Steady, Quasi-Periodic, and Chaotic Regimes of Natural Convection Around a Heated Cylinder in a Square Cavity

Technical Summary: Steady, Quasi-Periodic, and Chaotic Regimes of Natural Convection Around a Heated Cylinder in a Square Cavity

Problem Statement
This study investigates the unsteady natural convection of air ($Pr = 0.71$) within a cold square enclosure containing a centrally heated circular cylinder of radius $R = 0.2L$. While the differentially heated square cavity is a standard benchmark, the addition of an internal heated cylinder introduces complex flow reorganization. Previous literature has largely focused on steady-state solutions or limited unsteady ranges, often imposing artificial symmetry constraints (e.g., clamping the stream function to zero on the cylinder) that suppress asymmetric flow modes. Consequently, the critical Rayleigh number ($Ra_c$) for the onset of unsteadiness, the specific route to chaos, and the full sequence of flow regimes for this configuration remained unresolved. This work aims to bridge these gaps by simulating the transition from steady laminar flow to chaos without enforcing symmetry, allowing asymmetric modes to grow naturally.

Methodology
The authors employed a pressure-based finite element method (FEM) on a body-conforming mesh. Key methodological features include:

• Governing Equations: The unsteady 2D Navier-Stokes and energy equations under the Boussinesq approximation were solved in primitive variables. No symmetry conditions were imposed on the vertical centerline.
• Mesh Generation: A body-conforming mesh was generated from a Cartesian background grid using a ray-cast projection followed by harmonic smoothing. This approach resolves the curved cylinder boundary without coordinate transformation or immersed boundary interpolation.
• Numerical Scheme: An explicit fractional-step (projection) algorithm with second-order Adams-Bashforth treatment for convection-diffusion terms was used. The pressure Poisson equation was solved using a warm-started Conjugate Gradient method with Element-By-Element (EbE) matrix-free products to handle unstructured meshes efficiently.
• Validation and Grid Independence: The solver was validated against the De Vahl Davis benchmark ($Ra = 10^6$). A Grid Convergence Index (GCI) study confirmed grid independence with a fine-grid uncertainty of 0.84% on the cylinder Nusselt number.
• Simulation Scope: Eight Rayleigh numbers were simulated, ranging from $10^4$ to $5 \times 10^7$. Time histories of the cylinder-averaged Nusselt number ($Nu_{cyl}$) were recorded, and energy spectral densities (ESD) and Poincaré maps were generated using data from eleven probe points distributed across the cavity.

Key Results
The study identifies three distinct flow regimes and characterizes the transition between them:

1. Steady Regime ($10^4 \le Ra \le 5 \times 10^6$):

   • The flow remains steady and symmetric about the vertical centerline, characterized by two mirror-image recirculation cells and a single thermal plume rising from the cylinder to the top wall.
   • At $Ra = 5 \times 10^6$, the first physical signature of symmetry breaking appears: the magnitude of the negative stream function core ($|\psi_{min}|$) exceeds the positive core ($\psi_{max}$) by approximately 3.6%, though the flow remains visually symmetric.
   • Heat transfer is dominated by the top wall (absorbing ~55% of total wall heat), while the bottom wall accounts for less than 2% due to thermal stratification in the lower cavity.

2. Onset of Unsteadiness ($5 \times 10^6 < Ra_c < 7.5 \times 10^6$):

   • The critical Rayleigh number for the onset of unsteadiness is bracketed between $5 \times 10^6$ and $7.5 \times 10^6$. This is determined by a sharp jump (over three orders of magnitude) in the standard deviation of $Nu_{cyl}$.
   • This critical value is higher than that implied by previous steady-only studies that enforced symmetry, confirming that the suppression of asymmetric modes artificially delays the transition.

3. Unsteady Regime ($Ra \ge 7.5 \times 10^6$):

   • Limit Cycle ($Ra = 7.5 \times 10^6$): The flow exhibits a single-frequency periodic oscillation (limit cycle). The disorder originates in the plume above the cylinder.
   • Quasi-Periodic ($Ra = 10^7$): A second incommensurate frequency emerges, placing the flow on a two-torus. The Poincaré sections show a closed curve, and the lower stratified pool begins to register the secondary frequency.
   • Chaos ($Ra = 5 \times 10^7$): The flow transitions to a broad-band strange attractor. The Poincaré sections reveal a scattered cloud of points, and the energy spectral density becomes continuous. The disorder propagates from the plume down the side walls, eventually reaching the lower stratified pool, which becomes active across a broad range of timescales.
   • Symmetry in Averages: Despite instantaneous snapshots being strongly asymmetric, the time-averaged fields recover bilateral symmetry. This is a statistical effect where the system visits mirror-image states equally often over time.

Heat Transfer Correlations

• A single power law, $Nu_{cyl} = 0.764 Ra^{0.226}$, fits the time-averaged cylinder Nusselt numbers across the entire range ($10^4 \le Ra \le 5 \times 10^7$) with a deviation of less than 6%.
• The exponent for the steady branch is $n \approx 0.224$, and for the unsteady branch, $n \approx 0.243$. Both values lie between the classical laminar boundary layer prediction ($n=0.25$) and the Castaing soft-turbulence scaling ($n \approx 0.286$), suggesting the flow is not fully turbulent but influenced by fluctuating fields.
• The bottom wall Nusselt number ($Nu_{bot}$) decreases as $Ra$ increases, a counter-intuitive trend attributed to the strengthening thermal stratification and the trapping of cold jets in bottom corner eddies.

Significance and Claims
The paper claims to provide the first comprehensive numerical characterization of the full transition sequence from steady to chaotic natural convection in a centrally heated cylinder configuration without artificial symmetry constraints. Key contributions include:

• Critical Rayleigh Number: Establishing a more accurate range for $Ra_c$ by allowing asymmetric modes to develop, correcting previous overestimations caused by symmetry constraints.
• Route to Chaos: Confirming that the transition follows the Ruelle-Takens-Newhouse route (Limit Cycle $\to$ Two-Torus $\to$ Strange Attractor) rather than a period-doubling cascade.
• Spatial Propagation of Disorder: Demonstrating that chaos initiates in the thermal plume and propagates downward, with the lower stratified pool remaining relatively quiescent until the highest Rayleigh numbers.
• Reference Data: Providing reliable benchmark data for the critical Rayleigh number, regime transitions, and global heat transfer correlations for configurations where internal geometry and unsteady dynamics interact.

The authors note that the study is limited to 2D, a single cylinder radius ($R=0.2L$), and $Pr=0.71$, suggesting extensions to 3D, off-center positions, and varying Prandtl numbers as future directions.