The body the outer flow actually sees

The two halves of this subject are usually taught as though they were separate. Outside, a potential flow, exact and closed-form and predicting nothing has any drag. Inside a thin layer, a viscous flow that supplies the friction and the separation. The outer solution provides the pressure the inner one runs in, and the traffic is one-way.

It is not one-way, and the return channel is the whole of this rung. The boundary layer tells the outer flow what shape the body is.

The definition, and what makes it more than a definition #

The layer slows the fluid near the wall, so less mass gets past a station than would have got past if the fluid were inviscid. Write the deficit as an equivalent thickness:

$$\rho U \delta^* = \int_0^\infty \rho\,(U - u)\,dy, \qquad \delta^* = \int_0^\infty\left(1 - \frac{u}{U}\right)dy .$$

That is arithmetic. What makes it a physical statement is the next line: the same reduced mass flow is carried by an inviscid flow past a wall moved out by $\delta^*$. So an outer problem posed on the displaced body reproduces what the viscous flow does to it, without containing any viscosity.

For the flat plate, integrating the Blasius profile gives $\delta^* = 1.72077\sqrt{\nu x/U}$, and the deficit computed directly from the profile in physical units agrees with $\rho U\delta^*$ to a part in $10^{15}$ — which is what a definition working looks like, and is worth checking because this collection has already been caught by a factor of $\sqrt2$ in the normalisation of the same solve.

Two routes to the same number, and neither of them expects the other #

There is a second way of handing the layer’s effect back, and it is where the confidence in the first comes from.

Instead of displacing the wall, keep the wall where it is and blow fluid through it at the rate that reproduces the deficit:

$$v_w = \frac{d}{dx}\left(U_e\,\delta^*\right).$$

For the flat plate that is $0.86039\,U/\sqrt{Re_x}$.

Now compute something else entirely: the Blasius solution’s own vertical velocity at the top of the layer. It is $\tfrac12\sqrt{\nu U/x}\,(\eta f' - f)$ as $\eta\to\infty$, a limit of the profile which involves no integral of it at all, and it comes to $0.86039\,U/\sqrt{Re_x}$.

The same number to seven figures, from two computations with nothing in common. One differentiates an integral of the profile; the other takes a limit of the profile’s second variable. Their being equal is the statement that a boundary layer pushes the outer flow outwards at exactly the rate its own deficit grows, and it is the closest thing available to a proof that the displaced-body picture is exact rather than a way of speaking.

Why a thin layer is worth so much #

The size of the correction is worth stating, because it decides which problems the coupling matters for.

On a wing chord of two metres at a hundred metres per second, $Re$ is about $1.3\times10^7$ and the laminar displacement thickness at mid-chord would be two-thirds of a millimetre — three parts in ten thousand of the chord. That is a change of shape nobody would draw.

And it is not negligible, for two reasons. The layer grows fastest at the tail, where the section is thin and a tenth of a millimetre is a large fraction of the local thickness; and its effect on the outer flow is through its gradient, so what matters is $d\delta^*/dx$, which is largest exactly where the pressure is rising and the layer is thickening fastest.

The result is a systematic reduction in the effective camber near the trailing edge, an effective incidence lower than the geometric one, and a lift-curve slope below $2\pi$. Every measured aerofoil section has a slope between about 5.5 and 6.1 per radian against the inviscid 2π, and this is where the difference comes from.

Why the coupling has to be an iteration #

The two halves depend on each other, and the direction of the dependence is worth being explicit about because it decides how the calculation has to be organised.

The layer needs the outer velocity distribution $U_e(x)$, which is what drives it: the pressure gradient along the surface is $-\rho U_e\,dU_e/dx$ and nothing else enters. The outer flow needs the body’s shape, which now includes $\delta^*(x)$, which came out of the layer.

So neither can be computed first, and the standard resolution is to compute the outer flow on the bare body, run the layer on that, displace, and go round again. On an attached flow that converges in two or three passes, because the correction is small and its effect on $U_e$ is smaller still.

That is the weak interaction, and its convergence is a statement about the size of the correction rather than about the structure of the problem — which is why it stops converging exactly where the correction stops being small.

Running the coupling #

The calculation is an iteration, and the cheapest honest way to do it uses a boundary-layer method that needs only the outer velocity.

