Turbulence & the Reynolds Number

1. The Reynolds Number

The Reynolds number Re compares the magnitude of inertial forces (which promote turbulence) to viscous forces (which suppress it):

2. Laminar–Turbulent Transition

In his 1883 dye-stream experiment, Reynolds injected coloured water into a glass pipe and varied the flow rate. Below a critical Re the dye flowed in a smooth thread; above it the dye mixed chaotically throughout the cross-section.

Transition is not a sharp threshold — it depends on inlet disturbances, pipe roughness, and free-stream turbulence intensity. The Orr-Sommerfeld equation describes the linear stability: small perturbations grow when the eigenvalues of the associated operator have positive imaginary parts.

3. Kolmogorov Energy Cascade

In 1941 Kolmogorov proposed a statistical theory of fully developed turbulence. Energy is injected at large scales (integral scale L), cascades through a hierarchy of eddies, and dissipates as heat at the Kolmogorov microscale η:

4. Kármán Vortex Street

Flow past a bluff body (cylinder, bridge pier, chimney) sheds alternating vortices downstream at a characteristic frequency governed by the Strouhal number:

5. Boundary Layers

Near a solid surface the no-slip condition forces velocity to zero. The boundary layer thickness δ grows with distance x from the leading edge:

6. DNS, RANS and LES

7. JavaScript Vortex Shedding Simulation

// 2D Kármán vortex street via point vortex method
class PointVortex {
  constructor(x, y, gamma) {
    this.x = x; this.y = y;
    this.gamma = gamma; // circulation (+ = CCW, - = CW)
  }
}

function inducedVelocity(vortices, px, py, core = 0.5) {
  let u = 0, v = 0;
  for (const vt of vortices) {
    const dx = px - vt.x, dy = py - vt.y;
    const r2 = dx*dx + dy*dy + core*core;
    u +=  vt.gamma * dy / (2*Math.PI * r2);
    v -=  vt.gamma * dx / (2*Math.PI * r2);
  }
  return {u, v};
}

function stepVortices(vortices, U_inf, dt) {
  const newPos = vortices.map(v => {
    const others = vortices.filter(w => w !== v);
    const {u, v: vel_v} = inducedVelocity(others, v.x, v.y);
    return {x: v.x + (U_inf + u) * dt, y: v.y + vel_v * dt};
  });
  newPos.forEach((p, i) => { vortices[i].x = p.x; vortices[i].y = p.y; });
}

// Kármán street: shed vortices alternately above/below cylinder
const vortices = [];
let time = 0, shedTimer = 0;
const U = 1, D = 20, St = 0.2;
const shedPeriod = D / (St * U); // T = D/(St·U)

function shed(sign) {
  const gamma = sign * 2 * Math.PI * U * D * 0.3;
  vortices.push(new PointVortex(0, sign * D * 0.6, gamma));
}

function update(dt = 0.5) {
  shedTimer += dt;
  if (shedTimer >= shedPeriod / 2) {
    shed(vortices.length % 2 === 0 ? 1 : -1);
    shedTimer = 0;
  }
  stepVortices(vortices, U, dt);
  time += dt;
  // Remove vortices that have left the domain
  while (vortices.length > 60) vortices.shift();
}

// Reynolds number monitor
function striationRe(U, D, nu = 1e-6) {
  return U * D / nu;
}

8. Engineering Applications

• Aircraft design: Wing lift is critically affected by boundary-layer separation; turbulators and vortex generators delay stall by energising the boundary layer.
• Pipelines: Turbulent friction factor (Moody chart) determines pumping power requirements. Smooth pipes follow the Colebrook-White equation.
• Heat exchangers: Turbulence enhances convective heat transfer (Nusselt number Nu ∝ Re^0.8 × Pr^0.4 — Dittus-Boelter correlation).
• Wind engineering: Vortex-induced vibration governs the design of tall buildings, bridges, offshore platforms and power lines.
• Weather prediction: Atmospheric turbulence is the primary mechanism mixing momentum, water vapour and heat from the surface upward.