Aerodynamics

Euler's equation is given as \[dp =  - \rho vdv\]This equation applies to an incompressible and in-viscid flow where \(\rho  = \,{\rm{constant}}\). In a streamline, in between two points 1 and 2, the above Euler equation can be integrated as \[\int\limits_{{p_1}}^{{p_2}} {dp =  - \rho \int\limits_{{v_1}}^{{v_2}} {vdv} } \]\[{p_2} - {p_1} =  - \rho \left( {\frac{{v_2^2}}{2} - \frac{{v_1^2}}{2}} \right)\]\[ \Rightarrow {p_1} + \frac{1}{2}\rho v_1^2 = {p_2} + \frac{1}{2}\rho v_2^2\]This can be written as\[p + \frac{1}{2}\rho {v^2} = {\rm{constant}}\] for a streamline.For an rotational flow the value of constant is changing from streamline to another.For irrotational flow,the constant is same for all streamlines and \[p + \frac{1}{2}\rho {v^2} = {\rm{constant}}\]throughout the flow. Physical significance of Bernoulli's equation is that when the velocity increases, the pressure decreases  and when the velocity decreases, the pressure increases. ...

This theorem states that the lift per unit span is directly proportional to circulation.\[{L^\prime} = {\rho _\infty }{V_\infty }\tau \]Example: Lift per unit span of a spinning circular cylinder in a free stream with velocity  of 50 m/s is 8 N/m at a standard sea level conditions.Find the circulation \('\tau '\) around the cylinder. Solution: Lift per unit span is given as \({L^\prime} = {\rho _\infty }{V_\infty }\tau \)\[\tau  = \frac{{{L^\prime}}}{{{\rho _\infty }{V_\infty }}} = \frac{8}{{\left( {1.23} \right)\left( {50} \right)}} = 0.13\,{m^2}/s\]