Thwaites’ method is one quadrature:

$$\theta^2 = \frac{0.45\nu}{U_e^6}\int_0^x U_e^5\,dx,\qquad \lambda = \frac{\theta^2}{\nu}\frac{dU_e}{dx},$$

with the shape factor from a correlation and separation at $\lambda = -0.09$. Feed it a potential-flow surface velocity and it returns $\delta^*(x)$; displace the body by it; re-solve the outer flow; repeat.

On a circular cylinder the first pass already produces the number the method is judged by. The potential distribution is $U_e = 2U\sin\theta$, and Thwaites separates at 103.2 degrees from the front stagnation point.

Where the coupling stops #

That last sentence is the honest content of this rung, and it deserves to be a section rather than an aside.

Run the same coupling on an aerofoil and it does not finish. At three million Reynolds number a laminar layer on a twelve per cent section separates at 45 per cent of the chord at zero incidence, 36 per cent at two degrees and 28 per cent at four. It never reaches the trailing edge, which is exactly where the displaced shape was going to matter.

And past separation the method does not merely stop being accurate. Thwaites’ shape-factor correlation has a pole at $\lambda = -0.14$, so a calculation carried through separation returns displacement thicknesses of any size at all, and a lift computed from the resulting camber line is arithmetic rather than aerodynamics. The first version of this computation returned a lift coefficient of $-50{,}430$ at six degrees, and drew a perfectly smooth camber line to go with it.

So no coupled lift-curve slope is quoted here. The check written for this module asserts that the coupling fails to close, which is a check that demands a failure and exists so that a later change cannot quietly produce a number.

What would be needed #

The gap is not conceptual, it is a missing model, and naming it is worth more than papering over it.

A real aerofoil at those Reynolds numbers has a turbulent layer over most of its chord, and a turbulent layer survives an adverse gradient far better than a laminar one — which is what the drag crisis is and why a golf ball has dimples. The price is paid in friction: a turbulent layer’s skin friction is several times a laminar one’s, so the trade is a certainty of higher friction against a possibility of avoided separation. Doing the coupling properly needs a transition criterion to say where the layer changes character and a turbulent integral method to carry it onwards.

Both exist and neither is in this collection. What is here is the laminar half, computed exactly, together with the measurement of where it stops — which is the more useful thing to know, because a turbulent method quietly applied from the leading edge would produce a plausible answer at every station and would be wrong in the first twenty per cent.

What the displaced shape actually looks like #

It is worth describing, because the effect is often stated as “the layer makes the aerofoil thicker” and that is not what it does.

The layer is thin at the leading edge and grows along the chord, and it grows fastest on the upper surface, where the pressure rise after the suction peak is steepest. So the displaced upper surface is lifted a little near the tail and the displaced lower surface is lowered less. The resulting mean line is bent upwards at the back, which is aerodynamically a small negative flap deflection.

That is decambering, and it produces exactly the effects a small upward flap does: less lift at a given incidence, so a shallower lift-curve slope; a nose-up change in pitching moment; and a zero-lift angle that moves. All three are measured on every real section, all three grow as the Reynolds number falls and the layer thickens, and all three are absent from any calculation that treats the outer flow as a one-way supplier of pressure.

The half of the coupling that happens after the body ends #

Every displacement thickness above is computed on a surface, and the layer does not stop when the surface does. The two layers leave the trailing edge, merge, and continue downstream as a wake with a displacement thickness of its own — and a coupled calculation that stops at the tail has thrown away the part of the correction that decides both of the numbers a designer wants.

The circulation is set there. A sharp-edged inviscid section takes its circulation from the Kutta condition, which is a statement about a mathematical point. A real section takes it from what the two merging layers do: the effective body the outer flow sees does not close at the trailing edge, it tapers away into the wake, and the streamline that leaves is the displaced one rather than the geometric one. So the circulation a section actually carries is fixed by the displacement thickness at and just behind the tail, which is why a section with a thickened trailing edge, or one whose upper-surface layer is close to separating, loses lift that no inviscid calculation would predict. Every practical coupled method carries a modified Kutta condition for exactly this reason, and its form is a modelling choice rather than a derivation.

And the drag is measured there. The profile drag of a section is the momentum thickness of its wake far downstream, times $\rho U^2$ — which is a statement about a station the calculation never reaches, since a boundary-layer march ends at the tail. The standard bridge is Squire and Young’s formula,

$$C_d = 2\,\frac{\theta_{\text{TE}}}{c}\left(\frac{U_e}{U_\infty}\right)^{(H+5)/2},$$

which extrapolates the trailing-edge momentum thickness to infinity using the fact that in the wake there is no wall, so the shape factor falls steadily towards one and the momentum thickness grows in a way the integral equation can be solved for. That exponent is where a coupled panel code’s drag number comes from, and it is doing a great deal of work: it converts a quantity computed at the last station of a march into a quantity defined at infinity, using a model of what happens in between.

