Effect of Prandtl number and rotation on vortex shedding behind a circular cylinder subjected to cross buoyancy at subcritical Reynolds number

International Communications in Heat and Mass Transfer 70 (2016) 1–8

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Effect of Prandtl number and rotation on vortex shedding behind a circular cylinder subjected to cross buoyancy at subcritical Reynolds number☆ Dipankar Chatterjee ⁎, Chiranjit Sinha Simulation & Modeling Laboratory, CSIR, Central Mechanical Engineering Research Institute, Durgapur 713209, India

a r t i c l e

i n f o

Available online 21 November 2015 Keywords: Stationary/rotating cylinder Thermal buoyancy Vortex shedding Numerical simulation Subcritical Reynolds number Prandtl number Stuart–Landau model

a b s t r a c t We perform a two-dimensional numerical simulation following a finite volume approach to understand the vortex shedding (VS) phenomena around a circular cylinder subjected to cross thermal buoyancy at a subcritical Reynolds number, Re = 40. The flow is considered in an unbounded medium. The cylinder may either be stationary or rotating about its centroidal axis. At the subcritical Reynolds number, the flow and thermal fields are steady without the superimposed thermal buoyancy (i.e. for pure forced flow). However, as the buoyancy parameter (Richardson number, Ri) increases, flow becomes unstable, and eventually, at some critical value of Ri, periodic VS is observed to characterize the flow and thermal fields. An extended Stuart–Landau model is used in this work for the accurate quantitative estimation of the critical Richardson number for the onset of VS. The above phenomena of VS with imposed buoyancy is strongly dependent on the type of the fluid being used. We quantify here the minimum heating requirement for the initiation of VS by choosing three different types of fluids having Prandtl numbers, Pr = 0.71, 7, and 100. The dimensionless rotational speed (Ω) ranges between 0 and 4. It is revealed that as Pr increases, heating requirement also increases for the initiation of VS. A possible explanation for the observation is provided. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction The control of boundary layer separation and vortex structure formation over bluff objects is important both fundamentally and from an engineering application point. A significant control of the wake structure can be achieved by subjecting the body to rotary oscillations [1]. Rotation to a bluff body such as a circular cylinder causes lift enhancement, drag reduction, and suppression of vortex-induced oscillations [2]. The thermal buoyancy is also believed to have a role of paramount importance on determining the wake dynamics. The buoyancy may sometimes trigger or suppress the vortex shedding (VS) depending upon the direction of the externally imposed flow vis-à-vis that of the buoyancy-induced flow. Three special cases with aiding, opposing, and cross-stream buoyancy were studied extensively for stationary cylinders [3]. A comprehensive numerical study was first reported by Biswas and Sarkar [4] where the effect of cross buoyancy on the VS phenomena behind a heated circular cylinder was demonstrated for the Reynolds number (Re) range 10 ≤ Re ≤ 45. Later on, Chatterjee and Mondal [5,6] and Chatterjee [7] carried out numerical studies on the effect of cross buoyancy for unconfined flow around circular and square cylinders. ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (D. Chatterjee).

http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.11.006 0735-1933/© 2015 Elsevier Ltd. All rights reserved.

They found that the critical Richardson number (Ri) for the onset of VS decreases and the dimensionless frequency of VS (Strouhal number) increases with Re. Recently, Chatterjee and Sinha [8] studied the effect of cross buoyancy on the unbounded flow around a rotating circular cylinder for the subcritical range of Reynolds number keeping the Prandtl number fixed. They reported that the rotating cylinder requires more heating compared to the stationary one for the initiation of VS. It is to be emphasized that the studies conducted so far incorporating the thermal buoyancy effect for flow around circular cylinders are all essentially for stationary bodies and few works are available pertaining to the forced convective effects on rotating circular body. Recently, Paramane and Sharma [9] examined the effect of cross-stream buoyancy on the VS around a rotating circular cylinder at Re = 40 and 100. Buoyancy-induced onset of VS was observed for stationary/rotating cylinder at Re = 40 and at higher Ω, buoyancy-induced secondary frequency was originated for both the Re. We aim here to numerically investigate the effect of Prandtl number on the initiation of VS for the free stream flow around a stationary and/or rotating circular cylinder subjected to crossstream buoyancy at the subcritical Re = 40. The subcritical Re is chosen so that the flow remains steady in the absence of thermal buoyancy. However, with the inception of thermal buoyancy, the flow changes drastically and the present effort aims to capture those phenomena. Specifically, the main objective is to obtain the critical Richardson numbers for the onset of VS in cross flow around a stationary/rotating circular cylinder for

