The force of getting going

Anyone who has swung a hand through water knows that a fluid resists being pushed about, and the exact theory of an ideal fluid says the resistance to steady motion is precisely zero. Both are true, and the reconciliation is that the resistance being felt is not drag.

A body moving steadily through an ideal fluid feels nothing. A body changing speed feels a force — not because the fluid rubs on it, but because the fluid around it has to change speed too, and something must supply the energy for that.

The frame the question has to be asked in #

Almost every figure on this site is drawn in the frame of the body, with a stream running past it. That frame is useless here, and the reason is worth stating because it is the first thing that goes wrong.

In the stream frame, the fluid far away is moving at $U$, so its kinetic energy density far away is $\tfrac{1}{2}\rho U^2$ — a constant, over an infinite plane. The total energy is infinite, and no statement about “the energy the body puts into the fluid” can be extracted from it.

Choosing a frame is not usually a decision with consequences — the equations are the same in any of them, as the account of what a flow is sets out — but the integral is not frame-invariant, and that is the whole difficulty.

In the frame where the fluid is at rest at infinity, and the body moves through it at $U$, the velocity dies away as $1/r^2$ and the energy density as $1/r^4$. Integrated over an area element that grows only as $r$, that converges. The whole question exists only in this frame, which is why added mass is a property of unsteady motion and cannot be inferred from any steady picture.

The field in that frame is a pure doublet: the uniform stream has been subtracted away, and what remains is the disturbance the body makes. Its speed is $Ua^2/r^2$ at distance $r$, exactly.

What the solver computed, and how it was checked #

The kinetic energy of the exterior flow is

$$T = \tfrac{1}{2}\rho \iint_{r>a} \left(\frac{Ua^2}{r^2}\right)^2 r \, dr \, d\theta$$

which is elementary and comes to $\tfrac{1}{2}\rho\pi a^2 U^2$. The site’s habit is not to trust an integral it has only done on paper, so the same integral is evaluated numerically on a logarithmic radial mesh — logarithmic because the integrand falls as $r^{-3}$ and a uniform mesh would spend every sample where there is nothing left to add.

With the outer limit at 200 radii, the numerical total is 1.570757 against the closed form 1.570796, and the implied added mass is 3.141513 against $\rho\pi a^2 = 3.141593$. That is 0.0025%, and the residue is the tail beyond the outer limit rather than the quadrature.

The convergence itself is the interesting part.

The fraction inside radius $r$ is exactly $1 - a^2/r^2$, so the truncation error of stopping at $R$ is $a^2/R^2$: 25% at two radii, 4% at five, 0.03% at sixty. The build asserts that the integrated mass agrees with $\rho\pi a^2$ to within 0.2%, and refuses a mass 1% away from it — which is a tolerance chosen so that a domain too small to converge would fail rather than pass quietly.

The mass of fluid the cylinder displaces #

The answer, $\rho \pi a^2$ per unit length, is exactly the mass of fluid occupying the volume the cylinder occupies. A cylinder accelerating through an ideal fluid behaves as though it had twice its mass if it is made of the same stuff as the fluid, and the extra is not an approximation or a fitted coefficient. It is an integral over the plane that happens to come out to the displaced mass.

That coincidence is a two-dimensional accident, and it is worth saying so before it becomes a rule. For a sphere the added mass is half the displaced fluid, not all of it. For a flat plate moving broadside on, of width $b$ per unit span, it is $\rho\pi b^2/4$ — a plate of no thickness whatsoever, displacing nothing, with a perfectly definite added mass. That last case is the one that kills any intuition about “fluid being carried along”: there is no volume to carry.

The honest statement is that added mass is a property of the field, not of the body’s volume. It measures how much fluid has to be moved to get the body out of the way, and a thin plate moving edge-on has almost none while the same plate broadside has a great deal.

Stretching the body, and watching the coincidence go #

Rather than assert that, the site computes it. The same energy integral runs on a family of Rankine ovals — bodies that are level sets of the streamfunction rather than shapes anybody drew — of increasing length at roughly constant width.

The generic integrator is calibrated first on the case whose answer is known: run on the circle it returns an added mass of 3.14293 against $\rho\pi a^2 = 3.14159$, which is 0.04% and is the mesh rather than the physics. Only then is it pointed at the ovals.

