S. RashidiDepartment of Mechanical Engineering,

Ferdowsi University of Mashhad,

Mashhad 91775-1111, Iran

M. BovandDepartment of Mechanical Engineering,

Ferdowsi University of Mashhad,

Mashhad 91775-1111, Iran

J. A. Esfahani1Department of Mechanical Engineering,

Ferdowsi University of Mashhad,

Mashhad 91775-1111, Iran

e-mail: abolfazl@um.ac.ir

H. F. €OztopDepartment of Mechanical

Engineering Technology,

Firat University Elazig,

Elazig 23119, Turkey

R. MasoodiSchool of Design and Engineering,

Philadelphia University,

4201 Henry Avenue,

Philadelphia, PA 19144

Control of Wake StructureBehind a Square Cylinder byMagnetohydrodynamicsIn this paper, a two-dimensional (2D) numerical simulation has been performed for anunsteady magnetohydrodynamics (MHD) flow around a solid square cylinder placed in achannel. Computational simulations were done for the ranges of Reynolds and Stuartnumbers of 1–250 and 0–10, respectively. Finite volume method (FVM) has been used tosolve the unsteady Navier–Stokes equations. The effects of streamwise magnetic field onthe flow separation and suppress of the vortex shedding are studied in detail for the aboveranges. Additionally, four new empirical equations for wake length and Stuart numberare suggested. Finally, a comparison is performed between the cases of with and withouta channel to study the effect of channel walls. The obtained results revealed that Strouhalnumber decreases linearly with increasing Stuart number. Also, the flow distribution pat-tern changes from time-dependent pattern to steady-state by increasing Stuart number.[DOI: 10.1115/1.4029633]

Keywords: flow stabilization, Strouhal number, streamwise magnetic field, finite volumemethod, Stuart number

1 Introduction

The study of the flow approaching obstacles in a channel has awide variety of engineering applications such as off-shore marinestructures, heat exchangers, skyscrapers, chimney stacks, etc., [1].Another important example is the heat sinks for electronic compo-nents, which are based on arrays of short cylindrical pin-finsplaced inside a channel [2]. Having such important applications,this type of fluid flow has been the subject of intensive studiessince last decade [3–6].

Suzuki and Suzuki [7] studied unsteady flow in a channelobstructed by an inserted square rod. They found that the momen-tum transfer is enhanced due to the shear stress. Large eddy simu-lation of turbulent flow past a square cylinder confined in achannel was performed by Kim et al. [8]. It was found that the ex-istence of channel walls increases the drag force and lift fluctua-tions on the square cylinder compared with those in thecorresponding infinite-domain case. Influences of suction andblowing on vortex shedding behind a square cylinder in a channelinvestigated by Layek et al. [9]. It was observed that the amplitudeof the lift coefficient decreases with increase in the blowing veloc-ity. Jafari et al. [10] investigated particle dispersion and deposi-tion in a channel with a square cylinder obstruction using thelattice Boltzmann method. Their simulation showed that theBrownian diffusion affects the deposition rate of ultrafine particlesin front and on the back of the block. Heat transfer and flow past asquare unit of four isothermal cylinders is studied by AbolfazliEsfahani and Vaselbehagh [11,12]. The flow around a pair ofside-by-side square cylinders placed in a 2D channel using thelattice Boltzmann method is computed by Burattini and Agrawal[1]. Nazari et al. [13] studied the power-law fluid flow and heat

transfer in a channel partially filled with an anisotropic porousblock.

The obstacle placed in the channel with different cross sectionswas also subject of some studies. Visco-elastic flow past circularcylinders mounted in a channel is studied experimentally by Ver-helst and Nieuwstadt [14]. Armellini et al. [2] performed an ex-perimental study on the separated flow structures around acircular cylinder in a liquid cooling channel. Their research estab-lished that the mean recirculation wakes are affected by a span-wise mass transport from the channel walls to the horizontalsymmetry plane. Confined flow and heat transfer across a triangu-lar cylinder in a channel is performed by Srikanth et al. [15].Also, some researcher investigated MHD flows for differentboundary conditions [16–21].

