THE interaction of the flow over two cylinders is a

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1 Proceedins of the Interntionl MultiConference of Enineers nd Computer Scientists Vol I IMECS, Mrch -,, Hon Kon Behvior of the Two-Dimensionl Viscous Flow over Two Circulr Cylinders with Different Rdii Surttn Sunnul nd Ekkchi Kunnwuttipreechchn Abstrct In this pper, we study numericl simultion of two-dimensionl viscous flow over two circulr cylinders with different rdii. The flow structure depends on the rte of rottion, the p-spcin nd the Reynolds number. The lorithm used to simulte the numericl solutions is bsed on the concept of projection method. A mthemticl model describin the flow over the two rottin cylinders is pplied by the cylindricl bipolr coordinte system. The min objective is to investite the chrcteristics of the fluid flow. This investition ives set of numericl simultions for the hydrodynmic chrcteristics, which cn be pplied to other relted problems. Index Terms numericl simultion, cylindricl bipolr coordinte, projection method. I. INTRODUCTION THE interction of the flow over two cylinders is topic of prime scientific interest with mny enineerin nd rel life pplictions. Most of reserches studied on two cylinders were concerned with two rottin nd nonrottin cylinders of n identicl dimeter see for exmple []- [] nd literture cited there. There re two different types of sttionry motion of bodies in fluid. The first type is towed body which in the sttionry motion reime externl forces must ffect the body. The second type is self-propelled body. Self-propelled mens tht body moves becuse of the interction between its boundry nd the surroundin fluid nd without the ction of n externl force. In the present work, flow structures were clculted between two circulr cylinders the left cylinder is nonrottin nd the riht cylinder is rottin in counterclockwise nulr velocity with different rdii nd uniform strem flow directed perpendiculr to the line connectin the cylinders centers. We show some results of numericl simultions t fixed Reynolds number nd moderte p spcin nd rte of cylinders rottion. II. MATHEMATICAL MODELLING The overnin eqution is the Nvier-Stokes equtions written in cylindricl bipolr coordinte. The coordinte system is movin toether with the cylinders. The cylindricl bipolr coordinte system cn be defined by the followin eqution x = sinh η cosh η cos ξ, y = sin ξ cosh η cos ξ, Mnuscript received December, 7; revised Jnury 5,. This reserch ws finncilly supported by the Fculty of Applied Science, Kin Monkut s University of Technoloy North Bnkok Contrct no. 5. Surttn Sunnul is lecturer t the Deprtment of Mthemtics, Kin Monkut s University of Technoloy North Bnkok,, THAILAND nd resercher in Centre of Excellence in Mthemtics, CHE, Bnkok,, THAILAND e-mil: surttn.s@sci.kmutnb.c.th Ekkchi Kunnwuttipreechchn is lecturer in the Deprtment of Mthemtics, Kin Monkut s University of Technoloy North Bnkok,, THAILAND e-mil: ekkchi.k@sci.kmutnb.c.th where ξ [, π, η,, is chrcteristic lenth in the cylindricl bipolr coordinte system which is positive. This trnsformtion mps the xy plne form which the domin occupied by the cylinders is excluded into the rectnle η η η, ξ < π nd η <, η >. The surfces of the cylinders re locted t η = η nd η = η. The cylinder s rdii r, r nd the distnce of their centers from the oriin d, d re iven by r i = csch η i, d i = coth η i, i =,. The center to center distnce between the cylinders is d = d + d. The Nvier-Stokes equtions - in the cylindricl bipolr coordinte system ξ, η [5] re t + h ξ + v η sinh η v η sin ξv η = h + ν { h h ξ + cosh η + cos ξ p ρ ξ + sinh η v η ξ sin ξ v η }, v η t + v η h ξ + v v η η + + sinh η sin ξ v η = h + ν { v η h h ξ + v η + cosh η + cos ξ p ρ + sinh η ξ sin ξ v η }, 3 hvξ h + hv η =, ξ where nd v η re the physicl components of velocity vector v =, v η, p is the pressure, ρ is density, ν is the kinemtic viscosity of the fluid nd h = cosh η cos ξ. The boundry conditions re no-slip requirement on cylinders = ω i r i, v η =, on η = η i, ξ [, π, i =,, 5 where ω i, i =, re constnt nulr velocities of the cylinders rottion. Positive vlues of ω i, i =, correspond to counterclockwise rottion. Upstrem nd downstrem boundry conditions t infinity re v x =, v y = U, s r = x + y, where v x nd v y re components of velocity vector in x nd y directions respectively nd U is the oncomin free ISBN: ISSN: 7-95 Print; ISSN: 7-9 Online IMECS

2 Proceedins of the Interntionl MultiConference of Enineers nd Computer Scientists Vol I IMECS, Mrch -,, Hon Kon strem velocity. The net force nd torque exerted by fluid on n immersed body with surfce Σ re F = τ ds, M = [r τ]ds, Σ where n is the unit vector norml to the Σ tht points outside the reion occupied by the fluid. The force per unit re exerted cross riid boundry element with norml n in n incompressible fluid is defined by Σ τ = p n µn ω where ω is vorticity defined s ω = curl v nd µ is the coefficient of viscosity. If F xi nd F yi, i =, re the lift nd dr on the cylinders, the lift nd dr coefficients re defined by C Li = F x i ρu D, C D i = F y i, i =,, 7 ρu D where D is dimeter of riht cylinder nd ech consists of components due to the friction forces nd the pressure. Hence C L = C Lf + C Lp, C D = C Df + C Dp. The problem of self-motion is to find solution of the Nvier-Stokes equtions - with boundry conditions 5 nd dditionl constrints F = M =. 9 Eqution 9 determines the bsic distinction between sttionry flow over self-propelled nd towed bodies. The numericl simultion of the flow pst self-movin bodies becomes more complicted s result of the nonloclity of constrints like 9. For such flows, the results depend not only on the Reynolds number, Re, but lso depend on the non-dimensionl p spcin between the two cylinders,, nd prmeters, α i representin the rtios of the rottionl velocities of the cylinder wlls to the oncomin flow velocity Re = U D, α i = Dω i, i =,, nd = d r r. ν U D/ III. NUMERICAL ALGORITHM AND VALIDATION The lorithm of the problem solution is bsed on the concept of projection methods Chorin, 9 []. The intermedite velocity components ṽ ξ, ṽ η re computed in first step by solvin finite difference pproximtion of the momentum equtions. Intermedite velocity vector ṽ which is not solenoidl is then decomposed into diverence free nd rottionl free vector fields by solvin Poisson eqution with homoeneous Neumnn boundry conditions. The finl pproximtion of the v nd p t time t n+ cn be found nd the stedy-stte computed solution is defined by θ n+ θ n t θ n+ ε, where θ =, v η, C D, C L ; t is the time step nd θ n refers to the numericl pproximtion t time n t. To vlidte the present numericl lorithm, the uniform flow pst rottin circulr cylinders with Re =,. α = α. nd lre p between cylinder surfces = hve been clculted nd the results compred with simultion dt for flow pst sinle cylinder. All the TABLE I DRAG COEFFICIENT OF FLOW OVER A ROTATING CIRCULAR CYLINDER