A generalization of the conditional-eddy concept is proposed in which the conditional event specifies the local kinematic state in terms of the velocity and the deformation. Results are presented for stochastically estimated conditional eddies given the local kinematics. The equation governing the probability density function of a kinematic state has been derived for constant-property incompressible flow, providing a link between coherent flow structures corresponding to the conditional eddies and the modelling of turbulent transport. The primary contributions to the second-quadrant and fourth-quadrant Reynolds-stress events in homogeneous shear flow are shown to come from flow induced through the 'legs' and close to the 'heads' of upright and inverted 'hairpins', respectively.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112088001442

Stochastic estimation of organized turbulent structure - Homogeneous shear flow

Known structures and self-sustaining mechanisms of wall turbulence are reviewed and explored in the context of the scale interactions implied by the nonlinear advective term in the Navier–Stokes equations. The viewpoint is shaped by the systems approach provided by the resolvent framework for wall turbulence proposed by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), in which the nonlinearity is interpreted as providing the forcing to the linear Navier–Stokes operator (the resolvent). Elements of the structure of wall turbulence that can be uncovered as the treatment of the nonlinearity ranges from data-informed approximation to analysis of exact solutions of the Navier–Stokes equations (so-called exact coherent states) are discussed. The article concludes with an outline of the feasibility of extending this kind of approach to high-Reynolds-number wall turbulence in canonical flows and beyond.

The engine behind (wall) turbulence: perspectives on scale interactions

Concurrent, surface-pressure and planar, particle image velocimetry (PIV) measurements were obtained in the separating/reattaching flow region downstream of an axisymmetric, backward-facing step at a Reynolds number of 8081, based on step height. The surface-pressure and PIV measurements were used to investigate the evolution of coherent structures in the flow field by employing proper orthogonal decomposition (POD) and multipoint, linear, stochastic estimation (mLSE) analysis techniques. POD was used to determine the dominant modes in the pressure signature, while mLSE was used to estimate the dominant flow structures above the wall from the wall-pressure POD modes over a series of time steps. It was found that a large-scale, coherent structure develops in place (i.e., temporally) at approximately half the reattachment distance. Once this structure reaches a height equivalent to the step, it sheds and accelerates downstream. This growth in place, and then shedding, resembles the evolution of the flow str...

Stochastic estimation of a separated-flow field using wall-pressure-array measurements

The study of turbulence near walls has experienced a renaissance in the last decade, largely owing to the availability of high-quality numerical simulations. The viscous and buffer layers over smooth walls are essentially independent of the outer flow, and there is a family of numerically exact nonlinear structures that account for about half of the energy production and dissipation. The rest can be modelled by their unsteady bursting. Many characteristics of the wall layer, such as the dimensions of the dominant structures, are well predicted by those models, which were essentially completed in the 1990s after the increase in computer power made the kinematic simulations of the late 1980s cheap enough to undertake dynamic experiments. Today, we are at the early stages of simulating the logarithmic (or overlap) layer, and a number of details regarding its global properties are becoming clear. For instance, a finite Reynolds number correction to the logarithmic law has been validated in turbulent channels. This has allowed upper and lower limits of the overlap region to be clarified, with both upper and lower bounds occurring at much larger distances from the wall than commonly assumed. A kinematic picture of the various cascades present in this part of the flow is also beginning to emerge. Dynamical understanding can be expected in the next decade.

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What are we learning from simulating wall turbulence

The past two decades (approximately 1990 to 2010) have witnessed an ever-quickening pace of new findings pertaining to the Reynolds number dependencies, scaling, and dynamics of turbulent boundary layer flows (and wall-bounded turbulent flows in general). Given this, an important objective of the present effort is to provide a review that enables researchers new to the field (e.g., graduate students) to gain an appreciation for, and an understanding of, the prevalent research themes currently under investigation. Thus, the emphasis is more on laying a contextual foundation rather than, for example, comprehensively reporting all of the research findings of the past 20 years. The review begins with a brief exposition of scaling concepts and the normalizing parameters used in exploring Reynolds number dependence. An overall focus of the effort is to describe the scaling problem in relation to the underlying behaviors of the governing transport equations. For this reason, a number of relevant equations are concisely presented. The technical challenges associated with reliably exploring Reynolds number dependence are nontrivial and are of central importance. Thus, a separate section is devoted to this topic. Similarly, since they factor importantly relative to understanding and organizing the data trends, the attributes, strengths, and weaknesses of the various theoretical approaches and models (both physical and mathematical) are briefly reviewed. The statistical data presented primarily focus on means and variances since these quantities most directly relate to the time-averaged equations. Recent results pertaining to the spatial structure of turbulent boundary layers provide a useful context for describing instantaneous dynamics, often involving coherent vortical motions and including the so-called inner/outer interaction. Overall, the cumulative evidence increasingly supports a paradigm in which the scaling behaviors of the statistical profiles stem from the existence of an internal hierarchy of motions that approach a dynamically self-similar state as the Reynolds number becomes large.

Reynolds Number Dependence, Scaling, and Dynamics of Turbulent Boundary Layers

It is shown that the ’’forcing function’’ (the right‐hand side) of Poisson’s equation for the mean or fluctuating pressure in a turbulent flow can be divided into two parts, one related to the square of the rate of strain and the other to the square of the vorticity.

Y. M. Koh

Papers

A note on Poisson’s equation for pressure in a turbulent flow