The theory with no memory in it

The theory that solves everything sets out ideal flow as a complete description: throw away viscosity, assume nothing is spinning, and fluid mechanics collapses into a linear problem with closed-form answers. That essay is about what the two assumptions buy and what they cost.

This one is about a property of the resulting theory that is easy to state and rarely stated: it has no memory at all, and the absence is exact rather than approximate.

The velocity field of an ideal, incompressible, irrotational flow is the solution of Laplace’s equation with the present boundary condition on it. Laplace’s equation contains no time derivative. So the field now is a function of the boundary now, and of nothing else — not of the boundary a moment ago, not of how the body got to where it is, not of anything.

The indicial kernel, and what it is for #

The clean way to ask how much memory a system has is to hit it once and watch.

Give a body’s velocity a step — from nothing to something, instantaneously — and record the force that follows. That record is the indicial response, and it is the whole of the system’s memory, because a linear system’s response to any input is the input convolved with it.

A system with no memory has an indicial response that is a spike at the instant of the step and zero afterwards. A system with a memory has a tail, and the length of the tail is how far back the input still matters. Over one crossing time the ideal kernel delivers 0.75 and then nothing; the compressible one delivers 2.027; the viscous one delivers 2.665, and its half-life is 1.62 crossing times.

For an ideal flow, the force is the added mass times the acceleration, which for a step in velocity is a delta function: all of it at the instant, none of it afterwards. Integrated over a six-τ window at a step of 0.015 τ, 100 per cent of the ideal kernel’s total has arrived in the first sample and the remaining 399 samples contribute nothing at all. That is not a limiting case or an idealisation of a short tail. It is an exactly empty tail, and the integral of the spike is the impulse — which is everything about the start, except one vector.

The two kernels beside it #

Two other models of the same body give tails, and their shapes are as different from each other as either is from the ideal case.

The viscous one falls as the inverse square root of time. That is the Basset history term of the unsteady Stokes force, and its shape has an unusual consequence: an inverse square root has no time constant. There is no moment at which the fluid has finished remembering, and the tail is not a remainder: over the same window the viscous kernel delivers 2.665 against the ideal kernel’s 0.75, and 61.5 per cent of that arrives after the first τ — more than half the answer is in the part a memoryless model calls the tail. Its origin is diffusion — vorticity generated at the surface at the instant of the step spreads outwards for ever, and the force it exerts falls only as fast as the diffusion is slow.

The compressible one holds and then settles. Immediately after the step the fluid has not yet been told the body is moving, so the body is pushing against a slab of fluid at the acoustic impedance — the piston result. Once the signal has crossed the body, in one crossing time, the flow reorganises into the steady one. It delivers 2.027 over the window, 48.8 per cent of it by the first crossing time and 85.1 per cent by the fourth, so this kernel has a definite duration and the duration is a length divided by the speed of sound.

On logarithmic axes the viscous kernel is a straight line, and the slope measured over the last two thirds of the window is −0.5000. That is the signature of a diffusive memory and it is worth recognising, because a measured response with that slope is a response whose mechanism is diffusion rather than any particular geometry — and it is the same −1/2 the wall kernel of the wall the fluid is listening to is built on, arrived at here from a force on a body rather than a velocity at a height.

How much has arrived, and when #

Reading the three as fractions delivered makes the practical difference plain.

The ideal flow is at one from the first instant. By half a crossing time the compressible kernel has delivered 24.4 per cent and the viscous one 26.2; by one, 48.8 and 38.5; by two, 69.7 and 56.1; by four, 85.1 and 80.9. The compressible one is ahead at every point after the first τ and finishes first, and the viscous one is still climbing at the end of the window and would be at the end of any window: its own half-life over this window is 1.62 τ, and that number is a property of the window rather than of the fluid, which is the whole difficulty.

The viscous kernel has delivered 61.5 per cent of its total after the first crossing time, which means 38.5 per cent of the force is still owed when the ideal model says the event is over. That last statement is the useful one for anybody choosing a model. A viscous memory cannot be truncated by waiting; it can only be truncated by deciding what accuracy is wanted and computing how far back that requires going. A compressible memory can be truncated by waiting a crossing time, and an ideal one requires no waiting at all.