AT Re = WITH GAP SPACING = Contribution C D α =. α =. α =. Present Bdr et l. [].99. Inhm nd Tn [7] Chun [9].3..3 TABLE II LIFT COEFFICIENT OF FLOW OVER A ROTATING CIRCULAR CYLINDER AT Re = WITH GAP SPACING = Contribution C L α =. α =. α =. Present Bdr et l. [].7.7 Inhm nd Tn [7] Chun [9] TABLE III DRAG COEFFICIENTS OF FLOW OVER TWO CIRCULAR CYLINDERS AT FIXED Re =, α =. AND =.,., 3. FOR. α. α C D C D C D simultions hve been performed in lre domin so s to reduce the influence of the outer boundry. Tbles I nd II list dr nd lift coefficients from our clcultion nd mkes comprison with Inhm et l. 99 [7], Bdr et l. 99 [] nd Chun [9]. It cn be seen tht the differences re cceptble for C D nd C L. The nlysis of the dt collected in Tbles I nd II indicte n cceptble level of reement between our computtionl results nd the experimentl nd numericl dt vilble in literture. IV. NUMERICAL RESULTS In this work, two cylinders re plced in strem of the uniform speed U t infinity. Numericl results hve been presented into two prts. In the first prt, the left cylinder with rdius is non-rottin, whiles the riht cylinder with rdius is rottin in nti-clockwise nulr velocity. The influence of the rottion rte α is demonstrted in Tbles III nd IV. The vlues of dr nd lift coefficients in cse of fixed Reynolds number, Re = nd vrious p spcin, =.,., 3. for. α. re shown in Tbles ISBN: ISSN: 7-95 Print; ISSN: 7-9 Online IMECS

3 Proceedins of the Interntionl MultiConference of Enineers nd Computer Scientists Vol I IMECS, Mrch -,, Hon Kon TABLE IV L IFT COEFFICIENT OF FLOW OVER TWO CIRCULAR CYLINDERS AT FIXED Re =, α =. AND =.,., 3. FOR. α..3. α CL CL.59 CL α=. α= III nd IV. The dr coefficients of both cylinders decrese with incresin α, t fixed, see in the third column of Tble III. The lift coefficients of both cylinders increse with incresin α, t fixed, s shown in the third column of Tble IV. We found tht in the cse of fixed Reynolds number, Re =, rte of rottion correspondin to zero dr force α increses when increses. For exmple, Tble III shows tht t Re =, α correspondin to zero dr force re α.5, =., α 5., =. nd α 5.5, = 3.. The stremline ptterns from the simultions re shown in Fi.- Fi.3. The stremline ptterns nd pressure contours in the cse rte of rottion correspondin to zero dr force α re occurred for =.,., 3. with respectively see in Fi.. In the second prt, the left cylinder is rottin α =. nd the riht cylinder is rottin in counterclockwise nulr velocity. The effect of the rottion rte α is demonstrted in Tbles V nd VI. The vlues of dr nd lift coefficients in the cse of fixed Reynolds number, Re = nd vrious p spcin, =.,., 3. for. α. re lso shown. The dr coefficients of both cylinders decrese with incresin α, t fixed, see in the third column of Tble V. The lift coefficients of both cylinders increse with incresin α, t fixed, s shown in the third column of Tble VI. The simultions of stremline ptterns re shown in Fi. - Fi.. This work is fundmentl problem which cn be pplied to some clsses of obstructed flow. ISBN: ISSN: 7-95 Print; ISSN: 7-9 Online α=..95 α= Fi.. stremline ptterns of flow over two circulr cylinders t Re =, =.,., 3. nd α =., α =.. TABLE V D RAG COEFFICIENTS OF FLOW OVER TWO CIRCULAR CYLINDERS AT FIXED Re =, α =. AND =.,., 3. FOR. α. α CD CD CD V. C ONCLUSION AND S UGGESTION Numericl results show the flow structure over two rottin circulr cylinders with different rdii depend on the rte of rottion, the p spcin nd the Reynolds number. We