Outflow boundary conditions for one-dimensional finite element modeling of blood flow and pressure waves in arteries

Flow and pressure waves, originating due to the contraction of the heart, propagate along the deformable vessels and reflect due to tapering, branching, and other discontinuities. The size and complexity of the cardiovascular system necessitate a “multiscale” approach, with “upstream” regions of interest (large arteries) coupled to reduced-order models of “downstream” vessels. Previous efforts to couple upstream and downstream domains have included specifying resistance and impedance outflow boundary conditions for the nonlinear one-dimensional wave propagation equations and iterative coupling between three-dimensional and one-dimensional numerical methods. We have developed a new approach to solve the one-dimensional nonlinear equations of blood flow in elastic vessels utilizing a space-time finite element method with GLS-stabilization for the upstream domain, and a boundary term to couple to the downstream domain. The outflow boundary conditions are derived following an approach analogous to the Dirichlet-to-Neumann (DtN) method. In the downstream domain, we solve simplified zero/one-dimensional equations to derive relationships between pressure and flow accommodating periodic and transient phenomena with a consistent formulation for different boundary condition types. In this paper, we also present a new boundary condition that accommodates transient phenomena based on a Green’s function solution of the linear, damped wave equation in the downstream domain.     

Response of the streaks, the active and passive eddies in an unsteady channel flow

Spanwise space–time correlations of the wall shear stress and the longitudinal velocity fluctuations in the low buffer layer of an unsteady channel flow are reported. The imposed amplitude is 20% of the centerline velocity and the imposed frequency covers a large range going from the quasi-steady limit to the bursting frequency of the corresponding steady flow. The unsteady spanwise correlation coefficient is investigated both through its own modulation characteristics (amplitude and phase shifts) and those of the resulting streak spacing. A good correspondence is found between the modulation of the streak spacing and that of the ejection period. The data is further analyzed by temporal filtering of the wall shear stress and streamwise velocity fluctuations. It is shown that the large outer-layer structures play a “passive” role in the unsteady response of the near wall turbulence. The inner wall eddies, in return, are amply responsible for the unsteady reaction of both the turbulent wall shear stress and the streamwise velocity intensities in the buffer layer.     

Correlations predicting frictional pressure drop and liquid holdup during horizontal gas-liquid pipe flow with a small liquid holdup

Experimental data and correlations available in the literature for the liquid holdup ε_L and the pressure gradient ΔP_TP/L for gas-liquid pipe flow, generally, do not cover the domain 0 < ε_L < 0.06. Reliable pressure-drop correlations for this holdup range are important for calculating flow rates of natural gas, containing traces of condensate. In the present paper attention is focused on reliable measurements of ε_L and ΔP_TPIL values and on the development of a phenomenological model for the liquid-holdup range 0 < ε_L < 0.06. This model is called the “apparent rough surface” model and is referred to as the ARS model. The experimental results presented in this paper refer to air-water and air-water + ethyleneglycol systems with varying transport properties in horizontal straight smooth glass tubes under steady-state conditions. The holdup and pressure gradient values predicted with the ARS model agree satisfactorily with both our experimental results and data obtained from the literature referring to small liquid-holdup values 0 < ε_L < 0.06. Further, it has been shown that in the domain 38 < < 72 mPa m the interfacial tension of the gas-liquid system has no significant effect on the liquid holdup. The pressure gradient, however, increases slightly with decreasing surface tension values.     

Development of a numerical wave tank for analysis of nonlinear and irregular wave field

A numerical wave-absorption filter has been developed for an open boundary condition in the analysis of nonlinear and irregular wave evolution. The filter is composed of a simulated sponge layer and Sommerfeld's radiation condition at the outer edge of the layer. The wave-absorption characteristics of the filter have been investigated by applying the linear potential theory and a two-dimensional nonlinear boundary element model. In both cases, the filter is found to he applicable for a wide range of wave parameters. In order to realize an idealized “numerical wave tank”, the present model also incorporates a nonreflective wave generator in the computational domain composed of a series of vertically aligned point sources. Numerous numerical experiments demonstrate that the present approach is effective in generating an arbitrary wave profile without reflection not only at the open boundaries but also at the wave generator.     

