Flow and heat transfer over a trapezoidal cylinder: steady and unsteady regimes

Research article

Flow and heat transfer over a trapezoidal cylinder: steadyand unsteady regimes

Amit Dhiman* and Mudassir Hasan

Department of Chemical Engineering, Indian Institute of Technology Roorkee, Roorkee, 247 667, India

Received 30 August 2011; Revised 12 June 2012; Accepted 23 July 2012

ABSTRACT: Two-dimensional unconfined flow and heat transfer across a long tapered trapezoidal bluff body are investi-gated for the range Re = 1 to 150 (thereby covering both steady and unsteady periodic regimes) and Pr = 0.7 (air). A numberof engineering parameters, e.g. drag and lift coefficients, Strouhal and Nusselt numbers, and others, is calculated for theabove range of conditions. No flow separation occurs from the surface of the trapezoidal cylinder for the range Re≤ 5;however, flow starts to separate from the rear surface of the cylinder at Re = 6. Therefore, the onset of flow separation exitsbetween Re = 5 and 6. The critical value of the Reynolds number (i.e. transition from steady to unsteady) exists betweenRe = 46 and 47. The drag coefficient decreases with increasing Reynolds number in the steady regime; however, the dragincreases with Reynolds number in the unsteady regime. The Strouhal number and the average Nusselt number increasewith increasing value of the Reynolds number. Finally, the simple correlation for the average Nusselt number is obtainedin the steady flow regime. © 2012 Curtin University of Technology and John Wiley & Sons, Ltd.

KEYWORDS: trapezoidal cylinder; critical Reynolds number; wake length; drag; Strouhal number; Nusselt number

INTRODUCTION

In the last 10 decades or so, flow past bluff body (e.g.cylinders) has been investigated by many researchersboth numerically and experimentally due to its varietyof engineering applications, such as electronic cooling,heat exchange systems, offshore structures, suspensionbridges, chimneys, probes and sensors and flowmetering devices, especially vortex-shedding meters,and others. Further, it is analogous to the flow profilesobtained in a jet from a square orifice. For instance,in the near field region of the orifice, information tothe exit geometry is still retained by the flow pattern;however, in the far field, the jet approaches to circularsymmetry and the flow closely resembles what wouldbe obtained from a round orifice.[1] In spite of suchwide applications, very limited information is availableon the flow over a long obstacle of trapezoidal crosssection. This work is concerned with the flow and heattransfer across a long tapered trapezoidal cylinder inthe steady and unsteady unconfined flow regimes.The flow and temperature fields here depend uponReynolds number (Re), Prandtl number (Pr), heightratio (i.e. the ratio of the height of the rear side to thatof the front side of the trapezoidal cylinder) and the

trapezoidal height in the axial direction, which aregiven as rU1b/m,mcp/k, a/b and b, respectively. Also,flow behind such bluff body, even at low values ofReynolds number, can lead to regular vortex sheddingand induce a considerable dynamic load on the body.Therefore, the proper understanding of flow andthermal characteristics induced by vortex shedding isimportant in various industrial problems. The relevantliterature for the problem under consideration isgiven succeedingly.Goujon-Durand et al.[2] experimentally investigated

the vortex shedding from a trapezoidal bluff body. Theyobtained the scaling laws for the evolution of the globalmode describing the envelope of the peak to peak ampli-tude velocity oscillation in the wake flow downstream ofthe body. Lee [3] numerically studied the early stages ofthe symmetrical wake flow developments around atapered trapezoidal cylinder for Re = 25, 50, 150, 250,500 and 1000. The flow starts with no separation, andthe symmetrical standing zone of recirculation developsaft of the trapezoidal cylinder with the advancement oftime. However, no information is provided on the onsetsof flow separation and transition. Chung and Kang [4]

studied the Strouhal number from tapered trapezoidalcylinders for three values of the Reynolds numbers of100, 150 and 200 and height ratios = 0.3 to 1. TheStrouhal number has minimum values at height ratiosof 0.7 and 0.85 for Reynolds numbers of 100 and 150,respectively. The movement of the flow separation point

*Correspondence to: Amit Dhiman, Department of ChemicalEngineering, Indian Institute of Technology Roorkee, Roorkee, 247667, India. E–mail: [email protected], [email protected]

© 2012 Curtin University of Technology and John Wiley & Sons, Ltd.Curtin University is a trademark of Curtin University of Technology

ASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERINGAsia-Pac. J. Chem. Eng. (2012)Published online in Wiley Online Library(wileyonlinelibrary.com) DOI: 10.1002/apj.1678

from the rear to front corners and the change ofsecondary vortex strength are important factors in deter-mining the shedding structures for Reynolds numbers of100 and 150. However, no information is available onthe drag and lift coefficients. Kahawita and Wang [1]

carried out the two-dimensional (2D) numericalsimulations of the Benard von Karman hydrodynamicinstability behind trapezoidal bluff bodies using thespline method of fractional steps. They reported thatthe influence of the trapezoidal height is dominant onthe value of Strouhal number, when compared with theeffect of the smaller trapezoidal base width. Chenet al.[5] performed the 2D flow around a porousexpanded trapezoidal cylinder using a finite volumemethod, based on the body-fitted, non-orthogonal gridsand multi-block technique. At large Darcy number, theReynolds number has to be higher before thevortex shedding phenomena occurs and the fluctuation-amplitude of drag coefficient decreases. The effects ofthe stress jump parameters are provided for Reynoldsnumber= 20 to 200.Thus, as far as known to us, it can be concluded

here that very limited information is available in theopen literature on the flow across a long taperedtrapezoidal cylinder in the steady and unsteady flowregimes. However, no study is available on the heattransfer from a tapered trapezoidal cylinder in boththe steady and unsteady flow regimes. Thus, the mainobjective of this work is to fill these gaps in theliterature on the momentum and heat transfer acrossa tapered trapezoidal cylinder. The other importantobjectives of this study are to explore the value ofthe Reynolds number, where the flow starts to sepa-rate from the trapezoidal cylinder (i.e. the onset offlow separation) and to explore the critical Reynoldsnumber (i.e. the transition from steady to unsteady)for the system considered.

