Periodic unsteady mixed convection in square enclosure induced by inner rotating circular cylinder with time-periodic pulsating temperature

International Journal of Heat and Mass Transfer 111 (2017) 1250–1259

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Periodic unsteady mixed convection in square enclosure induced by inner rotating circular cylinder with time-periodic pulsating temperature Tongsheng Wang a, Zhiheng Wang a, Guang Xi a, Zhu Huang a,b,⇑ a b

Department of Fluid Machinery and Engineering, School of Energy and Power Engineering, Xi’an Jiaotong University, No. 28 West Xianning Rood, Xi’an 710049, Shaanxi, PR China Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK

a r t i c l e

i n f o

Article history: Received 30 December 2016 Received in revised form 13 March 2017 Accepted 17 April 2017

Keywords: Unsteady mixed convection Time pulsating temperature Inner rotating cylinder Heat transfer enhancement Backward heat transfer

a b s t r a c t The periodic unsteady mixed convective flow and heat transfer in a square enclosure driven by a concentric rotating circular cylinder is investigated numerically. The top and bottom walls of the enclosure are adiabatic and the side walls keep lower constant temperature, while the temperature of the inner circular cylinder fluctuates periodically with time at higher averaged value. The mixed convection is driven by the temperature difference and the velocity of the inner cylinder. The two-dimensional mixed convection is simulated with high accuracy temporal spectral method and local radial basis functions method. The Richardson number is studied in the range 1
Ri
20, the temperature pulsating period ranges from 1 to 10,000 and the temperature pulsating amplitudes are d = 0.5, 1.0 and 1.5. The numerical results show that the fluid flow and heat transfer are strongly dependent on the pulsating temperature of inner cylinder. Comparing with the steady state mixed convection, the heat transfer is enhanced generally for the time-periodic unsteady mixed convection. Moreover, the phenomenon of backward heat transfer is discussed quantitatively. Also, the influence of pulsating temperature on the unsteady fluid flow and heat transfer are discussed and analysed. Ó 2017 Published by Elsevier Ltd.

1. Introduction Fluid flow and heat transfer in an enclosure with a rotating obstacle have been a subject of considerable interest to the scientists and researchers for several decades due to its numerous engineering applications such as drilling of oil wells [1,4,7,9]. In addition, applications for this case of combined natural convection and rotation may be extended to design of solar collectors [2], rotating condensers for sea water distillation [5] and transpiration cooling [3]. However, when the convective flow is excited by some techniques, the related physics and phenomena would be very interesting and helpful in many engineering designs such as the rotating shafts, rotating-tube heat exchangers [1,7], nuclear reactor fuel rods [1,6], rotary machines placed in confined regions such as silencers [3], steel suspension bridge cables [1,7,9], lubrication of journal bearings rotary machine [2], melting of a phase change

⇑ Corresponding author at: Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK. E-mail addresses: [email protected] (T. Wang), wangzhiheng@ mail.xjtu.edu.cn (Z. Wang), [email protected] (G. Xi), [email protected] (Z. Huang). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.04.075 0017-9310/Ó 2017 Published by Elsevier Ltd.

material in a rotating crucible [5], cooling of electrical components [6,8] and so on. Therefore, a large number of investigations on this issue have received much attention of the researchers and engineers. When the inner cylinder rotates, the forced convection would be induced and results in mixed convection which is of course more complicated and attractive than natural convection in the thermodynamics system. The effect of the rotating of the cylinder on fluid flow and heat transfer has been studied widely. Ghaddar et al. [3] simulated the fluid flow and heat transfer in an isothermal rectangular enclosure driven by a rotating cylinder with constant heat flux inside with spectral element method. The rotation of the obstacle enhanced the heat transfer at low Rayleigh number while weakened the heat transfer at high Rayleigh number. Kimura et al. [10] investigated the effect of rotating cylinder on heat transfer within a differentially heated square cavity experimentally. They concluded that the rotating cylinder suppressed the heat transfer rate of the enclosure in the low rotating speed. However, obvious heat transfer enhancement could be achieved in the high rotating speed. Lewis [11] and Fu et al. [12] investigated the steady flow induced by rotating cylinder within an enclosure. Furthermore, Selimefendigil et al. [13–15] comprehensively studied the