On the centrifugal separation of a bulk mixture

The centrifugal separation of a mixture of particles and fluid in an axisymmetric container is examined. The flow consists of three distinct regions—mixture, sediment and purified fluid—with Ekman boundary layers at the interfaces and walls. In the settling process, the mixture and pure fluid acquire retrograde and prograde rotations relative to the tank. This flow pattern, and the shape and locus of the interface which are easily determined, provide another simple means to compare mixture theory and experiment. It is shown that when the Coriolis force is important, the pure fluid layer on the “outwardly” inclined wall is not thin. Moreover the interface between the mixture and the pure fluid is not perpendicular to the centrifugal force. Both features contrast those of the gravitational Boycott effect. As a consequence, there is no obvious enhancement of settling due to geometrical configuration.     

Sources of variability in traction data

A 18.4R38 tyre was tested at 124 kPa inflation pressure, approximately 24 kN axle load in a firm and in a tilled Yolo-loam soil using (i) constant slip, (ii) constant draught, (iii) varying slip and (iv) varying draught tyre testing procedures. The results indicated that the constant slip test procedure leads to repeatable and consistent results whereas a variable slip test procedure leads to considerable scatter in the data. The constant draught test procedure yielded acceptable results. Varying slip appeared to influence the system dynamics much more than varying draught during tyre testing. An accurate method of predicting “true rolling radius” and “true slip” for an assumed zero condition is presented. The concept of motion resistance, its variability due to assumed zero conditions, and possible interpretations are discussed. The traction test data indicates that the motion resistance is not constant but varies with slip.     

Investigation of a combustion driven oscillation in a refinery flare: Part B: Visualisation of a periodic flow instability in a bifurcating duct following a contraction

A flow visualisation study was performed to investigate a periodic flow instability in a bifurcating duct within the tip of the flares at the Shell refinery in Clyde, NSW, to verify the trigger of a combustion-driven oscillation proposed in Part A of this study, and to identify its features. The model study assessed only the flow instability, uncoupled from the acoustic resonance and the combustion that are also present in the actual flare. Three strong, coupled flow oscillations were found to be present in three regions of the fuel line in the flare tip model. A periodic flow separation was found to occur within the contraction at the inlet to the tip, a coupled, periodic flow oscillation was found in the two transverse “cross-over ducts” from the central pipe to the outer annulus and an oscillating flow recirculation was present in the “end-cap” region of the central pipe. The dimensionless frequency of these oscillations in the model was found to match that measured in the full-scale plant for high fuel flow rates. This, and the strength of these flow oscillations, gives confidence that they are integral to the full-scale combustion-driven oscillation and most likely the primary trigger. The evidence indicates that the periodic flow instability is initiated by the separation and roll-up of the annular boundary layer at the start of the contraction in the fuel section of the flare tip. The separation generates an annular vortex which interacts with the blind-ended pipe downstream, leading to a pressure wave which propagates back upstream, initiating the next separation event and repeating the cycle. The study also investigated flow control devices with a view to finding a practical approach to mitigate the oscillations. The shape of these devices was constrained to allow installation without removing the tip of the flare. This aspect of the study highlighted the strength and nature of the coupled oscillation, since it proved to be very difficult to mitigate the oscillation in this way. An effective configuration is presented, comprising of three individual components, all three of which were found to be necessary to eliminate the oscillation completely.     

Similarity solution of thermal boundary layers for laminar narrow axisymmetric jets

In this work, the similarity equation describing the thermal boundary layers of laminar narrow axisymmetric jets is derived based on boundary layer assumptions. The equation is solved exactly. Some properties of the thermal jet are discussed. By introducing new-defined non-dimensional coordinates, the similarity solution results in a “universal” format. The results can also be applied in the boundary layer problem of species diffusion.     

