Forced Convection over a Cylinder in Crossflow
Theory: correlations and when to use them
Overview
I want to estimate heat loss from an outdoor hot-water pipe. How do I get the heat transfer coefficient in wind?
The standard approach is a correlation for crossflow over a cylinder: get the Nusselt number from Reynolds and Prandtl numbers and convert to a heat transfer coefficient. But there are several correlations, differing by about 10% for the same conditions. And in light wind, radiation and natural convection matter, so forced convection alone badly misjudges heat loss. Check what dominates across your range of wind speeds.
Churchill–Bernstein correlation
Valid over the broad range $Re \cdot Pr > 0.2$, with properties at the film temperature (average of surface and fluid). Hilpert's and Zukauskas's correlations switch coefficients by Reynolds-number range.
Related dimensionless numbers
A Richardson number $Ri$ near 1 means natural and forced convection are comparable; below 0.1 forced convection dominates. $f_{shed}$ is the Kármán vortex shedding frequency.
Singing wires and the hot-wire anemometer
Power lines whistle in strong wind because of periodic forces from vortices shed behind them — studied by Strouhal in the 19th century. Cylinder heat transfer was studied from the same era, and early in the 20th century the hot-wire anemometer appeared: an electrically heated fine wire whose cooling by the wind measures speed. King gave a heat transfer law for fine wires in 1914, a foundation of turbulence measurement. Cylinder heat transfer underlies not only pipes and tube banks but measuring instruments too.
Worked example: a 25 mm hot-water pipe
A 25 mm pipe with surface at 80 °C sits in 20 °C air. Air properties at the 50 °C film temperature: kinematic viscosity 1.82×10⁻⁵ m²/s, conductivity 0.0280 W/mK, Prandtl number 0.704.
| Wind | Re | Nu (CB / Hilpert / Zukauskas) | Spread | h (CB) | Heat loss (CB) | Shedding |
|---|---|---|---|---|---|---|
| 0.5 m/s | 687 | 13.2 / 12.8 / 11.7 | 12% | 14.8 W/m²K | 70 W/m | 4 Hz |
| 2 m/s | 2,747 | 26.8 / 24.3 / 26.4 | 10% | 30.0 W/m²K | 141 W/m | 16 Hz |
| 5 m/s | 6,868 | 43.5 / 40.4 / 45.8 | 13% | 48.8 W/m²K | 230 W/m | 40 Hz |
| 10 m/s | 13,736 | 63.8 / 61.9 / 69.4 | 11% | 71.5 W/m²K | 337 W/m | 80 Hz |
| 20 m/s | 27,473 | 95.2 / 95.1 / 105.2 | 10% | 106.6 W/m²K | 502 W/m | 160 Hz |
Correlations differ by over 10%. Which one is right?
All are fits to experimental data, so treat the ~10% differences as their uncertainty. Rather than trusting one, check the spread across several and build margin into the design. Much bigger are heat paths other than forced convection. At 0.5 m/s, comparing radiation and natural convection as below shows forced convection alone isn't enough.
Example 2: radiation and natural convection
| Heat path | Coefficient | Heat loss |
|---|---|---|
| Forced convection (0.5 m/s) | 14.8 W/m²K | 70 W/m |
| Radiation (emissivity 0.9, painted) | 6.95 W/m²K | 33 W/m |
| Radiation (emissivity 0.1, bare aluminium) | 0.77 W/m²K | 3.6 W/m |
| Natural convection (still air) | 7.64 W/m²K | 36 W/m |
The Richardson number is 1.14 at 0.2 m/s, 0.18 at 0.5 m/s and 0.01 at 2 m/s. Below 0.5 m/s natural convection can't be ignored, and radiation varies about ninefold with surface finish.
Analysis workflow
- From the wind-speed range, compute Reynolds and Richardson numbers to see what dominates.
- Check the range of coefficients across several correlations.
- Estimate radiation from surface emissivity and add it to convection.
- If checking with CFD, resolve vortex shedding transiently and time-average the heat transfer.
- For tube rows, consider wake effects from upstream tubes (use tube-bank correlations).
“CFD heat transfer at 60% of the correlation”
A CFD calculation of cylinder heat transfer gave only about 60% of the correlation. It had been solved with steady RANS, so no vortex shedding occurred and rear-face heat transfer barely appeared. Re-solving transiently and time-averaging brought it within 10% of the correlation. Rear-face heat transfer is sustained by periodic shedding. When a steady solution won't converge, it may be a sign the flow is inherently unsteady.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Forced convection only | Heat loss underestimated in light wind | Add radiation and natural convection |
| Trusting a single correlation | ~10% uncertainty missed | Check spread across correlations |
| Properties at fluid temperature | Coefficient off | Use film temperature |
| Steady CFD ignoring shedding | Rear heat transfer underestimated | Transient with time averaging |
| Single-cylinder correlation for tube banks | Wake effects misjudged | Tube-bank correlations |
I'd like to try related calculations.
The Nusselt correlation tool, flow around a cylinder simulator and Strouhal and vortex shedding tool let you check heat transfer and flow. For related geometries see flow over a sphere, external flow over a flat plate, pin-fin arrays and compact heat exchangers.
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