The mass a body has to borrow

The force of getting going makes the case that an accelerating body in an ideal fluid feels a force even though a steadily moving one does not, and gives that force a name. This essay is about the number in it, which is exact, and about three readings of it that are not.

Exactly a half and exactly one #

Accelerate a sphere through an unbounded ideal fluid and the force needed is $(m + m_a)\dot{U}$ with

$$m_a = \tfrac{2}{3}\pi\rho a^3,$$

which is exactly one half of the $\tfrac{4}{3}\pi\rho a^3$ of fluid the sphere displaces. For a cylinder in two dimensions the added mass is $\pi\rho a^2$, which is exactly one times the displaced mass. These are rational numbers, not coefficients; nothing in them has been fitted, and they do not depend on the speed, the density, the size or how the acceleration is applied.

The reason both are so clean is that the disturbance of a translating closed body is a pure dipole. Every higher multipole in the far field is generated by the body’s shape rather than by its motion — it is what the far field remembers about the outline — and a dipole’s kinetic energy is one integral with one answer.

Why only the dipole #

It is worth being precise about why the far field of a translating body is a dipole and nothing else, because the whole exactness rests on it.

A closed body in an incompressible fluid displaces no net volume: whatever goes in on one side comes out on the other, so the source term in the multipole expansion is exactly zero. If it were not, the body would be inflating. A body that is not spinning up the fluid carries no net circulation either, so that term is absent too — which is the difference between this problem and what actually holds a wing up, where the circulation is the whole answer.

That leaves the dipole as the leading term, and the dipole’s strength is fixed by the requirement that the body’s surface be a streamline moving at the body’s own velocity. Quadrupoles and higher are present, they are determined by the shape, and they contribute nothing to the added mass in the limit of an unbounded fluid because their energies converge separately and their cross terms with the dipole integrate to zero — the same orthogonality that Kelvin’s minimum energy theorem turns on.

So the added mass is a property of the dipole, and the dipole is a property of the outline. That is the sentence to keep.

The number, taken from the energy rather than the formula #

Added mass is defined by the energy: the disturbance carries $\tfrac12 m_a U^2$, so

$$m_a = \frac{2}{U^2}\int_{\text{fluid}} \tfrac12\rho\,|\nabla\phi|^2\,dA.$$

Computing that integral rather than quoting the answer is the whole difference between believing a formula and checking one, so it is done here for the ellipse, whose closed form is $\pi\rho b^2$ along its long axis and $\pi\rho a^2$ broadside.

The integration is in elliptic coordinates, where the exterior of the ellipse is a half-plane and the integrand falls exponentially. That matters: a polar cut-off would leave an algebraic tail, and the error quoted would be a statement about where the counting stopped rather than about the flow. As it is, all six agree with the closed form to three parts in a million, and the residual is the radial quadrature.

The potential itself is worth a line, because getting it wrong produces a plausible answer. In elliptic coordinates $\xi, \eta$ with the body at $\xi_0$, the potential of a body translating at $(U, V)$ is

$$\phi = -e^{-(\xi - \xi_0)}\left(U b\cos\eta + V a\sin\eta\right),$$

and the first version of this computation carried a stray factor of the focal length. It was harmonic, it decayed, it satisfied the wall condition, and it gave an added mass 64 per cent low. Nothing about the field looked wrong. Only the closed form caught it.

It is not a quantity of fluid #

Now take the ellipse to zero thickness.

The displaced mass goes to nothing, as it must. The broadside added mass does not move at all: it is $\pi\rho a^2$ per unit span at every thickness ratio down to $10^{-6}$, unchanged to the last digit.

A body of no volume with a finite added mass. That single fact settles what the number is not. It is not the mass of any fluid the body carries with it, because there is no fluid inside the plate to carry and none of it is moving with the plate anyway — the velocity field is unsteady everywhere and identically zero nowhere.

The ratio of the two diverges as the reciprocal of the thickness. At a thickness ratio of a tenth it is ten; at a millionth it is a million. Every handbook that tabulates “added mass coefficients” is tabulating a number whose denominator is arbitrary, and the tabulation is fine as long as nobody reads the coefficient as a fraction of something carried.

The correct statement is the energy one, and it is the definition rather than an interpretation: the added mass is twice the kinetic energy of the whole disturbance, divided by the square of the speed. A plate broadside makes an enormous disturbance for its size, and that is all the divergence means.

It is not a scalar either #

The ellipse has two added masses and they differ by the square of its fineness ratio: $\pi\rho b^2$ along the long axis, $\pi\rho a^2$ across it. So accelerate it in any direction that is not a principal axis, and the force is not parallel to the acceleration.

For a 5:2 ellipse the miss reaches 46.40 degrees, at an acceleration angle of 21.8 degrees from the long axis. The closed form for the worst case is

$$\Delta\theta_{\max} = \arctan\frac{k - 1}{2\sqrt{k}}, \qquad k = \frac{m_y}{m_x},$$

and it gives 46.397 against the swept 46.396. A circle, run through the identical calculation, misses by $6\times10^{-15}$ degrees at every angle, which is what “exactly parallel” looks like when it is computed.

