numerical study of vortex shedding in viscoelastic flow past an … · 2015-08-20 · bluff bodies....
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© 2015 The Korean Society of Rheology and Springer 213
Korea-Australia Rheology Journal, 27(3), 213-225 (August 2015)DOI: 10.1007/s13367-015-0022-z
www.springer.com/13367
pISSN 1226-119X eISSN 2093-7660
Numerical study of vortex shedding in viscoelastic flow past an
unconfined square cylinder
Mahmood Norouzi1,*, Seyed Rasoul Varedi
2 and Mahdi Zamani
3
1,2Department of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran3Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
(Received October 16, 2014; final revision received May 3, 2015; accepted May 9, 2015)
In this paper, the periodic viscoelastic shedding flow of Giesekus fluid past an unconfined square cylinderis investigated numerically for the first time. The global quantities such as lift coefficient, Strouhal numberand the detailed kinetic and kinematic variables like normal stress differences and streamlines have beenobtained in order to investigate the flow pattern of viscoelastic flow. The effects of Reynolds number andpolymer concentrations have been clarified in the periodic viscoelastic flow regime. Our particular interestis the effect of mobility parameter on the stability of two dimensional viscoelastic flows past an unconfinedsquare cylinder. To fulfill this aim, the mobility parameter has been increased from 0 to 0.5 for differentpolymer concentrations. Results reveal that mobility factor noticeably affects the amplitude of lift coeffi-cient and shedding frequency more strongly at higher polymer concentrations.
Keywords: square cylinder, vortex shedding, viscoelastic flow, mobility factor, polymer concentration
1. Introduction
Vortex shedding from bluff cylinders has received an
increasing amount of attention since it is associated with
many cases of flow-induced structural and acoustic vibra-
tions. Vortex shedding from circular cylinders has been
extensively studied. The behavior of such flows, when
Reynolds number (Re) is increased, presents several pro-
gressive categorized behaviors.
For example in the case of square cylinder in uniform
cross flows, at very low Re, the flow is laminar, steady
and does not separate from the cylinder. By increasing Re,
the flow is separated from the trailing edge but remains
steady and laminar up to Re of about 50 (Sohankar et al.,
1999). Beyond this Re, the flow develops into a time
dependent periodically oscillating wake. With a further
increase in Re, localized regions of high vorticity are
shedding alternatively from either side of the cylinder and
are convected downstream. In this regime, the wake zone
consists of pairs of vortices which shed alternately from
the upper and lower parts of the rear surface, and stag-
gered rows of vortices behind a blunt body (Versteeg et
al., 2007) are generated.
The theoretical investigation of vortex pattern which is
observed in the wake of the cylinder was originated by
Von Kármán who considered double rows of vortices in a
two–dimensional (2D) flow. Note that the flow is still
laminar and 2D. By increasing the Re value, the flow
undergoes a further bifurcation at around Re = 150-200
and becomes 3D but remains time periodic (Robichaux et
al., 1999; Sohankar et al., 1999; Saha et al., 2003; Luo et
al., 2007). By increasing the Re further, the flow becomes
chaotic and eventually transition to turbulence occurs. A
similar sequence of bifurcations also occurs for other cross-
sections like square, elliptical, and so on (Jackson, 1987;
Williamson, 1996; Balachandar et al., 2002; Zhang et al.,
2006; Franke et al., 1990).
In this regard, most of research on flow past a cylindri-
cal object has been carried out for a circular cylinder
rather than a cylinder with a square cross section. The
main difference between the two is that separation points
are fixed at some edges of a square cylinder, while they
are time-dependent on the surface of a circular cylinder.
Let us consider some investigations relevant to the current
issue. Franke et al. (1990) employed the finite volume
method to analyze numerically the problem of laminar
vortex shedding from a square cylinder for Re ≤ 300.
Time dependence of a number of flow parameters such as
drag, lift and Strouhal number (St) were studied in their
research. To clarify the extent of end effects, Tamura et al.
(1990) simulated 2D and 3D flows past a square cylinder
for various length-to-diameter ratios at high Re. Saha et
al. (1999) have also numerically analyzed the force coef-
ficients and the frequency of vortex shedding in the wake
of a square cylinder. The Re was in the range of 250-1500.
The spatial evaluation of vortices and transition of three-
dimensionality in the wake of a square cylinder for the
range of Re 150-500 has been subsequently presented by
them in another research (Saha et al., 2003). The numer-
ical analysis of the flow structure and heat transfer char-
acteristics for an isolated square cylinder also was investi-
gated by Sharma et al. (2004). They presented their work
for both steady and unsteady periodic laminar flows in the*Corresponding author; E-mail: [email protected]
Mahmood Norouzi, Seyed Rasoul Varedi and Mahdi Zamani
214 Korea-Australia Rheology J., 27(3), 2015
2D regime for the range of Re of 1-160 and a Prandtl
number of 0.7. For Re ≤ 40, the flow showed a steady
regime as expected, presenting a transition to unsteadiness
for the range of 40 ≤ Re ≤ 50 and a stationary periodic
unsteady regime for Re ≥ 50. Considering present works,
it could be attained that studies upon characteristics of
Newtonian fluids seem to be recognized of particular
importance till now.
