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Next: Axisymmetric Incompressible Inviscid Flow Up: Two-Dimensional Potential Flow Previous: Blasius Theorem

Exercises

  1. Demonstrate that a line source of strength $ Q$ (running along the $ z$ -axis) situated in a uniform flow of (unperturbed) velocity $ {\bf V}$ (lying in the $ x$ -$ y$ plane) and density $ \rho$ experiences a force per unit length

    $\displaystyle {\bf F} =- \rho\,Q\,{\bf V}.$

  2. Demonstrate that a vortex filament of intensity $ {\mit\Gamma}$ (running along the $ z$ -axis) situated in a uniform flow of (unperturbed) velocity $ {\bf V}$ (lying in the $ x$ -$ y$ plane) and density $ \rho$ experiences a force per unit length

    $\displaystyle {\bf F} = -\rho\,{\mit\Gamma}\,{\bf V}\times{\bf e}_z.
$

  3. Show that two parallel line sources of strengths $ Q$ and $ Q'$ , located a perpendicular distance $ r$ apart, exert a radial force per unit length $ \rho\,Q\,Q'/(2\pi\,r)$ on one another, the force being attractive if $ Q\,Q'>0$ , and repulsive if $ Q\,Q'<0$ .

  4. Show that two parallel vortex filaments of intensities $ {\mit\Gamma}$ and $ {\mit\Gamma}'$ , located a perpendicular distance $ r$ apart, exert a radial force per unit length $ \rho\,{\mit\Gamma}\,{\mit\Gamma}'/(2\pi\,r)$ on one another, the force being repulsive if $ {\mit\Gamma}\,{\mit\Gamma}'>0$ , and attractive if $ {\mit\Gamma}\,{\mit\Gamma}'<0$ .

  5. A vortex filament of intensity $ {\mit\Gamma}$ runs parallel to, and lies a perpendicular distance $ a$ from, a rigid planar boundary. Demonstrate that the boundary experiences a net force per unit length $ \rho\,{\mit\Gamma}^{\,2}/(4\pi\,a)$ directed toward the filament.

  6. Two rigid planar boundaries meet at right-angles. A line source of strength $ Q$ runs parallel to the line of intersection of the planes, and is situated a perpendicular distance $ a$ from each. Demonstrate that the source is subject to a force per unit length

    $\displaystyle \frac{3\sqrt{2}\,\rho\,Q^{\,2}}{8\pi\,a}
$

    directed towards the line of intersection of the planes.

  7. A line source of strength $ Q$ is located a distance $ b$ from an impenetrable circular cylinder of radius $ a<b$ (the axis of the cylinder being parallel to the source). Demonstrate that the cylinder experiences a net force per unit length

    $\displaystyle \frac{\rho\,Q^{\,2}}{2\pi\,b}\,\frac{a^{\,2}}{(b^{\,2}-a^{\,2})}
$

    directed toward the source.

  8. A dipole line source consists of a line source of strength $ Q$ , running parallel to the $ z$ -axis, and intersecting the $ x$ -$ y$ plane at $ (d/2)\,(\cos\alpha$ , $ \sin\alpha)$ , and a parallel source of strength $ -Q$ that intersects the $ x$ -$ y$ plane at $ (d/2)(-\cos\alpha$ , $ -\sin\alpha)$ . Show that, in the limit $ d\rightarrow 0$ , and $ Q\,d\rightarrow D$ , the complex velocity potential of the source is

    $\displaystyle F(z) = \frac{D\,{\rm e}^{\,{\rm i}\,\alpha}}{2\pi\,z}.
$

    Here, $ D\,{\rm e}^{\,{\rm i}\,\alpha}$ is termed the complex dipole strength.

  9. A dipole line source of complex strength $ D\,{\rm e}^{\,{\rm i}\,\alpha}$ is placed in a uniformly flowing fluid of speed $ V_0$ whose direction of motion subtends a (counter-clockwise) angle $ \theta_0$ with the $ x$ -axis. Show that, while no net force acts on the source, it is subject to a moment (per unit length) $ M= \rho\,D\,V_0\,\sin(\alpha-\theta_0)$ about the $ z$ -axis.