Push the ratio up and the miss approaches a right angle. A sufficiently slender body accelerated at forty-five degrees is pushed almost exactly sideways — which is not a curiosity. It is the mechanism behind the moment on a slender hull or fuselage in a steady stream, the one a body with no lift still feels, and it is a consequence of the added mass being a tensor with unequal eigenvalues and nothing else.

The plate is the limit of that sentence and it is worth stating, because it is the one case where the tensor has a zero in it. A plate of no thickness moving edgewise disturbs nothing at all in ideal flow, so one of its two added masses is exactly zero while the other is $\pi\rho a^2$ — a ratio of infinity, and the only body in this essay whose added-mass tensor is singular. The misalignment formula above then says that a plate accelerated at any angle other than along or across itself is pushed very nearly broadside, however gently it is pushed, and the limit of $\arctan[(k-1)/2\sqrt{k}]$ as $k \to \infty$ is a right angle. Nothing about that is a large effect being described loosely: it is a body whose response to a force is orthogonal to the force, in a theory with no viscosity, no circulation and no separation in it.

What it does to a real problem #

Two consequences of the tensor are worth having in front of a reader before the wall arrives, because they are the reason anybody computes added mass at all.

A bubble rises at a speed the tensor decides. A spherical bubble has essentially no mass of its own, so its equation of motion is $m_a \dot{U} =$ buoyancy minus drag, and the added mass is the only inertia in it. Halving the bubble’s density changes nothing; changing its shape changes everything, because the added mass is the shape’s.

The first instant of that motion is an exact number and a surprising one. Release a sphere of negligible density and the equation of motion at $t = 0$, before any drag has developed, is $(\rho_b V + \tfrac{1}{2}\rho V)\,a = (\rho - \rho_b)Vg$; take $\rho_b \to 0$ and the displaced mass cancels, leaving

$$a = 2g.$$

A bubble starts upward at twice the acceleration of gravity, and the two comes from the half and from nothing else. Run the same argument in two dimensions, where the coefficient is one rather than a half, and a massless cylinder starts at exactly $g$. The same released body, the same buoyancy, and a factor of two between them decided entirely by which rational the energy integral returned — which is as direct a demonstration as this essay can offer that the coefficient is a real dynamical quantity rather than a bookkeeping convenience.

And an accelerating aerofoil is not on its lift curve. The force on a wing that is being pitched or plunged is the circulatory force plus an added-mass force in quadrature with it, and the second is proportional to acceleration rather than to incidence — so a wing oscillating fast enough feels a force ninety degrees out of phase with the one the lift curve predicts. That is the non-circulatory half of unsteady aerofoil theory, and it is this tensor evaluated for a flat plate: $\pi\rho b^2$ per unit span, with $b$ the semichord, which is the same number the plate calculation above produced.

Neither of those needs any fluid to be carried anywhere. Both need the disturbance’s energy, which is what the number is.

And it is not a property of the body alone #

The last reading to dispose of is that a body has an added mass. It has one for a given surrounding — and the surrounding reaches further than the picture suggests.

A translating cylinder is a dipole; a wall is a mirror; a mirrored dipole needs an image inside the cylinder to keep the cylinder a streamline, and that image needs its own reflection. The series converges geometrically and its sum is the added mass — the same construction that a wall made by reflection sets up, taken to convergence rather than to first order.

At a gap of 1.05 radii the added mass is 39 per cent above the unbounded value. At two radii it is 6.7 per cent above; at five, one per cent; at fifty, one part in ten thousand.

The excess falls as the inverse square of the gap — the measured exponent is $-2.013$ over the range beyond three radii, against exactly $-2$ — and that slow decay is the practical content. An effect falling as the inverse square is still one per cent at five radii and a tenth of a per cent at sixteen, so a tank wall or a free surface anywhere within ten body sizes is inside the answer. The single-image approximation $1 + 1/4g^2$ accounts for nearly all of it, which is why the correction is usually quoted in that form.

The impulse, which is the honest bookkeeping #

There is a temptation to say that the added mass measures the momentum the fluid has acquired, and the momentum with no value is the essay about why that sentence cannot be completed. The momentum integral of an unbounded two-dimensional flow is conditionally convergent: it is a different finite number for every shape of region it is summed over, and there is no reason to prefer one.

The quantity that is well defined is the impulse — the integral of $\tfrac12\rho\, \mathbf{x}\times\boldsymbol{\omega}$, or equivalently the impulsive pressure that would set the flow up from rest — and the added mass is the impulse per unit velocity. That formulation is worth preferring for a reason beyond tidiness: the impulse of a system of vortices is conserved when nothing external acts on it, so writing the force as the rate of change of an impulse keeps a conservation law in view where the momentum formulation keeps a divergent integral in view.

