APTEFF, 43, 1-342 (2012) UDK: 532.54:66.011:004.4 DOI: 10.2298/APT1243169M BIBLID: 1450-7188 (2012) 43, 169-178
Original scientific paper
169
USING THE ANSYS FLUENT FOR SIMULATION OF TWO-SIDED LID-DRIVEN FLOW IN A STAGGERED CAVITY
Jelena Đ. Marković*, Nataša Lj. Lukić, Jelena D. Ilić, Branislava G. Nikolovski,
Milan N. Sovilj and Ivana M. Šijački
University of Novi Sad, Faculty of Technology, Bulevar Cara Lazara 1, 21000 Novi Sad, Serbia
This paper is concerned with numerical study of the two-sided lid-driven fluid flow in a staggered cavity. The ANSYS FLUENT commercial software was used for the simulation, In one of the simulated cases the lids are moving in opposite directions (antiparallel motion) and in the other they move in the same direction (parallel motion). Calculation results for various Re numbers are presented in the form of flow patterns and velocity profiles along the central lines of the cavity. The results are compared with the existing data from the literature. In general, a good agreement is found, especially in the antiparallel motion, while in the parallel motion the same flow pattern is found, but the velocity profiles are slightly different. KEY WORDS: cavity benchmark; fluid flow; two-sided lid driven cavity; parallel mo-
tion; antiparallel motion
INTRODUCTION In the past decades, flow in a lid-driven cavity has been studied extensively as one of the most popular fluid problems in the computational fluid dynamics (CFD). This classical problem has attracted considerable attention because the flow configuration is relevant to a number of industrial applications. ANSYS FLUENT uses conventional algorithms for calculation of macroscopic variables. Computational advantages of this commercial software are simplicity of the problem setup, parallel computing and higher precision. Two-sided lid-driven staggered cavity appears to be a synthesis of two benchmark problems: a lid-driven cavity and backward facing step. Furthermore, it has all the main features of a complex geometry. Nonrectangular two-sided lid-driven cavities have been recently introduced and investigated as a potential benchmark problem by Zhou et al. (1), Nithiearasu and Liu (2) and Tekic et al. (3). Zhou et al. Presented a solution for the flow in a staggered cavity obtained by using wavelet-based discrete singular convolution. Nithiarasu and Liu solved the same problem using the artificial compressibility-based
* Corresponding author: Jelena Marković , University of Novi Sad, Faculty of Technology, Bulevar Cara Laza-
ra 1, 21000 Novi Sad, Serbia, e-mail: [email protected]
APTEFF, 43, 1-342 (2012) UDK: 532.54:66.011:004.4 DOI: 10.2298/APT1243169M BIBLID: 1450-7188 (2012) 43, 169-178
Original scientific paper
170
characteristic-based split scheme. Tekic et al. solved this problem by using the lattice-Boltzmann method. The aim of this work was to study two-sided lid-driven staggered cavity utilizing the commercial software package FLUENT. Solutions are presented in the parallel and antiparallel motion of the lid and the flow pattern which develops under these conditions.
Figure 1. Schematic diagram of two-sided lid-driven staggered cavity: (a) antiparallel;
(b) parallel motion.
MATHEMATICAL FORMULATION
General Scalar Transport Equation: Discretization and Solution - ANSYS FLU-ENT uses a control-volume-based technique to convert a general scalar transport equa-tion to an algebraic equation that can be solved numerically. This control volume techni-que consists of the integration of the transport equation about each control volume, yiel-ding a discrete equation that expresses the conservation law on a control-volume basis. Discretization of the governing equations can be illustrated most easily by considering the unsteady conservation equation for transport of a scalar quantity Φ. This is demon-strated by the following equation written in integral form for an arbitrary control volume V as follows:
VV
dVSAdAdvdVt
[1]
where ρ is the density, v - velocity vector; A - surface area vector; - diffusion coefficient for Φ, S source of Φ per unit volume. Equation [1] is applied to each control volume, or cell, in the computational domain. The two-dimensional, triangular cell shown in Figure 1 is an example of such a control volume. Discretization of Equation [1] on a given cell yield
faces facesN
f
N
f
ffffff VSAAvVt
[2]
APTEFF, 43, 1-342 (2012) UDK: 532.54:66.011:004.4 DOI: 10.2298/APT1243169M BIBLID: 1450-7188 (2012) 43, 169-178
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where Nfaces represents the number of faces enclosing the cell, Φf is the value of con-
vected through the face f, fA is the area of the face f and V is the cell volume. The equations solved by ANSYS FLUENT take the same general form as the one given above and apply readily to multi-dimensional, unstructured meshes composed of arbitrary polyhedra.
