Four Bernoullis and one name

“Bernoulli’s equation” names at least four different statements. They have four different constants, three different domains of validity and one shared reputation for being misapplied, and the question “does Bernoulli apply here?” is ill-posed until it is said which one is meant.

This collection has an essay on where the equation applies. This one is about the fact that there is more than one equation.

One flow separates the first three #

Take a parallel shear flow: $u = U(1 + Sy)$, $v = 0$. It is an exact steady solution of the Euler equations, and the momentum equation requires its static pressure to be uniform — every term in it vanishes identically, so there is nothing to balance a pressure gradient and there is none.

So the total pressure is entirely the dynamic pressure, and it varies with the speed. Across this layer it varies by 3.2 dynamic heads.

Statement one — $p + \rho q^2/2$ is constant along a streamline — holds. Each streamline is a straight line along which nothing changes, so the constant on it is a constant.

Statement two — the same quantity is constant everywhere — is false here, by a factor of eighty-one between the fastest streamline and the slowest.

And the difference between them is not a subtlety. The static pressure is the same everywhere, so a reader applying “fast means low pressure” across this flow would predict a pressure difference that is exactly zero.

It is worth dwelling on that, because the shear layer is the cleanest counter-example available and it is not a contrived one. A parallel shear flow is what exists downstream of a splitter plate, across a mixing layer, in the outer part of a jet, and in any velocity profile at all before the viscous terms are considered. It is the ordinary state of affairs rather than an exception.

And the ordinary state of affairs has a uniform static pressure. That is worth restating because it contradicts the sentence most people carry: a flow can have a fast side and a slow side at exactly the same pressure. Nothing has been violated. The two sides are on different streamlines, they carry different constants, and the constants differ by exactly the amount their speeds do.

The reason a reader finds this surprising is that the everyday demonstrations of the equation are all irrotational. A Venturi, a wing, a nozzle, a tube with a constriction: in each the flow is irrotational to a good approximation, so one constant does hold everywhere, and comparing across streamlines works. The exceptions are the flows with vorticity in them, which is most real flows once a boundary layer has appeared.

The statement that relates them #

Statement three is the general one, and it says precisely how much the constant varies from streamline to streamline:

$$\nabla H = \rho\,\mathbf{u}\times\boldsymbol{\omega},$$

so the total pressure is constant along a streamline always, constant everywhere if and only if the vorticity vanishes, and otherwise varies at a rate the vorticity fixes. Checked at forty-one heights across the layer, the two sides agree to ten figures.

That is the useful form, because it turns “does Bernoulli apply across streamlines?” into a question with a computable answer: how much vorticity is there?

In the flow past a cylinder — irrotational everywhere outside the body — the total pressure round three circles of different radii is the same number to 10⁻¹⁶. That is the case most textbook uses are quietly relying on, and it is a stronger hypothesis than the one usually stated.

The two corrections #

Statement four is the unsteady one, and it is the one whose size surprises.

Take a contraction with the flow rate rising. At any instant the velocity field looks exactly like a steady one — same shape, same magnitudes, nothing to distinguish it in a snapshot — and the pressure difference along it is not the steady one.

The unsteady term is a hundred and eighty-five per cent of the steady one here. It is not a correction; it is the larger of the two, and nothing in a picture of the velocity field says so.

Statement five is the compressible one, and this collection has an essay on the energy form that replaces the pressure form.

The correction to the dynamic pressure is $1 + M^2/4$ to leading order: 2.25 per cent at Mach 0.3, 6.4 at 0.5, and 17.0 at 0.8. An airspeed indicator that ignores it reads low, and the correction is a property of the air rather than of the instrument.

Where each one is the right one #

It is worth being constructive about this rather than only cautionary, because each statement has cases where it is exactly the tool required.

Along a streamline is the form for any steady inviscid flow with vorticity in it — a rotational core, a vortex, a swirling flow in a pipe, the flow behind a curved shock. It is weaker than the global form and it is available where the global form is not, and it is what the spin a shock leaves behind has to be analysed with.

The global form is the workhorse of external aerodynamics, because a body in a uniform stream generates no vorticity outside its boundary layer and every streamline arrives with the same constant. That single fact is what makes a pressure distribution computable from a velocity field, and it is why potential flow is worth as much as it is.

Crocco’s relation is the tool for the cases in between and for compressible flow with entropy gradients, where it acquires a temperature-times-entropy-gradient term and becomes the statement that relates vorticity to entropy production — which is how the vorticity behind a curved shock is computed.

The unsteady form is what any starting, stopping or oscillating problem requires, from the pressure that depends on the past to a valve closure to an oscillating aerofoil.

