One formula for both ends

The first essay on this anchor asked what “of order one” is worth — where a dimensionless group’s threshold actually sits, as against where the two terms it compares happen to be equal. This one asks the next question. Given two limits, each with a good description and neither valid everywhere, how are they joined, and what does the joining cost?

The answer is the machinery of matched asymptotic expansions, and it is worth measuring rather than describing, because two things about it are usually stated and are not quite true.

A problem with an exact answer #

The model problem is the standard one, and it is chosen because every claim below can be checked against the answer rather than against another approximation:

$$\varepsilon y'' + y' + y = 0,\qquad y(0) = 0,\ y(1) = 1.$$

Its solution is two exponentials with $r = (-1 \pm \sqrt{1-4\varepsilon})/2\varepsilon$, and the characteristic roots satisfy $\varepsilon r^2 + r + 1 = 0$ to a part in 10¹⁶ — which is the check that costs nothing and catches a sign or a factor of two in the quadratic.

Setting ε = 0 drops the highest derivative and one boundary condition with it. What survives is $y' + y = 0$ with $y(1) = 1$, giving the outer solution $e^{1-x}$, which is wrong at $x = 0$ by a whole unit. Rescaling $X = x/\varepsilon$ recovers the lost derivative and gives the inner solution $e(1 - e^{-x/\varepsilon})$, which is wrong everywhere else.

Neither is a poor approximation. Each is an excellent approximation to a different part of the answer, and that is the situation the whole method exists for.

The errors, which do not depend on ε #

The first thing worth measuring is how bad each is, and the answer is a surprise the first time.

The outer solution’s worst error is e = 2.71828, and the inner solution’s is e − 1 = 1.71828, and neither moves with ε at all: the measured slopes are 0 and −10⁻⁵. Making the small parameter smaller does not make either of them better, because each is failing in the region the other owns and that region does not shrink — it moves.

That is what “singular” means in singular perturbation. In a regular perturbation problem every term improves as ε falls; here the leading term’s worst error is a constant, and no amount of care with the next term will change it.

The composite, and its exponent #

The composite is the sum of the two less the part they agree about:

$$y_c = e^{1-x} - e^{1-x/\varepsilon},$$

and its worst error falls as ε^0.987, measured on the thin half of the sweep. First order, uniformly, which is the entire content of the method.

Measured over the whole sweep, out to ε = 0.1, the exponent comes back as 0.869 and is still drifting — the same difficulty the slender-body error has and for a related reason: there is an $O(\varepsilon^2)$ correction sitting under the leading term, and an exponent read over a range that includes ε = 0.1 is a property of the range.

At ε = 10⁻⁴ the composite’s worst error is 2.7 × 10⁻⁴ against the outer solution’s 2.72. Four orders of magnitude, for one subtraction.

The region that justifies it #

Now the part that is usually stated and is not quite true.

The matching principle needs a region in which both descriptions hold — an overlap — in which the inner solution’s outer limit and the outer solution’s inner limit are the same thing. The construction above subtracts that common part, and the argument that the result is uniformly valid runs through the overlap.

Measured, it is 1.67 decades wide at ε = 10⁻⁴, running from x = 4 × 10⁻⁴ to x = 0.019. It is 0.69 decades at 10⁻³, 0.27 at 3 × 10⁻³, and at ε = 10⁻² it does not exist: there is no x at which both approximations are within five per cent.

And at ε = 10⁻² the composite’s worst error is 2.4 per cent.

So the construction works long before its own justification does. That is not a paradox and it is not a criticism of the method: the overlap argument is a statement about the limit ε → 0, and the formula it produces is simply better than the argument that produced it. It is worth knowing, because the usual presentation implies the opposite — that the method is licensed by the overlap and therefore fails when the overlap closes.

The picture nobody draws #

Part of why the overlap is misunderstood is that it is invisible in the standard figure.

On a linear x axis at ε = 10⁻³ the entire layer and the entire overlap occupy the leftmost pixel. On a logarithmic axis they are three separate regions with the overlap between them, and everything the method is about is visible.

That is a general point about drawing perturbation problems and this collection has made it before, in the Knudsen layer and in the wall region of a turbulent layer. A boundary layer is a structure in the logarithm of a coordinate, and drawing it on the coordinate itself hides the structure.

What the composite does not buy #

It is worth being precise about what the extra accuracy is.

The composite is not more accurate than the outer solution in the outer region. Out there its error is the same $O(\varepsilon)$ the outer solution already had, because the correction it added is exponentially small away from the layer. Inside the layer it is the same $O(\varepsilon)$ the inner solution had.

