Scale-by-scale budget equation and its self-preservation in the shear-layer of a free round jet

Kolmogorov related the mean energy transport to both turbulent advection and molecular diffusion (K41), which, for moderate Reynolds numbers encountered under laboratory conditions, is not balanced. The main reason for this imbalance is the inhomogeneous and anisotropic large-scales, which are not sufficiently separated from the smallest dissipation scale. Various types of inhomogeneities have been examined in different turbulent flows such as grid turbulence, fully developed channel flow and along the jet centreline. The inhomogeneities associated with large-scales are examined in the shear-layer of a round jet since several large-scale phenomena coexist in this region. A new generalized form of Kolmogorov's equation is proposed for the off-centerline (shear-layer) of round jet flows, where the main sources of the inhomogeneity are the streamwise decay of turbulent energy, normal stress production, shear stress production, and axial and lateral diffusion. The validity of this equation is investigated using hot-wire data obtained for a round turbulent jet at ReD=50,000. The effect of radial distance is investigated on each term in this equation. It is found that the magnitude of the decay, normal stress production and axial diffusion terms deceases with increasing radial distance from the centreline for almost all scales while the shear stress production and lateral diffusion terms first increase in magnitude and then decreases with the radial distance. The similarity of the energy structure functions is also investigated in the shear portion of the jet using an equilibrium similarity analysis. On the basis of the current analysis, it is suggested that the turbulence kinetic energy, ⟨q2⟩, follows a power-law decay along the streamwsie direction. In addition, the second-order structure functions of u, v, q and uv are found to collapse approximately downstream of the self-similar jet when they are normalized by ⟨u2⟩, 〈v2〉, ⟨q2⟩ and 〈uv〉, respectively, and the Taylor microscale λ.