A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions
A second-order numerical technique is developed to study the steady laminar fluid motion in a straight two-dimensional hard-walled duct with two axisymmetric rectangular constrictions. In this technique, the governing relations are solved via deriving their integral analogs, performing a discretiz...
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| Cite this: | A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions / A.O. Borysyuk // Доповіді Національної академії наук України. — 2022. — № 1. — С. 48-57. — Бібліогр.: 5 назв. — англ. |
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Borysyuk, A.O. 2022-08-26T14:02:12Z 2022-08-26T14:02:12Z 2022 A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions / A.O. Borysyuk // Доповіді Національної академії наук України. — 2022. — № 1. — С. 48-57. — Бібліогр.: 5 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2022.01.048 https://nasplib.isofts.kiev.ua/handle/123456789/184929 532.542 A second-order numerical technique is developed to study the steady laminar fluid motion in a straight two-dimensional hard-walled duct with two axisymmetric rectangular constrictions. In this technique, the governing relations are solved via deriving their integral analogs, performing a discretization of these analogs, simplifying the obtained (after making the discretization) coupled nonlinear algebraic equations, and the final solution of the resulting (after making the simplification) uncoupled linear ones. The discretization consists of the spatial and temporal parts. The first of them is performed with the use of the TVD-scheme and a two-point scheme of discretization of the spatial derivatives, whereas the second one is made on the basis of the implicit three-point asymmetric backward differencing scheme. The above-noted uncoupled linear algebraic equations are solved by an appropriate iterative method, which uses the deferred correction implementation technique and the technique of conjugate gradients, as well as the solvers ICCG and Bi-CGSTAB. Розроблено чисельний метод розв’язування задачі про стаціонарний ламінарний рух рідини у прямому плоскому жорсткому каналі з двома осесиметричними прямокутними звуженнями. Цей метод має другий порядок точності. У ньому співвідношення, що описують зазначений рух, розв’язуються шляхом одержання їхніх інтегральних аналогів, дискретизації цих аналогів, зведення зв’язаних нелінійних алгебраїчних рівнянь (одержаних внаслідок дискретизації) до відповідних незалежних лінійних і подальшого розв’язування останніх. Зазначена дискретизація складається із просторової та часової частин. Перша з них виконується на основі використання TVD-схеми, а також двоточкової схеми дискретизації просторових похідних. При проведенні ж другої частини дискретизації застосовується неявна триточкова несиметрична схема з різницями назад. Що стосується методу розв’язування вказаних незалежних лінійних рівнянь, то це — відповідний ітераційний метод, який використовує методи відкладеної корекції та спряжених градієнтів, а також солвери ICCG та Bi-CGSTAB. en Видавничий дім "Академперіодика" НАН України Доповіді НАН України Механіка A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions Чисельний метод розв’язування задачі про рух рідини у прямому плоскому жорсткому каналі з двома осесиметричними прямокутними звуженнями Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions |
| spellingShingle |
A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions Borysyuk, A.O. Механіка |
| title_short |
A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions |
| title_full |
A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions |
| title_fullStr |
A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions |
| title_full_unstemmed |
A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions |
| title_sort |
numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions |
| author |
Borysyuk, A.O. |
| author_facet |
Borysyuk, A.O. |
| topic |
Механіка |
| topic_facet |
Механіка |
| publishDate |
2022 |
| language |
English |
| container_title |
Доповіді НАН України |
| publisher |
Видавничий дім "Академперіодика" НАН України |
| format |
Article |
| title_alt |
Чисельний метод розв’язування задачі про рух рідини у прямому плоскому жорсткому каналі з двома осесиметричними прямокутними звуженнями |
| description |
A second-order numerical technique is developed to study the steady laminar fluid motion in a straight two-dimensional
hard-walled duct with two axisymmetric rectangular constrictions. In this technique, the governing relations
are solved via deriving their integral analogs, performing a discretization of these analogs, simplifying the obtained
(after making the discretization) coupled nonlinear algebraic equations, and the final solution of the resulting
(after making the simplification) uncoupled linear ones. The discretization consists of the spatial and temporal parts.
The first of them is performed with the use of the TVD-scheme and a two-point scheme of discretization of the
spatial derivatives, whereas the second one is made on the basis of the implicit three-point asymmetric backward
differencing scheme. The above-noted uncoupled linear algebraic equations are solved by an appropriate iterative
method, which uses the deferred correction implementation technique and the technique of conjugate gradients,
as well as the solvers ICCG and Bi-CGSTAB.
