Technical Briefs

Wave Propagation in Tapered Vessels: New Analytic Solutions That Account for Vessel Distensibility and Fluid Compressibility

[+] Author and Article Information
George Papadakis

Department of Aeronautics,
Imperial College London,
Exhibition Road,
South Kensington Campus,
London SW7 2AZ, UK
e-mail: g.papadakis@ic.ac.uk

Contributed by the Pressure Vessel and Piping Division of ASME for publication in the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received March 31, 2013; final manuscript received August 27, 2013; published online October 29, 2013. Assoc. Editor: Samir Ziada.

J. Pressure Vessel Technol 136(1), 014501 (Oct 29, 2013) (9 pages) Paper No: PVT-13-1057; doi: 10.1115/1.4025447 History: Received March 31, 2013; Revised August 27, 2013

The central aim of this paper is to contribute to the theoretical analysis and understanding of the effect of vessel tapering on the propagation of pressure and velocity wave forms. To this end, it presents new analytic expressions for the temporal and spatial variation of these two variables that account for weak fluid compressibility. It extends previous work in which only the effect of wall deformation (i.e., vessel distensibility) was taken into account. The solutions are derived in the frequency domain and can account for the steady solution component (d.c. component) obtained by taking the asymptotic limit for very low frequencies. It is shown that the effect of compressibility makes the equations more complex but it is still possible to derive closed form analytic solutions in terms of Bessel functions of orders 1/3 and 4/3. The analytical solutions are compared with full 3D fluid structure interaction (FSI) simulations for the case of propagation of a step pressure variation at the inlet of a tapered vessel. Good agreement is observed between the 1D analytical and 3D numerical solutions.

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Lighthill, J., 1975, “Pulse Propagation Theory, in Mathematical Biofluid Dynamics,” CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 17, SIAM, Philadelphia, PA.
Fung, Y. C., 1996, Biomechanics, Circulation, 2nd ed., Springer, New York.
Pedley, T. J., 1980, The Fluid Mechanics of Large Blood Vessels, Cambridge University Press, Cambridge, UK.
Papadakis, G., 2011, “New Analytic Solutions for Wave Propagation in Flexible, Tapered Vessels With Reference to Mammalian Arteries,” J. Fluid Mech., 689, pp. 465–488. [CrossRef]
Wiggert, D. C., and Tijsseling, A., 2001, “Fluid Transients and Fluid-Structure Interaction in Flexible Fluid-Filled Piping,” Appl. Mech. Rev., 54(5), pp. 455–481. [CrossRef]
Ghidaoui, M. S., Zhao, M., McInnis, D. A., and Axworthy, D. H., 2005, “A Review of Water Hammer Theory and Practice,” Appl. Mech. Rev., 58, pp. 49–76. [CrossRef]
Tijsseling, A., 2003, “Exact Solution of Linear Hyperbolic Four-Equation System in Axial Liquid-Pipe Vibration,” J. Fluids Struct., 18, pp. 179–196. [CrossRef]
Wu, T., and Ferng, C.-C., 1999, “Effect of Non-Uniform Conduit Section on Water Hammer,” Acta Mech., 137, pp. 137–149. [CrossRef]
Warsi, Z. U. A., 1999, Fluid Dynamics: Theoretical and Computational Approaches, 2nd ed., CRC Press, Boca Raton, FL.
Flügge, W., 1960, Stresses in Shells, Springer-Verlag, Berlin, Germany.
LeVeque, R. J., 2002, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge.
Råde, L., and Westergren, B., 1999, Mathematics Handbook for Science and Engineering, 4th ed., Springer, New York.
Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P., 1997, Numerical Recipes in Fortran 77; The Art of Scientific Computing, 2nd ed., Cambridge University Press, Cambridge, UK.
Papadakis, G., 2008, “A Novel Pressure-Velocity Formulation and Solution Method for Fluid-Structure-Interaction Problems,” J. Comput. Phys., 227, pp. 3383–3404. [CrossRef]
Papadakis, G., 2009, “Coupling 3D and 1D Fluid-Structure-Interaction Models for Wave Propagation in Flexible Vessels Using a Finite Volume Pressure-Correction Scheme,” Commun. Numer. Methods Eng., 25, pp. 533–551. [CrossRef]


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Fig. 1

Sketch of the geometry under consideration in plane (s,θ) and definition of basic variables

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Fig. 5

Time variation of axial (a) pressure and (b) velocity at the middle of the vessel; comparison between 3D FSI simulations and 1D model

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Fig. 2

Variation of (a) |PFR(ω,s)| and (b) |UFR(ω,s)| against ω at a point located at the middle of the vessel

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Fig. 3

Effect of modulus of elasticity on the variation of (a) |PFR(ω,s)| and (b) |UFR(ω,s)| against ω at a point located at the middle of the vessel

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Fig. 4

Effect of the cone angle on |PFR(ω,s)| for (a) constant inlet and outlet radii (varying length) and (b) constant inlet radius and length (and varying outlet radius)

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Fig. 6

Axial profiles of (a) pressure and (b) velocity at several time instants (shown in μs)

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Fig. 7

Variation of Riemann variables (a) w1 and (b) w2 along the tube for different time instants (shown in μs)

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Fig. 8

Isopressure contours (in Pa) at different time instants (Tramp = 100 μs)

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Fig. 9

Isopressure contours (in Pa) at different time instants (Tramp = 0)



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