NUMERICAL INVESTIGATION OF STROUHAL FREQUENCIES OF TWO STAGGERED BLUFF BODIES

NUMERICAL INVESTIGATION OF STROUHAL FREQUENCIES OF TWO STAGGERED BLUFF BODIES Eswaran M 1, P. Goyal, Anu Dua, G.R. Reddy, R. K. Singh and K.K. Vaze Bhabha Aomic Research Cenre, Mumbai, India. 1 Corresponding auhor email: eswarm21@gmail.com ABSTRACT: In his paper, he 2-D unseady viscous flow around wo cylinders is sudied by numerical soluions of he unseady Navier-Soes equaions wih a finie elemen formulaion. The resuls of a numerical invesigaion of he Srouhal frequencies of wo idenical, saionary, parallel circular cylinders arranged in saggered configuraions is presened in his paper. A simple wo cylinder andem arrangemen is validaed for a cerain range of values of spacing raio (L/D) wih few previously published resuls. Resuls of measuremens of he Srouhal frequencies of circular cylinders arranged in andem and in some seleced saggered configuraions are also presened. In he case of wo circular cylinders, he invesigaion is performed a saggered angles (α) of 10, 15, 20 and 25 in he range of L/D =2.0 o 4.0 and Re = 100 o 15000. The findings in his sudy for wo circular cylinders are: (i) criical spacing raio is simulaed successfully and (ii) Lif force and drag force variaion for differen saggered angle. Keywords: Tandem cylinders; Srouhal frequency; Vorex shedding; Lif and drag force; Wind loads. INTRODUCTION: In engineering, fluid forces and Srouhal numbers are he primary facors considered in he design of muliple slender srucures subjeced o cross flow, e.g., chimney sacs, ube bundles in hea exchangers, overhead power-line bundles, bridge piers, says, mass, chemical reacion owers, offshore plaforms and adjacen syscrapers. The simples arrangemen of muliple slender srucures is wo cylinders in eiher andem or side-by-side arrangemen. Though he saggered arrangemen is perhaps he configuraion mos commonly found in engineering applicaions, numerous invesigaions on he flows pas wo circular cylinders in side-by-side and andem arrangemens have been performed, bu invesigaions peraining o wo saggered cylinders are relaively few, paricularly invesigaion of he vorex-shedding processes in erms of he Srouhal frequencies of wo saggered cylinders. As has been revealed in he previous sudies, for a small value of L/D (see Fig. 1 for definiions of symbols), i.e., when he downsream cylinder is fully or parially submerged ino he upsream cylinder wae, he wo cylinders behave lie a single body and he frequencies of vorex shedding behind he wo cylinders are he same. For an inermediae value of L/D, a which narrower and wider waes are formed behind he upsream and downsream cylinders, respecively, he frequencies of vorex shedding behind he upsream and downsream cylinders are of a higher and lower magniude, respecively. For a higher value of L/D, i.e., when each cylinder sheds vorices independenly, he frequency of vorex shedding behind each cylinder is he same as ha in he case of a single, isolaed cylinder. Flow around wo andem cylinders of idenical diameers is in general classified ino hree major regimes [1, 2]: (i) he exended-body regime (L/D < 0.7), where he wo cylinders are so close o each oher ha he free shear layers deached from he fron cylinder overshoo he rear one, and he flow in he gap of he cylinders is sagnan; (ii) he reaachmen regime (L/D = 0.7 o 3.5), where he shear layers separaed from he fron cylinder reaach on he rear cylinder and he flow in he gap is sill insignifican; (iii) he co-shedding regime (L/D > 3.5), where he shear layers roll up alernaely in he gap beween he cylinders and hus he flow in he gap is significan. From he fundamenal poin of view, aerodynamic inerference beween wo closely separaed cylinders may

give rise o flow separaion, reaachmen, vorex impingemen, recirculaion and quasi-periodic vorices, involving mos generic flow feaures associaed wih muliple srucures. Thus, flow around wo andem cylinders provides a good model o undersand he physics of flow around muliple cylindrical srucures. NUMERICAL METHODOLOGY: (i) Governing equaions Based on he Navier-soes ime averaged equaions and using Bousssinesq approximaion for Renolds sresses, differenial equaions governing viscous urbulen flow field can be wrien as div( u ) 0 (1) ( u) p div( u u) div( eff gradu) f x x (2) ( v) p div( u v) div( eff gradv) f y y (3) where is he fluid densiy, eff he effecive viscosiy, u he mean flow velociy field, p he pressure and u, v are he mean componens of flow field in he x and y direcions, respecively. The erms f x, f y denoe body forces for he uni volume of fluid. The urbulence is modelled by he wo equaion model which is easily implemen and has a wide applicaion in engineering problems. Thus, he urbulence of flow field is expressed in urbulence ineic energy () and dissipaion rae ( ), using following equaions, ( ) div( u) div( grad) P (4) 2 ( ) div( u ) div( grad ) c1 P c2 (5) 2 C ; (6) where / and / are diffusion coefficiens for producion rae by urbulen shear sresses, is he molecular viscosiy and eff and, respecively, P is he is he urbulen viscosiy. The erms (urbulen Prandl number) and (he urbulen Schmid number ), C 1, C 2 are experimenal consans of he model. For his model, C 1, C 2, and, and of 1.44, 1.92, 1, 1.3, 0.09 respecively. The sandard model along wih Boussinesq equaion, performs well for he broad range of engineering problems, however in he problems which include unbalanced effecs, ec., finally his model reaches o responses which are over diffused, i.e., he values prediced by his model will be large. C have he values in his sudy, he fluid is assumed incompressible. As here is no free surface, he body forces can be ignored (f x=0, f y =0). Equaions (1) hrough (3) is for laminar flow, plus Equaions (4) hrough (6s ) for urbulen flow are governing he flow field which will be solved numerically. (ii) Compuaional domain and boundary condiions Two cylinders of equal diameer are placed in a 35D x 20D recangular domain, in which he cener of upsream cylinder is locaed 10D from he inle boundary and cener of he downsream cylinders is placed a a disance of 25D from he oule boundary, where D denoes he cylinder diameer.

