DOI: 10.1007/s13367-015-0022-z
Numerical study of vortex shedding in viscoelastic flow past an unconfined square cylinder
Mahmood Norouzi1,*, Seyed Rasoul Varedi2 and Mahdi Zamani3
1,2Department of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran
3Department 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 cylinder is investigated numerically for the first time. The global quantities such as lift coefficient, Strouhal number and the detailed kinetic and kinematic variables like normal stress differences and streamlines have been obtained in order to investigate the flow pattern of viscoelastic flow. The effects of Reynolds number and polymer concentrations have been clarified in the periodic viscoelastic flow regime. Our particular interest is the effect of mobility parameter on the stability of two dimensional viscoelastic flows past an unconfined square cylinder. To fulfill this aim, the mobility parameter has been increased from 0 to 0.5 for different polymer 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]
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
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 Initial Conditions
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∇ α We ---β
+ τ2 = 2β D
D = (∇u ∇u+ T)/2 τ∇
τ∇ ∂τ
--- + U∂t ⋅∇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 fluids Fig. 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 ηηo
--- = ξ
λ--- + 1 ξλ----⎝⎛ – ⎠⎞ (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 --- = λ 1
---- 1β–
⎝ ⎠
⎛ ⎞
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 number
Table 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/ρUin2)
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/ρUin2
( )
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/ρUin2)
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.
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