Розроблено чисельний метод розв’язування задачі про стаціонарний ламінарний рух рідини у прямому
плоскому жорсткому каналі з двома осесиметричними прямокутними звуженнями. Цей метод має другий
порядок точності. У ньому співвідношення, що описують зазначений рух, розв’язуються шляхом одержання їхніх інтегральних аналогів, дискретизації цих аналогів, зведення зв’язаних нелінійних алгебраїчних
рівнянь (одержаних внаслідок дискретизації) до відповідних незалежних лінійних і подальшого
розв’язування останніх. Зазначена дискретизація складається із просторової та часової частин. Перша з
них виконується на основі використання TVD-схеми, а також двоточкової схеми дискретизації просторових похідних. При проведенні ж другої частини дискретизації застосовується неявна триточкова несиметрична схема з різницями назад. Що стосується методу розв’язування вказаних незалежних лінійних рівнянь, то це — відповідний ітераційний метод, який використовує методи відкладеної корекції та
спряжених градієнтів, а також солвери ICCG та Bi-CGSTAB.
|
| issn |
1025-6415 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/184929 |
| citation_txt |
A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions / A.O. Borysyuk // Доповіді Національної академії наук України. — 2022. — № 1. — С. 48-57. — Бібліогр.: 5 назв. — англ. |
| work_keys_str_mv |
AT borysyukao anumericaltechniquetosolveaproblemofthefluidmotioninastraightplanerigidductwithtwoaxisymmetricrectangularconstrictions AT borysyukao čiselʹniimetodrozvâzuvannâzadačíproruhrídiniuprâmomuploskomužorstkomukanalízdvomaosesimetričnimiprâmokutnimizvužennâmi AT borysyukao numericaltechniquetosolveaproblemofthefluidmotioninastraightplanerigidductwithtwoaxisymmetricrectangularconstrictions |
| first_indexed |
2025-11-27T01:48:58Z |
| last_indexed |
2025-11-27T01:48:58Z |
| _version_ |
1850791901719953408 |
| fulltext |
48 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2022. № 1: 48—57
Ц и т у в а н н я: Borysyuk A.O. A numerical technique to solve a problem of the fluid motion in a straight plane
rigid duct with two axisymmetric rectangular constrictions. Допов. Нац. акад. наук Укр. 2022. № 1. С. 48—57.
https://doi.org/10.15407/dopovidi2022.01.048
Study of flows in ducts is an actual problem in the gas-oil industry, architecture, medicine,
municipal economy, etc. Among others, a significant interest is related here to studying the flows
in ducts with local constrictions. That is explained by the fact that such irregularities in the duct
geometry cause local changes in the flow structure and/or character, etc. Those changes can result
in the corresponding consequences not only in a vicinity of, but also far from the irregularities [1].
As analysis of the scientific literature shows, the study of flows in ducts with local constric-
tions has been paid much attention to. In those studies, straight hard-walled ducts and their
constrictions of the simplest geometries were considered. The basic flow (i.e., the flow upstream
of a (first) constriction) was laminar, axisymmetric, and steady. As for fluids, they were assumed
to be homogeneous, incompressiblem and Newtonian (the other types of ducts, their constric-
https://doi.org/10.15407/dopovidi2022.01.048
UDC 532.542
A.O. Borysyuk, Corresponding Member of the NAS of Ukraine,
https://orcid.org/0000-0002-3878-3915
Institute of Hydromechanics of the NAS of Ukraine, Kyiv
E-mail: aobor@ukr.net
A numerical technique to solve a problem
of the fluid motion in a straight plane rigid duct
with two axisymmetric rectangular constrictions
A second-order numerical technique is developed to study the steady laminar fluid motion in a straight two-dimen-
sional hard-walled duct with two axisymmetric rectangular constrictions. In this technique, the governing relations
are solved via deriving their integral analogs, performing a discretization of these analogs, simplifying the obtained
(after making the discretization) coupled nonlinear algebraic equations, and the final solution of the resulting
(after making the simplification) uncoupled linear ones. The discretization consists of the spatial and temporal parts.
The first of them is performed with the use of the TVD-scheme and a two-point scheme of discretization of the
spatial derivatives, whereas the second one is made on the basis of the implicit three-point asymmetric backward
differencing scheme. The above-noted uncoupled linear algebraic equations are solved by an appropriate iterative
method, which uses the deferred correction implementation technique and the technique of conjugate gradients,
as well as the solvers ICCG and Bi-CGSTAB.
Keywords: fluid motion, flat duct, rectangular constriction, technique.
