On the laminar-turbulent transition of the rotating-disk flow: The role of absolute instability

© 2014 Cambridge University Press. This paper describes a detailed experimental study using hot-wire anemometry of the laminar-turbulent transition region of a rotating-disk boundary-layer flow without any imposed excitation of the boundary layer. The measured data are separated into stationary and unsteady disturbance fields in order to elaborate on the roles that the stationary and the travelling modes have in the transition process. We show the onset of nonlinearity consistently at Reynolds numbers, R, of ∼ 510, i.e. at the onset of Lingwood's (J. Fluid Mech., vol. 299, 1995, pp. 17-33) local absolute instability, and the growth of stationary vortices saturates at a Reynolds number of ∼ 550. The nonlinear saturation and subsequent turbulent breakdown of individual stationary vortices independently of their amplitudes, which vary azimuthally, seem to be determined by well-defined Reynolds numbers. We identify unstable travelling disturbances in our power spectra, which continue to grow, saturating at around R=585, whereupon turbulent breakdown of the boundary layer ensues. The nonlinear saturation amplitude of the total disturbance field is approximately constant for all considered cases, i.e. different rotation rates and edge Reynolds numbers. We also identify a travelling secondary instability. Our results suggest that it is the travelling disturbances that are fundamentally important to the transition to turbulence for a clean disk, rather than the stationary vortices. Here, the results appear to show a primary nonlinear steep-fronted (travelling) global mode at the boundary between the local convectively and absolutely unstable regions, which develops nonlinearly interacting with the stationary vortices and which saturates and is unstable to a secondary instability. This leads to a rapid transition to turbulence outward of the primary front from approximately R=565 to 590 and to a fully turbulent boundary layer above 650.

On the generation mechanisms of short-scale unbalanced modes in rotating two-layer flows with vertical shear

We report on the results of a laboratory investigation using a rotating two-layer annulus experiment, which exhibits both large-scale vortical modes and short-scale divergent modes. A sophisticated visualization method allows us to observe the flow at very high spatial and temporal resolution. The balanced long-wavelength modes appear only when the Froude number is supercritical (i.e. $F\,{>}\,F_\mathrm{critical}\,{\equiv}\, \upi^2/2$), and are therefore consistent with generation by a baroclinic instability. The unbalanced short-wavelength modes appear locally in every single baroclinically unstable flow, providing perhaps the first direct experimental evidence that all evolving vortical flows will tend to emit freely propagating inertia–gravity waves. The short-wavelength modes also appear in certain baroclinically stable flows. We infer the generation mechanisms of the short-scale waves, both for the baro-clinically unstable case in which they co-exist with a large-scale wave, and for the baroclinically stable case in which they exist alone. The two possible mechanisms considered are spontaneous adjustment of the large-scale flow, and Kelvin–Helmholtz shear instability. Short modes in the baroclinically stable regime are generated only when the Richardson number is subcritical (i.e. $\hbox{\it Ri}\,{<}\,\hbox{\it Ri}_\mathrm{critical}\,{\equiv}\, 1$), and are therefore consistent with generation by a Kelvin–Helmholtz instability. We calculate five indicators of short-wave generation in the baroclinically unstable regime, using data from a quasi-geostrophic numerical model of the annulus. There is excellent agreement between the spatial locations of short-wave emission observed in the laboratory, and regions in which the model Lighthill/Ford inertia–gravity wave source term is large. We infer that the short waves in the baroclinically unstable fluid are freely propagating inertia–gravity waves generated by spontaneous adjustment of the large-scale flow.

