Time-Dependent Flow with Convective Heat Transfer through a Curved Square Duct with Large Pressure Gradient

A numerical study is presented for the fully developed two-dimensional laminar flow of viscous incompressible fluid through a curved square duct for the constant curvature δ = 0.1. In this paper, a spectral-based computational algorithm is employed as the principal tool for the simulations, while a Chebyshev polynomial and collocation method as secondary tools. Numerical calculations are carried out over a wide range of the pressure gradient parameter, the Dean number, 100 ≤ Dn ≤ 3000 for the Grashof number, Gr, ranging from 100 to 2000. The outer wall of the duct is treated heated while the inner wall cooled, the top and bottom walls being adiabatic. The main concern of the present study is to find out the unsteady flow behavior i.e. whether the unsteady flow is steady-state, periodic, multi-periodic or chaotic, if Dn or Gr is increased. It is found that the unsteady flow is periodic for Dn = 1000 at Gr = 100 and 500 and at Dn = 2000, Gr = 2000 but steady-state otherwise. It is also found that for large values of Dn, for example Dn = 3000, the unsteady flow undergoes in the scenario “periodic→chaotic→periodic”, if Gr is increased. Typical contours of secondary flow patterns and temperature profiles are also obtained, and it is found that the unsteady flow consists of single-, two-, threeand four-vortex solutions. The present study also shows that there is a strong interaction between the heating-induced buoyancy force and the centrifugal force in a curved square passage that stimulates fluid mixing and consequently enhance heat transfer in the fluid.


Introduction
Fluid flow and heat transfer in curved ducts have been studied for a long time because of their fundamental importance in engineering and industrial applications.Today, the flows in curved non-circular ducts are of increasing importance in micro-fluidics, where lithographic methods typically produce channels of square or rectangular cross-section.These channels are extensively used in many engineering applications, 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 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 the pioneering work in this field by Dean [1].After that, many theoretical and experimental investigations have been done; for instance, the articles by Berger et al. [2], Nandakumar and Masliyah [3], and Ito [4] may be referenced.
One of the interesting phenomena of the flow through a curved duct is the bifurcation of the flow because generally there exist many steady solutions due to channel curvature.Studies of the flow through a curved duct have been made, experimentally or numerically, for various shapes of the cross section by many authors.However, an extensive treatment of the bifurcation structure of the flow through a curved duct of rectangular cross section was presented by Winters [5], Daskopoulos and Lenhoff [6] and Mondal [7].
Unsteady flows by time evolution calculation of curved duct flows was first initiated by Yanase and Nishiyama [8] 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 in a curved rectangular duct of large aspect ratio was investigated, in detail, by Yanase et al. [9] numerically.They performed time-evolution calculations of the unsteady solutions with and without symmetry condition and found that periodic oscillations appear with symmetry condition while aperiodic time variation without symmetry condition.Wang and Yang [10] [11] performed numerical as well as experimental investigation on fully developed periodic oscillation in a curved square duct.Flow visualization in the range of Dean numbers from 50 to 500 was carried out in their experiment.Recently, Yanase et al. [12] performed numerical investigation of isothermal and non-isothermal flows through a curved rectangular duct and addressed the time-dependent behavior of the unsteady solutions.In the succeeding paper, Yanase et al. [13] extended their work for moderate Grashof numbers and studied the effects of secon-dary flows on convective heat transfer.Recently, Mondal et al. [14] [15] performed numerical prediction of the unsteady solutions by time-evolution calculations for the flow through a curved square duct and discussed the transitional behavior of the unsteady solutions.
One of the most important applications of curved duct flow is to enhance the thermal 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 [16] presented numerical calculations to describe the secondary flow characteristics in the flow through curved ducts of aspect ratios ranging from 1 to 8 that were heated on the outer wall, where they studied for small Dean numbers and compared the numerical results with their experimental data.Yanase et al. [13] studied time-dependent behavior of the unsteady solutions for curved rectangular duct flow and showed that secondary flows enhance heat transfer in the flow.Mondal et al. [17] performed numerical prediction of the unsteady solutions by time-evolution calculations of the thermal flow through a curved square duct and studied convective heat transfer in the flow.Recently Norouzi et al. [18] [19] investigated fully developed flow and heat transfer of viscoelastic materials in curved square ducts under constant heat flux.Very recently, Chandratilleke and Narayanaswamy [20] numerically studied vortex structure-based analysis of laminar flow and thermal characteristics in curved square and rectangular ducts.To the best of the authors' knowledge, however, there has not yet been done any substantial work studying the transitional behavior of the unsteady solutions for thermal flows through a curved square duct for combined effects of large Grashof number and large Dean number, which has very practical applications in fluids engineering, for example, in internal combustion engine, gas turbines etc.Thus from the scientific as well as engineering point of view it is quite interesting to study the unsteady flow behavior in the presence of strong buoyancy and centrifugal forces.Keeping this issue in mind, in this paper, a comprehensive numerical study is presented for fully developed two-dimensional (2D) flow of viscous incompressible fluid through a curved square duct and studied effects of secondary flows on convective heat transfer in the flow.