found tht t fixed p spcin, the dr coefficient of both cylinders decrese with incresin α nd the lift coefficient of both cylinders increse with incresin α. In ddition we obtin the rte of rottion correspondin to zero dr force α increses when increses t fixed Re =. However these results α re not the numericl solutions of self-motion reime becuse its stisfy only CD but CL is not close to zero. Numericl solutions of flow α=. α=. over multiple cylinders cn be used the projection method with the finite difference method but the overnin equtions IMECS

4 Proceedins of the Interntionl MultiConference of Enineers nd Computer Scientists Vol I IMECS, Mrch -,, Hon Kon TABLE VI L IFT COEFFICIENTS OF FLOW OVER TWO CIRCULAR CYLINDERS AT FIXED Re =, α =. AND =.,., 3. FOR. α. α CL CL CL cylinders for self-motion reime with different rdii nd in cses of hiher Reynolds numbers my be investited for future works α=5. α= α=..9 α =. α=. α= α=. α= α=. α=5. α=..37 α=. α=. α= Fi. 3. stremline ptterns of flow over two circulr cylinders t Re =, α = 5. =., α =. =., α =. = 3. nd α = α=..3 α=5.5.5 ACKNOWLEDGMENT α=. α= We would like to thnks the Fculty of Applied Science, Kin Monkut s University of Technoloy North Bnkok for supportin us in doin this work Fi.. Stremline ptterns nd Pressure contours of flow over two circulr cylinders t Re =, α =.5 =., α = 5. =., α = 5.5 = 3. nd α =. hve to trnsform to nother coordinte system. In ddition, numericl solutions of fluid flow over two rottin circulr ISBN: ISSN: 7-95 Print; ISSN: 7-9 Online R EFERENCES [] M. M. Zdrvkovich, Review of Flow Interference between Two Circulr Cylinders in Vrious Arrnements, ASME J. Fluids En., vol. 99, pp. 33, 977. [] S. Kn, Chrcteristics of Flow over Two Circulr Cylinders in Side-by-Side Arrnement t Low Reynolds Numbers, Phys. of Fluids, vol. 5, no. 9, pp. -9, 3. [3] S. Sunnul nd N. P. Moshkin, Numericl Simultion of Flow over Two Rottin Self-Movin Circulr Cylinders, Recent Advnces in Computtionl Sciences, World Scientific Publishin-Imperil Collee Press, pp. 7-9,. IMECS

5 Proceedins of the Interntionl MultiConference of Enineers nd Computer Scientists Vol I IMECS, Mrch -,, Hon Kon α= α= α=. α= α=. α= α= α= α=. -. α=. α= α= Fi.. stremline ptterns of flow over two circulr cylinders t Re =, =.,., 3. nd α =., α = [] [7] α= α=. α= α= [] [9] Fi.. stremline ptterns of flow over two circulr cylinders t Re =, =.,., 3. nd α =., α =.. - in Cylindricl Bipolr Coordinte System, in Proceedins in Annul Ntionl Symposium on Computtionl Science nd Enineerin 5, pp S. J. Chorin, Numericl Solution of the Nvier-Stokes Equtions, Mth. Comp., vol., pp. 75-7, 9. D. B. Inhm nd T. Tn, A Numericl Investition into the Stedy Flow pst Rottin Circulr Cylinder t Low nd Intermedite Reynolds Numbers, Journl of Computtionl Physics, vol. 7, pp. 9-7, 99. D. Bdr, S. C. R. Dennis nd P. J. S. Youn, Stedy nd Unstedy Flow pst Rottin Cylinder t Low Reynolds Numbers, Computer nd Fluids, vol. 7, no., pp. 5799, 99. M-H. Chun, Crtesin Cut Cell Approch for Simultin Incompressible Flows with Riid Bodies of Arbitrry Shpe, Computer nd Fluids, vol. 35, no., pp. 73, α=. α= Fi. 5. stremline ptterns of flow over two circulr cylinders t Re =, =.,., 3. nd α =., α =.. [] S. K. Pnd, Two-Dimensionl Flow of Power-Lw Fluids over Pir of Cylinders in Side-by-Side Arrnement in the Lminr Reime, Brz. J. Chem. En., vol. 3, no., pp , 7. [5] S. Sunnul, On the Representtion of the Nvier-Stokes Equtions ISBN: ISSN: 7-95 Print; ISSN: 7-9 Online IMECS