The gas beside a hypersonic body remembers its nose

A hypersonic body leaves a line explosion behind it read a blunt body in hypersonic flight as a line of charges set off one after another along its path. Each slice of air it passes through is struck once, with the energy the body’s drag puts into it — a radius that gives the energy away is the spherical version — and left to expand as Sedov’s cylindrical blast wave. That analogy gives the bow shock’s width and the pressure on the afterbody with no Mach number in either, and on both it agrees well enough with measured shocks to be useful. It also puts the body in the blast’s core, and the fireball is hollow: at the centre of a blast wave the density falls to nothing and the temperature rises without bound. The analogy’s own essay said so, and ended on the calculation that would replace the core with something real. This essay does it.

The real gas beside the body is not the inside of an explosion. It is air that flowed past the nose, and every parcel of it carries one thing from that encounter that nothing afterwards can take away: the entropy it gained crossing the bow shock. Downstream the parcel’s pressure changes, its speed changes, its temperature changes, but in an inviscid flow its entropy does not, and entropy and pressure together fix its temperature. So the question the analogy cannot answer — how hot is the gas beside the body — reduces to two others it can: what entropy each streamline took from the shock, and what pressure it has reached.

Following a streamline from the shock #

The body is a hemisphere-nosed cylinder of diameter $d$. Its bow shock is Billig’s correlation of measured shocks on spheres, a hyperbola anchored at a measured standoff and a measured radius of curvature at the nose and running out to the free stream’s Mach cone. The shock’s angle to the stream, $\beta$, follows from the hyperbola’s equation in closed form: it is 90° on the axis, where the shock is normal, and it leans over to the Mach angle far out, where the shock has weakened to a Mach wave.

A streamline starting a distance $y$ from the axis upstream travels straight until it meets the shock at that radius, and crosses it as a normal shock of Mach number $M\sin\beta$ would be crossed — the jump the conservation laws allow, applied to the normal component alone; its entropy jumps by the normal-shock amount for that Mach number. Downstream, beside the cylinder, the pressure is taken to be uniform across the layer and equal to the afterbody pressure, and each streamline’s temperature is then its own isentrope’s at that pressure,

$$\frac{T}{T_\infty} = \left(\frac{p}{p_\infty}\right)^{(\gamma-1)/\gamma} e^{\Delta s/c_p}.$$

Its speed follows from its total enthalpy, which the shock does not change, and its distance from the body from the mass it carries: the streamlines between the axis and $y$ carry $\pi y^2 \rho_\infty U$, and that mass has to fit in an annulus of the local density and speed. The gas is a perfect gas with $\gamma = 1.4$.

The afterbody pressure is the analogy’s, with one repair. Its central pressure is a strong blast’s, which knows nothing of the air it expands into, and at Mach 15 the analogy’s shock stops being strong about four diameters behind the nose, beyond which it predicts pressures below ambient. The pressure used here is the ambient pressure plus the analogy’s overpressure: the analogy near the nose, relaxing to the free stream far behind it.

The temperature across the layer #

The first figure is the answer at Mach 15. Two diameters behind the nose, the gas next to the body is at 16.7 times the free stream’s temperature — about 3,700 K for air at 220 K, in a perfect gas — and the temperature falls outwards across the layer to a few times the free stream’s in the gas that crossed the oblique part of the shock. Twenty diameters back the pressure has relaxed and the whole profile has cooled, to 10.7 next to the body; two hundred back, to 9.3. The profile keeps its shape. What sets it is not the pressure, which is nearly uniform across the layer, but the entropy, which is not.

The analogy’s temperature at two diameters, dashed, leaves the top of the frame within a fraction of a diameter of the body. At the body’s radius it is 1,400 times the free stream’s, and closer to the axis it climbs without limit. The analogy has the right pressure and a density that goes to zero at its centre, and a temperature is a pressure divided by a density.

Why the analogy’s core heats up and the real layer cools down #

The second figure follows the temperature next to the body down its whole length, and the two models move in opposite directions. In the analogy the blast keeps growing, as the square root of the distance behind the nose, while the body stays the same size, so the body sits ever deeper in the blast’s core, where the temperature is ever higher: past ten million times the free stream’s a thousand diameters back. That is not a prediction; it is the analogy telling where it stops.

