CONJUGATE HEAT TRANSFER IN SUDDENLY EXPANDING FLOW
5.1 Introduction
CHAPTER 5
CONJUGATE HEAT TRANSFER IN SUDDENLY
heat generating parameters and length to diameter ratio for a fluid of Prandtl number 0.005 on steady state CHT for a vertical cylinder. Juncu [71] numerically investigated unsteady CHT for a circular cylinder at low Reynolds numbers (Re= 2,20). He [72] also presented a numerical study of steady state CHT from a circular cylinder with a heat source embedded at the center of the cylinder. Li et. al. [102] presented an experimental quantification of the effects of boundary conditions, both conjugate as well as convective, on heat transfer characteristics for the case of pin fin channel in trailing edge of gas turbine blade. Significant work has been done for CHT involving jet flows, for instance the works of Kanna and Das [126], Mondal et.al. [113], and Paulraj et.al. [123]. In recent years, the lattice Boltzmann method (LBM) has also been applied to problems of CHT, most notably by Seddiq et.al. [138], Pareschi et.al. [121], Patel et.al. [122] etc.
Sudden expansion flows are encountered in many industrial applications such as ejector systems [14], gas turbine combustors [50, 133], heat exchangers, and cooling of electronic equipments [134]. The fundamental nature of the flow physics encountered in the process of flow separation and reattachment makes it an invaluable topic of research. The addition of CHT introduces another layer of computational complexity to relatively simple geome- tries. In this chapter, we consider two commonly encountered geometries in suddenly expanding flows viz. flow over a backward facing step (BFS), and suddenly expanding flow in a symmetric channel with large expansion ratios.
Kanna and Das [127] first proposed the CHT case for BFS flow as a benchmark problem. They employed an alternate direction implicit (ADI) method in order to obtain steady state results via streamfunction-vorticity (ψ-ω) form of the N-S equations. Their numerical study was focused on investigating the effect ofRe,P r,k, and slab thickness on the heat transfer characteristics. However, subsequent investigations of the same problem by Teruel and Fogliatto [153], and Ramšak [128] revealed discrepancies with the results presented by Kanna and Das [127]. Ramšak [128] remarked that "Professor Kanna has confirmed in personal communication that their results published are probably wrong."
The numerical study by Ramšak [128] was also based on the ψ-ω formulation and the solution was obtained using the multidomain Boundary element method. However, both the studies are limited to a single set of parameters. There is a need for an investigation into the influence of all the parameters on the heat transfer characteristics, which we have
5.1. Introduction 103
endeavored to accomplish in this work.
On the other hand, much work has been done in the case of flow in a suddenly expand- ing symmetric channel. The experimental investigations conducted by Durst et.al. [42], Cherdron et.al. [31], Fearn et.al. [48] demonstrated that the two-dimensional steady state flow phenomena in a suddenly expanding channel at low Reynolds numbers (Re) contains two symmetric recirculation zones equal in size, and the recirculation length increases linearly with Re. These steady symmetric states exist only up to a critical Reynolds number (Rec). For Re > Rec, the flow, while remaining steady, loses its symmetry and evolves into an asymmetric state with two separation zones of different lengths attached to either walls of the channel. Subsequent numerical investigations of Durst et.al. [43], Alleborn et.al. [4], Battaglia et.al. [17], and Drikakis [41] have also confirmed these exper- imental findings. At higher Re’s the flow becomes time dependent as three-dimensional effects set in [43]. Rusak and Hawa carried out a weakly non-linear analysis of the sym- metry breaking pitchfork bifurcation [136]. Hawa and Rusak investigated the effect of slight asymmetry of the channel in flow phenomena [58], and carried out a numerical in- vestigation to study the interaction between effects of viscous dissipation and convection perturbations on the flow stability [59]. Revuelta et.al. [133] investigated the flow of round laminar jets with coaxial confinement when both the jet Reand the expansion ratio are large. Revuelta [132] presented a numerical study of 2D laminar flow of an incompressible viscous jet through a channel with large expansion ratios. It was shown that when the jet Re is sufficiently large, the structure and scale of the resulting flow as well as the critical conditions for asymmetric solutions to appear are determined by the momentum exchange between the incoming jet and the recirculating fluid. This flow behavior in a suddenly expanding channel with large expansion ratio thus presents an opportunity of studying the phenomenon of conjugate heat transfer.
From the preceding literature review, one can distinctly identify the gap in the study of conjugate heat transfer in suddenly expanding flow. Firstly, the results provided by Kanna and Das [127] for the benchmark problem of CHT in BFS flow was found to be erroneous [128]. However, the subsequent study by Ramšak [128] failed to address the physics behind the flow and heat transfer phenomena considering all the parameters involved in the benchmark problem. Additionally, benchmark result was established for
one set of parameters only. Secondly, to the best of our knowledge, CHT in a suddenly expanding symmetric channel at low Re’s has not been studied in detail previously. In this chapter, we first consider the problem of CHT in BFS flow and explore the physics behind the flow and heat transfer phenomena, in addition to establishing new benchmark results. Next, we study the phenomenon of CHT in a suddenly expanding symmetric channel at lowRe’s and large expansion ratios. The break in flow symmetry in a suddenly expanding channel is a particularly unique phenomenon, and interesting results are arrived while studying the phenomena of conjugate heat transfer. This chapter is organized as follows: In section 5.2 we simulate CHT in BFS flow. Sections 5.2.2 and 5.2.3 describe in brief the computational grid, and the algorithm for the solution of system of equations respectively. In section 5.2.4 code validation is carried out by comparing the present results with established results in the literature, and section 5.2.5 deals with grid independence of the computed data. The results and discussion for CHT in BFS flow case are presented in section 5.2.6. Next, in section 5.3 we simulate CHT in a suddenly expanding symmetric channel with large expansion ratios. This section follows a similar structure to section 5.2. Finally, in section 5.4 we present our conclusions for this chapter.