SIMPLE Method on Non-staggered Grids - Department of ...

<strong>SIMPLE</strong> METHOD FOR THE SOLUTION OF INCOMPRESSIBLE
FLOWS ON NON-STAGGERED GRIDS
I. Sezai - Eastern Mediterranean University, Mechanical Engineering Department, Mersin 10 -Turkey
Revised in January, 2011
1. Introduction
There are usually two kinds of grid arrangements used to solve the fluid flow problems:
staggered grids and non-staggered grids. For the nonstaggered grids, vector variables and
scalar variables are stored at the same locations, while for the staggered grids, vector
components and scalar variables are stored at different locations, shifted half control
volume in each coordinate direction. Staggered grids are popular because of their ability
to prevent checkerboard pressure in the flow solution as discussed in Chapter 6.
However, programming of the staggered grid method is tedious since u and v-momentum
equations are discretized at different control volumes shifted in different directions from
the main control volumes. The programming difficulties increase when one deals with
curvilinear or unstructured grids. As a result, nearly all codes written on curvilinear or
unstructured grids use non-staggered grid arrangement for the solution of fluid flow
problems.
On the other hand non-staggered grids are prone to produce a false pressure field -
checkerboard pressure if special precautions is not taken. For this reason, in the early
1980s and before, non-staggered were rarely used for incompressible flow. However,
since 1983 the nonstaggered grid (or collocated grid) has been used more widely, after
Rhie and Chow (1983) proposed a momentum interpolation method to eliminate the
checkerboard pressure problem.
1. Mathematical Formulation
The governing equations for two dimensional incompressible laminar heat and fluid flow
in Cartesian coordinates with constant properties are:
continuity equation
1

∂ u
∂ x
+
∂ v
∂ y
= 0
(1)
u-momentum equation
( ρ ) ( ρ ) ( ρ )
∂ u ∂ uu ∂ vu ∂p ∂ ⎛ ∂u⎞
∂ u
μ
⎛ ∂
+ + =− + ⎜ ⎟+ ⎜μ
⎞
⎟+
s
∂t ∂x ∂y ∂x ∂x⎝ ∂x⎠ ∂y⎝ ∂y
⎠
v-momentum equation
( ρ ) ( ρ ) ( ρ )
∂ v ∂ uv ∂ vv ∂p ∂ ⎛ ∂v⎞
∂ v
μ
⎛ ∂
+ + =− + ⎜ ⎟+ ⎜μ
⎞
⎟+
s
∂t ∂x ∂y ∂y ∂x⎝ ∂x⎠ ∂y⎝ ∂y⎠
Energy equation
( ) ( ) ( )
∂ ρT ∂ ρuT ∂ ρvT ∂ ⎛ k ∂T ⎞ ∂ ⎛ k ∂T
⎞
+ + = + + s
∂t ∂x ∂y ∂x⎜c ∂x ⎟ ∂y⎜c ∂y
⎟
⎝ p ⎠ ⎝ p ⎠
Equations (2) – (4) can be expressed in a general form as:
( ) ( u ) ( v )
∂ ρφ ∂ ρ φ ∂ ρ φ ∂ ⎛ ∂φ ⎞ ∂ ⎛ ∂φ
⎞
+ + = ⎜Γ ⎟+ ⎜Γ ⎟+
sφ
∂t ∂x ∂y ∂x⎝ ∂x ⎠ ∂y⎝ ∂y
(5)
⎠
where u and v are the velocity components, T is temperature and φ is any dependent
variable (i.e. u, v and T), and t, ρ , Γ , and s φ
are time, density, diffusion coefficient, and
source term per unit volume, respectively. Also,s u and s v terms represents source terms
per unit mass, ie. drag forces, buyancy forces etc. Note that for the continuity equation,
φ = 1, Γ= 0 , and s φ
= 0 .
In numerical calculations, the source terms in the above equations are written in
linearized form as
s = sc + spφ
p
For example, for natural convection problems, the right hand side of the v-momentum
equation contain an additional term ρ g
− β ref
( T − T ∞)
, for which the source terms are
sc
=−ρrefgβ ( T −T∞
) and s p = 0. Similarly, for problems involving heat generation the
right hand side of the energy equation have an additional term q which is the energy
generation per unit volume. In this case, s c = q , s p = 0.
T
v
u
(2)
(3)
(4)
2

