A round jet hands its wall jet a strength proportional to its throw

A jet striking a plate leaves a stronger wall jet the further it travelled found an inheritance. A plane jet that strikes a wall splits into two wall jets, and each keeps not the jet’s momentum — the wall takes that away as drag — but the quantity Glauert showed a wall jet conserves: the flux, through each height, of the momentum flux carried above that height. Turned without loss at the stagnation point, each half of the jet hands on exactly the momentum it brought times the fluid it had gathered, $3KQ/32$. A free jet gathers fluid as it travels, so the further the nozzle stood from the wall, the stronger the wall jet, by the cube root of the distance, though the jet arriving was slower.

The round jet does the same thing with different exponents, and the exponents are where the surprise is. A round jet gathers fluid at a rate that does not depend on how strong it is, so the invariant it hands on is in exact proportion to its throw. And the radial wall jet it leaves turns out to be the plane wall jet in disguise, mapped onto a disc by a change of variables that Mangler found for boundary layers on bodies of revolution.

A jet that gathers fluid at a fixed rate #

Schlichting solved the laminar round jet in 1933 as an exact similarity solution of the boundary-layer equations, and Landau and Squire later as an exact solution of the full Navier–Stokes equations, of which it is the slender, strong-jet limit. A jet of kinematic momentum flux $K$, a distance $h$ from its source, has a centreline speed and a profile

$$U = \frac{3K}{8\pi\nu h}, \qquad u = \frac{U}{(1 + \xi^2/4)^2}, \qquad \xi = \sqrt{\frac{3K}{16\pi}}\,\frac{\rho}{\nu h},$$

and the volume flux it carries, integrated over its cross-section, is

$$Q = 8\pi\nu h.$$

The momentum flux $K$ is not in it. A round jet gathers fluid at a rate set by the viscosity alone — eight π times ν per metre of travel, whether the jet is a trickle or a torrent. A stronger jet is narrower in the same proportion as it is faster, and the two cancel in the flux. A plane jet does not do this: its flux grows as $(36K\nu h)^{1/3}$, in which the momentum appears.

The same cancellation makes the round jet’s own Reynolds number, its centreline speed times its distance from the source over the viscosity, independent of the distance: $Uh/\nu = 3K/(8\pi\nu^2)$. A plane jet’s grows as the two-thirds power of the distance.

That number decides whether the jet is laminar, and since it does not change with distance, a round jet’s character is fixed at its nozzle. A laminar round jet with this Reynolds number is unstable to helical waves and would not stay laminar over a long throw in practice; the similarity solution describes it while it does, and the essay that found the round jet’s exact form in a turbulent one’s shape explains why the same profile, with an eddy viscosity in place of ν, describes it after.

The turn, and what the wall jet inherits #

At a plate the jet turns, and at a high enough Reynolds number the turn is quick and loses nothing: along each streamline the total head is kept, the pressure relaxes to ambient once the turned flow runs along the plate, and each streamline leaves with the speed it arrived with — the narrower statement of Bernoulli’s equation, along a streamline whether or not the flow is irrotational. The turn cannot mix streamlines, so it hands the wall jet the free jet’s profile as a function of the stream function, with the jet’s axis now on the plate.

For the round jet that profile has a simple form. Writing $\psi$ for the flux within a radius divided by $2\pi$, Schlichting’s profile is $u = U(1 - \psi/\Psi)^2$ with $\Psi = 4\nu h$. A stream tube carrying $2\pi\,d\psi$ in the jet carries $d\psi$ per radian of the spreading wall jet, so the radial wall jet starts with $u(\psi) = U(1 - \psi/\Psi)^2$, $\psi$ now the flux per radian below a height.

Glauert’s invariant for a radial wall jet is the plane one weighted by the radius at both integrals, $F = \int ru\left(\int_y^\infty ru^2\,dy'\right)dy$, and in terms of the flux per radian it takes exactly the plane form, $F = \int_0^\Psi \psi\,u(\psi)\,d\psi$. So

$$F = \frac{U\Psi^2}{12} = \frac{K\nu h}{2\pi}.$$

Each radial wall jet inherits the jet’s momentum flux times the viscosity times the throw — and since the gathered flux grew linearly with the throw, so does the inheritance. The plane jet’s $3KQ/32$ grew as the cube root. The figure at the top puts the two side by side: from 10 to 40 centimetres the round jet’s invariant quadruples where a plane jet’s grows by 1.59, while the round jet’s arriving speed falls to a quarter where a plane jet’s falls by 1.59 the other way.

