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Matthias Heil - One of the best experts on this subject based on the ideXlab platform.

  • chapter 8 flow in flexible Collapsible Tubes
    2015
    Co-Authors: Matthias Heil, Andrew L Hazel
    Abstract:

    Many fluid-conveying vessels in the human body are elastic and can undergo significant flow-induced deformations, making physiological fluid mechanics a rich source of large-displacement fluid–structure interaction problems. The aim of this chapter is to demonstrate the effects of wall elasticity on three canonical internal flows that arise in physiological applications. First, we discuss low-Reynolds number flows in the Starling resistor—the elastic-walled equivalent of Hagen–Poiseuille flow—as a generic model for single-phase flows in elastic vessels. This requires coupled solution of the Stokes equations and the equations of large-displacement elasticity. Next, we extend the theoretical framework to incorporate the presence of air–liquid interfaces and study the propagation of an air finger into a fluid-filled, non-axisymmetrically collapsed vessel—the fluid–structure interaction equivalent of the “Bretherton problem”, a model of pulmonary airway (re-)opening. Finally, we examine the effect of wall elasticity on the Rayleigh–Plateau instability and show that fluid–structure interaction facilitates the formation of occluding liquid bridges in liquid-lined elastic vessels—a scenario of relevance to the physiological problem of pulmonary airway closure. Throughout this chapter, we focus on the study of idealised model problems whose relative simplicity allows us to identify the primary physical mechanisms that underlie the observed behaviour.

  • Self-excited oscillations in three-dimensional Collapsible Tubes: simulating their onset and large-amplitude oscillations
    Journal of Fluid Mechanics, 2010
    Co-Authors: Matthias Heil, Jonathan Boyle
    Abstract:

    We employ numerical simulations to explore the development of flow-induced self-excited oscillations in three-dimensional Collapsible Tubes which are subject to boundary conditions (flow rate prescribed at the outflow boundary) that encourage the development of high-frequency oscillations via an instability mechanism originally proposed by Jensen & Heil (J. Fluid Mech., vol. 481, 2003, p. 235). The simulations show that self-excited oscillations tend to arise preferentially from steady equilibrium configurations in which the tube is buckled non-axisymmetrically. We follow the growing oscillations into the large-amplitude regime and show that short Tubes tend to approach an approximately axisymmetric equilibrium configuration in which the oscillations decay, whereas sufficiently long Tubes develop sustained large-amplitude limit-cycle oscillations. The period of the oscillations and the critical Reynolds number beyond which their amplitude grows are found to be in good agreement with theoretical scaling estimates.

  • How rapidly oscillating Collapsible Tubes extract energy from a viscous mean flow
    Journal of Fluid Mechanics, 2008
    Co-Authors: Matthias Heil, Sarah L. Waters
    Abstract:

    We present a combined theoretical and computational analysis of three-dimensional unsteady finite-Reynolds-number flows in Collapsible Tubes whose walls perform prescribed high-frequency oscillations which resemble those typically observed in experiments with a Starling resistor. Following an analysis of the flow fields, we investigate the system's overall energy budget and establish the critical Reynolds number, Re crit , at which the wall begins to extract energy from the flow. We conjecture that Re crit corresponds to the Reynolds number beyond which Collapsible Tubes are capable of performing sustained self-excited oscillations. Our computations suggest a simple functional relationship between Re crit and the system parameters, and we present a scaling argument to explain this observation. Finally, we demonstrate that, within the framework of the instability mechanism analysed here, self-excited oscillations of Collapsible Tubes are much more likely to develop from steady-state configurations in which the tube is buckled non-axisymmetrically, rather than from axisymmetric steady states, which is in agreement with experimental observations.