Why a kernel is the right object to compare #

Setting three models side by side is only meaningful if there is something they are all answers to, and the indicial response is that thing. It is worth saying why, because the choice does real work.

A frequency response would do as well in principle and much worse in practice. It contains the same information, and it hides the property this essay is about: a kernel with a long tail and a kernel with none have frequency responses that differ mostly at low frequency, where measurements are worst and where the difference reads as a small phase shift rather than as an absence.

A force history for one particular motion is worse still, because it conflates the system with the input. Two models compared on a gentle ramp look similar for the reason everything about the start, except one vector gives — a slow input hides the differences between fast responses.

The step is the input that separates them, and it does so because it contains every frequency at once. The tolerance the comparison is made at is two per cent of the step’s own amplitude, and by that standard the ideal kernel is exhausted at t = 0, the compressible one at about a crossing time, and the viscous one not within the window at all. That is the standard argument for indicial testing and it applies here with unusual force, since one of the three responses is literally a spike.

The three doors #

The negative result — that an ideal flow has no memory — is the useful one, because it forces an inventory. Every memory this collection has found in a fluid must have entered through some relaxation of ideal, incompressible, irrotational, and there are only three of them.

Vorticity. In an ideal flow vorticity is a material label: it is carried by the fluid and cannot be created or destroyed in the interior. So a flow with vorticity in it is carrying a record of whatever put the vorticity there, and the record travels with the fluid. That is the door nothing in the present picks the flow walks through, and the one that gives a wing its circulation — because the vortex a wing leaves behind is a piece of the start-up still out there in the fluid.

A free surface. A boundary whose own position is part of the unknown is a state variable, and it stores potential energy. That is what makes water waves possible at all, and it is why a drift made of two things that average to zero has a memory to correlate against.

Compressibility. A finite signal speed means that what happens here now was decided somewhere else then — which is exactly the retarded time an acoustic calculation is written in, and which pressure has no speed computes the price of assuming away.

And viscosity, which is the first door with a different key. Viscosity does not give the fluid a new state variable; it gives the vorticity a way of being created at a wall and a way of diffusing, so it is the mechanism that operates the first door rather than a fourth one. That is why the viscous kernel is a diffusion law and why its slope is a half.

What the solver computed, and how it was checked #

The three kernels are closed forms rather than solutions of anything, and this figure set is deliberately a comparison of shapes rather than of magnitudes: the three are not to a common scale and could not be, since they belong to three different models with three different dimensional groups.

What is checked is the shape. That the ideal kernel is zero after the step — not small, zero — and non-zero at it. That the viscous kernel never reaches zero anywhere in the window. And that the viscous kernel falls by exactly a half over a factor of four in time, which is the inverse-square-root law stated as a testable ratio and reads within two per cent.

The refusals are exercised: the check is offered a tolerance of zero on that ratio and refuses it.

What the absence buys, and what it costs, in one comparison #

The clearest way to feel the difference is to ask each model the same practical question: how much of the force on a body accelerated ten seconds ago is still being felt now?

Ideal: none. Not a small amount — none. The force ended when the acceleration did, and the field now depends on the velocity now.

Compressible: none, after a crossing time. For a metre-scale body in air that is about three milliseconds, so at ten seconds the answer is also none, and the model has become the ideal one.

Viscous: a definite fraction, and it depends on how the question is asked. Because the kernel is an inverse square root, the force at ten seconds after a step is proportional to one over the root of ten — about a third of what it was at one second, and about a tenth of what it was at a tenth. There is no “has finished” in that sentence anywhere.

The last of the three is why particle-tracking codes carry a history integral, why they are expensive, and why the usual approximation — truncating the history at a few time steps — is a decision about accuracy rather than a simplification. Truncating an exponential tail is safe once a few time constants have passed; truncating a power-law tail throws away a fraction that has to be computed.

Why a memoryless theory is worth having #

It would be easy to read all this as a list of the ideal theory’s failures, and it is closer to the opposite.

A memoryless model is enormously cheaper. There is no history to store, no convolution to evaluate, and no question of how far back to go. A panel method solves a linear system once per instant and is done, which is why unsteady aerodynamics is done that way even now — with the memory added back as a separate wake model rather than by abandoning the potential solver.

