arXiv:1609.07960v1 [physics.flu-dyn] 26 Sep 2016 n hi utain.Teaalbeha xhnemodels exchange heat available The exchange momentum fluctuations. and their heat and an the geometrical on of parameters influence There- buoyancy the pattern. on wake ordered focused more is a attention rearrangement fore, in to flow and field due coefficients near occurs the transition force of the the fact of In flux. deviation heat standard the in tion h hneo pcn ai from ratio spacing of change the epc oteRcado ubri eotd o Ri For reported. with is number number Richardson Nusselt the and to de- coefficients respect the force of the the analysis in parametric of transitions a pendence the Several and in found are pattern described. patterns are flow flow field the far on and spacing near and buoyancy of effects e ocne pcn (s spacing center to ter otefre ovcini ouae ihteRichardson the with between modulated ranging Ri, is respec number, convection with convection forced natural the of to buoyancy effect opposing The an or considered. aiding is an orientation, array the to fsxcrua yidr taRyod ubrof number Reynolds of a number at Prandtl cylinders circular array an six in of spacing for and field buoyancy flow of the combination of particular character oscillatory the in transition a hyaeaindwt h at ceeain( acceleration Earth the with aligned are they nti ae efud ymaso ueia simulations, numerical of means by found, we paper this In ∗ orsodn author. Corresponding uynyefc nteflwptenadthe and pattern flow the on effect Buoyancy hra efrac fa ra fcircular of array an of performance thermal mi:[email protected] Email: NNsz ec,710Lce Italy Lecce, 73100 Lecce, sez. INFN 0 . 7 i rbn ,715Br,Italy Bari, 70125 4, Orabona via ahmtc n Management and Mathematics h yidr r s-hra and iso-thermal are cylinders The . = eateto Mechanics, of Department 3 rnec Fornarelli Francesco . oyehi fBari, of Polytechnic 6 − d eirpost-doc Senior 1 − and 3 4 . )aecniee.The considered. are d) 6 to 1 w auso cen- of values Two . 4 NNsz ec,710Lce Italy Lecce, 73100 Lecce, sez. INFN nue transi- a induces i rbn ,715Br,Italy Bari, 70125 4, Orabona via ahmtc n Management and Mathematics mi:[email protected] Email: g eateto Mechanics, of Department .According ). oyehi fBari, of Polytechnic sitn Professor Assistant 100 cylinders ∗ = al Oresta Paolo − and 1 a d t , C U α Nomenclature studied. the here of cases prediction the accurate in not number Nusselt a provide array cylinders for H re Letters Greek u F T T q s ν T κ d k g p f ∆ L H x d ⋆ ⋆ ⋆ ⋆ , mi:[email protected] Email: , i rbn ,715Br,Italy Bari, 70125 4, Orabona via ahmtc n Management and Mathematics F nln yidrspacing, cylinder in-line udtemlconductivity, thermal fluid C rnvra yidrspacing, cylinder transversal at acceleration, Earth ieai viscosity, kinematic hra diffusivity, thermal yidrdiameter, cylinder iesols udvlct vector velocity fluid dimensionless iesols pressure dimensionless frequency, hra xaso coefficient, expansion thermal ettase coefficient, transfer heat yidriflwtmeauedifference, temperature cylinder-inflow y temperature, no temperature, inflow yidrtemperature, cylinder l no velocity, inflow eateto Mechanics, of Department oc components, force rgadlf coefficients lift and drag oyehi fBari, of Polytechnic noi Lippolis Antonio ulProfessor Full s − 1 K m m m / m m s / 2 K 2 s / / 2 K N s m s W W / ( m / m ( K mK 2 K − 1 ) ) K ; ( T H ⋆ − T L ⋆ ) ρ fluid density, kg/m3 the flow. In this case the flow is not influenced by temper- Dimensionless Numbers ature in the hypothesis of small temperature differences be- Gr Grashof number; gα∆d3/ν2 tween the bluff bodies and the flow temperature with respect Nu Nusselt number; Hd/k to the dominant velocity convection. In Fornarelli et al. [6] Pr Prandtl number; ν/κ the authors investigated the flow field and the heat exchange Re Reynolds number; U ⋆d/ν around six circular cylinders by means of numerical simula- Ri Richardson number; Gr/Re2 tion. The tests have been done in case of forced convection