A thin vortex filament carrying circulation $\Gamma$ induces velocity throughout the surrounding fluid. For an ideal filament following a curve $C$, the induced velocity at a field point $\mathbf x$ is given by the Biot-Savart law

$$\boxed{ \mathbf v(\mathbf x) =\frac{\Gamma}{4\pi} \int_C \frac{d\boldsymbol\ell\times\mathbf r}{|\mathbf r|^3} },$$

where

$$\mathbf r=\mathbf x-\mathbf x'$$

points from the filament element at $\mathbf x'$ to the field point, and $d\boldsymbol\ell$ is tangent to the filament in the direction of positive circulation according to the right-hand rule.

The cross product determines the direction of the induced velocity; the inverse-square spatial scaling is contained in

$$\frac{|d\boldsymbol\ell\times\mathbf r|}{|\mathbf r|^3}.$$

A filament therefore influences the velocity field nonlocally.

Contribution of one filament element

A small element produces

$$\boxed{ d\mathbf v =\frac{\Gamma}{4\pi} \frac{d\boldsymbol\ell\times\mathbf r}{r^3} }.$$

If the field point lies on the extension of the filament element, then

$$d\boldsymbol\ell\parallel\mathbf r,$$

so the cross product vanishes. The induced velocity comes from the component of separation perpendicular to the filament.

Reversing the sign of $\Gamma$ reverses the induced velocity everywhere.

Infinite straight filament

Consider an infinitely long straight filament along the $z$ axis. Let the field point be a perpendicular distance $h$ from the filament.

Take

$$d\boldsymbol\ell=dz,\hat{\mathbf z}.$$

For a field point on the positive $x$ axis,

$$\mathbf r=h\hat{\mathbf x}-z\hat{\mathbf z}.$$

Then

$$d\boldsymbol\ell\times\mathbf r =h,dz,\hat{\mathbf y}.$$

The induced speed is therefore

$$v =\frac{\Gamma}{4\pi} \int_{-\infty}^{\infty} \frac{h,dz}{(h^2+z^2)^{3/2}}.$$

The integral is

$$\int_{-\infty}^{\infty} \frac{h,dz}{(h^2+z^2)^{3/2}} =\frac{2}{h},$$

so

$$\boxed{v=\frac{\Gamma}{2\pi h}}.$$

The direction is tangential around the filament according to the right-hand rule. This recovers the familiar two-dimensional free-vortex velocity field.

Semi-infinite straight filament

If the same straight filament begins at the perpendicular foot and extends only from

$$z=0$$

to

$$z\to+\infty,$$

then exactly half of the infinite-filament integral remains:

$$\boxed{v=\frac{\Gamma}{4\pi h}}.$$

This factor of two is important in finite-wing theory, where wake filaments begin at the lifting line and extend downstream rather than extending infinitely in both directions.

Worked example

A straight vortex filament has circulation

$$\Gamma=0.80,\mathrm{m^2/s}.$$

At perpendicular distance

$$h=0.20,\mathrm m,$$

an infinite filament induces speed

$$v=\frac{0.80}{2\pi(0.20)} \approx\boxed{0.637,\mathrm{m/s}}.$$

A semi-infinite filament with the same circulation and geometry induces half as much:

$$v=\frac{0.80}{4\pi(0.20)} \approx\boxed{0.318,\mathrm{m/s}}.$$

Singular core and model scope

The ideal formula diverges as the field point approaches the filament centerline. A real vortex has a finite core in which vorticity is distributed over a nonzero area, so the ideal filament model is not valid arbitrarily close to the core.

The filament approximation is useful when the core is small compared with the geometric distances of interest. It is a central building block for vortex rings, wakes, horseshoe vortices, lifting-line theory, vortex-lattice methods, and many other reduced-order flow models.

Biot-Savart supplies the missing link between vorticity geometry and induced velocity: once the location, orientation, and circulation of vortex filaments are specified, their velocity influence can be computed by superposition.