Experiment: Fluid Flow and Turbulence

Contents

Efflux from a vessel: Torricelli's law (1738)
Poiseuille: flow in fine capillaries (1840)
Reynolds: the dye filament and the critical number (1883)
Bénard: cellular convection (1900)
Prandtl: the boundary layer made visible (1904)
What the record establishes, and what it does not

Fluid Dynamics sets out the equations of fluid motion and, in the same breath, concedes how much of their content is out of reach: the transition to turbulence is not derived from them, the turbulent state past it is organized by similarity hypotheses rather than deduced, and the existence of smooth three-dimensional solutions is an open problem [Fefferman:2006] [Leray:1934] catalogued with the rest in What We Observe but Do Not Understand. A theory in that condition is held to account by measurement more strictly than usual. This chapter reports the measurements, chosen to span the range in which the equations are trusted and the range in which they are not: efflux from a vessel, where the ideal theory is right to within a factor the ideal theory cannot compute; Poiseuille's capillaries, where the viscous theory is right to viscometric accuracy and thereby measures the no-slip condition; Reynolds's dye filament, where the laminar solution loses its authority at a definite value of a single dimensionless number; Bénard's convection cells, where a threshold computed from the linearized equations meets a measured one; and Prandtl's water channel, where the thin layer that reconciles the inviscid limit with the observed drag was made visible and then manipulated.

The chapter is deliberately not a vindication. Two of the regularities reported below — the value of the pipe transition threshold and the statistical structure of the flow past it — are not reproduced by any derivation from the Navier–Stokes equations, and they are set down here as constraints that any future theory of turbulence must meet rather than as confirmations of the present one. That distinction is the point of the closing section, and it is why an experiment chapter is owed to this part of mechanics more than to most others.

Efflux from a vessel: Torricelli's law (1738)

Tests Equations (36.2), (36.3) and (36.6). Assuming Equation (36.1).

The oldest quantitative statement about a fluid in motion is that a liquid escapes from a hole in a vessel at the speed a body would acquire in falling freely from the free surface to the hole. It is the simplest possible test of the ideal theory of Fluid Dynamics, it can be read off with a ruler, and — importantly for what follows — it is right about the speed and wrong about the discharge, by a factor that the ideal theory has no way to supply.

Apparatus

An open vessel of horizontal cross-section $A$, large against the area $a$ of a sharp-edged orifice pierced in a thin wall at depth $h$ below the free surface. Two variants are used. In the first the head is held constant by a supply and an overflow weir, so that a steady jet issues; in the second the vessel is simply allowed to empty and the falling level is recorded against time. For the range measurement the orifice is in a vertical side wall at a height $z$ above a horizontal floor carrying a scale. Bernoulli's Hydrodynamica works throughout with vessels of exactly this kind, efflux from them being its central subject [Bernoulli:1738].

A third mouthpiece is fitted for the discharge measurement alone: a thin-walled tube of the same bore $a$, projecting a short way inward from the wall into the body of the liquid rather than being flush with it. Its length is a few bore diameters — long enough that the tube's own walls stand clear of the wall of the vessel, short enough that the jet does not reattach inside it. Nothing else is changed: the same vessel, the same head, the same bore.

Procedure

The efflux speed is obtained twice, by independent routes, so that neither reading depends on trusting the other.

The first route is kinematic. A horizontal jet leaving the side orifice is a projectile: it falls the height $z$ in a time $\sqrt{2z/g}$ and lands at a horizontal range $x=v\sqrt{2z/g}$, so that

\begin{equation}\tag{36.1} v=x\sqrt{\frac{g}{2z}}\ec \end{equation}

which reduces the speed measurement to two lengths and the local $g$ — the same kinematics measured directly in Experiment: Free Fall and Projectile Motion. The second route is integral: with the supply shut off, the time taken to empty the vessel from an initial head $H$ is recorded and compared with the prediction derived below.

The discharge is measured separately, by collecting and weighing the liquid delivered in a timed interval, and is compared with the product of the orifice area and the measured jet speed. The jet is also examined a short distance downstream of the plane of the orifice. The whole discharge measurement is then repeated with the re-entrant tube substituted for the flush orifice at the same head and the same bore, which changes the geometry of the approaching flow and nothing else.

Observations and data

Three facts are found, and they are not all favourable to the ideal theory.

First, the jet speed varies as the square root of the head. In the range measurement this appears in a particularly clean form: combining Equation (36.1) with $v=\sqrt{2gh}$ gives $x=2\sqrt{hz}$, so the landing point is fixed by the two lengths alone and does not depend on $g$ at all. A vessel with orifices at several depths therefore delivers jets whose landing points are predicted without any measurement of gravity, and the prediction is met.

Second, the vessel empties in the time predicted from that speed by mass conservation, which tests the square-root law over the whole range of heads at once rather than at isolated values.

Third, the volume delivered per unit time is less than the product of the orifice area with the measured jet speed. The jet is seen to contract downstream of the orifice, reaching a minimum section — the vena contracta — within about half an orifice diameter of the wall, and for a sharp-edged orifice in a thin plate the ratio of that minimum section to the orifice area is in the neighbourhood of $0.6$. The speed law is exact; the discharge law is not.

Fourth — and this is the observation that makes the third one tractable — the contraction is stronger, and markedly so, when the liquid leaves through the inward-projecting tube instead of through the flush hole. Head, bore and jet speed are unchanged; the delivery is not. The contraction is therefore a property of the approach to the orifice and not of the orifice itself, and the two mouthpieces bracket the effect: the re-entrant tube is the geometry in which the approaching flow is most nearly radial, and the discharge is least.

Phenomenon 36.1 (Torricelli's law).

A liquid escaping through a small orifice at depth $h$ below the free surface of a large vessel open to the atmosphere issues with the speed

\begin{equation}\tag{36.2} v=\sqrt{2gh}\ec \end{equation}

independent of the density of the liquid, of the size of the orifice and of the direction in which the orifice faces. The volume discharged per unit time is smaller than $a\sqrt{2gh}$, by a factor near $0.6$ for a sharp-edged orifice, because the emerging stream contracts to a section smaller than the hole [Bernoulli:1738] [Euler:1757].

Derivation. Derives Phenomenon 36.1. Take the vessel large enough that the free surface descends slowly and the flow may be treated as steady. Follow a streamline from a point on the free surface, where the pressure is atmospheric and the speed negligible because $a\ll A$, to a point in the jet just outside the orifice, where the pressure is again atmospheric because the stream is surrounded by air and is straight enough there to carry no transverse pressure gradient. Bernoulli's theorem along that streamline gives

\[ p_{0}+\rho g h=p_{0}+\tfrac{1}{2}\rho v^{2}\ec \]

whence Equation (36.2). The density cancels, which is why the law holds equally for water and for mercury, and the orifice area has not entered, which is why it holds for any small hole. The result is exactly the free-fall speed from the head, because the argument is the energy balance of a particle descending the same height.

The emptying time follows by mass conservation. With the instantaneous head $h$, the free surface descends at $\dd h/\dd t=-a\,v/A$, so

\[ \dv{h}{t}=-\frac{a}{A}\sqrt{2gh} \quad\Longrightarrow\quad t=\frac{A}{a}\int_{0}^{H}\frac{\dd h}{\sqrt{2gh}} =\frac{A}{a}\sqrt{\frac{2H}{g}}\ec \]

which is $A/a$ times the free-fall time from the initial head. Its agreement with observation tests Equation (36.2) at every head simultaneously.

What the derivation does not give is the discharge. Bernoulli's theorem was applied along one streamline and says nothing about the area of the bundle of streamlines that reaches the jet: the fluid approaches the orifice from all directions across the inner face of the wall, so the streamlines arrive with an inward transverse velocity that they carry past the plane of the hole, and the stream keeps narrowing until that transverse motion has been arrested. The measured contraction ratio is therefore a statement about the geometry of the approaching flow, not about the speed, and the two must be tested separately — which is why the third observation above falsifies nothing.

∎

There is exactly one mouthpiece for which the ideal theory does supply the missing factor, and it is the re-entrant one. The reason is worth stating before the calculation: the contraction is undetermined because the pressure distribution over the wall around the orifice is undetermined, and the re-entrant tube is the geometry contrived so that no wall carries an unknown pressure at all.

Proposition 36.2 (The contraction coefficient of a re-entrant mouthpiece).

