Spectral energy transfer in high Reynolds number turbulence

Abstract

The spectral energy transfer of turbulent velocity fields has been examined over a wide range of Reynolds numbers by experimental and empirical methods. Measurements in a high Reynolds number grid flow were used to calculate the energy transfer by the direct Fourier-transform method of Yeh & Van Atta. Measurements in a free jet were used to calculate energy transfer for a still higher Reynolds number. An empirical energy spectrum was used in conjunction with a local self-preservation approximation to estimate the energy transfer at Reynolds numbers beyond presently achievable experimental conditions.Second-order spectra of the grid measurements are in excellent agreement with local isotropy down to low wavenumbers. For the first time, one-dimensional third-order spectra were used to test for local isotropy, and modest agreement with the theoretical conditions was observed over the range of wavenumbers which appear isotropic according to second-order criteria. Three-dimensional forms of the measured spectra were calculated, and the directly measured energy transfer was compared with the indirectly measured transfer using a local self-preservation model for energy decay. The good agreement between the direct and indirect measurements of energy transfer provides additional support for both the assumption of local isotropy and the assumption of self-preservation in high Reynolds number grid turbulence.An empirical spectrum was constructed from analytical spectral forms of von Kármán and Pao and used to extrapolate energy transfer measurements at lower Reynolds number to R_λ = 10⁵ with the assumption of local self preservation. The transfer spectrum at this Reynolds number has no wavenumber region of zero net spectral transfer despite three decades of $k^{-\frac{5}{3}}$. behaviour in the empirical energy spectrum. A criterion for the inertial subrange suggested by Lumley applied to the empirical transfer spectrum is in good agreement with the $k^{-\frac{5}{3}}$ range of the empirical energy spectrum.