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-, three- and 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.
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 [
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 [
Unsteady flows by time evolution calculation of curved duct flows was first initiated by Yanase and Nishiyama [
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 [
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
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,
Then the basic equations for
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
and the temperature
In the present study, Dn and Gr vary while Pr and
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, [
where
are expanded in terms of
where
The resistant coefficient
where, quantities with an asterisk (*) denote dimensional ones,
Since
where,
Time evolution of the resistance coefficient l are performed for
the contours for the stream lines of the secondary flow patterns
Then, we investigated time-dependent solutions of l for
asymmetric two-vortex solution.
We then performed time evolution of l for Dn = 1500 and
flows are asymmetric two-vortex solution.
Finally, the results of the time-dependent solutions for
and
the time-dependent flow creates multiple orbits, which suggests that the flow is multi-periodic. Typical contours
of secondary flow patterns and temperature profiles are shown in
Finally, the distribution of the time-dependent solutions, obtained by the time evolution calculations of the curved square duct flows, is shown in
A numerical study is presented for the time-dependent solutions of the flow through a curved square duct of constant curvature
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.
Rabindra NathMondal,Poly RaniShaha,Md. Nurul AminHelal,Nayan KumarPoddar, (2015) Time-Dependent Flow with Convective Heat Transfer through a Curved Square Duct with Large Pressure Gradient. Open Journal of Fluid Dynamics,05,238-255. doi: 10.4236/ojfd.2015.53026