Second-order spectral local isotropy in turbulent scalar fields

This work was motivated by recent experimental results on the spectra of fluctuating temperature gradients in a heated turbulent boundary layer obtained by Sreenivasan, Danh & Antonia. Standard techniques of turbulence theory are used herein to derive expressions relating the individual one-dimensional spectra of each of the three components of the spatial gradient ∂θ/∂xiin a locally isotropic turbulent scalar field. The results of the isotropic theory explain all of the new observed features of the temperature-gradient spectra. The spectra of ∂θ/∂yand ∂θ/∂zdecrease monotonically with increasing wavenumber, in contrast to the well-known behaviour of the spectrum of ∂θ/∂x, which reaches a maximum value at roughly one-tenth the Kolmogorov wavenumber. The spectra of ∂θ/∂yand ∂θ/∂zare relatively rich in low frequency energy and relatively poor in high frequency energy compared with the spectrum of ∂θ/∂x. The absolute magnitudes of the spectra of ∂θ/∂yand ∂θ/∂zcalculated from the spectrum of ∂θ/∂xusing the isotropic relations are in generally good agreement with the corresponding measured spectra for a large range of wavenumbers, indicating second-order spectral local isotropy of the fine-scale scalar structure for sufficiently large wavenumbers. The form of the spectra of ∂θ/∂yand ∂θ/∂zin the inertial subrange is derived analytically.