Undeniably, most fluids with industrial applications
such as high molecular weight polymers and their solu-
tions, suspensions and thin liquid mixtures foams and
froths present complex rheological behaviors unlike New-
tonian fluids with a quite predictable manner. In other
words, non-Newtonian fluids used in industry, indisput-
ably show unique characteristics like shear-dependent vis-
cosity, yield stress, viscoelasticity, normal stress differences
and so on which practically make them important. It is
readily acknowledged that shear-thinning is probably the
most common type of non-Newtonian fluid behavior
encountered in industrial applications. The effective vis-
cosity (i.e., shear stress divided by shear rate) of a shear-
thinning substance can decrease from a very high value at
low shear rates (relevant to rest conditions) to a vanish-
ingly small value at high shear rates such as that encoun-
tered in pipe or pump flows, mixing vessels, and bluff
body flows. Obviously, 2D flow over a cylinder (irrespec-
tive of its cross-section) gives rise to a flow field in which
the effective rate of deformation varies from point to point
in a complex fashion. Conversely, unlike in the case of a
Newtonian fluid whose viscosity is independent of the
shear rate, the effective viscosity of a shear-thinning fluid
can vary enormously in the vicinity of the bluff body
depending upon the local value of the deformation rate.
Needless to say, this in turn, is expected to have signif-
icant influence on the detailed structure of the velocity as
well as on the gross parameters of engineering signifi-
cance such as wake phenomena, etc. Therefore, the inter-
est in studying such model configurations is not only of
intrinsic theoretical relevance, but is also of overwhelming
pragmatic significance such as in tubes of various cross-
sections in tubular, pin-type and in other novel designs of
compact heat exchangers, in novel designs of mixing
impellers and also rake filters used for non-Newtonian
slurries.
To the best of our knowledge, only few studies have
been reported on the flow of non-Newtonian fluids past
bluff bodies. Also, no prior numerical study exists in the
literature related to the vortex-shedding characteristics of
a square cylinder in non-Newtonian flow except two
works done by Sahu et al. (2009; 2010). In these two
investigations, the power law model has been utilized to
clarify the shedding flow of generalized Newtonian fluid
(GNF) past a square cylinder. The first study deals with
the flow around an unconfined square cylinder in the
range of 60-160 for Re and 0.5-2 for n index (Sahu et al.,
2009).
The effect of parameters such as Re and power-law
index on flow structure, drag and lift coefficients and St
has specifically been studied. It is shown that similar to
the Newtonian fluids, shear-thickening and shear-thinning
fluids exhibit vortex shedding over the range of condi-
tions. They showed that transition values of Re denoting
the onset of leading edge separation in shear thinning flu-
ids is lower than the value for Newtonian fluids and the
drag coefficient is decreased by increasing Re in shear-
thickening fluids. Furthermore, in the present range of
conditions, the flow of shear-thickening fluids is truly
periodic in nature while in the case of shear-thinning flu-
ids, it becomes pseudo-periodic at high-Re and/or at small
values of power law index, i.e., in highly shear-thinning
fluids.
The second work Sahu et al. (2010) involves the study
of flow around the cylinder in a channel. The effect of
blockage ratio (B = 1/6, 1/4, 1/2) on the cross flow of
power-law fluids over a square cylinder confined in a pla-
nar channel has been studied for a range of power-law
index of 0.5 ≤ n ≤ 1.8 and Re of 60 ≤ Re ≤ 160 in the 2D
laminar flow regime. For n > 1, the flow was either truly
periodic or steady for all values of blockage ratios and Re
considered there. The presence of the walls at B = 1/2 led
to smaller recirculation zones over the top/bottom faces of
the cylinder than at B = 1/4 and 1/6. Irrespective of the
type of the fluid, enhancements in drag coefficient, St, the
root-mean-square values of drag and lift coefficients were
observed with an increasing in blockage ratio. It is shown
that in shear-thickening fluids, total drag coefficient decreases
with increasing Re for all three values of B while in shear-
thinning fluids at B = 1/4 and 1/6, the drag is increased
with increasing Re which is similar to the trend found for
the unconfined case.
According to the knowledge of authors and reports of
other researchers (such as, Sahu et al., 2009; Coelho and
Pinho, 2003a), there is no other study on viscoelastic flow
around the square cylinder and few experimental and
numerical works are only available about the viscoelastic
flow around the circular cylinders. Therefore, it is perhaps
useful to review briefly some experimental and numerical
literature for the flow over a circular cylinder. Usui et al.
(1980) investigated changing the frequency of PEO solu-
tion at concentrations of 100, 200, and 400 ppm for
100 ≤ Re ≤ 300. They found that increasing polymer con-
centration leads to reduction in the frequency of vortex
shedding. Furthermore, they developed an empirical cor-
relation between St and Weissenberg number (We). Coelho
and Pinho (2003a; 2003b) used two polymeric additive
solutions with high and low elastic properties (e.g., carboxy
methyl cellulose and tylose). The Re between 50 and 9000
covered the laminar vortex shedding regime, the transition
Numerical study of vortex shedding in viscoelastic flow past an unconfined square cylinder
Korea-Australia Rheology J., 27(3), 2015 215
regime and the shear-layer transition regime. The fluid
elasticity was found to reduce critical Re marking the
onset or the end of flow regimes.