For steady efflux under a head $h$ through a thin tube of bore $a$ projecting into the vessel, the ideal theory gives the jet section exactly:

\begin{equation}\tag{36.3} C_{c}:=\frac{a_{j}}{a}=\frac{1}{2}\ec\qquad Q=\tfrac{1}{2}a\sqrt{2gh}\ec \end{equation}

with no empirical input. For a sharp-edged orifice flush with a plane wall the same theory gives a larger coefficient, computed in Proposition 36.3 below. Rests on Equation (36.2).

Proof.

Derives Proposition 36.2. Apply the steady momentum theorem to the liquid inside the vessel, resolved along the axis of the tube. Because the tube projects inward, it is wetted on both faces, so whatever pressure acts on its outer surface acts equally on its inner one and it transmits no net axial force to the liquid. Everywhere else on the interior the liquid is nearly at rest — the vessel is large, and only in the immediate neighbourhood of the mouth does it acquire appreciable speed — so the pressure on the walls is hydrostatic. A hydrostatic pressure distribution on a closed surface exerts no net horizontal force, and the surface here fails to be closed in exactly one place: the mouth of the tube, of area $a$ at depth $h$, where there is no wall. Equivalently the patch of the opposite wall facing the mouth is unbalanced. The net axial force on the liquid is therefore

\begin{equation}\tag{36.4} F=\rho g h\,a\ep \end{equation}

In the steady state that force equals the rate at which axial momentum leaves the vessel. The jet carries mass at $\rho a_{j}v$ and moves at $v$, so the momentum flux is $\rho a_{j}v^{2}$, and with Equation (36.2) for $v$,

\begin{equation}\tag{36.5} \rho a_{j}\left(2gh\right)=\rho g h\,a\ec \end{equation}

From Equations (36.2) and (36.4) (equating the momentum flux of the jet to the unbalanced hydrostatic force and substituting $v^{2}=2gh$). whence $a_{j}=a/2$, which is Equation (36.3). Note what has and has not been used: Bernoulli's theorem supplied only the speed, and the geometry supplied the rest; $\rho$, $g$ and $h$ have all cancelled, so the prediction is a pure number, testable by weighing a delivery against the flush orifice at the same head.

The argument also shows precisely why the flush orifice resists it. There the wall surrounding the hole is wetted on one side only, and the liquid accelerates along it, so the pressure over it is not hydrostatic and Equation (36.4) acquires an unknown correction. Fixing that correction means knowing the shape of the free surface bounding the jet, which is the free-streamline problem below.

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Proposition 36.3 (Kirchhoff's free-streamline value for the flush slit).

Let an ideal incompressible liquid discharge under a head $h$ through a sharp-edged slit of width $a$ in a plane wall, the jet being bounded by free streamlines on which the pressure is that of the atmosphere and the speed therefore $\sqrt{2gh}$ by Equation (36.2). Then the free-boundary problem has a closed solution and the asymptotic width of the jet is

\begin{equation}\tag{36.6} C_{c}=\frac{a_{j}}{a}=\frac{\pi}{\pi+2}=0.611\ec \end{equation}

with no empirical input [Helmholtz:1868] [Kirchhoff:1869]. The corresponding problem for a circular orifice in three dimensions is not soluble in closed form, and the measured coefficient lies near the same figure. Rests on Equations (36.2) and (36.3).

Full derivation in Appendix A.

Derives Proposition 36.3.

What the appendix supplies, and what makes the problem tractable at all, is a change of unknown. The difficulty is that the boundary of the jet is not given: it is a streamline whose position is part of the answer, so the domain of the problem depends on its solution. The free-streamline construction removes that by working in the hodograph plane, where the velocity rather than the position is the coordinate: on a free streamline the speed is known and only the direction varies, and on a solid wall the direction is known and only the speed varies, so an unknown boundary in the physical plane becomes a known one in the hodograph plane. A conformal map then carries that known region onto a half plane, the complex potential is written there by inspection, and Equation (36.6) follows on integrating back. The one input this treatise does not carry is the conformal mapping itself: Complex Analysis builds the Cauchy theory and the residue calculus but no theory of conformal maps, and the appendix names that debt where it uses it.

Interpretation

The experiment establishes the ideal theory in its own domain and marks that domain's edge in the same measurement. Where the flow is essentially inviscid and no wall is being followed closely — along the interior streamlines, and in the jet — the Bernoulli relation is quantitatively exact. Where the geometry of the flow must be computed rather than assumed, the ideal theory is silent: it predicts the speed and not the area, and the missing factor of roughly $0.6$ is supplied by measurement or by the free-streamline theory of discontinuous motions begun by Helmholtz [Helmholtz:1868] [Kirchhoff:1869], not by the Bernoulli argument. This pattern — an exact statement along a streamline combined with an undetermined global geometry — recurs at every level of the subject, and in its severe form it is the closure problem of turbulence discussed in Section 36.6.

Proposition 36.2 sharpens the diagnosis rather than softening it. The re-entrant mouthpiece is not an easier case of the same calculation; it is a different kind of case, in which the unknown geometry has been arranged out of the problem by making every wetted surface carry a pressure that is either hydrostatic or self-cancelling. Once that is done a momentum balance settles the discharge exactly, with no free constant and no reference to the shape of the jet. The lesson generalizes and is used twice more in this chapter: where an integral conservation law can be applied to a control volume whose boundary is entirely known, it delivers an exact answer without solving the flow — the same move that gives the drag from the wake in Equation (36.25).

Primary references

[Bernoulli:1738] [Euler:1757]. The law is customarily attributed to Torricelli's memoir on the motion of heavy bodies of 1644 [Torricelli:1644], which is not among the sources verified for this treatise; it is therefore cited here to the Hydrodynamica, where efflux from vessels is treated at length, and to Euler's memoir, which first gives the modern local statement from which Equation (36.2) follows in one line [Euler:1757]. A reader checking the customary attribution should know that the change of source is deliberate and not an error.

Poiseuille: flow in fine capillaries (1840)

Tests Equations (36.7) and (36.8). Assuming Equation (31.32).

Poiseuille was a physician, and his question was the circulation of blood in the smallest vessels; his answer was the most accurate measurement of a viscous flow made in the nineteenth century, and it remains the calibration standard for viscometry. The law he found was established empirically, before it was derived, and its most exacting consequence is not the flux formula itself but what the formula's accuracy implies about the fluid's behaviour at the wall.

Apparatus

Glass capillaries drawn to fine bore, mounted horizontally, with bores ranging from a few tens of micrometres up to a few hundred — a span of more than an order of magnitude in diameter, which is what makes a power law in the diameter distinguishable from any other dependence. The driving pressure is supplied by a reservoir of compressed air held at a measured value, rather than by a gravitational head, so that the pressure difference can be varied independently of the geometry and held steady for the duration of a run. The capillary and its supply sit in a bath of controlled temperature, because the coefficient sought is strongly temperature-dependent; for water near room temperature the dynamic viscosity is close to $10^{-3}\,\mathrm{kg}/\mathrm{m}/\mathrm{s}$ and changes by a few percent per kelvin. Delivered volume is collected and timed [Poiseuille:1840]. Hagen, working independently in brass tubes, used the same logic with a gravitational head [Hagen:1839].

Procedure

One variable at a time. With the tube, the temperature and the length fixed, the delivered volume per unit time $Q$ is measured against the applied pressure difference $\Delta p$. With $\Delta p$ and the tube fixed, $Q$ is measured against the length $L$, shortened by cutting the same capillary so that the bore is not changed between readings — this is the step that removes the largest systematic error, since two nominally identical capillaries are not identical to the fourth power. With $\Delta p$, $L$ and the temperature fixed, $Q$ is measured across the family of capillaries, whose bores are determined separately by weighing a thread of mercury of known length drawn into the tube. Finally the bath temperature is varied to isolate the coefficient.

Observations and data

The delivered flux is found to be strictly proportional to the applied pressure difference and inversely proportional to the length of the tube, over the whole range in which the flow remains steady. Against the diameter it rises as the fourth power: halving the bore at fixed pressure difference divides the delivery by sixteen. The coefficient depends on the liquid and on the temperature and on nothing else — not on the material of the tube wall, not on the shape of the entry, not on the pressure level, only on the difference [Poiseuille:1840] [Hagen:1839].

The law fails in two recognizable ways, and the failures are as informative as the law. In tubes made short enough, the measured flux falls below the prediction, because a length of tube is consumed in establishing the flow profile before it becomes uniform along the axis. And above a sufficiently large flux the proportionality to $\Delta p$ is lost, the flux rising much more slowly than the pressure difference thereafter; that is the transition of Section 36.3, met here from the other side.