PROBLEM STATEMENT AND MATHEMATICALFORMULATION

The 2D unconfined flow across a long tapered trape-zoidal cylinder (front face width, b = 1, rear facewidth, a = 0.5b and cylinder height in axial direction,b = 1) is flowing from left to right, as shown in Fig. 1.The upstream distance from the inlet plane to the frontsurface of the trapezoidal cylinder (Xu) is taken as 12b,height of the trapezoidal cylinder is b, the downstreamdistance between the rear surface of the trapezoidalcylinder and the exit plane (Xd) is taken as 20b,with the total length of the computational domain(L) of 33b in the axial direction is utilized afterthorough experimentations. However, the height ofthe computational domain (H) is used as 30b in thelateral direction.

The governing continuity, x-component andy-component of Navier–Stokes and energy equations intheir dimensionless form can be written as follows:

Continuity equation

@Vx

@xþ @Vy

@y¼ 0 (1)

x-Momentum equation

@Vx

@tþ @ VxVxð Þ

@xþ @ VyVx

� �@y

¼ � @p

@xþ 1Re

@2Vx

@x2þ @2Vx

@y2

� �(2)

y-Momentum equation

@Vy

@tþ @ VxVy

� �@x

þ @ VyVy

� �@y

¼ � @p

@yþ 1Re

@2Vy

@x2þ @2Vy

@y2

� �(3)

Energy equation

@θ@t

þ @ Vxθð Þ@x

þ @ Vyθ� �@y

¼ 1RePr

@2θ@x2

þ @2θ@y2

� �(4)

In Eqns (1)–(4), the dimensionless Reynolds andPrandtl numbers are defined as Re = bU1r/m andPr =mcp/k, respectively. In the present study, fluid prop-erties (e.g. density, viscosity, thermal conductivity, etc.)are assumed to be constant, and the viscous dissipationin the energy equation is also neglected. Hence, thepresent results are applicable to situations where thetemperature difference is not too large and for moderate

a=0.5b

L

H/2Trapezoidal cylinderVx = 0, Vy = 0, Tw

x

y

H

Slip boundary

Slip boundary

= 0

Xu

Xd

Vy

Vx = 1b

b=1

Figure 1. Schematic of the flow around a taperedtrapezoidal cylinder.

A. DHIMAN AND M. HASAN Asia-Pacific Journal of Chemical Engineering

© 2012 Curtin University of Technology and John Wiley & Sons, Ltd. Asia-Pac. J. Chem. Eng. (2012)DOI: 10.1002/apj

viscosity. The resulting momentum and energyequations have been solved by treating them asdecoupled. For instance, the pressure used is constant[Eqns (2) and (3)] as a result of which Eqns. (1)–(3) aresolved first and thereafter, the thermal equation (4) isresolved for the temperature distribution.The dimensionless boundary conditions for the uncon-

fined isothermal flow and heat transfer across a (heated)trapezoidal cylinder can be written as (see Fig. 1)

• At the inlet boundary, Vx= 1, Vy= 0 (uniform flow)and θ = 0

• On upper and lower boundaries, @ Vx/@ y = 0, Vy= 0and @ θ/@ y= 0

• On the surface of the trapezoidal cylinder, Vx= 0,Vy= 0 (no slip) and θ= 1

• At the exit boundary, the default outflow boundarycondition in Fluent, which assumes a zero diffusionflux for all flow variables, is used. This is similar toNeumann boundary condition as @ Vx/@ x = 0, @ Vy/@x = 0 and @ θ/@ x = 0.

NUMERICAL METHODOLOGY

In the present study, the governing continuity, compo-nents of Navier–Stokes and energy equations along

x

y

0 5 10 15 20 25 300

5

10

15

20

25

30

Figure 2. Non-uniform computational grid struc-ture around a tapered trapezoidal cylinder.

Table 1. Validation of present trapezoidal cylinderresults of Strouhal number with literature values.

Source St

Re= 100Present work 0.1370Chung and Kang[4] 0.138

Re = 150Present work 0.1458Chung and Kang[4] 0.149

Table 2. Validation of present circular cylinder resultswith literature values at Re=100 and Pr = 0.7.

Source CD St Nu

Present work 1.3063 0.1626 5.0866Baranyi[6] 1.3500 0.1630 5.1320Rajani et al.[7] 1.3349 0.1569 —Lange et al.[8] — 0.1650 5.1270

(c) Re = 1 (e) Re = 30

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17(b) Re = 6

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14.8

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15.1

15.2

12.9 13 13.1 13.2 13.3

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15.1

15.2

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12 13 14 15

12.9 13 13.1 13.2 13.3

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17 (f) Re = 45

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17(d) Re = 20

12 13 14 15

12 13 14 15

12 13 14 15

12 13 14 15 16

(a1) Re = 5

(b1) Re = 6

Figure 3. a), (a1), (b) and (b1) depict the onset of flowseparation, and (c) – (f) represent the streamline contoursfor varying Reynolds number in the steady flow regime.

