particle method

J. of Thermal Science, Vol.13, No.3

Modified Moving Particle Semi-Implicit Meshless Method for Incompressible

Fluids

Jun GUO Zhi TAO

Division 402, Department of Jet Propulsion, Beijing University of Aeronautics and Astronautics, Beijing 100083, P.R. China

A modified moving particle semi-implicit method (MPS) is presented for incompressible fluids. Modification is on the removal of imaginary nodes to enforce the boundary conditions. Comparison with the original imaginary- node scheme has proved the validity of the proposed method. Performance of MPS method in general is also investigated by comparing the results of Lid-driven and natural convection problems with finite volume method (FVM). It is found that MPS method shows relatively strong numerical diffusion, and for convection problems, MPS method appears to be less robust than FVM. Though better results could be obtained with different kernel functions, such improvement is quite limited.

Keywords: meshless method; incompressible fluids; numerical diffusion; kernel function.

CLC number: O351.3 Document code: A Article ID: 1003-2169(2004)03-0226-09

Introduction

Numerical simulation of fluid flow and heat transfer process still remains a challenging problem, especially for complex geometrical problems and large deformation problems. The sticking point is that conventional computational methods require either structured or non-structured grids for the discretization of governing equations. Need of grids or meshes causes two major difficulties, i.e. the generation of grids and the complexity of applying governing equations upon them.

Recent developments in meshless methods have gained significant attention in the computation community for not relying on either structured or non-structured meshes. Several meshless methods were proposed and substantially improved, such as Smoothed Particle Hydrodynamics (SPH), Element-Free Galerkin method (EFG), Reproducing Kernel Particle method (RKPM), etc. A comprehensive review of these methods could be found in Belytschko et al. Ill and Liu 12].

The moving particle semi-implicit (MPS) method was developed for incompressible fluids I3]. In MPS method, governing equations are discretized by particle interaction models and the arbitrary Lagrangian-Eulerian (ALE) technique is implemented using the fractional step method for time integration. MPS method has been

Received 2003

Jun GUO: Associate Professor

successfully applied in some areas, such as solid mechanics, incompressible fluid dynamics, fluid-structure interaction, gas-liquid two-phase flow, especially flud fragmentation, which is difficult to be treated by traditional methods [3-71. However, in its original formation, it still has its disadvantages, especially the need of imaginary nodes to enforce boundary conditions, which restrain its application to complex geometries. This paper is twofold in that it firstly studies the feasibility of discarding imaginary nodes, and secondly the performance of MPS method with different weight functions is investigated by comparing simulation results with those of FVM for natural convection and lid-driven flows within square cavities.

MPS Meshless Method

Basic ideas In MPS method a point interacts with its vicinities

covered with a weight function w(r, re), where r is the distance between two discrete points, re is weight radius within which weight function is non-zero, and is usually optimally determined by the user. Suppose a point j possesses physical quantities #j at ~j, then a smooth quantity at any location ~ could be expressed as weighted average of its neighbors:

Jun GUO et al. Modified Moving Particle Semi-Implicit Meshless Method for Incompressible Fluids 227

Nomenclature a thermal diffusivity g gravitational acceleration H characteristic scale p pressure Pr Prandtl number, Pr = via Ra Rayleighnumber, Ra= gf lH3AT /va Re Reynolds number, Re : uH/v S source term T temperature T c cool wall temperature T h hot wall temperature AT temperature difference, AT = T h - T c At time increment U, V dimensionless velocity component u,v dimensional velocity component

I? fluid velocity vector 17c velocity of a computing point

Greek symbols fl coefficient of volumetric expansion 0 dimensionless temperature, 0 = (T - T c)/(T h - T~) p dynamic viscosity v kinematic viscosity p fluid density

Superscripts * temporary value of computational step 1 L temporary value of computational step 2 n time-step value

Subscripts i the current computing point j neighbor of the current point

w(i ' -~t,~e) (1)

Typical weight functions are as follows (for two- dimensional problems):

(1) quadratic polynomial (denoted as W2)

2 - 4(r / re ) 2

w(r, re )= l~r / re -2 ) 2

(O

228 Journal of Thermal Science, Vol. 13, No.3, 2004

2d (V2~), =-T-Y'.{(Oj - ~,)w(t fj -5 [,r~)} (7)

"~i j~i

_ y - 16_ 12 w(l j-;,I,re)

J (8)

2, = ~--~{] ~j -~ 12 w(lfj -f~ I,re)} (9) j~i

where d is space dimension number, here d=2.

