2.4 Vorticity and circulation

Definition of vorticity

Vorticity quantifies the local rotation of a fluid element at any point in a flow field. It's a vector quantity defined as the curl of the velocity field, so it captures both the direction and magnitude of that rotation.

Mathematical representation

Vorticity is defined as:

$\vec{\omega} = \nabla \times \vec{u}$

where $\vec{\omega}$ is the vorticity vector and $\vec{u}$ is the velocity field.

In Cartesian coordinates, the three components are:

• $\omega_x = \frac{\partial w}{\partial y} - \frac{\partial v}{\partial z}$
• $\omega_y = \frac{\partial u}{\partial z} - \frac{\partial w}{\partial x}$
• $\omega_z = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}$

The magnitude is $|\vec{\omega}| = \sqrt{\omega_x^2 + \omega_y^2 + \omega_z^2}$.

For two-dimensional flows in the $xy$-plane, only the $z$-component survives, so vorticity reduces to the scalar $\omega_z = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}$. This simplification comes up constantly in practice.

Physical interpretation

Vorticity equals twice the angular velocity of a fluid element spinning about its own axis. A small paddle wheel placed in the flow would rotate at a rate proportional to the local vorticity.

High-vorticity regions correspond to intense local rotation, like the core of a tornado or the wake behind a cylinder. Low-vorticity (or zero-vorticity) regions behave like irrotational or potential flow, where fluid elements translate and deform but don't spin.

Vorticity vs. rotation

These two concepts are closely related but distinct.

Similarities

• Both describe rotational aspects of fluid motion and are vector quantities with magnitude and direction.
• Both matter across aerodynamics, turbomachinery, and geophysical flows.

Key differences

• Rotation refers to the angular velocity of a fluid element relative to a fixed reference frame. Vorticity is an intrinsic, local property of the velocity field itself.
• Rotation can be imposed externally (e.g., a rotating tank), whereas vorticity arises from velocity gradients within the flow.
• Vorticity varies from point to point. Solid-body rotation, by contrast, can be uniform throughout a domain.
• A fluid in a uniformly rotating reference frame can have non-zero rotation but zero relative vorticity if the flow within that frame is irrotational.

The precise relationship is $\vec{\omega} = 2\vec{\Omega}$ for solid-body rotation at angular velocity $\vec{\Omega}$, but in general flows, vorticity and rotation decouple because velocity gradients vary spatially.

Vorticity equation

The vorticity equation governs how vorticity evolves in time and space. It's derived directly from the Navier-Stokes equations.

Derivation

1. Start with the incompressible Navier-Stokes equations: $\frac{\partial \vec{u}}{\partial t} + (\vec{u} \cdot \nabla)\vec{u} = -\frac{1}{\rho}\nabla p + \nu \nabla^2 \vec{u}$

2. Take the curl of both sides. The pressure gradient term vanishes (for barotropic flow) because $\nabla \times \nabla p = 0$.

3. Apply the vector identity for the convective term to obtain:

$\frac{\partial \vec{\omega}}{\partial t} + (\vec{u} \cdot \nabla)\vec{\omega} = (\vec{\omega} \cdot \nabla)\vec{u} + \nu \nabla^2 \vec{\omega}$

This assumes incompressible flow ($\nabla \cdot \vec{u} = 0$), which eliminates the dilatation term $\vec{\omega}(\nabla \cdot \vec{u})$.

Terms and their meanings

• $\frac{\partial \vec{\omega}}{\partial t}$: Local rate of change of vorticity at a fixed point.
• $(\vec{u} \cdot \nabla)\vec{\omega}$: Advection of vorticity by the flow. Vorticity is carried along with the fluid.
• $(\vec{\omega} \cdot \nabla)\vec{u}$: Vortex stretching/tilting. When velocity gradients stretch or tilt vortex lines, vorticity intensifies or changes direction. This term is zero in 2D flows, which is why 3D turbulence behaves so differently from 2D.
• $\nu \nabla^2 \vec{\omega}$: Viscous diffusion. Viscosity smooths out vorticity gradients over time.

The left side is the material derivative $\frac{D\vec{\omega}}{Dt}$, so the equation says: the rate of change of vorticity following a fluid element equals the combined effects of stretching and diffusion.

Vortex lines and tubes

Vortex lines and tubes are geometric tools for visualizing the vorticity field, analogous to how streamlines visualize velocity.

Definitions

A vortex line is a curve everywhere tangent to the local vorticity vector. It satisfies:

$\frac{dx}{\omega_x} = \frac{dy}{\omega_y} = \frac{dz}{\omega_z}$

A vortex tube is the surface formed by all vortex lines passing through a given closed curve. Think of it as a "bundle" of vortex lines forming a tubular structure. The cross-sectional area of a vortex tube is inversely proportional to the local vorticity magnitude, since the tube's strength (circulation) must be conserved.

