Two totals, one of which a shock cannot touch

Energy instead of pressure establishes the compressible form of Bernoulli’s equation: what is conserved along a streamline is not a pressure but a total enthalpy, and the pressure version is the incompressible limit of it.

This essay is about the pair of stagnation quantities that follow, and about the fact that they are different kinds of object. One of them is a conserved quantity in the strict sense. The other is a running record of everything irreversible that has happened, and it only ever falls.

What each of them is #

The total temperature is the temperature the gas would reach if brought to rest adiabatically. It is the total enthalpy in temperature units, so it is an energy: it changes if and only if energy is added to or removed from the gas.

The total pressure is the pressure the gas would reach if brought to rest isentropically. That extra word is the whole difference. It is not an energy; it is a measure of how much of the gas’s energy is available to do work, and it falls whenever anything irreversible happens.

A shock is adiabatic and irreversible, so it is the sharpest possible test of the distinction.

The measurement #

Across a normal shock, computed from the static states on both sides rather than assumed:

The total temperature ratio is 1.000000000000000 at Mach 1.2, at Mach 8 and at every Mach number in between. That is not a small change; it is the number one, to the last bit of double precision, because the total enthalpy is exactly what the energy equation conserves and the shock adds no energy.

The total pressure ratio falls from 0.9928 to 0.0085 over the same range. At Mach 8 a shock has destroyed 99.15 per cent of the gas’s ability to do work while leaving its energy untouched.

The total pressure is the entropy, written differently #

The two are one quantity. For a perfect gas the entropy rise across any adiabatic process is minus the gas constant times the logarithm of the total pressure ratio, exactly — so a total-pressure measurement is an entropy measurement, and the loss of total pressure is not a consequence of the entropy rise but the same statement in engineering units.

Seven shocks make the pair’s asymmetry a table:

One column is 1.000000000 in every row and the other falls by a factor of 117. The entropy column is the total-pressure column read through a logarithm, and it rises by 660 while the total temperature does not move in the ninth decimal place.

That identification is worth having because it explains why the total pressure behaves as it does. It cannot rise in an adiabatic flow, for the same reason the entropy cannot fall. It falls in a shock, in a friction-dominated duct, in a mixing region and in a separation, all by the same mechanism. And it is unaffected by anything reversible: a nozzle, a bend with no separation, an isentropic compression.

Two memories of different kinds #

The pair is a clean example of the two species of memory this collection has been separating.

The total temperature is a conserved label. Across seven shocks from Mach 1.2 to Mach 8 it reads 1.000000000 every time. It is set when energy is added and it is then carried by the gas unchanged, exactly as a parcel’s entropy is in an inviscid flow or a material loop’s circulation is in the drift was the instrument. A measurement of it recovers the reservoir the gas came from and nothing about the route.

The total pressure is an accumulator. It records every irreversibility since the reservoir, integrated, with no way to distinguish one from another. A Mach 3 shock costs 67 per cent of it and a Mach 1.2 shock costs 0.72; three Mach 1.2 shocks in a row cost 2.1 per cent, which is why an intake spreads its compression over several weak shocks rather than one strong one. A measurement of it recovers the total loss and nothing about where it happened.

So the two together are a separation: one says what was added and the other says what was wasted, and neither can substitute for the other. That is why every gas-turbine measurement plane carries both, and why the pair is the basis of component efficiency.

Reading the pair down a duct #

In an adiabatic duct with friction — the flow of a duct that cannot be run backwards — the total temperature is flat and the total pressure falls: 451.0 J/kg·K of entropy over a march from Mach 0.3 to Mach 0.937, which is a total-pressure ratio of 0.207. So the pair says immediately that the duct is doing nothing to the energy and something to the availability, which is the definition of friction.

Add heat instead, in a frictionless duct, and the total temperature rises — it is recording the heat — while the total pressure still falls, because adding heat to a moving gas is itself irreversible. That second fact surprises people and it is the useful one: a combustor loses total pressure even with perfect mixing and no friction at all, and the loss is a thermodynamic consequence of heating a flow rather than a defect of the design.

With both measurements, the two effects separate. The rise in total temperature gives the heat; the fall in total pressure gives the total loss; and subtracting the loss that the heating alone would have caused gives the friction. Two numbers, two mechanisms, and an inversion that works.

