Skip to main content

Lifting flow over a cylinder

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 lifting flow over a circular cylinder for which lifting coefficient  is 70.
Solution: Maximum velocity for lifting flow over a circular cylinder occurs at top surfaces and is equal to summation of non-lifting value of maximum flow velocity,\( - 2{V_\infty }\) and vortex \(\frac{{ - \tau }}{{2\pi R}}\) therefore,\[V =  - 2{V_\infty } - \frac{\tau }{{2\pi R}}\]coefficient of lift is \[{c_l} = \frac{\tau }{{R{V_\infty }}} = 7\]therefore, \(\frac{\tau }{R} = 7{V_\infty }\); On substitution the value of \(\frac{\tau }{R}\) in the velocity equation, we get \[V =  - 2{V_\infty } - \frac{7}{{2\pi }}{V_\infty } =  - 3.11{V_\infty }\]therefore peak coefficient of pressure will be \[{C_p} = 1 - {\left( {\frac{V}{{{V_\infty }}}} \right)^2} = 1 - {\left( { - \frac{{3.11{V_\infty }}}{{{V_\infty }}}} \right)^2}\]\[ = 1 - {\left( { - 3.11} \right)^2} =  - 8.67\]Example: Calculate the lift per unit span for a lifting flow over a circular cylinder with a diameter of 0.8 m, having a free stream velocity of 35 m/s and maximum velocity on the surface of the cylinder being 80 m/s ,considering freestream sea level conditions.
Solution: At sea level, density of air = \(1.225\,kg/{m^3}\).The maximum velocity for a lifting flow over a circular cylinder occurs at top surface of the cylinder, where \(\theta \) equals to \({90^ \circ }\).\[{V_\theta } =  - 2{V_\infty }\sin \theta  - \frac{\tau }{{2\pi R}}\]\[ \Rightarrow \tau  =  - 2\pi R\left( {{V_\theta } + 2{V_\infty }} \right)\]\({V_\theta }\) is negative in clockwise direction and \('\tau '\) is positive, according to sign conventions.Therefore, \[\tau  =  - 2\pi R\left( {{V_\theta } + 2{V_\infty }} \right) =  - 2\pi \left( {0.4} \right)\left[ { - 80 + 2\left( {35} \right)} \right]\]\[\tau  =  - 2\pi \left( {0.4} \right)\left[ { - 10} \right]\]\[\tau  = 25.133\,{m^2}/s\]Therefore, lift per unit span is \[{L^\prime} = {\rho _\infty }{V_\infty }\tau \]\[{L^\prime} = 1.225 \times 35 \times 25.133\]\[ = 1077.577\,N/m\]Example: If the velocity \({V_\infty }\) of a flow is doubled in the case of lifting flow over a circular cylinder of a given radius and with a given circulation, does the shape of the streamline change on keeping the circulation same.
Solution: Since for the case of lifting flow over a circular cylinder \[\left( {\frac{{{V_r}}}{{{V_\infty }}}} \right) = \left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right)\cos \theta \]and\[\left( {\frac{{{V_\theta }}}{{{V_\infty }}}} \right) =  - \left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right)\sin \theta  - \frac{\tau }{{2\pi r{V_\infty }}}\]Since\(\left( {\frac{{{V_\theta }}}{{{V_\infty }}}} \right)\) is a function of \({V_\infty }\), so as \({V_\infty }\) changes, the direction of the resultant velocity at a given point also changes.Therefore as \({V_\infty }\) is changing the shape of streamlines also changes.
Labels:

Comments

Post a Comment

Popular posts from this blog

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 ...
Post a Comment
Read more

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. ...
Post a Comment
Read more

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...
Post a Comment
Read more