INTRODUCTION
The flow through a pipe with sudden enlargement and contraction occurs in many
industrial applications and is characterized by increased pressure losses caused
by flow separation close to the change in the cross sectional area. This increasing
in pressure losses will increase the erosion rates and the heat in the regions
where separated flow occurs (Mahdi, 2001).
The most important flow characteristics are the nature of flow (laminar or
turbulent), velocity and pressure. Normally the nature of flow is circular and
depended on the Reynolds Number (Re) and for most engineering applications if
(Re>2300) the flow is laminar (Saleh, 1978).
Also, the fluid flowing through the pipe may impose pressures on the pipe walls
which deflect the pipe, where at a high velocity flow through a thin wall pipe
it can either buckle the pipe or cause it to fail. In certain applications involving
very high velocity flows through flexible thin wall pipes combined with vibration
such as (the feed lines to rockets and water turbines) the pipe may become susceptible
to resonance and fatigue failure if its natural frequency falls below certain
limits (Blevins, 1979).
Pittard (2003) studied the flowinduced vibration caused
by fully developed pipe flow under turbulent conditions. This study focuses
on the development of a numerical FluidStructure Interaction (FSI) model that
will help define the relationship between pipe wall vibration and the physical
characteristics of turbulent flow. The results show that a strong relationship
between pipe vibration and flow rate exists.
A cantilevered pipe subjected to external transverse (or lateral) force is
investigated by (LilkovaMarkova and Lolov, 2004).
The major findings are the variations in frequency with flow velocity and displacements
at different points and times.
Lolov and LilkovaMarkova (2006) studied the curved
pipes conveying fluids. Methods of numerical solution of the dynamic stability
of a pipe in its plane are developed. An example of a curved pipe is solved
by these methods. A nondimensional parameter of flow velocity and a nondimensional
circular frequency are obtained.
Experimental and numerical investigations of laminar flow in a pipe with a
sudden contraction in the crosssectional area were given by Durst
and Loy (1985). Investigations were carried out to understand the increased
pressure loss generated in this region. In addition, the detailed experimental
velocity profile measurements permit comparison with numerical predictions.
To yield reliable data, a laserdoppler anemometer together with a test section
containing a liquid with the same refractive index as the test section wall
materials, were employed. Numerical predictions of the flow employing finite
difference computer code were also undertaken. A small separated flow region
exists in the concave corner of the contraction.
A pipe conveying fluid with a sudden enlargement and with the effect of heat
flux combined with vibration on these pipes was studied by Hammoudi
(2007). Several end pipe supports (simply, flexible, fixed) were adopted.
A mathematical model was developed using the transfer matrix method to show
the effect of vibration and the effect of implementing different values of heat
flux on pipes conveying fluid with a sudden enlargement. The most important
result of this investigation is that the natural frequencies of the vibrated
system decreased when the flowing fluid and thermal forces were taken in consideration.
This reduction increased as the applied heat flux was increased. While increasing
fluid velocity without applying heat flux did not affect the values of the natural
frequencies.
To design safe and reliable piping systems free from excessive vibrations, the piping designer needs to know the frequencies of excitation forces in the piping and must be able to calculate the mechanical natural frequencies of the pipeline system.
The objective of the present research is to study the effect of different fluid
densities in a pipe conveying fluid with sudden enlargementsudden contraction.
Also, evaluate the natural frequencies and their corresponding mode shapes due
to external effect when the pipe is supported by flexible type of supports.
The study has been conducted by developing mathematical model solved in MATLAB
environment and a simulation model solved in ANSYS 14 commercial software. The
case was studied at Reynolds value of 1500.
METHODOLOGY
The case of pipe conveying fluid was modeled mathematically and solved by Transfer Matrix Method (TMM). The construction of the mathematical model is implemented as below.
Governing equation of motion: Consider a straight pipe conveying uniform internal flow as shown in Fig. 1. The straight pipe, supported at both ends, has dimensions given by the Length (L), the crosssectional outer Diameter (D) and the Thickness (th). It is assumed that the pipe is sufficiently slender, that is, (D/L)<<0.1. This diameter to length ratio makes it considered as a beam. Moreover, the fluid in the pipe is assumed to be incompressible. The flow regime in this study is assumed laminar, where the secondary flow effects are negligible.

Fig. 1: 
Pipe conveying fluid 
The equation of motion for free vibration of pipe conveying fluid derived and may be written as (Eq. 1):
So, for forced vibration, the equation of motion must be equated to the excitation harmonic force F (x,t) (Eq. 2):
where, F(x,t): is the external harmonic force:
Stiffness term:
Curvature term:
Coriolis force term:
Inertia force term:
Now the equation of motion for forced vibration of pipe conveying fluid may be written as (Eq. 3):
If there is no fluid, (u, P, ρ, m_{f}) equal to zero and the equation of motion will reduce to (Eq. 4):
The following dimensionless variables are adopted which are already used by
(Reddy and Wang, 2004).
Then, the equation for forced vibration becomes (Eq. 6):
where, F (
, τ) is the nondimensional external force applied normal to the pipe axis
in (ydirection).
Investigation of the flow stream: Since, the fluid discharge to atmosphere; therefore, the out let pressure of the pipe (P_{3}) = 1 atm and the inlet pressure to the pipe (P_{1}) can be found from the energy Eq. 6 as follows:
For horizontal pipe (z_{1} = z_{3} = 0) substitute in above equation gives (Eq. 7):
Where:
Losses 
= 
P_{L1}+P_{Le}+P_{L2}+P_{Lc}+P_{L3} 
P_{L1} 
= 
Losses for the first part of pipe (before enlargement) 
P_{L2} 
= 
Losses for the second part of pipe (after enlargement) 
P_{L3} 
= 
Losses for the third part of pipe (after contraction) 
P_{Le} 
= 
Losses at enlargement: 
P_{Lc} 
= 
Losses at contraction: 

