Pressure coefficient

Pressure coefficient Cp is defined as  $${C_p} = \frac{{p - {p_\infty }}}{{{q_\infty }}}$$ ${q_\infty } = \frac{1}{2}{\rho _\infty }v_\infty ^2$, let for any point in the flow where pressure and velocity are 'p' and 'v', respectively and free-stream pressure and velocity be ${{p_\infty }}$ and ${v_\infty }$. From Bernoulli's equation $${p_\infty } + \frac{1}{2}\rho v_\infty ^2 = p + \frac{1}{2}\rho {v^2}$$ $$\Rightarrow \left( {p - {p_\infty }} \right) = \frac{1}{2}\rho \left( {v_\infty ^2 - {v^2}} \right)$$ $${C_p} = \frac{{p - {p_\infty }}}{{{q_\infty }}} = \frac{{\frac{1}{2}\rho \left( {v_\infty ^2 - {v^2}} \right)}}{{\frac{1}{2}\rho v_\infty ^2}}$$ $$\Rightarrow {C_p} = 1 - {\left( {\frac{v}{{{v_\infty }}}} \right)^2}$$ This equation is valid for incompressible flow only.

Example: Find pressure coefficient at a point on an  airfoil where velocity is 220 ft/s, which is in a free stream flow of 100 ft/s.

Solution: pressure coefficient is given as  $${C_p} = 1 - {\left( {\frac{v}{{{v_\infty }}}} \right)^2}$$ $$= 1 - {\left( {\frac{{220}}{{100}}} \right)^2}$$ $$= 1 - 4.84$$ $$=  - 3.84$$ Example: Pressure coefficient at a certain point on an airfoil is -4.2.Calculate velocity at this point assuming the flow over the airfoil to be inviscid and incompressible is (a) 50 ft/s and (b) 200 ft/s.

Solution: (a) For inviscid and incompressible flow coefficient of pressure is given as  $${C_p} = 1 - {\left( {\frac{v}{{{v_\infty }}}} \right)^2}$$ $$v = \sqrt {v_\infty ^2\left( {1 - {C_p}} \right)}$$ $$= \sqrt {{{\left( {50} \right)}^2}\left( {1 - \left( { - 3.84} \right)} \right)}$$ $$= \sqrt {2500\left( {1 + 3.84} \right)}$$ $$= 110\,{\rm{ft/sec}}$$ also (b) for ${{v_\infty }}$ = 200 ft/sec  $${C_p} = 1 - {\left( {\frac{v}{{{v_\infty }}}} \right)^2}$$ $$v = \sqrt {v_\infty ^2\left( {1 - {C_p}} \right)}$$ $$= \sqrt {{{\left( {200} \right)}^2}\left( {1 - \left( { - 3.84} \right)} \right)}$$ $$= \sqrt {40000\left( {1 + 3.84} \right)}$$ $$= 440\,{\rm{ft/sec}}$$ Example: The velocity of an aircraft is 100 m/sec. At a given point on the surface of wing flow velocity is 150 m/sec. What is the pressure coefficient at this point?
Solution: Considering the flow to be inviscid and incompressible, Pressure coefficient ${{\rm{C}}_{\rm{p}}} = 1 - {\left( {\frac{v}{{{v_\infty }}}} \right)^2}$ $$= 1 - {\left( {\frac{{150}}{{100}}} \right)^2}$$ $$=  - 1.25$$