unsteady forced convection heat transfer over a semicircular cylinder at low reynolds numbers

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Unsteady Forced Convection HeatTransfer Over a Semicircular Cylinder atLow Reynolds NumbersDipankar Chatterjee a , Bittagopal Mondal a & Pabitra Halder aa Simulation and Modeling Laboratory, CSIR – Central MechanicalEngineering Research Institute , Durgapur , IndiaPublished online: 28 Jan 2013.

To cite this article: Dipankar Chatterjee , Bittagopal Mondal & Pabitra Halder (2013) UnsteadyForced Convection Heat Transfer Over a Semicircular Cylinder at Low Reynolds Numbers, NumericalHeat Transfer, Part A: Applications: An International Journal of Computation and Methodology, 63:6,411-429, DOI: 10.1080/10407782.2013.742733

To link to this article: http://dx.doi.org/10.1080/10407782.2013.742733

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UNSTEADY FORCED CONVECTION HEAT TRANSFEROVER A SEMICIRCULAR CYLINDER AT LOWREYNOLDS NUMBERS

Dipankar Chatterjee, Bittagopal Mondal, and Pabitra HalderSimulation and Modeling Laboratory, CSIR – Central MechanicalEngineering Research Institute, Durgapur, India

An unsteady two-dimensional numerical simulation is performed to investigate the laminar

forced convection heat transfer for flow past a semicircular cylinder in an unconfined

medium. The Reynolds number considered in this study ranges from 50 to 150 with a fixed

Prandtl number (Pr¼ 0.71). Two different configurations of the semicircular cylinder are

considered; one when the curved surface facing the flow and the other when the flat surface

facing the flow. Fictitious confining boundaries are chosen on the lateral sides of the com-

putational domain that makes the blockage ratio B¼ 5% in order to make the problem

computationally feasible. A finite volume-based technique is used for the numerical compu-

tation. The flow and heat transfer characteristics are analyzed with the streamline and

isotherm patterns at various Reynolds numbers. The dimensionless frequency of vortex

shedding (Strouhal number), drag coefficient, and Nusselt numbers are presented and

discussed. Substantial differences in the global flow and heat transfer quantities are

observed for the two different configurations of the obstacle chosen in the study. It is

observed that the heat transfer rate is enhanced substantially when the curved surface is

facing the flow in comparison to the case when the flat surface is facing the flow.

1. INTRODUCTION

Fluid flow and heat transfer around bluff obstacles have strong engineeringapplications and tremendous fundamental importance. In particular, the shape ofthe obstacle has significant influence on the transport processes and the resultingwake dynamics. In the present effort, fluid flow and heat transfer are analyzed overtwo different configurations of a semicircular cylinder with respect to the incomingflow. It should be mentioned here that the transport processes over semicircularcylinders has received very little attention in the contemporary literature. Such trans-port processes are relevant in the design and development of novel heat exchangerswith improved heat transfer characteristics and without a severe pressure drop [1],thermal processing of foodstuffs [2], electronics cooling, underwater vehicles witha flat bottom, polymer shaping operations, etc.

Received 24 July 2012; accepted 1 October 2012.

Address correspondence to Dipankar Chatterjee, Simulation and Modeling Laboratory, CSIR –

Central Mechanical Engineering Research Institute, Durgapur 713209, India. E-mail: d_chatterjee@

