Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Closure — where it appears

An assumption added to the averaged equations of motion to make up the terms averaging destroyed. Every practical turbulence calculation contains one, its constants are measured rather than derived, and a figure produced from it is a consequence of the assumption rather than of the equations.

Named by 19 essays across 5 fields — each of them below, with the objects they name alongside it.

A steady pressure field, from a flow with no steady part. The time-averaged pressure round a cylinder in a stream that oscillates as U₀cos ωt. The mean velocity is exactly zero at every point — the flow spends as long going one way as the other — and the mean pressure is not, because pressure depends on the square of the speed and a square has no sign. The mean coefficient reaches -2.00 at the shoulders and averages -1.00 over the surface, and its resultant is 6.6e-16: a real field with no force in it. The pale lines are the instantaneous streamlines, which reverse every half cycle.

The mean is not the flow

Average an unsteady flow and the result is a new object with its own properties, and it is not a solution of anything. An inviscid stream oscillating about zero has a mean velocity of exactly nothing everywhere, a mean pressure that reaches minus two dynamic pressures at the shoulders, and a missing term in its own momentum equation that can be written down in closed form.

kinematics · Averaging The section the pressure asked for. The designed section over the one it started from, both drawn to their own chords. Asking for 28 per cent more speed over the forward 62 per cent of the upper surface produces a section 15.0 per cent thick against the original's 10, with the extra thickness forward and the camber changed — none of which was asked for, and all of which is what that pressure distribution is. The pale outline is the baseline. The one thing the method cannot be told is where along the chord any of it happens: the speed is prescribed against the circle's parameter, and where a given station ends up is an output of the same solve that produces the shape.

Ask for the pressure, and see what shape that is

A designer knows what the pressure distribution has to do long before knowing what shape does it. Running the problem that way round is possible, it is exact, and it refuses more asks than it grants.

inviscid · Inverse design One number instead of a porous medium. The velocity through the bottom of a channel whose lower wall is a porous block of permeability 1e-4. Inside the block the flow decays over the pore scale √K = 1.0e-2 to Darcy's seepage velocity; above it the channel profile arrives at the interface with a slip velocity rather than at rest. A channel told nothing but u = √K du/dy at a flat wall reproduces that profile to 0.058 per cent of the flow rate, against 3.03 per cent for a wall told to hold the fluid still. The grid solve of the coupled problem agrees with the closed form to 0.0077 per cent.

A wall that is not quite there

A porous surface has structure on every scale below the pore, and no calculation resolves it. The whole of it can be replaced by one length — the square root of the permeability — and the replacement is exact to first order, with what it leaves out identifiable as the flow the wall itself carries.

kinematics · Boundary conditions A millionth, doubling every three-quarters of a second. The separation of two trajectories started a millionth apart, on log axes against time. It grows as a straight line until it saturates at the size of the attractor, and the slope of that line is the largest Lyapunov exponent — measured here on the system by renormalising a nearby pair, not quoted. Sensitive dependence is what the straightness of the line means.

The randomness that is not in the equations

Turbulence is described in the language of statistics — means, variances, spectra, probability distributions — and none of that language appears in the equations it is a description of. The Navier–Stokes equations have no random term anywhere in them. What is random is the observer's ignorance of the initial data, and the flow's habit of amplifying it.

misconceptions · Randomness The plateau that is the log law. y⁺ du⁺/dy⁺ across a channel, at five Reynolds numbers. Millikan's argument says this quantity must be constant wherever neither the viscous length nor the channel width may appear, and its value there is 1/κ. At Re_τ = 180 there is no flat part at all; at Re_τ = 100,000 it is flat over 2.16 decades and gives κ = 0.4120. The log law is a statement about a limit, and this is the picture of the flow approaching it.

The layer with no length in it

The logarithm in a turbulent wall profile does not come from any model of turbulence. It comes from a region where neither of the flow's two lengths is allowed to appear, and where a velocity gradient therefore has nothing to depend on but the distance to the wall. The constant in it has never been derived from anything.

turbulence · Wall law Where the first grid point may go. The error in the friction a wall treatment infers, against the height of the first grid point, at Re_τ = 20,000. The velocity fed to each treatment is the closure's own, so what is plotted is the modelling of the boundary condition with nothing else in it. The log-law function is exact between y⁺ 30 and 10261 and is 64 per cent wrong at y⁺ = 1; the sublayer treatment is exact below y⁺ = 5 and hopeless above it; and the blend that most codes ship is within a few per cent everywhere and exact nowhere.