The wake’s own displacement thickness matters to the outer flow as well, for a chord or two downstream, and neglecting it is one of the standard small errors in a coupled method — it slightly changes the pressure near the tail, which slightly changes the layer, which changes the drag. On an attached section that loop is small. On one near separation it is not, and it is one of the reasons a coupled method’s predictions degrade so sharply as the trailing edge approaches stall.

Which sharpens this essay’s own conclusion. The laminar coupling computed here fails because it separates before the trailing edge, and the trailing edge is where the lift is decided. The wake argument says the difficulty goes one step further: even a method that reaches the tail has not finished, because both of the section’s characteristic numbers are defined downstream of it — the lift by the displaced streamline that leaves, the drag by the momentum deficit that arrives at infinity — and a march that stops at the last panel has to be handed both by a model.

The strong interaction, where the traffic reverses #

There is a regime in which the coupling is not a correction at all, and it is worth knowing that it exists because it is where the whole scheme breaks down.

Near separation, and near a trailing edge, the displacement thickness’s effect on the outer flow is comparable with the pressure gradient driving the layer. The two halves then have to be solved simultaneously: the layer cannot be marched from the leading edge because what happens downstream influences what happens upstream through the outer flow.

That is the triple-deck structure, worked out in the late 1960s, and it is one of the more impressive pieces of asymptotic analysis in the subject. It explains why a boundary layer can feel a corner before it arrives at one, which the ordinary marching theory forbids outright.

What the picture cannot show #

The displaced body is not drawn on the aerofoil figures. At the thicknesses computed here it would be a line thickness, and drawing it to scale would show nothing while drawing it exaggerated would suggest the effect is geometric rather than a matter of gradients.

Thwaites is a correlation. The 0.45 and the shape-factor fits are calibrated against exact solutions of the boundary-layer equations, not derived, so the separation angle it produces is good to a degree or two and its confidence interval is not something the method reports.

The layer is drawn as though it had an edge, and it does not — the profile approaches the free stream asymptotically and every thickness drawn is a convention.

And the whole scheme is an asymptotic expansion in a parameter that is never quite small enough. The displaced-body picture is the first term in $Re^{-1/2}$; the second term is known and is what matters near the tail; and an aerofoil’s Reynolds number is high enough for the first term and not always high enough for the series.

One more consequence, and it is the one that makes the whole scheme respectable rather than a convenience. The displaced-body picture is not an engineering approximation invented to patch two theories together; it is the first term of a matched asymptotic expansion in the inverse square root of the Reynolds number. The outer expansion is a potential flow on a body of order-one size; the inner one is the boundary layer; and the matching condition between them, worked out properly, produces exactly the displacement rule. That is why the transpiration velocity and the layer’s own outward velocity are the same number: they are the two sides of the match.

Who found it, and when #

Prandtl introduced the boundary layer and the displacement idea in the same 1904 paper. The systematic coupling — outer flow, layer, displaced outer flow, iterate — is from the 1930s, and the transpiration form is Lighthill’s, from 1958. Thwaites published his method in 1949. The triple deck arrived independently from Stewartson, Messiter and Neiland around 1969.

The surprising connection is with how the idea is used outside aerodynamics. A displacement thickness is a boundary condition standing in for a region, and the same device appears wherever a thin region of complicated physics abuts a large region of simple physics: the sheath in a plasma, handed to the bulk as a modified potential; the space-charge layer at a semiconductor junction, handed to the device model as an effective width; the Knudsen layer at a rarefied wall, handed to the continuum as a slip velocity. In every case the thin region is not solved by the outer problem and is not ignored by it either — it is replaced by a correction to where the boundary is.

Where the ladder goes next #

Below this rung are the layer’s thicknesses and the outer theory the correction is made to.

Beside it are where the flow lets go, which is where the coupling stops being valid, and how much uphill a layer can take, which is the criterion that decides it.

And after it, the direction the whole ladder has been travelling: the outer problem’s missing information arrives here from the one place potential theory cannot look, which is the inside of the layer it threw away.