Layer 23

Fluids, Flow & Turbulence

Continuity, pressure, Reynolds number and vorticity reveal how continuum motion differs from particle-by-particle intuition.

interactivemodel assumptions visibleestablished science separated from analogy

Story step: Many-particle motion can become a continuum. Fluids turn local conservation laws into flow, pressure, vorticity and sometimes turbulence.

1 · Continuity: geometry sets speed

For steady incompressible flow, the same volume flow rate passes every cross-section.

A₁v₁ = A₂v₂ = Q

area A₁

area A₂

inlet speed v₁

—v₂

—flow rate Q

—speed ratio v₂/v₁

2 · Bernoulli along an ideal streamline

In steady, inviscid, incompressible flow along a streamline, pressure, kinetic and gravitational terms trade off.

P + ½ρv² + ρgh = constant

density ρ (kg/m³)

v₁ (m/s)

v₂ (m/s)

h₂ − h₁ (m)

—P₂ − P₁

—kinetic pressure change

—gravitational term

3 · Reynolds number

The Reynolds number compares inertial and viscous effects. Thresholds depend on geometry; the labels below use smooth circular pipe flow as a familiar example.

Re = ρvL / μ

speed v

length L (m)

dynamic viscosity μ (Pa·s)

—Re

—pipe-flow heuristic

—inertia / viscosity scale

4 · Vorticity & viscous decay

A 2D Taylor–Green vortex is a clean incompressible solution whose vortices decay as viscosity dissipates kinetic energy.

u = U cos x sin y · e^(−2νt), v = −U sin x cos y · e^(−2νt)

kinematic viscosity ν

time t

Scientific boundary: Continuity and Bernoulli results assume specific idealizations; Reynolds thresholds are geometry dependent. The Taylor–Green vortex is an idealized exact flow, not a model of fully developed real-world turbulence.