The governing equations are discretized by using the finite-volume method on a
nonstaggered grid system in which all variables are stored at the center of the control
volume (Figure 1).
Integrating Eq. (5) over the control volume with bounded cell faces e, w, n, and s
surrounding center P, we have
ρΔxΔy
o
( φP − φP) + ⎡( ρuφ) −( ρuφ) ⎤Δ y+ ⎡( ρvφ) −( ρvφ)
⎤Δ x=
e w n s
Δt
⎣ ⎦ ⎣ ⎦
⎡ Γe Γ ⎤ ⎡
w
Γn Γ ⎤
s
( φE −φP) − ( φP −φW ) Δ y+ ( φN −φP) − ( φP −φS)
Δx
(6)
⎢ ⎥ ⎢ ⎥
⎣δ xe δ xw ⎦ ⎣δ yn δ ys
⎦
+ s ΔxΔ y+ s φ ΔxΔy
c P P
where the superscript o refers to the previous time level and Δ x , Δ y , δ x , δx , δy
and
δ y s
are geometric lengths as shown in Figure 1.
e w n
Figure 1. Nonstaggered Grid Arrangement
The discretized continuity equation for incompressible flow is
( ρue) Δy−( ρuw) Δ y+ ( ρvn) Δx−( ρvs) Δ x= 0
(7)
The cell face values φ e , φ w etc are calculated using QUICK or any other high order
scheme in a deferred correction manner. Substituting these values into Eq. (6) and
3

earranging, we obtain the final discretized equation for any general variable φ as
follows:
o o
aPφP = aEφE + aWφW + aNφN + aSφS + aPφP
+ b+ pterm
(8)
where
aE
=
ΓΔ
e
y
+ max ⎡−( ρu ) Δy, 0⎤
e
Δx
⎣
⎦
e
aW
=
ΓΔ
w
y
+ max ⎡( ρu)
Δy, 0⎤
w
Δx
⎣ ⎦
w
aN
=
ΓΔ
n
x
+ max ⎡−( ρv)
Δx
, 0⎤
n
Δy
⎣
⎦
n
aS
=
ΓΔ
s
x
+ max ⎡( ρv)
Δx
, 0⎤
s
Δy
⎣
⎦
s
o ρΔΔ
x y
aP
=
Δt
o
aP = aE + aW + aN + aS + aP − Sp
+ΔF
Δ F = ( ρu) Δy−( ρu) Δ y+ ( ρv) Δx−( ρv)
Δx
e w n s
b= ( sc )
eqn
ΔxΔ y+ b1 b1
= Sdc + ( Sc)
bc
Sp = ( sp) eqnΔxΔ y+
( Sp)
bc
(9)
Sdc = −max ⎡
⎣( ρu) Δy, 0⎤( ) max ( ) , 0 ( )
e ⎦ φe − φP + ⎡
⎣
− ρu Δy
⎤
e ⎦ φe −φE
−max ⎡
⎣
−( ρu) Δy, 0⎤( w P) max ( ) , 0 ( w W)
w ⎦ φ − φ + ⎡
⎣ ρu Δy
⎤
w ⎦ φ −φ
−max ⎡
⎣( ρv) Δx , 0⎤( φn φP) max ( ρv) x, 0 ( φn φN)
n ⎦
− + ⎡
⎣
− Δ ⎤
n ⎦
−
−max ⎡
⎣
−( ρv) Δx, 0⎤( φs − φP) + max ⎡( ρv) Δx, 0⎤( φs −φS)
s ⎦ ⎣ s ⎦
⎧=−( pe
−pw) Δy
for x-momentum equation
⎪
pterm= ⎨=−( pn
− ps) Δx
for y-momentum equation
⎪
⎩ = 0 for all other equations
( s ) , ( s ) = source terms per unit volume in the differential equation (buoyancy, drag for
p eqn c eqn
( Sp) bc, ( S
c) bc=
source contributions of near-boundary points
where b represents all discretized source terms, excluding the pressure term. The term S dc
is the source contribution, which results from the adoption of the deferred-correction
o
procedure in which the face values of the independent variable, φ
P
is the value of φ
P
from the previous time step. φ e , φ w , φ n , φ s are calculated from a suitable high order
4

