A wake that keeps the drag and forgets the body

What a jet keeps, and what it collects establishes the shape of a self-similar free shear flow: a momentum flux that is conserved exactly, a profile that becomes a fixed shape scaled by a width, and an entrainment that follows.

That is the destination. This essay is about the journey, and about how long it takes — because the usual statement that a wake becomes self-similar “a few tens of diameters downstream” turns out to be generous by an order of magnitude, and the reason is that the approach is a power law.

Two wakes with one drag #

The experiment is two initial velocity deficits with identical integrals. One is a slab — roughly what a bluff body leaves. The other is a pair of separated lobes — roughly what a body with a jet through the middle of it leaves, or a pair of bodies close together.

Their integrals are the same, so by the momentum theorem the two bodies have exactly the same drag. Everything else about them is different.

Both are then marched down the far-wake equation with a constant eddy viscosity, which makes the problem a diffusion in the streamwise coordinate: the profile spreads, and the only thing that cannot change is its integral.

What is kept #

The momentum deficit is conserved to four parts in 10¹¹ by the march. That is the drag of the body that made the wake, and it is the one thing about the body that the far wake still carries — which is the whole basis of the wake-survey method for measuring drag, and is what four profiles, one drag is careful about.

And what is lost, slowly #

A hundred initial half-widths downstream the two profiles are recognisably the same shape and are not the same profile: the difference is still about a fifth of the deficit.

The difference falls as the 0.96 power of the distance — essentially one over x — which is a power law and not an exponential. The distinction decides everything about how far the forgetting takes.

Twenty per cent agreement needs 217 initial half-widths. Ten per cent needs 457. Five per cent needs 933. Each halving of the tolerance roughly doubles the distance, which is what a power law gives and what an exponential would not: an exponential would need a fixed extra distance per halving, and the tolerance could be tightened almost for free.

Why a power law and not an exponential #

The mechanism is the same one that makes a duct’s memory a single mode, run in a domain with no walls in it.

A duct that forgets everything but one number finds an exponential decay, because the duct is bounded across the flow: the diffusion operator has a discrete spectrum, there is a slowest mode, and everything decays at its own rate.

A wake is unbounded. The diffusion operator on an infinite line has a continuous spectrum with no slowest mode, so there is no exponential to be had. What is left is the algebraic decay of the higher moments of the initial profile relative to the spreading Gaussian — and the first moment that can decay is the second, which decays as one over the width squared, which is one over x. The measured exponent is 0.9572, and a decay that slow is why the three thresholds sit at 217, 457 and 933 half-widths rather than within a factor of two of one another: halving the tolerance roughly doubles the distance, for ever.

A bounded diffusion has a slowest mode and an unbounded one does not, and every difference between the two essays follows from that sentence.

What the solver computed, and how it was checked #

The far-wake equation with a constant eddy viscosity of 0.02 in units of the free stream and the initial half-width, marched explicitly on 1,201 points across sixty half-widths either side, to two thousand initial half-widths downstream. That is a linear diffusion, so the answer is known in closed form for a Gaussian initial condition and the march is a way of carrying an arbitrary one.

Four checks. That the momentum deficit is conserved, to two per cent, which is what makes the comparison a comparison of two wakes with the same drag rather than of two different flows. That the two profiles converge to twenty, ten and five per cent. That the tighter tolerances are reached later, which would catch a non-monotonic error. And that the approach is a power law with an exponent between 0.4 and 1.2, which is the essay’s structural claim.

One failure is recorded because it was silent. The explicit march is stable only while the diffusion number is below a half, and at the first set of parameters it was 0.56: the profiles went to not-a-number, the deficit was reported as not-a-number, and nothing else complained. An unstable march produces a result that looks like a result until it does not, and the stability parameter is now checked before the first step.

The two wakes, read as two records #

Putting this essay beside its namesake in the circulation field is worth doing, because the two say opposite things about the same object and both are right.

A wake that says what made it inverts a trailing vortex pair back into the aircraft’s weight and span, and finds the inversion exact and well conditioned. Here two wakes with the same drag become indistinguishable, so nothing about the body can be recovered.