The entropy layer does the opposite. The gas next to the body keeps the entropy of the nose’s normal shock, and as the afterbody pressure falls towards ambient that gas expands and cools along its isentrope. At Mach 15 it goes from 16.7 at two diameters to 10.7 at twenty and 9.5 at a hundred, levelling off at a floor it cannot go below.

The floor is the temperature the nose’s gas reaches when its pressure is back to the free stream’s: $T_\infty e^{\Delta s_0/c_p}$, with $\Delta s_0$ the normal shock’s entropy jump. It is the one number in the problem that owes nothing to the afterbody at all. At Mach 15 it is 9.1; at Mach 10, 5.2; at Mach 20, 13.6. A parcel that crossed the normal shock at the nose is, for as long as nothing mixes it, a parcel at least nine times hotter than the air it is travelling through.

Where the heat comes from: the shock’s curvature #

The third figure is the source of the layer. A streamline on the axis crosses the normal part of the shock and takes the largest entropy, which at Mach 15 multiplies its temperature at any pressure by 9.1. A streamline starting half a diameter out — level with the body’s edge — crosses the shock where it has already leaned to about 50° and takes a factor of 6.3; one diameter out, 3.7; one and a half, 2.5. Far out the shock is a Mach wave and the factor is one.

So the entropy layer is made by the shock’s curvature, and it is made from a larger stream than the body blocks.

That the source tube is wider than the body is not a paradox. The bow shock stands well ahead of the nose and is still steep where it passes the body’s radius — about 50° half a diameter out — so the air that will flow past the body’s shoulders and even outside them has already been through a strong shock by the time it gets there. The streamlines that start between half a diameter and 1.2 diameters from the axis never touch the body; they pass beside it, a little way off, and it is they that give the layer its width. The hottest gas, from the axis itself, flows over the nose and hugs the body’s surface, and the layer is arranged in the order the streamlines crossed the shock: the most-shocked gas innermost. The gas whose entropy is at least half the axis’s comes from within 1.2 diameters of the axis at Mach 15, a tube of air nearly six times the body’s frontal area. On a sharp-nosed body the shock is oblique from the start, the entropy is small and nearly uniform, and there is no layer; blunting the nose, which designers do to keep its heating bearable, is exactly what creates one.

The hot layer thickens as it cools #

The fourth figure is the layer’s thickness, measured out to the streamline that carries half the axis’s entropy. The mass inside it is fixed — it is the air that started within 1.2 diameters of the axis — but that air is hot and therefore thin, and as the afterbody pressure falls it expands. At Mach 15 the layer is 0.9 of a diameter thick two diameters back and 2.2 a thousand back: wider than the body it surrounds. At Mach 20 it reaches three diameters.

The thickening is the reason the entropy layer matters for heating long after the nose. The body’s own boundary layer grows along the afterbody, feeding on the gas at its edge. While it is thinner than the entropy layer, that gas is the hot, low-density, low-velocity gas that came through the normal shock, not the cooler, denser gas further out, and the boundary layer’s heat transfer and its transition to turbulence are both set by those edge conditions. The distance at which the boundary layer has grown through the entropy layer and “swallowed” it is a standard quantity in hypersonic design, and it is long precisely because the entropy layer thickens as it goes.

A sharp nose makes no layer, and cannot be flown #

The layer exists because the shock is curved, and the comparison that shows it most plainly is a body whose shock is not. A sharp cone of 10° half-angle at Mach 15 carries a shock that lies close along its surface at about 13° to the stream, and every streamline that reaches the cone crosses it at the same angle. Each takes the same entropy — a factor of about 1.5 on its temperature at any pressure — and the gas beside the cone is simply warm, uniformly, with no hot sheath and no floor worth the name. Against the blunt nose’s factor of 9.1 on the axis, the difference is the whole of what this essay is about.

The sharp cone is not flown, for a reason that is the other face of the same arithmetic. A nose of radius $r$ in hypersonic flow receives a stagnation-point heat flux that grows as $1/\sqrt{r}$: the boundary layer at a sharp tip is infinitely thin, and it carries heat into the wall without limit. Every body that has to survive re-entry is therefore blunt, on purpose, and its bow shock stands off the nose and curves — a wedge that is too blunt for its shock to stay attached is the two-dimensional version of the same choice. The price of a bearable nose is a shock that makes an entropy layer, and the layer carries the nose’s choice down the whole body.

