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. ...

When there is a superposition of a uniform flow with a doublet (Which is a source-sink pair of flows), a non-lifting flow over a circular cylinder can be analysed. Stream function $'\psi '$ for a uniform flow is  $$\psi  = \left( {{V_\infty }r\sin \theta } \right)\left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right)$$ velocity field is obtained by differentiating above equation $${V_r} = \left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right){V_\infty }\cos \theta$$ $$\,{\rm{and}}\,{{\rm{V}}_\theta } =  - \left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right){V_\infty }\sin \theta$$ Stagnation points can be obtained by equating ${V_r}$ and ${{\rm{V}}_\theta }$ to zero.Considering an incompressible and inviscid flow pressure coefficient over a circular cylinder is  $${C_p} = 1 - 4{\sin ^2}\theta$$ Example: Calculate locations on the surface of a cylinder where surface pressure equals to free-stream pressure considering a non-lifting flow. Solution: Pressure coefficient ...

Lifting flow over a circular cylinder can be synthesised as superposition of a vortex flow of strength $'\tau '$ with non lifting flow over a cylinder. The stream function for the flow can be given as $$\psi  = \left( {{V_\infty }r\sin \theta } \right)\left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right) + \frac{\tau }{{2\pi }}\ln \frac{r}{R}$$ and velocity components in the radial and tangential direction can be obtained as  $${V_r} = \left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right){V_\infty }\cos \theta$$ $${V_\theta } =  - \left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right){V_\infty }\sin \theta  - \frac{\tau }{{2\pi r}}$$ The velocity on the surface of the cylinder can be given as   $$V =  - 2{V_\infty }\sin \theta  - \frac{\tau }{{2\pi R}}$$ and coefficient of lift as  $${c_l} = \frac{\tau }{{R{V_\infty }}}$$ Lift per unit span is given as  $${L^\prime} = {\rho _\infty }{V_\infty }\tau$$ Example: Calculate the peak coefficient of pressure for a...

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 ${L^\prime} = {\rho _\infty }{V_\infty }\tau$ $$\tau  = \frac{{{L^\prime}}}{{{\rho _\infty }{V_\infty }}} = \frac{8}{{\left( {1.23} \right)\left( {50} \right)}} = 0.13\,{m^2}/s$$