What the solver computed, and how it was checked #

The jump conditions are this collection’s own, and both totals are computed from the static states on each side rather than from a stagnation relation — so the total temperature’s constancy is a result rather than an identity.

Three checks. That the total temperature ratio is one to a part in 10¹² at every Mach number, which is the essay’s central claim and would catch an error in the energy book-keeping. That the strongest shock keeps under a fifth of the total pressure, so the contrast is present. And that the weakest keeps over 98 per cent, which bounds the effect from the other side and confirms that the loss is a strong function of the shock’s strength rather than a constant.

What a component efficiency actually measures #

The whole apparatus of turbomachinery performance rests on this pair, and it is worth unpacking one definition to show how.

A compressor’s isentropic efficiency is the total-temperature rise an ideal isentropic compression would have needed to reach the measured total-pressure ratio, divided by the total-temperature rise actually observed. Both quantities in it are totals, and the definition works precisely because the two carry different information.

The numerator is what the pressure ratio was worth. It is computed from the total-pressure ratio through the isentropic relations, so it is a statement about the useful output.

The denominator is what it cost. The total-temperature rise is the work put in, exactly, by the energy equation.

So the efficiency is output over input, and the reason it is less than one is that some of the work went into entropy rather than into pressure. A machine with an efficiency of 0.9 has put a tenth of its work into raising the gas’s temperature without raising its ability to do anything with it.

That is the same separation the essay is about, applied to a machine rather than to a shock, and it is why both instruments are on every measurement plane in every test facility.

Why the loss is so sudden #

The total pressure ratio at Mach 1.2 is 0.9928 and at Mach 2 it is 0.72, which is a startlingly steep dependence and is worth explaining because it decides how supersonic intakes are designed.

Near Mach one a shock is weak, and a weak shock’s entropy rise goes as the cube of the strength — which is the classical result that a weak shock is very nearly isentropic. So a Mach 1.2 shock costs almost nothing.

By Mach 2 the shock is no longer weak, the cubic behaviour has been left behind, and the loss is growing quickly. By Mach 3 the ratio is a third and by Mach 5 a sixteenth.

The engineering consequence is the reason supersonic intakes have several shocks rather than one. Two weaker shocks in series cost far less than one strong shock producing the same overall deceleration, because the cost is convex — and an intake with a well-designed series of oblique shocks followed by a weak normal one recovers most of what a single normal shock would have thrown away.

What each measurement is worth #

The practical statement is about instrumentation, and the two quantities are unequal in an important way.

Total temperature is easy and robust. A thermocouple in a stagnation probe reads it to within a recovery correction, and it is unaffected by whatever the flow has been through — which is exactly what makes it useful as a reference.

Total pressure is easy and fragile. A pitot probe reads it directly, and it is affected by everything: a probe in a boundary layer, a wake or a separation reads a total pressure that belongs to that region rather than to the stream. That sensitivity is what makes a total-pressure survey a diagnostic — a map of total pressure is a map of where the losses are — and it is also why a single reading is nearly meaningless without knowing where it was taken.

And the difference between them is the dynamic head, so measuring both gives a Mach number. That is how nearly every flight instrument works, and it depends on the total temperature being uncorrupted — which is the property this essay is about.

The surface that carries the entropy this shock made is two essays back, on the same machinery.

Why a shock adds no energy #

The constancy of the total temperature across a shock is the essay’s headline and it deserves a sentence of derivation rather than an assertion, because it looks like something that ought to be approximate.

Draw a control volume straddling the shock. Mass in equals mass out. The energy equation says that the total enthalpy flux in equals the total enthalpy flux out provided no heat is added and no work is done, and across a shock neither is: there is no moving boundary and no heat source inside the control volume.

Since the mass flux is the same, the total enthalpy per unit mass is the same, and for a perfect gas that is the total temperature. The result is therefore a consequence of the energy equation alone and carries no assumption about what happens inside the shock — which is exactly the property that makes the jump conditions work at all, and is the jump does not ask what made it’s subject.

A shock rearranges energy between forms and does not change the total, and the entropy it produces is a rearrangement rather than a loss of anything conserved.

Where the total temperature does change #

The constancy has conditions and it is worth listing them, because each is a real case.

Work. A compressor or turbine changes the total temperature, and that is what it is for. The change is the work per unit mass, exactly.