МЕХАНІКА
MECHANICS
49ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2022. № 1
A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct....
tions, fluidsm and the basic flow are not considered in this paper, because they were studied
much less intensively compared with the noted ones). These allowed one, on the one hand, to
study (within the framework of appropriate models chosen and with acceptable accu racy) the
influence of the basic parameters of a duct, its constriction(s) and the basic flow on the flow not
only nearm but also far downstream of the constriction(s), and, on the other hand, to simplify
significantly solutions to the corresponding problems of interest [1-4].
Among the results obtained in those studies, numerical methods, which have been develo ped
to investigate flows around duct constrictions, are of a particular interest. One of the latest of
them was presented in [1]. It has been devised to solve a problem of the flow in a straight hard-
walled two-dimensional duct with two rigid constrictions of a rectangular axisymmetric shape.
That method allows one to study the fluid motion in the noted duct in the stream function–vor-
ticity–pressure variables, has high stability of a solution and the second order of accuracy in the
spatial co-ordinates. However, its first order of accuracy in the temporal coord inate should ap-
parently stimulate researchers either to develop more accurate appropriate computational tech-
niques or to improve the method in such a way to make its temporal accu racy higher.
In this study, an alternative technique is presented to solve the same problem. This technique
uses the fluid velocity and the pressure as the basic variables, has nearly the same stability of a
solution, the same order of accuracy in the spatial coordinates and higher (i.e., the second) order
of accuracy in the temporal coordinate. However, due to the large amount of mathematical
operations used in this technique, it needs a more computational time to obtain a solution com-
pared to the above one.
Statement of the problem. A straight hard-walled plane duct of dimensionless width 1,
having two rigid constrictions of a rectangular axisymmetric shape, is considered (Fig. 1). The
constrictions are situated at the distance 12L from each other, and have the diameters iD and the
lengths iL ( 1, 2i = ). In this duct, a viscous homogeneous Newtonian fluid moves. The fluid has
mass density ρ and kinematic viscosity ν . Its flow is characterized by a small Mach number and the
rate Q per unit depth of the duct. In addition, the flow upstream of the first constriction (i.e., the
basic flow) is steady and laminar. It is necessary to study the flow around the constrictions.
The fluid motion in the duct is governed by the dimensionless momentum and continuity
equations, viz.
1
Re
i i i
j
j i j j
U U UP
U
T X X X X
⎛ ⎞∂ ∂ ∂∂ ∂
+ = − + ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠
, 1,2i = , (1)
Fig. 1. Geometry of the problem and the computational domain
50 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2022. № 1
A.O. Borysyuk
/ 0i iU X∂ ∂ = . (2)
The boundary conditions consist in the absence of a fluid motion at the channel wall, chS , and on
both constrictions, jS , ( 1, 2)j = . Also, the flow rate Q must be invariable along the duct axis, viz.
,
0
ch j
i S S
U = , 1/ 0Q X∂ ∂ = , 1Q = , , 1, 2i j = . (3)
Apart from these, the parabolic velocity profile is specified outside the disturbed flow region
due to the constrictions 1, viz.
1 1 12 2
2
1 2– ,
1.5(1 4 )
u dX L L L L L
U X
= + + +
= − ,
1 1 12 2
2 – ,
0
u dX L L L L L
U
= + + +
= . (4)
As for the pressure P , it is assumed to be constant both sufficiently far upstream of the
first constriction
1 –
( )
u
uX L
P P
=
= , and far downstream of the second one
1 1 12 2
( )
d
dX L L L L
P P
= + + +
= .
In addition, the corresponding pressure drop, 0u dP P PΔ = − > , should ensure the existence of
the given laminar regime of the basic flow. Also, without loss of generality, the pressure dP is
taken to be zero 2, and the magnitude uP , like the pressure in the whole duct, needs to be found.
Apart from these, the normal pressure derivative is zero on the rigid walls of the channel and
both its con strictions, viz.
,( / ) 0
ch jS SP n∂ ∂ = , 1, 2j = . (5)
Regarding the initial conditions, they are in the absence of a fluid motion in the channel at
the time instant 0T = [1], viz.
0
0
T
P
=
= ,
0
0i T
U
=
= . (6)
In (1)-(6), 1 2 3, ,X X X are the rectangular Cartesian coordinates shown in Fig. 1 (here,
the axis 3X is normal to the plane 1 2X X and directed to us); T the time; iU the local fluid
velocities in the directions iX ; Re /aU D= ν the Reynolds number of the cross-sectionally ave-
raged basic flow; aU its velocity; and the values of the distances uL and dL are given in the
next section. In addition, hereinafter, the vector n
denotes the outward unit normal to the
appro priate surface, and the summation over repeated indices is assumed throughout the paper.