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Nomenclature B CD d f g Gr h H k Ld Lu Pr Re Ri St T u v x y

Blockage ratio (−) Drag coefficient (−) Cylinder diameter, m Frequency, Hz Acceleration due to gravity, m/s2 Grashof number (−) Local heat transfer coefficient, W/m2K Width of computational domain, m Thermal conductivity of fluid, W/mK Downstream length, m Upstream length, m Prandtl number (−) Reynolds number (−) Richardson number (−) Strouhal number (−) Temperature, K Dimensionless velocity (−) Dimensionless velocity (−) Dimensionless coordinate (−) Dimensionless coordinate (−)

Greek symbols α Thermal diffusivity of fluid, m2/s β Volume expansion coefficient, 1/K ν Kinematic viscosity of fluid, m2/s Θ Dimensionless temperature (−) θ Polar angle (−) Amplification rate (−) σr Angular frequency, rad/s σi ω Rotational speed, rad/s Ω Dimensionless rotational speed ξ Vorticity, s−1 Subscripts avg Average cr Critical W Wall ∞ Free stream

different Prandtl numbers at a subcritical Re which is yet to be reported in the literature. The critical Re here signifies the transition from the steady twin vortex regime to the unsteady periodic regime in the absence of any rotational or buoyancy effects. 2. Mathematical formulation The problem under consideration is shown schematically in Fig. 1. A two-dimensional horizontal circular cylinder of diameter d (which is also the length scale) heated to a constant temperature Tw is exposed to a uniform free stream of velocity u∞ (considered as the velocity scale) and temperature T∞. The cylinder is either stationary or made to rotate in the counterclockwise direction about its axis at a uniform speed ω (the corresponding dimensionless rotational speed is computed from Ω = ω/(2u∞/d)). The overall transport process is a combined effect of forced flow along horizontal, cylinder rotation in angular and buoyancy along vertically upward directions. In order to make the problem computationally feasible, artificial confining boundaries are placed on the lateral sides of the computational domain making the blockage ratio B = d / H = 0.05 (where H is the width of the computational domain). The upstream and downstream lengths of the computational domain are fixed as Lu = 10d and Lu = 30d, respectively (a detailed study

Fig. 1. Schematic diagram of the computational domain.

