3.10 Lift due to circulation

Example: force on a vortex in a uniform stream.

$$\phi = Ux + \frac{\Gamma }{2\pi }\theta$$

Consider a control surface in the form of a circle radius R centered at the point vortex. Then according to Newton's law:

$$\underbrace{0}_{\begin{array}{c}\mbox{\small {For}} \\ \mbo... ... \frac{d}{dt}M_{cv} = \left( {F_{cs} + F_v } \right) + M_{IN}$$

$M_{CV}$ = Total (linear) momentum of control volume
$F_{CS}$ = Hydrodynamic force on CS by surrounding fluids
$F_{V}$ = Hydrodynamic force on CS by vortex = -(force on vortex by fluid)
$M_{IN}$ = net flux of momentum in CV through CS

Therefore, force on vortex $F = -F_{V} = F_{CS}+M_{IN}$.
Momentum flux:

$$\begin{align}u = & U - \frac{\Gamma }{2\pi r}\sin \theta , \notag \\ v = & \fr... ...riptscriptstyle\rightharpoonup$ }}\over {V}} \cdot \hat {n} \notag \end{align}$$

Evaluation of $M_{IN}$.

$$\begin{align}(M_x )_{IN} = & - \rho \int\limits_0^{2\pi } {d\theta ruV_r = } - ... ...^{2\pi } {\cos ^2\theta d\theta = - \frac{\rho U\Gamma }{2}} \notag \end{align}$$

Pressure:

$$p = - \frac{1}{2}\rho \left\vert \mathord{\buildrel{\lower3p... ...ptscriptstyle\rightharpoonup$ }}\over {v}} \right\vert^2 + C$$

Velocity vector:

$$\begin{align}\left\vert \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\r... ... r}U\sin \theta + \left( {\frac{\Gamma }{2\pi r}} \right)^2 \notag \end{align}$$

Force on the control surface - $F_{CS}$:

$$\begin{array}{l} \left( {F_{CS} } \right)_x = \int\limits_0... ...2\theta } }_\pi = - \frac{1}{2}\rho U\Gamma \\ \end{array}$$

Finally,

$$\begin{array}{l} F_x = \left( {F_{CS} } \right)_x + \left( ... ...left( {M_y } \right)_{IN} = - \rho U\Gamma \\ \end{array}$$

i.e. the fluid exerts a downward force $- \rho U\Gamma$ on the vortex

Kutta-Joukowski law:

$$\begin{array}{l} F = - \rho U\Gamma \\ 3D: \mathord{\bui... ...style\rightharpoonup$ }} \over {\Gamma }} \\ \end{array}$$

Generalized Kutta-Joukowski Law:

$$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightha... ...riptstyle\rightharpoonup$ }}\over {\Gamma }} _i } } \right),$$

where $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$ }}\over {F}}$ is the total force on a system of n vortices in a free stream with speed $\vec{U}$.

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