cmeri.res.in

Numerical Heat Transfer, Part A, 63: 411–429, 2013

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407782.2013.742733

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A gamut of literature is available pertaining to the heat transfer and fluid flowover a circular cylinder [3–12]; however, the corresponding body of knowledge for asemicircular cylinder is much less extensive. Kiya and Arie [13] performed a numeri-cal simulation for the 2-D viscous shear flow over a semicircular and semi-ellipticalprojection attached to a plane wall for the Reynolds number range of 0.1–100. Theydocumented extensive results on the geometrical shape of the front and rear vortices,drag coefficients, pressure, and shear distributions as a function of Reynolds num-ber. Forbes and atleagues [14, 15] demonstrated the dependence of the flow behaviorof a semicircular obstacle in the ideal 2-D-critical free surface flow. They reportedthat the downstream velocity increases with the radius of the semicircular obstacle.An experimental study was conducted by Boisaubert et al. [16] for the 3-D confinedflow over a semicircular cylinder for Reynolds numbers ranging between 60 to 600for the flat and curved surfaces facing the flow using a solid tracer visualization tech-nique. They found that the critical values of the Reynolds numbers for the onset ofvortex shedding as 140 and 190 for the two configurations of the obstacle, respect-ively. Furthermore, they introduced a splitter plate behind the rounded fore-bodyconfiguration to understand its effect on the flow behavior, and suggested the suit-ability of this arrangement as a flow-controlling device [17, 18]. Kotake and Suwa[19] studied the flow behavior and wake dynamics for a semicircular cylinder exposedto an uniform shear flow using the hydrogen bubble method (visualizationtechnique), and reported that the vortex forms only on the side of the body wherethe fluid velocity is lower. Iguchi and Terauchi [20] reported that the K�aarm�aan vortexstreet is significantly influenced by the angle of attack measured from the flowdirection. Koide et al. [21] experimentally investigated the influences of circular,

NOMENCLATURE

B blockage ratio, (¼ d=H)

CD drag coefficient

CL lift coefficient

D cylinder size, m

F vortex shedding frequency, Hz

H local heat transfer coefficient, W=m2K

H height of computational domain, m

K thermal conductivity of fluid, W=mK

L width of computational domain, m

Ld downstream length, m

Lr dimensionless recirculation length,

ð¼ Lr=dÞLu upstream length, m

Nu local Nusselt number, (¼ hd=k)

N normal direction

p dimensionless pressure, ð¼ �pp=q u21ÞPr Prandtl number, (¼g=a)Re Reynolds number, (¼ u1d=g)S spacing between cylinders

St Strouhal number, (¼ fd=u1)

t dimensionless time, ¼ ðu1�tt=dÞT temperature, (K)

T1 free stream temperature, K

TW cylinder temperature, K

u1 free stream velocity, m=s

u dimensionless axial velocity,

(¼ �uu=uinfty)

v dimensionless normal velocity,

(¼ �vv=u1)

x dimensionless axial coordinate,

(¼ �xx=d)

y dimensionless normal coordinate,

(¼ �yy=d)

a thermal diffusivity of fluid, m2=s

g kinematic viscosity of fluid, m2=s

h dimensionless temperature, ¼ T�T1TW�T1

q density of fluid, kg=m3

Subscripts

W cylinder surface

1 free stream

Superscripts

- dimensional quantity

412 D. CHATTERJEE ET AL.

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semicircular and triangular shaped cylindrical bodies on the lock-in (when the fre-quency of vortex shedding coincides with that of free oscillating cylinder) phenom-enon of K�aarm�aan vortex shedding using a mechanical oscillator for producing crossflow oscillations. They observed that the separation point moves freely for a circularcylinder; its movement is restricted to the upstream curved surface for a semicircularcylinder and fixed at the rear vertices for a triangular cylinder (apex facing up-stream). Sophy et al. [22] observed that the flow is unsteady at Reynolds numberRe¼ 65 for the flow past a semicircular cylinder (curved surface facing the flow)and the corresponding Strouhal number was found to be 0.166, which is 7% largerthan the case of a circular cylinder at the corresponding transition.