And the compressible form is what anything above a Mach number of 0.3 requires, and increasingly badly above 0.5.

So the four are not competitors and none of them is a simplification of another to be preferred where possible. They are five different statements about five different situations, and the skill is identifying the situation.

Why the four get confused #

There is a reason this particular equation attracts misapplication, and it is worth naming rather than deploring.

The four statements have the same symbols. $p + \rho q^2/2$ appears in all of them, and what changes is what the equality is with and where it holds. So a reader who has learned the expression has learned something that is nearly right in every case and exactly right in one, and the cases are distinguished by hypotheses that are not visible in the formula.

The hypotheses are also of different kinds. “Along a streamline” is a statement about where the equation is applied. “Irrotational” is a statement about the flow. “Steady” is a statement about time, and “incompressible” about the fluid’s response. Nothing groups them, and a reader checking one has no prompt to check the others.

What a Pitot tube is measuring #

The distinction has a direct instrumental form, and it is the one that turns the abstraction into something a reader can hold.

A Pitot tube reads the total pressure at the point it is put — the pressure the flow reaches when brought to rest on the streamline passing through the probe. In an irrotational flow that is one number everywhere, so where the probe is put does not matter and a single reading gives the free stream. In a rotational flow it is a different number on every streamline.

So a traverse across the shear layer above gives a reading that changes by a factor of three, with no losses anywhere, no viscosity acting, and a static pressure that never moves. Nothing has been dissipated; the streamlines simply arrived carrying different amounts.

That is exactly what a Pitot rake across a wake is measuring, and it is why a wake survey works: the deficit in total pressure marks the streamlines that have passed through the body’s boundary layer and lost some, and the integral of the deficit is the drag. The instrument is reading statement one on each streamline separately, and the variation between them is the measurement.

Which also settles a question that troubles people about the shear layer. If the total pressure varies across it and nothing has been dissipated, where did the variation come from? It came from wherever the layer was made — a splitter plate, a nozzle lip, a history the flow has had — and Crocco’s relation says only that it is conserved along each streamline thereafter, not that it was ever equal between them.

What to check, and in what order #

Four questions, each of which takes a moment.

Is it steady? If not, the unsteady term is present and may dominate. It is present whenever anything is starting, stopping, oscillating or being pulsed, and it does not appear in a snapshot.

Is the comparison along a streamline or across streamlines? Along, the equation holds for any steady inviscid flow. Across, it needs the next question.

Is the flow irrotational between the two points? Not “is it irrotational somewhere” — the path between them matters. A streamline that has passed through a boundary layer, a wake, a shock or a region of shear carries a different constant from one that has not, which is the whole content of a Pitot rake’s readings across a wake.

And is the Mach number small? Below 0.3 the correction is under two and a quarter per cent; above it, the compressible form is needed.

The first three are the ones that go wrong, and they go wrong in that order of frequency: unsteady flows treated as steady, comparisons made across streamlines without checking the vorticity, and paths that have crossed a viscous region treated as if they had not.

The one that is not on the list #

There is a sixth statement that goes under the same name and deserves separating out, because it is the one most often reached for and the one with the fewest hypotheses attached to it in practice.

“The pressure difference across a body is $\tfrac{1}{2}\rho(U_2^2 - U_1^2)$.” That is statement two applied between two points on a body’s surface, and it requires the flow to be steady, inviscid, incompressible and irrotational, with both points connected to the same upstream conditions by streamlines that have stayed outside the boundary layer.

On the upper surface of an attached aerofoil ahead of separation, all of that holds, and the calculation is correct — which is why the lowest pressure is on the body and why a computed pressure distribution matches a measured one over most of a section. Aft of separation none of it holds, the pressure there is set by the wake rather than by the local speed, and the same arithmetic gives a badly wrong answer.

So the equation’s most common use is legitimate over most of a body and illegitimate over the rest of it, with no warning at the boundary between them. That is the honest summary of its reputation: it is not that the equation is unreliable, it is that its hypotheses are satisfied in a region whose extent has to be determined by something other than the equation.

The habit that avoids all of it #

There is a single habit that disposes of most of the difficulty, and it is worth stating because it is not the one usually taught.

Ask what is conserved along what. The equation is not a relation between a pressure and a speed; it is the statement that a particular quantity is carried unchanged by a particular thing. Statement one says the total pressure is carried along a streamline. Statement two says it is the same on all of them. Statement three says how much it differs when it is not.

Framed that way, every question about applicability becomes a question about a path: what has this fluid been through since it carried a known value? A streamline from the free stream that has stayed outside every viscous region carries the free-stream constant. One that has crossed a boundary layer, a wake, a shear layer or a shock does not, and the amount it has lost is what a survey measures.

