Natural-Convection Flows in Air-Filled, Differentially Heated Cavities with Adiabatic Horizontal Walls

Abstract

Two-dimensional natural convection in air-filled differentially heated cavities with adiabatic horizontal walls has been revisited by using stability analysis algorithms such as Newton's iteration (steady-state solving), Arnoldi's method, and the continuation method. We are particularly interested in computing Hopf bifurcation points characterizing the onset of time-dependent flows in these cavities.

Aspect ratios of 1–7 have been investigated, and accurate critical points of several unstable modes which were not fully available in the literature have been provided for each cavity. The critical values agree well with the results available and confirm previous observations: for A= 1–3 the onset of time-dependent flows is due to the detached flow structure near the exiting corners of vertical boundary layers; for A= 4–8 the onset of time-dependent flows results from traveling waves in vertical boundary layers. Investigating mesh dependence showed that the critical values obtained possess four significants figures for A= 1–3 and six for A= 4–8.

Steady-state solution by Newton's iteration allowed us to discover multiple steady-state solutions at A about 3, which have never been reported before. The first branch, B ₁, of steady-state solutions exists up to a turning point where is born the second branch, B ₂. It ends at another turning point at lower Rayleigh number, and one observes the third branch, B ₃, for increasing Rayleigh number. Solutions on B ₁ do not exhibit any detached flow structure, but those on B ₂ and B ₃ do. The detached flow structure is more pronounced on B ₃ than on B ₂. For A= 3, B ₃ becomes unstable because of the detached flow structure. With increasing A, branch B ₁ becomes unstable to traveling waves of vertical boundary layers and the stable part of B ₃ decreases and disappears completely. Due to different instability mechanisms observed on branches B ₁ and B ₃, there exist multiple time-dependent flows of quite different frequencies (for A= 3.4 at Ra = 1.15 × 10⁸, for example), which is also a novel phenomenon. Multiple steady-state solutions disappear between A= 2.8 and 2.9 and are believed to be closely linked to the behavior of the detached flow and the limiting effect of increasing A.

The authors thank Dr. L. S. Tuckerman for fruitful discussions and suggestions. Computations have been performed at CINES under research project lim2072 and at IDRIS under research project 50326.

Notes

Critical angular frequency, ω _c , is in the unit of κ/H ²Ra^0.5. CS, centrosymmetry; ACS, anticentrosymmetry. For A ≤ 3, flow structure near the exiting corners of the vertical boundary layers becomes unstable; it is low frequency. For A ≥ 4, vertical boundary layers become unstable to traveling waves of higher frequencies; 𝒩 denotes the number of wave structures that each unstable eigenmode contains. Results for A = 8 are reported from [21].

The results show that the values presented in Table 2 have at least four significant figures for A ≤ 3 and six for A ≥ 4. Results for A = 8 are reported from [21].

Computations have been performed with a unique mesh of 50 × 80. The numerical values should be considered as approximate since no mesh dependence nor accurate search of turning points were performed.