The pressure that depends on the past

Bernoulli’s equation for an unsteady potential flow is

$$\frac{p}{\rho} + \frac{\partial\phi}{\partial t} + \tfrac12 q^2 = F(t),$$

and two things about it are usually skipped.

The first is $\partial\phi/\partial t$, which is exactly zero once a flow has settled and is the whole of the flow while it is settling. So it never appears in a steady answer, and it is never negligible in getting to one.

The second is the right-hand side. It is a function of time, not a constant: the same everywhere in the fluid at each instant, and different from instant to instant. It is allowed to be a function because the potential is only defined up to one, and it is the accounting entry that makes the whole equation consistent.

Two ways the past gets in #

The term carries history into the pressure, and it does so in two distinct ways which are worth separating because they look nothing alike.

As a pressure impulse. Start a body impulsively and integrate the equation across the instant. Everything bounded integrates to nothing across an interval of zero length; what survives is

$$\int p\,dt = -\rho\phi,$$

so the pressure impulse is the velocity potential, times the density, with a sign. That is a striking statement about a quantity normally introduced as a mathematical convenience with an arbitrary constant in it: hit the fluid, and the bruise is shaped like $\phi$.

As a finite time to start. In a flow that is being established rather than struck, the term is what takes up the difference between the driving pressure and the kinetic energy the flow does not yet have, and it sets the timescale of the transient.

The bruise #

For a cylinder started instantaneously to speed $U$, the potential on its surface is $-Ua\cos\theta$, so the pressure impulse is $\rho U a\cos\theta$ — largest at the front and the back, zero at the shoulders, of opposite sign fore and aft.

Integrating that over the surface gives $\rho\pi a^2 U$, which is the added mass times the speed, which is the hydrodynamic impulse computed by a completely different route in a completely different essay. Two integrals of two different quantities, agreeing to nine figures.

The practical version of this is slam. A hull dropping onto a wave surface, a wave hitting a sea wall, a water-hammer front arriving at a bend: in each case the pressure is enormous and the duration is short, and what the structure cares about is the integral rather than the peak. The integral is a potential flow quantity and can be computed exactly, while the peak depends on compressibility, on entrained air, and on the local shape, and cannot.

The tube that takes two seconds #

Take a tube of length $L$ and area $A$ leading from the bottom of a reservoir of head $h$, full of water, with a valve at the end. Open it.

Unsteady Bernoulli from the free surface to the exit, with the flow uniform along the tube, gives

$$L\frac{dU}{dt} + \tfrac12 U^2 = gh ,$$

whose solution from rest is

$$U = U_\infty\tanh\frac{U_\infty t}{2L},\qquad U_\infty = \sqrt{2gh}.$$

The time constant is $2L/\sqrt{2gh}$, set by the length of the tube — not its diameter, not the viscosity, not the roughness, none of which appears anywhere. For $L = 2$ m and $h = 1$ m that is 0.90 seconds, and reaching within one per cent of Torricelli’s 4.43 m/s takes 2.4 seconds.

The two terms in the equation are equal at 0.80 seconds. Before that the flow is dominated by the term that is usually deleted; after it, by the term that is usually kept.

Reading the time constant #

The expression $2L/\sqrt{2gh}$ deserves reading rather than glancing at, because everything about it is a little surprising.

The area does not appear. A wide tube and a narrow one under the same head take the same time to start, because the driving pressure and the inertia both scale with the area and it cancels. What matters is the length of the fluid column being accelerated.

A longer tube starts more slowly and ends at the same speed. The final velocity is Torricelli’s and knows nothing about $L$; the approach to it takes proportionally longer. So a long pipeline is not slower to deliver — it is slower to become fast.

And the higher the head, the shorter the transient. Doubling $h$ multiplies the final speed by $\sqrt2$ and divides the time constant by the same factor, so the acceleration goes up by two. Every part of that is the algebra of the equation and none of it is obvious in advance.

The shape of the solution is worth a note too. A hyperbolic tangent is what an exponential approach becomes when the resisting term is quadratic rather than linear, and that is the signature of a transient limited by convective inertia rather than by friction. A friction-limited pipe fills exponentially; an inertia-limited one fills as a tanh, and the difference is measurable.

What a quasi-steady calculation gets wrong #

It is worth being precise about the failure, because “quasi-steady” is often taken to mean “accurate except during a brief transient”.