  10. Consider a dipole line source of complex strength $ D_1\,{\rm e}^{\,{\rm i}\,\alpha_1}$ running along the $ z$ -axis, and a second parallel source of complex strength $ D_2\,{\rm e}^{\,{\rm i}\,\alpha_2}$ that intersects the $ x$ -$ y$ plane at $ (x$ , $ 0)$ . Demonstrate that the first source is subject to a moment (per unit length) about the $ z$ -axis of

    $\displaystyle M = \frac{\rho\,D_1\,D_2}{2\pi\,x^{\,2}}\,\sin(\alpha_1+\alpha_2),
$

    as well as a force (per unit length) whose $ x$ - and $ y$ -components are

    $\displaystyle X$ $\displaystyle = - \frac{\rho\,D_1\,D_2}{\pi\,x^{\,3}}\,\cos(\alpha_1+\alpha_2),$    
    $\displaystyle Y$ $\displaystyle =\frac{\rho\,D_1\,D_2}{\pi\,x^{\,3}}\,\sin(\alpha_1+\alpha_2),$    

    respectively. Show that the second source is subject to the same moment, but an equal and opposite force.

  11. A dipole line source of complex strength $ D\,{\rm e}^{\,{\rm i}\,\alpha}$ runs parallel to, and is located a perpendicular distance $ a$ from, a rigid planar boundary. Show that the boundary experiences a force per unit length

    $\displaystyle \frac{\rho\,D^{\,2}}{8\pi\,a^{\,3}}
$

    acting toward the source.

  12. Demonstrate that a conformal map converts a line source into a line source of the same strength, and a vortex filament into a vortex filament of the same intensity.

  13. Consider the conformal map

    $\displaystyle z ={\rm i}\,c\,\cot(\zeta/2),
$

    where $ z=x +{\rm i}\,y$ , $ \zeta=\xi+{\rm i}\,\eta$ , and $ c$ is real and positive. Show that

    $\displaystyle x$ $\displaystyle =\frac{c\,\sinh \eta}{\cosh\eta-\cos\xi},$    
    $\displaystyle y$ $\displaystyle = \frac{c\,\sin\xi}{\cosh \eta-\cos\xi}.$    

    Demonstrate that $ \xi=\xi_0$ , where $ 0\leq \xi_0\leq \pi$ , maps to a circular arc of center $ (0$ , $ c\,\cot \xi_0)$ , and radius $ c\,\vert{\rm cosec}\,\xi_0\vert$ , that connects the points $ (\pm c$ , $ 0)$ , and lies in the region $ y>0$ . Demonstrate that $ \xi=\xi_0+\pi$ maps to the continuation of this arc in the region $ y<0$ . In particular, show that $ \xi=0$ maps to the region $ \vert x\vert>c$ on the $ x$ -axis, whereas $ \xi=\pi$ maps to the region $ \vert x\vert<c$ . Finally, show that $ \eta=\eta_0$ maps to a circle of center $ (c\,\coth\,\eta_0$ , $ 0)$ , and radius $ c\,\vert{\rm cosech}\vert\,\eta_0$ .

  14. Consider the complex velocity potential

    $\displaystyle F(z) = -\frac{2\,c\,{\rm i}\,V_0}{n}\,\cot(\zeta/n),
$

    where

    $\displaystyle z ={\rm i}\,c\,\cot(\zeta/2).
$

    Here, $ V_0$ , $ n$ , and $ c$ are real and positive. Show that

    $\displaystyle -\frac{dF}{dz} = \frac{4V_0}{n^2}\,\frac{\sin^2(\zeta/2)}{\sin^2(\zeta/n)}.
$

    Hence, deduce that the flow at $ \vert z\vert\rightarrow \infty$ is uniform, parallel to the $ x$ -axis, and of speed $ V_0$ . Demonstrate that

    $\displaystyle \psi(\xi,\eta) = -\frac{2\,V_0\,c}{n}\,\frac{\sin(2\,\xi/n)}{\cosh(2\,\eta/n)-\cos(2\,\xi/n)}.
$

    Hence, deduce that the streamline $ \psi=0$ runs along the $ x$ -axis for $ \vert x\vert>c$ , but along a circular arc connecting the points $ (\pm c$ , $ 0)$ for $ \vert x\vert<c$ . Furthermore, show that if $ 1<n<2$ then this arc lies above the $ x$ -axis, and is of maximum height