What the plate itself feels #

The plate feels something simpler. A jet that is turned through a right angle, its momentum flux now running parallel to the plate, pushes on the plate with a force equal to the momentum flux it brought, $\rho K$, and the jet keeps its momentum flux exactly as it travels, so the force on the plate does not depend on the throw. What changes is how it is spread. The pressure at the stagnation point is the centreline’s dynamic pressure, $\tfrac12\rho U^2$, which falls as $1/h^2$, and the jet’s width grows as $h$, so the same force is spread over an area growing as $h^2$. A jet held four times further from the plate presses on it with a sixteenth of the peak pressure over sixteen times the area.

That is three behaviours from one jet, each scaling differently with the throw: the force on the plate is constant, the peak pressure falls as $h^{-2}$, and the wall jet’s invariant grows as $h$. The first is momentum, the second is momentum spread over a width, and the third is momentum times the fluid gathered. A measurement of any one of them would not predict the others, and the question “is the jet stronger or weaker further away” has three answers depending on which of the three is being asked about.

The stagnation point also carries the gradient that heat transfer depends on. The velocity along the plate grows linearly from the stagnation point at a rate set by the centreline speed over the jet’s width, both falling with $h$, so the stagnation rate falls as $1/h^2$ and the heat transfer there, which goes as its square root, as $1/h$ — the same square-root dependence a re-entry nose’s heating has on its stagnation gradient. A cooling jet held further away cools its stagnation point less and drives a stronger wall jet further out: the throw moves the cooling outward.

The plane wall jet in disguise #

What happens to the radial wall jet downstream needs a solution of the radial boundary-layer equations, and they do not have to be solved afresh. In 1948 Mangler showed that the boundary layer on a body of revolution is a two-dimensional boundary layer in disguise: with the distance along the surface stretched to $\bar x = \int r^2\,dx/L^2$ and the distance from it to $\bar y = ry/L$, the axisymmetric equations become the plane ones exactly, with the same viscosity. A radial wall jet is a boundary layer on a “body” whose radius is the distance from the axis itself, so $\bar x = r^3/3L^2$, and the radial wall jet at radius $r$ is the plane wall jet at $\bar x$. The invariant carries over unchanged, since the weighting by $r$ is exactly what turns $dy$ into $d\bar y$.

So the far radial wall jet is Glauert’s plane wall jet at $\bar x = r^3/3$. Its peak speed goes as $\sqrt{F/\nu\bar x}$, which is $r^{-3/2}$; the flux per radian it carries grows as $(F\nu\bar x)^{1/4} \propto r^{3/4}$; and its thickness, $\bar y/r$ with $\bar y \propto \bar x^{3/4}$, grows as $r^{5/4}$. Glauert found those exponents for the radial wall jet directly in 1956; Mangler’s map says they are the plane jet’s, read on a disc. And since the far wall jet’s speed goes as $\sqrt F$, the throw that multiplies the invariant by four multiplies the far wall jet’s speed by two, at every radius.

The turned profile’s shape differs from the plane jet’s in one way that matters for how the wall jet develops. Written in Mangler’s coordinate the turned round jet is $U/(1 + U\bar y/\Psi)^2$, which falls off only as the inverse square of the distance from the plate, where the turned plane jet’s $\mathrm{sech}^2$ falls exponentially. The round jet arrives with fluid spread far from its axis moving slowly, and that fluid lands far from the plate in the wall jet. A real wall jet must shed that tail into the surrounding fluid as it develops; the invariant, which weights each layer by the momentum flux above it, counts the tail at its full value.

The far wall jet, in millimetres #

For the water jet from a two-millimetre nozzle at fifteen centimetres a second, striking a plate twenty centimetres away, the far wall jet’s peak runs at 7.9 millimetres a second three centimetres out, 3.7 at five and 1.3 at ten, its peak 2.6, 4.9 and 11.6 millimetres above the plate. Those are small speeds — the radial wall jet spreads its momentum flux round a growing circumference as well as up into a thickening layer, and it slows much faster than the plane wall jet’s $x^{-1/2}$. The invariant it inherits is of order $10^{-15}$ in its own units, metres to the seventh over seconds cubed, which is why such a jet is a laboratory object: the radial wall jets used for cooling, from nozzles a centimetre across at metres a second, are turbulent from the turn.

Why the radial wall jet slows so much faster #

The plane wall jet slows as $x^{-1/2}$ and the radial one as $r^{-3/2}$, and the extra power is geometry that conservation alone explains. A wall jet keeps the flux of its momentum flux, and a radial wall jet spreads that flux round a circumference $2\pi r$. Per unit length of the circumference the invariant falls as $1/r^2$ — once for the circumference in each of its two integrals — and a wall jet of that local strength, at that distance from its origin, slows as the plane wall jet does with an extra factor of $1/r$. Mangler’s map does the same bookkeeping by stretching the distance as $r^3$.

