Round jet: self-preservation confirmed at every scale

In 2016, I co-authored this Journal of Fluid Mechanics paper with Lyazid Djenidi, Robert Antonia (University of Newcastle, Australia) and Jean Lemay (Université Laval, Canada). The topic: showing that, far from the nozzle, turbulence on the axis of a round jet is self-preserving at every scale, and deriving its decay laws from that.

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

• A flow is self-preserving when its statistics keep the same shape as it develops: only a length scale and a velocity scale change.
• On the axis of a round jet, the mean velocity U is taken to decay as x⁻¹, the turbulent energy k as x⁻² and the dissipation ε as x⁻⁴.
• These laws are usually obtained from momentum conservation or by assuming that the parameter C_ε is constant. Our question: can they be derived from self-preservation alone, scale by scale?

The method

• Theory: we apply self-preservation to the transport equation of the second-order structure function, that is to the scale-by-scale energy budget, under local isotropy. On the axis, turbulent and pressure diffusion are negligible.
• Wind tunnel: an air jet leaves a nozzle of diameter D = 55 mm, after an 85:1 contraction, at a Reynolds number Re_D of about 10⁴ to 10⁵.
• Measurements: I placed a single 1.2 µm hot wire on the jet axis. The turbulence intensity there reaches 25%, so I apply a corrected form of Taylor's hypothesis to the signal to capture the small scales, and hence ε, without bias.

The theory yields all the decay laws

• Self-preserving solutions: U ∼ (x − x_o)⁻¹, u'² ∼ (x − x_o)⁻², length scales ∼ (x − x_o), ε ∼ (x − x_o)⁻⁴, with x_o a virtual origin.
• No extra assumption: the x⁻¹ decay of U becomes an outcome of the analysis, not an input.
• The Taylor Reynolds number Re_λ must be constant, and the constancy of C_ε follows from complete self-preservation.

• The mean velocity does follow the x⁻¹ law: U_j/U is linear in x/D, with a virtual origin x_o ≈ 4.5D.

The measurements confirm complete self-preservation

• Re_λ reaches a plateau once Re_D ≥ 2 × 10⁴. At Re_D = 10⁴, it does not settle: self-preservation is not reached.
• The plateau starts further downstream as the Reynolds number increases: around x/D ≈ 15 at Re_D = 3.4 × 10⁴, around x/D ≈ 40 at 1.3 × 10⁵. Measurements must therefore be taken far enough from the nozzle.

• The dissipation follows the x⁻⁴ law in the far field.
• The prefactor A_ε is calculated from the decay of U and the turbulence intensity: about 36, within 8% of the measurement, against 48 for the often-quoted value of Friehe et al. (1971).
• The second- and third-order structure functions measured at different distances collapse once normalised: self-preservation holds at every scale.

Key takeaways

• On the axis of a round jet, self-preservation is enough to recover the decay laws of U, k and ε, with no other equation.
• The measurements confirm it, provided they are taken far enough from the nozzle, and further still as the Reynolds number increases.
• The prefactor formula gives a simple way to estimate ε on the axis, from measurements of mean velocity and turbulence intensity.

Further reading

L. Djenidi, R. A. Antonia, N. Lefeuvre, J. Lemay, Complete self-preservation on the axis of a turbulent round jet, Journal of Fluid Mechanics 790, 57–70 (2016), PDF · DOI