[유체] Chap 7. Simple Potential Flow & Singularities

Chap 7. Simple Potential Flows & Singularities

7.1 Simple flows

Uniform flow

$u=U,\quad v=V,\quad w=W$

$\phi=Ux+Vy+Wz$

2D Source and sink

\[\phi=\frac{m}{2\pi}\ln r\]

$m>0 $ : source, $m<0$ : sink

Polar coordinate

\[(u_r,u_\theta)=\left(\frac{\partial\phi}{\partial r},\frac{1}{r}\frac{\partial\phi}{\partial\theta}\right)\] \[u_r=\frac{m}{2\pi r},\quad u_\theta=0\]

$m$ : source / sink strength

\[\int_0^{2\pi}u_rR\,d\theta=\int_0^{2\pi}\frac{m}{2\pi R}\cdot R\,d\theta=m\]

3D Source & Sink

\[\phi=-\frac{m}{4\pi}\frac{1}{r}\] \[u_r=\frac{\partial\phi}{\partial r}=\frac{m}{4\pi r^2} \quad u_\theta = 0\]

2D Dipole / Doublet

\[\phi = {m \over 2\pi} \ln{\sqrt{(x+\varepsilon)^2+y^2}}-{m \over 2\pi} \ln{\sqrt{(x-\varepsilon)^2+y^2}}\]

Let dipole monent $\mu=2m\varepsilon=\mathrm{const.}$

\[\phi=\frac{m}{2\pi}\left(\ln\sqrt{(x+\varepsilon)^2+y^2}-\ln\sqrt{(x-\varepsilon)^2+y^2}\right)\] \[=\frac{\mu}{4\pi\varepsilon}\left(\ln\sqrt{(x+\varepsilon)^2+y^2}-\ln\sqrt{(x-\varepsilon)^2+y^2}\right)\] \[\lim\limits_{\varepsilon\to 0}\phi =\lim\limits_{\varepsilon\to 0}{\mu \over 4\pi \varepsilon}{2\varepsilon x \over x^2+y^2} = {\mu x \over 2\pi r^2}\]

For dipole(doublet),

\[\phi_D = \mu\frac{\partial G}{\partial x}\] \[G=\frac{\mu r}{2\pi}\quad\text{or}\quad-\frac{1}{4\pi r}\]

→ $G$ : Unit Velocity potential (with strength $m=1$)

3D Doublet / Dipole

\[\begin{aligned}\mu\frac{\partial}{\partial x}\left\{-\frac{1}{4\pi}\frac{1}{\sqrt{x^2+y^2+z^2}}\right\}&=-\frac{\mu}{4\pi}\cdot -\frac{1}{2}\cdot \frac{2x}{r^3} \\ &= \frac{\mu x}{4\pi r^3} = \frac{\mu r\cos\theta}{4\pi r^3} \\ &= \frac{\mu\cos\theta}{4\pi r^2}\end{aligned}\]

7.2 Circulation, Point Vortex

• Circulation $\oint_C\vec{u}\cdot d\vec{l}=\Gamma$

Stoke’s Theorem

\[\int_S(\nabla\times\vec{F})\cdot\vec{n}\,ds = \oint_C\vec{F}\cdot d\vec{s}\]

for irrotational flow, $\Gamma = 0 (\because \nabla \times \vec u = 0)$

2D Point vortex

$\phi=\mu\theta$, $(u_r,u_\theta)=\left(\frac{\partial\phi}{\partial r},\frac{1}{r}\frac{\partial\phi}{\partial\theta}\right)=\left(0,\frac{\mu}{r}\right)$

$\Gamma=\mu\cdot2\pi$

\[\oint\vec{u}\cdot d\vec{s}=\Gamma=\int(\nabla\times\vec{u})\cdot\vec{n}\,ds=\int_0^{2\pi}\frac{\mu}{R}\cdot R\,d\theta=2\pi\mu\] \[\therefore\phi=\frac{\Gamma}{2\pi}\theta\]

$\Gamma$ : Circulation, Strength of vortex