Dean-Taylor flow with convective heat transfer through a coiled duct

Accepted Manuscript

Dean-Taylor Flow with Convective Heat Transfer through a Coiled Duct Md. Zohurul Islam , Rabindra Nath Mondal , M.M. Rashidi PII: DOI: Reference:

S0045-7930(17)30073-7 10.1016/j.compfluid.2017.03.001 CAF 3412

To appear in:

Computers and Fluids

Received date: Revised date: Accepted date:

20 August 2015 15 January 2017 2 March 2017

Please cite this article as: Md. Zohurul Islam , Rabindra Nath Mondal , M.M. Rashidi , Dean-Taylor Flow with Convective Heat Transfer through a Coiled Duct, Computers and Fluids (2017), doi: 10.1016/j.compfluid.2017.03.001

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Highlights: Role of secondary vortices on convective heat transfer.

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Hydrodynamic instability and vortex generation for rotating curved duct flow.

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Strong interaction between buoyancy force and centrifugal-Coriolis instability.

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Two- to multi-vortex solutions for rotating and co-rotating cases.

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Dean-Taylor Flow with Convective Heat Transfer through a Coiled Duct

1)

Department of Mathematics and Statistics, Jessore University of Science and Technology Jessore-7408, Bangladesh 2)

Department of Mathematics, Jagannath University, Dhaka-1100, Bangladesh

Shanghai Key Lab of Vehicle Aerodynamics and Vehicle Thermal Management Systems,

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3)

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Md. Zohurul Islam1*, Rabindra Nath Mondal2 and M. M. Rashidi3, 4

4800 Cao An Rd., Jiading, Shanghai 201804, China 4)

ENN-Tongji Clean Energy Institute of Advanced Studies, Shanghai, China *

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Email: [email protected]

Abstract

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Due to engineering application and its intricacy, flow in a rotating coiled duct has become

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one of the most challenging research fields of fluid mechanics. In this paper, a spectralbased numerical study is presented for the fully developed two-dimensional flow of viscous

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incompressible fluid through a rotating coiled rectangular duct. The emerging parameters

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controlling the flow characteristics are the rotational parameter i.e. the Taylor number, Tr; the Grashof number, Gr; the Prandtl number, Pr and the pressure-driven parameter i.e. the Dean number, Dn. The rotation of the duct about the centre of curvature is imposed in both the positive and negative direction and combined effects of the centrifugal, Coriolis and buoyancy forces are investigated, in detail, for two cases of the Dean Numbers, Case I: Dn = 1000 and Case II: Dn = 1500. For positive rotation, we investigated unsteady solutions 2

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for 0  Tr  500 , and it is found that the chaotic flow turns into steady-state flow through periodic or multi-periodic flows, if Tr is increased in the positive direction. For negative rotation, however, unsteady solutions are investigated for 700  Tr  50 , and it is found that the unsteady flow undergoes through various flow instabilities, if Tr is increased in the

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negative direction. Typical contours of secondary flow patterns and temperature distributions are obtained at several values of Tr, and it is found that the unsteady flow consists of asymmetric two- to eight-vortex solutions. The present study demonstrates the role of secondary vortices on convective heat transfer and it is found that convective heat

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transfer is significantly enhanced by the secondary flow; and the chaotic flow, which occurs at large Dn’s, enhances heat transfer more effectively than the steady-state or periodic solutions. This study also shows that there is a strong interaction between the heating-

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induced buoyancy force and the centrifugal-Coriolis instability in the curved channel that

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stimulates fluid mixing and consequently enhances heat transfer in the fluid.

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Keywords: Rotating curved duct; secondary vortex; unsteady solutions; Dean number;

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Taylor number; time evolution.

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1. Introduction

Flow and heat transfer through curved ducts and channels have attracted much attention

because of their enormous applications in fluids engineering, such as in turbo-machinery, refrigeration, air conditioning systems, heat exchangers, rocket engine, internal combustion engines and blade-to-blade passages in modern gas turbines. In a curved duct, centrifugal forces are developed in the flow due to channel curvature causing a counter rotating vortex 3

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motion applied on the axial flow through the channel. This creates characteristics spiraling fluid flow in the curved passage known as secondary flow. At a certain critical flow condition and beyond, additional pairs of counter rotating vortices appear on the outer concave wall of curved fluid passages which are known as Dean Vortices, in recognition of

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the pioneering work by Dean [1]. After that, many theoretical and experimental investigations have been done by keeping this flow in mind; for instance, the articles by Berger et al. [2], Nandakumar and Masliyah [3], Ito [4] and Rashidi et al. [5] may be referenced.

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The fluid flow in a rotating curved duct generates centrifugal and Coriolis force. Such rotating passages are used in many engineering applications e,g. in cooling system for conductors of electrical generators. For isothermal flows of a constant property fluid, the

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Coriolis force tends to generate vortices while centrifugal force is purely hydrostatic (Zhang

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et al. [6]). When a temperature induced variation of fluid density occurs for non-isothermal flows, both Coriolis and centrifugal type buoyancy forces can contribute to the formation of

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vortices. These two effects of rotation either enhance or counteract each other in a non-

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linear manner depending on the direction of wall heat flux and the flow domain. Therefore, the effect of system rotation is more subtle and complicated, and yields new richer features

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of flow and heat transfer for non-isothermal flow in a rotating curved channel. Ishigaki [7] examined the flow structure and friction factor numerically for both the counter-rotating and co-rotating curved circular pipe with a small curvature. Rashadi et al. [8, 9] examined thermal-diffusion effects on combined heat and mass transfer of a steady MHD convective and slip flow due to a rotating disk with viscous dissipation and mixed convective flow about an inclined flat plate in a porous medium. They obtained approximate analytic solutions by

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the combination of the differential transform method (DTM) and the Pade´ approximants for the problems. Selmi et al. [10] and Dennis and Ng [11] examined the combined effects of system rotation and curvature on the bifurcation structure of two-dimensional flows in a rotating curved duct with square cross section. Hocking [12] developed a solution based on

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the momentum integral method for rotating curved duct. Miyazaki [13] examined the solution when the rotation is in the same direction as the Coriolis force emphasizing the centrifugal force caused by the duct curvature, which is known as co-rotating case. Wang and Cheng [14] and Daskopoulos and Lenhoff [15] carried out a bifurcation study of the

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flow through a circular pipe and employed the finite volume method. They examined the flow characteristics and heat transfer in curved square ducts for positive rotation and found reverse secondary flow for the co-rotation cases.