scheme such as QUICK. The term S bc is the source contribution of the near boundary
points. For example, for a west boundary, if a Dirihlet boundary condition φ = φ w exists,
then, for a point near a west boundary, the source contribution is S bc = a W φ w , where a W
would be the coefficient if that point were an interior point. Similarly, (S p ) bc is the source
contribution of the near boundary points.
Also, note that for source terms small letters correspond to per unit mass of the source.
That is, S = sΔV, S c = s c ΔV, and S p = s p ΔV, where ΔV is the volume of the control
volume.
It can be observed that the coefficients of the discretized x- and y-momentum equations
are the same in collocated grid system (excluding the near boundary points), provided
that the diffusion coefficients Γ are the same in x and y-directions.
In order to slow down the changes of dependent variables in consecutive
solutions, an underrelaxation factor is introduced into the discretized equation as follows:
( 1−αφ
)
−1
aP
φ = a φ + a φ + a φ + a φ + a φ + b+ pterm+ a φ
α
(10)
φ
o o n
P E E W W N N S S P P P P
αφ
where the superscript (n-1) refers to the previous iteration. In that case, the terms in Eq.
(9)are redefined as
⎧ aP
n 1
b ← b + (1 −
−
α)
φP
⎪
α
⎨
⎪ aP
aP
←
⎪⎩ α
Equation (10) can be written as
α
α pterm
1
( a a a a b a
o o ) ( 1 ) n φ
φ = φ + φ + φ + φ + + φ + − αφ
φ − + (11)
a
or
φ
P E E W W N N S S P P P P
aP
α
αφpterm
φ = ( a φ + a φ + a φ + a φ + B ) + (12)
a
φ
P E E W W N N S S P
aP
where
( 1−αφ
)
−1
B = b + a φ + a φ
(13)
o o n
P P P P P P
αφ
P
P
5

4. Momentum Interpolation
4.1. Momentum interpolation for steady-state problem
BP = bP + ⎡⎣ 1−αu αu⎤⎦ aPuP
[Eq.(13)]. For the
n−1
Note that for the steady problems ( )
velocity component u at nodes P and E, Eq. (12) can be written as
u
u
P
E
( )
( a )
P
P
( )
( a )
αu ∑
i
au
i i
+ Bp α
P uΔy pe − pw
P
= − (14)
( )
( a )
P
E
P
P
( )
( a )
αu ∑
i
au
i i
+ Bp α
E uΔy pe − pw
E
= − (15)
Mimicking the formulation of u
E
interface velocity at the cell face e.
u
e
( )
( a )
P
e
P
E
and u
P
, we can obtain the following expression for the
( )
( a )
αu ∑
i
au
i i
+ Bp α
e uΔy pE − pP
= − (16)
P
e
where the terms on the right-hand side with subscript e should be interpolated in an
appropriate manner. The interface velocity at cell faces w, n, and s can be obtained
similarly.
In Rhie and Chow’s momentum interpolation, the first term and 1 ( aP ) in second
e
term of the Eq.(16) are linearly interpolated from their counterparts in Eqs.(14) and (15):
⎛∑ au + B ⎞ ⎛∑ au + B ⎞ ⎛∑ au + B ⎞
⎜ ⎟ ⎜ ⎟ ( 1 fe
) ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
i i i p + i i i p + i i i p
= fe
+ −
aP a
e P
a
E P P
(17)
1 + 1 + 1
= f + − (18)
a a a
e
( ) ( )
( 1 fe
) ( )
P e P E P P
where f + e
is a linear interpolation factor defines as
+ Δx
P
f
e
= (19)
2δ x
e
In order to have a better understanding of Eq. (16), substituting ( au B a )
Eq. (17) and ( ∑ au + B a ) , ( au B a )
i i i p P P
o o
(16) and omitting the term au , we obtain
P
P
i i i p P E
∑ + from
i i i p P e
∑ + from Eqs. (14) and (15) into Eq.
6

