Long enough to make a wake

Every regime number in this collection so far has been built from a speed and a length, and every one of them is silent about the same question. Take a cylinder in a flow that reverses — a pile in a wave, a riser in a swell, a heat-exchanger tube in a pulsating duct — and ask whether the fluid gets round it before the flow turns and comes back. A Reynolds number cannot answer, because it has no time in it.

$$\mathrm{KC} = \frac{U_m T}{D}$$

is the answer: the distance a fluid particle travels in one oscillation, over the width of the thing it is travelling past. Two exact statements follow from that definition and both are computed here rather than asserted.

The distance swept in a half cycle is $U_mT/\pi$, by quadrature over $|\cos|$, so KC/π is the number of diameters swept and KC = π is the flow that sweeps exactly one. And a wake needs several diameters of travel to form at all, which is why the observed sequence of regimes is laid out along this axis and not along a Reynolds number.

Below about KC = 1 the flow reverses before it reaches the shoulder and there is no separation at all. Between 2 and 3 a symmetric pair of vortices forms and is reabsorbed each half cycle. Above about 7 a pair is shed to one side and a transverse force appears; above 15 there is a transverse street; and above 30 the flow has forgotten the previous stroke by the time it returns and each half cycle grows a wake of its own. That last regime is where a steady-flow drag coefficient starts to be a reasonable thing to use, and it is a long way up the axis.

Two terms, and one number that says which #

Morison’s equation, from 1950, writes the force on a slender cylinder as a sum:

$$F = \tfrac12\rho C_D D\, U|U| \;+\; \rho C_M \frac{\pi D^2}{4}\,\dot U.$$

A drag term in phase with the velocity and going as its square, and an inertia term in phase with the acceleration and ninety degrees ahead of it. It is not derived from anything — it is a two-term fit, proposed because a purely inviscid calculation gives the second term exactly and a steady-flow calculation gives the first — and it is what every offshore structure in the world is designed against.

With $U = U_m\cos\omega t$ the ratio of the peaks is

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

so one number decides which term is the force. Bisection puts the crossing at $\mathrm{KC} = 16.4493$ for $C_M = 2$ and $C_D = 1.2$, and the closed form $\pi^2C_M/C_D$ agrees to a part in 10⁹.

Below sixteen a structure is designed against an inertia load and above it against a drag load, and the two are different problems: the first scales with volume and the second with area, the first is in phase with the water surface and the second with its slope, and the first can be computed from potential flow while the second cannot be computed at all.

The term that carries the force and none of the energy #

The two terms differ in a way the peak-force picture hides completely.

$$\oint \dot U\,U\,\mathrm{d}t = \tfrac12\big[U^2\big] = 0$$

over a closed cycle, exactly. So the inertia term does no work at all, and every joule the structure takes out of the wave is taken by the drag term.

That is integrated here rather than argued, and it comes out at $-5.6\times10^{-16}$ against the drag term’s $8C_D/3 = 3.2$: a ratio of $1.7\times10^{-16}$, which is round-off. The drag work is also exactly $8C_D/3$ per cycle in these units, independent of KC, because the average of $|\cos|^3$ over a period is $4/3\pi$ and everything else cancels.

The consequence is worth stating slowly. At KC = 3 the inertia term supplies three quarters of the peak force and none of the damping. A structure whose response is dominated by inertia has no hydrodynamic damping from that term whatever, and its motion is limited by whatever else is available — structural damping, radiation damping, the drag term’s small contribution. This is the same distinction the Stokes layer draws between the part of an oscillating flow that stores momentum and the part that dissipates it, and it is why an added-mass coefficient and a drag coefficient are not two numbers of the same kind.

What a peak force cannot tell apart #

Morison’s equation is a fit, so the coefficients have to come from a measurement, and the measurement everybody has is a force record. What can be got out of it?

The answer is: not the coefficients. For each drag coefficient in the range 0.5 to 2.0 there is an inertia coefficient that restores the observed maximum, found here by bisection, and every member of that family produces the same peak force to fifteen decimal places. The energy each dissipates is proportional to its own $C_D$, so the family spans a factor of exactly 4 in dissipation.

And over the lower half of the range it is worse than a curve. Below about $C_D = 1.3$ the inertia coefficient does not move at all — the fitted values are bit-identical — because when the inertia term dominates the peak occurs at the instant the velocity is zero, and the drag term contributes precisely nothing there. The peak force carries no information about the drag coefficient whatever.

