A blunted nose delays its cone's transition only so far, and the correlation decides how far

A blunted cone eats its nose’s sheath in a few hundred radii followed the hot, slow gas a blunted tip leaves along a slender hypersonic cone. The curved part of the bow shock near the nose is nearly normal — the blast that a hypersonic body leaves behind it — so the gas that passes through it is heated more and keeps less of its speed than the gas through the cone’s oblique shock further out, and that gas forms a sheath along the body. The laminar boundary layer growing under the sheath eats into it, and on a cone, whose widening perimeter feeds the layer fast, it swallows the sheath in tens to hundreds of nose radii — far sooner than on the cylinder whose sheath outlasts the body.

It ended on the consequence that motivates the subject. Inside the sheath the boundary layer’s edge is not the sharp cone’s: its Mach number and its Reynolds number per metre are lower. Past the swallowing distance the layer meets the cone’s own gas and both jump. That is the mechanism usually given for the observation that blunting a cone’s tip moves its transition to turbulence aft. The question was how far.

The answer is: only so far. Read the layer’s edge through the sheath and every correlation of the usual form gives a delay that grows with the nose radius, peaks, and falls again. The peak is where the transition point has moved inside the sheath, and its size depends on whether the correlation divides by the edge Mach number. At Mach 6, five times the sharp cone’s distance for the momentum-thickness Reynolds number alone; 1.4 times for its ratio to the Mach number.

The layer’s own Reynolds number #

The calculation is the earlier essay’s, extended by one quantity. The sheath — the gas beside a hypersonic body that remembers its nose — comes from Billig’s correlation for a sphere-cone’s bow shock: a hyperbola whose far asymptote is the sharp cone’s own shock, so that each streamline crossing it at a given distance from the axis carries a known entropy rise. Every streamline is brought to the sharp cone’s surface pressure with its own entropy, which fixes its temperature, speed and density. The laminar boundary layer is locally similar, in Lees and Dorodnitsyn’s form with Mangler’s transformation for the cone’s widening perimeter, and the streamline at its edge is the one whose upstream mass it has swallowed.

The extension is the momentum thickness. The same similarity variables that give the layer’s mass flux give its momentum deficit, and dividing by the edge’s density and speed gives the momentum thickness; multiplied by the edge’s unit Reynolds number it is the momentum-thickness Reynolds number $\mathrm{Re}_\theta$ — the Reynolds number the layer itself has grown to, measured with its own thickness rather than with the distance it has run. On a sharp cone the edge is uniform and $\mathrm{Re}_\theta$ grows exactly as the square root of the distance; the march reproduces that exponent to eight parts in a million.

Transition is then read from a correlation, and this is the essay’s one stated choice. Hypersonic design practice correlates the onset of transition on cones and re-entry bodies with $\mathrm{Re}_\theta$ alone, or with $\mathrm{Re}_\theta/M_e$, its ratio to the edge Mach number, or with forms in between. None is a theory; each is a summary of measurements. All three are written here as $\mathrm{Re}_\theta / M_e^{\,n} = C$ with $n = 0$, $\tfrac12$ and 1, and the constant is not taken from any data set. It is fixed by asking that a sharp cone of the same angle, in the same stream, transition at a stated distance, 0.3 metres, so every result below is a ratio: how far bluntness moves the transition point from where the sharp cone’s would be.

The case is a Mach 6 stream at 2.6·10⁷ per metre over an 8° cone — the scale of a conventional hypersonic tunnel — with the sharp cone’s edge at Mach 5.24 and its momentum-thickness Reynolds number 1,293 at 0.3 metres.

The delay peaks #

The first figure is the result. Below a nose Reynolds number of about $10^4$ — noses under a third of a millimetre — nothing moves: the sheath is swallowed within a few centimetres of the tip, long before the layer has grown enough to matter. Above it the transition point moves aft under every criterion, faster for $n = 0$ than for $n = 1$. And then each curve turns over.

The momentum-thickness Reynolds number alone gives the largest delay, 4.96 times the sharp cone’s distance at a nose of 44 millimetres, nose Reynolds number $1.1\cdot10^6$. The intermediate form peaks at 2.57, at 20 millimetres. The Mach ratio gives the smallest, 1.41 at 7.3 millimetres, and falls fastest after it: by a 30-millimetre nose its transition point is back near the sharp cone’s, and past a nose Reynolds number of about $9\cdot10^5$ it falls within ten nose radii of the tip, where the model’s treatment of the nose region is least trustworthy and the curve is not to be read.

