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Finite Element Solution of the Two-dimensional Incompressible
Navier-Stokes Equations Using MATLAB
1*Endalew Getnet Tsega and 2V.K. Katiyar
Department of Mathematics
Indian Institute of Technology Roorkee
Uttarakhand, India [email protected], [email protected]
*Corresponding Author
Received: July 5, 2017; Accepted: May 28, 2018
Abstract
The Navier–Stokes equations are fundamental in fluid mechanics. The finite element method
has become a popular method for the solution of the Navier-Stokes equations. In this paper,
the Galerkin finite element method was used to solve the Navier-Stokes equations for two-
dimensional steady flow of Newtonian and incompressible fluid with no body forces using
MATLAB. The method was applied to the lid-driven cavity problem. The eight-noded
rectangular element was used for the formulation of element equations. The velocity
components were located at all of 8 nodes and the pressure variable is located at 4 corner of
the element. From location of velocity components and pressure, it is obvious that this
element consists of 16 unknowns for velocities and 4 unknowns for pressure. As a result, the
unknown variables for velocities and pressure are 20 per each element. The quadratic
interpolation functions represent velocity components while bilinear interpolation function
represents pressure. Finite element codes were developed for implementation. The numerical
results were compared with benchmark results from the literature.
Keywords: Navier–Stokes equations; finite element method; steady-state solution; eight
node rectangular element; MATLAB
MSC 2010 No.: 35Q30, 35Q35, 76D05, 76M10
Available at
http://pvamu.edu/aam
Appl. Appl. Math.
ISSN: 1932-9466
Vol. 13, Issue 1 (June 2018), pp. 535 - 565
Applications and Applied
Mathematics:
An International Journal
(AAM)
536 Endalew Getnet Tsega et al.
1. Introduction
The Navier-Stokes equation is a set of nonlinear partial differential equations that describe
the flow of fluids, which represent conservation of linear momentum. It is the cornerstone of
fluid mechanics as noted by Cengel et al. (2010). It is solved jointly with continuity equation.
These equations cannot be solved exactly. So, approximations and simplifying assumptions
are commonly made to allow the equations to be solved approximately. Recently, high speed
computers have been used to solve such equations by replacing with a set of algebraic
equations using a variety of numerical techniques like finite difference, finite volume, and
finite element methods.
Finite element method is the most powerful numerical technique for computational fluid
dynamics which is readily applicable to domains of complex geometrical shape and provides
a great freedom in the choice of numerical approximations. It reduces a partial differential
equation system to a system of algebraic equations that can be solved using traditional linear
algebra techniques. In finite element method, the domain of interest is subdivided into small
subdomains called finite elements. Over each finite element, the unknown variable is
approximated by a linear combination of approximation functions called shape functions
which are associated with the node of the element characterize the element. The piecewise
approximations for elements are assembled together to obtain a global system to the whole
domain. One of the major advantages of the finite element method is that a general purpose
computer program can be developed easily to analyze various kinds of problems as noted by
Kwon et al. (1997). In particular, any complex shape of problem domain with prescribed
boundary conditions can be handled with ease using the finite element method.
Jiajan (2010) discussed the Galerkin finite element formulation of two dimensional unsteady
incompressible Navier-Stokes equations using the quadratic triangular element (6-nodes).
The pressure variable was located at the corner nodes and the velocity components were
located at all of the six nodes. Ghia et al. (1982) studied high Reynolds number solutions for
incompressible flow using the Navier-Stokes equation and the multigrid method. Persson
(2002) implemented a finite element based solver of the incompressible Navier-Stokes
equations on unstructured two dimensional triangular meshes. He solved the lid-driven cavity
flow problem for four different Reynold’s numbers: 100, 500, 1000 and 2000.
Glaisner et al. (1987) discussed finite-element procedures for the Navier-Stokes equations in
the primitive variable formulation and the vorticity stream-function formulations. Steady-
state solution of lid-driven cavity flow was obtained by the velocity-pressure formulation
using the nine-node rectangular element in their work. Taylor et al. (1981) and Smith et al.
(2014) used eight-noded rectangular element mesh to solve two-dimensional incompressible
Navier-Stokes Equations with FORTRAN programming language. Rhaman et al. (2014)
presented Galerkin finite element method to simulate the motion of fluid particles which satisfies the
unsteady Navier-Stokes equations through a programming code developed in FreeFem++.
Simpson (2017) used nine noded rectangular elements with two degree of freedom on each
node for finite element simulation of a coupled reaction-diffusion problem using MATLAB.
