Small-scale statistics of the passive scalar field convected by turbulent flow of an incompressible fluid under a uniform mean scalar gradient are studied theoretically and numerically. The flow is assumed to be statistically homogeneous and isotropic. Attention is given to the effects of the mean scalar gradient on the anisotropy of the statistics, in particular, the anisotropy of the second- and third-order mixed velocity-scalar structure functions, $\langle \delta u_j\delta \psi \rangle$ and $\langle \delta u_j{\left(\delta \psi \right)}^2\rangle$, where $u_j(\mathit{x},t)(j=1,2,3)$ is the $j\mathrm{th}$ component of the velocity field, $\psi (\mathit{x},t)$ is the fluctuating part of the scalar field, $\delta \xi =\xi (\mathit{x}+\mathit{r},t)-\xi (\mathit{x},t), \xi =\psi ,u_j, \langle ·\rangle$ is the ensemble average, $\mathit{x}$ is the position, and $t$ is time. A theory of the effects is proposed by extending the idea of the linear response theory of turbulence. Its predictions are examined by direct numerical simulation (DNS) in a periodic box. In the DNS, the Schmidt number is unity, and the Taylor microscale Reynolds number is about 260 at a fully developed state of the turbulent velocity field. It is found that the predictions of the anisotropy are generally consistent with the DNS results.

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• Homogeneous turbulence
• Turbulence
• Turbulent mixing