Malcolm R. Davidson

VOLUME-OF-FLUID CALCULATION OF HEAT OR MASS TRANSFER ACROSS DEFORMING INTERFACES IN TWO-FLUID FLOW

A new volume-of-fluid (VOF)-based numerical method for calculating heat transfer or mass transfer of a species within and between fluids with deforming interfaces is described. The algorithm is tested first against an analytical solution for diffusion from a sphere, and good agreement between theory and calculation is shown. The method is then demonstrated by predicting (a) heat transfer from a rising bubble when the bubble forms a toroidal shape, and (b) mass transfer from a rising drop when the drop phase controls diffusion. The method is shown to be a viable approach for complex interfacial heat/mass transfer.

A simple, versatile valve model for use in lumped parameter and one-dimensional cardiovascular models.

Lumped parameter and one-dimensional models of the cardiovascular system generally employ ideal cardiac and/or venous valves that open and close instantaneously. However, under normal or pathological conditions, valves can exhibit complex motions that are mainly determined by the instantaneous difference between upstream and downstream pressures. We present a simple valve model that predicts valve motion on the basis of this pressure difference, and can be used to investigate not only valve pathology, but a wide range of cardiac and vascular factors that are likely to influence valve motion.

The reservoir-wave paradigm introduces error into arterial wave analysis: a computer modelling and in-vivo study

Objectives: Arterial wave reflection has traditionally been quantified from pressure and flow measurements using wave separation and wave intensity (WI) analysis. In the recently proposed reservoir-wave paradigm, these analyses are performed after dividing pressure into ‘reservoir’ and ‘excess’ components, yielding a modified wave intensity (WIRW). This new approach has led to controversial conclusions about the nature and significance of arterial wave reflection. Our aim was to assess whether WI or WIRW more accurately represent wave phenomena. Methods: We studied two computer models (a simple network and a full model of the systemic arterial tree) in which all systolic forward waves and reflection properties were known a priori. Results of these models were compared with haemodynamic measurements in the ascending aorta of five adult sheep at baseline and after incremental arterial constriction. Results: The key findings of model studies were that the reservoir-wave approach markedly underestimated or eliminated reflected compression waves, overestimated or artefactually introduced forward and backward expansion waves, and displayed nonphysical interactions between distal reflection sites and early systolic waves. These errors arose because, contrary to a key assumption of the reservoir-wave approach, reservoir pressure was not spatially uniform during systole. In-vivo results were qualitatively similar to model results, with baseline WI and WIRW suggesting that the arterial network was dominated by positive and negative wave reflection, respectively, while under all conditions, reflected WIRW compression waves were substantially smaller than corresponding WI waves. Conclusion: We conclude that the reservoir-wave paradigm introduces error into arterial wave analyses.

Numerical calculations of two-phase flow in a liquid bath with bottom gas injection: The central plume

Abstract The two-phase flow resulting from the bottom injection of gas into a liquid bath, constrained to be axisymmetric, is modelled by governing equations that incorporate the virtual mass and particle lift forces and a diffusive interfacial force. The equations are solved numerically by using the transient, two-fluid-flow program K-FIX. Both the bath and the top space are included in the calculation domain, but attention is focused on the central plume. Inclusion of terms describing the virtual mass force together with a force due to microscopic bulk pressure differences are found to ensure formal stability of a simplified case for all admissible parameter values; this also ensures numerical stability in practice. The particle lift force, together with a diffusive interfacial force, is shown to account for the observed spreading of the plume, the vertical variation in the centerline void fraction, and the magnitude of the bubble rise velocity in the upper half of the plume; however, although the void fraction is adequately predicted away from the centerline, the off-center bubble velocity remains poorly determined despite only small errors in the calculated gas flow rate at different heights.

Numerical calculations of flow in a hydrocyclone operating without an air core

Abstract Steady flow in a hydrocyclone operating without an air core is modelled by a finite difference solution of the Navier-Stokes equations, following the approach of Pericleous and Rhodes, in which the shear stress due to tangential motion is derived from the familiar Prandtl momentum transport theory, applied to angular momentum. In this application, the Prandtl model breaks down very near the axis of symmetry and a boundary condition, corresponding to zero shear, must be imposed at a radius of about one mixing length to ensure realistic flow predictions. Calculations are based on a commonly used solution procedure for velocity and pressure which uses the SIMPLE algorithm of Patankar and Spalding. Hybrid first-order upwind/central differencing is used, and calculated flow velocities are obtained which agree with both published data and analytical predictions. The corresponding transport equation for the dispersed (particle) phase is solved similarly, and the predicted efficiency curve and distributions of particle concentration are shown. Finally, predicted velocity and particle distributions are compared with corresponding results based on quadratic upstream differencing of the governing equations; it is concluded that numerical diffusion on has little effect on the predicted fluid or particle movement in this case.