Figure 2. Control volume used to illustrate discretization of a scalar transport equation.
For relatively uncomplicated problems (laminar flows with no additional models acti-vated) in which convergence is limited by the pressure-velocity coupling, a converged so-lution can often be obtained more quickly using SIMPLEC. With SIMPLEC, the pres-sure-correction under-relaxation factor is generally set to 1.0, which aids in convergence speedup. In the present study, a slightly more conservative under-relaxation value was used, and it is equal to 0.7 .Special practices related to the discretization of the momen-tum and continuity equations and their solution by means of the pressure-based solver is most easily described by considering the steady-state continuity and momentum equati-ons in the integral form:
0 Adv [3]
V
dVFAdAdpIAdvv [4]
where I is the identity matrix, is the stress tensor, and F is the force vector. Discretization of the Momentum Equation - previously described a discretization scheme for a scalar transport equation is also used to discretize the momentum equations. For example, the x-momentum equation can be obtained by setting u :
SiApuauanb
fnbnbP ^
[5]
If the pressure field and face mass fluxes are known, Equation [5] can be solved in the previously outlined manner, and a velocity field can be obtained. However, the pressure field and face mass fluxes are not known a priori and have to be obtained as a part of the solution. There are important issues with respect to the storage of pressure and the discretization of the pressure gradient term. ANSYS FLUENT uses a co-located scheme, whereby pressure and velocity are both stored at cell centers. However, Equation [5] requires the value of the pressure at the face between cells c0 and c1, shown in Figure 2.
APTEFF, 43, 1-342 (2012) UDK: 532.54:66.011:004.4 DOI: 10.2298/APT1243169M BIBLID: 1450-7188 (2012) 43, 169-178
Original scientific paper
172
Therefore, an interpolation scheme is required to compute the face values of pressure from the cell values. Discretization of continuity equation- Equation [1] may be integrated over the control volume to yield the following discrete equation
facesN
fff AJ 0 [6]
where Jf is the mass flux through the face nv . In order to proceed further, it is necessary
to relate the face values of the velocity, nv , to the stored values of velocity at the cell
centers. Linear interpolation of cell-centered velocities to the face results in an unphysical checker-boarding of pressure. ANSYS FLUENT uses a procedure similar to that outlined by Rhie and Chow (4) to prevent checkerboarding. The face value of velocity is not ave-raged linearly; instead, momentum-weighted averaging, using weighting factors based on the aP coefficient from the equation [5], is performed. Using this procedure, the face flux, Jf, may be written as:
)()))(())((( 1
^
10
,,
,,,,
01100
10
1100
ccffccccf
cpcp
cncpcncp
ff ppdJrpprppdaa
vavaJ
[7]
where 0cp ,
1cp and 0,cnv ,
1,cnv , are the pressures and normal velocities, respectively,
within the two cells on either side of the face, and ^
fJ contains the influence of velocities
in these cells (Figure 2). The term fd is a function of Pa , the average of the momentum
equation of the Pa coefficients for the cells on either side of the face f.
Spatial Discretization - By default, FLUENT stores discrete values of the scalar at
the cell centers (c0 and c1 in Figure 2). However, the face values f are required for the
convection terms in Equation [2] and they have to be interpolated from the cell center values. This is accomplished using an upwind scheme. Upwinding means that the face value f is derived from quantities in the cell upstream, or „upwind“, relative to the di-rection of the normal velocity vn in Equation [2]. The diffusion terms are central-differenced and are always second-order accurate. When second-order accuracy is desired, the quantities at cell faces are computed using a multidimensional linear reconstruction approach (5,6). In this approach, higher-order accuracy is achieved at cell faces through a Taylor series expansion of the cell-centered solution about the cell centroid. Thus, when second-order upwinding is selected, the face value f is computed using the following expression:
rSOUf , [8]
where and are the cell-centered value and its gradient in the upstream cell, and r
is the displacement vector from the upstream cell centroid to the face centroid. This for-mulation requires the determination of the gradient in each cell. Finally, the gradient
is limited so that no new maxima or minima are introduced.
