Reynolds Number and Flow Transition, Through Visualization

What Is the Reynolds Number?#

One of the most important dimensionless numbers in fluid flow is the Reynolds number (Re). It expresses the ratio of inertial to viscous forces.

$Re = \frac{\rho U L}{\mu} = \frac{U L}{\nu}$

Where:

• $\rho$: fluid density $[\text{kg/m}^3]$
• $U$: characteristic velocity $[\text{m/s}]$
• $L$: characteristic length $[\text{m}]$
• $\mu$: dynamic viscosity $[\text{Pa·s}]$
• $\nu = \mu/\rho$: kinematic viscosity $[\text{m}^2/\text{s}]$

Flow Regimes#

The Physics of Transition#

The path from laminar to turbulent flow begins with the Kelvin–Helmholtz instability. At a boundary with a velocity gradient, small disturbances grow and roll into vortices, which cascade energy down the scales.

The governing equation:

$\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} = -\frac{1}{\rho}\nabla p + \nu \nabla^2 \mathbf{u}$

In the Navier–Stokes equations, the nonlinear term $(\mathbf{u} \cdot \nabla)\mathbf{u}$ on the left carries inertia, while $\nu \nabla^2 \mathbf{u}$ on the right damps via viscosity. The Reynolds number measures the relative size of these two.

Velocity-Field Visualization: How Flow Changes with Re#

Try the simulation below to see velocity fields shift as Reynolds number changes:

What to look for:

• Low Re (Re ≈ 1–50): vectors smooth and orderly — viscosity damps disturbances immediately
• Mid Re (Re ≈ 100–500): asymmetry appears in obstacle wakes — inertia rivals viscosity
• High Re (Re > 1000): vector directions become irregular and vortex structures develop

Streamlines Past a Cylinder: the Kármán Vortex Street#

For flow past a cylinder, increasing Reynolds number produces a periodic vortex shedding called the Kármán vortex street.

The shedding frequency is normalized by the Strouhal number $St$:

$St = \frac{f D}{U_\infty}$

Where $f$ is shedding frequency, $D$ is cylinder diameter, $U_\infty$ is freestream velocity. For a cylinder, $St \approx 0.2$ stays roughly constant over $100 < Re < 10^5$.

Check the streamline pattern in the cylinder wake below:

What to look for:

• The stagnation point at the front of the cylinder, where streamlines split
• Symmetry breaking in the wake region as periodic vortex shedding starts
• Increasing speed (higher freestream) intensifies wake instability

Reynolds Number in Numerics: Grid Requirements#

Direct numerical simulation (DNS) of turbulence must resolve down to the smallest scale, the Kolmogorov microscale:

$\eta = \left(\frac{\nu^3}{\varepsilon}\right)^{1/4}$

Total grid count scales with Reynolds number as:

$N \sim Re^{9/4}$

So a 10× increase in $Re$ demands roughly 178× more grid points. This is why practical turbulence simulations rely on RANS or LES models.

Summary#

• The Reynolds number is the inertial-to-viscous ratio—the key dimensionless number for flow behavior
• Above critical Re, flow follows laminar → transitional → turbulent
• Cylinder wakes develop the Kármán vortex street, characterized by Strouhal number
• DNS grid cost scales as $Re^{9/4}$, so high-Re flows need turbulence models

Next week we cover how to discretize these Navier–Stokes equations with the finite volume method (FVM) and compare upwind vs. central-difference schemes for accuracy.