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Sandbox Physics

Fluid dynamics · vorticity transport

Lid-Driven Cavity

A steadily moving lid transfers momentum into a transparent glass cavity through viscous shear. Orbit the apparatus, then change Reynolds number and lid speed to follow luminous material tracers, streamline ribbons, the primary vortex core, and corner recirculation together.

Interactive modelLid-Driven Cavity
Simulation time0.00 s0.00\,\mathrm{s}
Reynolds number Re\mathrm{Re}400400
Kinematic viscosity ν\nu2.00×10−3 m2 s−12.00\times10^{-3}\,\mathrm{m^2\,s^{-1}}
Primary vortex center(0.50, 0.70)L\left(0.50,\,0.70\right)L
Probe speed ∥u∥\lVert\boldsymbol{u}\rVert0.000 m s−10.000\,\mathrm{m\,s^{-1}}
Net circulation Γ\Gamma0.000 m2 s−10.000\,\mathrm{m^2\,s^{-1}}

Physics tutorial

Lid-Driven Cavity: how one wall moves an entire fluid

BackgroundThree walls of a square container remain fixed while only the lid moves right. The no-slip condition gives adjacent fluid the same horizontal speed, then viscosity diffuses that momentum into the interior. The view layers the same two-dimensional solution through a shallow depth so it can be inspected from different angles.

Why it mattersThis spare geometry still contains wall shear, vorticity production, primary recirculation, and secondary corner eddies, so it has long served as a benchmark for incompressible-flow algorithms.

Start with the essentials

Focus question
If the lid only travels right, why does one giant clockwise circulation emerge in the cavity?
One-sentence intuition
The lid continuously injects negative vorticity. Kinematic viscosity ν\nu diffuses it while advection transports it, and Reynolds number sets the competition.

Core mathematical model

Incompressible Navier–Stokes equations

∂u∂t+(u⋅∇)u=−1ρ∇p+ν∇2u,∇⋅u=0\frac{\partial\boldsymbol{u}}{\partial t}+\left(\boldsymbol{u}\cdot\nabla\right)\boldsymbol{u}=-\frac{1}{\rho}\nabla p+\nu\nabla^2\boldsymbol{u},\qquad \nabla\cdot\boldsymbol{u}=0

Inertia, pressure, and viscosity determine the velocity field. Zero divergence means fluid is neither created nor destroyed inside the cavity.

Vorticity–streamfunction form

∂ω∂t+u⋅∇ω=ν∇2ω,u=(∂ψ∂y,−∂ψ∂x),∇2ψ=−ω\frac{\partial\omega}{\partial t}+\boldsymbol{u}\cdot\nabla\omega=\nu\nabla^2\omega,\qquad \boldsymbol{u}=\left(\frac{\partial\psi}{\partial y},-\frac{\partial\psi}{\partial x}\right),\qquad \nabla^2\psi=-\omega

The two-dimensional model transports scalar vorticity, then recovers a streamfunction and an automatically divergence-free velocity field from a Poisson equation.

Reynolds number

Re=ULν\mathrm{Re}=\frac{UL}{\nu}

Low Reynolds number means stronger viscous diffusion. Raising it thins the wall-shear layer and concentrates recirculation.

Common difficulties

A streamline ribbon is not a particle path

Typical misconceptionThe luminous ribbons and tracer particles show exactly the same information.

Better mental modelA ribbon is tangent to the velocity field at one instant; the luminous dots are material markers carried through time. They need not coincide in an evolving flow, and depth layers are only a visual extrusion of the same two-dimensional solution.

Color shows rotation, not speed

Typical misconceptionThe deepest blue region must be moving fastest.

Better mental modelVorticity measures local rotation and shear, while speed is ∥u∥\lVert\boldsymbol{u}\rVert. A thin wall layer can carry intense vorticity without being the fastest part of the whole flow.

Run the experiment

  1. 01

    Watch vorticity enter from the lid

    Choose Viscous flow, reset the fluid, and watch the blue layer spread downward from the moving lid.

    What to observe: Strong clockwise vorticity appears at the top first, then a broad and smooth primary circulation develops in the interior.
  2. 02

    Track the primary vortex core

    Switch to Classic benchmark and compare the vortex-center coordinates with the white material tracers.

    What to observe: The core does not stay at the geometric center; lid shear and the three fixed walls break that simple symmetry.
  3. 03

    Increase inertia

    Choose Inertial flow, enable velocity arrows, and let the field continue to develop.

    What to observe: The shear layer beneath the lid becomes more concentrated, the main recirculation changes shape, and small corner return flows become easier to form.