The wind takes the air and not the cloth

Everyone knows what wind does to a coat, and the usual explanation is that it “goes through” the fabric. That explanation is right about the sensation and wrong about the arithmetic, in a way that decides what a garment should be made of.

A muslin’s own thermal resistance is 0.0108 m²K/W. The still-air layer clinging to any surface out of the wind is about 0.12 — eleven times as much. So a person in a shirt is wearing 0.131 m²K/W, of which the shirt contributes eight per cent and the air outside it contributes ninety-two.

Wind attacks both. It attacks the larger one first, by a route that has nothing to do with the fabric at all.

The claim #

A garment’s insulation is a quantity of still air, held partly inside the fabric and mostly on its outside, and wind removes it in two ways that are both about air and neither of which is about the cloth’s own conductivity.

The first mechanism is the boundary layer thinning. The second is air being pushed through the fabric, which short-circuits the fabric’s resistance in parallel rather than reducing it. Both are computed here and one of them is much the larger — but the one that is larger is not the one the sensation suggests.

The argument #

The boundary layer. A surface out of the wind carries a film of still air held by viscosity, and that film is the insulation. Its thickness falls as the flow over it speeds up — as the square root of the speed, which is the standard flat-plate result and is quoted here rather than derived. So the 0.12 m²K/W at rest becomes 0.085 at one metre per second, 0.054 at four and 0.029 at sixteen.

The through-flow. Air driven through a permeable fabric carries heat with it, and the heat it carries is a conductance in parallel with the fabric’s own. The mass flux per unit area is the fabric’s permeability at the wind’s own dynamic pressure — re-solved at that pressure rather than scaled from the hundred-pascal figure, because the relation is not linear — times the air’s heat capacity. The permeability itself is computed from the construction rather than measured, by the two-path arithmetic that gives a cloth’s channels and its threads separately.

At four metres per second a muslin passes 822 mm/s of air, which is a conductance of about a kilowatt per square metre per kelvin. Set against the fabric’s own resistance of 0.0108 — a conductance of ninety-three — it wins by a factor of eleven. The fabric’s own resistance is annihilated.

And it barely matters, because the fabric’s own resistance was eight per cent of the total to begin with. The boundary layer has fallen from 0.120 to 0.054 over the same interval, and that loss is six times larger than the entire fabric contribution that was destroyed.

What was counted, and how #

The boundary layer at rest is taken as 0.12 m²K/W, which is the standard figure for a vertical surface in still indoor air, and it thins as 1/√(1+u). Both are quoted.

The through-flow conductance is ρ c_p q, with q the face velocity the permeability gives at that wind’s dynamic pressure. It is written that way deliberately: it is an upper bound on the damage, because it assumes the air passing through leaves none of its heat behind in the fabric, and real air passing through a fabric is warmed by it. The bound is the honest thing to compute, since the alternative needs a heat-exchanger effectiveness that nobody has measured for cloth.

The whole is asserted monotone as the sweep runs — more wind is never more insulation — which would catch a sign error in either term, and which is the only assertion the model deserves, since both terms are quoted forms rather than derived ones.

Read the last two columns together. By four metres per second — a light breeze — the through-flow has reduced the fabric’s contribution to almost nothing, and the total is within two per cent of the boundary layer alone. After that point the garment is a boundary layer and the cloth is scenery.

The wind at which a fabric stops being a resistance #

Every cloth has a wind speed at which the air coming through it carries more heat than conduction through it does, and for the eight cloths of this collection’s table that speed is under two metres per second.

One and eight tenths of a metre per second is a quiet day: it is the speed at which smoke drifts and a leaf turns over, and it is slower than anybody walks. The closest cloth in the table is overtaken by the air moving at less than walking pace.

The duck is worth a second look. It has the largest own resistance of the eight — it is the thickest — and it is overtaken earlier than a batiste half its thickness, because it is also more permeable. Being thicker helps the resistance and hurts the crossover, and the two effects are not weighed against each other by anything in a fabric specification.

