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Numerical Heat Transfer, Part B:FundamentalsAn International Journal of Computation andMethodologyPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713723316

An Efficient and Robust Numerical Scheme for theSIMPLER Algorithm on Non-Orthogonal CurvilinearCoordinates: CLEARERY. P. Cheng a; T. S. Lee a; H. T. Low a; W. Q. Tao ba Laboratory of Fluid Mechanics, Department of Mechanical Engineering, NationalUniversity of Singapore, Singaporeb State Key Laboratory of Multiphase Flow in Powering Engineering, School ofEnergy & Power Engineering, Xian, Shaanxi, People's Republic of China

Online Publication Date: 01 May 2007To cite this Article: Cheng, Y. P., Lee, T. S., Low, H. T. and Tao, W. Q. (2007) 'An Efficient and Robust NumericalScheme for the SIMPLER Algorithm on Non-Orthogonal Curvilinear Coordinates: CLEARER', Numerical Heat Transfer,Part B: Fundamentals, 51:5, 433 - 461To link to this article: DOI: 10.1080/10407790601009115URL: http://dx.doi.org/10.1080/10407790601009115

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AN EFFICIENT AND ROBUST NUMERICAL SCHEMEFOR THE SIMPLER ALGORITHM ON NON-ORTHOGONALCURVILINEAR COORDINATES: CLEARER

Y. P. Cheng, T. S. Lee, and H. T. LowLaboratory of Fluid Mechanics, Department of Mechanical Engineering,National University of Singapore, Singapore

W. Q. TaoState Key Laboratory of Multiphase Flow in Powering Engineering, School ofEnergy & Power Engineering, Xian, Shaanxi, People’s Republic of China

In this article, an Improved SIMPLER (CLEARER) algorithm is formulated to solve the

incompressible fluid flow and heat transfer on the nonstaggered, nonorthogonal curvilinear

grid system. By virtue of a modified momentum interpolation method in calculating the

interface contravariant velocity in both the predictor step and the corrector step, the coup-

ling between pressure and velocity is fully guaranteed, and the conservation law is also

satisfied. A second relaxation factor is introduced in the corrector step, of which the con-

vergent solution is independent. By setting the second relaxation factor less than the under-

relaxation factor for the velocity to some extent, both the convergence rate and robustness

can be greatly enhanced. Meanwhile, the CLEARER algorithm can also overcome the

severe grid nonorthogonality. With the simplified pressure-correction equation, the conver-

gent solution can still be obtained even when the intersection angle among grid lines is as low

as 1�, which may provide valuable guidance in studying the fluid flow in complex geometries.

INTRODUCTION

The fluid flow and heat transfer in complex geometries are often encounteredin engineering situations; hence curvilinear body-fitted coordinates are widelyadopted to deal with these problems. In this method the complex physical domainis mapped into a simple computational domain with orthogonal grids, hence well-established numerical methods on orthogonal coordinates, such as SIMPLE-familyalgorithms and high-order schemes, can be easily extended to computationaldomain.

In solving the incompressible Navier-Stokes equations, two grid arrangementscan be adopted to discretize the governing equations, the staggered grid and the non-staggered grid. On the staggered grid the coupling between the velocity and pressurecan be naturally guaranteed, and the unreasonable checkerboard pressure field canbe successfully avoided. However, the staggered arrangement may cause great

Received 15 July 2006; accepted 31 July 2006.

Address correspondence to Y. P. Cheng, Laboratory of Fluid Mechanics, Department of Mechanical

Engineering, National University of Singapore, 119260, Singapore. E-mail: [email protected]

433

Numerical Heat Transfer, Part B, 51: 433–461, 2007

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7790 print=1521-0626 online

DOI: 10.1080/10407790601009115

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geometric and mathematical complexity associated with three sets of grids for two-dimensional problems and four sets of grids for three-dimensional problems. Fur-thermore, these complexities are particularly overwhelming in curvilinearcoordinate systems. It is therefore desirable to calculate the pressure and velocitycomponents on the nonstaggered grid but in such a way that unrealistic fields arenot predicted.

In 1983, Rhie and Chow [1] proposed a momentum interpolation method(MIM) to eliminate the checkerboard pressure field on the nonstaggered grid; sub-sequently it is refined by Peric [2] and Majumdar [3]. Majumdar [4] and Millerand Schmidt [5] pointed out independently that the solution using the original Rhieand Chow scheme is underrelaxation factor-dependent; then Majumdar [4] proposed

NOMENCLATURE

a thermal diffusivity

A surface area

AP, AE , AW ,

AN , AS

coefficients in the discretized

equation

b source term

B, C coefficient in pressure or

pressure-correction equation

D diffusion conductivity

E time-step multiple

F flow rate

Flowch characteristic flow rate

g gravitational acceleration

J geometric factor

L length of cavity

Nu Nusselt number

p pressure

p� temporary pressure

p0 pressure correction

Pr Prandtl number

Ra Rayleigh number

Re Reynolds number

Rmax maximum relative mass flow

rate unbalance of control

volume

S source term

T temperature

u, v velocity component

u�, v� temporary velocityeuu, evv pseudo-velocity

u0, v0 velocity correction

U , V contravariant velocity at

interface, nondimensional

velocity

U�, V� temporary contravariant

velocityeUU , eVV pseudo-contravariant velocity

U 0, V 0 contravariant velocity

correction

ULid moving velocity of lid

x, y coordinates in physical domain

xn, xg, yn, yg geometric factor

X , Y dimensionless coordinates

a underrelaxation factor,

geometric factor

b relaxation factor, geometric

factor, volume expansion

coefficient

c geometric factor

C nominal diffusion coefficient

dn, dg distance between two adjacent

grid points in n andg directions

DT temperature difference

Dn, Dg distance between two adjacent

interfaces in n and g directions

g dynamic viscosity

n kinematic viscosity

n, g coordinates in computational

domain

q density

/ general variable

Subscripts

e, w, n, s cell face

max maximum

nb neighboring grid points

p refers to pressure

P, E, W , N, S grid points

T refers to temperature

u, v refers to u and v velocities

Superscripts

u, v coefficients related to u and v

velocities

0 resolution of the previous

iteration

� intermediate value

434 Y. P. CHENG ET AL.

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008 a modified momentum interpolation method (MMIM) to overcome this depen-

