Above a warm wire the plume speeds up as it rises

A plate laid flat draws its layers in from the edges and sends them up from the middle as a plume, and the essay ended on what such a plume would be if nothing below it mattered: a sheet of warm fluid rising above a line of heat, with no wall to slow it and no edge to start from. That flow has fewer inputs than anything else in this collection. There is no speed anybody chose, no length, and — far enough above the source — not even a temperature, because the plume cools as it rises. What is left is the heat it carries, and that heat is the whole of the problem.

The answer contradicts the picture most people have of a plume, which is of something that thins out and slows as it mixes with the air around it. Above a warm wire in still air the plume does the opposite. It speeds up.

One invariant, and the speed it fixes #

The heat a plume carries upward is the same at every height, because nothing outside the plume takes heat from it. Written as a buoyancy flux per unit length of the source,

$$B = \frac{g\beta Q'}{\rho c_p},$$

with $Q'$ the power per metre, it has dimensions of length cubed over time cubed. A laminar plume’s width grows by viscous diffusion, $\delta \sim \sqrt{\nu x/u}$, and its momentum balance is buoyancy against inertia, $u^2/x \sim g\beta\Delta T$. Carrying the heat, $u\,\Delta T\,\delta$ fixed, then gives

$$u \sim \left(\frac{B^2 x}{\nu}\right)^{1/5},\qquad \delta \sim \nu^{3/5}x^{2/5}B^{-1/5},\qquad g\beta\Delta T \sim \frac{u^2}{x} \propto x^{-3/5}.$$

The speed grows as the fifth root of the height. The temperature excess falls faster than the plume widens, so the buoyancy per unit length, $\Delta T\,\delta \propto x^{-1/5}$, falls slowly — but it is still positive, it still acts on every metre of the rise, and it adds momentum to fluid whose width grows only as fast as viscosity can spread it. The momentum flux, $u^2\delta$, grows as $x^{4/5}$: the plume is a jet whose momentum is being supplied from inside.

Four plumes, four exponents #

The same argument run for the other three common plumes gives the other three exponents. Above a point source the heat is spread round a circle rather than along a line, so the plume’s area rather than its width carries it, $u\,\Delta T\,\delta^2$ fixed, and with laminar diffusion the height drops out entirely: a laminar plume above a point rises at a constant speed, $(B/\nu)^{1/2}$ with $B$ now in $\mathrm{m^4/s^3}$. A turbulent plume widens in proportion to its height, because its eddies entrain at a rate set by its own speed rather than by viscosity, and then the line plume’s speed is constant and the point plume’s falls as $x^{-1/3}$ — the behaviour a speed nobody imposed quoted for the plume above a fire.

So the picture of a plume that slows as it spreads is the turbulent round one, and it is one case of four. The laminar line plume is the only one that accelerates, because it combines the two features that favour speed: a source whose heat is spread along a line rather than diluted round a circle, and a width that grows by diffusion, as the square root of the time the fluid has had, rather than by entrainment in proportion to the height.

The comparison with a jet is exact and instructive. What a jet keeps is its momentum flux, and Bickley’s plane jet slows as $x^{-1/3}$ because a fixed momentum is shared among more and more fluid. The plume keeps its heat instead, and its momentum is not conserved but manufactured. A jet is a machine for acquiring fluid it did not start with; a plume is the same machine with its fuel on board.

The equations, and the closed form that pins them #

With $\eta = y/\delta$, a stream function $\psi = u_s\delta f(\eta)$ on the speed scale $u_s = (B^2x/\nu)^{1/5}$, and $g\beta(T - T_\infty) = (u_s^2/x)\,h(\eta)$, the boundary-layer equations for a symmetric plume become

$$f''' + \tfrac35 ff'' - \tfrac15 f'^2 + h = 0,\qquad h' = -\tfrac35\,\mathrm{Pr}\,f h,\qquad 2\int_0^\infty f'h\,\mathrm{d}\eta = 1,$$

with $f(0) = f''(0) = 0$ on the axis and $f'$ vanishing far away. The energy equation has been integrated once already, using the symmetry; the last condition is the heat flux, and it is what sets the scale of everything else. It is carried in the box scheme as one more unknown, the running integral of $f'h$, with zero at the axis and a half at the far end, so that the solver imposes the invariant rather than checks it afterwards.

At one Prandtl number the equations can be solved by hand. Try $f = A\tanh(c\eta)$ and $h = D\,\mathrm{sech}^2(c\eta)$: the energy equation then needs $A = 10c/(3\mathrm{Pr})$, the momentum equation’s $\tanh^2$ terms need $A = 6c$, and the two agree only at $\mathrm{Pr} = 5/9$. The heat flux then fixes $c = (5/768)^{1/5}$. This is Fujii’s solution of 1963, and the box scheme reproduces its centreline speed, axis temperature and entrainment to three parts in a million. Fujii found a second one: with $h$ proportional to $\mathrm{sech}^4(c\eta)$ instead, the $\mathrm{sech}^2$ terms of the momentum equation cancel when $A = 10c/3$, and the energy equation then needs $\mathrm{Pr} = 2$. The box scheme lands on that one to eight parts in a million. Two closed forms at two Prandtl numbers, a decade of the range apart, pin the solver at both ends of the fluids that matter most, air and the light oils. At that Prandtl number velocity and temperature are the same curve: momentum and heat diffuse at rates whose ratio makes the plume’s two profiles one.

