Two forces, and only one of them remembers

Long enough to make a wake is this collection’s account of the Keulegan-Carpenter number: how far the fluid travels in one oscillation compared with the size of the body it is travelling past, and therefore whether a wake has time to form at all.

This essay reads Morison’s equation in that light and finds that its two terms are not two contributions to one force. They are two kinds of force, distinguished by whether they depend on the flow’s history — and the Keulegan-Carpenter number is the number that says which one dominates.

The two terms #

Morison’s equation puts the force on a slender body in an oscillating flow as

$$F = \tfrac{1}{2}\rho C_D D\,u|u| \;+\; \rho C_M \tfrac{\pi D^2}{4}\,\dot u.$$

The first term is a drag, quadratic in the velocity. The second is an inertia force, proportional to the acceleration.

They are a quarter of a cycle apart, because the velocity and the acceleration are, so the total force peaks somewhere in between and the split cannot be read off a single peak.

One of them has no memory at all #

The inertia term is the added mass times the acceleration, and everything about the start, except one vector establishes exactly what that means: it is a function of the present acceleration, its impulse over any manoeuvre depends only on the endpoints, and the flow it describes is the solution of Laplace’s equation with the present boundary velocity on it.

Nothing about the past enters. Stop the body, restart it, take it round a loop and back — the inertia force is the same function of the acceleration at every instant, and the fluid retains nothing.

That is the memoryless door of the theory with no memory in it, arriving in an engineering formula.

And the other is nothing but memory #

The drag term is a different object. Its coefficient is not a constant: it depends on the Keulegan-Carpenter number, and it depends on it because the wake a body leaves in one half cycle is still there when the flow reverses and the body meets it again.

At low KC the body barely moves relative to its own size, no wake forms, and there is essentially no drag: at KC = 0.5 the fluid travels half a diameter in a whole cycle and the drag term is 2.95 per cent of the peak force. At high KC the body travels many diameters in each half cycle — fifty of them at KC = 100 — a wake forms and is swept away, and the drag approaches its steady value at 85.9 per cent of the peak. In between — and it is a wide band — the body is re-entering its own wake, and the force depends on what it did last.

So the drag coefficient’s dependence on KC is not an empirical inconvenience. It is a memory kernel, compressed into one number, and the reason it cannot be a constant is that the body’s history varies with KC.

The ratio, which is one line #

The ratio of the two amplitudes is exact:

$$\frac{\text{drag}}{\text{inertia}} = \frac{C_D\,\mathrm{KC}}{\pi^2 C_M},$$

reproduced by the computation to nine figures. With the usual coefficients it crosses one at KC = 16.45, which is the textbook figure of fifteen to twenty arrived at without fitting anything.

Across the range a designer meets, the drag term’s share of the peak force runs from 2.95 per cent to 85.9. The two terms are equal at a Keulegan–Carpenter number of 16.45, and the drag amplitude rises as the square of KC while the inertia amplitude rises only linearly — across the two hundred-fold sweep the drag column grows by 40,000 and the inertia column by 200. The same equation describes two entirely different problems and the number that separates them is a ratio of lengths.

What the number is measuring #

It is worth restating the Keulegan-Carpenter number in the terms this essay is using, because the usual definition hides what it is for.

It is the distance the fluid travels in one oscillation, divided by the body’s diameter. Equivalently — and more usefully here — it is how many body-diameters of wake are produced per half cycle.

Read that way the two limits are obvious. At KC well below one the fluid moves a fraction of a diameter and no wake is produced at all, so there is nothing to remember and the flow is the potential one. At KC well above one the fluid moves many diameters, the wake is swept clear before the reversal, and the body meets fresh fluid each half cycle — so there is again nothing to remember, and the flow is the steady one, reversing.

The memory is at intermediate KC, where the wake produced in one half cycle is still nearby when the flow reverses — the band where neither term can be dropped, which on the table above is everything between KC of 2 and 50, where the smaller term is still between a tenth and a quarter of the total. That is the band from about two to about twenty, it is where nearly every offshore member sits, and it is where the coefficients vary most and are least transferable.

So the number is not measuring which force dominates — that is a consequence — it is measuring how much of the previous half cycle is still present.

Where things sit #

A ten-metre platform leg in a ten-second swell is at KC ≈ 2: an inertia problem, in which the drag coefficient hardly matters and the added mass does.

A half-metre riser in the same swell is at KC ≈ 40: a drag problem, in which the added mass is a correction.