In this regard, a few numerical investigations are avail-
able in the literature. As an example, Oliveira (2001) and
Sahin et al. (2004) utilized the modified FENE-CR rhe-
ological model to compute the shedding frequency of vis-
coelastic flow behind a cylinder at finite Re. In the study
of Oliveira (2001), attenuation of vortex shedding fre-
quency, reduction of the lift and drag coefficients and
increasing the recirculation region by elasticity are observed.
Also, the effects of elongation viscosities have been inves-
tigated by raising the extensibility factor of the viscoelas-
tic model. He found that this would enhance the length of
the recirculation region even further. These results are in
agreement with most previous experiments.
Sahin et al. (2004) investigated the effect of polymer
additives on linear stability of 2D viscous flow past a con-
fined cylinder. The results revealed that as the maximum
extensibility was greater, the larger value of the critical Re
marking the onset of the vortex shedding occurred. Also,
the effects of the elasticity on shedding frequency, drag
and lift coefficient in the vortex–shedding regime and
recirculation length have been studied. The results are also
found to be in good agreement with numerical work of
Oliveira (2001). Richter et al. (2010) studied the effects of
polymer extensibility on wake transitions of circular cyl-
inders. Two distinct Re (100 and 300) were used in their
works. The results showed that polymer extensibility has
a qualitative effect on the shedding frequency. Also, the
ability of viscoelasticity was investigated to stabilize the
flow to 3D instabilities.
Kim et al. (2009) numerically investigated the effect of
viscoelasticity on 2D laminar vortex dynamics in flows
past a single rotating cylinder at Re = 100. Their results
illustrated that the vortex shedding in the flow around a
rotating cylinder can be more effectively suppressed for
viscoelastic fluids than Newtonian fluids. Norouzi et al.
(2013) studied viscoelastic shedding flow around circular
cylinder at Re = 100 and We = 80. The numerical results
of inertial viscoelastic flow behind a circular cylinder
illustrate the significant effect of the fluid elasticity on the
flow structure.
In this research, 2D laminar viscoelastic flow around a
square cylinder is studied using a parallelized finite vol-
ume method, running on a cluster of workstations. The
parallelization of the program is performed by a domain
decomposition strategy. All of the algebraic equations are
solved sequentially using the semi implicit method for
pressure linked equations revised (SIMPLER) iteration
procedure with the Gauss–Seidel point solver (Courant et
al., 1952; Bird et al., 1995). Under–relaxation technique is
used to deal with non–linearity of the equations. The com-
putational domain size was selected so that the simula-
tions would represent the unbounded flow around a square
cylinder.
The schematic geometry of current study is shown in
Fig. 1. The physical problem investigated here is the
unsteady viscoelastic shedding flow of Giesekus fluid past
a long square cylinder of size B, placed in a uniform stream
having velocity U∞. According to our knowledge, there is
a serious dearth in literature of viscoelastic flow around
the square cylinder and the present study is the first inves-
tigation in this field. The main innovative aspects of the
current research are i) shedding flow of Giesekus fluid
around an unconfined square cylinder and comparison
with Newtonian flow, ii) effects of Re and We for certain
values of the parameters such as polymer concentration
and mobility factor of viscoelastic fluid, iii) polymer con-
centration (β ) on shedding frequency and lift amplitude of
vortex shedding, and iv) the effect of increasing mobility
over the range of 0 ≤ α ≤ 0.5 at various β from low to high
values.
The current paper is structured as follows. In section 2,
the governing equations for the unsteady flow of an
incompressible Giesekus fluid are presented. In section 3,
the numerical procedure and the algorithm used for the
solution of time–dependent equations are briefly described.
Also initial and boundary conditions are represented in
this section. Grid study and validation of the code are
investigated in section 4. Then, the results of this work are
given in section 5 and the main conclusions have been
represented in section 6.
2. Governing Equations
Consider the flow of an incompressible viscoelastic fluid
in the 2D domain. The dimensionless equations governing
the transient fluid motion are mathematical statements of
the conservation of momentum and mass
Re , (1)
. (2)
DU
Dt-------- = ∇– p + 1 β–( )ΔU + ∇ τ⋅
∇ U = 0⋅
Fig. 1. Schematic shape of the computational domain for the
flow past a square cylinder.
Mahmood Norouzi, Seyed Rasoul Varedi and Mahdi Zamani
216 Korea-Australia Rheology J., 27(3), 2015
The viscoelastic stress response of the fluid to deforma-
tions is described by the Giesekus constitutive equation
. (3)
In Eqs. (1)-(3), U is the velocity, p is the pressure, τ is
the polymeric stress, and is the rate of
deformation tensor. Furthermore, α is a mobility factor
and denotes the upper-convected derivative of τ defined
by
= . (4)
The dimensionless quantities are We, Re, and the ratio of
polymer to total viscosity β. The origin of the term involv-
ing α can be associated with anisotropic Brownian motion
and/or anisotropic hydrodynamic drag on the constitutive
polymer molecules (Malvandi et al., 2014). It should be
mentioned that for α = 0, the Giesekus model is reduced
to the Oldroyd-B model (Bird et al., 1995).
