A fourth order compact formulation for the steady 2D incompressible
NavierStokes equations is presented. The formulation is in the same
form of the NavierStokes equations such that any numerical method
that solve the NavierStokes equations can also be applied to this
fourth order compact formulation. The formulation is solved with an
efficient numerical method that requires the solution of tridiagonal
systems using a fine grid mesh of 601×601. Using this
formulation, the steady 2D incompressible flow in a driven cavity
is solved up to Reynolds number of 20,000 with fourth order spatial
accuracy. Detailed solutions are presented.
High Order Compact (HOC) formulations are becoming more popular in
Computational Fluid Dynamics (CFD) field of study. Compact
formulations provide more accurate solutions in a compact stencil.
In finite differences, a standard three point discretization
provides second order spatial accuracy and this type of
discretization is very widely used. When a high order spatial
discretization is desired, ie. fourth order accuracy, then a five
point discretization have to be used. However in a five point
discretization there is a complexity in handling the points near the
boundaries.
High order compact schemes provide fourth order spatial accuracy in
a 3×3 stencil and this type of compact formulations do not
have the complexity near the boundaries that a standard wide (five
point) fourth order formulation would have.
Dennis & Hudson [1], MacKinnon & Johnson
[7], Gupta et. al. [5], Spotz & Carey
[8] and Li et. al. [6] have demonstrated the
efficiency of the high order compact schemes on the streamfunction
and vorticity formulation of 2D steady incompressible NavierStokes
equations.
In the literature, it is possible to find numerous different type of
iterative numerical methods for the NavierStokes equations. These
numerical methods, however, could not be easily used in HOC schemes
because of the final form of the HOC formulations used in
[1], [6], [4], [7] and
[5]. This fact might be counted as a disadvantage of HOC
formulations that the coding stage is rather complex due to the
resulting stencil used in these studies. It would be very useful if
any numerical method for the solution of NavierStokes equations
described in books and papers could be easily applied to high order
compact (HOC) formulations.
In this study, we will present a new fourth order compact
formulation. The difference of this formulation with [1],
[6], [4], [7] and [5] is not
in the way that the fourth order compact scheme is obtained. The
main difference, however, is in the way that the final form of the
equations are written. The main advantage of this formulation is
that, any iterative numerical method used for NavierStokes
equations, can be easily applied to this new HOC formulation, since
the final form of the presented HOC formulation is in the same form
with the NavierStokes equations. Moreover if someone already have a
second order accurate (O Dx^{2}) code for the
solution of steady 2D incompressible NavierStokes equations, using
the presented formulation, they can easily convert their existing
code to fourth order accuracy (O Dx^{4}) by just
adding some coefficients into their existing code. With this new
compact formulation, we have solved the steady 2D incompressible
driven cavity flow at very high Reynolds numbers using a very fine
grid mesh to demonstrate the efficiency of this new formulation.
In nondimensional form, steady 2D incompressible NavierStokes
equations in streamfunction (y) and vorticity (w) formulation are given as
¶^{2}y
¶x^{2}
+
¶^{2}y
¶y^{2}
= w
(1)
1
Re
¶^{2}w
¶x^{2}
+
1
Re
¶^{2}w
¶y^{2}
=
¶y
¶y
¶w
¶x

¶y
¶x
¶w
¶y
(2)
where x and y are the Cartesian coordinates and Re is
the Reynolds number. For first order and second order derivatives
the following discretizations are fourth order accurate
¶f
¶x
=
f_{x} 
Dx^{2}
6
¶^{3}f
¶x^{3}
+ O (Dx^{4})
(3)
¶^{2}f
¶x^{2}
=
f_{xx}
Dx^{2}
12
¶^{4}f
¶x^{4}
+ O (Dx^{4})
(4)
where f_{x} and f_{xx} are standard second order central
discretizations such that
f_{x}
=
f_{i+1}f_{i1}
2 Dx
(5)
f_{xx}
=
f_{i+1}2 f_{i}+f_{i1}
Dx^{2}
(6)
If we apply these discretizations in equations (3 and 4) to
equations (1 and 2), we obtain the following equations
y_{xx} + y_{yy} 
Dx^{2}
12
¶^{4}y
¶x^{4}