Khennane (2013) developed MATLAB codes for 4-nodded and 8-noded quadrilateral
elements for the linear elastic static analysis of a two dimensional problem using finite
element method.
AAM: Intern. J., Vol. 13, Issue 1 (June 2018) 537
2. Governing Equations
For steady Newtonian incompressible fluid with no body forces, the governing equations for
two-dimensional flow are:
Continuity equation:
.0
y
v
x
u (1)
Navier-Stokes equation:
x-component
,2
2
2
2
y
u
x
u
x
p
y
uv
x
uu (2)
y-component
.2
2
2
2
y
v
x
v
y
p
y
vv
x
vu (3)
where u and v are the x, y components of the velocity vector, p is static pressure, is density
and dynamic viscosity of the flowing fluid.
Using L and V as a characteristics (reference) length and velocity respectively, we define the
dimensionless variables
.*and,*,*,*,*2V
pp
V
vv
V
uu
L
yy
L
xx
The governing equations, Equation(1), Equation (2),and Equation (3) can be written in their
dimensionless form (ignoring the astrix ‘*’) as:
0
y
v
x
u, (4)
,Re
12
2
2
2
y
u
x
u
x
p
y
uv
x
uu (5)
,Re
12
2
2
2
y
v
x
v
y
p
y
vv
x
vu (6)
where
VLRe
is the Reynolds number.
538 Endalew Getnet Tsega et al.
u,v u,v,p
u,v
u,v
u,v
u,v,p
u,v,p u,v,p
3. Formulation of Element Equations
To describe the structure of finite element programming of a steady-state solution of the
Navier–Stokes equations, let us consider a flow confined to a rectangular cavity driven by a
uniform horizontal velocity at the top. The velocities at the other three boundaries are set to
zero. Eight-noded rectangular elements are used to model the flow. We use all the 8 nodes of
each element to model velocities components u and v and the 4 corner nodes to model the
pressure p. Meshes are numbered in x-direction. Freedoms numbered are in the order u−p−v
as used by Smith et al. (2014) and Taylor et al. (1981).
Figure 1. Lid-driven cavity
Let us denote an element by . Shape functions for the rectangular elements are expressed in
terms of local coordinates and where
=2(x-xc)/lx, =2(y-yc)/ly
Here, (xc, yc) is the centroid of the
lx, ly represent its element, and
length in x and y-direction.
AAM: Intern. J., Vol. 13, Issue 1 (June 2018) 539
Figure 2. Eight-noded rectangular element
Figure 3. Eight-noded rectangular elements mesh
Suppose the nodes 1, 2, 3, 4, 5, 6, 7, 8 have coordinates (-1, -1), (0, -1), (1, -1), (1, 0), (1, 1),
(0, 1), (-1, 1), (-1, 0) in the local coordinate system. Then, the general form of shape functions
for 4-noded bilinear rectangular element (considering corner nodes) using local coordinates is
. dcbaM (7)
The general form of shape functions for 8-noded quadratic rectangular element (considering
all nodes) using local coordinates is
.2222 hgfedcbaN (8)
Using Kronecker-delta property of shape functions, from Equation (7) and Equation (8) shape
functions for 4-noded and 8-noded rectangular elements are
540 Endalew Getnet Tsega et al.
).1(4
1
),1(4
1
),1(4
1
),1(4
1
4
3
2
1
M
M
M
M
(9)
).1)(1(
),1)(1)(1(
),1)(1(
),1)(1)(1(
),1)(1(
),1)(1)(1(
),1)(1(
),1)(1)(1(
2
21
8
41
7
2
21
6
41
5
2
21
4
41
3
2
21
2
41
1
N
N
N
N
N
N
N
N
(10)
Thus, the quadratic interpolation functions are used for velocity components while bilinear
interpolation functions for pressure. As a result, the unknown variables for velocities and
pressure are 20 per each element. Thus, the dependent variable u, v, and p are expressed as
.
,
,
4
1
8
1
8
1
i
ii
i
ii
i
ii
pMp
vNv
uNu
(11)
where iii pvu and, are velocity and pressure values at the nodes.