dency. Kobayashi and Pereira [6] also solved this problem by simply setting theunderrelaxation factor equal to 1 before the momentum interpolation method isimplemented, but this method may lower the robustness of the algorithm. Choi [7]found that the original Rhie and Chow scheme is also time-step size-dependent,and he proposed a new scheme to overcome this problem. However, Yu et al. [8]observed that Choi’s scheme does not solve the time-step size-dependent problem,and they further reported that a checkerboard pressure field might still be obtainedfor small time-step size and the underrelaxation factor when Rhie and Chow’smethod is used. Later, Yu et al. [9] discussed the role of the interface velocity onthe nonstaggered grid, and recommended that all the interface velocities be obtainedwith the momentum interpolation method.

For curvilinear nonorthogonal coordinates, the physical and geometric conser-vation law should be satisfied as much as possible. Shyy and Vu [10] recommendedusing the Cartesian velocity components as the primary variables and contravariantvelocity components as cell face velocities to satisfy the conservation law. AlthoughRhie and Chow’s scheme can suppress the unrealistic pressure field by introducingan additional pressure gradient correction term, the local mass conservation isdemolished, as proved by Acharya and Moukalled [11] and Qu et al. [12]. In [11],for the SIMPLER formulation, the residuals of the momentum equation are foundnot to decrease to acceptably low values, hence Acharya and Moukalled proposedthe SIMPLEM algorithm, which can show reasonable convergence behavior andis superior to both the SIMPLE and SIMPLER algorithms. However the effect ofthe underrelaxation factor is not eliminated in the SIMPLEM algorithm. Choi etal. [13] formulated a calculation procedure for the SIMPLE algorithm to overcomethis problem without introducing an additional correction term. However, it wasfound that if the contravariant velocity is selected as the cell face velocity, unstableconvergence history might occur when the grid nonorthogonality is significant, sothe covariant velocity components are chosen as the cell face velocities. However,this cannot guarantee the geometric conservation law in the discrete form. In thisarticle, focus is concentrated on a new algorithm which can satisfy the conservationlaw and is also underrelaxation factor-independent.

One crucial problem encountered in curvilinear nonorthogonal coordinates isthe pressure-correction equation. On orthogonal grids, the structure of the governingequation usually corresponds to 5-point or 7-point computational molecules in twoor three dimensions, respectively. For nonorthogonal grids, the structure becomesmore complex and corresponds to a 9-point or 19-point computational moleculein two or three dimensions, hence the storage and solution procedure for such kindsof matrices become too complex or too expensive, especially for the three-dimensional case. Therefore, it is common practice to simplify the equations byneglecting the nonorthogonal terms entirely, to obtain a 5-point or 7-point moleculeagain. However, this simplification may make the convergence behavior of thepressure-correction equation deteriorate. Peric [14] studied the properties of thepressure-correction equation by solving two-dimensional flow in a skewed cavity,and claimed that the simplified pressure-correction equation can work well onslightly nonorthogonal grids because the contribution of the cross-derivatives inthe pressure-correction equation is small. However, when the intersection angle

EFFICIENT NUMERICAL SCHEME: CLEARER 435

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008 among grid lines approaches 45�, the simplified pressure equation becomes inef-

ficient; and it usually fails to converge when the intersection angle is below 30�. Inthis case the full pressure-correction equation must be adopted, but it is impracticalfor three-dimensional problems to solve the full pressure-correction equation with a19-point molecule. To overcome this problem, Cho and Chung [15] adopted a newtreatment to decompose the nonorthogonal terms in the pressure correction intoexplicit and implicit parts. In this method, the convergence solution can be obtainedeven when the intersection angle is 30�. Unfortunately, the performance of theirapproach depends on an additional parameter, which needs to be decided by the user.

Based on the discussion above, we can see that for an efficient numerical algor-ithm, four fundamental requirements must be satisfied, as stated by Qu et al. [12]: (1)the algorithm can suppress the checkerboard pressure distribution; (2) the conver-gent solution is independent of the underrelaxation factor; (3) the algorithm shouldpossess the required robustness; and (4) the conservation law, both physical and geo-metric, should be satisfied as much as possible. Based on these requirements, Chenget al. [16] proposed an Improved SIMPLER (CLEARER) algorithm on the orthog-onal nonstaggered grid, which is much more robust and efficient than theCLEARER algorithm. Recently, Qu et al. [12] formulated the SIMPLERM algor-ithm on curvilinear coordinates, which shows great convergence performance underlow-value underrelaxation factor. However, when the underrelaxation factorincreases, the iteration number will increase sharply or the algorithm will diverge.In this article, the CLEARER algorithm is extended to nonorthogonal curvilinearcoordinates, which can overcome the disadvantages of SIMPLERM and exhibitexcellent performance even on highly nonorthogonal grids.

In the following presentation, the mathematical formulation of CLEARER isfirst provided, then both lid-driven flow and natural convection in an inclined cavitywith available solutions are used to validate the CLEARER algorithm. The perform-ance of CLEARER and SIMPLERM is also compared, and the minimum intersec-tion angle among grid lines is investigated under which the convergent solution canstill be obtained. Finally, some conclusions are arrived.

MATHEMATICAL FORMULATION OF CLEARER

Governing Equations and Discretization

For simplicity, here we take two-dimensional steady incompressible laminarflow in Cartesian coordinates as our example. The governing equations can beexpressed in a general way as

q qu/ð Þqx

þ q qv/ð Þqy

¼ qqx

Cq/qx

� �þ qqy

Cq/qy

� �þ Sðx; yÞ ð1Þ

where / is the general variable and S is the source term, shown in Table 1.The nonorthogonal curvilinear coordinates are introduced in the following

way:

x ¼ xðn;gÞ y ¼ yðn;gÞ ð2Þ


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Then the irregular physical domain is mapped into the regular computationaldomain with orthogonal gridlines, as seen in Figure 1.