As for the scaling factors used in (1)-(6), these are the duct width D as the length scale, the
velocity aU as the velocity scale, the product aU D as the flow rate scale, the ratio / aD U as the
time scale, and the product 2
aUρ as the pressure scale.
1 This is the region before the constrictions, where the flow is still undisturbed by them, and far behind
them, where the flow is already undisturbed (i.e., where the flow disturbances disappear, and it becomes like
the basic one).
2 A choice of the value of dP always can be compensated for by the choice of the corresponding value of uP
in such a way that the corresponding pressure drop PΔ (which governs fluid motion in the duct) remains
unchangeable.
51ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2022. № 1
A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct....
Computational domain. The domain, in which a solution to the problem should be found,
is shown in Fig. 1. It is restricted by the duct sections 1 uX L= − , 1 1 12 2 dX L L L L= + + + and
3 3aX X= , 3 3 3daX X X= + (where 3d 1X << , and 3aX is an arbitrary value of the coordinate
3X ). Herewith, the boundary 1 uX L= − is taken upstream of the first constriction, where the
flow is still undisturbed by it, and the boundary 1 1 12 2 dX L L L L= + + + behind the second con-
striction, where the flow disturbances already disappear, and the flow redevelops into the basic
state at 1 uX L= − . As for the distances uL and dL , for the basic flow velocities considered in
this study, their values should vary in the ranges 0.5uL and 12dL [1, 2].
The chosen computational domain is divided into the small volumes Vnm by the duct sec-
tions 1 1nX X= and 2 2mX X= (where 1 1( 1) 1dn nX X X−= + , 1d 1X << , and 2 2( 1) 2dm mX X X−= + ,
2d 1X << ), as shown in Fig. 2. Herewith, in order to have a smooth velocity profile in an arbit-
rary duct cross-section, the steps 1dX and 2dX are reduced in an appropriate manner as one
approaches either the duct or constrictions’ walls.
Integral equations and their discrete analogs. Integral analogs of Eqs. (1) and (2) are
obtained by their integrating over the control volumes Vnm (in making this operation, the
appropriate conservation laws take place in each volume Vnm ). It gives
V V V V
1
d d d d
Re
nm nm nm nm
i i
i j
j i j j
U UP
U V U V V V
T X X X X
⎛ ⎞∂ ∂∂ ∂ ∂
+ = − + ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠
∫∫∫ ∫∫∫ ∫∫∫ ∫∫∫ , (7)
V
( / )d 0
nm
i iU X V∂ ∂ =∫∫∫ . (8)
The application (wherever possible) of the Gauss theorem to the terms of Eqs. (7) and (8),
and/or the expansion (wherever needed) of their integrands (which are denoted by ( )f r
) in the
Fig. 2. A scheme of fragmentation of the computational domain into small
volumes
52 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2022. № 1
A.O. Borysyuk
Taylor series around the mass center Cnm of the volume3 Vnm (Fig. 2), further use of the first
two terms of these series (i.e., ( ) ( ) ( ) ( )
nm nm
cnm
c cr r
f r f r f r r
=
= +∇ ⋅ −
), making the discretization
of the temporal and spatial derivatives on the basis of the implicit three-point non-symmetric
backward differencing and two-point schemes [5], respectively, viz.
1 2( , ) 1.5 2 0.5
nm nm nm nm
k k k
c c c cf r T f f f
T T
− −∂ − +
=
∂ Δ
,
( )( )
( / )
jcj nmcnm
i i r rr r
f e f X ==
∇ = ∂ ∂
,
( 1)
(1)
1 1
( ) ( )
d
n m nm
cnm
c c
r r
f r f rf
X X
+
=
−∂
=
∂
, ( 1)
(2)
1 1
( ) ( )
d
nm n m
cnm
c c
r r
f r f rf
X X
−
=
−∂
=
∂
,
( 1)
(3)
2 2
( ) ( )
d
n m nm
cnm
c c
r r
f r f rf
X X
+
=
−∂
=
∂
, ( 1)
(4)
2 2
( ) ( )
d
nm n m
cnm
c c
r r
f r f rf
X X
−
=
−∂
=
∂
,
as well as the application (wherever necessary) of the following TVD-scheme [5]
( )
( ) ( ) ( )
1 2 1( ) ( )j
nm
j j j
c
f r f f f= +Φ −
,
( )
1
( ),
( ),
nm
j
cj
c
f r
f
f r
⎧
⎪= ⎨
⎪⎩
( )
( )
0,
0,
j
nm
j
nm
F
F <
1 ( 1)n mC C += , 2 ( 1)n mC C −= , 3 ( 1)n mC C += , 4 ( 1)n mC C −= ,
( )
2 ( ) (1 ) ( )
nm j
j
j c j cf f r f r= α + −α
, ( ) /j j nm jnm
j c c cc
r r r rα = − −
,
( ) max(0, min(4 ,1))j jΦ η = η , ( ) ( )( ) ( ) / ( ) ( )j jnm jnm nm
j c cc c
U r U r U r U rη = − −
,
allows one to proceed to considering the discrete analogs of Eqs. (7) and (8), viz.
