Consider the nonlifting flow over a circular cylinder. Derive an expression for the pressure coefficient at an arbitrary point \((r, θ)\) in this flow, and show that it reduces to Equation \(Cp = 1 − 4 sin^{2} θ\) on the surface of the cylinder.
Consider the nonlifting flow over a circular cylinder. Derive an expression for the pressure coefficient at an arbitrary point \((r, θ)\) in this flow, and
show that it reduces to Equation \(Cp = 1 − 4 sin^{2} θ\) on the surface of the cylinder.
For the nonlifting flow over a circular cylinder stream function \[\psi=\left ( V_{\infty}r sin\theta \right )\left ( 1-\frac{R^{2}}{r^{2}} \right )\]
\[V_{r}=\frac{1}{r}\frac{\partial \psi}{\partial \theta}=V_{\infty}cos\theta \left ( 1-\frac{R^{2}}{r^{2}} \right )\]
\[V_{\theta}=-\frac{\partial \psi}{\partial r}=-\left ( 1+\frac{R^{2}}{r^{2}} \right )V_{\infty}sin\theta\]
\[V^{2}=V_{r}^{2}+V_{\theta}^{2}=\left ( 1-\frac{R^{2}}{r^{2}} \right )V_{\infty}^{2}cos^{2}\theta+\left ( 1+\frac{R^{2}}{r^{2}} \right )^{2}V_{\infty}^{2}sin^{2}\theta\]
\[C_{p}=1-\frac{V^{2}}{V_{\infty}^{2}}=1-\left ( 1-\frac{R^{2}}{r^{2}} \right )cos^{2}\theta-\left ( 1+\frac{R^{2}}{r^{2}} \right )^{2}sin^{2}\theta\]
At the surface of the cylinder ; \(r=R\)
Therefore \[C_{p}=1-4sin^{2}\theta\]
