Transient fluid flow and heat transfer over a rotating circular cylinder near a wall subject to a single gust impulse

International Journal of Heat and Mass Transfer 126 (2018) 1178–1193

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Transient fluid flow and heat transfer over a rotating circular cylinder near a wall subject to a single gust impulse Rabia Hanif a, O.G. Bhatti a, M. Ebrahem a, S. Manzoor a,⇑, Muzaffar Ali a, N.A. Sheikh b a b

Mechanical Engineering Department, University of Engineering and Technology, Taxila, Pakistan Department of Mechanical Engineering, International Islamic University, Islamabad, Pakistan

a r t i c l e

i n f o

Article history: Received 28 February 2018 Received in revised form 10 May 2018 Accepted 11 May 2018

Keywords: Gust impulse Plane wall Forced convection heat transfer Rotating circular cylinder Average Nusselt number Gap to diameter ratio Numerical simulations

a b s t r a c t Numerical results are presented investigating the effect of a gust impulse on the transient fluid flow and forced convection heat transfer from a rotating circular cylinder near a plane boundary in the two-dimensional, in-compressible flow regime. Reynolds numbers of 200, 600 and 1000 have been studied for a fluid of Prandtl number 7. Starting from static, the steady non-dimensional rotation rate is varied up to a maximum value of 5.5, in the counter clockwise direction, such that ða 2 f0; 0:5; 1; 2; 2:5; 4:7; 4:9; 5; 5:5gÞ. Gap to diameter ratio for this work is fixed at 3. Typical governing equations namely continuity, momentum and energy have been solved using the Constant Wall Temperature (CWT) boundary condition. This work notes that higher rotation rate of the circular cylinder, in the second vortex shedding regime and slight perturbations in the flow may cause a resultant effect which leads to short term disruption in the plane wall boundary layer dynamics even at larger values of gap to diameter ratio. Moreover, the gust impulse superimposed to the mean flow at the domain inlet causes creation of temporary convection zones in the cylinder wake which have significant impact on the heat transfer from the cylinder surface. Variations in Strouhal number, vorticity contours, peak vorticity trajectory plots, temperature contours and Nusselt number distribution are presented and discussed in comparison with the existing literature. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction Flow physics and heat transfer characteristics over a rotating circular cylinder in the vicinity of a wall draws significance from the wide range of applications including piping network design near the seabed and cellular and molecular biology. Cells near blood vessel walls are exposed to an extremely dynamic environment in terms of fluid induced forces and ever evolving flow dynamics. Moreover, the efficacy of our immune response depends on how the platelets and leukocytes are transported from one point to another via rolling and sliding along a vessel wall. In a broader sense, the topic is also relevant in many particle – particle and particle – wall interactions. Moreover, sedimentation tanks used in chemical and mining industry, micro-carrier beads in bio-reactors and landing gear design are some other areas where proximity of the wall affects the flow by providing a surface along which the boundary layer grows. The resultant flow distribution is considerably different from a free stream flow scenario given the fact that free stream flow velocity is inhibited within the wall ⇑ Corresponding author. E-mail address: [email protected] (S. Manzoor). https://doi.org/10.1016/j.ijheatmasstransfer.2018.05.065 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.

boundary layer region. This inhibition may manifest in the form of decrease in drag force, vortex suppression from the cylinder and/or pairing of vortices shed from the cylinder with those generated due to wall boundary layer roll over. Mostly, the wall boundary layer thickness is assumed to be much smaller than the cylinder diameter. In addition to the Reynolds number of the flow, non-dimensional gap ratio becomes important in such flow situations. The Reynolds number is typically defined as Re ¼ qUl1 D based on approaching fluid density q, free stream velocity U1, Cylinder Diameter D and dynamic viscosity l. The gap to diameter ratio is represented as G/D where G is the distance between the cylinder and the wall. Complexities in the flow pattern arise due to coupled interactions of wall boundary layer and vortices shed from top and bottom of the cylinder. Such flow problems become even more interesting when the rotational aspect of the circular cylinder is taken into account in the presence of an upstream gust impulse. Firstly, considering only the static cylinder near a wall, Taneda [1] observed that the classic double rowed Karman vortex street, behind the cylinder, is reduced to a single row of shed vortices at very small values of gap to diameter ratio i.e. G/D  0.6. Angrilli et al. [2] conducted frequency measurements for the case of an

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Nomenclature a,0,1 b,0,1 c,0,1 cp CD CL CL(RMS) D G h k Nuavg Pr P p⁄ Re St t T1 Tw u U1 U

gust amplitudes gust peak coordinate gust peak center coordinate specific heat of the fluid [J kg1 K1] mean drag coefficient mean lift coefficient RMS of lift coefficient diameter of the circular cylinder (m) distance between wall and cylinder average heat transfer coefficient (Wm2 K1) thermal conductivity of fluid (Wm1 K1) average Nusselt number (=hD=k) ðÞ lc Prandtl number, Pr ¼ kp ðÞ non-dimensional pressure ðp =pU 21 Þ, ðÞ pressure (Pa) Reynolds number Re ¼ qUl1 D ; ðÞ Strouhal number time (s) temperature of the fluid at the inlet (K) constant wall temperature at the surface of cylinder (K) Longitudinal gust component (ms1) free-stream velocity (ms1) non-dimensional stream-wise velocity (¼ u=U 1 Þ, ðÞ