St Strouhal number; fd/U ⋆ (Ri = 0), and a transition in the flow patterns and in the heat exchange has been identified. The flow pattern transition oc- curred for a spacing ratio between 3.6 and 4. The flow is 1 Introduction unsteady and the heat exchange of each cylinder is strongly The flow around multiple bluff bodies is a prototype influenced by the vorticity dynamics. The influence of the of many engineering problems ranging from heavy-duty to buoyancy force on the flow field is expected to be important micro-devices applications. Offshore pipelines, electrical in order to change the flow and heat transfer dynamics. The power lines, electronic and bio-tech devices are just few ex- buoyancyforce influences both the near and the far field with amples of applications in which flow interacts with multi- respect to a solid obstacle immersed in a flow affecting the ple bluff bodies. Among them the heat exchangers involve boundary layer separation and the onset of the vortex shed- a wide range of engineering applications. In general, they ding in the wake [7, 8]. In a two cylinders configuration, consist of solid surfaces at a certain temperature immersed mixed convection with aided buoyancy, aligned to the free in a cross flow at a different temperature. In particular tube stream velocity, has a stabilizing effect on the flow pattern, bundles heat exchangers are common in several micro ap- vice versa the opposed buoyancy anticipates the boundary plications such as in heat exchange control in Li-ion batter- layer separation at the cylinder surface and makes the flow ies [1] or in biomedical devices. For instance, the thermal more unstable [9]. Nevertheless a simple two bodies model performance of lab-on-chip devices assumes a key role in a is not able to predict the multiple cylinders configuration be- wide range of biological applications, such as the study of haviour because of a more complex wake interference phe- tumor cells under constant temperature [2]. The small di- nomenon that affects the downstream cylinders. In literature, mensions and the low flow velocity induce unsteady laminar the in-line configuration of multiple heated cylinders consid- regimes [3]. The oscillations induced by the flow patterns af- ering the effect of buoyancy force has not been extensively fects the force and thermal response of such devices that have investigated. Khan et al. [10] studied the thermal response to be taken into account in the design process [4,5]. Indeed of isothermal tube bundle of circular cylinders in in-line and the prediction of the performance of these devices is still the staggered configuration over a wide range of Reynolds num- subject of study. Numerical simulation of the flow field and ber but only in case of forced convection(Ri = 0) usingan in- heat exchange aids to give a detailed overview of the flow tegral solution of the boundary layer equations. Multiple row quantities involved in such a flow. In the present study the configurations have been studied focusing on the characteri- flow field around six circular cylinders has been investigated zation of the mean value of the force and the heat transfer co- by means of numerical simulations. Three dimensionless pa- efficients [11,12]. The analytical results are able to model a rameters are involved in this type of problem: the Reynolds wide parameter range in the hypothesis of an infinite number (Re), Prandtl (Pr) and Richardson (Ri) number defined as: of rows, but the unsteady characteristics cannot be extrap- olated. Moreover at a Reynolds number of 100 the effects of wake interference on the heat exchange is not easily pre- ⋆ U d ν Gr dictable by means of simplified models [13,14]. The aim of Re = Pr = κ Ri = (1) ν Re2 the present work is to shed light on the oscillatory charac- teristic of the force and heat transfer coefficients in case ofa where U ⋆, d, ν and κ are , respectively, the inflow velocity, single in-line array of six circular cylinders. In order