( ) ( )
1 2 4 4
( ) ( )
1 1
1.5 2 0.5 1
V
Re
nm nm nm
j j
nm nm
k k k
ic ic ic j k k k j
nm nm j nmic ic
j j
U U U
F U U n S
T
− −
= =
− +
+ − ∇ ⋅ =
Δ ∑ ∑
( )
4
( )
1
( / ) V
j
nm
nm
k j
ji nmc
j
k
i C nm
P n S
P X
=
⎧
⎪⎪= −⎨
⎪ ∂ ∂⎪⎩
∑
, (9)
3 Since the fluid in the duct is homogeneous (see the problem formulation), the mass center of the volume Vnm
coincides with its geometrical center. The analogous situation is with the mass center of each side face of the
volume Vnm .
53ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2022. № 1
A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct....
( ) ( )
4 4 4
( ) ( ) ( )
1 1 1
0j j
nm nm
k j k j j k
j nm ji nm nmc ic
j j j
U n S U n S F
= = =
⋅ = = =∑ ∑ ∑
,
(10)
which have the second order of accuracy. Here ∇
is the
gradient; i ir X e=
and
nm nmc ic ir X e=
the position vectors of
an arbitrary point in the region Vnm and its mass cen ter
nmC , respecti vely; the point in the Taylor series in dicates
the scalar product of the corresponding magnitudes; ie
the unit directivity vector of the axis iX ; TΔ the small
time step;
nm
k
cf
and ( )j
nm
k
c
f
the values of the function f
at
the points nmC and ( )j
nmC at the time instant T k T= Δ ;
( )j
nmC the mass center of the side face ( )j
nmS of the volume
Vnm ; 1
nm
k
cf
−
and 2
nm
k
cf
−
the known values of the function f
at the point nmC at the moments ( 1)T k T= − Δ and ( 2)T k T= − Δ , respectively; Vnm the vol-
ume of the region Vnm; ( )j
nmS and j ji in n e=
the area and the outward unit normal to the face ( )j
nmS
( 1 1n e=
, 2 1n e= −
, 3 2n e=
, 4 2n e= −
, 5 3n e=
, 6 3n e= −
; Fig. 3); and ( )
( ) ( )
j
nm
j k k j
nm j nmc
F U n S= ⋅
the fluid flow
across the face ( )j
nmS at the moment T k T= Δ .
Discrete analogs of conditions (3)-(6) and their application to Eqs. (9) and (10). The
discrete analogs of conditions (3)-(5) on the boundary of the computational domain are as
follows:
1 1 12 2
2
1 2,
1.5(1 4 )
u d
k
X L L L L L
U X
=− + + +
= − ,
1 1 12 2
2 ,
0
u d
k
X L L L L L
U
=− + + +
= ,
,
0
ch j
k
i S S
U = , 1/ 0kQ X∂ ∂ = , 1kQ = , 0.5uL , 12dL , , 1, 2i j = ,
(1)
1 1 12 2
0
nm
d
k
c X L L L L
P
= + + +
= , 2( / ) 0
ch
k
SP X∂ ∂ = , ( / ) 0
j
k
SP n∂ ∂ = .
They allow one to find ( )j k
nmF and ( )j
nm
k
ic
U∇
on the noted boundary in Eqs. (9) and (10), viz.