Now, expressing Equation (4), Equation (5), and Equation (6) using these shape functions we
get
08
1
8
1
j
j
j
j
j
jv
y
Nu
x
N, (12)
,Re
1 8
12
28
12
2
4
1
8
1
8
1
8
1
8
1
j
j
j
j
j
j
l
l
l
j
j
j
k
kk
j
j
j
k
kk
uy
Nu
x
N
px
Mu
y
NvNu
x
NuN
(13)
AAM: Intern. J., Vol. 13, Issue 1 (June 2018) 541
.Re
1 8
12
28
12
2
4
1
8
1
8
1
8
1
8
1
j
j
j
j
j
j
l
l
l
j
j
j
k
kk
j
j
j
k
kk
vy
Nv
y
N
py
Mv
y
NvNv
x
NuN
(14)
Employing Galerkin weighted residual approach on Equation (13), we get
0Re
1 8
12
28
12
2
4
1
8
1
8
1
8
1
8
1
dAuy
Nu
x
N
px
Mu
y
NvNu
x
NuNN
j
j
j
j
j
j
l
l
l
j
j
j
k
kk
j
j
j
k
kki
(15)
Using Gauss-Divergence Theorem, we have
8
1
8
1
8
1
8
12
28
12
2
Re
1
Re
1
Re
1
j
j
j
i
j
j
ji
j
j
ji
j
j
j
i
j
j
j
i
dSun
NN
dAuy
N
y
NdAu
x
N
x
NdAu
y
NNu
x
NN
where is the boundary of the element , yx nnn , is the unit outward normal vector to
the element and y
j
x
jjn
y
Nn
x
N
n
N
is the directional derivative of
jN in the direction
normal to the boundary . Hence, we have
8
1
8
1
8
1
4
1
8
1
8
1
8
1
8
1
.0Re
1
Re
1
j
j
j
i
j j
j
ji
j
ji
l
l
l
i
j
j
j
k
kki
j
j
j
k
kki
dSun
NNdAdAu
y
N
y
Nu
x
N
x
N
px
MNu
x
NvNNu
x
NuNN
i, j=1,2, ..., 8.
For Dirichilet boundary condition, we have
.0Re
1 8
1
8
1
4
1
8
1
8
1
8
1
8
1
dAdAuy
N
y
Nu
x
N
x
N
px
MNu
x
NvNNu
x
NuNN
j j
j
ji
j
ji
l
l
l
i
j
j
j
k
kki
j
j
j
k
kki
(16)
i, j=1,2, ..., 8.
542 Endalew Getnet Tsega et al.
Applying similar procedure for Equation (14), we get
.0Re
1 8
1
8
1
4
1
8
1
8
1
8
1
8
1
dAdAvy
N
y
Nv
x
N
x
N
py
MNv
x
NvNNv
x
NuNN
j j
j
ji
j
ji
l
l
l
i
j
j
j
k
kki
j
j
j
k
kki
(17)
i, j=1,2, ..., 8.
Employing Galerkin weighted residual approach on Equation (12) using the weight functions
Ml , we get
,08
1
8
1
dAvy
Nu
x
NM
j
j
j
j
j
j
l
or
.08
1
8
1
dAvy
NMu
x
NM
j
j
j
l
j
j
j
l (18)
l = 1, 2, 3, 4 .
Due to the nonlinearity, the set of algebraic equations that will be obtained here cannot be
solved in a single shot, but an iterative solution is necessary. In such an iterative solution
nonlinear terms can be linearized in a number of different ways. The simplest possibility,
which will be used in this paper, is known as Picard linearization, in which the nonlinear
terms are replaced by
,y
uv
x
uu
y
uv
x
uu
.y
vv
x
vu
y
vv
x
vu
where u and v are approximate values for the velocity components.
We assume starting values 080201080201 .,..,,.,..,, vvvanduuu for the element and
.
,
8
1
0
8
1
0
k
kk
k
kk
vNv
uNu
The iteration process continues by replacing k
u0 and k
v0 , k =1, 2, ..., 8, by the average of
velocity component values from the previous two iterations until tolerance is satisfied.