Dy^{2}
12
¶^{4}y
¶y^{4}
+ O (Dx^{4}, Dy^{4}) = w
(7)
1
Re
w_{xx} +
1
Re
w_{yy} 
1
Re
Dx^{2}
12
¶^{4}w
¶x^{4}

1
Re
Dy^{2}
12
¶^{4}w
¶y^{4}
+ O (Dx^{4}, Dy^{4}) = y_{y} w_{x}  y_{x}w_{y}

Dy^{2}
6
w_{x}
¶^{3}y
¶y^{3}

Dx^{2}
6
y_{y}
¶^{3}w
¶x^{3}
+
Dx^{2}
6
w_{y}
¶^{3}y
¶x^{3}
+
Dy^{2}
6
y_{x}
¶^{3}w
¶y^{3}
+O (Dx^{4}, Dx^{2} Dy^{2}, Dy^{4})
(8)
In these equations we have third and fourth derivatives
(¶^{3}/¶x^{3}, ¶^{3}/¶y^{3}, ¶^{4}/¶x^{4} and ¶^{3}/¶y^{4}) of streamfunction
and vorticity (y and w) variables. In order to find an expression for these derivatives
we use equations (1 and 2). For example, when we take the first and
second xderivative (¶/¶x and ¶^{2}/¶x^{2}) of the streamfunction
equation (1) we obtain
¶^{3}y
¶x^{3}
=

¶w
¶x

¶^{3}y
¶x ¶y^{2}
(9)
¶^{4}y
¶x^{4}
=

¶^{2}w
¶x^{2}

¶^{4}y
¶x^{2}¶y^{2}
(10)
And also, by taking the first and second yderivative
(¶/¶y and ¶^{2}/¶y^{2}) of the streamfunction equation (1) we obtain
¶^{3}y
¶y^{3}
=

¶w
¶y

¶^{3}y
¶x^{2}¶y
(11)
¶^{4}y
¶y^{4}
=

¶^{2}w
¶y^{2}

¶^{4}y
¶x^{2}¶y^{2}
(12)
Using standard second order central discretizations given in Table I,
these equations ((9), (10), (11) and (12)) can be written as the
followings
¶^{3}y
¶x^{3}
=
 w_{x} y_{xyy} + O (Dx^{2}, Dy^{2})
(13)
¶^{4}y
¶x^{4}
=
 w_{xx} y_{xxyy} + O (Dx^{2}, Dy^{2})
(14)
¶^{3}y
¶y^{3}
=
 w_{y} y_{xxy} + O (Dx^{2}, Dy^{2})
(15)
¶^{4}y
¶y^{4}
=
 w_{yy} y_{xxyy} + O (Dx^{2}, Dy^{2})
(16)
When we substitute equations (14 and 16) into equation (7) we obtain
the following finite difference equation.
y_{xx} + y_{yy} =  w
Dx^{2}
12
w_{xx} 
Dy^{2}
12
w_{yy} 
æ è
Dx^{2}
12
+
Dy^{2}
12
ö ø
y_{xxyy} + O (Dx^{4}, Dx^{2} Dy^{2},Dy^{4})
(17)
We note that, the solution of this equation (17) is also a solution
to streamfunction equation (1) with fourth order spatial accuracy.
Therefore if we numerically solve equation (17), the solution we
obtain will satisfy the streamfunction equation up to fourth order
accuracy.
In order to obtain a fourth order approximation for the vorticity
equation, we follow the same procedure. When we take the first and
second derivatives of the vorticity equation (2) with respect to x and ycoordinates and we obtain
¶^{3}w
¶x^{3}
=
Re
¶^{2}y
¶x ¶y
¶w
¶x
+ Re
¶y
¶y
¶^{2}w
¶x^{2}
 Re
¶^{2}y
¶x^{2}
¶w
¶y
 Re
¶y
¶x
¶^{2}w
¶x ¶y