on the domain size is outlined later). These values are chosen to reduce the effect of outlet and inlet boundary conditions on the flow patterns in the vicinity of the obstacle. The study is undertaken for Re = 40, Pr = 0.71, 7, and 100 (the Prandtl numbers are so chosen because air and water are very common fluids in engineering practice and further to understand the behavior of a higher Pr fluid such as the mixture of glycerin and water is considered) subjected to varying Richardson number and dimensionless rotational speed (0 ≤ Ω ≤ 4). The Reynolds number based on the cylinder dimension can be given as Re = u∞d/v, the Richardson number as Ri = Gr/Re2 with Gr = gβ(Tw − T∞)d3/v2) being the Grashof number and the dimensionless temperature as Θ = (T − T∞)/(TW − T∞). A laminar, incompressible flow of a Newtonian fluid with constant thermophysical properties along with Boussinesq approximation is assumed. We consider that the free stream is at the room temperature (≈300 K) and the maximum temperature of the cylinder is O(≈302 K) for air, O(≈307 K) for water, and O(≈310 K) for glycerin solution (since we simulate up to Ri ≈ 2, 7, 10 for air, water, and glycerin solution that corresponds to TW − T∞ = 2, 7, 10). This ensures that the temperature difference between the cylinder and the free stream is reasonably small such that the Boussinesq approximation can effectively be employed [10] and also the constant thermophysical property assumption is justified. Additionally, since the temperature variations of the thermophysical properties are not taken into account, the change in the Prandtl number with temperature is also neglected. The governing equations in the dimensionless form are available in [8] and not repeated here for the purpose of brevity (for various characteristic scales along with definitions of dimensionless parameters one can refer to [8]). The boundary conditions applied in this investigation can be described as follows: at the inlet a uniform flow (free stream) having Cartesian velocity components, u = 1, v = 0 with temperature Θ = 0, i.e., a Dirichlet type boundary condition is prescribed. At the exit plane, the outflow boundary where the diffusion fluxes in the direction normal to the exit surface are zero for all variables (∂u/∂x = ∂v/∂x = ∂Θ/∂x = 0) is proposed. Although, an Orlanski (convective) boundary condition for the unsteady flow would be more appropriate, however, since the outlet is located far downstream the outflow boundary condition would produce negligible error. On the artificial confining boundaries, a symmetry boundary condition i.e., a zero normal velocity and a zero normal gradients of all variables are prescribed. Finally, the dimensionless rotational speed (Ω) is prescribed along with a uniform temperature (Θ = 1) on the cylinder surface. The Cartesian velocity components on the cylinder surface can be obtained from u = −Ωsinθ, v = Ωcosθ where θ is the polar angle and θ = 0 coincides with the positive x-axis. Pressure boundary conditions are not explicitly required since the solver extrapolates the pressure from the interior. The aerodynamic responses and the heat transfer characteristics of the cylinder can be represented by the lift and drag coefficients and Nusselt number. The details regarding the computations of these quantities are outlined elsewhere [6].

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3. Numerical technique and verification

3

Table 2 Variation of drag coefficient and surface average Nusselt number with Ω (Pr = 0.71, Ri = 0).

The numerical simulation is performed by using the commercial CFD package Ansys Fluent [11]. The details in this regard is available in [5]. A O-type non-uniform grid distribution having a close clustering of grid points in the vicinity of the solid cylinder wall is used. The following different mesh sizes are adopted for the grid sensitivity analysis so as to check for the self-consistency of the present problem: (a) 56,900 quadrilateral elements, 57,490 nodes; (b) 93,000 quadrilateral elements, 93,570 nodes; (c) 128,800 quadrilateral elements, 129,640 nodes. Computations are performed for Ri = 0 and 2 and Ω = 0 and 4 for Pr = 0.71 and 100. It is observed that the simulation results show a maximum difference of about 4.92% between first and second, while about 1.93% between second and third types, in terms of the time average CD and time and surface average Nu (The surface average Nu is obtained by integrating the local Nusselt number Nuθ = hd/k = − ∂Θ/∂n along the cylinder surface, with n being the unit outward normal). Accordingly, type (b) mesh is preferred keeping in view the accuracy of the results and computational convenience in the simulations. In order to find an appropriate size of the computational domain such that it approximately mimics the real condition of unconfined flow, the upstream, downstream lengths and width are varied. Computations are once again performed for Ri = 0 and 2 and Ω = 0 and 4 for Pr = 0.71 and the results are depicted in Table 1 (a, b). From the table, it is evident that H = 20d, Lu = 10d, and Ld = 30d are the most optimum choices. For numerical verification, present results are compared with that of Stojkovic et al. [12] and Paramane and Sharma [9] when there is no superimposed thermal buoyancy. The comparisons are presented in Table 2. The comparison shows fairly good agreement with [12,9]. By looking into Table 2, it is ascertained that the rotation can be used as a drag reduction and heat transfer suppression technique. 4. Results and discussion The main purpose of this study is to investigate the influence of Prandtl number (i.e., different types of fluids) on the shedding phenomena around a stationary/rotating circular object in cross buoyancy situation at a subcritical Reynolds number Re = 40. However, to understand the interaction among the forced flow, rotation-induced flow, and buoyancyinduced flow, it is necessary to have a brief discussion on the global flow and thermal fields under such circumstances. Also, the reason for flow instability with the introduction of thermal buoyancy should be explained along with a discussion on the effect of rotation and buoyancy on major hydrodynamic and thermal transport quantities.