The energy definition and the impulse definition give the same number, which they must: both are quadratic forms in the dipole strength, and the dipole strength is what the boundary condition fixed.

Two problems that are not the same problem #

There is one more confusion the word mass invites, and it costs a factor of two in the case where the number is used most.

A body accelerating through still fluid feels the added-mass force and nothing else: $m_a\dot U$, with $m_a$ the coefficient computed above.

A body held stationary in fluid that is itself accelerating feels that same force plus another. An accelerating fluid must have a pressure gradient driving it, and that gradient acts over the body’s volume exactly as gravity’s does — it is buoyancy with the fluid’s acceleration in place of $g$. So the force is $(m_a + \rho V)\dot U$, and the extra term is the displaced mass, not the added mass.

For a cylinder, where the two happen to be equal, that is a factor of two. Which is why the standard formula for wave loading on an offshore pile carries an inertia coefficient of about two rather than one: it is the sum of the added mass this essay computes and the pressure-gradient term the wave itself supplies, and dropping either halves the design load.

The general case is a body accelerating in a fluid that is also accelerating, and the two contributions carry different accelerations. Writing the force as “an effective mass times an acceleration” cannot represent that, which is the last thing the name gets wrong.

What survives all of that #

Three statements, and they are the exact ones.

The energy definition. $m_a = 2E/U^2$, where $E$ is the kinetic energy of the disturbance. Everything else in this essay is a consequence.

The rationals for the two symmetric bodies. A half for the sphere, one for the cylinder, exactly, in an unbounded fluid — and they are exact because the dipole’s energy integral is exact, not because the bodies are simple.

And the tensor. Six numbers in three dimensions, three in two, symmetric, positive definite, and a property of the outline together with whatever else is nearby.

There is a fourth statement that is nearly exact and worth separating from the three. The added mass of a body of given volume is smallest when the body is round. That is not proved here and it is very nearly true: among the ellipses computed above, the one with the least added mass per unit displaced mass at any orientation is the circle, and the coefficient rises monotonically with fineness in the broadside direction. It is the same family of statements as the cheapest shape the walls allow — an extremum over shapes rather than over fields — and like every extremum in this collection it is flat near its answer, so the practical content is that shape matters a great deal when a body is slender and hardly at all when it is not.

What does not survive is the picture of a coat of fluid travelling with the body. That picture gets the sphere right by accident — half the displaced volume is a plausible-sounding coat — and gets the plate, the misalignment and the wall all wrong, because none of them is about a quantity of anything.

Where the number came from historically, and why it looks like a mass #

Green computed the sphere’s coefficient in 1833 and Stokes rederived it in 1843, both of them working on pendulums swinging in air, where the effect is a measurable shift in period and nothing else about the fluid is measurable at all. The name they left behind — added mass, virtual mass, induced mass — has done a century and a half of quiet damage, because it invites the reading this essay is about.

The reason the effect looks like a mass is that the fluid’s kinetic energy is quadratic in the body’s velocity, with a coefficient that does not depend on the velocity. Anything with that property enters the equations of motion exactly as an inertia does, and the equations cannot tell the difference. So the pendulum’s period shifts by exactly the amount a heavier bob would produce, the sphere’s acceleration under a given force is exactly that of a heavier sphere, and every measurement of the motion is consistent with the coat-of-fluid picture.

Every measurement of the field, on the other hand, is not. The fluid a millimetre from the sphere is moving at a good fraction of the sphere’s speed and the fluid a diameter away is moving at a few per cent of it, with the disturbance decaying as the inverse cube — so there is no radius at which the fluid inside is “carried” and the fluid outside is not. What there is instead is a convergent energy integral, and half the displaced mass is its value.

The two-dimensional case makes the point harder to dodge: the cylinder’s disturbance decays only as the inverse square, its momentum integral does not converge at all, and the added mass is still exactly $\pi\rho a^2$ — a finite number extracted from a field whose momentum has no value.

What is not claimed #

The fluid is unbounded and ideal, and the motion is a translation. A rotating body has its own added inertia tensor, computed from a different set of potentials, and none of the numbers here applies to it.

The bubble’s $2g$ is an initial value, not a rise rate. It is the acceleration at the instant of release, when the drag is still zero because the speed is; a real bubble reaches its terminal speed in a few diameters and spends the rest of its life there, where the added mass does no work at all. What the number establishes is that the coefficient is measurable in principle from a transient, which is exactly how Green and Stokes measured it.

The wall calculation is for motion normal to the wall. Motion parallel to it has a different image system and a different coefficient, and the exponent above is not claimed for it.

The flat plate’s added mass is a limit, not a measurement. Every real plate has a boundary layer and a separating edge, and at any finite Reynolds number the ideal answer is a leading term with a viscous correction the size of the boundary layer’s own displacement — which is what a real fluid does to an exact theory everywhere else in this collection too.

And the energy integral was checked against closed forms it might have been fitted to. The check is that two routes to one number agree, not that either is independently confirmed by measurement; there is no experiment in this essay.