Everything about the start, except one vector

The momentum with no value establishes that the momentum of an unbounded ideal flow is not a well-defined quantity — the integral converges conditionally and gives a different answer for every shape of region — and that the object which replaces it is Kelvin’s impulse: the vector whose rate of change is the force on the body.

The mass a body has to borrow computes the added mass that sets its size, and finds exactly a half for a sphere and exactly one for a cylinder.

This essay asks the question those two leave standing. The impulse is the time integral of the force, and the force during a start-up depends on the whole manoeuvre. Does the impulse?

It does not, and the demonstration is worth doing rather than asserting, because the negative result is the theme these essays share arriving from the wrong end: an ideal fluid with no vorticity in it has no memory of a manoeuvre at all.

Six routes #

All six histories start from rest and reach exactly the same speed at the same moment. Two are ramps, one ten times as sharp as the other, and their peak forces are 3.142 and 31.42. Two are eased with a raised cosine, again at different sharpnesses. One overshoots the final speed and comes back down to it. And one goes backwards before it goes forwards.

The last two are the interesting members. They are not perverse: a body being manoeuvred does exactly this, and so does a wing in a gust, and so does anything under feedback control. Their point here is that they visit states the others do not — the reversing law takes the body backwards before it goes forwards, so it does negative work on the fluid for part of its journey and its peak force, 32.84, is more than ten times the slow ramp’s 3.142. Any quantity that accumulated along the path would notice them.

The forces are entirely different. The sharpest start peaks at 123.4 and the slowest at 3.14, a factor of 39.3, and two of the histories require a negative force for part of the journey. A strain gauge on the mounting would record six unrelated signals.

And one number #

The impulse is the time integral of that force, and it comes out at 3.1415926536 for every one of the six, which is the added mass times the final speed. The largest departure from the closed form anywhere in the comparison is 2.8·10⁻¹³ of itself.

The reason is one line of calculus and it is worth writing out, because the generality of the result is exactly the generality of that line. In an ideal fluid the force on a body moving in a straight line is its added mass times its acceleration. The impulse is therefore the added mass times the integral of the acceleration, which is the added mass times the change in velocity — a function of the endpoints and of nothing in between.

That is a path-independence statement of the same kind as a potential energy’s, and it has the same cause: the integrand is an exact derivative.

The energy does the same thing #

The work done against the fluid is the integral of the force times the velocity, which is a more promising candidate for path-dependence because it multiplies two things that both vary.

Six histories, one impulse:

The peak-force column spans a factor of 39.3 and the impulse column is π to ten figures in every row — the worst departure across the six is 2.8·10⁻¹³. The energy comes out as half the added mass times the square of the final speed, to nine figures, in all six cases — including the law that goes backwards and therefore does negative work for part of its journey, recovering energy from the fluid before putting it back.

Again the reason is that the integrand is an exact derivative: force times velocity is added mass times acceleration times velocity, which is the derivative of half the added mass times the velocity squared.

So the fluid’s state at the end of the manoeuvre is completely specified by the body’s velocity at the end of the manoeuvre. Not approximately, not to leading order: completely — six routes whose peak forces differ by a factor of thirty-nine end at one impulse and one energy, agreeing to twelve decimal places. The flow field is the solution of Laplace’s equation with that boundary velocity on it, and there is only one.

Sharpening the start without limit #

The impulse’s indifference to the route has a limit worth pointing at, because it is where the mathematics says something a strain gauge would find surprising.

Make the start twice as abrupt and the peak force doubles. Make it a hundred times as abrupt and it is a hundred times larger. In the limit of an instantaneous start the force is unbounded — it is an impulse in the mathematical sense, a delta function — and the time integral of it is still 3.1416.

That is the origin of the phrase impulsive start, and it is why the added mass is sometimes introduced as “the momentum a body has to give the fluid to start moving at all” — here π times the final speed, for a cylinder whose added mass is the mass of the fluid it displaces. The fluid’s resistance to being started is finite even when the force required to start it instantaneously is not.

What a measurement of any of this would actually see #

It is worth walking through what an experiment would record, because the three quantities separate in a way that decides which of them is measurable at all.