Non-ergodicity of inviscid two-dimensional flow on a beta-plane and on the surface of a rotating sphere

It is shown that, for a sufficiently large value of β, two-dimensional flow on a doubly-periodic beta-plane cannot be ergodic (phase-space filling) on the phase-space surface of constant energy and enstrophy. A corresponding result holds for flow on the surface of a rotating sphere, for a sufficiently rapid rotation rate Ω. This implies that the higher-order, non-quadratic invariants are exerting a significant influence on the statistical evolution of the flow. The proof relies on the existence of a finite-amplitude Liapunov stability theorem for zonally symmetric basic states with a non-vanishing absolute-vorticity gradient. When the domain size is much larger than the size of a typical eddy, then a sufficient condition for non-ergodicity is that the wave steepness ε < 1, where ε = 2[surd radical]2Z/βU in the planar case and $\epsilon = 2^{\frac{1}{4}} a^{\frac{5}{2}}Z^{\frac{7}{4}}/\Omega U^{\frac{5}{2}}$ in the spherical case, and where Z is the enstrophy, U the r.m.s. velocity, and a the radius of the sphere. This result may help to explain why numerical simulations of unforced beta-plane turbulence (in which ε decreases in time) seem to evolve into a non-ergodic regime at large scales.

Mean zonal flows generated by librations of a rotating spherical cavity

Longitudinal librations represent oscillations about the axis of a rotating axisymmetric fluid-filled cavity. An analytical theory is developed for the case of a spherical cavity in the limit when the libration frequency is small in comparison with the rotation rate, but large in comparison with the inverse of the spin-up time. It is shown that longitudinal librations create a steady zonal flow through the nonlinear advection in the Ekman layers. The theory can be applied to laboratory experiments as well as to solid planets and satellites with a liquid core. Copyright © Cambridge University Press 2010.

Contribution of thickened drinks, food and enteral and parenteral fluids to fluid intake in hospitalised patients with dysphagia

Background Studies amongst older people with acute dysphagic stroke requiring thickened fluids have assessed fluid intakes from combinations of beverage, food, enteral and parenteral sources, but not all sources simultaneously. The study aimed to comprehensively assess total water intake from food, beverages, enteral and parenteral sources amongst dysphagic adult in-patients receiving thickened fluids. Methods Patients requiring thickened fluid following dysphagia diagnosis were recruited consecutively from a tertiary teaching hospital’s medical and neurosurgical wards. Fluid intake from food and beverages was assessed by wastage, direct observation and quantified from enteral and parenteral sources through clinical medical records. Results No patients achieved their calculated fluid requirements unless enteral or parenteral fluids were received. The mean daily fluid intake from food was greater than from beverages whether receiving diet alone (food 807±363mL, food and beverages 370±179mL, p<0.001) or diet with enteral or parenteral fluid support (food 455±408mL, food and beverages 263±232mL, p<0.001). Greater daily fluid intakes occurred when receiving enteral and parenteral fluid in addition to oral dietary intake, irrespective of age group, whether assistance was required, diagnosis and whether stage 3 or stage 2 thickened fluids were required (p<0.05). After enteral and parenteral sources, food provided the most important contribution to daily fluid intakes. Conclusions The greatest contribution to oral fluid intake was from food, not beverages. Designing menus and food services which promote and encourage the enjoyment of fluid dense foods, in contrast to thickened beverages, may present an important way to improve fluid intakes of those with dysphagia. Supplemental enteral or parenteral fluid may be necessary to achieve minimum calculated fluid requirements.

Computational fluid dynamics analysis of a flat plate photocatalytic reactor for storm and wastewater reuse

A computational fluid dynamics (CFD) analysis has been performed for a flat plate photocatalytic reactor using CFD code FLUENT. Under the simulated conditions (Reynolds number, Re around 2650), a detailed time accurate computation shows the different stages of flow evolution and the effects of finite length of the reactor in creating flow instability, which is important to improve the performance of the reactor for storm and wastewater reuse. The efficiency of a photocatalytic reactor for pollutant decontamination depends on reactor hydrodynamics and configurations. This study aims to investigate the role of different parameters on the optimization of the reactor design for its improved performance. In this regard, more modelling and experimental efforts are ongoing to better understand the interplay of the parameters that influence the performance of the flat plate photocatalytic reactor.