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Numerical solutions of steady incompressible flow around a circular cylinder
up to Reynolds number 500
Article · October 2018
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International Journal of Mechanical Engineering and Technology (IJMET)
Volume 9, Issue 10, October 2018, pp. 1368–1378, Article ID: IJMET_09_10_140
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=9&IType=10
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication
Scopus Indexed
NUMERICAL SOLUTIONS OF STEADY
INCOMPRESSIBLE FLOW AROUND A
CIRCULAR CYLINDER UP TO REYNOLDS
NUMBER 500
Ercan Erturk
Bahcesehir University, Mechatronics Engineering Department
Besiktas, Istanbul, Turkey
Orhan Gokcol
Bahcesehir University, Computer Education and Instructional Technologies Department
Besiktas, Istanbul, Turkey
ABSTRACT
In this study, steady incompressible viscous flow around a circular cylinder at high
Reynolds numbers are solved numerically. Using a very efficient numerical method and
a very large mesh, numerical solutions are calculated up to 𝑅𝑒=500. It is found that the
solution change behavior around 𝑅𝑒=100 and 𝑅𝑒=300. At the center of the wake
bubble, around 𝑅𝑒=300 the vorticity starts to increase fast linearly and the width of the
wake bubble starts to increase rapidly. Good agreement is found with the results found
in the literature. Detailed results are presented.
Key words: Steady flow around a circular cylinder, high Reynolds numbers
Cite this Article: Ercan Erturk and Orhan Gokcol, Numerical Solutions of Steady
Incompressible Flow Around a Circular Cylinder Up To Reynolds Number 500,
International Journal of Mechanical Engineering and Technology, 9(10), 2018,
pp. 1368–1378.
http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=9&IType=10
1. INTRODUCTION
The flow around a circular cylinder is one of the classical flow problems in fluid mechanics. It
is well known that the two-dimensional incompressible flow around a circular cylinder is steady
for Reynolds number up to approximately 40 and beyond that Reynolds number, there appears
Karman vortex street at downstream of the cylinder. However, for this flow case, the twodimensional incompressible flow around a cylinder, it is possible to obtain a steady solution
mathematically when Re goes to infinity as the limiting solution (Kirchoff–Helmholtz solution,
see Schlichting and Gersten [1]). Smith [2] and Peregrine [3] have done detailed mathematical
analysis on two-dimensional steady incompressible flow around a circular cylinder at high
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Reynolds numbers. Apart from these mathematical studies, in the literature there are also
computational numerical studies on the same flow problem. Fornberg [4,5], Son and Hanratty
[6], Tuann and Olson [7] and Dennis and Chang [8], Christov, Marinova and Marinov [9] have
presented numerical solutions of steady flow past a circular cylinder at Re=300, 600, 500, 100,
100 and 200 respectively. Most probably, the studies of Fornberg [4,5] are the most
comprehensive studies on the subject of two-dimensional steady incompressible flow around a
circular cylinder in the literature. At high Reynolds numbers, for the considered flow problem
although it is unstable, steady solution still exists along with the transient time dependent
solutions. For the theory of the Navier-Stokes model it is important to study the stationary
solution at high Reynolds numbers even when it loses stability. In the literature Fornberg [5]
have presented the largest Reynolds number solutions of the steady flow around a circular
cylinder and he [5] has presented solutions up to 𝑅𝑒=600. The results of Fornberg [5] show that
above 𝑅𝑒=300 the width of the wake bubble change behavior and starts to increase rapidly. In
the literature the numerical results of the steady flow around a circular cylinder at high
Reynolds numbers presented by Fornberg [5] have never been verified.
The aim of this study is then, to solve the 2-D steady incompressible flow around a circular
cylinder. In this study, the efficient numerical method that allows numerical solutions at high
Reynolds numbers described in Erturk, Corke and Gökçöl [10] and Erturk, Haddad and Corke
[11] is applied to solve to the streamfunction and vorticity form of the Navier-Stokes equations.
With this, the steady flow around a circular cylinder is solved up to high Reynolds number of
𝑅𝑒=500 using a very large mesh. We compare our results in detail with the numerical study of
Fornberg [5]. Detailed quantitative numerical results are presented for future references.