( − p ) +
fe
( aP)
e
α y( p p
+
)
e
( a )
( − p )
( a )
⎧ αuΔy p
αuΔy p
E P e w
⎫
E
⎪− +
⎪
+ + ⎪
P E ⎪
ue = ⎡fe uE ( 1 fe ) u ⎤
⎣
+ −
P⎦ +⎨ ⎬
⎪
uΔ e
−
w P
⎪
⎪
+ ( 1−
f )
⎪
⎩
P P
⎭
Equations (16) and (20) are essentially equivalent. However, Eq. (20) separates
the interfacial velocity into two parts: a linear interpolation part and the additional one.
The term in first set of brackets of Eq. (20) is the arithmetic-averaged values (linear
interpolation method) of two neighbor nodal velocities. The term in braces can be
regarded as a correction term, which has the function of smoothing the pressure field, and
it is this term that may remove the unrealistic pressure field.
Equations (16) with (17) and (18) or equation (20) constitute the Rhie and Chow’s
Original Momentum Interpolation <strong>Method</strong> (OMIM). Majumdar (1988) reported that
solutions of steady-state problems from Rhie and Chow OMIM are dependent on the
underrealxation factor. To eliminate this underrelaxation factor dependency, an iteration
algorithm was proposed by him to calculate the cell-face velocity for steady-state
problem as follows:
( )
( )
( a )
α ∑ au + b α Δy p − p
u u f u f u
( 1 α ) ⎡
( 1 )
u i i i p
e u E P n− 1 + n− 1 + n−1
e
= − + −
u e e E e P
( aP)
⎣
+ − −
e
P e
where superscript n-1 refer to previous iteration values. This iterative implementation
algorithm can achieve a unique solution that is independent of underrelaxation factors.
⎤
⎦
(20)
(21)
4.2. Momentum interpolation for unsteady problems
Choi (1999) reported that, the solution using the original Rhie and Chow scheme
is time step size dependent. He proposed a modified Rhie and Chow scheme for an
unsteady problem as follows, which is quite similar to Majumdar’s scheme for a steady
problem:
( )
( )
⎛∑ au + b ⎞ α Δy p − p α a
u u u
o
i i i p u E P n−1
u e o
e
= αu⎜
⎟ − + ( 1− αu)
e
+
e
⎝ aP ⎠ aP ( aP)
e e e
with
a
o
e
ρδ xeΔy
=
Δt
(22)
(23)
7

It is to be noted that body-force term is neglected for simplicity of presentation. By a
similar substitution process as the one for the Majumdar’s interpolation in Eq. (21), the
Eq. (22) can be written equivalently as
( 1 )
u ⎡f u f u ⎤
⎣
⎦
( − p ) +
fe
( aP)
e
α Δy( p − p
+
)
e
( a )
⎧ α Δy p
⎪− +
⎪
⎪
⎪+ ( 1−
f )
⎪
( − p )
( a )
u E P u e w E
u e w P
α Δy p
+ +
P P
e
=
e E
+ −
e P
+⎨ ⎬
n− 1 + n− 1 + n−1
( 1 α ) ⎡
u
ue fe uE ( 1 fe ) u ⎤
⎣
P
⎦
o
o
⎡α
α ( a )
+ +
⎢
( 1 )
⎪+ − + − −
⎪
⎪
o
a α ( a
⎪ u f u f
)
+ − − −
u
⎪ ⎢( a ) a a
⎩ ⎣
u e o u P
E o u P
P o
e e E e P
P ( P)
( P)
e E P
P
E
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎤⎪
⎥⎪
⎥⎪
⎦⎭
(24)
According to Yu et al. (2002), solutions by using this scheme are still time step
size dependent, though the dependence is quite small. They proposed a different
interpolation technique for the terms appearing in equation (22) which appears to be both
under relaxation factor and time step size independent. In this method the first term on
the right-hand side of Eq. (22) is interpolated as follows:
∑ au + b
( ∑ +
1) + ( 1− )( ∑ + )
E
1
+ +
fe ( sc) ( 1 fe )( sc)
⎤δ
x
E
P e
y
( ∑ ) + ( 1− )( ∑ )
f au b f au b
+ +
e i i i e i i i
+ ⎡ + − Δ
⎛ i i i p ⎞ ⎣ ⎦
⎜ ⎟ =
+ +
⎝ aP ⎠ f
e e i
ai f
E e i
ai
P
where b
1
is defined in Eq. (9).
( ) ( 1 )( )
− ⎡f s + − f s ⎤δ
x Δ y+
a
⎣
⎦
+ +
o
e p
E
e p
P
e e
Also, the denominator of the second and third terms in Eq. (22) is interpolated as follows:
+ +
( a ) = f ( ∑ a ) + ( 1− f )( ∑ a )
P e e i i E e i i P
( ) ( 1 )( )
− ⎡f s + − f s ⎤δ
x Δ y+
a
⎣
⎦
+ +
o
e p
E
e p
P
e e
Equation (22) combined with Eq. (25) and Eq. (26) is Yu et al.’s (2002) new
scheme. Substituting Eq. (25) into Eq. (22) the following equation is obtained:
P
(25)
(26)
8