Several researchers studied flow around objects placed ininfinite-domain [22–27]. Dhiman et al. [28] performed a numeri-cal study on the flow and heat transfer across a confined squarecylinder. Their results showed that as Reynolds number increases,the length of the recirculation bubble increases. Valipour et al.[29] carried out a numerical study of convection around a squareporous cylinder using Al2O3–H2O Nanofluid. The flow patternsand vortex shedding behavior behind a square cylinder is studiedby Yen and Yang [30].

Control of the vortex shedding leads to a reduction in theunsteady forces acting on the bluff bodies and can significantlyreduce their vibration. Several kinds of control can be imple-mented to control bluff-body flows such as a modifying geometryof the surface (by putting tabs, streaks, or dimples on the surface),blow and suction from body’s surface, base bleed, controlling theboundary layer, etc. A new approach is using a magnetic field[31]. Many researches are available about magnetohydrodynam-ics. Barletta et al. [32] studied the mixed convection with heatingeffects in a vertical porous annulus with a radially varying mag-netic field. The MHD effect, represented by a horizontal magneticfield, on heat transfer from a circular cylinder embedded in aninfinitely thick porous layer was studied numerically by ZareGhadi et al. [33]. Also, Valipour et al. [34] applied the least square

1Corresponding author.Contributed by the Fluids Engineering Division of ASME for publication in the

JOURNAL OF FLUIDS ENGINEERING. Manuscript received October 12, 2014; finalmanuscript received January 20, 2015; published online March 9, 2015. Assoc.Editor: Shizhi Qian.

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method [35–37] and suggested two equations for the averageNusselt number.

Yoon et al. [38] presented the results of a numerical study onthe fluid flow and heat transfer parameters around a circular cylin-der in the presence of a magnetic field. Effect of aligned magneticfield on the steady viscous flow around a circular cylinder is pre-sented by Sekhar et al. [39]. Their results show that the down-stream base pressure is more affected by the magnetic field thanupstream base pressure. An analogy between the effects of thestream wise and transverse magnetic field on flow and heat trans-fer around a porous obstacle has been done by Rashidi et al. [40].They found that the effects of transverse magnetic field on flowand heat transfer characteristics are more than that of stream wisefield. Note that above two studies [39,40] are performed for steadyregime that the flow is time-independent and there is no any shed-ding phenomena in the wake. Also, another analogy between theeffects of the stream wise and transverse magnetic field on flowcharacteristics around a circular cylinder has been performed byGrigoriadis et al. [41].

The main aim of this paper is to investigate the streamwise mag-netic field is used for controlling the flow parameters around an ob-stacle in a rectangular channel. Above literature review shows thatthis subject of study has received many attentions due to the impor-tant in different applications. As we know, this is for first time thatthe control of wake structure behind a square cylinder placed in thechannel by magnetohydrodynamics is investigated.

2 Problem Description

Figure 1 shows the geometry and computational domain for theconsidered problem. A 2D flow of fluid with density q anddynamic viscosity l is considered. The fluid flow with a parabolicinlet velocity from left to right is used to pass through a rectangu-lar channel. An obstacle with square cross section is mounted inthis channel. The height of the channel is H and D is the height ofthe cylinder. In the figure, Ld and Lu are downstream and upstreamdistances of the cylinder, respectively. Note that in this figure, theorigin point is located at the center of the cylinder. In this study,the following assumptions are made:

• Flow is 2D, laminar, incompressible, viscous, and conductive.• In order to reduce the influence of inflow and outflow bound-

ary conditions, the length of the channel was set to L/D¼ 50.• Inflow length of channel was set to Lu¼ 12.5D [42].• The channel has conductive walls.