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Nomenclature g L NT Nu Nul n P Pr R Ra Re Ri t T u, v U, V

gravitational acceleration length of square enclosure number of equispaced time intervals overall Nusselt number local Nusselt number unit normal vector dimensionless pressure Prandtl number radius of circular cylinder Rayleigh number Reynolds number Richardson number dimensionless time temperature velocity components dimensionless velocity components

convective fluid flow and heat transfer of different fluids at different flow regions with a rotating cylinder, and compared the Nusselt number with different control parameter to investigate the pronounced enhancement of heat transfer. The above studies focused on the heat transfer enhancement or decreasement caused by the rotating cylinder. More recently, Wang et al. [16] discussed the entropy generation of the mixed convection using a local radial basis function method and concluded that the total entropy generation increases with Reynolds number and Richardson number generally. The convective flow and heat transfer in enclosure can be very interesting driven by other excitation which might induce new phenomena. One of the most simple and widely employed excitation is sinuosiodally varying temperature, which has been investigated extensively in unsteady natural convection in a cavity since first studied by Kazmierczak et al. [17]. They studied the effects of the amplitude and period of the hot wall temperature pulsation and found that the transient buoyancy-driven flow is periodic in time and back heat transfer through the warm sidewall is observed. Kwak et al. [18] investigated the unsteady natural convection and concluded that the maximum heat transfer rate can be achieved. Kalabin et al. [19] numerically studied the periodic natural convection in a tilted square cavity, in which the temperature of cold wall oscillates and the hot wall keeps constant. Cheikh et al. [20] investigated the aspect ratio effects on the convective flow, and found that the period-averaged heat transfer rate reaches a peak at certain period, and the back heat transfer from the fluid to the hot sidewall is noticed close to the enclosure top wall during part of the period. Zhang et al. [21–23] studied the conjugate conduction-natural convection in a 3-D enclosure with timeperiodic sidewall temperature. Numerical results revealed that the backward heat transfer exists in region close to the corners formed by either the top or bottom walls and the enclosure hot sidewall, and disappears gradually as the pulsating period increases. Huang et al. [24,25] discussed the natural convection with time-periodic boundary temperature comprehensively. The maximal heat transfer rate is observed in the case of large Rayleigh number and amplitude of pulsating sidewall temperature. The numerical results showed that backward heat transfer rate first decreases rapidly with s, then tends to zero or small constant value forecasted that the sidewall temperature pulsating amplitude is respectively smaller or larger than the period averaged sidewalls temperature difference. Inspired by the previous interesting

x, y X, Y

directions of Cartesian coordinate dimensionless Cartesian coordinates

Greek symbols thermal diffusivity of fluid b coefficient of volume expansion d temperature pulsating amplitude m kinematic viscosity of the fluid q density h dimensionless temperature s temperature pulsating period

a

Subscripts c cold h hot l local ref reference parameter

studies, more complex and realistic model has been investigated, Roslan et al. [26] investigated the unsteady natural convection induced by a sinusodially heated circular cylinder in a square enclosure numerically with COMSOL. Within their considered parameters, the fluid flow and heat transfer is definitely time periodic and the heat transfer rate follows a sinusoidal law; moreover, the enhancement of heat transfer is observed. Huang et al. [27] also studied the periodic unsteady natural convection numerically with high accuracy temporal pseudo spectral method and a local radial basis function method. They found the heat transfer enhancement and discussed the backward heat transfer quantitatively. The steady mixed convection in the configuration employed in present studied has been extensively investigated [3,4,28,29], while the time-periodic unsteady mixed convection induced by rotating cylinder and pulsating temperature boundary in square enclosure has yet been investigated to the authors’ best knowledge. Additionally, the heat transfer enhancement and backward heat transfer have been observed and analysed quantitatively for the mixed convection induced by rotating cylinder for the first time. To explore the fluid flow and heat transfer process, this paper presents a comprehensive study on the time periodic mixed convection. The high accuracy temporal pseudo spectral method which is very suitable for time periodic problem and the local radial basis function method are employed for the numerical simulations. Furthermore, the instantaneous distribution of isotherms and streamlines confirm the heat transfer enhancement and backward heat transfer and the fluid flow pattern changes in one period. The parameters are chosen as following, the Richardson number is 1
Ri
20, the temperature pulsating period and amplitude are 1
s
10,000 and d = 0.5, 1.0 and 1.5 respectively, while the Prandtl number and Reynolds number are fixed at Pr = 0.71 and Re = 40 respectively and the ratio between the radius of circular cylinder and length of the square enclosure is kept as 0.2.

