An equilibrium similarity analysis is applied to the transport equation for <(deltatheta)(2)>, the second-order temperature structure function, for decaying homogeneous isotropic turbulence. A possible solution is that the temperature variance <theta(2)> decays as x(n), and that the characteristic length scale, identifiable with the Taylor microscale lambda, or equivalently the Corrsin microscale lambda(theta), varies as x(1/2). The turbulent Reynolds and Peclet numbers decay as x((m+1)/2) when m<-1, where m is the exponent which characterizes the decay of the turbulent energy <q(2)>, viz., <q(2)>similar tox(m). Measurements downstream of a grid-heated mandoline combination show that, like <(deltaq)(2)>, <(deltatheta)(2)> satisfies similarity approximately over a significant range of scales r, when lambda, lambda(theta), <q(2)>, and <theta(2)> are used as the normalizing scales. This approximate similarity is exploited to calculate the third-order structure functions. Satisfactory agreement is found between measured and calculated distributions of <deltau(deltaq)(2)> and <deltau(deltatheta)(2)>, where deltau is the longitudinal velocity increment.
Antonia, R. A., Smalley, R. J., Zhou, T., Anselmet, F., & Danaila, L. (2004). Similarity solution of temperature structure functions in decaying homogeneous isotropic turbulence. Physical Review E, 69(1), online - approx 5-20pp. https://doi.org/10.1103/PhysRevE.69.016305