Which mechanism a windproof layer defeats #

This is where the arithmetic earns its keep, because the two mechanisms want opposite remedies and only one of them is available.

Thickening the cloth raises the flat line and does nothing to the curve. Choosing a warmer fibre moves the flat line by less than the width of the bracket the model carries. Neither touches the boundary layer, which is not in the fabric.

A windproof outer layer does two things at once and they are both about air. It stops the through-flow, which restores the fabric’s own resistance underneath it — a small gain. And it provides a surface whose boundary layer the wind can thin but cannot penetrate, so whatever still air is held inside the assembly stays still. A fleece under a shell keeps its thirty millimetres of trapped air; the same fleece alone loses it to the first breeze. The thirty millimetres is not a figure of speech — it is what ten tog of anything near air’s conductivity costs in thickness.

That is why a thin nylon shell over a thick pile is warmer than either alone by far more than the sum of their resistances. The shell contributes 0.01 m²K/W of its own and preserves 0.3 of somebody else’s.

Why it feels like the wind is coming through #

The sensation and the arithmetic disagree, and the disagreement is worth resolving rather than dismissing, because the sensation is reporting something real.

What a person feels is not a thermal resistance. It is a rate of heat loss at the skin, and the skin’s own boundary layer is inside the garment. Air moving through a fabric arrives at the skin cold and moving, and moving air at the skin is felt directly as a draught in a way that a slow decline in a resistance is not. The two-per-cent change in the total resistance at four metres per second is spread over the whole garment; the draught is local, sudden and exactly where the leakage path is.

So the everyday explanation — it goes through — is a correct account of the sensation and a bad account of the heat budget. Both halves matter for a garment, and they want different remedies: the heat budget wants a still-air layer preserved, and the draught wants a barrier at the point of leakage, which is usually a seam or a closure rather than the middle of a panel.

Why the crossover speeds are all so close together #

The eight cloths’ crossover speeds run from 0.3 to 1.8 metres per second, which is a factor of six across constructions that differ by far more than that in almost everything else. The narrowness is worth explaining, because it is what makes the conclusion general rather than a property of the table.

The crossover is where the through-flow conductance equals the fabric’s own, which is where ρ c_p q equals the thickness over the conductivity. Both sides move with the cloth and they move the same way.

A closer cloth is thicker, so its own resistance rises — and it is far less permeable, so its through-flow falls. Both changes push the crossover to a higher wind, and they are the same two changes that make it a better fabric. So the crossover is a ratio of two quantities that a construction moves together, and a ratio of two co-moving quantities is much steadier than either.

The steadiness is not exact, and where it fails is the duck. Being thick helps the numerator and being open hurts the denominator, and a duck is both — a heavy canvas at a coarse sett — so its two effects disagree and it lands earlier than its thickness alone would put it. A cloth that is thick and open is the worst case for this crossover, and a heavy loosely woven fabric is exactly what a traditional outdoor cloth is.

That is the practical form of the whole essay. No woven construction moves the crossover past walking pace, because the two levers a weaver has are tied together by a ratio, and the only route out is to remove the through-flow term entirely — which is a coating and not a construction. The narrowness of the range is not a fact about these eight cloths; it is a fact about what a sett can do to two quantities at once.

The same tying explains why a knit is no better placed. A knitted fabric is thicker per unit weight than a woven one, which raises its own resistance, and it is also far more permeable — a jersey passes several times what a shirting does — so both terms move again and the crossover lands in the same neighbourhood. The two mechanisms are tied by the fabric being mostly air, which is the finding the rung below arrived at from the conduction side, and nothing that is mostly air can be both thick and closed.

That is the sharpest statement of why a barrier layer is a different kind of object rather than a better fabric. A coating is not mostly air, so it is not on this curve at all: it has almost no thermal resistance of its own and it removes the through-flow term entirely, which is precisely the combination no cloth can offer.