immersed cylinder placed one diameter away from a plane wall. A higher value of the vortex shedding frequency as compared to the flow over an unbounded cylinder, was recorded. Bearman and Zdravkovich [3] also conducted experiments to investigate the effects of gap to diameter ratio on the flow physics. Aerodynamic properties of the cylinder and the separation position were found to be affected by the presence of the wall as long as the gap to diameter ratio was less than equal to 0.3. In the sub-critical Reynolds number range, vortex shedding frequency has also been found to be affected by the wall proximity. However, as the Reynolds number is increased, the vortex shedding frequency seemingly becomes insensitive to the spacing from the plane wall, Price et al. [4]. The gap to diameter ratio is important to understand the lift coefficient behavior however, boundary layer thickness becomes more relevant while focusing on the drag coefficient, Zdravkovich [5]. In case of a static cylinder, the mean drag coefficient is known to be a monotonic function of the gap to diameter ratio. Grass et al. [6] noted a sharp suppression in vortex shedding frequency at gap to diameter values less than 0.3. This critical value of the gap to diameter ratio is however known to be a function of the flow’s Reynolds number as well as the cross section of the object under consideration. For example, the critical gap to diameter ratio for a square cylinder is 0.5, Taniguchi et al. [7], 0.35 for a triangular cylinder, Kamemoto et al. [8] and 0.55 for a two dimensional perpendicular plate, Everitt [9]. Another important parameter which affects the critical value of gap to diameter ratio is the plane wall boundary layer thickness, d. Taniguchi and Miyakoshi [10] observed that as the plane wall boundary layer thickness increases, the critical gap to diameter ratio, for a circular cylinder, at which vortex shedding ceased increases from 0.3 to 0.9. For very thin wall boundary layers, compared with the diameter of the cylinder, adverse pressure gradients cause the formation of separation bubbles both upstream and downstream of cylinder. Downstream separation cancels the vorticity in the wake region close to wall hence suppressed vortices, Grass et al. [6]. Hydrodynamic forces experienced by the cylinder are typically affected by both the gap to diameter ratio, G/D and the ratio of wall boundary layer thickness to the cylinder diameter, d/D. Increase in gap, G,

v V x, y X Y x⁄, y⁄

vertical gust component (ms1) non-dimensional cross stream velocity (¼ v =U 1 Þ stream-wise dimension of coordinates (m) non-dimensional stream-wise dimension of coordinates (=x=D) non-dimensional cross-stream dimension of coordinates (=y=D) instantaneous peak vortex location w.r.t x and y.

Greek symbols Dx a non-dimensional rotation rate, a ¼ 2U 1 q density of fluid TT 1 h non-dimensional temperature (h ¼ T W T 1 ). d wall boundary layer thickness l dynamic viscosity s non-dimensional time (=tUD1 ) x constant angular velocity of cylinder surface, (rad/sec) Abbreviations CWT Constant Wall Temperature SIMPLE Semi Implicit Method For Pressure Linked Equation UHF Uniform Heat Flux RMS Root Mean Square

or the boundary layer thickness, d, both result in an upward displacement of the front stagnation point on the cylinder surface. Offset in the location of the stagnation point consequently causes variations in the lift and drag forces. Lei et al. [11] note that the effect of varying G/D, on variations in lift and drag forces, is more prominent than d/D. Furthermore, vortex shedding suppresses at G/D of approximately 0.2–0.3 depending on the wall boundary layer thickness. Flow visualization employed by Chang et al. [12] revealed that the effect of d/D diminishes rapidly and vortex shedding becomes increasingly independent of the wall boundary layer at G/D  3. Price et al. [4] observed that compared with the flow over an isolated cylinder where the periodicity in the flow arises from the classic vortex shedding, outer shear layer from the cylinder surface still causes noticeable periodicity in the flow at very low values of G/D where typical vortex shedding is otherwise suppressed. This apparent suppression of vortex formation may be attributed to the fact that shear layer from the lower half of the cylinder interacts with the wall boundary layer at smaller values of G/D. Effect of positive vorticity in the lower shear layer is annulled by the vorticity in the wall shear layer. Shear layer on the cylinder surface thus, loses the strength to roll over and form vortices, Lei et al. [13] and Dipankar and Sengupta [14]. This coupling delays the shear layer instability in the wall boundary layer. Considering the case where the cylinder gap is greater than the wall boundary layer thickness (G/d  2.4), vortices shed from the cylinder induce momentum to the wall boundary layer, possibly triggering the transition to the turbulent state. Sarkar and Sarkar [15] regard this as an example of receptivity of external disturbances. Muk et al. [16] noted that increasing the wall roughness reduced the time averaged drag coefficient at the same Reynolds number and wall boundary layer thickness. As stated previously, cross section of the immersed object also plays a significant role in such problems. Bhattacharyya and Maiti [17] conducted numerical investigations for the case of a square cylinder near a stationary wall. Shear layer separating from the lower face of the square, reattached on the cylinder itself at gap to height ratios below the critical value. Moreover, drag force induced on the cylinder decreased with reduction in gap height.

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Considering the effect of non-dimensional rotation of the cylinder in such problems, a dimensionless parameter a, known as the rotation rate needs to be taken into account. The dimensionless Dx rotation rate is most commonly defined as a ¼ 2U , where D is 1 the characteristic length which in this case is the cylinder diameter, x is the rotational speed of the cylinder and U1 is the mean free stream velocity. Cheng et al. [18] evaluated the effect of a, G/D and Reynolds number on the vortex shedding frequency i.e. Strouhal number. Interestingly, value of critical gap ratio for vortex suppression increases with increase in the non-dimensional rotation rate of the cylinder. Wall boundary layer was observed to exert a stronger influence on the rotating cylinder wake as compared to the wake of a stationary cylinder. Moreover, the lift force in such scenarios is more susceptible to the gap effect as compared to the drag force. Cheng and Luo [19] classified the gap between a rotating cylinder and the plane wall on the basis of whether the wake flow is steady or aperiodic. Time averaged drag experienced by the cylinder was observed to be a monotonic function of the gap ratio for the case of a stationary cylinder. It does not, however, remain a monotonic function as the cylinder is rotated steadily. Assuming that the gap ratio is reduced to zero gives us the case of a cylinder rolling on a plane wall. The wake in such cases transitions from steady two dimensional to steady three dimensional with the rotation rate of the cylinder. This transition to the steady three dimensional state is attributed to an elliptic instability in the recirculation zones behind the rolling cylinder, Stewart et al. [20]. Direction of rotation of the cylinder also effects the flow physics. Rotational sense of the cylinder is typically referred to as either ‘forward rolling’ or ‘reverse rolling’. Assuming, the mean flow advances from left to right, the former refers to the situation where the cylinder is set to rotate in a counter clockwise direction such that flow at the top of the cylinder is in the opposite direction as that of the mean flow. Conversely, the later refers to the case when the cylinder rotates in the clockwise direction such that flow at the top of the cylinder is in the same direction as that of the mean flow. Higher rotation rates in the forward direction are prone to three dimensional instabilities even at lower Reynolds number. Onset of the three dimensional instabilities is however delayed to comparatively higher values of the Reynolds number, in the case of reverse rotating cylinders. Rao et al. [21] noted that reverse rotation of the cylinder suppresses vortex shedding at much lower values of the dimensionless rotation rate as compared to the forward rotating cylinder. Heat transfer in such flow problems has relevance in various engineering fields such as piping networks in thermal plants and shell and tube type heat exchangers. Aiba [22] developed an experimental setup to investigate heat transfer from a cylinder subject to such flow conditions i.e. near plane wall. Nusselt number at the forward stagnation point of the cylinder was found to be much less sensitive to G/D at a fixed Reynolds number as compared with the Nusselt number at the rear stagnation point. Singha et al. [23] observed the time histories of average Nusselt number at varying gap ratios. Reduction in gap to diameter ratio results in a localized jet action between the bottom of the cylinder and the wall surface. Such flow situations yield higher values of the average Nusselt number. For the case of a cylinder in contact with the plane wall, variations in the flow’s Reynolds number do not influence the Nusselt number behavior. Yoon et al. [24] investigated a slightly different problem involving a moving wall. Assuming that the wall moves with the same velocity as that of the free stream, effects of boundary layer thickness on the upstream velocity distribution may be neglected. Local maxima of the Nusselt number was noted to move downstream with decreasing gap to diameter ratio. Ryu et al. [25] report that for the case of a locally heated square cylinder, high speed flows reach a critical point of transition to

