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Dou, H. S., Ben，A.Q., Simulation and Instability Investigation of the Flow around a Cylinder between Two Parallel Walls, Journal of
Thermal Science，Vol.24, No.2, March, 2015，140-148.
*Corresponding author : huashudou@yahoo.com
Hua-Shu Dou: Professor. - 1 -
Simulation and Instability Investigation of the Flow around a Cylinder between Two
Parallel Walls
Hua-Shu Dou*, An-Qing Ben*
Faculty of Mechanical Engineering and Automation, Zhejiang Sci-Tech University,, Hangzhou, Zhejiang 310018, China
The two-dimensional flows around a cylinder between two parallel walls at Re=40 and Re=100
are simulated with computational fluid dynamics (CFD). The governing equations are
Navier-Stokes equations. They are discretized with finite volume method (FVM) and the solution
is iterated with PISO Algorithm. Then, the calculating results are compared with the numerical
results in literature, and good agreements are obtained. After that, the mechanism of the formation
of Karman vortex street is investigated and the instability of the entire flow field is analyzed with
the energy gradient theory. It is found that the two eddies attached at the rear of the cylinder have
no effect on the flow instability for steady flow, i.e., they don‟t contribute to the formation of
Karman vortex street. The formation of Karman vortex street originates from the combinations of
the interaction of two shear layers at two lateral sides of the cylinder and the absolute instability in
the cylinder wake. For the flow with Karman vortex street, the initial instability occurs at the
region inner a vortex downstream of the wake and the center of a vortex firstly loses its stability
inner a vortex. For pressure driven flow, it is confirmed that the inflection point on the
time-averaged velocity profile leads to the instability. It is concluded that the energy gradient
theory is potentially applicable to study the flow stability and to reveal the mechanism of turbulent
transition.
Keywords：numerical simulation, cylinder, energy gradient theory, stability, inflection point
Introduction
The flow around a cylinder and the wake behind the
cylinder are classical issues of fluid mechanics [1]. This
phenomenon is common in nature, such as the water flow
past the bridge pier, the wind past a building and the air
flow past the airfoil etc. It‟s also closely relevant to a large
number of other practical applications, such as submarines,
off shore structures, and pipelines etc. [2]. An important
characteristic of this type of flow is the periodical vortex
shedding in the cylinder wake at a sufficient high Reynolds
number, and the wake can induce longitudinal and
transverse unsteady loads acting on the object and excite
vibrating response to the structure leading to seriously
damage. Thus, the study on the mechanisms of flow
separation and the shedding of vortexes is of great
significance.
It has taken a long time in studies of the flow around a
bluff body. Von Karmam firstly studied and analyzed the
phenomenon, and he found the relationship between the
structure of vortex and the drag acting on the cylinders in
1912. Later, a large number of researchers devoted to
revealing the physic feature of the flow around the cylinders.
Roshko [3] is the first one who found the existence of
transition regime of the flow around a cylinder with
experimental method, and he determined the existence of
three different stages, i.e., linear flow, transition flow and
disordered turbulence, in the cylinder wake between the low
Reynolds and Middle Reynolds. Taneda [4-5] investigated
- 2 -
the characteristic of eddies behind the cylinder by
experiments. It was found that the Fopple eddies behind the
cylinder occur when the Reynolds is 5, and the two eddies
become longer and the frequency of vortex shedding
accelerates as the Reynolds number increases. It was also
found that Fopple eddies steadily attach on the rear of the
cylinders when the Reynolds is smaller than 45, then the
Karman vortex street is formed after the vortex shedding
alternatively and moving downstream. He observed the
formation of secondary vortex street after the vortex moving
a long distance. Mathis [6] confirmed the existence of
several modes proposed by Roshko [3], with experiment
and in particular, he observed and interpreted a
three-dimensional motion that occurs when two modes
co-exist. Williamson [7] found with experiment that the
three dimensional transition occurs, at Reynolds number
180-260. It is shown that there exist two modes related to
three dimensional transitions near the cylinder wall.
Zdravkovich [8] partitioned the flow into different
modes according to the Reynolds, and they are as follows:
The flow begins to separate at lateral sides of the cylinder
when 5<Re<40 and a pair of steady eddies occurs in the
wake. The shedding vortex street forms at laminar state in
the wake of the cylinder as 40<Re<200. The inner of the
vortex begins to transit to turbulence when 200≤Re<300.