What a code says to a wall

A calculation that cannot afford to resolve the viscous sublayer has to tell the wall something else instead, and what it tells it is the law of the wall — an asymptotic result, applied at one grid point, on the assumption that the point lies in a region the calculation has not checked exists. Where it does, the answer is exact. Where it does not, the friction is out by tens of per cent, and refining the grid makes it worse.

turbulence · Wall law Ballistic first, diffusive later. The mean square displacement of a parcel carried by a fluctuating velocity, from Taylor's 1921 integral with an exponential correlation of time scale 1. Below the correlation time it follows the straight-line law u²t² exactly — the parcel has not yet changed its mind — and above it the curve joins the diffusion law 2u²T·t, offset by the head start the ballistic phase gave it. The eddy diffusivity is u²T = 1.000, and it is a property of the correlation rather than of any equation of motion.

How far a parcel gets

Turbulent transport is one integral. A parcel carried by a fluctuating velocity goes in a straight line while it still remembers its own motion and performs a random walk once it has forgotten, and the crossover is the correlation time — so an eddy diffusivity is not a property a flow has, it is the limit of a measurement that has run for long enough.

turbulence · Mixing Most of the fluctuation is the switching. What a hot wire at the edge of a shear layer records, taken apart. The conditional intensity — the fluctuation inside the turbulent fluid — is flat at 0.16 by construction. The measured intensity peaks 20 per cent above it, and the excess is the third curve: the variance of a signal that keeps switching between two mean velocities, which peaks where the switching is most even and is not turbulence at all. A model calibrated against the measured curve is being fitted to an artefact of averaging.

Turbulent some of the time

At the edge of a jet or a wake a probe is inside turbulent fluid for part of the time and in perfectly smooth flow for the rest, and an ordinary time average mixes the two. Most of the fluctuation it records there is not turbulence at all — it is the switching between two states, and it peaks where the switching is most even rather than where the turbulence is strongest.

turbulence · Intermittency Two exact solutions of the same problem. Two steady Euler flows in the same square cell with the same boundary condition, differing only in the function relating vorticity to streamfunction. On the left the vorticity is proportional to the streamfunction, which is the textbook cellular flow; on the right it is uniform. Both satisfy the equations exactly. Nothing in the ideal theory prefers either, and at the same peak streamfunction their kinetic energies differ by thirty-one per cent.

The vorticity nothing decides

A streamline that comes from upstream carries its vorticity with it. A closed one comes from nowhere, so nothing determines what it carries — the ambiguity is not one number per body but a whole function. What closes it is a limit, and setting the viscosity to zero gives a different answer from letting it go to zero.

inviscid · Euler rotational And the net drift, which is where they disagree. The Stokes drift plus each return flow. All three drift forward at the surface, because the Stokes drift there swamps any return current of the right size. Below that they part company completely: the uniform current has most of the column moving upstream, and the two that satisfy no slip have almost none of it. The reversal depths span 77 per cent of the water column.

The drift a closed box will not allow

A wave in a wave tank carries mass forward, and the tank has nowhere to put it. So a return current appears carrying exactly the opposite transport — exactly, from mass conservation and nothing else. Which fixes a total and leaves the answer anybody wants entirely open.

kinematics · Stokes drift A normal stress the closure makes negative. The Boussinesq closure's first normal stress in a plane strain, against the strain measured in units of the turbulence's own time scale. It crosses zero at S k/eps = 1/(3 C_mu) = 3.704 — eleven per cent above the value the constant was calibrated at — and goes on falling. A variance below zero is not a small error; it is a statement that cannot be true.

The constant that makes a variance negative

Every engineering turbulence calculation in the world rests on one number, C-mu equals 0.09. It is not a property of turbulence. It is the assertion that a particular ratio is ten thirds, which is true in one flow — and eleven per cent above that flow the same closure reports a mean square below zero.

turbulence · Closure Two flows with one mean profile. The time-averaged velocity of a plain shear and of the same shear carrying a zero-mean disturbance. There is one line on this plot: the largest difference anywhere across the channel is four parts in 10¹⁷.

What a mean profile cannot tell anybody

Two flows are built here with mean velocity profiles that agree to four parts in 10¹⁷. One of them carries momentum across the shear and dissipates forty per cent more energy; the other carries nothing. Everything that distinguishes them is second order in a disturbance the mean cannot see at all.

kinematics · Averaging Four wall models, through the buffer layer. Van Driest's damped mixing length, Reichardt's fit, Spalding's implicit law, and a control with no buffer layer at all — the viscous sublayer joined straight to the logarithm where they cross, at y+ = 11.6. The three fitted models agree with each other to a per cent and a half; the control is nineteen per cent above them at y+ = 10.

