ALBERT

Description: 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, UL0, is not to be identified with the U0appearing 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 (6) 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 a. 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. © 1990, Cambridge University Press. All rights reserved.

Description: A turbulence model to compute bulk properties is presented in which a prescribed source function is used to provide the rate of energy input into the turbulent flow, and the EDQNM model is used to treat the nonlinear transfer in the Navier-Stokes equations. The predictions of the model are tested against (1) the measured Nusselt number versus Rayleigh number relation in turbulent laboratory convection, and (2) the measured bulk kinetic energies and dissipation rates in turbulent channel flow for Reynolds numbers Re = 12,300 and 30,800. In addition, a sensitivity study is performed with respect to the choice of Kolmogorov constant. Generally, the predictions of the model are in reasonable accord with the available experimental data.

Description: A method for calculating mean square velocity fluctuations, mean square temperature fluctuations, and convective flux for a turbulent rotating fluid with externally applied magnetic field is presented. A new spectral model of large scale turbulence is used which requires, as the sole ingredient, the growth rate of the instability generating the turbulence. Results are presented for the convective flux with rotation and magnetic field for a range of parameters of astrophysical interest. This new formula presented here can be viewed as an extension of the mixing length theory to include magnetic fields and rotation.

Description: 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 non-linear equation; the two models employ the same source function but different closures. The source function is characterized by a rate n sub s (k) which is derived from the complex eigenvalues of the Orr-Sommerfeld (OS) equation in which the basic flow is taken to be of a Poiseuille type. The O-S 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, U0 sup L, is not to be identified with the U0 appearing in the experimental Reynolds number. The theoretical results are compared with two types of experimental data: (1) turbulence bulk properties, and (2) properties that depend strongly on the structure of the turbulence spectrum at low wave numbers. The only existing analytical expression for Pi (k) cannot be used in the present case because it applies to the case of a flat plate, not a finite channel.

Description: In view of the present discussion, the Heisenberg-Kolmogoroff (HK) model of turbulence that is often used for turbulent phenomena on all scales is actually valid for a turbulent spectrum wavelength band typically much smaller than the size of the system, and cannot describe phenomena at large scales in astrophysical systems. The results of mixing length theory cannot be accommodated within the HK model's framework without the adoption of an unreasonable coupling constant. It is also noted that the use of the observed velocity-size relationship in molecular clouds with turbulent velocity values of about l to the 1/2-power, within the HK model, gives rise to a growth rate that does not correspond to any known physical processes suspected of operating in molecular clouds.

Description: Theoretical modeling of the wealth of experimental data on propagation of electromagnetic radiation through turbulent media has centered on the use of the Heisenberg-Kolmogorov (HK) model, which is, however, valid only for medium to small sized eddies. Ad hoc modifications of the HK model to encompass the large-scale region of the eddy spectrum have been widely used, but a sound physical basis has been lacking. A model for large-scale turbulence that was recently proposed is applied to the above problem. The spectral density of the temperature field is derived and used to calculate the structure function of the index of refraction N. The result is compared with available data, yielding a reasonably good fit. The variance of N is also in accord with the data. The model is also applied to propagation effects. The phase structure function, covariance of the log amplitude, and variance of the log intensity are calculated. The calculated phase structure function is in excellent agreement with available data.

Description: 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 non-linear equation; the two models employ the same source function but different closures. The source function is characterized by a rate n sub s (k) which is derived from the complex eigenvalues of the Orr--Sommerfeld (OS) equation in which the basic flow is taken to be of a Poiseuille type. The O--S 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, U0 sup L, is not to be identified with the U0 appearing in the experimental Reynolds number. The theoretical results are compared with two types of experimental data: (1) turbulence bulk properties, and (2) properties that depend stongly on the structure of the turbulence spectrun at low wave numbers. The only existing analytical expression for Pi (k) cannot be used in the present case because it applies to the case of a flat plate, not a finite channel.