Where the model stops #

The boundary-layer figure is for a flat vertical surface and a person is not one. A real garment’s outer film varies enormously with posture, geometry and body movement, and the pumping action of a walking body drives air through cuffs and openings far more effectively than any steady pressure drives it through fabric.

The through-flow term is an upper bound and the true one is smaller. Air passing through a fabric exchanges heat with it, and a fabric with a large internal surface is a decent heat exchanger. Including that would raise the total resistance at every non-zero wind and would not change the ordering of the two mechanisms.

The fabric’s own conductivity carries the bracket the mixture rule leaves, which is wide enough to swallow the fibre, so every number in the flat line here is a mid-band figure quoted at a precision the model does not have. It does not matter, and saying why is the point: the flat line could be twenty per cent out and the conclusion would be unchanged, because it is eight per cent of the total.

Radiation is absent. At body temperatures the radiative exchange across a still gap is a substantial fraction of the conductive one, which is why a reflective liner is worth anything at all, and nothing here computes it.

Nothing here is wet. A damp fabric is a different mixture — water conducts twenty-three times better than air — and evaporation from a wet surface removes heat at rates that dwarf everything computed here. That is the other thing water does to a cloth, and it is the reason wet clothing is dangerous in a way that thin clothing is not.

The still air inside the cloth is treated as still. It is not, once there is a pressure gradient across the fabric, and the transition between air held in the fabric and air passing through the fabric is the whole subject of the through-flow term rather than something it resolves.

And a garment leaks. The fabric arithmetic is a necessary condition, in the same way that a filter cloth’s two criteria are necessary and not sufficient for a filter. Openings, seams, closures and the neck of a jacket are almost always the dominant leakage path, and no fabric property fixes them.

The generalisation #

When a series chain of resistances has one dominant term, an intervention that destroys a small term feels dramatic and changes almost nothing, while an intervention that preserves the large one changes everything and is invisible.

The wind here destroys the fabric’s resistance completely — annihilates it, by a factor of eleven — and the total moves by two per cent, because the fabric was eight per cent to start with. Meanwhile the thing that matters is a film of air that nobody is making, selling or specifying.

The diagnostic is to write the series sum out before optimising any of its terms. It is a triviality and it is skipped constantly, because the term that is easy to change is usually the term that has a supplier attached to it. A fabric has a specification, a price and a data sheet; a boundary layer has none of those, and it is the answer.

The second lesson is about parallel paths inside a series chain. The through-flow does not reduce the fabric’s resistance — it puts a conductance beside it, and a large conductance beside a small one erases the small one. That is why the loss is so complete and so sudden: parallel paths do not degrade a resistance gracefully, they replace it.

Who found it, and when #

That clothing insulation is very nearly proportional to the thickness of still air, and that the still-air layer on a garment’s surface is a large part of it, comes from the military clothing physics of the 1940s, and the clo unit was defined in that work.

The thinning of a boundary layer with the square root of the flow is standard forced-convection theory. The treatment of through-flow as a parallel convective conductance is standard building physics, where it is used for wind-washing of insulation in a wall and where the same conclusion is reached: a permeable insulant needs a wind barrier and does not need a better insulant.

What appears to belong to this collection is joining the second of those to a computed fabric permeability rather than a measured one, so that the wind at which the through-flow overtakes the fabric can be read off a construction. For an ordinary shirting it is under two metres per second, which is a quiet day.

Where the ladder goes next #

The air is the answer here and the air is also what carries water, which is the other thing a cloth is asked to keep out and let out. That question has a much sharper form, because water’s pressure scale comes from a pore radius rather than from a flow: the pore that wicks is the pore that leaks, and the two demands turn out to be one expression read at the two ends of a contact angle.

Sideways, everything above assumed the holes were the ones the cloth was woven with. A knit’s are not — a knit has no hole to lose, because its loops are already over-full — which changes what the through-flow term is even about.