Three buffer layers, one friction

Four wall models are put through a pipe. The one with no buffer layer at all is nineteen per cent wrong where the turbulence production peaks and two and a half per cent wrong in the friction; the three respectable ones are within two per cent of each other in the buffer layer and spread by seven in the friction. The answer is not where it was expected.

turbulence · Wall law One shape, at every station and for every jet. The axial velocity of Schlichting's round jet at four distances, each scaled on its own centreline speed and its own half-width. The four curves are one curve: the shape function (1 + eta²/4)⁻² has no parameter in it at all, and every station of every jet — laminar or modelled, weak or strong — collapses onto it exactly.

Exactly similar, and one number short

The round jet has an exact solution of the Navier–Stokes equations, and a turbulent jet is modelled by the same formula with the viscosity replaced by a fitted constant. The shape is identical. Nothing a measurement of the profile can do will tell the two apart.

viscous · Free shear The production jumps and the dissipation does not. Production and dissipation against time, through a step change in the strain rate. The production follows the strain immediately — it is the strain squared times an eddy viscosity — and the dissipation moves by less than one per cent at the instant of the step, because it is set by a cascade that has not been told yet.

A dissipation that lags its production

Change the strain rate on a patch of turbulence and the production of energy follows instantly — it is the strain squared. The dissipation moves by less than one per cent, because it is the far end of a cascade that has not been told yet, and the two are out of balance by a factor of six for the next turnover.

turbulence · Dissipation The stress a closure predicts, and the stress there is. The Reynolds-stress anisotropy through a step change in the strain rate, against what an eddy viscosity gives — which is the equilibrium value at every instant. The real stress takes about a turnover to get there, and during that turnover the closure is wrong by up to sixty per cent.

A closure with no memory at all

An eddy viscosity says the Reynolds stress is the mean strain rate times a number, now. The stress it is standing in for takes a turnover to arrive, so the closure is the zero-frequency limit of a response that has a lag in it — and the curve it is the limit of is the same shape as an aerofoil's lift deficiency.

turbulence · Closure A window one core wide reads the vortex 7.5 per cent slow. The tangential velocity across a Lamb–Oseen vortex, in units of its core radius and of Γ/2π divided by it, as it is and as particle image velocimetry reports it with square interrogation windows of three widths — the average of the velocity over each window, which is what a correlation over the window returns to first order. The true peak is 0.6382 at 1.1209 core radii. A window 0.5 core radii wide reports 98.0 per cent of it, 1.021 times as far out, a window 1 core radii wide reports 92.5 per cent of it, 1.084 times as far out and a window 2 core radii wide reports 77.1 per cent of it, 1.334 times as far out. The instrument that measures velocity directly still reports a slower, fatter vortex than the one there, by an amount set entirely by the window against the core.

The window every vector is averaged over

Particle image velocimetry is the one flow-visualisation technique that reports the velocity itself, and it still applies an operator: every vector is an average over an interrogation window. A window is a filter with a transfer function, and it makes a vortex slower and fatter, a thin shear layer exactly as thick as the window, and some features smaller than the window point the wrong way.

misconceptions · Visualisation The coefficient that was a constant, against the number it is said not to depend on. The dissipation coefficient Cε = eps·l/u³ along two decays, plotted against the Taylor-scale Reynolds number they pass through. One is flat because it was put in flat; the other falls as the reciprocal of the Reynolds number, which is what is measured in the near field of a grid. Neither line is a derivation. What is exact is the relation between them, Cε = 15(ℓ/λ)/Reλ, which is a rearrangement of two definitions and holds along both curves to 5·10⁻¹⁶.

The constant that travels

Every decay law in the subject rests on the dissipation being some constant times u³ over a length. The constant is not one. Letting it move the way grid measurements say it moves changes the decay exponent by a quarter — and lands one of the answers five per cent from another that is entirely different physics.

turbulence · Decay The plateau everybody looks for is a summit, and a low one. −Dₗₗₗ/((4/5)εr) against separation in decaying turbulence at five Taylor-scale Reynolds numbers. None has a plateau at one. Each rises through the viscous range and turns over, peaking at 0.49, 0.63, 0.75, 0.85, 0.90 for Reλ = 50, 100, 200, 500, 1000. A measurement of ε that takes the largest value of this curve as four-fifths reads each of those shortfalls as a smaller dissipation.

The decay inside the four-fifths law

The four-fifths law gives the dissipation of turbulence from one measured moment with no constant in it, which makes it the obvious way to measure how the dissipation coefficient travels during a decay. But the law is exact only in a limit, and a decaying flow is not in it. The decay itself takes a quarter off the moment at the Reynolds numbers grids reach — and the bias moves as the flow decays, by a sixth, in the direction opposite to the effect being looked for.

turbulence · Decay

Named alongside it

The objects these essays reach for when they reach for this one.

MeasurementTurbulenceModel validityAveragingDissipationEddy viscosityBoundary conditionModel limitRegimeReynolds stressStatisticsBoundary layer

All concepts