Fluid Dynamics and Diffusion Laboratory, College of Engineering, Colorado State University
Abstract
CER68-69JTL21.February 1969.Originally presented as the author's thesis, Colorado State University.Includes bibliographical references (pages 94-97).Prepared under Office of Naval Research, project no. NR 062-414/6-6-68(Code 438), U.S. Department of Defense.A generalized eddy-viscosity approximation is used to study the turbulence spectra of thermally stratified shear flows. For a stationary process in the wave number range investigated--the buoyancy subrange--under the assumption of local homogeneity of the flow, two governing spectral equations with six unknowns are derived from the equations of motion and energy. In order to reduce the number of unknowns to two so that the spectral equations can be solved, a generalized eddy-viscosity is used for expressing the integrated forms of the inertial transfers of energy and temperature inhomogeneity, the shear stress and vertical heat flux in terms of velocity spectrum ¢(k) and temperature spectrum ¢TT(k). Asymptotic solutions are obtained in the buoyancy subrange where the local production and local dissipation of turbulent energy is negligible as compared to the inertial transfer and vertical heat flux terms when the flow conditions satisfy the criterion ε|dT'/dz| << N g/T' or g/T'. |dT'/dz| << N/ε.(g/T)^2. In the buoyancy subrange of stably stratified turbulent flow, the power law for the velocity and temperature spectra is not universal but varies with the flow conditions in the way ¢(k) ~ k^n and ¢TT(k) ~ k^m where 11/5 ≥ n ≥ -3 and -1 ≥ m ≥ -7/5. According to the measurements of velocity spectra in the atmosphere (Pinus and Schcherbakova, 1966; Myrup, 1968), the dependence of the power law on the flow conditions was confirmed. The solutions of Bolgiano (1959) and Luialey-Shur (1964) are only two particular cases of the present results under cert ain flow conditions. In the case of the unstably stratified turbulent flow, the velocity spectrum exhibits a hump in the buoyancy subrange as a result of the energy input from the temperature field to the velocity field. In the left side of this hump the velocity spectrum approaches a +1 slope and the temperature spectrum shows a -3 slope. The measurements of the velocity spectra in the atmosphere (Ivanov and Ordanovich, 1967) confirms this tendency.Under contract no. N00014-68-A-0493-0001