Fig. 2: 
Pipe with discrete elements and masses 
Transfer matrix method: The adopted solution method of the mathematical model is the transfer matrix method, known as T.M.M. In this method the vibrating system is arranged in a line and the behavior at any point in the system is only influenced by the behavior at neighboring points as shown in Fig. 2.
To fully describe the situation at each node, four quantities must be known;
the deflection, Y, slope, Φ, moment, M and shear forces, Q. In the present
study, two fluid physical properties have been added, which are velocity, U
and pressure, P. Accordingly, six variables are affecting the node status and
can be arranged in a state vector as:
The matrices below show the adopted field and point matrices in this investigation.
• 
Field matrix: The field matrix [F] for a pipe element
may be written in matrix notation in dimensionless form, as (Eq.
8): 
Or, in detailed matrix format:
Where:

Fig. 3: 
Schematic diagram of pipe system with flexible support 
• 
Point matrix at the supported node: 
Boundary conditions: The boundary conditions in the transfer matrix method give the description of the state vector parameters at the supported ends of the pipe, which is selected as flexible support for this analysis. For flexible supported the moment and the shear forces are equal to zero and all other parameters have a value more than zero as shown in Fig. 3.
RESULTS AND DISCUSSION
Effect of fluid density on natural frequencies: To study the effect
of fluid density on the natural frequencies, two types of fluids were selected
which are water and oil. Their physical properties are listed in Table
1. Where, the deflections at mid length of copper pipe conveying fluid with
Reynolds number (1500) and same supports (flexible) with various excitation
frequencies for different kinds of fluid density are presented in (Fig.
4ac). The values of three lowest natural frequencies
from the peaks of these figures are given in Table 2. It can
be noticed from the results in (Fig. 4b) that the values of
the 1st, 2nd and 3rd natural frequencies for the case of vibrated pipe system
conveying water at Reynolds numbers (1500) are less than the values of the 1st,
2nd and 3rd natural frequencies for the case of vibrated pipe system without
fluid (Fig. 4a). This can be related to the effect of the
fluid mass which is added to the mass of the system and it is inversely proportional
to the natural frequencies. Also, the values of 1st, 2nd and 3rd natural frequencies
(Fig. 4c) of vibrated pipe system conveying oil are higher
than those of pipe conveying water due to the lower density.
Table 2: 
Comparison of three lowest natural frequencies values of (Sudden
enlargementsudden contraction) copper pipe conveying different fluid with
different densities 

Table 3: 
Results of comparison between the transfer matrix method by
using MATLABR2012 program and finite element method by using ANSYS14 software
for flexible support copper pipe without fluid 

Effects of density on the mode shape: Figure 5ac
and 6ac represent 1st, 2nd and 3rd mode
shapes, respectively for flexible support copper pipe and for different fluid
density with Reynolds number equal 1500. It can be observed that there is changing
in the phase of the mode shape at the natural frequency for some of these figures
like (Fig. 6a, c) where the mode shapes
for flexible support pipe conveying oil is positive, while the mode shapes for
flexible support pipe conveying water is negative see (Fig. 5a,
c) that’s because the effect of changing system properties
at the different density. Also, in these figures, new phenomenon could be noticed,
whereas a change in the sequence of maximum amplitude values at changing of
fluid density for the same supports type.
Comparison between the mathematical and simulation results: The results
obtained from transfer matrix method by adopting MATLABR2012 program were compared
with prediction results obtained from the finite element method using ANSYS14
software. The comparisons are presented in Table 35.
The comparisons between the results for these two programs show a good agreement.

Fig. 4(ac): 
Deflection at mid length of flexible supports copper pipe
(a) Without fluid, (b) Conveying water and (c) Conveying oil with various
excitation frequencies represents three lowest natural frequency, using
MATLABR2012 program 

Fig. 5(ac): 
(a)1st, (b) 2nd and (c) 3rd mode shapes for flexible support
pipe conveying water using ANSYS14 software 

Fig. 6(ac): 
(a) 1st, (b) 2nd and (c) 3rd mode shapes for flexible support
pipe conveying oil using ANSYS14 software 
Table 4: 
Results of comparison between the transfer matrix method by
using MATLABR2012 program and finite element method by using ANSYS14 software
for flexible support copper pipe conveying water 

Table 5: 
Results of comparison between the transfer matrix method by
using MATLABR2012 program and finite element method by using ANSYS14 software
for flexible support copper pipe conveying oil 

CONCLUSION
For the (Sudden enlargementsudden contraction) pipe conveying fluid with different density and exposed to vibration, it can be concluded:
• 
The natural frequencies for pipe system conveying fluid is
less than the natural frequencies for pipe system without fluid 
• 
Increasing the density at the same values of Reynolds number
leads to decrease the values of the natural frequencies 
• 
The change of fluid density effect on maximum amplitude values
for the same supports type 
• 
The results of the transfer matrix method by using (MATLBR2012)
program and finite element method by using (ANSYS14) software show a good
agreement with percentage for a maximum difference of (0.767%) and a minimum
difference of (0.032%) 