Asia-Pacific Journal of Chemical Engineering FLOW AND HEAT TRANSFER OVER A TRAPEZOIDAL CYLINDER


with appropriate boundary conditions are solvedby using a finite volume method based commercialcomputational fluid dynamics (CFD) solver Fluent.The computational grid structure is generated by usingGambit, as shown in Fig. 2. This grid consists of120,520 cells with each side of the trapezoidal cylinderhaving 100 control volumes and a very fine grid of cell

size of 0.004b is used near the cylinder; however, thelargest grid size of 0.4b is used away from the obstacle.The second-order upwind scheme is used to discretizeconvective terms; whereas, the diffusive terms arediscretized by central difference scheme. The first-orderimplicit time-integration method is used here and thedimensionless time step is set to 0.01 as the smaller valueof the time step did not produce any significant change inthe values of the physical parameters considered here.The resulting algebraic equations are solved by Gauss–Siedel iterative scheme. The residuals of the continuity,x-component and y-component and energy equations ofthe order of 10�15 in the steady state regime and of10�20 in the unsteady state regime are used.For the grid independence study, three non-uniform

grids (80,550, 120,520 and 146,830 cells with 50,100 and 150 control volumes on each side of thetrapezoidal cylinder, respectively) are tested for theReynolds number of 150 and the Prandtl number of0.7. For the last two grid sizes (i.e. 120,520 and146,830 cells), the percentage relative changes in thevalues of the mean drag coefficient, Strouhal numberand the mean Nusselt number are found to be less than1.2%, less than 1.0% and less than 0.75%, respectively.However, the corresponding changes in the values ofthe mean drag coefficient, Strouhal number and themean Nusselt number are found to be less than 4.8%,less than 3.5% and less than 1.75%, respectively for

Re5 10 15 20 25 30 35 40 45

0

0.5

1

1.5

2

2.5

3

Square cylinder

Trapezoidal cylinder

Lr /b

Figure 4. Variation of the wake length alongwith the literature values of square cylinder(Dhiman et al.[10,12]) in the steady regime.

12

13

14

15

16

17

18

(f) Re=47

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16

17

18

(d) Re=47

12

13

14

15

16

17

18

(e) Re = 46

12

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16

17

18

(c) Re = 46

t

CL

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

(b) Re=47

CL

12 13 14 15 16 1712 13 14 15 16 17

12 13 14 15 16 1712 13 14 15 16 17

1000 1500 2000 2500

450 500 55 600 650

-1E-05

-5E-06

0

5E-06

1E-05 (a) Re=46

Figure 5. (a) and (b): temporal variation of the lift coefficient; (c) and (d): contours of streamline; and (e)and (f) isotherm contours for Re = 46 and 47.



the grids of 80,550 and 120,520 cells. Thus, the grid,consisting of 120,520 cells with each side of thetrapezoidal cylinder of 100 points, is used here.The upstream distance is varied from 12b to 14b to

fix the upstream distance. For this, two values of theupstream distances (Xu= 12b and 14b) are tested atthe lowest value of the Reynolds number of unity andthe Prandtl number of 0.7. The percentage relativechanges in the values of the drag coefficient are foundto be about 1.0%. However, the corresponding changesin the values of the mean Nusselt number are found tobe about 0.20%. Thus, the upstream distance of 12b isused in this study.Similarly, to fix the downstream distance, two values

of the downstream distance are checked, i.e. Xd= 20band 22b for the highest value of the Reynolds numberof 150 used in this study. At the Reynolds number of150 and the Prandtl number of 0.7, the percentagerelative changes in the values of the drag coefficientand the Strouhal number are found to be less than0.05% and almost negligible, respectively. The percen-tage change in the value of the mean Nusselt number isfound to be less than 0.15%. Therefore, the downstreamdistance of 20b is used here.

The effect of the domain height is also carried out forthe computational domain heights of 22b and 30b inthe lateral direction for the Reynolds number of 150and Prandtl number of 0.7. The relative changes inthe values of the drag coefficient and the Strouhalnumber are found to be less than 0.65% and about0.30%, respectively. The percentage change in thevalue of the mean Nusselt number is found to be lessthan 0.25%. Thus, the domain height of 30b in thelateral direction is used.

RESULTS AND DISCUSSION

In the present study, 2D numerical simulations areperformed in the full computational domain to investigatethe flow and heat transfer around a long cylinder oftapered trapezoidal cross section in the unconfined steadyand unsteady flow regimes. The various engineeringparameters such as drag and lift coefficients, Strouhalnumber, local and average Nusselt numbers, and others,are calculated for Reynolds number ranging from 1 to150 (thereby covering both steady and unsteady periodicregimes) for a Prandtl number of 0.7 (air).

12

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17

18(g) Re=100, t=2T/4

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18(h) Re=100, t=3T/4

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18

(a) Re=50, t=T

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18

(b) Re=50, t=T/4

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18

(c) Re=50, t=2T/4

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(d) Re=50,t=3T/4

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18(e) Re=100, t=T

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18(f) Re=100, t=T/4

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18(i) Re=150, t=T

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18(j) Re=150, t=T/4

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18(k) Re=150, t=2T/4

12 13 14 15 16 17 12 13 14 15 16 17

12 13 14 15 16 17 12 13 14 15 16 17 12 13 14 15 16 17 12 13 14 15 16 17

12 13 14 15 16 17 12 13 14 15 16 17

12 13 14 15 16 17 12 13 14 15 16 17 12 13 14 15 16 17 12 13 14 15 16 1712

13

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17

18(l) Re=150, t=3T/4

Figure 6. Instantaneous streamline profiles for (a)–(d) Re = 50, (e)–(h) Re = 100, and (i)–(l) Re = 150 in theunsteady regime.