The wake becomes fully turbulence as 300<Re<3×105, but
the boundary layer remains as laminar flow.
The physical mechanism of the Karman vortex street is
still not fully understood, and there exist four different
descriptions about the formation of the vortex street:
Gerrard [9] described the mechanism of vortex shedding
and the formation of vortex street, and he found that the
determined factor leading to vortex shedding is the
interaction of two separating shear layer at the lateral sides
of the cylinder, i.e., interacting mode of shear layers.
Coutanceau [10] meticulously investigated the formation of
vortex with the secondary vortex oscillation mode by
experiment, i.e. secondary vortex oscillation mode. Perry
[11] found that once the vortex-shedding process begins, a
so-called „closed‟ cavity becomes open, and instantaneous
„alleyways‟ of fluid are formed which penetrate the cavity,
i.e., open mode of the wake eddy. Trianiafyliou [12]
Monkewitz and Nguyen [13] and Ortel [14] believed that
the origin of vortex street is from the absolutely instability
in the near wake, i.e., absolute instability mode.
The understandings on the mechanism of the formation
of vortex street and the stability of flow field are deepened
and promoted undoubtedly with all these studies. However,
there still exist many problems to be resolved. For example,
the reason why the center of a vortex firstly loses its
stability and transits to turbulence in the cylinder wake is
not known. The mechanism of the formation of vortex street
is still controversial as discussed above. For example, Shi
[15] pointed out that the instability in the cylinder wake
originates from two inflection points at rear of the cylinder,
but it doesn‟t always right for parallel shear driven flow, e.g.
there always exists inflection point for plane
Poiseuille-Couette flow, but the flow is stable when
Re<2000.
With the aim to clarify these problems, the parameter
distributions in the entire flow field for Re=40 and Re=100
are simulated with CFD in this paper. Then, the calculating
results are compared with numerical results of Zovatto [16].
At last, the stability of the flow field is analyzed and the
mechanism of the formation of Karman vortex street is
investigated with the energy gradient theory.
Briefly Introduction of the Energy Gradient
Theory
Fig.1 Movement of a particle around its original equilibrium
position in a cycle of disturbance
Dou et al. [17-20] proposed a new theory—the energy
gradient theory, which is based on Newtonian mechanics
and compatible with Navier-Stokes equations, to study the
turbulent transition and the flow stability. The theory has
been used to determine the flow stability and turbulent
transition and good agreement with experiments has been
obtained. The theory describes that: for a giving base
parallel flow, the fluid particle moves ahead with oscillation
when it is subjected to disturbance (see Fig.1). The fluid
particle gains energy
E
leading to amplification of a
disturbance, and it also loses energy
H
in streamwise
direction which tends to absorb this disturbance and to keep
the original laminar flow. The transition to turbulence
depends on the relative magnitude of the two roles of energy
gradient amplification and viscous friction damping under
given disturbance. When the ratio reaches a critical value,
the flow instability may be exited. So the determining
criterion of instability can be written as follows:
u
A
Ku
ws
HA
n
E
H
E
Fd
d
2
22
- 3 -
Const
u
v
Km '
2
2
(1)
sH nE
K
(2)
Here, F is a function of coordinates which expresses the
ratio of the energy gained in a half-period by the particle
and the energy loss due to viscosity in the half-period; K is a
dimensionless field variable and expresses the ratio of
transversal energy gradient and the rate of the energy loss
along the streamline;
2
2
1UpE
is the total
mechanical energy per unit volumetric fluid; s is along the
streamwise direction, and n is along the transversal direction.
H is the energy loss per unit volumetric fluid, u is the
steamwise velocity of main flow;
A
is the amplitude of
the disturbance distance,
d
is the frequency of
disturbance, and
dm Av
'
is the amplitude of the
disturbance of velocity.
Two criterions are proposed for pressure driven flow
and shear driven flow based on the theory, and especially
the criterion to determine the flow stability of pressure
driven flow is described as: the necessary and sufficient
condition for turbulent transition is the existence of an
inflection point on the velocity profile in the averaged flow
[19].