From Equation (16), Equation (17) and Equation (18), we get a system of equations in matrix
form as
AAM: Intern. J., Vol. 13, Issue 1 (June 2018) 543
,bAh (19)
where
)9(
88
)9(
87
)9(
86
)9(
85
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84
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83
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82
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83
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81
)9(
78
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75
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71
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68
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67
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61
)8(
64
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63
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62
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)9(
58
)9(
57
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56
)9(
55
)9(
54
)9(
53
)9(
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51
)8(
54
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53
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48
)9(
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45
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43
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41
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44
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42
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38
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37
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35
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31
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34
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32
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28
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27
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12
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48
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47
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36
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37
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28
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23
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21
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28
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26
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24
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23
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22
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21
)6(
18
)6(
17
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16
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15
6(
14
)6(
13
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12
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11
)4(
18
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16
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14
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12
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12
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11
)2(
84
)2(
83
)2(
82
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81
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88
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87
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86
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85
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84
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83
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82
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74
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72
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71
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78
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54
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22
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21
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)2(
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)2(
12
)2(
11
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11
00000000
00000000
00000000
00000000
00000000
00000000
00000000
00000000
0000
0000
0000
0000
00000000
00000000
00000000
00000000
00000000
00000000
00000000
00000000
aaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaa
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aaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaa
A
,
8
7
6
5
4
3
2
1
4
3
2
1
8
7
6
5
4
3
2
1
v
v
v
v
v
v
v
v
p
p
p
p
u
u
u
u
u
u
u
u
v
p
u
h
b = 0 (20X1 zero vector)
544 Endalew Getnet Tsega et al.
Here,
,8,...,2,1,
,Re
1)1(
ji
dAy
N
y
N
x
N
x
N
y
NvN
x
NuNa
jijij
i
j
iij
,)1()9(
ijij aa
4,3,2,18,...,2,1
,)2(
ji
dAx
MNa
j
iij
4,3,2,18,...,2,1
,)8(
ji
dAy
MNa
j
iij
,8,...,2,1,4,3,2,1
,)4(
ji
dAx
NMa
j
iij
,8,...,2,14,3,2,1
.)6(
ji
dAy
NMa
j
iij
4. Derivatives and Integrals Using Local Coordinates
Let N(,) be a shape function in terms of local coordinates. If x and y are the global
coordinates, then
.
,
y
y
Nx
x
NN
y
y
Nx
x
NN
.
y
Nx
N
yx
yx
N
N
,
y
Nx
N
JN
N
where
.
yx
yx
J
AAM: Intern. J., Vol. 13, Issue 1 (June 2018) 545
is the Jacobian matrix of the transformation of the global coordinate system to local
coordinate system.
Then, we have
.1
N
N
J
y
Nx
N
(20)
5. Computing the Jacobian Matrix
Let N1(,), N2(,), . . . , Nn(,), be the shape functions for an element in local
coordinates. If (x1, y1), (x2, y2),. . . , (xn, yn) are the global coordinates of the nodes of the
element and (x , y) is the global coordinate of a point on the element, then
x = N1(,)x1+N2(,)x2+. . . + Nn(,)xn ,
y = N1(,)x1+N2(,)x2+. . . + Nn(,)xn .
Then,
....
,...
,...
,...
21
21
21
21
21
21
21
21
n
NNNy
n
NNNx
n
NNNy
n
NNNx
yyy
xxx
yyy
xxx
n
n
n
n
Hence, the Jacobian matrix of the transformation J is
n
i
iNi
n
i
i
N
n
i
i
Nn
i
i
N
nn
NNN
NNN
yx
yx
yx
yx
yx
yx
yx
Ji
ii
n
n
11
1122
11
21
21
(21)
Formation of discrete finite element system requires evaluation of integrals over elements.
Except for simplest of element geometries, this integral cannot be evaluated analytically.
Hence, numerical integrations is the only alternatives. Gaussian quadrature is mostly
employed. For example, using calculus for coordinate transformation, a typical integral for a
two dimensional rectangular element can be evaluated as
546 Endalew Getnet Tsega et al.
.)det(),(),,(),(,),(
)det(),(),,(
)det(),(),,(),(
1 1
1
1
1
1
'
Jyxfffww
ddJyxf
ddJyxfdxdyyxf
m
i
n
j
jiji
where J is the Jacobian matrix of the transformation, i and j are Gaussian quadrature
abscissa, and wi and wj are corresponding weights.
6. Global System of Equations
After developing governing equations for each element, assembly of the element equations
was performed in order to establish global system of equations for the whole domain. In
addition to the element equations, Global coordinates to nodes, elements connectivity, and
global degree of freedom for nodes are also used develop the global system equations.
Applying the boundary conditions, the modified global system of equations is obtained. The
MATLAB codes used for solution process are indicated in the appendix.
7. Results and Discussion
To illustrate the finite element method algorithm discussed in this paper, we considered a
square lid-driven cavity flow of length 1 unit. The boundary conditions are such that the flow
is driven by a unit horizontal velocity at the top boundary. The velocities at the other three
boundaries are set to zero. Eight-node elements are used to model the vector field of
velocities, and 4-node elements are used to model the scalar field of pressures. The flow is
simulated with Reynolds numbers 1, 10, 50, 100, 200, 500 and 1000 using the same mesh of
100 elements (803 nodes). The numerical results are presented here in terms of velocity
quiver plot and pressure contour at the Reynolds numbers. The results in this work were
generated by the series of finite element codes we developed. The computations had been
carried out with the convergence check of 610 (tolerance).