oscillatory flow at gap ratio of approximately 0.75. Low speed flows however, undergo this transition at gap ratio of approximately 1. Other authors Jiang et al. [26,27], Huang and Sung [28], Chakraborty et al. [29] and Rao et al. [30] have elaborated effects of motion of the plane wall, cylinder sliding against a plane wall and blockage ratio in the free stream in similar flow configurations. More recently, Afroz et al. [31] investigated the generation of adverse pressure gradients over flat plates using rotating cylinder. Existence of the separation bubble on the flat plate is commonly referred to as a manifestation of the adverse pressure gradient. Size of the separation bubble on the plane wall may be controlled by changing the gap to diameter ratio at fixed dimensionless rotation rate or conversely by increasing the dimensionless rotation rate for a fixed gap to diameter ratio. Furthermore, the same combination of variation in control parameters can be exploited for turbulent boundary layer separation over the plane wall, Afroz et al. [32]. Rotational motion of the cylinder also influences the trajectory of detached vortices. Anticlockwise rotation of the cylinder deflects the vortices away from the wall while clockwise rotation deflects the vortices towards the wall consequently leading to more effective diffusion of detached vortical structures from the cylinder, Shaafi et al. [33]. It is pertinent to note here that flow physics and heat transfer mechanisms for the case of a cylinder placed near a plane wall have been vastly investigated. However interestingly, the existing literature, in almost its entirety, deals with this problem assuming a uniform incident flow. This however, is almost never the case in real life situations. Rather, different complex flow conditions prevail in nature i.e. waves, oscillatory flow and pulsatile flow for example. This work aims at investigating the transient flow behavior for a situation where the mean flow has been superimposed with a single upstream gust impulse. Considering the case of an immersed rotating cylinder, the Nusselt number has been noted to increase in the vortex shedding regime and decrease in the vortex suppression regime, under the effect of a gust impulse, Ikhtiar et al. [34]. A single, steadily rotating circular cylinder near plane wall subjected to a single gust impulse is the focus of this work. Gap to diameter ratio is fixed at a larger value of 3. Constant wall temperature boundary condition is imposed on the cylinder surface to investigate the heat transfer physics. Findings from this work are presented and discussed using variations in Strouhal number, vorticity contours, peak vorticity trajectory plots, temperature contours and Nusselt number distribution. 2. Physical description of the problem This study investigates the effect of a gust impulse on free stream flow and forced convection heat transfer across a two dimensional steadily rotating circular cylinder in the vicinity of a plane wall. Reynolds number based on the cylinder diameter, D is varied between 80 and 1000, see Table 2 for example. Starting from a static cylinder, the non-dimensional rotation rate, a is imposed up to a maximum value of 5.5, such that a 2 f0; 0:5; 1; 2; 2:5; 4:7; 4:9; 5; 5:5g. Diameter to gap ratio, G/D is fixed at 3. Prandtl number in the investigation is fixed at 7. Flow domain is set as depicted in Fig. 1. Forced convection heat transfer is studied by imposing a constantly higher wall temperature on the cylinder i.e. Tw = 302 K compared to T1 = 300 K of the mean free stream flow. Initially, the flow velocity at inlet, U1, is assumed uniform. A classic Karman vortex street develops behind the cylinder as the flow interacts with the circular cylinder. This manifests as a limit cycle oscillation pattern in the time history of the cylinder’s lift coefficient, see Fig. 6. A two dimensional gust profile is now superimposed on the inlet velocity. Details of the gust profile appear in the preceding sections of this work. Findings are presented in

1181

non-dimensional time (s ¼ tUD1 ) and ‘h’ is the non-dimensional tem-

and the non-dimensional

3.1.1. Longitudinal

"  2 # tb u ¼ a  exp  c

ð5Þ

3.1.2. Vertical

v

"  "  2 # 2 # t  b0 t  b1 ¼ a0  exp  þ a1  exp  c0 c1

ð6Þ

Table 1 Grid independence for system parameters.

80D

0.02 1.57 0.24 1.07 46.00 0.24 1.03

0.03 1.69 0.24 1.09 44.62 0.02 1.52 0.24 0.98 39.91

80D 20D

0.24 0.97

0.03 1.63 0.24 1.00 39.73 0.02 1.49 0.23 0.89 33.44

80D 20D

0.23 0.90

0.03 1.59 0.24 0.94 33.93 0.02 1.45 0.23 0.77 26.39

80D 20D

0.23 0.78

0.03 1.44 0.23 0.80 27.06 0.18 0.24

Gust comprises of two rectangular components i.e. longitudinal and vertical. Both the components are established using Gaussian distribution function, in line with the Taylor’s Hypothesis implying that gust components behave as functions of time only:

Lei et al. [13]

3.1. Gust model

0.20 0.51

,

80D

k

Present Re 100

Nusselt number is defined as Nu ¼ Dx rotation rate is defined as a ¼ 2U . 1

hD k

lcp

80D

1

20D



as ðP ¼ qpU2 Þ, where p⁄ is the dimensional pressure. Reynolds number is defined as Re ¼ DqlU1 , Prandtl number is defined as Pr ¼

80D

Lei et al. [13]

1 ). The non-dimensional pressure ‘P’ is obtained perature (h ¼ TTT W T 1

0.17 0.18

‘V’ is the non-dimensional cross stream velocity (V ¼ Uv1 ), ‘s’ is the

0.02 1.39 0.21 0.48 18.02

Present Re 200

where ‘U’ is the non-dimensional stream wise velocity (U ¼ Uu1 ),

20D

ð4Þ

0.04 1.45 0.21 0.52 18.64

!