Validation of results

The benchmarking of the present flow results on thetapered trapezoidal bluff body is conducted with theresults of Chung and Kang [4] for the values of theReynolds number of 100 and 150 (Table 1). An excellentagreement is found between the present results and that ofChung and Kang.[4] The maximum deviations in thevalues of Strouhal number are found to be less than1.0% and about 2.15% for Re=100 and 150, respectively.This validates the present numerical methodology.Because no heat transfer results are available on this

bluff body, the present numerical solution procedure isapplied on the flow past a long circular cylinder to vali-date the present heat transfer results and flow results aswell. Table 2 presents the validation of the present flowand heat transfer results for the Reynolds number of100 and Prandtl number of 0.7. The maximum changesin the present values of the mean drag coefficients forthe Reynolds number of 100 are found to be less than3.25% and less than 2.15% with respect to the results

of Baranyi [6] and Rajani et al.,[7] respectively. Thecorresponding changes in the values of Strouhal numberare found to be less than 0.25%, less than 3.65% and lessthan 1.5% as compared with the results of Baranyi [6],Rajani et al.[7] and Lange et al.,[8] respectively. How-ever, the maximum differences in the values of theaverage Nusselt number for the Reynolds number of100 and Prandtl number of 0.7 are found to be less than1.0% between the present results and that of Baranyi[6]

and of Lange et al.[8]. This further validates the presentnumerical solution procedure.

Flow patterns

Figure 3 presents the streamline profiles near the taperedtrapezoidal cylinder for the varying values of theReynolds number in the steady flow regime (Re≤ 46).It can be stated here that the thermal distribution has noeffect on the flow hydrodynamics for the range ofsettings. It is found that no flow separation occurs fromthe surface of the trapezoidal cylinder for the range ofvalues of the Reynolds number, Re≤ 5 [Fig. 3(a), (a1)and (c)]. However, the flow starts to separate from therear surface of the tapered trapezoidal cylinder at Re= 6[Fig. 3(b) and (b1)]. The zoomed views of vector andx-velocity profiles at Re= 5 and 6 are also presented toshow the onset of flow separation in Fig. 3(a1) and(b1). Thus, the onset of flow separation exits betweenRe= 5 and 6 for the flow across an unconfined taperedtrapezoidal cylinder. For another similar bluff body, i.e.long square cylinder, the onset of flow separation occursbetween Re= 1 and 2 (Sharma and Eswaran[9]; Dhimanet al.[10]). However, the onset of flow separation isdelayed for trapezoidal bluff as compared with theregular square block (Sharma and Eswaran [9], Dhimanet al.[10]) owing to the decrease in the width of the rearface of the tapered trapezoidal cylinder. In the presentstudy, where the rear width to front width ratio is fixedat 0.5, flow separation occurs between Re = 5 and 6.The change in the onset of flow separation can occur atother values of Reynolds number also, depending uponthe rear width to front width ratio of the trapezoidal bluffbody (Chung and Kang [4] and O’Connor.[11])As the value of the Reynolds number gradually

increases (6≤Re≤ 46), flow separates from the rearsurface of the trapezoidal cylinder and two vortices areformed behind the small base width of the trapezoidalcylinder [Figs 3(d) and (e)]. However, for Re> 20, theflow not only separates from the rear side but also fromthe top and bottom sides of the trapezoidal cylinder[Fig. 3(e) and (f)]. Similar to square cylinder, the size ofthese vortices behind the rear surface of the trapezoidalcylinder increases with increasing value of the Reynoldsnumber in the steady flow regime. The variation of thewake length with Reynolds number in the steady regimeis presented in Fig. 4. The wake length increases withincreasing value of the Reynolds number. It can also be

13

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16

17 (f) Re = 45

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17

(c) Re = 10

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17

(e) Re = 30

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14

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17

(b) Re = 5

13

14

15

16

17(d) Re = 20

12 13 14 15 1612 13 14 15

12 13 14 1512 13 14 15

12 13 14 1512 13 14 1513

14

15

16

17(a) Re = 1

Figure 7. Isotherm profiles for (a) Re = 1, (b) Re = 5,(c) Re = 10, (d) Re = 20, (e) Re = 30 and (f) Re = 45 in thesteady regime.



seen from Fig. 4 that the wake length is smaller for thetrapezoidal cylinder than that of the square cylinder(Sharma and Eswaran[9]; Dhiman et al.[10,12]) as the rearside of the trapezoidal cylinder is half of the side of thesquare cylinder for the range of conditions simulated.On further increasing the value of the Reynolds

number (i.e. Re> 46), the flow changes from steadyregime to unsteady regime. Figure 5 (a)–(d) shows thevariation of the lift coefficient with time and the corre-sponding streamline patterns for Reynolds numbers of46 and 47. It can be seen from these figures that the valueof the lift coefficient decays with time very slowly at aReynolds number of 46 [Fig. 5(a)] and is expected toreach a steady state after a long time [Fig. 5(c)]. On theother hand, for the Reynolds number of 47, the valueof the lift coefficient increases with time [Fig. 5(b)],and the wake loses its steady state [Fig. 5(d)]. Theseobservations suggest that the critical Reynolds number(i.e. transition to periodic unsteady) exits between theReynolds numbers of 46 and 47 for the unconfinedtapered trapezoidal cylinder. It also implies that thetransition from steady to unsteady regime occursbetween previously mentioned Reynolds numbers, andit depends upon the rear width to front width ratio ofthe tapered trapezoidal cylinder[4]. For the square

cylinder, the critical Reynolds number lies between Re=45 and 50 (Sharma and Eswaran[9]; Dhiman et al.[10,12]).The flow patterns in the unsteady flow regime are

presented by instantaneous streamline profiles at differ-ent values of the Reynolds number. Figures 6 (a)–(l)represent the instantaneous streamline profiles forRe = 50, 100 and 150 for four successive moments oftime (i.e. T, T/4, 2T/4 and 3T/4), which span over thewhole period. It is also observed that from the nextmoment onwards (i.e. after t= 3T/4), the behavior getsrepeated. It is interesting to see here that the flowseparation occurs at the front corners of the trapezoidalcylinder, whereas for the square cylinder (of side b)flow separation occurs at the rear corners in theunsteady flow regime. It can also be seen that vorticesdisappear behind the rear surface of the trapezoidalcylinder in the far flow field. Similar to the squarecylinder case, it is also observed here that the waveringmotion for the trapezoidal cylinder increases as thevalue of the Reynolds number increases.