Correspondingly, the governing equation is transformed into

1

J

q qU/ð Þqn

þ 1

J

q qV/ð Þqg

¼ 1

J

qqn

CJ

aq/qn� b

q/qg

� �� þ 1

J

qqg

CJ�b

q/qnþ c

q/qg

� �� þ Sðn;gÞ

ð3Þ

where U and V are contravariant velocity components and a, b, c, J are geometricparameters, defined as follows:

U ¼ uyg � vxg V ¼ vxn � uyn ð4Þ

a ¼ x2g þ y2

g b ¼ xnxg þ ynyg c ¼ x2n þ y2

n J ¼ xnyg � xgyn ð5ÞThe source terms for the transformed u and v equations can be expressed as

qp

qx¼ 1

Jðygpn � ynpgÞ

qp

qy¼ 1

Jð�xgpn þ xnpgÞ ð6Þ

Here the Cartesian velocities are arranged in the center of the cell, while the contra-variant velocities are arranged at the interfaces. The transformed governing equa-tions are discretized in the computational domain with the finite-volume method(FVM) [17, 18]. The final discretized equations are as follows.

Continuity equation:

ðqDgUf Þe � ðqDgUf Þw þ ðqDnVf Þn � ðqDnVf Þs ¼ 0 ð7Þ

Momentum equations with underrelaxation factors incorporated:

AuP

auuP ¼

XAu

nbunb � BuP

qp

qn� Cu

P

qp

qgþ bu þ 1� au

auAu

Pu0P ð8aÞ

AvP

avvP ¼

XAv

nbvnb � BvP

qp

qn� Cv

P

qp

qgþ bv þ 1� av

avAv

Pv0P ð8bÞ

here

qp

qn¼ ðpeÞP � ðpwÞP

Dnqp

qg¼ ðpnÞP � ðpsÞP

Dgð9Þ

Table 1. Correspondence among /; C, and S

/ C S

Continuity equation 1 0 0

u-Momentum equation u g ð�qp=qxÞv-Momentum equation v g ð�qp=qyÞEnergy equation T g=Pr 0


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Figure 1. Computational grid and the definition of parameters.


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008 The interface pressure ðpeÞP; ðpwÞP; ðpnÞP; and ðpsÞP can be obtained by linear

interpolation from the values on the main nodes:

ðpwÞP ¼ðdnÞw�ðdnÞw

pP þðdnÞwþðdnÞw

pW ðpeÞP ¼ðdnÞe�ðdnÞe

pE þðdnÞeþðdnÞe

pP ð10aÞ

psð Þp¼ðdgÞs�ðdgÞs

pp þðdgÞsþðdgÞs

ps ðpnÞp ¼ðdgÞn�ðdgÞn

pN þðdgÞnþðdgÞn

pp ð10bÞ

The coefficients in Eq. (8) can be calculated as

BuP ¼

qy

qg

� �P

Dn Dg CuP ¼ �

qy

qn

� �P

DnDg BvP ¼ �

qx

qg

� �P

Dn Dg

CvP ¼

qx

qn

� �P

Dn Dg

ð11Þ

AE ¼ DeA PDej jð Þ þ ½�Fe; 0� AW ¼ DwA PDwj jð Þ þ ½Fw; 0� ð12aÞ

AN ¼ DnA PDnj jð Þ þ ½�Fn; 0� AS ¼ DsA PDsj jð Þ þ ½Fs; 0� ð12bÞ

AP ¼ AE þ AW þ AN þ AS ð12cÞ

bu ¼ SJ DnDg� CJ

bqu

qgDg

� ��ew

þ CJ

bqu

qnDn

� ��ns

� �ð13aÞ

bv ¼ SJ Dn Dg� CJ

bqv

qgDg

� ��ew

þ CJ

bqv

qnDn

� ��ns

� �ð13bÞ

where F and D are the flux and diffusion conductivity at the interface, respectively,

Fe ¼ qUDgð Þe Fw ¼ qUDgð Þw Fn ¼ qVDnð Þn Fs ¼ qVDnð Þs ð14Þ

De ¼aJ

CDgDn

� �e

Dw ¼aJ

CDgDn

� �w

Dn ¼cJ

CDnDg

� �n

Ds ¼cJ

CDnDg

� �s

ð15Þ


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008 The Predictor Step of CLEARER

In the SIMPLE-like pressure-correction algorithms, there are two steps at eachiterative level, which are the predictor step and corrector step. In the predictor stepthe velocity is predicted by solving the momentum equations based on a guessedpressure field or a field obtained from a given velocity field. Because the predictedvelocity cannot satisfy the continuity equation, in the corrector step it is improvedwith the solved pressure correction. With this process the velocity field graduallysatisfies the momentum equations and continuity equation, hence a convergentsolution is obtained. In curvilinear coordinates the contravariant velocity at theinterface plays a crucial role in accelerating the convergence rate. According to Shyy[19], in terms of the conservation law, in the computational domain the contravar-iant velocity is recommended to be adopted in the continuity equation, and theCartesian velocity in the momentum equations. This choice is widely used in mostof the literature, and so also in this article.

For the conventional SIMPLE-like algorithms, in the predictor step the under-relaxation factor is incorporated in the momentum equation to relax the velocity,and in the corrector step the underrelaxation factor is incorporated in the press-ure-correction equation. The adoption of the same underrelaxation factor for onevelocity component in the predictor and corrector steps may cause some inconsis-tency, so the convergence rate will be lowered. Furthermore, in the corrector stepon the nonstaggered grid, both the velocity on the main nodes and the contravariantvelocity at the interfaces will be improved with the solved pressure correction. Chenget al. [16] pointed out that it is appropriate to improve the interface velocity with thepressure correction, but the velocity on the main nodes is overcorrected, so, it isnecessary to introduce a second relaxation factor in the corrector step, which shouldbe less than the underrelaxation factor for velocity. Then the convergence perform-ance can be improved greatly, and furthermore, a convergent solution can still beobtained even when the grid lines are severely nonorthogonal.

