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

Power law — where it appears

A constitutive relation in which stress goes as a power of the strain rate, with an index below one for a shear-thinning fluid and above for a shear-thickening one. It is a description over a stated range rather than a mechanism, and it diverges at both ends of that range.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

A limit that exists and is never reached. The exponent of the best power law fitted across the overlap layer, against the friction Reynolds number. A logarithm is the zero-exponent member of that family, so the log law is what this sequence is heading for — and it heads there as 1/ln Re_τ, which is the slowest useful way of approaching anything. The exponent is still 0.102 at Re_τ = 10⁶, and driving it to a hundredth needs a Reynolds number with a hundred and fourteen in its logarithm.

A limit nothing reaches

A dimensional argument that succeeds says a variable has dropped out of the answer. The Blasius profile has no Reynolds number in its shape at any Reynolds number; the overlap layer's power-law exponent is still 0.102 at Re_τ of a million and falls as a logarithm, so the limit exists and nothing ever gets there.

regimes · Dimensional Seven fluids in one pipe at one pressure gradient. Velocity profiles of power-law fluids in a round pipe, all at the same pressure gradient and the same consistency, scaled to the fastest. A shear-thinning fluid (n below one) is flatter in the middle and steeper at the wall; a shear-thickening one is the reverse. The flattening is often called plug-like, which invites the reading that the fluid is moving more freely — what has actually happened is that all the shear has been pushed into a thin annulus at the wall, which is the expensive place to put it.

A viscosity that depends on the question

Blood, paint, molten polymer and drilling mud have no viscosity. They have a relation between stress and strain rate that is not proportional, so the ratio of the two depends on how hard they are being sheared — and an instrument that reports one number is reporting a property of itself.

viscous · Non-newtonian Two decay laws from two invariants, and nothing in the equations to choose. The energy of a decaying turbulence against time, integrated from dK/dt = −A K^(3/2)/l with the large scales conserving u² l³ in one case and u² l⁵ in the other. The exponents come out at 1.1997 and 1.4282 against the closed forms 6/5 and 10/7. Which invariant holds is decided by the shape of the spectrum at the very largest scales, at the moment the stirring stops.

What decay never forgets

Stir a box of fluid and stop. The turbulence decays, at a rate with no viscosity in it — so the rate cannot come from the fluid. It comes from an invariant of the very largest scales, fixed at the moment the stirring stops, and never revisited.

turbulence · Decay Model spectra at four Reynolds numbers, compensated. The spectrum multiplied by k^(5/3) and divided by eps^(2/3), so that a true inertial range is a horizontal line at the Kolmogorov constant. What a finite Reynolds number has instead is a single maximum: it reaches 1.4996 at the highest and 1.49 at the lowest, and the band over which it is flat to one per cent goes from a third of a decade to two.

The range a real Reynolds number does not have

Kolmogorov's minus five thirds is a statement about a band of scales that has forgotten the forcing and does not feel the viscosity. Both conditions are about separation, and separation is exactly what a finite Reynolds number does not have much of.

turbulence · Spectrum The condensate, and the limit of no friction which does not exist. In a steady state the friction must remove everything the forcing puts in, so the energy is eps/(2 alpha) and the coherent velocity is sqrt(eps/alpha) — exactly a minus one half power, checked to 10⁻¹². As the friction is weakened the condensate grows without bound: the limit alpha to zero is not a flow with a weak condensate, it is a flow with no steady state at all.

Where the inverse cascade stops

Two-dimensional turbulence sends its energy upward in scale, and the upward direction has an end: the box. Without something to remove the energy before it arrives, it accumulates there in a pair of vortices filling the domain, and the limit of no friction has no steady state at all.

turbulence · Two-dimensional Four fluids under the same stress, and the four profiles that result. Each fluid carries the same linear stress and answers it differently: a Newtonian fluid with a parabola, a shear-thinning one with a blunter profile, a shear-thickening one with a sharper one, and a Bingham plastic with a plug in the middle where the stress is below its yield point.

The stress a pipe knows

A capillary viscometer measures a pressure drop and a flow rate and reports a viscosity. The first half of that inference is a force balance and is exact for any fluid there is; the second needs the slope of a whole flow curve, which is the experiment the instrument was bought to avoid.

viscous · Non-newtonian Five spectra, five exponents, one linear equation. The energy of a decaying turbulence after the nonlinear term has stopped mattering, computed by integrating the exact modal solution E(k,0)exp(−2 nu k² t) at five different shapes of the spectrum at the origin. Each is a straight line on these axes and no two have the same slope: the exponent is (m+1)/2, where k^m is the spectrum's behaviour at wavenumbers smaller than any eddy. The 5/2 that is quoted as the final period's universal exponent is the m = 4 line and one of five.

Universal, and one of five

A turbulence that has run its Reynolds number down stops being turbulent, the equations go linear, and the decay picks up a new exponent. That exponent is quoted everywhere as 5/2 and as universal. It is neither: it is the same corner of the same spectrum deciding the answer a second time.

turbulence · Decay 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

Named alongside it

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

MeasurementModel limitReynolds numberDissipationDecayIntegral scaleInvariantSelf-similaritySpectrumAsymptoticsConstitutive lawFinal period

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