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

Considerable confusion surrounds the longstanding question of what constitutes a vortex, especially in a turbulent flow. This question, frequently misunderstood as academic, has recently acquired particular significance since coherent structures (CS) in turbulent flows are now commonly regarded as vortices. An objective definition of a vortex should permit the use of vortex dynamics concepts to educe CS, to explain formation and evolutionary dynamics of CS, to explore the role of CS in turbulence phenomena, and to develop viable turbulence models and control strategies for turbulence phenomena. We propose a definition of a vortex in an incompressible flow in terms of the eigenvalues of the symmetric tensor ${\bm {\cal S}}^2 + {\bm \Omega}^2$ are respectively the symmetric and antisymmetric parts of the velocity gradient tensor ${\bm \Delta}{\bm u}$. This definition captures the pressure minimum in a plane perpendicular to the vortex axis at high Reynolds numbers, and also accurately defines vortex cores at low Reynolds numbers, unlike a pressure-minimum criterion. We compare our definition with prior schemes/definitions using exact and numerical solutions of the Euler and Navier–Stokes equations for a variety of laminar and turbulent flows. In contrast to definitions based on the positive second invariant of ${\bm \Delta}{\bm u}$ or the complex eigenvalues of ${\bm \Delta}{\bm u}$, our definition accurately identifies the vortex core in flows where the vortex geometry is intuitively clear.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112095000462

On the identification of a vortex

The two main principles underlying the use of isentropic maps of potential vorticity to represent dynamical processes in the atmosphere are reviewed, including the extension of those principles to take the lower boundary condition into account. the first is the familiar Lagrangian conservation principle, for potential vorticity (PV) and potential temperature, which holds approximately when advective processes dominate frictional and diabatic ones. the second is the principle of ‘invertibility’ of the PV distribution, which holds whether or not diabatic and frictional processes are important. the invertibility principle states that if the total mass under each isentropic surface is specified, then a knowledge of the global distribution of PV on each isentropic surface and of potential temperature at the lower boundary (which within certain limitations can be considered to be part of the PV distribution) is sufficient to deduce, diagnostically, all the other dynamical fields, such as winds, temperatures, geopotential heights, static stabilities, and vertical velocities, under a suitable balance condition. the statement that vertical velocities can be deduced is related to the well-known omega equation principle, and depends on having sufficient information about diabatic and frictional processes. Quasi-geostrophic, semigeostrophic, and ‘nonlinear normal mode initialization’ realizations of the balance condition are discussed. an important constraint on the mass-weighted integral of PV over a material volume and on its possible diabatic and frictional change is noted.

Some basic examples are given, both from operational weather analyses and from idealized theoretical models, to illustrate the insights that can be gained from this approach and to indicate its relation to classical synoptic and air-mass concepts. Included are discussions of (a) the structure, origin and persistence of cutoff cyclones and blocking anticyclones, (b) the physical mechanisms of Rossby wave propagation, baroclinic instability, and barotropic instability, and (c) the spatially and temporally nonuniform way in which such waves and instabilities may become strongly nonlinear, as in an occluding cyclone or in the formation of an upper air shear line. Connections with principles derived from synoptic experience are indicated, such as the ‘PVA rule’ concerning positive vorticity advection on upper air charts, and the role of disturbances of upper air origin, in combination with low-level warm advection, in triggering latent heat release to produce explosive cyclonic development. In all cases it is found that time sequences of isentropic potential vorticity and surface potential temperature charts—which succinctly summarize the combined effects of vorticity advection, thermal advection, and vertical motion without requiring explicit knowledge of the vertical motion field—lead to a very clear and complete picture of the dynamics. This picture is remarkably simple in many cases of real meteorological interest. It involves, in principle, no sacrifices in quantitative accuracy beyond what is inherent in the concept of balance, as used for instance in the initialization of numerical weather forecasts.

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On the use and significance of isentropic potential vorticity maps

The role of coherent structures in the production and dissipation of turbulence in a boundary layer is characterized, summarizing the results of recent investigations. Coherent motion is defined as a three-dimensional region of flow where at least one fundamental variable exhibits significant correlation with itself or with another variable over a space or time range significantly larger than the smallest local scales of the flow. Sections are then devoted to flow-visualization experiments, statistical analyses, numerical simulation techniques, the history of coherent-structure studies, vortices and vortical structures, conceptual models, and predictive models. Diagrams and graphs are provided.