Dropping $\partial\phi/\partial t$ leaves $\tfrac12 U^2 = gh$, which says $U = U_\infty$ at every instant, including $t = 0$. It is not a slightly wrong answer during the transient; it is the answer to a different question, and it has the flow at full speed before any fluid has moved.

The check written for this calculation therefore demands a disagreement: it requires the true solution to be below fifteen per cent of the final speed a tenth of a time constant in, and refuses a computation in which the two agree early.

Where quasi-steady is legitimate is where the timescale of the forcing is long compared with $L/U$ — which is what a reduced frequency measures, and is the number to compute before assuming it.

The oscillation that is nothing else #

The cleanest demonstration that the term is not a small correction is a flow in which it is the entire dynamics.

Fill a U-tube with liquid, displace it, and release. The column of length $L$ is accelerated by the imbalance $2\rho g x$, so $L\ddot x = -2gx$: simple harmonic motion with period $2\pi\sqrt{L/2g}$, with no amplitude in it.

There is no viscosity in that calculation and no vorticity anywhere in the flow. The fluid is ideal, the motion is irrotational, and every steady-state theory in this collection would say nothing is happening — and it oscillates, at 1.42 seconds for a metre of liquid, because the whole of the physics is in the term that vanishes from every steady answer.

The same term with a gas spring instead of gravity #

The U-tube oscillates because a column of fluid has inertia and gravity supplies a restoring force. Replace the gravity with the compressibility of a trapped volume of gas and the same term produces the most familiar acoustic object there is.

A Helmholtz resonator is a cavity with a short open neck. The fluid in the neck moves as a plug — its inertia is the $\partial\phi/\partial t$ term integrated along the neck — and pushing it inwards compresses the gas in the cavity, which pushes back. Two elements, an inertia and a spring, and a frequency

$$f = \frac{c}{2\pi}\sqrt{\frac{A}{V L'}}$$

with $A$ the neck’s area, $V$ the cavity’s volume and $L'$ its effective length. For a wine bottle — three quarters of a litre, a neck a centimetre across and eight centimetres long — that is about 110 hertz, which is the note a bottle hums when it is blown across, and it is set by a volume and a neck rather than by any dimension a wavelength would care about.

The primed length is where this essay’s argument is doing its work. The fluid that has to be accelerated is not only the fluid geometrically inside the neck: the flow converges into the mouth from outside and diverges out of the other end, and that surrounding fluid moves too. The correction — about 0.8 of a neck radius at each end — is an added mass, computed by exactly the surface integral of $\phi$ this essay identifies as the pressure impulse, and it is a substantial fraction of a short neck’s own length. A bottle’s note cannot be predicted without it.

Once the two elements are named the analogy runs the whole way. Pressure plays the part of voltage, volume flow rate the part of current, the neck’s fluid inertia is an inductance — the inertance, $\rho L'/A$ — and the cavity’s compressibility a capacitance. Any duct network is then a circuit, its resonances are the circuit’s, and the whole apparatus of impedance carries over unchanged. That is how an exhaust silencer is designed, how a vented loudspeaker cabinet is tuned, and how the resonances of the human vocal tract are computed.

It is also why an engine’s intake is a length rather than merely a pipe. An inlet runner has an inertance and the cylinder a capacitance, and tuning the pair so that a pressure wave arrives at the valve just as it closes is worth several per cent of an engine’s torque at the speed it is tuned for — which is why variable-length intake manifolds exist, and why an engine’s torque curve has bumps in it that have nothing to do with combustion.

And it is why a car with one rear window open at speed produces an unbearable low-frequency buffeting. The cabin is the volume, the open window is the neck, the shear layer across the opening supplies the excitation, and the resulting Helmholtz frequency is fifteen or twenty hertz — below hearing and squarely in the range the inner ear objects to. The remedy is to open a second window, which changes the neck and moves the frequency, and the fact that it works is a demonstration that the term dropped from every steady Bernoulli equation is the one doing all of it.

Where the term hides in plain sight #

Three places where it is doing work under another name.

Added mass. The force of getting going is the surface integral of the pressure that the $\partial\phi/\partial t$ term produces when the body accelerates. Writing it as an effective mass is a way of avoiding ever mentioning the term.

Water hammer. Stopping a column of water costs more than moving it, and the pressure rise is $\rho c\,\Delta U$ rather than $\rho L\,dU/dt$ only because the fluid’s compressibility limits how much of the column participates. In an incompressible idealisation the pressure rise on instantaneous closure is infinite, and the infinity is this term.

