Continuity equations

Continuity equation is used to describe the transport of some quantities. It is based on the law of conservation of mass which states that mass can neither be created nor destroyed. For a flow process through a control volume where the stored mass does not change, fluid enters and leaves the controlled volume through its surface called controlled surface. Inflow of fluid equals to outflow which is net mass flow out of control volume through surface S = time rate of decrease of mass inside control volume \(v\).\[\frac{\partial }{{\partial t}}\mathop{{\int\!\!\!\!\!\int\!\!\!\!\!\int}\mkern-31.2mu \bigodot}\limits_v
{\rho dv + \mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc}\limits_S
{\rho v.ds = 0} } \]

Continuity equation in the form of partial differential equation is \[\frac{{\partial \rho }}{{\partial t}} + \nabla .\left( {\rho u} \right) = 0\] For a steady flow,\[\frac{\partial }{{\partial t}} = 0\]\[\nabla \cdot \left( {\rho v} \right) = 0\]For a flow with constant density on applying continuity equation, Area x velocity = constant.\[{A_1}{V_1} = {A_2}{V_2}\]

Example: Water is flowing through a pipe, having inlet inside diameter of 80 mm at 8m/sec. Calculate the velocity at exit if inside diameter is reduced to 30 mm.

Solution : From continuity equation, we know that \[{A_1}{V_1} = {A_2}{V_2}\] where 1 , 2 represents inlet and exit conditions\[{A_1} = \pi {\left( {\frac{{80}}{2}} \right)^2}\]\[{V_1} = 8\,m/s\]\[{A_2} = \pi {\left( {\frac{{30}}{2}} \right)^2}\]\[{V_2} = \,\,?\]\[ \Rightarrow \pi {\left( {\frac{{80}}{2}} \right)^2} \times 8 = \pi {\left( {\frac{{30}}{2}} \right)^2} \times {V_2}\]\[ \Rightarrow {V_2} = \frac{{80 \times 80 \times 8}}{{30 \times 30}}\]\[ = 56.88\,m/s.\]Therefore water is flowing at exit of pipe at 56.88 m/s.

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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 ...

Bernoulli's equation

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. ...

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...

Aerodynamics

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