A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions. An alternative approach
A numerical technique is devised to solve a problem of the steady laminar fluid motion in a straight plane hard-walled duct with two local axisymmetric rectangular constrictions. It uses the stream function, the vorticity and the pressure as the basic variables, has the second order of accuracy in...
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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. An alternative approach / A.O. Borysyuk // Доповіді Національної академії наук України. — 2022. — № 4. — С. 55-65. — Бібліогр.: 5 назв. — англ. |
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Borysyuk, A.O. 2022-12-08T19:14:01Z 2022-12-08T19:14:01Z 2022 A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions. An alternative approach / A.O. Borysyuk // Доповіді Національної академії наук України. — 2022. — № 4. — С. 55-65. — Бібліогр.: 5 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2022.04.055 https://nasplib.isofts.kiev.ua/handle/123456789/187174 532.542 A numerical technique is devised to solve a problem of the steady laminar fluid motion in a straight plane hard-walled duct with two local axisymmetric rectangular constrictions. It uses the stream function, the vorticity and the pressure as the basic variables, has the second order of accuracy in the spatial and the first order of accuracy in the temporal coordinates, provides high stability of a solution, and needs significantly less computational time to obtain a result compared to appropriate techniques available in a scientific literature. The technique consists in: a) introducing the stream function and the vorticity, and subsequent transiting from the non-dimensional governing relations for the fluid velocity and the pressure to the corresponding non-dimensional relations for the stream function, the vorticity and the pressure; b) deriving their discrete counterparts in the nodes of the chosen space-time computational grid; c) integrating the systems of linear algebraic equations obtained after making the discretization. The discretization is based on applying appropriate differencing schemes to the terms of the equations for the basic variables. These are the two-point temporal onward difference for the unsteady term of the vorticity equation, as well as the two-point backward differences (for its convective term) and the five-point approximations (for its diffusive term and for the Poisson’s equations for the stream function and the pressure) in the axial and cross-flow coordinates. As for the velocity components, the appropriate central differences are applied to discretize their expressions. The above-mentioned systems of linear algebraic equations for the stream function and the pressure are integrated by the iterative successive over-relaxation method. The algebraic relation for the vorticity does not need application of any method to be solved, because it is a computational scheme to find this quantity based on the known magnitudes computed at the previous instant of time. Розроблено чисельний метод розв’язування задачі про стаціонарний ламінарний рух рідини у прямому плоскому жорсткому каналі з двома локальними осесиметричними прямокутними звуженнями. У цьому методі як основні змінні використовуються функція течії, завихореність і тиск. Він має другий порядок точності по координатах і перший порядок точності по часу, забезпечує високу стійкість розв’язку і потребує значно менше комп’ютерного часу для одержання результату порівняно з відповідними методами, описаними в науковій літературі. За цим методом сформульована задача розв’язується шляхом: а) введення функції течії і завихореності та подальшого переходу від безрозмірних співвідношень для швидкості і тиску до відповідних безрозмірних співвідношень для функції течії, завихореності і тиску; б) виведення дискретних аналогів цих співвідношень у вузлах вибраної просторово-часової решітки; в) інтегрування систем лінійних алгебраїчних рівнянь, одержаних внаслідок проведення зазначеної дискретизації. Дискретизація ґрунтується на застосуванні відповідних різницевих схем до членів рівнянь для введених змінних. Це — одностороння різниця вперед для нестаціонарного члена рівняння переносу завихореності, а також односторонні різниці проти потоку (для конвективного члена цього рівняння) та п’ятиточкові шаблони (для його дифузійного члена та рівнянь Пуассона для функції течії і тиску) по осьовій та поперечній координатах. Що стосується компонент швидкості, то для дискретизації їхніх виразів застосовуються відповідні центральні різниці. Зазначені вище системи лінійних алгебраїчних рівнянь для функції течії і тиску інтегруються ітераційним методом послідовної верхньої релаксації. Алгебраїчне ж співвідношення для завихореності не потребує застосування ніякого методу розв’язування, оскільки вже є розрахунковою схемою для визначення цієї величини на основі відомих величин, знайдених у попередній момент часу. 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. An alternative approach Чисельний метод розв’язування задачі про рух рідини у прямому плоскому жорсткому каналі з двома осесиметричними прямокутними звуженнями. альтернативний підхід 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. An alternative approach |
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A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions. An alternative approach 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. An alternative approach |
| title_full |
A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions. An alternative approach |
| title_fullStr |
A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions. An alternative approach |
| 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. An alternative approach |
| title_sort |
numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions. an alternative approach |
| 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. альтернативний підхід |
| description |
A numerical technique is devised to solve a problem of the steady laminar fluid motion in a straight plane hard-walled
duct with two local axisymmetric rectangular constrictions. It uses the stream function, the vorticity and the pressure
as the basic variables, has the second order of accuracy in the spatial and the first order of accuracy in the temporal
coordinates, provides high stability of a solution, and needs significantly less computational time to obtain a result
compared to appropriate techniques available in a scientific literature. The technique consists in: a) introducing the
stream function and the vorticity, and subsequent transiting from the non-dimensional governing relations for the fluid
velocity and the pressure to the corresponding non-dimensional relations for the stream function, the vorticity and the
pressure; b) deriving their discrete counterparts in the nodes of the chosen space-time computational grid; c) integrating
the systems of linear algebraic equations obtained after making the discretization. The discretization is based on applying
appropriate differencing schemes to the terms of the equations for the basic variables. These are the two-point temporal
onward difference for the unsteady term of the vorticity equation, as well as the two-point backward differences (for its
convective term) and the five-point approximations (for its diffusive term and for the Poisson’s equations for the stream
function and the pressure) in the axial and cross-flow coordinates. As for the velocity components, the appropriate
central differences are applied to discretize their expressions. The above-mentioned systems of linear algebraic equations
for the stream function and the pressure are integrated by the iterative successive over-relaxation method. The algebraic
relation for the vorticity does not need application of any method to be solved, because it is a computational scheme to
find this quantity based on the known magnitudes computed at the previous instant of time.