Fig.1 Noaion for saggered configuraion. A he inle, he Dirchle boundary condiions are applied (u=u, v=0, =0.03U 2 2 and 0. 005l, where U is he free sream velociy, and l is he radius of compuaional domain of flow).the pressure is prescribed a a poin a he inle. A he oule, he Neumann-ype condiions are employed (,, and. No slip boundary condiion is applied a he u x 0 v x 0 x 0 u 0 v 0 boundary of a fixed cylinder, i.e., velociy componens on he boundaries zero (, ). The op and boom boundaries of he compuaional domain are considered as symmery boundary o negae he propagaion wall effecs inside he domain over he ime period. (iii) Grid generaion and discreizaion In his sudy, non uniform saggered Caresian grid is used and he enclosed area of wo cylinders is refined wih respec o oher areas of he flow field. Boundary layers have also been creaed around he solid boundary. In such a grid, he velociy componens (u and v) are calculaed for he poins locaed on he main conrol volume surfaces, i.e., he saggered poins, while pressure, ineic energy and dissipaion rae of urbulen energy (p, and are calculaed for he poins locaed on he main grid. The flow close o a solid wall is for a urbulen flow, is very differen compared o he free sream. This means ha he assumpions used o derive he model are no valid close o walls. While i is possible o modify he model so ha i describes he flow in wall regions such as exended and RNG models. Insead, analyical expressions are used o describe he flow a he walls. These expressions are nown as wall funcions. The disance y w is auomaically compued so ha, where C 1 4 y w y /. w is he fricion velociy, which becomes 11.06. This corresponds o he disance from he wall where he logarihmic layer mees he viscous sub-layer (or o some exen would mee if here was no a buffer layer in beween). Temporal discreizaion is performed based on second order implici mehod which causes much less damping and is hereby more accurae. Fig.2 Grid arrangemen Fig.3 Boundary layers around he cylinder. Resuls and discussion: The dimensionless flow parameers are defined as follows, 2F C D D 2, 2F C L L 2 U D U D where F L and F D denoe life and drag forces, respecively. Dimensionless flow parameers are also given by

UD fvd U Re, S, T U D where Re is he Reynolds number, S is he Srouhal number, T is he dimensionless ime, is he fluid viscosiy, D is he cylinder diameer, is he ime a each ime sep and f v is he frequency of verex shedding which can be calculaed from he oscillaing frequency of lif force. Case 1: Flow pas wo andem cylinders a Re =200 and α =0. In order o beer undersand he flow characerisics and wae inerference beween wo andem cylinders, iniially, flow pas a saionary circular cylinder is solved for laminar and urbulen cases. And hen, simple andem cylinder arrangemen cases were analysed and compared wih Zdravovich e al. (1987) and Alam and Zhou (2007) resuls. The presen resuls are very closely maching wih heir experimenal resuls and numerical resuls. Afer he validaion, he differen saggered angles are sudied in deail. Downsream cylinder is submerged ino he upsream cylinder wae for he lower L/D raio. The negaive drag will disappear, while increasing he saggered angle α. Here he Fig. 4 and 5 show he andem arrangemen wih α =0 and L/D =3.0 drag and α =15 and L/D =2.0 drag respecively. Fig. 4 Sreamline plo for andem cylinder α =0 Fig. 5 velociy conour plo for andem cylinder α =0 and L/D =4.0 and L/D =4.0 Srouhal number is an significan feaure of fluid which has a srong dependence on boh Reynolds number and spacing. This non-dimensional number, specify how cylinders response o hydrodynamic forces and when heir oscillaion frequency reaches o he poin near naural frequency which can lead o damage of he srucure. Thereby, comparisons of he Srouhal numbers and mean drag coefficiens in he presen wor wih oher daa are presened for Re = 100, 200 and 300 in Table 1. Obviously, he Srouhal numbers are he same and hey are smaller han hose of a single cylinder, which signifies he effec of wae inerference in he variaion of Srouhal number. Frequency of he verex shedding which can be calculaed from he oscillaion frequency of lif force. Table 1. Comparisons of flow wo andem cylinders a Re =200 and α =0. Mean drag coefficien Srouhal Frequency Parameric cylinder Slaoui and Meneghini Slaoui and Meneghini Resuls Presen Presen sansby e al. Sansby e al. UC 1.0641 0.89 1.03 0.1401 0.13 0.13 L/D =2 DC -0.1628-0.21-0.17 0.1401 0.13 0.13 UC 1.0685 0.87 1.0 0.1465 NA 0.125 L/D =3 DC -0.1578-0.16-0.08 0.1465 NA 0.125 UC 1.1052 1.11 1.18 0.1650 0.19 0.174 L/D =4 DC 0.7991 0.88 0.38 0.1650 0.19 0.174 UC= Upsream cylinder ; DC = Downsream cylinder.