to retain the cylinder diameter, the kinematic viscosity of the fluid and the two-dimensional character of the flow field, our simula- its thermal diffusivity. Gr = gα∆d3/ν2 is the Grashof num- tions have been carried out at Re = 100, being in literature, ber where g, α and ∆ are, respectively, the Earth accelera- for the case of a single cylinder, Re = 200 the threshold for tion modulus, thermal expansion coefficient and the temper- the transition from two to three-dimensional flow [15] and ature difference between the cylinder surface and the cross also for two identical in-line cylinders, in a wide range of flow free stream temperature. The Richardson number rep- in-line spacings, the two dimensional character of the flow resents the importance of the natural convection with respect is retained at Re = 100, as reported in the works of Carmo to the forced convection. Usually the range in which both et al. [16,17]. They state that the onset of three-dimensional effects are present is characterized by values of −1 ≤ Ri ≤ 1 instabilities, for a spacing ratio between 3.6 and 4, occurs and it is called mixed convection. The higher is the abso- for Re ≃ 150. Three-dimensional instabilities induced by the lute value of the Richardson number the smaller is the effect buoyancy force are limited being the buoyancy force mod- of the convection forced by the inlet velocity with respect to ulated in the range −1 < Ri < 1 [18,19,20]. A detailed the natural convection. Forced convection, Ri = 0, is the first description of the flow patterns and temperature distribution step to study the heat exchange between the bluff bodies and have been reported. Moreover the dependence of the dimen- δ(u,T) sionless force and heat transfer coefficients (Cd,Cl ,Nu) have =0 v=0 been reported with a quantitative analysis of their mean and δy oscillating components. 20 d s
T=1 40 d u=1 δ(u,v,T) =0 δx 2 Numerical setup v=0 The incompressible two-dimensional Navier-Stokes T=0 d equations and the heat transfer equation are considered. Here δ(u,T) follows the governing equations in dimensionless form: =0 v=0 y δy ∂ x 120 d u ∇ ∇ 1 ∇2 ∂ + u u = − p + u + RiT (2) t Re Fig. 1. Outline of the numerical setup
∂ ∂ ∇ · u = 0. (3) T/ x = 0. Symmetry boundary conditions are imposed on the walls (v = 0, ∂u/∂y = 0 and ∂T /∂y = 0). The cylin- ders surfaces are kept at a constant dimensionless tempera- ture equal to 1 whereas the inlet flow temperature is 0. The ∂T 1 spacing between the cylinders is constant and two different + u∇T = ∇2T (4) ∂t RePr values are considered, s = 3.6d and s = 4.0d. All the quan- tities reported in the results section are averaged over 1000 where lengths are scaled by the cylinder diameter (d) and the dimensionless time units (t = d/U ⋆) in order to achieve the velocities by the free-stream velocity (U ⋆). The temperature convergence of the statistics. To ensure initial condition in- ⋆ is scaled with respect to the free-stream temperature (TL ) and dependence of the results statistics are collected after 800t. ⋆ the constant temperature of the cylinders (TH ) as follows: The numerical results of the force coefficients and heat trans- fer coefficient are reported. Here, Cd is the drag coefficient, C is the lift coefficient and Nu is the Nusselt number, which T = (T ⋆ − T ⋆)/(T ⋆ − T ⋆). (5) l L H L are defined as:
In the momentum equation ( 2), in order to take into account F Fy Hd the buoyancy force, the Boussinesq approximation has been C = x , C = , Nu = , (6) d ρ ⋆2 l ρ ⋆2 k considered . Ri represents the ratio between the buoyancy 0.5 U d 0.5 U d and the inertial force. The gravity vector g is aligned with the streamwise direction x. The Reynolds number and the where Fx and Fy are the drag and lift force per unit length, Prandtl number have been kept fixed, Re = 100 and Pr = 0.7, respectively. H is the local heat transfer coefficient and k is respectively. Direct numerical simulations have been per- the thermal conductivity. formed using a