Phenomenon 36.4 (The measured fourth-power law and the no-slip condition).

Steady flow of a viscous liquid along a straight circular tube of radius $R$ and length $L$ delivers a volume flux

\begin{equation}\tag{36.7} Q=\frac{\pi R^{4}}{8\mu}\,\frac{\Delta p}{L} \end{equation}

with no adjustable constant beyond the viscosity $\mu$ of the liquid, the agreement holding over more than an order of magnitude in $R$ [Poiseuille:1840] [Hagen:1839]. Because the coefficient in Equation (36.7) is fixed by the condition that the liquid be at rest against the wall, the accuracy of the fit is a quantitative measurement of that condition, and a liquid retaining any appreciable tangential speed at the wall is excluded. Rests on Equation (36.2).

Derivation. Derives Phenomenon 36.4. Fluid Dynamics derives Equation (36.7) from the Navier–Stokes equations with the no-slip condition $v(R)=0$. What is needed here is the sensitivity of that result to the boundary condition itself, since that is the quantity the experiment actually bounds. Replace no-slip by the general linear slip condition, in which the tangential speed at the wall is proportional to the local shear rate with a coefficient $\lambda$ of the dimensions of a length, the slip length:

\[ v(R)=\lambda\left(-\dv{v}{r}\right)_{r=R}\ep \]

The interior solution is unchanged, because the differential equation is untouched; only the constant of integration moves. Writing $v(r)=\left(\Delta p/4\mu L\right)\left(R^{2}-r^{2}\right)+v_{s}$ and evaluating the shear rate of that profile at the wall, $\left(-\dd v/\dd r\right)_{R}=\Delta p\,R/2\mu L$, the slip condition fixes

\[ v_{s}=\frac{\lambda\,\Delta p\,R}{2\mu L}\ec \]

a plug velocity added uniformly across the section. Integrating,

\begin{equation}\tag{36.8} Q=\frac{\pi R^{4}}{8\mu}\,\frac{\Delta p}{L} +v_{s}\pi R^{2} =\frac{\pi R^{4}}{8\mu}\,\frac{\Delta p}{L} \left(1+\frac{4\lambda}{R}\right)\ep \end{equation}

Three consequences follow, and they are the experimental content of this section. The correction is a pure enhancement, so slip can only make a tube deliver more than Equation (36.7), never less — the observed shortfall in short tubes cannot be slip and must be an entry effect. The correction scales as $1/R$, so it is the finest capillaries that carry the information: a family of tubes obeying the same fourth-power law across a wide range of $R$ is direct evidence that $\lambda/R$ is negligible throughout, which no single tube could establish. And the bound is quantitative: agreement with Equation (36.7) to a fractional accuracy $\epsilon$ in a tube of radius $R$ requires $\lambda\le\epsilon R/4$, so agreement at the percent level in a capillary of radius $10\,\mu\mathrm{m}$ bounds the slip length at a few tens of nanometres — a molecular distance. The liquid is at rest against the wall, and Equation (36.7) is how we know it.

∎

Interpretation

Two separate things are established. The first is the Navier–Stokes equations themselves in their simplest nontrivial exact solution: the inertial term vanishes identically for this geometry, so Equation (36.7) is a consequence of the viscous stress model alone, and its confirmation to viscometric accuracy is a direct confirmation of Newton's linear relation between stress and rate of strain for these liquids. The second is the boundary condition, which is logically independent of the equations and had to be settled by measurement: Navier's own formulation admitted slip, and it is the capillary data, through Equation (36.8), that reduced the slip length to a molecular scale for ordinary liquids on ordinary walls.

The molecular origin of the coefficient $\mu$ is not touched here; it belongs to Kinetic Theory of Gases, whose sharpest prediction — that the viscosity of a gas is independent of its density — was confirmed by Maxwell in a separate measurement [Maxwell:1866].

Primary references

[Poiseuille:1840] [Hagen:1839].

Reynolds: the dye filament and the critical number (1883)

Tests Equation (36.9). Assuming Equation (36.7).

This is the experiment that gave the subject its organizing parameter and its central unsolved problem in the same afternoon. Its apparatus is a glass tube and a thread of coloured water; its result is that the laminar solution of Section 36.2 ceases to describe the flow above a definite value of one dimensionless combination, and that the value of that threshold is not a property of the equations alone.

Apparatus

A horizontal glass tube leading out of a large glass-walled tank of water, entering the tank through a smoothly rounded bell mouth so that the fluid is not disturbed on the way in. Three such tubes were used, their diameters spanning a factor of several, so that the same transition could be sought at different sizes. A fine nozzle on the axis of the bell mouth introduces a filament of dyed water drawn from a separate reservoir, at a rate adjusted to match the local flow so that the filament neither spreads nor is drawn out. A valve at the far end of the tube sets the discharge, which is measured by collection and timing; a thermometer in the tank gives the temperature, and hence the viscosity. For the study of the broken filament the room is darkened and the tube illuminated by an electric spark, whose brief flash resolves the structure that continuous light averages into a haze [Reynolds:1883].

The tank water is left standing for hours before a run, because the result depends on how quiet it is — a fact discovered in the course of the experiment and not anticipated by it.

Procedure

The valve is opened progressively and the filament watched down the length of the tube. At each setting the discharge is measured, the temperature read, and the dimensionless group

\begin{equation}\tag{36.9} \mathrm{Re}=\frac{\rho\,\bar{v}\,D}{\mu} \end{equation}

formed from the mean speed $\bar{v}$, the tube diameter $D$, and the density and viscosity of the water at that temperature. The procedure is then repeated with a different tube and, since the viscosity is a strong function of temperature, at a different bath temperature, so that the same value of Equation (36.9) is reached by three independent routes: by changing the speed, by changing the size, and by changing the fluid property.

A second, independent series measures the pressure difference required to drive a given flux along a fixed length of pipe, over a range straddling the transition observed optically.

Observations and data

At low speed the dye filament runs the whole length of the tube as a sharp unbroken thread, and the water around it stays clear: the fluid moves in layers that do not interchange. As the speed is raised the thread persists until, at a definite discharge, it breaks up abruptly at some point down the tube and the colour spreads across the whole section beyond that point. Raising the speed further moves the point of breakdown upstream towards the inlet. Under the spark the region past the breakdown is not a uniform mixture but a mass of distinct eddies, which the continuous-light view had smeared into a haze — the flow is not disordered at the molecular level but structured at scales comparable to the tube.

The transition occurs at a value of Equation (36.9) of order $2000$ for ordinary conditions of entry and supply, and at substantially the same value in all three tubes and at all bath temperatures — which is the essential observation, since the speed, the diameter and the viscosity individually differ by large factors between those runs. But the threshold is not sharp and not a constant of nature: with the tank water left undisturbed for long enough and the bell mouth kept clean, the unbroken filament survives to values several times larger, and no upper limit was established.

In the pressure series the two regimes obey different laws. Below the transition the pressure gradient required is proportional to the flux, as Equation (36.7) demands. Above it the gradient rises much faster, as a power of the flux near $1.7$ — close to, but measurably below, the square. The change of law occurs at the same place as the optical breakdown, so the two observations are of one event [Reynolds:1883].

Modern work has replaced the vague “of order $2000$” with a sharp number, and in doing so changed what the number means. Below a critical Reynolds number a patch of turbulence deposited in the pipe decays; above it, patches split faster than they decay and the turbulence is sustained indefinitely. That critical point — a statistical threshold between two competing rates, not the value at which a filament happens to break — is $\mathrm{Re}_{\mathrm{c}}=2040(10)$ [Avila:2011].

Quantity	Value	Source
$\mathrm{Re}$ at which the dye filament breaks, ordinary inlet	$\sim2\times 10^{3}$	[Reynolds:1883]
$\mathrm{Re}$ above which turbulence is sustained rather than transient	$2040(10)$	[Avila:2011]
Exponent of the flux in the turbulent resistance law	$\approx1.7$	[Reynolds:1883]

The pipe transition as it is currently known. The first entry is what Reynolds's filament shows under ordinary conditions of supply and is not a constant of nature; the second is a sharply defined critical point, being the Reynolds number at which the decay rate of a turbulent patch equals its splitting rate. All three quantities are dimensionless.

Phenomenon 36.5 (One parameter controls the transition).