APTEFDOI: 1
Sifinem0.001densilid vesired for thof mfirst-oused
Inlid drresulsgeomresult
Figu
Ttours velocFigur
FF, 43, 1-342 (2012)10.2298/APT1243169
imulation setup -ment adjacent to th1 Pas. Reynolds nity; μ is dynamic velocity in the x dirRe number. Boun
he upper and bottomoving depending
order upwind scheas starting conditi
Validatio
n order to validateriven square cavist in the available
metric center of thts of Chen et al. (6
ure 3. Velocity pro
he results for antat various Re nu
city u – and v-prore 5. For comparis
M
Mesh was createdhe walls. Density
number was calcuvisocity of the flurection. The velocndary conditions wom moving lid as
on the case (pareme were taken aions for the secon
RESULTS
on or results of o
e the simulation mity is simulated fliterature. Figure
he cavity. The o6) and Ghia et al (
ofiles u – and v- asq
Antiparall
tiparallel motion oumbers are presentofiles through the son sake, the resul
d with 140x140 ny of the fluid was lated as Re = uL
uid; L is the characcity of the movingwere set as no-slimoving walls wit
rallel or antiparals 0.5 velocity of t
nd-order upwind sc
AND DISCUSSI
one-sided lid –dri
mehod, a popular bfor different Re n
3 shows the u- anobtained results a(7).
along the vertical aquare cavity.
lel motion of the l
of lids are listed ited in Figure 4, wmid-section of th
lts obtained by Te
UDK: 53BIBLID: 1450-7188 (
Origin
number of elemenset to 1 kg/m3, a
L/. where ρ reprecteristical length og lid was calculateip for the left and th defined velocitllel). Starting conthe moving lids, acheme.
ION
iven square cavit
benchmark problenumbers and comnd v-velocity profare in good agree
and horizontal cen
lids
in Table 1. Streamwhile the results ohe staggered caviekic et al. (3) are a
32.54:66.011:004.4 (2012) 43, 169-178 nal scientific paper
173
nts with grid re-and viscosity to esents the fluid of cavity, and u ed based on de-right wall, and
ty and direction nditions for the nd results were
ty
em of one-sided mpared with the file, through the ement with the
nterlines of the
mfunction con-obtained for the ity are given in also presented.
APTEFDOI: 1
174
Itcompcompstron Abehavpattermoreferredcally locate
Wthe excreasmarycorne
Figu
VThe m
FF, 43, 1-342 (2012)10.2298/APT1243169
t is evident that wponents also increpared to the viscoger for higher Re.
As previously menvior for Re numbrns are achieved e
e precisely there ard to as secondary aligned along the
ed in the left and r
Figure 4. Streamf
With the increase oxpense of the prim
se of the Re numby vortices along ters next to the mov
ure 5. Velocity prostaggered c
Velocity profiles almost notable diff
M
with the increase iease in magnitudous ones. As a re. ntioned, three studbers above 1000. even at Re numbere three primary vfor easier compa
e mid-section of thright bottom corne
function contours
of Re, the primarymary vortex locat
ber, the bottom lefhe long diagonalving lid.
ofiles u – and v- acavity –antiparalle
long the vertical cference is for Re
in the Re numberde. Furthermore, tesult, the gradien
dies on staggered In the present stu
ers lower than 100vortices, although arison of the resulthe cavity. Opposeer of the cavity.
at various Re num
y vortex located inted in the upper rft corner vortex dl of the cavity, se
along the vertical ael motion (Rea – T
centerline of the c=100. While the
UDK: 53BIBLID: 1450-7188 (
Origin
r, extreme values the inertial forcesnts close to the m
cavity, (1)-(3) shudy, symmetric a00. Multiple vorticin Table 1 the thi
ts. Primary vorticed to this, seconda
mbers – antiparall
n the left bottom cright corner. Withisappears, and theecondary vortices
and horizontal cenTekic et al. results
cavity differ for soresults of Tekic
32.54:66.011:004.4 (2012) 43, 169-178 nal scientific paper
of the velocity s are dominant
moving lids are
howed unsteady and asymmetric ces are formed, ird vortex is re-es are all verti-ary vortices are
el motion.
corner grows at h the further in-ere are two pri-s appear in the
nterlines of the (3))
ome Re values. et al. (3) show
APTEFDOI: 1
moremore(Re=Consof Ghprevifoundtions TTablevortic Tabl
Acompdiffer
Itsheargular
Re
50a
50b
50c
100a
100b
100c
FF, 43, 1-342 (2012)10.2298/APT1243169
e flattened profilese similar to the p1000). The veloc
sidering that therehia et al. (6) andiously mentioned d in different Re caused by the difo summarize the e 1 and comparedces are in good ag
le 1. Locations an
As expected, paralpared to the antiprent Re numbers.
