Applications of second-moment turbulent closure hypotheses to geophysical fluid problems have developed rapidly since 1973, when genuine predictive skill in coping with the effects of stratification was demonstrated. The purpose here is to synthesize and organize material that has appeared in a number of articles and add new useful material so that a complete (and improved) description of a turbulence model from conception to application is condensed in a single article. It is hoped that this will be a useful reference to users of the model for application to either atmospheric or oceanic boundary layers.

http://fap.if.usp.br/~hbarbosa/uploads/Teaching/Modclim2010a/MellorYamada1982.pdf

Development of a turbulence closure model for geophysical fluid problems

The paper develops proposals for a model of turbulence in which the Reynolds stresses are determined from the solution of transport equations for these variables and for the turbulence energy dissipation rate E. Particular attention is given to the approximation of the pressure-strain correlations; the forms adopted appear to give reasonably satisfactory partitioning of the stresses both near walls and in free shear flows. Numerical solutions of the model equations are presented for a selection of strained homogeneous shear flows and for two-dimensional inhomogeneous shear flows including the jet, the wake, the mixing layer and plane channel flow. In addition, it is shown that the closure does predict a very strong influence of secondary strain terms for flow over curved surfaces.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112075001814

Progress in the development of a Reynolds-stress turbulence closure

The three-dimensional, primitive equations of motion have been integrated numerically in time for the case of turbulent, plane Poiseuille flow at very large Reynolds numbers. A total of 6720 uniform grid intervals were used, with sub-grid scale effects simulated with eddy coefficients proportional to the local velocity deformation. The agreement of calculated statistics against those measured by Laufer ranges from good to marginal. The eddy shapes are examined, and only the u-component, longitudinal eddies are found to be elongated in the downstream direction. However, the lateral v eddies have distinct downstream tilts. The turbulence energy balance is examined, including the separate effects of vertical diffusion of pressure and local kinetic energy.It is concluded that the numerical approach to the problem of turbulence at large Reynolds numbers is already profitable, with increased accuracy to be expected with modest increase of numerical resolution.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112070000691

A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers

A review is presented of the models of magnetic field line merging defined as the process whereby plasma flows across a surface which separates regions including topologically different magnetic field lines. The models examined are characterized by uniform and antiparallel external magnetic fields. An attempt is made to simplify the presentation of the models, to clarify some doubtful mathematical points, or to extend the results to a different range of physical parameters. The models are described from a hydromagnetic point of view, with the configuration in any given case being determined by the boundary conditions. It is shown that the models developed by Sweet (1958), Parker (1957, 1963), Petschek (1964), Sonnerup (1970), and by Yeh and Axford (1970) are basically consistent, describing different aspects of the same problem; however, there is not a single model that would account for all the cases considered. The singular models and the compressible similarity models are physically not feasible.

Theoretical models of magnetic field line merging

Publisher Summary This chapter presents an overview of the theory of homogeneous turbulence. Homogeneous isotropic turbulence is an idealized concept of turbulence, assumed to be governed by a statistical law that is invariant under arbitrary translation (homogeneity), rotation, or reflection (isotropy) of the coordinate system. This is an idealization of real turbulent motions, which are observed in nature or produced in a laboratory, and generally have much more complicated structures. This idealization was first introduced by Taylor to the theory of turbulence and used to reduce the formidable complexity of the statistical expression of turbulence, and thus make the subject feasible for theoretical treatment. Turbulence combines the difficulty of a strongly nonlinear dynamical system with that of a nonconservative system. The presence of energy dissipation due to viscosity deprives turbulence of the opportunity of being in a thermal equilibrium state. In this chapter, concepts of quasi-normality of large-scale motions are explained. A brief discussion on zero cumulant approximation and modified zero cumulant approximation is presented. Turbulence of other dimensions is also explained.

Theory of Homogeneous Turbulence

The effect of wind-tunnel contraction on free-stream turbulence is determined by passing a well-defined turbulence through three contractions of ratios 4 : 1 , 9 :1 , and 16:1. Turbulent velocity measurements show that, in absolute magnitudes, the longitudinal component decreases and the lateral component increases as the flow accelerates through the contraction. Postand precontraction spectra are measured to get an idea about the distortion of turbulence structure.