A warm core inside a moving sheet #

Away from five ninths the profiles separate, and the dial shows which way. In water, at a Prandtl number of seven, the temperature has fallen to one per cent of its axis value at $\eta = 1.7$ and the velocity at 5.9: a warm core a third the width of the fluid it is moving. At a thousand the core is a twentieth. The buoyancy acts only on the core, and the viscosity carries its effect outward into a broad sheet of fluid that is moving but was never warmed. At low Prandtl number the order reverses, and the temperature spreads wider than the motion, as it does on the vertical plate in mercury.

The crossing is exact rather than approximate, because at five ninths the two profiles are the same function, and it sits close to the Prandtl number of air. So a plume in air is the case where heat and motion travel together, and a smoke-laden plume in a room shows the moving fluid and the warm fluid at once; in water a dye released at the source marks only the warm core, and the moving sheet beside it is invisible. The point is the one what a photograph shows makes about every visualisation: a tracer reports the field it is carried in, and two fields can differ in width by a factor of twenty while the photograph shows one.

What the Prandtl number moves #

In the similarity units the centreline speed coefficient $f'(0)$ moves only from 0.36 to 0.93 across six decades of Prandtl number. The axis temperature moves from 0.029 to 10.2, a factor of 350, and the entrainment falls from 24 to 1.4. The speed is fixed mostly by the heat flux, through $u_s$, and only weakly by the fluid; the temperature is fixed by how narrow a core the fluid allows, and a viscous fluid allows a very narrow one, so the same heat flux concentrates into a much hotter centreline. That is why plumes in oils and in glycerine are visible by refraction as sharp threads, and plumes in air are diffuse.

A wire in still air #

The dimensional numbers make the acceleration concrete. A wire dissipating ten watts per metre in air at 20 °C — a heating element, a de-misting strand — has a buoyancy flux of $2.8\times10^{-4}\ \mathrm{m^3/s^3}$. A centimetre above it the plume’s centreline rises at 11 cm/s and is 22 kelvin above the room; at five centimetres, 15 cm/s and 8 K; at twenty, 20 cm/s and 3.6 K, warm to about two and a half centimetres either side of its axis. A tenth of a watt per metre still sends air up at nearly two centimetres a second a centimetre above the source. The speed goes as the two-fifths power of the heat, so a hundredfold change in power is a sixfold change in speed.

Those speeds are not small beside the flows a heated sensor is often asked to measure. A wire or film anemometer is a heated line, and in a flow of a few centimetres a second its own plume is a comparable velocity, pointing upward whatever the flow is doing — a speed nobody imposed, added to the one being measured. The similarity solution does not describe the region within a few wire diameters, where the plume is not yet thin against its height, but it gives the scale the anemometer’s own heating sets, and it says that scale is bigger the further above the wire the disturbance is felt.

Where the heat and the momentum go #

The bookkeeping closes in a way worth stating. The heat flux, $u\,\Delta T\,\delta$, is constant: $x^{1/5}\cdot x^{-3/5}\cdot x^{2/5}$. The volume flux, $u\delta$, grows as $x^{3/5}$, which is the entrainment — fluid drawn sideways into the plume from the room. The momentum flux, $u^2\delta$, grows as $x^{4/5}$, supplied by the buoyancy acting on the warm core. Nothing is lost; the one thing the plume conserves is the thing it started with, and everything else grows. It is the conservation argument a jet makes with a different invariant, and the difference between a slowing jet and an accelerating plume is entirely in which quantity the source fixed.

The volume flux is worth a number, because it is what a room’s ventilation sees. Ten watts per metre of wire draws in about 3.6 litres a second per metre of its length by twenty centimetres above it, and the flux keeps growing as the three-fifths power of the height. A row of warm pipes, or a strip of electronics, is a line plume of this kind, and in a room ventilated from below, where the air is meant to rise past its occupants and leave at the ceiling, it is the plumes rather than the fans that decide how much air passes each height.

A Reynolds number the plume makes for itself #

A plume has no imposed speed, so its Reynolds number, like the vertical plate’s in a speed nobody imposed, is an output. On the centreline speed and the height it is

$$\mathrm{Re}_x = \frac{u_c x}{\nu} = f'(0)\,\mathrm{Gr}^{*\,2/5},\qquad \mathrm{Gr}^* = \frac{Bx^3}{\nu^3},$$

the Grashof number written on the heat flux rather than on a temperature difference, because the heat is what the source fixes. Twenty centimetres above the ten-watt wire it is about 2,700, and it grows as the six-fifths power of the height, faster than the height itself, because the plume speeds up as it climbs. A laminar jet’s Reynolds number on its own speed and distance grows only as the two-thirds power, since it slows; a turbulent round plume’s grows as the two-thirds power too. The laminar line plume’s is the fastest-growing of them, and a Reynolds number that grows fastest with height is the one that leaves any laminar range soonest.

