spatially movesaccording to the flow field solved in the grid system. Equa-tions (6) and (7) are solved separately.

2.3.2 Advection

The advection equation (Eq. 6) governs time evolution ofparticle position. To calculate the particle movement, veloc-ity information is required at particle position, but velocityis defined only on grid. Thus, the velocity field calculated onthe grid system is mapped onto to each particle by means ofthemoving least square [30] with theweight function definedas

w(x) =⎧⎨

⎩

1

|x|/re + ε− 1

1 + ε0 ≤ |x| < re

0 re ≤ |x|, (8)

where re is the effective radius, ε is a small value to avoidthe singularity encountered at |x| = 0. The effective radiusre is set as 1.8�x (�x is grid spacing), and ε is 10−6. Theparticle position r is updated based on a two-stage algorithm(so-called Heun’s method, illustrated in Fig. 2) as

r(t + �t) = r(t) + �t

2{u(t, r(t)) + u(t + �t, r(t))} , (9)

where r(t) is an intermediate position

r(t) = r(t) + �t u(t, r(t)), (10)

and u(t, x) is a velocity vector approximated by the movingleast square.

To preserve the accuracy of the particle advection calcu-lation, subdivision of the time advancement is recommendedso that the local Courant number

Co(t, x) = |u(t, r(t))|�t

�x(11)

r(t+Δt)

r(t)

r(t)˜

Fig. 2 Calculation of particle advection using Heun’s method

does not exceed some specified limit, e.g. Comax = 0.5.

2.3.3 Diffusion

The Laplacian operator in the diffusion equation (Eq. 7) isdiscretized by means of the standard LSMPS scheme type-A [55], which is based on the Taylor series expansion andthe weighted least squares method. The formulation of thediscretization scheme is briefly explained below. The Taylorseries expansion of scalar function φ around ri with nearbypoint r j yields

φ(r j ) = φ(ri )

+n∑

m=1

[1

m!(ri j · ∇)m

φ(r)]

r=ri

+ O(|ri j |n+1) (12)

⇔n∑

m=1

[rmsm!

(r∗i j · ∇

)mφ(r)

]

r=ri

= φ(r j ) − φ(ri )

+O(|ri j |n+1) (13)

where ri j = r j − ri , r∗i j = ri j/rs, rs is scaling parameter.

Equation (13) can be written in the form:

pT(r∗i j ) · H−1

rs δ = φ(r j ) − φ(ri ) + O(|ri j |n+1), (14)

where namely in two-dimensional space

r∗i j = r j − ri

rs=

(x j − xi

rs,y j − yi

rs

)T

, (15)

p(r) =(x, y, x2, xy, y2, . . . , xn, xn−1y, . . . , yn

)T, (16)

Hrs = diag

(1!r1s

,1!r1s

,2!r2s

,1!1!r2s

,2!r2s

,

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Comp. Part. Mech. (2015) 2:233–246 237

. . . ,n!rns

,(n − 1)!1!

rns, . . . ,

n!rns

)

, (17)

δ =(

∂φ

∂x,∂φ

∂y,∂2φ

∂x2,

∂2φ

∂x∂y,∂2φ

∂y2,

. . . ,∂nφ

∂xn,

∂nφ

∂xn−1∂y, . . . ,

∂nφ

∂yn

)T

. (18)

One minimizes the functional

J(H−1rs δ

)

=∑

j �=i

w(ri j )[pT(r∗

i j ) · H−1rs δ − {

φ(r j ) − φ(ri )}]2

,

(19)

which leads to the normal equations

⎧⎨

⎩

∑

j �=i

w(ri j )p(r∗i j )p

T(r∗i j )

⎫⎬

⎭·(H−1rs δ

)

=∑

j �=i

w(ri j )p(r∗i j )

{φ(r j ) − φ(ri )

}. (20)

Equation (20) is expressed in matrix notation as

Mi

(H−1rs δ

)= bi (21)

with

Mi =∑

j �=i

w(ri j )p(r∗i j )p

T(r∗i j ), (22)

bi =∑

j �=i

w(ri j )p(r∗i j )

{φ(r j ) − φ(ri )

}. (23)

If the moment matrix Mi is not singular, solutions δ areuniquely determined by solving the normal equations,

δ = Hrs

(M−1

i bi)

, (24)

and the Laplacian ∇2φ can, in two-dimensional case, be cal-culated as

∇2φ = ∂2φ

∂x2+ ∂2φ

∂y2= (0, 0, 1, 0, 1, 0, . . . , 0)T · δ. (25)

In this study, the effective radius is set as re = 2.5�0 (�0 is theparticle spacing) for the weight function w(ri j ), and n = 3is chosen, which gives 2nd order accuracy for the Laplacianoperator.