Table 1 shows comparisons of flow wo andem cylinders a Re =200 and α =0. I can be seen from Table 1 ha he negaive drag has compleely eliminaed in he co-shedding regime while increasing he L/D raio. The reaachmen of upsream shear layer ono he second cylinder is observed. Due o his, negaive drag is found on he downsream cylinder which resuls from he pressure difference beween fron and bac sides of his cylinder. Case 2: Flow pas wo andem cylinders a Re =200 and a differen saggered angles. The numerical invesigaion is performed a differen saggered angles (α) of 10, 15, 20 and 25 in he range of L/D =2.0 wih Re = 200; where a is he angle beween he free-sream flow and he line connecing he ceners of he cylinders, L is he gap widh beween he cylinders, and D is he diameer of a cylinder. While increasing he saggered angle, he drag of he downsream cylinder is increasing. Influence of upsream cylinder on downsream in coninuing up o saggered angle 25. Thereafer, i will behave lie flow pas a single cylinder. I can be seen from Table 1 ha he drag coefficien is increasing drasically wih saggered angle. Fig. 6 Sreamline plo for andem cylinder α =10, Re= 200 and L/D =2.0 Fig. 7 Force coefficien on cylinders of α =10, Re= 200 and L/D =2.0 Table 2. Flow pas wo andem cylinders a Re =200 and L/D =2. Parameric Mean drag coefficien Srouhal frequency cylinder resuls C d S UC 1.169542 0.1625 α =10 DC 0.391452 0.166 UC 1.135565 0.1455 α =15 DC 0.611852 0.1456 UC 1.246236 0.1325 α =25 DC 1.125252 0.132 Case 3: Flow pas wo andem cylinders a Re =15000. In andem cylinder arrangemen, here are several ineresing races can be seen which is showing he effec of he verex shedding. In addiion, he lower ime averaged drag of he rear cylinder reflecs is being immersed in he wae of he fron cylinder. Here, he urbulen flow over he andem is modelled o show he effec of he flow separaion around he bluff body. The flow behaviour along wih force coefficiens of flow around andem cylinders in laminar flow were discussed in above. Here model is used along wih wall funcions. The flow characerisics for he Re=15000 are shown for α = 0 and α = 15 wih L/D = 2.0 case in Fig. 8 and 9. In he urbulen region, he lif force ime hisories for hese wo groups show nearly smooh and relaed variaions and drag values are negaive lie laminar cases for he low gap raios.

(a) (e) (b) (f) (c) (g) (d) (h) Fig. 8 Sreamline plo for andem cylinder Re= 1.5E4 and L/D =2.0 (a-d for α =0 and e-h for α =15 ). Conclusions In his paper, flow over wo in-line and differen saggered angle is simulaed successfully. Differen Reynolds numbers were used in order o invesigae he effec of various parameers on flow characerisics and hydrodynamic forces acing on he bodies. During he low Reynolds number and low gap spacing, he rear cylinder highly disurbed he flow behind his cylinder. Drag values are negaive. in he oher hand when he gap raio is more, hen vorices are shed couner wise from boh cylinders and no synchronized. In he urbulen region, he lif force ime hisories for hese wo groups show nearly smooh and relaed variaions and drag values are negaive lie laminar cases. References [1] Singh and Sinhamahapara KP, 2010, High resoluion numerical simulaion of low Reynolds number incompressible flow abou wo cylinders in andem, journal of fluids Engineering, ASME, 2010, 132(1): 011101. [2] Alam MM and Zhou Y, 2007, Dependence of srouhal number, drag and lif on he raio of cylinder diameers in a wo-andem cylinder wae, 16h Ausralasian Fluid Mechanics Conference, Ausralia. [3] Meneghini JR and Saara F, 2001, Numerical simulaion of flow inerference beween wo circular cylinders in andem and side by side arrangemens, Journal of fluids and srucures, 15(2), pp. 327-350. [4] Slaoui and Sansby,1992, Flow around wo circular cylinders by he random verex mehod. journal of fluids and srucures, 6(6) pp. 641-670. [5] Zdravovich MM, 1987, The effecs of inerference beween circular cylinders in cross flow, J. Fluids and Srucure, 1, 239-261.