Governing equations and convection scheme If the coordinates are moving with a velocity I y~ ,

the continuity, momentum and energy equations for incompressible viscous flows are:

V.V=O (10)

01~ +(V- ~ 'Q.V=- Ivp+WZf +S (11) 0t p

OT - -+ (V - Vc) VT = aVZT (12) Ot

An arbitrary calculation is allowed between fully Lagrangian (pc = I? ) and Eulerian (~c = 0 ).

One-dimensional flow-directional interpolation scheme is adopted for convection term, which consists of three steps Is]. Firstly, a local grid is generated at each computing_ point along the flow direction of Vf =V~-Vf as shown in Fig.2. Distance between interpolation points is At. Detailed configuration depends on interpolation scheme. Then the values (~)k of local grid points are determined according to equation (13), while the region for weighted averaging is limited by a circle of radius re and lines vertical to the flow- directional local grid to avoid numerical diffusion (shadow area in Fig.2) [3]. At last, the convection term is solved according to the adopted difference scheme. In the present study, first-order upwind scheme is applied, thus the convection term could be written as:

(f~ _~.C). v#, :~.ia ~b, -(~)_, (13) Ar

With such method, local interpolation point may locate outside the computing region for points near boundaries. In this case, local interpolation point should be changed to the intersection of the flow direction and the boundary upstream, its physical parameters are interpolated from nearby boundaries similar to the skew upstream difference scheme. The value of parameter Ar of the corresponding point should be adjusted to the real distance between the computing point and the local interpolation point.

(0)+:..

Interpolation Region

Fig.2 Convection scheme based on flow-directional local grid (QUICK scheme) [sl

~ Current computing point i

fj Other points

O (r)k Local gridpointsforpointi, k=-2,-l,+l. The negative sign means upwind, while positive sign means downstream.

Boundary conditions and model parameters Because of lack of interpolation characteristics,

disposal of essential boundary condition is more problematic in meshless method. Several remedies have been proposed, including Lagrange multipiers method, penalty function method, combination with finite elements, transformation method, etc [1'9]. Besides, R-function meshless method is even more promising, which could exactly satisfy all given boundary conditions [11.

So far, boundary conditions in MPS haven't been well treated yet. In the original MPS method, imaginary nodes are introduced to model simple boundary conditions [6l. The values of those imaginary nodes are interpolated from inner points as conventional finite difference method did. Fig.3 shows an example of using imaginary nodes to deal with essential boundary conditions and no-slip wall conditions, where the superscripts indicate the layer number of nodes departing from the wall. Governing equations are solved at the boundaries in the same way as at the inner nodes [61. It is easy to know that the values of imaginary nodes' parameter are difficult to be determined for irregular/ curved surfaces (see Fig.4).

In this study, no image nodes are used, weighted regions for points near boundaries only include those inside the interesting region. Taking the boundary point i in Fig.4 for example, its approximation is

E4w(i i,r )

Zw(i l, re) J


where j denote only those inside the computing domain within the weight radius. The default thermal boundary condition is adiabatic. Any nodes on surfaces are effectively adiabatic, since there are no particles beyond, with which they can exchange energy. While for Dirichlet boundaries, the prescribed values are simply imposed to the corresponding nodes, and approximation calculations for these quantities are skipped.