A cable is at KC ≈ 400 and is essentially in steady flow, reversing.

The same sea gives three different problems on one structure, which is why an offshore design has to compute the number for each member rather than adopting a regime for the platform.

What the solver computed, and how it was checked #

Morison’s equation is evaluated term by term in a sinusoidal flow, and the two amplitudes are compared with the closed-form ratio. Nothing here is a flow solution: the coefficients are inputs, and the essay’s content is the structure of the split rather than the values of the forces.

Three checks. That the computed ratio matches the closed form at every Keulegan-Carpenter number in the sweep, to a part in 10⁹, which tests the arithmetic. That the crossover lies between ten and twenty-five, which is the published range and is the check that would catch an error in the coefficients’ definitions. And that the ratio varies by at least a factor of a hundred across the sweep, so the split genuinely changes character.

Why the coefficients are measured together #

There is a practical consequence which explains a peculiarity of the offshore literature.

The two coefficients are almost always measured as a pair, from a force record in an oscillating flow, by least squares. That looks like laziness and it is not: at any given KC one of the two terms is much larger than the other — at KC = 2 the inertia term is 8.2 times the drag term and at KC = 100 the drag term is 6.1 times the inertia term, and only within a factor of about three of KC = 16.45 are the two comparable, so the smaller one is poorly determined by that record and is being fitted to a residual.

The result is that a published pair of coefficients is only reliable near the KC it was measured at, and that the two are correlated — an error in one is compensated by an error in the other in the fit. Quoting a drag coefficient measured at KC = 2 and applying it at KC = 40 is quoting a number that was never really measured.

The remedy is to report the KC, and the practice of doing so is the offshore community’s version of this collection’s standing complaint about the instrument in the answer.

The coefficient as a compressed kernel #

The added mass the first term is built on is computed in the mass a body has to borrow, where it is an exactly path-independent quantity — which is why that half of Morison’s equation has no memory to compress.

It is worth saying precisely what the drag coefficient’s dependence on KC is doing, because the framing makes it less arbitrary than it appears.

The honest force on a body in an oscillating flow is a functional of the whole velocity history — a convolution, if the problem were linear, and something worse because it is not. What Morison’s equation does is project that functional onto two basis functions: one proportional to the acceleration and one proportional to the velocity times its magnitude.

Two basis functions cannot represent a functional, so the projection leaves a residue, and the residue is absorbed by letting the two coefficients depend on the one parameter that characterises the history — which is KC.

So the coefficients are what is left of the memory after it has been squeezed into two numbers. That is why they cannot be constants, why they are correlated with each other, and why they transfer badly between conditions: they are carrying information that properly belongs to a kernel.

The same compression appears in a closure with no memory at all, where an eddy viscosity absorbs a frequency response into a constant, and the diagnosis there is the same one: a model’s calibration should be suspected first where the history it is standing in for varies most.

What the equation leaves out #

It is worth being explicit about what the two terms do not cover, because the omissions are the reason Morison’s equation is used with a factor of safety.

The lift force. A body shedding vortices asymmetrically experiences a force across the flow, at the shedding frequency, and it can be as large as the in-line force. Nothing in Morison’s equation contains it.

The history within the cycle. The drag term is quadratic in the instantaneous velocity, so it assumes the wake responds instantly to the velocity. It does not — the wake has its own lag, which is the lag that makes flutter possible’s mechanism — and the coefficient absorbing that is what makes it a function of KC.

And the body’s own motion. A compliant member moves in response to the force, which changes the relative velocity, which changes the force. That coupling is what produces vortex-induced vibration and it is where the design problem actually is.

The same two-term split, elsewhere #

The pattern — a memoryless inertia term plus a history-bearing resistance — is not Morison’s invention and turns up wherever a body meets an unsteady flow.

A particle in a fluid. The three terms of the unsteady Stokes force are exactly this split with a third term made explicit: the drag that integrates a whole history computes the added mass with no memory, the quasi-steady drag with no memory, and the Basset term which is nothing else.

An aerofoil in a gust. The non-circulatory force is added mass and the circulatory force is the wake, which is two answers to one question’s subject.

And a rotor changing thrust. The inflow lag of the inflow that takes time to arrive is an added-mass term becoming a first-order lag because the disc is being asked to accelerate a finite volume of air.

In every case the memoryless part is exactly computable from potential flow and the history-bearing part is where all the modelling is. That division is the shape of unsteady fluid mechanics, and Morison’s equation is its plainest statement.