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Unsteady solutions of fully developed curved duct flows were initiated by Yanase and

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Nishiyama [16] for a rectangular cross section. In that study, they investigated unsteady solutions for the case where dual solutions exist. The time-dependent behavior of the flow

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in a curved rectangular duct of large aspect ratio was investigated, in detail, by Yanase et al.

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[17] numerically. They performed time-evolution calculations of the unsteady solutions with and without symmetry condition. Wang and Yang [18] performed numerical as well as

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experimental investigations of periodic oscillations for the fully developed flow in a curved square duct. They showed, both experimentally and numerically, that a temporal oscillation takes place between symmetric/asymmetric 2-cell and 4-cell flows when there are no stable steady solutions. Mondal et al. [19] performed numerical prediction of non-isothermal flows through a rotating curved square duct and revealed some of such new features. Recently, Mondal et al. [20, 21] investigated combined effects of centrifugal and Coriolis 5

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instability of the isothermal/non-isothermal flows through a rotating curved rectangular duct numerically. The secondary flow characteristics in a curved square duct were investigated experimentally by using visualization method by Yamamoto et al. [22]. Three-dimensional incompressible viscous flow and heat transfer in a rotating U-shaped square duct were

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studied numerically by Nobari et al. [23]. However, transient behavior of the unsteady solution is not yet resolved, in detail, for the flow through a rotating curved rectangular duct with large pressure gradients, which motivated the present study to fill up this gap.

One of the most important applications of curved duct flow is to enhance the thermal

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exchange between two sidewalls, because it is possible that the secondary flow may convey heat and then increases heat flux between two sidewalls. Chandratilleke and Nursubyakto [24] presented numerical calculations to describe the secondary flow characteristics in the

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flow through curved ducts of aspect ratios ranging from 1 to 8 that were heated on the outer

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wall, where they studied for small Dean numbers and compared the numerical results with their experimental data. Yanase et al. [25] and Mondal et al. [26] studied time-dependent

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behavior of the unsteady solutions for curved rectangular/square duct flow and showed that

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secondary flows enhance heat transfer in the flow. Recently, Norouzi et al. [27, 28] investigated inertial and creeping flow of a second-order fluid with convective heat transfer

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in a curved square duct by using finite difference method. The effect of centrifugal force due to the curvature of the duct and the opposing effects of the first and second normal stress difference on the flow field were investigated in that study. Chandratilleke et al. [29] presented a numerical investigation to examine the secondary vortex motion and heat transfer process in fluid flow through curved rectangular ducts of aspect ratios 1 to 6. The study formulated an improved simulation model based on 3-dimensional vortex structures 6

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for describing secondary flow and its thermal characteristics. To the best of the authors' knowledge, however, there has not yet been done any substantial work studying the unsteady flow behavior through a curved rectangular duct in the presence of buoyancy induced centrifugal-Coriolis force. But from the scientific as well as engineering point of

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view it is quite interesting to study such kind of flows, because this type of flow is often encountered in engineering applications such as in rotating machinery, centrifugal pumps. In this paper, we investigate transient flow of viscous incompressible fluid through a coiled rectangular duct of aspect ratio 2 by using a spectral-based numerical scheme, and

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show an enhancement of convective heat transfer by secondary flows. Studying the effects of rotation on the unsteady flow characteristics, caused by the combined action of the centrifugal, Coriolis and buoyancy forces, is an important objective of the present study.

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Nomenclature Dn : Dean number Tr : Taylor number Gr : Grashof number h : Half height of the cross section d : Half width of the cross section L : Radius of the curvature Pr : Prandtl number t : Time

T : Temperature u : Velocity components in the x  direction v : Velocity components in the y  direction

: Velocity components in the z  direction : Horizontal axis : Vertical axis z : Axis in the direction of the main flow  : Resistance coefficient w x y

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Greek letters  : Kinematic viscosity  : Curvature of the duct  : Viscosity  : Density  : Thermal diffusivity  : Sectional stream function

2. Geometrical Model and Mathematical Formulations

Consider a hydro-dynamically and thermally fully developed two-dimensional (2D) flow of viscous incompressible fluid through a rotating coiled rectangular duct, whose height and width are 2h and 2d, respectively. The coordinate system with the relevant notation is 7

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shown in Fig. 1, where x′ and y′ axes are taken to be in the horizontal and vertical directions respectively, and z’ is the axial direction. The system rotates at a constant angular velocity around the y′ axis. It is assumed that the outer wall of the duct is heated while the inner wall cooled, the top and bottom walls being thermally insulated. The temperature of the

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outer wall is T0  T and that of the inner wall T0  T , where T > 0. It is also assumed that the flow is uniform in the axial direction, which is driven by a constant pressure gradient along the center-line of the duct as shown in Fig. 1. y'

Pressure-driven main flow

y

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x

L

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z

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x'

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Upper wall

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y'

Inner wall (Cold)

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2d

g x

z

Lower wall

Outer wall (Hot)

z'

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(a) (b) Figure 1. (a) Coordinate system of the rotating curved channel (b) Cross section of the curved channel.