Whether flow along a straight circular pipe remains laminar is determined, for a given standard of inlet quality, by the single dimensionless combination Equation (36.9) and not by the speed, the diameter, the density or the viscosity separately. Flows agreeing in $\mathrm{Re}$ break down at the same place and carry the same dimensionless resistance, however far apart their dimensional parameters lie [Reynolds:1883]. Rests on Equations (36.7) and (36.9).

Derivation. Derives Phenomenon 36.5. Measure lengths in units of the diameter $D$, speeds in units of the mean speed $\bar{v}$, times in $D/\bar{v}$ and pressures in $\rho\bar{v}^{2}$. The incompressible Navier–Stokes system then contains no dimensional quantity whatever, its viscous term carrying the single coefficient $1/\mathrm{Re}$ with $\mathrm{Re}$ as in Equation (36.9), and the boundary conditions on a geometrically similar pipe are identical in those units. Two experiments agreeing in $\mathrm{Re}$ therefore pose the identical mathematical problem, and every dimensionless property of their solutions — including whether the laminar solution is realized and including the dimensionless resistance — must agree. The observed collapse of three tubes and several temperatures onto one threshold is that statement measured, and it is the licence for all model-scale hydraulic and aerodynamic testing.

Note precisely what the argument does and does not deliver. It shows that if there is a threshold it must lie at a fixed value of $\mathrm{Re}$, and it therefore explains the collapse completely. It says nothing about where that value is. And it does not survive the observation that the threshold moves when the inlet is made quieter: the disturbance amplitude entering the pipe is a further dimensionless datum of the experiment which the nondimensionalization of the interior equations does not see, so a flow specified only by $\mathrm{Re}$ is not fully specified at all. The similarity argument is exactly as strong as the control of the boundary data.

∎

The similarity argument explains the collapse and stops there. One further thing can be settled exactly, and it is negative: the breakdown is not the laminar solution losing stability to an infinitesimal disturbance in the way a heated layer does.

Proposition 36.6 (Rayleigh's inflexion criterion).

Let $U(y)$ be a steady parallel flow of an inviscid fluid between rigid walls at $y_{1}$ and $y_{2}$. A two-dimensional normal-mode disturbance $\propto\ee^{\ii k(x-ct)}$ can have $\operatorname{Im}c\ne0$ — that is, can grow — only if $U''$ vanishes somewhere in $\left(y_{1},y_{2}\right)$ [Rayleigh:1880]. The Poiseuille profile has $U''$ constant and non-zero, so no inviscid mode of it grows. Rests on Equation (36.7).

Proof.

Derives Proposition 36.6. Write the disturbance through a streamfunction $\psi$, with $u'=\pp_{y}\psi$ and $v'=-\pp_{x}\psi$, so that continuity is automatic. Linearizing the Euler equations about $U(y)$ and taking the curl removes the pressure, leaving the transport of perturbation vorticity by the base flow against the gradient of the base vorticity $-U'$:

\begin{equation}\tag{36.10} \left(\pp_{t}+U\pp_{x}\right)\nabla^{2}\psi -U''\,\pp_{x}\psi=0\ep \end{equation}

Putting $\psi=\varphi(y)\ee^{\ii k(x-ct)}$ and cancelling $\ii k$ gives Rayleigh's equation,

\begin{equation}\tag{36.11} \left(U-c\right)\left(\varphi''-k^{2}\varphi\right) -U''\varphi=0\ec\qquad \varphi(y_{1})=\varphi(y_{2})=0\ec \end{equation}

From Equation (36.10) (substituting the normal mode and dividing by the common factor $\ii k$). the boundary conditions expressing that no fluid crosses either wall.

Suppose $c_{i}:=\operatorname{Im}c\ne0$. Then $U-c$ never vanishes on the real interval, so Equation (36.11) may be divided by it. Multiplying by the complex conjugate $\bar{\varphi}$, integrating across the channel and integrating $\int\bar{\varphi}\varphi''$ by parts — the boundary term dies on the wall conditions —

\begin{equation}\tag{36.12} \int_{y_{1}}^{y_{2}} \left(\abs{\varphi'}^{2}+k^{2}\abs{\varphi}^{2}\right)\dd y +\int_{y_{1}}^{y_{2}} \frac{U''\abs{\varphi}^{2}}{U-c}\,\dd y=0\ep \end{equation}

The first integral is real. In the second, $\left(U-c\right)^{-1}=\left(U-\bar{c}\right)/\abs{U-c}^{2}$, whose imaginary part is $c_{i}/\abs{U-c}^{2}$, so the imaginary part of Equation (36.12) reads

\begin{equation}\tag{36.13} c_{i}\int_{y_{1}}^{y_{2}} \frac{U''\abs{\varphi}^{2}}{\abs{U-c}^{2}}\,\dd y=0\ep \end{equation}

From Equation (36.12) (taking the imaginary part, the first integral being real). If $c_{i}\ne0$ the remaining integral must vanish. Its weight $\abs{\varphi}^{2}/\abs{U-c}^{2}$ is non-negative and not identically zero, so $U''$ cannot keep one sign: it must vanish somewhere. For the parabolic profile of Equation (36.7), $U''$ is a non-zero constant, and the condition fails.

∎

Remark 36.7 (What that does and does not prove).

Three limits must be stated, or the proposition will be read as more than it is. It is a criterion for the inviscid problem, and viscosity does not merely damp — for some profiles it destabilizes, so an inviscidly stable flow may still have a growing viscous mode. It is proved here for a plane channel and for two-dimensional disturbances. And the corresponding statement for the pipe, that the Hagen–Poiseuille solution has no growing eigenmode at any Reynolds number, is a computational result which this treatise cites no primary source for; it is recorded here as uncited, and nothing below rests on more than the plane case proved above. What survives is the conclusion that matters: the observed breakdown of Section 36.3 is of a different kind from the onset of Section 36.4, where an infinitesimal disturbance really does grow at a computable threshold. Rests on Proposition 36.6.

If no disturbance grows, how does a disturbance ever get large? The question sounds rhetorical and is not: decay of every eigenmode forbids growth only in the long run, and the linearized operator here is not normal — its eigenfunctions are not orthogonal — so a superposition of decaying modes may become very large before it becomes small. The mechanism has a name and a simple physical form.

Proposition 36.8 (Algebraic amplification by the lift-up mechanism).

A streamwise-invariant disturbance of a shear flow $U(y)$ — a pair of vortices whose axes lie along the flow — decays in its own cross-stream components while forcing the streamwise component linearly in time. Before viscosity stops it, the streamwise velocity perturbation exceeds the cross-stream one that created it by a factor of order $\mathrm{Re}$:

\begin{equation}\tag{36.14} \frac{u_{\max}}{v}\sim\frac{U_{0}R}{\nu}\sim\mathrm{Re}\ec \end{equation}

with $R$ the pipe radius and $U_{0}$ the centreline speed. The growth is algebraic and transient, and occurs although every eigenvalue of the problem has negative real part. Rests on Equation (36.9) and Proposition 36.6.

Proof.

Derives Proposition 36.8. Linearize about $U(y)\vect{\hat{x}}$ and take a disturbance independent of $x$. The cross-stream components $(v,w)$ then satisfy a problem in the cross-stream plane alone: the base flow does not appear in it, because $U$ has no component in that plane and no gradient along $x$ to be advected. That sub-problem is purely viscous and simply decays, on the time

\begin{equation}\tag{36.15} \tau_{\nu}\sim\frac{1}{\nu k_{\perp}^{2}}\sim\frac{R^{2}}{\nu}\ec \end{equation}

for a vortex filling the pipe, so $k_{\perp}\sim1/R$. The streamwise component does not decouple. Its equation retains one coupling term, the advection of the base profile across its own gradient:

\begin{equation}\tag{36.16} \left(\pp_{t}-\nu\nabla^{2}\right)u=-v\,U'(y)\ep \end{equation}

This is a forced equation, and that asymmetry — $u$ driven by $v$, with no reaction of $u$ back on $v$ — is the non-normality, written out. Over times short compared with $\tau_{\nu}$ the forcing is essentially steady and

\[ u\approx-U'(y)\,v\,t\ec \]

growing linearly, not exponentially. Viscosity ends it at $t\sim\tau_{\nu}$, so with $U'\sim U_{0}/R$ and Equation (36.15),

\[ u_{\max}\sim\frac{U_{0}}{R}\,v\,\frac{R^{2}}{\nu} =v\,\frac{U_{0}R}{\nu}\ec \]

which is Equation (36.14). The physical content is plain: a streamwise vortex carries slow fluid from the wall towards the axis and fast fluid the other way, and because the profile it is stirring is steep, a small transverse motion redistributes a large streamwise velocity. What it leaves behind are alternating fast and slow streaks, elongated along the pipe.

