Effects of Inhomogeneity and of Shear Flow in Weak Turbulent Fields

Abstract

Spectra for weak turbulence are found for: (1) turbulence inhomogeneous in the direction of a uniform mean velocity; (2) an initially homogeneous turbulence becoming inhomogeneous in a transverse direction because of the presence of boundaries; and (3) turbulence homogeneous with a uniform mean transverse velocity gradient. The treatment is based on two‐point correlation and spectral equations. The effect of inhomogeneity is to cause a diffusion of turbulence in the direction of decreasing turbulent intensity. Longitudinal inhomogeneity produces an accumulation of energy mainly in the high wave‐number portions of the energy spectrum, whereas the transverse inhomogeneity causes a depletion of energy at a point in the fluid and does not alter the shape of the spectrum. For homogeneous turbulence with a uniform transverse velocity gradient, energy is transferred from the mean flow into the turbulence by a turbulent production term in the spectral equation. At high velocity gradients the production spectrum shifts toward the low wave‐number region and the dissipation spectrum toward the high wave‐number region. The pressure‐force term, which is dependent on the velocity gradient, transfers energy between the directional components in such a way as to oppose local isotropy in the high wave‐number region. Although triple correlations are neglected (low‐turbulence Reynolds number), a term containing the mean velocity gradient occurs in the spectral equation which is interpreted as transferring energy between wave numbers.