The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

We survey the newly developed Hilbert spectral analysis method and its applications to Stokes waves, nonlinear wave evolution processes, the spectral form of the random wave field, and turbulence. Our emphasis is on the inadequacy of presently available methods in nonlinear and nonstationary data analysis. Hilbert spectral analysis is here proposed as an alternative. This new method provides not only a more precise definition of particular events in time-frequency space than wavelet analysis, but also more physically meaningful interpretations of the underlying dynamic processes.

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A new view of nonlinear water waves: the Hilbert spectrum

Small-scale turbulence has been an area of especially active research in the recent past, and several useful research directions have been pursued. Here, we selectively review this work. The emphasis is on scaling phenomenology and kinematics of small-scale structure. After providing a brief introduction to the classical notions of universality due to Kolmogorov and others, we survey the existing work on intermittency, refined similarity hypotheses, anomalous scaling exponents, derivative statistics, intermittency models, and the structure and kinematics of small-scale structure—the latter aspect coming largely from the direct numerical simulation of homogeneous turbulence in a periodic box.

/pdf/the-phenomenology-of-small-scale-turbulence-3a5uzrupts.pdf

The phenomenology of small-scale turbulence

A review of classical percolation theory is presented, with an emphasis on novel applications to statistical topography, turbulent diffusion, and heterogeneous media. Statistical topography involves the geometrical properties of the isosets (contour lines or surfaces) of a random potential $\ensuremath{\psi}(\mathrm{x})$. For rapidly decaying correlations of $\ensuremath{\psi}$, the isopotentials fall into the same universality class as the perimeters of percolation clusters. The topography of long-range correlated potentials involves many length scales and is associated either with the correlated percolation problem or with Mandelbrot's fractional Brownian reliefs. In all cases, the concept of fractal dimension is particularly fruitful in characterizing the geometry of random fields. The physical applications of statistical topography include diffusion in random velocity fields, heat and particle transport in turbulent plasmas, quantum Hall effect, magnetoresistance in inhomogeneous conductors with the classical Hall effect, and many others where random isopotentials are relevant. A geometrical approach to studying transport in random media, which captures essential qualitative features of the described phenomena, is advocated.

/pdf/percolation-statistical-topography-and-transport-in-random-1ddemlupey.pdf

Percolation, statistical topography, and transport in random media

The Structure of Turbulent Shear Flow By Dr. A. A. Townsend. Pp. xii + 315. 8¾ in. × 5½ in. (Cambridge: At the University Press.) 40s.

/pdf/the-structure-of-turbulent-shear-flow-2yf3szhaqb.pdf

The Structure of Turbulent Shear Flow

A total of 21 planar fractal grids pertaining to three different fractal families have been used in two different wind tunnels to generate turbulence The resulting turbulent flows have been studied using hot wire anemometry Irrespective of fractal family, the fractal-generated turbulent flows and their homogeneity, isotropy, and decay properties are strongly dependent on the fractal dimension Df≤2 of the grid, its effective mesh size Meff (which we introduce and define) and its ratio tr of largest to smallest bar thicknesses, tr=tmax∕tmin With relatively small blockage ratios, as low as σ=25%, the fractal grids generate turbulent flows with higher turbulence intensities and Reynolds numbers than can be achieved with higher blockage ratio classical grids in similar wind tunnels and wind speeds U The scalings and decay of the turbulence intensity u′∕U in the x direction along the tunnel’s center line are as follows (in terms of the normalized pressure drop CΔP and with similar results for v′∕U and w′∕U)

https://www.imperial.ac.uk/media/imperial-college/research-centres-and-groups/turbulence-mixing-and-flow-control-group/hurst_2007.pdf

Scalings and decay of fractal-generated turbulence

Space-filling fractal square grids fitted at the entrance of a wind tunnel’s test section generate unusually high Reynolds number homogeneous isotropic turbulence which decays locked into a single length-scale l. Specifically, during turbulence decay along the streamwise coordinate x, E11(k1,x)=u′2lf(k1l) over the entire range of wavenumbers, where l and the function f are about the same for all the grids tried here. As a result, this fractal-generated turbulence has the following properties which we have also observed in the decaying region: L∕λ is constant, independent of the x grid and Reλ; ϵ∼Reλ−1u′3∕Lu; and E11(k1)∼(u′3∕Lu)2∕3k1−5∕3 instead of E11(k1)∼ϵ2∕3k1−5∕3 in the observed range of wavenumbers where f(k1l)∼(k1l)−5∕3.

/pdf/dissipation-and-decay-of-fractal-generated-turbulence-3nqo5w3dx5.pdf

Dissipation and decay of fractal-generated turbulence

In this survey we consider the impact of turbulence on cloud formation from the cloud scale to the droplet scale. We assess progress in understanding the effect of turbulence on the condensational and collisional growth of droplets and the effect of entrainment and mixing on the droplet spectrum. The increasing power of computers and better experimental and observational techniques allow for a much more detailed study of these processes than was hitherto possible. However, much of the research necessarily remains idealized and we argue that it is those studies which include such fundamental characteristics of clouds as droplet sedimentation and latent heating that are most relevant to clouds. Nevertheless, the large body of research over the last decade is beginning to allow tentative conclusions to be made. For example, it is unlikely that small-scale turbulent eddies (i.e. not the energy-containing eddies) alone are responsible for broadening the droplet size spectrum during the initial stage of droplet growth due to condensation. It is likely, though, that small-scale turbulence plays a significant role in the growth of droplets through collisions and coalescence. Moreover, it has been possible through detailed numerical simulations to assess the relative importance of different processes to the turbulent collision kernel and how this varies in the parameter space that is important to clouds. The focus of research on the role of turbulence in condensational and collisional growth has tended to ignore the effect of entrainment and mixing and it is arguable that they play at least as important a role in the evolution of the droplet spectrum. We consider the role of turbulence in the mixing of dry and cloudy air, methods of quantifying this mixing and the effect that it has on the droplet spectrum. Copyright © 2012 Royal Meteorological Society and British Crown Copyright, the Met Office

http://research.me.udel.edu/lwang/reprints/Devenish_etal_QJRMS_review_2012.pdf

Droplet growth in warm turbulent clouds

We study turbulence generated by low-blockage space-filling fractal square grids [5]. This device creates a multiscale excitation of the fluid flow. Such devices have been proposed as alternative and complementary tools for the investigation of turbulence fundamentals, modelling and applications [3, 5, 6]. New insights on the fundamentals of homogeneous turbulence have been found, showing in particular that the small scales are not universal beyond small corrections caused by intermittency, finite Reynolds number and anisotropy. The unprecedented possibilities offered by these devices also open new attractive perspectives in applications involving mixing, combustion and flow management and control.

https://www.imperial.ac.uk/media/imperial-college/research-centres-and-groups/turbulence-mixing-and-flow-control-group/MV_2010.pdf

Turbulence without Richardson-Kolmogorov Cascade

Kinematic simulations are non-Markovian Lagrangian models of dispersion that incorporate turbulentlike flow structure. We investigate the conditions for two-particle dispersion to be local in a turbulentlike flow, and the dependence of the Richardson constant ${G}_{\ensuremath{\Delta}}$ on the topology of individual realizations of the flow.

/pdf/two-particle-dispersion-in-turbulentlike-flows-3vrrci7shk.pdf

John Christos Vassilicos

Papers

Scalings and decay of fractal-generated turbulence

Dissipation and decay of fractal-generated turbulence

Droplet growth in warm turbulent clouds

Turbulence without Richardson-Kolmogorov Cascade

Two-particle dispersion in turbulentlike flows