-
aaya45,
Keywords:Heated curved ductsDean vorticesSecondary owFlow
instability
ericid imp
es is a, such
neering work in this eld by Dean [1].Dean [1] relates the
secondary ow behaviour to a single
parameter K, called the Dean number, which is dened asK
Dh=R1=2Re. It has been identied that the Deans instability
isculminated through the interaction between the centrifugal
forces
also dependent on the aspect ratio and curvature ratio along
withthe Dean number.
Literature reports many numerical studies among which somekey
studies are indicated below. Hoon et al. [10] used a 2-dimensional
numerical model for uid and thermal behaviour ineccentric curved
pipes and discussed the relationship among Deannumber, friction
factor and Nusselt number in the range0 K 900 and Grashof 12.5 Gr
12,500. Yamamoto et al. [11]numerically examined the ow eld within
a helical pipe and
* Corresponding author.
Contents lists available at
International Journal
w.e
International Journal of Thermal Sciences 59 (2012) 75e86E-mail
address: [email protected] (T.T. Chandratilleke).air
conditioning, heat exchangers and rocket engine coolantpassages. In
a curved passage, centrifugal forces are developed inthe ow due to
channel curvature causing a counter rotating vortexmotion applied
on the axial ow through the passage. This createscharacteristic
spiralling uid ow in the curved passage known assecondary ow. At a
certain critical ow condition and beyond,additional pairs of
counter rotating vortices appear on the outerconcave wall of curved
uid passages. This ow condition isreferred to as Deans Hydrodynamic
Instability and the additionalvortices are known as Dean vortices,
in recognition of the pio-
requirements for Deans instability.White [4] performed ow
visualisation on a coiled pipe and
reported that, the ow curvature alters the laminar
regimecompared to straight channels allowing
laminar-to-turbulenttransition to occur at reduced ow velocities.
Early analytical andexperimental investigations, such as Baylis
[5], Humphrey et al. [6],concluded that Dean number was solely
responsible for secondaryow and Dean instability in curved
passages. However later studieswith curved rectangular ducts by
Cheng et al. [7], Ghia and Sokhey[8] and Sugiyama et al. [9] have
shown that the Dean instability is1. Introduction
Fluid ow through curved passaga vast range of industrial
applications1290-0729/$ e see front matter 2012 Elsevier
Masdoi:10.1016/j.ijthermalsci.2012.04.014curvatures and external
wall heat uxes. The ow conditions leading to hydrodynamic
instability andDean vortex generation in curved passages are
carefully analysed, identifying the ow and geometricalparametric
inuences. Active interaction of buoyancy force on uid motion
arising from wall heating isconsidered where aspects of boundary
layer separation is used in recognising thermal enhancement dueto
secondary ow. The simulated results are validated with the
available experimental data. Surpassingprecision and reliability of
current practise, the study develops two numerical approaches
offeringimproved capabilities for capturing the onset of
hydrodynamic instability and resulting Dean vortices.These
techniques are compared and assessed for their merits of
application in determining the onset ofsecondary ow hydrodynamic
instability with and without external wall heating.
2012 Elsevier Masson SAS. All rights reserved.
common occurrence inas in gas turbine blades,
and lateral uid pressure gradient at the outer wall acting
towardsthe duct centre of curvature. This instability has been
illustratedwith respect to duct velocity in the work of Moffatt [2]
and Eustice[3] who experimentally observed and veried the critical
velocityAvailable online 19 May 2012 curved rectangular ducts, the
analysis performs a detailed parametric study involving the
contours ofhelicity and outer duct wall pressure gradient for a
range of ow rates, duct aspect ratios, duct ow13 April 2012Accepted
13 April 2012
describing secondary ow and its thermal characteristics. For
developing laminar uid ow throughVortex structure-based analysis of
lamincharacteristics in curved ducts
Tilak T. Chandratilleke*, Nima Nadim, Ramesh NarDepartment of
Mechanical Engineering, Curtin University, GPO Box U1987, Perth
WA68
a r t i c l e i n f o
Article history:Received 18 July 2011Received in revised
form
a b s t r a c t
This paper presents a numheat transfer process in uthe study
formulates an
journal homepage: wwson SAS. All rights reserved.r ow behaviour
and thermal
naswamyAustralia
al investigation for examining the secondary vortex motion and
associatedow through curved passages. Overcoming current modelling
limitations,roved simulation model based on 3-dimensional vortex
structures for
SciVerse ScienceDirect
of Thermal Sciences
lsevier .com/locate/ i j ts
-
urndemonstrated the role of centrifugal forces in the formation
ofsecondary ow. Ko et al. used ow entropy generation as a
tech-nique to identify secondary ow instability and performed
thermaloptimisation on rectangular passages in both laminar and
turbulentregimes [12e14].