The consequence for the experiment is a matter of arithmetic. At $\mathrm{Re}=2\times 10^{3}$ the factor in Equation (36.14) is of order a thousand, so a disturbance entering the bell mouth at a thousandth of the mean speed — far below anything the dye filament would reveal — can produce streaks comparable to the mean flow itself, at which point the nonlinear terms that the linear analysis discarded are no longer small. That is why the threshold moves when the tank is left to settle: what the experimenter controls by waiting is not $\mathrm{Re}$ but the amplitude the amplifier is fed, and Equation (36.14) says the amplifier's gain rises with $\mathrm{Re}$. A transition governed jointly by a Reynolds number and a disturbance level is exactly what Equation (36.14) predicts and exactly what Table 36.1 reports.

∎

Remark 36.9 (Where the derivation stops, and why nothing further is owed).

Proposition 36.8 carries the account exactly as far as a linear argument can go, and it is worth stating plainly what lies beyond it, so that the boundary is not mistaken for an omission. What follows the algebraic amplification is the finite-amplitude stage: the streaks left by the lift-up develop a secondary instability of their own, and the breakdown of the streaks regenerates the streamwise vortices that made them, closing a cycle that sustains itself without any growing eigenmode anywhere in it. That picture is a description of solutions found numerically and observed experimentally, not a derivation from the Navier–Stokes equations; the scaling of the critical disturbance amplitude with a negative power of $\mathrm{Re}$ is measured, and the near-$1.75$ power by which the turbulent resistance grows with the flux is an empirical correlation.

No derivation of the threshold from the equations of motion exists. That is a statement about the subject and not about this book, and it is why nothing in this chapter asserts one: the phenomenon stated here is Phenomenon 36.5, the collapse onto a single parameter, which is derived in full above, and the honest account of the threshold itself is given in the interpretation of this experiment below, as two available states whose boundary no theory predicts [Avila:2011]. An unwritten derivation would be owed only if a claim rested on it, and none does.

Interpretation

Three distinct claims must be separated, because they are of very different strength.

The collapse onto one parameter is explained, completely, by the derivation above; it is a consequence of the equations and needs nothing else. The existence of a transition is an observation the equations are believed to contain but from which it has never been extracted: by Proposition 36.6 the profile supports no growing inviscid mode, and the laminar solution is not found to be unstable to infinitesimal disturbances at any Reynolds number, so the observed breakdown cannot be the linear instability that the analogous calculation supplies for the convecting layer of Section 36.4. And the value of the threshold is not a property of the equations at all in the form usually quoted, since it depends on the disturbances present — Reynolds's own runs establish this by moving the threshold with nothing but patience.

That last point deserves emphasis because the number $2000$ is so often quoted as though it were a constant. What the experiment establishes is a lower bound on the laminar range under realistic conditions, and an upper bound that has never been reached. The honest statement of the result is that the pipe has two available states, that which of them is realized depends jointly on $\mathrm{Re}$ and on the level of disturbance admitted [Avila:2011], and that no theory currently predicts the boundary between them.

What is understood is why the two must enter together. Proposition 36.8 supplies an amplifier whose gain grows with $\mathrm{Re}$ and which needs no instability to run, so the product of gain and input — not either alone — decides whether the disturbance reaches the amplitude at which the neglected nonlinear terms take over. A threshold in that product is a curve in the $(\mathrm{Re},\text{amplitude})$ plane, not a number; quoting $\mathrm{Re}\approx2000$ is quoting one point of it, at whatever inlet quality the quoter had. The sharply defined $2040(10)$ of Table 36.1 escapes that objection only because it is defined by a different question — whether turbulence, once present, persists — which does not refer to the disturbance that started it [Avila:2011].

Reynolds returned to the problem a decade later with the decomposition of the velocity field into mean and fluctuating parts, which produces the averaged equations and the Reynolds stresses [Reynolds:1895]; that route reorganizes the problem without closing it, and is taken up in Section 36.6.

Primary references

[Reynolds:1883] [Reynolds:1895].

Bénard: cellular convection (1900)

Tests Equations (36.21) and (36.22). Assuming Equation (36.17).

Where the pipe gives a transition no calculation predicts, the layer heated from below gives one that a calculation predicts exactly. The two sit in this chapter side by side deliberately: they show that the difficulty in the pipe is specific and not a general failure of hydrodynamic stability theory.

Apparatus

A shallow horizontal layer of liquid spread on a levelled metal plate that is heated from below and held at a uniform temperature, the upper surface left free and open to the air. Bénard used molten spermaceti and paraffin, liquids whose viscosity is high enough to keep the motion slow and orderly and which solidify on cooling, so that the pattern can be fixed and examined at leisure. Layer depths are of the order of a millimetre. The layer is observed from above, photographically and by the optical distortion of a reflected grid, which registers the deformation of the free surface accompanying the flow [Benard:1900].

Procedure

The plate temperature is raised slowly and the layer watched for the onset of motion. Once cells appear, the temperature difference across the layer is increased further and the pattern followed; the layer depth is then changed and the whole sequence repeated, so that the cell size can be plotted against the depth — the measurement that distinguishes a pattern imposed by the apparatus from one selected by the fluid.

Observations and data

Below a threshold temperature difference nothing moves: the liquid transports heat by conduction alone, and the layer is optically featureless. Above the threshold a steady circulation appears, and it is not disordered but cellular, the layer dividing into a tessellation of nearly regular hexagons with liquid rising in the interior of each cell and descending along the shared boundaries. The cells are stable, persist indefinitely while the heating is maintained, and vanish when the temperature difference is lowered back through the threshold.

The horizontal size of a cell is proportional to the depth of the layer and independent of the lateral extent of the apparatus, which is far larger. This is the decisive quantitative observation, because it identifies the depth as the only length the fluid uses: the pattern is selected by the physics of the layer and not by its container [Benard:1900].

Phenomenon 36.10 (Convective onset at a threshold, with a selected horizontal scale).

A horizontal fluid layer heated from below remains at rest, carrying heat by conduction, until the imposed temperature difference exceeds a threshold; above it a steady cellular circulation sets in whose horizontal periodicity is a fixed multiple of the layer depth and independent of the size of the vessel [Benard:1900]. The threshold is controlled by the single dimensionless combination

\begin{equation}\tag{36.17} \mathrm{Ra}=\frac{g\alpha\,\Delta T\,d^{3}}{\nu\kappa}\ec \end{equation}

in which $\alpha$ is the thermal expansion coefficient, $\nu$ the kinematic viscosity, $\kappa$ the thermal diffusivity and $d$ the depth, and it occurs at a critical value of order a thousand [Rayleigh:1916]. Rests on Equation (36.17).

Derivation. Derives Phenomenon 36.10. Work in the Boussinesq approximation, in which the density variation is kept only where it multiplies $g$ and the fluid is otherwise treated as incompressible with constant $\nu$ and $\kappa$. The base state is motionless conduction, with the temperature falling linearly at the rate $\beta=\Delta T/d$ from the hot lower plate to the cold upper one. Write the perturbation velocity $\vect{u}$, its vertical component $w$, and the temperature perturbation $\theta$. Linearizing about that state and taking the double curl of the momentum equation — which removes the pressure and, using $\nabla\cdot\vect{u}=0$, leaves an equation for $w$ alone — gives the pair

\begin{equation}\tag{36.18} \pp_{t}\nabla^{2}w=g\alpha\nabla_{h}^{2}\theta+\nu\nabla^{4}w\ec \qquad \pp_{t}\theta=\beta w+\kappa\nabla^{2}\theta\ec \end{equation}

with $\nabla_{h}^{2}=\pp_{x}^{2}+\pp_{y}^{2}$ the horizontal Laplacian. The first equation says that buoyancy, acting only where the temperature varies horizontally, drives vertical motion against viscosity; the second says that a rising parcel carries the conduction gradient with it and loses the excess by diffusion. The two effects compete, and $\mathrm{Ra}$ of Equation (36.17) is the ratio in which they stand.