Effect of Wind-Tunnel Contraction on Free-Stream Turbulence

Axial flow in the core of a laminar steady trailing vortex from the tip of a semi‐infinite wing is analyzed assuming small departure of the axial velocity from the free‐stream velocity. It is further assumed that the axial pressure gradient is determined by the swirl velocities of an ideal infinite line vortex in which the radial and the associated axial velocity variations are neglected in the equation for the angular momentum. The axial and lateral variations of the axial velocity depend on the strength of the vortex and initial axial velocity distribution which must be specified at some station behind the wing except at the virtual origin of the vortex where a nonintegrable singularity exists. Numerical solutions for the axial velocity are obtained using the axial pressure gradient given by the line vortex and analytical solutions are obtained using an equivalent axial pressure gradient with good agreement between the two sets of axial velocity distributions. Resolution of the previous uncertainties in this field is given which were due to the unrecognized singularity at the virtual origin of the vortex. Using the calculated axial velocity the neglected radial and the associated axial fluxes of angular momentum are determined and the limits of validity of the theory presented here in terms of a suitably defined vortex Reynolds number and a nondimensional distance measured from the virtual origin of the vortex are given.

Axial flow in trailing line vortices

Axisymmetric turbulent wake behind a sphere in an incompressible fluid has been experimentally investigated from 50 to 300 diam downstream from the sphere at Reynolds numbers from 4000 to 150 000. Mean and turbulent velocity measurements show that the region of self‐preservation starts 50 sphere diam downstream, and the virtual origin of the wake is 12 sphere diam downstream. Detailed measurements were made in the self‐preserving region of the wake. The three components of the turbulent velocity, turbulent shear, and the mean velocity defect were measured across the wake using the hot‐wire anemometer. From the measurements the turbulent energy production, dissipation, convection, and diffusion across the wake were determined. The one‐dimensional energy spectra of the three components of the turbulent velocity, their dependence on distance along and across the wake and on Reynolds number have been measured. The large scale motion (low wavenumbers) is dynamically similar and the small scale motion (large wavenumbers) exhibits Kolmogoroff's universal equilibrium. The present spectral measurements are compared with other published measurements of the universal equilibrium spectra in various turbulent flows.

Turbulent energy balance and spectra of the axisymmetric wake

Energy transfer from large to small eddies at three stations in turbulence behind a square mesh is determined by measuring the rates of change and viscous dissipation of the spectrum and the results are compared with a theoretical prediction. Large eddies for which viscous dissipation is negligible satisfy a similarity relation which agrees with the fact that the total energy decays as some negative power of time. Small eddies which are in approximate statistical equilibrium satisfy local similarity according to Kolmogoroff. Various terms in the vorticity equation are also determined and the quantities representative of small scale motion are universal constants when expressed in terms of Kolmogoroff parameters.

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Energy Transfer in Isotropic Turbulence

The tip vortex of a laminar flow wing was studied at a sectional lift‐to‐drag ratio of 60. The vortex Reynolds number was Γ0/ν=7.8×104, where Γ0 is the total circulation and ν is the kinematic viscosity. At and near the wing the vortex core was turbulent with an axial jet. Downstream of the wing the jet rapidly dissipated and a wake developed in the core and intensity of turbulent velocities decreased. From 13 to 40 chord length periodic oscillations dominated the velocity fluctuations with little background turbulence. These instabilities had a symmetric and a helical mode with wavelength of the same order as the core diameter. In this range of distances along the vortex core the maximum axial, swirl, and fluctuating velocities vary slowly. At 40 chord lengths behind the wing there is a rapid change in these velocities. This change of state of the vortex core is accompanied by change of velocity fluctuations from periodic to turbulent. The core showed spatial excursions. Measurements up to 80 chord lengths downstream showed no self‐similar decay.

Mahinder S. Uberoi

Papers

Effect of Wind-Tunnel Contraction on Free-Stream Turbulence

Axial flow in trailing line vortices

Turbulent energy balance and spectra of the axisymmetric wake

Energy Transfer in Isotropic Turbulence

Experiments on vortex stability