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The continuity, Navier-Stokes and energy equations, in terms of dimensional variables, are expressed as: Continuity equation:

u  v  u     0, r  y  r 

(1)

Momentum equations:   2u   2u  1 u  u   u  u  u  w 2 1 P   u  v       , t   r y  r  r   r  2 y  2 r  r  r  2 

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(2)

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  2 v  1 v   2 v   v  v  v  1 P    u v       gT , 2 2   t   r y   y  r  r  r  y  

(3)

  2 w  2 w 1 w w  w w w u w 1 1 P  u  v     2    , t   r y  r  r  z  y  2 r  r  r  2   r 

(4)

(5)

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  2T  1 T   2T   T  T  T  Energy equation:  u  v  2   , t   r y  r  r  y  2   r 

where r   L  x , and u , v  and w are the dimensional velocity components in the x , y  and z  directions respectively and these velocities are zero at the wall. Here, P  is the

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dimensional pressure, T  is the dimensional temperature and t  is the dimensional time. In the above formulations  ,  ,  ,  and g are the density, the kinematic viscosity, the coefficient of thermal expansion, the coefficient of thermal diffusivity and the gravitational

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acceleration, respectively. Thus in Eqs. (1) to (5) the variables with prime denotes the dimensional quantities. The dimensional variables are then non-dimensionalized by using

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the representative length d and the representative velocity U 0   / d . We introduce the non-

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dimensional variables defined as:

u v 2 y z  x 1  , v , w w, x     , y  , z  , U0 U0 Uo d d d 

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u

T

U T d P , t  0 t ,   , P  T  d L  U 02

where u, v and w are the non-dimensional velocity components in the x, y and z directions, respectively; t is the non-dimensional time, P is the non-dimensional pressure,

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 is the non-dimensional curvature and temperature is non-dimensionalized by T . Henceforth, all the variables are non-dimensionalized if not specified. A new coordinate variable y is then introduced in the y direction as y  a y , where a  (h / d )  2 is the aspect ratio of the duct cross section. Since the flow field is assumed to

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be uniform in the z -direction, the sectional stream function  is introduced in the x and y directions as

1  1  (6) , v . 1  x  y 1  x x Then the basic equations for the axial velocity w, the stream function  and the temperature T

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u

are derived from the Navier-Stokes equations and the energy equation as, (7)

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w 1 ( w, )  2w 1   w  (1   x)  Dn    (1   x) w  w   Tr 2 t 2 ( x, y) 1   x 2 (1   x) y x y

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   ( 2 , ) 1     1 1  3   2        2 2    2   1   x x  t 2 (1   x) ( x, y) 2 1   x 2  y  1   x x x 2  

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  2 3 2    2    3      x xy  1   x 2  x 2 1   x x

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

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1 w T 1   w  Gr (1   x)  Tr , x 2 y 2 y

T 1   T  1 (T , )   2T   t Pr  1   x x  (1   x) ( x, y) where,  2 



2

x

2

4



2

y

2

,

( f , g )  ( x, y )

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f g x y

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f g y x

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 2   2  22  1   x  x 

(8)

(9)

.

(10)

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The non-dimensional parameters Dn , the Dean number; Tr , the Taylor number; Gr, the Grashof number and Pr, the prandtl number, which appear in equations (7) to (9) are defined as:

The rigid boundary conditions for w and  are used as

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2 2 WT d 3 Gd 3 2d  g Td 3  Dn  , Tr  , Gr  , Pr   L   2

  ( 1, y)  ( x, 1)  0 x y and the temperature T is assumed to be constant on the walls as w(1, y)  w( x, 1)   (1, y)   ( x, 1) 

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T (1, y)  1, T (1, y)  1, T ( x, 1)  x

(11)

(12)

(13)

In the present study, Dn and Tr are varied while Gr,  and Pr are fixed as Gr =100,

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  0.1 and Pr = 7.0 (water).

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3. Numerical Calculations

3.1. Method of numerical calculation

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In order to solve the Equations (2) to (4) numerically, the spectral method is used. This

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is the method which is thought to be the best numerical method for solving the NavierStokes as well as energy equations (Gottlieb and Orszag [30]). By this method the variables

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are expanded in a series of functions consisting of the Chebyshev polynomials. That is, the expansion functions n (x) and  n (x) are expressed as

n ( x)  (1  x 2 ) C n ( x), 

  n ( x)  (1  x 2 ) 2C n ( x)

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



where Cn ( x)  cos n cos1( x) is the n-th order Chebyshev polynomial. w( x, y, t ),  ( x, y, t ) and T ( x, y, t ) are expanded in terms of n (x) and  n (x) as

M N  w( x, y, t )    wm n (t ) m ( x) n ( y ),  m0 n0   M N   ( x, y, t )     m n (t ) m ( x) n ( y),  m0 n0   M N T ( x, y, t )    Tm n  m ( x) n ( y )  x,   m0 n0

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(15)

where M and N are the truncation numbers in the x and y directions respectively. The

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expansion coefficients wmn , mn and Tmn are then substituted into the basic Eqs. (7) - (9) and the collocation method is applied. As a result, the nonlinear algebraic equations for wmn , mn

   

  , M  2  i

   

yi  cos  1 

  N  2  i

(16)

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xi  cos  1 

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and Tmn are obtained. The collocation points are taken to be

where i  1,..., M  1 and j  1,..., N  1 . The convergence is assured by taking  p < 10 10 ,

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where subscript p denotes the iteration number and  p is defined as:



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M N  p1  w p  p     wmn mn m0 n0 

   mn p1  mnp   Tmn p1  Tmnp  . 2

2

2

(17)

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3.2. Numerical Accuracy The accuracy of the numerical calculations is investigated for the truncation numbers M

and N used in this study. For good accuracy of the solutions, N is chosen equal to 2M . The grid sizes are taken as 14  20, 14  28, 16  32, 18  36 and 20  40 . Table 1 shows that M = 16 and N = 32 gives sufficient accuracy of the numerical solutions. 12

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Table 1. The values of  and w(0,0) for various M and N at Dn  1500 and Tr  195 .

 0.36057358 0.36089734 0.36090702 0.36090845 0.36090815

N 20 28 32 36 40

3.3. Time-evolution Calculation

w(0,0) 180.0593 180.2898 180.4822 180.5519 180.6031

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M 14 14 16 18 20

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In order to calculate the unsteady solutions, we use the Crank-Nicolson and AdamsBashforth methods together with the function expansion (15) and the collocation method. Details of this method are discussed in Gottlieb and Orszag [30]. By applying the Crank-

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Nicolson and the Adams-Bashforth methods to the non-dimensional basic equations (7)-(9), and rearranging, we get

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 1 2   1 2  w(t )  w(t  t )    w(t  t )     w(t )   x 2  2  t  t  t  2 w(t ) w(t )  Dn   x 2 w(t )   1  x x 3 1 P (t )  P (t  t ), 2 2

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

(18)

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 1 2   1 2   1   (t )  (t  t )       2 (t  t )      2 (t )    2  2  1   x t   x x   t  t     2 (t ) 3 2  (t )    2 1   x   2 (t )  3  x (1   x)  x   x2 1   x 2    Gr (1   x)

T (t ) 3 1  Q(t )  Q(t  t ) x 2 2

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(19)

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 1 2   1 2  T (t )  T (t  t )    T (t  t )     T (t )   x t  t 2 Pr   t 2 Pr  1  T (t )     x 2T (t )  Pr  x  3 1  R (t )  R (t  t ). 2 2

(20)

Then applying the Adams-Bashforth method for the second term of R.H.S of Eqs. (18), (19)

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and (20) and simplifying we calculate w(t  t ) ,  (t  t ) and T (t  t ) by numerical computation.