Heat transfer. A cooled turbine blade or a heated duct changes it too.

And a moving frame. In a rotating machine the total temperature in the rotating frame — the rothalpy — is the conserved quantity rather than the absolute one, which is why turbomachinery is analysed in two frames and why the transformation between them is where the work appears.

Outside those, the total temperature is a label attached to the gas at its reservoir and carried thereafter. A measurement of it anywhere in an adiabatic system identifies the reservoir, which is a genuinely useful thing in a machine with several streams.

The pair as an inverse problem #

Reading the two together is an inversion, and it is worth setting it out as one because it is unusually well behaved.

Two measurements, two unknowns. The total temperature at a station, relative to the reservoir, gives the energy added; the total pressure, relative to the reservoir, gives the entropy produced. Between them they determine the thermodynamic state’s history in the two respects that matter.

And the inversion is exact and stable. Both relations are algebraic and monotone, and neither amplifies an error: a one per cent error in a total pressure is a one per cent error in the recovered loss, and no more. Compare the exponential degradation of a scalar is a record of where its fluid was or the algebraic one of a wake that keeps the drag and forgets the body.

What it cannot do is localise. The total pressure records the sum of the losses and says nothing about where they happened, so a single plane gives a total and a pair of planes gives a difference. That is why test facilities instrument many planes and why component-by-component accounting is the whole discipline.

The general shape is familiar from this collection: a conserved quantity is a perfect record of one thing and no record of anything else, and the way to get more is to add stations rather than precision.

What the picture cannot show #

The total temperature is drawn as a horizontal line at one, and a line at one is not a figure. It is drawn because the point of the essay is that it stays there, and a reader looking for structure in it will find none — which is correct and is difficult to make visually interesting.

Nothing here draws a probe. Both quantities are defined by a hypothetical process — bringing the gas to rest, adiabatically or isentropically — and the relation between that hypothesis and what an instrument does involves a recovery factor that the wall that heats itself computes.

Why the asymmetry is not an accident #

The currency the second total is spent in is priced in what a shock costs, which is the same loss read as a number a designer pays rather than as a conservation statement.

It is worth saying why one total survives and the other does not, because stated as a pair of facts it looks arbitrary and it is not.

Total temperature is an energy statement, and the shock does no work on the flow and adds no heat to it. Energy is conserved across any adiabatic process whatever happens inside it, so the total enthalpy is the same on both sides for the same reason a control volume’s energy balance always closes.

Total pressure is not a conserved quantity at all. It is a statement about how much of the flow’s energy is available as ordered motion, and the shock’s interior is precisely where ordered motion is turned into disordered motion by viscosity and heat conduction over a few mean free paths.

So the pair is one conserved quantity and one availability, and only the first is protected. The same asymmetry appears at every irreversible feature in a compressible system — a friction duct, a mixing layer, a heat exchanger — and in each the temperature bookkeeping is easy and the pressure bookkeeping is where the design margin goes.

Who found it, and when #

The stagnation quantities are as old as compressible flow and their distinction is Stodola’s and Prandtl’s. The identification of total-pressure loss with entropy production is a consequence of the Gibbs relation and was standard by the 1930s.

What is more recent is the systematic use of the pair as a diagnostic. Component efficiency defined through total pressures and total temperatures is the language of gas-turbine engineering, and it exists because those two measurements separate the two things a designer can do wrong.

Limits recorded rather than smoothed over #

A perfect gas with constant specific heats. The identification of the total pressure ratio with the entropy uses it, and at high temperature it fails — when gamma stops being a number is the correction.

Adiabatic, throughout. Every statement about the total temperature’s constancy assumes no heat transfer. In a real duct there is some.

The heat-addition figure is schematic. The Rayleigh flow’s total-pressure loss is drawn from a plausible dependence rather than computed, and the essay says so; what is computed is the shock case and the friction case.

The shocks here are normal ones. An oblique shock’s totals behave identically — the total temperature is unchanged and the total pressure falls with the normal component of the Mach number — which is exactly why an intake uses them, and is a shock that leans’s subject.

And a total is a hypothetical. Neither quantity is a property of the gas as it is; both are properties of a process that might be applied to it. That is why they behave so differently from static quantities and why both need the process specified before a number means anything.