Numerical study of vortex shedding in viscoelastic flow past an … · 2015-08-20 · bluff bodies. Also, no prior numerical study exists in the literature related to the vortex-shedding

© 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.



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∞



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)



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



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.



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).



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( )



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( )



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( )



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.



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.

References

Balachandar, S. and S. Parker, 2002, Onset of vortex shedding in

an inline and staggered array of rectangular cylinders, Phys.

Fluids 14, 3714-3732.

Bird, R.B. and J.M Wiest, 1995, Constitutive equations for poly-

meric liquids, Ann. Rev. Fluid Mech. 27,169-193.

Coelho, P. and F. Pinho, 2003a, Vortex shedding in cylinder flow

of shear-thinning fluids: I. Identification and demarcation of

flow regimes, J. Non-Newtonian Fluid Mech. 110, 143-176.

Coelho, P. and F. Pinho, 2003b, Vortex shedding in cylinder flow

of shear-thinning fluids: II. Flow characteristics, J. Non-

Newtonian Fluid Mech. 110, 177-193.

Courant, R., E. Isaacson, and M. Rees, 1952, On the solution of

nonlinear hyperbolic differential equations by finite differ-

ences, Communications Pure Appl. Math. 5, 243-255.

Franke, R., W. Rodi, and B. Schönung, 1990, Numerical calcu-

lation of laminar vortex-shedding flow past cylinders, J. Wind

Eng. Ind. Aerodynamics 35, 237-257.

Giesekus, H., 1994, Phänomenologische rheologie: eine Ein-

führung, Springer, Berlin.

Jackson, C.P., 1987, A finite-element study of the onset of vortex

shedding in flow past variously shaped bodies, J. Fluid Mech.

182, 23-45.

Kim, J.M., K.H. Ahn, and S.J. Lee, 2009, Effect of viscoelasticity

on two-dimensional laminar vortex shedding in flow past a

rotating cylinder, Korea-Aust. Rheol. J. 21, 27-37.

Leweke, T. and M. Provansal, 1995, The flow behind rings: bluff

body wakes without end effects, J. Fluid Mech. 288, 265-310.

Luo, S.C., X.H. Tong, and B.C. Khoo, 2007, Transition phenom-

ena in the wake of a square cylinder, J. Fluids Structures 23,

227-248.

Malvandi, A. and D. Ganji, 2014, Brownian motion and thermo-

phoresis effects on slip flow of alumina/water nanofluid inside

a circular microchannel in the presence of a magnetic field,

International J. Thermal Sci. 84, 196-206.

McKinley, G.H., R.C. Armstrong, and R.A. Brown, 1993, The

wake instability in viscoelastic flow past confined circular cyl-

inders, Phil. Trans. Royal Soc. London, Series A: Phys. Eng.

Sci. 344, 265-304.

Norouzi, M., S.R. Varedi, M.J. Maghrebi, and M.M. Shahmar-

dan, 2013, Numerical investigation of viscoelastic shedding

flow behind a circular cylinder, J. Non-Newtonian Fluid Mech.

197, 31-40.

Oldroyd, J.G., 1950, On the formulation of rheological equations

of state, Proc. Royal Soc. London, Series A: Math. Phys. Sci.

200, 523-541.

Oliveira, P.J., 2001, Method for time-dependent simulations of

viscoelastic flows: vortex shedding behind cylinder, J. Non-

Newtonian Fluid Mech. 101, 113-137.

Patankar, S.V. and D.B. Spalding, 1972, A calculation procedure

for heat, mass and momentum transfer in three-dimensional

parabolic flows, International J. Heat and Mass Transfer 15,

1787-1806.

Phan-Thien, N., 2002, Understanding viscoelasticity: basics of

rheology, Springer, Berlin.

Richter, D., G. Iaccarino, and E.S. Shaqfeh, 2010, Simulations of

three-dimensional viscoelastic flows past a circular cylinder at

moderate Reynolds numbers, J. Fluid Mech. 651, 415-442.

Robichaux, J., S. Balachandar, and S.P. Vanka, 1999, Three-

dimensional Floquet instability of the wake of square cylinder,

Phys. Fluids 11, 560-578.

Saha, A., G. Biswas, and K. Muralidhar, 1999, Influence of inlet

shear on structure of wake behind a square cylinder, J. Eng.

Mech. 125, 359-363.

Saha, A., G. Biswas, and K. Muralidhar, 2003, Three-dimen-

sional study of flow past a square cylinder at low Reynolds

numbers, International J. Heat Fluid Flow 24, 54-66.

Sahin, M. and R.G. Owens, 2004, On the effects of viscoelasticity

on two-dimensional vortex dynamics in the cylinder wake, J.

Non-Newtonian Fluid Mech. 123, 121-139.

Sahu, A.K., R.P. Chhabra, and V. Eswaran, 2009, Two-dimen-

sional unsteady laminar flow of a power law fluid across a

square cylinder, J. Non-Newtonian Fluid Mech. 160, 157-167.

Sahu, A.K., R.P. Chhabra, and V. Eswaran, 2010, Two-dimen-

sional laminar flow of a power-law fluid across a confined

square cylinder, J. Non-Newtonian Fluid Mech. 165, 752-763.

Sharma, A. and V. Eswaran, 2004, Heat and fluid flow across a

square cylinder in the two-dimensional laminar flow regime,

Numerical Heat Transfer, Part A: Applications 45, 247-269.

Sheard, G.J., M.J. Fitzgerald, and K. Ryan, 2009, Cylinders with

square cross-section: wake instabilities with incidence angle

variation, J. Fluid Mech. 630, 43-69.

Sohankar, A., C. Norberg, and L. Davidson, 1999, Simulation of

three-dimensional flow around a square cylinder at moderate

Reynolds numbers, Phys. Fluids 11, 288-306.

Tamura, T. and K. Kuwahara, 1990, Numerical study of aerody-

namic behavior of a square cylinder, J. Wind Eng. Ind. Aero-

dynamics 33, 161-170.

Usui, H., T. Shibata, and Y. Sanu, 1980, Karman vortex behind

a circular cylinder in dilute polymer solutions, J. Chem. Eng.

Japan 13, 77-79.

Versteeg, H.K. and W. Malalasekera, 2007, An introduction to

computational fluid dynamics: the finite volume method, Pear-

son Education, Harlow.

Williamson, C., 1988, Defining a universal and continuous Strou-

hal–Reynolds number relationship for the laminar vortex shed-

ding of a circular cylinder, Phys. Fluids 31, 2742-2744.

Williamson, C.H. 1996, Three-dimensional wake transition, J.

Fluid Mech. 328, 345-407.

Xue, S.-C., N. Phan-Thien, and R.I. Tanner, 1995, Numerical

study of secondary flows of viscoelastic fluid in straight pipes

by an implicit finite volume method, J. Non-Newtonian Fluid

Mech. 59, 191-213.

Zhang, L. and S. Balachandar, 2006, Onset of vortex shedding in a

periodic array of circular cylinders, J. Fluids Eng. 128, 1101-1105.

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Numerical study of vortex shedding in viscoelastic flow past an … · 2015-08-20 · bluff bodies. Also, no prior numerical study exists in the literature related to the vortex-shedding