3 Mathematical Model

3.1 Governing Equations. Governing equations, which aremomentum and continuity equations, are derived to simulate thisproblem. The continuity equation is written as

@u�

@x�þ @v�

@y�¼ 0 (1)

Also, momentum equations in x and y directions are written as

q@u�

@t�þ u�

@u�

@x�þ v�

@u�

@y�

� �¼ � @p�

@x�þ l

@2u�

@x�2þ @

2u�

@y�2

� �(2)

q@v�

@t�þ u�

@v�

@x�þ v�

@v�

@y�

� �¼ � @p�

@y�þ l

@2v�

@x�2þ @

2v�

@y�2

� �� rv�B2

x

(3)

where superscript * denotes dimensional variables. Also, Bx and rare uniform magnetic field strength in horizontal direction andelectrical conductivity of the fluid, respectively.

The governing equations are converted to dimensionless formsby using the following dimensionless variables:

x ¼ x�

H; y ¼ y�

H; u ¼ u�

U0

; v ¼ v�

U0

; p ¼ p�

qU20

; t ¼ t�U0

H(4)

Then, the governing equations reduce to the following dimension-less equations:

The continuity equation is

@u

@xþ @v

@y¼ 0 (5)

The momentum equations in the x and y directions are

@u

@tþ u

@u

@xþ v

@u

@y

� �¼ � @p

@xþ 1

Re

@2u

@x2þ @

2u

@y2

� �(6)

@v

@tþ u

@v

@xþ v

@v

@y

� �¼ � @p

@yþ 1

Re

@2v

@x2þ @

2v

@y2

� �� Nv (7)

where Re and N are the Reynolds and Stuart numbers, respec-tively, which are defined by

Re ¼ qU0H

l; N ¼ rB2H

qU0

(8)

Note that the Reynolds and Stuart numbers is defined based on thehydraulic equivalent diameter of the channel (H) and the velocityat the centerline of inlet (U0).

3.2 The Boundary and Initial Conditions. At the inlet sec-tion of the channel, the flow is assumed to be parabolic distribu-tion. Note that the parabolic velocity distribution is used forhaving the fully developed laminar flow. Parabolic inlet velocityis very common for studying the flow in a channel [7,15,42]. Thisboundary condition is given by

Fig. 1 Definition of the geometry and computational domain

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u ¼ u�

U0

¼ 1� 2y

H

� �2 !

; v ¼ 0 (9)

where U0 is the maximum inlet velocity (velocity on the axis).At the outlet of the channel, zero gradient boundary conditions

are used [15]. The flow at the exit boundary is close to fully devel-oped condition. It means flow characteristics are constant in thatdirection. Besides, since a definite condition is needed for outletand there is no other logical outlet condition, this boundary condi-tion is applied for outlet. Note that applying of this boundary con-dition just affects the last two or three nodes that are notcalculated in the final postprocessing [43]. Therefore, a zero gradi-ent condition is appropriate for the present simulation [44,45].

This boundary condition is similar to Neumann boundary condi-tion as given

@u

@x¼ 0;

@v

@x¼ 0 (10)

For the channel walls (upper and downer walls) and the cylindersurface, no slip boundary condition is considered as

u ¼ 0; v ¼ 0 (11)

As the initial condition, there is no flow inside the channel at theinitial time (t¼ 0).

4 Computational Model

4.1 Numerical Method. In the present simulation, the equa-tions on a 2D staggered grid are solved numerically using FVM. Instaggered grid system, the velocity components are stored at cellfaces and pressure is stored at the cell center. The SIMPLE algo-rithm [46] is utilized for the coupling between continuity and mo-mentum equations. For time discretization, the first-order implicitscheme has been used and a third-order accurate QUICK scheme(quadratic upwind interpolation for convective kinematics) was usedfor the convective referenced [46]. Convergence criterion isassumed to be achieved when the summation of residuals decreasedto �10�7 for all equations. The proper numerical computations

Fig. 2 Mesh distribution and subcomputational domains

Table 1 The effect of grid number on Strouhal number atRe 5 100

Grid number

No. D1(n�m) D2(n�m) D3(n�m) D4(n�m) D5(n�m) St

1 80� 30 5� 30 150� 30 80� 5 150� 5 0.1362 95� 40 15� 40 180� 40 95� 15 180� 15 0.13783 110� 50 25� 50 210� 50 110� 25 210� 25 0.1394 125� 60 35� 60 240� 60 125� 35 240� 35 0.1393

Fig. 3 Variation of wake length against Reynolds numberFig. 4 Comparison of computed Strouhal number with pub-lished literatures

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were applied to determine the flow behavior, Strouhal number, con-tours of streamline and vorticity, and wake length.

