On the skewness of the temperature derivative in turbulent flows

This note provides some explanation of the fact that, contrary to the requirements of local isotropy, the skewness S of the streamwise temperature derivative ∂θ/∂ x 1 has been observed to be a non-zero constant of magnitude of about unity in high-Reynolds-number and high-Peclet-number turbulent shear flows. Measurements in slightly heated homogeneous shear flows and in unsheared grid turbulence suggest that S is non-zero only when the mean shear dU 1 / dx 2 and the mean temperature gradient dT / dx 2 are both non-zero. The sign of S is given by –sgn ( dU 1 / dx 2 ).sgn ( dT / dx 2 ). For fixed dU 1 / dx 2 , S is of the form tanh (α dT / dx 2 ), α being a constant, while for fixed dT / dx 2 , it is of the form S / S * = 1 − β 1 exp (− β 2 τ), where S * is a characteristic value of S , β 1 and β 2 are positive constants, and τ can be interpreted as a ‘total strain’. The derivative skewness data in other (inhomogeneous) shear flows are also compatible with the latter relation. Predictions from a simplified transport equation for $\overline{(\partial\theta/\partial x_1)^3}$ , derived in the light of the present experimental observations, are in reasonable agreement with the measured values of S. A possible physical mechanism maintaining S is discussed.