In order to guarantee the interconnection between velocity and pressure, thepressure equation is solved based on the velocity field at the previous iterative level,hence the first step is to derive the pressure equation.

We rewrite Eq. (8) in the explicit manner

uP ¼ aufu0

Pu0P � B0

uP

qp

qn

� �P

�C0uP

qp

qg

� �P

� �þ 1� auð Þu0

P ð16aÞ

vP ¼ avev0Pv0P � B0

vP

qp

qn

� �P

�C0vP

qp

qg

� �P

� �þ 1� avð Þv0

P ð16bÞ

where fu0Pu0P and ev0

Pv0P are called pseudo-velocities,

fu0Pu0P ¼

PA0u

nbu0nb þ b0u

P

A0uP

; ev0Pv0P ¼

PA0v

nbv0nb þ b0v

P

A0vP

ð17Þ


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008 The superscript 0 denotes the values from the previous iteration. The source terms b

are calculated as

b0uP ¼ SJ DnDg� C

Jbqu0

qgDg

� ��ew

þ CJ

bqu0

qnDn

� ��ns

� �ð18aÞ

b0vP ¼ SJ Dn Dg� C

Jbqv0

qgDg

� ��ew

þ CJ

bqv0

qnDn

� ��ns

� �ð18bÞ

The coefficients in Eq. (16) are determined by

B0uP ¼

BuP

A0uP

� �P

¼yg

� �P

Dn Dg

A0uP

� �P

C0uP ¼

CuP

A0uP

� �P

¼ �yn

� �P

DnDg

A0uP

� �P

ð19aÞ

B0vP ¼

BvP

A0vP

� �P

¼ �xg� �

PDn Dg

A0vP

� �P

C0vP ¼

CvP

A0vP

� �P

¼xn� �

PDn Dg

A0vP

� �P

ð19bÞ

By mimicking Eq. (16), the interface Cartesian velocities can be expressed as

ue ¼ aueu0eu0e � B0

u

qp

qn

� �e

� C0u

qp

qg

� �e


e ð20aÞ

ve ¼ avev0ev0e � B0

v

qp

qn

� �e

� C0v

qp

qg

� �e


e ð20bÞ

un ¼ aueu0nu0n � B0

u

qp

qn

� �n

� C0u

qp

qg

� �n


n ð20cÞ

vn ¼ avev0nv0n � B0

v

qp

qn

� �n

� C0v

qp

qg

� �n


n ð20dÞ

The coefficients can be calculated as

B0u

� �e¼

yg� �

ednð ÞeDg

A0uP

� �e

C0u

� �e¼ �

yn� �

ednð ÞeDg

A0uP

� �e

ð21aÞ

B0v

� �e¼ �

xg� �

ednð ÞeDg

A0vP

� �e

C0v

� �e¼

xn

� �e

dnð ÞeDg

A0vP

� �e

ð21bÞ

B0u

� �n¼

yg� �

nDn dgð Þn

A0uP

� �n

C0u

� �n¼ �

yn� �

nDn dgð Þn

A0uP

� �n

ð21cÞ


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B0v

� �n¼ �

xg� �

nDn dgð Þn

A0vP

� �n

C0v

� �n¼

xn

� �nDn dgð Þn

A0vP

� �n

ð21dÞ

The pseudo-velocities eu0eu0e , ev0

ev0e , eu0

nu0n, ev0

nv0n and the coefficients at the interfaces A0u

P

� �e,

A0vP

� �e, A0u

P

� �n, A0v

P

� �n

can be linearly interpolated from the values on the main nodesP, E, and N.

eu0eu0e ¼

dnð Þe�dnð Þe

fu0Eu0E þ

dnð Þeþdnð Þe

fu0Pu0P

ev0ev0e ¼

dnð Þe�dnð Þe

ev0Ev0E þ

dnð Þeþdnð Þe

ev0Pv0P ð22aÞ

eu0nu0n ¼

dgð Þn�dgð Þn

fu0Nu0N þ

dgð Þnþdgð Þn

fu0Pu0P

ev0nv0n ¼

dgð Þn�dgð Þn

fv0Nv0N þ

dgð Þnþdgð Þn

ev0Pv0P ð22bÞ

A0uP

� �e¼ dnð Þe�

dnð ÞeA0u

P

� �Eþ dnð Þeþ

dnð ÞeA0u

P

� �P

A0vP

� �e¼ dnð Þe�

dnð ÞeA0v

P

� �Eþ dnð Þeþ

dnð ÞeA0v

P

� �P

ð23aÞ

ðA0uP Þn ¼

ðdgÞn�ðdgÞn

ðA0uP ÞN þ

ðdgÞnþðdgÞn

ðA0vP ÞP

ðA0vP Þn ¼

ðdgÞn�ðdgÞn

ðA0vP ÞN þ

ðdgÞnþðdgÞn

ðA0vP ÞP ð23bÞ

The interface contravariant velocities are defined as

Ue ¼ uqy

qg� v

qx

qg

� �e

Vn ¼ vqx

qn� u

qy

qn

� �n

ð24Þ

Substituting Eq. (20) into Eq. (24) yields

Ue ¼ ðauygeu0u0 � avxg

ev0v0Þe � ðauygB0u � avxgB0

vÞeqp

qn

� �e

þ ð�auygC0u þ avxgC0

v Þeqp0

qg

� �e

þð1� auÞðygÞeu0e

� ð1� avÞðxgÞev0e ð25aÞ

Vn ¼ ðavxnev0v0 � auyn

eu0u0Þn þ ðauynB0u � avxnB0

vÞnqp0

qn

� �n

þ ðauynC0u � avxnC0

v Þnqp

qg

� �n

þð1� avÞðxnÞnv0n

� ð1� auÞðynÞnu0n ð25bÞ


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008 Here, in order to enhance the robustness of the CLEARER algorithm, the underre-

laxation factors are involved in Eq. (25). To guarantee that the convergent solution isindependent of the underrelaxation factor, the underrelaxation factors for u and vmomentum equations are set to be identical, i.e., au ¼ av ¼ a 6¼ 1. Then Eq. (25)can be recast as