Розроблено чисельний метод розв’язування задачі про стаціонарний ламінарний рух рідини у прямому
плоскому жорсткому каналі з двома локальними осесиметричними прямокутними звуженнями. У цьому
методі як основні змінні використовуються функція течії, завихореність і тиск. Він має другий порядок
точності по координатах і перший порядок точності по часу, забезпечує високу стійкість розв’язку і потребує значно менше комп’ютерного часу для одержання результату порівняно з відповідними методами,
описаними в науковій літературі. За цим методом сформульована задача розв’язується шляхом: а) введення функції течії і завихореності та подальшого переходу від безрозмірних співвідношень для швидкості і
тиску до відповідних безрозмірних співвідношень для функції течії, завихореності і тиску; б) виведення
дискретних аналогів цих співвідношень у вузлах вибраної просторово-часової решітки; в) інтегрування
систем лінійних алгебраїчних рівнянь, одержаних внаслідок проведення зазначеної дискретизації. Дискретизація ґрунтується на застосуванні відповідних різницевих схем до членів рівнянь для введених змінних. Це — одностороння різниця вперед для нестаціонарного члена рівняння переносу завихореності, а також односторонні різниці проти потоку (для конвективного члена цього рівняння) та п’ятиточкові шаблони (для його дифузійного члена та рівнянь Пуассона для функції течії і тиску) по осьовій та поперечній
координатах. Що стосується компонент швидкості, то для дискретизації їхніх виразів застосовуються відповідні центральні різниці. Зазначені вище системи лінійних алгебраїчних рівнянь для функції течії і тиску інтегруються ітераційним методом послідовної верхньої релаксації. Алгебраїчне ж співвідношення для завихореності не потребує застосування ніякого методу розв’язування, оскільки вже є розрахунковою схемою для визначення цієї величини на основі відомих величин, знайдених у попередній момент часу.
|
| issn |
1025-6415 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/187174 |
| citation_txt |
A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions. An alternative approach / A.O. Borysyuk // Доповіді Національної академії наук України. — 2022. — № 4. — С. 55-65. — Бібліогр.: 5 назв. — англ. |
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55ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2022. № 4: 55—65
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. An alternative approach. Допов. Нац. акад. наук
Укр. 2022. № 4. С. 55—65. https://doi.org/10.15407/dopovidi2022.04.055
https://doi.org/10.15407/dopovidi2022.04.055
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.
An alternative approach
A numerical technique is devised to solve a problem of the steady laminar fluid motion in a straight plane hard-walled
duct with two local axisymmetric rectangular constrictions. It uses the stream function, the vorticity and the pressure
as the basic variables, has the second order of accuracy in the spatial and the first order of accuracy in the temporal
coordinates, provides high stability of a solution, and needs significantly less computational time to obtain a result
compared to appropriate techniques available in a scientific literature. The technique consists in: a) introducing the
stream function and the vorticity, and subsequent transiting from the non-dimensional governing relations for the fluid
velocity and the pressure to the corresponding non-dimensional relations for the stream function, the vorticity and the
pressure; b) deriving their discrete counterparts in the nodes of the chosen space-time computational grid; c) integrating
the systems of linear algebraic equations obtained after making the discretization. The discretization is based on applying
appropriate differencing schemes to the terms of the equations for the basic variables. These are the two-point temporal
onward difference for the unsteady term of the vorticity equation, as well as the two-point backward differences (for its
convective term) and the five-point approximations (for its diffusive term and for the Poisson’s equations for the stream
function and the pressure) in the axial and cross-flow coordinates. As for the velocity components, the appropriate
central differences are applied to discretize their expressions. The above-mentioned systems of linear algebraic equations
for the stream function and the pressure are integrated by the iterative successive over-relaxation method. The algebraic
relation for the vorticity does not need application of any method to be solved, because it is a computational scheme to
find this quantity based on the known magnitudes computed at the previous instant of time.