Now, the theory has been successfully applied to
Taylor-Couette flow, plane Couette flow, plane Poiseuille
flow and pipe Poiseuille flow, and it is found that the results
show good agreement with experiments. These results
demonstrate that the critical value of Kmax at subcritical
turbulent transition for wall bounded parallel flows
including both pressure driven and shear driven flows is Kc
=370-389 [17]. It means that the flow transition won‟t occur
when the dimensionless parameter Kmax is less than Kc in
the flow field, otherwise it depends on the disturbance.
Position with Kmax in a flow field firstly loses its stability,
and position with large value of K will lose its stability
earlier than that with small value of K.
Physical Model and the Numerical Method
Governing Equations and Numerical Method
In this paper, the flow is two-dimensional and keeps as
laminar and the fluid used is water. Thus, the governing
equations for incompressible fluid are as follows:
0
v
(3)
vpvv
t
v2
1
(4)
Here, ρ is the density,
v
is the velocity director. The
governing equations are discretized with finite volume
method (FVM). The coupling of pressure and velocity is
done using PISO algorithm. The Reynolds number based on
the cylinder diameter is defined as Re=
UD
, and
U
is
the average velocity at the inlet. D is the diameter of the
cylinder,
is the kinematic viscosity.
Geometric Model and the Meshing
Fig.2 Computational domain
The computational domain is shown in Fig.2. The flow
field is modeled in two dimensional with the axes of the
cylinder perpendicular to the direction of flow. The diameter
of the cylinder is D=0.02m. The computational domain is
35D×5D. The upstream and downstream lengths are 10D
and 25D from the center of the cylinder, respectively. The
domain is partitioned into nine blocks, and the structured
mesh is used for each block (see Fig.3).
Fig.3 The grid around the cylinder
Characteristic Parameter of the Flow
The lift coefficient and drag coefficient are important
characteristic parameters describing the fluid acting on the
cylinder. And the Strouhal number is an important
characteristic parameter describing the unsteady feature of
vortex shedding. They are defined as follows:
AU
F
Cl
l2
2
1
(5)
- 4 -
AU
F
Cd
d2
2
1
(6)
U
fd
St
(7)
Here, Fl and Fd represents the drag and the lift force
respectively, A is the area projected in the flow direction,
and the magnitude of A is determined by the diameter of the
cylinder in two-dimensional flow,
U
is the average velocity,
and
f
is the frequency of vortex shedding.
Boundary Conditions
As shown in Fig.2, the velocity inlet boundary
condition is applied at the upstream of the cylinder. At the
downstream, a pressure outlet boundary condition is defined.
No-slip boundary condition is applied on the walls and the
cylinder, i.e., u=0, v=0.
Grid Independence Test
In this paper, the condition of Re=100 is taken as an
example to examine the grid independence. Three meshes
are used and they are marked as M1, M2 and M3
respectively. The minimum size of the three meshes is
ΔX1min=3.924×10-4m， ΔX2min=1.963×10-4m and ΔX3min=
2.797×10-5m respectively. The average drag coefficient
D
C
is selected as a reference to validate the calculating
results. The simulation results of present study and the data
in literature are shown in Fig.4. It can be found that good
agreement is obtained with the experimental results. Finally,
M2 is selected as the calculating grid.
Fig.4 Grid independence test
Simulating Results and Analysis
Comparison of the Calculating Results with those in
Literature
The Strouhal number and the drag coefficient of
present study and those in reference are shown in Table.1. It
can be found that good agreement has been achieved with
reference. As such, the calculating method used in this paper
is reliable.
Table.1 The results of present study and those in reference
Strouhal Number
Drag Coefficient
Present
0.2814
2.68
Reference[20]
0.2739
2.77
Relative Error
2.74%
3.36%
Calculation of the Energy Gradient Function K
According to the energy gradient theory [18], the
equation of the dimensionless parameter K can be written
as:
sH nE
K
(8)
Here the total mechanical energy E can be written as
2
2
1UPE
, and the total velocity can be written as
22 vuU
. Here
u
is the velocity along the X
direction,
v
is the velocity along the Y direction, P is the
static pressure of flow field,
n
is the normal direction of
the streamline, and s is the streamwise direction along the
streamline.