2. PROBLEM FORMULATION AND NUMERICAL METHOD
For the numerical solution of steady incompressible viscous flow around a circular cylinder, we
use the streamfunction (πœ“) and vorticity (πœ”) formulation of the steady Navier-Stokes equations
given as the following
πœ•2 πœ“
πœ•π‘₯ 2
πœ•2 πœ”
πœ•π‘₯ 2
πœ•2 πœ”
+ πœ•π‘¦ 2 βˆ’
πœ•2 πœ“
+ πœ•π‘¦ 2 + πœ” = 0
𝑅𝑒 πœ•πœ“ πœ•πœ”
2
( πœ•π‘¦
(1)
πœ•πœ“ πœ•πœ”
βˆ’ πœ•π‘₯
πœ•π‘₯
πœ•π‘¦
)=0
(2)
where π‘₯ and 𝑦 are the Cartesian coordinates and 𝑅𝑒 is the Reynolds number based on the
π‘ˆπ‘‘
diameter of the cylinder which is defined as 𝑅𝑒 =
where 𝑑 is the diameter, π‘ˆ is the free
𝜈
stream velocity and 𝜈 is the kinematic coefficient of viscosity. Following Fornberg [5] we
transform the physical domain to the computational domain using the following conformal
mapping
𝑍 = βˆšπ‘‹ +
1
(3)
βˆšπ‘‹
where both 𝑋 and 𝑍 are complex numbers defined as the following
𝑋 =π‘₯+𝑖𝑦
(4)
𝑍 =πœ‰+π‘–πœ‚
(5)
With this conformal mapping, the governing equations in the complex 𝑍-plane in the
computational domain become as the following
πœ•2 πœ“
πœ•πœ‰ 2
πœ•2 πœ”
πœ•πœ‰ 2
πœ•2 πœ”
+ πœ•πœ‚2 βˆ’
πœ•2 πœ“
+ πœ•πœ‚2 +
πœ”
𝐽
𝑅𝑒 πœ•πœ“ πœ•πœ”
2
( πœ•πœ‚
πœ•πœ‰
=0
βˆ’
πœ•πœ“ πœ•πœ”
πœ•πœ‰ πœ•πœ‚
(6)
)=0
(7)
where 𝐽 is Jacobian of the mapping which is defined as the following
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𝑑𝑍 2
𝐽 = |𝑑𝑋|
(8)
The physical domain and also the computational domain are shown in Figure 1.
Figure 1 Schematics of the flow problem and the solution domain,
a) physical domain (top), b) computational domain (bottom)
2.1. Boundary Conditions
For external flows such as the flow around a cylinder, the boundary conditions are very
important for the accuracy of the solution. The boundary conditions used in this study are
described briefly below.
2.1.1. Boundary conditions for streamfunction (𝝍)
Left Boundary (πœ‰ = 0 , 0 ≀ πœ‚ ≀ πœ‚π‘šπ‘Žπ‘₯ );
As seen in Figure 1-a and 1-b, left boundary in the (πœ‰, πœ‚) corresponds to the symmetry line
upstream of the cylinder. The symmetry line itself is a streamline and its value is chosen as
zero. Therefore, the value of streamfuction on this symmetry line is
πœ“=0
(9)
Bottom Boundary (πœ‚ = 0 , 0 ≀ πœ‰ ≀ πœ‰π‘šπ‘Žπ‘₯ );
The bottom boundary in Figure 1-b corresponds to the wall of the cylinder and the
symmetry line downstream of the cylinder. On this boundary the streamfunction value is the
same with that of the symmetry line upstream of the cylinder such that
πœ“=0
(10)
Top Boundary (πœ‚ = πœ‚π‘šπ‘Žπ‘₯ , 0 ≀ πœ‰ ≀ πœ‰π‘šπ‘Žπ‘₯ );
The top boundary corresponds the to the free stream boundary and at sufficiently far away
from the cylinder the 𝑒-velocity should be equal to 1 (𝑒 β†’ 1). This states that at the free stream
πœ•πœ“
boundary πœ•π‘¦ β†’ 1. Therefore, at the free stream, the value of the streamfunction should
approach to πœ“ β†’ 𝑦. Using the 𝑍-plane coordinates the value of the streamfunction becomes
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πœ•πœ“
πœ“ β†’ 2πœ‰πœ‚. Hence at the free stream boundary
πœ•πœ‚
β†’ 2πœ‰. In external flows such as the flow
around a cylinder, the free stream boundary condition is the driving boundary condition of the
numerical solution. For this reason, the numerical solution is usually sensitive to the location of
the free stream boundary. In order to decrease this sensitivity, the free stream boundary
condition must be chosen very away from the cylinder. At the boundaries, one can either use
the boundary condition or use both the governing equations and boundary conditions in order to
calculate the flow variables. After extensive numerical experimentation on the flow around a