Ue ¼ aðygeu0u0 � xg

ev0v0Þe � a B0uf

qp

qn

� �e

þa C0uf

qp0

qg

� �e

þð1� aÞU0e ð26aÞ

Vn ¼ aðxnev0v0 � yn

eu0u0Þn þ a B0vf

qp0

qn

� �n

�a C0vf

qp

qg

� �n

þð1� aÞV 0n ð26bÞ

where

B0ufe ¼

y2g

A0uP

þx2

g

A0vP

!e

ðdnÞeDg C0ufe ¼

ynyg

A0uP

þ xnxg

A0vP

� �e

ðdnÞeDg ð27aÞ

B0vfn ¼

ynyg

A0uP

þ xnxg

A0vP

� �e

ðdgÞnDn C0vfn ¼

y2n

A0uP

þx2

n

A0vP

!n

ðdgÞnDn ð27bÞ

U0e ¼ u0

eðygÞe � v0eðxgÞe V0

n ¼ v0nðxnÞn � u0

nðynÞn ð28Þ

Equation (26) can be regarded as the extension of the idea of the modifiedmomentum interpolation method (MMIM) in orthogonal coordinates proposedby Majumdar [4]. When the iteration converges, Ue and Vn will approach U0

e andV 0

n , respectively. Then Eq. (26) will equivalent to

Ue ¼ ðygeu0u0 � xg

ev0v0Þe � B0uf

qp

qn

� �e

þ C0uf

qp0

qg

� �e

ð29aÞ

Vn ¼ ðxnev0v0 � yn

eu0u0Þn þ B0vf

qp0

qn

� �n

� C0vf

qp

qg

� �n

ð29bÞ

Hence the effect of the underrelaxation factors will be eliminated completely.By introducing the pseudo-contravariant velocity,

fU0eU0e ¼ aðyg

eu0u0 � xgev0v0Þe þ a C0u

f

qp0

qg

� �e

þ ð1� aÞU0e ð30aÞ

fV0nV0n ¼ aðxn

ev0v0 � yneu0u0Þn þ a B0v

f

qp0

qn

� �n

þ ð1� aÞV 0n ð30bÞ


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008 Eq. (26) can be rewritten as

Ue ¼ fU0eU0e � a B0u

f

qp

qn

� �e

Vn ¼ fV 0nV 0n � a C0v

f

qp

qg

� �n

ð31Þ

By substituting Eq. (31) into the continuity Eq. (7), we can get the pressure equation,

A0Pp�P ¼

XA0

nbp�nb þ b ð32Þ

where

ðA0EÞP ¼ ðA0

W ÞE ¼qDg B0u

f

dn

!e

ðA0NÞP ¼ ðA0

SÞN ¼q Dn C0v

f

dg

!n

ð33Þ

A0P ¼ A0

E þ A0W þ A0

N þ A0S ð34Þ

b ¼ ðqDg fU0U0Þw � ðq Dg fU0U0Þe þ ðq Dn fV 0V 0Þs � ðq Dn fV 0V 0Þn ð35Þ

To further improve the robustness of CLEARER algorithm, the pressure isunderrelaxed when the pressure equation is solved. Incorporating the underrela-xation factor, we can get the final form of the pressure equation in the predictor step:

A0P

aPp�P ¼

XA0

nbp�nb þ bþ 1� aP

aPA0

Pp0P ð36Þ

After the pressure equation is solved, the pressure gradient source terms in themomentum Eqs. (8) can be obtained, then the discretized momentum equationscan be solved to update the Cartesian velocities at the main node, and the resultingvelocities are defined as u�; v�. As they cannot satisfy the continuity equation, correc-tion is needed to improve them.

The Corrector Step of CLEARER

In order to avoid the additional pressure gradient term used by Rhie and Chow[1], which will destroy the conservation constraint, the modified momentum interp-olation method is adopted here to calculate the contravariant velocity at the inter-face, as suggested by Yu et al. [9] on the orthogonal nonstaggered grid. To set anextra access for controlling the convergence process, an additional relaxation factorb is introduced in the determination of the contravariant velocity at the interface.The details are as follows.


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008 Based on the intermediate velocities, similar to Eq. (17) the pseudo-Cartesian

velocities on the main nodes can be recalculated as

fu�Pu�P ¼P

A0unbu�nb þ b0u

P

A0uP

ev�Pv�P ¼P

A0vnbv�nb þ b0v

P

A0vP

ð37Þ

Then the corresponding interface pseudo-Cartesian velocities can be linearly interpo-lated as

eu�eu�e ¼ðdnÞe�ðdnÞe

fu�Eu�E þðdnÞeþðdnÞe

fu�Pu�P ev�ev�e ¼ðdnÞe�ðdnÞe

ev�Ev�E þðdnÞeþðdnÞe

ev�Pv�P ð38aÞ

eu�nu�n ¼ðdgÞn�ðdgÞn

fu�Nu�N þðdgÞnþðdgÞn

fu�Pu�P ev�nv�n ¼ðdgÞn�ðdgÞn

fv�Nv�N þðdgÞnþðdgÞn

ev�Pv�P ð38bÞ

Similar to the derivation of Eq. (26), the intermediate contravariant velocities aregained with the modified momentum interpolation method:

U�e ¼ bðyg eu�u� � xg ev�v�Þe � b B0uf

qp�

qn

� �e

þb C0uf

qp�

qg

� �e

þð1� bÞU0e ð39aÞ

V �n ¼ bðxn ev�v� � yn eu�u�Þn þ b B0vf

qp�

qn

� �n

�b C0vf

qp�

qg

� �n

þð1� bÞV0n ð39bÞ

The coefficients B0uf ;C

0uf ;B

0vf ;C

0vf are the same as those in Eq. (26), and can be cal-

culated according to Eq. (27). When the iteration converges, U�e and V �n willapproach U0

e and V0n , respectively, hence the convergent solution is also independent

of the second relaxation factor b. Because the additional pressure gradient term isdiscarded in this method, the CLEARER algorithm can guarantee the mass conser-vation of control volumes.

