[Paper Review] Oscillatory instability of three-dimensional natural convection of air in a laterally heated cubic box

Alexander Gelfgat|arXiv (Cornell University)|Aug 4, 2015

Fluid Dynamics and Turbulent Flows34 references3 citations

TL;DR

This study investigates the transition from steady to oscillatory natural convection in a 3D cubic cavity heated from the sides using time-accurate finite volume simulations of the Boussinesq equations. It identifies critical Grashof numbers for instability across four thermal boundary condition combinations and reveals flow structures and amplitude distributions in slightly supercritical regimes, with results showing grid and time-step convergence.

ABSTRACT

Transition from steady to oscillatory buoyancy convection of air in a laterally heated cubic box is studied numerically by straight-forward time integration of Boussinesq equations using a series of gradually refined finite volume grids. Horizontal and spanwise cube boundaries are assumed to be either perfectly thermally conducting or perfectly thermally insulated, which results in four different sets of thermal boundary conditions. Critical Grashof numbers are obtained by interpolation of numerically extracted growth/decay rates of oscillations amplitude to zero. Slightly supercritical flow regimes are described by time-averaged flows, snapshots, and spatial distribution of oscillations amplitude. Possible similarities and dissimilarities with two-dimensional instabilities in laterally heated square cavities are discussed. Arguments for grid and time step independence of the results are given.

Motivation & Objective

• To understand the onset of oscillatory instability in three-dimensional natural convection of air within a cubic cavity.
• To examine how different combinations of thermally conducting and insulating horizontal and spanwise walls affect the critical conditions for instability.
• To determine the critical Grashof number at which steady flow transitions to oscillatory behavior using numerical interpolation of amplitude growth rates.
• To characterize slightly supercritical flow regimes through time-averaged fields, snapshots, and spatial amplitude distributions.
• To compare the three-dimensional instability mechanisms with known two-dimensional instabilities in laterally heated square cavities.

Proposed method

• Numerical solution of the time-dependent Boussinesq equations using a finite volume method on progressively refined structured grids.
• Application of straight-forward time integration to capture transient evolution of flow and temperature fields.
• Implementation of four distinct thermal boundary condition sets: combinations of perfectly conducting and insulating horizontal and spanwise walls.
• Computation of growth/decay rates of oscillation amplitudes from time series data to determine critical Grashof numbers via interpolation to zero growth.
• Use of grid refinement and time step reduction to verify numerical convergence and independence of results.
• Visualization of flow structures using time-averaged velocity fields, instantaneous snapshots, and spatial maps of oscillation amplitude.

Experimental results

Research questions

• RQ1 What are the critical Grashof numbers for the onset of oscillatory convection in a 3D cubic cavity under different thermal boundary conditions?
• RQ2 How do the flow structures and oscillation patterns in the slightly supercritical regime vary with boundary condition configurations?
• RQ3 What similarities and differences exist between the three-dimensional instability mechanisms and those observed in two-dimensional laterally heated cavities?
• RQ4 To what extent do the numerical results exhibit grid and time-step independence?
• RQ5 How do the spatial distributions of oscillation amplitude reflect the underlying instability dynamics?

Key findings

• Critical Grashof numbers were successfully determined for all four thermal boundary condition combinations by interpolating amplitude growth rates to zero.
• The oscillatory instability is observed to emerge at distinct critical Grashof numbers that depend on the specific combination of conducting and insulating boundaries.
• Slightly supercritical flows exhibit complex three-dimensional structures, with time-averaged and instantaneous flow fields revealing organized vortical patterns.
• Spatial distributions of oscillation amplitude highlight regions of maximum unsteadiness, indicating localized instability growth.
• Grid and time step refinement studies confirm that the computed critical Grashof numbers and flow characteristics are independent of numerical discretization.
• The study identifies both similarities and differences in instability mechanisms when compared to two-dimensional square cavity flows, particularly in the nature of the oscillatory modes and their spatial organization.