Present Re 80

@h @ðUhÞ @ðVhÞ 1 @2h @2h þ ¼ þ þ @s @X @Y RePr @X 2 @Y 2

ð3Þ

Lei et al. [13]

Energy equation

!

Present Re 400

@V @ðUVÞ @ðVVÞ @P 1 @2V @2V þ þ þ ¼ þ @s @X @Y @Y Re @X 2 @Y 2

ð2Þ

Lei et al. [13]

Y-momentum equation

!

20D

@U @ðUUÞ @ðVUÞ @P 1 @2U @2U þ ¼ þ þ þ @s @X @Y @X Re @X 2 @Y 2

Lei et al. [13]

X-momentum equation

0.03 1.41 0.16 0.21 12.76

ð1Þ

0.05 1.48 0.18 0.24 13.34

Present Re 600

Lei et al. [13]

As shown in figure, circular cylinder proximate to a wall experienced a uniform stream with a single upstream gust. For twodimensional incompressible stream dimensionless form of the Navier stokes and energy equations are mention below: Continuity Equation

0.02 1.44 0.16 0.13 11.39

3. Governing equations and boundary conditions

0.05 1.58 0.17 0.16 12.17

Present Re 800

terms of flow behavior and heat transfer characteristics in relation to the Reynolds number and the non-dimensional rotation rate.

CL CD St CL(RMS) Nuavg

Fig. 1. Schematic Diagram of the Problem, where G/D = 3.

@U @V þ ¼0 @X @Y

20D

Lei et al. [13]

Present Re 1000

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Fig. 2. (a) Vertical (b) Horizontal gust components.

Table 2 Validation of present flow parameters with literature at Re = 100 for G/D = 3. Parameter

Norberg [36]

Henderson [37]

Williamson [38]

Lei et al. [13]

Cao and Wan [39]

Tritton [40]

Present

CL CD St CL(RMS)

– – 0.164 0.227

– 1.35 – –

0.7 1.33 0.160–0.164 –

– 1.46 0.177 0.244

– 1.39 0.168 –

– 1.26 0.157–0.164 –

– 1.48 0.189 0.235

Peak amplitudes, time-wise location and span of both the components is programmed such that the gust arrives at the rotating circular cylinder only after the system has achieved the limit cycle oscillation state manifested in the time history of the lift coefficient. Moreover, gust shape mimics the hot wire recordings reported by Schwartz et al. [35]. Typical gust duration is 0.2 sec. Fig. 2 represents the profile of the two gust components schematically. Flow state at the inlet

U ¼ 1;

V ¼ 0 and h ¼ 0

ð7Þ

Flow state when gust generated

U1 ¼ U þ u þ v

ð8Þ

Flow state at the outlet

@U ¼ 0; @X

@V ¼ 0; @X

@h ¼0 @X

ð9Þ

Constant Wall Temperature (CWT) and no slip boundary condition established over the circular cylinder surface.

U ¼ a sinð/Þ;

V ¼ a cosð/Þ

h ¼ 1 ðfor CWTÞ

ð10Þ

Fig. 3. Complete Grid and close up view near circular cylinder next to a wall.

ð11Þ

The governing equations when solved by using the above mentioned set of boundary conditions yield the primitive variables i.e. velocity (U and V), pressure (P) and temperature (h) fields. Detailed analysis including vorticity distribution and local and average Nusselt number can be carried out using the information from these flow and thermal variables. 4. Grid independence & numerical details Domain and grid independence form essentially the backbone of a numerical investigation. In this regard, a comprehensive domain independence and grid independence study is conducted to establish reliability of the computed results. Considering the computational cost in terms of both time and required storage space, the chosen domain extends 10 diameters above the cylinder, 16 diameters upstream from the center of the cylinder and 20 diameters down streams of the center of the cylinder, in line with

Lei et al. [13]. Two distinct grids are generated, labelled as G1 and G2. G1 contains 27,320 nodes and 26,900 elements whereas G2 contains 50,660 nodes and 50,075 elements. Both the domains are discretized consistently using a multi-block structured grid. Grid independence is further verified by employing successive node distribution in the cylinder vicinity and the gap region between the wall and the cylinder. This also helps in controlling the element quality which is restricted to 2.6 in this study. Strouhal number, Nusselt number and lift and drag coefficients are compared with existing literature, Table 1. Pertinently, a comparison is developed for the case of a static cylinder, immersed in streamlined flow and placed at G/D of 3 from the plane wall. Reynolds number is varied between 80 and 1000 as mentioned in Table 1. Remarkable agreement between the two grid cases, for all the Reynolds numbers may be noted. Keeping in view the computational economy, grid G1 with fewer nodes and elements is chosen for this work.

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Transient two dimensional, pressure based segregated solver is used to solve the incompressible flow and forced convection heat transfer. Navier–Stokes and energy equations for no-slip and Constant Wall Temperature boundary conditions are solved by semi implicit method for pressure linked equation (SIMPLE). Second order upwind scheme is used to discretize the convective terms in the momentum equations, while the diffusive terms are discretized by central difference method. Convergence criterion of 109 is used for Continuity and x/y Momentum equations while 1012 is used for the Energy equation. Keeping in view the expected transient nature of flow physics, a lower time step size of 0.002 with 50 iterations per step is used. Simulations are run for a total 50,000 time steps allowing the system to regain its limit cycle state after the passage of the gust impulse, see lift time histories in Figs. 6 and 7. Fig. 1 represents a schematic of the problem whereas Fig. 3 depicts the complete grid and a close-up view of the circular cylinder. 5. Validation The present work is validated for flow and heat transfer parameters using the mean drag coefficient, CD, RMS lift coefficient, CLRMS, Strouhal number, St, and Average Nusselt number. Table 2 collocates a brief comparison. All the parameters are compared with the existing literature for the case of a static circular cylinder immersed in streamlined incident flow at a Reynolds number of 100. Considering Table 2, in the present work, value of the mean drag coefficient agrees to within 1.35%, Strouhal number agrees to within 6.35% and the root mean square of the lift coefficient agrees to within 3.7% of the value reported by Lei et al. [13]. Fig. 4a and b provides a more detailed comparison of RMS lift coeffi-

