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\[\int {f(x)} \]

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Importance of Coefficients (Lift and Drag)

Aerodynamic coefficients plays an important role in performance analysis as well as  design of aeroplanes.Coefficient of lift is defined as Lift divided by dynamic pressure, \(L/{q_\infty }S\) where as drag coefficient is defined as Drag divided by dynamic pressure, \(D/{q_\infty }S\) . Coefficient of lift,maximum or \({C_{L,\max }}\) is the determining factor for stalling velocity of aircraft.The higher is the \({C_{L,\max }}\) the lower is the stalling velocity.\[{V_{stall}} = \sqrt {\frac{{2W}}{{{\rho _\infty }S{C_{L,\max }}}}} \] However,\({C_{L,\max }}\) can be increased by the use of mechanical devices like high-lift devices.High lift devices include flaps,slats and slots on the wing. Air plane flying at given altitude with maximum thrust \({T_{\max }}\), the maximum value of \({V_\infty }\) ,corresponds to flight at \({C_{D,\min }}\).\[{V_{\max }} = \sqrt {\frac{{2{T_{\max }}}}{{{\rho _\infty }S{C_{D,\min }}}}} \] The actual value of \({C_L}\) and...
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Non Lifting flow over a circular cylinder

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 ...
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Streamlines

Streamlines: Streamlines represents field lines in a fluid flow. It is a path traced out by massless fluid particles moving with the flow, which is tangential to the instantaneous velocity direction. Different streamlines at same instant in a flow do not intersect or flow across it, because a fluid particle cannot have two different velocities at the same point. For a unsteady flow, streamline pattern is different at different times because the velocity vectors are fluctuating with time both in magnitude and direction. In a fluid flow, a bundle of streamlines is called a streamtube. Walls of an ordinary garden hose form a streamtube for the water flowing through the hose. Differential equation for a streamline in two dimensions is \[vdx - udy = 0\] Question: x and y components of a velocity field are given as \(u = \frac{{4x}}{{\left( {{x^2} + {y^2}} \right)}}\) and \(v = \frac{{4y}}{{\left( {{x^2} + {y^2}} \right)}}\), what is the equation of streamlines.   ...
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