fractional step projection method to enforce the continuity equation with a pressure correction approach [21]. The advection terms are treated by means of a Go- 3 Results dunov procedure using a second order upwind method. For 3.1 Effect of spacing and buoyancy force on tempera- the viscous terms an implicit Crank-Nicholson scheme has ture spatial distribution been implemented. The Poisson equation for the pressure The flow and thermal behaviour of six in-line cylinders correction step is iterated until the local error on the conti- configuration as function of the Richardson number for two nuity equation is greater than 10−6. In fig. 1 a schematic values of spacing ratio are investigated. First, a qualitative representation of the computational domain and boundary comparison of the temperature distribution contours at the conditions is shown. The numerical domain is 120d long same instant is reported. In fig. 2 opposing buoyancy cases in the streamwise direction and 40d wide in the transversal (Ri < 0) are shown; in these cases, due to the complex spa- direction. An adaptive mesh refinement approach has been tial distribution of the temperature contours, a close up of the implemented in the numerical code [21]. The refinement near field is also reportedin fig. 3. In fig. 4 forced convection strategies include the limiting of the velocity and tempera- and aiding buoyancy cases (Ri ≥ 0) are shown. The figures ture difference between two adjacent grid cells every time- are oriented according to the direction of the gravity vector, step. A detailed and comprehensive grid independence study g. Thus, the free stream velocity is oriented downward or up- and a sensitivity analysis on the domain dimension has been ward for negative or positive values of Ri, respectively. The carried out as reported in a previous work of the authors [6]. counter-oriented buoyancy induces a thermal wake widen- At the inflow uniform flow is imposed with u = 1, v = 0 ing, instead of the aiding buoyancy cases, where the wake and T = 0, while at the outflow the spatial variation of the appears narrower. This phenomenon affects both the spac- velocity components and temperature in the streamwise di- ings remarking that the buoyancy influences the boundary rection are imposed as follows: ∂u/∂x = 0, ∂v/∂x = 0 and layer separation around bluff obstacles as already described -20y 20 -20 − s − s − s − s Ri = 1, d = 3.6 Ri = 0.75, d = 3.6 Ri = 0.5, d = 3.6 Ri = 0.25, d = 3.6
~g ~g ~g ~g
x
a) b) c) d) 100
− s − s − s − s Ri = 1, d = 4.0 Ri = 0.75, d = 4.0 Ri = 0.5, d = 4.0 Ri = 0.25, d = 4.0
~g ~g ~g ~g
T 1
0.75
0.5
0.25 e) f) g) h) 0
Fig. 2. Comparison of the dimensionless temperature distribution at t = 1800 for opposing buoyancy Ri < 0. The domain is placed in vertical position with the free stream velocity oriented downward. in previous analogous studies [9,7]. In case of forced con- e) a phase shift in the shedding behaviour of two succes- vection, Ri = 0, the s/d = 3.6 configuration does not show sive cylinders produces alternate shedding structures on the a secondary instability of the wake, whereas at s/d = 4 the right and left hand side with respect to the array centerline. wake evolves in a meandering configuration as described in Whereas, at s/d = 4.0 (fig.5f-i) the flow structures are shed Fornarelli et al. [6] (fig. 4a-e). This behaviour is related to in phase. In case of mixed convection, the opposing buoy- the flow structures shed by the cylinders. In fig.5 the temper- ancy widens the wake according to the enhanced boundary ature distributions of about one oscillating cycle at Ri = 0 layer separation at the cylinders surface. Indeed for Ri 6= 0 for both the spacings are reported. At s/d = 3.6 (fig.5a- the buoyancy force adds its contribution as source term in the -4 0 4 0
5
10
15
20
25 a) b) c) d) − s − s − s − s Ri = 1, d = 3.6 Ri = 0.75, d = 3.6 Ri = 0.5, d = 3.6 Ri = 0.25, d = 3.6
T 1
0.75
0.5
0.25 e) f) g) h) − s − s − s − s Ri = 1, d = 4.0 Ri = 0.75, d = 4.0 Ri = 0.5, d = 4.0 Ri = 0.25, d = 4.0 0