  • steady finite reynolds number flows in three dimensional Collapsible Tubes
    Journal of Fluid Mechanics, 2003
    Co-Authors: Andrew L Hazel, Matthias Heil
    Abstract:

    A fully coupled finite-element method is used to investigate the steady flow of a viscous fluid through a thin-walled elastic tube mounted between two rigid Tubes. The steady three-dimensional Navier–Stokes equations are solved simultaneously with the equations of geometrically nonlinear Kirchhoff–Love shell theory. If the transmural (internal minus external) pressure acting on the tube is sufficiently negative then the tube buckles non-axisymmetrically and the subsequent large deformations lead to a strong interaction between the fluid and solid mechanics. The main effect of fluid inertia on the macroscopic behaviour of the system is due to the Bernoulli effect, which induces an additional local pressure drop when the tube buckles and its cross-sectional area is reduced. Thus, the tube collapses more strongly than it would in the absence of fluid inertia. Typical tube shapes and flow fields are presented. In strongly collapsed Tubes, at finite values of the Reynolds number, two ’jets‘ develop downstream of the region of strongest collapse and persist for considerable axial distances. For sufficiently high values of the Reynolds number, these jets impact upon the sidewalls and spread azimuthally. The consequent azimuthal transport of momentum dramatically changes the axial velocity profiles, which become approximately $\uTheta$-shaped when the flow enters the rigid downstream pipe. Further convection of momentum causes the development of a ring-shaped velocity profile before the ultimate return to a parabolic profile far downstream.

  • stokes flow in an elastic tube a large displacement fluid structure interaction problem
    International Journal for Numerical Methods in Fluids, 1998
    Co-Authors: Matthias Heil
    Abstract:

    SUMMARY Viscous flow in elastic (Collapsible) Tubes is a large-displacement fluid-structure interaction problem frequently encountered in biomechanics. This paper presents a robust and rapidly converging procedure for the solution of the steady three-dimensional Stokes equations, coupled to the geometrically non-linear shell equations which describe the large deformations of the tube wall. The fluid and solid equations are coupled in a segregated method whose slow convergence is accelerated by an extrapolation procedure based on the scheme’s asymptotic convergence behaviour. A displacement control technique is developed to handle the system’s snap-through behaviour. Finally, results for the tube’s post-buckling deformation and for the flow in the strongly collapsed tube are shown. © 1998 John Wiley & Sons, Ltd.

T J Pedley - One of the best experts on this subject based on the ideXlab platform.

  • flow and oscillations in Collapsible Tubes physiological applications and low dimensional models
    Sadhana-academy Proceedings in Engineering Sciences, 2015
    Co-Authors: T J Pedley, Draga Pihlerpuzovic
    Abstract:

    The motivation for this subject comes from physiology: Air-flow in the lungs, where flow limitation during forced expiration is a consequence of large-airway collapse, and wheezing, which is a manifestation of self-excited mechanical oscillations; Blood flow in veins, such as those of giraffes, in which the return of blood to the heart from the head must be accompanied by partial venous collapse, and in arteries, which exhibit self-excited oscillations (Korotkov sounds) when compressed by a blood-pressure cuff. Laboratory experiments are frequently conducted in a Starling resistor, a finite length of flexible tube, mounted between two rigid Tubes and contained in a pressurised chamber. Steady conditions upstream and downstream give rise not only to steady flows, but also to a rich variety of self-excited oscillations, which theoreticians have been seeking to understand for at least five decades. Some of the observations have been reproduced in full Navier–Stokes computations for a two-dimensional model, but these do not provide physical understanding. We seek a self-consistent mathematical model for the oscillations. We concentrate first on 1D models, in which the key dependent variables are the cross-sectional area A and the cross-sectionally averaged velocity u and pressure p, all taken to be functions of longitudinal coordinate x and time t. The governing equations are those of conservation of mass and momentum and a tube law representing the elastic properties of the vessel. In the momentum equation, the viscous resistance term is conventionally modelled either as a linear function of fluid velocity, accurate at low Reynolds number, or with an ad hoc representation of the energy loss at flow separation. Even with such crude approximations, the predictions of 1D models agree quite well both with observations in the giraffe and with some of the 2D computations and 3D experiments. For a more rational model, we examine a 2D model problem, in which part of one wall of a parallel sided channel is replaced by a membrane under tension. One approach, for large Reynolds-number flow, and a long membrane, is to consider small deflections of the membrane and use interactive boundary-layer theory. This leads to interesting predictions, such as the impossibility of simultaneously prescribing the flow rate and the upstream pressure, but not to oscillations, except in cases where wall inertia is important (flutter). Another approach is to assume a parabolic velocity profile everywhere, leading to a rational choice for the inertia and viscous terms in the 1D momentum equation. If, further, the undisturbed membrane is taken to be flat, by a suitable choice of external pressure distribution, the system leads to an oscillatory instability even without wall inertia. Whether these oscillations have the same physics as those computed numerically at lower Reynolds number remains to be seen.

