Random Vortex-Street Model for a Self-Similar Plane Turbulent Jet

We ask what determines the (small) angle of turbulent jets. To answer this question we first construct a deterministic vortex-street model representing the large-scale structure in a self-similar plane turbulent jet. Without adjustable parameters the model reproduces the mean velocity profiles and the transverse positions of the large-scale structures, including their mean sweeping velocities, in a quantitative agreement with experiments. Nevertheless, the exact self-similar arrangement of the vortices (or any other deterministic model) necessarily leads to a collapse of the jet angle. The observed (small) angle results from a competition between vortex sweeping tending to strongly collapse the jet and randomness in the vortex structure, with the latter resulting in a weak spreading of the jet.

Figure 1

Profiles of the normalized streamwise mean velocity: Black squares—experimental points [7]; black dashed line—profile (9) obtained from the Prandtl closure with constant turbulent viscosity approximation; red dash-dotted line—Eq. (7) ($\alpha \to 0$ limit) with $a=0.747$, $c\approx 1.1$. Black solid line—profile (8) with $\alpha =0.1$ and the same values of $a$ and $c$.

Figure 2

Velocity field lines showing the vortex structure (a fragment from experimental proper orthogonal decomposition, Ref. 7).

Figure 3

Experimental profile [7] of the mean square of the cross-stream component of the fluctuating velocity (dots connected by line). In a continuous line we present the profile predicted by the random model. The dashed line is the profile of the regular contribution. The difference at $a$ is the source of spreading of the jet, cf. Eq. (13).