In the context of heat transfer, Nada and Mowad [23] demonstrated experi-mental observations on free convection heat transfer from a semicircular cylinder con-figured at different orientations. Subsequently, they reported experimental andnumerical results for forced convection and fluid flow around a semicircular cylinderfor the range Re¼ 2.2� 103� 4.5� 104 and orientation angle of 30, 60, and 90 fromflow direction [24]. They also formulated correlations for the average Nusselt numberin terms of Reynolds number and angle of attack. Hocking and Vanden-Broeck [25]have explored the effect of gravity on the wake region for the flow past a semicircularcylinder, when the flow is parallel to the flat surface. They observed narrow and longwake regions for the upward facing curved surface configuration and a wide and shortwake region for the downward facing curved surface configuration. Recently, Chandraand Chhabra [1, 26] in two back-to-back articles discussed fluid flow and forcedconvection heat transfer for different Prandtl numbers for Newtonian flow aroundsemicircular cylinder and the influence of power-law index on transitional Reynoldsnumbers for the non-Newtonian flow over a semicircular cylinder. Finally, someresults are also available for the size of the recirculation zone and formation of vorticeson a hollow semicircular cylinder with the curved surface facing the flow [27, 28].

From the above discussion, it can be concluded that very little information isavailable for the hydrodynamics and thermal transport around semicircular cylinderand no systematic effort has so far been dedicated to document the fluid flow andheat transfer characteristics for different configurations of the semicircular cylinder.Furthermore, an important agenda of the present work is to examine whether heattransfer enhancement is possible for two different configurations of the obstacle withrespect to the incoming flow under consideration. Accordingly, we aim here to inves-tigate numerically the unsteady fluid flow and heat transfer phenomena for flowaround a semicircular cylinder in an unconfined medium for the range of Reynoldsnumbers 50�Re� 150 with a fixed Prandtl number Pr¼ 0.71, and for two differentconfigurations of the obstacle: the curved surface facing the flow and the flat surfacefacing the flow. The numerical simulations are performed using a finite volume-based commercial CFD software Fluent.

2. PHYSICAL PROBLEM AND MATHEMATICAL FORMULATION

The geometry of the physical problem considered in this study along with thecoordinate system used is shown schematically in Figure 1. The system of interestconsists of a semicircular cylinder of diameter d (which is also the length scale)heated to a constant temperature TW and exposed to a uniform fluid stream of

HEAT TRANSFER OVER SEMICIRCULAR CYLINDER 413

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velocity u1 (considered as the velocity scale) and temperature T1. Two differentconfigurations (marked as C1 and C2 in Figures 1a and 1b respectively) are con-sidered; one when the flow impinging the curved surface (C1), and the other whenthe flow strikes the flat surface (C2). In order to make the problem computationallyfeasible, artificial confining boundaries are placed on the lateral sides of the compu-tational domain making the blockage ratio B¼ d=H¼ 0.05 (where H is the width ofthe computational domain). The upstream and downstream lengths of the computa-tional domain in this study are fixed as Lu¼ 10 d and Ld¼ 30 d, respectively. Thesevalues are chosen to reduce the effect of inlet and outlet boundary conditions onthe flow patterns in the vicinity of the cylinders.

2.1. Governing Equations

The dimensionless governing equations for the two-dimensional, laminar,incompressible flow, and heat transfer with constant thermophysical propertiesand negligible dissipation effect can be expressed in the following forms.

Continuity

quqx

þ qvqy

¼ 0 ð1Þ

Momentum

quqt

þ qðuuÞqx

þ qðuvÞqy

¼ � qpqx

þ 1

Re

q2 uqx2

þ q2 uqy2

!ð2aÞ

qvqt

þ qðuvÞqx

þ qðvvÞqy

¼ � qpqy

þ 1

Re

q2 vqx2

þ q2 vqy2

!ð2bÞ

Energy

qhqt

þ qðuhÞqx

þ qðvhÞqy

¼ 1

RePr

q2hqx2

þ q2hqy2

!ð3Þ

Figure 1. Schematic diagram of the computational domain. (a) C1: curved surface facing the flow, and (b)

C2: flat surface facing the flow.


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Where u, v are the dimensionless velocity components along x and y directions ofa Cartesian coordinate system, respectively, p is the dimensionless pressure, Re(¼ u1d=g) is the Reynolds number based on the cylinder dimension, h is the dimen-sionless temperature, Pr¼g=a is the Prandtl number, and t is the dimensionlesstime. The fluid properties are described by the density q, kinematic viscosity g,and thermal diffusivity a. The dimensionless variables are defined as follows.

u ¼ �uu

u1; v ¼ �vv

u1; x ¼ �xx

d; y ¼ �yy

d; p ¼ �pp

q u21; h ¼ T � T1

TW � T1; t ¼ u1�tt

dð4Þ

The corresponding dimensional quantities are denoted by �uu, �vv, �xx, �yy, �pp, and �tt,respectively.

