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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

ω=∇×v=2Ω\boldsymbol{\omega}=\nabla\times\mathbf v=2\boldsymbol{\Omega}

Ω∈[0.1,80]\Omega \in \left[0.1, 80\right]

where

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

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FLD-20 · orbit
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①

Governing law — red

ω=∇×v=2Ω\boldsymbol{\omega}=\nabla\times\mathbf v=2\boldsymbol{\Omega}

Ω∈[0.1,80]\Omega \in \left[0.1, 80\right]

Ω=0.1\Omega = 0.1ω=0.2\omega = 0.2

②

Domain of the variable — blue

Ωmin⁡=0.1,Ωmax⁡=80\Omega_{\min} = 0.1,\quad \Omega_{\max} = 80

Ω∈[0.1,80]\Omega \in \left[0.1, 80\right]

Ω=0.1\Omega = 0.1Ω=0.1\Omega = 0.1

③

Response curve — green

ω=∇×v=2Ω\boldsymbol{\omega}=\nabla\times\mathbf v=2\boldsymbol{\Omega}

Ω∈[0.1,80]\Omega \in \left[0.1, 80\right]

Ω=0.1\Omega = 0.1ω=0.2\omega = 0.2

④

Rate of change — amber

dωdΩ\frac{\mathrm{d}\omega}{\mathrm{d}\Omega}

Ω∈[0.1,80]\Omega \in \left[0.1, 80\right]

Ω=0.603\Omega = 0.603ω′=2\omega^{\prime} = 2

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: Ω\Omega →\rightarrow ω\omega. ω = ∇×v. Kelvin and Helmholtz constrain how vortex lines move in an inviscid barotropic fluid.

Purpose

Purpose: compute ω\omega from Ω\Omega in Fluid physics via ω=∇×v=2Ω\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 Ω=4.000 rad/s\Omega = 4.000\,\mathrm{rad/s}, the governing relation ω=∇×v=2Ω\boldsymbol{\omega}=\nabla\times\mathbf v=2\boldsymbol{\Omega} yields ω=8.00 1/s\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.

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Inputs

Outputs

  • Vorticity \omega8.00 1/s
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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