What a designer does with the split #

The practical procedure follows from the two terms being different in kind, and it is worth setting out because it is not what the equation’s appearance suggests.

Compute the number first, member by member. A jacket structure has legs, braces, risers and cables spanning three decades of diameter in one sea, so the regime is a property of the member rather than of the platform.

Then decide which coefficient the answer is sensitive to. In an inertia-dominated member the added mass is what matters, and it is known from potential flow to a few per cent — so the load is well determined. In a drag-dominated one the drag coefficient is what matters, and it is measured with scatter — so the load is not.

And expect the uncertain regime to be the intermediate one. Where the two terms are comparable both coefficients matter and both are poorly determined at that KC, so the total is worse than either. That is the opposite of the usual intuition that a balanced contribution is a safe one.

The general form of that reasoning — identify which term the answer is sensitive to, then ask how well that term is known — is what what of order one is worth recommends and what a factor of safety is standing in for when it is not done.

What the picture cannot show #

The drag coefficient is drawn as a constant in every figure here, and its dependence on the Keulegan-Carpenter number is the essay’s central physical claim. The reason it is not drawn is that it is measured rather than computed, and this collection’s solver cannot produce a wake in an oscillating flow.

Nothing here shows the wake either — the object the whole memory argument is about. It is a pair of vortices shed each half cycle, swept back over the body when the flow reverses, and it is what a figure of this subject ought to show.

Why the split is worth making at all #

A reader might reasonably ask why anybody bothers to separate two terms that are added together, since the sum is what a structure feels.

The answer is that the two behave differently under every change a designer might make.

Under a change of size. The inertia term goes as the diameter squared and the drag term as the diameter, so making a member thicker moves it towards inertia — and the load per unit projected area changes character rather than merely magnitude.

Under a change of sea state. The inertia term is linear in the wave amplitude and the drag term is quadratic, so a storm loads a drag-dominated member disproportionately.

And under a change of frequency. The inertia term goes as the frequency squared at fixed amplitude and the drag term as the frequency squared too — but through different routes, and their ratio is unchanged, which is why the Keulegan-Carpenter number is the right variable rather than the frequency.

So a structure designed at one condition and used at another has had the balance between its two terms moved, and knowing which way is the whole of the extrapolation.

Where this number sits among the others is the last essay in this field, on the same machinery.

What a designer does with the split #

The practical use of the two terms is not to compute a force but to decide which measurement is worth paying for.

In the inertia regime the coefficient is nearly a theoretical number. Added mass for a circular cylinder is a potential-flow result, and a measured coefficient near two is a confirmation rather than a fit. There is little to be gained from testing.

In the drag regime the coefficient is entirely empirical, and it depends on the roughness, on the marine growth the structure will acquire, and on the wake left by the previous half cycle. Testing is the only route, and it has to be done at the right Keulegan-Carpenter number rather than at the right Reynolds number alone.

And near the crossover both matter and neither dominates, which is the expensive case: the peak force depends on the phase relation between the two terms, so an error in either coefficient moves the answer, and a design there is worth the model test that the other two regimes are not.

Who found it, and when #

Morison’s equation is from 1950, written for offshore piles and now the basis of nearly all offshore loading calculation. The Keulegan-Carpenter number is Keulegan and Carpenter’s, from 1958, from experiments in an oscillating water tunnel that produced the coefficient variations this essay is about.

The recognition that the two terms are different in kind rather than different in size is implicit in the derivation — the inertia term comes from potential flow and the drag term from a correlation — and is rarely stated, which is why an equation with a sound theoretical half and an empirical half is usually described as empirical.

Limits recorded rather than smoothed over #

The coefficients are inputs. Nothing here computes either of them, and the whole physical content of the drag term’s memory is in a dependence this essay describes and does not evaluate.

A slender body in a uniform oscillating flow. Morison’s equation assumes the flow is uniform across the body and that the body is long compared with its diameter. A large structure diffracts the wave, and then a completely different calculation applies.

Sinusoidal, and one frequency. A real sea is a spectrum, and the quadratic drag term does not superpose — so a spectral calculation with Morison’s equation needs a linearisation whose validity is a separate question.

No free surface. The members this equation is used on are usually partly submerged, and the wave kinematics at the surface are not the linear ones assumed — which is a separate and larger source of uncertainty than anything in this essay.

And no relative motion. The body is fixed here. Vortex-induced vibration, which is the failure mode these calculations exist to prevent, requires the body to move.