Buoyancy effect on the flow pattern and the thermal performance of an array of circular cylinders Francesco Fornarelli∗ Antonio Lippolis Senior post-doc Full Professor Department of Mechanics, Department of Mechanics, Mathematics and Management Mathematics and Management Polytechnic of Bari, Polytechnic of Bari, via Orabona 4, 70125 Bari, Italy via Orabona 4, 70125 Bari, Italy INFN sez. Lecce, 73100 Lecce, Italy Email: [email protected] Email: [email protected] Paolo Oresta Assistant Professor Department of Mechanics, Mathematics and Management Polytechnic of Bari, via Orabona 4, 70125 Bari, Italy INFN sez. Lecce, 73100 Lecce, Italy Email: [email protected] In this paper we found, by means of numerical simulations, for cylinders array provide a not accurate prediction of the a transition in the oscillatory character of the flow field for a Nusselt number in the cases here studied. particular combination of buoyancy and spacing in an array of six circular cylinders at a Reynolds number of 100 and Prandtl number of 0.7. The cylinders are iso-thermal and Nomenclature they are aligned with the Earth acceleration (g). According Cd, Cl drag and lift coefficients to the array orientation, an aiding or an opposing buoyancy Fx, Fy force components, N is considered. The effect of natural convection with respect H heat transfer coefficient, W/(m2K) arXiv:1609.07960v1 [physics.flu-dyn] 26 Sep 2016 to the forced convection is modulated with the Richardson T ⋆ temperature, K ⋆ number, Ri, ranging between −1 and 1. Two values of cen- TL inflow temperature, K ⋆ ter to center spacing (s = 3.6d − 4d) are considered. The TH cylinder temperature, K effects of buoyancy and spacing on the flow pattern in the U ⋆ inflow velocity, m/s near and far field are described. Several transitions in the d cylinder diameter, m flow patterns are found and a parametric analysis of the de- f frequency, s−1 pendence of the force coefficients and Nusselt number with g Earth acceleration, m/s2 respect to the Richardson number is reported. For Ri = −1, k fluid thermal conductivity, W /(mK) the change of spacing ratio from 3.6 to 4 induces a transi- p dimensionless pressure tion in the standard deviation of the force coefficients and q transversal cylinder spacing, m heat flux. In fact the transition occurs due to rearrangement s in-line cylinder spacing, m of the near field flow in a more ordered wake pattern. There- u dimensionless fluid velocity vector fore, attention is focused on the influence of geometrical and Greek Letters ∆ ⋆ ⋆ buoyancy parameters on the heat and momentum exchange cylinder-inflow temperature difference, K; (TH − TL ) and their fluctuations. The available heat exchange models α thermal expansion coefficient, K−1 κ thermal diffusivity, m2/s ν kinematic viscosity, m2/s ∗Corresponding author. 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.