Coherent Motions in the Turbulent Boundary Layer

This paper describes a theoretical investigation into (i) the response of a spherical particle to a one-dimensional fluid flow, (ii) the motion of a spherical particle in a uniform two-dimensional fluid flow about a circular cylinder and (iii) the motion of a particle about a lifting aerofoil section. In all three cases the drag of the particle is allowed to vary with (instantaneous) Reynolds number by using an analytical approximation to the standard experimental drag-Reynolds-number relationship for spherical particles.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/5B1FF01A248EBF4988C156246EFF844A/S0022112072001806a.pdf/div-class-title-an-investigation-of-particle-trajectories-in-two-phase-flow-systems-div.pdf

An investigation of particle trajectories in two-phase flow systems

The geometry of solution trajectories for three first‐order coupled linear differential equations can be related and classified using three matrix invariants. This provides a generalized approach to the classification of elementary three‐dimensional flow patterns defined by instantaneous streamlines for flow at and away from no‐slip boundaries for both compressible and incompressible flow. Although the attention of this paper is on the velocity field and its associated deformation tensor, the results are valid for any smooth three‐dimensional vector field. For example, there may be situations where it is appropriate to work in terms of the vorticity field or pressure gradient field. In any case, it is expected that the results presented here will be of use in the interpretation of complex flow field data.

A general classification of three-dimensional flow fields

The structure of round jets in cross-flow was studied using flow visualization techniques and flying-hot-wire measurements. The study was restricted to jet to freestream velocity ratios ranging from 2.0 to 6.0 and Reynolds numbers based on the jet diameter and free-stream velocity in the range of 440 to 6200.Flow visualization studies, together with time-averaged flying-hot-wire measurements in both vertical and horizontal sectional planes, have allowed the mean topological features of the jet in cross-flow to be identified using critical point theory. These features include the horseshoe (or necklace) vortex system originating just upstream of the jet, a separation region inside the pipe upstream of the pipe exit, the roll-up of the jet shear layer which initiates the counter-rotating vortex pair and the separation of the flat-wall boundary layer leading to the formation of the wake vortex system beneath the downstream side of the jet.The topology of the vortex ring roll-up of the jet shear layer was studied in detail using phase-averaged flying-hot-wire measurements of the velocity field when the roll-up was forced. From these data it is possible to examine the evolution of the shear layer topology. These results are supported by the flow visualization studies which also aid in their interpretation.The study also shows that, for velocity ratios ranging from 4.0 to 6.0, the unsteady upright vortices in the wake may form by different mechanisms, depending on the Reynolds number. It is found that at high Reynolds numbers, the upright vortex orientation in the wake may change intermittently from one configuration of vortex street to another. Three mechanisms are proposed to explain these observations.

An experimental study of round jets in cross-flow

©Cambridge University Press. Perry, A.E. & Marusic, Ivan. (1995) A wall-wake model for the turbulence structure of boundary layers. Part 1. Extension of the attached eddy hypothesis. Journal of fluid mechanics, 298, 361-388. http://journals.cambridge.org/action/displayJournal?

/pdf/a-wall-wake-model-for-the-turbulence-structure-of-boundary-1s3fyc1595.pdf

A wall-wake model for the turbulence structure of boundary layers. Part 1. Extension of the attached eddy hypothesis

Turbulent boundary layer experiments have been conducted at various Reynolds numbers on smooth walls and also on ‘k-type’ and ‘d-type’ rough walls. Both the spectral results and the broadband turbulence intensity results strongly support the Townsend (1976) attached eddy hypothesis and the Perry & Chong (1982) model. The spectral results obtained using the ‘flying’ hot-wire technique show the errors involved when using Taylor's (1938) hypothesis for converting the spectra from the frequency domain to the wavenumber domain. If the viscous dissipation spectral region is taken into account, the broadband turbulence intensity results agree well with the attached eddy hypothesis. The inconsistency of the various constants given in Perry, Lim & Henbest (1987) for the smooth and rough walls has been explained and removed. Lack of spatial resolution of the hot wires explains to some extent the scatter in the turbulence intensity of the component normal to the wall. This spatial resolution effect is most pronounced in the near-wall region at high Reynolds number and has been corrected by using the method of Wyngaard (1968).

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112090001057

Experimental support for the attached-eddy hypothesis in zero-pressure-gradient turbulent boundary layers

©Cambridge University Press. Marusic, Ivan and Perry A.E. (1995) A wall-wake model for the turbulence structure of boundary layers. Part 2. Further experimental support. Journal of fluid mechanics, 298, 389-407. http://www.jfm.damtp.cam.ac.uk/

/pdf/a-wall-wake-model-for-the-turbulence-structure-of-boundary-449ct28k16.pdf

A. E. Perry

Papers

A general classification of three-dimensional flow fields

An experimental study of round jets in cross-flow

A wall-wake model for the turbulence structure of boundary layers. Part 1. Extension of the attached eddy hypothesis

Experimental support for the attached-eddy hypothesis in zero-pressure-gradient turbulent boundary layers

A wall-wake model for the turbulence structure of boundary layers. Part 2. Further experimental support