Categories: Fluid dynamics, Fluid mechanics, Physics.

# Vorticity

In fluid mechanics, the vorticity $$\va{\omega}$$ is a measure of the local circulation in a fluid. It is defined as the curl of the flow velocity field $$\va{v}$$:

\begin{aligned} \boxed{ \va{\omega} \equiv \nabla \cross \va{v} } \end{aligned}

Just as curves tangent to $$\va{v}$$ are called streamlines, curves tangent to $$\va{\omega}$$ are vortex lines, which are to be interpreted as the “axes” that $$\va{v}$$ is circulating around.

The vorticity is a local quantity, and the corresponding global quantity is the circulation $$\Gamma$$, which is defined as the projection of $$\va{v}$$ onto a close curve $$C$$. Then, by Stokes’ theorem:

\begin{aligned} \boxed{ \Gamma(C, t) \equiv \oint_C \va{v} \cdot \dd{\va{l}} = \int_S \va{\omega} \cdot \dd{\va{S}} } \end{aligned}

## Ideal fluids

For an inviscid, incompressible fluid, consider the Bernoulli field $$H$$, which is defined as:

\begin{aligned} H \equiv \frac{1}{2} \va{v}^2 + \Phi + \frac{p}{\rho} \end{aligned}

Where $$\Phi$$ is the gravitational potential, $$p$$ is the pressure, and $$\rho$$ is the (constant) density. We then take the gradient of this scalar field:

\begin{aligned} \nabla H &= \frac{1}{2} \nabla \va{v}^2 + \nabla \Phi + \frac{\nabla p}{\rho} \\ &= \va{v} \cdot (\nabla \va{v}) - \Big( \!-\! \nabla \Phi - \frac{\nabla p}{\rho} \Big) \end{aligned}

Since $$-\nabla \Phi = \va{g}$$, the rightmost term is the right-hand side of the Euler equation. We substitute the other side of said equation, yielding:

\begin{aligned} \nabla H &= \va{v} \cdot (\nabla \va{v}) - \frac{\mathrm{D} \va{v}}{\mathrm{D} t} = \va{v} \cdot (\nabla \va{v}) - \pdv{\va{v}}{t} - (\va{v} \cdot \nabla) \va{v} \end{aligned}

We isolate this equation for $$\pdv*{\va{v}}{t}$$, and apply a vector identity to reduce it to the following:

\begin{aligned} \pdv{\va{v}}{t} = \va{v} \cdot (\nabla \va{v}) - (\va{v} \cdot \nabla) \va{v} - \nabla H = \va{v} \cross (\nabla \cross \va{v}) - \nabla H \end{aligned}

Here, the definition of the vorticity $$\va{\omega}$$ is clear to see, leading us to an equation of motion for $$\va{v}$$:

\begin{aligned} \boxed{ \pdv{\va{v}}{t} = \va{v} \cross \va{\omega} - \nabla H } \end{aligned}

More about this later. Now, we take the curl of both sides of this equation, giving us:

\begin{aligned} \nabla \cross \pdv{\va{v}}{t} = \nabla \cross (\va{v} \cross \va{\omega}) - \nabla \cross (\nabla H) \end{aligned}

On the left, we swap $$\nabla$$ with $$\pdv*{t}$$, and on the right, the curl of a gradient is always zero. We are thus left with the equation of motion of the vorticity $$\va{\omega}$$:

\begin{aligned} \boxed{ \pdv{\va{\omega}}{t} = \nabla \cross (\va{v} \cross \va{\omega}) } \end{aligned}

Let us now return to the equation of motion for $$\va{v}$$. For steady flows where $$\pdv*{\va{v}}{t} = 0$$, in which case Bernoulli’s theorem applies, it reduces to:

\begin{aligned} \nabla H = \va{v} \cross \va{\omega} \end{aligned}

If a fluid has $$\va{\omega} = 0$$ in some regions, it is known as irrotational. From this equation, we see that, in that case, $$\nabla H = 0$$, meaning that $$H$$ is a constant in those regions, a fact sometimes referred to as Bernoulli’s stronger theorem.

Furthermore, irrotationality $$\va{\omega} = 0$$ implies that $$\va{v}$$ is the gradient of a potential $$\Psi$$:

\begin{aligned} \va{v} = \nabla \Psi \end{aligned}

This fact allows us to rewrite the Euler equations in a particularly simple way. Firstly, the condition of incompressibility becomes the well-known Laplace equation:

\begin{aligned} 0 = \nabla \cdot \va{v} = \nabla^2 \Psi \end{aligned}

And second, the main equation of motion for $$\va{v}$$ states that the quantity $$H + \pdv*{\Psi}{t}$$ is spatially constant in the irrotational region:

\begin{aligned} \pdv{\va{v}}{t} = \nabla \pdv{\Psi}{t} = - \nabla H \quad \implies \quad \nabla \Big( H + \pdv{\Psi}{t} \Big) = 0 \end{aligned}

## References

1. B. Lautrup, Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition, CRC Press.

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