The second moments, spectra and correlation functions of velocity and temperature fluctuations in the gradient sublayer of a retarded boundary layer

Abstract

The turbulent structure of velocity and temperature fields in moving equilibrium retarded boundary layers is analyzed. Most attention is given to ‘the gradient sublayer’, where, according to Ginevskii and Solodkin [ Prikl. Mat. Mech. (Appl. Math. Mech.) 22, 819–825 (1958)], Stratford [ J. Fluid Mech. 5, 1–16; 17–35 (1959)] and Perry et al. [ J. Fluid Mech. 25, 299–320 (1966)], the mean velocity and temperature profiles are described by the half-power and inverse half-power laws. Kader [ Dokl. Akad. Nauk U.S.S.R. 279, 323–327 (1984); Int. J. Heat Mass Trans. 34, 2837–2857 (1991)] deduced formulas for spectra and cospectra of velocity components and temperature in the gradient sublayer for the mesoscale range of wave numbers k by dimensional analysis and then compared them with available experimental data. It is shown that accurate determination of velocity variances and Reynolds stresses requires taking into account the contribution of large-scale turbulent disturbances corresponding to small values of k. It is not so for determination of the temperature variance and vertical heat flux evaluation. An analysis of low wave number parts of velocity and temperature spectra and cospectra is given, and its results are used to determine the correlation functions of turbulent fluctuations in the gradient sublayer. The formulas for one-point second-order moments (variances 〈 t 2〉, 〈 u 2) and 〈 v 2〉, temperature flux 〈 vt〉, and Reynolds stress 〈 −uv〉) in the gradient sublayer of quasi-equilibrium flows are also derived and compared with the available data. Comparison of calculated and experimental spectra of non-equilibrium retarded flows uncovers disagreement in the mesoscale wave number part of the spectra for vertical velocity and Reynolds stress fluctuations. At the same time longitudinal fluctuation spectra and one-point variance 〈 u 2) prove to be less sensitive to non-equilibrium conditions.