Categories: Fluid dynamics, Fluid mechanics, Physics.

Vorticity

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

ω⃗≡∇×v⃗\begin{aligned} \boxed{ \va{\omega} \equiv \nabla \cross \va{v} } \end{aligned}

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

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

Γ(C,t)≡∮Cv⃗⋅dl⃗=∫Sω⃗⋅dS⃗\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 HH, which is defined as:

H≡12v⃗2+Φ+pρ\begin{aligned} H \equiv \frac{1}{2} \va{v}^2 + \Phi + \frac{p}{\rho} \end{aligned}

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

∇H=12∇v⃗2+∇Φ+∇pρ=v⃗⋅(∇v⃗)−( ⁣− ⁣∇Φ−∇pρ)\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 −∇Φ=g⃗-\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:

∇H=v⃗⋅(∇v⃗)−Dv⃗Dt=v⃗⋅(∇v⃗)−∂v⃗∂t−(v⃗⋅∇)v⃗\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 ∂v⃗/∂t\ipdv{\va{v}}{t}, and apply a vector identity to reduce it to the following:

∂v⃗∂t=v⃗⋅(∇v⃗)−(v⃗⋅∇)v⃗−∇H=v⃗×(∇×v⃗)−∇H\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 ω⃗\va{\omega} is clear to see, leading us to an equation of motion for v⃗\va{v}:

∂v⃗∂t=v⃗×ω⃗−∇H\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:

∇×∂v⃗∂t=∇×(v⃗×ω⃗)−∇×(∇H)\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 ∂/∂t\ipdv{}{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 ω⃗\va{\omega}:

∂ω⃗∂t=∇×(v⃗×ω⃗)\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 v⃗\va{v}. For steady flows where ∂v⃗/∂t=0\ipdv{\va{v}}{t} = 0, in which case Bernoulli’s theorem applies, it reduces to:

∇H=v⃗×ω⃗\begin{aligned} \nabla H = \va{v} \cross \va{\omega} \end{aligned}

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

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

v⃗=∇Ψ\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:

0=∇⋅v⃗=∇2Ψ\begin{aligned} 0 = \nabla \cdot \va{v} = \nabla^2 \Psi \end{aligned}

And second, the main equation of motion for v⃗\va{v} states that the quantity H+∂Ψ/∂tH + \ipdv{\Psi}{t} is spatially constant in the irrotational region:

∂v⃗∂t=∇∂Ψ∂t=−∇H  ⟹  ∇(H+∂Ψ∂t)=0\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.