Chandratilleke [15e17] report an extensive parametric
studyexamining the effects of curvature ratio and aspect ratio
aswell as thewall heat ux. The validation of numerical work has
been performedagainst theirownexperimental data [15,16]. Their
numericalmethod,that was effectively a 2-dimensional formulation,
used toroidalcoordinates and utilised a stream function approach
with dynamicsimilarity in axial direction. Intersecting stream
function contourswere deployed as a qualitative criterion for
detectingoccurrenceowinstability and Dean vortices. This approach
is clearly adequate for 2-dimensional ow systems and cannot be
extended for real owsituations. They discussed results in a
comprehensive range of Deannumber 25 K 500, aspect ratio 1 Ar 8 and
Grashof number
Nomenclature
Ar Aspect Ratio a/ba Height of cross section (mm)b Width of
cross section (mm)Cp Specic Heat (j/kg k)Dh Hydraulic diameter (mm)
2ab=a bFc Centrifugal force (N)g Gravity (m/s2)H* Dimensionless
Helicityh heat transfer coefcienti^; j^; k^ unit vectors in x,y,z
directionsK Dean number Dh=R1=2Rek Thermal Conductivity (W/m k)Nu
Nusselt numberp Static pressure (Pa)p* Dimensionless static
pressure p=1=2rU2inq Heat ux(W/m2)R Radius of curved channel
(mm)
T.T. Chandratilleke et al. / International Jo7612.5 Gr 12,500.
It has been illustrated that the onset of Deaninstability would
vary with the duct aspect ratio and curvature ratiowhile the
application of wall heat ux radically changes the owpatterns.
Subsequently, other numerical work such as those carriedout by
Fellouah et al. [20,21] have validated the results of
Chan-dratilleke [16]. Yanase et al. [18,19] investigated ow in
curved ductusing a so called spectral method and by which classied
ow rangeinto three different regimes; stable, periodic and chaotic.
They usedsuch a method to see eld response against perturbation
anddiscovered that while for low ow rate system is condently
stableagainst perturbation it will turn into periodic and even
chaoticbehaviours for higher ow rates. Guo et al. [22] used a
laminarincompressible three-dimensional numerical model to explore
theinteractive effects of geometrical and ow characteristic on
heattransfer and pressure drop. They applied entropy generation asa
hydrothermal criterion and reported the inuence of Reynoldsnumber
and curvature ratio on the ow prole and Nusselt number.
Most published work focus on 2-dimenasional numericalmodels.
Accurate 3-dimensional simulations are clearly scarce inliterature
largely due to the computational complexities arisingfrom secondary
ow perturbations superimposed on themain axialow through curved
passages. H. Fellouah et al. [20,21] haveattempted to developed an
elementary 3-dimensional simulationcovering duct curvature ratios
of 5.5 g 20 and aspect ratios of0.5 Ar 12. With water and air as
working uids, this modelshowed reasonable agreement with their own
experimentsinvolving a semi-circular channel test section that
permitted visu-alisation of vortex formation along channel
locations for variousDean numbers. They presented a quantitative
criterion for identi-fying the Dean instability in curved channels
using the radialgradient of the axial velocity in channels.
Secondary ow dynamicssuggest inadequacy of this approach-it is not
the axial ow velocityin channel but the radial uid velocity
component directed towardsouter wall is responsible for
hydrodynamic instability and thestagnation pressure build-up at the
outer curved wall.
The study presented in this paper formulates amuch improved
3-dimensional computational model using helicity function
thataccurately describes the secondary ow vortex structures in
thedeveloping uid ow through curved passages, thus overcoming
thelimitations of mostly 2-dimensional previous studies. It
incorporatesa curvilinear mesh system for much compliant tracking
of secondaryow path, facilitating more effective grid denition for
capturing
Re Reynolds number UinDh=yS_
Coordinate along duct cross section for deningsecondary ow
direction
T Temperature(K)Tin Temperature at duct inlet (K)u,v,w
Velocities component (m/s)u*,v*,w* Dimensionless velocity u;
v;w=UinUin Velocity at duct inlet (m/s)Vr Axial velocity (m/s)x,y,z
Coordinates (m)
Greek symbolsq Angular position of cross section (deg)g
Curvature ratio R=bn Kinematic Viscosity (m2/s)m Dynamic viscosity
(Ns/m2)r Density (kg/m3)u Vorticity (1/s)
al of Thermal Sciences 59 (2012) 75e86intricate details of
vortex generation. In addition, a curvilinear coor-dinate system is
dened along the outer duct wall permitting preciseand efcient
evaluation of local uid pressure and its gradient in thatvicinity.
Simulation results are validated against the availableexperimental
data. Veriedagainst available data, the studyproposestwo novel
approaches for computational schemes in determining theonset of
hydrodynamic ow instability, reected by Dean vortexgeneration, in
curved passages. These criteria demonstrate a higherdegree of
reliability for detecting the inception of Dean instabilitywhile
being able to identify the boundary layer separation resultingfrom
buoyancy-centrifugal force interaction induced by the
ductwallheating. The analysis also examines the effects of material
propertyvariation on Nusselt number.
2. Model description and numerical analysis
Fig. 1 illustrates the geometry used for the 3-dimensional
modeldevelopment. The model consists of a semi-circular curved
ducttted with straight inlet and outlet passages for ensuring
fullydeveloped ow at entry and smooth outow at exit. The workinguid
air having temperature dependent properties ows throughthe passage
under steady and laminar ow conditions and isassumed to be an
incompressible Newtonian uid. The analysisfocuses on the curved
duct that is designated by 0 at inlet to 180
at outlet. Fig. 1 also shows the coordinate system used,
whose
-
nel
ournal of Thermal Sciences 59 (2012) 75e86 77origin is pegged at
curved duct outlet and the key geometricalparameters, duct height,
width and radius of curvature consideredin the study.