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3.4. Resistance coefficient

The resistance coefficient  is used as the representative quantity of the flow state. It is also called the hydraulic resistance coefficient, and is generally used in fluids engineering, defined as

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P1  P2  1     w  2  z dh 2

stands for the mean over the

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where quantities with an asterisk denote dimensional ones,

(21)

calculated by

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cross section of the duct and d h is the hydraulic diameter. The mean axial velocity  w  is

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 w  



1

4 2 d



1

dx  w( x, y, t )dy.

1

1

(22)

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Since P1  P2 z  G ,  is related to the mean non-dimensional axial velocity w as 

16 2 Dn , 2 3w

(23)

where  w  2 d  w   . Equation (23) will be used to find the resistance coefficient of the flow state by numerical calculations. 4. Results and Discussion 14

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We take a coiled rectangular duct of finite aspect ratio and rotate it around the centre of curvature with an angular velocity WT in both the positive and negative directions. In this paper, time evolution calculations of  are performed for the non-isothermal flow with stream wise analysis of the secondary flows over a wide range of Tr for two cases of the

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Dean numbers, Case I: Dn = 1000 and Case II: Dn = 1500. 4.1 Case I: Dean Number, Dn =1000 4.1.1 Positive Rotation

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In order to study the nonlinear behavior of the unsteady solutions, time evaluation calculations of  are performed for positive rotation of the duct (0  Tr  500) at Dn  1000 and Gr = 100. Figure 2(a) shows time evolution result for Tr = 0 and Dn = 1000. It is found

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that the unsteady flow at Tr = 0 oscillates irregularly that means the flow is chaotic which is

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well justified by drawing the phase space as shown in Fig. 2(b) in the    plane, where    dxdy and this quantity is zero at the cross-section. As seen in Fig. 2(b), the

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flow creates multiple orbits, which shows that the flow is chaotic. Figure 2(c) shows typical

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contours of secondary flow patterns and temperature profiles for Tr = 0, where it is observed that the streamlines of the secondary flow consist of two opposite vortices; one is

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an outward flow (anticlockwise direction) shown by solid line and the other one inward flow (clockwise direction) shown by dotted lines. The flow is accelerated due to combined action of the centrifugal, Coriolis and buoyancy forces; centrifugal force is created due to the motion through a curved channel, Coriolis force due to the rotation of the duct around the vertical axis while buoyancy forces because of the thermal gradient. To draw the

15

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contours of secondary flow patterns (stream function) and temperature profiles (isotherms), the increments   0.8 and ∆T = 0.1 are used. The same increments of  and T are used for all the figures in this study unless specified. In the figures of the secondary flows, solid lines (  0 ) show that the secondary flow is in the counter clockwise direction while the

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dotted lines   0 in the clockwise direction. As seen in Fig. 2(c), the unsteady flow is a two- and four-vortex solution. Then we perform time evolution of  for Tr = 100 as shown in Fig. 3(a). It is found the unsteady flow oscillates in an irregular pattern, but the mode of

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frequency is low which shows that the flow is a transitional chaos (Mondal et al. [31]).

(b)

(b)

9.00

10.00

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ED

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(a)

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CE



(c)

T

t

7.00

8.00

16

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Figure 2. Unsteady solution for Dn = 1000 and Tr = 0. (a) Time evolution of , (b) Phase

(b)

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ED



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(a)

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space, (c) Streamlines (top) and isotherms (bottom) for 7.0  t  10.0

T

AC

CE

PT

(c)

t 33.50 34.00 34.50 35.00 Figure 3. Unsteady solution for Dn = 1000 and Tr = 100. (a) Time evolution, (b) Phase space, (c) Streamlines (top) and isotherms (bottom) for 33.50  t  35.00

Then in order to study the nature of the flow, in more detail, we draw the phase space of the time evolution result for Tr = 100 as shown in Fig. 3(b). Figure 3(b) shows that the flow 17

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oscillates in an irregular orbit that means the flow at Tr = 100 is a transitional chaos that turns into multi-periodic or periodic flows as Tr is increased. Then contours of streamlines and isotherms are obtained for Tr = 100 as shown in Fig. 3(c). It is found that the unsteady flow is an asymmetric two-vortex solution where the streamlines of the temperature

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contours are distributed symmetrically. It should be noted that the patterns of secondary flows are fundamentally different from those in a straight channel; even at low flow rate (low Dean number), the flow profile has two large counter-rotating vortices. This vortex flow is developed consequent to the centrifugal and buoyancy forces induced by the duct

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stream-wise curvature and thermal gradient. We perform time history of  for Dn = 1000 and Tr = 105 as shown in Fig. 4(a). Figure 4(a) shows that the unsteady flow is a periodic oscillation, which is well justified by drawing the phase space of the time history result as

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shown in Fig. 4(b). Figure 4(b) shows that the orbit of the oscillation is same at a certain

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time interval that assures that the fluid particles move through the same phase. Streamlines of the secondary flows and isotherms of temperature distributions are shown in Fig. 4(c) for

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Tr = 105 at 7.50  t  10.60 where we observe that the streamlines are symmetrically

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distributed and produce asymmetric two-vortex solutions. The streamlines of temperature profiles are found to be consistent with the secondary flow distribution. The rotational

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speed is then increased and it is found that the unsteady flow is a steady-state solution for any value of Tr in the range 105  Tr  500 . Figures 5(a) and 6(a) respectively show, for example, time evolution results for Tr = 120 and Tr = 200 at Dn = 1000, where steady-state solutions are observed. Since the flow is steady-state, a single contour of each of the secondary flow patterns and isotherms is shown in Figs. 5(b) and 6(b) for Tr = 120 and Tr =

18

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200 respectively, where we see that the steady-state flow is an asymmetric two-vortex

(a)

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(b)

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solution.