Mathematical Formulations
Consider an incompressible viscous fluid streaming through a curved duct with square cross section whose width or height is 2d.The coordinate system is shown in Figure 1.It is assumed that the temperature of the outer wall is 0 T T + ∆ and that of the inner wall is 0 T T − ∆ , where 0 T ∆ > .The x, y, and z axes are taken to be in the horizontal, vertical, and axial directions, respectively.It is assumed that the flow is uniform in the axial direction, and that it is driven by a constant pressure gradient G along the center-line of the duct, i.e. the main flow in the axial direction as shown in Figure 1.The variables are non-dimensionalized by using the representative length d and the representative velocity 0 U v d = .We introduce the non-dimensional variables defined as 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 the non-dimensional pressure, δ the non-dimensional curvature, and temperature is non-dimensionalized by T ∆ .Henceforth, all the variables are nondimensionalized if not specified.The stream function ψ is introduced in the x-and y-directions as Then the basic equations for , w ψ and T are derived from the Navier-Stokes equations and the energy equation under the Boussinesq approximation as, ( ) where, ( ) ( ) . , The Dean number Dn, the Grashof number Gr, and the Prandtl number Pr, which appear in Equations ( 2) to (4) are defined as The rigid boundary conditions for w and ψ are used as and the temperature T is assumed to be constant on the walls as ( ) In the present study, Dn and Gr vary while Pr and δ are fixed as Pr = 7.0 (water) and curvature 0.1.

Method of Numerical Calculation
In order to solve the Equations ( 2) to (4) numerically the spectral method is used.This is the method which is thought to be the best numerical method to solve the Navier-Stokes equations as well as the energy equation (Gottlieb and Orazag, [21]).By this method the variables are expanded in a series of functions consisting of the Chebyshev polynomials.That is, the expansion functions where is the nth order Chebyshev polynomial.( ) where M and N are the truncation numbers in the x and y directions respectively.In this study, for necessary accuracy of the solutions, we use M = 20 and N = 20.In order to calculate the unsteady solutions, the Crank-Nicolson and Adams-Bashforth methods together with the function expansion (10) and the collocation methods are applied.

Resistance Coefficient
The resistant coefficient λ 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 where, quantities with an asterisk ( * ) denote dimensional ones, stands for the mean over the cross section of the duct and * h d is the hydraulic diameter.The main axial velocity w * is calculated by In the present study, λ is used to perform time evolution of the unsteady solutions.14(c) respectively, where we see that the periodic flow oscillates between asymmetric two-vortex solutions.The temperature distribution is well consistent with the secondary vortices, and it becomes so entan-gled when the secondary vortices become stronger.In this regard, it should be worth mentioning that irregular oscillation of the non-isothermal and isothermal flows has been observed experimentally by Wang and Yang [10] for a curved square duct flow and by Ligrani and Niver [22] for flow through a curved rectangular duct of large aspect ratio.

Conclusion
A numerical study is presented for the time-dependent solutions of the flow through a curved square duct of constant curvature 0.1 δ = .Numerical calculations are carried out by using a spectral method, and covering a wide range of the Dean number 100 3000 Dn ≤ ≤ and the Grashof number 100 2000.Gr ≤ ≤ A temperature difference is applied across the vertical sidewalls, where the outer wall is heated and the inner wall cooled.In order to study the non-linear behavior of the unsteady solutions, we performed time evolution calculations and it is found that the unsteady flow is a steady-state solution for Dn = 100, 500 and 1500 for all values of the Gr investigated in this study.However, the unsteady flow is periodic for 1000 Dn = at 100,500 Gr = and for Dn = 2000 at Gr = 2000.The unsteady flow is also periodic/multi-periodic for Dn = 2500 at Gr = 1500 and 2000.For large Dean numbers, e.g.Dn = 3000, on the other hand, the unsteady flow undergoes in the scenario "period-ic→chaotic→periodic", if Gr is increased.Typical contours of secondary flow patterns and temperature profiles are also obtained, and it is found that periodic or multi-periodic solution oscillates between asymmetric two-, and four-vortex solutions, while for chaotic solution, there exist only asymmetric two-vortex solution.The temperature distribution is consistent with the secondary vortices and it is found that the temperature distribution occurs significantly from the heated wall to the fluid as the secondary flow becomes stronger.The present study also shows that there is a strong interaction between the heating-induced buoyancy force and the centrifugal force in the curved passage which stimulates fluid mixing and thus results in thermal enhancement in the flow.

Figure 1 .
Figure 1.Coordinate system of the curved square duct.
λ is related to the mean non-dimensional axial velocity w

Figure 7 (
c) shows typical contours of secondary flow patterns and temperature profiles for the steady-state solutions at 100 1500 Gr ≤ ≤ and Figure 7(d) shows those for 2000 Gr = , for one period of oscillation at time 20.68 21.01 t ≤ ≤ , and we find that both the time-periodic and steady-state solutions are asymmetric two-vortex solutions.Then we show the results of the time-dependent so-As seen in Figure 8(a), the time-dependent flow for 2500 Dn = is a steady-state solution for 100 1000 Gr ≤ ≤ but periodic oscillating flow for 1500 Gr = and multi-periodic (or transitional chaos) flow for 2000.Gr = Figures 8(b)-(d) respectively show those of the time-dependent solutions for the steady-state solutions at Gr = 100, 500 and 1000.Since the flow is steady-state at 100,500 Gr = and 1000, a single contour of each of the secondary flow patterns and temperature profiles is shown in Figures 8(b)-(d) respectively.