Kolmogorov inertial range for inhomogeneous turbulent flows

Catalysis by Imaging: From Meso- to Nano-scale

In-situ imaging of catalytic reactions has provided insights into reaction front propagation, pattern formation and other spatio-temporal effects for decades. Most recently, analysis of the local image intensity opened a way towards evaluation of local reaction kinetics. Herein, our recent studies of catalytic CO oxidation on Pt(hkl) and Rh(hkl) via the kinetics by imaging approach, both on the meso- and nano-scale, are reviewed. Polycrystalline Pt and Rh foils and nanotips were used as µm- and nm-sized surface structure libraries as model systems for reactions in the 10–5–10–6 mbar pressure range. Isobaric light-off and isothermal kinetic transitions were visualized in-situ at µm-resolution by photoemission electron microscopy (PEEM), and at nm-resolution by field emission microscopy (FEM) and field ion microscopy (FIM). The local reaction kinetics of individual Pt(hkl) and Rh(hkl) domains and nanofacets of Pt and Rh nanotips were deduced from the local image intensity analysis. This revealed the structure-sensitivity of CO oxidation, both in the light-off and in the kinetic bistability: for different low-index Pt surfaces, differences of up to 60 K in the critical light-off temperatures and remarkable differences in the bistability ranges of differently oriented stepped Rh surfaces were observed. To prove the spatial coherence of light-off on nanotips, proper orthogonal decomposition (POD) as a spatial correlation analysis was applied to the FIM video-data. The influence of particular configurations of steps and kinks on kinetic transitions were analysed by using the average nearest neighbour number as a common descriptor. Perspectives of nanosized surface structure libraries for future model studies are discussed.

Phenomenological friction equation for turbulent flow of Bingham fluids

Most discussions in the literature on the friction coefficient of turbulent flows of fluids with complex rheology are empirical. As a rule, theoretical frameworks are not available even for some relatively simple constitutive models. In the present work, a formula is proposed for the evaluation of the friction coefficient of turbulent flows of Bingham fluids. The developments combine a fresh analysis for the description of the microscales of Kolmogorov and the phenomenological turbulence model of Gioia and Chakraborty [G. Gioia and P. Chakraborty, Phys. Rev. Lett. 96, 044502 (2006)PRLTAO0031-900710.1103/PhysRevLett.96.044502]. The resulting Blasius-type friction equation is tested against some experimental data and shows good agreement over a significant range of Hedstrom and Reynolds numbers. Comments on pressure measurements in yielding fluids are made. The limits of the proposed model are also discussed.

Phenomenological Blasius-type friction equation for turbulent power-law fluid flows

We propose a friction formula for turbulent power-law fluid flows, a class of purely viscous non-Newtonian fluids commonly found in applications. Our model is derived through an extension of the friction factor analysis based on Kolmogorov's phenomenology, recently proposed by Gioia and Chakraborty. Tests against classical empirical data show excellent agreement over a significant range of Reynolds number. Limits of the model are also discussed.

Spatially coupled catalytic ignition of CO oxidation on Pt: mesoscopic versus nano-scale

Spatial coupling during catalytic ignition of CO oxidation on μm-sized Pt(hkl) domains of a polycrystalline Pt foil has been studied in situ by PEEM (photoemission electron microscopy) in the 10(-5) mbar pressure range. The same reaction has been examined under similar conditions by FIM (field ion microscopy) on nm-sized Pt(hkl) facets of a Pt nanotip. Proper orthogonal decomposition (POD) of the digitized FIM images has been employed to analyze spatiotemporal dynamics of catalytic ignition. The results show the essential role of the sample size and of the morphology of the domain (facet) boundary in the spatial coupling in CO oxidation.

Transitional flow analysis in the carotid artery bifurcation by proper orthogonal decomposition and particle image velocimetry

Blood flow instabilities in the carotid artery bifurcation have been highly correlated to clot formation and mobilization resulting in ischemic stroke. In this work, PIV-measured flow velocities in normal and stenosed carotid artery bifurcation models were analyzed by means of proper orthogonal decomposition (POD). Through POD analysis, transition to more complex flow was visualized and quantified for increasing stenosis severity. While no evidence of transitional flow was seen in the normal model, the 50%-stenosed model started to show characteristics of transitional flow, which became highly evident in the 70% model, with greatest manifestation during the systolic phase of the cardiac cycle. By means of a model comparison, we demonstrate two quantitative measures of the flow complexity through the power-law decay slope of the energy spectrum and the global entropy. The more complex flow in the 70%-stenosed model showed a flatter slope of energy decay (-0.91 compared to -1.34 for 50% stenosis) and higher entropy values (0.26 compared to 0.17). Finally, the minimum temporal resolution required for POD analysis of carotid artery flow was found to be 100 Hz when determined through a more typical energy-mode convergence test, as compared to 400 Hz based on global entropy values.