At marginal stability the disturbance neither grows nor decays, so the time derivatives vanish. Eliminating $\theta$ between the two steady equations — solve the second for $\theta$ and substitute —

\begin{equation}\tag{36.19} \nu\kappa\,\nabla^{6}w=g\alpha\beta\,\nabla_{h}^{2}w\ep \end{equation}

From Equation (36.18) (setting the time derivatives to zero, applying $\kappa\nabla^{2}$ to the first equation and using $\kappa\nabla^{2}\theta=-\beta w$ from the second). Now separate. Horizontally, take any planform $f(x,y)$ with $\nabla_{h}^{2}f=-a^{2}f/d^{2}$, so that $a$ is the horizontal wavenumber measured in units of $1/d$; a roll, a square and a hexagon all satisfy this, which is already the statement that the linear problem fixes the periodicity and not the plan form. Vertically, take the boundaries stress-free and perfectly conducting, so that $w=\pp_{z}^{2}w=\theta=0$ at $z=0$ and $z=d$. The single half-wave $w=W_{0}\sin\left(\pi z/d\right)f(x,y)$ meets every one of those conditions, together with the further conditions $\pp_{z}^{4}w=0$ that Equation (36.19) inherits, and it is the gravest such mode. For it,

\[ \nabla^{2}w=-\frac{\pi^{2}+a^{2}}{d^{2}}\,w\ec\qquad \nabla_{h}^{2}w=-\frac{a^{2}}{d^{2}}\,w\ec \]

so that Equation (36.19) becomes an identity between numbers,

\[ \nu\kappa\,\frac{\left(\pi^{2}+a^{2}\right)^{3}}{d^{6}} =g\alpha\beta\,\frac{a^{2}}{d^{2}}\ep \]

Multiplying by $d^{6}/\nu\kappa$ and using $\beta d=\Delta T$ to recognize $g\alpha\Delta T d^{3}/\nu\kappa$ as the Rayleigh number of Equation (36.17),

\begin{equation}\tag{36.20} \mathrm{Ra}(a)=\frac{\left(\pi^{2}+a^{2}\right)^{3}}{a^{2}}\ec \end{equation}

From Equations (36.17) and (36.19) (substituting the half-wave mode and collecting the dimensionless group). which is the marginal-stability relation [Rayleigh:1916]. Note that no property of the fluid survives in it: the whole of the material physics has gone into the single number $\mathrm{Ra}$.

The layer becomes unstable as soon as the imposed Rayleigh number exceeds Equation (36.20) for some wavenumber, so the threshold is the minimum of that curve. Writing $u=a^{2}$,

\[ \dv{\mathrm{Ra}}{u} =\frac{3\left(\pi^{2}+u\right)^{2}u-\left(\pi^{2}+u\right)^{3}}{u^{2}} =\frac{\left(\pi^{2}+u\right)^{2}\left(2u-\pi^{2}\right)}{u^{2}}\ec \]

which vanishes only at $u=\pi^{2}/2$ and changes sign from negative to positive there, so the minimum is unique. Substituting,

\begin{equation}\tag{36.21} a_{\mathrm{c}}=\frac{\pi}{\sqrt{2}}\approx2.221\ec \qquad \mathrm{Ra}_{\mathrm{c}}=\frac{27\pi^{4}}{4}\approx657.5\ep \end{equation}

The critical wavenumber fixes the horizontal periodicity of the pattern that appears at onset:

\begin{equation}\tag{36.22} \lambda_{\mathrm{c}}=\frac{2\pi d}{a_{\mathrm{c}}}=2\sqrt{2}\,d \approx2.83\,d\ec \end{equation}

so that one rising and one descending region together span a little under three depths, and a single cell is roughly one and a half depths across. This is the measured proportionality of cell size to depth, with the constant predicted and no parameter free. Note that the container never entered the calculation: the only length available to the linearized problem is $d$, which is why the selected scale cannot depend on the lateral extent. The numerical threshold does depend on the boundary conditions assumed — for the physically commoner case of two rigid boundaries the same calculation gives a critical Rayleigh number near $1708$, with a correspondingly larger critical wavenumber [Chandrasekhar:1961] — so the value quoted must always be paired with the boundary conditions it belongs to.

∎

Remark 36.11 (Why the rigid case is not done the same way).

The half-wave $\sin\left(\pi z/d\right)$ is what makes the calculation above three lines long, and it works only because a stress-free boundary imposes $\pp_{z}^{2}w=0$. A rigid boundary imposes $\pp_{z}w=0$ instead — no slip — which the half-wave violates, and the vertical problem no longer separates into a single trigonometric mode: the sixth-order operator must be solved with six boundary conditions, the solution is a combination of one trigonometric and two hyperbolic vertical structures, and the eigenvalue condition is a transcendental equation whose lowest root gives the $\mathrm{Ra}_{\mathrm{c}}\approx1708$ quoted above [Chandrasekhar:1961]. The physics is unchanged and the architecture of the argument is unchanged; only the vertical eigenfunction is no longer elementary. That is why the stress-free value is the one derived in full here and the rigid value is cited. Rests on Equation (36.20).

Interpretation

The onset of convection is the case in which hydrodynamic stability theory works completely: a threshold and a length scale are computed from the linearized equations with no adjustable parameter, and both are observed. Set beside Section 36.3, this is the useful comparison — the failure to predict the pipe threshold is a failure of a particular flow to lose stability in the linear way, not a failure of the method.

Two honest qualifications are owed. The first concerns Bénard's own configuration. His layers were thin, with a free upper surface, and in that geometry the temperature dependence of the surface tension drives a competing instability with its own threshold and its own preferred scale; later analysis — not among the primary sources verified for this treatise — attributes the cells Bénard actually photographed largely to that mechanism rather than to buoyancy [Block:1956] [Pearson:1958]. The buoyancy instability that Equation (36.21) describes is properly tested in a layer confined between two plates, where the free surface is absent. The name attached to the phenomenon is therefore, strictly, attached to the wrong experiment, and it is recorded here so that a reader comparing the 1900 photographs with the 1916 calculation is not misled into thinking they are the same system.

The second qualification is that the calculation predicts only the onset. It gives the threshold and the initial pattern; it does not give the amplitude of the motion above threshold, the hexagonal rather than striped plan form, or anything about the sequence of further instabilities that follows as the heating is increased. That sequence — steady cells, then oscillations, then a cascade of period doublings into aperiodic motion — was measured in convecting liquid metal [Libchaber:1982] and in the closely analogous rotating-cylinder flow [Gollub:1975] [Taylor:1923], and it is the experimental material of Nonlinear Dynamics and Chaos. Taylor's rotating cylinders deserve a separate mention as the first case in which a hydrodynamic stability calculation and an experiment agreed quantitatively [Taylor:1923].

Primary references

[Benard:1900] [Rayleigh:1916].

Prandtl: the boundary layer made visible (1904)

Tests Equations (36.23) and (36.24). Assuming Equation (36.9).

The sharpest wrong answer in classical physics is d'Alembert's: steady potential flow of an inviscid fluid past a finite body exerts no drag whatever [dAlembert:1752], and the limit of vanishing viscosity therefore contradicts every observation of a real fluid however thin. Prandtl's resolution was not a calculation but a photograph, followed by a calculation. What he showed was that the limit is singular: the viscous term never becomes negligible near a wall, however small the viscosity, because the gradients there grow as fast as the coefficient shrinks.

Apparatus

A small hand-cranked water channel, of the order of a metre long, with a glass side wall for observation and a paddle arrangement returning the water so that a steady stream can be maintained by hand for as long as a run requires. The water is seeded with suspended particles and illuminated from the side, so that in a photograph of finite exposure each particle draws a streak along its path and the streamline pattern is recorded directly. Models are placed in the working section: a flat plate aligned with the stream, a circular cylinder, and a channel with diverging walls. One cylinder is pierced with a narrow slit connected to a suction line [Prandtl:1904].

Procedure

The flow past each model is photographed at increasing stream speed, and the streak pattern examined for the line at which the streamlines cease to follow the surface. The diverging channel is used because the same effect occurs there without a curved body, in a geometry where the pressure is known to rise along the flow direction. The decisive run is the last: with the cylinder in place and the flow separated, suction is applied through the slit on one side only, removing the fluid nearest the wall along a single line, and the photograph is repeated.

Observations and data

Outside a thin region adjacent to the surface, the streamline pattern over the forward part of a body agrees with the ideal-flow solution; the fluid there behaves as the inviscid theory says it should. The region in which it does not is thin, and thins further as the speed is raised.