So the claim that a blunter tip always moves transition further back is false in this model under every criterion. Bluntness buys a delay up to a nose size, and beyond it a blunter nose gives some of the delay back. And the criterion is not a detail: it changes the largest delay by a factor of three and a half and the nose that gives it by a factor of six.

What the sheath does to the edge #

The second figure shows the two numbers the correlations are made of, along the cone. At 0.3 metres the sharp cone’s layer has a momentum-thickness Reynolds number of 1,293 and an edge Mach number of 5.24. A 1-millimetre nose’s sheath is swallowed at 6.7 centimetres, and by 0.3 metres the layer has nearly forgotten it: 1,210 and 5.02. A 3-millimetre nose’s sheath reaches to 0.3 metres, and there the numbers are 824 and 3.83. A 10-millimetre nose’s sheath reaches 1.5 metres, and at 0.3 metres the layer sees 609 and 2.85.

Both numbers fall inside the sheath, and they fall for the same reason. The sheath’s gas went through a nearly normal shock, so it is hot, and hot gas is both slower in Mach number and more viscous; its unit Reynolds number is lower, so the layer grows to a smaller $\mathrm{Re}_\theta$ in the same distance. The criteria differ in what they do with the fall in Mach number. A correlation that ignores it, $n = 0$, sees only the lower Reynolds number and is the most impressed by bluntness. One that divides by the Mach number sees the lower Mach number as raising its ratio, which partly cancels the lower Reynolds number. Hence the ordering of the three peaks: the more weight the edge Mach number has, the less a sheath can delay.

Past the swallowing distance both curves climb back towards the sharp cone’s, and the momentum-thickness Reynolds number catches up only slowly. A layer that grew its first stretch in the sheath carries less momentum deficit than one that grew in the cone’s gas throughout, and it keeps that history.

Where the transition point lies #

The third figure follows the Mach-ratio criterion along the cone and shows why its curve turns over. The dashed line is the sharp cone’s ratio, which reaches the criterion’s level, 247, at 0.3 metres by construction. With a 1-millimetre nose, at the dial’s first setting, the blunted ratio runs just below it, and transition moves only to 0.314 metres. With 3 millimetres the ratio is held down further and transition moves to 0.382 metres, with the sheath swallowed at 0.3.

At 7 millimetres, the peak, the ratio reaches the level at 0.424 metres while the sheath lasts to 0.945: the transition point has moved inside the sheath. At 15 millimetres it is deeper inside, the sheath lasting to 2.6 metres, and transition comes back to 0.393 metres. At 30 millimetres the sheath lasts the whole cone, and transition is back at 0.318.

The mechanism of the reversal is visible in the dial. Inside a long sheath, the layer’s edge gas is the nose’s gas, nearly the same whatever the nose’s size: it went through a nearly normal shock and was brought to the cone’s pressure. The one thing a bigger nose then changes is its own Reynolds number, which raises the layer’s $\mathrm{Re}_\theta$ at a given number of nose radii as its square root. So once the transition point is inside the sheath, a bigger nose transitions in fewer nose radii, and the distance in metres stops growing and falls.

The peak is where the sheath outlasts the transition point #

The fourth figure collapses the argument into one picture. It plots each criterion’s transition distance against the swallowing distance, both scaled by the sharp cone’s 0.3 metres, with the diagonal on which the two are equal. While the sheath is swallowed well ahead of the transition point — to the left of the diagonal’s crossing — every curve rises: the layer reaches the cone’s gas before it transitions, and its history in the sheath delays it. As the swallowing distance passes the transition distance, the curves bend, and each peaks somewhere to the right of the diagonal: for the Mach ratio when the sheath is swallowed 2.4 times further back than the transition point, for the intermediate form at 5.1 times, for the momentum-thickness Reynolds number alone at 7.5.

So the rule of thumb the figure supports is a qualitative one, and a useful one. Bluntness delays transition while its sheath is shorter than the transition distance, and stops delaying it once the sheath is several times longer. Where between those the peak lies depends on the correlation. The swallowing distance is the number the earlier essay computed; the sharp cone’s transition distance is whatever the stream and the surface give. Their ratio says which side of the peak a nose is on.

Faster streams, larger delays #

The fifth figure repeats the search at five free-stream Mach numbers, keeping the cone, the unit Reynolds number and the sharp cone’s 0.3 metres. The largest delay grows with the Mach number under both criteria: for the Mach ratio from 1.25 at Mach 5 to 2.74 at Mach 15, for the intermediate form from 2.0 to 8.6. A faster stream’s nose shock heats its sheath more, against a sharp cone’s edge that stays cooler, so the contrast between sheath and cone gas — and with it the Reynolds number the sheath can take away — grows.

