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/∂x₁ has been observed to be a non-zero constant of magnitude of about unity in high-Reynolds-number and high-Péclet-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 dU₁/dx₂ and the mean temperature gradient dT/dx₂ are both non-zero. The sign of S is given by –sgn (dU₁/dx₂).sgn (dT/dx₂). For fixed dU₁/dx₂, S is of the form tanh (αdT/dx₂), α being a constant, while for fixed dT/dx₂, it is of the form S/S* = 1 − β₁ exp (− β₂τ), where S* is a characteristic value of S, β₁ and β₂ 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.