The phase does. A force record’s shape over a cycle separates the two terms uniquely, because one is even about the velocity peak and the other is odd; a least-squares fit to the whole trace is well-conditioned where a fit to its maximum is singular. That is a general moral about fitting and it has appeared in this collection before, in the single exponential through a relaxation spectrum and in reading an exponent off a log–log plot: a fit that succeeds is not the same as a fit that identifies.

The second number, which will not go away #

There is a further residual and it is not about fitting.

$C_D$ and $C_M$ are not constants of the body. They depend on KC — which is the whole point of the number — and also on

$$\beta = \frac{\mathrm{Re}}{\mathrm{KC}} = \frac{D^2}{\nu T},$$

a frequency parameter with no amplitude in it at all. It is a Stokes number: the ratio of the body’s size to the thickness of the oscillating viscous layer on it, squared.

At KC = 2 the drag coefficient falls by a factor of 10.5 over two decades of β, as $\beta^{-0.511}$. The exponent is derivable: in the attached limit the whole dissipation is a Stokes layer, whose wall stress Stokes’ second problem gives exactly, and equating its work per cycle to that of a quadratic drag law gives

$$C_D = \frac{3\pi^{3/2}}{4\,\mathrm{KC}\sqrt\beta},$$

so $\beta^{-1/2}$, to machine precision, for a flat plate. Wang’s low-KC expansion for a cylinder — quoted rather than derived — has the same exponent with a coefficient the plate calculation cannot supply.

So the collapse onto KC alone is false, and the second axis is exactly the one the Womersley number uses for internal oscillating flow: a ratio of a size to a diffusion depth. The two numbers are the same number in two geometries, and the pair (KC, β) is what Sarpkaya’s tables are indexed by for that reason.

What the number looks like in a real sea #

The abstract axis is worth grounding, because the regimes above are not evenly represented in engineering.

A wave of ten-second period and two metres of orbital amplitude gives a water-particle excursion of about six metres. Past a two-metre-diameter jacket leg that is KC ≈ 3 — deep in the inertia regime, no persistent wake, a force ninety degrees out of phase with the water velocity and in phase with its acceleration. Past a 0.3-metre riser in the same wave it is KC ≈ 20, on the far side of the crossover, with a wake each half cycle and a drag-dominated load. The same wave, the same sea state, two members of the same structure, and two different physical problems.

That is the practical content of the collapse and it is why the number is worth having. It also explains an asymmetry in how the two loads are treated: the inertia coefficient of a smooth cylinder is close to its potential-flow value of 2 and varies by tens of per cent, so it is nearly predictable, while the drag coefficient varies by a factor of ten with β and with surface roughness and is not predictable at all. Marine growth on a leg changes $C_D$ by more than doubling the diameter does.

The design consequence is that the inertia-dominated members are computed and the drag-dominated members are measured, and the boundary between those two engineering cultures sits at KC ≈ 16.

The regimes, and why they are not thresholds #

It is worth being careful about the boundaries on the axis at the top of this essay, because they are not thresholds in the sense the crossover essay means.

Each of them is a bifurcation of an observed flow pattern, measured in a laboratory, and each has its own dependence on β. The onset of separation is not at KC = 1.1 for every cylinder; it is at KC = 1.1 for the β at which somebody looked. The transverse-street boundary moves by several units across Sarpkaya’s range. What is robust is the ordering and the mechanism — more travel means more of the flow gets round the body, so the sequence runs from no separation to a full wake and never backwards.

That is a weaker claim than a threshold and it is the honest one. The number orders the regimes; it does not locate them.

What the quadratic term does to a real sea #

Everything above prescribes a single sinusoid, and a sea is a spectrum. That matters more than it sounds, because the drag term is quadratic and a quadratic term does not treat a spectrum as the sum of its parts.

Feed $U|U|$ a flow containing two frequencies and the output contains their sum and their difference as well as the originals. So a narrow-banded sea — waves clustered around, say, a ten-second period — produces a force with a component at a few tenths of a hertz and another at periods of a minute or more, neither of which is present in the water motion at all.