1 2
( )
, ,
0
ch
j k
nm S S S
F = ,
1 1 12 2 1 1 12 2
(3) (4)
, ,
0
u d u d
k k
nm nmX L L L L L X L L L L L
F F
=− + + + =− + + +
= = ,
1
(2) 2
2 2 31.5(1 4 )d d
u
k
nm X L
F X X X
=−
= − − ,
1 1 12 2
(1) 2
2 2 31.5(1 4 )d d
d
k
nm X L L L L
F X X X
= + + +
= − ,
(3)
2
2 2
1/2
/ d
nmnm
k k
icic X
U e U X
=
∇ = −
, (4)
2
2 2
1/2
/ d
nmnm
k k
icic X
U e U X
=−
∇ =
,
1 0uL X− , 1 1 1 12L X L L+ , 1 12 2 1 1 12 2 dL L L X L L L L+ + + + + ,
Fig. 3. The small volume Vnm, its side faces
( )i
nmS and their outward unit normals in
( 1, ..., 6)i =
54 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2022. № 1
A.O. Borysyuk
(2)
1
2 21
12
nm
u
k
c X L
U X e
=−
∇ = −
, (1)
1 1 12 2
2 21
12
nm
d
k
c X L L L L
U X e
= + + +
∇ = −
,
(2)
1
2
0
nm
u
k
c X L
U
=−
∇ =
, (1)
1 1 12 2
2
0
nm
d
k
c X L L L L
U
= + + +
∇ =
,
(1)
0, /2 1/2, 1/2 1/2 /21 1 2 2 1
, /2 1/2, 1/2 1/2 /21 1 12 2 2 2 2
1 1/ d
nmnm
X D X X D
X L L D X X D
k k
icic
U e U X
= − − +
= + − − +
∇ = −
,
(2)
1 1 1 2 2 1
1 1 12 2 2 2 2 2
1 1, /2 1/2, 1/2 1/2 /2
, /2 1/2, 1/2 1/2 /2
/ d
nmnm
k k
icX L D X X Dic
X L L L D X X D
U e U X= − − +
= + + − − +
∇ =
,
(3)
1 1 2 1 1 12 1 1 12 2 2 2
2 2
0 , /2; , /2
/ d
nmnm
k k
icic X L X D L L X L L L X D
U e U X
= + + + =
∇ = −
,
(4)
1 1 2 1 1 12 1 1 12 2 2 2
2 2
0 , 1/2 /2; , 1/2 /2
/ d
nmnm
k k
icic X L X D L L X L L L X D
U e U X
=− + + + + =− +
∇ =
.
As for the discrete analogs of conditions (6) (i.e., 0 0k
iU = = , 0 0kP = = ), they give one the possi-
bility to compute the appropriate terms in (9) and (10) at the initial time instant in the compu-
tational domain, viz. ( ) 0 0j k
nmF = = , ( )
0 0j
nm
k
ic
U =∇ =
, 0( / ) 0
nm
k
i CP X =∂ ∂ = .
A method of solution to Eqs. (9) and (10). The system of equations (9), (10) is solved
numerically. In making this, one comes across the two significant problems. The first of them is
connected with a nonlinearity of Eq. (9), whereas the second one is due to the absence of an equa-
tion for the pressure which is available in the right part of (9).
In order to solve the first problem, the flow ( )j k
nmF is modified in the appropriate way. More
specifically, the velocity components in it are initially replaced by their values found at the
pre vious time step. After that, the components are replaced by their known previous appro-
xima tions. These replacements allow one to proceed from solving the coupled systems of non-
linear algebraic equations to the corresponding uncoupled linear ones.
The second problem is solved via introducing the pressure in Eq. (10) and the subsequent
agreeing of the velocity and the pressure with each other, when making the noted modification
of the flow ( )j k
nmF . The velocity and pressure values, which are obtained in making this, are cor-
rected at each step by performing the appropriate operations. Let us demonstrate the above-
said in more details.
If one formally solves (9) with respect to
nm
k
icU , one obtains the equation
( ) ( )
4
( )
0
1
1/ V ,
( / ) .
j
nm
nm nm nm nm
nm
k j
nm ji nmck k p
jic ic ic ic
k
i C
P n S
U A A A
P X
=
⎧
⎪⎪= + − ⎨
⎪ ∂ ∂⎪⎩
∑
(11)
55ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2022. № 1
A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct....
Here 0
nmicA is a rational function whose numerator contains the known values 1
nm
k
icU − and 2
nm
k
icU − .
Its denominator involves ( )j k
nmF . The term
nm
k
icA also is a rational function whose denominator
only differs from that of the function 0
nmicA in the multiplier V /nm TΔ . Its numerator has the
unknown quantities
j
k
icU and ( )
j
j k k
nm icF U . As for the fractional multiplier
nm
p
icA , its numerator
only consists of the time step TΔ , whereas the denominator coincides with that of the func-
tion 0
nmicA .
From relation (11), one can obtain an equation for ( )j
nm
k
ic
U . That equation, after substituting
into the expression for ( )j k
nmF and then the obtained relation into (10), yields an equation for
the pressure, viz.
