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Wave propagation in fluid lines

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Abstract

The wave propagation in a fluid line consisting of a circular tube, initially stressed, viscoelastic, orthotropic, surrounded by external materials, containing a compressible linear elastic fluid is considered under isentropic conditions. The dispersion equation is derived and a number of simplifications are discussed. The impedances of the line and the transfer matrix are given.

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Abbreviations

a :

radius of the tube

A :

integration constant in (2.11)

A′ :

factor in A defined by (5.16)

B′ :

tube parameter defined by (9.6)

B (ij) (i, j=1, 2):

coefficients of linear stress-strain relations

B′ ij :

dimensionless parameters defined in (5.8)

B 011 , B 012 , B 021 :

dimensionless parameters defined in (5.10)

c :

complex wave propagation velocity

c i :

phase velocity of the i-th mode

c MK :

Moens-Korteweg reference velocity (4.4)

c p :

specific heat capacity at constant pressure

c T :

thermodynamic velocity of sound (∂ρ/c) 1/2 s

c 0 :

reference velocity defined in (4.1)

C :

dimensionless parameter defined in (4.7)

D 1, D 2 :

undetermined constants of solution

Dn:

dissipation number (10.12)

E :

Young's modulus

F :

function defined in (9.4)

F n (λ y):

functions defined in (2.12)

F 1α , F 1β :

functions defined in (5.4) and (5.5)

F nα (r/a)F nβ (r/a):

functions defined in (6.4)

h :

wall thickness

J n (y):

Bessel functions of the n-th order and first kind

k :

c 0/c

K :

bulk modulus of the compressible fluid

K r :

dimensional coefficient defined in (4.2)

K′ j (j=x, r):

dimensionless coefficient defined in (5.8)

l :

length of tube section

L j :

damping of external wall material

m :

ρ w h/ρa (5.11)

M j :

mass of external wall material

N j :

elasticity of external wall material

p :

fluid pressure in excess of steady state pressure

p 1(a), p 2(a):

fluid pressure at the wall at x=0 and x=l

Pn:

propagation number (10.12)

P :

total fluid pressure

P 1, P 2 :

P at x=0 and x=l

Pr:

Prandtl number

r :

radial coordinate in r, θ, x coordinate system

s :

specific entropy

s + :

perturbation longitudinal wall stress

S :

initial tension

t :

time coordinate

t + :

perturbation circumferential wall stress

T :

thermodynamic temperature

u :

axial liquid velocity

v :

radial liquid velocity

x :

axial coordinate in r, θ, x coordinate system

X j :

stresses exerted on the wall

Z c :

characteristic impedance (10.8)

Z 1, Z 2 :

impedances at x=0 and x=l

α, α′ :

dimensionless parameters defined by (2.13)

α T :

cubic expansion coefficient

β, β′, β 0T :

dimensionless parameters defined by (2.13)

β 0n , β 1n :

non negative roots of the Bessel function J 0, J 1

γ :

propagation constant (10.9)

γ e :

parameter defined by (3.5)

Δ i :

logarithmic decrement of the i-th mode

ζ :

parameter defined by (2.13)

η :

radial wall displacement

θ :

angular coordinate in r, θ, x coordinate system

κ :

bulk viscosity

λ:

thermal conductivity; parameter in F n (λy)

λ i :

wave length of the i-th mode

μ :

dynamic viscosity

ν :

kinematic viscosity

ν′ :

dimensionless kinematic viscosity (7.3)

ξ :

axial wall displacement

ρ :

density of the fluid

ρ w :

density of the wall

ρ 0 :

density of the fluid in the steady state

σ :

Poisson's ratio

φ 1, φ 2, φ 3, φ 4 :

parameters defined in (2.13), (5.8), (5.10)

Φ:

volume flux of the fluid

Φ1, Φ2 :

Φ at x=0 and x=l

ω :

circular frequency

^:

amplitude of an oscillating quantity

′:

dimensionless quantity

0:

reference values

θ :

circumferential direction

r :

radial direction

x :

axial direction

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Authors and Affiliations

  1. Laboratory for Aero- and Hydrodynamics, Department of Mechanical Engineering, Delft University of Technology, The Netherlands

    G. D. C. Kuiken

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Kuiken, G.D.C. Wave propagation in fluid lines. Applied Scientific Research 41, 69–91 (1984). https://doi.org/10.1007/BF00419360

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