Stability analysis of annular flow structure, using a discrete form of the “two-fluid model”

The differential form of the “two-fluid model” for annular flow, neglecting surface tension, is ill-posed, and it is not suited for examining the stability of the steady-state solutions with respect to the average film thickness. It is shown here that a discrete (difference) representation of the two-fluid model may lead to an appropriate criterion for the stability of the steady-state solutions. Exactly the same criterion is obtained from the requirement that the kinematic waves will propagate in the downstream direction. The suggested discrete form of the “two-fluid model” is used to perform transient simulation and for examining the system response to finite disturbances.     

On the boundary layer/riblets interaction mechanisms and the prediction of turbulent drag reduction

Turbulent drag reduction experienced by ribletted surfaces is the result of both (1) the interaction between riblet peaks and the coherent structures that characterize turbulent near-wall flows, and (2) the laminar sublayer flow modifications caused by the riblet shape, which can balance, under appropriate conditions, the drag penalty due to the increased wetted surface. The latter “viscous” mechanism is investigated by means of an analytical model of the laminar sublayer, which removes geometrical restrictions and allows us to take into account “real” shapes of riblet contours, affected by manufacturing inaccuracies, and to compute even for such cases a parameter, called protrusion height, related to the longitudinal mean flow. By considering real geometries, riblet effectiveness is clearly shown to be related to the difference between the longitudinal and the transversal protrusion heights. A simple method for the prediction of the performances of ribletted surfaces is then devised. The predicted and measured drag reduction data, for different riblet geometries and flow characteristics, are in close agreement with each other. The soundness of the physical interpretation underlying this prediction method is consequently confirmed.     

Kinematics and dynamics of small-scale vorticity and strain-rate structures in the transition from isotropic to shear turbulence

We applied a technique that defines and extracts “structures” from a DNS dataset of a turbulence variable in a way that allows concurrent quantitative and visual analysis. Local topological and statistical measures of enstrophy and strain-rate structures were compared with global statistics to determine the role of mean shear in the dynamical interactions between fluctuating vorticity and strain-rate during transition from isotropic to shear-dominated turbulence. We find that mean shear adjusts the alignment of fluctuating vorticity, fluctuating strain-rate in principal axes, and mean strain-rate in a way that (1) enhances both global and local alignments between vorticity and the second eigenvector of fluctuating strain-rate, (2) two-dimensionalizes fluctuating strain-rate, and (3) aligns the compressional components of fluctuating and mean strain-rate. Shear causes amalgamation of enstrophy and strain-rate structures, and suppresses the existence of strain-rate structures in low-vorticity regions between enstrophy structures. A primary effect of shear is to enhance “passive” strain-rate fluctuations, strain-rate kinematically induced by local vorticity concentrations with negligible enstrophy production, relative to “active,” or vorticity-generating strain-rate fluctuations. Enstrophy structures separate into “active” and “passive” based on the level of the second eigenvalue of fluctuating strain-rate. We embedded the structure-extraction algorithm into an interactive visualization-based analysis system from which the time evolution of a shear-induced hairpin enstrophy structure was visually and quantitatively analyzed. The structure originated in the initial isotropic state as a vortex sheet, evolved into a vortex tube during a transitional period, and developed into a well-defined horseshoe vortex in the shear-dominated asymptotic state.     

Counter current gas-liquid vertical flow, model for flow pattern and pressure drop

Counter-current flow pattern transition and pressure drop are modeled. Emphasis is placed on the understanding of the transition mechanisms from a mechanistic point of view.

Unlike the case of co-current flow, in counter-current flow, the situations of “no solution” as well as “multiple solutions” for the flow pattern and pressure drop exist. These possibilities are discussed and criteria for the actual flow pattern that will take place are suggested.