Fig. 3. Near field comparison of the dimensionless temperature distribution at t = 1800 for opposing buoyancy Ri < 0. The domain is placed in vertical position with the free stream velocity oriented downward. momentum equation, eq. (2), affecting the streamwise com- tion in the near field of the array. Hence, consequenceson the ponent of the velocity, especially near the cylinders, where performance of the array in terms of force and heat exchange the temperature is the highest. Being the cylinders the only are expected and will be detailed below. On the other hand source of vorticity, the flow patterns and the temperature dis- the aiding buoyancy, Ri > 0, stabilizes the flow field. With tribution are very sensitive to Ri. In particular for Ri < 0 respect to the case of forced convectionthe wake is narrower. the opposing buoyancy force induces the boundary layer in- The flow field around the cylinders appears very stable and stabilities in the near field as reported in detail in fig.3. It no sheddingstructures appear behind the array. For Ri = 0.25 is worth to note that for both the spacings, in the cases of and s/d = 3.6 the temperature distribution in the flow field Ri ≤−0.5 (fig. 2-a,b,c,e,f,g), the opposing buoyancy affects reveals a stable wake pattern without any secondary instabil- the vortex shedding of the array. The more is the opposing ity, even in the far field (see fig. 4-b). The increasing of the buoyancy the more chaotic the temperature pattern appears, buoyancy force triggers a Kelvin-Helmholtz instability of the except for the case with s/d = 3.6 at Ri = −1 (see fig. 2- wake in the far field. For s/d = 4 the increasing of Ri from a). Although at Ri = −1 a more chaotic configuration is ex- 0 to 0.25 is not sufficient to suppress the wake oscillation pected, at s/d = 3.6, enhancing the opposing buoyancy from (see fig. 4-f). Increasing the buoyancy force at Ri = 0.5, the Ri = −0.75 to Ri = −1, a well-ordered wake pattern can be wake oscillation is suppressed and there is only a secondary recognized, due to a reorganization of the vorticity interac- oscillation in the far field (see fig. 4-g). ~g = 0 ~g ~g ~g
a)s b)s c)s d) s Ri = 0, d = 3.6 Ri = 0.25, d = 3.6 Ri = 0.5, d = 3.6 Ri = 1, d = 3.6 100
~g = 0 ~g ~g ~g
T x 1
0.75
0.5
0.25
s s s s Ri = 0, d = 4.0 Ri = 0.25, d = 4.0 Ri = 0.5, d = 4.0 Ri = 1, d = 4.0 e) f) g) h) -20 0 -20y 20
Fig. 4. Comparison of the dimensionless temperature distribution at t = 1800 for aiding buoyancy Ri ≥ 0. The domain is placed in vertical position with the free stream velocity oriented upward.
3.2 Drag and lift forces analysis tive values, hCd i = −0.778 for s/d = 3.6 and hCdi = −0.914 for s/d = 4. Indeed, at Ri = −1, the drag coefficient are The drag coefficient of the entire array, C (t) = d hC i = −0.401 at s/d = 3.6 and hC i = 1.095 at s/d = 4. For ∑6 C (t) has been measured for each value of Ri and s/d. d d i=1 di the forced convection and the aiding buoyancy cases, Ri ≥ 0, Being C a function of time, its time average, hC i, and stan- d d the time averaged drag coefficient increases linearly: dard deviation, σ(Cd), havebeencalculated. In fig.6 thetime averaged drag coefficient is reported. The opposing buoy- ancy reduces the drag of the array with respect to the force hCd i = 18.126Ri + 3.288 Ri ≥ 0. (7) convection case for both the spacings. At Ri = −0.75, a min- imum time average drag coefficient is found, reaching nega- On the other hand, in fig. 8 the standard deviation of the 25
20
15
10
5
0 a) b) c) d) e) -4 0 4 T 1
0.75
0.5
0.25
f) g) h) i) 0
Fig. 5. Snapshots of the dimensionless temperature distribution over one oscillation cycle in case of forced convection Ri = 0, for s/d = 3.6 at t = 1800.0 (a), t = 1802.5 (b), t = 1805.0 (c), t = 1807.5 (d), t = 1810.0 (e); and s/d = 4.0 at t = 1800.0 (a), t = 1802.5 (b), t = 1805.0 (c), t = 1807.5 (d). The domain is placed in vertical position with the free stream velocity oriented upward.