The trend is monotone, and the build asserts that it is: a set of ovals in which a longer body carried more added mass per unit displaced would be refused, as would any ratio above one, since the circle is the worst case in this family.

This is what the word streamlined means when it is said about acceleration rather than about drag. The usual justification for a long, tapered shape is that the boundary layer stays attached and the pressure drag is small — a viscous argument. The result here is inviscid: even with no viscosity anywhere, a stretched body is cheaper to accelerate, because the fluid it has to push aside has a gentler path to take.

Why the paradox is not contradicted #

d’Alembert’s paradox is the statement that the drag on a body in steady potential flow is zero, and it survives everything on this page intact. The force computed here is not a drag: it is in phase with acceleration rather than with velocity, it vanishes the instant the motion becomes steady, and it does no net work over a cycle that returns the body to its original speed.

The distinction matters because the two are constantly conflated in ordinary language. A hand pushed through water feels resistance that is mostly real drag, from separation and viscosity, and a small unsteady component when it is speeding up. An ideal fluid supplies only the second.

And the second is not small in every application. For a body much lighter than the fluid it moves in — a bubble, an airship, a balloon — the added mass dominates the body’s own, and the equations of motion are almost entirely about the fluid. A bubble rising in water accelerates as though its mass were half that of the water it displaces, its own contents being negligible, which is why bubbles reach terminal velocity so quickly.

Where the energy actually goes #

There is a temptation to picture the added mass as a lump of fluid glued to the body and dragged about with it. The energy map in the first figure refutes that directly: the energy is not concentrated in a shell around the surface but spread out, with a quarter of it beyond two radii.

The flow pattern is the giveaway. Fluid in front of the cylinder is being pushed forward and sideways; fluid behind is closing in forwards to fill the space; fluid at the shoulders is sliding backwards relative to the body. Nothing travels with it. What propagates is a pattern of displacement, and the pattern has energy in it.

It is not a number, it is a matrix #

The plate that has almost no added mass edge-on and a great deal broadside is not an oddity; it is the first sign that the quantity is not a scalar at all.

The kinetic energy of the fluid is a quadratic form in the body’s motion, and a body has six ways to move — three translations and three rotations. So

$$T = \tfrac12\, m_{ij}\,U_i U_j,$$

with $m_{ij}$ a six-by-six array. It is symmetric, and for the same reason Munk’s reciprocal theorem is: a quadratic form’s matrix has no antisymmetric part to contribute, so $m_{ij} = m_{ji}$ identically, with nothing in the arithmetic arranging it.

For a body with three planes of symmetry the matrix is diagonal, and the six numbers are the added masses for surge, sway and heave and the added moments of inertia for roll, pitch and yaw. A sphere is the extreme case: all three translational entries are half the displaced fluid, and all three rotational ones are zero, because a sphere spinning in an ideal fluid moves no fluid at all.

Off the diagonal is where the surprises are. A body without symmetry, accelerated straight ahead, feels a force sideways and a moment as well — because the energy’s cross terms are not zero, and the force is the gradient of the energy. That is why a submarine’s or an airship’s equations of motion carry a full matrix rather than three numbers, and why an asymmetric hull accelerating in a straight line does not stay in one.

And it supplies the mechanism for a result this collection reaches elsewhere by a different route. A slender body translating steadily at incidence feels a pitching moment of magnitude $\tfrac12(m_{22}-m_{11})U^2\sin 2\alpha$ — the difference between the transverse and longitudinal added masses. For a slender hull the first is nearly the displaced fluid and the second is nearly nothing, so the moment is of order $\rho V U^2\alpha$ and it is destabilising. That is exactly the Munk moment, and it is a steady force arising from an added-mass coefficient — which is a useful corrective to reading the whole subject as being about acceleration.

Where it is actually felt #

Four places, in rough order of how much of the total force it accounts for.

Ships and offshore structures. A hull heaving in a swell is accelerating continuously, and its added mass is comparable with its real one — for some motions considerably larger. The natural periods of a moored structure are set by the sum, so a calculation that omitted the fluid would predict resonance at the wrong frequency and put the answer in the middle of the wave spectrum instead of outside it.

Anything lighter than what it moves in. A bubble, an airship, a balloon. Here the added mass dominates the body’s own by an enormous factor, and the equation of motion is almost entirely about the fluid. A rising bubble reaches its terminal velocity in a fraction of the time a naive calculation predicts, because most of the inertia being accelerated is water.