The numerical model involves the solution of the
followingfundamental governing equations: Steady-state continuity
equation;
VV! 0 (1)the momentum and energy conservation equations;
V!$VrV! Vp mV2 V! rm g! F
!c (2-a)
V!$VrcpT
kV2T ST (2-b)The centrifugal body force term along radial
directionwould be,
Fc rV2rr
ru2 w2x2 z2
p (3)Considering position and alignment of curved part of
geometry,
centrifugal source term in Cartesian coordinate system would
be,
F!
c r1 signz2
u2 w2x2 z2 x^i zk^ (4)
In Equation (4), a Sign Function has been included to ensure
the
Fig. 1. Geometry of curved chan
T.T. Chandratilleke et al. / International Jcentrifugal source
term is applied only on the curved side of thegeometry (i.e. z 0).
To be used with the dimensionless parame-ters, the characteristics
length, velocity and pressure are chosen tobe Dh, Uin, 1=2rU2in,
respectively.
In 3-dimensional simulation of ow through curved ducts,
owpatterns are illustrated by the helicity function dened as,
H uvwvy
vvvz
vvuvz
vwvx
w
vv
vx vuvy
(5)
Which is non-dimensionalised as,
HzU2inDh0H* H
DhU2in
!(6)
For identifying the inception of hydrodynamic ow instabilityand
to explain the boundary layer separation due to secondaryow, the
non-dimensionalised wall pressure gradient in thedirection of
secondary uid motion is dened by Equation (7) interms of the
coordinate Sh along duct cross section. A signconvention is
incorporated in Equation (7) to designate theopposite rotational
directions of the vortices in upper and lowerhalf of duct cross
section.
dp*
d S!
8>>>>>: Dh1=2rU2in
dpdy
at below half of cross section
Dh1=2rU2in
dpdy
at above half of cross section
(7)
In solving the numerical model, a constant velocity condition
isapplied to the inlet of the straight duct section attached to
thecurved duct. The length of this inlet duct section is chosen
toprovide fully developed ow at entry to the curved duct section.
Apressure outlet condition is applied to the outlet of the
ductgeometry. The air temperature is taken to be constant at 300 K
atthe inlet. The duct walls are assumed to have no slip
boundarycondition. For the cases of heated ducts, a constant heat
ux isapplied on outer wall. Table 1 provides the geometrical
parametricrange considered with the values of aspect ratio Ar and
curvatureratio g. The buoyancy force arising fromwall heating is
introducedto themodel through a temperature-dependent density
expression.The Boussinesq approximationwas found to be inadequate
becausethe linearity in the density function caused some
discrepancy in thecomputation of wall pressure gradient and the
detection ofboundary layer separation. Therefore, a sixth order
polynomialrepresenting density temperature dependency was used
with
with rectangular cross section.coefcients obtained through
published data. In improving theoverall computational accuracy, all
uid and thermal properties arealso dened as temperature dependent
quantities.
For the analysis, the local Nusselt number is dened asNu
hDh=kwhere the heat transfer coefcient h, is determined
byconsidering the grid cell temperature difference between
theheatedwall and the adjacent uid cell. The Average Nusselt
numberis obtained from a surface integral dened by, Nu R
ANudA=A:.
The nite volume-based CFD model is formulated using
thecommercial CFD package FLUENT where SIMPLE algorithm is usedfor
pressure-velocity coupling. The momentum and energy
Table 1Geometrical values for computational model (length
dimensions in mm).
a Dh R Ar g
25 25 125 1 555 33.33 100 2 475 37.5 3100 40 4125 41.67 5150
42.86 6
-
equations are discretized by rst and second order
schemes,respectively. Since the model incorporates both buoyancy
andcentrifugal source terms, pressure discretization was
performedwith body force weighted approach. The stability of
solution ismonitored in terms of continuity, velocity, energy and
dimensionlesshelictywhere the convergence is achievedwith values
are not higherthan 105. Grid independency has been carefully
checked withparticular attention to boundary layer modelling at the
outer wall.
The solution is performed with respect to a suitably dened
much closer steps of K to establish the exact point of
instability. Thisprocedure was repeatedly applied for all
permutations of aspectratio and curvature ratio combinations
whereby the critical Deannumber for each test case was ascertained.
Dashed and solid linerepresent negative and positive dimensionless
helicity value,respectively. Negative dimensionless helicity value
shows contour-clockwise rotation (in channel view) while the
positive value indi-cates opposite direction for vortex
rotation.
For the purpose of validating results against published
work,Fig. 3 shows the axial ow velocity in x and y directions
predictedby the current 3-dimensional model and those from the
analyses ofGhia and Sokhey [8] and Fellouah et al. [20]. It is seen
that bothmagnitudes and trends of axial velocity are very
favourablycompared conrming the integrity of the numerical
process.
