FLD-20

Vorticity (rigid rotation)

ω = 2 Ω. For rigid-body rotation the vorticity is twice the angular velocity.

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FluidsHelmholtz

Governing equation

\boldsymbol{\omega}=\nabla\times\mathbf v=2\boldsymbol{\Omega}

where

\Omega: Angular velocity (rad/s)
\omega: Vorticity (1/s)

Lecture brief

Historical brief

Bernoulli, Navier–Stokes, Reynolds (1883) and Stokes drag made continuum flow a dimensionless craft. The sheets compute head, drag, Re and a pedagogical NS snapshot. This sheet (FLD-20 — Vorticity (rigid rotation)) is the form associated with Helmholtz. Working symbols: $\Omega$ $\rightarrow$ $\omega$ . ω = ∇×v. Kelvin and Helmholtz constrain how vortex lines move in an inviscid barotropic fluid.

Purpose

Purpose: compute

\omega

from

\Omega

in Fluid physics via

\boldsymbol{\omega}=\nabla\times\mathbf v=2\boldsymbol{\Omega}

ω = 2 Ω. For rigid-body rotation the vorticity is twice the angular velocity. Use it when a real fluid physics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\Omega = 4.000\,\mathrm{rad/s}

, the governing relation

\boldsymbol{\omega}=\nabla\times\mathbf v=2\boldsymbol{\Omega}

yields

\omega = 8.00\,\mathrm{1/s}

. A spinning cylinder of fluid, a vorticity vector. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Angular velocity \Omega4 rad/s

Outputs

Vorticity \omega8.00 1/s

Reading speed

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Narration of this film

A spinning cylinder of fluid, a vorticity vector.

ω = ∇×v. Kelvin and Helmholtz constrain how vortex lines move in an inviscid barotropic fluid.

Reading speed

Watch on YouTube