MODELS AND METHODS IN MODERN SCIENCE
International scientific-online conference
63
NUMERICAL ANALYSIS OF FLUID FLOW AND PRESSURE
DISTRIBUTION IN A PIPE UNDER ISOTHERMAL CONDITIONS
Khozhikulov Sh.Sh.
101325, Uzbekistan
Begimov O.M.
101325, Uzbekistan
Institute of Mechanics and Seismic Stability of Structures named after
M.T.Urazbaev, AS RUz, Tashkent, Uzbekistan
*corresponding author: oybek.begimov@mail.ru
https://doi.org/10.5281/zenodo.15240772
Abstract:
This study proposes a numerical method for investigating the
isothermal state of an elementary section of a pipe with a given inlet pressure,
outlet mass flow rate, and known initial distributions of mass flow and pressure.
A quasi-one-dimensional model reflects the linear relationship between fluid
density and pressure, considering the quadratic resistance law and the
volumetric compressibility coefficient of the fluid. The governing equations and
conditions were modified through the introduction of dimensionless variables
and opposing wave propagation. To solve the nonlinear equations of moving
waves, a first-order accuracy scheme and an anti-flow scheme were applied. The
nonlinear boundary condition was satisfied with high accuracy using the
Newton’s iteration method. The study identifies the characteristics of the
transition process to a new operating regime of the linear pipe section, where
the mass flow rate, pressure, and velocity are represented by multi-joint curves.
The results show the influence of sharp boundary condition changes on the
system's response and provide a detailed analysis of the flow's behavior.
Keywords:
N.E. Zhukovsky equations , isothermal state, wave propagation,
numerical method, pressure, speed, mass flow rate, nonlinear boundary
conditions, quasi-one-dimensional equations
Introduction
This thesis investigates the unsteady isothermal flow in a relief-type pipe
segment using a numerical approach. The model is based on known initial
distributions of mass flow and pressure, with boundary conditions defined at
the inlet and outlet. A quasi-one-dimensional formulation accounts for quadratic
resistance and a linear relationship between pressure and density via fluid
compressibility. By introducing dimensionless variables and counter-
propagating waves, the nonlinear system is solved using first-order accurate and
upwind schemes. Results show multi-segmented behavior of mass flow,
pressure, and velocity, capturing the system’s transition to a new steady state.
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Main part
We consider a pipeline segment characterized by its length
𝑙
, inner
diameter
𝐷
, resistance coefficient
𝜆
, and variable elevation profile
𝑦
(
𝑥
). The
slope of the pipe relative to the horizontal is defined as
sin
/
dy x
dx
The problem involves determining the isothermal flow behavior of the fluid
in a relief-type pipeline element, where the inlet pressure varies with time, and
the outlet mass flow rate is also time-dependent. Initial conditions assume a
constant mass flow rate. The flow dynamics are governed by the quasi-one-
dimensional forms of the momentum and mass conservation equations, based
on N.E. Zhukovsky’s approach.
| |
sin ,
2
.
u
u
p
u
u u
g
t
x
x
D
u
t
x
(1)
The correlation between pressure and density of the liquid is defined via
the modulus of elasticity of water:
0
0
0
.
p
p
k
(2)
To solve the non-stationary problem, a numerical method is developed by
introducing counter-propagating waves. The stationary flow regime with a
constant mass flow rate is assumed as the initial condition for the problem.
0
( ,0)
,0
,0
.
M x
f
x
u x
M
const
(3)
To determine the initial pressure distribution, we assume:
0,
0
u
t
t
and refer to the stationary form of the momentum conservation equation:
2
p
c
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1
2
2
0
0
2
2
2
2
sin
1
2
M c
M c
dp
pg
dx
f p
c
D
f p
(4)
To transform the system into a form suitable for analyzing moving wave
analogs, we normalize the equations by dividing both sides by the reference
pressure
𝑝
.
As a result, the following equation is obtained:
2
*
*
*
ln( /
)
| |
sin ,
2
ln( /
)
ln( /
)
0.
u
u
u
c
u u
g
t
x
x
D
u
u
t
x
x
(5)
Here, we introduce a new variable:
*
ln
Using this substitution, the equation transforms into the following form:
2
| |
sin ,
2
0.
u
u
u
c
u u
g
t
x
x
D
u
u
t
x
x
(6)
To solve the equation using numerical methods, we first convert the
quantities (such as distance, velocity, density, and time) into dimensionless
form:
*
,
,
/ ,
x
l x u
cu t
t l c
The following equation is obtained:
2
|
|
sin ,
2
0.
u
u
l
lg
u
u u
t
x
x
D
c
u
u
t
x
x
(7)
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In solving the tasks,
0
*
,
0
101325
p
Pa
,
0
293.15
T
K
,
2120
k
МPа
and
according to the pressure scale,
*
100000
p
Pa
Taking this information into account, the initial conditions will be modified.
0
0
0
*
*
( ,0)
( ) / ,
( ,0)
( ) /
( ,0)
( ) /
u x
u x
c
p x
p x
p
x
x
and the boundary conditions:
0
0
( )
(0, ) 1
,
t
p
t
k
*
( )
(1, ) (1, )
( ).
l
l
Q t
t u
t
Q t
f
c
From the final system, we introduce analogs of the propagating waves:
1
f
u
(8)
2
f
u
(9)
Using the conditions of the system of equations, we obtain the first
expression by addition.
1
1
2
(1
)
|
|
sin ,
2
f
f
l
lg
u
u u
t
x
D
c
(10)
Using the conditions of the system of equations, we obtain the second
expression by subtraction.
2
2
2
(
1)
|
|
sin .
2
f
f
l
lg
u
u u
t
x
D
c
(11)
According to the introduced modifications, the boundary conditions are
changed.
New initial conditions:
1
( , 0)
( , 0)
ln ( , 0)
f x
u x
x
2
( ,0)
( ,0)
ln ( ,0)
f x
u x
x
The methods for implementing these boundary conditions are described in
the process of developing the algorithm for solving the problem with respect to
oppositely propagating waves and their analogs.
Conclusion
The modification of the N.E. Zhukovsky model accounts for the time-
dependent changes in inlet pressure, outlet mass flow rate, and the initial
distribution of density and velocity, considering volumetric compression of the
fluid during its transmission through the pipe. A numerical method for solving
the fluid transmission equations, which include the propagation of two opposing
waves, is proposed. The non-linearity and convergence of the equations require
the use of small time and distance steps and consistent integration phases. The
non-linear boundary condition is implemented using Newton's iteration method.
MODELS AND METHODS IN MODERN SCIENCE
International scientific-online conference
67
Algorithms and computational programs were developed in the Pascal ABC
environment. A series of calculations related to the sharp change in boundary
conditions at one or both ends of the pipe were performed. Results show that
the mass flow rate, pressure, and velocity demonstrate qualitative repeatability,
with significant variations occurring at the first boundary condition period and
exponential transitions with new distributions as the boundary conditions
change.
For small segment lengths and initial time intervals, frictional forces are
negligible, allowing the use of a short-pipe approximation. However, for medium
and long segments, it is necessary to account for the full spectrum of force
factors, as frictional forces become a significant consideration.
References:
1.
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3.
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4.
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