⎧ + +
⎡fe ( aP) uE+ ( 1−
fe )( aP)
u ⎤
P
⎪⎣
E
P ⎦
⎪ ⎡ + + +
( fe ( sc) ( 1 fe )( sc)
) δ xe y fe ( sc)
x
E P E E
y⎤
⎪
+ − Δ − Δ Δ
α ⎢
⎥
⎪+ u ⎢ +
( 1 fe )( sc)
x
P P
y
⎥
⎪ ⎣
− − Δ Δ
⎦
1
⎪
u = ⎨+ α ⎡ y p p f y p p f y p p
E
( aP
) ⎣
−Δ − + Δ − + − Δ −
e ⎪
n− 1 + n− 1 + n−1
⎪+ ( 1−αu) ⎡( aP) ue − fe ( aP) uE −( 1−
fe )( aP)
u ⎤
e E P
P
⎪ ⎣
⎦
⎪ o o + o o + o
+ α ⎡
o
u
au
e e
− fe ( aP) uE −( 1−
fe )( aP)
u ⎤
⎪ ⎣
E
P
P⎦
⎪
⎩ ⎪
+ +
( ) ( ) ( 1 ) ( )
e u E P e e w e e w
where ( a )
P
e
is found from equation (26). It should be noted that the term in the
parenthesis, which is multiplied by (1–α u ) is incorrect in the paper of Yu et al.’s (2002).
It should also be noted that the cell face velocities found from the momentum
interpolation method are used to determine the mass fluxes across the cell faces.
They should not be used for the cell face value of the independent variable φ in the
deferred correction term b dc in eq. (9) in the case φ stands for u or v in the x- or y-
momentum equations. The face values of the independent variable φ are calculated
using a suitable convection scheme, such as UPWIND or QUICK.
P
⎫
⎪
⎪
⎪
⎪
⎪
⎤
⎪
⎦⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪ ⎭
(27)
The <strong>SIMPLE</strong> Algorithm
For a guessed pressure field p * the corresponding face velocity
Eq. (22) as
( )
( )
* * *
o
* u i i i p
e u E P n−1
u e o
e ( 1 αu)
e e
( aP)
( aP)
( aP)
e e e
*
u
e
can be written using
au b y p p a
u α ∑ + α Δ −
= − + − u + α u
(28)
*
A similar equation can be written for the face velocity v
n
. Now we define the correction
p' as the difference between the correct pressure field p and the guessed pressure field p *
so that
*
p = p + p′
(29)
Similarly we define velocity corrections u' and v' as
u = u + u′
(30)
*
e e e
9

v = v + v′
(31)
*
n n n
Subtracting Eq. (28) from Eq. (22) gives
e
( ′ )
( a )
P
e
( − )
( a )
αu ∑
i
au
i i
+ bp α
e uΔy p′ E
p′
P
u′ = − (32)
P
e
As an approximation, in <strong>SIMPLE</strong> method the first term in the above equation is neglected
giving
( )
u′ = d p′ − p′
(33)
u
e e P E
where
d
α A
= (34)
u u e
e
( aP
)
e
where A e = Δy is the area of the control volume at the east face. Similarly,
( )
v′ = d p′ − p′
(35)
v
n n P N
where
d
α A
= (36)
v v n
n
( aP
)
n
Then the corrected velocities become
( ′ ′ )
u = u + d p − p
(37)
* u
e e e P E
( ′ ′ )
v = v + d p − p
(38)
* v
n n n P N
Substituting the corrected face velocities such as that given by Eq.s (37)and (38) into the
discretized continuity equation (7) gives
a p′ = a p′ + a p′ + a p′ + a p′
+ b
(39)
P P W W E E S S N N
where
a = ( ρ Ad) a = ( ρAd) a = ( ρAd) a = ( ρAd)
E e W w N n S s
* * * *
( ρ ) ( ρ ) ( ρ ) ( ρ )
b= u A − u A + v A − u A
w e s n
After solving the p' field from Eq. (39) the face velocities are corrected using Eq.'s (37)
and (38) and the pressure field is corrected by using
= + α ′
(41)
*
p p
p
p
where α p is the pressure under-relaxation factor which is chosen to be between 0 and 1.
(40)
10