Figure 6. Stream
t can be noticed thr layer forms betwr cavities, where t
Primary vortex
(xc1,yc1)
(0.9781, 1.1600) (0.4219, 0.2518)
(0.9637, 1.1551) (0.4494, 0.2543)
(1.10383,1.17135) (0.41911,0.343648)
(1.0172, 1.1091) (0.3828, 0.2889)
(1.0031, 1.1382) (0.4082, 0.2693)
(1.20251, 1.25965) (0.452649,0.360417)
M
s, ANSYS FLUENprofiles which Tcity profiles alon is a very good ag
d Chen et al. (7), authors, the reasonumber definition
fferent numerical aresults, the loca
d with (1) and (3greement with the
nd secondary vortiet al.(3
Par
llel motion of theparallel motion. F
mfunction contou
hat two primary coween them. Compathe free shear laye
First secondary vortex R(xc2,yc2)
(1.3556, 0.4405) (0.0444, 0.9595)
40
(1.3484, 0.4476) (0.0460, 0.9543)
40
(1.27995, 0.69013) (0.032408, 0.8355)
40
(1.3556, 0.4486) (0.0444, 0.9514)
10
(1.3502, 0.4457) (0.0460, 0.9543)
10
(0.90263, 0.89427) (0.04669, 0.84673)
10
NT results show thTekic et al. (3) s
g the horizontal greement betweenand also between
ons for disagreemn and different boapproach.
ations of the cent3). It can be notiresults in the avai
ices – antiparallel 3), c present study
rallel motion
e opposite lids deFigure 6 shows t
urs at various Re n
ounter-rotating voared to the previoer is formed along
Re Primary vortex
(xc1,yc1)
00a (0.7000, 0.7000)
00b (0.6822, 0.6859)
00c (0.472214,0.445062)
000a (0.7000, 0.7000)
000b (0.7000, 0.7000)
000c (0.6934, 0.6972)
UDK: 53BIBLID: 1450-7188 (
Origin
he existence of a sshowed for highecenterline are al
n the present studyn results of Tekicents with present oundary condition
ers of the vorticeced that the resuilable literature.
motion, aZhou et
evelops a differenthe streamfunctio
numbers – parallel
ortices are presentous studies of flowg a horizontal cen
First secondary vortex (xc2,yc2)
(1.3500, 0.4656) (0.0500, 0.9344)
(1.3522, 0.4607) (0.0554, 0.9506)
) (1.25479, 0.60078) (0.05631, 0.60016)
(1.3250, 0.4844) (0.0750, 0.9063)
(1.3250, 0.4844) (0.0750, 0.9063)
(1.3371, 0.4851) (0.0722, 0.9280)
32.54:66.011:004.4 (2012) 43, 169-178 nal scientific paper
175
sine-like curve, er Re numbers most identical. y and the work c et al. (3) and study could be
ns implementa-
es are listed in lts for primary
al. (1), b Tekic
nt flow pattern on contours for
motion.
t and that a free w inside rectan-nterline (6), (8),
Second secondary vortex (xc3,yc1)
(0.4703, 1.625) (0.9219, 0.2375)
(0.4382, 1.1345) (0.9824, 0.2862)
(0.42548, 1.07007)
(0.7256,0.2000) (0.5339, 1.1907)
(0.8811,0.2167) (0.5301, 1.1962)
(0.82157, 0.13123) (0.61360, 0.96692)
APTEFDOI: 1
176
in theis no movecornestrengcauseshowze, sothe cabody velocwith Furthincreaboutresult
Figu
Aprofildifferhorizwhileles obnounned, the di
FF, 43, 1-342 (2012)10.2298/APT1243169
e staggered cavitylonger symmetric
es towards the offer of the right wagth at the cost of es splitting of the
wn in Figure 6. Boo that viscous effavity (9). As menwith a constant a
city profiles alongthe increase in th
her, the free shearase in the Re nut the horizontal cets of Tekic et al., i
ure 7. Velocity prstaggered
As in the case of anles, while the presrences occur at lo
zontal centerline she for the Re=100 tbtained by simulaced minimum andcould be a result ifferent boundary
M
y the free shear laycal due to the uppfset. At low Re nuall and offset. Athe upper primary
e primary vortex oth primary vorticfects are confined ntioned by Sahin aangular velocity ag mid sections of he Re number, extr layer formed be
umber due to turbenterline of the cavit can be seen that
rofiles u – and v- acavity – parallel m
ntiparallel motionsent study shows twer values of Re how relatively gothere is a more sig
ation in the presend maximum veloc
of the different Rconditions.