Keywords: fluid motion, plane duct, rectangular constriction, technique.
МЕХАНІКА
MECHANICS
Study of fluid motions in ducts is an actual problem in medicine, architecture, municipal economy,
etc. Among others, here a significant interest is shown for investigating flows in ducts with local
constrictions, such as wall deposits, welding joints, stenoses, etc. That is due to the fact that such
inhomogeneities in the duct geometry cause local changes in the flow structure and/or character,
etc., and those changes can result in the corresponding consequences not only in the vicinity of,
but also far from the inhomogeneities [1].
56 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2022. № 4
A.O. Borysyuk
As analysis of the scientific literature shows, study of flows in ducts with local constrictions
has been paid much attention to. In those studies, straight rigid ducts and their constrictions of
the simplest geometries were of interest. The basic flow (i. e., the flow upstream of a (first) con-
striction) was laminar, axisymmetric and steady. As for fluids, they were homogeneous, incom-
pressible and Newtonian (the other types of ducts, their constrictions, fluids 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 investigate (within the framework of ap-
propriate models chosen and with acceptable accuracy) the influence of the basic parameters of a
duct, its constriction(s) and the basic flow on the flow not only near but also far downstream of
the constriction(s), and, on the other hand, to simplify significantly solutions to the correspon-
ding problems of interest [1-4].
Among the results obtained in those investigations, numerical techniques, which allow study-
ing the fluid motion near duct constrictions, are of a great importance. Recently, a technique of
such kind was described in [1]. It has been developed to investigate the flow in a straight flat rigid
duct having two rectangular axisymmetric constrictions. That technique has a second order of ac-
curacy both in the temporal and spatial co-ordinates, and allows making appropriate investigation
in the variables velocity-pressure. However, due to the huge amount of mathematical operations,
it needs a lot of computational time to obtain a result.
In this work, an alternative technique is described to solve the same problem. It uses the
stream function, the vorticity and the pressure as the basic variables, and has nearly the same
order of accuracy as that noted above. However, this technique has higher stability of a solution
and, due to the use of less powerful mathematical apparatus, needs much less computational time
to obtain a result.
Statement of the problem. A detailed statement of the initial and boundary problem to be
solved, is available in [1]. Proceed from this, now we shall only briefly remind it.
A straight flat rigid duct of dimensionless width 1 is considered (Fig.1) in which fluid moves.
In this duct, two narrowed axisymmetric segments of a rectangular form are available. They have
the cross-flow, iD , and axial, iL , dimensions ( 1, 2i = ), respectively, and are separated by the dis-
tance 12L from one another. The fluid is viscous Newtonian and homogeneous, and has mass den-
sity ρ and kinematic viscosity ν . Its flow is characterised by a small Mach number and the rate
Q per unit depth of the duct. Also, the flow upstream of the first narrowed segment (i. e., the basic
flow) is steady and laminar. It is necessary to investigate the fluid motion around the segments.
The fluid motion 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)
/ 0i iU X∂ ∂ = . (2)
The boundary conditions are in the absence of a fluid motion at the duct wall, chS , and on both
its narrowed segments, jS ( 1, 2j = ). Also, since the mass conservation in the duct is assumed, the
flow rate Q must not vary along the duct, viz.
,
0
ch j
i S S
U = , 1/ 0Q X∂ ∂ = , 1Q = , , 1, 2i j = . (3)
57ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2022. № 4
A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct...
Apart from these, due to the laminar basic flow regime, the parabolic velocity profile is speci-
fied outside the disturbed flow region due to the narrowed segments1, 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
narrowing (
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 ad-
dition, the difference 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 can be taken to be zero2, and the
magnitude uP , like the pressure in the whole duct, should be found. Apart from these, the normal
pressure derivative is zero on the rigid walls of the duct and both its narrowed segments, viz.
,( / ) 0
ch jS SP n∂ ∂ = , 1, 2j = . (5)
Regarding the initial conditions, they are in absence of the fluid motion in the duct at the
time instant 0T = [1], viz.
0
0
T
P = = ,
0
0i T
U = = . (6)
In (1)-(6), iX ( 1, 2, 3i = ) are the rectangular Cartesian coordinates depicted in Fig. 1 (here
the axis is normal to the plane 1 2X X and directed to us); T the time; iU the local fluid veloc-
ities in the directions iX ; Re /aU D= ν the Reynolds number of the cross-sectionally averaged
basic flow; aU its velocity; and the values of the distances uL and dL are given in section “Com-
putational domain and computational mesh”. In addition, hereinafter the vector n
denotes the
outward unit normal to appropriate surface, and the summation on repeated indices is assumed
throughout the paper. As for the scaling factors used in this paper, these are the duct width D as
the length scale, the velocity aU as the velocity scale, the product aU D as the scale for both the
flow rate and the stream function, the ratios / aD U and /aU D as the time scale and the vorti-
city scale, respectively, and the product 2
aUρ as the pressure scale (the stream function and the
vorticity are introduced in the next section).