2.2. Boundary Conditions

At the inlet a uniform flow is prescribed (u¼ 1, v¼ 0, h¼ 0). The exit boundaryis located sufficiently far downstream from the region of interest; hence, an outflowboundary condition (anticipating a fully developed flow situation) (qu=qx¼ qv=qx¼qh=qx¼ 0) is proposed at the outlet. A symmetry boundary condition considering africtionless wall (qu=qy¼ v¼ 0) and zero heat flux (qh=qy¼ 0) are imposed on theartificial confining boundaries, and a no-slip boundary condition with a uniform pre-scribed temperature (h¼ 1) are imposed on the cylinder surfaces. Pressure boundaryconditions are not explicitly required, since the solver extrapolates the pressure fromthe interior. The flow is assumed to start impulsively from rest.

The lift and drag coefficients are computed from the following.

CL ¼ CLP þ CLV ¼ 2FL

qu21 dð5Þ

CD ¼ CDP þ CDV ¼ 2FD

qu21 dð6Þ

Where CLP and CLV represent the lift coefficients due to pressure and viscous forces,respectively. Similarly, CDP and CDV represent the drag coefficients due to pressureand viscous forces. FL and FD are the lift and drag forces, respectively, acting on thecylinder surface. The periodicity in the flow field is characterized by the Strouhalnumber, which is the dimensionless vortex shedding frequency and is defined asSt¼ f d=u1, where f is the dimensional vortex shedding frequency.

The heat transfer between the cylinder and the surrounding fluid is calculatedby the Nusselt number. The local Nusselt number based on the cylinder dimension isgiven by the following.

Nu ¼ hd

k¼ � qh

qnð7Þ

Where h is the local heat transfer coefficient, k is the thermal conductivity of thefluid, and n is the direction normal to the cylinder surface. Surface average heat


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transfer is obtained by integrating the local Nusselt number along the cylinder face.The time average Nusselt number is computed by integrating the local value over alarge time period.

3. METHOD OF SOLUTION

The numerical simulation is performed by using the commercial CFD packageFLUENT [29]. FLUENT uses a control volume-based technique to solve thegoverning system of partial differential equations in a collocated grid system by con-structing a set of discrete algebraic equations with conservative properties. Thepressure-based numerical scheme, which solves the discretized governing equationssequentially, is selected. The laminar viscous model is selected to account for thelow Reynolds number flow consideration. An implicit scheme is applied to obtainthe discretized system of equations. The sequence updates the velocity field throughthe solution of the momentum equations using known values for pressure and velo-city. Then, it solves a Poisson-type pressure correction equation obtained by combin-ing the continuity and momentum equations. A quadratic upstream interpolationconvective kinetics (QUICK) scheme is used for spatial discretization of the convec-tive terms, and a central difference scheme is used for the diffusive terms of themomentum and energy equations. Pressure-implicit with splitting of operators(PISO) is selected as the pressure-velocity coupling scheme. The standard pressureinterpolation technique is used to interpolate the face pressure from the cell centervalues, and the time discretization is carried out by a second order accurate fullyimplicit scheme. Finally, the algebraic equations are solved by using the Gauss-Siedelpoint-by-point iterative method in conjunction with the algebraic multigrid (AMG)solver. The time step size is varied from 0.005 to 0.01 to determine an optimum valuethat results in less computational time, but produces sufficiently accurate results. Adimensionless time step size of 0.008 is finally used in the computation. The conver-gence criteria for the inner (time step) iterations are set as 10�8 for the discretizedcontinuity and momentum equations and 10�12 for the discretized energy equation.The convergence has been declared within a time step if the scaled residuals (basedon the relative error criteria) become lower than the predefined values of 10�8 and10�12 for the respective equations.

