[Paper Review] Natural convection in eccentric spherical annuli

Domingo Gallegos, Carlos Málaga|arXiv (Cornell University)|Sep 21, 2015

Lattice Boltzmann Simulation Studies14 references3 citations

TL;DR

This study investigates natural convection in eccentric spherical annuli using a three-dimensional two-component thermal lattice Boltzmann method (LBM) to simulate flow and heat transfer. It reveals that eccentricity reduces the critical Rayleigh number for convection onset, enhances flow instability, and leads to complex unsteady patterns with transverse and horizontal oscillations of isothermal surfaces, resulting in lower Nusselt numbers and earlier transition to unsteady flow compared to concentric configurations.

ABSTRACT

A fluid between two spheres, concentric or not, at different temperatures will flow in the presence of a constant gravitational force. Although there is no possible hydrostatic state, energy transport is dominated by diffusion if temperature difference between the spheres is small enough. In this conductive regime the average Nusselt number remains approximately constant for all Rayleigh numbers below some critical value. Above the critical Rayleigh number, plumes appear and thermal convection takes place. We study this phenomenon, in particular the case where the inner sphere is displaced from the centre, using a two-component thermal lattice Boltzmann method to characterize the convective instability, the evolution of the flow patterns and the dependence of the Nusselt number on the Rayleigh number beyond the transition.

Motivation & Objective

• To investigate the influence of eccentricity on natural convection in spherical annuli, particularly the transition from conductive to convective regimes.
• To characterize flow patterns, instability mechanisms, and heat transfer behavior (Nusselt number) in eccentric configurations compared to concentric ones.
• To validate the two-component thermal lattice Boltzmann method against known concentric case results and extend its application to asymmetric geometries.
• To explore the dependence of the Nusselt number on Rayleigh number in eccentric annuli and identify power-law correlations.
• To analyze the emergence of unsteady, periodic, and oscillatory flow structures due to symmetry breaking in the system.

Proposed method

• A D3Q19 two-component thermal lattice Boltzmann method (LBM) is used to solve the Oberbeck-Boussinesq equations for thermal convection in three dimensions.
• The method employs distribution functions for density $ f_k $ and temperature $ g_k $, evolving via collision and streaming steps with relaxation times $ au $ and $ au_g $ linked to kinematic viscosity and thermal diffusivity.
• Buoyancy is incorporated via a body force term $ G_k = -3eta w_k (T - T_0) m{e}_k m{ullet} m{g} $, modeling gravity-driven flow.
• Macroscopic fields—velocity $ m{u} $, temperature $ T $, and density $ ho $—are reconstructed from distribution functions using moment-based relations.
• Boundary conditions are applied on curved spherical surfaces using a simple, efficient scheme compatible with LBM’s lattice structure.
• Simulations are validated against experimental and numerical data for concentric spherical annuli before extending to eccentric configurations with varying eccentricity $ eta $, offset $ heta $, and Rayleigh number $ Ra $.

Experimental results

Research questions

• RQ1 How does eccentricity affect the critical Rayleigh number for the onset of natural convection in spherical annuli?
• RQ2 What are the characteristics of the resulting flow patterns—steady or unsteady—beyond the transition to convection in eccentric configurations?
• RQ3 How does the Nusselt number depend on Rayleigh number in eccentric spherical annuli, and how does this compare to the concentric case?
• RQ4 What instabilities emerge in the isothermal surfaces and how do they correlate with temporal oscillations in the Nusselt number?
• RQ5 How does the asymmetry of the eccentric configuration influence heat transfer efficiency and flow structure compared to symmetric concentric cases?

Key findings

• The critical Rayleigh number for the onset of convection is lower in eccentric spherical annuli compared to concentric ones, indicating earlier transition to convective heat transfer.
• The average Nusselt number increases at a reduced rate in eccentric configurations due to enhanced mixing and reduced thermal gradients, resulting in lower overall heat flux.
• Unsteady, periodic flow behavior emerges at higher Rayleigh numbers, with the Nusselt number exhibiting time-dependent oscillations correlated with deformation of isothermal surfaces.
• Isothermal surfaces develop transverse instabilities (flapping tails) in the vertical plane of symmetry, followed by horizontal oscillations at higher $ Ra $, indicating complex three-dimensional instabilities.
• Flow patterns show alternating current structures from the front of the inner sphere during Nusselt number peaks, with streamlines revealing complex recirculation and vortical structures.
• The thermal LBM model successfully captures the transition from conductive to convective regimes and reproduces known trends in concentric cases, validating its use for eccentric configurations.