1 This is the region before the segments, 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 Pd always can be compensated for by the choice of the corresponding value of Pu in
such a way that the corresponding pressure drop ΔP (which governs fluid motion in the duct) remains un-
changeable.
0
Lu L1 L12 L2 Ld
X1
lD1
X2
U1
Q
D2
Fig. 1. Geometry of the
problem and the computa-
tional domain
58 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2022. № 4
A.O. Borysyuk
Transition to the variables stream function-vorticity-pressure. We begin to solve the prob-
lem formulated in the previous section with introducing the stream function Ψ (which satisfies
(2)) and the vorticity Ω , viz.
1
2
U
X
∂Ψ=
∂
, 2
1
U
X
∂Ψ= −
∂
, 2 1
1 2
U U
X X
∂ ∂
Ω = −
∂ ∂
. (7)
This allows one to rewrite the initial and boundary problem (1)-(6) in the variables Ψ , Ω and
P instead of iU , P. More specifically, making the differentiation of the equations for 1U and 2U
in (1) with respect to 2X and 1X , respectively, subsequent taking the difference of the second
and the first of the obtained relationships and using the third relation in (7) yields the vorticity
transfer equation, viz.
1 2
2
1 2 ( , )
1 2
1
Re X XU U
T X X
∂Ω ∂Ω ∂Ω+ + = ∇ Ω
∂ ∂ ∂
, (8)
in which
1 2
2 2
2
( , ) 2 2
1 2
X X
X X
∂ ∂∇ = +
∂ ∂
is the Laplace operator in the plane 1 2,X X .
Further use of the first two representations in (7) in the third one gives one the Poisson’s
equation for the stream function, viz.
1 2
2
( , ) –X X∇ Ψ = Ω . (9)
Taking the derivatives of the equations for 1U and 2U in (1) with respect to 1X and 2X , re-
spectively, making the summation of the obtained relations and the use of the continuity equation
(2) results in the Poisson’s equation for the pressure, viz.
1 2
2 2
2 1 1 2 2
( , )
1 2 1 2
2X X
U U U U
P
X X X X
⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂
∇ = − − −⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠
. (10)
Regarding the boundary and initial conditions for the quantities Ψ , Ω and P , they are easily
derived from (3)-(6). Indeed, based on (4) and (7) one can obtain the following conditions for Ψ
and Ω on the upper, 1 – uX L= , and lower, 1 1 12 2 dX L L L L= + + + (in the flow direction) bounda-
ries of the disturbed flow region due to the narrowed segments, viz.
1 1 12 2
2
2 2– ,
3 4
1
2 3u dX L L L L L
X X= + + +
⎛ ⎞Ψ = −⎜ ⎟⎝ ⎠
,
1 1 12 2
2– ,
12
u dX L L L L L
X= + + +Ω = . (11)
A zero normal component of the fluid velocity on the walls of the duct and both its narrowed
segments (see (3)) together with the first two representations in (7) give one the corresponding
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A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct...
constancy of the stream function, viz.
,
const
ch jS S+ + +Ψ = , Ψ =– – –,
const
ch jS S
, 1, 2j =
(here the index “+” indicates the upper walls of the duct and the segments, whereas “–” their
lower walls). This constancy together with the first relation in (11) yields
,
1
2ch jS S+ +Ψ = , Ψ =– –,
1
–
2ch jS S
, 1, 2j = . (12)
The absence of the tangential fluid motion on chS and jS (see (3)) results in absence of the
first-order normal derivatives and the second-order mixed derivative of the function Ψ there, viz.
h2 ,
0
ch jS S
X
∂Ψ =
∂
,
v1
0
jS
X
∂Ψ =
∂
,
2
1 2 ,
0
ch jS S
X X
∂ Ψ =
∂ ∂
, 1, 2j = (13)
(in (13), h
jS and v
jS are the horizontal and vertical parts of jS , respectively).
Relations (13) together with (9) provide the following boundary conditions for the vorticity
h
h
2
2,
2 ,
ch j
ch j
S S
S S
X
∂ ΨΩ = −
∂
, v
v
2
2
1
j
j
S
S
X
∂ ΨΩ = −
∂
, 1, 2j = . (14)
As for boundary conditions for the pressure, their set (5) is completed with the following
two more ones
1 2 vv
2 1
( , ) 1
1
1
Re
jj
X X
SS
UP
U
X T
∂∂ ⎡ ⎤= ∇ −⎢ ⎥∂ ∂⎣ ⎦
,
1 2 hh
2 2
( , ) 2
2 ,,
1
Re
ch jch j
X X
S SS S
UP
U
X T
∂∂ ⎡ ⎤= ∇ −⎢ ⎥∂ ∂⎣ ⎦
, (15)
which are obtained from (1) and the first condition in (3).
Finally, the initial conditions for Ψ , Ω and P are derived from (6) after the use of (7)
there, viz.