The long-period half of that is where the consequences are. A moored floating structure has natural periods in surge and sway of one to three minutes — placed there deliberately, well outside the wave band, so that the mooring is not resonantly excited. The difference frequencies of a wave spectrum land squarely in that window, and the resulting slow drift oscillation is what most mooring systems are actually designed against. It is a second-order effect exciting a first-order resonance, and a linear analysis of the same sea contains no term for it whatever.

The quadratic term also changes the statistics. A Gaussian sea passed through a linear transfer function stays Gaussian; passed through $U|U|$ it does not, and the force distribution acquires heavier tails than the wave. So the extreme force in a three-hour storm is not the extreme wave multiplied by a transfer function, and estimating it that way is unconservative in exactly the case that matters.

The standard working compromise is to replace $U|U|$ by an equivalent linear damping, chosen so that the energy dissipated per cycle matches. That is the right choice if the question is a response amplitude and the wrong one if the question is an extreme, and it is the same trade the peak-fitting argument above makes: matching one integral of a record while discarding its shape.

The limits of the whole picture #

Morison’s equation is a fit and does not become a theory. It has the right inviscid limit at small KC and the right steady limit at large KC and no justification in between, which is precisely the range of every real structure. Its residual against a measured record is typically ten to twenty per cent, and that residual is not noise: it is a transverse force at the shedding frequency, which is a different physical effect and cannot be represented by any pair of in-line coefficients. It is the transverse force a shedding wake produces, arriving in a flow that reverses before the street has finished forming.

The transverse force is often the design case. For KC between about 8 and 25 the lift force perpendicular to the flow is comparable with the in-line force and is at two or three times the wave frequency, which is nearer most structural resonances. Nothing above computes it, and Morison’s equation has no term for it.

The Stokes-layer derivation is a plate. It gives the exponent, not the coefficient, and it applies in the attached regime only. The moment separation occurs the layer is not the whole dissipation and the $\beta^{-1/2}$ stops holding.

And everything here is for a fixed cylinder. A cylinder free to move responds to its own wake, locks in over a band of frequencies, and is the problem the frequency a wake chooses ends on. The coefficients of a moving cylinder are different from those of a fixed one at the same KC and β, by factors rather than percentages.

A third quantity, and the reason it is not here #

There is one more thing the pair (KC, β) leaves out, and naming it is the honest end of the identification argument.

Morison’s equation is local: it gives the force per unit length from the flow at that point, so it assumes the force at one height on a pile is decided by the velocity at that height. For a long slender member in a wave that is nearly true, and it is what makes the equation usable at all. It stops being true when the wake at one station is correlated with the wake at another — which is exactly what happens when the cylinder is free to move, and is why lock-in raises the fluctuating load so much: a correlated force over twenty diameters is twenty times a force that decorrelates every diameter.

The measure of that is a correlation length, which is neither KC nor β and is not dimensionless at all until it is divided by something. Sarpkaya’s tables do not contain it. A structural analysis handles it by assuming full correlation and accepting the conservatism, or by a spectral model with a coherence function fitted to measurements.

So the honest count is three: two dimensionless groups that index the coefficients, and one length that indexes how much of the structure feels the same force at once — the same correlation length that decides whether a spanwise-varying load adds or cancels along a lifting surface. The essay’s title question — whether the flow has time to make a wake — is answered by the first. Whether the wake is the same wake along the member is answered by none of them.

Where the number came from #

Garbis Keulegan and Lloyd Carpenter measured forces on cylinders and plates in an oscillating water tunnel at the National Bureau of Standards, and published in 1958. The number is theirs, and so is the observation that the two Morison coefficients extracted from their own records varied systematically with it rather than being constants.

The interesting part of that history is that the equation came first, in 1950, and the number that says when it works came eight years later. Sarpkaya’s tables from the 1970s and 1980s added the second axis and are still what the offshore codes are built on — which is to say that seventy-five years after the equation was proposed, the coefficients in it are a look-up table indexed by two dimensionless numbers, and there is no theory for either of them.

What this leaves #

The collapse onto KC is real and useful: it orders the regimes and it says which term is the force. What it discards is the phase, which is the only thing that separates the two coefficients, and the second number β, which moves one of them by an order of magnitude.

The next essay is about a group that is not a hypothesis at all — the dimensionless number that is an answer, which cannot be set by anybody and appears on the wrong side of the equals sign in every regime diagram it is drawn on.