2.5 2.5 Cyl 1 Cyl 1 Cyl 2 Cyl 2 Cyl 3 Cyl 3 2 Cyl 4 2 Cyl 4 Cyl 5 Cyl 5 Cyl 6 Cyl 6 1.5 1.5 (Cd) (Cd) σ σ 1 1
0.5 0.5
0 0 a) -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 b) -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 Ri Ri
Fig. 7. Standard deviation of the drag coefficient (σ(Cd)) of each cylinder with respect to the Richardson number (Ri) for s/d = 3.6 (a) and s/d = 4.0 (b). s/d = 3.6 8 s/d = 4.0 s/d = 3.6 20
6 15
5 2
0
-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 0 Ri -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 Ri
Fig. 6. Time averaged drag coefficient (hCd i) of the array with re- Fig. 8. Standard deviation of the drag coefficient (σ(C )) averaged spect to the Richardson number (Ri) for s/d = 3.6 and s/d = 4.0. d over six cylinders with respect to the Richardson number (Ri) for The linear fitting of the aiding buoyancy cases (0 ≤ Ri ≤ 1) is re- s d 3 6 and s d 4 0. ported. / = . / = .
8 s/d = 3.6 s/d = 4.0 drag coefficient shows an increase of its oscillating ampli- 6 tude in opposing buoyancy cases. The aiding buoyancy sup- presses the drag coefficient oscillation, as described in the
qualitative description of the temperature distribution in the (Cl) 4 σ previous section. A sudden transition, at Ri = −1, changing the spacing between the cylinders is found. The decreasing of the spacing from 4.0 to 3.6 causes a limiting effect on 2 the oscillation amplitude of Cd. The standard deviation of the drag force coefficient (σ(Cd)) of the array has been de- 0 tailed on each cylinder in fig. 7. It is worth to note that all -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 the cylinders are involved in the transition at Ri = −1. This Ri result is strictly connected with the flow pattern transition σ depicted in fig. 2. The standard deviation of the lift coeffi- Fig. 9. Standard deviation of the lift coefficient ( (Cl )) averaged over six cylinders with respect to the Richardson number (Ri) for cient, σ(Cl ), confirms the transition at Ri = −1 (see fig. 9). Moreover, the temperature distribution in the near field re- s/d = 3.6 and s/d = 4.0. veals that for s/d = 3.6 the boundary layer separations at the cylinder surfaces are reduced (see fig. 3-a), compared to the s/d = 4 spacing (see fig. 3-e), inducing a dumping in the os- 0.2 cillation amplitude of C and C . At Ri = 0 and Ri = 0.25 a s/d=3.6 d l s/d=4.0 difference in the value of σ(Cl ) can be recognized. Indeed it is related to the wake oscillation of the s/d = 4.0 case with 0.15 respect to the stable configuration of s/d = 3.6 remarking the qualitative discussion about the instantaneous tempera- ture distribution reported in section 3.1. σ(C ) is more sen- l St 0.1 sitive than σ(Cd ) to the wake oscillation in the near field. In figure 10 the Strouhal number (St = fd/U ⋆, where f is the oscillation frequency) of the maximum peak of the fft 0.05 of the Cl for the first cylinder is reported. The higher dif- ferences in the oscillation frequency correspond to the cases at Ri = −1 and Ri = 0, where different temperature patterns 0 have been recognized changing the spacing ratio between the -1 -0.5 0 0.5 1 Ri cylinders. At Ri = −1 the sudden jump of St is related to the above mentioned boundary layer separation reduction that Fig. 10. Strouhal number (St) of the first cylinder with respect to the influences the shedding behaviour at s/d = 3.6 compared to Richardson number (Ri) for s/d = 3.6 and s/d = 4.0. the case at s/d = 4. 7 7 6
6 5
4 5 Nu Nu 3 4
2 3
1 1700 1725 1750 1775 1800 2 t 1700 1725 1750 1775 1800 t Cyl 1 Cyl 3 Cyl 5 Cyl 2 Cyl 4 Cyl 6 Cyl 1 Cyl 3 Cyl 5 Cyl 2 Cyl 4 Cyl 6 Fig. 11. Surface averaged Nusselt number (Nu) of each cylinder with respect to the dimensionless time t, for s/d = 3.6 and Ri = 0. Fig. 13. Surface averaged Nusselt number (Nu) of each cylinder with respect to the dimensionless time t, for s/d = 3.6 and Ri = −1.