Fig. 3 also illustrates the secondary ow effect on the axial
uidvelocity in the curved passage. The prole on the left (along
x-axis)shows a skewed peak towards the outer wall arising from
thecentrifugal action and is characteristically different to axial
velocitydistribution in straight ducts. This peak gradually spreads
towardsthe centre with increasing ow rate because of the radial
pressurebuild at the outer wall. The prole on the right (along
y-axis) showsa dip in the centre and two marginal peaks on other
side. Thesepeaks essentially correspond to the upper and lower eye
ofsecondary vortices (Fig. 4). Increased ow rate shows a
marginalimpact on the prole.
3.1. Effect of ow rate
A typical ow prole in terms of helicity contours in the
curvedduct cross section is shown in Fig. 4. The corresponding
experi-
0.5
Ar 2.
T.T. Chandratilleke et al. / International Journ78curvilinear
mesh. In capturing the intricate characteristics ofhydrodynamic
instability, a progressively reducing mesh isconsidered in the
analysis, facilitating a much ner mesh near theouter wall where the
onset of instability is anticipated. Thisapproach has not been
attempted in previous studies[11e13,18e20] arguing that a mesh size
less than 1 mm did notsufciently improve accuracy, but only led to
increased computa-tional time. The mesh renement approach adopted
in the currentanalysis clearly demonstrated that a ner mesh near
the wall iscritical for detecting the onset of Dean vortices as
accurately aspossible. For testing grid dependency, the study used
ve meshschemes indicated in Table 2. In this, columns A, B and C
representthe number of grids over duct width, height and length,
respec-tively, while the column D indicates the progressive
reduction ofgrid size over duct width towards the outer wall.
Fig. 2 illustrates the grid dependency test through two
separatevariables. Fig. 2(a) considers the velocity derivative in
x-direction atthemid-point of duct cross sectionwhile Fig. 2(b)
shows the velocityderivative in y-direction at outer wall. Fig.
2(a) indicates that all veschemes behave very similarly and are
equally acceptable. However,it is clear from Fig. 2(b) that the
Schemes 4 and 5 having mesh sizeless than 1 mm show much better
suitability than the other threeschemes. Nonetheless, Scheme 5 is
taken to be the optimumbecauseof its slightly larger cell volume
arising from progressively variedmesh size. This approach provided
a remarkable ability to capturethe onset of vortex generation in
the solution domain withoutcausing excessive computational demand.
As such, the present studyperformed all computations with Scheme 5
using mesh size lessthan 1 mm, hence achieving much higher accuracy
than anyprevious reported work.
3. Results and discussion
Using the 3-dimensional numerical model developed, the airow
through a semi-circular curved rectangular duct was analysedand the
results were generated over an extensive range ofgeometrical and ow
parameters. The duct height and width werevaried, as given by Table
1, to obtain the aspect ratio Ar in the rangeof 1e6. Severaluidow
rateswere chosen to achieveDeanNumberK in the range of 80e500 while
the constant wall heat ux at theouter wall ranged from 100 to 3750.
This selection enables a directcomparison tobemadewith the
available experimental data [17,20].For analysis, the ow proles
were obtained at the exit plane (180
position) of curved duct. The test cases indicating
hydrodynamicinstabilitywere further rened byand running the
simulationswith
Table 2Parametric selection for mesh schemes.
Scheme Number of grids
A B C D
Mesh1 26 51 305 1Mesh2 31 64 305 1Mesh3 43 84 305 1Mesh4 50 98
305 1
Mesh5 26 51 305 1.05Velocity derivative in x-direction at a
mid-plane in cross section
Velocity derivative in y-direction at outer wall
-3.5
-2.5
-1.5
-0.5-0.5 -0.3 -0.1 0.1 0.3 0.5x/b Mesh1
Mesh2Mesh3Mesh4Mesh5
-2
-1.5
-1
-0.5
0
0.5
1
-0.5 -0.3 -0.1 0.1 0.3 0.5
du/d
x
y/a
Mesh1Mesh2Mesh3Mesh4Mesh5
a
b
Fig. 2. Grid independancy test for different grid schemes at
curved duct exit, K 130,1.5
2.5
du/d
x
al of Thermal Sciences 59 (2012) 75e86mental ow pattern is also
shown there for comparison. It is readily
-
ourn0.0
0.5
1.0
1.5
2.0
-0.5 -0.3 -0.1 0.1 0.3 0.5
Vr
x/b
Ghia and Sokhey K=55 Present Study K=55
0.20.40.60.81.01.21.41.61.8
Vr
T.T. Chandratilleke et al. / International Jnoted that these
patterns are fundamentally different from those instraight
channels. Even at low ow rates (or low K), the ow prolehas two
large counter-rotating vortices. This vortex ow is devel-oped
consequent to the centrifugal forces induced by the ductstream-wise
curvature.