The computational domain size was selected so that the
simulations would represent the unconfined flow around a
square cylinder. The height of the computational domain,
upstream length and downstream length of the domain are
nominated H, Lu and Ld, respectively (see Fig. 1). These
values are necessary to obtain the results which are free
from the domain effects. Based on the previous studies
(Sharma et al., 2004; Sahu et al., 2009), the values of H,
Lu and Ld used in this work are 20B, 8.5B and 16.5B,
respectively.
3. Numerical Method with Boundary and InitialConditions
In general, all terms are discretized by means of central
differences, except for the convection terms which are
approximated by the linear-upwind differencing scheme
(LUDS) Xue et al. (1995). This is a generalization of the
well-known upwind differencing scheme (UDS), where
the value of a convected variable at a cell face location is
given by its value at the first upstream cell center. In the
LUDS scheme, the value of that convected variable at the
same cell face is given by a linear extrapolation based on
the values of variable at the two upstream cells. It is, in
general, second-order accurate, as compared with first-
order accuracy of UDS, and thus, its use reduces the prob-
lem of numerical diffusion (Phan-Thien, 2002).
To create an equation for the pressure, continuity equa-
tion is utilized using a semi–discretized form of Eq. (1). It
is then solved by SIMPLER iterative algorithms (Courant
et al., 1952; Bird et al., 1995) using under relaxation
method. The viscoelastic stress has been decomposed and
entered into an implicit component due to numerical sta-
bilization and momentum–stress coupling.
The convergence of solution is verified by calculating
the residuals of each equation. The absolute tolerance for
pressure was 1.0×10−7 and 1.0×10−6 for velocity and
stress. The iteration is controlled by monitoring the con-
vergence history so that all the magnitude of variables
reach values lower than a prescribed tolerance. The solu-
tion procedure can be divided into four stages:
• Calculating the pressure gradient and stress divergence
by substituting initial fields for velocity, pressure and
stress. Accordingly, the momentum equation is solved
implicitly for each velocity component and a new
velocity field U* is estimated.
• Estimating the pressure field by using the new velocity
field U*. Then the velocity field is corrected subse-
quently. The new velocity field, U** and the new pres-
sure p*, can be calculated by SIMPLER algorithms.
The corrected velocity field U**, satisfies the continu-
ity equation.
• Estimating the new stress field τ* by substituting the
corrected velocity field U** in the constitutive equation.
• Recursively iterating the above mentioned stages to
obtain the more accurate solution.
The flow is characterized by three dimensionless num-
bers, Re, We, and St:
(5)
where is the free upstream velocity, fs is the shedding
frequency, ρ is the density, and η0 is the summation of
polymer and solvent viscosities at zero shear rate.
Boundary conditions consist of a uniform velocity at
inlet on the left side with zero pressure gradient and zero
stress tensor components. At the domain outlet, pressure is
set to the atmospheric pressure. At this boundary, the
velocity gradient and stress tensor components are also
considered to be zero. For two far–field boundaries, referred
to upper and lower boundaries in Fig. 1, a zero flux slip
condition is used for all variables since the boundaries are
adequately far from the cylinder flow to be assumed par-
allel and unaltered by the internal dynamics. Along the
cylinder wall, a no–slip condition is imposed for the fluid
velocity.
Initial conditions are considered completely symmetric
so that the fluid assumed at rest and the shedding flow is
produced itself naturally. At the beginning of the simula-
tion, the generated eddies get longer due to no-slip bound-
ary conditions by time-marching. Note that the attached or
standing eddies, appeared behind the cylinder, are com-
pletely symmetric. Then pairs of vortices are shed alter-
nately from the upper and lower parts of the rear surface
so that the staggered row of vortices behind a cylinder is
generated. The passage of regular vortices causes velocity
measurements in the wake to have a dominant periodicity.
From this initial condition, transient simulation begins,
using the first order scheme in time to ease convergence.
τ +Weτ∇ αWe
β------------+ τ
2 = 2βD
D = ∇u ∇uT
+( )/2
τ∇
τ∇ ∂τ
∂t----- + U ∇τ⋅ ∇U( )T τ τ ∇U( )⋅–⋅–
We = λU∞/B, Re = ρU∞B/η0, St = fsB/U∞
U∞
Numerical study of vortex shedding in viscoelastic flow past an unconfined square cylinder
Korea-Australia Rheology J., 27(3), 2015 217
Once the drag coefficient on cylinder has reached a quasi–
periodic regime, the time discretization is switched to sec-
ond order Crank Nicholson scheme.
4. Grid Study and Validation
In the present study, we used a non-uniform grid struc-
ture similar to previous researches (Sahu et al., 2009;
2010). Fig. 2 shows the computational grid with 534×474
grid points and enlarged view of grid near the square cyl-
inder. The grid is divided into five separate zones in both
directions so that cell size around the square cylinder was
made fine to much better resolve the gradients near the
solid surfaces and wake zones of the cylinder in advance.
Zones far away from the cylinder are constructed with
uniform coarse cells. The size of coarse cells (∆) is 0.2B
for all meshes. The hyperbolic tangent function has been
used to stretch the cell sizes between fine and coarse
meshes.
To check for grid independency, we performed numer-
ical computations for four sets of grid points with 336×
276, 384×324, 534×474 and 1434×1374 mesh points in
x and in y-directions, respectively. Finally, a grid, which
represents a suitable precision and computational cost,
was selected. Characteristics of grids are given in Table 1.