4.2 Grid-Indecency Study. Figure 2 shows mesh distributionand subcomputational domains along the x- and y-directions nearthe surface of square cylinder and channel wall. The whole com-putational domain is divided into five subdomains, D1, D2, D3, D4,and D5. Nonuniform grids were chosen and these grids are refinednear the cylinder and channel walls where the velocity gradient islarge compared to other regions. To obtain fine grid, the finestcells were generated with dimensions 0.01� 0.01 and 0.01� 0.02in the adjacent of the cylinder and channel walls, respectively.

These cells are located around the cylinder walls and at the cornerof D1 region.

A grid independence study was also done. The test is performedfor four different grid numbers at Re¼ 100. Table 1 shows thecomparisons of Strouhal number for various grid numbers. Notethat in table 1, n�m refers to the number of mesh numbers in thehorizontal (x) and vertical (y) directions, respectively. The differ-ence in Strouhal number between cases 3 and 4 is 0.21%. There-fore, the grid of case 4 is chosen for the rest of simulations.

4.3 Validation. In order to demonstrate the validity of thepresent simulation, the results are compared with some data fromthe literature. Figure 3 presents variation of wake length withReynolds number for a square cylinder placed in a channel. Ascan be seen in Fig. 3, a good agreement is found between the pres-ent results and that of Breuer et al. [42] for Re< 60. Also, this fig-ure shows that the wake length linearly increases with Reynoldsnumber for Re< 60 and decreases for higher Reynolds numbers.For Re> 150, the wake length has a constant value in differentReynolds numbers. A curve fit of the results leads to an empiricalrelationship for wake length as a function of Reynolds number.This equation is given by

LR ¼ �0:0830þ 0:0563Re (12)

This equation is valid for Reynolds number in the range of6<Re< 60. For comparison, wake length for case of cylinderwithout channel is presented in this figure. It is worth mentioningthat existence of channel walls decreases the size of the recirculat-ing wake behind the cylinder compared with those in the corre-sponding infinite-domain case. This reduction is more obvious for

Fig. 5 Variation of Strouhal number as a function of Stuartnumber at Re 5 100

Fig. 6 Contours of streamline and vorticity for different Reynolds number (N 5 0)

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higher Reynolds numbers. For example, this reduction is in thevicinity of 20% for Re¼ 50.

Figure 4 presents the variation of Strouhal number versus Reyn-olds numbers. The computed result is compared with the resultsof Breuer et al. [42]. This figure shows that the curve has a maxi-mum value at Re> 150 and the Strouhal number increases forRe< 150 and decreases for Re> 150.

Note that Breuer et al. [42] investigated the 2D flow past a squarecylinder placed in a channel for the Reynolds number in the rangeof 0<Re< 300. In order to generate reliable numerical results, twototally different approaches were applied including the lattice Boltz-mann and FVMs. Their results show an excellent agreementbetween the LBA and FVM computations. Therefore, this work pro-vided reliable and accurate results for the confined cylinder flowwhich is coincident with our paper. Also, this work [38] had mostconsistent with current study among all the available literatures.

5 Results and Discussion

A computational work has been done to see the effects of mag-netohydrodynamics in a channel with square obstruction. Themain parameters affecting the fluid flow in this problem are theReynolds and Stuart numbers. In this research, the Reynolds andStuart numbers are in the ranges 1–250 and 0–10, respectively.Also, the blockage ratio (S¼D/H) is taken as 0.8 for allsimulations.

