HYDRODYNAMICS OF POROUS AND NON-POROUS CYLINDERS

Abstract

In this research work, a thorough analysis of qualitative and quantitative aspects of the dynamics of a Newtonian and non-Newtonian fluid has been carried out in a viscous dominant flow field. The domain is confined with horizontal walls with porous (square as well as circular shaped) cylinders positioned side-by-side at the mid-plane. A substantial modification in the flow field is achieved by varying the permeability levels and transverse gap between the side-by-side porous cylinders. A further complication is encountered when the flow comprises of non-Newtonian fluids due to the effect of viscosity stratification. This is best realized for a shear-thinning fluid near the walls of the square cylinders. Moreover, triangular array of cylinders in presence of porous media has been investigated. The presence of temperature gradient creates cross-buoyancy in the flow field. The flow and thermal fields (together with possible recirculating wake), heat transfer enhancement and the resistance offered by the periodic array (total pressure drop) at various cylinders spacing in the presence of thermal buoyancy is quite intriguing. Based on these intriguing questions, the following studies have been carried out in the laminar regime: 1. Analysis of Wake Structure Interaction between Permeable Side-by-Side Square Cylinders We have analyzed the laminar flow through permeable side-by-side bars of a square cross section in a channel-confined domain. Vorticity generation on the leeward sides of the permeable bodies further necessitates the study for a better understanding of underlying physics. Reynolds number (Re) and Darcy number (Da) are varied from 5 to 150 and 10-6 to 10-2, respectively, at transverse gap ratios s/d=2.5–10. In the perspective of periodic unsteady flow, critical Re for the onset of vortex shedding is analyzed. Streamlines, vorticity, pressure coefficient distribution, and velocity profiles are discussed to identify the wake patterns. In lower permeability level, vortex-shedding from the permeable square cylinders is observed either in synchronized antiphase mode or a single large vortex street with a synchronized in-phase pattern in the near wake. A steady-state wake pattern symmetric and flocked toward the centerline is observed for all s/d at a higher permeability level regardless of Re. Wake patterns are not altered for Da=10-6 to 10-3; instead, prompt extermination of the two vortex streets downstream is observed at Da=10-3 as compared to Da=10-6. The impact of s/d, Re, and permeability on the drag is examined. A jump in the flow characteristics and drag forces is noticed at higher Re for the midrange Da remarkably at lower s/d. For the extent of high permeability, the drag coefficient asymptotically gets closer to zero. 2. Numerical Investigation of Two-dimensional Power-Law Fluid Flow across Porous Square Cylinders In this study, we investigated the flow physics associated with power-law fluids through two identical porous square cylinders in a side-by-side configuration within a channel. The effects of three critical parameters, power-law index (n), Darcy number (Da), and gap ratio (g/W) on the flow behavior are explored for ranges of g/W=0.5–5, n=0.4-0.8, and Da=10-6–10-2, respectively. Two flow conditions are considered: first, for a creeping flow (unseparated flow) at Re=1 where only Darcy’s law is applicable; second, for a viscous dominant flow at Re=100, where Darcy–Forchheimer extended model is exercised. The flow patterns behind the porous square cylinders are analyzed using streamlines, velocity profiles, pressure distribution curves, and vorticity structural parameters (Г). In low permeability levels, the flow exhibits an irregular non-periodic vortex shedding characterized by a single large vortex street far-off the downstream for g/W=0.5. However, synchronized wake patterns were observed in either anti-phase or in-phase modes for higher gap ratios. Depending on the power-law index, leading-edge separation with two side recirculation was observed, leading to quasi-periodicity in the flow for all g/W. It was found that the leading edge separation can be prevented by increasing the permeability. Additionally, a transition from anti-phase to in-phase mode takes place when the permeability is altered while keeping the flow-time constant. The presence of a jet-like flow in the cylinders’ gap section significantly influences unsteady wake patterns. The impact of g/W, power-law index, and permeability on drag is also examined. A jump in some flow parameters was observed at higher Re for the midrange Darcy number, but no such increase was noted for the high shear-thinning behavior (n˂0.6). The study findings provide valuable insights into flow behavior and offer a potential approach for improving the design of fluidic systems that involve porous cylinders. 3. Newtonian-Fluid Flow Past a Pair of Porous Side-by-Side Circular Cylinders In this study, the dynamics of incompressible flow around two identical porous cylinders for a side-by-side configuration in a closed channel has been carried out. The formation of various flow patterns behind permeable cylinders is more intriguing and further compelling to assimilate the underlying flow physics. The effects of three critical parameters, gap ratio (s/d), Reynolds number (Re), and the Darcy number (Da), on the flow behavior are investigated for the ranges of s/d=1.5-6, and Da=10-6- 10-2 at Re=100. In an unsteady flow regime, jet-like flow in the gap section mainly governs the unsteady wake patterns. In the low range of Darcy numbers (10-6-10-3), asymmetric flip-flopping patterns are observed for s/d=1.5 and 2; and synchronized wake patterns either in anti-phase or in-phase mode are observed for higher gap ratios. The velocity profiles in the gap and free sides of the cylinders and pressure distribution along the porous surface are also discussed to facilitate the understanding of different wake patterns. Surprisingly, a case of pattern shifting from anti-phase to in-phase mode is observed when permeability is altered for the same flow-time. A symmetric and clustered strands of vorticity near the centerline is observed for all cases of s/d at Da=10-2. The effects of s/d and Da on the drag coefficient and critical Reynolds number are also discussed. A jump in the drag values, a maximum of 13.9% for s/d=3.5, is witnessed for the mid-range of Da at higher Re. 4. Analysis of Laminar Flow with Mixed Convection in a Triangular Periodic Array of Heated Cylinders Immersed in Porous Media In this study, an investigation of the laminar flow and enhanced rate of heat transfer through a triangular array of circular cylinders embedded in a fluid-saturated porous media is considered. Mixed convection regime together with forced convection cases have been studied using Local Thermal Equilibrium (LTE) model and Darcy-Brinkman-Forchheimer momentum equations coupled with certain assumptions; particles diameter constituting the porous matrix is constant, solid-fluid thermal conductivity ratio is considered as unity and the cylinders are at a constant temperature. The study is constrained to the low and intermediate Peclet number (Re =5 to 40, Pr=50) and high permeability level (Da=10-3 to 10-1) for different cylinder spacings (0.7 ≤ φ ≤ 0.99). It is anticipated that porous media would enhance the degree of heat transfer, but it derives a multiple order in pressure drop indubitably not desirable in many heat applications. Investigation reveals that both mean drag coefficient (CD) and Nusselt number (Nu) are strong functions of Da and φ, however, the influence of buoyancy parameter is mainly witnessed in higher permeability level and high Peclet number. The study also details the effect of governing parameters on mean gap velocities and pressure coefficient. Surprisingly, recirculation wakes exist for the lowest cylinder spacing (φ =0.7) in high fluid momentum. It is found that low φ and high permeability are desirable in a forced convection regime, whereas both φ and Da should be high in the case of mixed convection. This article also quantifies the combined effect of Darcy number and buoyancy parameter on heat-transfer enhancement ratio.