One has two options for the time discretization schemeof the diffusion equation (Eq. 7), namely the explicit Eulerscheme

c(t + �t) − c(t)

�t= 1

Pe∇2c

∣∣∣∣t

(26)

and the implicit Euler scheme

c(t + �t) − c(t)

�t= 1

Pe∇2c

∣∣∣∣t+�t

. (27)

Note that the Laplacian operator is evaluated at the old posi-tion r(t) in the explicit Euler scheme (Eq. 26), while it isevaluated at the new position r(t + �t) in the implicit Eulerscheme (Eq. 27). Therefore, when the explicit Euler schemeis applied, the time advancement of the particle position(Eq. 9) is calculated after the diffusion equation; on the otherhand, for the implicit Euler scheme the calculation order isthe other way around.

2.3.4 Solid boundary

In contrast to the grid method, in which the implementa-tion of boundary condition is relatively straightforward, theparticle method has no entrenched rigorous boundary treat-ment. One commonly-used technique to implement the solidboundary is to fill the solid domain with fixed dummy parti-cles as illustrated in Fig. 3a. This dummy particle techniqueis generally successful to prevent interior fluid particles fromunphysical penetration through the boundary, while it suffersfrom several drawbacks. In general, the technique requires alarge number of dummy particles to create solid boundaries.That is because the solid phase represented by the dummyparticles needs sufficient thickness (at least thicker than theeffective radius) to prevent the unphysical effect caused byinsufficient particle density for the interior particle near theboundary. This fact leads to increase of the problem size andthus the computational cost more than is necessary. More-over, the requirement for the solid domain thickness resultsin the limitation of the computational geometry; for example,the zero thickness obstacle (baffle) can not be implemented.In the present hybrid method, the solid boundary conditionin the particle system is implemented by means of the virtualmirror particle technique [10,29,41] depicted in Fig. 3b. Thevirtualmirror particle is the reflection of interior particle withrespect to the boundary plane, where the physical quantity ofthe mirror particle is obtained by the corresponding interiorparticle and the boundary condition.

In order to solve the diffusion equation (Eq. 7) with theparticle system, one needs to impose appropriate boundaryconditions for calculation of the Laplacian operator. In thisstudy, only the zero-flux boundary condition (Eq. 28) is takeninto account for the concentration field,

∂c

∂n

∣∣∣∣

= 0, (28)

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238 Comp. Part. Mech. (2015) 2:233–246

Interior partilce

Dummy particle

(a)

Point reflection

Line reflection

Line reflection

Interior partilce

Virtual mirror particle

(b)

Fig. 3 Solid boundary implemented by dummy particle (a) and virtualmirror particle (b)

where n denotes normal unit vector with respect to theboundary plane. The zero-flux condition is awidely-acceptedmathematical model of the impermeable boundary for chem-ical species in fluid mixing problem. The zero-flux conditionis imposed by considering the virtual mirror particles. Thevirtual mirror particles are included as nearby particles forthe discretization of the Laplacian operator, wherein the con-centration of the mirror particle is the same value as thecorresponding interior particle.

2.3.5 Inlet and outlet boundaries

The inlet boundary for the particle system is implementedusing the particle injectors as drawn in Fig. 4. Those injec-tors are uniformly distributed with spacing �0 and fixed onthe inlet boundary plane to inject new fluid particles at afrequency given by the inlet flow rate

V inj(t) = �20n · u(t, rinj). (29)

If the inlet velocity is time-independent, the injection fre-quency is simply given by

V inj(t)

V0, (30)

where V0 = �30, but that is not always the case. Therefore,determination of the injection timing makes use of the totalflow rate

l0

Injector

Inlet boundary

Fluid particle

V0

Vinj

Fig. 4 Inlet boundary implemented by particle injectors

V inj(t) = V02

+∫ t

0V inj(t)dt, (31)

the time advancement of which is calculated as

V inj(t + �t) = V inj(t)

