A vortex sheet is an idealized surface across which the tangential component of velocity has a finite jump while the normal component remains continuous. In two-dimensional flow, the sheet reduces to a curve.

Consider a straight sheet along the $x$ axis. Let

$$u^+(x)$$

be the tangential velocity immediately above the sheet and

$$u^-(x)$$

the tangential velocity immediately below it. Define the sheet strength by

$$\boxed{\gamma(x)=u^+(x)-u^-(x)}.$$

Thus

$$[\gamma]=\mathrm{m/s}.$$

The quantity $\gamma$ is not itself a total circulation. It is a circulation density along the sheet.

Relation to circulation

Take a very thin rectangular contour enclosing a short sheet segment from $x$ to $x+dx$, oriented counterclockwise.

The lower side of the contour runs in the positive $x$ direction, contributing approximately

$$u^-dx.$$

The upper side runs in the negative $x$ direction, contributing

$$-u^+dx.$$

The short end segments vanish as the contour thickness tends to zero. Therefore the counterclockwise circulation around the segment is

$$d\Gamma =(u^- -u^+)dx.$$

Using the definition of $\gamma$,

$$\boxed{d\Gamma=-\gamma(x),dx}.$$

Hence the total counterclockwise circulation associated with a sheet extending from $x=a$ to $x=b$ is

$$\boxed{\Gamma=-\int_a^b\gamma(x),dx}.$$

With this convention, positive $\gamma$ corresponds to locally clockwise sheet circulation.

Why a sheet produces a velocity jump

A point vortex produces a tangential velocity that changes direction across its center. A continuous row of infinitesimal point vortices produces a finite jump in tangential velocity across the row.

The sheet idealization compresses a thin region of concentrated vorticity into zero thickness while preserving its integrated circulation.

In a real viscous flow, a shear layer has finite thickness and a smoothly varying velocity. A vortex sheet is the inviscid limiting model in which that thickness is neglected.

Vorticity concentrated on the sheet

Away from an ideal vortex sheet, the velocity field can be irrotational:

$$\boldsymbol\omega=\mathbf0.$$

The circulation is then carried by a singular concentration of vorticity on the sheet itself.

This makes a vortex sheet useful when the surrounding fluid is modeled by potential flow but the model still needs a distributed circulation source.

Discretizing a sheet into point vortices

A small segment of length $\Delta x$ has counterclockwise circulation approximately

$$\boxed{\Delta\Gamma\approx-\gamma(x)\Delta x}.$$

It can therefore be approximated by a point vortex of that circulation placed on the segment.

As the segments become finer, the collection of point vortices approaches the continuous sheet.

This is the conceptual basis of vortex-panel and thin-airfoil discretizations.

Worked example

Suppose a straight vortex sheet extends over

$$0\le x\le2.0,\mathrm m$$

with constant strength

$$\gamma=3.0,\mathrm{m/s}.$$

Its counterclockwise circulation is

$$\Gamma =-\int_0^{2.0}3.0,dx =-6.0,\mathrm{m^2/s}.$$

Thus

$$\boxed{\Gamma=-6.0,\mathrm{m^2/s}}.$$

The negative sign means the net circulation is clockwise under the convention that positive circulation is counterclockwise.

If the sheet is divided into four equal panels of width

$$\Delta x=0.50,\mathrm m,$$

then each panel carries approximately

$$\Delta\Gamma=-\gamma\Delta x =-(3.0)(0.50) =-1.5,\mathrm{m^2/s}.$$

The four discrete vortex strengths sum to the same total circulation $-6.0,\mathrm{m^2/s}$.

Where vortex sheets are used

Vortex sheets appear in idealized models of

• lifting surfaces;
• free shear layers and wakes;
• interfaces with tangential velocity jumps;
• vortex-panel methods;
• thin-airfoil and lifting-line theory.

The central idea is reusable: a finite tangential velocity jump can be represented as circulation distributed continuously along a curve or surface.