7
6 7
5 6 4 Nu 5 3 Nu
2 4
1 3 1700 1725 1750 1775 1800 t 2 Cyl 1 Cyl 3 Cyl 5 1700 1725 1750 1775 1800 Cyl 2 Cyl 4 Cyl 6 t Cyl 1 Cyl 3 Cyl 5 Fig. 12. Surface averaged Nusselt number (Nu) of each cylinder Cyl 2 Cyl 4 Cyl 6 with respect to the dimensionless time t, for s/d = 4.0 and Ri = 0. Fig. 14. Surface averaged Nusselt number (Nu) of each cylinder with respect to the dimensionless time t, for s/d = 4.0 and Ri = −1. 3.3 Heat exchange performance The thermal performance of the cylinders array with respect to the buoyancy force modulation is evaluated by 7 means of the Nusselt number. The instantaneous surface averaged Nusselt numbers, Nu, for each cylinder have been 6 sampled during the simulations. It is obtained averaging the ′ ∂ ∂ local Nusselt number, Nu = T / n. The forced convec- 5 tion case (Ri = 0) shows a monochromatic response of the Nu surface averaged Nusselt number for each cylinder for both 4 the spacing ratio, s/d = 3.6 (fig.11) and s/d = 4.0 (fig.12). The opposing buoyancy (Ri = −1) induces more complex 3 oscillations of Nu as reported in fig. 13 for s/d = 3.6 and in fig.14 for s/d = 4.0. On the other hand, the aiding buoy- 2 ancy, for Ri = 1, suppresses the oscillations on all the cylin- 1700 1725 1750 1775 1800 t ders as reported in fig. 15 for s/d = 3.6 and in fig. 16 for Cyl 1 Cyl 3 Cyl 5 s/d = 4.0. Thus, the time averaged Nusselt number, hNui, Cyl 2 Cyl 4 Cyl 6 and its standard deviation, σ(Nu), have been extracted. First, the time-average Nusselt number of the entire array is ana- Fig. 15. Surface averaged Nusselt number (Nu) of each cylinder lyzed (see fig. 17). The results are compared to the predic- with respect to the dimensionless time t, for s/d = 3.6 and Ri = 1. tion of time-averaged heat exchange in case of tube banks of Table 1. Parameters used in eq. 8 according to the Zukauskas [13] 7 model for the prediction of the Nusselt number of a tube bundle.