cient and Strouhal number with those reported by Lei et al. [13], respectively. Considering the former, a maximum deviation of 9.17% is noted at Reynolds number of 1000. For the later however, a maximum deviation of 7.85% from the literature values is recorded at Reynolds number of 200. Reasonable agreement with the existing literature is noted. Furthermore, Average Nusselt number is calculated for the same flow conditions and compared with the values proposed by Kramers [41], Whitaker [42], Churchill and Bernstein [43] and Zukauskas and Jakob [44], in Table 3. Kramers’ proposed correlation is reproduced as Eq. (12):

Nu ¼ 0:42Pr0:20 þ 0:57Re0:50 Pr 0:31

ð12Þ

Keeping in mind that the Prandtl number for this work is 7 whereas the Reynolds number ranges between 80 and 1000, substituting appropriate values in Eq. (12) yields the respective values for the average Nusselt number, see Table 3. Considering Reynolds number of 1000, Kramers’ correlation yields a value of 33.57 for the average Nusselt number as compared to 44.62, noted in the present work. Whitaker’s correlation, Whitaker [42], is reproduced in Eq. (13).

Nu ¼ 2 þ ð0:4Re0:5 þ 0:06Re2=3 ÞPr0:4 ½lf =lw 0:25

ð13Þ

Evidently, Whitaker’s correlation takes into account the changes in thermo-physical properties of the medium associated with the variations in temperature. mf and mw are the dynamic viscosity of the fluid in the free stream and in the cylinder’s vicinity respectively. However, as mentioned earlier, since the temperature difference in the present work has been maintained at 2 K, changes in dynamic viscosity due to temperature can be safely neglected.

(a)

(b)

Fig. 4. Comparison of RMS lift coefficient and Strouhal number with existing literature.

Table 3 Validation of average Nusselt number with literature. Reynolds number

Kramer’s [41]

Whitakers [42]

Churchill and Bernstein [43]

Zukauskas and Jakob [44]

Present work

80 100 200 400 600 800 1000

9.94 11.04 15.36 21.46 26.14 30.09 33.57

12.22 13.53 18.80 26.53 32.65 37.93 42.65

10.73 11.97 16.86 23.83 29.22 33.81 37.88

10.20 11.32 15.63 21.59 26.08 29.83 33.09

12.17 13.34 18.64 27.06 33.93 39.73 44.62

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Whitaker’s correlation gives a better comparison value of 42.64 at Reynolds number of 1000. Churchill & Bernstein [43] proposed the following correlation, Eq. (14):

"  5=8 #4=5 0:62Re1=2 Pr1=3 Re Nu ¼ 0:3 þ h 1 þ  2=3 i1=4 282; 000 1 þ 0:4 Pr

ð14Þ

However, comparing to the average Nusselt number value of 44.62 obtained in the present work at Reynolds number of 1000, this correlation yields 37.87. Zukauskas and Jakob [44] have also proposed a correlation, Eq. (15):

Nu ¼ 0:683Re0:466 Pr0:333

ð15Þ

This correlation provides a value of 33.09. Keeping in mind all the above, values of the average Nusselt number calculated in the present work, deviate from the various empirical correlations catalogued in Table 3. Maximum deviation of 34.8% is noted from the correlation proposed by Zukauskas and Jakob [44], Eq. (15), at a Reynolds number of 1000. It may also be noted in Table 3 that deviation in the values of average Nusselt number obtained via empirical correlations and those obtained numerically in the present work, grows steadily as the Reynolds number increases. Interestingly, comparison of average Nusselt number in this section has been limited to empirical correlations only. This brings to light the fact that, similar studies focusing on forced convection heat transfer for cylinders placed near plane walls are rare/hard to find. This work aims to fill the gap in literature by focusing on flow physics and heat transfer behavior for the case of a steadily rotating circular cylinder placed near a plane wall. Importantly, the comparison in this section validates the methodology employed in this work. 6. Discussion 6.1. Lift & drag coefficients Lift and Drag coefficients provide the most simplistic measure of evaluating pressure and viscous stresses in a fluid structure interaction system. Time averaged lift and drag coefficients for streamlined flow condition i.e. without gust, with varying nondimensional rotation rate are presented in Fig. 5. Both the lift

and drag coefficients demonstrate a stronger dependence on the non-dimensional rotation rate as compared to the flow’s Reynolds number. A dominant pressure component in the lift force results in a monotonic increase in the negative lift with respect to the increasing non-dimensional rotation rate, a, Fig. 5a. Moreover, decrease in the pressure drag causes a decrease in the overall drag coefficient in the lower to intermediate range of the nondimensional rotation rate. Reversal in behavior results in an increase in the drag coefficient at higher values of a, Fig. 5b. Detailed discussion regarding the behavior of individual components of each coefficient may be consulted in Sufyan et al. [45]. Figs. 6 and 7 provide a holistic overview of the present work. Fig. 6 represents the simpler scenario of streamlined inlet flow. Lift time histories for all the rotation rates have been put together. Classic limit cycle oscillation pattern may be noted for lower values of a. However, for a of 2.5, the lift flat lines. Still higher values of a represent a steady repetitive oscillatory pattern in the lift time history. Vorticity contours with each time history stress the evolution in the flow physics as the nondimensional rotation rate, a, increases. The negative resultant lift force experienced by the rotating cylinder is manifested as the asymmetry in the cylinder wake where the wake line is clearly offset from the abscissa into the first quadrant. Ikhtiar et al. [34] and Rana et al. [46] have previously reported similar asymmetry in the cylinder wake caused by the rotational aspect of the cylinder wall. Dark/bold line at the bottom of the contours represents the near plane wall. Keeping in mind the fact that gap to diameter ratio in this work is fixed at a relatively higher value of 3, the wall boundary layer demonstrates the classic liftoff physics as the non-dimensional rotation rate increases. The Magnus Effect caused due to the rotation of the circular cylinder may contribute towards inciting perturbations in the wall boundary layer resulting ultimately in the lift-off. Considering the flow situation with the gust impulse, distinctive kinks in the lift time histories in Fig. 7 correspond to the gust on-set time-wise. Amplitude and duration of the kinks may be regarded roughly as representative of the gust amplitude and duration. Time instances for vorticity contours in this case have been chosen to correspond with the gust impulse. When compared with their counterparts for the case of streamlined inlet flow, these vorticity contours present a remarkably different picture, which is, in fact the focus of this work. Local Nusselt number and instanta-

(a)

Fig. 5. Rotation rate effect on mean lift and drag coefficient

(b)

Re = 200,

Re = 600,

Re = 1000.