For pressure driven flows, Eq.(8) can be written as
sE nE
K
The energy gradient in the normal direction of a
streamline can be written as [20]:
n
UP
n
E
2
2
1
||
)(
||
)( 2
nd
nd
U
nd
nd
U
(9)
The energy loss in the streamwise direction along a
streamline for pressure driven flow can be written as [18]
[20]:
s
UP
s
E
2
2
1
||
)(
||
)( 2
sd
sd
U
sd
sd
U
(10)
Thus, the dimensionless parameter K of the energy
- 5 -
gradient function can be written as:
s
En
E
K
2
2
2
||
)(
||
)(
||
)(
||
)(
sd
sd
U
sd
sd
U
nd
nd
U
nd
nd
U
(11)
Fig.5 Geometric relationship of physical qualities
As is shown in Fig.5, a streamline passes by point O, i.e.,
the origin of X axis and Y axis. From Fig.5, it can be
obtained that:
U
u
cos
,
U
v
sin
(12)
||
nd
nd
= (
sin
,
cos
) (13)
||
sd
sd
= (
cos
,
sin
) (14)
dy
du
dx
dv
z
,
U
= (
,
z
v
z
u
) (15)
Then the equation (11) can be written as:
s
En
E
K
cos)(sin)(
sin)(cos)(
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
yu
xu
yv
xvyu
xu
yv
xv
cos)(sin)(
cossin
2
2
2
2
2
2
2
2
yu
xu
yv
xvuv zz
(16)
Calculating Results and Analysis
Flow at Re=40
(a) Streamline
(b) Vorticity contour
(c) K contour
Fig.6 Distribution of Streamline, Vorticity and K at Re=40
As shown in Fig.6 (a) and (b), it can be found that there
is no Karman vortex street in the cylinder wake and there
are two eddies attached at the rear of the cylinder. The flow
is laminar for plane-Poiseuille flow, when the Reynolds
number is 40. The K contour of this case is shown in Fig.6
(c), and it can be found that the distribution of K ahead of
the cylinder (inlet region) is in accordance with Dou‟s [18]
equation
h
y
KRe75.0
)1( 2
2
h
y
(here h is half of
the height between two walls). The detailed numerical
distribution is shown in Fig.7. It can be seen from Fig.7 (b)
that the magnitude of K is about 57, which is in accordance
with theoretical value. The Kmax of this profile is located at
58.0
h
y
, which is in accordance with experiments [21].
(a) Velocity (b) K
- 6 -
Fig.7 The distribution of velocity and K at section 1
Comparing the velocity contour with K contour, it can be
found that the K at the position of the two eddies is very low.
It represents that the two eddies at the rear of the cylinder
have no effect on flow stability according to the energy
gradient theory. To clarify this problem clearly, the detailed
distributions of parameters at section 2 are shown in Fig.8.
In Fig.8 (c), it can be found that the value of K at lateral
sides of two eddies is large and the value of K at the
position with two eddies is low, which numerically validates
the conclusion that the two eddies at the rear of the cylinder
have no effect on the flow stability. It can also be found that
the Kmax at this section is located at the position where the
second derivative of velocity is zero, i.e., the inflection
point.
(a)Velocity (b) Vorticity (c) K
Fig.8 Distribution of physical quantities at section 2
It can also be found that K at lateral sides of the cylinder
is large. To study the mechanism, the detailed distributions
of parameters at section 3 are shown in Fig.9. As mentioned
above, Kmax at this section is also located at the position
with inflection point in velocity profile.
(a) Velocity (b) Vorticity (c) K
Fig.9 Distribution of physical quantities at section 3
According to the energy gradient theory, the position with
Kmax in the entire flow field will firstly lose its stability. At
Re=40 condition, it can be found that the Kmax of the entire
flow field occurs at section 4 (see Fig.6). To investigate the
reason why the K here is largest, the detailed distributions
of parameters at section 4 are shown in Fig.10. As shown in
Fig.10, it can be found that Kmax occurs at the position with
inflection point of velocity and maxima of vorticity.
According to the definition of vorticity in two-dimensional
flow,
y
u
x
v
z
. To get the maxima of vorticty, let
0
yz
, i.e.,
0
2
22
y
u
yx v
yz
. For the flow around a
cylinder between two parallel walls,
yx v
2
is very low.
Thus, the equation
0
yz
is proximately equal to
0
2
2
y
u
. Therefore ， the maxima of the vorticity
corresponds to the inflection point of velocity and the
location of Kmax at each section.