CE

PT

(c)

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

T

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t 7.50 8.20 9.90 10.60 Figure 4. Unsteady solution for Dn = 1000 and Tr = 105. (a) Time evolution, (b) Phase space, (c) Contours of secondary flow patterns (top) and temperature profiles (bottom) for 7.50  t  10.60.

19

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

T

(a) (b) Figure 5. Unsteady solution for Dn = 1000 and Tr = 120. (a) Time evolution, (b)

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Streamlines (left) and isotherm (right) at t  10.



T

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(a) (b) Figure 6. Unsteady solution for Dn = 1000 and Tr = 200. (a) Time evolution, (b)

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4.1.2 Negative Rotation

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Streamlines (left) and isotherm (right) at t  8.0

Then we investigated unsteady solutions for the case of negative rotation

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( 700  Tr  50 ) at Dn = 1000. Negative rotation means that the rotational direction is

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opposite to the main flow direction. Figure 7(a) shows time dependent flow for Tr = -50 at Dn = 1000. It is found that the time dependent flow at Tr = -50 is a non-periodic oscillating flow, which oscillates in an irregular pattern that means the flow is chaotic. This chaotic oscillation is well justified by tracing out the phase space of the time evolution result as shown in Fig. 7(b). Streamline contours and isotherms for the corresponding flow parameters are shown in Fig. 7(c), where it is found that the chaotic oscillation at Tr = -50 is 20

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a four- to six-vortex solution. It is observed that the streamlines of the secondary flows and isotherms are significantly distributed that generates more heat which is transferred from the outer wall (heated wall) to the fluid. Then we perform time history of  for Tr = -250 and Dn = 1000 as shown in Fig. 8(a). It is found that the unsteady flow at Tr = -250 is a

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periodic oscillation, which is well justified by drawing the phase space as shown in Fig. 8(b). Typical contours of secondary flow patterns and temperature profiles for the

corresponding flow parameters are shown in Fig. 8(c), where it is found that the periodic oscillation at Tr = -250 is an asymmetric four-vortex solution. We then performed time

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history of  for Tr = -300 and Dn = 1000 as shown in Fig. 9(a). It is found that the unsteady flow at Tr = -300 is a periodic solution. This periodic oscillation is well justified by drawing

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the phase space of the time history result as shown in Fig. 9(b).

(b)

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CE

PT

(a)



(c)

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T t 6.00 6.50 7.00 7.50 Figure 7. Unsteady solution for Dn = 1000 and Tr = -50. (a) Time evolution, (b) Phase

(b)

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

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(a)

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space, (c) Streamlines of secondary flows (top) and isotherms (bottom) for 6.00  t  7.50

CE

PT

(c)

T

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t 15.90 15.95 16.00 16.05 Figure 8. Unsteady solution for Dn = 1000 and Tr = -250. (a) Time evolution, (b) Phase space, (c) Streamlines (top) and isotherms (bottom) for 15.90  t  16.05

(a)

(b) 22

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

T

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(c)

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t 4.92 5.22 5.44 5.65 Figure 9. Unsteady solution for Dn = 1000 and Tr = -300. (a) Time evolution, (b) Phase

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PT

space, (c) Streamlines (top) and isotherms (bottom) for 4.92  t  5.65.

Figure 9(b) shows that the flow creates a couple of orbits instead of a single orbit, so

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that the unsteady flow at Tr = -300 is a multi-periodic solution. Streamlines of secondary flows and isotherms of temperature distributions for Tr = -300 are shown in Fig. 9(c), where it is found that the multi-periodic oscillation at Tr = -300 is a four-vortex solution. If the rotational speed is increased in the negative direction up to Tr =-700, it is found that the flow remains steady-state. Figures 10(a) and 11(a) respectively show, for example, time evolution results for Tr = -340 and Tr = -500 at Dn = 1000, where steady-state flows are 23

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observed. Corresponding contours of secondary flow patterns and isotherms are shown in Figs. 10(b) and 11(b) for Tr = -340 and Tr = -500 respectively for a single contour of each, where we see that the secondary flow is a four-vortex solution. Temperature distribution is found to be consistent with the secondary vortices and a strong interaction is observed

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between the heating-induced buoyancy force and the centrifugal instability, which stimulates fluid mixing and thus results in thermal enhancement in the flow. In this study, it is found that, if the rotation is increased in the positive direction, the number of secondary vortices decreases and consequently heat transfer does not occur substantially; however in

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the case of negative rotation, heat transfer occurs more frequently because of strong

secondary vortices at the outer wall, and consequently heat transfer increases remarkably in

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the fluid.

T

CE

PT



(a) (b) Figure 10. Unsteady solution for Dn = 1000 and Tr = -340. (a) Time evolution, (b)

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Streamlines (left) and isotherm (right) for t  8.0



24

T

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(a) (b) Figure 11. Unsteady solution for Dn = 1000 and Tr = -500. (a) Time evolution, (b)

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Streamlines (left) and isotherm (right) at t  10.

4.2 Case II: Dean Number, Dn =1500 4.2.1 Positive Rotation

We perform time dependent analysis of flow for a strong centrifugal force

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at Dn  1500 for the positive rotation of the duct at 0  Tr  500 . Figure 12(a) shows time evolution result for Dn = 1500 at Tr = 0. It is found that the unsteady flow at Tr = 0 is a strongly chaotic solution, which is well justified by drawing the phase space as shown in

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Fig. 12(b). Figure 12(c) shows typical contours of secondary flow patterns and temperature profiles for Dn = 1500 and Tr = 0, where it is found that the unsteady flow is a four-vortex

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solution. Then we perform time history of  for Tr = 100 as shown in Fig. 13(a). It is found

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that the unsteady flow is a multi-periodic oscillation for Tr = 100. The multi-periodic oscillation is well justified by tracing out the phase space of the time evolution result as

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shown in Fig. 13(b). Typical contours of secondary flow patterns and temperature profiles