The simulations for grid independency study are performed
for a Newtonian case (We = 0) at Re = 100 because no
experimental results about viscoelastic shedding flow
around the square cylinder are available for comparison.
In Table 2, the results of CFD simulation for Newtonian
flow has been presented. It comprises the values of St, the
mean of the absolute lifting force and drag force. The
effect of grid size on major parameters characterizing the
flow (i.e., CD, CL, and St) are shown in Table 2. The per-
centage changes of these parameters for grid M-1 and the
finest grid M-4 are 0.45, 9.53, and 0.55%, respectively.
The corresponding changes between the grid M-2 and M-
4 are 0.1, 8.55, and 0.037%. Also, the percentage changes
of these parameters for grids M-3 and M-4 are 0.008,
0.74, and 0.005%, respectively. Grid M-4 (1434×1374)
has thrice as many cells as M-3 (534×474) in both x- and
y-directions. The computation time with grid M-4 is nearly
seven times bigger than that with grid M-3. Therefore one
can conclude that the grid M-3 denotes a good compro-
mise between accuracy and the computational effort. For
the sake of independency of solution to the grid, grid M-
3 is used in all further computations.
The numerical method used here has been validated and
benchmarked by an experimental work done by Robi-
chaux et al. (1999) and numerical work done by Sahu et
al. (2009) for the flow of Newtonian fluids in the unsteady
flow regime. The present values of the key parameters
including drag coefficient CD and St, in the unsteady flow
regime are compared with those of Robichaux et al.
(1999) and Sahu et al. (2009) in Table 3. As expected, an
excellent match is seen to exist between the present and
previous published work.
We also used the results of Sahu et al. (2009) for val-
idation. They presented a numerical solution for flow of
power-law fluid around a square cylinder at θ = 0. In order
to prepare an identical condition for viscous response, we
should estimate the constants of Giesekus model so that
Table 1. Non- uniform grids used for grid independency study.
S. No.
No. of uniform control
volume on each face of
cylinder
Cell size
(δ)Grid size
1 17 0.06 336×276
2 25 0.04 384×324
3 50 0.02 534×474
4 200 0.005 1434×1374
Fig. 2. Non-uniform computational mesh with 534×474 grid
points; (inset) enlarged view of mesh near the square cylinder.
Table 2. Effect of mesh refinement on flow parameter.
Grid St*CL
**CD CD min CD max
M-1 0.147529 0.274365 1.523185 1.51575 1.53062
M-2 0.148405 0.271918 1.517935 1.51045 1.52542
M-3 0.148342 0.252355 1.51648 1.50957 1.52339
M-4 0.14835 0.2505 1.51635 1.50925 1.52345
*CL = 0.5(|CL max| + |CL min|), **CD = 0.5(CD max + CD min)
Mahmood Norouzi, Seyed Rasoul Varedi and Mahdi Zamani
218 Korea-Australia Rheology J., 27(3), 2015
the viscosity of this model in steady shear test is fitted to
the results of power-law fluid. The viscous response of
Giesekus model in steady shear test is in following frac-
tional form
(6)
where
, (7a)
, (7b)
. (7c)
Here, we consider the results of Sahu et al. (2009) for m
= 1 Pa·sn and n = 0.8 ( ). Using the least square
method, the constant of Giesekus model is calculated as,
ηs = 0.091 Pa·s, ηp = 0.909 Pa·s, λ = 2.5 s, ξ = 0.2503 s,
and α = 0.1.
Based on the above constants, the viscous response of
Giesekus model has a suitable agreement with results of
power-law model for power-law region of viscometric
test. Here, we calculated the generalized Re, given by
. (8)
The effective viscosity is calculated by substituting
in viscous response of Giesekus model (Eq. (6)).
In Table 4, the value of CL and CD for generalized Re are
presented for shear thinning power-law fluids at n = 0.8
reported by Sahu et al. (2009) for θ = 0o and Giesekus
fluid of present study. Careful inspection shows that sim-
ilar treatment for both power-law and Giesekus fluids. The
differences observed between the results presented here
are due to the structural differences between the models.
Giesekus model is a nonlinear viscoelastic model that pre-
dicts elastic force and normal stress differences. Elongational
viscosity and its nonlinear viscosity not only depend on
the second invariant of shear rate tensor but also on the
third invariant.
5. Results and Discussion
5.1 Comparison viscoelastic and Newtonian fluidsFig. 3 displays the representative instantaneous stream-
lines in the vicinity of the square cylinder for Newtonian
and viscoelastic flows at Re = 100 and We = 20. Polymer
concentration of the viscoelastic flow is con-
sidered as β = 0.05 and the mobility parameter is thought
0.1 as well. Streamlines presented in Figs. 3a-3f are the
vortex shedding phenomenon at six sequential moments
of time history in a way that the first moment will be
repeated after the sixth moment for the next cycle of vor-
tex shedding for viscoelastic and Newtonian flows. Vortex
forming develops on the top of rear face in both flows; it
is broken off the back of cylinder (Figs. 3a-3c) and con-
vects along the flow in two cases afterwards. The same
event occurs in the next half of the vortex-shedding cycle
at the bottom of the rear face (Figs. 3c-3f).