And it is the right model for the fast part. The added-mass force arrives immediately and the viscous and compressible corrections do not, so at short times after a disturbance the ideal answer is the leading term. What the kernel comparison adds is the warning that “short” has to be measured against a crossing time or a diffusion time, and that the viscous correction never becomes negligible in the way a decaying one would.

The trap is the other reading. It is often said that ideal flow is valid at short times because viscosity has not had time to act. That is right about the magnitude and wrong about the character: the viscous force at short times is larger relative to the total than at long times, because an inverse square root is largest at the start.

Where the kernel is the whole answer #

The convolution picture is not a way of talking about the problem; it is how the problem is solved in practice, in at least three places in this collection.

Unsteady aerofoil theory. The lift following an arbitrary motion is the motion convolved with Wagner’s indicial function, which is the lift that arrives late — and the kernel there is the wake, which is vorticity, which is the first door.

Particle dynamics in a fluid. The force on a small sphere is a sum of an added-mass term with no memory, a Stokes drag with no memory, and a Basset term which is entirely memory. Which of the three dominates is a question about the particle’s density and its Stokes number.

And aeroacoustics. The sound at a listener is an integral over the source at a retarded time, which is a kernel that is a delta function at a delay rather than at zero — the compressible door, with the memory carried entirely in the delay.

How to tell which door a measured memory came through #

The three kernels have distinguishable shapes, and that makes the inventory usable rather than merely tidy. Given a measured response, three questions settle it.

Does the tail have a time constant? If the response decays exponentially, or settles after a definite interval, the memory has a length scale and a speed in it — which points at compressibility, or at a convection time, or at a structure being swept past. If it decays as a power, it points at diffusion.

Is the power a half? Diffusion of vorticity from a wall gives an inverse square root, exactly, and the exponent is a strong fingerprint: it is set by the diffusion equation rather than by geometry, so it is the same for a sphere, a plate and a cylinder.

Does the response depend on the fluid’s density alone, or on its viscosity? An added-mass response scales with density and not at all with viscosity, so a test in two fluids of similar density and very different viscosity separates the memoryless part from the rest immediately.

None of those needs a model of the body, which is what makes them worth having: they identify the mechanism before anybody has to choose an equation to fit.

What the picture cannot show #

The ideal kernel is a delta function and a delta function cannot be drawn. What appears in the first figure is a tall narrow rectangle of finite width, and its width is a drawing decision: the true object is infinitely tall and infinitely narrow with unit area. A reader comparing its height with the others’ is comparing an artefact.

The three curves are also, as stated, not to a common scale. Their relative heights carry no information whatever, and only their shapes and their extents do — which is an unusually severe restriction for a figure on this site, and is the reason the comparison of fractions delivered exists as a separate view.

Who found it, and when #

The absence of time in Laplace’s equation has been evident since the equation, and the added-mass result is Green’s and Stokes’. The Basset term is Basset’s and Boussinesq’s, independently, in the 1880s, and its inverse-square-root form was regarded as a curiosity for most of a century before particle-laden flow made it a practical necessity.

The indicial framing — treating a fluid’s response as a kernel to be convolved — is Wagner’s and von Kármán’s, from the 1920s and 1930s, and it arrived from aeroelasticity rather than from fluid mechanics proper, because that is where somebody needed the response to an arbitrary motion rather than to a sinusoid.

Limits recorded rather than smoothed over #

The three kernels are models, not solutions. Each is the standard closed form for its case, written down here to be compared; none is computed from a flow solver, and the compressible one in particular is a schematic of the piston-then-settle behaviour rather than a solution of the wave equation about a particular body.

Linearity is assumed throughout. A kernel is only the whole of a system’s response if the system is linear, and a body at large amplitude in a real fluid separates, which is not.

The viscous kernel is the unsteady Stokes one, valid at small Reynolds number. At large Reynolds number the memory is in the boundary layer’s own history instead, which has a different shape and a different origin, and is the subject of the viscous essays in this collection.

Nothing here computes a flow field. This essay is a comparison of response shapes, and the flows those responses belong to are computed elsewhere in the collection — the ideal one in the theory that solves everything, the viscous layer in the wall that shakes, and the compressible signal in pressure has no speed.

And the three doors are an inventory of this model’s relaxations, not of physics. A fluid with a suspended phase, a chemical reaction, a polymer or a temperature-dependent viscosity has memories of its own that none of the three describes.