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are shown in Fig. 13(c) for Tr = 100. It is found that the multi-periodic oscillation is a fourvortex solution. If the rotational speed is increased, for example Tr = 195, the flow is still multi-periodic. Figures 14(a) and 14(b) respectively show time evolution result and its phase space for Tr = 195. Though Fig. 14(a) shows that the flow is periodic, however, its phase space (Fig. 14(b)) shows that it is a multi-periodic rather than periodic which creates multiple orbits. In this regard, it should be noted that, the occurrence of the chaotic state, as 25

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presented in the present study, is related with destabilization of the periodic or quasiperiodic solutions which reminds us the case of Lorenz attractor [32]. It may be possible that the transition in the present study is caused by a similar mechanism as that of RuelleTakens scenario [33] in the laminar flow. If we increase the rotational speed in the same

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direction up to Tr = 250, it is found that the flow becomes steady-state as we predict in the general scenario of fluid dynamics. Figure 15(a) shows time evolution result for Tr = 250 at Dn = 1500, and it is found that the unsteady flow is a steady-state solution for both the cases. Since the flow is steady-state, a single contour of the secondary flow pattern and

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isotherms is shown in Fig. 15(b) for Tr = 250 at Dn = 1500, and it is found that the unsteady flow for Tr = 250 is an asymmetric two-vortex solution. In this study, it is found that combined action of the centrifugal, Coriolis and buoyancy force help to increase the number

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of secondary vortices, and as the flow becomes chaotic, the number of secondary vortices

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increases and consequently heat is transferred substantially from the heated wall to the

(b)

AC

(a)

CE

PT

fluid.



26

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(c)

T

(b)

ED



M

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(a)

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t 5.00 5.50 6.00 6.50 Figure 12. Unsteady solution for Dn = 1500 and Tr = 0. (a) Time evolution, (b) Phase space, (c) Streamlines (top) and isotherms (bottom) for 5.00  t  6.50.

T

t

5.00

5.75

6.00

AC

CE

PT

(c)

6.54

Figure 13. Unsteady solution for Tr = 100 and Dn = 1500. (a) Time evolution, (b) Phase space, (c) Streamlines of secondary flows (top) and isotherms (bottom) for 5.00  t  6.54

(a)

(b) 27

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

ED

T

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(c)

t 10.50 11.00 11.50 12.00 Figure 14. Unsteady solution for Tr = 195 and Dn = 1500. (a) Time evolution, (b) Phase

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PT

space, (c) Streamlines of secondary flow (top) and isotherms (bottom) for 10.50  t  12.00



(a)

T

(b)

28

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Figure 15. Unsteady solution for Dn = 1500 and Tr = 250. (a) Time evolution, (b)

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Streamlines of secondary flow (left) and isotherm (right) at t  6.0

4.2.2 Negative Rotation

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Finally, we performed time evolution calculation for the negative rotation of the duct at 700  Tr  50 for Dn = 1500. Figure 16(a) shows time dependent solution for Tr = -100.

It is found that the unsteady flow at Tr = -100 is a chaotic oscillation, which is well justified

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by drawing the phase space as shown in Fig. 16(b). As seen in Fig. 16(b), the flow creates chaotic orbits in its path, so that the unsteady flow at Tr = -100 is a chaotic solution. Typical

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contours of streamlines and isotherms for the corresponding flow parameters are shown in

PT

Fig. 16(c), where it is found that the chaotic oscillation at Tr = -100 is a three- and fourvortex solution. Here we also observe that streamlines of temperature profiles are consistent

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with secondary vortices and that convective heat generation is stronger in the whole

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position of the contour. We continue this process and perform unsteady solution for Tr = 620 and show in Fig. 17(a). As seen in Fig. 17(a), the unsteady flow at Tr = -620 is also chaotic oscillation but its frequency has been decreased. To observe the mode of the oscillation of the unsteady flows, we trace out phase space for Tr = -620 as shown in Fig. 17(b). It is found that the unsteady flow at Tr = -620 is a weak chaotic solution. Typical contours of secondary flow patterns and temperature profiles for the corresponding flow 29

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parameters are shown in Fig. 17(c), where it is found that the chaotic oscillation at Tr = 620 is an asymmetric three- and four-vortex solution. Contours of temperature profiles show that the streamlines of the heat flow is uniformly distributed to all parts of the contour transferring heat from outer wall to the fluid, and the contribution of the rotation and

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pressure on secondary flows significantly change and increase the number of secondary vortices. It is clearly evident that heating the outer wall causes the temperature contours to become asymmetrical in comparison to isothermal cases. This essentially arises from the interaction between the heating-induced buoyancy force and the centrifugal force that drives

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secondary vortices. In this regard, it should be noted that the centrifugal force due to the channel curvature creates two effects; it generates a positive radial fluid pressure field in the duct cross section and induces a lateral fluid motion driven from inner wall towards the

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outer wall. This lateral fluid motion occurs against the radial pressure field generated by the

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centrifugal effect and is superimposed on the axial flow to create the secondary vortex flow structure. As the flow through the curved duct is increased (larger Dn), the lateral fluid

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motion becomes stronger and the radial pressure field is intensified. In the vicinity of the

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outer wall, the combined action of adverse radial pressure field and viscous effects slows down the lateral fluid motion and forms a stagnant flow region. Beyond a certain critical

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value of Dn, the radial pressure gradient becomes sufficiently strong to reverse the flow direction of the lateral fluid flow. A weak local flow re-circulation is then established creating an additional pair of vortices in the stagnant region near the outer wall. This flow situation is known as Dean’s hydrodynamic instability while the vortices are termed as Dean vortices.

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(b)

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(a)

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 (c)

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T

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t 6.00 6.50 7.00 7.50 Figure 16. Unsteady solution for Dn = 1500 and Tr = -100. (a) Time evolution, (b) Phase

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CE

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space, (c) Streamlines of secondary flows (top) and isotherms (bottom) for 6.00  t  7.50.

