Self-similarity of passive scalar flow in grid turbulence with a mean cross-stream gradient

Abstract

Scaling of grid turbulence with a constant mean cross-stream temperature gradient is investigated using a combination of theoretical predictions and experimental data. A novel nanoscale temperature probe (T-NSTAP) was used to acquire temperature data. Conditions for self-similarity of the governing equations and the scalar spectrum are investigated, which reveals necessary conditions for the existence of a self-similar solution. These conditions provide a theoretical framework for scaling of the temperature spectrum as well as the temperature flux spectrum. One necessary condition, predicted by the theory, is that the characteristic length scale describing the scalar spectrum must vary as $\sqrt{t}$ in the case of a zero virtual origin for a self-similar solution to exist. As predicted by the similarity analysis, the data show the variance growing as a power law with streamwise position. When scaled with the similarity variable, as found through the theoretical analysis, the temperature spectra show a good collapse over all wavenumbers. A new method to determine the quality of the scaling was developed, comparing the coefficient of variation. The minimum coefficient of variation, and thus the best scaling, for the measured spectra agrees well with the similarity requirements. The theoretical work also reveals an additional requirement related to the scaling of the scalar flux spectrum.