Thermal patterns

Isotherms close to the trapezoidal obstacle are presentedfor Re= 1, 5, 10, 20, 30 and 45 for the fixed value of

(j) Re=150, t=T/4 (k) Re=150, t=2T/4

(f) Re=100, t=T/4 (g) Re=100, t=2T/4

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18

(a) Re=50, t=T (b) Re=50, t=T/4 (c) Re=50, t=2T/4 (d) Re=50, t=3T/4

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(e) Re=100,t=T (h) Re=100, t=3T/4

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18(i) Re=150,t=T

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12 13 14 15 16 17 12 13 14 15 16 1712 13 14 15 16 17 12 13 14 15 16 17

12 13 14 15 16 17 12 13 14 15 16 17

(l) Re=150, t=3T/4

Figure 8. Instantaneous isotherm profiles for (a)–(d) Re = 50, (e)–(h) Re = 100 and (i)–(l) Re = 150 in theunsteady regime.



the Prandtl number of 0.7 in the steady flow regime, asshown in Fig. 7. Because the flow is steady for the range1≤Re≤ 46, the symmetry in the temperature field aboutthe mid plane (i.e. at y=15b) can be seen in Figs. 5(e) and7. It can also be seen that isotherms are more pronouncedfor the low values of the Reynolds numbers as the con-duction is more dominant here; however, with increasingvalue of the Reynolds number, temperature fields decayin the steady flow regime [Figs. 5(e) and 7]. The turningof isotherms towards the rear surface of the trapezoidalcylinder is also observed for the range 40≤Re≤ 46[Figs. 5(e) and 7(f)] in the steady flow regime.Because the flow field becomes unsteady periodic for

Re> 46, the instantaneous isotherm profiles for foursuccessive moments of time, which span over thewhole period, are presented for Reynolds number of50, 100 and 150 for the fixed value of the Prandtlnumber of 0.7 in Fig. 8. It is observed that a temperaturestreet is formed behind the trapezoidal cylinder, whichshows one row in the unsteady flow regime for the rangeof the conditions studied. Similar to the flow field, it can

be seen from these figures that the wavering motion ofisotherms increases with increasing value of theReynolds numbers. Also, the disturbance on the smallbase width of the trapezoidal cylinder increases withincreasing value of the Reynolds number (Fig. 8). Similarto square cylinder, the maximum crowding of isothermson the front surface of the trapezoidal cylinder can alsobe seen in Fig. 8, as compared with other surfaces of thetrapezoidal cylinder. This results in the higher value ofthe Nusselt number on the front surface of the trapezoidalcylinder as compared with the value of the Nusseltnumber on the other surfaces of the trapezoidal cylinder.

Time history

Figure 9 (a)–(i) presents the temporal variations of thedrag [Figs. 9(a)–(c)] and lift [Fig. 9(d)–(f)] coefficientsand Nusselt number [Fig. 9(g)–(i)] for the Reynoldsnumbers of 50, 100 and 150. The instantaneous valuesof the drag and lift coefficients and the Nusselt numberare calculated at each time step and plotted opposed

Nu

3.785

3.79

3.795

3.8

3.805

3.81

(h) Re = 100

1.66

1.665

1.67

1.675

1.68

1.685

1.69 (b) Re = 100

1.6068

1.607

1.6072

1.6074

1.6076

1.6078 (a) Re = 50

CD

CD

CD

-0.6

-0.4

-0.2

0

0.2

0.4

0.6 (e) Re = 100

CL

CL

CL

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

(d) Re = 50

Nu

2.721

2.7211

2.7212

2.7213(g) Re = 50

t

Nu

4.6

4.65

4.7

4.75

(i) Re = 150

t

-1

-0.5

0

0.5

1(f) Re = 150

t

360 380 400 420 440360 380 400 420 440

560 580 600 620 640

340 360 380 400 420 440

520 540 560 580 600 620 640 520 540 560 580 600 620 640

160 180 200100 120 140 160 180 200160 180 2001.86

1.88

1.9

1.92

1.94

1.96

(c) Re = 150

Figure 9. Time history of drag and lift coefficients and Nusselt number for (a), (d), (g) Re = 50; (b), (e), (h)Re = 100; and (c), (f), (i) Re = 150.



to time. These instantaneous values of the drag and liftcoefficients and the Nusselt number fluctuate initiallyand finally stabilize (Fig. 9) with time for the range ofsettings studied. The frequency of the drag and theNusselt number is found to be the same; whereas, itfound to be double the frequency of the lift. Further,these stabilized values of the drag and lift coefficientsand the Nusselt number are utilized to calculate thetime-averaged values of the drag coefficient, lift coeffi-cient and the Nusselt number by averaging 10 cyclesbeyond the time the asymptotic shedding frequencyof Karman vortex is achieved.