The fluid it carries behaves the other way. The radial wall jet’s flux per radian — and so its total flux round the whole circumference, $2\pi$ times that — grows as $r^{3/4}$: it gathers fluid from above three times faster in the exponent than the plane jet’s $x^{1/4}$, because the boundary it gathers across grows with the radius as well as the distance travelled. Per unit length of the circumference the flux falls, as $r^{-1/4}$, which is the thinning-out of a flow spread round a widening ring, but the ring’s growth more than pays for it. So the radial wall jet is slow, thick and still entraining strongly where a plane jet at the same distance would be thinner and faster — a flow that is mostly gathered fluid soon after it leaves the turn, which is why its starting profile’s tail may matter less than the plane jet’s start did.

How far the thin-layer picture reaches #

A boundary layer is thin against the distance along the surface, and a radial wall jet’s distance along the surface is its radius. Its thickness grows as $r^{5/4}$, faster than the radius, so its thickness over its radius grows as $(\nu^3 r/F)^{1/4}$: the wall jet gets relatively fatter as it spreads, and the thin-layer equations, Glauert’s similarity and Mangler’s map with them, hold only out to a radius of order $F/\nu^3$. For the twenty-centimetre throw the peak sits at a tenth of the radius five centimetres out and a sixth fifty centimetres out; a throw of forty centimetres, with twice the invariant, keeps it a little thinner at every radius. The plane wall jet’s thickness over its distance falls as $x^{-1/4}$, the other way. A longer throw does the radial wall jet a second service: it makes it thin enough for its own theory to apply further out.

The convention, and what the turning leaves out #

The invariant is per radian and in Glauert’s units, metres to the seventh over seconds cubed, with $K$ the kinematic momentum flux of the jet in metres to the fourth over seconds squared. The standoff $h$ is measured from the jet’s virtual source, which for a laminar round jet sits close to the nozzle once the jet is many nozzle diameters long; near the nozzle Schlichting’s solution, which has a point source, overstates the speed. The radial distance $r$ is measured from the stagnation point.

The turn is taken as lossless and instantaneous, which needs the stagnation region to be small against the radii of interest and inviscid inside: at a Reynolds number of thousands the viscous layer at the stagnation point is thin, but the region where the jet bends is a few jet widths across, and the wall jet’s similarity form is reached only well beyond it. Nothing here computes how far beyond — the march that checked the plane jet’s inheritance was not repeated in Mangler’s variables, and the round jet’s algebraic tail, which the plane jet does not have, may well lengthen the approach. The wall is smooth, flat, infinite and normal to the jet, and the fluid at rest round it.

Who found the pieces #

Schlichting published the laminar round jet in 1933 and Bickley the plane one in 1937. Squire found the round jet as an exact solution of the Navier–Stokes equations in 1951, after Landau had in 1944. Glauert’s 1956 paper on the wall jet treated the plane and the radial cases together and found the invariant for each; Mangler’s transformation is from 1948, written for compressible boundary layers on bodies of revolution. That the impinging jet’s two halves inherit the free jet’s momentum times its gathered flux follows from putting them together, and the round jet’s version, with its flux free of the momentum, is the one in which the inheritance is simplest.

How it was checked #

Integrated over its radius, Schlichting’s profile carries a volume flux of $8\pi\nu h$ to three parts in a hundred thousand and a momentum flux of $K$ to six parts in a hundred million. The invariant computed as $U\Psi^2/12$ equals $K\nu h/(2\pi)$ to rounding, and computed again from Glauert’s definition on the turned profile in Mangler’s coordinate — the inner integral in closed form, the outer by quadrature on a grid uniform in the logarithm, with the algebraic tail’s remainder added — to one and a half parts in a billion. The far wall jet’s peak speed, flux per radian and peak height, from Glauert’s plane jet at $\bar x = r^3/3$, scale with the radius as $-3/2$, $3/4$ and $5/4$ to ten figures. The invariant quadruples between standoffs of 10 and 40 cm, and a plane jet’s grows by $4^{1/3}$, to rounding. And the model refuses a jet with no momentum, a negative standoff and a wall jet asked for at the stagnation point itself.

Still open: the approach to similarity on a disc #

The far wall jet is Glauert’s, but how far out it becomes Glauert’s was measured only for the plane jet, where it took a distance of about the Reynolds number in jet widths. Mangler’s map makes the radial question the plane one with a stretched distance, $\bar x = r^3/3$, and a different starting profile, the round jet’s algebraic tail. The next calculation marches the plane wall-jet equations in Mangler’s variables from the turned round jet, pre-growing the no-slip layer at a starting radius of a few jet widths, and asks at what radius the peak speed comes within five per cent of Glauert’s — whether the cube in $\bar x$ makes the radial jet forget its origin in a far shorter radius than the plane jet needed in distance, or whether its slowly decaying tail makes it remember for longer.