Dimensional analysis, the Reynolds number and the scaling symmetry

Physical laws cannot depend on the choice of units. Measure lengths in units of a typical size \(L\), velocities in units of a typical speed \(U\), times in units of \(L/U\) and pressure in units of \(\rho U^2\). Writing \(\mathbf{x} = L\mathbf{x}'\), \(\mathbf{u} = U\mathbf{u}'\), \(t = (L/U)t'\), \(p = \rho U^2 p'\) in the Navier-Stokes equations and dropping the primes gives \[ \frac{\partial\mathbf{u}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \frac{1}{\mathrm{Re}}\Delta\mathbf{u}, \qquad \mathrm{Re} = \frac{UL}{\nu}. \] Everything about the fluid, the speed and the size has collapsed into one number, the Reynolds number, named after Osborne Reynolds, whose 1883 dye experiments in glass pipes showed that the transition from smooth to turbulent flow is governed by this combination.

The Reynolds number compares inertia with viscosity: \(|(\mathbf{u}\cdot\nabla)\mathbf{u}| \sim U^2/L\) against \(|\nu\Delta\mathbf{u}| \sim \nu U/L^2\), whose ratio is \(UL/\nu\). A bacterium swimming (\(\mathrm{Re} \sim 10^{-4}\)) lives in a world ruled by viscosity; a whale (\(\mathrm{Re} \sim 10^{8}\)) in one ruled by inertia. Water at 2 m/s in a 10 cm pipe has \(\mathrm{Re} = 2\times0.1/10^{-6} = 2\times10^5\). Two geometrically similar flows with the same Reynolds number are the same dimensionless flow: this dynamic similarity is why a scale model in a water tunnel can predict a full-size aircraft. More generally the Buckingham Pi theorem (1914) says a relation among \(n\) dimensional quantities built from \(k\) independent units is a relation among \(n-k\) dimensionless groups; the drag on a sphere must take the form \(F = \rho U^2 L^2\,g(\mathrm{Re})\).

On all of space, with no boundaries to fix a length, the equations have an exact scaling symmetry. If \((\mathbf{u},p)\) solves the unforced Navier-Stokes equations on \(\mathbb{R}^3\), then so does \[ \mathbf{u}_\lambda(\mathbf{x},t) = \lambda\,\mathbf{u}(\lambda\mathbf{x},\lambda^2 t), \qquad p_\lambda(\mathbf{x},t) = \lambda^2 p(\lambda\mathbf{x},\lambda^2 t), \] for every \(\lambda \gt 0\). To check it, count powers of \(\lambda\): \(\partial_t\mathbf{u}_\lambda\), \((\mathbf{u}_\lambda\cdot\nabla)\mathbf{u}_\lambda\), \(\nabla p_\lambda\) and \(\nu\Delta\mathbf{u}_\lambda\) each carry exactly \(\lambda^3\). Time scales like length squared, as in the heat equation, and velocity like inverse length.

The scaling tells you how any norm behaves at small scales. For the \(L^q\) norm in space, the change of variables \(\mathbf{y} = \lambda\mathbf{x}\) gives \[ \|\mathbf{u}_\lambda(\cdot,t)\|_{L^q(\mathbb{R}^3)} = \lambda^{1 - 3/q}\,\|\mathbf{u}(\cdot,\lambda^2 t)\|_{L^q}. \] The kinetic energy \(\int|\mathbf{u}_\lambda|^2\,dx\) scales like \(\lambda^{-1}\): zooming in on small scales (\(\lambda\) large) makes the energy look small even when the velocity gradients are huge. The \(L^3\) norm is unchanged. Norms that scale like \(\lambda^0\) are called critical, and this bookkeeping is the organising principle of the modern theory in the last lessons.

Picture it: a flow zoomed in by a factor \(\lambda\) and played \(\lambda^2\) times faster, with velocities multiplied by \(\lambda\), is still a Navier-Stokes flow. Think it: the local Reynolds number of the rescaled flow, speed times length over viscosity, is unchanged by the scaling; a critical norm is one that measures a flow the same way at every scale.