yer is formed alonper lid moving froumbers, a secondas the Re numbery vortex. At higеand formation o
ces have become mto the thin bound
and Owens (10), fat high Re numbe
the cavity. As ditreme velocity vaetween the two prbulence. The profvity, as previouslyt the obtained prof
along the vertical motion (Rea – Tek
n, the results of Tethe existence of anumber (50 and od agreement for gnificant differencnt study are more sity pitch. These diRe calculation pr
UDK: 53BIBLID: 1450-7188 (
Origin
ng the shorter diagom the offset, whilary vortex is preser increases, this vеr Re numbers, sef a second seconmore prominent adary layers close fluid begins to rotrs. Figure 7 showiscussed in the pralues also increaserimary vortices shfiles confirm asymy mentioned. Comfiles are quite sim
an horizontal cenkic et al. results (3
ekic et al. (3) givea minimum veloci100). Velocity prothe Re values 50,
ce. In general, thesymmetrical and hifferences, as prev
rocedure, and imp
32.54:66.011:004.4 (2012) 43, 169-178 nal scientific paper
gonal. The flow le the lower lid ent close to the vortex gains in condary vortex
ndary vortex as and larger in si-
to the walls of tate like a rigid
ws the u- and v-revious section, e in magnitude. hrinks with the mmetrical flow mpared with the
milar.
nterlines of the 3)).
more flattened ity pitch. These ofiles along the , 400 and 1000, e velocity profi-have more pro-viously mentio-plementation of
APTEFF, 43, 1-342 (2012) UDK: 532.54:66.011:004.4 DOI: 10.2298/APT1243169M BIBLID: 1450-7188 (2012) 43, 169-178
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CONCLUSION
Results of the ANSYS FLUENT commercial software simulation of two-sided lid-driven flow inside a staggered cavity are presented in this article. Both antiparallel and parallel motions of two facing lids are investigated. The benchmark results obtained with ANSYS FLUENT are in good agreement with the results available in the literature. For antiparallel motion of lids in a staggered cavity results show symmetrical and asy-mmetrical flow patterns. Velocity profiles along the horizontal centerline are in a good agreement with existing data from the literature, while the profiles along the vertical cen-terline are slightly different from those used for comparison, especially for Re=50 and Re=100. These differences could be explained by the different Re calculation procedures and different boundary conditions implementation methods, considering the different numerical approach. The situation is quite similar in case of parallel motion of lids. Un-like for antiparallel motion, steady-state asymmetric patterns are obtained for all investi-gated Re numbers. It can be noticed that a free shear layer is formed along the short dia-gonal of the staggered cavity. All the main features of the flow are shown, streamline contours, horizontal and vertical velocity components along the mid sections of the cavity are visually presented, while the location of vortices is presented in Table 1.
Acknowledgement This research was financially supported by the Ministry of Science and Technological Development of the Republic of Serbia (Project No. 46010)
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СИМУЛАЦИЈА ТОКА У ДВОСТРАНО ВОЂЕНОМ ПОКРЕТНОМ КАНАЛУ ПОМОЋУ ANSYS FLUENT ПРОГРАМСКОГ ПАКЕТА
Јелена Ђ. Марковић, Наташа Љ. Лукић, Јелена Д. Илић, Бранислава Г. Николовски,
Милан Н.Совиљ и Ивана М. Шијачки
Универзитет у Новом Саду, Технолошки факултет, Булевар цара Лазара 1, 21000 Нови Сад, Србија
Рад се бави проблематиком нумеричке анализе струјања флуида у каналима у којима струјање флуида настаје услед кретања горње и доње странице канала. Ко-мерцијални софтвер ANSYS FLUENT је коришћен за симулацију двострано вође-ног струјања флуида. Симулација је урађена за два случаја, први – када се горња и доња страна крећу у супротним смеровима (антипаралелно струјање) и други – када се горња и доња страна крећу у истом смеру. Резултати прорачуна за низ вредности Рејнолдсовог броја приказани су у виду путања струјања флуида и профила брзина дуж хоризонталне и вертикалне централне линије канала. Добијени резултати су упоређени са потојећим подацима у литератури. Генерално уочено је добро слагање са резултатим претходних истраживања, нарочито када се ради о антипаралелном струјању. У случају паралелног струјања, визуелно ток флуида је исти, али потоји мала разлика у профилима брзина. Кључне речи: симулација, Ansys Fluent, струјање флуида, двострано вођени
покретни канали
Received: 6 July 2012 Accepted: 14 September 2012