Alternate vortex forming in the top and bottom rear
faces of the square cylinder brings about periodic flows.
As can be seen in Fig. 3, the vortex forming and shedding
flow phenomena in viscoelastic fluids are qualitatively
the same as those in Newtonian fluids. Scanning more
precisely, it could be detected that vortex forming is
occurred more rapidly in Newtonian case in a way that
vortices are detached and convected into downstream so
that according to Fig. 3A-d, vortex growth is faster in the
bottom surface of the cylinder in comparison with visco-
elastic fluids. It is broken off the rare face of the square
cylinder.
Effect of viscoelasticity on the flow structure is shown
more clearly in Fig. 4, which depicts the variation of time
history of the lift coefficient for the Newtonian case (dashed
curve) and viscoelastic case (solid curve). The time origin
was chosen arbitrarily at a moment within the fully-devel-
oped oscillatory regime in which CL reaches a minimum;
both curves exhibit a perfectly sinusoidal behavior, main-
taining the period and amplitude. It is evident that the fre-
quency of vortex shedding is reduced due to the fluid
elasticity, so that it is decreased from 0.1483 in Newtonian
flow to 0.06575 in viscoelastic flow. Not only the fre-
quency of the shedding in the viscoelastic flow is lower
compared to the Newtonian case, but also the amplitude of
the lift coefficient is less around 30%. In fact, damping of
maximum values of lift by fluid elasticity is stronger than
damping the frequency. It is revealed that fluid elasticity
tends to decrease in vortex frequency. Also, the similar
η
ηo
----- = ξ
λ--- + 1
ξ
λ----–⎝ ⎠
⎛ ⎞ 1 f–( )2
1 1 2α–( )f+-----------------------------
f = 1 x–
1 1 2α–( )x+-----------------------------
x2 =
1 16α 1 α–( ) λγ·( )2
+[ ]1/2
1–
8α 1 α–( ) λγ·( )2
---------------------------------------------------------------
ξ = ληs
ηp
----- = λ1
β---- 1–⎝ ⎠⎛ ⎞
η = mγ·n 1–
ReG = ρU∞B/ηeff
γ· = U∞/B
ηp/ ηp ηs+( )( )
Table 3. Comparison of CD and St values in unsteady flow
regime with previous studies at Re = 100.
Source CD St
Present work 1.51 0.1483
Robichaux et al. (1999) 1.53 0.1540
Sahu et al. (2009) 1.48 0.1485
Table 4. Comparison of CL and CD between Giesekus viscoelastic
fluids (present work) and power-law fluids (Sahu et al., 2009).
ReG
CL (present
work)
CL (Sahu et al.,
2009)
CD (present
work)
CD (Sahu et al.,
2009)
60 0.0426 0.08 1.5586 1.5094
80 0.1143 0.1415 1.4639 1.4515
100 0.1683 0.1957 1. 4137 1.4311
120 0.2116 0.24 1.3905 1.4328
Numerical study of vortex shedding in viscoelastic flow past an unconfined square cylinder
Korea-Australia Rheology J., 27(3), 2015 219
effect is reported in experimental observations of Coelho
and Pinho (2003a) for a circular cylinder.
5.2 Effects of elasticity and Reynolds numberTable 5 gives a summary of the main results obtained
when elasticity is increased, by increasing We from 0 to 20
at Re = 80. We see that as We goes from 0 to 0.1 the fre-
quency values measured by the St and lift coefficient are
reduced by 54.8% and 32.4%, respectively. In fact, as
elasticity of the flow past an unconfined square cylinder is
increased, a progressive modification to the velocity field
around the cylinder is accrued.
The polymer molecules of viscoelastic flow near the
centerline relaxed their configuration at the upstream of
cylinder similar to Newtonian flow. Because of viscoelas-
ticity, the macromolecules moving very close to the cyl-
inder edges experience progressively stronger deformation
rates which lead to the development of large molecular
extensions and high elongation stresses. This deformation
is remembered by the fluid and the configuration of the
molecules as they enter the downstream of the flow changes
with increasing elasticity (refer to Fig. 3).
Fig. 3. Instantaneous streamline contours near the cylinder for (A) Newtonian and (B) viscoelastic (We = 20, β = 0.05, α = 0.1) during
one cycle of vortex shedding behind a square cylinder at Re = 100.
Fig. 4. Comparison of lift coefficient of viscoelastic flow (We =
20, β = 0.05, α = 0.1) and Newtonian flow at Re = 100.
Mahmood Norouzi, Seyed Rasoul Varedi and Mahdi Zamani
220 Korea-Australia Rheology J., 27(3), 2015
On the other hand, by increasing elasticity, fluid velocity
moving away from the cylinder recovers more slowly.
Therefore, cylinder wake is extended downstream with
increasing We, equivalent to a downstream shift of the
streamlines around the cylinder (McKinley et al. 1993). In
this regard, Experimental studies by Usui et al. (1980) and
numerical work done by Oliveira (2001) have also revealed
that even small amounts of a dissolved polymer, compared
to the purely Newtonian solvent, lead to a reduction in fre-
quency of vortex shedding. For We larger than around 1-
5, no further reduction in the St and lift coefficients is
observed. This may be explained by noting that relaxation
time of the fluid is then larger than the period of vortex
shedding (TL ≈ 15.846) and the controlling time scale
becomes the latter.
