Dr. A.B.M. Toufique Hasan. Lecture-13

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1 7/5/8 ME 45: Aeodynamics D. A.B.M. Toufique Hasan Pofesso Depatment of Mechanical Engineeing g Bangladesh Univesity of Engineeing & Technology BUET, Dhaka Lectue-3 7/5/8 Doublet & Flow Ove a stationey ti Cicula Cylinde teache.buet.ac.bd/toufiquehasan/ bd/toufiquehasan/ toufiquehasan@me.buet.ac.bd D. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeodynamics Jan. 8 Souce + Sink Flows Conside a souce and sink placed at disce b to the left and ight of the oigin,. Futhe, conside souce and sink ae of equal stengths. In this case, the steam function is; soucesink souce sink soucesink ik souce sink soucesink soucesink angulal positionof point itpelativeto lti t souce angula positionof point p elativeto sink in genealized coodinate system,location of p p, x cos y sin x y D. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeodynamics Jan. 8

2 7/5/8 Souce + Sink Flows b sin b souce sink soucesink b sin b soucesink sin Again, cos b sin cos b The esult steamlines fo this type of flow souce + sink ae shown in the above figue. D. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeodynamics Jan. 8 3 Doublet, b b Fo small values of disce, b the gent of an angle appoaches the value of the angle fo small angle soucesink b sin b b sin b b b sin The so - called doublet is fomed by letting the souce and sink ikappoach one anothe b while inceasing the stength b ; so that the poduct b emains const. In this case, so the doublet above expession sin comes as, whee stength of doublet b D. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeodynamics Jan. 8 4

3 7/5/8 Doublet doublet sin const. The plots of lines of const ψ eveal that the steamlines fo a doublet ae cicles though the oigin gent to x-axis. const. The coesponding velocity potential cos doublet Flow net of doublet In this case, the souce and sink ae placed along the x-axis. b const. Howeve, if the souce and sink ae placed along y-axis, the esult t doublet will be oiented as follows: D. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeodynamics Jan. 8 5 Flow ove a stationey cicula cylinde y,θ x Unifom flow Doublet Flow ove a cicula cylinde Conside the addition of a unifom flow with velocity eoc y and a doublet of stength κ as shown in figue. The diection of the doublet is upsteam, facing into the unifom flow. The combined steam function fo this type of flow is: unifomflowdoublet unifom flow sin sin sin doublet ; whee sin Steam function fo the flow ove a cicula cylinde of adius D. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeodynamics Jan

4 7/5/8 Flow ove a stationey cicula cylinde The velocity field can be obtained as: cos sin To locate the stagnation points in the flow field: cos sin Simuleously solving the above two equations fo and θ gives two stagnation points, namely, A and B which ae located at:,, A, B, Two Stagnation points A, B B D. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeodynamics Jan. 8 7 Flow ove a stationey cicula cylinde On the suface of the cylinde = cos sin sin And the esult velocity on the cylinde suface is s sin The maximum velocity occus at the top and bottom of the cylinde θ = π/, - π/ and has a magnitude of twice the upsteam velocity,. D. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeodynamics Jan

5 7/5/8 Flow ove a stationey cicula cylinde Pessue distibution on the cylinde suface can be obtained fom the Benoulli s equation witten fo a point fa fom the cylinde whee pessue is p and velocity is : p ps s ; ps is thesufacepessue p ps sin ; s sin p s p sin p p s 4sin c p suface 4sin D. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeodynamics Jan. 8 9 Flow ove a stationey cicula cylinde The esult foce pe unit length developed on the cylinde can be detemined by integating the pessue ove the suface as: Dag foce, F x ps d cos F D [ p F D 4sin ]cos d since p s p 4sin Lift foce, F y ps d sin F L [ p F L 4sin ]sin d These indicate thatt both the dag and lift as pedicted d by potential ti flow theoy inviscid, i id iotational ti fo a fixed stationay cylinde in a unifom steam ae zeo. These esults ae consistent with the pessue distibution which is symmetic fo a fixed cylinde in a unifom steam. D. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeodynamics Jan. 8 5

6 7/5/8 Flow ove a stationey cicula cylinde Howeve, we know fom expeience that thee is a significant dag developed on a cicula cylinde when it is placed in a moving fluid. This discepancy in the evaluation of dag is known as d Alembet s dalembetspaadox. The paadox is named afte Jean le ond d Alembet , 783 a Fench mathematician who fist showed that the dag on bodies immesed in inviscid fluids is zeo. Howeve, this paadox is explained while fluid viscosity it is consideed d as shown in figue: cous isc paa s sep ation Pessue distibution is symmetic along y-axis F D = Pessue distibution is NOT symmetic along y-axis F D actual/eal flow D. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeodynamics Jan. 8 6