Natural convection of Bingham fluids in rectangular cross-sectional cylindrical annuli with differentially heated vertical walls

Abstract

Purpose

This paper aims to numerically analyse natural convection of yield stress fluids in rectangular cross-sectional cylindrical annular enclosures. The laminar steady-state simulations have been conducted for a range of different values of normalised internal radius (r_i/L 1/8 to 16, where L is the difference between outer and inner radii); aspect ratio (AR = H/L from 1/8 to 8 where H is the enclosure height); and nominal Rayleigh number (Ra from 10³ to 10⁶) for a single representative value of Prandtl number (Pr is 500).

Design/methodology/approach

The Bingham model has been used to mimic the yield stress fluid motion, and numerical simulations have been conducted for both constant wall temperature (CWT) and constant wall heat flux (CWHF) boundary conditions for the vertical side walls. The conservation equations of mass, momentum and energy have been solved in a coupled manner using the finite volume method where a second-order central differencing scheme is used for the diffusive terms and a second-order up-wind scheme is used for the convective terms. The well-known semi-implicit method for pressure-linked equations algorithm is used for the coupling of the pressure and velocity.

Findings

It is found that the mean Nusselt number based on the inner periphery Nu¯_i increases (decreases) with an increase in Ra (Bn) due to augmented buoyancy (viscous) forces irrespective of the boundary condition. The ratio of convective to diffusive thermal transport increases with increasing r_i/L for both Newtonian (i.e. Bn = 0) and Bingham fluids regardless of the boundary condition. Moreover, the mean Nusselt number Nu¯_i normalised by the corresponding Nusselt number due to pure conductive transport (i.e. Nu¯_i/(Nu¯_i)_cond) shows a non-monotonic trend with increasing AR in the CWT configuration for a given set of values of Ra, Pr, L_i for both Newtonian (i.e. Bn = 0) and Bingham fluids, whereas Nu¯_i/(Nu¯_i)_cond increases monotonically with increasing AR in the CWHF configuration. The influences of convective thermal transport strengthen while thermal diffusive transport weakens with increasing AR, and these competing effects are responsible for the non-monotonic Nu¯_i/(Nu¯_i)_cond variation with AR in the CWT configuration.

Originality/value

Detailed scaling analysis is utilised to explain the observed influences of Ra, BN, r_i/L and AR, which along with the simulation data has been used to propose correlations for Nu¯_i.

Citation

Yigit, S. and Chakraborty, N. (2019), "Natural convection of Bingham fluids in rectangular cross-sectional cylindrical annuli with differentially heated vertical walls", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 29 No. 1, pp. 43-77. https://doi.org/10.1108/HFF-06-2017-0242