Because the contravariant velocities at the interfaces in Eq. (39) do not satisfythe continuity equation, they need to be improved. Based on the original momentumequations without the underrelaxation factor incorporated, the velocity correctionson the main nodes are obtained omitting the neighboring velocity correction.

u0P ¼ �B0uP

qp0

qn

� �P

�C0uP

qp0

qg

� �P

v0P ¼ �B0vP

qp0

qn

� �P

�C0vP

qp0

qg

� �P

ð40Þ

Then the improved velocities at the main nodes are expressed as

uP ¼ u�P þ u0P vP ¼ v�P þ v0P ð41Þ

According to the definition of contravariant velocities in Eq. (4), the correspondingcontravariant velocity corrections on the main nodes are

U 0P ¼ ðu0yg � v0xgÞP V 0P ¼ ðv0xn � u0ynÞP ð42Þ


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008 Substituting Eq. (41) into Eq. (42), the following equations can be obtained:

U 0P ¼ ð�ygB0uP þ xgB0

vPÞp0n þ ð�ygC0uP þ xgC0

vPÞp0g ð43aÞ

V 0P ¼ ð�xnC0vP þ ynC0

uPÞp0g þ ð�xnB0vP þ ynB0

uPÞp0n ð43bÞ

To obtain the pressure-correction equation with a 5-point computational molecule,the cross pressure-correction derivatives, p0g in Eq. (43a) and p0n in Eq. (43b), areneglected; then the equations can be simplified into

U 0P ¼ ð�ygB0uP þ xgB0

vPÞp0n V 0P ¼ ð�xnC0vP þ ynC0

uPÞp0g ð44Þ

By mimicking Eq. (44), the contravariant velocity corrections at the interfaces areexpressed as

U 0e ¼ �B0ufe p0n V 0n ¼ �C0v

fn p0g ð45Þ

where the coefficients B0ufe and C0v

fn are calculated according to Eq. (27). Then theimproved interface contravariant velocities can be expressed as

Ue ¼ U�e þU 0e ¼ U�e � B0ufe p0n Vn ¼ V �n þ V 0n ¼ V �n � C0v

fn p0g ð46Þ

Substituting Eq. (46) into the continuity Eq. (7), the pressure-correction equation inthe corrector step is derived:

A0Pp0P ¼

XA0

nbp0nb þ b ð47Þ

The coefficients A0P;A

0E ;A

0W ;A

0N , and A0

S are the same as those in the pressure equa-tion in the predictor step, as seen in Eq. (33) and (34); the only difference lies in thecalculation of source term b, which is obtained from the intermediate interface com-travariant velocities.

b ¼ ðqDg U�f Þw � ðq Dg U�f Þe þ ðq Dn V �f Þs � ðq Dn V �f Þn ð48Þ

Solution Procedure of CLEARER Algorithm

The computational steps of the CLEARER algorithm can be summarized asfollows.

1. Assume the initial velocity field on both the main nodes and interfaces, u0P; v

0P;U

0f ;

and V0f .

2. Based on the interface contravariant velocities and Cartesian velocities on themain nodes, calculate the coefficients of the momentum equation [Eqs. (11)–(15)] and pseudo-contravariant velocityies fU0

eU0e and fV 0

nV 0n [Eq. (30)].

3. Calculate the coefficients of the pressure equation [Eqs. (33)–(35)] and solve it;the pressure field p� is obtained.

4. Calculate the pressure source terms in the momentum equations [Eq. (9)], andsolve them to obtain the intermediate velocities u�P and v�P.


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008 5. Calculate the intermediate interface contravariant velocities [Eq. (39)].

6. Calculate the source term b [Eq. (48)] in the pressure-correction equation andsolve it; the pressure correction term p0 is obtained.

7. Obtain the improved interface contravariant velocities Ue and Vn [Eq. (46)], andthe improved Cartesian velocities at the main nodes uP and vP [Eq. (41)].

8. Solve the discretized equations of other scalar variables if necessary.9. Return to step 2 and repeat the process until a convergent solution is obtained.

Discussion of the Second Relaxation Factor b

In the SIMPLE-like pressure-correction method, the velocities are first pre-dicted with the momentum equations, and they are corrected with the continuityequation, then the improved values are taken as the solution of the current iterationlevel to start the next iteration. However, the same underrelaxation factor is usuallyadopted for one velocity component in both the predictor step and the correctorstep, which may cause inconsistency in the relaxation for the variable in both steps.In the CLEARER algorithm, a second relaxation factor b is introduced in the cor-rector step to overcome this inconsistency.

Rewriting Eq. (39),

U�e ¼ b yg eu�u� � xg ev�v� � B0uf

qp�

qnþ C0u

f

qp�

qg

� �e

þ ð1� bÞU0e ð49aÞ

V�n ¼ b xn ev�v� � yn eu�u� þ B0vf

qp�

qn� C0v

f

qp�

qg

� �n

þ ð1� bÞV 0n ð49bÞ

We can see that the intermediate interface pseudo-velocities include two parts; one isobtained by the momentum interpolation method and satisfies the momentum equa-tion; the other part is the convergent values in the previous iteration which satisfiedthe continuity equation. To guarantee the convergence of the solution, the dependentvariables cannot vary much, especially for the problems in nonorthogonal curvilinearcoordinates, where nonlinearity is very severe. In the CLEARER algorithm, the relax-ation factor b can take a low value, thus the part from the continuity equation willbecome dominant in constituting the intermediate contravariant velocities at the inter-faces. Thus only a small value of correction is needed to add to the intermediate valuesto make them satisfy the continuity equation, which avoids the great variation of thecorrection values, hence the stability is increased during the iteration, and the robust-ness of the algorithm can also be enhanced, even in cases where the grid lines areseverely nonorthogonal. It is notable that when the underrelaxation factors in the pre-dictor step and corrector step are equal, i.e., a ¼ b, the CLEARER algorithm becomesthe SIMPLERM algorithm proposed by Qu et al. [12].