T 2 T 1 T O T-1 T-2 /22 /11 U 0 /2 1 /1-2

V 2 V 1 V 0 V I V 2

0 0 ~ @ @ 0 0 I '~ @ @ 0 0 Side wall

T=Tw, u =v=0

70 =T~, T -l =2T . -T , T z =2T. T 2 U 0 :0 , I1-1 = 111 , ll 2 : _112

V 0 =0, V 1 = _V 1 , V -2=_V 2

Boundary Imaginary node O Fluid

Fig. 3 Modeling of essential boundaries using imaginary nodes 16]

Boundary Imaginary O Fluid

'x:;I o 0 "0

0

"o o~

"~'U0

*2 o

e~,-~

Support of node P

Support of boundary node using imaginary nodes

__@

Boundary

Fig.4 The support of nodes (that is the region for weighted averaging of nodes)

Weight radius also has significant influence on model's precision. On the one hand, it should be large enough to ensure there are enough neighbors for interpolation. On the other hand it should be small enough to keep local characteristics. In the present study, these parameters are determined according to reference TM. That is, weight radius is 2.0r~vg for the gradient and divergence models, 3.0r~vg for the Laplacian model, while it's 1.5r~g for advection calculation. The distance of local interpolation point Ar is also 1.5ravg

except for those points near boundaries mentioned above. Favg is the average distance of the scattered points.

Numerical Implementat ion Fractional step method is adopted for time integration,

which consists of three steps in each time step: (1) The diffuse terms and source terms of momentum

equations are calculated explicitly as:

15"- v-------L" - (v v~+s~. At

(2) Pressure Poisson equation is solved as:

(14)

then implicitly

V2p.+~ =v./5" (15) At

The temporal velocities are updated by adding the pressure gradient term,

~L _~. 1 - - - Vp n+l (16)

At p

The computing points are moved to ~L = ~-. + ~-LAt ' and the diffuse term in energy equation is explicitly solved:

T L - T" - - - aVeT" (17)

At

(3) The new-time positions are determined by adjusting the configuration of computing points, and getting the velocity of computing points ~c = (?n+l _ F n) / At. Then the new-time values V"+land T"+lare obtained through calculating convection terms. Here fully Eulerian computation is used by assigning ~,+1 = F".

vn+l _ ~ 'L

At ~ (/?_/?c).v/SL =0 (18)

T,,+I _ TL ~ (V-/?c).VTL =0 (19)

At

Detailed description could be found in reference [6]. Other numerical algorithm may also be used, such as SIMPLE or artificial compressibility approach, to deal with incompressible flow. Present fractional step method could easily implement ALE calculation which is in favor of simulating large deformation problems.

Numer ica l Exper iments

In this section, the feasibility of discarding imaginary nodes and the performance of the method are discussed with Lid-driven and natural convection problems in square cavities as numerical examples. All calculation is conducted with uniform configuration of


41x41 diverse points if there is no special statement, fully Eulerian calculation is adopted, and dimensionless equations are solved. (Pr=0.733 for all of the calculations). For natural convection problem, Boussinesq's approximation is used to represent the buoyancy force.

Feasibility of discarding imaginary nodes The feasibility of discarding images is tested for

Lid-driven and natural convection problems in square cavities. Fig.5 and Fig.6 compare the numerical results using imaginary nodes with those without using. Fig.5(a) and Fig.5(b) are results using W5 as kernel function for Re=I.OE2 of Lid-driven problems at central sections X=0.5 and Y=0.5, while Fig.5(c) and Fig.5(d) are those of W2. From these comparisons, it can be seen that the resolutions of the same weight function with imaginary nodes or not are nearly the same, indicating that the effect of asymmetry of weight function's region near boundaries is minimal. Fig.6 are results for Ra=I.OE2 of natural convection in a square cavity at sections X=0.5 and Y=0.5 using W5 and W2 as kernels, which appear to have the same characteristic.

More numerical tests for different Re and Ra numbers of the problems have shown the same feature.

Numerical performance of MPS In this section, numerical results predicted by the

present MPS method (no imaginary nodes are introduced) are compared with those of traditional FVM, while FVM calculations are conducted using first-order UDS with 40 40 meshes consisting with those of the present method.

