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

Reynolds stress — where it appears

The correlation of two velocity fluctuations, which enters the averaged momentum equation as though it were a stress. It is not a stress but a momentum flux, it is what the closure problem is a problem about, and it exists in any averaged nonlinear flow.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

Ten unknowns, four equations. What is left after the Navier–Stokes equations are averaged. The mean velocities and mean pressure were there before; the six Reynolds stresses are new, and they arrived from the one term that does not average away. Nothing in the count is an approximation — the averaged equations are exact — and that is what makes the gap uncomfortable.

What averaging costs

Split the velocity into a mean and a fluctuation, average the equations, and the result is exact. It is also short of six equations, because the one nonlinear term does not average away and leaves six new unknowns behind that nothing determines.

turbulence · Closure Every rung of the hierarchy is worse than the last. Writing a transport equation for the Reynolds stress does not close the system: that equation contains the triple correlation, whose equation contains the quadruple. The bars are the number of independent components at each order and the number of new unknowns its own equations introduce. The second is always larger, and the gap widens.

The ladder that never closes

Being six equations short is a problem with an obvious remedy — derive six more. The remedy works, produces an exact equation for the Reynolds stress, and leaves ten new unknowns behind. The gap does not narrow at any level, and the counting says why.

turbulence · Closure 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 An oscillation with no mean, and the steady flow it drives. The steady second-order velocity through a Stokes layer, in units of U U′/ω. The first-order flow averages to zero at every height; the average of its own nonlinear term does not, and the pale curve is that forcing. Integrating it twice across the layer, with no slip at the wall and no stress at the top, gives a steady velocity that rises through the layer and settles at -0.749998 — Rayleigh's −3/4, which was not put in anywhere. Beyond about five layer thicknesses nothing more happens, which is why the number is a boundary condition for the flow outside.

An oscillation with somewhere to go

Shake a fluid back and forth over a body and it develops a steady circulation that never reverses. The driving flow has no mean at all; the mean of its own nonlinear term does, and integrating that twice across the oscillatory layer gives a slip velocity of exactly three-quarters of U dU/dx over the frequency.

viscous · Streaming What a probe in a flame sees. Half the time hot light fluid at a hundred metres a second, half the time cold heavy fluid at twenty. That is what intermittency in a jet flame looks like at a point, and it is the simplest field in which the two averages of the velocity are different numbers.

Two averages of one flow

In a flow whose density varies there are two mean velocities, they are both correct, and across a flame they differ by a factor of two. One of them is what a hot wire returns; the other is what every compressible turbulence model is written in; and the mass flux is the single product they agree on.

kinematics · Averaging 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 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 band of growing waves that opens at 5772 and narrows as the viscosity goes. The wavenumbers at which a two-dimensional wave on plane Poiseuille flow neither grows nor decays, against the Reynolds number on a logarithmic axis. Inside the tongue waves grow; outside they decay. The tongue's tip is the critical point. Both edges slope downward and towards each other in wavenumber as the Reynolds number rises, so the band of unstable waves shrinks towards long waves — the direction in which the inviscid problem, which has no growing wave at all, is reached.

The profile Rayleigh cleared and viscosity did not

Flow between two plates has no inflection point, so without viscosity no wave on it can grow. With viscosity one does, above a Reynolds number of 5772. Taking the viscosity away again slows that wave and narrows the band it grows in, because the stress that feeds it is made by viscosity in the first place.

turbulence · Instability

Named alongside it

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

AveragingTurbulenceClosureDissipationNonlinearityClosure problemMeasurementCorrelationEddy viscosityMemory kernelModel validityMomentum flux

All concepts