Ω

0 1 2 4

CD

Nu

Stojkovic et al. [12]

Present

Paramane and Sharma [9]

Present

1.51989 1.50230 1.30864 0.84088

1.5011 1.4982 1.3011 0.8408

3.25121 3.21739 3.03140 3.04831

3.28552 3.25125 3.14089 3.09556

4.1. Effect of rotation and buoyancy on flow and thermal fields At the subcritical Re for the unconfined flow around a stationary or rotating circular cylinder, the flow is inherently steady and stable in absence of thermal buoyancy effect. However, with increased heating of the obstacle, the buoyancy parameter (i.e., the Richardson number, Ri) increases and imparts instability to the flow and thermal fields which eventually leads to the initiation of VS at the subcritical Re. The Ri value for which the VS process initiated is considered as the critical Richardson number (Ricr). We now propose an extended Stuart–Landau (SL) model to estimate Ricr for the onset of VS. The development of unsteady periodic flow emanating from a steady flow, causing periodic VS, is a Hopf bifurcation that can be modeled by the SL equation. Traditionally, the SL theory has been applied to estimate the critical Re for the onset of VS in an isothermal flow past stationary obstacles [13–15]. Here we extend the same theory for non-isothermal flow, i.e., the flow past a stationary/rotating body with superimposed thermal buoyancy. We start with the truncated form of the Landau equation [16] for modeling the onset from steady to periodic flow (a Hopf bifurcation):   dA ¼ σ ðbÞA−λM 2 A þ O M 5 ; M ¼ jAj; A ¼ M exp½iΦðt Þ dt

ð1Þ

where A(t) is a characteristic complex amplitude associated with the fundamental frequency component, σ = σr + iσi and λ = λr + iλi are both complex constants. In Eq. (1), b represents a bifurcation parameter such as Ri for the present case of non-isothermal flow. Physically, σr and σi represent the amplification rate and angular frequency respectively of oscillations having infinitesimal amplitudes. An amplitude equation

Table 1 Domain size study (Pr = 0.71). (a) Domain width (H)

Nu

CD Ri = 0

10d 20d 30d

Ri = 2

Ri = 0

Ri = 2

Ω=0

Ω=4

Ω=0

Ω=4

Ω=0

Ω=4

Ω=0

Ω=4

1.6895 1.6141 1.5963

1.1235 0.9344 0.9231

1.1695 1.0283 1.0056

−1.6589 −1.4113 −1.3986

3.3654 3.3343 3.3146

3.1986 3.0911 2.9856

3.4256 3.3972 3.3965

3.1245 3.0761 2.9789

(b) Domain length

Nu

CD Ri = 0

Lu = 5d,Ld = 20d Lu = 10d,Ld = 30d Lu = 15dLd = 40d

Ri = 2

Ri = 0

Ri = 2

Ω=0

Ω=4

Ω=0

Ω=4

Ω=0

Ω=4

Ω=0

Ω=4

1.7012 1.6141 1.5835

1.1562 0.9344 0.9102

1.2345 1.0283 0.9965

−1.6426 −1.4113 −1.4012

3.5602 3.3343 3.3201

3.1645 3.0911 3.0723

3.4856 3.3972 3.3862

3.2564 3.0761 2.9865

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and a phase equation can be constructed from Eq. (1) which are basically the real and imaginary parts of the equation: dM ¼ σ r M−λr M 3 dt

ð2aÞ

dΦ ð2bÞ ¼ σ i −λi M2 dt This model has applicability if the onset of VS at the critical bifurcation parameter behaves as a Hopf bifurcation. In a linear regime, the second term on the right-hand side of Eq. (2) is negligible and the disturbances grow in the usual exponential manner at a rate that depends on the magnitude of σr. As the amplitude increases, nonlinear effects become more important and the amplitude saturates to a certain level. The saturation state is given by the condition dM/dt = 0 and the corresponding saturation value becomes Msat = (σr/λr)0.5. Following Landau and Stuart, σr and σi can be expanded in terms of the bifurcation parameter (Ri) in a certain nontrivial neighborhood of the critical Ri as: σ r ðRiÞ ¼

  dσ r ðRi−Ricr Þ þ O jRi−Ricr j2 dRi

σ i ðRiÞ ¼ σ i ðRicr Þ þ

ð3aÞ

  dσ i ðRi−Ricr Þ þ O jRi−Ricr j2 dRi

ð3bÞ

where Ricr is defined by σr(Ricr) = 0. Eq. (3) shows that the growth or decay rate is a linear function of Ri in the vicinity of Ricr. An expression for saturation or limit-cycle amplitude can be obtained as:  Msat ¼

