Natural convection problem in a Bingham fluid using the operator-splitting method

The heat transfer characteristics from an isothermal heated plate in a quiescent yield stress fluid in a cavity was investigated over a wide range of parameters (Rayleigh number, 102 ≤ Ra ≤ 105, Prandtl number, 10 ≤ Pr ≤ 100, and Bingham number, Bn ≥ 0) where the flow is known to be laminar and steady. The coupled momentum and energy equations have solved here numerically within the framework of Boussinesq approximation to capture the temperature-dependent fluid density. The results demonstrate that for a given value of the Rayleigh number, there exists a critical value of the Bingham number, above which the fluid is completely unyielded and heat transfer occurs solely by conduction. In order to delineate the effect of domain geometry on the conduction limit, the study was extended over a range of geometrical aspects by varying the aspect ratio (λ = diameter of the cavity/ a length of the plate), 2 ≤ λ ≤ 5. This work shows that the critical value of the Bingham number can be described as a function of geometry of domain, Ra and Pr. The value of critical Bingham number increases with the increasing aspect ratio and Rayleigh number in order to approach the conduction limit. The yield surfaces show that the increasing values of Rayleigh number induce fluid-like behaviour whereas Bingham number opposes this propensity. The average Nusselt number decreases with the increasing Bingham number due to the suppression of the advective component of heat transfer.

Heat transfer in steady free convection from differentially heated cylinders enclosed in a rectangular duct filled with Bingham plastic fluids has been solved numerically for the ranges of the dimensionless groups as, Rayleigh number, 10 2 Ra 10 6 ; Prandtl number, 10 Pr 100 and, Bingham number, 0 Bn 50 for aspect ratios AR = 0.5, 0.6, 0.7, 0.8, 0.9 and 2. The streamlines, isotherm contours, yield surfaces, local and average Nusselt numbers were analysed and discussed.It is found that as the aspect ratio of the enclosure increases from 0.5 to 0.9, the average Nusselt number on the surface of the hot cylinder increases as a larger amount of fluid takes part in convection.Moreover, at sufficiently large Bingham numbers, yield stress forces dominate over buoyancy causing the flow to cease and thus the Nusselt number approaches its conduction limit.Finally, the Nusselt number approaches its conduction limit once the maximum Bingham number is reached.

Lattice Boltzmann simulation of viscoplastic fluids on natural convection in an inclined enclosure with inner cold circular/elliptical cylinders (Part I: One cylinder)

In many situations, heat transfer occurs primarily by free convection, such as that in confined and restricted spaces and/or where it is not possible to induce fluid motion by using a pump or impeller. In this work, natural convection from an isothermal cone in power law and Bingham plastic fluids has been investigated numerically by solving the coupled momentum and energy equations over wide ranges of conditions such as the Grashof number (), the Prandtl number (5–100), the Bingham number (1–50), and the power-law index (0.4–1). The structure of the flow and temperature fields is visualized in terms of streamlines and isotherm contours, yielded and unyielded regions, and the variation of the local Nusselt number along the surface of the cone. The location and shape of the yielded and unyielded regions are influenced by the values of the Grashof and Bingham numbers. The size of the unyielded regions progressively decreases with the increasing Bingham number, whereas the Grashof number promotes the growth of fluidlike regions. Shear-thinning fluid behavior promotes heat transfer with reference to that in Newtonian fluids. On the other hand, the Bingham yield stress has an adverse influence on the overall heat transfer. The present values of the average Nusselt number are correlated in terms of the pertinent dimensionless numbers to enable its prediction in a new application. Finally, a criterion for the onset of convection in Bingham fluid is also reported in terms of ; above which, conduction is the main mode of transport. Thus, the relative magnitudes of the buoyancy and viscous (including the contribution of yield stress) forces determine the degree of deterioration in the rate of heat transfer with reference to that in Newtonian fluids otherwise under identical conditions. This information is valuable: not only in designing heat transfer equipment, but also in the processing of temperature-sensitive materials.

Laminar natural convection of Bingham fluids in a square enclosure with differentially heated side walls

Turbulent Rayleigh–Bénard convection in low Prandtl–number fluids

Lattice Boltzmann simulation of viscoplastic fluids on natural convection in inclined enclosure with inner cold circular/elliptical cylinders (Part III: Four cylinders)

