Four times the speed, still 19.47 degrees — where water depth blows the Kelvin wedge open to 90
The Kelvin half-angle in deep water stays at 19.47° even if speed quadruples. A shallow bottom opens it toward 90° once the depth Froude number reaches 1.
Sending the smoke down 1 m cost 7 Pa of draft — Franklin's stove and the buoyancy budget
One metre of down-leg removes 6.16 Pa and the longer chimney gives back less, 38.32 Pa instead of 39.15. Net draft falls from 44.72 Pa to 37.72 Pa.
Viscosity Is Not What Kills a Vortex — Newton's 1/r and the Lamb–Oseen Core
At r=0.5 the speed is unchanged to four decimals after 50 diffusion times. At n=1 the viscous force is zero; only a wall removes circulation.
Four Shocks Beat One — The Oblique Shock Train in a Supersonic Inlet
At Mach 3 one normal shock recovers 0.3283 of stagnation pressure. Four 8° ramps plus a terminal shock lift that figure to 0.8235.
How Far Must Water Flow Before It Becomes a Parabola — Entrance Length and Hagen–Poiseuille
In laminar pipe flow a flat inlet needs about 0.05 Re diameters to become a parabola, and 64/Re is not valid until that station.
A Single Ripple Outruns Its Own Group — Wave Dispersion and Group Velocity
Deep-water gravity waves move energy at half the phase speed. The slowest ripple is near 1.73 cm and 0.232 m/s, where gravity balances surface tension.
Why a Spinning Ball Curves — the Magnus Effect and Circulation
Random spin drifts a musket ball sideways. Circulation Γ=2πa²ω sets the side force, and spin 2 misses by 6.90 m, not twice the 3.71 m printed at spin 1.
Waves Hanging in the Sky — The Kelvin-Helmholtz Instability
On air and water the first wave shows up near 6.6 m/s, about 1.69 cm long, growing at 4.89 per second. Below that wind the surface stays calm.
Chaos That Begins at the Bottom of a Pot — Rayleigh–Bénard Convection and the Lorenz Equations
Rolls switch on at Rayleigh number 1708 with rigid walls. In the Lorenz truncation that heating, written as rho, stops settling once rho passes 24.74.
The Shock Wave in Your Kitchen Sink — Froude Number and the Hydraulic Jump
At Fr 2.86 a 0.2 m supercritical sheet jumps to Fr 0.42 and a depth of 0.714 m, scattering 0.238 m of specific energy into the roller.
Why Gasoline Engines Stop at a Compression Ratio of 12 — Otto vs Diesel
Same compression ratio: Otto is 56.5% at r=8 and 63.0% at r=12, Diesel at cutoff 2 is 49.0% and 56.7%. Knock, not the heat input, stops gasoline near 12.
Measure the Drag, Get the Heat Transfer — The Reynolds Analogy
At Re_x = 1e5 on a flat plate, Cf/2 stays at 1.050e-3, and St times Pr to the 2/3 matches that value at Pr = 0.7, 1, and 7.
To Go Supersonic, Widen the Pipe — The De Laval Nozzle and Choking
Area ratio 4 designs to Mach 2.940 and pe/p0 of 0.0298. Gamma 1.4 locks the choked mass-flow factor near 0.5787, and back pressure stays out of it.
Why Droplets Are Round and Water Climbs the Glass — Surface Tension and Contact Angle
A 5 µm droplet shows a 28800 Pa pressure jump, and a 0.1 mm tube at 30 degrees lifts water 127.1 mm under Jurin's law here.
Heating It Hotter Made It Cooler — The Boiling Curve and Critical Heat Flux
Zuber's CHF for water at 1 atm is 1.259 MW/m². Rohsenow's nucleate flux reads 1.094 at 20°C and 3.692 at 30°C, already past that ceiling.
Why a Rougher Ball Flies Farther — The Drag Crisis and Boundary-Layer Separation
At 70 m/s and Re = 1.99e5 the same ball gives 2.049 N smooth and 0.488 N dimpled. Separation at 82 degrees versus 120 opens that gap.
Fluid That Warms Up While Standing Still — Natural Convection and Rayleigh 1708
A 0.30 m wall at 60 C in 20 C air prints Ra 7.03e7, Nu 54.9, and a heat flux of 197.6 W/m^2, still on the laminar side of 1e9.
Turbulence Runs Three Rulers at Once — The Kolmogorov Cascade and the -5/3 Law
A tenfold rise in Reynolds shrinks η/L to about 0.18 of its value, and a lab tunnel at Re=10^4 already asks a 3D DNS for about 10^9 cells.
Three Portraits of a Flow — Why Streamlines, Pathlines, and Streaklines Diverge
Drop the contraction to 0.25 and throat speed jumps 4x while the local term stays at 0. With V0=0.6 and omega=2, streamline y never leaves ±0.6.
Why Smokestack Plumes Curl Up Instead of Lying Flat — The Jet in Crossflow and Its Counter-Rotating Vortex Pair
With A=1.6 and B=0.33, the centerline at x=10d sits at 5.44d, 8.66d, and 13.78d for r=2, 4, and 8. Doubling r raises height by about 1.59, not by 2.
Two Functions Added, and a Cylinder Appeared — Complex Potential and d'Alembert's Paradox
Adding a doublet to a uniform stream draws a cylinder without sketching one. With zero circulation the pressure integral leaves only roundoff for drag.
Above Mach 1, You Must Widen to Speed Up — The Laval Nozzle Paradox
Area ratio 2.5 exits near Mach 2.44 on the supersonic branch, static pressure at 6.3 percent of stagnation. Raise back pressure and a shock stands inside.
When the Stream Breaks — Reynolds' Critical Number and the Layered Turbulent Boundary
Pipe transition is usually quoted near Re 2300, yet a quiet pipe can stay laminar to 50,000. The wall then splits into sublayer, buffer, and log layer.
Water Boils at 20°C Too — The Physics of Cavitation
At 20°C vapor pressure is 2.34 kPa. If a venturi throat drives static pressure under that line, water boils and the cavitation number turns negative.
Two Faces of a Whirl — Free Vortex, Forced Vortex, and Helmholtz
A free vortex revolves with zero vorticity except at the center. The Rankine core spins at 2U/R and meets the 1/r tail smoothly at the core radius r = R.
Why Water Streams Break into Droplets — Plateau–Rayleigh Instability and Breakup Regimes
The fastest Plateau–Rayleigh wave sits near 9.02 radii. Weber and Ohnesorge then decide whether that jet stays in fat drops or turns to mist.
Bugs Swim in Honey — The World of Flow Changed by Reynolds Number
A whale at Re 2.5e8 and a bacterium at 3e-5 share the same water. The factor 1/Re on the viscous term decides which balance actually survives.