0
0
T =Ψ = ,
0
0
T =Ω = , = =
0
0
T
P . (16)
Computational domain and computational mesh. The domain, in which a solution to the prob-
lem should be found, is bounded by the duct sections 1 – uX L= and 1 1 12 2 dX L L L L= + + + (see
Fig. 1). The first of them is chosen before the first narrowed segment, where the flow is still undis-
turbed by it, whereas the second section downstream of the second segment, where the flow redevel-
ops into the basic state at 1 – uX L= . Herewith, for the basic flow velocities considered in this study,
the values of uL and dL should not exceed 0.5 and 12, respectively, viz. 0.5uL and 12dL [1, 2].
In this domain, one introduces a uniform rectangular computational mesh with the small
steps iΔ in the directions iX ( 1, 2i = , Fig. 2), viz.
1 1( –1) 1n nX X= +Δ , 2 2( –1) 2m mX X= +Δ , const 1i iΔ = << . (17)
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A.O. Borysyuk
Regarding the integration time, it is divided into the
small constant intervals TΔ , viz.
–1k k T TT T k= +Δ = Δ , const 1T TΔ = << , 0 0T = . (18)
After that, one performs a discretization of the appro-
priate relationships of the previous subsection at the nodes
1 2, ,n m kX X T of the grid (17), (18). In making this, one
denotes the values of an arbitrary quantity f at the point
1 2, ,n m kX X T by ,
k
n mf , viz.
1 1 2 2
, 1 2 , ,
( , , )
n m k
k
n m X X X X T T
f f X X T = = == .
Discrete forms of the equations for Ψ , Ω and P . Discrete forms of the first two relations
in (7) are resulted from the use of the appropriate central differences [5], viz.
, 1 , –1
1 ,
2
( )
2
k k
n m n mk
n mU +Ψ −Ψ
=
Δ
, 1, –1,
2 ,
1
( )
2
k k
n m n mk
n mU +Ψ −Ψ
= −
Δ
. (19)
Expressions (19) are of the second order of accuracy.
A discrete analog of the third relation in (7) is not given here, since it is not used in this article.
A discretization of the vorticity transfer equation (8) is performed on the basis of the ap-
propriate differencing schemes. More specifically, the non-steady term in (8) is discretized on the
basis of the two-point temporal onward difference, viz.
1
, ,
,
k kk
n m n m
Tn mT
+Ω −Ω∂Ω⎛ ⎞ =⎜ ⎟∂ Δ⎝ ⎠
. (20)
Representation (20) has the first order of accuracy.
The application of the two-point backward difference schemes in the coordinates 1X and
2X to the convective term of Eq. (8) yields its discrete counterpart of the second order of ac-
curacy, viz.
, –1,
1 , 1 ,
1
1
1 , 1, ,
1 , 1 ,
1
( ) ; ( ) 0,
( ) ; ( ) 0,
k k
n m n mk k
k n m n m
k k
n m n m n mk k
n m n m
U U
U
X
U U+
⎧ Ω −Ω
⎪
Δ⎛ ⎞∂Ω ⎪=⎨⎜ ⎟∂ Ω −Ω⎝ ⎠ ⎪
<⎪ Δ⎩
, , –1
2 , 2 ,
2
2
2 , , 1 ,
2 , 2 ,
2
( ) ; ( ) 0,
( ) ; ( ) 0.
k k
n m n mk k
k n m n m
k k
n m n m n mk k
n m n m
U U
U
X
U U+
⎧ Ω −Ω
⎪
Δ⎛ ⎞∂Ω ⎪=⎨⎜ ⎟∂ Ω −Ω⎝ ⎠ ⎪
<⎪ Δ⎩
(21)
X2(m+2)
X2(m+1)
X2m
X2(m–1)
X1(n–1) X1n X1(n+1) X1(n+2)
�2
�1
Fig. 2. Computational mesh
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A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct...
And the last, the diffusive term of Eq. (8) has the following discrete form
2
1, , –1,
2 2
1 1,
2
k k k k
n m n m n m
n m
X
+⎛ ⎞ Ω − Ω +Ω∂ Ω =⎜ ⎟⎜ ⎟∂ Δ⎝ ⎠
, +⎛ ⎞ Ω − Ω +Ω∂ Ω =⎜ ⎟⎜ ⎟∂ Δ⎝ ⎠
2
, 1 , , –1
2 2
2 2,
2
k k k k
n m n m n m
n m
X
. (22)
Expressions (22) are of the second order of accuracy and derived on the basis of the five-point
differencing scheme (see Fig. 2).
Relations (20)-(22) together with (19) give one a discrete analog of Eq. (8), viz.
1
, , , –1, –1, 1, 1, , –1 , –1 , 1 , 1
k k k k k k k k k k k
n m n m n m n m n m n m n m n m n m n m n mC C C C C+
+ + + +Ω = Ω + Ω + Ω + Ω + Ω . (23)
Here the factors ...