At the same time, by adjusting the second relaxation factor, the convergence ratecan also be enhanced greatly, which can be seen from Figure 2. Here we take the lid-driven flow in the skewed cavity as our example. When b ¼ a, the iteration number isnot the optimum. By decreasing the value of b, the iteration number will decrease,and the lowest iteration number is about 88% of that at b ¼ 0:2. But if b is deceased


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further, the iteration number will increase greatly; meanwhile, if b > a, the iterationnumber will also increase, hence b is recommended to take a value less than a, althoughit cannot be too small. In the following section, the optimum value of b is obtained bytrial and error, and the performance of the CLEARER and SIMPLERM algorithms arecompared with two cases with benchmark solutions.

NUMERICAL VALIDATION AND COMPARISON

In order to verify the feasibility of the CLEARER algorithm on nonorthogonalcurvilinear coordinates, two typical numerical examples with available solutions arecomputed: (1) lid-driven flow in an inclined cavity; and (2) natural convection in aninclined cavity. The benchmark solutions for the two cases are provided byDemirdzic et al. [20]. Peric [14] pointed out that the SIMPLE algorithm will failto converge when the intersection angles among grid lines are less than 30� if the non-orthogonal term is omitted in the derivation of the pressure-correction equation.Hence, in their calculation [20], the cross-derivatives were treated implicitly at severenonorthogonality, which leads to a pressure correction with a 9-point computationalmolecule for a two-dimensional problem. However, in the present CLEARER algor-ithm, the nonorthogonal terms are dropped to gain a 5-point pressure correctionequation no matter how severe the grid nonorthogonality is.

Table 2. Some correspondence between a and E

a 0.1 0.2 0.4 0.6 0.8 0.9 0.95

E 0.11 0.25 0.67 1.5 4 9 19

Figure 2. Influence of second relaxation factor b on the iteration number at a ¼ 0:2.


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To make the comparison between the CLEARER and SIMPLERM algo-rithms meaningful, the numerical treatments of all other aspects should be keptthe same. In both algorithms, the stability-guaranteed second-order differencescheme (SGSD) [21] is adopted to discretize the convection term; the algebraic equa-tions are solved by the alternating direction implicit (ADI) method [22]incorporating the block-correction technique [23]. To show well the convergenceperformance in the high-value region of underrelaxation factor, the time-step mul-tiple E is introduced, which relates to the underrelaxation factor a by

E ¼ a1� a

ð0 < a < 1Þ ð50Þ

The correspondence between a and E is presented in Table 2.The same convergence criterion is also used for the two algorithms, as indi-

cated below:

Rmax ¼MAX

"ðqU�f AÞw � ðqU�f AÞe þ ðqV�f AÞs � ðqV �f AÞn

Flowch

#< 1:0� 10�8 ð51Þ

where Rmax is the maximum relative mass flow rate imbalance of all the controlvolumes in the computational domain; Flowch is the characteristic flow rate throughthe centerline of the cavity. U�f and V �f are the intermediate contravariant velocities.

Figure 3. Geometry and boundary condition for lid-driven cavity.

Figure 4. Grid system used in lid-driven cavity.


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A uniform grid system 51� 51 is used for the two cases, and the underrelaxa-tion factor for pressure ap ¼ 0:9; for the natural-convection problem in Case 2, theunderrelaxation factor for temperature aT ¼ 0:8.

In the following two cases the computation conditions are introduced first,then numerical results with the CLEARER algorithm are compared with the bench-mark solution to test its accuracy, followed by a comparison of iteration number androbustness between the CLEARER and SIMPLERM algorithms. Furthermore, theratio of the iteration number between the two algorithms is also provided. Becausethere is the same computational effort at every iterative level, the ratio the of the iter-ation numbers is also that of the computational time. Finally, the investigation ofminimum intersection angle is conducted under which CLEARER can still converge.

Case 1. Lid-Driven Flow in Inclined Cavity

Computations are conducted when the inclination angle h ¼ 45� andRe ¼ 1,000, which is defined as

Re ¼ ULidL

nð52Þ

Figure 5. Streamlines at inclined cavity at Re ¼ 1,000.


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Figure 6. Comparison of centerline velocity profiles.


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Figure 7. Comparison of iteration number between SIMPLERM and CLEARER.

Figure 8. Ratio of iteration number of CLEARER over SIMPLERM.


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here ULid is the moving velocity of the top lid, and L stands for the length of the cav-ity. The schematic diagram is shown in Figure 3, where CL1 and CL2 are the cen-terlines in two directions along the wall of the cavity. In Figure 4, the coarse gridsare shown, in which the grid lines are parallel to the walls.

Figure 9. Geometry and boundary condition for natural convection in inclined cavity.

Figure 10. Streamlines in inclined cavity at Rd ¼ 106.


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In Figure 5 the streamlines in the inclined cavity with the present CLEARERalgorithm are compared with those provided by Demirdzic et al. [20], from which wecan see that they agree quite well. For accurate comparison, the velocity distribu-tions along the centerlines are also compared with the benchmark solutions, whereX ;Y are nondimensional coordinates, normalized by the cavity length L, andU ;V are the nondimensional velocities, normalized by ULid. From Figure 6 wecan see that the present results agree quite well with the benchmark solutions, whichproves the accuracy of the CLEARER algorithm.

The iteration numbers of CLEARER and SIMPLERM are compared underdifferent underrelaxation factors in Figure 7, from which it can be found that underlow underrelaxation factor, the iteration number of CLEARER is only slightly lowerthan that of SIMPLERM, but when the underrelaxation a � 0:5, i.e., E � 1, the iter-ation number of SIMPLERM increases sharply, and a convergent solution is notobtained when a ¼ 0:6, i.e., E ¼ 1:5. However, for the CLEARER algorithm, withincreasing underrelaxation factor the iteration number decreases greatly, and it canstill converge even at a ¼ 0:95. It is notable that the iteration number of CLEARERat high underrelaxation may be only one-tenth that at low underrelaxation factor.