PurposeThe purpose of this paper is to analyze two-dimensional steady-state Rayleigh–Bénard convection within rectangular enclosures in different aspect ratios filled with yield stress fluids obeying the Herschel–Bulkley model.Design/methodology/approachIn this study, a numerical method based on the finite element has been developed for analyzing two-dimensional natural convection of a Herschel–Bulkley fluid. The effects of Bingham number Bn and power law index n on heat and momentum transport have been investigated for a nominal Rayleigh number range (5 × 103 < Ra < 105), three different aspect ratios (ratio of enclosure length:height AR = 1, 2, 3) and a single representative value of nominal Prandtl number (Pr = 10).FindingsResults show that the mean Nusselt number Nu¯ increases with increasing Rayleigh number due to strengthening of convective transport. However, with the same nominal value of Ra, the values of Nu¯ for shear thinning fluids n < 1 are greater than shear thickening fluids n > 1. The values of Nu¯ decrease with Bingham number and for large values of Bn, Nu¯ rapidly approaches unity, which indicates that heat transfer takes place principally by thermal conduction. The effects of aspect ratios have also been investigated and results show that Nu¯ increases with increasing AR due to stronger convection effects.Originality/valueThis paper presents a numerical study of Rayleigh–Bérnard flows involving Herschel–Bulkley fluids for a wide range of Rayleigh numbers, Bingham numbers and power law index based on finite element method. The effects of aspect ratio on flow and heat transfer of Herschel–Bulkley fluids are also studied.

Lattice Boltzmann simulation of viscoplastic fluids on natural convection in inclined enclosure with inner cold circular/elliptical cylinders (Part II: Two cylinders)

Natural convection in a square cavity filled with a porous medium: Effects of various thermal boundary conditions

Natural convection within trapezoidal enclosures finds significant practical applications. The natural convection flows play a prominent role in the transport of energy in energyrelated applications, in case of proper design of enclosures to achieve higher heat transfer rates. In the present study, a two-dimensional cavity with adiabatic right side wall is studied. The left side vertical wall is maintained at the constant hot temperature and the top slat wall is maintained at cold temperature. The dimensionless governing partial differential equations for vorticity-stream function are solved using the finite difference method with incremental time steps. The parametric study involves a wide range of Rayleigh number, Ra, 103 ≤ Ra ≤ 105 and Prandtl number (Pr = 0.025, 0.71 and 10). The fluid flow within the enclosure is formed with different shapes for different Pr values. The flow rate is increased by enhancing the Rayleigh number (Ra = 104 ). The numerical results are validated with previous results. The governing parameters in the present article, namely Rayleigh number and Prandtl number on flow patterns, isotherms as well as local Nusselt number are reported.

Abstract. We use a four-dimensional variational data assimilation (4D-VAR) algorithm to observe the growth of 2-D plumes from a point heat source. In order to test the predictability of the 4D-VAR technique for 2-D plumes, we perturb the initial conditions and compare the resulting predictions to the predictions given by a direct numerical simulation (DNS) without any 4D-VAR correction. We have studied plumes in fluids with Rayleigh numbers between 106 and 107 and Prandtl numbers between 0.7 and 70, and we find the quality of the prediction to have a definite dependence on both the Rayleigh and Prandtl numbers. As the Rayleigh number is increased, so is the quality of the prediction, due to an increase of the inertial effects in the adjoint equations for momentum and energy. The horizon predictability time, or how far into the future the 4D-VAR method can predict, decreases as Rayleigh number increases. The quality of the prediction is decreased as Prandtl number increases, however. Quality also decreases with increased prediction time.

\nThis study focuses on the numerical simulation of heat transfer by natural convection in a rectangular enclosure, filled with a liquid metal (low Prandtl number) partially heated from below with a sinusoidal temperature. The value of the study lies in its involvement in the crystal growth for the manufacture of semiconductors and electronics cooling. Indeed, the occurrence of convection during crystal growth can lead to in homogeneities that lead to striations and defects that affect the quality of the crystals obtained by the Bridgman techniques or Chochrawlski. Temperature of the oscillations, due to the instabilities of the convective flow in the liquid metal, also induces non-uniform cooling in the solidification front. Convection is then studied in order to reduce it. A modelling of the problem in two dimensions was conducted using Comsol computer code that is based on the finite element method, by varying the configuration of the control parameters, namely, the Rayleigh number, the nature of fluid (Prandtl number) and amplitude of temperature on heat transfer rate (Nusselt number) on convective structures that appear.\n

Heat transfer induced by buoyancy from a pipe buried in a semi-infinite porous medium with a superimposed fluid layer has been numerically examined in this study. Due to the complexity involved, finite difference method along with body-fitted coordinate systems has been employed. The Brinkman-extended Darcy equations are used to model flow in the porous medium while Navier-Stokes equations are used for the fluid layer. The conditions applied at the interface between the fluid and porous layers are the continuity of temperature, heat flux, normal and tangential velocity, shear stress and pressure. A parametric study has been performed to investigate the effects of Rayleigh number, Prandtl number, Darcy number, and fluid layer thickness on the flow patterns and heat transfer rates. The results show that heat transfer increases with the Rayleigh number, but the convective strength decreases with the Darcy number. The heat transfer rate is smaller when the superimposed fluid is air instead of water. For a porous layer with Da ≤ 0.0005 and an overlaying fluid layer thickness of L/ri ≥ 1, convection is initiated in the fluid layer and it may develop into multiple recirculating cells at a moderate Rayleigh number (i.e., Ra ≤ 104), and may further develop into a single cell at a higher Rayleigh number of 105.

Numerical study of two-dimensional natural convection in a horizontal fluid layer heated from below, by finite-element method: influence of Prandtl number