Properties

• Vortex lines cannot start or end inside the fluid. They either form closed loops or extend to boundaries.
• Vortex lines cannot cross, because that would require the vorticity vector to point in two directions at the same point.
• The strength of a vortex tube (the circulation around any cross-section) is constant along its length.
• In inviscid flow, vortex lines are material lines: they're always composed of the same fluid particles. This follows from Helmholtz's theorems.

Helmholtz's vortex theorems

These theorems describe how vorticity behaves in inviscid, barotropic flows (where density depends only on pressure). They place powerful constraints on vortex dynamics.

Kelvin's circulation theorem

Kelvin's theorem states that the circulation around a closed material loop (a loop that moves with the fluid) is constant in time:

$\frac{D}{Dt} \oint_C \vec{u} \cdot d\vec{l} = 0$

This means that in an inviscid, barotropic flow with conservative body forces, vorticity cannot be created or destroyed. Any circulation present must have existed from the start or been introduced at a boundary.

Vortex tube evolution

Helmholtz's second theorem: a vortex tube moves with the fluid and retains its strength, even as the flow stretches or deforms it. The circulation around any cross-section of the tube stays constant. If the tube gets thinner, the vorticity magnitude increases (and vice versa), but the product of vorticity and cross-sectional area remains fixed.

Vortex lines and material lines

Helmholtz's third theorem: in inviscid, barotropic flow, vortex lines are material lines. Fluid particles that start on a vortex line stay on that vortex line forever. This is a Lagrangian statement: vorticity is "frozen into" the fluid.

These theorems break down when viscosity, baroclinic effects ($\nabla \rho \times \nabla p \neq 0$), or non-conservative body forces are present.

Circulation

Circulation is the scalar measure of macroscopic rotation along a closed curve. Where vorticity tells you about rotation at a point, circulation tells you about rotation over a region.

Definition and properties

Circulation is the line integral of velocity around a closed curve $C$:

$\Gamma = \oint_C \vec{u} \cdot d\vec{l}$

• Positive $\Gamma$ corresponds to counterclockwise rotation (by convention).
• Circulation is a global quantity that depends on which closed curve you choose.

The connection to vorticity comes through Stokes' theorem:

$\Gamma = \oint_C \vec{u} \cdot d\vec{l} = \int_S \vec{\omega} \cdot d\vec{A}$

This says the circulation around a curve equals the total vorticity flux through any surface bounded by that curve. If there's no vorticity inside the loop, the circulation is zero.

Circulation vs. vorticity

Stokes' theorem is the bridge: it converts the global measure (circulation) into an integral of the local measure (vorticity).

Kutta-Joukowski theorem

This theorem provides the direct link between circulation and aerodynamic lift, making it one of the most practically important results in fluid dynamics.

Lift generation

The Kutta-Joukowski theorem states that the lift per unit span on a body in a uniform stream is:

$L' = \rho_{\infty} U_{\infty} \Gamma$

where $\rho_{\infty}$ is the freestream density, $U_{\infty}$ is the freestream velocity, and $\Gamma$ is the circulation around the body.

The circulation doesn't appear spontaneously. For an airfoil, it's established by the Kutta condition, which requires the flow to leave the sharp trailing edge smoothly with finite velocity. Without this condition, the inviscid solution would predict infinite velocity at the trailing edge, which is physically unrealistic.

Circulation around airfoils

• The Kutta condition selects a unique value of circulation that produces a smooth departure at the trailing edge.
• In potential flow models, circulation is represented using vortex elements (point vortices, vortex sheets, or vortex panels distributed along the airfoil surface).
• Circulation increases with angle of attack: higher angles produce stronger circulation and more lift, up to the stall angle where the flow separates and the model breaks down.
• The theorem applies to any 2D body shape, not just airfoils, though it's most useful for streamlined bodies where potential flow is a reasonable approximation.

Potential flow

Potential flow assumes the fluid is inviscid, incompressible, and irrotational. Despite these simplifications, it captures surprisingly useful physics, especially far from solid boundaries.

Irrotational vs. rotational flow

An irrotational flow has $\vec{\omega} = \nabla \times \vec{u} = 0$ everywhere. This means a velocity potential $\phi$ exists such that:

$\vec{u} = \nabla \phi$

A rotational flow has $\vec{\omega} \neq 0$ in at least part of the domain and cannot be fully described by a velocity potential alone.

Most real flows have both regions: irrotational flow away from boundaries, and rotational flow inside boundary layers and wakes where viscosity generates vorticity.

