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: T.Chandratilleke@curtin.edu.au (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

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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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[18] S. Yanase, R.N. Mondal, Y. Kaga, Numerical study of non-isothermal ow withconvective heat transfer in a curved rectangular duct, International Journal ofThermal Sciences 44 (2005) 1047e1060.

[19] S. Yanase, T. Watanabe, T. Hyakutake, Traveling-wave solutions of the ow ina curved-square duct, Physics of Fluids 20 (2008) 124101.

[20] H. Fellouah, A. Ould El Moctar, H. Peerhossaini, A criterion for detection of theonset of Dean instability in Newtonian uids, European Journal of MechanicsB/Fluids 25 (2006) 505e531.

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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