( ) ( ) ( )( )
4 4
( ) 0 ( )
1 1
( / ) ( )j j jj
nm nm nmnm
p k j k j
i ji nm ji nmC ic icic
j j
A P X n S A A n S
= =
∂ ∂ = +∑ ∑ . (12)
Then we begin to solve (11), (12) with finding the first approximations of the velocities
*
nm
k
icU . For this purpose, the unknown quantities ( )j
nm
k
c
P in (11) are replaced with the known
ones ( )
1
j
nm
k
c
P − , and the functions ...
nmicA are modified by replacing the unknown velocities in the flow
( )j k
nmF with their known values 1
nm
k
icU − . This results in the following system of linear algebraic
equations for *
nm
k
icU :
( )
4
* 0 1 ( )
1
( / V ) jnm nm nm nm nm
k l kl pl k j
ic ic ic nm ji nmic c
j
U A A A P n S−
=
= + − ∑ (13)
in which ...
nm
l
icA are the modified functions ...
nmicA .
Once the quantities *
nm
k
icU are found from (13) (the method of solution of this system is
described below), they are further used to obtain the corresponding values of the operators
( )
...
j
nmic
A , which are then substituted into (12). This yields the system of linear algebraic equa-
tions for the first approximation of the pressure, viz.
( ) ( ) ( )( )
4 4
* * ( ) 0* * ( )
1 1
( / ) ( )j j jj
nm nm nmnm
p k j k j
i ji nm ji nmC ic icic
j j
A P X n S A A n S
= =
∂ ∂ = +∑ ∑ (14)
in which ( )
...
j
nmic
A denote the values of the operators ( )
...
j
nmic
A found with the use of *
nm
k
icU .
Further, one applies a procedure which is similar to the just described one. More speci-
fically, the first approximations of the pressure, ( )
*
j
nm
k
c
P , found from (14) are substituted into (11)
in stead of ( )j
nm
k
c
P . Also, in the functions ...
nmicA in (11), the flow ( )j k
nmF is modified by replacing the
unknown velocities in it with their first approximations obtained from (13). This results in
the systems of linear algebraic equations for the second approximations (or the first correc-
tions) of the velocities **
nm
k
icU , viz.
( )
4
** 0 * * * * ( )
1
( / V ) jnm nm nm nm nm
k l kl pl k j
ic ic ic nm ji nmic c
j
U A A A P n S
=
= + − ∑ (15)
56 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2022. № 1
A.O. Borysyuk
(here ... *
nm
l
icA are the functions ...
nmicA in which the just noted flow modification has been performed).
After that, the second approximations of the velocities, obtained from (15), are used to
obtain the values **
( )
...
j
nmic
A of the operators ( )
...
j
nmic
A . The subsequent substitution of these values into
(12) instead of ( )
...
j
nmic
A allows one to write a system of equations for the second approximation
(or the first correction) of the pressure, ( )
**
j
nm
k
c
P , which is similar to (14) viz.
( ) ( ) ( )( )
4 4
** ** ( ) 0** ** ( )
1 1
( / ) ( )j j jj
nm nm nmnm
p k j k j
i ji nm ji nmC ic icic
j j
A P X n S A A n S
= =
∂ ∂ = +∑ ∑ . (16)
If the accuracy of the second approximations of the velocities and the pressure is not satisfac-
tory, then the just-described procedure must be carried out until the accuracy becomes as desired.
Solution of Eqs. (13)-(16). The systems of linear algebraic equations (SLAEs) (13)-(16)
can be rewritten in the generalized form, with the unknown quantities
nm
k
cξ and
i
k
cξ
4
1
nm nm i i nm
k k k k k
c c c c c
i
a a b
=
ξ + ξ =∑ . (17)
Such systems are solved either by direct or iterative methods. Usually, the direct methods are
applied to small systems of equations and give good results. However, when one deals with big
SLAEs (especially with systems whose matrices are rarified), the direct methods need a huge
amount of time to obtain their solutions. Therefore, their application is unreasonable here. The
iterative methods, when applied to big SLAEs, need much less computational memory and
time, save the rarefaction degree of their matrices (when the matrices are rarified) and give sa-
tisfactory results.
Proceed from the just-said, as well as from the dimension and the rarefaction degree of the
matrix of system (17), an iterative method is chosen in this paper to solve the system. Within
its framework, an initial approximation of the solution is chosen initially, which is then improved
by making iterations until its accuracy reaches the desired value. Herewith, the attention is paid
to the following two features. The first of them concerns with the necessity of providing do-
mi nation of the diagonal terms in the matrix of system (17). In this study, it is realized by ap-
plying the deferred correction implementation method [5] to the convective term. In accordance
with this method, the part of the convective term, which corresponds to the scheme for ( )
1
jf
(it is
given before (9)), is inserted into the matrix, whereas its remainder is placed into the right part
of SLAE (17).