+�t

2

{V inj(t) + V inj(t + �t)

}. (32)

Injection of particle is executed (repeatedly if needed) as longas a condition

V inj(t) − V0Ninj ≥ V0 (33)

satisfies, where N inj denotes the total count of the injectedparticles which is incremented each after the injection. Posi-tion of the injected particle is determined as

rinj + V inj(t) − V0(N inj + 1)

V inj(t) + V inj(t + �t)

×{u(t, rinj) + u(t + �t, rinj)

}. (34)

Implementation of the outlet boundary is more simple.Through the outlet boundary, particles are moving out fromthe computational domain. Thus, those particles are detectedto be deleted or marked as a ghost particle.

2.3.6 Particle redistribution

As the particle positions evolve in time, the particle distri-bution is becoming more distorted. The highly anisotropicparticle spacing eventually causes the numerical instability[27]. To remedy this issue, we employ the particle redis-tribution based on the particle shifting approach [59] whichimproves particle spacinguniformity by slightly shifting eachparticle position according to

r′i = ri + δri , (35)

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Comp. Part. Mech. (2015) 2:233–246 239

δri = −αUmax�t∑

j∈Λi

R2i

|ri j |2 ni j , (36)

Ri = 1

|Λi |∑

j∈Λi

|ri j |, (37)

ni j = ri j|ri j | , (38)

Λi = {j |0 ≤ |ri j | < re

}, (39)

where r′i is particle position after the redistribution, α is

relaxation parameter,Umax is maximum velocity magnitude,and �t is time step size. The relaxation parameter is set asα = 0.01, and the effective radius is re = 2.5�0. This par-ticle redistribution is operated once in each time step afterthe particle advection. Calculation of the particle shiftingalso requires appropriate boundary treatment. For the particlenear the solid boundary, the virtual mirror particle techniqueis applied and the mirror particles are included to calcu-late Eq. (36). In the vicinity of inlet or outlet boundary, theparticle redistribution with the virtual mirror technique maycause numerical instability. Thus, if the shortest distance toinlet/outlet boundary is smaller than the effective radius ofthe particle shifting (here 2.5�0), the particle redistributionis switched off or set α = 0.

2.4 Calculation procedure

The present hybrid method performs the fluid mixing sim-ulation in the following procedure as described in Fig. 5.After the initial setup, the velocity field of the next time stepis firstly solved with the grid method and the concentrationfield is then solved in the particle system. In the particlephase, solution of the concentration field is proceed in threesteps: (a) particle advection including injection (inlet) andejection (outlet), (b) solution of the diffusion equation, (c)particle redistribution, where the order of the calculationsteps changes depending on the adopted time discretizationscheme of the diffusion equation, namely the explicit Eulerscheme (Eq. 26) or the implicit Euler scheme (Eq. 27), asstated previously. The particle phase is proceed as (b) → (a)→ (c) in the explicit case, or (a) → (c) → (b) in the implicitcase.

3 Numerical simulations

3.1 Unsteady one-dimensional diffusion problem

In the stationary one-dimensional domain x = [0, 1], theunsteady diffusion problem governed by

∂c

∂t= ∂2c

∂x2(40)

Grid phase

NO

YES

Particle phase

Time advancement

START

END

Input initial conditions

Update velocity and pressure

Solution of diffusion equation(explicit case)

Solution of diffusion equation(implicit case)

Particle advection

Injection and ejection

Particle redistribution

Termination?

Fig. 5 Flow chart of the calculation procedure in the hybrid grid-particle method

is considered under the initial condition

c0(x) =⎧⎨

⎩

0 x < 0.50.5 x = 0.51 x > 0.5

(41)

and the boundary conditions

∂c

∂x

∣∣∣∣x=0

= ∂c

∂x

∣∣∣∣x=1

= 0. (42)

The analytical solution for the time evolution of the concen-tration field is given by

cext(t, x) = 2∞∑

n=0

An cos (nπx) exp(−n2π2t

), (43)

An =∫ 1

0c0(x) cos (nπx) dx . (44)