...C are written as
12 , , 2 , , 1 1 2 1 , 2 ,
12 , , 2 , , 1 1 2 1 , 2 ,
,
12 , , 2 , , 1 1 2 1 , 2 ,
12 , , 2
1 (( ) ( ) ) 2 2 ; ( ) 0, ( ) 0;
1 (( ) ( ) ) 2 2 ; ( ) 0, ( ) 0;
1 (( ) ( ) ) 2 2 ; ( ) 0, ( ) 0;
1 (( )
k k k k
n m n m n m n m
k k k k
n m n m n m n mk
n m k k k k
n m n m n m n m
k
n m
U U
U U
C
U U
−α ΔΨ − ΔΨ − α − α
−α ΔΨ + ΔΨ − α − α <
=
+α ΔΨ + ΔΨ − α − α <
+α ΔΨ −
, , 1 1 2 1 , 2 ,( ) ) 2 2 ; ( ) 0, ( ) 0;k k k
n m n m n mU U
⎧
⎪
⎪⎪
⎨
⎪
⎪
ΔΨ − α − α < <⎪⎩
12 , , 2 1 1 , 2 ,
12 , , 2 1 1 , 2 ,
–1,
1 1 , 2 ,
1 1 , 2 ,
( ) ; ( ) 0, ( ) 0;
( ) ; ( ) 0, ( ) 0;
; ( ) 0, ( ) 0;
; ( ) 0, ( ) 0;
k k k
n m n m n m
k k k
n m n m n mk
n m k k
n m n m
k k
n m n m
U U
U U
C
U U
U U
⎧α ΔΨ +α
⎪
⎪α ΔΨ +α <⎪= ⎨
α <⎪
⎪
α < <⎪⎩
1 1 , 2 ,
1 1 , 2 ,
1,
12 , , 2 1 1 , 2 ,
12 , , 2 1 1 , 2 ,
; ( ) 0, ( ) 0;
; ( ) 0, ( ) 0;
( ) ; ( ) 0, ( ) 0;
( ) ; ( ) 0, ( ) 0;
k k
n m n m
k k
n m n mk
n m k k k
n m n m n m
k k k
n m n m n m
U U
U U
C
U U
U U
+
⎧ α
⎪
⎪ α <⎪= ⎨
−α ΔΨ +α <⎪
⎪
−α ΔΨ +α < <⎪⎩
12 , , 1 2 1 , 2 ,
2 1 , 2 ,
, –1
12 , , 1 2 1 , 2 ,
2 1 , 2 ,
( ) ; ( ) 0, ( ) 0;
; ( ) 0, ( ) 0;
( ) ; ( ) 0, ( ) 0;
; ( ) 0, ( ) 0;
k k k
n m n m n m
k k
n m n mk
n m k k k
n m n m n m
k k
n m n m
U U
U U
C
U U
U U
⎧−α ΔΨ +α
⎪
⎪ α <⎪= ⎨
−α ΔΨ +α <⎪
⎪
α < <⎪⎩
2 1 , 2 ,
12 , , 1 2 1 , 2 ,
, 1
2 1 , 2 ,
12 , , 1 2 1 , 2 ,
; ( ) 0, ( ) 0;
( ) ; ( ) 0, ( ) 0;
; ( ) 0, ( ) 0;
( ) ; ( ) 0, ( ) 0;
k k
n m n m
k k k
n m n m n mk
n m k k
n m n m
k k k
n m n m n m
U U
U U
C
U U
U U
+
⎧ α
⎪
⎪α ΔΨ +α <⎪= ⎨
α <⎪
⎪
α ΔΨ +α < <⎪⎩
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A.O. Borysyuk
in which the quantities 1α , 2α , 12α are the ratios of the steps of the computational grid (17),
(18), viz.
1 2
1
1
Re
TΔ
α =
Δ
, 2 2
2
1
Re
TΔ
α =
Δ
, 12
1 22
TΔ
α =
Δ Δ
,
and , ,( )k
n m iΔΨ the increase of the stream function in the direction iX of the grid ( 1, 2i = ), viz.
, , 1 1, –1,( )k k k
n m n m n m+ΔΨ = Ψ −Ψ , , , 2 , 1 , –1( )k k k
n m n m n m+ΔΨ = Ψ −Ψ .
Regarding the discrete forms of the Poisson’s equations (9) and (10), they are similar, viz.
1, , –1, , 1 , , –1
,2 2
1 2
2 2
–
k k k k k k
n m n m n m n m n m n m k
n m
+ +Ψ − Ψ +Ψ Ψ − Ψ +Ψ
+ = Ω
Δ Δ
, (24)
1, , –1, , 1 , , –1
2 2
1 2
2 2
1 1 2 2
1 2 1 2
,
2 2
– 2 .
k k k k k k
n m n m n m n m n m n m
k
n m
P P P P P P
U U U U
X X X X
+ +− + − +
+ =
Δ Δ
⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂⎢ ⎥= + +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
(25)
Expressions (24), (25) are resulted from the use of the five-point differencing scheme (see Fig. 2)
in Eqs. (9), (10), and have the second order of accuracy.