(a)

(b)

31

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 (c)

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T

t 6.50 7.20 7.90 8.60 Figure 17. Unsteady solution for Dn = 1500 and Tr = -620. (a) Time evolution, (b) Phase space, (c) Streamlines of secondary flows (top) and isotherms (bottom) for 6.50  t  8.60

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If the rotational speed is accelerated in the negative direction at Tr = -680, time

evolution result (Fig. 18(a)) shows that the unsteady flow passes in a regular pattern in a certain time interval, that means the flow is periodic, which is well confirmed by drawing

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the phase space as shown in Fig. 18(b). As seen in Fig. 18(b), the orbit of the flow does not

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intersect and moves in a regular pattern creating a single orbit. The streamlines of the secondary flows and isotherms (Fig. 18(c)) show that they lie in an asymmetric two-vortex

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solution. In fact, the periodic oscillation, which is observed in the present study, is a

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traveling wave solution advancing in the downstream direction which is well-justified in the recent investigation by Yanase et al. [34] for three-dimensional (3D) travelling wave

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solutions as an appearance of 2D periodic oscillation. If we raise the simulation for strongly negative rotation of the duct, for example, Tr = -700 or more, as shown in Fig. 19(a) for Tr = -700 it is found the periodic flow changes to a steady-state solution, which creates asymmetric two-vortex solution as shown in Fig. 19(b). In this study, it is found that the number of secondary vortices increases for the chaotic solution, which is occurred at large Dn’s compared to that of the periodic solutions, and consequently it is suggested that 32

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chaotic solutions enhance heat transfer more effectively than the periodic or steady solutions.

(b)

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(a)



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(c)

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T

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t 15.00 15.50 16.00 16.48 Figure 18. Unsteady solution for Dn = 1500 and Tr = -680. (a) Time evolution, (b) Phase

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space, (c) Streamlines of secondary flow (top) and isotherms (bottom) for

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15.00  t  16.48

(a)

(b)



33

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Figure 19. Unsteady solution for Dn = 1500 and Tr = -700. (a) Time evolution, (b) Phase space, (c) Streamlines of secondary flow (left) and isotherm (right) t = 8.0. Validation of the present study

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Here, we will discuss the validity of our numerical results with the experimental studies conducted by some authors. By using visualization method, Yamamoto et al. [35]

performed experimental investigations of the flow through a rotating curved square duct of

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curvature  = 0.03, where three of the duct walls, except the outer wall, rotate around the center of curvature at a constant rotational speed for both positive (Tr =150) and negative (Tr = -150) rotation of the duct walls. In the present study, however, we investigate flow characteristics for rotating the whole system (not the three walls only), and to show the

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validity of the present study we use the same curvature and rotational speed as Yamamoto

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et al. [35] used. Figures 20 and 21 show comparative study of the experimental vs. numerical results for the rotating curved square duct flow at different values of Dn. Figure

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20 shows experimental investigations by Yamamoto et al. [35] (left) and numerical results

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by Mondal [36] (right) for Tr = 150, while Fig. 21 shows experimental results by Yamamoto et al. [35] (left) and numerical results by Mondal [36] (right) for Tr = - 150. For

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the flow through a curved rectangular duct of curvature  = 0.032, Chandratilleke [37, 38] conducted experimental investigations of the flow for Gr = 0 without rotation. Figure 22 shows a comparative study of the experimental investigation obtained by Chandratilleke [37, 38] and the numerical result by Mondal [36] for Gr = 0. We see that in both the cases our numerical results have a good agreement with the experimental investigations. Note

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that, till now no experimental investigations are available for rotating curved rectangular

De = 114

(a)

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duct flows.

Dn = 115

De = 176.2

(b)

Dn = 415

Figure 20. Experimental vs. numerical results for rotating curved square duct flow at Tr =

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150. (a) & (b) Experimental result by Yamamoto et al. [35] (left) and numerical result by

De = 153

(a)

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Mondal [36] (right).

Dn = 156

De = 45.6

(b)

Dn = 476

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Figure 21. Experimental vs. numerical results for rotating curved square duct flow at Tr = -

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150. (a) & (b) Experimental result by Yamamoto et al. [35] (left) and numerical result by

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Mondal [36] (right).

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Dn = 262

(a)

Dn = 265

Dn = 321

(b)

Dn = 520

Figure 22. Experimental vs. numerical result for curved rectangular duct flow of aspect ratio 2. (a) & (b) Experimental results by Chandratilleke [36, 37] (left) and numerical result

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by Mondal [36] (right).

5. Discussions

In this section, a brief discussion on recent studies as well as plausibility of applying 2D calculations on curved duct flows will be given. First, we will discuss recent studies,

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both numerical and experimental, on curved duct flows. Yamamoto et al. [35] used

visualization method to perform experimental investigations of the flow through a rotating curved square duct and compared their results with the experimental data, where they

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obtained a good agreement between the two investigations. Chandratilleke et al. [29]

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presented extensive 3D computational study using helicity function that describes the secondary vortex structure and thermal behavior in the fluid flow through curved

PT

rectangular ducts of aspect ratios ranging from 1 to 6. A curvilinear coordinate system was used in that study facilitating effective grid definition for capturing vortex generation and

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permitting efficient evaluation of local pressure gradient. They obtained maximum eight-

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vortex solution for aspect ratio 4. Their simulation results were also validated against available experimental data. Norouzi and Biglari [39] performed, for the first time, an analytical solution of Dean flow inside a curved rectangular duct, where perturbation method was used to solve the governing equations. The main flow velocity (axial flow), vector plots of lateral velocity (secondary flows) and flow resistance ratios were obtained in that study. However, their study was limited to low Reynolds number and obtained 36

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maximum four-vortex solutions. Wu et al. [40] performed numerical study of the secondary flow characteristics in a curved square duct by using the spectral method, where the walls of the duct except the outer wall rotate around the centre of curvature and an azimuthal pressure gradient was imposed. In that study, multiple solutions with two-, four-, eight-

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vortex and even non-symmetric vortices were obtained at the same flow condition. Recently, Kun et al. [41] performed experimental investigations on laminar flows of pseudoplastic fluids in a square duct of strong curvature using an ultrasonic Doppler velocimetry and microphones, where streamwise velocity in the cross-section of the duct

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and the fluctuating pressure on the walls were measured for different flow rates. The velocity contours and their development along the duct were presented and compared with benchmark experiments by Taylor, Whitelaw and Yianneskis [42].