  • flow past highly compliant boundaries and in Collapsible Tubes proceedings of the iutam symposium held at the university of warwick united kingdom 26 30 march 2001
    2003
    Co-Authors: P W Carpenter, T J Pedley, Past Other Highly Compliant Boundaries, In Collapsible Tubes
    Abstract:

    Preface. Scientific Committee. Acknowledgement. 1. Introduction P.W. Carpenter, T.J. Pedley. A: Flow in Collapsible Tubes. 2. Flows in deformable Tubes and channels: Theoretical models and biological applications M. Heil, O.E. Jensen. 3. Experimental studies of Collapsible Tubes C.D. Bertram. B: Instability of Flow Past Compliant Walls. 4. Convective and absolute instabilities of flows over compliant walls C. Davies. 5. Hydrodynamic stability of flow through compliant channels and Tubes V. Kumaran. 6. Wave excitation on flexible wails in the presence of a fluid flow A.D. Lucey, N. Peake. 7. Propagation of waves across junctions between rigid and compliant walls P.W. Carpenter, P.K. Sen. 8. Rotating fiows over compliant walls P.W. Carpenter, P.J. Thomas, M. Nagata. C: Drag Modification and Turbulence Modification. 9. Drag reduction using compliant walls M. Gad-el-Hak. 10. Theoretical approaches to the effect of wall compliance on turbulent flow D. Rempfer, L. Parsons, S. Xu, J. Lumley. 11. Flow-induced waves on compliant surfaces subject to a turbulent boundary layer K.S. Yeo. 12. Experimental research on turbulent flows over compliant walls K.-S. Choi. 13. Dolphin hydrodynamics V.V. Babenko, P.W. Carpenter.

  • a model for time dependent flow in giraffe jugular veins uniform tube properties
    Journal of Biomechanics, 2002
    Co-Authors: Bindi S Brook, T J Pedley
    Abstract:

    Abstract Computations are reported for a one-dimensional model of time-dependent flow in Collapsible Tubes representing long mammalian veins. The Tubes are taken to have uniform intrinsic properties and we concentrate on the effect of longitudinal gravity. The main application is to the jugular vein of the upright giraffe, with given inflow rate from the head, a given pressure, slightly above the external, atmospheric pressure, at the downstream (vena caval) end, and a variety of initial conditions. We show that: (i) previously calculated steady flows are the long time limits of unsteady computations, although only after a considerable time in which slowly-decaying waves and elastic jumps propagate up and down, (ii) steady flows are indeed not found when the steady-flow analysis shown them not to exist, although the consequent unsteadiness is of such small amplitude as to be practically unimportant, (iii) the time taken for the flow to become steady when the neck is raised from the horizontal or the head-down position can be several seconds longer than the neck-raising time itself (3– 7 s ). We also find that roll-waves do not develop despite having been previously predicted for long Collapsible Tubes. Further application is made to the effect of postural changes on human neck and leg veins.