cylinder flow problem, we think that the latter approach is less stringent on the numerical
solution and therefore the numerical solution is less sensitive to the location of the free stream
boundary condition. Thus, at the free stream boundary we use both the streamfunction equation
and the boundary condition for streamfunction in order to calculate the streamfunction value. At
the free stream boundary, the following equations are valid
πœ•2 πœ“
πœ•πœ‰ 2
πœ•2 πœ“
+ πœ•πœ‚2 +
πœ”
𝐽
=0
and
πœ•πœ“
πœ•πœ‚
β†’ 2πœ‰
(11)
The discretization of the streamfunction equation requires a ghost point outside the
computational domain. The value of this ghost point is calculated using the boundary condition
πœ•πœ“
for the streamfunction, i.e. πœ•πœ‚ β†’ 2πœ‰. We note that, with this approach, at the free stream
boundary we satisfy both the boundary condition and also the governing streamfunction
equation.
Right Boundary (πœ‰ = πœ‰π‘šπ‘Žπ‘₯ , 0 ≀ πœ‚ ≀ πœ‚π‘šπ‘Žπ‘₯ );
The right boundary is the outflow boundary. At this outflow boundary we used a nonreflecting boundary condition such that any wave generated in the computational domain could
pass through the exit boundary and leave without any reflection back into the computational
domain. For details on the subject the reader is referred to the study of Engquist and Majda [12]
in which the nonreflecting boundary condition concept is first introduced in the name of
β€œAbsorbing Boundary Condition”, and also to Jin and Braza [13] for a review of non-reflecting
boundary conditions. In their study Liu and Lin [14] attached a buffer region to the physical
domain to damp erroneous numerical fluctuations. In this region they [14] added a buffer
function to the streamwise second order derivatives in the momentum equations such that the
reflected outgoing waves from an artificially truncated outlet are thus absorbed. This approach
for the exit boundary condition has been used in various similar studies [11,15] successfully.
Therefore, at the right boundary we solve the streamfunction equation without the elliptic
πœ•2
terms, i.e. (πœ•πœ‰2 ) terms, such as
πœ•2 πœ“
πœ•πœ‚ 2
+
πœ”
𝐽
=0
(12)
In the computational domain near the outflow boundary, in order to have smooth transition
πœ•2 πœ“
πœ•2 πœ“
from the streamfunction equation ( πœ•πœ‰2 + πœ•πœ‚2 +
πœ”
𝐽
= 0) used inside the solution domain to the
πœ•2 πœ“
streamfunction equation without the elliptic terms used at the outflow boundary ( πœ•πœ‚2 +
πœ”
𝐽
= 0)
πœ•2
we used a buffer zone as it was done in [11,15]. In this buffer zone we kill the elliptic (πœ•πœ‰2 )
terms in the governing equations gradually. To accomplish this, these elliptic terms are
multiplied by a weighting factor 𝑠. At the beginning of the buffer zone, we set 𝑠=1 and at the
end of the buffer zone, it is zero, 𝑠= 0. In between, the weighting factor 𝑠 changes as
𝑠=
tanh(4)+tanh(π‘Žπ‘Ÿπ‘”)
2 tanh(4)
(13)
where
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2(π‘–βˆ’π‘–π‘π‘’π‘“)
π‘Žπ‘Ÿπ‘” = 4 (1 βˆ’ (π‘–π‘šπ‘Žπ‘₯βˆ’π‘–π‘π‘’π‘“))
(14)
where 𝑖 is the numerical streamwise index, π‘–π‘šπ‘Žπ‘₯ is the numerical index of the last grid point in
streamwise direction and 𝑖𝑏𝑒𝑓 is the index 𝑖 of the first grid point at the beginning of the buffer
zone. We used 20 grid points in this buffer zone. Therefore, near the outflow boundary in the
computational domain, in this buffer zone we basically solve
πœ•2 πœ“
πœ•2 πœ“
𝑠 πœ•πœ‰2 + πœ•πœ‚2 +
πœ”
𝐽
=0
(15)
where 𝑠 is defined above. The non-reflecting boundary condition together with a buffer zone
provides very smooth solutions. This approach also decrease the sensitivity of the numerical
solution to the location of the outflow boundary.