Drag coefficient

Drag coefficient is one of the most important flow para-meters. The total drag coefficient around any bluffbody is calculated as CD=CDF+CDP. The variationof the individual and overall values of the drag coeffi-cients with varying values of the Reynolds number isshown in Fig. 10(a)–(c) in the steady flow regime. Thisfigure also includes the variation of the individual andtotal drag coefficients for the square cylinder in thesteady regime (Dhiman et al.[10,12]). Furthermore, thezoomed views of pressure and overall drag coefficients

are also shown on separate frames in the top-right partsof Fig. 10(b) and (c). The values of the friction, pressureand total drag coefficients decrease with increasingvalues of the Reynolds number for both the sharp edgedbluff bodies in the steady flow regime. The friction dragcoefficient is observed to be lower for the trapezoidalcylinder than the square cylinder in the steady regime(1≤Re≤ 46); however, the pressure drag is observedto be higher for the range 10≤Re≤ 46 and lower forthe range Re< 10 than that of the square cylinder case.Figure 10(c) also shows the variation of the mean dragcoefficient for the square cylinder with Reynolds numberin the unsteady flow regime (Sharma and Eswaran.[9]) Itcan be seen from Fig. 10(c) that the value of the meandrag coefficient for the trapezoidal cylinder increaseswith Reynolds number beyond the value of the criticalReynolds number and the minimum drag coefficient isobtained at the Reynolds number of 50. The mean dragcoefficient for the trapezoidal cylinder is found to belower for the range 1≤Re≤ 50 and higher for the range50<Re≤ 150 than the square cylinder (Sharma andEswaran;[9] Dhiman et al.;[10,12] Sahu et al.[13]) This isdue to the trapezoidal shape of the cylinder. Further,the variation of the root mean square (RMS) value ofthe drag coefficient for the trapezoidal cylinder

0

0.005

0.01

0.015

0.02

0.025

0.03

11.5

22.5

33.5

44.5

55.5

66.5

77.5

88.5

9

Re

23456789

101112131415

(c)

CD

50 75 100 125 150

5 10 15 20 25 30 35 40 45

25 50 75 100 125 150

5 10 15 20 25 30 35 40 450

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Square cylinder

25 50 75 100 125 1501.4

1.6

1.8

2

2.2

2.4

2.6

2.8

5 10 15 20 25 30 35 40 451.3

1.5

1.7

1.9

2.1

2.3

2.5

2.7

2.9

3.1

3.3


(a)

CDF

Re

CDrms

(d)

(b)

CDP

Figure 10. Variation of (a) and (b) the individual and (c) overall drag coefficients and(d) the root mean square value of the drag coefficient along with the results of Sharmaand Eswaran[9] (unfilled symbols) for the square cylinder with Reynolds number.



along with the results of Sharma and Eswaran [9] for thesquare cylinder with Reynolds number is presented inFig. 10 (d). These RMS values give the measure of theamplitude of the unsteady cylinder wake oscillations.Similar to the square cylinder case, the RMS value ofthe drag coefficient increases with increasing value ofthe Reynolds number in the unsteady periodic regime.The RMS value of the drag coefficient for the trapezoidalcylinder is observed to be higher for 50<Re≤ 150;however, the slight variation is observed at Re= 50 thanthat of the square cylinder (Sharma and Eswaran[9]). Thehigher RMS value of the drag coefficient for thetrapezoidal cylinder is due to the higher amplitudes ofthe unsteady wake oscillations than that of the squarecylinder. The experimental results have led to the estab-lishment of the fact that a trapezoidal shape is found tobe more desirable than at other shapes because a well-defined vortex emission due to clean separation at itssharp edges is assured (Kahawita and Wang [1]). Also,the saturation amplitude of self sustained oscillations inthe wake has a well defined maximum and correspond-ing downstream location, which are functions of theReynolds Number (Zieilinska and Wesfreid [14]).Further, the difference between the RMS values of thedrag coefficient of trapezoidal and square cylinders isalso increasing with Reynolds number in the unsteadyperiodic regime.

Lift coefficient

Similar to the time-averaged drag, the time-averagedlift coefficient is also calculated by averaging 10 cyclesbeyond the time the asymptotic shedding frequency ofKarman vortex is accomplished. The time-averaged liftcoefficient remains approximately zero (in the order ofabout 10�5 to 10�4) in the unconfined unsteady peri-odic regime. The variation of the RMS value of the liftcoefficient for the trapezoidal cylinder along with theresults of Sharma andEswaran [9] for the square cylinderwithReynolds number is presented in Fig. 11(a). Similar tothe RMS drag, the RMS value of the lift coefficientincreases with increasing value of the Reynolds numberfor the Reynolds number range 47≤Re≤ 150 and theRMS values for the trapezoidal cylinder areobserved to be higher than the square cylinder(Sharma and Eswaran[9]; Sahu et al.[13]). This isdue to the higher amplitudes of the unsteady wakeoscillations for the trapezoidal cylinder than thatof the square cylinder. Similarly, the differencebetween the RMS values of trapezoidal and squarecylinders is also increasing with Reynolds numberin the unsteady periodic regime.

Strouhal number

The vortex shedding frequency is exploited todetermine the Strouhal number (St) in the unsteady

periodic flow regime and the Strouhal number is definedas St= fb/U1. Figure 11(b) presents the variation of theStrouhal number with Reynolds number for therange 47≤Re≤ 150. This figure also shows the compar-ison of present Strouhal number results for the trapezoi-dal cylinder with that of the results of Sharma andEswaran [9] for the square cylinder. It can be seen thatthe Strouhal number for the trapezoidal cylinderincreases with increasing value of the Reynolds numberfor the range of conditions embraced. The Strouhal num-ber for the long trapezoidal obstacle is found to be lowerthan that of the square cylinder in the unsteady periodicflow regime (Sharma and Eswaran; [9] Sahu et al. [13])Obviously, the lower value of the Strouhal number forthe trapezoidal cylinder than that of the square cylinderis due to the fact that the rear side of the trapezoidal cylin-der is half of the side of the square cylinder. Thedifference between the values of Strouhal number oftrapezoidal and square cylinders is also increasing withReynolds number in the unsteady periodic regime.