Discrete forms of the boundary and initial conditions. Discrete forms of the boundary and
initial conditions (5), (11)-(15) and (16) are as follows
1 1 12 2
2
, 2 2– ,
3 4
1
2 3u d
k
n m X L L L L L
X X
= + + +
⎛ ⎞Ψ = −⎜ ⎟⎝ ⎠
, , ,
1
2ch j
k
n m S S+ +Ψ = , Ψ =
– –, ,
1
–
2ch j
k
n m S S
,
h, 1 , –1 ,
( ) 0
ch j
k k
n m n m S S+Ψ −Ψ = , v1, –1,( ) 0
j
k k
n m n m S+Ψ −Ψ = ,
1 1 12 2
, 2– ,
12
u d
k
n m X L L L L L
X
= + + +
Ω = ,
h
h
, 1 , , –1
, 2,
2 ,
2
ch j
ch j
k k k
n m n m n mk
n m S S
S S
+Ψ − Ψ +Ψ
Ω = −
Δ
,
v
v
1, , –1,
2
1
2
j
j
k k k
n m n m n m
S
S
+Ψ − Ψ +Ψ
Ω = −
Δ
,
, ,
0
ch j
k
n m S S
P
n
∂⎛ ⎞ =⎜ ⎟∂⎝ ⎠
, 1, 2j = . (26)
v
1 1, 1 , 1 –1,
2
1 1,
( ) 2( ) ( )1
Re
j
k k k k
n m n m n m
n m S
U U UP
X
+
⎡ ⎛ − +⎛ ⎞∂ ⎢ ⎜= +⎜ ⎟ ⎜∂ ⎢ Δ⎝ ⎠ ⎝⎣
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A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct...
v
1
1 , 1 1 , 1 , –1 1 , 1 ,
2
2
( ) 2( ) ( ) ( ) –( )
j
k k k k k
n m n m n m n m n m
T
S
U U U U U+
+
⎤⎞− +
⎥⎟+ −
⎟ Δ ⎥Δ ⎠ ⎦
,
h
2 1, 2 , 2 –1,
2
2 1, ,
( ) 2( ) ( )1
Re
ch j
k k k k
n m n m n m
n m S S
U U UP
X
+
⎡ ⎛ − +⎛ ⎞∂ ⎢ ⎜= +⎜ ⎟ ⎜∂ ⎢ Δ⎝ ⎠ ⎝⎣
h
1
2 , 1 2 , 2 , –1 2 , 2 ,
2
2 ,
( ) 2( ) ( ) ( ) ( )
ch j
k k k k k
n m n m n m n m n m
T
S S
U U U U U+
+
⎤⎞− + −
⎥⎟+ −
⎟ Δ ⎥Δ ⎠ ⎦
,
, 0
0k
n m k =
Ψ = , , 0
0k
n m k =
Ω = , , 0
0k
n m k
P
=
= (27)
(in the conditions for the pressure in (26), it is necessary to use representations (19)). Expres-
sions (26) and (27) allow one to compute the values of all the terms of Eqs. (19), (23)-(25) on the
boundary of the computational domain and at the initial time instant in this domain, respectively.
Solution of Eqs. (19), (23)-(25). The preliminary study of Eqs. (23), (24) indicates that at
first sight Eq. (23) is nonlinear (because its right part is the function of ΨΩ ) and these equations
are coupled. However, more detailed analysis of the right part of (23) shows that all its terms
are computed at the previous time moment, kT T= , whereas at the initial time instant they are
established. It means that the right part of (23) is the known quantity, and hence, (23) is just a
computational scheme (rather than an equation) to find 1
,
k
n m
+Ω . Also, (23) and (24) is not a system
of coupled algebraic equations.
Once the vorticity at all the points of the grid (17), (18) can be computed on the basis of
the scheme (23), it is possible to solve the system of linear algebraic equations (i. e., SLAE) (24),
which now has the known right part.
Usually, SLAEs are solved either by direct or iterative methods. The first ones are applied to
small SLAEs and give satisfactory results. However, in case of big SLAEs (especially those having
the rarefied matrices) the direct methods require a huge amount of computational time, and their
application is unreasonable here. The iterative methods, when applied to SLAEs of big dimen-
sions, require significantly less computational time, save the rarefaction degree of their matrices
(when the matrices are rarefied) and give good results [5].
Proceed from these, as well as from the dimension and the rarefaction degree of the matrix of
system (24), we choose the successive over-relaxation iteration method [5] to solve the system.
The corresponding computational scheme for the discussed SLAE is of a second order of accu-
racy and looks as
1 2 2 2
, , 1, –1, , 1 , –1 1 ,2
(1 ) ( )
2(1 )
k k k k k k k
n m n m n m n m n m n m n m
+
+ +
γΨ = − γ Ψ + Ψ +Ψ +β Ψ +β Ψ +Δ Ω
+β
(28)
(here 1 2< γ and 1 2/β = Δ Δ ). Since all the terms on the right in (28) are known, the quantities
1
,
k
n m
+Ψ on the left can be easily obtained.