A nonuniform grid distribution having a close clustering of grid points in thevicinity of the cylinder (i.e., the regions with higher gradients of flow and thermalvariables) have been used in the present computation. Grids are generated usingthe grid generation package GAMBIT. Figure 2a shows the grid distribution inthe entire computational domain; whereas, an expanded view near the obstacle isdepicted in Figure 2b for configuration C1. Similar strategy is also, adopted for con-figuration C2, but not shown for the purpose of brevity. 100 and 150 grid points are,respectively, used on the flat and curved surfaces. The following different mesh sizesare adopted for the grid sensitivity analysis so as to check for the self consistency ofthe present problem.

. (M1) 82,000 quadrilateral cells, 82,580 nodes.




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In order to carry out the grid sensitivity study, computations are performed forRe¼ 100 and for both C1 and C2 configurations. It is observed that the simulationresults show a maximum difference of about 1.44% between (M1) and (M2), whileabout 0.54% between (M2) and (M3) types in terms of the time average CD and timeand surface average Nu. Accordingly, type (M2) mesh is preferred, keeping in viewthe accuracy of the results and computational convenience in the simulations. All thecomputations are carried out in an Intel XeonTM (2.40GHz) workstation computer.

4. RESULTS AND DISCUSSION

Air (Pr¼ 0.71) is considered the working fluid for the present study. Simula-tions are carried out for Re¼ 50–150 with a dimensionless time step size of 0.008,and results are presented at a dynamic steady state condition (30,000 time iterationsthat correspond to a dimensionless time instant of 240).

4.1. Numerical Verification

Since a commercial code is used in the present study, a rigorous numerical vali-dation is performed in order to establish the authenticity of the method. As a firstcase of validation, the flow and thermal characteristics are obtained for the steadyseparated flow over a circular cylinder. Figure 3 shows the dimensionless recircula-tion length, separation angle, overall drag coefficient, and surface average cylinderNusselt number under the forced convective condition and Re in the range10�Re� 40. The recirculation length is the streamwise distance from the rear stag-nation point of the cylinder to the re-attachment point for the near closed streamlinealong the wake centerline (i.e., the wake stagnation point). The separation angle (u)is the angle measured from the rear stagnation point on the cylinder where thevorticity vanishes. It is evident from Figure 3a that the recirculation length increaseslinearly with increasing Reynolds number over the range of interest. The separationangle also increases with increasing Re, as shown in Figure 3b. The above observa-tions are in close agreement with Takami and Keller [30] and Dennis and Chang [31].The overall drag coefficient decreases with Re (refer to Figure 3c) and the variation is

Figure 2. Computational mesh: (a) entire domain, and (b) expanded view near the obstacle for configur-

ation C1.


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in good agreement with Henderson [32] and Posdziech and Grundmann [33],although they have computed for slightly different blockage ratios. The surface aver-age cylinder Nusselt number (Nuavg) (Figure 3d) is seen to increase monotonicallywith Re, and the results are in excellent agreement with Dennis et al. [34], Jafroudiand Yang [35], and Apelt and Ledwich [36].

As a further verification, the transport quantities obtained for flow over a semi-circular cylinder with the curved surface facing the incoming flow are compared withChandra and Chhabra [1] in Table 1. It can be observed from Table 1 that the dragcoefficient decreases with Re in the chosen range, the dimensionless recirculationlength and the average Nusselt number increases monotonically with Re. Again,the matching is quiet satisfactory.

4.2. Flow and Thermal Fields

Figures 4–6 shows the instantaneous streamlines, vorticity, and isotherm con-tours recorded at a dimensionless time instant of 240 for Re¼ 50, 100, and 150 andfor both C1 and C2 configurations. The positive and negative vortices are shown by

Figure 3. Variation of (a) dimensionless recirculation length (Lr), (b) separation angle (u), (c) overall dragcoefficient (CD), and (d) surface average cylinder Nusselt number (Nuavg) with Reynolds number (Re) for

the steady separated flow over a circular cylinder.