The formation of Karman vortex street has been
investigated by many researchers, but the mechanism is still
not clear and controversial conclusions exist in literature. As
mentioned above, it is found that the Kmax is located at
section 4. According to the concept of absolute instability
[12] [13] [14], the instability here will spread to the
upstream and affect the stability of the two shear layers
which are located at lateral sides of two eddies and are with
large K. Then, the interaction of two shear layers leads to
vortex shedding and formation of the Karman vortex street.
This is consistent with the conclusion obtained by Gerrard
[9] that the determined factor leading to vortex shedding is
the interaction of two separating shear layers at the lateral
sides of the cylinder. In summary, vortex shedding is
originated from the combination of the interaction of two
shear layers and absolute instability in the cylinder wake,
which leads to the formation of Karman vortex street.
(a) Velocity (b) Vorticity (c) K
Fig.10 Distribution of physical quantities at section 4
Flow at Re=100
(a) Vorticity contour
(b) K contour
Fig.11 Distributions of vorticity and K at Re=100
- 7 -
The flow at Re=100 is calculated and the vortcity contour
and K contour are shown in Fig.11. As described above, the
flow ahead of the cylinder and after the velocity inlet is
laminar, and the distribution of K is in accordance with
Dou‟s [18] equation mentioned above. The detailed
distribution of K is shown in Fig.12. The magnitude of K is
about 144, which is in accordance with theoretical value.
The Kmax of this profile is located at
58.0
h
y
, which is
also in accordance with experiments [21].
(a) Velocity (b) K
Fig. 12 Distribution K velocity and K at section 1
The stability of the cylinder wake is investigated with the
energy gradient theory below. Through further investigation,
it is found that Kmax of the entire flow field is located at each
vortex in the cylinder wake. To study the mechanism, the
detailed distributions of parameters at sections 1 and 2 are
shown in Figs.13 and 14, respectively.
Comparing the velocity, vorticity and K contours in
Fig.13 and Fig.14, it can also be found that the location of
Kmax at each cross section corresponds to the inflection point
of the velocity profile and the maxima of vorticity as
mentioned above. It can also be found that, Kmax is located
in a vortex center for each vortex, which represents that the
flow will firstly lose its stability in a vortex center in the
cylinder wake. In fact, the finding is in agreement with the
conclusion in Zdravkovich [8]: the inner of the vortex
begins to transit to turbulence when 200≤Re<300. In this
part, the results validate and deepen the conclusion: the
inner of the vortex will firstly lose its stability in entire flow
field, and the center of a vortex will firstly lose its stability
at each vortex. The reason is that it is the inflection point on
the velocity profile that leads to the instability.
(a) Velocity (b) Voticity (c) K
Fig.13 Distribution of physic quantities at section 2
(a) Velocity (b) Voticity (c) K
Fig.14 Distribution of physic quantities at section 3
Conclusions
In this paper, the flows around a cylinder between two
parallel walls at Re=40 and Re=100 are simulated with
computational fluid mechanics (CFD). The simulating
results are compared with the numerical results in literature
and good agreement is obtained. Then the instabilities of
flow field are investigated with the energy gradient theory.
The formation of Karman vortex street is studied with
Re=40. The initial instability of entire flow field with vortex
street in the cylinder wake is investigated with Re=100.
Several conclusions are obtained as follows:
1. The two eddies at the rear of the cylinder have no
effect on the flow instability for steady flow, i.e., they don‟t
contribute to the formation of Karman vortex street.
2. The formation of Karman vortex street originates from
the combinations of the interaction of two shear layers at
two lateral sides of the cylinder and the absolute instability
in the cylinder wake.
3. For the flow with Karman vortex street, the initial
instability occurs at the region inner a vortex and the center
of a vortex firstly loses its stability inner a vortex.
4. For pressure driven flow, it is confirmed that the
inflection point on the time-averaged velocity profile leads
to the instability.
5. The energy gradient theory is potentially applicable to
study the flow stability and it can profoundly reveal the
mechanism of turbulent transition.
Acknowledgement
This investigation is supported by the Special Major
Project of Science and Technology of Zhejiang province
(No. 2013C 01139), and the Science Foundation of Zhejiang
Sci-Tech University (No. 11130032661215).
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