2.1.2. Boundary conditions for vorticity (𝝎)
Left Boundary (πœ‰ = 0 , 0 ≀ πœ‚ ≀ πœ‚π‘šπ‘Žπ‘₯ );
As the same with the streamfunction variable, at the symmetry line the vorticity value is
zero.
πœ”=0
(16)
Bottom Boundary (πœ‚ = 0 , 0 ≀ πœ‰ ≀ πœ‰π‘šπ‘Žπ‘₯ );
On the bottom boundary, for vorticity calculations there are two regions. On the cylinder
where πœ‚ = 0 , 0 ≀ πœ‰ ≀ 2, we used the wall condition for vorticity such as
βˆ’2πœ“1
2
1 βˆ’πœ‚0 )
πœ”0 = 𝐽0 (πœ‚
(17)
where 0 denotes the points on the wall and 1 denotes the first grid points adjacent to the wall.
Also, at the symmetry line downstream of the cylinder where πœ‚ = 0 , 2 ≀ πœ‰ ≀ πœ‰π‘šπ‘Žπ‘₯ , the
vorticity value is zero
πœ”=0
(18)
Top Boundary (πœ‚ = πœ‚π‘šπ‘Žπ‘₯ , 0 ≀ πœ‰ ≀ πœ‰π‘šπ‘Žπ‘₯ );
At the free stream the vorticity should be equal to zero (πœ” β†’ 0). Instead of using a Drichlet
type boundary conditions for vorticity we decided to use a Neumann type boundary condition.
At sufficiently far away from the cylinder since πœ” β†’ 0, the derivative of the vorticity should
πœ•πœ”
also be equal to zero ( β†’ 0). As the same done for the streamfunction variable, at the free
πœ•πœ‚
stream boundary we use both the vorticity equation and the Neumann boundary condition for
vorticity in order to calculate the vorticity value. At the free stream boundary, the following
equations are valid
πœ•2 πœ”
πœ•πœ‰ 2
πœ•2 πœ”
πœ•πœ”
+ πœ•πœ‚2 βˆ’ 𝑅𝑒 πœ‰ πœ•πœ‰ = 0
where
πœ•πœ”
πœ•πœ‚
β†’0
and
πœ•πœ“
πœ•πœ‚
β†’ 2πœ‰
(19)
The discretization of the above substituted vorticity equation requires a ghost point outside
the computational domain. The value of this ghost point is calculated using the boundary
πœ•πœ”
condition for the vorticity, i.e. πœ•πœ‚ β†’ 0. We note that, with this approach, at the free stream
boundary we both satisfy the boundary conditions and also the governing vorticity equation.
Right Boundary (πœ‰ = πœ‰π‘šπ‘Žπ‘₯ , 0 ≀ πœ‚ ≀ πœ‚π‘šπ‘Žπ‘₯ );
Similar to the explained approach earlier, we used a non-reflecting boundary condition at
πœ•2
the outflow boundary and thus we solve the vorticity equation without the elliptic terms (πœ•πœ‰2 )
such as
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πœ•2 πœ”
πœ•πœ‚ 2
βˆ’
𝑅𝑒 πœ•πœ“ πœ•πœ”
2
( πœ•πœ‚
πœ•πœ‰
βˆ’
πœ•πœ“ πœ•πœ”
πœ•πœ‰ πœ•πœ‚
)=0
(20)
We note that the discretization of this equation requires a ghost point outside the
computational domain. The value of this ghost point is calculated using the fact that on this
boundary
πœ•2 πœ“
πœ•πœ‰ 2
= 0 and
πœ•2 πœ”
πœ•πœ‰ 2
= 0 such that πœ“ and πœ” change linearly.
