A circular cylinder and a sphere are in a free stream flow with their axis perpendicular to the flow. At the top of the sphere there is a pressure tap which is connected by a tube to one side of a manometer. There is a pressure tap on the surface of the cylinder which is connected to the other side of the manometer. Find the location of this tap on cylindrical surface such that there is no deflection of fluid in the manometer.
A circular cylinder and a sphere are in a free stream flow with their axis perpendicular to the flow. At the top of the sphere there is a pressure tap which is connected by a tube to one side of a manometer. There is a pressure tap on the surface of the cylinder which is connected to the other side of the manometer. Find the location of this tap on cylindrical surface such that there is no deflection of fluid in the manometer.
Pressure distribution for a flow over a sphere is given by \[\left ( C_{p} \right )_{Sphere}= 1 – \frac{9}{4}sin^{2}\theta\]For a cylinder, coefficient of pressure is given as \[\left ( C_{p} \right )_{Cylinder} = 1 – 4sin^{2}\theta\]At the top of the sphere, \(\theta = \frac{\pi}{2}\)
Therefore, coefficient of pressure is \[\left ( C_{p} \right )_{Sphere}=\left ( 1 – \frac{9}{4}sin^{2}\theta \right )=1-\frac{9}{4}sin^{2}\left ( \frac{\pi}{2} \right )=-1.25\]In order that there is no deflection of the manometer fluid \(\left ( C_{p} \right )_{Sphere}\) should be equal to the \(\left ( C_{p} \right )_{Cylinder}\)
Therefore,
\[\left ( C_{p} \right )_{Sphere}=\left ( C_{p} \right )_{Cylinder}\]\[\Rightarrow -1.25=1-4sin^{2}\theta\]\[\Rightarrow sin^{2}\theta = \frac{-1.25-1}{-4}=0.5625\]\[\Rightarrow sin\theta=\sqrt{0.5625}=0.75\]\[\Rightarrow \theta = sin^{-1}\left ( 0.75 \right )=48.59^{\circ}\]Therefore, tap should be located at \(48.59^{\circ}\) above or below the stagnation point on the cylinder.
