Question

3. (4 pts) Write down the velocity potential for a lifting circular cylinder in a uniform flow of velocity V. Find the velocity components in this flow field. For what value of the circulation will the stagnation point be located on the surface of the cylinder at: (a) (r, 8) = (RO), (b) (r,®) = (R, -*/2).

Answer 1

The potential flow descoibes the velocity field as the gradient of a scalar function the velocity potential. we know that stream function of a cylindeo of radius R as 4.- Vooo sing (1-R -O before Vortex

is consider the se soltant velocity at any point on the suo face of cylindeo L ive ausing - here the poessure will be identified E there will be no lift on cylinder let.us.consider constant circulation ir asoound the cylindco. peripheral velocity is the speed that a point in the ciocumfeoence moves peo second. ② & ③ from the equations composity velocity 2 v= ausine+ 2TR Voo

The stocam function is defined for incompressible flows in two dimensions, - 42 = 1 4 Anotc. = I lng The above equation is the stream function for voster where ci is constant, take ci as fln Rso that the supeo impose V = 4, +42 a) The stoeam lines with ciocle radius R, when r=o. if D-R, 4-0 for all values of o! The velocity factoo also called the wave poopagation speed oo velocity of propagation. (l- en ) Vos cose [: W = Witwa dirrerentiate] vo che = (1 + R2 ha) vas sino of the

[: sino = coso Stagnation point location VG = VO FO 10-(-) voo cost sino ---Cuivoor) 0--Citieke od we sin e cost BER 2118 :: O lies between a & air b) when 8=R, O = -11 4 TTVOR Here the stagnation point be located on the surface of cylinder at Guvere) =!, UT VOOR