From Figure 8 we can see that the iteration number ratio of CLEARER overSIMPLERM ranges from 0.24 to 1 in the variation range of underrelaxation factorunder which both algorithms can converge. From the analysis above, we can see that

Figure 11. Isothermals in inclined cavity at Rd ¼ 106.


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Figure 12. Comparison of Nu along the hot wall between present results and benchmark solution.

Figure 13. Comparison of iteration number between SIMPLERM and CLEARER.


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compared with the SIMPLERM algorithm, CLEARER can not only enhance theconvergence rate greatly, it also improves the robustness significantly, which provesits superior performance over the SIMPLERM algorithm.

Case 2. Natural Convection in Inclined Cavity

Natural convection is studied in a cavity when inclination angle h ¼ 45�, withtop and bottom walls adiabatic while the left and right walls are at a constant butdifferent temperature, as seen in Figure 9. The calculation is conducted at Pr ¼ 10and Ra ¼ 106, defined as

Ra ¼ qgbL3 DT

agð53Þ

Figures 10 and 11 show the streamlines and isothermals in the inclined cavity, andthe results with the CLEARER algorithm are compared with those provided byDemirdzic et al. [20], showing good agreement between them. For accurate compari-son, the Nu distribution along the hot wall is also compared in Figure 12. Due to thelimitation of grid size near the top and bottom, the current grid cannot catch the Nunumber in positions very near the top and bottom walls, while in the interior regionthe present results agree quite well with the benchmark solutions.

Figure 15. Streamlines in lid-driven cavity flow at inclination angle h ¼ 5�.

Figure 14. Ratio of iteration number of CLEARER over SIMPLERM.


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In Figure 13 the iteration numbers of CLEARER and SIMPLERM are com-pared with variation of underrelaxation factor. At a ¼ 0:1, i.e., E ¼ 0:11, the iter-ation numbers for the two algorithms are almost identical. With increasing

Figure 16. Velocity distribution along the centerlines at inclination angle h ¼ 5�.


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underrelaxation factor, the iteration number of SIMPLERM increases greatly, whilethat of CLEARER does not vary much. When a > 0:5, i.e., E > 1, SIMPLERMdiverges, but CLEARER can still converge until a ¼ 0:9, i.e., E ¼ 9. From the aboveanalysis we can see that the robustness of CLEARER is much higher than that ofSIMPLERM. The ratio of iteration number of CLEARER over SIMPLERM is alsoprovided in Figure 14, from which we can see that the ratio varies from 0.29 to 0.98,which proves that CLEARER has much better convergence performance thanSIMPLERM.

Investigation of Minimum Intersection Angle Among Grid Linesto Guarantee Convergence

In curvilinear coordinates the gridlines have to be nearly orthogonal to ensurethat the discretization of the governing equations is as accurate as possible.

Figure 17. Natural convection in inclined cavity at inclination angle h ¼ 5�.

Figure 18. Nu distribution along the hot wall at inclination angle h ¼ 5�.


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008 However, in some practical applications such grids cannot be generated, due to lim-

itations of the geometry. Then a nonorthogonal grid is used. In general, in theSIMPLE-like algorithms the pressure-correction equation has a very low conver-gence rate, and most of the computational time is devoted to solving for it. On aslightly nonorthogonal grid the contribution of the cross pressure-correction deriva-tive is not so large that the cross-terms can be neglected in order to obtain a simpli-fied 5-point pressure-correction equation in two dimensions and a 7-point pressure-correction equation in three dimensions. However, when the grid is severely non-orthogonal, the contribution of the cross pressure-correction derivative becomesdominant and cannot be neglected arbitrarily, otherwise a convergent solution can-not be obtained. In this case the full pressure-correction equation has to be solved,which is 9-point in two dimensions and 19-point in three dimensions in thecomputational molecule, thus the computational effort in solving the equations willbecome quite difficult or even prohibitive. In the present CLEARER algorithm, thesimplified pressure-correction equation is still adopted. By reducing the second relax-ation factor b, say 0.05, we can still get a convergent solution even when the inter-section angle among grid lines is as low as 1�. For convenience of display, onlythe cases when the intersection angle is 5� are shown below.

From Figure 15 we can see that a vortex dominates the right half-region of thecavity, while several weak vortices align at the left sharp corner. As the vortex in theleft region is usually several orders lower in magnitude than the dominant vortex, theflow there is nearly stagnant, hence the velocities in the left part are nearly zero, whichcan be seen from Figure 16. It can also be found that the vertical velocity is much lowerthan the horizontal velocity, which is caused by the severe inclination of the cavity.

Figure 17 shows the natural convection in the inclined cavity at h ¼ 5�, fromwhich we can see that a dominant vortex is located in the center of the cavity withtwo weak vortices in the left and right corners, and the temperature gradient ismainly focused in the center of the cavity. From Figure 18 we can see that at thelower part of the left wall, due to the very weak flow there, the heat transfer rateis nearly zero, while near the top of the left wall, due to the strong impulsion ofthe flow, the Nusselt number increases greatly, and it can be as high as 50 there.

CONCLUSION

In this article, the CLEARER algorithm has been extended to the curvilinearnonorthogonal grid, then two numerical examples with benchmark solutions havebeen calculated to validate the new algorithm. Furthermore, the performance ofCLEARER and SIMPLERM have been compared. The major conclusions are sum-marized as follows.

1. The CLEARER algorithm can satisfy the four fundamental requirements for anefficient numerical algorithm.

2. The CLEARER algorithm can predict the fluid flow and heat transfer accurately.3. Compared with the SIMPLERM algorithm, both the robustness and conver-

gence rate of the CLEARER algorithm are greatly enhanced, and the ratio ofiteration number of CLEARER over SIMPLERM can be as low as 0.24.


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008 4. With the simplified pressure-correction equation, a convergent solution can still

be obtained on a severely nonorthogonal grid with the CLEARER algorithm,even when the intersection angle among grid lines is as low as 1�.

ACKNOWLEDGEMENT

The fourth author thanks for the support from the National Natural ScienceFoundation of China (50476046).

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