What it does
Flows that look nothing alike behave the same when their dimensionless numbers match: that is why wind tunnels work with models and why honey and air can be compared. Each number is a ratio of two effects, and its size tells you which one wins. This page works out the common ones from one set of inputs and says in a line what each means here.
How to use it
- Pick a fluid. Its rounded properties near 20 °C fill in under Fluid properties; open that box to change them, or to describe your own fluid.
- Pick the shape: the Reynolds number’s regime limits depend on it (a pipe turns turbulent near 2300, a flat plate’s boundary layer near 5 × 10⁵).
- Give the speed and the length that sets the flow: the pipe diameter, the distance along the plate, the body’s diameter or the water depth.
- Optionally add a temperature difference (for natural convection: Grashof, Rayleigh and Richardson) and a frequency (for the Strouhal number).
- For a gas, the molar mass, pressure and temperature give the Knudsen number, which says when the gas stops behaving like a continuum.
The numbers
- Reynolds (inertia ÷ viscosity): laminar or turbulent.
- Mach (speed ÷ speed of sound): below 0.3 the flow can be treated as incompressible.
- Froude (speed ÷ surface wave speed): sub- or supercritical channel flow, ship waves.
- Weber, capillary, Bond, Ohnesorge: surface tension against inertia, viscosity and gravity: drops, bubbles, jets and inkjets.
- Prandtl and Péclet: how heat spreads compared with momentum, and whether the flow or conduction carries it.
- Grashof, Rayleigh, Richardson: buoyancy-driven flow and whether it beats the forced flow.
- Strouhal: unsteadiness; vortex shedding from a cylinder sits near 0.2.
- Knudsen: mean free path ÷ size; continuum, slip or free-molecular flow.
Worked example
The page opens on water at 20 °C flowing at 1 m/s through a 5 cm pipe, with a 10 K temperature difference and a 4 Hz oscillation. Re = 49900: turbulent pipe flow (above about 4000). The Mach number is 0.000676 (incompressible), Froude 1.428, Weber 685, Prandtl 6.990 and Péclet 348799. With ΔT = 10 K, Grashof is 2.56 × 10⁶ and Rayleigh 1.79 × 10⁷ (laminar natural convection), but Richardson is only 0.001030, so the forced flow wins. The 4 Hz oscillation gives Strouhal 0.2, typical of vortex shedding. Switch to air and a 1 µm gap and Knudsen is 0.0649: slip flow.
Tips
- Use the same L the regime limits were written for: diameter for pipes, distance from the leading edge for plates.
- Viscosity changes a lot with temperature (water’s halves between 20 °C and 50 °C); edit μ when you are far from 20 °C.
- For non-circular ducts, use the hydraulic diameter, 4 × area ÷ wetted perimeter.
FAQ
Why is my “laminar” flow turbulent? The limits are typical values; quiet lab pipes stay laminar far above 2300, rough or disturbed flows go turbulent earlier.
Where do the property values come from? They are rounded, well-known values near 20 °C, given so the page works straight away. Use measured values for real work.
Can I use it for design? It is for study and planning, not a substitute for professional judgement.
Where the numbers come from
- Definitions of the dimensionless groups: standard results, for example White, F. M., Fluid Mechanics, and Incropera et al., Fundamentals of Heat and Mass Transfer. Mean free path from viscosity: first-order kinetic theory, λ = (μ/p)√(πkT/2m).
- Regime limits (pipe Re 2300 and 4000, plate 5 × 10⁵, cylinder shedding near 47, Rayleigh-Bénard 1708, Knudsen 0.01, 0.1 and 10, Mach 0.3, 0.8 and 1.2) are the usual rough textbook guides.
- Fluid properties are rounded, general values near 20 °C (water, air, sea water, glycerol; the light oil is a made-up typical value), editable on the page.
- Checked against an independent NumPy script using scipy.constants.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.