Theoretical study of turbulent channel flow: bulk properties, pressure fluctuations, and propagation of electromagnetic waves

Abstract

In this paper, we apply two theoretical turbulence models, DIA and the recent GISS model, to study properties of a turbulent channel flow. Both models provide a turbulent kinetic energy spectral function E(k) as the solution of a nonlinear equation; the two models employ the same source function but different closures. The source function is characterized by a rate ns(k) which is derived from the complex eigenvalues of the Orr–Sommerfeld equation in which the basic flow is taken to be of a Poiseuille type. The Orr–Sommerfeld equation is solved for a variety of Reynolds numbers corresponding to available experimental data. A physical argument is presented whereby the central line velocity characterizing the basic flow, U0L, is not to be identified with the U0 appearing in the experimental Reynolds number. A renormalization is suggested which has the effect of yielding growth rates of magnitude comparable with those calculated by Orszag & Patera based on their study of a secondary instability. From the practical point of view, this renormalization frees us from having to solve the rather time-consuming equations describing the secondary instability. This point is discussed further in §13. In the present treatment, the shear plays only the role of a source of energy to feed the turbulence and not the possible additional role of an interaction between the shear of the mean flow and the eddy vorticity that would give rise to resonance effects when the shear is equal to or larger than the eddy vorticities. The inclusion of this possible resonance phenomenon, which is not expected to affect the large-eddy behaviour and thus the bulk properties, is left for a future study. The theoretical results are compared with two types of experimental data: (a) turbulence bulk properties, table 4, and (b) properties that depend strongly on the structure of the turbulence spectrum at low wavenumbers (i.e. large eddies), tables 5 and 6. The latter data are taken from recent experiments measuring the changes in the propagation of an electromagnetic wave through a turbulent channel flow. The fluctuations in the refractive index of the turbulent medium are thought to be due to pressure fluctuations whose spectral function Π(k) is contributed mostly by the interaction between the mean flow and the turbulent velocity. The spectrum Π(k) must be computed as a function of the wavenumber k, the position in the channel x2, and the width of the channel Δ. The only existing analytical expression for Π(k), due to Kraichnan, cannot be used in the present case because it applies to the case x2 = 0 and Δ = ∞, which corresponds to the case of a flat plate, not a finite channel. A general expression for Π(k, x2; Δ) is derived here for the first time and employed to calculate the fraction of incoherent radiation scattered out of a coherent beam. In §11, we treat anisotropy and show how to extend the previous results to include an arbitrary degree of anisotropy α in the sizes of the eddies. We show that the theoretical one-dimensional spectra yield a better fit to the data for a degree of anisotropy (α ≈ 4) that is within the range of experimental values. We also extend the expression for Π(k, x2; Δ) to Π(k, x2; Δ, α) and compute the pressure fluctuations for different values of α. Similarly, we evaluate the fraction of electromagnetic energy scattered by an anisotropic turbulent flow and find a good fit to the laboratory data for a value of α ≈ 4–6. Scaling formulae for the scattered fraction are presented in §12. These formulae reproduce the calculated results, both with and without the addition of anisotropy, to better than 5%.Theoretical problems however remain which will require further study: among them, lack of backscatter (i.e. the transfer of energy from large to small wavenumbers) in the GISS model, possible resonance effects between the shear and eddy vorticity, behaviour of the one-dimensional spectral function at low wavenumbers, and the role of the secondary instability. These topics are now under investigation.