As it was done for streamfunction variable, in order to have smooth transition from the
πœ•2 πœ”
πœ•2 πœ”
vorticity equation ( πœ•πœ‰2 + πœ•πœ‚2 βˆ’
𝑅𝑒 πœ•πœ“ πœ•πœ”
2
( πœ•πœ‚
πœ•πœ‰
βˆ’
πœ•πœ“ πœ•πœ”
πœ•πœ‰ πœ•πœ‚
πœ•2 πœ”
) = 0) to the vorticity equation without the
elliptic terms used at the outflow boundary ( πœ•πœ‚2 βˆ’
𝑅𝑒 πœ•πœ“ πœ•πœ”
2
( πœ•πœ‚
πœ•πœ‰
βˆ’
πœ•πœ“ πœ•πœ”
πœ•πœ‰ πœ•πœ‚
) = 0) we use a buffer
zone and in this buffer zone we solve
πœ•2 πœ”
πœ•2 πœ”
𝑠 πœ•πœ‰2 + πœ•πœ‚2 βˆ’
𝑅𝑒 πœ•πœ“ πœ•πœ”
2
( πœ•πœ‚
πœ•πœ‰
βˆ’
πœ•πœ“ πœ•πœ”
πœ•πœ‰ πœ•πœ‚
)=0
(21)
where 𝑠 is defined the same as explained earlier.
We think that the outflow boundary conditions used in this study provides much smooth and
accurate solutions compared to the Oseen approximation at the outflow boundary as it was used
in Fornberg [5].
3. RESULTS AND DISCUSSIONS
Using the boundary conditions described above, we solved the governing equations using a
very efficient numerical method. The numerical method used is described briefly in Erturk,
Corke and Gökçöl [10] and Erturk, Haddad and Corke [11]. With this numerical method the
governing equations (6) and (7) are solved up to very low residuals. We assume that the
convergence is achieved when the streamfunction πœ“ and the vorticity πœ” values satisfies the
governing equations (6) and (7) with a maximum residual of 10βˆ’10 at every grid point inside
the computational domain.
In order to capture the physical phenomenon more accurately, more grid points were located
near the wall in πœ‚-direction, and near the cylinder in πœ‰-direction. Also, in order to eliminate the
effect of the far field and outflow boundary condition on the interior solution, the far field and
the outflow boundary should be chosen sufficiently away from the cylinder. This was done
using Robert's stretching transformation of the original uniform grid (Anderson, Tannehill and
Pletcher [16]). The formula of the transformation is
𝑦=β„Ž
Μ…
(𝛽+1)βˆ’(π›½βˆ’1)[(𝛽+1)⁄(π›½βˆ’1)]1βˆ’π‘¦
Μ… +1
[(𝛽+1)⁄(π›½βˆ’1)]1βˆ’π‘¦
(22)
where 𝑦̅ represents the original uniformly spaced grid points, 𝑦 represents the stretched grid
points and 𝛽 is the stretching parameter. In order to place the far field boundary and the outflow
boundary sufficiently away from the cylinder we used πœ‰π‘šπ‘Žπ‘₯ =20 and πœ‚π‘šπ‘Žπ‘₯ =15 and also we have
used 1800 and 300 grid points in πœ‰- and πœ‚-directions respectively. Figure 2-a shows the grid
points used in this study. In this figure since we have used such large number of grid points, we
have plotted only one out of every 20 grid points in both πœ‰- and πœ‚-directions. We note that, as it
is seen in this figure, the outflow boundary is almost 600 diameters away from the cylinder. We
also note that in this figure the cylinder is located inside the small red rectangle shown in the
Figure 2-a around π‘₯=0 and 𝑦=0 location. The grid points and also the cylinder inside this red
rectangle is shown with magnification in Figure 2-b. In this Figure 2-b again for a clear view
one out of every 4 grid points is plotted in both πœ‰- and πœ‚-directions.