What it buys is that there is no region where the error is $O(1)$, and that is worth having for a reason that is about use rather than about accuracy: a single formula can be differentiated, integrated, plotted and handed to somebody else. Two formulas with a note about where each applies cannot.

There is also a choice in it that is rarely mentioned. The additive composite subtracts the common part; a multiplicative composite divides by it. Both are uniformly valid, they differ at $O(\varepsilon^2)$, and which is better depends on the problem. That the construction is not unique is a sign that its content is the uniform validity rather than the particular formula.

The fluid case, which needed the whole apparatus #

The reason any of this belongs in a collection about fluids is a drag calculation that was wrong for seventy years.

Stokes solved the creeping flow past a sphere in 1851 and got $C_D = 24/\mathrm{Re}$, which is right. Attempting the same thing for a cylinder gives no solution at all — Stokes’ paradox — and attempting the next term for the sphere gives a term that grows without bound far from the body. That is Whitehead’s paradox, and it defeated everybody until the 1950s.

The diagnosis is the one this essay is about. Stokes’ solution is an inner solution. Inertia is negligible near the sphere and is never negligible far enough away, because the neglected term goes as $\mathrm{Re}\,r$ relative to the retained one: at $r \sim a/\mathrm{Re}$ the expansion has broken down whatever the Reynolds number is. There is an outer region, Oseen’s linearised equation describes it, and the two have to be matched.

Stokes’ drag is within two per cent of the measured curve up to $\mathrm{Re} = 0.055$; Oseen’s $24/\mathrm{Re}\,(1 + 3\mathrm{Re}/16)$ up to $\mathrm{Re} = 0.817$. A factor of 14.9 in Reynolds number, for one term.

And the term that carries a logarithm #

The next term in the properly matched expansion is

$$C_D = \frac{24}{\mathrm{Re}}\left(1 + \frac{3\mathrm{Re}}{16} + \frac{9\mathrm{Re}^2}{160}\ln\frac{\mathrm{Re}}{2} + \cdots\right),$$

and the $\mathrm{Re}^2\ln\mathrm{Re}$ is the signature of the whole business. No regular expansion in the Reynolds number could produce it, because a regular expansion produces powers, and the logarithm comes from the matching itself — the same mechanism that puts $\ln(1/\varepsilon)$ into a slender body’s added mass.

A logarithm also changes sign. Below Re = 2 the term reduces the drag and above it increases it, which is computed here as −0.94 at Re = 1 and +6.18 at Re = 5 in units of the correlation. A term that changes sign inside the range it is supposed to correct is not something an intuition about “the next small correction” would have suggested.

The cylinder is worse still: its drag coefficient has $1/\ln(1/\mathrm{Re})$ at leading order, which approaches zero so slowly that no experiment has ever been done in the asymptotic regime. That is the same measurability problem the slender body has, and it is why Stokes’ paradox took a century to resolve into a formula anybody could use.

Where the two limits are in this collection #

The pattern recurs and it is worth listing where, because in each case the “inner” and “outer” are different physical descriptions rather than different regions of one formula.

A boundary layer is the archetype: the inviscid outer flow and the viscous inner layer, matched through the requirement that the outer flow sees the displacement thickness. Prandtl’s 1904 paper is a matched asymptotic expansion written twenty years before the technique had a name.

The wall law and the defect law in a turbulent boundary layer overlap in a region where both hold, and the only function consistent with both is a logarithm. That is matching used to derive a functional form rather than to join two known solutions, which is the most powerful thing the method does.

A shock’s internal structure is an inner solution to the outer problem in which the shock is a discontinuity — and the outer problem is the one everybody solves, with the jump conditions standing in for the whole of the inner solution.

Where the overlap goes when it goes #

The overlap region is the part of the method most often stated and least often measured, so it is worth being explicit about what the measurement above found and what it did not.

At ε = 10⁻⁴ the region in which the inner and outer expansions agree with each other to five per cent is 1.67 decades wide. At ε = 10⁻² it is zero decades wide — the two expansions never agree with each other anywhere to that tolerance. And the composite at ε = 10⁻² is still good to two and a half per cent.

Both halves of that matter. The first says the classical justification is not a formality: the overlap really does close, and at a value of the small parameter that is not small in any practical sense. The second says the composite does not stop working when it closes.