What the nose charges, and when it is paid #

The floor is a record of what the shock cost. A normal shock turns part of the flow’s kinetic energy into internal energy irreversibly, and the entropy it creates is the measure of the part that cannot be turned back into speed by any expansion. Along the afterbody the gas beside the body re-expands and gives back what it can, which is why its temperature falls from the nose’s; what it keeps is the floor. The energy that went into the floor is energy the body spent through its drag, and the blast analogy was right to see the drag as energy deposited in the air — its error was in where it put it. The energy is not at the centre of a blast; it is in a sheath of hot gas wrapped round the body and trailing behind it, and the sheath’s heat is the part of the drag the air has not yet been able to spread out.

The floor, against Mach number #

The fifth figure plots the floor against Mach number. At high speed it grows as $M^{2/\gamma}$ — the normal shock’s pressure rises as $M^2$ and its density ratio levels off, so the entropy grows as the logarithm of $M^2$, and its exponential as a power. The total temperature, the temperature the gas reaches only where it is brought to rest, grows as $M^2$ and is far higher: 46 times the free stream’s at Mach 15 against the floor’s 9.1. The difference is the part of the nose’s heating that the gas gives back as it re-expands, and the floor is the part it cannot give back, because it is entropy.

What was checked #

The sixth figure is the ledger. The axis streamline, expanded back to the pitot pressure instead of the afterbody’s, has exactly the total temperature, at three Mach numbers, to $10^{-16}$. The entropy jump reproduces Rayleigh’s pitot pressure formula through $p_{02}/p_{01} = e^{-\Delta s/R}$. A streamline ten thousand diameters out crosses a shock weakened to a Mach wave and takes no entropy. The closed-form shock angle matches the slope of Billig’s shock radius, differentiated numerically, to $3\cdot10^{-10}$ radians. And $10^5$ diameters downstream the gas next to the body sits on its floor to four parts in a hundred thousand.

What the streamline picture leaves out #

Real air. At Mach 15 the gas behind the normal shock is hot enough to dissociate, and $\gamma$ stops being a number: the real stagnation temperature is far below the perfect gas’s 10,000 K, and the entropy layer’s temperatures with it. Every number here is a perfect gas’s, and the pattern — a floor set by the nose, a layer thicker than the body — is what survives into real air.

Uniform pressure across the layer. Near the nose the pressure varies across the shock layer, and the outer streamlines, which crossed the oblique shock, sit at higher pressure than the afterbody’s. Their temperatures are the least reliable part of the first figure; the gas next to the body, at the afterbody pressure by construction, is the most.

The afterbody pressure. It is the analogy’s plus ambient, a patch rather than a solution, and it carries the analogy’s error near the nose and a guess where the two meet.

Viscosity and mixing. The layer is followed as inviscid. The boundary layer eats into it from the wall and the shear across it mixes it slowly outwards; the floor is where it would settle if neither happened.

The convention the numbers depend on #

Temperatures are fractions of the free stream’s; distances are in body diameters, behind the nose along the body and outwards from the axis. The body is a hemisphere-nosed cylinder. The entropy layer’s edge is the streamline carrying half the axis streamline’s entropy jump. The gas is perfect, with $\gamma = 1.4$; the bow shock is Billig’s measured correlation for a sphere, not a computed one.

Who found it, and when #

The entropy layer was identified in the 1950s, as the first hypersonic blunt bodies were studied: the gas that had crossed the strong part of a curved shock was seen to form a layer of high entropy along the body, and Ferri and others pointed out its effect on the heating downstream. The blast-wave analogy it corrects is Lin’s and Cheng’s work of the same decade, and Billig’s shock correlation is from 1967. The swallowing of the entropy layer by the boundary layer has been estimated since the 1960s, most simply by equating the mass the boundary layer has entrained to the mass the entropy layer carries.

Still open: where the boundary layer swallows it #

The mass argument that ends the section on thickness has a definite form. The entropy layer carries the air that started within about a diameter of the axis; a laminar boundary layer along the afterbody entrains mass at a rate set by the density, speed and viscosity at its edge, which are the entropy layer’s own. The next calculation grows the boundary layer along the body with those edge conditions, updated as it eats inwards through the layer, and asks at what distance the two masses are equal — how that swallowing distance scales with the Mach number and the Reynolds number, and whether a body of practical length ever sees the cooler gas outside the layer at all.