Considering a non-lifting flow over a circular cylinder derive an expression for the pressure coefficient.
Consider a non-lifting flow over a circular cylinder and derive an expression for the pressure coefficient at any arbitrary point \(\left( {r,\theta } \right)\) in the flow. On the surface of the cylinder show that it reduces to \({C_p} = 1 – 4{\sin ^2}\theta \).
The stream function for a non lifting flow over a circular cylinder is given as\[\psi = \left( {{V_\infty }r\sin \theta } \right)\left( {1 – \frac{{{R^2}}}{{{r^2}}}} \right)\] \[{V_r} = \frac{1}{r}\frac{{\partial \psi }}{{\partial \theta }} = \left( {{V_\infty }\cos \theta } \right)\left( {1 – \frac{{{R^2}}}{{{r^2}}}} \right)\]\[{V_\theta } = – \frac{{\partial \psi }}{{\partial r}} = – \left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right){V_\infty }\sin \theta \]\[{V^2} = V_r^2 + V_\theta ^2 = \left( {V_\infty ^2{{\cos }^2}\theta } \right){\left( {1 – \frac{{{R^2}}}{{{r^2}}}} \right)^2} + {\left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right)^2}V_\infty ^2{\sin ^2}\theta \]Coefficient of pressure \({C_p}\) will be \[{C_p} = 1 – \frac{{{V^2}}}{{V_\infty ^2}} = 1 – {\cos ^2}\theta {\left( {1 – \frac{{{R^2}}}{{{r^2}}}} \right)^2} + {\left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right)^2}{\sin ^2}\theta \]\[{C_p} = 1 – {\left( {1 – \frac{{{R^2}}}{{{r^2}}}} \right)^2}{\cos ^2}\theta – {\left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right)^2}{\sin ^2}\theta \]This represents coefficient of pressure at any arbitrary point in the flow.At, the surface of cylinder r = R, therefore, \[{C_p} = 1 – \frac{{{V^2}}}{{V_\infty ^2}} = 1 – 4{\sin ^2}\theta \]