The relative difference between the Newtonian St (StN)
and the viscoelastic St (StV) is reported in a Table 6. At
present, simulations are limited to the moderate Re (60 <
Re < 120) due to the large amount of computation time
required to probe higher Re. In general, cylinder flow is
rich in physical effects such as shear layers, recirculation
regions, boundary layers and vortex dynamics, thus mak-
ing this problem ideal for studying complex viscoelastic
effects. Furthermore, as the Re is increased, we know that
the flow type changes dramatically, starting from steady
laminar flow, changing to unsteady 2D vortex shedding,
then going through several stages of 3D transition before
finally reaching full turbulence (Williamson, 1996). As a
result, these different stages also present opportunities to
investigate the effect of viscoelasticity under many differ-
ent circumstances.
In Fig. 5, the variation of St versus Re for Newtonian
and viscoelastic cases are shown. According to the figure,
increasing Re enhances the St for both cases. The same
effect on shedding frequency for Newtonian flow around
the square (Williamson, 1988) and circular cylinders (Leweke
et al., 1995) was reported in literature. In Fig. 6, the vis-
coelastic data lie below the Newtonian, reflecting the ten-
dency for vortex suppression induced by the elastic forces.
As shown in Fig. 6, there is a progressive reduction in lift
amplitude like St by increasing the fluid elasticity. It is not
only the frequency of vortex shedding which is decreased
by elasticity effects, but also more strongly reduction hap-
pens in the amplitude of the lift coefficient so that its dec-
rement doubles in amount from 0.046 at Re = 60 to 0.092
at Re = 120 compared to the Newtonian flows. Similar
results are offered by Oliveira (2003) for the circular cyl-
inder. This feature is reflected on sudden variations of the
average recirculation length behind the cylinder.
5.3 Effect of polymer concentration As reported by Coelho and Pinho (2003b), the effect of
Table 5. Effect of increasing We (Re = 80, β = 0.05, α = 0.1) on
flow parameters.
We TL λ St Cl
0 7.138 0 0.14010 0.1904
0.1 15.810 1 0.06325 0.1287
1 15.836 10 0.06315 0.1281
5 15.846 50 0.06310 0.1278
10 15.872 100 0.06300 0.1278
20 15.872 200 0.06300 0.1278
Table 6. Comparison between Newtonian and viscoelastic flows
(We = 20, β = 0.05, α = 0.1) for different Re.
Re StN − StV CLN − CLV
60 0.0702 0.0460
70 0.0742 0.0562
80 0.0771 0.0626
100 0.0825 0.0737
120 0.0846 0.0926
Fig. 5. Strouhal number vs. Reynolds number for Newtonian and
viscoelastic cases (We = 20, β = 0.05, α = 0.1).
Fig. 6. Lift coefficient vs. Reynolds number for Newtonian and
viscoelastic cases (We = 20, β = 0.05, α = 0.1).
Numerical study of vortex shedding in viscoelastic flow past an unconfined square cylinder
Korea-Australia Rheology J., 27(3), 2015 221
shear thinning viscometric functions on viscoelastic cyl-
inder flow is to increase the vortex shedding frequency,
while fluid elasticity tends to decrease it. Since the Giesekus
model exhibits shear thinning behavior, simulations were
performed for the range of 0 ≤ β ≤ 1 at Re = 100 and We
= 20 in order to probe the effect of increasing the shear
thinning contribution to the total viscosity. For this pur-
pose, mobility parameter is assumed to be α = 0.1. This is
entirely consistent with the present results since the para-
meter β effectively controls the polymer concentration.
Table 7 shows the effect of polymeric concentration on
lift amplitude and St. According to the Table, St and lift
amplitude are increased 8.89% and 15.5% by enhancing
Table 7. Effect of increasing the polymeric concentration (Re =
100, We = 20, α = 0.1) on flow parameters.
β TL St CL
St Comparison
with the base
case %
CL Comparison
with the base
case %
0.05 15.20 0.0657 0.1786 - -
0.1 15.00 0.0666 0.1797 1.37 0.61
0.3 14.78 0.0676 0.1846 2.81 3.36
0.5 14.58 0.0685 0.1904 4.26 6.60
0.7 14.36 0.0695 0.2000 5.78 11.9
0.9 14.16 0.0.071 0.2062 7.38 15.5
0.95 13.94 0.0716 0.2062 8.89 15.5
Fig. 7. Instantaneous first normal stress differences during one cycle of vortex shedding behind a square cylinder at
Re = 100, We = 20, and α = 0.1 for β = 0.05 and 0.95.
N1/ρUin
2( )
Mahmood Norouzi, Seyed Rasoul Varedi and Mahdi Zamani
222 Korea-Australia Rheology J., 27(3), 2015
the viscosity ratio from 0.05 to 0.95. In order to explain
the origin of the effect of polymeric concentration on vis-
cous shedding, we studied this effect on viscometric func-
tions. Figs. 7-9 show the effect of viscosity ratio on normal
stress differences and shear stress. In spite of increasing
the effective viscosity and decreasing the generalized Re,
the intensifying the flow instability could be attributed to
enhancing the first normal stress difference up to 102 times
by increasing the viscosity ratio from 0.05 to 0.95.