Re

0.1

0.11

0.12

0.13

0.14

0.15

0.16

St

(b)

50 75 100 125 150

50 75 100 125 1500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Square cylinder


CLrms

(a)

Figure 11. Variation of (a) the root mean squarevalue of the lift coefficient and (b) the Strouhalnumber along with the literature values ofSharma and Eswaran[9] for square cylinder withReynolds number.



Local and average Nusselt numbers

The local Nusselt number is defined here as � @ θ/@ n,where n is the cylinder surface normal direction.Figure 12 presents the variation of the local Nusselt num-ber around the four surfaces (front: f, top: t, rear: r and bot-tom: b) of the trapezoidal cylinder for Re=1, 20 and 40[Fig. 12(a)–(c)] in the steady regime. However, the instan-taneous variation of the local Nusselt number for the four

successive moments of time, which span over the wholeperiod, for Reynolds numbers of 50, 100 and 150 in theunsteady periodic regime is shown in Fig. 12 (d)–(f).The variation of the local Nusselt number on the surfacesof the trapezoidal cylinder is observed to be qualitativelysimilar to that of the square cylinder. The local Nusseltnumber is found to be symmetric about the mid planefor the top and bottom halves (i.e. about y=15b) of thetrapezoidal cylinder. The value of the local Nusselt

Nu

(a) Re = 1

Nu

Nu

(b) Re = 20

A

B

C

D

f

t

r

b

1

2

3

4

5

6

7

8

9

10

11

12

13

14

t=T/4t=T/4t=2T/4t=3T/4

(d) Re = 50

123456789

101112131415161718 (e) Re = 100

Position around the trapezoidal cylinder

(c) Re = 40

Position around the trapezoidal cylinder

2

4

6

8

10

12

14

16

18

20 (f) Re = 150

A B C D A

A B C D A

A B C D A

A B C D A

A B C D A

A B C D A

2

4

6

8

10

0.2

0.6

1

1.4

1.8

2.2

2.6

3

1

3

5

7

9

1

2

3

4

5

6

7

8

9

10

11

12

13

Figure 12. Variation of the local Nusselt number around the surfaces of the trapezoidalcylinder for (a) Re=1, (b) Re=20, (c) Re=40, (d) Re=50, (e) Re=100 and (f) Re=150.



number decreases on the top and bottom surfaces (i.e.from B to C and from A to D) and then increasessharply towards the rear corners C and D, respectively.On the front and rear surfaces of the trapezoidalcylinder, the local minimum in the value of the Nusseltnumber exists near the middle of these surfaces. It canbe seen from these figures that the distribution of localNusselt number on the rear surface of the trapezoidalcylinder affects more than that of the top and bottomsurfaces of the trapezoidal obstacle with time due tovortex shedding in the unsteady regime [Fig. 12(d)–(f)].The disturbance in the value of the local Nusselt numberon the rear, top and bottom surfaces of the trapezoidalcylinder increases with increasing value of the Reynoldsnumber.The variation of the average Nusselt number with

Reynolds number in both steady and unsteady flowregimes for the Reynolds number range Re = 1 to150 and Prandtl number = 0.7 is presented in Fig. 13(a)(1<Re≤ 46) and Fig. 13(b) (47≤Re≤ 150) respec-tively. For the range of conditions examined, the averageNusselt number for the trapezoidal obstacle increaseswith increasing value of the Reynolds number. Thevariation of the average Nusselt number for the longsquare cylinder (Sharma and Eswaran[9]; Dhimanet al.[10,12]) is also shown in Fig. 13(a) and (b). Theaverage Nusselt number for the trapezoidal cylinder isfound to be higher for the range Re< 10 and lower forthe range 10≤Re≤ 150 than that of the square cylinder.The Nusselt number is less for the range 10≤Re≤ 150,due to the smaller wake length for the trapezoidalcylinder[3], than that of the square cylinder; however,for the range Re< 10, the average Nusselt number ismore for the trapezoidal cylinder, due to the delayed flowseparation, than that of the square cylinder (see subsec-tion on Flow Patterns).The present heat transfer results are further correlated

[Eqn. (5)] to calculate the average Nusselt number forthe intermediate values of the Reynolds number in thesteady flow regime (1≤Re≤ 46).

Nu� ¼ 0:0758þ 0:6523Re0:3572 (5)

The above correlation has a maximum deviation ofless than 1.25% for Re = 1; however, the deviation isless than 1.0% for 1<Re≤ 46 with the presentcomputed heat transfer results for Pr = 0.7.Furthermore, the RMS value of the Nusselt number is

calculated and found that the RMS Nusselt number forthe trapezoidal cylinder increases with increasing valueof the Reynolds number [Fig. 13(c)]. For instance, theRMS value of the Nusselt number is found to increasefrom 8.1�10�6 to 4.75�10�2 with Reynolds numberfor the range 47≤Re≤ 150 and Pr = 0.7.If one takes the front side smaller than the

rear side (as in the case of expanded trapezoidal

cylinder), then the obstacle will behave like astreamlined body and will produce less drag thanthat of the obstacle having rear side small (or in

Square cylinder

2.5

3

3.5

4

4.5

5

(b)

Nu-

Re

0

0.01

0.02

0.03

0.04

0.05

Nurms

(c)

50 75 100 125 150

50 75 100 125 150

5 10 15 20 25 30 3 40 450.5

1

1.5

2

2.5

3


(a)

Nu-

Figure 13. Variation of the average Nusselt number(a) steady flow with the literature values of squarecylinder (Dhiman et al.[10,12]), (b) unsteady flowwiththe literature values of square cylinder (Sharma andEswaran[9]) and (c) the root mean square value ofthe Nusselt number with Reynolds number.



the case of tapered trapezoidal cylinder). In contrast,the heat transfer rate will be higher for the expandedcylinder than the tapered one.