We believe that with the use of large number of grid points and with having the free stream
boundary and the outflow boundary sufficiently away from the cylinder and using a stretching
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Numerical Solutions of Steady Incompressible Flow Around a Circular Cylinder
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function that allows to concentrate more points close to the cylinder, and also the very low
residuals used for convergence criteria assure the accuracy of our numerical solutions.
a) computational mesh
(one out of every 20 grids is shown)
b) enlarged view near the cylinder
(one out of every 4 grids is shown)
Figure 2 Computational mesh used in the numerical solution
We numerically solve the steady incompressible viscous flow around a circular cylinder
starting from 𝑅𝑒=10 up to 𝑅𝑒=500. Figure 3 shows the formation and the growth of the wake
bubble as the Reynolds number increases up to 𝑅𝑒=100.
Figure 3 Growth of the wake bubble as the Reynolds number increases
In Figure 3, one can notice that at 𝑅𝑒=10 the wake bubble is very small and its height and
length grows bigger as the Reynolds number increases. Since Reynolds number range
considered in this study is very big, we have plotted the streamfunction (πœ“) and vorticity (πœ”)
contours in Figure 4 with βˆ†π‘…π‘’=50 increments up to 𝑅𝑒=500.
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a) streamfunction contours
b) vorticity contours
Figure 4 Streamfunction (πœ“) and vorticity (πœ”) contours as Reynolds number increases
In Figure 4 one can easily notice that after 𝑅𝑒=300 the wake bubble behave differently and
it starts to grow bigger in height faster. Figure 5 shows the schematic view of the wake bubble.
Figure 5 Schematic view of the wake bubble
(𝐹: front stagnation point, 𝑆: separation point, 𝑅: rear stagnation point, π‘Š: wake stagnation point)
As shown in the schematics we will analyze several parameters such as the length of the
wake bubble measured from the center of the cylinder (𝐿), the maximum height of the wake
bubble (𝐻 max), the separation angle measured from the front stagnation point (πœƒπ‘  ), the π‘₯- and 𝑦location of the vortex centers (π‘₯center , 𝑦center) and the streamfunction (πœ“center) and vorticity
(πœ”center) values at the vortex center. Table 1 tabulates all these values as a function of the
Reynolds number.
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Table 1 Solutions at different Reynolds numbers
Reynolds
π‘₯center
Number (from the center)
10
1.1956
20
1.6771
30
2.0655
40
2.4272
50
2.7807
60
3.1339
70
3.5931
80
4.0611
90
4.5401
100
5.1438
150
8.7585
200
13.9923
250
20.7981
300
26.8167
350
30.2251
400
33.4746
450
38.2300
500
42.7574
𝑦center
πœ“center
πœ”center
πœƒπ‘ 
𝐿
𝐻 max
0.2400
0.4370
0.5203
0.5919
0.6331
0.6841
0.7189
0.7598
0.8054
0.8302
0.9858
1.1060
1.2202
1.4227
1.7976
2.4378
3.5491
4.9593
-3.4341 × 10βˆ’4
-8.3209 × 10βˆ’3
-2.2279 × 10βˆ’2
-3.7126 × 10βˆ’2
-5.1189 × 10βˆ’2
-6.4214 × 10βˆ’2
-7.6210 × 10βˆ’2
-8.7411 × 10βˆ’2
-9.7946 × 10βˆ’2
-1.0803 × 10βˆ’1
-1.5470 × 10βˆ’1
-2.0158 × 10βˆ’1
-2.6237 × 10βˆ’1
-3.7410 × 10βˆ’1
-5.6899 × 10βˆ’1
-9.3001 × 10βˆ’1
-1.6384
-2.6442
-0.04271
-0.18848
-0.27752
-0.34596
-0.38477
-0.42071
-0.44045
-0.45822
-0.47409
-0.47513
-0.47286
-0.45285
-0.44952
-0.43156
-0.37248
-0.31852
-0.26638
-0.22572
2.635
2.382
2.276
2.206
2.152
2.109
2.072
2.041
2.013
1.989
1.900
1.841
1.798
1.765
1.737
1.714
1.692
1.672
1.4774
2.8099
4.1465
5.4792
6.8037
8.1168
9.4179
10.7090
11.9926
13.2718
19.7024
26.3917
33.5003
41.1125
49.4163
58.7430
69.0971
78.7269