Fig.7 are velocity distributions at central sections X=0.5 and Y=0.5 for lid-driven problem. It can be seen that, generally speaking, the present method agrees well with conventional FVM. Among the kemels used, W5 predicts better results, while W2 is the worst. This may be explained from the weight curves in Fig. 1, where W5 is the steepest and W2 the gentlest, indicating W5 could keep better local characteristics. It's clear that the model's accuracy is, to some extent, weight function dependent. Fig.7 also indicates that MPS results are not as plump as those of FVM, showing the characteristics of numerical diffusion.

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

i I i i i i

Re=1.0E2 w5,x=0.5 ~ J

[] Imaginray Vj~ ~ f

010 ' 0'.2 014 0:6 ' 018 1'.0

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

Y (a) W5 results at section X = 0.5

0.12 0.2

0.10 0.1

0.08 0.0

0.06 .~ 0

0.04 "~ -0.1 >

0.02 -0.2

0.00 -0.3

i i i i i

Re=1.0E2 W2o X=0.5

[] Imaginary. V ~t~

010'0'.2'014 0'.6'0'.8'1.0 Y

(c) W2 results at section X = 0.5

0.10

0.08

0.06

0.04

0.02

0.00

0.2

0.1

0.0

-0.1 >

-0.2

. . . . Re=I.0E2' ' W5, Y=0.5

[] Imaginary Non-Imaginary

I i I

0'.0 0.2 0.4 016 018 110 X

(b) W5 results at section Y= 0.5

i , i . i i i . i

Re=I.0E2 ' W2, Y=0.5

[] Imaginary Non-Imaginar.

' 0'.2 ' 0'.4 ' 016 018

X (d) W2 results at section Y= 0.5

-0.3 0'.0

Fig.5 Velocity profiles for Lid-driven problem using imaginary nodes or not using (Re = 1.0E2, At = 1.0E-3)

I

1.0


0.4

0.2

~ 0.0 _o >

-0.2

-0.4 I i I i

0.0 012 0.4 016 018 ll.O

Xor Y

(a) W5 results at sections X = 0.5, and Y = 0.5

>

0.4

0.2

0.0

-0.2

-0.4

i i i i r

I i !

010 ' 012 0.4 016 018 1.0

Xor Y

(b) W2 results at sections X = 0.5, and Y= 0.5

Fig.6 Comparison of results for natural convection in a square cavity using imaginary nodes or not using (Ra = 1.0E2, At = 1.0E-4)

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

i i . i . i . i i

....... W2 o F \ /

0'.0 ' 0:2 ' 0'.4 ' 0:6 ' 0'.8 ' l'.O

Y

0.10

0.08

0.06

0.04

0.02

0.00

(a) at section X = 0.5

0.2

0.1

0.0

"~ o0.1 >

-0.2

-0.3

i , i i . t . i . i

....... W2

W5

010 ' 012 014 016 ' 018 ll.0 X

(b) at section Y= 0.5

Fig.7 Results of Lid-driven problem (Re = 1.0E2, At = 1.0E 3)

Natural convection problems in square cavities were also analyzed which are much sensitive to conservation. Fig.8 and Fig.9 are results of natural convection in a square cavity for Ra= 1.0E2. Fig.8 are velocity profiles at different sections X,Y=0.2, 0.4, 0.6, 0.8 with W5 as kernels. At these sections, results of the present method agree well with those of FVM. Fig.9.1 is the distribution of major velocity component, while Fig.9.2 is the secondar 3, velocity component at central sections )(=0.5 and Y=0.5. It can be seen that for the secondary velocity component, results of the present method diverge from those of FVM, and the plot of the V component of the velocity vector at X=0.5 even appears to have oscillatory behavior near boundaries. This oscillation, however, decreases greatly and the resolutions become much better with finer diverse points, 81 81 nodes for example.

Fig.10 are results for Ra=1.0E5 at central sections. It shows that MPS results of the major velocity are reasonable compared with FVM, though it also shows false diffusion characteristics. The secondary velocity component appears to have stronger oscillatory behavior,

oscillation going deeper into the cavity. Analysis shows that at the points where oscillation taking place, buoyancy term are always much larger than diffuse term. Finer point configuration (8181 nodes) again reduces the oscillation greatly, that is, f'mer point configuration is needed to get smooth resolutions compared with FVM, indicating that the method's accuracy needs to be improved. Special treatment must be introduced to stabilize the resolution and improve its accuracy.