A construction that outlives its justification is not a paradox, but it is a warning. What the overlap guarantees is that the composite is uniformly valid as ε → 0 — a statement about a limit, with no claim at all about any particular ε. Every use of an asymptotic result at a finite parameter is outside what the derivation promised, and the only thing that can license it is a comparison with something independent. The exact solution supplies that here. In the problems the method is actually used for, nothing does, which is why the honest practice is to compute a second approximation with different assumptions and read the gap — the procedure the lift-slope essay is entirely about.

Knowing which end the layer is at #

Every step above began by putting the layer at $x = 0$, and nothing in the method chose that. It is a decision made before any matching happens, it can be made wrongly, and a wrong choice does not announce itself.

The rule comes from the fast root. With $\varepsilon y'' + a\,y' + \cdots$, that root is $r \approx -a/\varepsilon$, so the rapidly varying part behaves as $e^{-ax/\varepsilon}$. For positive $a$ it decays moving right, so it can be attached at the left-hand boundary and will be invisible at the other end — a layer at $x=0$. Reverse the sign of $a$ and that exponential grows to the right, so it cannot live at the left end at all and the layer is at $x=1$ instead. One sign decides which boundary condition the outer solution is allowed to satisfy and which one the layer repairs.

What makes it a hazard rather than a detail is what the wrong choice produces. Put the layer at the wrong end and the recipe still runs: an outer solution carrying the other boundary condition, an inner solution repairing it, a composite that is smooth, that satisfies both boundary conditions exactly, and that is nowhere near the answer. Nothing grows, nothing fails to converge, and the formula looks exactly like the one on this page. It is the same shape of failure a lift slope computed outside its own range has, arriving in the machinery rather than in the physics.

And when $a$ changes sign inside the interval, neither end is right: the layer is interior, its position is part of the answer, and it is the same object as the discontinuity an equal-area rule places — which is where the linear model problem stops being a model of anything simple.

What is left out #

Everything here is the leading term. Higher-order matched expansions require care about which variables are held fixed in which limit, and the composite of two two-term expansions is not the sum of two composites.

The model problem is linear. Nonlinear singular perturbations can have layers whose position is part of the unknown, which is a different and much harder problem — a free boundary rather than a fixed one.

And the five per cent in the overlap measurement is a choice. A tighter tolerance closes the overlap sooner and a looser one keeps it open longer; what does not depend on the choice is that the overlap closes at a value of ε where the composite is still good, which is the essay’s claim.

What this says about when to stop #

There is a practical question the measurements above answer and the usual presentation does not: given a problem with a small parameter, how small does it have to be before an asymptotic result is worth having?

The instinct is to look at the next term and require it to be small, which is the standard convergence test and is the wrong test here — asymptotic series do not converge, and for many of them the terms eventually grow. The right question is how the composite behaves, and the answer measured above is that it is within a per cent at ε = 0.005 and within two and a half at ε = 0.01, which is far more generous than the overlap argument would suggest.

The sphere gives the same reading from the other direction. Stokes’ drag is within two per cent to Re = 0.055, so the naive statement “creeping flow applies when Re ≪ 1” is about a factor of twenty too optimistic — which is the same kind of gap the fourteen-group survey found between where two terms are equal and where an observable moves. Adding one term moves the limit to Re = 0.82, which is where the folklore said the first term applied.

So the honest rule is not about ε at all. It is: compute the composite, compare it with something independent over the range of interest, and quote the range. Every figure in this essay is that procedure, done for a problem where the something-independent happens to be exact.

Where the method came from #

Prandtl had the idea in 1904 without the language. Friedrichs, Kaplun and Lagerstrom gave it the language in the 1950s, and it was Kaplun and Lagerstrom’s work on the sphere and the cylinder — with Proudman and Pearson arriving at the sphere independently in 1957 — that produced the $\mathrm{Re}^2\ln\mathrm{Re}$ and closed Whitehead’s paradox.

Van Dyke’s 1964 book is what made it a technique rather than a set of results. His matching principle — the m-term inner expansion of the n-term outer expansion equals the n-term outer expansion of the m-term inner expansion — is the operational statement of what the overlap region above is a picture of, and it is applied by most people who use it without any overlap being checked at all.

Which, given the measurement in this essay, is a defensible thing to do.

What this leaves #

Two limits joined by a construction that outlives its own justification, an error that is uniformly first order, and a logarithm that a regular expansion could not have produced.

That closes the regimes half of the argument. The essays that follow leave dimensionless numbers for a different kind of collapse: a whole distribution of circulation, replaced by the single scalar that lifts a wing — starting with the half of a section that carries no lift.