2. Mathematical model and numerical methods 2.1. Governing equations and boundary conditions The non-dimensional governing equations for the twodimensional incompressible mixed convective flow based on Boussinesq approximation can be expressed in the primitive variables form as

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successful application [16,27,30,31]. Radial basis function (RBF) is a powerful tool for approximating multivariate functions on scattered data. Franke et al. [30] found that multi quadric (MQ) offers better interpolation for two dimensional scattered data than other RBFs. The multi quadric radial function is considered on in this paper and written as follow:

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 þ c2

uðrÞ ¼

ð4Þ

where r denotes the distance between the reference node and the supporting node [16], and c is shape parameter. For the given scattered nodes xi e Rd and the corresponding values u(xi), i = 1, 2, . . ., n, the RBF interpolation s(x) can be written as a linear combination form generally.

sðxÞ ¼

n X kj uðkx  xj kÞ

ð5Þ

j¼1

where n is the number of supporting nodes, {kj} is the vector of coefficient and |||| is the Euclidean norm in the equation. With the interpolation conditions: s(xi) = u(xi), i = 1, 2, . . ., n. The coefficients are obtained by solving the following linear matrix: Fig. 1. Schematic diagram of the physical problem.

@U @X

þ @V ¼0 @Y

@U @t

@P 1 þ U @U þ V @U ¼  @X þ Re @X @Y

 2 þ @@YU2
 @V @P 1 @2 V @2 V þ U @V þ V @V ¼  @Y þ Re þ Rih 2 þ 2 @t @X @Y @X @Y
2  @h @h @h 1 @ h @2 h þ U @X þ V @Y ¼ RePr @X2 þ @Y 2 @t

Ak ¼ u




@2 U @X 2

ð1Þ

ð6Þ

where Ai,j = u(||xi  xj||), i, j = 1, . . ., n. Since the iteration of coefficients kj is time-consuming. Sanyasiraju et al. [32] have combined the interpolations of Lagrange and RBF, which results that the original variable values are updated by the iteration. Lagrange interpolation is showed in Eq. (7) with Lagrange basis function l.

LðxÞ ¼

n X lj ðxÞuðxj Þ;

 lj ðxk Þ ¼ djk ¼

j¼1

where U, V, X, Y, P and h are dimensionless parameters from the Cartesian coordinates by using characteristic scales R, uref = xR, DT ¼ T h  T c and Th = DT + DT  d  sin (2pt/s)

X¼

P¼

x ; R

Y¼

p þ qgy ; qu2ref

y ; R

U¼

Re ¼

u ; uref

ðxRÞR

t

;

V¼

v uref

Ri ¼

;

Ra PrRe2

h¼

T  Tc Th  Tc

Prandtl number is defined as Pr = m/a, and Rayleigh number is defined as Ra = gbR3 (T h  Tc)/ ma. Here a, m, b and g are the fluid thermal diffusivity, kinematic viscosity, volume expansion coefficient and gravitational acceleration respectively. Ri represents the ratio of the buoyancy term to the flow gradient term which is known as Richardson number. Fig. 1 shows the computational domain and the Cartesian coordinating system (x and y). The employed boundary conditions are described as follows. The dimensionless temperature of the left and right walls of the square enclosure is kept constant hc = 0, while the top and bottom walls are kept adiabatic oh/oy = 0. The temperature of the heated circular cylinder is sinusoidally varying as hh = 1.0 + d  sin (2pt/s). No-slip boundary condition for velocity components is imposed on all the walls of enclosure. The dimensionless velocity on the inner rotating cylinder surface [(U2 + V2)1/2] equals to 1 and we can easily obtain X and Y direction velocity from U = sin f, V = cos f, where f is the polar angle and f = 0 coincides with positive X direction. 2.2. Numerical approaches 2.2.1. Spatial dicretization The spatial discretization is conducted with the proposed local radial basis function (RBF) method. The multiquadric radial basis function is employed in this paper due to the extensive and

ð7Þ

The derivative of L(x) can be written by differentiating Eq. (7).