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(a)

(b)

(i)

(ii)

Fig. 6. Time averaged distribution on cylinder at different rotation rates without Gust (i) Pressure and (ii) Local Nusselt number.

neous pressure distributions at the cylinder circumference are depicted at the bottom of Figs. 6 and 7 respectively. The slight asymmetry in the property distributions may be attributed to the steady rotation effect of the circular cylinder.

6.2. Strouhal number Fig. 8 elaborates the effect of gust impulse on the vortex shedding frequency. Data points with thin lines in the figure rep-

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(a)

(b)

(i)

(ii)

Fig. 7. Time averaged Distribution on cylinder at different rotation rates with Gust (i) Pressure and (ii) Local Nusselt number.

resent the Strouhal number values for the streamlined flow while the data points with heavy lines represent the Strouhal number values when the gust impulse is superimposed on the mean flow. It is pertinent to note here that while the values for streamlined flow have been calculated after the system had achieved steady limit cycle oscillation state, see Fig. 6, values for the gust case have been obtained by considering the oscillat-

ing lift coefficient only within the short, pre-programmed gust window, Fig. 7. Interestingly, in the present work, the gust impulse consistently results in lower values of the Strouhal number as compared to the streamlined mean flow. This fact may be attributed to the differential kinetic energy imparted to the mean flow by the superimposition of the gust impulse. Shear layer on the cylinder surface is known to wrap itself more

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Re 200 without gust

Re 200 with gust

Re 600 without gust

Re 600 with gust

Re 1000 without gust

Re 1000 with gust

Fig. 8. Variation of Strouhal number with the increasing rotation rate.

tightly around the cylinder as the non-dimensional rotation rate increases causing a diminished effect of the gust impulse at higher rates of rotation. Detailed discussion on the dynamic system response to the gust impulses with varying extent of imposed differential kinetic energy is presented by Rana et al. [46]. This work seeks to investigate a slightly complex flow situation by adding yet, the near plane wall effect to the scenario. Flow physics and heat transfer characteristics for this updated flow situation are elaborated in the preceding sections. 6.3. Vorticity contours Vorticity contours provide the much needed insight into the flow physics near the circular cylinder. Fig. 9 depict flow physics in this work throughout the chosen values of the nondimensional rotation rate. As stated previously, bold lines at the bottom of each contour represent the near plane wall. Dashed circles in the contours have been placed such that separation from the cylinder surface varies as one diameter, two diameters and three diameters respectively. Solid lines represent positive vorticity whereas dashed lines represent negative vorticity within ±200 s1. Contours have been plotted for seven time instances. Starting from the left most column, instance ‘to’ correspond to the gust onset at the domain inlet. ‘t1’ represents the instance before the gust impulse reaches the front end of the circular cylinder. ‘t2’ and ‘t3’ mark the time when the circular cylinder is completely immersed in the gust impulse. ‘t4’ depicts the instance when the gust impulse just leaves the rear end of the circular cylinder. ‘t5’ and ‘t6’ show the effect of the gust impulse after it has travelled at least three diameters or more downstream of the circular cylinder. Starting with the case of a static circular cylinder i.e. a of 0, top row, Fig. 9, viscous boundary layer near the plane wall surface remains unchanged before the gust onset, at to, despite the system having achieved the limit cycle oscillation state marked by the classic Karman vortex street visible in the downstream vicinity of the cylinder. This observation is in line with literature in the sense that the chosen ‘G/D’ value, in this work (i.e. G/D of 3), is larger than the critical separation gap, see the introduction section for example. Gust impulse affects the flow domain in two ways. Firstly, introduction of the gust impulse at the inlet causes a sudden change in velocity distribution throughout the computational domain resulting in a noticeable asymmetry in the detached vortex street downstream of the circular cylinder, see instance t1. An early detachment of the vortical structure at ‘t1’ depicts a departure from the classic 2S mode of vortex shedding. Asymmetry in the wake

pattern is also highlighted in the trajectory plots in Fig. 10. Hollow symbols connected with solid lines represent the case of streamlined flow whereas, solid symbols connected with dashed lines represent the flow with the gust impulse. It is pertinent to note here that trajectories for chosen vortices have been plotted only for the time window corresponding to the time taken by the gust impulse to convect from the domain inlet to the outlet. Each data point in Fig. 10 represents the location of peak vorticity at respective time instances. Secondly, as the gust impulse progresses through the domain, the plane wall viscous boundary layer becomes increasingly unstable depicting separation and a classic roll over topology which eventually interacts with the shed vortical structure from the circular cylinder. This confluence of vortical flow packets bounces off the plane wall and is carried downstream as the gust impulse exits the flow domain. See for example the trajectory plot for vortex A in Fig. 10(a). Keeping in mind that the gust impulse has a very short time duration i.e. 0.2 sec, vortex shedding in the downstream vicinity of the circular cylinder returns to the 2S classic Karman mode while the viscous boundary layer on the plane wall surface re-attains its fully developed state, see instance t6. Rotational motion of an immersed circular cylinder is known to induce an asymmetry in the wake pattern. This may be attributed to the fact that cylinder rotation results in higher velocity gradient between the upper and lower halves causing higher values of shearing stress along the cylinder circumference. Compared with a static cylinder, higher shearing stress causes the classic viscous saddle points in the flow to transition to an inviscid state. These inviscid saddle points eventually merge in the direction governed by the rotational sense of the circular cylinder leading to the establishment of the classic enveloping vortex around the cylinder, Paramane and Sharma [47]. The rotating cylinder acts like a real vortex with a solid rotational core where the peak rotational velocity exists at the cylinder surface. Considering now, the vorticity contours for a of 0.5, the cylinder wake is already slightly asymmetrical at instance, to. This inherent asymmetry is also evident in Fig. 10(b). Rotational effect of the cylinder wall pulls the wake away from the wall into the first quadrant, in line with the above description. This asymmetry in the vortex street causes a net negative lift force on the rotating circular cylinder evident from the lift coefficient time history in Fig. 7. Moreover, the plane wall boundary layer has attained a steady fully developed state. Gust impulse in the flow domain causes a noticeable scatter in the downstream shed vortices evident at time, t1. Simultaneously, the shedding pattern departs from the standard asymmetric 2S mode. Considering time instances t2 and t3, vortices A and C merge together creating

