Consider the lifting flow over a circular cylinder. The lift coefficient is 7. Calculate the peak coefficient of pressure.

Maximum velocity for lifting flow over a circular cylinder occurs at top surfaces and is equal to summation of non-lifting value of maximum flow velocity,$– 2{V_\infty }$ and vortex $\frac{{ – \tau }}{{2\pi R}}$ therefore, $$V =  – 2{V_\infty } – \frac{\tau }{{2\pi R}}$$ coefficient of lift is  $${c_l} = \frac{\tau }{{R{V_\infty }}} = 7$$ therefore, $\frac{\tau }{R} = 7{V_\infty }$; On substitution the value of $\frac{\tau }{R}$ in the velocity equation, we get  $$V =  – 2{V_\infty } – \frac{7}{{2\pi }}{V_\infty } =  – 3.11{V_\infty }$$ therefore peak coefficient of pressure will be  $${C_p} = 1 – {\left( {\frac{V}{{{V_\infty }}}} \right)^2} = 1 – {\left( { – \frac{{3.11{V_\infty }}}{{{V_\infty }}}} \right)^2}$$ $$= 1 – {\left( { – 3.11} \right)^2} =  – 8.67$$