6
Re C1 C2 m n 5 Nu 4 0 − 100 0.9 0.945 0.4 0.36 3 100 − 1000 0.52 0.945 0.5 0.36
2 1700 1725 1750 1775 1800 t ratio: Cyl 1 Cyl 3 Cyl 5 Cyl 2 Cyl 4 Cyl 6 2 2 Nu = fa 0.3 + Nu + Nu (9) lam turb Fig. 16. Surface averaged Nusselt number (Nu) of each cylinder q with respect to the dimensionless time t, for s/d = 4.0 and Ri = 1. where
0.5 0.33 Nulam = 0.664Reλ Pr , 6 0.8 0.037Reλ Pr Nuturb = , −0.1 2/3 5 1 + 2.443Reλ (Pr − 1)
4
0.5 0.5 0 0 a) -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 b) -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 Ri Ri Fig. 19. Standard deviation of the Nusselt number (σ(Nu)) of each cylinder respect to the Richardson number (Ri) for s/d = 3.6 (a) and s/d = 4.0 (b). 0.75 σ(Nu) of each cylinder is reported in fig. 19. At Ri = −1 the s/d = 3.6 s/d = 4.0 spacing ratio increase produces higher oscillations of Nus- selt number on each cylinder. At Ri = −0.25 the spacing in- crease enhances the σ(Nu) of the first cylinder. At Ri = 0.25 0.5 the higher value of the averaged σ(Nu) reported in fig. 18 at s/d = 4 is mainly related to the oscillating amplitude of the (Nu) trailing cylinders. σ 0.25 4 Conclusions We can infer, by means of numerical simulations, that 0 the buoyancy force and the spacing between the cylinders, -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 arranged in a single six elements row, affect their perfor- Ri mance in terms of forces and heat exchange. Generally, the Fig. 18. Standard deviation of the Nusselt number (σ(Nu)) of the aiding or the opposing buoyancy induces a stable or unsta- array with respect to the Richardson number (Ri) for s/d = 3.6 and ble behaviour of the flow, respectively. However, in case of s/d = 4.0. opposing buoyancy a flow transition has been recognized. In case of aiding buoyancy the spacing influences the sec- ondary instability of the wake in the far field, therefore the amplitude of the Nu oscillation with respect to the Ri and the differences in terms of force and heat exchange between the spacing ratio. The buoyancy effect on the flow pattern in- fluid and the cylinders are small because they are linked with fluences the amplitude of the averaged Nusselt number. The the near field behaviour. On the other hand, in case of op- behaviour of the standard deviation of the Nusselt number posing buoyancy the flow instabilities in the near field affect shows the stabilization effect of the aiding buoyancy except the array performance. The oscillation amplitudes of Cd, Cl for s/d = 4 and Ri = 0.25. The wake oscillation depicted and Nu become relevant with respect to the mean quantities in the instantaneous temperature distribution for s/d = 4 and for both the spacing here investigated. At Ri = −1 the spac- Ri = 0.25, reported in fig. 4-f, causes the onset of Nusselt ing affects heavily the oscillation amplitude of the measured amplitude even in the case of aiding buoyancy. The oppos- quantities. At Ri = −1 and s/d = 4 the standard deviation ing buoyancy increases the oscillation amplitude of the heat of the performance coefficients, σ(Cd), σ(Cl ) and σ(Nu) in- exchange. From Ri = −0.25 to Ri = −0.75 a linear increase creases with Ri. However, for a spacing ratio s/d = 3.6, even of the standard deviation of Nu for both spacings has been with a strong opposing buoyancy (Ri = −1), the flow is able found. The small differences of σ(Nu) for the two spacings to rearrange itself in a more ordered wake pattern configu- are influenced by the wider gap between the cylinders of the ration limiting the oscillation amplitude of the performance s/d = 4 case. The further increase of opposing buoyancy coefficients. In the range of parameters here presented, the force, Ri = −1 shows a transition of the oscillation ampli- heat exchange and force coefficients of the cylinders array tude of Nusselt number for the two spacing. The s/d = 4, appear very sensitive with respect to the value of Richardson Ri = −1 case holds an increase of the amplitudeof Nu reach- number. Thus, this work highlights the care that should be ing σ(Nu)= 0.66, otherwise at s/d = 3.6 σ(Nu)= 0.25. 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