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Turbulence

solid pillar 1 · Physical modeling and domain theory

In one minute

  • Turbulence is a multiscale problem: energy enters at the integral scale \(L\), cascades without loss, and dissipates at the Kolmogorov scale \(\eta\).
  • \(L/\eta \sim \mathrm{Re}^{3/4}\), so resolving everything costs \(\mathrm{Re}^{9/4}\) points in 3D and \(\mathrm{Re}^{3}\) in work. This single scaling explains why RANS and LES exist.
  • Averaging the equations creates more unknowns than equations — the closure problem. Every model is a guess at the missing term.
  • The \(k^{-5/3}\) inertial range is the one universal result, and it holds only at high \(\mathrm{Re}\), away from walls, over a limited band.
  • Choose the method by what you need: RANS for mean loads, LES for unsteady structure, DNS for physics at modest \(\mathrm{Re}\).

Key results

Scale separation. With dissipation rate \(\varepsilon \sim u'^3/L\):

\[ \eta = \left(\frac{\nu^3}{\varepsilon}\right)^{1/4}, \qquad \frac{L}{\eta} \sim \mathrm{Re}_L^{3/4}, \qquad N_{\text{3D}} \sim \mathrm{Re}_L^{9/4}, \qquad \text{work} \sim \mathrm{Re}_L^{3} \]

The Taylor microscale \(\lambda = \sqrt{15\nu u'^2/\varepsilon}\) sits between them — useful as a correlation length, but not a dissipation scale, a common misreading.

Kolmogorov spectrum in the inertial range \(1/L \ll k \ll 1/\eta\):

\[ E(k) = C_K\,\varepsilon^{2/3} k^{-5/3}, \qquad C_K \approx 1.5 \]

Reynolds decomposition \(u_i = \overline{u}_i + u_i'\) gives the RANS equations:

\[ \overline{u}_j\frac{\partial \overline{u}_i}{\partial x_j} = -\frac{1}{\rho}\frac{\partial \overline{p}}{\partial x_i} + \nu\nabla^2\overline{u}_i - \frac{\partial \overline{u_i'u_j'}}{\partial x_j} \]

The last term, the Reynolds stress \(-\rho\overline{u_i'u_j'}\), is six new unknowns with no equation. Deriving one introduces triple correlations, and so on — the closure problem does not terminate.

Turbulent kinetic energy budget:

\[ \frac{Dk}{Dt} = \underbrace{-\overline{u_i'u_j'}\frac{\partial \overline{u}_i}{\partial x_j}}_{\text{production }P} - \underbrace{\varepsilon}_{\text{dissipation}} + \underbrace{\text{transport}}_{\text{redistribution}} \]

Near equilibrium \(P \approx \varepsilon\). How badly a model violates that is a good first diagnostic.

Cost comparison, wall-bounded flow:

Method Resolves Cost scaling Use when
RANS mean only \(\mathrm{Re}^{0}\) (grid set by geometry) mean forces, design sweeps
Wall-modelled LES energetic eddies \(\sim\mathrm{Re}^{1.0\text{–}1.3}\) unsteady loads, mixing, acoustics
Wall-resolved LES to near-wall streaks \(\sim\mathrm{Re}^{1.8}\) transition, separation detail
DNS everything \(\sim\mathrm{Re}^{3}\) model development, canonical physics

Mental model

Energy is injected by the mean shear at the scale of the geometry, handed down through a cascade of vortex stretching — each generation smaller and faster — until the strain rate is large enough for viscosity to convert it to heat. The cascade itself is inviscid; viscosity only sets where it stops, not how much is dissipated. \(\varepsilon\) is fixed by the large scales.

That is why the \(\mathrm{Re}\)-dependence is so brutal. Raising \(\mathrm{Re}\) does not change the energy flux; it just adds more generations before viscosity can act, and each generation needs its own grid points.

Numerics / practice

  • Check \(P/\varepsilon\) in a converged RANS solution. Far from 1 in an equilibrium boundary layer means the model is being used outside its calibration.
  • Compute the local \(\eta\) before claiming DNS, and the local integral scale before claiming LES resolves the energetic eddies.
  • Average long enough. Statistics need \(O(100)\) integral timescales \(L/u'\). Shorter records give confident-looking but unconverged means.
  • Spectra need care: window the signal, and remember that the highest resolved wavenumbers are contaminated by the scheme's own dissipation long before the grid cutoff.
Failure modes

Assuming isotropy where there is none. The \(k^{-5/3}\) law and most model constants are calibrated on homogeneous isotropic turbulence. Near walls, in strong curvature, under rotation or stratification, the anisotropy is the physics — and the model has averaged it away.

Too-short averaging. Means and especially second moments converge slowly. Symptom: statistics that shift noticeably when you extend the window, or that differ between the upper and lower half of a symmetric geometry. Always report the averaging interval in integral timescales.

Reading the numerical cutoff as physics. The tail of a computed spectrum rolls off because of the scheme's dissipation, not Kolmogorov's. Fitting \(\eta\) from that roll-off gives a number that depends on your grid.

Taylor microscale used as a dissipation scale. It is a correlation length that happens to appear in the dissipation formula. Sizing a grid on \(\lambda\) rather than \(\eta\) under-resolves by roughly \(\mathrm{Re}^{1/4}\).

"DNS" that is not. Resolving to \(\Delta x \approx \eta\) is necessary but not sufficient — the timestep, domain size (must contain several integral scales) and run length all bind. An under-sized periodic box artificially constrains the largest eddies and biases every statistic.

Worked example

The cost scaling is worth feeling numerically rather than remembering as an exponent:

Re = [1e4, 1e6, 1e8]
for r in Re:
    n_per_dim = r**0.75
    print(f"Re={r:.0e}   L/eta={n_per_dim:.2e}   "
          f"3D points={n_per_dim**3:.2e}   work~{r**3:.1e}")

Re=1e+04   L/eta=1.00e+03   3D points=1.00e+09   work~1.0e+12
Re=1e+06   L/eta=3.16e+04   3D points=3.16e+13   work~1.0e+18
Re=1e+08   L/eta=1.00e+06   3D points=1.00e+18   work~1.0e+24

A full-scale aircraft at \(\mathrm{Re}=10^8\) would need \(10^{18}\) points. Modelling is not a convenience.

Pages in this section

  • RANS — closures, eddy viscosity, wall treatment.
  • LES — filtering, subgrid models, resolution requirements.

Connections

Sources

  • Pope, Turbulent Flows — the standard graduate reference.
  • Tennekes & Lumley, A First Course in Turbulence — scales and the cascade, concisely.
  • Archive: Others/Turbulence_JCM, Others/UMD_Turb — turbulence notes.
  • Archive: see the course archive.