1 dσ r ðRi−Ricr Þ λr dRi

0:5 ð4Þ

Combining Eqs. (2) and (3) and noting that dΦ/dt = 2πf with f being the frequency of oscillations in the disturbed state, one can write: S¼

  1 dM dð ln MÞ ¼ ¼ σ r −λr M 2 ¼ σ r 1−R2 ; M dt dt

R¼

M M sat

  dΦ λ ¼ 2π f ¼ σ i −λi M 2 ¼ σ i − σ r i R2 λr dt

ð5aÞ

ð5bÞ

Eq. (5) implies that the instantaneous growth rate (S) and the instantaneous frequency (dΦ/dt) are linear functions of the square of

(a) Instantaneous growth rate vs. normalized amplitude squared at Re = 40, Ri = 1 and

the instantaneous amplitude (M). It is to be mentioned that from transient numerical simulation at Ri close to the onset, giving different values of σr, the onset value of Ri can be estimated by interpolation with Eq. (3). The following procedure is adopted for estimating the Ricr when the VS commences. The lift coefficient (a global variable in the flow) can be used as the signal (M) i.e., the Landau model variable in the analysis. As pointed out by Kumar and Biswas [15], the transverse velocity signal (local variable) at some sampling location can also be used as the Landau model variable. Following Kumar and Biswas [15], we use here the transverse velocity signal as the Landau model variable. At a particular Re, for each Ri close to the onset of VS, the value of growth or decay rate (σr) is found from Eq. (5a) by fitting a straight line in the plot of the dimensionless instantaneous growth rate S vs. dimensionless instantaneous amplitude squared R2. Fig. 2 is plotted in this regard at a representative Re = 40 and Ω = 1. Fig. 2a shows the S vs. R2 plot for Ri = 1. Similar plots can be obtained for other values of Ri also. The value of σr is equal to the value of S at a location where the fitting line crosses the S axis (where R = 0). By this method, the values of σr are obtained for different Ri close to its critical value and at a particular Re. The same procedure can be used to obtain the critical frequency and other model constants in the SL equation. Finally, the σr values obtained for different Ri are plotted in Fig. 2b and the critical Ri is determined from the condition σr(Ricr) = 0. Thus, by fitting a line to data in Fig. 2b and finding the corresponding value of Ri where σr = 0, the critical Ri is obtained as Ricr = 0.7. For a visual appreciation of the shedding phenomena by the introduction of thermal buoyancy, it is instructive to look into the flow and thermal fields as depicted in Fig. 3 where the vorticity and isotherm are plotted for the stationary (Ω = 0) and rotating (Ω = 4) cylinder at the proximity of the critical Richardson number and for Pr = 0.71. The wake structure as observed is found to be steady for Ri b Ricr, whereas it shows unsteady periodic nature for Ri = Ricr. From the steady flow regime (i.e., for Ri b Ricr) as Ri increases from the zero value, the point of separation moves toward the leading edge causing an early separation and the recirculation region decreases. This can be attributed by the fact that as Ri increases, the thermal buoyancy or the natural convection effect will be more pronounced and the fluid will start moving toward the upward direction in the wake zone of the cylinder. Consequently, the incoming fluid will be accelerated underneath the cylinder owing to the mass conservation principle. Accordingly, when the flow approaches the cylinder, most of the fluid flows underneath the cylinder. Hence the mass flow rate becomes more beneath the cylinder than

(b) Linear global growth rate vs. Richardson 1

number at Re = 40 and

Fig. 2. Stuart–Landau analysis at Re = 40, Ω = 1, and Pr = 0.71.