Similarly the nodal velocities are corrected using
( ′ ′)
u = u + d p − p
(42)
* u
P P P w e
( ′ ′)
v = v + d p − p
(43)
* v
P P P s n
where
d
u
P
αuAe v αvAn
= and d = (44)
P
( a ) ( a )
P
P
P
P
The pressure corrections at the cell faces in Eqs. (42) and (43) are calculated by linear
interpolation from the nodal values as
+ +
p′ = f p′ + (1 − f ) p′
(45)
w w P w W
+ +
p′ = f p′ + (1 − f ) p′
(46)
e e E e P
+ +
p′ = f p′ + (1 − f ) p′
(47)
s s P s S
+ +
p′ = f p′ + (1 − f ) p′
(48)
n n N n P
Boundary Conditions for Pressure
Since there is no equation for the pressure, no boundary conditions are needed for the
pressure at the boundary points. The pressure values at the boundary points can be
calculated by linear extrapolation using the two near-boundary node pressures.
Boundary Conditions for Pressure Correction Equation
When the velocities at the boundaries are known, there is no need to correct the velocities
at the boundaries in the derivation of the pressure correction equation. For example if the
velocity at the west boundary is known then for a control volume near the west boundary:
( ′ ′ )
u = u + d p − p
(49)
u
* u
e e e P E
w
= u
(50)
wall
( ′ ′ )
v = v + d p − p
(51)
* v
n n n P N
( ′ ′ )
v = v + d p − p
(52)
* v
s s s S P
Substituting equations (49)-(52) into the discretized continuity equation (7) we obtain the
following pressure correction equation for a control volume near the west boundary
a p′ = a p′ + a p′ + a p′ + a p′
+ b
(53)
P P W W E E S S N N
11

where
a = ( ρ Ad) a = 0 a = ( ρAd) a = ( ρAd)
a = a + a + a + a
E e W N n S s
P W E S N
* * *
( ρ ) ( ρ ) ( ρ ) ( ρ )
b= uA − u A + v A − u A
wall e s n
Comparing Eq.'s (53)-(54) with (39)-(40) for a near boundary control volume the same
definition of the coefficients as used for the interior points can be used for a near
boundary control volume by setting the corresponding coefficient (a w in this case) to zero
and using u wall in the b term.
The above formulation corresponds to Neumann boundary condition ∂p´/∂n = 0, where n
is normal to the boundary. As a result no value of pressure correction at the boundary
( p′
w
) is involved in this formulation. However, the value of the pressure correction is
needed for correcting the nodal velocities near boundaries. For example, for correcting
the u-velocity at a nodal point P near a west boundary, p′
w
at the west boundary is needed
in accordance with equation (42). This value can be obtained by using ∂p´/∂n = 0 at the
boundary, that is using p′ (1, j) = p′
(2, j)
.
References
Choi S. K., "Note on the Use of Momentum Interpolation <strong>Method</strong> for Unsteady Flows",
Numerical. Heat Transfer Part, A, vol. 36, pp. 545-550, 1999.
Majumdar S., "Role of Underrelaxation in Momentum Interpolation for Calculation of
Flow with Nonstaggered Grids", Numerical Heat Transfer, Part B, vol. 13, pp. 125-132,
1988.
Rhie C. M. and Chow W. L., "Numerical Study of the Turbulent Flow Past an Airfoil
with Trailing Edge Separation", AIAA Journal, vol. 21, no 11, pp. 1525-1535, 1983.
Yu B., Tao W., and Wei J., "Discussion on Momentum Interpolation <strong>Method</strong> for
Collocated Grids of Incompressible Flow", Numerical. Heat Transfer Part B, vol. 42, pp.
141-166, 2002.
(54)
12