Manoeuvring flight and control response. An aircraft rolling or pitching quickly is accelerating air as well as aluminium, and the apparent-mass terms appear in the flight-dynamics equations as additions to the moments of inertia. For a conventional aeroplane they are small; for a very light, large-area vehicle — a glider, a hang glider, an airship — they are not.

Slamming and the first instant of impact. When a hull, a float, or a wave-piercing bow strikes water, the force in the first milliseconds is added mass changing as the wetted area grows. It is the entire content of the impact load, and it happens before any wake exists, which is precisely the window in which the ideal theory is quantitatively right.

What the picture cannot show #

The energy field is drawn at one instant of a motion that is, in the mathematics, steady in the body’s frame. Nothing in the picture moves, and the whole argument is about what happens when the speed changes — an event that cannot be drawn in a still frame at all.

Nor does the picture show the reaction. The force the fluid exerts back on the body is distributed over the surface as a pressure, and in the accelerating case that distribution is fore-and-aft antisymmetric, which is exactly what makes it sum to a force rather than to nothing. It is the mirror image of the steady case, where the distribution is symmetric and sums to zero.

The force, written down #

It is worth having the equation, because it is short and because its shape says more than the number does.

For a cylinder accelerating at $\dot{U}$ through an unbounded ideal fluid, the force the fluid exerts on it is

$$F = -\rho \pi a^2 \dot{U}$$

per unit length, opposing the acceleration. Three features of that expression are the whole argument. It is proportional to $\dot{U}$ and not to $U$, so it vanishes for steady motion, which is the paradox restated. It is proportional to $\rho$ and not to viscosity, so it survives in an ideal fluid where drag does not. And it is linear, unlike drag, which goes as $U^2$ — so at low speeds the unsteady force dominates and at high speeds the drag does.

The crossover between the two is worth an estimate. Comparing $\rho \pi a^2 \dot{U}$ against a drag of order $\tfrac{1}{2}\rho U^2 (2a) C_D$ says that the added-mass term wins whenever $\dot{U} a / U^2$ exceeds about $C_D/\pi$ — that is, whenever the body changes its speed appreciably within the time it takes to travel its own diameter. Slow manoeuvres are drag problems; sudden ones are added-mass problems. A hand pushed steadily through water is the first; the same hand slapped onto the surface is the second.

Where the model stops #

Three boundaries.

Viscosity is absent. A real accelerating body sheds vorticity as well as displacing fluid, and at high enough accelerations the shed vorticity dominates. The clean result here belongs to the first instant of motion, before a wake has developed — which is exactly when the ideal theory is at its best, and is one of the very few circumstances in which it is quantitatively right about a real flow.

The body is rigid and the fluid unbounded. Added mass depends on the boundaries: the same cylinder accelerating near a wall has a different added mass, because the fluid displaced has less room to go round.

The motion is small. For a body oscillating with a large amplitude in a real fluid, the separated flow from the previous half-cycle is still there when the next one begins, and the force acquires a history term that no potential theory produces. The complete unsteady equation of motion for a small sphere — added mass, steady drag, and a memory integral over the whole past — is the Basset–Boussinesq–Oseen equation, and it is a long way beyond anything drawn here.

Who found it, and when #

The concept is Green’s, from 1833, and Stokes gave it its modern form in the 1840s while studying pendulums swinging in air — a problem in which the added mass of the bob is a measurable correction to the period, and was the first place anybody needed the number.

That origin is worth keeping. Added mass entered physics not as an aerodynamic curiosity but as an error term in the most precise instrument of the age, and Stokes’ paper on it is also where the viscous correction to a pendulum’s damping first appears. The two effects — one from the ideal flow, one from the boundary layer — were separated in the same work.

Where the ladder goes next #

The rung below is the paradox itself, which says the steady force is zero and is right. This rung is the fine print: zero steady force is not no force.

Sideways from here lies what a parcel actually does, which is the same distinction seen from the fluid’s point of view — a steady flow in which every particle is accelerating violently. The two essays are the same observation about the material derivative, one applied to the body and one to the fluid.

Above it, the unsteady thread continues into the vortex a wing sheds when it starts, where the fluid’s memory of the acceleration is not an energy but a piece of circulation left behind.