The answer to the method's low accuracy may be found from the MPS model itself. The basic approximation equation (that is Eq.(1)) of MPS method corresponds to using Shepard function as shape function, which is only Co continuum.

Fig.11 are temperature profiles for Ra=I.0E2 and Ra = 1.0E5 at central sections. It is clear that the temperature distributions predicted by the present method at different Ra number agree well with those of FVM. This indicates that velocity distribution is more sensitive to the performance of numerical method, especially the V component of velocity vector, to which equation the buoyance term is enforced.


0.4

0.3.

0.2-

0.1

~ 0.0

~ -0.1

-0.2

-0.3

-0.4

0.4 ~

0.3

0.2

0.1

~ 0.0 0

"~ -o.1 - > . -0.2-

-0.3 -~ -o.41

i i , i , i , i , i

f "~- - o V(X-0.6) \ ,,~ + u(x-o.4) k .~ a V(X=0.4)

- -MRS

0:0 0:2 0'4 0:6 0:8 l:o

0.5

0.4-

0.3:

0.2-"

.~ 0.1 i

o.o "-6 > -0.1.

-0.2!

-0.3-"

-0 .4 -

i , i i i i i

D u(x=o.2) o V(X=0.2 ) + u(x-o.8) .o~ \

v(x=o 8)/ \

0.0 0.2 0.4 0.6 0.8 1.0

(a) at sections X = 0.4, 0.6 (b) at sections X = 0.2, 0.8

i ) I i O I i

U(Y=0.4) o V(Y=O.4) V U(Y=0.6)

+ v(r=0.6)

i, i , i , i , i i

0 0 0.2 0.4 0.6 0.8 1.0

0.5

0.4~ 0.3

0.2

0.1

0.0

-0.1i -0.2- -0.3-' -0.4-

, uo ,=o ~) . . . . [ ] o v(r=0.2) a u(r=o.8) ~ ~, .

+__ v(r 8) y -,,.

' 0 ' .2 . . . . 0.0 0.4 0.6 0.8 1.0

X X

(c) at sections Y = 0.4, 0.6 (d) at sections Y = 0.2, 0.8

Fig.8 Velocity profiles for natural convection in a square cavity (Ra = 1.0E2, At = 1.0E-4)

Scattered points are results of FVM, while lines are those of MPS (W5, no imaginary nodes)

0.4

0.2

"~ 0.0 0

> -0.2

-0.4

I i i i i i

0'.0 012 0'.4 ' 016 ' 0'.8 1'.0

Xor Y

(a) major velocity component

O >

0.012 I 0.010 I

0.008 I 0.006 I

0.004

0.002

0.000

-0.002

-0.004

0.0

, , . ,~ I~VM ' -' . . . . . . . W5 41"41~

- ' " - .---. o W5 81"81" / f

ii t %%% I j " ~1

' ' o ' . ' '.8 ' 0.2 4 0.6 0 1.0

Xor f

(b) secondary velocity component

Fig.9 Velocity profiles for natural convection in a square cavity at sections X = 0.5 & Y= 0.5 (Ra = i.0E2, At = 1.0E 4)


60-

401

201

0 .

-20-

-40-

-60-

i ~" i i i i i

** V ( Y-O, 5 )

. / u(x=o.5) i l l l l / "

I in

l I I

mm

[] W3 * L.W* - - W5 ~,

0 0.2 0.4 0.6 0.8 1.0

Xor Y

(a) major velocity component

O >

4

0

-4

-8 0.0 0:2 014 0:6 018

. . . . . . . '----:- w'2 " V(X=0.5) n W3 i *** . U(Y=0.5) W5

* . ~ o W5 8t'81 i I I l l d - -

mm ~

: FVM_V **"* - FVMU

I

1.0 1.2

X

(b) secondary velocity component

Fig.10 Velocity profiles for natural convection in a square cavity at center sections X = 0.5 & Y= 0.5 (Ra = 1.0E5, At = 1.0E-5 for 4l )


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