0.4822
0.7502
0.9129
1.0270
1.1158
1.1911
1.2590
1.3221
1.3819
1.4406
1.7228
2.0100
2.3245
2.7595
3.5564
4.8906
7.1177
9.8780
Fornberg [5] have plotted the length and the maximum height of the wake bubble with
respect to the Reynolds number. Following Fornberg [5], we plotted the same figure using the
values given in Table 1. Figure 6 shows the variation of the length of the wake bubble measured
from the center of the cylinder (𝐿) and also the maximum height of the wake bubble (π»π‘šπ‘Žπ‘₯ ) as
a function of the Reynolds number. Figure 6 also shows the same results of Fornberg [5] for
comparison. We note that the values are scanned and digitized from Fornberg [5] and included
in Figure 6. In Figure 6 we can see that our results agree with that of Fornberg [5] up to 𝑅𝑒=300
very good, however above 𝑅𝑒=300 there is a slight difference. We believe that our results are
more accurate.
b) maximum height of the wake bubble
π‘šπ‘Žπ‘₯ )bubble
Figure 6 Length and the maximum height of the(𝐻
wake
a) length of the wake bubble (𝐿)
As seen in Figure 6-a, the length of the wake bubble increases almost linearly up to
𝑅𝑒=300. Above this Reynold number the length starts to increase with a little faster rate. Also,
in Figure 6-b, except 𝑅𝑒=10 where the wake bubble is very small and developing, while the rate
of the increase in the maximum height of the wake bubble is gradual before 𝑅𝑒=300, the
increase is fast at higher Reynolds numbers. In Figure 7 we plot the variation of the π‘₯- and 𝑦location of the vortex centers and also the streamfunction (πœ“) and vorticity (πœ”) values at the
vortex center as a function of the Reynolds number. As it can be seen in Figure 7-a, the π‘₯location of the vortex center change almost linearly in the Reynolds number regions of 0-100,
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Ercan Erturk and Orhan Gokcol
100-300 and >300. The colored lines in Figure 7 show the linear regression line fitted to the
values between the Reynolds number range of 0-100, 100-300 and β‰₯ 300. The change in the
behavior of the solution is best seen in the vorticity value at the vortex center. In Figure 7-d
there is a clear change in the behavior of the vorticity at the vortex center at 𝑅𝑒=100 and
𝑅𝑒=300, where in the range between 100 β‰₯ 𝑅𝑒 β‰₯ 300 and also 𝑅𝑒 β‰₯ 300 the vorticity change
almost linearly with different rates.
a) π‘₯-location
b) 𝑦-location
c) streamfunction value (πœ“)
d) vorticity value (πœ”)
Figure 7 Values at the vortex center
4. CONCLUSIONS
In this study the steady incompressible viscous flow around a circular cylinder is solved
numerically at high Reynolds numbers. Using a very efficient numerical method with large
number of grid points, highly accurate numerical solutions are obtained with very low residuals.
The presented results agree well with the results found in the literature. Our results show that
the solution of the steady incompressible flow around a circular cylinder change behavior at
𝑅𝑒=100 and 𝑅𝑒=300 as the Reynolds number increases.
REFERENCES
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[2]
Schlichting H, Gersten K. β€œBoundary Layer Theory (8th revised and enlarged edn)”.
Springer: Berlin, 2000
Smith FT. β€œA structure for laminar flow past a bluff body at high Reynolds number”.
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http://www.iaeme.com/IJMET/index.asp
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[email protected]
Numerical Solutions of Steady Incompressible Flow Around a Circular Cylinder
up to Reynolds Number 500
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