DðLðxÞÞ ¼

n X Dðlj ðxÞÞuðxj Þ

ð8Þ

j¼1

ð2Þ

ð3Þ

0; k–j 1; k ¼ j

where lij = lj(xi) and D() denotes a linear differential operator. Using RBF interpolation on the Lagrange basis function lj(x) can obtain the following equation:

li ðxÞ ¼

n X

ki;j uðkx  xj kÞ i ¼ 1; . . . ; n

ð9Þ

j¼1

The linear matrix can be expressed in the form:

uk ¼ I T

ð10Þ T

T

With the condition of uk = I in Eq. (10) and D(l) = D(u)k from Eq. (9), it is easy to obtain D(l)u = D(u). The known matrices u and D(u) can be obtained by multi quadric radial function (4), the coefficient matrix D(l) can be obtained by solving the linear matrix D(l) u = D(u). With the known matrices u, D(u) and D(l), the spatial discretization of Navier-Stokes equation is easily performed with their supporting nodes in local RBF interpolation. 2.2.2. Temporal dicretization The time derivatives in governing equations are discretized by the high accuracy temporal pseudo spectral method [21–25,27], which is designed to solving the time periodic unsteady problems. Assumed that all the computed variables of the governing equation are pulsating periodically in time, and denotes the solution in one period at NT (NT is odd) equal time intervals:

  / ¼ /t0 ; /t1 ; . . . ; /tNT1 ;

ti ¼ i

s NT

ð11Þ

The direct and inverse Fourier transforms between the time domain variables and frequency domain variables are given by

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1 XNT1 tj ik2spjDt /^k ¼ / e j¼0 NT /tj ¼

XðNT1Þ=2 k¼ðNT1Þ=2

ð12Þ

2p /^k eik s jDt

ð13Þ

Substituting Eq. (12) into Eq. (13), the time derivatives of /tj are computed in the time domain involving the same variable at all the time levels

d/tl XNT1 t tj dlj / ¼ j¼0 dt

ð14Þ

where dtlj is the element of Fourier differentiation matrix that can be derived from Eqs. (12) and (13)

( t

dlj ¼

0 2p

s






1 ð1Þlj cosec 2

pðljÞ NT



l¼j l–j

ð15Þ

By discretizing the time derivatives with temporal pseudospectral method [33,34], the original time-varying unsteady problem is decomposed into several interrelated steady-state problems, and the values of the variables at different time levels are connected by the Fourier differentiation matrix during the iteration. Therefore, the solutions strictly satisfy the periodicity after the iteration converged. 2.2.3. Numerical solvers The discretized equations are solved by a semi-implicit solver based on pseudo time-stepping philosophy, which includes two basically steps in one iteration. First, the convective terms, time derivatives and source terms are explicitly marched in pseudo time by four-stage Runge-Kutta method. Second, the diffusive terms are implicitly calculated. The studies [21–25,27] describe the detail of the solver, please to refer to them if you are interested. 2.3. Definitions of characteristic parameters The heat transfer rate is represented by the Nusselt number. The local and surface averaged Nusselt numbers are defined as

Nul ðtÞ ¼

 Z @hðtÞ 1 Nul ðtÞdS ; NuðtÞ ¼ @n wall W

ð16Þ

where n is the normal direction with respect to the walls and W is the surface area of the walls. The periodic averaged Nusselt number is calculated as

Nu ¼

1

s

Z s

NuðtÞdt

ð17Þ

0

Comparing the steady state case, the ratio of heat transfer is defined as

Nu E¼ Nus

ð18Þ

3. Results and discussions Cheikh et al. [20] studied the unsteady periodic natural convection with the pulsating period of boundary temperature ranges from 1 to 1600 and Kwak et al. [18] investigated natural convection with time-varying temperature whose non-dimensional period varies from 0.2p to 200p. With these periods, the heat transfer enhancement has been observed. Kalabin et al. [19] simulated the inclined enclosure with periodic temperature fluctuating and found backward heat transfer when the pulsating amplitude is larger than 1. Zhang et al. [22] and Huang et al. [25] have already investigated the natural convection with time-periodic boundary temperature as the pulsating period ranges from 1 to 10,000. They