¶^{3}w
¶x ¶y^{2}
(18)
¶^{4}w
¶x^{4}
=
Re
¶^{3}y
¶x^{2} ¶y
¶w
¶x
+Re
¶^{2}y
¶x¶y
¶^{2}w
¶x^{2}
+ Re
¶^{2}y
¶x ¶y
¶^{2}w
¶x^{2}
+ Re
¶y
¶y
¶^{3}w
¶x^{3}
 Re
¶^{3}y
¶x^{3}
¶w
¶y
 Re
¶^{2}y
¶x^{2}
¶^{2}w
¶x¶y
 Re
¶^{2}y
¶x^{2}
¶^{2}w
¶x ¶y
 Re
¶y
¶x
¶^{3}w
¶x^{2}¶y

¶^{4}w
¶x^{2} ¶y^{2}
(19)
¶^{3}w
¶y^{3}
=
Re
¶^{2}y
¶y^{2}
¶w
¶x
+ Re
¶y
¶y
¶^{2}w
¶x ¶y
 Re
¶^{2}y
¶x ¶y
¶w
¶y
 Re
¶y
¶x
¶^{2}w
¶y^{2}

¶^{3}w
¶x^{2} ¶y
(20)
¶^{4}w
¶y^{4}
=
Re
¶^{3}y
¶y^{3}
¶w
¶x
+Re
¶^{2}y
¶y^{2}
¶^{2}w
¶x ¶y
+ Re
¶^{2}y
¶y^{2}
¶^{2}w
¶x ¶y
+ Re
¶y
¶y
¶^{3}w
¶x ¶y^{2}
 Re
¶^{3}y
¶x ¶y^{2}
¶w
¶y
 Re
¶^{2}y
¶x ¶y
¶^{2}w
¶y^{2}
 Re
¶^{2}y
¶x ¶y
¶^{2}w
¶y^{2}
 Re
¶y
¶x
¶^{3}w
¶y^{3}

¶^{4}w
¶x^{2} ¶y^{2}
(21)
If we substitute equations (18 and 20) for the third derivatives of
vorticity (¶^{3}w/¶x^{3} and ¶^{3}w/¶y^{3}) into equations (8, 19
and 21) and also if we substitute equations (13 and 15) for the
third derivatives of streamfunction (¶^{3}y/¶x^{3} and ¶^{3}y/¶y^{3}) in equations (8, 19 and 21) and finally if we substitute
equations (19 and 21) for the fourth derivatives of vorticity
(¶^{4}w/¶x^{4} and ¶^{4}w/¶y^{4}) into equation (8),
then equation (8) become the following
w_{xx} + w_{yy} Re
Dx^{2}
6
y_{xy}w_{xx} +Re
Dy^{2}
6
y_{xy}w_{yy} +Re^{2}
Dx^{2}
12
y_{y}y_{y}w_{xx} +Re^{2}
Dy^{2}
12
y_{x}y_{x}w_{yy} =
Rey_{y}w_{x}Rey_{x}w_{y} +Re
æ è
Dx^{2}
12
+
Dy^{2}
12
ö ø
y_{xxy}w_{x} Re
æ è
Dx^{2}
12
+
Dy^{2}
12
ö ø
y_{xyy}w_{y}
Re^{2}
Dx^{2}
12
y_{y}y_{xy}w_{x} +Re^{2}
Dy^{2}
12
y_{x}y_{yy}w_{x} +Re^{2}
Dx^{2}
12
y_{y}y_{xx}w_{y} Re^{2}
Dy^{2}
12
y_{x}y_{xy}w_{y}
+Re
æ è
Dx^{2}
12
+
Dy^{2}
12
ö ø
y_{y}w_{xyy} Re
æ è
Dx^{2}
12
+
Dy^{2}
12
ö ø
y_{x}w_{xxy} Re
Dx^{2}
6
y_{xx}w_{xy}
+Re
Dy^{2}
6
y_{yy}w_{xy} +Re^{2}
æ è
Dx^{2}
12
+
Dy^{2}
12
ö ø
y_{x}y_{y}w_{xy} Re
æ è
Dx^{2}
12

