Abstract

An analytic solution for the classical water hammer problem for unsteady laminar pipe flow involving Zielke's frequency-dependent friction model is derived. By means of separation of variables, the solution of the initial boundary value problem is compiled in the form of an infinite series representing its modal expansion. In this way, solving the damped wave equation reduces to root-finding of scalar algebraic equations for the complex eigenfrequencies. An analysis of this dispersion relation underlines the frequency-dependent nature of the solution in comparison with less sophisticated friction models.

References

1.
Zielke
,
W.
,
1968
, “
Frequency-Dependent Friction in Transient Pipe Flow
,”
ASME J. Basic Eng.
,
90
(
1
), pp.
109
115
.10.1115/1.3605049
2.
Holmboe
,
E. L.
, and
Rouleau
,
W. T.
,
1967
, “
The Effect of Viscous Shear on Transients in Liquid Lines
,”
ASME J. Basic Eng.
,
89
(
1
), pp.
174
180
.10.1115/1.3609549
3.
Trikha
,
A. K.
,
1975
, “
An Efficient Method for Simulating Frequency-Dependent Friction in Transient Liquid Flow
,”
ASME J. Fluids Eng.
,
97
(
1
), pp.
97
105
.10.1115/1.3447224
4.
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
(
1
), pp.
49
76
.10.1115/1.1828050
5.
Urbanowicz
,
K.
,
Firkowski
,
M.
, and
Bergant
,
A.
,
2018
, “
Comparing Analytical Solutions for Unsteady Laminar Pipe Flow
,”
13th International Conference on Pressure Surges
, BHR Group, Bordeaux, France, pp.
283
303
.
6.
Tijsseling
,
A. S.
, and
Anderson
,
A.
,
2007
, “
Johannes von Kries and the History of Water Hammer
,”
J. Hydraul. Eng.
,
133
(
1
), pp.
1
8
.10.1061/(ASCE)0733-9429(2007)133:1(1)
7.
Wylie
,
E. B.
, and
Streeter
,
V. L.
,
1978
,
Fluid Transients
,
McGraw-Hill
,
New York
.
8.
D'Souza
,
A. F.
, and
Oldenburger
,
R.
,
1964
, “
Dynamic Response of Fluid Lines
,”
ASME J. Basic Eng.
,
86
(
3
), pp.
589
598
.10.1115/1.3653180
9.
Viersma
,
T. J.
,
1980
,
Analysis, Synthesis and Design of Hydraulic Servosystems and Pipelines
,
Elsevier
,
Amsterdam
.
10.
Ham
,
A. A.
,
1982
, “
On the Dynamics of Hydraulic Lines Supplying Servosystems
,” Doctoral thesis,
Delft University of Technology
,
Delft, The Netherlands
.
11.
Goodson
,
R. E.
, and
Leonard
,
R. G.
,
1972
, “
A Survey of Modeling Techniques for Fluid Line Transients
,”
ASME J. Basic Eng.
,
94
(
2
), pp.
474
482
.10.1115/1.3425453
12.
Gradshteyn
,
I. S.
, and
Ryzhik
,
I. M.
,
2000
,
Table of Integrals, Series, and Products
, 6th ed.,
Academic Press
,
San Diego, CA
.
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