What a round source does instead #

Above a point source the same reasoning gives a laminar plume of constant speed, and the difference from the line source is geometry alone. The heat is spread round a circle whose area grows as the square of the plume’s width, so the temperature excess falls faster — as $x^{-1}$ instead of $x^{-3/5}$ — and the buoyancy per unit height, $\Delta T\,\delta^2$, stays constant; it adds momentum at a steady rate to a plume whose mass flux grows in proportion to its height, and the speed neither rises nor falls. The plume above a small warm object on a desk — a phone charger, a cup — is of this kind while it stays laminar, rising at a steady speed and widening only as the square root of its height. The line plume’s acceleration is what it gets for keeping all its heat in one plane.

When a warm jet becomes a plume #

The same invariant settles a question about sources that blow as well as heat. A slot that issues warm air has a momentum flux of its own, $U_0^2 d$ per unit length, which a jet would keep for ever, and a buoyancy flux, $g\beta\Delta T\,U_0 d$, which a plume turns into momentum at a rate that grows with height. Near the slot the flow is a jet that happens to be warm; far above, the buoyancy’s momentum has overtaken the nozzle’s and the flow is a plume that happens to have started fast. Setting the plume’s own momentum flux, $2\int f'^2\, \mathrm{d}\eta$ in its units, equal to the slot’s gives the height of the change,

$$x_c = \left(\frac{U_0^2 d}{M\,B^{3/5}\nu^{1/5}}\right)^{5/4},\qquad M = 2\int_0^\infty f'^2\,\mathrm{d}\eta = 2.22 \text{ in air},$$

the laminar counterpart of the length Morton defined for turbulent forced plumes. A two-millimetre slot blowing air twenty kelvin above the room at 0.3 m/s becomes a plume about four centimetres up; at 1 m/s, about 36. The length grows as the 1.75 power of the speed, so a vent that blows only gently is a plume almost from its mouth, and the question of whether a warm outlet’s air will reach the ceiling or spread along the floor is decided by buoyancy within a hand’s breadth of it.

The convention behind the numbers #

The plume is symmetric about a vertical plane through a line source of infinite length and negligible size, in fluid at rest and at one temperature far away, with constant properties and the Boussinesq approximation. Heights are measured from the source, which is where the similarity solution places its origin; a real source of finite width has a virtual origin some distance below it, and every height here would be measured from that. Widths are where a profile has fallen to one per cent of its axis value. The speed, temperature and entrainment coefficients are in the units $u_s = (B^2x/\nu)^{1/5}$ and $u_s^2/x$, and every dimensional number is for air at 20 °C with $\beta = 1/293$ per kelvin.

What the picture cannot show #

The source. Within a few source widths the plume is not thin against its height and the boundary-layer approximation fails; the solution says nothing about how the heat leaves the wire. The sway. Laminar plumes in rooms wander and flap long before they become turbulent, because any draught or stratification of the room deflects a flow this slow; the local Grashof number on the heat flux, $Bx^3/\nu^3$, is $10^7$ five centimetres above a ten-watt wire and $7\times10^8$ at twenty, and the higher of those is a statement about the model rather than a promise about the plume. The ceiling and the room. A real plume meets a ceiling or a warm layer of its own making and spreads. Three dimensions. A finite wire’s ends feed the plume from the sides, and its middle is a line source only if it is long against the height.

Who solved it #

Zel’dovich gave the similarity exponents for laminar and turbulent plumes in 1937. Yih solved the laminar plumes above line and point sources in the early 1950s, and Spalding and Cruddace computed the line plume across Prandtl numbers in 1961. Fujii found the closed forms in 1963, at five ninths and at two for the line source, and Gebhart and his collaborators measured laminar plumes above heated wires through the 1960s and 1970s, including their swaying and their transition. The solution here repeats Fujii’s at five ninths and Spalding and Cruddace’s elsewhere; the box scheme it uses is the one that solved the vertical plate, with the heat flux added as an unknown.

Still open: whether a strip’s plume forgets its strip #

Every number here is for a line source, and the plume the flat plate sends up from its middle does not start at one: it starts as two layers meeting, already moving sideways, carrying their heat in a sheet as wide as the layers are thick. The question is how many plate widths above the strip the plume takes the line source’s form, with a virtual origin some distance below the plate. That needs the plume’s equations marched upward from the strip’s own outflow rather than solved for their self-similar end state, and it would say whether the handbook’s warm-plate correlation has been measuring the plate, or the plume above it.