The difference is which quantities are conserved. A vortex pair carries circulation and impulse, both conserved, and the aircraft’s weight and span are functions of exactly those two — so the record is complete for those two questions and empty for every other. A momentum wake carries one conserved quantity, the deficit, and the body’s drag is a function of exactly that.

So both wakes are perfect records of a small number of things and useless records of everything else, and the number of things is the number of conserved quantities the wake carries. A wake remembers precisely its own invariants, which is a short and transferable statement and is the reason the two essays do not contradict each other.

It also says which questions to ask of a wake. Ask for a conserved quantity and the answer is exact whatever the distance; ask for anything else and the answer is a race between the measurement’s precision and the algebraic forgetting computed here.

What this means for a measurement #

Three consequences, and the first reverses a common practice.

A drag measured from a wake survey is reliable long before the wake is self-similar. The momentum integral of both profiles holds at 1.000000000 over the whole march, drifting by 4.2·10⁻¹¹ across two thousand half-widths, so the deficit is conserved from the first instant, so the integral can be taken anywhere the survey covers the whole wake — which is exactly why the method works close behind a body where the wake is still recognisably the body’s.

A profile measured from a wake is not a property of wakes until very far downstream. The two profiles differ by 197 per cent of the peak deficit where they start; the gap is still 28.6 per cent at 143 half-widths, 15.4 at 287, 10.5 at 430, 8.0 at 573 and 5.4 at 860, and it takes 933 before it is under a twentieth. Reports of a self-similar wake profile measured at fifty diameters are reports of a profile that still carries a large fraction of its body’s signature.

And two facilities can disagree without either being wrong. Wake profiles measured at the same distance behind different models converge to the same shape eventually and differ measurably at any practical station, which is one of the reasons free-shear-flow data scatter more than the measurements’ own precision.

Why self-similarity is a statement about forgetting #

The word “self-similar” describes a solution that looks the same at every station once it is scaled, and it is worth saying what that means in the language used here, because the connection is exact.

A self-similar solution has no memory of its initial condition except through its invariants. Its shape is fixed by the equation and its amplitude and width by the conserved quantities, so two flows starting differently and sharing their invariants end up identical.

That is why similarity solutions are found by dimensional analysis: if the only surviving information is a set of conserved quantities and the fluid’s properties, the answer can be assembled from them, and the initial condition has nowhere to enter.

The corollary is the useful one. A similarity solution is an attractor, and everything about how long it takes to be reached is outside it. The solution itself is silent about its own basin — it describes the end state and says nothing about the approach — which is exactly why this essay has to march a computation rather than evaluate a formula.

Exactly similar, and one number short is the same point in a jet: the similarity form is exact and one constant in it is not determined by the similarity at all, and has to come from the initial condition.

Why the far wake is not universal after all #

There is a stronger version of this result in the recent literature and it is worth flagging, because it goes further than anything computed here.

The classical picture says all wakes become the same eventually, differing only in a length scale set by the drag. Careful experiments in the 2010s found wakes from different generators that were still distinguishable in their scaling — not merely in their profiles — many hundreds of diameters downstream, and argued that the initial conditions are never entirely forgotten.

That is a claim about a limit that this computation cannot address: a constant eddy viscosity has no mechanism for a non-universal end state, so its wakes converge by construction and only the rate is at issue. What the computation does establish is that even in the most favourable possible model, the rate is algebraic, so the observation that real wakes take a very long time to forget is expected before any question of non-universality arises.

What a body’s wake starts as #

The two initial profiles are constructions and it is worth saying what a real one looks like, because the answer says which of them is nearer.

A bluff body sheds two shear layers which roll into a vortex street, and what a survey a few diameters downstream finds is a broad deficit with a good deal of structure in it — closer to the slab than to the lobes, and much less tidy than either.

A body with a propeller or a jet in the middle of it leaves a deficit with a surplus in the centre, which is the two-lobe case with the middle filled in. Such wakes are common — every self-propelled body has one — and they are the ones for which the forgetting matters most, because their profiles are furthest from the eventual shape.