Table 1. Computational performance of five simulations performed for the cavity flow
Re 1 10 50 100 200 500 1000
Number of
iterations 21 22 26 29 35 47 147
Time Spent for
Iterations (sec.) 4.04 4.14 4.72 5.26 6.21 7.62 21.06
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0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
x
Re = 1, Velocity
y
Re = 1
x
y
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
pre
ssure
-10
-8
-6
-4
-2
0
2
4
6
8
10
As seen from Table1, high Reynolds numbers require more iteration and elapsed time to
converge.
(a)
548 Endalew Getnet Tsega et al.
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
x
Re = 10, Velocity
y
Re = 10
x
y
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
pre
ssure
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
(b)
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0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
x
Re = 50, Velocity
y
Re = 50
x
y
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
pre
ssure
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
(c)
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Re = 100
x
y
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
pre
ssure
-0.1
-0.05
0
0.05
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
x
Re = 100, Velocity
y
(d)
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Re = 200
x
y
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
pre
ssure
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
x
Re = 200, Velocity
y
(e)
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0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
x
Re = 500, Velocity
y
Re = 500
x
y
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
pre
ssure
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
(f)
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0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
x
Re = 1000, Velocity
y
Re = 1000
x
y
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
pre
ssure
-0.015
-0.01
-0.005
0
0.005
0.01
(g)
Figure 4. (a) - (g) Velocity quiver and pressure contour plots at different
Reynolds numbers
554 Endalew Getnet Tsega et al.
8. Conclusion
In this paper, we discussed finite element solution of the two-dimensional incompressible
Navier-Stokes equations by the benchmark of square lid-driven cavity. Dirichlet boundary
conditions were imposed on every boundary of the domain. The finite element programming
codes were constructed to solve these equations. These programming codes are written using
MATLAB 7.10.0 (R2010a). The finite element programs consist of one main program and
nine sub programs. These programs with modification can be used to solve related fluid flow
problems. The codes for the geometry and the boundary conditions are original and very efficient.
The numerical results from finite element programming agreed with the numerical results
obtained from finite element analysis done by Ghia et al. (1982).
Acknowledgement
The authors would like thank the reviewers for their valuable comments to improve our
paper. We also want to express our sincere thanks and appreciation to the chief editor for his
prompt reply, careful work and clear instructions.
REFERENCES
Cengel, Yunus A. and Cimbala, John M. (2014). Fluid Mechanics: Fundamentals and
Applications, Third edition, McGraw-Hill.
Chung, T. J. (2010), Computational Fluid Dynamics, Second edition, Cambridge University
press.
Ghia U., Ghia, K. N., and Shin, C. T. (1982). High Reynolds number solutions for
Incompressible Flow Using the Navier-Stokes Equation and
the Multigrid Method, Computational Physics, vol. 48, pp. 387–411.
Glaisner, F. and Tezduyar, T.E. (1987). Finite Element Techniques for the Navier-Stokes
Equations in the Primitive Variable Formulation and the Vorticity Stream-
function Formulation, Department of Mechanical Engineering, University of
Houston.
Jiajan, Wanchai (2010). Solution to incompressible Navier- Stokes equations by using finite
element method, MSc thesis, University of Texas at Arlington.
Khennane, Amar (2013). Introduction to Finite Element Analysis Using MATLAB and
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Sert, Cüneyt (2012). Finite Element Analysis in Thermofluids, Middle East
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Simpson, Guy (2017). Practical Finite Element Modelling in Earth Science using Matlab.
Smith, I.M., Griffiths, D. V. and Margets, L. (2014). Programming the Finite Element
Method, John Wiley & Sons Ltd
Taylor, C. and Hughes, T.G. (1981). Finite Element Programming of Navier-Stokes
Equations, Swansea, U.K., Pineridge Press Ltd.
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556 Endalew Getnet Tsega et al.
APPENDIX
The Finite Element Programming Codes
1. The Main program
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2. Mesh Generating Code
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3. Global coordinates to nodes
4. Global node numbers Code
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5. Create global degree of freedom
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6. Calculate local matrix for each element
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7. Calculate local vector for each element
8. Assemble local matrices into the global
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9. Assemble local vectors into the global
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10. Identify boundary nodes