For this test case, we examine two kinds of the parti-cle distribution, the regular arrangement and the irregulararrangement. In the regular arrangement, the particles aredistributed uniformly

x regi =(

i + 1

2

)

�0, (45)

where the particle spacing is �0 = 1/Np (Np denotes theparticle number). In the irregular arrangement, the parti-

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240 Comp. Part. Mech. (2015) 2:233–246

cle position is determined using the uniform pseudorandomnumber δx ∈ [−0.3�0, 0.3�0] as

x irri = x regi + δx . (46)

The explicit Euler scheme is chosen for the time discretiza-tion of the diffusion equation, and the time step size �t isdetermined based on the condition

Di = Pe · �t

�20≤ 0.1, (47)

where Di is the diffusion number, and Pe = 1. As the currentproblem assumes absence of fluid flow, the particle redistri-bution is not employed.

For the comparison with the present hybrid method whichutilizes theLSMPSLaplacian discretization scheme,wehavealso tested the MPS Laplacian model[28]

⟨∇2φ

⟩

i= 2d

n0λ0∑

j �=i

w′(rij)(φ j − φi

), (48)

where d is the number of space dimensions, n0 and λ0 areconstant parameters calculated with uniformly arranged par-ticles as

n0 =∑

j �=i

w′(rij), (49)

λ0 =

∑

j �=i

w′(rij)|rij|2

∑

j �=i

w′(rij)(50)

with the MPS weight function

w′(x) ={ re

|x| − 1 0 ≤ |x| < re

0 re ≤ |x|. (51)

Figure 6 shows the concentration distributions simulatedusing the present hybrid method employing the LSMPSLaplacian scheme. The results showgood agreementwith theanalytical solutions for both regular (Fig. 6a) and irregularparticle arrangements (Fig. 6b). Figure 7 shows convergenceof the nodal supreme error defined as

emax(t) = maxxi

|c(t, xi ) − cext(t, xi )| (52)

for the spatial resolution range from Np = 16 to 256 att = 0.025. In the irregular arrangement case, we plot themean value of 20 samples, where the error bar size is equalto the standard deviation. For the regular arrangement, bothMPS and LSMPS schemes show the 2nd order convergence

x0.0 0.2 0.4 0.6 0.8 1.0

x0.0 0.2 0.4 0.6 0.8 1.0

c

0.0

0.2

0.4

0.6

0.8

1.0t = 0

t = 0.001

t = 0.01t = 0.025

t = 0.1

t = 1

(a)

c

0.0

0.2

0.4

0.6

0.8

1.0

(b)

Fig. 6 Comparison of concentration profile of the one-dimensionaldiffusion problem between calculated (symbols) and analytic (lines)solutions; a regular arrangement, b irregular arrangement

and almost the same nodal supreme error. Quality of theMPSscheme is degraded for the irregular arrangement, whereasthe LSMPS scheme shows the same calculation accuracyas the regular arrangement case and the consistency is con-firmed.

3.2 Mixing in a lid-driven cavity

We have calculated the fluid mixing problem in a two-dimensional square lid-driven cavity illustrated in Fig. 8,where the upper wall moves rightward at a constant veloc-ity and other walls are fixed. For the boundary condition ofthe concentration field, the zero-flux condition applies at allwalls. The time evolution of the flow field and the concen-tration field are solved under the initial conditions

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Comp. Part. Mech. (2015) 2:233–246 241

Particle spacing, l010-3 10-2 10-1

Max

imum

err

or

10-6

10-5

10-4

10-3

10-2

10-1

100

l02

l00.3

MPS LSMPSRegularIrregular

Fig. 7 Convergence of the nodal supreme error for the unsteady one-dimensional diffusion problem at t = 0.025

x

y

1

0

0 0.5 1

c = 0 c = 1

Fig. 8 Simulation condition of the lid-driven cavity flow

u0(x, y) = 0, (53)

c0(x, y) =⎧⎨

⎩

0 x < 0.50.5 x = 0.51 x > 0.5

. (54)

The uniform grid system is used for solution of the fluid flow,and the particles are also uniformly distributed at the initialtime. The implicit Euler scheme is adopted for the time dis-cretization of the diffusion equation, by which the diffusionnumber limit of the time step size for stability condition isavoided and the computational cost can fairly be reduced byusing relatively large time step at high spatial resolutions.The time step size �t is set as �t = �0, and subdivision ofthe particle advection is employed based on the local Courantnumber limit Comax = 0.5. The Reynolds number based onthe upper wall velocity and the edge length is set as Re = 1,and the Péclet number is Pe = 104. As a reference, we alsocarried out the simulation using FVM, where the convectiveterm is discretized by the 2nd order upwind scheme with thegradient limiter.