Dy^{2}
12
ö ø
w_{x}w_{y}

æ è
Dx^{2}
12
+
Dy^{2}
12
ö ø
w_{xxyy} + O (Dx^{4}, Dx^{2} Dy^{2}, Dy^{4})
(22)
Again we note that, the solution of this equation (22) satisfy the
vorticity equation (2) with fourth order accuracy.
As the final form of the HOC scheme, we prefer to write equations
(17 and 22) as the following
We note that, the finite difference Equations (23) and (24) are
fourth order accurate (O (Dx^{4},Dx^{2}Dy^{2},Dy^{4})) approximation of the
streamfunction and vorticity equations (1) and (2). In Equations
(23) and (24), however, if A, B, C, D, E and F are
chosen to be equal to zero then the finite difference Equations (23)
and (24) simply become
y_{xx} + y_{yy} = w
(26)
1
Re
w_{xx} +
1
Re
w_{yy} = y_{y} w_{x}  y_{x} w_{y}
(27)
Equations (26) and (27) are the standard second order accurate
(O (Dx^{2},Dy^{2}) )
approximation of the streamfunction and vorticity equations (1) and
(2). When we use Equations (23) and (24) for the numerical solution
of 2D steady incompressible NavierStokes equations, we can easily
switch between second and fourth order accuracy just by using
homogeneous values for the coefficients A, B, C, D, E and
F or by using the expressions defined in Equation (25) in the
code.
In Equations (23), (24) and (25) instead of finite difference
discretizations, if we substitute for partial derivatives we obtain
the following differential equations
¶^{2}y
¶x^{2}
+
¶^{2}y
¶y^{2}
= w+ A
(28)
1
Re
(1+B)
¶^{2}w
¶x^{2}
+
1
Re
(1+C)
¶^{2}w
¶y^{2}
=
æ è
¶y
¶y
+D
ö ø
¶w
¶x

æ è
¶y
¶x
+E
ö ø
¶w
¶y
+F
(29)
where
A
=

Dx^{2}
12
¶^{2}w
¶x^{2}

Dy^{2}
12
¶^{2}w
¶y^{2}

æ è
Dx^{2}
12
+
Dy^{2}
12
ö ø
¶^{4}y
¶x^{2}¶y^{2}
B
=
 Re
Dx^{2}
6
¶^{2}y
¶x ¶y
+ Re^{2}
Dx^{2}
12
¶y
¶y
¶y
¶y
C
=
Re
Dy^{2}
6
¶^{2}y
¶x¶y
+ Re^{2}
Dy^{2}
12
¶y
¶x
¶y
¶x
D
=
æ è
Dx^{2}
12
+
Dy^{2}
12
ö ø
¶^{3}y
¶x^{2} ¶y
 Re
Dx^{2}
12
¶y
¶y
¶^{2}y
¶x ¶y
+ Re
Dy^{2}
12
¶y
¶x
¶^{2}y
¶y^{2}
E
=
æ è
Dx^{2}
12
+
Dy^{2}
12
ö ø
¶^{3}y
¶x ¶y^{2}
 Re
Dx^{2}
12
¶y
¶y
¶^{2} y
¶x^{2}
+ Re
Dy^{2}
12
¶y
¶x
¶^{2}y
¶x ¶y
F
=
æ è
Dx^{2}
12
+
Dy^{2}
12
ö ø
¶y
¶y
¶^{3}w
¶x ¶y^{2}

æ è
Dx^{2}
12
+
Dy^{2}
12
ö ø
¶y
¶x
¶^{3}w
¶x^{2} ¶y

Dx^{2}
6
¶^{2} y
¶x^{2}
¶^{2}w
¶x ¶y
+
Dy^{2}
6
¶^{2}y
¶y^{2}
¶^{2}w
¶x ¶y
+ Re
æ è
Dx^{2}
12
+
Dy^{2}
12
ö ø
¶y
¶x
¶y
¶y
¶^{2}w
¶x ¶y