Some of the results are supported by data (from the literature), others are somewhat tentative suggesting future experimental verification is needed.     

DYNAMICS AND STABILITY OF PINNED–CLAMPED AND CLAMPED–PINNED CYLINDRICAL SHELLS CONVEYING FLUID

The paper examines the dynamics and stability of fluid-conveying cylindrical shells having pinned–clamped or clamped–pinned boundary conditions, where “pinned” is an abbreviation for “simply supported”. Flügge's equations are used to describe the shell motion, while the fluid-dynamic perturbation pressure is obtained utilizing the linearized potential flow theory. The solution is obtained using two methods — the travelling wave method and the Fourier-transform approach. The results obtained by both methods suggest that the negative damping of the clamped–pinned systems and positive damping of the pinned–clamped systems, observed by previous investigators for any arbitrarily small flow velocity, are simply numerical artefacts; this is reinforced by energy considerations, in which the work done by the fluid on the shell is shown to be zero. Hence, it is concluded that both systems are conservative.     

Tests of a new numerical scheme on a “non-reflection and free-transmission” open-sea boundary for longwaves

A new numerical scheme of a “non-reflection and free-transmission” boundary for longwave equations proposed by Hino (1987) has been tested for a variety of cases. The test results verify the effectiveness of the method for (a) a single progressive wave train on a horizontal bottom, (b) two wave trains each propagating in opposite directions on a horizontal bottom, (c) a single wave train propagating on a sloping bottom with friction, (d) oscillatory flood waves in an open channel flow, (e) two-dimensional waves travelling obliquely to open boundaries and (f) water surface oscillation in a harbor by waves incident through an opening.     

Modelling of particle-wall collisions in confined gas-particle flows

This paper demonstrates that numerical simulations of confined particulate two-phase flows require a detailed modelling of particle—wall collisions which includes the wall surface structure and the particle shape. These effects are taken into account by “irregular bouncing” models which are based on the statistical treatment of the collision process. In this study, results obtained using various “irregular bouncing” models based on the impulse equations for a particle—wall collision are considered and compared with experimental observations. The wall roughness is simulated by assuming that the particle collides with a virtual wall which has a randomly distributed inclination with respect to the plane, smooth wall. A Gaussian distribution for this random inclination showed the best agreement with experimental results. Numerical predictions of a turbulent two—phase flow in a vertical channel, where the particle phase is treated using a Lagrangian approach, showed that the different models applied for a particle-wall collision have a strong effect on the particle velocity fluctuations and the mass flux profiles in the region of fully developed flow. The numerical simulations using the irregular bouncing models yielded considerably higher values for the particle velocity fluctuations, which also agreed better with the experimental values. This effect was most pronounced for large particles, where the distance they need to respond to the fluid flow is larger than the characteristic dimension of the confinement. On the other hand, the motion of small particles is less affected by the choice of the wall-collision model. These effects of the wall roughness on the velocity fluctuations of the dispersed phase have not been considered in previous studies using irregular bouncing models.     

Off-site calibration of a two-phase pattern recognition flowmeter

A technique is presented for the measurement of individual phase flowrates in two-phase flow without separation. The technique is based on signal analysis and pattern recognition methodologies to extract, classify and identify stochastic features from turbulent pressure waveforms. Experiments in a horizontal air-water loop have shown that a set of features is uniquely related to the individual phase flowrates. It is shown that the “flowmeter” can be isolated from installation effects by means of a homogenizer.     

Acoustics and dynamics of coaxial interacting vortex rings

Using a contour dynamics method for inviscid axisymmetric flow we examine the effects of core deformation on the dynamics and acoustic signatures of coaxial interacting vortex rings. Both “passage” and “collision” (head-on) interactions are studied for initially identical vortices. Good correspondence with experiments is obtained. A simple model which retains only the elliptic degree of freedom in the core shape is used to explain some of the calculated features.