At a definite line on the surface — always in the part of the body where the external pressure is rising along the flow — the streamlines leave the wall. Downstream of that line the photograph shows fluid moving backwards near the surface, and beyond the body a broad wake in place of the smooth closing streamlines that potential theory requires. The same departure appears in the diverging channel, where no curvature is involved, showing that the rising pressure and not the geometry of the body is the operative condition.

With suction applied through the slit, the flow on that side remains attached past the point where it had separated, and the wake becomes strongly asymmetric. Removing a thin sheet of fluid — a small fraction of the discharge, taken from immediately against the wall — changes the entire flow around the body. Nothing else was altered.

Phenomenon 36.12 (The thin viscous layer and its separation).

At large Reynolds number the effect of viscosity in the flow past a body is confined to a layer against the surface whose thickness decreases as the Reynolds number rises, the flow outside it being described by the inviscid equations. The layer detaches from the surface where the external pressure rises along the flow, leaving a wake of reversed flow behind the body; the detachment can be suppressed by removing the fluid within the layer, which demonstrates that the whole downstream flow is controlled by a sheet of fluid of negligible volume [Prandtl:1904]. Drag is therefore not zero in the limit of vanishing viscosity, and the d'Alembert paradox is a statement about a limit that is not attained uniformly [dAlembert:1752]. Rests on Equation (36.9).

Derivation. Derives Phenomenon 36.12. Consider steady flow of speed $U$ past a body of streamwise extent $L$, and suppose the viscous effects confined to a layer of thickness $\delta\ll L$. Within it the streamwise velocity must rise from zero at the wall to the external value over the distance $\delta$, so transverse gradients exceed streamwise ones by the factor $L/\delta$ and the dominant viscous term is $\nu\,\pp_{y}^{2}u$, of order $\nu U/\delta^{2}$. The streamwise inertia is $u\,\pp_{x}u$, of order $U^{2}/L$. A layer in which viscosity acts at all exists precisely where these balance:

\begin{equation}\tag{36.23} \frac{\nu U}{\delta^{2}}\sim\frac{U^{2}}{L} \quad\Longrightarrow\quad \frac{\delta}{L}\sim \left(\frac{\nu}{UL}\right)^{1/2} =\mathrm{Re}^{-1/2}\ep \end{equation}

This settles the paradox as a matter of principle. The layer thins as $\mathrm{Re}^{-1/2}$ and so disappears in the inviscid limit, which is why the potential solution describes the outer flow ever better; but the velocity gradient across it grows as $U/\delta\sim \mathrm{Re}^{1/2}$, so the viscous stress $\mu\,U/\delta$ on the surface falls only as $\mathrm{Re}^{-1/2}$ while the area over which it acts is fixed. The dissipation does not vanish with the viscosity, because the region in which it occurs sharpens at exactly the compensating rate. A limit of that kind cannot be taken by setting the small parameter to zero in the equation, which is what d'Alembert's argument does.

The scaling is confirmed in detail by the one case that solves completely, the flat plate aligned with a uniform stream, for which the boundary-layer equations reduce to a single ordinary differential equation in the similarity variable $y\sqrt{U/\nu x}$ [Blasius:1908]. Its solution gives the thickness at which the velocity has reached $99\,\mathrm{\%}$ of the free stream as $\delta_{99}\approx5.0\,x\,\mathrm{Re}_{x}^{-1/2}$ and the local wall stress coefficient as $0.664\,\mathrm{Re}_{x}^{-1/2}$, whence a friction drag coefficient $1.328\,\mathrm{Re}_{L}^{-1/2}$ for one side of a plate of length $L$. The numbers are worth making concrete: a plate $1\,\mathrm{m}$ long in water of kinematic viscosity $10^{-6}\,\mathrm{m}^{2}/\mathrm{s}$ at $1\,\mathrm{m}/\mathrm{s}$ has $\mathrm{Re}_{L}=10^{6}$ and a layer $5\,\mathrm{mm}$ thick at the trailing edge — half a percent of the plate. That is the region which, when removed by suction, changes the flow everywhere.

∎

The scaling settles the paradox; the two further observations — where the layer leaves the wall, and how much drag it leaves behind — are settled by two arguments that need nothing from the layer equations beyond the thinness just established.

Proposition 36.13 (Separation requires a rising pressure).

In steady flow along a solid wall the curvature of the velocity profile at the wall is fixed by the streamwise pressure gradient alone,

\begin{equation}\tag{36.24} \mu\left(\pp_{y}^{2}u\right)_{y=0}=\dv{p}{x}\ec \end{equation}

and consequently the wall shear can vanish — which is what separation is — only at a station where $\dd p/\dd x>0$. Wherever the external pressure falls along the flow, the boundary layer stays attached. Rests on Equation (36.23).

Proof.

Derives Proposition 36.13. Evaluate the streamwise component of the steady incompressible Navier–Stokes equation on the wall itself. There $u=v=0$, so both convective terms $u\,\pp_{x}u$ and $v\,\pp_{y}u$ vanish identically and what remains is

\[ 0=-\frac{1}{\rho}\pp_{x}p +\nu\left(\pp_{x}^{2}u+\pp_{y}^{2}u\right)_{y=0}\ep \]

By Equation (36.23) the profile varies across the layer over the distance $\delta$ and along it over the distance $L$, so $\pp_{y}^{2}u$ exceeds $\pp_{x}^{2}u$ by the factor $\left(L/\delta\right)^{2}\sim\mathrm{Re}$ and the streamwise derivative may be dropped; the same thinness makes the pressure variation across the layer negligible, so $p$ is the external pressure and depends on $x$ alone. That is Equation (36.24), and note that it is exact at the wall — no approximation has been made to the form of the equation, only to the relative size of two of its terms.

Now suppose the flow separates at some station, meaning $\left(\pp_{y}u\right)_{y=0}=0$ there. Expanding the profile about the wall, where $u$ and its first derivative both vanish,

\[ u(y)=\tfrac{1}{2}\left(\pp_{y}^{2}u\right)_{y=0}y^{2} +O\!\left(y^{3}\right)\ec \]

so the sign of the curvature at the wall is the sign of $u$ just above it. The fluid a little way out from the wall is still moving forward, towards the external stream $U>0$, so $\left(\pp_{y}^{2}u\right)_{y=0}>0$, and by Equation (36.24) $\dd p/\dd x>0$ at that station. Contrapositively, no station at which the external pressure is falling can be a separation point. That is exactly what the photographs report: the streamlines leave the wall only over the rear of the body, and they do so in the diverging channel as well, where there is no curvature at all and the rising pressure is the only thing the two geometries share.

∎

Proposition 36.14 (The drag is the momentum deficit of the wake).

For a body of unit span held in a uniform stream $U$ of an incompressible fluid at constant pressure far from it, the streamwise force exerted on the body equals the momentum flux missing from the wake,

\begin{equation}\tag{36.25} F=\rho\int u\left(U-u\right)\dd y=\rho U^{2}\vartheta\ec \qquad \vartheta:=\int\frac{u}{U} \left(1-\frac{u}{U}\right)\dd y\ec \end{equation}

the integrals taken across a plane downstream of the body and $\vartheta$ being the momentum thickness. A single traverse of the velocity profile behind a body therefore measures its drag, with no access to the stress at its surface. Rests on Equation (36.23).

Proof.

Derives Proposition 36.14. Take a control volume of unit span bounded below by the surface, above by a plane $y=Y$ lying wholly outside the wake, upstream by a plane where the flow is still uniform at $U$, and downstream by the traverse plane. Steady mass conservation fixes what crosses the top: the volume entering upstream is $UY$ per unit span and that leaving downstream is $\int_{0}^{Y}u\,\dd y$, so the flux through the upper plane is $\rho\left(UY-\int_{0}^{Y}u\,\dd y\right)$, outward. That fluid crosses the top with streamwise velocity $U$, because $y=Y$ is outside the wake.