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To determine hydrodynamic instability and convective heat transfer, recently Mondal et

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al. [43] performed a spectral-based numerical study for the flow through a rotating curved rectangular duct of small curvature, where they investigated unsteady solutions for small

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values of the flow parameters with positive rotation of the duct, and presented some

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preliminary results of the flow evolution. Very recently, Razavi et al. [44] employed control volume method to investigate flow characteristics, heat transfer and entropy generation in a

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rotating curved duct. The effects of Dean number, wall heat flux and force ratio on the entropy generation were presented in that paper. However, complete behavior of the unsteady solutions with flow transition is still absent in literature for both rotating and corotating cases; which has been described in the present paper very clearly. Furthermore, hydrodynamic instability and vortex generation, which is presented in the present paper, gives a clear conception about the convective heat transfer through a coiled duct via 37

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periodic, multi-periodic and chaotic flows. As of now, a reliable technique for Dean instability is not known in literature for such flows. The present study also shows that there is a strong interaction between the heating-induced buoyancy force and the centrifugalCoriolis instability in the curved channel, which stimulates fluid mixing and thus results in

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thermal enhancement in the flow, because heated fluid is transported into the bulk fluid by the secondary vortices; this process is preciously demonstrated by the temperature contours as shown in the present study.

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Now we will discuss the plausibility of applying 2D calculations to study curved duct flows in the present study. It has been shown by many experimental and numerical studies, for example, Finlay [45], Belaidi [46], Arnal [47] and Bara [48], that curved duct flows easily attain asymptotic fully developed 2D states (uniform in the main flow direction) at

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most 2700 from the inlet. In this concern, a question may arise that 2D analysis is valid if

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the asymptotic state is steady, but its applicability is doubtful if it is not steady, such as periodic. However, the recent work by Wang and Yang [18] shows that even periodic flows

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can be analyzed by 2D calculations. They showed that for an oscillating flow, there exists a

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close similarity between the flow observation at 2700 and 2D calculation. Mees et al. [49] observed traveling wave solutions in the study of curved duct flows. In the paper by Mees

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et al. [50], they observed the change of secondary flow pattern far downstream using a spiral duct, where Dn increases in the downstream due to an increase of the curvature as the radius of the duct becomes small. It was found that the flow exhibits an oscillation between two-vortex and four-vortex flows first, but turns into a steady two-vortex flow downstream; which was observed in the present study. In fact the periodic oscillation, observed in the

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cross section of their duct, was a traveling wave advancing in the downstream direction that was justified by a recent 3D study of curved square duct flows by Yanase et al. [34]. Therefore, it is assumed that 2D calculations can accurately predict the existence of 3D traveling wave solutions by showing an appearance of 2D periodic oscillation. There is

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some other evidence showing that the occurrence of chaotic or turbulent flow may be predicted by 2D analysis. Yamamoto et al. [51] investigated helical pipe flows with respect to 2D perturbations and compared the results with their experimental data. There was a good agreement between the numerical results and the experimental data, which shows that

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even the transition to chaos which resembles to turbulence in real flows, can be predicted by 2D analysis. The transition from periodic oscillation to chaotic state, obtained by the 2D calculation in the present paper, may correspond to the destabilization of travelling waves in

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the curved duct flows like that of Tollmien-Schlichting waves in a boundary layer.

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In the present study, only 2D velocity and temperature field, uniform along the duct, is treated mainly for two reasons. Firstly, 2D field is used as a basis for 3D analysis, and

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secondly, various types of 2D solutions give good sign of 3D behavior of the field; stable

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steady solutions may correspond to steady state realized in the downstream of the duct, periodic solutions to 3D oscillating solutions, and chaotic solution indicates more intricate

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3D turbulence. Our 2D analysis, therefore, may contribute to the study of curved duct flows. Firstly, it gives an outline of the behaviors for not only fully developed but also developing curved duct flows. Secondly, the asymptotic behavior is obtained by only 2D analysis and without the knowledge of an asymptotic behavior of the flow; it is difficult to have a good physical insight into the curved duct flows. Finally, transitional behavior of the

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unsteady solutions, which is hardly possible in 3D analysis, may give a firm framework for the study of 3D analysis of steady, periodic and chaotic flows by the present study. However, to observe real characteristics of the flow, a 3D analysis is a must, and ours is a 2D simulation of the curved duct flows which is not only a basis for 3D analysis but also a

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qualitative or semi-quantitative estimation of the condition for the appearance of travellingwave solutions. At the moment, 3D calculation is much expensive for numerical

computation; however our next target is to do a 3D simulation of the flow to observe real

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scenario of the traveling wave solutions through a rotating curved rectangular channel. 6. Conclusion

A spectral-based numerical study is presented for the fully developed two-dimensional

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flow of viscous incompressible fluid through a rotating coiled rectangular duct of aspect

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ratio 2 and curvature 0.1. Numerical calculations are carried out for the Dean numbers, Dn = 1000 and Dn = 1500, over a wide range of the Taylor number, Tr. A temperature

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difference is applied across the vertical sidewalls for the Grashof number Gr = 100, where

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the outer wall is heated and the inner wall cooled, the top and bottom walls being thermally insulated. The rotation of the duct about the centre of curvature is imposed in both the

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positive and negative direction and combined effects of the centrifugal, Coriolis and buoyancy forces are investigate. Time evolution calculations as well as their phase spaces show that the unsteady flow undergoes in the scenario ‘Chaotic  multi-periodic  periodic  steady-state’ if Tr is increased in the positive rotation. For negative rotation, however, time evolution calculations show that the unsteady flow undergoes through various flow instabilities, if Tr is increased in the negative direction. Phase spaces were 40

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found to be very fruitful to justify the transition of the unsteady flow characteristics. Typical contours of secondary flow patterns and temperature profiles are also obtained at several values of Tr, and it is found that there exist two- to multi-vortex solutions if the duct rotation is involved in both the positive and negative direction. It is found that the

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temperature distribution is consistent with the secondary vortices, and secondary flows play a significant role in convective heat transfer from the heated wall to the fluid. It is also found that chaotic flow enhances heat transfer substantially than the steady-state or periodic solutions generating multi-vortex solutions at the outer concave wall. The present study also

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shows that there is a strong interaction between the heating-induced buoyancy force and the centrifugal-Coriolis instability in the rotating coiled channel that stimulates fluid mixing

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and consequently enhances heat transfer in the fluid.

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