  • numerical solutions for unsteady gravity driven flows in Collapsible Tubes evolution and roll wave instability of a steady state
    Journal of Fluid Mechanics, 1999
    Co-Authors: Bindi S Brook, S A E G Falle, T J Pedley
    Abstract:

    Unsteady flow in Collapsible Tubes has been widely studied for a number of different physiological applications; the principal motivation for the work of this paper is the study of blood flow in the jugular vein of an upright, long-necked subject (a giraffe). The one-dimensional equations governing gravity- or pressure-driven flow in Collapsible Tubes have been solved in the past using finite-difference (MacCormack) methods. Such schemes, however, produce numerical artifacts near discontinuities such as elastic jumps. This paper describes a numerical scheme developed to solve the one-dimensional equations using a more accurate upwind finite volume (Godunov) scheme that has been used successfully in gas dynamics and shallow water wave problems. The adapatation of the Godunov method to the present application is non-trivial due to the highly nonlinear nature of the pressure–area relation for Collapsible Tubes. The code is tested by comparing both unsteady and converged solutions with analytical solutions where available. Further tests include comparison with solutions obtained from MacCormack methods which illustrate the accuracy of the present method. Finally the possibility of roll waves occurring in Collapsible Tubes is also considered, both as a test case for the scheme and as an interesting phenomenon in its own right, arising out of the similarity of the Collapsible tube equations to those governing shallow water flow.

  • Modelling Flow and Oscillations in Collapsible Tubes
    Theoretical and Computational Fluid Dynamics, 1998
    Co-Authors: T J Pedley, X.y. Luo
    Abstract:

    Laboratory experiments designed to shed light on fluid flow through Collapsible Tubes, a problem with several physiological applications, invariably give rise to a wide variety of self-excited oscillations. The object of modelling is to provide scientific understanding of the complex dynamical system in question. This paper outlines some of the models that have been developed to describe the standard experiment, of flow along a finite length of elastic tube mounted at its ends on rigid Tubes and contained in a chamber whose pressure can be independently varied. Lumped and one-dimensional models have been developed for the study of steady flow and its instability, and a variety of oscillation types are indeed predicted. However, such models cannot be rationally derived from the full governing equations, relying as they do on several crude, ad hoc assumptions such as that concerning the energy loss associated with flow separation at the time-dependent constriction during large-amplitude oscillations. A complete scientific description can be given, however, for a related two-dimensional configuration, of flow in a parallel-sided channel with a segment of one wall replaced by a membrane under longitudinal tension T. The flow and membrane displacement have been calculated successively by lubrication theory, Stokes-flow computation, steady Navier–Stokes computation and unsteady Navier–Stokes computation. For a given Reynolds number, Re, steady flow becomes unstable when T falls below a critical value (equivalently, when Re exceeds a critical value for fixed T), and the consequent oscillations reveal at least one period-doubling bifurcation as T is further reduced. The effect of wall inertia has also been investigated: it is negligible if the flowing fluid is water, but leads to an independent, high frequency flutter when it is air. The computations require very large computer resources, and a simpler model would be desirable. Investigation of the streamlines of the flow and the distribution of viscous energy dissipation reveals how the one-dimensional model might be improved; but such improvement is as yet incomplete

Riccardo Materassi - One of the best experts on this subject based on the ideXlab platform.

C D Bertram - One of the best experts on this subject based on the ideXlab platform.

  • the onset of flow rate limitation and flow induced oscillations in Collapsible Tubes
    Journal of Fluids and Structures, 2006
    Co-Authors: C D Bertram, J Tscherry
    Abstract:

