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Next: Flow Past a Cylindrical Up: Two-Dimensional Incompressible Inviscid Flow Previous: Two-Dimensional Vortex Filaments

Two-Dimensional Irrotational Flow in Cylindrical Coordinates

In a two-dimensional flow pattern, we can automatically satisfy the incompressibility constraint, $ \nabla\cdot{\bf v}=0$ , by expressing the pattern in terms of a stream function. Suppose, however, that, in addition to being incompressible, the flow pattern is also irrotational. In this case, Equation (5.10) yields

$\displaystyle \nabla^{\,2}\psi = 0.$ (5.60)

In cylindrical coordinates, because $ \psi=\psi(r,\theta,t)$ , this expression implies that (see Section C.3)

$\displaystyle \frac{1}{r}\,\frac{\partial}{\partial r}\!\left(r\,\frac{\partial...
...right) + \frac{1}{r^{\,2}}\,\frac{\partial^{\,2}\psi}{\partial\theta^{\,2}} =0.$ (5.61)

Let us search for a separable solution of Equation (5.61) of the form

$\displaystyle \psi(r,\theta) = R(r)\,{\mit\Theta}(\theta).$ (5.62)

It is easily seen that

$\displaystyle \frac{r}{R}\,\frac{d}{dr}\!\left(r\,\frac{dR}{dr}\right)= - \frac{1}{{\mit\Theta}}\,\frac{d^{\,2}{\mit\Theta}}{d\theta^{\,2}},$ (5.63)

which can only be satisfied if

$\displaystyle r\,\frac{d}{dr}\!\left(r\,\frac{dR}{dr}\right)$ $\displaystyle =m^{\,2}\,R,$ (5.64)
$\displaystyle \frac{d^{\,2}{\mit\Theta}}{d\theta^{\,2}}$ $\displaystyle = -m^{\,2}\,{\mit\Theta},$ (5.65)

where $ m^{\,2}$ is an arbitrary (positive) constant. The general solution of Equation (5.65) is a linear combination of $ \exp(\,{\rm i}\,m\,\theta)$ and $ \exp(-{\rm i}\,m\,\theta)$ factors. However, assuming that the flow extends over all $ \theta $ values, the function $ {\mit\Theta}(\theta)$ must be single-valued in $ \theta $ , otherwise $ \nabla\psi$ --and, hence, $ {\bf v}$ --would not be be single-valued (which is unphysical). It follows that $ m$ can only take integer values (and that $ m^{\,2}$ must be a positive, rather than a negative, constant). The general solution of Equation (5.64) is a linear combination of $ r^{\,m}$ and $ r^{-m}$ factors, except for the special case $ m=0$ , when it is a linear combination of $ r^{\,0}$ and $ \ln r$ factors. Thus, the general stream function for steady two-dimensional irrotational flow (that extends over all values of $ \theta $ ) takes the form

$\displaystyle \psi(r,\theta) = \alpha_0+ \beta_0\,\ln r + \sum_{m>0} (\alpha_m\,r^{\,m}+\beta_m\,r^{-m})\, \sin[m\,(\theta-\theta_m)],$ (5.66)

where $ \alpha_m$ , $ \beta_m$ , and $ \theta_m$ are arbitrary constants. We can recognize the first few terms on the right-hand side of the previous expression. The constant term $ \alpha _0$ has zero gradient, and, therefore, does not give rise to any flow. The term $ \beta_0\,\ln r$ is the flow pattern generated by a vortex filament of intensity $ 2\pi\,\beta_0$ , coincident with the $ z$ -axis. (See Section 5.6.) The term $ \alpha_1\,r\,\sin(\theta-\theta_1)$ corresponds to uniform flow of speed $ \alpha_1$ whose direction subtends a (counter-clockwise) angle $ \theta _1$ with the minus $ x$ -axis. (See Section 5.4.) Finally, the term $ \beta_1\,\sin(\theta-\theta_1)/r$ corresponds to a dipole flow pattern. (See Section 5.5.)

The velocity potential associated with the irrotational stream function (5.66) satisfies [see Equations (4.89) and (5.7)]

$\displaystyle \frac{\partial\phi}{\partial r}$ $\displaystyle =\frac{1}{r}\,\frac{\partial\psi}{\partial\theta},$ (5.67)
$\displaystyle \frac{1}{r}\,\frac{\partial\phi}{\partial\theta}$ $\displaystyle =-\frac{\partial\psi}{\partial r}.$ (5.68)

It follows that

$\displaystyle \phi(r,\theta) = \alpha_0-\beta_0\,\theta+\sum_{m>0}(\alpha_m\,r^{\,m}-\beta_m\,r^{-m})\,\cos[m\,(\theta-\theta_0)].$ (5.69)

Figure: Streamlines of the flow generated by a cylindrical obstacle of radius $ a$ , whose axis runs along the $ z$ -axis, placed in the uniform flow field $ {\bf v}= V_0\,{\rm e}_x$ . The normalized circulation is $ \gamma =0$ .
\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter05/cylinder.eps}}
\end{figure}


next up previous
Next: Flow Past a Cylindrical Up: Two-Dimensional Incompressible Inviscid Flow Previous: Two-Dimensional Vortex Filaments
Richard Fitzpatrick 2016-01-22