The goal of this study is to determine the effect of the Reynoldsand Stuart numbers on some hydrodynamic parameters, such aswake structure, Strouhal number, streamlines, and vorticitycontours.

Figure 5 shows variation of Strouhal number as a function ofStuart number at Re¼ 100. It can be observed that the Strouhalnumber decreases linearly with increasing Stuart number. As

shown in Eqs. (3) and (7), if the magnetic field is applied in thehorizontal direction, the vertical force acts in the negativey-direction, and this tends to retard the motion of the fluid [20].This resistive force is called Lorentz force, which is experiencedby a fluid carrying a current density J in a magnetic field B andcalculates as follows [34]:

F!

Lorentz ¼ J!� B

!¼ rðV!� B!Þ � ~B (13)

Therefore, this force tends to retard the motion of the fluid. Theflow is stabilized as the strength of applied magnetic fieldsincreases and this leads to decrease in Strouhal number. Thisreduction is in the vicinity of 80% for N¼ 0.14.

The streamline and vorticity contours for square cylinders atdifferent Reynolds number and N¼ 0 are shown in Fig. 6. Notethat in this paper, N¼ 0 represents the absence of an external mag-netic field. It can be observed that for Re¼ 1, the flow is fullyattached to the cylinder and there is no recirculating wake andflow separation behind the cylinder. At Re¼ 40, a recirculatingwake appears clearly because the pressure starts to increase in thesurface of the cylinder and the flow experiences an adverse pres-sure gradient. Also, the flow is still steady for these Reynoldsnumbers (Re¼ 1–40) because the flow is symmetric about thewake centerline. At higher Reynolds number (i.e., Re¼ 75), vor-tex shedding begins to take place and eddies are shed continu-ously from each side of the body. In this Reynolds number, theflow is time-dependent. The strength of the shed vortices increasesby further increase in Reynolds number. Note that shedding maycause large fluctuation in the pressure forces leading to structuralvibrations, acoustic noise, or resonance. Therefore, controlling ofwake is important to get energy efficiency.

Contours of streamline and vorticity for different Stuart numberat Re¼ 100 are shown in Fig. 7. This figure shows that the flow

Fig. 7 Contours of streamline and vorticity at different Stuart number (Re 5 100)

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distribution changes from the time-dependent pattern to thesteady-state by increasing Stuart number. The flow is steady forN¼ 4. However, if N is further increased, the recirculating wakeweakens and it completely disappears when the Stuart numberreaches 8.

The critical Stuart number is plotted versus Reynolds numberin Fig. 8. Here Ncr is a value of the Stuart number that makes thesteady fluid flow (the flow pattern changes from the time-dependent pattern to the steady-state one). This critical numberdepends on the Reynolds number. It is observed that the criticalStuart number increases with increasing Reynolds number. Thisfigure presents a map for detection of steady and unsteady regions.It is found that the flow is steady in the domains located above thedash line and it is unsteady in the domains located under this line.Also, in this figure, an empirical relationship is presented for criti-cal Stuart number as a function of the Reynolds number. Thisequation is given as

Ncr ¼ �2:227þ 0:302 lnðReÞ (14)

This equation is valid for 60<Re< 250 (unsteady region). Notethat control of vortex shedding leads to a reduction in the unsteadyforces acting on the bluff bodies and can significantly reduce theirvibrations. Therefore, the present of this equation is necessary.

Figure 9 shows the variation of disappearance Stuart number asa function of the Reynolds number for steady and unsteadyregions. The disappearance Stuart number Ndis, is defined to mea-sure the strength of the inserted magnetic field required for disap-pearing the recirculating wake. It is found that the disappearanceStuart number increases with Reynolds number for both steadyand unsteady regions. Note that this reduction is in the vicinity of55% for Re¼ 250 and unsteady flow. Also, in this figure, two em-pirical relationships are presented for disappearance Stuart num-ber as a function of the Reynolds number as

Ndis¼� 0:946þ 0:338� Re0:611 (15)

Ndis¼10:96Re

81:75þ Re(16)

Equations 15 and 16 are valid in the ranges of 5<Re< 60 and75<Re< 250, respectively. Note that bluff bodies experiencepressure drag due to the existence of a large wake and so it is criti-cal to control this wake.

