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 is given as \[{C_p} = \frac{{p - {p_\infty }}}{{{q_\infty }}}\]Since \(p = {p_\infty }\) therefore \({C_p} = 0\).Also, \[{C_p} = 1 - 4{\sin ^2}\theta = 0\]\[ \Rightarrow \sin \theta = \pm \frac{1}{2}\]\[ \Rightarrow \theta = {30^ \circ },{150^ \circ },{210^ \circ },{330^ \circ }\]are the locations on the surface of cylinder.
Example: Consider a non-lifting flow over a circular cylinder and derive an expression for the pressure coefficient at any arbitrary point \(\left( {r,\theta } \right)\) in the flow.On the surface of the cylinder show that it reduces to \({C_p} = 1 - 4{\sin ^2}\theta \).
Solution: The stream function for a non lifting flow over a circular cylinder is given as\[\psi = \left( {{V_\infty }r\sin \theta } \right)\left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right)\] \[{V_r} = \frac{1}{r}\frac{{\partial \psi }}{{\partial \theta }} = \left( {{V_\infty }\cos \theta } \right)\left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right)\]\[{V_\theta } = - \frac{{\partial \psi }}{{\partial r}} = - \left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right){V_\infty }\sin \theta \]\[{V^2} = V_r^2 + V_\theta ^2 = \left( {V_\infty ^2{{\cos }^2}\theta } \right){\left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right)^2} + {\left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right)^2}V_\infty ^2{\sin ^2}\theta \]Coefficient of pressure \({C_p}\) will be \[{C_p} = 1 - \frac{{{V^2}}}{{V_\infty ^2}} = 1 - {\cos ^2}\theta {\left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right)^2} + {\left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right)^2}{\sin ^2}\theta \]\[{C_p} = 1 - {\left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right)^2}{\cos ^2}\theta - {\left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right)^2}{\sin ^2}\theta \]This represents coefficient of pressure at any arbitrary point in the flow.At, the surface of cylinder r = R, therefore, \[{C_p} = 1 - \frac{{{V^2}}}{{V_\infty ^2}} = 1 - 4{\sin ^2}\theta \]Example: Consider the non lifting flow over a circular cylinder of a given radius, where free stream velocity is \({V_\infty }\).If \({V_\infty }\) is doubled,explain if there is any change in the shape of the stream lines.
Solution: \[{V_r} = \frac{1}{r}\frac{{\partial \psi }}{{\partial \theta }} = \left( {{V_\infty }\cos \theta } \right)\left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right)\] and \[{V_\theta } = - \frac{{\partial \psi }}{{\partial r}} = - \left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right){V_\infty }\sin \theta \] therefore, \[\frac{{{V_r}}}{{{V_\infty }}} = \left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right)\cos \theta \]\[\frac{{{V_\theta }}}{{{V_\infty }}} = - \left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right)\sin \theta \]So, at any given point \(\left( {r,\theta } \right)\), \({V_r}\) and \({{V_\theta }}\) are both directly proportional to\({{V_\infty }}\).Therefore, the direction of the resultant,\(\mathop V\limits^ \to \) is the same, for any value of \({{V_\infty }}\).This infers that the shape of the streamlines remains the same.
Example: Consider a non-lifting flow over a circular cylinder and derive an expression for the pressure coefficient at any arbitrary point \(\left( {r,\theta } \right)\) in the flow.On the surface of the cylinder show that it reduces to \({C_p} = 1 - 4{\sin ^2}\theta \).
Solution: The stream function for a non lifting flow over a circular cylinder is given as\[\psi = \left( {{V_\infty }r\sin \theta } \right)\left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right)\] \[{V_r} = \frac{1}{r}\frac{{\partial \psi }}{{\partial \theta }} = \left( {{V_\infty }\cos \theta } \right)\left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right)\]\[{V_\theta } = - \frac{{\partial \psi }}{{\partial r}} = - \left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right){V_\infty }\sin \theta \]\[{V^2} = V_r^2 + V_\theta ^2 = \left( {V_\infty ^2{{\cos }^2}\theta } \right){\left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right)^2} + {\left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right)^2}V_\infty ^2{\sin ^2}\theta \]Coefficient of pressure \({C_p}\) will be \[{C_p} = 1 - \frac{{{V^2}}}{{V_\infty ^2}} = 1 - {\cos ^2}\theta {\left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right)^2} + {\left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right)^2}{\sin ^2}\theta \]\[{C_p} = 1 - {\left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right)^2}{\cos ^2}\theta - {\left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right)^2}{\sin ^2}\theta \]This represents coefficient of pressure at any arbitrary point in the flow.At, the surface of cylinder r = R, therefore, \[{C_p} = 1 - \frac{{{V^2}}}{{V_\infty ^2}} = 1 - 4{\sin ^2}\theta \]Example: Consider the non lifting flow over a circular cylinder of a given radius, where free stream velocity is \({V_\infty }\).If \({V_\infty }\) is doubled,explain if there is any change in the shape of the stream lines.
Solution: \[{V_r} = \frac{1}{r}\frac{{\partial \psi }}{{\partial \theta }} = \left( {{V_\infty }\cos \theta } \right)\left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right)\] and \[{V_\theta } = - \frac{{\partial \psi }}{{\partial r}} = - \left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right){V_\infty }\sin \theta \] therefore, \[\frac{{{V_r}}}{{{V_\infty }}} = \left( {1 - \frac{{{R^2}}}{{{r^2}}}} \right)\cos \theta \]\[\frac{{{V_\theta }}}{{{V_\infty }}} = - \left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right)\sin \theta \]So, at any given point \(\left( {r,\theta } \right)\), \({V_r}\) and \({{V_\theta }}\) are both directly proportional to\({{V_\infty }}\).Therefore, the direction of the resultant,\(\mathop V\limits^ \to \) is the same, for any value of \({{V_\infty }}\).This infers that the shape of the streamlines remains the same.
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