æ è
Dx^{2}
12

Dy^{2}
12
ö ø
¶w
¶x
¶w
¶y

1
Re
æ è
Dx^{2}
12
+
Dy^{2}
12
ö ø
¶^{4}w
¶x^{2} ¶y^{2}
(30)
We note that the numerical solutions of Equations (28) and (29),
strictly provided that second order discretizations in Table I are
used and also strictly provided that a uniform grid mesh with
Dx and Dy is used, are fourth order accurate to
streamfunction and vorticity equations (1) and (2). We prefer to
call Equations (28) and (29) Fourth Order NavierStokes (FONS)
equations. The only difference between FONS equations (28) and (29)
and NavierStokes (NS) equations (1) and (2) are the coefficients
A, B, C, D, E and F. We note that FONS equations (28)
and (29) are in the same form with NavierStokes (NS) equations (1)
and (2), therefore any iterative numerical method (such as SOR, ADI,
factorization schemes, pseudo time iterations and etc.) used to
solve streamfunction and vorticity equations (1) and (2) can also be
easily applied to fourth order equations (28) and (29). Moreover,
any existing code that solve the streamfunction and vorticity
equations with second order accuracy can easily be modified to
provide fourth order accuracy just by adding the coefficients A,
B, C, D, E and F into the existing code to obtain the
solution of FONS equations. Of course, when the coefficients A,
B, C, D, E and F are added into a second order accurate
code to obtain fourth order accuracy, evaluating these coefficients
would require extra CPU work. This might be considered as the cost
of increasing accuracy from second order to fourth order.
Recently Erturk et al. [2] have presented a new,
stable and efficient numerical method that solve the streamfunction
and vorticity equations. The numerical method solve the governing
steady equations through iterations in the pseudo time. In this
study, we will apply the numerical method Erturk et al.
[2] have proposed, to FONS equations (28) and (29) and
solve the steady driven cavity flow with fourth order accuracy. For
details about the numerical method, the reader is referred to Erturk
et al. [2]. When we apply the numerical method to
Equations (28) and (29), we obtain the following equations
æ è
1  Dt
¶^{2}
¶x^{2}
ö ø
æ è
1  Dt
¶^{2}
¶y^{2}
ö ø
y^{n+1} = y^{n} + Dt w^{n}  Dt A^{n}+
æ è
Dt
¶^{2}
¶x^{2}
ö ø
æ è
Dt
¶^{2}
¶y^{2}
ö ø
y^{n}
(31)
æ è
1  Dt (1+B^{n})
1
Re
¶^{2}
¶x^{2}
+ Dt
æ è
¶y
¶y
+ D
ö ø
n
¶
¶x
ö ø
æ è
1  Dt (1+C^{n})
1
Re
¶^{2}
¶y^{2}
 Dt
æ è
¶y
¶x
+ E
ö ø
n
¶
¶y
ö ø
w^{n+1} = w^{n} Dt F^{n}
+
æ è
Dt (1+B^{n})
1
Re
¶^{2}
¶x^{2}
 Dt
æ è
¶y
¶y
+ D
ö ø
n
¶
¶x
ö ø
æ è
Dt (1+C^{n})
1
Re
¶^{2}
¶y^{2}
+ Dt
æ è
¶y
¶x
+ E
ö ø
n
¶
¶y
ö ø
w^{n}
(32)
The solution methodology of these two equations are quite simple.
First the streamfunction equation (31) is solved in two steps. For
streamfunction equation, a new variable f is defined as the
following
æ è
1  Dt
¶^{2}
¶y^{2}
ö ø
y^{n+1} = f
(33)
Using this variable in Equation (31) we obtain the following
equation
æ è
1  Dt
¶^{2}
¶x^{2}
ö ø
f = y^{n} + Dt w^{n}  DA^{n}+
æ è
Dt
¶^{2}
¶x^{2}
ö ø
æ è
Dt
¶^{2}
¶y^{2}
ö ø
y^{n}
(34)
In this equation, the only unknown is the variable f. We first
solve this equation for f by solving a tridiagonal system. After
this, when we obtain the value of f at every grid point we solve
Equation (33) for streamfunction (y^{n+1}) by solving another
tridiagonal system.
After solving the streamfunction equation (31), we solve the
vorticity equation (32). For this, similarly, we introduce a new
variable g which is defined as the following
æ è
1  Dt (1+C^{n})
1
Re
¶^{2}
¶y^{2}
 Dt
æ è
¶y
¶x
+ E
ö ø
n
¶
¶y
ö ø
w^{n+1} = g
(35)
Using this variable in Equation (32), we obtain the following
equation
æ è
1  Dt (1+B^{n})
1
Re
¶^{2}
¶x^{2}
+ Dt
æ è
¶y
¶y
+ D
ö ø
n
¶
¶x
ö ø
g = w^{n}  DtF^{n}
+
æ è
Dt (1+B^{n})
1
Re
¶^{2}
¶x^{2}
 Dt
æ è
¶y
¶y
+ D
ö ø
n
¶
¶x
ö ø
æ è
Dt (1+C^{n})
1
Re
¶^{2}