As a preliminary test, we have investigated effect ofthe particle redistribution to the particle distribution. Fig-ure 9 shows the particle distributions at t = 0.25 comparedsolutions without and with particle redistribution utilizingthe particle shifting technique, where diffusion of chemicalspecies concentration is neglected.Without the particle redis-tribution, calculation of the concentration field was divergedfor Pe = 104. This is due to the highly anisotropic particlespacing. In the LSMPS scheme, evaluation of the Laplacianoperator makes use of the weighted least square approach,and the approximation quality is sensitive to the distribu-tion of neighboring particles; for example, the calculationaccuracy can be deteriorated in scarce region and particlestoo close to each other cause strong numerical instabilities.Appropriate particle redistribution technique can remedy

Fig. 9 Particle distributions att = 0.25; a without particleredistribution, b with particledistribution

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242 Comp. Part. Mech. (2015) 2:233–246

Fig. 10 Simulatedconcentration distribution in thelid-driven cavity atRe = 1, Pe = 104, and t = 10by FVM (a) and the presenthybrid method (b)

c0.0 0.2 0.4 0.6 0.8 1.0

y

0.0

0.2

0.4

0.6

0.8

1.0

Present

FVM

32x3264x64128x128256x256

32x3264x64128x128256x256

(a)

c0.7 0.8 0.9 1.0

y

0.60

0.64

0.68

0.72

0.76

(b)

Fig. 11 Comparison of resultant concentration profiles between FVMand the present hybridmethod in lid-driven cavity at t = 10 and x = 0.5

those issues by preserving the particle uniformity and numer-ical simulation was stably carried out in the present study.

Figure 10 is the calculated concentration distributions att = 10, where the spatial resolution is set as 64 × 64 for

0

0 1 62 3 4 5 7

1

2

Inlet

Outlet

Baffle

y

x

(a)

Velocity magnitude5.10

(b)

Fig. 12 Fluid mixing problem in a zig-zag channel; a geometry andboundary conditions, b flow field at Re = 1

both grid and particle systems. It is shown that solution ofthe present hybrid method qualitatively agrees with that ofFVM. In Fig. 11, the concentration profile at the middleplane x = 0.5 for varying spatial resolution from 32× 32 to256×256. The results show that both computationalmethodsconverges to the same solution. It is noteworthy that insuffi-cient spatial resolution in FVM leads to overestimation of thediffusion rate, i.e. numerical diffusion, whereas in the hybridmethod the perturbation appears instead of the numericaldiffusion. This perturbative error is resulted by the particleredistribution; the particle shifting generates dissipative par-ticlemovement to improve uniformity, and as a consequence,particles are scattered in a random manner. It is also impor-tant to be noted that the error due to the particle redistribution

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Fig. 13 Concentration distributions in a zig-zag channel calculated by FVM (upper), backward random-walk Monte Carlo method (middle), andthe present hybrid method (lower); a Pe = 103, b Pe = 104, c Pe = 105, d Pe = ∞

fairly vanishes as the spatial resolution increases, and thehybrid method achieves the convergence with lower spatialresolution than FVM.

The hybrid method requires some additional computa-tional cost for utilizing the particle system with respect toFVM which uses only the grid system since the particlesystem generally entails more expensive cost than the gridsystem. However, in the current numerical experiment, thecalculation time of the present hybrid method is only about15% longer than that of FVM in case that one uses the samenumber of particles as that of cells. This is because the domi-nant cost of the fluid mixing simulation is the part of solutionof the fluid flow, accounting for more than 80% of the overallcalculation. Therefore, the additional computational expense

for using particle system in solution of the concentration fieldshould not be a severe penalty.