The force is easy and is the least informative. The six laws here reach peaks of 3.142, 31.42, 4.935, 123.4, 13.67 and 32.84 — a spread of 39.3 for one impulse. A load cell on the mounting records that peak directly, at whatever bandwidth the mounting allows — and the mounting is the problem: a sharp start excites the structure’s own modes, so the recorded signal is the fluid force convolved with the rig. The peak is exactly the quantity most corrupted by that.

The impulse is harder and is much more robust. The six peaks here span 3.142 to 123.4 and the six impulses agree to 2.8·10⁻¹³, which is the same statement an experimenter meets as a ringing load cell. It is the integral of the same signal, and integration suppresses the ringing that corrupts the peak: a structural oscillation about the true force integrates to nearly nothing over a few of its own periods. So the quantity that is path-independent in the theory is also the quantity least sensitive to the apparatus, which is a pleasant coincidence and not one.

The energy is hardest. It needs the force and the velocity together, in phase, and a small timing error between two channels produces an error in their product that does not integrate away.

The general shape of that — that the quantities which are exactly conserved tend also to be the quantities that survive a bad measurement — recurs through this collection. It is why a control-volume answer is worth more than a local one, and it is the argument a force without the flow that makes it makes in a completely different setting.

Why this is the theme in the negative #

The force whose integral this is is computed in the force of getting going, and the point of the pair is that the force depends on the whole acceleration history while its integral does not.

Everything else in this collection is about a flow that remembers something. This one is about a flow that does not, and it is the more useful of the two statements because it says where the memory has to come from.

An ideal, incompressible, irrotational flow has no state of its own. Its velocity field is the solution of Laplace’s equation with the present boundary condition, and Laplace’s equation has no time in it. So the fluid at the end of a manoeuvre carries no record of the manoeuvre, and cannot: there is nowhere for a record to be kept.

Every memory this collection has found in a fluid therefore enters through something the description above excludes. Vorticity is a material label that the fluid carries with it, which is why a wing needs a vortex left behind before it can carry any circulation at all. A free surface has its own position, which is a state. Compressibility gives a finite signal speed, so what happens here now was decided elsewhere then.

The theory with no memory in it is the essay that takes that inventory seriously; this one is the measurement that motivates it.

What a starting vortex changes #

The contrast with a lifting body is exact and is worth making, because it is where the negative result turns into a positive one.

A cylinder has no sharp edge, so an ideal flow past it sheds nothing, and everything above holds. A wing has a trailing edge, at which the ideal flow would have to turn a corner at infinite speed — which is nothing turns a sharp corner — so vorticity leaves the body and stays in the fluid.

Once that has happened the fluid does have a state. The circulation shed during the start-up is still out there, it is not a function of the wing’s present velocity, and the lift consequently takes time to arrive: the lift that arrives late is the measurement of how long. The difference between that essay and this one is one sharp edge.

What the solver computed, and how it was checked #

The added mass of a circular cylinder in two dimensions is the mass of the fluid it displaces, and the force is that times the acceleration. Each history is differentiated numerically at twenty thousand samples and the force is integrated back, so the impulse is a round trip through the arithmetic rather than an evaluation of the closed form.

Three checks. That every impulse matches the closed form, to a part in 10¹⁰ of itself — which is the essay’s claim and is the check that would catch a history whose endpoint was not what it was supposed to be. That every energy matches, to a part in a million, which is a separate statement because the work integral multiplies two varying quantities. And that the peak forces genuinely spread, by at least a factor of twenty — because a comparison between six nearly identical histories would prove nothing, and the check refuses one.

That last check earned its place. An early version of the fifth and sixth laws had a kink in the velocity at the moment they rejoined the others, which put a spike in the numerically differentiated force and moved their computed energies by six per cent. The laws are now built from a smooth carrier so that velocity and acceleration are both continuous at both ends, and the discrepancy is gone. The six per cent was the difference scheme meeting a corner, not a physical result, and it is recorded because it looked exactly like one.

The same statement, three times in this collection #

Path-independence is a structural property rather than a fluid-mechanical one, and this collection has now met three instances of it that are worth reading together.

Here, the impulse. The force is an exact derivative of a quantity that depends only on the state, so its integral depends only on the endpoints.