The second feature is related to a desire to have the as minimal as possible number of itera-
tions. Here, it is made by the use of the method of conjugate gradients [5]. This method allows one
to solve a SLAE via the iterations’ whose number does not exceed the number of its unknown
values. Herewith, if a successful choice of the initial approximation is made, the number of ite-
rations sharply decreases. Also, the preconditioning results in a significant reduction of the num-
ber of iterations. For this purpose, the solvers ICCG and Bi-CGSTAB [5] are used.
Conclusions
1. A second-order numerical technique has been developed to study the steady laminar fluid
motion in a straight flat hard-walled duct with two axisymmetric rectangular constrictions.
57ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2022. № 1
A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct....
2. In this technique, the governing relations are solved via deriving their integral analogs,
performing a discretization of these analogs, simplifying the obtained (after making the discre-
ti zation) coupled nonlinear algebraic equations, and the final solution of the resulting (after mak-
ing the simplification) uncoupled linear ones.
3. The discretization consists of the spatial and temporal parts. The first of them is perfor-
med with the use of the TVD-scheme and the two-point scheme of discretization of the spatial
derivatives, whereas the second one is made on the basis of the implicit three-point asymmetric
backward differencing scheme.
4. The above-noted uncoupled linear algebraic equations are solved by an appropriate ite-
rative method, which uses the deferred correction implementation technique and the technique
of conjugate gradients, as well as the solvers ICCG and Bi-CGSTAB.
REFERENCES
1. Borysyuk, A. O. (2019). Flow modelling in a straight rigid-walled duct with two rectangular axisymmetric
narrowings. Part 1. A theory. Bulletin of V.N. Karazin Kharkiv National University, series “Mathematical
Modeling. Information Technology. Automated Control Systems”, 44, pp. 4-15.
https://doi.org/10.26565/2304-6201-2019-44-01
2. Borysyuk, A. O. & Borysyuk, Ya. A. (2017). Wall pressure fluctuations behind a pipe narrowing of various
shapes. Science-Based Technologies, 32, No. 2, pp. 162-170. https://doi.org/10.18372/2310-5461.34.11615
3. Brujatckij, E. V., Kostin, A. G. & Nikiforovich, E. V. (2011). Numerical investigation of the velocity and pres-
sure fields in a flat channel having a square-shape obstacle on the wall. Appl. Hydromech., 13, No. 3, pp. 33-47
(in Russian).
4. Young, D. F. (1979). Fluid mechanics of arterial stenosis. J. Biomech. Eng., 101, No. 3, pp. 157-175. https://
doi.org/10.1115/1.3426241
5. Ferziger, J. H. & Peri´c, M. (2002). Computational methods for fluid dynamics, 3rd ed. Berlin: Springer.
Received 22.11.2021
А. О. Борисюк, https://orcid.org/0000-0002-3878-3915
Інститут гідромеханіки НАН України, Київ
E-mail: aobor@ukr.net
ЧИСЕЛЬНИЙ МЕТОД РОЗВ’ЯЗУВАННЯ ЗАДАЧІ ПРО РУХ РІДИНИ
У ПРЯМОМУ ПЛОСКОМУ ЖОРСТКОМУ КАНАЛІ З ДВОМА
ОСЕСИМЕТРИЧНИМИ ПРЯМОКУТНИМИ ЗВУЖЕННЯМИ
Розроблено чисельний метод розв’язування задачі про стаціонарний ламінарний рух рідини у прямому
плоскому жорсткому каналі з двома осесиметричними прямокутними звуженнями. Цей метод має другий
порядок точності. У ньому співвідношення, що описують зазначений рух, розв’язуються шляхом одержан-
ня їхніх інтегральних аналогів, дискретизації цих аналогів, зведення зв’язаних нелінійних алгебраїчних
рівнянь (одержаних внаслідок дискретизації) до відповідних незалежних лінійних і подальшого
розв’язування останніх. Зазначена дискретизація складається із просторової та часової частин. Перша з
них виконується на основі використання TVD-схеми, а також двоточкової схеми дискретизації просто-
рових похідних. При проведенні ж другої частини дискретизації застосовується неявна триточкова не-
симетрична схема з різницями назад. Що стосується методу розв’язування вказаних незалежних ліній-
них рівнянь, то це — відповідний ітераційний метод, який використовує методи відкладеної корекції та
спряжених градієнтів, а також солвери ICCG та Bi-CGSTAB.
Ключові слова: рух рідини, плоский канал, прямокутне звуження, метод.
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