    Abstract Experiments were mounted to investigate the onset in a ‘Starling resistor’ of Collapsible-tube oscillation, at the lowest possible Reynolds number so as to facilitate matched numerical simulations. The protocol adopted was to set pressure outside the tube and inside the tube at the upstream end, constant and equal to each other, then to progressively lower the downstream pressure past the point of tube collapse and, when this occurred, of oscillation onset. The working fluid was a glycerine/water mixture, and the silicone-rubber tube was suspended horizontally in air. Measurements were made of pressures and flow-rates and of the cross-sectional area at the approximate location of maximum oscillation; separately, the cross-sectional area of the tube in relation to transmural pressure was measured. Parameters varied in the flow experiments were the length of rigid pipe downstream of the collapsing tube, and the fluid viscosity. The pressure/flow-rate coordinates of both the point of peak flow-rate achieved before flow-rate limitation, and the point of oscillation onset, were satisfactorily independent of the pipe length downstream. Both points occurred at flow-rates that decreased with increasing fluid viscosity, so that the corresponding Reynolds numbers decreased more so. Oscillation did not break out below a Reynolds number of about 290 unless there was external mechanical agitation of the apparatus. The amplitude of oscillation decreased progressively towards zero at this point as viscosity was raised. After the flow-rate peak, flow limitation causes a local flow-rate minimum. When oscillation occurred, it started just before this minimum, and died away at the minimum.

  • flow limitation in uniform thick walled Collapsible Tubes
    Journal of Fluids and Structures, 1999
    Co-Authors: C D Bertram, R J Castles
    Abstract:

    Abstract To investigate the flow-rate limitation behaviour of the same thick-walled silicone-rubber Tubes with aqueous flow as have previously been characterized by this laboratory in terms of their pressure-drop limitation behaviour, we measured how the pressure drop along the tube varied with flow rate, when both the upstream head and the external pressure were varied in such a way as to keep the transmural pressure at the upstream end to the tube at a series of constant values. Flow limitation with ‘negative effort dependence’ occurred, and it was found that the flow rate depended not just on the upstream transmural pressure but also on its history, in a hysteretic manner. Furthermore, when external pressure was being reduced to the required point, either flow limitation or absence of collapse could be obtained for the same values of upstream transmural pressure. The reductions in flow rate when flow limitation came into effect were typically much greater, relative to the flow-limited flow rates themselves, than has been reported by others using thinner Tubes and lower flow rates. Large-amplitude self-excited oscillation was confined to this reducing-flow-rate transition when external pressure was being increased to set the required point, and largely confined to this transition when it was being reduced. Flow limitation was mostly associated with only small-amplitude noise-like fluctuations of the downstream pressure. The transition was analysed and explained by reference to modified control-space diagrams, which show explicitly all oscillatory and divergent instabilities as closed regions. The prominence of the transition in these results forced consideration of whether flow limitation occurs when the flow rate ceases to increase or when it becomes substantially independent of the pressure drop. In adopting the latter definition, we were led to hypothesize that the initial collapse-inducing instability is not the result of choking.

  • A study of the bifurcation behaviour of a model of flow through a Collapsible tube
    Bulletin of Mathematical Biology, 1996
    Co-Authors: J. P. Armitstead, C D Bertram, O. E. Jensen
    Abstract:

    Most of the elastic Tubes found in the mammalian body will collapse from a distended circular cross section and when collapsed may undergo flow-induced oscillations. A mathematical model describing fluid flow in a Collapsible tube is analysed using the software package AUTO-86. AUTO-86 is used for continuation and bifurcation problems in systems of non-linear ordinary differential equations. The model is a third-order lumped-parameter type and is based on the classical “Starling resistor”; it describes the unsteady flow behaviour and, in particular, the experimentally observed self-excited oscillations, in a way which is simple enough to give physical understanding, yet still firmly based on fluid mechanical principles. Some of the bifurcation types found in this model bear close resemblance to the types suggested by experimental observations of self-excited oscillations in Collapsible Tubes; they thus shed some light on the various topological changes which occur in practice, particularly in view of the fact that some of the points found numerically are diffcult to achieve experimentally, while the existence of others can only be inferred indirectly and uncertainly from experiment.

Mario R Tredici - One of the best experts on this subject based on the ideXlab platform.