5.4 Effect of mobility factor (α)The equations to describe the viscoelastic polymer part
of the extra stress tensor τp as a function of the rate of
deformation tensor, can be classified in models which are
linear or nonlinear in the extra stress tensor. For example,
the Oldroyd eight constants fluid (Oldroyd, 1950) and the
Oldroyd with the Giesekus extension fluid (Giesekus,
1994) are mentioned. Furthermore, linear models, such as
Maxwell, Oldroyd or Jeffrey fluids, are able to reproduce
some rheological behaviors. However, linear models show
some weakness in describing fluids like polymer melts.
In order to model real fluid behavior for high deforma-
tion rates, additional nonlinear quantities in Eq. (3) are
necessary. Because of restrictions in the experimental
identification, we prefer to describe material behaviors
with models that include as few parameters as possible.
Here the new material parameter α is able to control the
influence of the nonlinearity, such as shear thinning
behavior in case of the Giesekus constitutive equation.
The differences of the material models are visible in the
Fig. 8. Instantaneous second normal stress differences during one cycle of vortex shedding behind a square cylinder at
Re = 100, We = 20, and α = 0.1 for β = 0.05 and 0.95.
N2/ρUin
2( )
Numerical study of vortex shedding in viscoelastic flow past an unconfined square cylinder
Korea-Australia Rheology J., 27(3), 2015 223
flow behavior under rheometric conditions. The mobility
factor of the Giesekus model, which accounts for Brown-
ian isotropic behavior in molecular hydrodynamics of vis-
coelastic flow, profoundly affects the shear and extensional
behavior of fluid flow. In fact, the α parameter affects
directly the order of non-linearity of viscometric functions
(Patankar et al., 1972).
Fig. 10 presents the effect of mobility parameter on flow
stability at Re = 100 and We = 20 for different polymeric
concentrations. As considered in this figure, the frequency
and amplitude of vortex shedding frequency is little
enhanced by increasing the mobility parameter at low vis-
cosity ratios such as 0.05 and 0.1 while the enhancement
is remarkable at large enough viscosity ratios (concen-
trated viscoelastic solutions). These effects are better shown
in Fig. 11 via diagrams of St and lift coefficient versus the
mobility factor for different viscosity ratios. It is important
to remember that increasing mobility factor intensifies the
shear thinning behavior of model so the effective viscosity
is decreased by increasing the mobility factor especially in
large viscosity ratios. Therefore, intensifying the flow
instability by increasing the mobility factor could be
attributed mostly to decreasing the stabilizing viscous forces.
6. Conclusions
The unsteady flow of Giesekus fluids past an uncon-
fined square cylinder is investigated numerically over the
range of conditions 60 ≤ Re ≤ 120, 0 ≤ β ≤ 1, 0 ≤ We ≤ 20
and 0 ≤ α ≤ 0.5. The global quantities such as lift coeffi-
Fig. 9. Instantaneous shear stress during one cycle of vortex shedding behind a square cylinder at Re = 100, We = 20, and
α = 0.1 for β = 0.05 and 0.95.
τxy/ρUin
2( )
Mahmood Norouzi, Seyed Rasoul Varedi and Mahdi Zamani
224 Korea-Australia Rheology J., 27(3), 2015
cient, St and the detailed kinematic variables like normal
stress differences and stream line have been obtained in
order to investigate the flow pattern of viscoelastic fluid
for the above range of conditions. The obtained results are
in good agreement with the recent numerical and experi-
mental results.
We conclude the followings from the present work.
First, fluid elasticity leads to decrease in the amplitude and
vortex shedding frequency. Second, strongly reduction
happens in the amplitude of the lift coefficient so that lift
coefficient decrement doubles in amount in Re = 60 to
Re = 120 compared to the Newtonian flows. Third, the St
increases by viscosity ratio increment. The same proce-
dure happens for lift amplitude. More specifically speak-
ing, the St. Number and lift amplitude both increase by
8.89% and 15.5% in order of appearance in viscosity ratio
0.95, compared to the base case showing a 0.05 value.
Finally, it is undoubtedly shown that increasing mobility
parameter, increase lift amplitude more tangibly in com-
parison with frequency at high polymer concentrations. It
Fig. 10. Effect of mobility parameter on flow pattern for various polymer concentrations β (a) 0.05, (b) 0.1, (c) 0.3, (d) 0.5, (e) 0.7,
and (f) 0.9, where α = 0.05, 0.1, 0.3, and 0.5 for solid, dashed, dashed dotted, and long dashed lines, respectively.
Fig. 11. Effect of mobility parameter for constant viscosity ratios (β) on (a) Strouhal number and (b) amplitude of lift for viscoelastic
flow at Re = 100 and We = 20.
Numerical study of vortex shedding in viscoelastic flow past an unconfined square cylinder
Korea-Australia Rheology J., 27(3), 2015 225
is also perceived that enhancing the mobility parameter
reduces the shedding frequency variation rate.
The wakes of viscoelastic flows behind square cylinders
with incidence variation can be examined later. The Sim-
ulation of vortex shedding for viscoelastic fluid past a
confined square cylinder or sphere can also be the subject
of future work in this context using Giesekus model.
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