found that the backward heat transfer disappears for large temperature pulsating period when the pulsating amplitude d
1. In order to explore both the heat transfer enhancement the backward heat transfer comprehensively for time-periodic unsteady mixed convection, the pulsating period varies from 1 to 10,000 and d = 0.5, 1.0, 1.5 have been chosen with carefully consideration of previous studies. 3.1. Code validation The present temporal pseudo spectral, local radial basis function method and the semi-implicit solver are validated through the mesh sensitivity study which is validated by two steps. First, the mesh sensitivity of local radial basis function method is conducted for the steady case of the mixed convection meaning that the pulsating temperature vanishes, i.e. d = 0. Actually, both the validation and mesh sensitivity of the local radial basis function method have been performed carefully in Fig. 4 of our previous research [16] which focused on entropy generation. Here we present the mesh sensitivity of heat transfer rate, i.e. the Nusselt number, in Table 1. The node number of each enclosure wall is set at 30, 50, 70, 90, 110 and 130 respectively. Total node number within the computational domain can easily get from the node number of the enclosure wall, which are set at 1001, 2638, 5178, 8152, 12,160 and 16,754 respectively. It is observed from Table 1 that node independence is achieved with the node number of enclosure wall being 90. Hence, for the rest of the calculations, the node number of enclosure wall is chosen to be 90 and total node number is 8152 correspondingly. The numerical method is further validated through temporal mesh sensitivity study as NT increases from 7 to 15 in the simulation of time-periodic unsteady mixed convection with the parameters Ri = 10, s = 1.0 and d = 0.5. As shown in Table 2, the Nusselt number obtained at NT = 11 are very close to those obtained at NT = 13 and 15. Therefore, in the rest of present study, all the simulations of time-periodic unsteady mixed convection in square enclosure are performed on the grid system of 8152 nodes and NT = 11. 3.2. Heat transfer enhancement The variations of the periodic averaged Nusselt number, i.e. the total heat transfer rate, with the inner temperature pulsating

Table 1 Average Nusselt number for steady mixed convection with respect to total node number at Ri = 10.0 and d = 0. Node number on the wall

Total node number

Nus

30 50 70 90 110 130

1001 2638 5178 8152 12,160 16,754

5.05718 5.05875 5.06089 5.06484 5.06709 5.06841

Table 2 The total heat transfer rate for time-periodic mixed convection at Ri = 10, s = 10.0 and d = 0.5. NT

Nu

7 9 11 13 15

5.13155 5.13154 5.13158 5.13156 5.13158

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Ri = 1

Ri = 5

Ri = 10

Ri = 20

Fig. 2. Variations of Nusselt number with respect to s at all investigated values of d and Ri.

Fig. 3. The ratio of the Nusselt number obtained at s = 100 of periodic unsteady case to that of steady case for different d and Ri.

Fig. 4. Variation of Nu(t) with time at representative temperature pulsating periods for Ri = 20 and d = 1.

period at different Richardson numbers are shown in Fig. 2. Generally, Nu increases with temperature pulsating period monotonically and tends to constant value for large temperature pulsating period, which means the heat transfer enhancement is achieved comparing with the steady case. For Ri = 1, it is interesting that Nu is slightly larger than that of steady case for d = 0.5 and 1.0, while Nu is larger than that of steady case for d = 1.5 obviously. The heat transfer enhancement is obvious for large Richardson numbers Ri  5, implies the buoyancy effect plays a significant role

on the heat transfer. It is noteworthy that the total heat transfer rate increases with the temperature pulsating amplitude. Moreover, the Nusselt number almost keeps constant as the period increases beyond 100. Therefore, focus on the maximal heat transfer enhancement of the configuration in many engineering applications, the Nusselt number obtained at s = 100 is most interesting and its ratio to Nusselt number of the steady case is shown in Fig. 3. The enhancement increases with temperature pulsating amplitude and Richardson number, and the maximum enhancement is about 27.2% for Ri = 20 and d = 1.5.

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Fig. 4 presents the time varying surface averaged Nusselt number for the representative temperature pulsating periods at d = 1 and Ri = 20. Generally, Nu(t) varies sinusoidally with time due to the sinusoidal varying temperature for the Richardson number and temperature pulsating periods. The pulsating amplitude of Nu(t) decreases with temperature pulsating period apparently. It should be noted that Nu(t) is negative during some time instants for small periods, indicating that the heat transfers from the induced air to the heated inner circular cylinder which is treated as backward heat transfer usually. However, the backward heat transfer disappears for the large period since Nu(t) is greater than zero for s  100. It is interesting that there is phase lag between the sinusoidally varying Nu(t) and temperature for small period. It might be attributed to that it has not sufficient time to let the induced fluid flow and heat transfer surrounding the inner circular cylinder varying synchronously with the sinusoidal temperature. Moreover, the variations of Nu(t) are almost indiscernible and follow sinusoidal law synchronously with the temperature as the period increases beyond 100, indicating the fluid flow and heat transfer have sufficient time to follow the variation of pulsating temperature and the heat transfer rate keeps constant.