The framing also generalises, which the pressure-and-speed framing does not. In a compressible flow the carried quantity is the stagnation enthalpy; in an unsteady one there is a term for the field changing under the particle; in a rotational one the quantity is still carried, just with different values on neighbouring paths.

That is the same discipline this collection applies to the constant a hole leaves behind and to what survives being wound up: identify the quantity, identify what carries it, and the rest is bookkeeping.

Why the unsteady term is the one that hides #

Of the four hypotheses, three announce themselves and one does not, and it is worth saying which.

Compressibility announces itself through a Mach number, which anybody computing a flow has. Rotation announces itself through a vorticity, which is visible in a velocity field and which a reader can usually reason about from where the flow has been. The distinction between along and across a streamline announces itself in the geometry of the question being asked.

Unsteadiness does not announce itself in a snapshot at all. The duct above is the demonstration: at the instant it is examined, its velocity field is identical to that of a steady flow through the same duct at the same flow rate. Every quantity a reader could compute from the picture — the speeds, the gradients, the streamlines, the strain rate — is the same. What differs is a time derivative, and a time derivative is not in a picture.

That is why the unsteady term is the one most often left out. It is not that people forget the hypothesis; it is that nothing in the data they are looking at reminds them of it. The correct prompt is not “does the field look unsteady” but “is anything about the boundary conditions changing”, which is a question about the problem rather than about the flow.

The size makes it worth asking. A hundred and eighty-five per cent is not a correction to be estimated and neglected, and it appears in every start-up transient, every valve closure, every oscillating surface and every pulsatile flow — which includes an artery, a reciprocating pump, and anything with a rotor passing a stator.

Where the collection uses each of them #

It is worth ending with a short audit, because this collection applies all five statements and it should be possible to say which is which.

The global form is used wherever a potential flow’s pressure is computed from its velocity — the exact theory, the flow past a cylinder, the surface pressure on an aerofoil. Every one of those is irrotational outside the body, which is what licenses it.

The streamline form is used in the compressible essays, where a flow behind a curved shock has vorticity in it and the total pressure varies from streamline to streamline — the spin a shock leaves behind is that variation, and Crocco’s relation is how it is computed.

The unsteady form appears wherever something starts or oscillates: the force of getting going is entirely an unsteady term, and so is the added mass it produces.

The compressible form is the whole of the compressible field’s machinery.

And the fact that the collection has to keep saying which is being used is the argument of this page. There is no default. Choosing one is choosing a set of hypotheses, and a figure that quotes a pressure without saying which set was assumed has left out the part that decides whether the number is right.

Why the equation is worth all this care #

It would be reasonable to conclude from all of this that the equation is more trouble than it is worth, and the opposite is true, which is why the care is warranted.

Bernoulli’s relation is the most useful single statement in this subject. It converts a velocity field into a pressure field with no further computation, which is what makes potential flow a predictive theory rather than a kinematic one; it is the reason a pressure measurement can be read as a speed; and it is the bridge between the two quantities every instrument in the subject measures.

The four versions are not four complications of one idea. They are the same conservation statement under four sets of circumstances, and each has been worked out because the circumstance is common enough to matter — rotational flows, unsteady flows and compressible flows are the ordinary state of affairs rather than exceptions.

So the care asked for here is not scepticism about the equation. It is the ordinary discipline of checking a theorem’s hypotheses before using it, applied to the one theorem in fluid mechanics that is used more often than any other and is therefore misapplied more often than any other. A statement that gets used ten thousand times a day will accumulate misuse in proportion, whatever its quality.

What is not claimed #

Nothing here is a new result. All five statements are in every text on the subject, and Crocco’s relation is a century old. What is computed is their sizes on flows where they differ, so that “Bernoulli does not apply here” can be replaced by a number.

The shear layer is inviscid. A real shear layer diffuses and is unstable, so the exact solution used here is a snapshot of an initial condition rather than a flow that persists. Nothing in the argument needs it to persist: it is an exact steady solution of the Euler equations at the instant it is examined, which is all the statements require.

The unsteady example is one-dimensional. A duct with a prescribed area and a prescribed flow rate is the simplest flow with an unsteady term in it, and the hundred and eighty-five per cent belongs to the acceleration chosen. What is general is that the term exists and is not small.

And the compressible correction assumes isentropic flow. Across a shock it is not isentropic, the stagnation pressure falls, and a Pitot tube in supersonic flow reads the pressure behind its own bow shock — which is a fifth complication and a well-known one.