Turbulent dissipation: which approximation to choose?

In 2014, while at the University of Newcastle (Australia), I published this paper in Physics of Fluids with Fabien Thiesset, Lyazid Djenidi and Robert Antonia. The topic: the turbulent kinetic energy dissipation rate ε, a key quantity in turbulence, and how to estimate it correctly from incomplete measurements.

My contribution: as first author, I developed the LBM code, ran the simulations, then handled the data post-processing and analysis.

Why it is hard

• ε involves 12 velocity derivatives. A simulation gives all of them, but an experiment only provides a few.
• So we use approximations ("surrogates") that assume some properties of the flow:
• ε_iso, if turbulence is locally isotropic (a single derivative is enough);
• ε_a,x and ε_a,y, if it is locally axisymmetric along x or y;
• ε_hom, if it is locally homogeneous (from the enstrophy);
• ε_4x, if it is homogeneous in the transverse plane (10 of the 12 terms, measurable with a 4 X-wire probe).
• Our question: which ones remain reliable, on average but also instant by instant?

The method

• Simulation: I used the lattice Boltzmann method (LBM), with a Smagorinsky model to keep the computation stable near the cylinder.
• Flow: wake of a square cylinder, Reynolds number Re_D = 2000. I chose a square cylinder because it gives a more isotropic wake than a circular one and is easy to model.
• Resolution: from 3.5η near the cylinder to 1η at x/D = 100 (η: Kolmogorov scale), comparable to reference direct simulations.
• Validation: I obtain a Strouhal number St = 0.129, close to the measured 0.13, and spectra in agreement with hot-wire measurements and a direct simulation of channel flow.

On average: everything converges far from the cylinder

• Beyond about 40 diameters, all approximations give the right mean dissipation.
• Closer (x = 20D), ε_iso and ε_a,y clearly depart near the axis, because of the flow anisotropy. ε_a,x, ε_hom and ε_4x remain correct.
• Notably, ε_iso works even when isotropy does not hold: the differences between the derivatives compensate each other.

Fluctuations: the isotropic approximation gets it wrong

• The standard deviation of ε is as large as its mean: dissipation fluctuates strongly, as Landau pointed out.
• ε_4x reproduces these fluctuations well. ε_iso overestimates them by almost a factor of 2.

Instant by instant: ε_4x stands out

• All probability densities have a nearly exponential tail, but they differ on the most probable values.
• Only the one of ε_4x closely follows that of ε.

• Correlation coefficients with ε on the axis: 0.87 for ε_4x, 0.68 to 0.76 for ε_iso, ε_a,x and ε_a,y, and 0.63 for ε_hom.
• ε_hom correlates less well than expected: pressure fluctuations play a significant role in the wake.

Key takeaways

• For the mean dissipation, far from the obstacle, all approximations are suitable.
• For fluctuations and instantaneous values, ε_4x is the best surrogate.
• It is a useful result for experimentalists: ε_4x can be obtained with a four-wire probe (two X-wires).

Further reading

N. Lefeuvre, F. Thiesset, L. Djenidi, R. A. Antonia, Statistics of the turbulent kinetic energy dissipation rate and its surrogates in a square cylinder wake flow, Physics of Fluids 26, 095104 (2014), PDF · DOI