Hybrid quadrature moment method for accurate and stable representation of non-Gaussian processes applied to bubble dynamics

Solving the population balance equation (PBE) for the dynamics of a dispersed phase coupled to a continuous fluid is expensive. Still, one can reduce the cost by representing the evolving particle density function in terms of its moments. In particular, quadrature-based moment methods (QBMMs) invert these moments with a quadrature rule, approximating the required statistics. QBMMs have been shown to accurately model sprays and soot with a relatively compact set of moments. However, significantly non-Gaussian processes such as bubble dynamics lead to numerical instabilities when extending their moment sets accordingly. We solve this problem by training a recurrent neural network (RNN) that adjusts the QBMM quadrature to evaluate unclosed moments with higher accuracy. The proposed method is tested on a simple model of bubbles oscillating in response to a temporally fluctuating pressure field. The approach decreases model-form error by a factor of 10 when compared with traditional QBMMs. It is both numerically stable and computationally efficient since it does not expand the baseline moment set. Additional quadrature points are also assessed, optimally placed and weighted according to an additional RNN. These points further decrease the error at low cost since the moment set is again unchanged. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.

On the lift-optimal aspect ratio of a revolving wing at low Reynolds number

Lentink & Dickinson (2009 J. Exp. Biol.212, 2705-2719. (doi:10.1242/jeb.022269)) showed that rotational acceleration stabilized the leading-edge vortex on revolving, low aspect ratio (AR) wings and hypothesized that a Rossby number of around 3, which is achieved during each half-stroke for a variety of hovering insects, seeds and birds, represents a convergent high-lift solution across a range of scales in nature. Subsequent work has verified that, in particular, the Coriolis acceleration plays a key role in LEV stabilization. Implicit in these results is that there exists an optimal AR for wings revolving about their root, because it is otherwise unclear why, apart from possible morphological reasons, the convergent solution would not occur for an even lower Rossby number. We perform direct numerical simulations of the flow past revolving wings where we vary the AR and Rossby numbers independently by displacing the wing root from the axis of rotation. We show that the optimal lift coefficient represents a compromise between competing trends with competing time scales where the coefficient of lift increases monotonically with AR, holding Rossby number constant, but decreases monotonically with Rossby number, when holding AR constant. For wings revolving about their root, this favours wings of AR between 3 and 4.

Effect of direct bubble-bubble interactions on linear-wave propagation in bubbly liquids

We study the influence of bubble-bubble interactions on the propagation of linear acoustic waves in bubbly liquids. Using the full model proposed by Fuster and Colonius [J. Fluid Mech. 688, 253 (2011)], numerical simulations reveal that direct bubble-bubble interactions have an appreciable effect for frequencies above the natural resonance frequency of the average size bubble. Based on the new results, a modification of the classical wave propagation theory is proposed. The results obtained are in good agreement with previously reported experimental data where the classical linear theory systematically overpredicts the effective attenuation and phase velocity.

Quasi-linear gradients for capillary liquid chromatography with mass and tandem mass spectrometry

Gradient elution, capillary liquid chromatography mass spectrometry was performed with linear, static gradients constructed by laminar flowing ten, 1.5 microL volume steps of decreasing organic concentration into tubing of small internal diameter. Sample loading, gradient formation, and sample elution were accomplished entirely by means of a commercially available micro-autosampler and single-syringe drive pump. The procedure was simple, fast, stable, and reproducible. Essentially linear gradients were produced without the use of additional valves, mixers, pumps or software. It took less than 10 minutes to form a gradient and less than 30 minutes to construct the set of individual buffer vials. The gradients were shown to be stable to storage. One hour after forming, peak retention times were reproduced to +/-0.5%. Long-term retention time reproducibility was found to vary by +/-2%. Chromatographic resolution was comparable or superior to that obtained by gradient elution with conventional dynamic mixing and split flow. The procedure was adapted with a 'peak parking' method which extended the time for generating peptide fragmentation data up to 10 minutes per peptide with the triple quadruple mass spectrometer. Using this technique, collision data were collected at the 25 femtomole level on nine of ten tryptic peptides in a single run.