¶y^{2}
+ Dt
æ è
¶y
¶x
+ E
ö ø
n
¶
¶y
ö ø
w^{n}
(36)
In this equation the only unknown is the variable g. By solving a
tridiagonal system, we obtain the value of g at every grid point.
Then we solve Equation (35) for vorticity (w^{n+1}) by
solving another tridiagonal system.
In a compact formulation, the stencil have 3×3 points. The
solution at the first diagonal grid points near the corners of the
cavity would require the vorticity values at the corner points.
However, the corner points are singular points for vorticity. Gupta
et al. [3] have introduced an explicit asymptotic
solution in the neighborhood of sharp corners. Similarly,
Störtkuhl et al. [8] have presented an
analytical asymptotic solutions near the corners of cavity and using
finite element bilinear shape functions they also have presented a
singularity removed boundary condition for vorticity at the corner
points as well as at the wall points. We follow Störtkuhl et
al. [8] and use the following expression for
calculating vorticity values at the wall
1
3 Dh^{2}
é
ê
ê
ê
ê
ê
ê
ê
ë
·
·
·
1
2,
4
1
2
1
1
1
ù
ú
ú
ú
ú
ú
ú
ú
û
y
+
1
9
é
ê
ê
ê
ê
ê
ê
ê
ë
·
·
·
1
2
2
1
2
1
4
1
1
4
ù
ú
ú
ú
ú
ú
ú
ú
û
w = 
V
h
(37)
where V is the speed of the wall which is equal to 1 for
the moving top wall and equal to 0 for the three stationary walls.
For corner points, we use the following expression for calculating
the vorticity values
1
3 Dh^{2}
é
ê
ê
ê
ê
ê
ê
ê
ë
·
·
·
·
2
1
2
·
1
2
1
ù
ú
ú
ú
ú
ú
ú
ú
û
y
+
1
9
é
ê
ê
ê
ê
ê
ê
ê
ë
·
·
·
·
1
1
2
·
1
2
1
4
ù
ú
ú
ú
ú
ú
ú
ú
û
w = 
V
2h
(38)
where again V is equal to 1 for the upper two corners
and it is equal to 0 for the bottom two corners. The reader is
referred to Störtkuhl et al. [8] for details.
The schematics of the driven cavity flow is given in Figure 1.
In this figure the abbreviations BR, BL and TL refer to bottom right,
bottom left and top left corners of the cavity, respectively. The
number following these abbreviations refer to the vortices that
appear in the flow, which are numbered according to size.
For every Reynolds number considered, we have continued our
iterations until, in the computational domain both the maximum
residual of Equations (23) and (24), which are given as
are less than 10^{10}. Such a low value is chosen to
ensure the accuracy of the solution. At these residual levels, the
maximum absolute change in streamfunction value between two time
steps, (max(y^{n+1}y^{n})), was in the order of
10^{16} and for vorticity, (max(w^{n+1}w^{n})), it
was in the order of 10^{14}. Obviously these convergence levels
are far more less than satisfactory, however such low values
demonstrate the efficiency of the numerical method used in this
study which was presented by Erturk et al. [2]..
Using an efficient numerical method, Erturk et al.
[2] have clearly shown that numerical solutions of driven
cavity flow is computable for Re > 10,000 when a grid mesh larger
than 256×256 is used. With a grid mesh of 601×601
Erturk et al. [2] have solved the cavity flow up to
Re=21,000 using the numerical method also used in this study. In
order to be able to obtain solutions at high Reynolds numbers,
following Erturk et al. [2], in this study we have
used a large grid mesh with 601×601 grids. With this many
number of grid points we obtained steady solutions of the cavity
flow up to Re=20,000 with fourth order accuracy.
Figures 2 to 6 show the streamfunction and vorticity contours of the
driven cavity flow between Re=1,000 and Re=20,000.
These figures
show the vortices that are formed in the flow field as the Reynolds
number increases. From these contour figures, we conclude that the
fourth order compact formulation provides very smooth solutions.
In Figure 7 we plot a very enlarged view of the top right corner
(where the moving wall moves towards the stationary wall) of the