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5

=

= Fig. 3. Contours of instantaneous vorticity and isotherm at different rotational speeds for Pr = 0.71.

that above it and the resulting flow structure becomes asymmetric. The vorticity contours show symmetric nature about the center-line for Ri b Ricr; however this symmetry breaks down for Ri ≥ Ricr due to the initiation of the vortex shedding phenomena. The isotherm lines also lose their symmetry due to the reason explained above. The initiation of VS with cylinder heating, at constant Re, is due to the increase in the vorticity production as a consequence of two buoyancy-induced effects: change in the flow and baroclinic vorticity production [17]. The baroclinic vorticity production (ξprod = Ri∂Θ/∂x) is the production of vorticity within the flow domain due to non-zero component of the temperature gradient perpendicular to the gravity vector. Fig. 4a shows the local vorticity production along the cylinder surface for Ω = 0, 4 and Ri = 0 and 1. This figure actually depicts the net effect of the interaction among the free stream, rotation-induced flow, and thermal buoyancy-induced flow. A symmetric variation about θ = 180° line is observed for Ri = 0 for both Ω = 0 and 4, whereas for Ri N 0, the variation becomes asymmetric. For the stationary cylinder (Ω = 0), the total positive and negative vorticity, obtained by integrating the local vorticity along the cylinder surface are 444 and −444 with a net value of 0 for Ri = 0 whereas for Ri = 1, the respective values are 589 and −295 with a net value of 294. For the cylinder rotating in the

a

counterclockwise direction (say at Ω = 4), the total positive and negative vorticity are 748 and −1309 with a net value of −561 for Ri = 0 whereas for Ri = 1, the respective values are 947 and − 1265 with a net value of −318. The buoyancy causes the flow to increase (decrease) just below (above) the cylinder which results in an increase (decrease) of the positive (negative) vorticity. It is also observed from Fig. 7a that for rotating cylinder at higher Ω, the maximum positive as well as negative vorticity is more for buoyancy-induced flow. Furthermore, the increase is more for the negative vorticity magnitude above the cylinder in comparison to the corresponding positive counterpart below the cylinder. Additionally, the discrepancy in the local vorticity production with the inception of thermal buoyancy is attributable to the baroclinic vorticity production as depicted in Fig. 4b. The baroclinic vorticity production is positive in the front portion and negative in the rear portion of the cylinder at Ri = 1. The magnitude of the total positive vorticity is larger than the negative counterpart due to the clustering of isotherms in the front portion of the cylinder. It can further be interpreted that at a given Ω, as Ri increases from 0 to 1, the total positive vorticity increases whereas the negative vorticity decreases and the total negative vorticity is greater than the total positive vorticity for the cylinder rotating in counterclockwise direction in contrast to the stationary

b

Fig. 4. Variation of (a) local surface vorticity and (b) baroclinic vorticity production along the surface of the cylinder for Pr = 0.71.

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Fig. 5. (a) Critical Richardson number as a function of rotational speed for different Prandtl numbers; (b) stability diagram for different rotational speeds and Prandtl numbers.

cylinder. The total positive and negative vorticity are equal for the stationary cylinder at Ri = 0. Hence, at a subcritical Re, due to the inception of thermal buoyancy, an imbalance between the net positive and negative vorticity is experienced which increases progressively with subsequent increase in the strength of buoyancy. This cross-stream buoyancy which is in a direction perpendicular to that of the undisturbed flow and axis of the cylinder eventually causes the wake instability that leads to the onset of VS. 4.2. Effect of Prandtl number on VS In order to understand the effect of Prandtl number on the VS phenomena, we choose three different fluids such as air (Pr = 0.71), water (Pr = 7), and a solution of 60% glycerin and 40% water (Pr = 100) and compute the critical Richardson numbers for the onset of VS at Re = 40. Fig. 5a shows the results for different Ω. It can be observed that as Pr increases, the critical Ri increases, i.e., the high Pr fluid requires more heating for the onset of VS. Furthermore, as the rotational speed increases, the increase in critical Ri is more for high Pr fluids. By looking into the physical properties of the fluids under consideration, it is to be noted that the kinematic viscosity increases and thermal diffusivity decreases from air to glycerin solution. Apart from being an agent for