The centrifugal body forces due to the duct ow
curvatureessentially create two effects. It generates a positive
radial uidpressure eld in the duct cross section and induces a
lateral uidmotion driven from inner duct wall towards outer duct
wall (left-sidewall in Fig. 4). This lateral uidmotion occurs
against the radialpressure eld generated by the centrifugal effect
and is super-imposed on the axial ow to create the secondary vortex
owstructure. As the ow through the curved duct is increased
(largerK), the lateral uid motion becomes stronger and the radial
pres-sure eld is intensied. In the vicinity of the outer duct wall,
thecombined action of adverse radial pressure eld and viscous
effectsslows down the lateral uid motion and forms a stagnant
owregion. Beyond a certain critical value of K, the radial
pressuregradient becomes sufciently strong to reverse the ow
direction ofthe lateral uid ow. A weak local ow re-circulation is
thenestablished creating an additional pair of vortices in the
stagnantregion near the outer wall. This ow situation is known as
Dean
0.0-0.5 -0.3 -0.1 0.1 0.3 0.5
x/b
Ghia and Sokhey K=55 Present Study K=100
0.00.20.40.60.81.01.21.41.6
-0.5 -0.3 -0.1 0.1 0.3 0.5
Vr
x/b
Ghia and Sokhey K=55 Present Study K=350
At middle plane (y = 0) along x-axisFig. 3. Comparison of
dimensionless axial velocity pro0.0
0.5
1.0
1.5
-0.5 -0.3 -0.1 0.1 0.3 0.5
Vr
y/aFellouah et al K=60 Present Study K=60
0.00.20.40.60.81.01.21.41.6
Vr
al of Thermal Sciences 59 (2012) 75e86 79Hydrodynamic
Instability while the vortices are termed DeanVortices. The
helicity contours for K 380 in Fig. 4 depicts the owpattern at the
hydrodynamic instability in the duct cross section.
3.2. Identication of hydrodynamic instability
Secondary ow Hydrodynamic Instability in curved passages
istraditionally identied through tedious experimental ow
visual-isation techniques or by trial-and-error in numerical
simulations. Inthe latter, repeated numerical computations are
performed in thevicinity of anticipated ow conditions by
continually narrowing therange to obtain the critical Dean number K
and the ow patternswithin the chosen tolerance limits. This
involves guesswork andrequires signicant computational time.
Chandratilleke et al. [16]successfully used the criterion of
zero-potential stream functioncontours to identify locations of
Dean vortex generation, which waspractically sufcient for
2-dimensional simulations, but is notapplicable for 3-dimensional
ows. The work of Fellouah et al. [20]used radial gradient of the
axial velocity as a measure of identifyingthe ow instability. It is
difcult for this selection to be rationalisedbecause the axial
velocity change in radial direction is not funda-mentally connected
with the vortex generation. This inadequacy is
-0.5 -0.3 -0.1 0.1 0.3 0.5y/a
Fellouah et al K=100 Present Study K=100
0.00.20.40.60.81.01.2
-0.5 -0.3 -0.1 0.1 0.3 0.5
Vr
y/a
Fellouah et al K=350 Present Study K=350
At middle plane (x = R) along y-axisle at curved duct exit plane
(q 180), Ar 1.
-
cur
urnal of Thermal Sciences 59 (2012) 75e86clearly reected in the
work of Fellouah et al. [20], where theirsimulation failed to
detect the ow instability for some basic owconditions. For example
around K 100, hydrodynamic instabilityis well observed in the
current study, yet the analyses of Ghia andSokhey [8] and Fellouah
et al. [20] did not identify this occurrence.
Based on the mechanics of secondary ow vortex
generationdiscussed in Section 3.1, the current study recognises
the strongafliation between the outer wall uid pressure prole and
the
Fig. 4. Helicity contours and experimental ow pattern [17]
at
T.T. Chandratilleke et al. / International Jo80hydrodynamic
instability. In ascertaining this, the pressuregradient prole along
the outer wall is obtained in terms of thedisplacement variable S
(in the direction of secondary owmotion)at the outer wall boundary.
For the curved duct exit a typicaldistribution of the wall pressure
gradient is illustrated in Fig. 5whereas aspect ratio is xed at
4.
Considering the case of K 380, it is observed that the
pressuregradient is sharply positive on either side of the duct
centreline(S_ 0). The ow patterns of the current study indicate the
peakpressure locations consistently coincide with the appearance
ofDean vortices at the outer wall. For lower K values, the
pressuregradient becomes gradually shallower with the
diminishingpossibility of ow instability, which also complies with
the currentresults. This suggests that the outer wall pressure
gradient isa useful parameter for recognising ow instability that
is exploredin the current study.
As of now, a reliable technique for detecting the
hydrodynamicinstability is not known in literature for
3-dimensional numericalsimulations of ow through curved ducts.
Formulation of a gener-alised approach is made further complicated
by the dependency ofpressure prole on the geometry of duct cross
section. Inaddressing these issues, the current work reports for
the rst time,two reliable methods for identifying and predicting
the hydrody-namic instability in curved uid passages.
3.2.1. Determination of hydrodynamic instability -
dimensionlesshelicity criterion
For detecting the onset of Dean vortices, this proposed
criterionassigns a value for the dimensionless helicity function
H*. Bydenition, helicty describes helix-like uid motion in a
three-dimensional domain while stream function represents the
trajec-tory of particles in two-dimensional plane depicting uid
owstreamlines. Although there is a similarity between stream
functionand helicity contours, the stream function only captures
two-dimensional aspects of the ow eld, whereas helicity is
inher-ently a three-dimensional representation of vortex ow
structure,which is therefore the preferred and accurate option for
depicting
ved duct exit plane (q 180) for different ow rates, Ar 4.the
spiralling motion of secondary ow. The vortex contours
cor-responding to this value is regarded as Dean vortices.