And the virtual mass of a bubble or a droplet, which is the same integral and is what makes a bubble in an accelerating liquid rise faster than gravity alone would take it.

The other place the constant of integration matters #

The right-hand side being a function of time rather than a constant is easy to skip past, and there is one situation in which skipping it produces a wrong answer rather than an untidy one.

Consider a body accelerating through a fluid, and work in the frame of the body. In that frame the fluid far away is accelerating too, so there is an apparent body force, and the “steady” Bernoulli equation applied between a distant point and the body’s surface is wrong by a term linear in distance.

The correct procedure is to work in the ground frame, where the fluid at infinity is at rest and $F(t)$ is fixed by conditions there. Getting this wrong is the standard route to a spurious pressure gradient in an accelerating frame, and it is the fluid version of forgetting a pseudo-force.

The same care is needed for a rotating frame, where the additional term is not a function of time alone and Bernoulli’s equation acquires a centrifugal potential. That is why the analysis of a rotating flow is written with a modified pressure from the start.

What the picture cannot show #

A transient is a sequence and a figure is a moment. The starting-tube figure is a curve of speed against time, which is the honest way to draw it, and it means the flow field itself — which is uniform along the tube and therefore uninteresting — is nowhere in the picture.

The pressure impulse is drawn as a distribution and it lasts no time. A figure of it is a figure of an integral over an instant, and nothing about the drawing conveys that the pressure during that instant was unbounded. What is finite is the area under it.

And the incompressible idealisation removes the mechanism that limits everything. In a real liquid an impulsive start sends a pressure wave along the tube at the speed of sound, and the “instantaneous” response of the whole column is really the response after that wave has traversed it. For a two-metre tube that is a millisecond, which is short compared with 0.9 seconds — and for a two-kilometre pipeline it is not, which is why water hammer is a wave problem and this is not.

Why it disappears from the textbooks #

Worth a paragraph, because the omission is systematic rather than accidental.

Bernoulli’s equation is introduced for steady flow, where it is an algebraic relation between two points and can be applied by inspection. The unsteady form is a differential relation along a path, needs the potential rather than just the velocity, and requires knowing that the flow is irrotational — a much stronger hypothesis than the steady form needs, since the steady equation holds along a streamline whatever the vorticity is while the unsteady one needs $\phi$ to exist at all.

So the term is dropped early, for good reasons, and rarely picked up again. The cost is a generation of engineers whose first instinct about a starting flow is Torricelli.

The last observation is worth making explicit because it ties two ends of the ladder together. Kelvin’s minimum-energy theorem says which field the boundary conditions select; the unsteady term says how long the fluid takes to get there and what it costs. The first is a statement about a state and the second about a history, and the pressure impulse is the quantity that belongs to both — a surface integral of the potential, which is the state, taken over the duration of the transient, which is the history.

A closing note on when the term can be dropped, since most of this essay has been about the cost of dropping it. The condition is that the flow’s own timescale be long compared with the time a fluid particle spends traversing the region of interest — the reduced frequency $fL/U$ small — and for most of aerodynamics it is: a wing in steady flight has no unsteady term at all, and one in a slow manoeuvre has a negligible one. The term matters at starts, at stops, at impacts, and in oscillations whose period is comparable with a transit time, which is a short list and contains most of the occasions on which anything breaks.

Who found it, and when #

Euler had the unsteady form in 1757, in the same paper as the equations; Lagrange gave the potential version in 1781. The pressure-impulse interpretation is Kelvin’s and is of a piece with his impulse and his circulation theorem — all three are statements about what an ideal fluid remembers. The tube transient is in every nineteenth-century hydraulics text and in rather few modern ones.

The surprising connection is with how the term is used in naval architecture. A ship’s added mass is measured by hitting it — a decay test, in which the hull is displaced and released and its natural period is compared with the period it would have with no water — and the difference is the same $\partial\phi/\partial t$ integral computed above for the U-tube. So a quantity introduced as an abstraction in a chapter nobody reads is the thing an experimenter measures with a rope and a stopwatch, and it has been measured that way since the 1860s.

Where the ladder goes next #

Below this rung are the steady relation this generalises and the added-mass force it produces.

Beside it is the impulse, which is the same surface integral seen as a momentum, and how slow is slow enough for quasi-steady to be true.