3.3 Mixing in a zig-zag channel

The continuousmixing process in a two-dimensional zig-zagchannel is illustrated inFig. 12a. Segregatedfluids are enteredfrom the inlet boundary and mixed in the channel bended ina zig-zag manner. Fully developed laminar velocity profile

uin(y) = 6y(1 − y) (55)

is applied at the inlet boundary, and a reference constantpressure is prescribed at the outlet. Figure 12b shows steady

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solution of the velocity field solved at Re = 1, where theReynolds number is based on the mean inlet velocity and thechannel height. In this problem,wehave calculated the steadysolution for the concentration field using three different com-putational approaches: FVM, backward random-walkMonteCarlo (Monte Carlo) method [57], and the present hybridmethod. In the hybrid method, the explicit Euler scheme isused for the time discretization of the diffusion equation, andthe time step size is determined in the same manner as thelid-driven cavity case. For all approaches, the computationalnode or particle spacing is fixed at 1/40 and uniform.

Figure 13 shows the concentration distributions calculatedfor Pe = 103, 104, 105, and ∞. The concentration field cal-culated by FVM agrees well with that of the Monte Carlomethod at Pe = 103; however, for Pe ≥ 104 the numer-ical diffusion causes overestimation of diffusion rate andappreciable discrepancies with respect to the Monte Carlomethod emerge. On the other hand, the calculation results ofthe present hybrid method shows good agreement with theMonte Carlo method for all Péclet numbers tested.

In order to compare state ofmixing quantitatively, we haveevaluated themixing quality based on theDanckwerts’ inten-sity of segregation [11], so-called intensity of mixing [6] ormixing index [22],

MI = 1 − σ

σmax, (56)

where σ indicates the standard deviation of concentration inthe measurement plane (or volume), and σmax is the standarddeviation at the inlet (here σmax = 0.5). The mixing index(MI) is defined to be 0 for a completely segregated state and 1for the completely mixed state (homogeneous concentrationdistribution). In Fig. 14, the mixing index measured at cross-section x = 6 is plotted for the range 102 ≤ Pe ≤ 106. In

Pe101 102 103 104 105 106 107

Mix

ing

inde

x

10-3

10-2

10-1

100

FVMPresentMonteCarlo

Fig. 14 Mixing index measured in a zig-zag channel at x = 6

case of FVM, the numerical diffusion dominates and mixingstate is not correctly evaluated for Pe ≥ 104. In contrast,the present hybrid method faithfully evaluates the diffusioneffect even for high values of Pe, and discrepancy with theMonte Carlo method is kept small.

4 Conclusion

We propose a hybrid grid-particle method for the numericalsimulation of fluid mixing problem. The current main targetis a liquid–liquid mixing problem in microsystem, in whichthe non-dimensional parameters are typically Re ∼ 102 andPe ∼ 105 (Sc ∼ 103). For such high Pe, conventional grid-based approaches suffer from severe numerical diffusion.The proposedmethod combines the grid-based approach andthe particle-based approach in such a way that the incom-pressible Navier–Stokes equations are solved using FVM,and the advection–diffusion equation of chemical speciesconcentration is solved using a particle method. With thiscalculation principle, the fluid mixing problem of interestcan efficiently be solved utilizing the characteristics of eachapproach; the flow field is stably solved even with long timestep, and the concentration field is accurately solved withminimal numerical diffusion.

The hybrid grid-particle method is examined throughthree test cases. In the unsteady one-dimensional diffusionproblem, convergence of the resulting nodal error norm ischecked, and consistency of the Laplacian discretizationschemeemployed in theparticle phase is verified.As a secondtest, the hybrid method is applied to the fluidmixing problemin a lid-driven cavity flow. Converged result of the concentra-tion field agreeswellwith that obtained byFVM, and it is alsofound that the hybrid method achieves the convergence withlower spatial resolution than FVM. The continuous mixingprocess in a two-dimensional zig-zag channel is calculatedusing three different approaches: present hybrid method,FVM, and a reliable Monte Carlo method. In this test, thecomputational node or particle spacing is fixed at 1/40 forall approaches. The resulting concentration field obtainedby FVM shows considerable discrepancies with respect tothe Monte Carlo method for Pe ≥ 104 due to the numeri-cal diffusion; on the other hand, the hybrid method showsgood agreement for Pe = 103, 104, 105, and ∞ in quali-tative assessment, and for 102 ≤ Pe ≤ 105 in quantitativeassessment.

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