streamfunction contour plot for the highest Reynolds number
considered, Re=20,000.
In this figure the dotted lines show the grid
lines. As it is seen in this enlarged figure, fourth order
streamfunction contours are very smooth even at the first set of
grid points near the corner.
Table 2 tabulates the streamfunction and vorticity values at the
center of the primary and secondary vortices and also the location
of the center of these vortices for future references. This table is
in good agreement with that of Erturk et al. [2].
Using Richardson extrapolation on the solutions obtained with
different grid meshes, Erturk et al. [2] have
presented theoretically fourth and sixth order accurate (O Dx^{4} and O Dx^{6}) streamfunction and
vorticity values at the center of the primary vortex. Table III and
IV compares the forth order compact scheme solutions of the
streamfunction and the vorticity values at the center of the primary
vortex with the fourth order (O Dx^{4}) Richardson
extrapolated solutions tabulated in Erturk et al.
[2]. The present solutions and the solutions of Erturk
et al. [2] agree with each other.
In this study a new fourth order compact formulation is presented.
The uniqueness of this formulation is that the final form of the HOC
formulation is in the same form of the NavierStokes equations such
that any numerical method that solve the NavierStokes equations can
be easily applied to the FONS equations in order to obtain fourth
order accurate solutions (O Dx^{4}). Moreover with
this formulation, any existing code that solve the NavierStokes
equations with second order accuracy (O Dx^{2}) can
be altered to provide fourth order accurate (O Dx^{4}) solutions just by adding some coefficients into the code at
the expense of extra CPU work of evaluating these coefficients.
In this study, the presented fourth order compact formulation is
solved with a very efficient numerical method introduced by Erturk
et al. [2]. Using a fine grid mesh of
601×601, as it was suggested by Erturk et al.
[2] in order to be able to compute for high Reynolds
numbers, the driven cavity flow is solved up to Reynolds numbers of
Re=20,000. The solutions obtained agree well with previous
studies. The presented fourth order accurate compact formulation is
proved to be very efficient.
Acknowledgement
This study was funded by Gebze Institute of Technology with project
no BAP2003A22. E. Ertürk is grateful for this financial
support.
S. C. Dennis, J. D. Hudson, Compact h^{4} Finite Difference
Approximations to Operators of NavierStokes Type, Journal of
Computational Physics, 85 (1989) 390416.
E. Erturk, T.C. Corke, C. Gokcol, Numerical Solutions of 2D Steady
Incompressible Driven Cavity Flow at High Reynolds Numbers,
International Journal for Numerical Methods in Fluids, 48
(2005) 747774.
M. M. Gupta, R. P. Manohar, J. W. Stephenson, A Single Cell High
Order Scheme for the ConvectionDiffusion Equation with Variable
Coefficients, International Journal for Numerical Methods in
Fluids, 4 (1984) 641651.
M. Li, T. Tang, B. Fornberg, A Compact ForthOrder Finite Difference
Scheme for the Steady Incompressible NavierStokes Equations,
International Journal for Numerical Methods in Fluids, 20
(1995) 11371151.
R. J. MacKinnon, R. W. Johnson, DifferentialEquationBased
Representation of Truncation Errors for Accurate Numerical
Simulation, International Journal for Numerical Methods in
Fluids, 13 (1991) 739757.
W. F. Spotz, G. F. Carey, HighOrder Compact Scheme for the Steady
Streamfunction Vorticity Equations, International Journal for
Numerical Methods in Engineering, 38 (1995) 34973512.
T. Stortkuhl, C. Zenger, S. Zimmer, An Asymptotic Solution for the
Singularity at the Angular Point of the Lid Driven Cavity,
International Journal of Numerical Methods for Heat Fluid Flow,
4 (1994) 4759.
File translated from
T_{E}X
by
T_{T}H,
version 3.63.