Balance streamwise momentum. The pressure is uniform on every face by hypothesis and contributes nothing, so the only force is the reaction $-F$ of the body on the fluid, and

\[ -F=\rho\int_{0}^{Y}u^{2}\,\dd y +U\rho\left(UY-\int_{0}^{Y}u\,\dd y\right) -\rho U^{2}Y\ec \]

the three terms being the momentum leaving downstream, that leaving through the top, and that entering upstream. The terms in $Y$ cancel identically — which is what makes the answer independent of where the upper plane was drawn — and

\[ -F=\rho\int_{0}^{Y}u^{2}\,\dd y-\rho U\int_{0}^{Y}u\,\dd y =-\rho\int_{0}^{Y}u\left(U-u\right)\dd y\ec \]

which is Equation (36.25), the integrand vanishing above the wake so that the upper limit may be sent to infinity. Two things are worth noting. The result is exact: it uses conservation of mass and momentum and the uniformity of the pressure, and nothing about viscosity, the layer or the shape of the body. And it makes the content of Phenomenon 36.12 quantitative — a drag that does not vanish with the viscosity is a wake that does not close, and Equation (36.25) is the identity between the two.

∎

The scaling argument of Equation (36.23) and the two exact identities just proved are as far as order-of-magnitude reasoning and control volumes can take the layer. The constants quoted with them — the $5.0$, the $0.664$ and the $1.328$ — come from somewhere else, and it is worth stating as a proposition what that is.

Proposition 36.15 (Prandtl's layer equations and the Blasius similarity solution).

At large Reynolds number the Navier–Stokes system for steady two-dimensional flow past a body reduces, in the layer, to Prandtl's equations

\begin{equation}\tag{36.26} u\,\pp_{x}u+v\,\pp_{y}u =-\frac{1}{\rho}\dv{p}{x}+\nu\,\pp_{y}^{2}u\ec\qquad \pp_{x}u+\pp_{y}v=0\ec\qquad \pp_{y}p=0\ec \end{equation}

with $u$ the streamwise and $v$ the wall-normal velocity component, in which the pressure is not an unknown of the layer but is handed to it by the external potential flow. For a flat plate aligned with a uniform stream $U$ the pressure gradient vanishes and Equation (36.26) admits the similarity solution $u=U f'(\eta)$ with $\eta=y\sqrt{U/\nu x}$, where $f$ solves the Blasius equation

\begin{equation}\tag{36.27} 2f'''+f f''=0\ec\qquad f(0)=f'(0)=0\ec\qquad f'(\infty)=1\ec \end{equation}

whose numerical solution has $f''(0)=0.3321$ and $f'=0.99$ at $\eta=5.0$ [Blasius:1908]. Those two numbers are what produce the thickness $\delta_{99}\approx5.0\,x\,\mathrm{Re}_{x}^{-1/2}$, the local friction coefficient $2f''(0)\,\mathrm{Re}_{x}^{-1/2} =0.664\,\mathrm{Re}_{x}^{-1/2}$ and, on integrating the latter along the plate, the coefficient $1.328\,\mathrm{Re}_{L}^{-1/2}$ quoted in Phenomenon 36.12. Rests on Equations (36.9) and (36.23).

Full derivation in Appendix A.

Derives Proposition 36.15.

The reduction is a matched asymptotic expansion, and the appendix does it as one: an outer expansion in which the viscous term is dropped and which cannot satisfy the no-slip condition, an inner expansion on the stretched variable $y\,\mathrm{Re}^{1/2}$ in which it cannot be dropped, and a matching condition in the overlap which is what transfers the outer pressure to the inner problem. Two things are worth knowing before reading it. The method of matched asymptotic expansions is not built anywhere in this treatise — Ordinary Differential Equations and Sturm–Liouville Theory carries the WKB approximation, which is a different singular perturbation with the same flavour, and nothing more general — so the appendix states the matching principle it uses explicitly rather than appealing to a theorem of Part II. And Equation (36.27) has no solution in closed form: the two constants above are quadratures of a numerical integration, quoted from [Blasius:1908] and not computed here.

Interpretation

The experiment does three things at once, and they are usually run together to the detriment of all three.

It rescues the inviscid theory, by showing where it is valid: outside the layer, over the forward part of a body, the potential solution is what the streaks record. It falsifies the inviscid theory's global conclusion, by exhibiting the mechanism — separation — through which a vanishingly thin region imposes a finite drag on the whole flow. And it demonstrates control: the suction run is not an illustration but an intervention, and its success is the strongest evidence that the causal chain runs from the layer outwards and not the other way about.

The practical consequences run through the rest of fluid mechanics. The layer itself undergoes the same laminar-to-turbulent transition as the pipe of Section 36.3, and a turbulent layer, carrying more momentum near the wall, separates later than a laminar one; that is why the measured drag coefficient of a sphere falls abruptly as the Reynolds number rises through the transition of its own boundary layer, and why a roughened sphere can be measurably less resistant than a smooth one [Eiffel:1912]. The drag crisis is an experimental consequence of Phenomenon 36.12, and its explanation was Prandtl's.

Primary references

[Prandtl:1904] [Blasius:1908].

What the record establishes, and what it does not

Table 36.2 collects the five results. Read as a whole, they divide sharply into those the equations of Fluid Dynamics explain and those they do not, and the division does not follow the historical order.

Experiment	Regime	Status of the result
Efflux, to 1738	ideal flow	speed derived exactly; discharge derived exactly for the re-entrant mouthpiece by a momentum balance, and undetermined for the flush orifice, where it needs the shape of the free streamline
Poiseuille 1840	$\mathrm{Re}$ small	derived exactly; its accuracy measures the no-slip condition and bounds the slip length at a molecular scale
Reynolds 1883	$\mathrm{Re}\sim10^{3}$	the collapse onto one parameter is derived; so is the negative result that no infinitesimal disturbance of the profile grows, and the algebraic amplification by $\mathrm{Re}$ that takes its place; the threshold value is not derived, and depends on the disturbance level
Bénard 1900	$\mathrm{Ra}$ near critical	threshold and cell scale derived in full from linear stability with no free parameter; the plan form and everything above onset are not
Prandtl 1904	$\mathrm{Re}$ large	the layer thickness scaling is derived and resolves the d'Alembert paradox; separation is shown to require a rising pressure, and the drag is identified with the wake's momentum deficit; the layer equations themselves are not derived

The five experiments of this chapter, with the status of each result relative to the equations of Fluid Dynamics. The last column is the point of the chapter: three of the five are derivations, two are constraints on a theory that does not yet exist.

The honest summary of the turbulent state is that there is no theory of it. What exists is a set of measured regularities, an averaging procedure that reorganizes the equations without closing them, and a similarity argument whose hypotheses are assumed rather than derived.

The averaging is Reynolds's own [Reynolds:1895]: split each field into a mean and a fluctuation and average the equations, and the nonlinear term leaves behind the correlations of the fluctuations acting as additional stresses on the mean flow. Those correlations are new unknowns. Writing an equation for them produces triple correlations, and so on without end: the hierarchy of moment equations never closes, and every practical calculation of a turbulent flow closes it by an assumption that is fitted to data rather than derived. This is a structural feature of the nonlinearity, not a temporary gap.

The similarity argument is Kolmogorov's [Kolmogorov:1941], with the spectral form due to Obukhov [Obukhov:1941], resting on Richardson's picture of energy passed down a cascade of scales [Richardson:1922]. Granted that the small scales forget how the turbulence was made and, in an intermediate range, do not feel the viscosity either, dimensional analysis fixes the energy spectrum as a power law of exponent $-5/3$ with a single constant — a result derived in Fluid Dynamics and confirmed across a wide range of scales by the measurements of Grant, Stewart and Moilliet in a tidal channel, at a Reynolds number no laboratory of the time could reach [Grant:1962]. The confirmation is real and the hypotheses remain hypotheses; measured departures at high order — the intermittency corrections — show that they are not exactly true.

Underneath all of it sits an unresolved mathematical question. It is not known whether smooth solutions of the three-dimensional Navier–Stokes equations exist for all time from smooth initial data; what is proved is the existence of weak solutions [Leray:1934], which are not known to be unique or smooth, and the gap is the substance of the Millennium problem statement [Fefferman:2006]. Every statement in this chapter about the turbulent regime therefore rests on equations whose well-posedness is open — a situation with no parallel elsewhere in this treatise, and one that What We Observe but Do Not Understand records as such.

The five results above are set down in that spirit. A future theory of turbulence, if there is one, will not be asked to predict something new before it has reproduced these: that the transition in a pipe is governed by one dimensionless number, that its threshold moves with the disturbance level and has no established upper bound, that the resistance law changes exponent there, that the flow past the transition is composed of eddies at scales comparable to the geometry rather than being structureless, and that its intermediate-scale energy spectrum follows the five-thirds law. The standard reference for the phenomenology as a whole is Batchelor's monograph [Batchelor:1967].