Session A16: Free Convection and Rayleigh-Benard II

Turbulence development of a vertical natural convection boundary layer

The present study considers the development of a turbulent natural convection boundary layer (NCBL) adjacent to a vertical isothermal wall with periodic boundary conditions. Results obtained from direct numerical simulations show evidence that the turbulent flow has two distinct stages depending on Grashof (or Reynolds) number. By investigating the turbulent statistics and energy spectra, it is shown that the near-wall region at relatively low Grashof number remains laminar while the outer boundary layer is fully turbulent (classical turbulent regime); whereas at higher Grashof number, the near wall is shown to become turbulent, indicating onset of an ultimate regime (Kraichnan 1962; Grossmann & Lohse 2011). It is found that the near-wall turbulence in the classical turbulent regime is predominantly sustained by the pressure transport from the outer shear layer; while in the ultimate turbulent regime the near-wall turbulence is generated and sustained locally via shear production. The transition from the classical to the ultimate turbulent regime is also clearly discerned using scaling and momentum balance arguments and by the appearance of near-wall streaks with constant spanwise spacing.   

Numerical computation of strongly nonlinear large-wavenumber steady states in Rayleigh–Bénard convection

Recent investigations confirm that steady convective flows share many structural features with turbulent Rayleigh–Bénard convection (RBC) and organize the turbulent dynamics. Previous computations of steady roll solutions in two-dimensional (2D) RBC between no-slip boundaries reveal that for fixed Rayleigh number Ra and Prandtl number Pr, the heat-flux-maximizing solution is always in the large-wavenumber regime. In this study, we numerically explore the large-wavenumber steady convection roll solutions that bifurcate supercritically from the motionless conductive state for 2D RBC between stress-free boundaries. Our new computations confirm the existence of a local heat-flux-maximizing solution in the large-wavenumber regime. To elucidate the asymptotic properties of this solution, we perform computations over eight orders of magnitude in the Rayleigh number, 10⁸ ≤ Ra ≤ 10^16.5, and two orders of magnitude in the Prandtl number, 10^-1 ≤ Pr ≤ 10^3/2. The numerical results indicate that as Ra → ∞, the local heat-flux-maximizing aspect ratio Γ_loc* ~ Ra^-1/4, the Nusselt number Nu(Γ_loc*) ~ Ra^3/10, and the Péclet number Re(Γ_loc*)Pr ~ Ra^2/5, where Re is the Reynolds number. Moreover, we demonstrate that the interior flow is accurately described by an analytical heat-exchanger solution, and discuss the connection to the large-wavenumber asymptotic solution given by Blennerhassett & Bassom (IMA J. Appl. Math., 1994). With a fixed aspect ratio 0.06 ≤ Γ ≤ π/5 at Pr = 1, however, our computations show that as Ra increases, the steady rolls converge to the semi-analytical asymptotic solutions constructed by Chini & Cox (Phys. Fluids, 2009), with scalings Nu ~ Ra^1/3 and RePr ~ Ra^2/3. Finally, we construct a phase diagram to demarcate distinct regimes of steady solutions in the large-Rayleigh-number-wavenumber plane.   

Asymptotic structure of strongly nonlinear large-wavenumber steady states in Rayleigh-Benard convection

Recent numerical studies (e.g., Wen et al. J. Fluid Mech. 2020, 2022) have revealed the existence of highly nonlinear steady states in two-dimensional (2D) Rayleigh-Benard convection (RBC) having a horizontal wavenumber k that increases in proportion to Ra^1/4, where Ra is the Rayleigh number. At O(1) Prandtl number Pr, these large-wavenumber states are local (in k) maximizers of the heat flux in steady 2D convection for both stress-free and no-slip walls. Here, we elucidate the asymptotic structure of these large-wavenumber equilibria as Ra→∞. Our construction thus complements the analysis of Blennherhassett & Bassom (IMA J. Appl. Math. 1994), who analyzed less strongly supercritical steady convective states having the same wavenumber scaling, and that by Chini & Cox (Phys. Fluids 2009), who analyzed steady cellular flows arising in stress-free RBC having O(1) wavenumber. We demonstrate the emergence of an intricate four-region wall-normal asymptotic structure and show that the Nusselt number Nu=O(Ra^3/10). These results are corroborated by numerical computations of coherent convective states in 2D RBC at extreme values of Ra and consistent with recent work by Deguchi (J. Fluid Mech. 2023) on steady axisymmetric Taylor-Couette flow.   

