Question

QUESTION 2 In an experiment to measure approaching uniform flow, U.to a non-rotating circular cylinder as shown in Figure Q2, a small hole is to be drilled at the stagnation point on the cylinder surface. (a) Determine the location of the drilled hole on the cylinder surface using potential flow theory by combining a doublet with a uniform flow. (3.5 marks) (b) Determine the theoretical pressure distribution expression on the cylinder surface, Pc and plot or sketch a graph of coefficient of pressure, Cp against location, on the cylinder surface for 0 = 0, 90 and 180°. (3.5 marks) (c) Show that dimensionless drag force or coefficient of drag, Cp and coefficient of pressure, Cp on the cylinder surface pressure is related with expression p cos Ꮎ dᎾ (2 marks) (d) During the experiment, the drilled hole is misaligned by an angle, a = 10' but still interpreted as the stagnation point. Is the measured free stream velocity, Uco measured is the same, lower or greater? Briefly explain your answer. (1 mark) Given the coefficient of pressure, Cp = 0 and coefficient of drag, Cp = 2 where D is the cylinder diameter Figure Q2

Answer 1

(PAGE NO: (DATE: Soln since the surface of the cylinder is to be approximated using potential flow the solution is as follows. combining a uniforme flow having velocity Has with a doublet Shaving strength K. In polar co-ordinates (210) the streane functions for a uniform flow is vorsino and for a doublet is uniform flow y = Vorsino Doublet 4 = -K sino nonliffins flow over a cylinder 4 = Vorsino-k suo 27 flow is Superposition of uniform and doublet gives nonlifting over a ylinder and the combined streams function ' Vo y sino k siue - Von Siro ( -.42) Then, 4 = Vorsino (182) Stagnation point is where the velocity is zero. Therefore calculating velocity from & and equating to zero to locate stagnation points.

Voy = 124 = I (vs r cose) (1-2²) 7 20 20 Vo = -24 =-{ voor sine 28² + (1-2) a sing? 3-614.02 ) Van sing Equating Vox of vo to zero gives (R² (1+R² t P2 Uxo sino =0 oo & =0 J ao caso 6.o) - Ro) and (RT) . These are the locations of a stagnation points on the cylinder which are represented by A4 B in the figure The frent stagnation point B at 6, 8) = (RO) is where the hole is to be drilled. a 827 b) The velocity field is given by the Vy = (1-R2) Uo caso No = -(1+R² ) o sino It the surface of the cylinder rart V = 0 and vo = - 2 Voo sino I now, the pressurse wefficient, le for incompres flow is given as p= 1-10 12 1-4 sino

DATE: at At At o=0 o= 90 o = 180 (p= cp = Cp 1-4X0 = 1 1-4x1= -3 1-4x0 = 4 Bottom hath of cylinder Top half of cylinder Under Rear haly of cylinder Theoretical pressure distribution, up vso on wybiades surface c) The drag coefficient cd is given by . E u Cp of upper Cola Les Capua Core) dy z Sus Subject TS Pie Cp of lower surface TE cada I f cp, udy - 1 f (pose dy In polar co-ordinates Rsino dy = R caso do cal = 1 Cpu loso do and C= 2R Cpre coso do - Cp, u coso do- f (p. e coso do

Both (p. u 4 pre are given by the u (p. 1-4 sinto a cola 2 - fcp loso do af up caso do/ - sa cp corso do op Cos o do. 5 000 Lideally the stagnation point should have been at B but due to misaligned operations it has been neoved 10° to B' At B the pressure is stagnation which is the man neum pressure with zero velocity and at & the presse is minimum with maximune velocity . Hence, the pressure decreases as one neoves frone B to & along the surface and relocity increases. This measured free streami velocity , o measured is not e not same as stagnation velocity are is greater a nd