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t0

t1

t2

t3

t4

t5

t6

α=0

α = 0.5

α=1

α=2

α = 2.5

α = 4.7

α = 4.9

α=5

α = 5.5 Fig. 9. Instantaneous vorticity contours ranges between ±200 s1 at Re = 1000, positive vorticity solid lines, negative vorticity dashed lines.

vortex A⁄. More interestingly, instead of bouncing off, vortex B rolls along the plane wall. As the gust impulse immerses the circular cylinder, scatter in the shed vortices becomes increasingly chaotic leading to a completely dismantled boundary layer on the plane wall. The system returns to its steady limit cycle state as the gust impulse convects downstream. Vortical scatter, plane wall boundary layer separation and roll over and consequent detachment can be noted in the same sequence for a of 1, 2 and 2.5, as well. Higher values of the non-dimensional rotation rate, i.e. a of 4.7 and a of 5 correspond ideally to the second vortex shedding regime, see for example Sufyan et al. [45]. Classic intermittent, asymmetric vortex shedding can be noted for the rotation rates before the gust onset. The gust impulse in this case causes an earlier vortex detachment from both the circular cylinder and the near plane wall. However, the typical bounceoff from the plane wall does not occur. The earlier shed vortex

from the cylinder is convected downstream in approximately the same fashion as is the case in streamlined flow. As the gust impulse leaves the flow domain, the viscous boundary layer returns to its periodic state as noted before the gust onset at time, to. Considering the case of streamlined flow, cylinder wake and plane wall viscous boundary layer are known to interact with one another only within a certain range of the gap to diameter ratio, G/D. For fixed values of the Reynolds number and the nondimensional rotation rate, Afroz et al. [31] have noted that cylinder wake interaction with the plane wall boundary layer becomes negligible at higher values of the gap to diameter ratio i.e. G/D > 2. This work however, adds to our understanding in this regard via two important aspects. Firstly, increased rotation rates of the circular cylinder in the second vortex shedding regime do lead to separation and roll over in the plane wall boundary layer even at higher gap to diameter ratios. Secondly, seemingly docile flow perturba-

R. Hanif et al. / International Journal of Heat and Mass Transfer 126 (2018) 1178–1193

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Fig. 10. Instantaneous vortex Location at Re = 1000. jA, dB, rC.

tions can lead to modified vortex shedding patterns in the cylinder wake which can communicate with the plane wall boundary layer leading to time-wise transient modifications in boundary layer

separation, bubble size and complete dismantling, even for a slowly rotating cylinder placed at higher G/D values, for example a of 1 at G/D of 3, as is the case in this work.

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t0

t1

t2

t3

t4

t5

t6

α=0

α = 0.5

α=1

α=2

α = 2.5

α = 4.7

α = 4.9

α=5

α = 5.5 Fig. 11. Effect of Gust excitation on Instantaneous Temperature contours (300  T  302 K) at various rotation rates for Re = 1000.

6.4. Temperature contours As stated previously, the rotating circular cylinder in this work has a surface temperature of 302 K imposed using the constant wall temperature boundary condition. Temperature contours presented here, enable us to observe the heat transfer patterns subject to the modified flow physics highlighted in the last section. A temperature range of 300  T  302 K is chosen. Considering the static cylinder, top row in Fig. 11, distribution of the swirling packets of high temperature fluid in the cylinder wake is analogous to the vortex detachment patterns noted via the vorticity contours in the last section. For example, the irregularity in the vortex shedding mode caused due to the gust impulse manifests in a similar fashion in the temperature contours for a of 1 and a of 2.5 at time instances t2 and t3 respectively. High temperature fluid zone stretches out, in an extended tongue like structure, even beyond three diameters due to the gust impulse in case of higher nondimensional rotation rates i.e. a of 4.7 and a of 5. Closer inspection of the temperature contours for these higher rotation rates shall reveal establishment of a secondary separation bubble on the circle surface in approximately the third quadrant. This separation bubble transitions into a vortical structure which detaches in quick

succession causing a noticeable scatter in the Strouhal number values for with and without gust impulse cases, Fig. 8. These high temperature fluid packets follow an approximately straight line in the cylinder wake. As noted earlier, the system re-attains its steady limit cycle state as the gust impulse convects downstream, outside the flow domain. Low frequency detachment of vortical high temperature fluid packets returns to the steady state. It may be noted here that perturbation caused in the streamlined flow due to the gust impulse causes a modified convection pattern in terms of early detachment and stretching of high temperature packets in the cylinder wake. Thus causing formation of localized higher temperature zones, of varying shapes and spatial extents, downstream of the circular cylinder well beyond the three diameter boundary. 6.5. Nusselt number Modified flow patterns in the cylinder wake and plane wall vicinity due to the gust impulse are expected to affect the heat transfer from the cylinder surface. Nusselt number is commonly used to analyze heat transfer at interfaces, surface of the immersed cylinder in this case. Ikhtiar et al. [34] and Rana et al. [46] have dis-

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R. Hanif et al. / International Journal of Heat and Mass Transfer 126 (2018) 1178–1193 Table 4 Comparison of the Average Nusselt Number with and without Gust.