CONCLUSION

In this paper, 2D flow and heat transfer around atapered trapezoidal cylinder is investigated numeri-cally in the unconfined steady and unsteady (periodic)flow regimes (1≤Re≤ 150 and Pr = 0.7). A numberof flow and heat transfer parameters such as dragand lift coefficients, Strouhal number, local and aver-age Nusselt numbers, and others. is studied for theabove range of conditions. It is found that no flowseparation occurs from the trapezoidal cylinder sur-face for Re≤ 5; however, flow starts to separate atRe = 6. Therefore, the onset of flow separation exitsbetween Re = 5 and 6. The wake length increases asthe Reynolds number increases; however, the totaldrag coefficient decreases with increasing value ofthe Reynolds number in the steady flow regime(1≤Re≤ 46). The transition from steady regime tounsteady regime occurs between Re = 46 and 47.Finally, the simple correlation of the average cylinderNusselt number is reported for the range 1≤Re≤ 46and Pr = 0.7.

NOMENCLATURE

a Rear side of a taperedtrapezoidal cylinder, m

b Front side of a taperedtrapezoidal cylinder, m

cp Specific heat of the fluid,J kg�1 K�1

CD Drag coefficient¼ 2FD=rU2

1b ¼ CDF þ CDP

� �CDF Friction drag coefficient

¼ 2FDF=rU21b

� �CDP Pressure drag coefficient

¼ 2FDP=rU21b

� �CDrms RMS value of the drag

coefficientCL Lift coefficient

¼ 2FL=rU21b

� �CLrms RMS value of the lift

coefficientf Vortex shedding frequency,

s�1

FD Drag force per unit length ofthe cylinder, Nm�1

FDF Frictional drag force per unitlength of the cylinder, Nm�1

FDP Pressure drag force per unitlength of the cylinder, Nm�1

FL Lift force per unit length ofthe cylinder, Nm�1

h Local heat transfer coefficient,Wm�2K�1

�h Average heat transfer coeffi-cient, Wm�2K�1

H Height of the computationaldomain, m

k Thermal conductivity of thefluid, Wm�2 K�1

L Length of the computationaldomain, m

Lr Wake length, mNu Local Nusselt number (=hb/k)Nu Average Nusselt number

¼ �hb=kð ÞNurms RMS value of the average

Nusselt numberp Pressure ¼ p�= rU2

1� ��

Pr Prandtl number (=mcp/k)Re Reynolds number (=rU1b/m)St Strouhal number (=fb/U1)T1 Temperature of the fluid at

the inlet, KT�w Constant wall temperature at

the surface of the cylinder, Kt Time (=t*/(b/U1))U1 Uniform velocity at the

inlet, m s�1

Vx Component of the velocity inthe x-direction ¼ V�

x =U1� �

Vy Component of the velocity inthe y-direction ð¼ V�

y =U1Þx Stream-wise coordinate

(=x*/b)Xd Downstream distance of the

cylinder, mXu Upstream distance of the

cylinder, my Transverse coordinate

(=y*/b)

GREEK SYMBOLS

θ Temperature(= T� � T1ð Þ= T�

w � T1� �

)m Viscosity of the fluid,

kgm�1 s�1

r Density of the fluid, kgm�3

SUPERSCRIPT

* Dimensional variable



REFERENCES

[1] R. Kahawita, P. Wang. Comput. Fluids, 2002; 31, 99–112.[2] S. Goujon-Durand, K. Renffer, J.E. Wesfreid. Phy. Rev.E,

1994; 50, 308–313.[3] T.S. Lee. Int. J. Numer. Methods Fluids, 1998; 26, 1181–1203.[4] Y.J. Chung, S.-H. Kang. Phys. Fluids, 2000; 12(5), 1251–1254.[5] X.B. Chen, P. Yu, S.H. Winoto, H.T. Low. Int. J. Numer.

Methods Heat Fluid Flow, 2009; 19(2), 223–241.[6] L. Baranyi. J. Comput. Appl. Mech., 2003; 4, 13–25.[7] B.N. Rajani, A. Kandasamy, S. Majumdar. Appl. Math.

Modelling, 2008; 33, 1228–1247.

[8] C.F. Lange, F. Durst, M. Breuer. J. Heat Mass Transfer,1998; 41, 3409–3430.

[9] A. Sharma, V. Eswaran. Numer. Heat Transf. A, 2004;45, 247–269.

[10] A.K. Dhiman, R.P. Chhabra, V. Eswaran. Chem. Eng. Res.Des., 2006; 84(A4), 300–310.

[11] L. O’Connor. Mech. Eng., 1991; 113, 46–49.[12] A.K. Dhiman, R.P. Chhabra, A. Sharma, V. Eswaran. Numer.

Heat Transf. A, 2006; 49, 717–731.[13] A.K. Sahu, R.P. Chhabra, V. Eswaran. J. Non-Newt. Fluid

Mech., 2009; 160, 157–167.[14] B.J.A. Zieilinska, J.E. Wesfreid. Phys. Fluids, 1996;

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Documents

Flow and heat transfer over a trapezoidal cylinder: steady and unsteady regimes