arXiv subjects

Patrick J. Stofanak

Publications and source records attributed to Patrick J. Stofanak.

3 recordsLinked to original sources

arXiv2025

Dynamical stability and flow regimes in a stably stratified valley-shaped cavity heated from below

We investigate the three-dimensional stability of a stably stratified fluid in a valley-shaped cavity heated from below using linear stability analysis and direct numerical simulations. We first describe the pure-conduction flow state and derive a dimensionless criterion that provides a lower bound for the onset of instability, valid for any slope angle. We then examine the sequence of flow regimes for a slope angle of $α= 30^{\circ}$ and Prandtl number $Pr = 7$, including two-dimensional steady states, the emergence of a Hopf bifurcation, and the formation of steady and oscillatory three-dimensional structures preceding the transition to fully unsteady, chaotic flow. Although the nonlinear governing equations depend on two dimensionless parameters, we find that the flow dynamics across a wide parameter range collapse to depend on a single parameter--the composite stratification parameter $Π_c$. However, as the system becomes more unstable, sensitivity to the second parameter, $Π_h$, increases. We construct a regime map of all observed flow states as a function of $Π_c$ and $Π_h$, and confirm the onset of chaos using Lyapunov exponents. Across all regimes, asymmetric circulation remains the dominant flow structure, persisting even in time-averaged fields of chaotic states. Finally, we characterize heat transfer in the cavity using the Nusselt number, which scales as $Nu \sim Π_c^{0.43}$ or equivalently $Nu \sim Ra^{0.275}$. This result further establishes $Π_c$ as a key dimensionless parameter governing the flow dynamics preceding the chaotic regime.

physics.flu-dyn↗

arXiv2024

Self-organization in a stably stratified, valley-shaped enclosure heated from below

We observe the spontaneous onset of three-dimensional motion from a quiescent, purely conductive state of a stably stratified fluid in a V-shaped enclosure heated from below, which ultimately self-organizes into a two-dimensional steady state without any external forcing to the initial configuration. We identify a dominant three-dimensional instability through modal stability analysis. Direct numerical simulations confirm this instability but also reveal that, after an initial period of spontaneous three-dimensional growth, the flow gradually self-organizes into a steady two-dimensional state without external intervention. This self-organization manifests consistently for any arbitrary infinitesimal three-dimensional disturbance to the initial quiescent configuration. We demonstrate that the mechanism driving this self-organization is the increasing dominance of viscous dissipation over buoyant production of disturbance kinetic energy at later stages of flow evolution from the initial quiescent state. Our investigation reveals a flow scenario in which the most natural transition pathway to the final state involves passing through an intermediate state with a higher dimension than the final state itself. Specifically, our final flow state is less complex than the three-dimensional most unstable eigenvector predicted by linear stability analysis. We demonstrate that the entire flow evolution remains non-turbulent throughout and closely aligns with results from linear stability analysis, distinguishing the present flow dynamics from transient chaos, which also features complex transient states that eventually converge to a less complex final state.

physics.flu-dyn↗

arXiv2024

An unusual bifurcation scenario in a stably stratified, valley-shaped enclosure heated from below

We delineate the structure of steady laminar flows within a stably stratified, valley-shaped triangular cavity heated from below through linear stability analysis and Navier-Stokes simulations. We derive an exact solution to the quiescent conduction state, and characterize the flow via the stratification perturbation parameter, $Π_s$, which is a measure of the strength of the surface heat flux relative to the background stable stratification. Beyond a threshold value of $Π_s$, two unstable eigenmodes appear, one marked by a dominant central circulation, and the other one exhibiting dual circulations of equal strength. Through Navier-Stokes simulations, we confirm that the central-circulation eigenmode generates a pair of asymmetric steady states, whereas the dual-circulation eigenmode leads to distinct upslope and downslope symmetric steady states. Linear stability analysis and Navier-Stokes simulations jointly confirm the instability of the two symmetric steady states, both of which transition to the asymmetric steady state under a perturbation. Thus, for a given set of dimensionless parameters, the Navier-Stokes equations admit at least five possible steady-state solutions. Two of these solutions, namely the quiescent, pure conduction state and the counter-intuitive symmetric downslope state, have previously been overlooked in heated, stably stratified, valley-shaped enclosures. These five flow solutions reveal an intriguing bifurcation structure, including both a perfect pitchfork bifurcation and a nested bifurcation that gives rise to two distinct states. The inner bifurcation, while resembling a pitchfork in some respects, does not break any symmetry of the valley due to the lack of any possible horizontal axis of symmetry. The categorization of this inner bifurcation remains an unresolved matter, as it does not conform to any established descriptions of canonical bifurcations.

physics.flu-dyn↗