momentum diffusion, the kinematic viscosity is always a dissipative agent responsible for the eventual conversion of kinetic energy into heat. Thus, the effect of viscosity has always been to stabilize a flow apart from the case of Tollmien–Schlichting waves. On the other hand, the role of thermal diffusivity is always destabilizing for gravity-induced buoyancy-driven flow. This may be the reason for which the high Pr fluid becomes more stable and accordingly it requires more heating for the initiation of unstable VS process. Furthermore, since the rotation has a stabilizing effect, additional heating would be required to destabilize the flow. In order to categorically depict the stable and unstable zones of operation, Fig. 5b is plotted to show the stability map for different rotational speeds and for three different Prandtl numbers. The flow field is found stable (i.e., steady without VS) below the critical Ri value (shaded area in the figure), and beyond the critical Ri, the flow becomes unstable with VS. To substantiate the above findings, Fig. 6a is plotted to show the distribution of local vorticity on the cylinder surface for different Prandtl numbers at Ω = 4 and Ri = Ricr. The figure shows asymmetric distribution of the surface vorticity about θ = 180° for all Pr values with a net negative vorticity. The total positive and negative vorticity as obtained from integration of the local vorticity along the cylinder surface are shown in Fig. 6b. It is observed that both the total positive and negative

Fig. 6. Variation of (a) local surface vorticity and (b) total positive and negative vorticity for different Prandtl numbers at Ω = 4 and Ri = Ricr.

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7

Fig. 7. (a) Strouhal number and (b) time and surface average Nusselt number as functions of rotational speed for different Prandtl numbers at Ri = Ricr.

vorticity decreases with increase in the Prandtl number and the corresponding net vorticity (which is a negative quantity since the cylinder is rotating in the counterclockwise direction) increases. This increase in the net vorticity causes the flow instability which eventually results in the initiation of VS. The dimensionless vortex shedding frequency (Strouhal number, St = fd/u∞, with f being the frequency of vortex shedding) is plotted as a function of rotational speed for various Prandtl numbers at Ri = Ricr in Fig. 7a. The Strouhal number is computed from the fast Fourier transform (FFT) of the transverse velocity signal obtained at some sampling location in the domain. A sampling location of 2.5d downstream from the rear face of the cylinder along the horizontal x-axis is considered here. The Strouhal numbers corresponding to the critical Richardson numbers increases with rotational speed, reaches a maximum at Ω = 2 and thereafter decreases. Another interesting thing that can be observed from Fig. 7a is that the shedding frequency increases from Pr = 0.71 to Pr = 7 and then decreases to Pr = 100 corresponding to Ri = Ricr. There are actually two competing physical phenomenon, one is the heating that causes the boundary layer along the cylinder surface to be accelerated resulting in a subsequent enhancement of the shedding frequency, and the other is the rotation that acts as a stabilizing agent resulting in the decrease of shedding frequency. Up to a moderate rotation rate (say, Ω = 2), the heating wins over the rotation and consequently the frequency increases, however, for higher rotation rate, although the corresponding critical heating parameter is also high, the high rotation wins over the heating resulting in a subsequent decrease of the shedding frequency. Since the higher Pr fluid is more stable as discussed earlier, the shedding frequency decreases at very high Prandtl number (at Pr = 100). Fig. 7b shows the effect of Prandtl number on the time and surface average Nusselt number for various rotational speeds and at Ri = Ricr. With increase in the Prandtl number, the heat transfer increases as usual; however, it decreases with rotational speed as explained earlier. Fig. 7b suggests that the average Nusselt is pretty independent of Ω at Ri = Ricr for low to moderate Pr (Pr = 0.71 and 7). This is because of the fact that the critical Richardson number is not varying significantly at lower Pr values. However, at higher Pr (=100), since the critical Ri changes significantly with Ω, the Nusselt number does also the same. 5. Conclusion The influences of cylinder rotation, cross buoyancy, and Prandtl number on VS around a circular cylinder in the two-dimensional laminar flow regime are discussed in this work. The Reynolds number is fixed in the subcritical regime Re = 40 and three different Prandtl numbers, Pr = 0.71, 7, and 100 are considered keeping the dimensionless

