Question

Consider a circular cylinder of radius a whose central axis is stationary. The cylinder is surrounded by a fluid that is moving with a uniform steady velocity Uk far from the cylinder; the cylinder axis is in the z-direction The cylinder is also spinning on its axis with constant angular velocity, Ω2, where x, у and z are the usual Cartesian unit vectors We wish to model this 2D flow using an inviscid approximation. This is achieved by first calculating the rotating steady flow of an inviscid fluid that is stationary far from the cylinder and whose angular velocity at the cylinder's surface is Ω2. The velocity field resulting solely from the impinging uniform flow, Ux, is then superposed onto the rotating flow to give the complete flow i. Calculate the velocity potential, velocity field and streamfunction for this complete flow, using the above approximation ii. Hence calculate the pressure and vector force acting on the cylinder. Express the force as a function of the circulation「--2παΏ iii. What are the drag and lift on the cylinder? How are these related to the cylinder's direction of rotation? iv. Non-dimensionalise the streamfunction and express it in terms of the dimensionless angular velocity of the cylinder Ω Ωα/U. Explain the physical significance of Ω. Hence plot the streamlines of the flow for a range of different values of Ω (using any numerical package, eg., Matlab, Mathematica, WolframAlpha etc). Connect the shape of the streamlines to the vector force acting on the cylinder, and thus provide a physical explanation for the underlying force v. What common occurrence (particularly in sport) does this flow (and force) represent? Hint: The same phenomenon occurs for rotating spheres. Does the above solution give you any insight into the physics underlying this common occurrence? Explain your answer

Answer 1

Drag and Lift on the cyL Q -inder: Pressy re in the cylinder Surfacet- Pressure becomes uniform at large distance fron the cylindes. the pressure b is knouN as Let, asSuMe- uniform velocit Uo well aS Bernoulli's equation betuween in finitd APbly cylindes the on the poin ts and PotentialEn ergy the variation of Neglectimg 2 2 Ub Pb P8 1 Po 28 28 P8 ( where, b- surfa ce on ) Hlind ea Up shouId be oniy in the transverse fluid eon't penetra te the direction' Becouse boundary) SOlid 2 Uo vo(at Wb - A/2 ( lat, 2 Uo sine z

Pg. 2 Se obtain egn (1 and (2) From U (2Uo sine P8 Pb 28 Pa Lift and Drag Lift:- Force acti ng on the clindeo (p er direcion normal Cinit length) in the uniform to flous Drag Force acting the cylinder (bea on Linit lenath) in the direchion parallel to flows 1Pb A e Uo 10.1 the Drag is calculated by the integraHng force components grising out of pressure the in the 2-direction boundard on the then, Drag Force 2TT DE- Ph coserde O rde =infinitsimal eng th on the circumference since, ( 12

-Po de COSe D I Uo 2TT (2Uosine) 2 1/2 Uo Po + D - P& Uo .28 COsOde 2T NO (1-4 U, sin?e) COSB.de lift force may be Similary, the catculated as - 2TT /2 J sine ( Uo From eq and aftes the intara is eaed aue LO D O 9 Cylinder wil aloays exprieme force drog Hlouo eue, Some Bernoullis equation calculate . be cab distri bution on the clind pressure surface- the Pb (e) Ub ) 28 Pa 20 Pa J 4sin2p pressure cofficient, Pble)- Po Cp is therefore the CI-4 sin2e Cp= /2 PUo 2