Reduced-order models for closed-loop wake control

We review a strategy for low- and least-order Galerkin models suitable for the design of closed-loop stabilization of wakes. These low-order models are based on a fixed set of dominant coherent structures and tend to be incurably fragile owing to two challenges. Firstly, they miss the important stabilizing effects of interactions with the base flow and stochastic fluctuations. Secondly, their range of validity is restricted by ignoring mode deformations during natural and actuated transients. We address the first challenge by including shift mode(s) and nonlinear turbulence models. The resulting robust least-order model lives on an inertial manifold, which links slow variations in the base flow and coherent and stochastic fluctuation amplitudes. The second challenge, the deformation of coherent structures, is addressed by parameter-dependent modes, allowing smooth transitions between operating conditions. Now, the Galerkin model lives on a refined manifold incorporating mode deformations. Control design is a simple corollary of the distilled model structure. We illustrate the modelling path for actuated wake flows.

Spectral theory of the turbulent mean-velocity profile

It has long been surmised that the mean-velocity profile (MVP) of pipe flows is closely related to the spectrum of turbulent energy. Here we perform a spectral analysis to identify the eddies that dominate the production of shear stress via momentum transfer. This analysis allows us to express the MVP as a functional of the spectrum. Each part of the MVP relates to a specific spectral range: the buffer layer to the dissipative range, the log layer to the inertial range, and the wake to the energetic range. The parameters of the spectrum set the thickness of the viscous layer, the amplitude of the buffer layer, and the amplitude of the wake.

Intermittency and rough-pipe turbulence

Recently, by analyzing the measurement data of Nikuradze [NACA Tech. Memo No. 1292 (1950)], it has been proposed [N. Goldenfeld, Phys. Rev. Lett. 96, 044503 (2006)] that the friction factor, f , of rough-pipe flow obeys a scaling law in the turbulent regime. Here, we provide a phenomenological scaling argument to explain this law and demonstrate how intermittency modifies the scaling form, thereby relating f to the intermittency exponent, eta . By statistically analyzing the measurement data of f , we infer a satisfactory estimate for eta ( approximately 0.02) , the inclusion of which is shown to improve the data-collapse curve. This provides empirical evidence for intermittency other than the direct measurement of velocity fluctuations.

Turbulent friction in rough pipes and the energy spectrum of the phenomenological theory

The classical experiments on turbulent friction in rough pipes were performed by Nikuradse in the 1930s. Seventy years later, they continue to defy theory. Here we model Nikuradse's experiments using the phenomenological theory of Kolmogórov, a theory that is widely thought to be applicable only to highly idealized flows. Our results include both the empirical scalings of Blasius and Strickler and are otherwise in minute qualitative agreement with the experiments; they suggest that the phenomenological theory may be relevant to other flows of practical interest; and they unveil the existence of close ties between two milestones of experimental and theoretical turbulence.

Localized turbulent flows on scouring granular beds

In many applications a sustained, localized turbulent flow scours a cohesionless granular bed to form a pothole. Here we use similarity methods to derive a theoretical formula for the equilibrium depth of the pothole. Whereas the empirical formulas customarily used in applications contain numerous free exponents, the theoretical formula contains a single one, which we show can be determined via the phenomenological theory of turbulence. Our derivation affords insight into how a state of dynamic equilibrium is attained between a granular bed and a localized turbulent flow.

Scaling and similarity in rough channel flows

We show that Manning's empirical formula for the mean velocity of turbulent flows in channels represents the power-law asymptotic behavior of a flow of incomplete similarity in the relative roughness. We then derive the formula based on the phenomenological theory of turbulence. Our derivation yields the correct similarity exponent; it justifies Manning's use of a single parameter, the hydraulic radius, to characterize the geometry of the cross section; and it affords insight into the mechanism of momentum transfer.