Turbulence

Lewis Fry Richardson supplied the physical picture in 1922, in a book otherwise devoted to forecasting weather by hand:

Big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity.

Energy enters at the scale of whatever stirs the flow — the pipe diameter, the aircraft, the width of the ocean current. Viscosity can only remove energy where velocity gradients are steep, which means at very small scales. The nonlinear term does the transport in between, breaking large motions into smaller ones. In the middle of that range, energy is neither added nor dissipated; it merely passes through at a rate $\varepsilon$.

G. I. Taylor made the statistics precise in 1935, defining turbulence by the correlation between velocities at two points and introducing homogeneous isotropic turbulence — an idealisation that no real flow satisfies and in which nearly all theory is still done.

Then Andrey Kolmogorov, in two short papers in 1941, extracted quantitative predictions from the cascade picture with almost no mathematics.

Kolmogorov's hypothesis is a statement about what the middle of the cascade can depend on. At separations $r$ much smaller than the stirring scale $L$ and much larger than wherever viscosity acts, the flow has forgotten how it was stirred and does not yet feel viscosity. The only quantity available is $\varepsilon$, the energy flux per unit mass, with dimensions

$$[\varepsilon] = \frac{\text{energy}}{\text{mass} \times \text{time}} = \frac{\mathrm{m^{2}}}{\mathrm{s^{3}}}.$$

Prediction one. The typical velocity difference $\delta u$ across a separation $r$ can only be built from $\varepsilon$ and $r$. The only combination with units of velocity is

$$\delta u \sim (\varepsilon r)^{1/3}.$$

Prediction two. The energy per unit wavenumber $E(k)$ has units of m³/s². Built from $\varepsilon$ and $k$ (units 1/m), the only possibility is

$$E(k) = C\,\varepsilon^{2/3} k^{-5/3}.$$

This is the five-thirds law, and $C \approx 1.5$ turns out to be very nearly the same in every flow measured.

Prediction three. Viscosity takes over where the local Reynolds number falls to one. With $\nu$ (units m²/s) now allowed, the only length is

$$\eta = \left(\frac{\nu^{3}}{\varepsilon}\right)^{1/4}.$$

Now put numbers on the last one, because it is the reason turbulence remains computationally out of reach. Estimate $\varepsilon \sim U^{3}/L$, as the cascade picture requires, and the ratio of the largest to the smallest scale becomes

$$\frac{L}{\eta} = \left(\frac{L^{4}U^{3}}{\nu^{3}L}\right)^{1/4} = \mathrm{Re}^{3/4}.$$

A simulation that resolves every scale needs a grid spacing of order $\eta$ across a box of size $L$, in three dimensions, so the number of grid points is

$$N \sim \left(\mathrm{Re}^{3/4}\right)^{3} = \mathrm{Re}^{9/4}.$$

For an aircraft wing at $\mathrm{Re} = 10^{7}$:

$$N \sim \left(10^{7}\right)^{9/4} = 10^{15.75} \approx 5.6\times10^{15} \text{ points}.$$

Storing five numbers per point in double precision is

$$5.6\times10^{15} \times 5 \times 8 \text{ bytes} \approx 2.2\times10^{17} \text{ bytes} = 220 \text{ petabytes},$$

which is more memory than exists in any machine. The time step must also shrink with the grid, adding a further factor of about $\mathrm{Re}^{3/4} \approx 2\times10^{5}$ steps per flow-through time, bringing the operation count for a single flow-through to the order of $10^{21}$. On a machine doing $10^{18}$ operations per second — exascale, achieved in 2022 — that is around a thousand seconds of computing per flow-through time, if memory were free and the efficiency perfect. Neither holds.

This is why engineering does not simulate turbulence directly, and why the closure problem is not a theoretical nicety. Every aircraft, engine and weather forecast computes a modelled turbulence whose coefficients were fitted to experiments, and the exponent $9/4$ says that no foreseeable computer changes that.

Lev Landau objected almost immediately, at a seminar, and the objection is subtle. Kolmogorov treats $\varepsilon$ as a constant. It is not: dissipation fluctuates violently in space and time, concentrated in thin sheets and filaments of intense vorticity with relatively quiet fluid between them. Averaging $\varepsilon^{2/3}$ is not the same as taking the two-thirds power of the average, so the predictions should be corrected — and the corrections should grow with the order of the statistic being measured.

Measurements confirm this. The five-thirds law for the energy spectrum, a second-order quantity, holds beautifully. Higher-order structure functions deviate systematically from the 1941 exponents, by amounts that increase with order, and the pattern is reproducible across very different flows. Kolmogorov published a refined theory in 1962 assuming a lognormal distribution of local dissipation; it is known to be internally inconsistent at high orders. Multifractal models describe the data without deriving it.

So the position after eighty years is this. A dimensional argument whose central assumption is demonstrably false makes a prediction that is right to within a percent or two, and the corrections to the false assumption have not been derived from the equations by anyone. The deviations are small, universal-looking, and unexplained — a combination that keeps the problem alive.

What turbulence looks like in a rotating, stratified fluid the size of a planet, where the cascade runs partly backwards and large structures organise themselves out of small ones, is the subject of geophysical fluid dynamics.