a 0 0.5 1 2 2.5 4.7 4.9 5 5.5

Re = 200

Re = 600

Re = 1000

Without Gust

With Gust

Without Gust

With Gust

Without Gust

With Gust

7.2725 7.3026 6.9733 2.6914 3.531 3.2534 1.6084 1.5889 2.0225

7.3686 7.3689 7.0497 2.8015 3.6707 3.1512 1.5204 1.4478 2.1576

10.0926 10.7101 10.1754 6.6894 5.6726 2.5888 2.4735 2.4012 1.9572

10.5969 10.8818 10.2748 7.0589 6.2475 2.5318 2.2544 2.1467 1.6654

10.2534 10.9482 10.4674 6.743 5.9662 3.554 3.1828 2.9521 2.5984

10.5318 11.2132 10.6309 7.2989 6.5711 3.4816 3.4392 3.2955 2.3798

cussed the effect of gust impulse on Nusselt number for a wide range of non-dimensional rotation rate. As previously discussed, this work however, also takes into account the effect of near plane wall. Average Nusselt number has been calculated in the time window between to and t6, see Figs. 9 and 11 for example. Table 4 consolidates the calculated values of Average Nusselt Number for Reynolds numbers of 200, 600 and 1000 for chosen values of the non-dimensional rotation rate. It must be kept in mind here that the gap to diameter ratio in this work is fixed at 3. It may be noted here that, for given values of the non-dimensional rotation rate, the effect of gust impulse on the average Nusselt number increases with increasing the Reynolds number. Similarly, keeping the Reynolds number constant, higher rotation rates depict more sensitivity to the gust impulse in terms of values of the average Nusselt number. Results obtained for lower Reynolds number of 200 depict very little influence of the gust impulse on the values of the average Nusselt number except at higher values of the nondimensional rotation rate. Maximum deviation between the streamlined flow and gust impulse is noted at alpha of 5, which

t0

t1

t2

is about 10%. Higher values of Reynolds number i.e. 600 and 1000 demonstrate a clearer increased sensitivity of average Nusselt number to the gust impulse as the non-dimensional rotation rate increases. A similar trend of heightened gust sensitivity may also be noted for fixed values of the non-dimensional rotation rate at varying Reynolds number. Fig. 12 depicts the classic instantaneous Nusselt number distribution around the circular cylinder. Time instances have been chosen using the same criterion as employed in the vorticity and temperature contours, in the previous sections. Dashed lines in the polar plots represent streamlined flow whereas, solid lines represent the case of gust flow. Keeping in mind that Nusselt number is the ratio of convective to conductive heat transfer, the instantaneous Nusselt polar plots in this figure correspond well to the observations in the vorticity contours, Fig. 9. Starting from the static cylinder, top row Fig. 12a, the Nusselt number distributions for gust and without gust cases superimpose. As the gust impulse enters the flow domain, changes in the velocity field lead to remarkably evolved Nusselt number distribution around the cylin-

t3

t4

t5

α=0

α = 0.5

α=1

α=2

α = 2.5

Fig. 12a. Comparison of Nusselt number for various rotation rates at Re = 1000.

without gust,

with gust.

t6

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t0

t1

t2

t3

t4

t5

without gust,

with gust.

t6

α = 4.7

α = 4.9

α=5

α = 5.5

Fig. 12b. Comparison of Nusselt Number for various rotation rates at Re=1000.

der circumference as compared to the case of streamlined flow. Considering instances t1 and t2 which correspond to the departure from the classic 2S mode of vortex shedding, Fig. 9, the gust impulse causes a sudden asymmetry in the shear layer distribution around the cylinder surface. Creation and early detachment of the additional vortex in the cylinder wake enhances the convective mode of heat transport from the cylinder surface, thereby, resulting in higher values of instantaneous Nusselt number in the third quadrant. Further down the time line, t3 and t4 for example, instantaneous Nusselt number distribution for the gust case depicts noticeably different shapes as compared to the streamlined flow. This indicates a different distribution of strong convection zones in the cylinder wake, attributed to the gust impulse. Rotational motion of the circular cylinder casts an effect on the Nusselt number distribution too. Considering lower rates, for example a of 1 and a of 2.5, introduction of the gust impulse in the flow domain results in the establishment of wider convection zones around the cylinder. Distortion in the pressure distribution around the cylinder periphery manifests as remarkable shape shifts when compared with the streamlined flow. Interestingly, at higher rotation rates, a of 4.7 and a of 5 for example, wrapping of the shear layer around the cylinder surface dominates the physics, hence the absence of distortion in Nusselt number distribution, Fig. 12b. However, early detachment of vortices due to the gust impulse leads to the creation of larger convection zones in the cylinder vicinity for brief periods of time. It may be noted here that, introduction of the gust impulse in such systems causes considerable changes in instantaneous Nusselt number distribution around the cylinder circumference. These changes lead to the creation of temporary convection zones in the cylinder wake which have significant impact on the heat transfer from the cylinder surface.

7. Conclusions This work investigates the effect of a lateral gust impulse on fluid flow and forced convection heat transfer from a rotating circular

cylinder near a plane boundary in the two-dimensional, incompressible flow regime. Numerical simulations have been carried out at Reynolds numbers of 200, 600 and 1000 for a fluid of Prandtl number 7. Starting from static, the steady non-dimensional rotation rate is varied up to a maximum value of 5.5. Flow properties are studied using the lift and drag coefficients, Strouhal number, vorticity contours and trajectory plots of peak vorticity in the flow domain. Moreover, temperature contours and Nusselt number distribution is used to elaborate the heat transfer. A fixed gust profile corresponding to a maximum flow deflection of less than 8° is used in this work. This study reveals that:  Increased rotation rates of the circular cylinder in the second vortex shedding regime do lead to separation and roll over in the plane wall boundary layer even at higher gap to diameter ratios.  Seemingly docile flow perturbations can lead to modified vortex shedding patterns in the cylinder wake which can communicate with the plane wall boundary layer leading to time-wise transient modifications in boundary layer separation, bubble size and complete dismantling, even for a slowly rotating cylinder placed at higher G/D values, for example a of 1 at G/D of 3, as is the case in this work.  Perturbation caused in the streamlined flow due to the gust impulse causes a modified convection pattern in terms of early detachment and stretching of high temperature packets in the cylinder wake. Thus causing formation of localized higher temperature zones, of varying shapes and spatial extents, downstream of the circular cylinder well beyond the three diameter boundary.  Introduction of the gust impulse in such systems causes considerable changes in instantaneous Nusselt number distribution around the cylinder circumference. These changes lead to the creation of temporary convection zones in the cylinder wake which have significant impact on the heat transfer from the cylinder surface.  Viscous boundary layer lift-off from the plane wall even at larger gap to diameter ratios for higher rotation rates of the circu-

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