pulsating period for d = 1. However, the exact heat transfer from the induced fluid to the excited hot circular cylinder is important to understand the convective heat transfer process. Therefore, this phenomenon related to backward heat transfer rate is analysed quantitatively in this section. Fig. 5 shows the variations of periodic averaged backward heat transfer rate, represented by Nur, with temperature pulsating period s at different temperature pulsating amplitudes and Richardson numbers. It can be seen that Nur decreases with s and increases with d. Nur is almost the same for all the Richardson numbers which indicates the flow patterns of them are similar. As the temperature pulsating amplitude d
1.0, Nur decreases rapidly to zero at certain small s for all the Ri, which indicates the phenomenon of backward heat transfer disappears for that temperature pulsating period. The period corresponding to Nur = 0 increases with temperature pulsating amplitude. However, Nur decreases remarkable with s and then keeps a small constant as s > 100 for d = 1.5, meaning that the phenomenon of backward heat transfer always occurs and the rate of backward heat transfer is independent on s for large temperature pulsating period.

3.3. Backward heat transfer

Heat transfer enhancement and backward heat transfer have been discussed. In order to explore these phenomena deeply, the isotherms and flow pattern induced by the pulsating temperature in the enclosure is discussed in detail in this section.

The interesting phenomenon of backward heat transfer is depicted in Fig. 4 during some time instants in small temperature

Ri = 1

Ri = 10

3.4. Temperature distribution and flow pattern

Ri = 5

Ri = 20

Fig. 5. Variations of backward heat transfer rate Nur with respect to s at all investigated values of d and Ri.

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t = 0/11τ

t = 2/11τ

t = 4/11τ

t = 6/11τ

t = 8/11τ

t = 10/11τ

τ=1

τ=5

τ = 10

τ = 100

τ = 10000 Fig. 6. Instantaneous isotherm and streamline distributions for representative instants during one period at at d = 1 and Ri = 20 for different values of temperature pulsating period.

Firstly, the pulsating period of the temperature on the heat transfer is discussed. The instantaneous isotherms and streamlines in the enclosure at d = 1 and Ri = 20 for different temperature pulsating period are presented in Fig. 6. For s = 1, the temperature varies very frequently with time, the fluid filled in the enclosure could not follow the variation of temperature synchronically as the obvious phase lag shown in Fig. 4. The temperature is confined

in the zone near the circular cylinder due to not sufficient time and the isotherms are cluster near the inner cylinder. At the instant t = 2/11s, the isotherms are very cluster near inner cylinder and results in large Nusselt number. However, at t = 6/11s and 8/11s, the temperature of the ambient fluid is higher than that of inner cylinder which indicates the heat transfers from the fluid to the cylinder, i.e. backward heat transfer phenomenon occurs. The

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t = 0/11τ

t = 2/11τ

t = 4/11τ

t = 6/11τ

t = 8/11τ

t = 10/11τ

δ = 0.5

δ = 1.0

δ = 1.5 Fig. 7. Instantaneous isotherm and streamline distributions for representative instants during one period at s = 10 and Ri = 20 for different values of d.

distribution of isotherms shows the large variation amplitude of Nu(t) for s = 1, which explains the corresponding variation shown in Fig. 4. The fluid pattern is dominated by a pair of vortices locate at the left and right side of the inner circular cylinder respectively and the induced flow pattern varies slightly in the whole period. For s = 5 and 10, the temperature varies in the whole enclosure and the pulsating temperature has relative sufficient time to penetrate through the enclosure. The backward heat transfer is also obtained at t = 6/11s and 8/11s and the isotherms are quite sparse comparing with s = 1, which shows the decrease of backward heat transfer rate with s. The flow pattern is dominated by a pair vortices in the left and right side of cylinder from 2/11s to 6/11s, while the vortices in the left side of the cylinder almost disappear at the time instant t = 10/11s and 0/11s. It might be this variation of flow pattern during one period result in the heat transfer enhancement. As the period increases beyond 100, the distribution of isotherms and streamlines are almost the same, which indicates the same heat transfer rate. The vortices in the left side of the cylinder are very small at t = 6/11s and disappear at the time instant t = 8/11s and 10/11s. It is interesting for the instant t = 8/11s, the temperature of the inner circular cylinder is nearly zero, the corresponding induced flow pattern is dominated by the rotating cylinder where the main vortex driven the cylinder in the enclosure and two small vortices in the upper and lower right corner of the enclosure. Fig. 7 presents the instantaneous distribution of the isotherms and streamlines for different temperature pulsating amplitude at s = 10 and Ri = 20. For d = 0.5, the dimensionless temperature of inner cylinder is larger than 0.5 in the whole period, the temperature of the cylinder is larger than its surrounding fluid hence the