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the solid and dashed lines, respectively. The flow and thermal fields downstream inthe wake region behind the cylinder show an unsteady periodic nature for the chosenReynolds number range. The unsteadiness occurs due to the shedding of vorticesfrom the rear portion of the cylinder. Distinct features of the flow and thermal fieldsare clearly visible in the figures. One interesting thing to observe from the streamlineplots is that the flow separates at the front end corners of the cylinder for theconfiguration when the flat surface faces the incoming flow (C2); however, the sep-aration occurs earlier for the configuration when the curved surface faces theincoming flow (C1). The separated shear layers at relatively higher Reynolds num-bers are found to stretch, bend, and finally, disconnect into the wake leading to

Table 1. Validation of flow quantities with available literature for different Reynolds

numbers (for configuration C1)

Present calculation Chandra and Chhabra [1]

Re CD Lr Nu CD Lr Nu

5 3.798 0.4420 1.5894 3.832 0.4475 1.5789

10 2.712 0.8375 2.0052 2.710 0.8220 2.1053

20 1.995 1.5820 2.6789 1.993 1.5799 2.6974

30 1.645 2.3755 3.1102 1.691 2.3744 3.0921

40 1.498 3.1089 3.5565 1.520 3.1142 3.5526

Figure 4. Instantaneous streamlines for different Reynolds numbers.


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Figure 5. Instantaneous vorticity contours for different Reynolds numbers.

Figure 6. Instantaneous isotherms for different Reynolds numbers.


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two horizontal layers of vortices (refer to Figure 5). However, at lower Re, a singlestream of shear layer moves to the downstream. The two vortices form at the rearend vertices of the cylinder shed alternately in the wake. Two types of inviscid criticalpoints such as center and saddle, can be observed from the streamline plots. The centeris the location where the slope of the streamline becomes indefinite with the point ofzero velocity, and saddle refers to the point of start or end corresponding to a closedstreamline. The instantaneous vorticity contours show the positive and negative vor-ticity corresponding to counterclockwise and clockwise motion, represented by thesolid and dashed lines, alternately engulf the rear face of the obstacle. These layersexhibit a flapping motion, and the vortices remain in the same half about the centerline from where they are shed. The isotherm contour (Figure 6) reveals the formationof a temperature street behind the cylinder, which is very similar to the formation of theK�aarm�aan vortex street. Since the flow separates behind the rear face of the cylinder, aclose clustering of the isotherms are observed in the vicinity of the frontal faces, whichresults in high temperature gradients on those surfaces as compared to the rear sur-faces. As the Reynolds number increases, the distribution of the isotherms on the rearsurface is more affected than the side faces due to vortex shedding.

Figures 7 and 8 show, respectively, the instantaneous vorticity and C contoursaround the cylinder for a representative Reynolds number, Re¼ 100 at four consecu-tive time instants during a vortex shedding cycle for the curved surface facing theflow (C1) and the flat surface facing the flow (C2). It should be mentioned thatthe parameter C describes the accumulation of vorticity and vortex structureformation and is defined following Kieft et al. [37].

C ¼ 1

2ðr2

1 þ r22 � x2Þ ð8Þ

Where r1 ¼ quqx � qv

qy and r2 ¼ qvqx þ qu

qy are the shear rates and x ¼ qvqx � qu

qy is the vorticity.

A negative C (C< 0) indicates that the fluid motion is elliptic in nature signifyingaccumulation of vorticity with increasing local circulation; whereas, a positive C(C> 0) corresponds to hyperbolic fluid motion signifying stretching of vorticity thatprevents the growth of a coherent structure [38]. Accordingly, in the elliptic regionscoherent vortex structures originate and in the hyperbolic regions the strain rate isdominant, causing stretching and deformation of fluid elements. The vortex sheddingprocess is actually initiated by the formation of coherent vortex blob at the tip of thestrand. This vortex blob subsequently detaches from the strand through the initiationof a constriction process upstream of the strand tip and, finally, the detached vortexblob from the constriction area moves downstream of the cylinder and the vortexshedding starts. It is observed in Figures 7, 8 that the constriction of the vorticitystrand occurs in the area of C> 0, indicating a dominant strain rate. At this stage,the vorticity structure reaches to a developed state and eventually it becomes discon-nected from the vorticity strand and the vortex shedding occurs.