Throughextensive simulation runs with innitesimal steps in K
andobserving the predicted ow patterns, the current study
identiesH* 0.01 provides an excellent detection ability for Dean
vorticesin all cases of ducts examined. This threshold for H*
essentiallydepends on the accuracy required similar to dening the
boundarylayer thickness in traditional uid ows. The ow
conditionsassociated with Deans instability is signied in
literature by the
-1.1
-0.6
-0.1
0.4
0.9
-0.5
dp*/
ds
y/a
K=180 K=260 K=300 K=380
Fig. 5. Pressure gradient on outer wall at curved duct exit (q
180) for Ar 4.
-
T.T. Chandratilleke et al. / International Journal of Thermal
Sciences 59 (2012) 75e86 81appearance of additional vortices and
the corresponding (critical)Dean number. The current study utilises
dimensionless helicity H*to characterise ow behaviour. By selecting
appropriate values forH*, the helicity contours can be rened
permitting the recognitionof contours that identify Dean vortices.
Continuous renement ofthese contours for all cases of study
indicates that H* 0.01 is thepractical lower limit that effectively
detect the onset of Deanvortices. A value less than 0.01 does not
change the critical Deannumber appreciable while any higher values
would develop thepossibility of missing Dean vortices altogether in
the ow. Similar
Fig. 6. Helicity contours before and at hydrodynamic instabi
Table 3Data label and contour values illustrated in Fig. 6.
Data label 1 2 3 4 5
Dimensionless Helicity 6 5.4 4.8 4.2 3Data label 12 13 14 15
Dimensionless Helicity 0.01 0.6 1.2 1.8to the boundary layer
thickness, the value of H* to be used dependson the detection
accuracy to be implemented.
The key advantage of this technique is that it can be
readilyintegrated into the computational process and performed
locallywithin the solution domain rather than as a cumbersome
post-simulation method. Hence, the determination of Dean vortices
ismore precise and less time consuming.
For several duct aspect ratios, Fig. 6 shows the helicity
contoursdepicting the rst appearance of Dean vortices identied by
thedimensionless helicity criterion and the ow patterns just prior
to
lity for g 5. (Top boundary repersents outer duct wall).
6 7 8 9 10 11
.6 3 2.4 1.8 1.2 0.6 0.0116 17 18 19 20 21 22
2.4 3 3.6 4.2 4.8 5.4 6
-
this critical condition of ow instability. Table 3 provides the
hel-icity contour labels and values for Fig. 6.
Evidently, this technique recognises the appearance of
vorticeswithin a very narrow range of K, demonstrating the
precision of the
outer wall. As K increases, the inection points of the prole
rstacquire positive values for dp=d s
_, creating (marked) regions of
adverse pressure gradients at the outer wall. These
localitiesdevelop ow reversal and represent critical conditions for
hydro-
Ar=4, =5
-0.2
-0.15
-0.1
-0.05
0
0.05
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
dp*/
ds
y/aK=128 K=136 K=147 K=155
Fig. 7. Variation of dimensionless pressure gradient on the
outer duct wall at exit (180).
T.T. Chandratilleke et al. / International Journal of Thermal
Sciences 59 (2012) 75e8682approach. Previous methods never provided
such vortex detectionability or sensitivity signifying that this
method is far superior tothe axial velocity gradient approach of
Ghia and Sokhey [8] andFellouah et al. [20].
3.2.2. Determination of hydrodynamic instability - adverse
pressuregradient criterion
This second proposed criterion is based on the fact that the
owreversal leading to Dean vortex generation is fundamentally
causedby the existence of adverse pressure gradient (dp=d s
_> 0) at the
outer duct wall in the direction of secondary uid motion. It is
usedfor Dean vortex detection, as follows.
For a typical case, Fig. 7 shows the variation of pressure
gradientdp=d s
_evaluated from Equation.(7) along the outer duct wall for
several chosen values of Dean number K. Direction of s_is dened
to
be secondary ow rotation which is upward for above side of
crosssection and downward for below side at the outer wall. For K
valuesunder 155, the pressure gradient remains negative over the
entireFig. 8. Comparison of two criteria for identifying the onset
of Dean instabdynamic instability with Dean vortex generation. K
155 is rec-ognised to be the critical Dean number. Reported for the
rst time,this criterion based adverse pressure gradient is
presented asa reliable mechanistic technique to identify the onset
hydrody-namic instability.
Dean vortices always form initially inside the ow domain closeto
the outer wall (not attached to it) at the onset of this
owcondition. The exact position of these vortices depends on the
ductaspect ratio and the cross sectional geometry. Fig. 8 allows
corre-lation of the pressure gradient and the corresponding
vortexposition based on helicity while outer wall is located at
left side.The Figurers indicates that, at low ow rates, the
helicity approachidenties the appearance of Dean vortices while the
pressuregradient remains negative, meaning that the latter approach
doesnot capture this existence of vortices. Also, it is noted that
Deanvortices appear not at the outer wall but in the bulk uid away
fromthe wall. However, with increased ow rate, Dean vortices
willgradually migrate towards the outer wall wherein the
pressureility curved duct exit plane (q 180) for different ow
rates, Ar 4.