64 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2022. № 4
A.O. Borysyuk
The obtained values of Ψ allow one to compute the velocity components in (19) and fur-
ther use them in (25). After that the above-noted successive over-relaxation method is applied
to (25), viz.
1 2 2 2
, , 1, –1, , 1 , –1 1 ,2
(1 ) ( )
2(1 )
k k k k k k k
n m n m n m n m n m n m n mP P P P P P S+
+ +
γ= − γ + + +β +β +Δ
+β
, (29)
2 2
1 1 2 2
,
1 2 1 2
,
2
k
k
n m
n m
U U U U
S
X X X X
⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂⎢ ⎥= + +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
.
One can see that all the terms on the right in (29) are known. This permits one to find the pressure
1
,
k
n mP + on the left in (29).
Conclusions.
1. A numerical technique has been devised to solve a problem of the steady laminar fluid mo-
tion in a straight plane hard-walled duct with two local axisymmetric rectangular constrictions.
2. This technique uses the stream function, the vorticity and the pressure as the basic vari-
ables, has the second order of accuracy in the spatial and the first order of accuracy in the temporal
coordinates, provides high stability of a solution, and needs significantly less computational time
to obtain a result compared to appropriate techniques available in a scientific literature.
3. The technique consists in: a) introducing the stream function and the vorticity, and subse-
quent transiting from the non-dimensional governing relations for the fluid velocity and the pressure
to the corresponding non-dimensional relations for the stream function, the vorticity and the pres-
sure; b) deriving their discrete counterparts in the nodes of the chosen space-time computational
grid; c) integrating the systems of linear algebraic equations obtained after making the discretization.
4. The discretization is based on applying appropriate differencing schemes to the terms of the
equations for the basic variables. These are the two-point temporal onward difference for the un-
steady term of the vorticity equation, as well as the two-point backward differences (for its convec-
tive term) and the five-point approximations (for its diffusive term and for the Poisson’s equations
for the stream function and the pressure) in the axial and cross-flow coordinates. As for the velocity
components, the appropriate central differences are applied to discretize their expressions.
5. The above-mentioned systems of linear algebraic equations for the stream function and the
pressure are integrated by the iterative successive over-relaxation method. Regarding the alge-
braic relation for the vorticity, it is a computational scheme to find this quantity on the basis of
the known magnitudes computed at the previous instant of time.
REFERENCES
1. Borysyuk, A. O. (2018). A method to solve a problem of flow in a channel with two axisymmetric narrowings.
Science-Based Technologies, 38, No. 2, pp. 270-278 (in Ukrainian).
https://doi.org/10.18372/2310-5461.38.12825
2. Borysyuk, A. O. & Borysyuk, Ya. A. (2017). Wall pressure fluctuations behind a pipe narrowing of various
shapes. Science-Based Technologies, 34, 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).
65ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2022. № 4
A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct...
4. Young, D. F. (1979). Fluid mechanics of arterial stenoses. J. Biomech. Eng., 101, No. 3, pp. 157-175.
https://doi.org/10.1115/1.3426241
5. Ferziger, J. H. & Perić, M. (2002). Computational methods for fluid dynamics. 3rd ed. Berlin: Springer.
Received 11.04.2022
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.
АЛЬТЕРНАТИВНИЙ ПІДХІД
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методі як основні змінні використовуються функція течії, завихореність і тиск. Він має другий порядок
точності по координатах і перший порядок точності по часу, забезпечує високу стійкість розв’язку і по-
требує значно менше комп’ютерного часу для одержання результату порівняно з відповідними методами,
описаними в науковій літературі. За цим методом сформульована задача розв’язується шляхом: а) введен-
ня функції течії і завихореності та подальшого переходу від безрозмірних співвідношень для швидкості і
тиску до відповідних безрозмірних співвідношень для функції течії, завихореності і тиску; б) виведення
дискретних аналогів цих співвідношень у вузлах вибраної просторово-часової решітки; в) інтегрування
систем лінійних алгебраїчних рівнянь, одержаних внаслідок проведення зазначеної дискретизації. Дис-
кретизація ґрунтується на застосуванні відповідних різницевих схем до членів рівнянь для введених змін-
них. Це — одностороння різниця вперед для нестаціонарного члена рівняння переносу завихореності, а
також односторонні різниці проти потоку (для конвективного члена цього рівняння) та п’ятиточкові ша-
блони (для його дифузійного члена та рівнянь Пуассона для функції течії і тиску) по осьовій та поперечній
координатах. Що стосується компонент швидкості, то для дискретизації їхніх виразів застосовуються від-
повідні центральні різниці. Зазначені вище системи лінійних алгебраїчних рівнянь для функції течії і тис-
ку інтегруються ітераційним методом послідовної верхньої релаксації. Алгебраїчне ж співвідношення для
завихореності не потребує застосування ніякого методу розв’язування, оскільки вже є розрахунковою схе-
мою для визначення цієї величини на основі відомих величин, знайдених у попередній момент часу.
Ключові слова: рух рідини, плоский канал, прямокутне звуження, метод.
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