4.3. Flow Parameters

The time response of the lift coefficient signal for different Reynolds numberand two different configurations of the semicircular cylinder is presented in


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Figures 9a and 9b. In general, it is observed that as the Reynolds number increasesthe amplitude of oscillation also increases. This is due to the fact that the flow insta-bility increases as the Reynolds number increases. It is interesting to note that theamplitude of oscillation of the lift coefficient at any Reynolds number when the flatsurface is facing the flow (C2) is almost double to the corresponding value when thecurved surface is facing the flow (C1). The total drag coefficient is found to be anincreasing function of Re for both the cylinder configurations in the chosen rangeof Re, as shown in Figure 10. It should be mentioned that at very low Reynolds num-ber, when there is no flow separation, the drag is predominantly a friction drag.When flow separation starts the drag becomes a combination of friction and pressuredrag. At this stage, the pressure drag starts to increase slowly; whereas, the frictiondrag decreases sharply resulting a continuous drop in the total drag coefficient withincreasing Re. At relatively higher Re (Re> 40� 50), the onset of vortex shedding

Figure 7. Instantaneous vorticity and C plot for Re¼ 100 at t (a, e), tþ s=4 (b, f), tþ s=2 (c, g), and

tþ 3s=4 (d, h) for configuration C1.


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results a change in the flow pattern. The recirculation or bubble length which wasfound to increase in the range 5�Re� 40 (refer to Table 1), now found to decreasebeyond this range due to vortex shedding. Consequently, the drag coefficientincreases. Furthermore, similar to the amplitude of oscillation of lift coefficient, itis again observed that the drag in the case of configuration C2 is more than the cor-responding value for C1. To understand the partial contributions of pressure andviscous components on the total drag, Figure 10 is plotted to show the variationsof pressure (CDP) and viscous (CDV) drag components. It is observed that the press-ure drag contributes more to the total drag in comparison to the viscous drag and thepressure drag is much higher for the C2 configuration compared to the correspond-ing value of its counterpart. This is attributable to the fact that as the flow strikes thecurved surface for C1 configuration, the center of the surface is marked as the stag-nation point, the flow accelerates along the surface on either side of the center linepassing through the stagnation point with a clear development of boundary layer

Figure 8. Instantaneous vorticity and C plot for Re¼ 100 at t (a, e), tþ s=4 (b, f), tþ s=2 (c, g), and

tþ 3s=4 (d, h) for configuration C2.


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and, finally, separates close to the corners. However, for the C2 configuration thepressure of the incoming fluid is entirely distributed over the flat surface, the bound-ary layer does not develop at all, and the flow separates at the sharp corners.Consequently, the viscous drag is more for C1; whereas, the pressure drag issignificantly higher for C2. As in Figure 10, DCDP is much higher than DCDV and,accordingly, the C2 configuration experiences higher drag force.

Figure 11 represents the variation of the dimensionless vortex shedding frequ-ency (Strouhal number, St) with a Reynolds number for two different configurations

Figure 10. Variation of drag coefficient with Reynolds number for two different configurations.

Figure 9. Lift coefficient signal for different Reynolds numbers: (a) configuration C1, and (b) configur-

ation C2.


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of the semicircular cylinder. The frequency is obtained from the fast fourier trans-form (FFT) of the lift coefficient signals recorded at every time step. The frequencyincreases as usual with the Reynolds number in the range considered. The Strouhalfrequency is found to be more for the C1 configuration than the C2 configuration.This is due to fact that the hydrodynamic boundary layer is accelerated over thecurved surface of the cylinder for the C1 configuration, making the flow moreunstable; whereas, for the C2 configuration the flow is stabilized more as a resultof the blocking effect on the incoming flow by the flat surface facing the flow.