Velocity potential

The velocity potential $\phi$ is a scalar function with components:

• $u = \frac{\partial \phi}{\partial x}$, $v = \frac{\partial \phi}{\partial y}$, $w = \frac{\partial \phi}{\partial z}$

For incompressible flow, $\nabla \cdot \vec{u} = 0$ combined with $\vec{u} = \nabla \phi$ gives Laplace's equation:

$\nabla^2 \phi = 0$

Laplace's equation is linear, which means you can superpose elementary solutions (uniform flow, source, sink, doublet, point vortex) to build complex flow fields. This superposition principle is what makes potential flow so powerful as an analytical tool.

Vorticity in viscous flows

Real fluids have viscosity, and viscosity fundamentally changes how vorticity behaves. It's responsible for both creating and destroying vorticity.

Diffusion of vorticity

The viscous term $\nu \nabla^2 \vec{\omega}$ in the vorticity equation acts like a diffusion operator: it spreads vorticity from regions of high concentration to regions of low concentration, smoothing out gradients over time.

• The rate of diffusion scales with the kinematic viscosity $\nu$. High-viscosity fluids (honey) diffuse vorticity quickly; low-viscosity fluids (air, water) retain sharp vorticity gradients longer.
• Vorticity diffusion is dissipative. It causes vortical structures to decay and eventually homogenizes the vorticity field if no new vorticity is supplied.

Vorticity generation at boundaries

In viscous flow, the no-slip condition (fluid velocity matches the wall velocity) creates steep velocity gradients near solid surfaces. These gradients are the primary source of vorticity in most flows.

The process works as follows:

1. The no-slip condition enforces zero relative velocity at the wall.
2. Sharp velocity gradients develop in the thin boundary layer adjacent to the surface.
3. These gradients produce vorticity, as dictated by the curl of the velocity field.
4. The generated vorticity diffuses away from the wall and is advected downstream by the flow.

The interaction between boundary-generated vorticity and the outer flow produces many important phenomena: separation bubbles, vortex shedding, and the transition from laminar to turbulent boundary layers.

Vortex shedding

Vortex shedding occurs when flow past a bluff body (like a cylinder) produces periodic detachment of vortices into the wake.

Mechanism

1. Boundary layers develop on the body surface due to viscosity.
2. An adverse pressure gradient behind the body causes the boundary layers to separate from the surface.
3. The separated shear layers roll up into discrete vortices.
4. These vortices detach alternately from the upper and lower sides of the body, creating an oscillating wake pattern.

von Kármán vortex street

The von Kármán vortex street is the characteristic pattern of two staggered rows of counter-rotating vortices in the wake. Its key features:

• Vortices in opposite rows have opposite circulation and are offset from each other.
• The shedding frequency is characterized by the Strouhal number: $St = \frac{fD}{U}$, where $f$ is the shedding frequency, $D$ is the body diameter, and $U$ is the freestream velocity. For a circular cylinder, $St \approx 0.2$ over a wide range of Reynolds numbers.
• The alternating vortex shedding produces oscillating lift and drag forces on the body, which can cause structural vibrations (a major concern for bridges, chimneys, and offshore platforms).
• The same phenomenon produces sound: an Aeolian harp generates tones from wind-induced vortex shedding on its strings.

Vorticity in turbulence

Turbulent flows contain a wide range of eddy sizes, and vorticity dynamics drive the transfer of energy across these scales.

Vortex stretching

Vortex stretching is the mechanism that transfers energy from large scales to small scales in 3D turbulence. Here's how it works:

1. Large-scale velocity gradients stretch existing vortex lines.
2. As a vortex line stretches, its cross-section shrinks (conservation of mass).
3. To conserve angular momentum, the vorticity magnitude increases as the tube gets thinner.
4. This creates smaller, more intense vortical structures.

The vortex stretching term $(\vec{\omega} \cdot \nabla)\vec{u}$ in the vorticity equation captures this process mathematically. It's the engine behind the energy cascade: energy flows from large eddies to progressively smaller ones until it reaches the Kolmogorov scale, where viscosity dissipates it as heat.

Vortex stretching is inherently three-dimensional. In strictly 2D flows, this term vanishes, which is why 2D turbulence behaves qualitatively differently (energy cascades to larger scales instead).

Enstrophy and energy dissipation

Enstrophy measures the total squared vorticity in a flow:

$\mathcal{E} = \frac{1}{2}\int_V |\vec{\omega}|^2 \, dV$

(Note: the factor of $\frac{1}{2}$ is included in some definitions; conventions vary by author.)

Enstrophy matters because it's directly tied to energy dissipation. The viscous dissipation rate of kinetic energy in an incompressible flow is:

$\epsilon = \nu \int_V |\vec{\omega}|^2 \, dV = 2\nu \mathcal{E}$

This relationship shows that regions of high vorticity are where kinetic energy is most rapidly converted to heat. In turbulent flows, enstrophy production (via vortex stretching) and enstrophy dissipation (via viscosity) are in approximate balance at the small scales, which is a hallmark of the turbulent cascade.