A Model Analysis of Arterial Oxygen Desaturation during Apnea in Preterm Infants

Rapid arterial O2 desaturation during apnea in the preterm infant has obvious clinical implications but to date no adequate explanation for why it exists. Understanding the factors influencing the rate of arterial O2 desaturation during apnea () is complicated by the non-linear O2 dissociation curve, falling pulmonary O2 uptake, and by the fact that O2 desaturation is biphasic, exhibiting a rapid phase (stage 1) followed by a slower phase when severe desaturation develops (stage 2). Using a mathematical model incorporating pulmonary uptake dynamics, we found that elevated metabolic O2 consumption accelerates throughout the entire desaturation process. By contrast, the remaining factors have a restricted temporal influence: low pre-apneic alveolar causes an early onset of desaturation, but thereafter has little impact; reduced lung volume, hemoglobin content or cardiac output, accelerates during stage 1, and finally, total blood O2 capacity (blood volume and hemoglobin content) alone determines during stage 2. Preterm infants with elevated metabolic rate, respiratory depression, low lung volume, impaired cardiac reserve, anemia, or hypovolemia, are at risk for rapid and profound apneic hypoxemia. Our insights provide a basic physiological framework that may guide clinical interpretation and design of interventions for preventing sudden apneic hypoxemia.

Spreading of an inviscid drop impacting on a liquid film

Abstract The evolution of the deforming liquid surface following the impact of a drop onto a film of the same liquid is analysed numerically using a boundary integral method assuming axisymmetric, inviscid flow. Surface tension and gravity are taken into account. At times comparable to, or larger than the impact time scale (based on initial drop radius and impact velocity), the section of the liquid surface bounded by the radially propagating crown or rim is predicted to approach a single central shape independent of film thickness. At times which are much smaller than the impact time scale, jetting behaviour is obtained in the neck region where the drop meets the film when the Weber number is large enough. The jet is found to move close to the film, and this suggests the possibility of bubble entrapment, confirming a previous report in the literature. The present results suggest the occurrence of a train of bubble rings from repeated near-reconnection events as the neck moves radially outwards under jetting conditions.

Flow patterns in models of small airway units of the lung

Quasi-steady creeping flow in models of small airway units of the lung is investigated. A respiratory unit of the lung is modelled by a sphere, an oblate and a prolate ellipsoid of revolution, and a circular cylinder of finite length. The solution of the Stokes equations for each of these geometries is indicated for general axi-symmetric boundary conditions. For particular cases consistent with the models, streamlines are plotted and some velocity profiles are shown. It is suggested that bulk 00w in the ha1 generations of the lung is significant for gas transport even though diffusion is the predominant mechanism there.

Boundary integral prediction of the spreading of an inviscid drop impacting on a solid surface

Abstract Axisymmetric spreading of an idealised inviscid liquid drop impinging on a horizontal solid surface is analysed (including surface tension) using a boundary integral method for Weber numbers ( We ), based on initial drop radius and impact velocity, ranging from 3 to 100. Progressive accumulation of liquid in a rim around the periphery of the spreading inviscid drop is predicted. The effect diminishes with increasing Weber number, and is negligible when We =50. It is concluded that the experimentally observed rim at Weber numbers exceeding this value is due solely to viscous retardation. For We ⩾10, the calculated reduction in drop height with time is found to be almost independent of Weber number, and agrees extremely well with experimental data despite the absence of viscous effects in the calculations. The inviscid spreading rate increases with increasing Weber number, and a simple model predicts a dimensionless limiting value of 2 at large times as We →∞. The viscous reduction in the radius of spreading, determined by subtracting the measured and calculated (inviscid) values, is found to be approximately linear in time during most of the primary deformation. Derived values of the slope m can be fitted by m=0.5We Re −0.5 for We less than about 40. Modification of the calculated inviscid spreading radius using a linear viscous correction provides an improved prediction of drop spreading.

Definition of a capture zone for shallow water table lakes

Abstract Lake-aquifer interaction is studied with the aim of developing simple relationships between easily measurable geometrical and aquifer parameters and the bulk behaviour of the flow system. Attention is focused on shallow flow-through lakes which receive groundwater along the up-gradient shoreline and discharge lake water along the down-gradient shoreline. Although the flow system is physically three-dimensional, two idealised two-dimensional geometries, in plan and in vertical section, are studied in detail. The resulting potential flow problems are solved using a boundary integral approach, based on a Greens function chosen to satisfy desired homogeneous boundary conditions on a semi-infinite strip. Results are presented for circular and elliptical lakes in plan and for lakes so shallow that they are adequately represented in vertical section by a line at the surface. The size of an upstream capture zone, in which all groundwater flow eventually passes through the body of the lake, is defined in terms of the size of the lake, inter-lake spacing, aquifer saturated thickness, an anisotropy ratio and the ratio of downstream to upstream hydraulic gradients. Even in cases where the net groundwater inflow to a lake is zero, it is shown that a substantial throughflow can occur.