-
d De
ournal of Thermal Sciences 59 (2012) 75e86 83gradient becomes
adverse (or positive) and both criteria are able tocapture Dean
vortices.
3.2.3. Three-dimensional vortex core representation
ofhydrodynamic instability
Fig. 9. Structure of secondary ow an
T.T. Chandratilleke et al. / International JFig. 9 illustrates a
visualisation method for hydrodynamicinstability development within
the curved duct from its inletto outlet. Fig. 9(i) provides the
view with duct inlet in theforeground while Fig. 9(ii) shows that
with duct outlet in theforeground. These proles were obtained from
a sophisticatedpost-processing technique integrated into the
current simula-tion, which has not been previously attempted in
publishedliterature.
The illustration on the left depicts the iso-value envelop
ofhelicity captured by applying the dimensionless helicity
criterionwith H* 0.01 discussed in Section 3.2.1. The envelop
representsthe 3-dimensional vortex structures (red and blue
indicate oppo-site vortex rotational directions) within the ow eld
in thecurved duct and accounts for the helicity magnitude,
swirlingstrength and vorticity. The growth sequence of secondary
vortices(blue and red) is clearly visible from the inlet up to a
locationwhere the Dean vortices also appear in the ow, indicated by
twosmaller envelops embedded in the main iso-value surface. At
theearly part of the curved section, weak vortices appear at
bothupper and lower duct sections. These cannot be recognised
asDean vortices because the ow in this region is developing
againstcurvature effect. With the disappearance of these initially
formedvortices, the ow could be considered as fully developed.
There-after, the extra vortices formed due to centrifugal force
areregarded as Dean vortices. This observation states the
owdevelopment concept in curved channels.
The illustration on the right depicts the contours of
constantouter wall pressure gradient, discussed in Section 3.2.2.
Theprole shows the presence of adverse pressure gradient
(whiteareas) in the vicinity of inlet. This region extends
downstream upto a location where the hydrodynamic instability is
triggered,where ow reversal occurs with an observed (colour) change
incontour prole.
It is noted that the vortex core iso-surfaces and
pressuregradient contours show a remarkable similarity in
formationwithin the duct and indentify hydrodynamic instability
in
an vortices at K 155, Ar 4, g 5.conformation with each
technique. This signies the consistencyand validity of both
criteria as means of detecting hydrodynamicinstability. However,
the helicity criterion evidently predictsa lower K for instability
than the adverse pressure gradientcriterion. For example, from Fig.
6 for Ar 4, g 5, thedimensionless helicity criterion predicts the
critical Deannumber to be 100 while it is noted to be 155 by the
adversepressure gradient approach in Fig. 7. This discrepancy
arisesbecause; the helicity criterion detects the formation of
Deanvortices within the uid as per assigned H* value while
theadverse pressure gradient approach can only detect vorticeswhen
they have migrated to the outer wall, which occurs ata slightly
increased ow rate (or K).
0
50
100
150
200
250
1 2 3 4 5 6
Criti
cal D
ean
nu
mbe
r
Aspect ratio
Cheng et al [7] (Present Study (Present Study (
Fig. 10. Variation of critical Dean number with duct aspect
ratio.
-
3.3. Effect of duct aspect ratio
The effect of aspect ratio is demonstrated in Fig. 10 for a
curva-ture ratio of 5. It is noted that the critical Dean number
initiallyincreases with the duct aspect ratio and then falls away
for higherK. This conforms to the previously reported experimental
andnumerical observations [7,16,18]. Figure also compares the
criticalDean number variation of the present numerical model with
theresults of Chen et al. [7]. It indicates a good agreement
although thecurrent model predicts lower ow conditions for
hydrodynamicinstability. This is attributed to the fact that the
current modelidenties (with helicity criterion) the hydrodynamic
instabilitywithin the uid mass before the Dean vortices appear at
the outer
wall boundary, whereas the experimentation of Chen et al.
[7]could detect the Dean vortices upon their migration to the
outerwall that occur at slightly higher K.
3.4. Wall heating effect and thermal analysis
Fig. 11 illustrates the dimensionless helicity contours
withexternal heating applied to the outer duct wall (left-side)
fordifferent ow Dean numbers along with the corresponding pres-sure
gradient proles at the outer wall for aspect ratio xed at 3.
It is clearly evident that the outer wall heating causes the
hel-icity contours to become asymmetrical in comparison to
isothermalcases depicted in Figs. 4 and 6. This essentially arises
from the
T.T. Chandratilleke et al. / International Journal of Thermal
Sciences 59 (2012) 75e8684Fig. 11. Dimensionless helicity at the
end of curvature and correspondent pressure gradient at other wall
in different Dean numbers (Ar 3, outer wall located at
left-side).
-
Fig. 12. Temperature on other wall at the end of curvature (q
180) in different Dean numbers (Top boundary repersents outer duct
wall).