4.4. Local and Average Nusselt Number and Heat TransferEnhancement

The distribution of the local Nusselt number for different Reynolds numbersand for two different configurations of the cylinder is shown in Figures 12a and12b. The higher the Reynolds number, the higher the local Nusselt number and,consequently, the higher the heat transfer rate. The front stagnation point (pointB) for the C1 configuration experiences the highest heat transfer and, accordingly,the highest Nusselt number. The Nusselt number then decreases along the curvedsurface and it is minimum on the rear face (AC). Hence, the bulk of the total heattransfer occurs from the curved surface, since the areas under the curve differs bya factor of p. For the reverse configuration (C2), the flat surface of the cylinder isfacing the flow and the highest heat transfer is found to occur at the corners ofthe flat surface (BD). The distribution of the local Nusselt number is symmetricalong the rear curved surface (BCD).

Figure 11. Strouhal number as a function of Reynolds number for two different configurations.


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The variation of time and surface average Nusselt numbers with Reynoldsnumber for the two configurations of the cylinder is depicted in Figure 13. The posi-tive dependence of Nu with Re is obvious. Apart from this, it is interesting to observethat the heat transfer rate (and, hence, Nu) is larger for the C1 configuration in com-parison to the C2 configuration. Since the flat surface for the C2 configuration facingthe flow creates the obstruction towards the flow, the thermal boundary layer thick-ness will be thicker; whereas, a thinner thermal boundary layer will be developed on

Figure 13. Variation of time and surface average Nusselt numbers with Reynolds number for two different

configurations. Inset: % increase in heat transfer for C1 configuration at different Reynolds numbers.

Figure 12. Distribution of local Nusselt number on the cylinder surface for different Reynolds numbers:

(a) configuration C1, and (b) configuration C2.


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the curved surface facing the flow for the C1 configuration. Accordingly, the heattransfer rate will be more for the C1 configuration. In the inset of Figure 13, the heattransfer enhancement for the configuration when the curved surface of the cylinderfacing the flow (C1) is depicted for varying Reynolds numbers. A substantial increasein the heat transfer rate is observed for the C1 configuration and this percnt increasein heat transfer further increases with increasing Reynolds number. Fitting nonlinercurves following the least square method gives the following two correlations for theNusselt number as a function of Reynolds number for the Re range considered in thepresent study.

Nu ¼ 0:39792 Re0:59163 ðfor C1Þ ðwith a correlation coefficient R2 ¼ 0:9998Þ ð9Þ

Nu ¼ 0:52215 Re0:50127 ðfor C2Þðwith a correlation coefficientR2 ¼ 0:99939Þ ð10Þ

5. CONCLUSION

Numerical simulation for the unsteady forced convection heat transfer arounda semicircular cylinder for two different configurations with respect to the incomingflow in an unconfined medium is performed. The simulation is carried out for theReynolds number range 50�Re� 150 with a fixed Prandtl number (0.71) and twodifferent configurations of the cylinder: when the curved surface facing the flow,and when the flat surface facing the flow. The numerical simulation is performedusing the commercial software Fluent. The results obtained are found in good agree-ment with the available results in the literature for such classes of problems. Thecharacteristic behavior of the hydrodynamic and thermal fields is pictorially repre-sented by the streamline, vorticity, and isotherm contours. The global flow and heattransfer quantities such as the drag and lift coefficients, Strouhal number, andNusselt number are computed and discussed. It has been found that the flow andthermal fields are more unstable when the flow impinges on the curved surface(C1) rather that in the flat surface (C2). The global flow quantities, such as the dragand lift coefficients, are found more for the C2 configuration. The frequency of vor-tex shedding and heat transfer rate are obtained more for the C1 configuration.Finally, simple heat transfer correlations depicting the functional dependence ofthe Nusselt number with the Reynolds number are obtained for the chosen rangeof conditions.

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unsteady forced convection heat transfer over a semicircular cylinder at low reynolds numbers

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