Is the Cε parameter universal? What self-preservation tells us

In 2017, I co-authored this Journal of Fluid Mechanics paper with Lyazid Djenidi, M. Kamruzzaman and Robert Antonia (University of Newcastle, Australia). The topic: the normalised dissipation parameter C_ε = εL/u'³, and whether it is a universal constant of turbulence.

My contribution: automating the laboratory's round jet (LabVIEW, Matlab), the hot-wire measurements along its axis, then the data post-processing and analysis.

Why it matters

• The dissipation ε is almost impossible to measure directly in the field (atmosphere, ocean, industrial processes): it requires 12 velocity derivatives.
• If C_ε is constant, measuring the turbulence intensity u' and the integral scale L, two easy quantities, is enough to deduce ε.
• This constancy is often taken for granted, on dimensional grounds. Yet Taylor (1935) and Kolmogorov (1941) assumed it in a specific setting: a self-preserving flow, whose statistics keep the same shape as it decays.

The method

• Theory: we start from the Navier–Stokes equations, through the scale-by-scale energy budget, with no closure model, and impose self-preservation.
• Two decaying flows: grid turbulence, close to homogeneous isotropic turbulence but not self-preserving, and the far-field axis of a round jet, locally isotropic and self-preserving.
• Measurements: three grids (two perforated plates, one woven mesh) in a wind tunnel and the round jet, which I measured at three Reynolds numbers, complemented by published data (fractal grids, plane jet, direct numerical simulations).

In the jet, Cε stays constant; behind a grid, it decreases

• The theory shows that if turbulence decays while remaining self-preserving, C_ε stays constant for a given initial condition.
• On the jet axis, which is self-preserving, C_ε is indeed nearly constant after a development stage.
• Behind the grids, which are not, C_ε decreases with distance. The constancy of C_ε requires self-preservation, not the other way round.

As the Reynolds number increases, Cε tends to a limit

• The ratio of the integral scale to the Taylor microscale, L_u/λ, grows roughly linearly with the Reynolds number Re_λ, whatever the flow.
• With the relation C_ε = 15 (L/λ)/Re_λ, we deduce that C_ε decreases towards a limiting value C_ε,∞.

• Measurements and theory agree for each flow.
• At high Reynolds numbers, the values for decaying flows join those of forced turbulence, around 0.5.
• The limit still seems to depend on the flow, but a universal constant cannot be ruled out.

Key takeaways

• The constancy of C_ε during decay stems from self-preservation: it holds in the round jet, not behind a grid.
• C_ε decreases as the Reynolds number increases, towards a value close to 0.5, possibly universal.
• To estimate ε from u' and L, my takeaway is that you must check that the flow is self-preserving and that its Reynolds number is high.

Further reading

L. Djenidi, N. Lefeuvre, M. Kamruzzaman, R. A. Antonia, On the normalized dissipation parameter C_ε in decaying turbulence, Journal of Fluid Mechanics 817, 61–79 (2017), PDF · DOI