The right-hand panel shows a regularity the model does not explain in a sentence. At every Mach number the largest delay falls when the transition point is forty to sixty-five nose radii from the tip: 46 to 64 for the Mach ratio, 37 to 40 for the intermediate form. The nose that gives the most delay is the one whose transition point lies about fifty of its own radii back, whatever the stream.

What the measurements say, and what they do not test #

Blunted cones in hypersonic tunnels have shown both halves of this picture: transition moving aft as a small tip is blunted, and, at larger bluntness, moving forward again — the behaviour known as transition reversal. The measurements’ numbers depend strongly on the tunnel, whose own noise sets the disturbances the layer starts with, and no correlation of the kind used here was fitted to them.

The model’s reversal is not evidence for any account of the measured one. Here the reversal follows from a correlation applied inside a sheath whose edge gas no longer changes with the nose; in the experiments the reversal is accompanied by disturbances the modal instabilities of a smooth laminar layer do not predict, and the current candidates — transient, non-modal growth, and instabilities of the entropy layer itself — are outside a similarity layer with a correlation on top. What the model does establish is narrower and firmer: under any correlation of this family, a blunted tip’s delay is bounded, and the bound is reached when the transition point moves inside the sheath.

The convention behind the numbers #

Lengths along the cone are in metres from the tip, and nose sizes are radii. The nose Reynolds number is the free stream’s unit Reynolds number times the nose radius. The momentum-thickness Reynolds number and the Mach number are those at the boundary layer’s edge — the streamline the layer has swallowed — and the momentum thickness is the compressible one, with the edge density. “Transition” is the station at which the stated correlation first reaches its constant, and the constant is set by the sharp cone of the same angle in the same stream transitioning at 0.3 metres. “Sharp” is a nose of a ten-thousandth of a millimetre, whose sheath is swallowed within a millimetre.

What the model leaves out #

The nose region. The layer starts at the sphere’s tangency with the cone’s surface pressure; the overexpansion behind the nose and its recovery are left out, so the edge within a few nose radii of the tip is approximate. Results that put transition within ten nose radii are flagged and not read.

Wall temperature. The wall is adiabatic in the similarity form, the wall that heats itself to its recovery temperature; a cold wall, as in flight, changes both the momentum thickness and the stability of the layer, and its effect on the bluntness delay is not computed.

The correlations themselves. Each is a summary, not a mechanism. A transition calculation that tracked the growth of disturbances — the second-mode waves that dominate a sharp cone’s transition at Mach 6, and the different ones that grow in a sheath — would replace the correlation, and where the straight line stops explains why transition here is read from a correlation rather than derived.

A real gas. At Mach 15 the nose’s normal shock heats the sheath enough to excite vibration and, in flight, dissociation, which when gamma stops being a number treats; the perfect gas here overstates the sheath’s temperature at the highest Mach numbers.

Who measured the delay #

Stetson’s experiments on blunted cones near Mach 6 in the 1980s are the classic study of the bluntness effect and of transition reversal; Softley and others before and since documented the aft movement in tunnels and on flight vehicles, and the explanation through the entropy layer’s lowering of the edge Reynolds number goes back to Moeckel’s analysis of bluntness and transition in the 1950s and Rotta’s of the swallowing in the 1960s. Correlations of transition with $\mathrm{Re}_\theta/M_e$ are standard in re-entry design. The finding here that a delay read through such correlations peaks, with the peak set by the transition point’s position relative to the swallowing distance and its size by the correlation’s Mach exponent, is this calculation’s, for the model above.

How it was checked #

On a sharp cone the edge is uniform, so the momentum-thickness Reynolds number must grow as the square root of the distance and the edge Mach number must not change: the march gives the exponent to eight parts in a million and holds the Mach number to two parts in ten million. Halving the march’s step moves a 7-millimetre nose’s transition distance by less than one part in a million. The swallowing distances are the earlier essay’s, from the same march, which was checked there against the closed form for a uniform edge.

Still open: the cold wall the flight vehicle has #

Every layer here sits over an adiabatic wall, which is what a tunnel model at equilibrium approaches and what a re-entry vehicle does not. A flight vehicle’s wall is far colder than its edge gas — the case a cold wall holds its layer on longer took up for separation — and cooling a hypersonic layer thins it, raises its density near the wall and, at Mach 6, destabilises the second mode while stabilising the first. The next calculation would give the similarity layer a wall temperature ratio, carry the cold wall’s momentum thickness through the sheath, and ask whether the bluntness delay grows or shrinks on a cold wall — and whether the nose that gives the largest delay still sits about fifty of its radii ahead of the transition point when the wall is a fifth of the edge temperature.