Figure 10 provides a snapshot for the contours of Lorentz forceat different Reynolds number. These snapshots were taken afterthe activation of the streamwise magnetic field in the x direction.In this figure, the solid line refers to the force in favor directionwith y and dashed line refers to the force in opposite direction

with y. As can be seen in this figure, the Lorentz force is symmet-ric about the centerline for low Reynolds number (i.e., Re¼ 40).For higher Reynolds numbers (i.e., Re¼ 75,100), where the flowis unsteady, the direction of Lorentz force changes continuously.The direction of velocity is changed continuously in unsteady re-gime and Lorentz force has a direct relation with velocity (see Eq.(13)). It is observed that the Lorentz force increases in the regionsaround cylinder and centerline that the velocity in vertical direc-tion is maximum. Also, Lorentz force decreases in other regionssuch as near the channel wall, where velocity is small. The Lor-entz force increases with increasing Reynolds number.

6 Conclusion

A numerical study has been carried out to study the effect ofmagnetic field on wake formation and suppressing the vortexshedding for a square cylinder placed in a rectangular channel.This problem is a classical problem with many important practicalapplications. The most important results found in this study arehighlighted as follows:

• Existence of channel walls decreases the size of the recircu-lating wake in the vicinity of 20% for Re¼ 50 compared withthose in the corresponding infinite-domain case.

• Strouhal number decreases linearly with increasing Stuartnumber. This reduction is in the vicinity of 80% for N¼ 0.14.

• If the Stuart number increases, the flow is stabilized and thedistribution changes from the time-dependent to the steady-state. An equation is investigated for value of the Stuart num-ber that makes the steady fluid flow.

• Critical Stuart number increases with increasing Reynoldsnumber.

Fig. 9 Variation of disappearance Stuart number against Reyn-olds number: (a) steady flow and (b) unsteady flow

Fig. 8 Variation of critical Stuart number against Reynoldsnumber

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• The disappearance Stuart number increases with Reynoldsnumber for both steady and unsteady regions. This reductionis in the vicinity of 55% for Re¼ 250 and unsteady flow.

• Lorentz force increases in the regions around cylinder and cen-terline, where the velocity in vertical direction is maximum.

Nomenclature

B ¼ uniform magnetic field strength (T)D ¼ height of the cylinder ðmÞf ¼ vortex shedding frequency (s�1)

F ¼ force (N)FVM ¼ finite volume method

H ¼ height of the channel ðmÞL ¼ channel length ðmÞ

Ld ¼ downstream distance of the cylinder ðmÞLR ¼ wake length ðmÞLu ¼ upstream distance of the cylinder ðmÞ

LBA ¼ lattice Boltzmann approachMHD ¼ magnetohydrodynamics

N ¼ Stuart number (—) (rB2H/(qU0))

P ¼ pressure (Pa)Re ¼ Reynolds number (—)¼U0H=�

S ¼ blockage ratio (—)¼D=HSt ¼ Strouhal number (—)¼ fD=U0

t ¼ time (s)U0 ¼ velocity on the axis (ms�1)

u; v ¼ velocity components in x- and y-directions (ms�1)x; y ¼ rectangular coordinates (m)

Greek Symbols

l ¼ dynamic viscosity ðkg=m�1s�1Þ� ¼ fluid kinematic viscosity ðm2s�1Þq ¼ fluid density ðkg=m�3Þr ¼ electrical conductivity of the fluid (1/Xm)

Subscripts

cr ¼ criticaldis ¼ disappearance

x ¼ horizontal direction* ¼ dimensional variable

Fig. 10 Contour of Lorentz force just after the activation of the magnetic field

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