T.T. Chandratilleke et al. / International Journal of Thermal
Sciences 59 (2012) 75e86 85interaction between the heating-induced
buoyancy force and thecentrifugal force that drives secondary
vortices. As shown inFig. 11(a) for low ow rate, the
heating-induced buoyancy ow(natural convection) is relatively
stronger and acts to assist therotation of secondary vortex in the
upper half of the duct whileopposing the rotation of the bottom
vortex. Consequently, uppervortex becomes larger with subdued
bottom vortex. Withincreasing ow rate, the centrifugal forces
become stronger and thesecondary vortices are inuenced to a lesser
extent by the presenceof buoyancy ow due to heating. Consequently,
the ow contoursgradually migrate towards a symmetrical pattern, as
progressivelyillustrated by Fig. 11(b)(c)(d), where the appearance
Dean vorticesis also evident.
The outer wall pressure gradient proles provide further
insightinto the formation of Dean vortices. Similar to isothermal
cases, thelocations of Dean vortices exactly coincide with the
positions1
2
3
4
5
6
7
8
9
-0.5 -0.3 -0.1 0.1 0.3 0.5
Nu
y/a
K=125 K=250
K=375 K=500
Fig. 13. Local Nusselt number at outer wall at duct exit.where
the pressure gradient undergoes value reversal (positive tonegative
change). The regions of adverse pressure gradient act astriggering
points for boundary layer near the outer wall to separateand form
into vortex mass. InFig. 11(a), only one such location isnoted in
the absence of Dean vortices while in Fig. 11(b)(c)(d),multiple
occurrences are observed at the outer wall.
The boundary layer separation process at the outer wall
stim-ulates ow mixing and contributes to thermal enhancement
incurved passages, because the heated uid is transported into
thebulk uid by the secondary vortices. This process is
clearlydemonstrated in Fig. 12 that shows uid temperature contours.
InFig. 12(b)(c)(d), hot plume of uid is seen to penetrate into the
bulkuid promoting heat transfer from the outer wall to uid.
Instraight passages, this uid mixing mechanism does not exist dueto
absence of secondary ow. As such, curved passages carryintrinsic
thermal enhancement capabilities, as illustrated by Fig. 13.3
8
13
50 150 250 350 450 550
Nu
Dean numberConstant Heat Flux
4.5
5.5
6.5
500 1500 2500 3500
Nu
Heat Flux (W/m2) Constant Dean number
Fig. 14. Average Nusselt number at outer wall at duct exit.
-
Fig. 13 depicts the variation of local Nusselt number at the
outerwall at duct exit for several values of Dean number. It is
seen thatthe Nusselt number rapidly increases with higher ow rate
(higherDean number) indicating the inuence of secondary ow and
theuid mixing effectiveness thereof. The peak points of the
prolecoincide with the locations of value reversal in wall
pressuregradient (or ow separation). Fig. 14 shows the variation of
averageouter wall Nusselt number with heat ux (at constant
Deannumber) and that with Dean number (constant heat ux). It
isevident that, increasedwall heat ux under constant ow rate
leadsto slightly reducing thermal performance, which is attributed
tostronger buoyancy ow that dampens uid separation at the
outerwall. Under constant heat ux conditions, increased ow rate
tendsto generate better ow separation at the outer wall
permittingbetter heat ow to uid from the heated wall.
4. Conclusions
The paper presents an extensive numerical study that
develops
[2] H.K. Moffatt, Viscous and resistive eddies near a sharp
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convection andentropy generation in a curved rectangular duct with
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T.T. Chandratilleke et al. / International Journal of Thermal
Sciences 59 (2012) 75e8686a new simulationmodel for secondary
owwithin curved ducts. Themodel incorporates a dimensionless
helicity (vortex structures)function to account for realistic
3-dimensional representation ofsecondary vortex ow and as such,
shows greater improvement overprevious numerical simulations. The
analysis accurately predicts theow and thermal behaviour in
compliance with experimental andnumerical work published in
literature. The results generated exten-sively examine the effects
of uidow rate, duct aspect ratio and heatux. Reporting for the rst
time in literature, the model introducestwo new concepts for
determining the onset of hydrodynamic insta-bility, one based on
assigned helicity threshold and the other onadverse pressure
gradient at outer duct wall. Unlike previousmethods, both
approaches can be readily integrated into a computa-tional scheme
for accurate and reliable identication of Dean vortexgeneration. It
is noted that the adverse pressure gradient method,
byvirtueover-predicts the onsetof instability. The analysis
indicates thatthe interaction between the heating-induced buoyancy
force and thecentrifugal force in curved passage produces outer
wall pressureprole that lead touidowseparation at the outerwall.
This processstimulates uid mixing that results in thermal
enhancement.
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42 (2002) 187e198.
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[19] S. Yanase, T. Watanabe, T. Hyakutake, Traveling-wave
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Vortex structure-based analysis of laminar flow behaviour and
thermal characteristics in curved ducts1. Introduction2. Model
description and numerical analysis3. Results and discussion3.1.
Effect of flow rate3.2. Identification of hydrodynamic
instability3.2.1. Determination of hydrodynamic instability -
dimensionless helicity criterion3.2.2. Determination of
hydrodynamic instability - adverse pressure gradient
criterion3.2.3. Three-dimensional vortex core representation of
hydrodynamic instability
3.3. Effect of duct aspect ratio3.4. Wall heating effect and
thermal analysis
4. ConclusionsReferences