backward heat transfer does not occur. For d = 1.0, as discussed above, the weak backward heat transfer is observed at t = 6/11s and 8/11s. The distribution of streamlines shows that the flow patterns are similar for d
1.0. For large temperature pulsating amplitude d = 1.5, the temperature of inner cylinder is negative at t = 8/11s, which is much lower than that of surrounding fluid. Therefore, the backward heat transfer appears and results in that the fluid flows in opposite direction comparing with the main stream flow in the left side of the cylinder. It is interesting that a secondary vortex is generated in the upper left corners of enclosure. Furthermore, a pair of vortices in opposite direction of the main stream flow in the left corner of the enclosure are generated at t = 10/11s. As d increases, the variation range of temperature increases and the flow pattern might be changed significantly during the period, which might result in increase of heat transfer rate. The effect of Richardson number on the time periodic unsteady mixed convection is also discussed. Fig. 8 shows the instantaneous isotherms and streamlines in one period for different Richardson numbers at d = 1 and s = 100. The instantaneous isotherms and stream function are very similar during the period since that the heat transfer and fluid flow in the enclosure has sufficient time to follow the varying temperature at this temperature pulsating period. For Richardson numbers Ri = 1 and 5, the flow patterns are similar which are kept with one pair vortices in the right corner of the enclosure in the whole period. However, the size of right corner vortices increases with Ri. As Ri increases to 10, the buoyancy effect is significant since a pair vortices in the left side of the cylinder are generated at the time period from 2/11s to 4/11s, while the pair left vortices occur from 0/11s to 6/11s for Ri = 20. The flow

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t = 0/11τ

t = 2/11τ

t = 4/11τ

t = 6/11τ

t = 8/11τ

t = 10/11τ

Ri = 1

Ri = 5

Ri = 10

Ri = 20 Fig. 8. Instantaneous isotherm and streamline distributions for representative instants during one period at d = 1 and s = 100 for different Richardson numbers.

pattern induced by the pulsating temperature and the rotating circular cylinder in the enclosure changes with Ri obviously, which results in the increase of the heat transfer rate. 4. Conclusions The time periodic unsteady mixed convection induced by the rotating circular cylinder with sinusoidally varying and averagedhigh temperature inside a square enclosure is numerically investigated using high accuracy temporal pseudospectral and local Radial Basis Function method. The top and bottom walls of the enclosure are adiabatic while the side walls keep lower constant temperature. The effects of Richardson number (1
Ri
20) and time sinusoidal varying temperature (0.5
d
1.5 and 1
s
10,000) on the fluid flow and heat transfer are extensively studied at Pr = 0.71 and Re = 40. The following conclusions are concluded from the numerical results: (1) The heat transfer of the time-periodic unsteady mixed convection is enhanced comparing with steady state case

generally. The heat transfer rate increases with temperature pulsating period and gradually tends to a constant. The enhancement increases with Richardson number and temperature pulsating amplitude. (2) The backward heat transfer rate is calculated, as the temperature pulsating period increases, it decreases to zero for d
1 indicating the phenomenon of backward heat transfer disappear while it decreases to small constant for d > 1 meaning the backward heat transfer always exists. The backward heat transfer increases with temperature pulsating amplitude. (3) The instantaneous distribution of temperature confirms the heat transfer enhancement and backward heat transfer and the flow pattern changes in one period.

Acknowledgement The authors gratefully acknowledge the financial supports provided by National Natural Science Foundation of China (Nos.

T. Wang et al. / International Journal of Heat and Mass Transfer 111 (2017) 1250–1259

51236006 and 51576153) and China Scholarship Council grant 201506285130.

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