rotational speed of the cylinder in the range 0 ≤ Ω ≤ 4 and subjected to varying Richardson numbers. The thermal buoyancy and the rotation are found to have competing effects. The buoyancy tries to make the flow unstable, whereas the rotation acts as a stabilizing agent. As a consequence of this competition, the flow which is inherently stable at the subcritical Reynolds number becomes unstable with initiation of VS at some critical heating parameter and that critical value increases with rotation rate. The critical Ri is computed from an extended Stuart–Landau model for non-isothermal flow. The initiation of shedding phenomena also strongly depends on the type of fluid being used. Overall, it can be concluded that the rotating cylinder needs more heating for the initiation of VS in comparison to a stationary cylinder. As such, the critical Richardson number for the onset of VS increases with the rotational speed of the cylinder. Furthermore, the high Prandtl number fluids requires more heating for the initiation of VS. The aerodynamic response shows that the shedding frequency increases at lower rotation rate and decreases at higher rotational speed. It also increases initially with Prandtl number and then decreases at higher Prandtl number. Finally, the heat transfer rate decreases with rotational speed at the critical Richardson number and it increases with increase in the Prandtl number.

References [1] P.T. Tokumaru, P.E. Dimotakis, Rotary oscillation control of cylinder wake, J. Fluid Mech. 224 (1991) 77–90. [2] V.J. Modi, Moving surface boundary-layer control: a review, J. Fluids Struct. 11 (1997) 627–663. [3] D. Chatterjee, Dual role of thermal buoyancy in controlling boundary layer separation around bluff obstacles, Int. Commun. Heat Mass Transfer 56 (2014) 152–158. [4] G. Biswas, S. Sarkar, Effect of thermal buoyancy on vortex shedding past a circular cylinder in cross-flow at low Reynolds numbers, Int. J. Heat Mass Transfer 52 (2009) 1897–1912. [5] D. Chatterjee, B. Mondal, Effect of thermal buoyancy on vortex shedding behind a square cylinder in cross flow at low Reynolds number, Int. J. Heat Mass Transfer 54 (2011) 5262–5274. [6] D. Chatterjee, B. Mondal, On the vortex shedding mechanism behind a circular cylinder subjected to cross buoyancy at low Reynolds numbers, Comput. Therm. Sci. 4 (2012) 23–38. [7] D. Chatterjee, Triggering vortex shedding by superimposed thermal buoyancy around bluff obstacles in cross-flow at low Reynolds numbers, Numer. Heat Transfer Part A 61 (2012) 800–806. [8] D. Chatterjee, C. Sinha, Influence of thermal buoyancy on vortex shedding behind a rotating circular cylinder in cross flow at subcritical Reynolds numbers, J. Heat Transfer Trans. ASME 135 (2014) 051704. [9] S.B. Paramane, A. Sharma, Effect of cross-stream buoyancy and rotation on the freestream flow and heat transfer across a cylinder, Int. J. Therm. Sci. 49 (2010) 2008–2025. [10] D.D. Gray, A. Giorgini, The validity of the Boussinesq approximation for liquids and gases, Int. J. Heat Mass Transfer 19 (1976) 545–551. [11] ANSYS Fluent 14.0 Theory Guide. , ANSYS Inc, 2009.

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[12] D. Stojkovic, M. Breuer, F. Durst, Effect of high rotation rates on the laminar flow around a circular cylinder, Phys. Fluids 14 (2002) 3160–3178. [13] T. Leweke, M. Provansal, Model for transition in bluff body wakes, Phys. Rev. Lett. 72 (1994) 3174–3177. [14] J. Dusek, P. Le Gal, P. Fraunie, A numerical and theoretical study of the first Hopf bifurcation in a cylinder wake, J. Fluid Mech. 264 (1994) 59–80. [15] N. Senthil Kumar, G. Biswas, A finite element study of the onset of vortex shedding in a flow past two-dimensional circular cylinder, Prog. Comput. Fluid Dyn. 8 (2008) 288–298.

[16] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Course of Theoretical Physics, vol.6, Pergamon Press, London, 1959. [17] R. Kieft, C. Rindt, A. van Steenhoven, G. van Heijst, On the wake structure behind a heated horizontal cylinder in cross-flow, J. Fluid Mech. 486 (2003) 189–211.