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Fluid flow in soft-walled tubes part 1: Steady flow

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Abstract

The channel-flow form of the equations describing fluid flow in soft-walled tubes is analysed. Several analytical solutions for the properties of the flow existing in tubes with various analytical forms of pressure/ area relationships are presented. The vital relationship of the tube mechanical stability to the flow Mach number is established. The necessary conditions for the formation of a hydraulic jump are established, and the resultant degradation in total pressure determined. Finally, an example numerical result for an hydraulic jump is given.

Sommaire

On analyse l'écoulement sous forme canalisée de l'équation découlement des fluides dans les tuyaux à paroi souple. Plusieurs solutions analytiques des propriétés de l'écoulement rencontré dans les tuyaux ayant différentes formes analytiques des relations pression surface, sont présentées. La relation vitale entre la stabilité mécanique du tuyau et le nombre de Mach de l'écoulement est établie. Les conditions nécessaires pour la formation d'un saut hydraulique sont établies et la dégradation résultante de la pression totale est déterminée. On donne enfin l'exemple d'un résultat numérique d'un saut hydraulique.

Zusammenfassung

Die Kanaldurchflußform von Gleichungen, die den Flüssigkeit durchfluß in weichwandigen Röhren beschreiben, wird analysiert. Es werden mehrere analytische Lösungen für die Durchflußmerkmale, die in Röhren mit verschiedenen analytischen Formen von Druckbereichsverhältnissen bestehen, vorgestellt. Das wichtige Verhältnis der mechanischen Festigkeit der Röhre zur Machzahl des Durchflusses wird aufgestellt, und auch die notwendigen Bedingungen für die Bildung eines hydraulischen Sprungs und die sich ergebende Abnahme im Gesamtdruck werden festgelegt. Schließlich wird ein numerisches Ergebnis als Beispiel angeführt.

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Abbreviations

A :

tube cross-sectional area

C :

wavelet speed

D :

mean hydraulic diameter=4A/perimeter

D/Dt :

substantial derivative=∂/∂t+u∂/∂x

E :

Young's modulus

f :

friction factor=wall shear stress/ρu 2/2

f x :

axial body force

g :

gravitational constant

H :

height above reference plane

h :

enthalpy

h o :

reference-tube wall thickness (equns. 8 and 9)

K, k :

elastic constants (eqns. 8 and 10)

M :

Mach number=u/C=U/S

p :

pressure

P :

dimensionless pressure=p−p 0/ρC 0 2

p f, pt :

fluid pressure and tube pressure (eqn. 29), respectively

P t :

total pressure=p+1/2ρu 2+ρgH

P T :

diemnsionless total pressure=p t−p0/ρC 0 2

q :

heat added per mass per second

R 0 :

reference-tube radius (eqns. 8 and 9)

s :

entropy

S :

dimensionless wavelet speed=C/C 0

t :

time

u :

velocity

U :

dimensionless velocity=u/C 0

W :

constant=U 2α2 (eqn. 33)

x :

axial direction

α:

dimensionless area=A/A 0

v :

Poisson's ratio

ρ:

density

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Authors and Affiliations

  1. Department of Aeronautics and Astronautics, University of Washington, 98195, Seattle, Washington, USA

    Gordon C. Oates

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  1. Gordon C. Oates
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Additional information

Superscript * refers to conditions at the location where the Mach number is unity Subscript 0 refers to conditions when tube is at reference pressurep 0 (see eqns. 8–10)

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Oates, G.C. Fluid flow in soft-walled tubes part 1: Steady flow. Med. & biol. Engng. 13, 773–779 (1975). https://doi.org/10.1007/BF02478077

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  • DOI: https://doi.org/10.1007/BF02478077

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