And a self-propelled body at exactly zero net force has a wake with zero momentum deficit: the drag and the thrust cancel, so the one conserved quantity is zero and the whole of the wake is in the quantities that decay. Such a wake decays much faster than an ordinary one and its far field is set by its next invariant rather than by its first, which is a genuinely different problem and a good example of what happens when the leading conserved quantity vanishes.

What is really being forgotten #

It is worth naming the quantity, because it is not the profile.

The initial condition is a whole function, and the wake retains its integral exactly. What it loses is every higher moment: the second moment, which is how spread out the deficit was; the fourth, which is how peaked it was; and so on, each decaying faster than the last relative to the growing Gaussian.

So the memory is a sequence of moments decaying at increasing rates, which is exactly the modal picture with the modes relabelled. The slowest of them decays as one over the distance rather than exponentially, and that slowest one is what the measured difference is.

The parallel with a layer that is an integral of everything upstream is worth drawing: there the layer keeps one number and forgets the profile, which is the same statement about a bounded diffusion in the cross-stream direction. Both are cases of a system whose state is much smaller than its input, and both explain why an integral method works.

How far is far enough, in practice #

The distances above are in initial half-widths and a reader wants them in diameters, so it is worth saying what the conversion costs.

A bluff body’s wake begins with a half-width of roughly its own size, so the two hundred half-widths for twenty per cent agreement is a couple of hundred diameters. That is much further than any wind tunnel and comparable with the whole length of most towing tanks.

A streamlined body’s wake begins much thinner — a boundary-layer thickness rather than a chord — so its wake in half-widths is far further downstream at the same physical distance, and it forgets sooner in diameters. That is a real and useful asymmetry: a thin wake forgets faster than a thick one at the same distance, because the distance measured in the units that matter is larger.

And a real turbulent wake spreads faster than this constant-viscosity model, because its eddy viscosity grows with its width, which shortens all three distances by a factor that depends on the closure. The ordering and the algebraic character survive; the numbers should be read as a bound rather than as a prediction.

What the picture cannot show #

The wake is drawn as a one-dimensional deficit profile, and a real wake is three-dimensional, turbulent and unsteady. The eddy viscosity standing in for all of that is a constant, which is the crudest possible closure and is chosen so that the diffusion is exactly linear — the point being to isolate the forgetting from everything else.

Nothing here shows the near wake at all. The initial profiles are imposed at a station that is already in the far field, and how a body produces one is a separate problem this collection treats elsewhere.

Who found it, and when #

The self-similar far wake is Schlichting’s, from the 1930s, and the conservation of the momentum deficit is older and is the momentum theorem. The rate of approach has had far less attention than the limit, which is the usual arrangement: a similarity solution is a clean result and its basin of attraction is a messy one.

The non-universality question is Vassilicos and colleagues’ from the 2010s, alongside the non-equilibrium dissipation work that a dissipation that lags its production describes. The two are the same argument in different quantities: that turbulent flows carry their initial conditions much further than the classical theory allows.

Limits recorded rather than smoothed over #

A constant eddy viscosity. This is the most favourable possible model for forgetting, because it is linear and its solution converges to a Gaussian for any initial condition with a finite integral. A real wake’s eddy viscosity varies across and along it, and its convergence is slower.

No vortex street. A real bluff-body wake begins as a periodic array of vortices, and this collection’s own account of what it cannot compute is the street this site cannot draw. The march here starts downstream of all of that, with a smooth deficit imposed.

Two dimensions, and no turbulence. The march is a linear diffusion. There is no turbulence in it, no entrainment model, and no mechanism by which the initial condition could affect the eventual state rather than merely the rate.

The distances are in initial half-widths. Converting them to body diameters requires knowing the relation between a body’s size and the width of the wake it leaves, which depends on the body. The ratios between the three distances do not.

No streamwise pressure gradient. A wake in a pressure gradient does not conserve its momentum deficit at all — the deficit grows in an adverse gradient and shrinks in a favourable one — so the one quantity this essay says is kept is kept only in a uniform stream. That is the same qualification the wake-survey drag method carries and is why such surveys are made in the free stream rather than near a body.

And the exponent is this construction’s. Two initial profiles differing in their second moment give one over x; profiles differing only at higher moments converge faster. What is robust is that the approach is algebraic, not the value 0.96.