    $\displaystyle h = c\left[\frac{\cos(\pi\,n/2)+1}{\sin(\pi\,n/2)}\right],
$

    but if $ 2<n<3$ then the arc lies below the $ x$ -axis, and is of maximum depth

    $\displaystyle d = c\left[\frac{\cos(\pi\,n/2)+1}{\vert\sin(\pi\,n/2)\vert}\right].
$

    Hence, deduce that if $ 1<n<2$ then the complex velocity potential under investigation corresponds to uniform flow of speed $ V_0$ , parallel to a planar boundary that possesses a cylindrical bump (whose axis is normal to the flow) of height $ h$ and width $ 2\,c$ , but if $ 2<n<3$ then the potential corresponds to flow parallel to a planar boundary that possesses a cylindrical depression of depth $ d$ and width $ 2\,c$ . Show, in particular, that if $ n=1$ then the bump is a half-cylinder, and if $ n=3$ then the depression is a half-cylinder. Finally, demonstrate that the flow speed at the top of the bump (in the case $ 1<n<2$ ), or the bottom of the depression (in the case $ 2<n<3$ ) is

    $\displaystyle v = \frac{2\,V_0}{n^{\,2}}\left[1-\cos(\pi\,n/2)\right].
$

  15. Show that $ z=\cosh(\pi\,\zeta/a)$ maps the semi-infinite strip $ 0\leq \eta\leq a$ , $ \xi\geq 0$ in the $ \zeta $ -plane onto the upper half ($ y\geq 0$ ) of the $ z$ -plane. Hence, show that the stream function due to a line source of strength $ Q$ placed at $ \zeta = (0$ , $ a/2)$ , in the rectangular region $ 0\leq \eta\leq a$ , $ \xi\geq 0$ bounded by the rigid planes $ \eta=0$ , $ \xi=0$ , and $ \eta = a$ , is

    $\displaystyle \psi(\xi,\eta) = \frac{Q\,\sinh(\pi\,\xi/a)\,\sin(\pi\,\eta/a)}{2\pi\,[\sinh^2(\pi\,\xi/a)+\cos^2(\pi\,\eta/a)]}.
$

  16. Show that the complex velocity potential

    $\displaystyle F(z)= - \frac{a\,\pi\,V_0}{\tanh(a\,\pi/z)}
$

    can be interpreted as that due to uniform flow of speed $ V_0$ over a cylindrical log of radius $ a$ lying on the flat bed of a deep stream (the axis of the log being normal to the flow). Demonstrate that the flow speed at the top of the log is $ (\pi^{\,2}/4)\,V_0$ . Finally, show that the pressure difference between the top and bottom of the log is $ \pi^{\,4}\,\rho\,V_0^{\,2}/32$ .

  17. Show that the complex potential

    $\displaystyle F(z)= -V\left(\zeta\,{\rm e}^{-{\rm i}\,\alpha}+\frac{c^{\,2}}{\zeta}\,{\rm e}^{\,{\rm i}\,\alpha}\right),
$

    where

    $\displaystyle z = \zeta+ \frac{l^{\,2}}{\zeta}.
$

    ($ l<c$ ) represents uniform flow of unperturbed speed $ V$ , whose direction subtends a (counter-clockwise) angle $ \alpha $ with the $ x$ -axis, around an impenetrable elliptic cylinder of major radius $ a=c+l^{\,2}/c$ , aligned along the $ x$ -axis, and minor radius $ b=c-l^{\,2}/c$ , aligned along the $ y$ -axis. Demonstrate that the moment per unit length (about the $ z$ -axis) exerted on the cylinder by the flow is

    $\displaystyle M = -\frac{\pi}{2}\,\rho\,V^{\,2}\,(a^{\,2}-b^{\,2})\,\sin(2\,\alpha).
$

    Hence, deduce that the moment acts to turn the cylinder broadside-on to the flow (i.e., $ \alpha=\pi/2$ is a dynamically stable equilibrium state), and that the equilibrium state in which the cylinder is aligned with the flow (i.e., $ \alpha=0$ ) is dynamically unstable.