In the wake, the drag. Four profiles, one drag has four wake profiles carrying identical momentum deficits and nothing else in common — a case where the integral is fixed and the integrand is free, which is the same arithmetic seen from the other side.

And in a strain history, the stretch. Two strainings, and the order they came in is the counter-example: a quantity that looks like an integral of instantaneous rates and is not, because deformation gradients multiply rather than add, and the order therefore matters by a factor of 2.16.

The test that separates the three is always the same one. Is the thing being accumulated an exact derivative of a function of the state? If it is, the path drops out; if it is not, no amount of care about the endpoints will recover it. That is not an analogy between the three cases — it is the same theorem, and the fluid mechanics only decides which case one is in.

Where the path-independence stops #

The result is exact within its model and it fails in three specific ways, each of which is worth naming because each is a real effect and none of them is small.

Viscosity. A real fluid started impulsively grows a boundary layer whose thickness depends on how long it has been growing, so the force history and the total impulse both depend on the route. The Basset history term in the unsteady Stokes force is precisely the path-dependence this model does not have, and it falls off as the inverse square root of time — so it never quite finishes.

Separation. A body that is accelerated hard enough separates, and separation sheds vorticity into the fluid, which is a state. Two histories reaching the same speed with different separation histories leave different flows.

And a free surface, or compressibility. Both give the fluid somewhere to store the past, and both are excluded here.

The practical form of the result #

For anybody sizing a structure or a control system, the statement is a useful separation of concerns.

The peak load depends entirely on how fast the manoeuvre is. It is bounded only by how abruptly the motion is commanded, and halving the manoeuvre time doubles it.

The impulse and the energy do not depend on the manoeuvre at all. They are set by where the motion started and where it finished, so a control law that shapes a command to reduce peak load costs nothing in either — which is why command shaping works and why it is nearly free.

And the two are different design problems. A structure is sized by the first, and a power system by the second, and optimising one has no effect whatever on the other in this model.

What the picture cannot show #

The force histories are drawn as curves, and two of them are not really curves. The sharp ramp’s force is a rectangle — constant while the ramp lasts and zero afterwards — and the drawing renders its edges as steep line segments rather than as the discontinuities they are. The sharper the start, the more the picture is a picture of the plotting resolution.

Nothing in the figures shows the flow field at any point during any of the six manoeuvres either, and it would be the most persuasive picture available: six fields at the final instant, all identical. It is left out because six copies of one picture is not a figure.

Who found it, and when #

Kelvin introduced the impulse in the 1870s precisely because the momentum integral does not converge absolutely, and the path-independence is immediate from his definition. The added-mass concept is older and belongs to Green and to Stokes, from work on pendulums swinging in air and in water — where the measured period is shifted by exactly this borrowed mass.

The engineering form of the result — that peak load and total impulse are independent design quantities — arrived much later and from structural dynamics rather than from fluid mechanics, and the two literatures still mostly quote it separately.

Limits recorded rather than smoothed over #

A cylinder in two dimensions. The added mass of a general body is a tensor, and for a body that is not symmetric the force is not parallel to the acceleration — the mass a body has to borrow measures a misalignment of 46.4 degrees. The path-independence survives that: the impulse is still the tensor applied to the final velocity.

Straight-line motion only. A body that rotates as well as translates has an angular impulse too, and the two are coupled. The result generalises and the arithmetic does not stay this simple.

The force is differenced, not derived. Each history is differentiated numerically, which is why the kinks in an earlier version mattered. The closed form is available and is used only as the check, on purpose: a computation that evaluates the answer it is checking has checked nothing.

The added mass is taken as a constant. For a rigid body of fixed shape in an unbounded fluid it is, and that is what makes the force a pure derivative. A body near a wall has an added mass that depends on its distance from the wall, so the force acquires a term proportional to the square of the velocity and the impulse stops being a function of the endpoint alone — which is worth knowing before applying any of this to a body manoeuvring near a boundary.

And “the fluid remembers nothing” is a statement about this model. Everything real that has been left out — viscosity, separation, a surface, compressibility — puts some of the memory back, and the rest of this collection is about how much.