  18. Consider a simply-connected region of a two-dimensional flow pattern bounded on the inside by the closed curve $ C_1$ (lying in the $ x$ -$ y$ plane), and on the outside by the closed curve $ C_2$ . Here, $ C_1$ and $ C_2$ do not necessarily correspond to streamlines of the flow. Demonstrate that the kinetic energy per unit length (in the $ z$ -direction) of the fluid lying between the two curves is

    $\displaystyle K= \frac{1}{2}\,\rho\left[\int_{C_2}\phi\,d\psi-\int_{C_1}\phi\,d\psi-\int_{C_3}[\phi]\,d\psi\right],
$

    where $ \rho$ is the fluid mass density, $ \phi$ the velocity potential, and $ \psi $ the stream function. Here, $ C_3$ is a curve that runs from $ C_1$ to $ C_2$ , and $ [\phi]$ denotes the amount by which the velocity potential increases as the argument of $ x+{\rm i}\,y$ increases by $ 2\pi$ .

  19. Show that the complex potential

    $\displaystyle F(z)= -V\left(c^{\,2}\,{\rm e}^{\,{\rm i}\,\alpha} -l^{\,2}\,{\rm e}^{-{\rm i\,\alpha}}\right)\,\zeta^{\,-1}
$

    where

    $\displaystyle z = \zeta+ \frac{l^{\,2}}{\zeta}.
$

    ($ l<c$ ) represents the flow pattern around an impenetrable elliptic cylinder of major radius $ a=c+l^{\,2}/c$ , aligned along the $ x$ -axis, and minor radius $ b=c-l^{\,2}/c$ , aligned along the $ y$ -axis, moving with speed $ V$ , in a direction that makes a counter-clockwise angle $ \alpha $ with the $ x$ -axis, through a fluid that is at rest far from the cylinder. Demonstrate that the kinetic energy per unit length of the flow pattern is

    $\displaystyle K = \frac{1}{2}\,\rho\,V^{\,2}\,\pi\left(a^{\,2}\,\sin^2\alpha+b^{\,2}\,\cos^2\alpha\right),
$

    where $ \rho$ is the fluid mass density. Hence, show that the cylinder's added mass per unit length is

    $\displaystyle m_{\rm added}= \pi\left(a^{\,2}\,\sin^2\alpha+b^{\,2}\,\cos^2\alpha\right)\rho.
$

  20. Demonstrate from Equation (6.110) that the equation of the free streamline $ BC$ , in the case of a liquid jet emerging from a two-dimensional orifice of semi-width $ d$ formed by a gap between two semi-infinite plane walls that subtend an angle $ 2\,\alpha$ , can be written parametrically as:

    $\displaystyle \frac{x}{d}$ $\displaystyle =\frac{2}{\pi}\int_{\sin^{-1}(\lambda)}^{\pi/2}\cos\left(\frac{2\...
...left(\frac{2\,\alpha}{\pi}\,\beta\right)\frac{d\beta}{\tan\beta}\right]^{\,-1},$    
    $\displaystyle \frac{y}{d}$ $\displaystyle =1-\frac{2}{\pi}\int_{\sin^{-1}(\lambda)}^{\pi/2}\sin\left(\frac{...
...left(\frac{2\,\alpha}{\pi}\,\beta\right)\frac{d\beta}{\tan\beta}\right]^{\,-1},$    

    where $ 0\leq\lambda\leq 1$ . Here, the orifice corresponds to the plane $ x=0$ , and the flow a long way from the orifice is in the $ +y$ -direction. Show that for the case of a two-dimensional Borda mouthpiece, $ \alpha=\pi$ , the previous equations reduce to

    $\displaystyle \frac{x}{d}$ $\displaystyle = \frac{1}{\pi}\left[-\ln(\lambda)-1+\lambda^{\,2}\right],$    
    $\displaystyle \frac{y}{d}$ $\displaystyle = \frac{1}{2}+\frac{1}{\pi}\left[\sin^{-1}(\lambda)+\lambda\,(1-\lambda^{\,2})^{1/2}\right].$    

    Finally, show that the previous equations predict that the free streamline is re-entrant, with $ x/d$ attaining its minimum value $ -(1-\ln 2)/(2\pi)=-0.153$ when $ y/d=3/4+1/(2\pi)= 0.909$ .

next up previous
Next: Axisymmetric Incompressible Inviscid Flow Up: Two-Dimensional Potential Flow Previous: Blasius Theorem
Richard Fitzpatrick 2016-03-31