Simulating convection at extreme parameters on a logarithmic Fourier lattice

Our ability to study numerically the scaling properties of turbulent convection is limited by the high cost of direct numerical simulations (DNS) as the Rayleigh number (Ra) increases. This motivates the exploration of alternatives to DNS which enable faster computation by using reduced models of the full dynamics. Here we explore the use of logarithmic Fourier lattices (LFL) to capture extreme dynamic ranges of spatial scales in Rayleigh-Benard convection (RBC) at high Ra. The LFL scheme uses a Fourier series with logarithmically rather than linearly distributed wavenumbers. This scheme exactly captures the dynamics of constant-coefficient linear operators, but approximates nonlinear operators with a finite number of lattice-supported triads. By combining an LFL horizontal discretization with a sparse Chebyshev method in the vertical, we can simulate RBC at a substantially reduced cost compared to DNS. We will discuss ongoing work to implement efficient mixed LFL-Chebyshev solvers in 2D and 3D, along with results from RBC simulations in various parameter regimes. We compare these simulations to DNS results and assess their suitability for extrapolation beyond the current capabilities of DNS.   

Fixed-flux Rayleigh-Benard convection in doubly periodic domains

This work studies two-dimensional fixed-flux Rayleigh-Benard convection with periodic boundary conditions in both horizontal and vertical directions and analyzes its dynamics using numerical continuation, secondary instability analysis and direct numerical simulation. The fixed-flux constraint leads to time-independent elevator modes with a well-defined amplitude. Secondary instability of these modes leads to tilted elevator modes accompanied by horizontal shear flow. For Pr=1, where Pr is the Prandtl number, a subsequent subcritical Hopf bifurcation leads to hysteresis behavior between this state and a time-dependent direction-reversing state, followed by a global bifurcation leading to modulated traveling waves without flow reversal. At high Rayleigh numbers, chaotic behavior dominated by modulated traveling waves appears. In the low Pr regime, relaxation oscillations between the conduction state and the elevator mode appear, followed by quasiperiodic and chaotic behavior as the Rayleigh number increases. At high Pr, the large-scale shear weakens, and the flow shows bursting behavior that can lead to significantly increased heat transport or even intermittent stable stratification.    

High-resolution simulation boundary layer studies in Rayleigh-Bénard convection

We analyse the near-wall statistics of Rayleigh-Bénard convection through high resolution direct numerical simulations of convection in a box with periodic sides and a modestly large aspect ratio of 4. The Prandtl number is Pr = 0.7, while the Rayleigh number is 10⁵ < Ra < 10¹⁰. We analyze the time-series and temporal statistics of volume averaged and wall Nusselt numbers, as well as mean kinetic energy. Additionally we present the velocity and temperature statistics in the viscous and thermal boundary layers. These new DNS data are also related to our previous high-Rayleigh number DNS in slender cells. We employ the latest GPU-accelerated spectral element solver, NekRS, which is derived from the widely used Nek5000 code, for our simulations. Weak and strong scaling performance of this new solver for simulations of thermal convection is also discussed in brief.   

Oscillatory motion in Rayleigh-Bénard convection confined in a thin rectangular container

Rayleigh-Bénard convections (RBCs) generated in a laterally thin fluid layer with a moderate thickness has been investigated recently from purposes to clarify relaxation of two-dimensional (2D), Hele-Shaw confinement and quasi-2D restriction of fully three-dimensional (3D) motion. We performed laboratory experiments using different thickener solutions adjusting Prandtl number Pr to investigate flow transition of the RBC in quasi-2D rectangular containers. The experimental setup was designed to modify the thickness of fluid layer by inserting acrylic plates, which also provides good thermal insulation. Rayleigh number Ra examined is distributed from 1 ⨯ 10⁵ to 4 ⨯ 10⁶ corresponding to the range in between quasi-Hele-Shaw regime and 3D convections according to the criteria recently proposed by Letelier et al. (2019) using permeability ε and Rayleigh-Darcy number, Ra_D, ε² Ra_D << 1. Velocity vector fields obtained by PIV represented the flow transition from elongated quasi-2D plumes to emergence of oscillatory corner rolls. The transition occurs around ε²Ra_D = 1 independent of Pr.   

Experimental measurements of spatiotemporal-resolved energy dissipation rate in turbulent Rayleigh–Bénard convection: Properties of the velocity gradient tensor and energy dissipation rate surrogates

We report a home-made velocity gradient tensor resolved particle image velocimetry (VGTR-PIV) system which spatially and temporally resolves all nine components of the highly fluctuating velocity gradient tensor. We applied this technique in the paradigmatic turbulent Rayleigh–Bénard convection in a cylindrical cell at three representative positions, i.e., center, side and bottom regions. The Rayleigh number varied in the range Ra=2×10⁸-8×10⁹ and the Prandtl number was fixed at Pr=4.34. The measured full velocity gradient tensor allows us to directly access, for the first time, the time-resolved energy dissipation rate and enstrophy in turbulent thermal convection. The probability density functions of the velocity gradient follow the exponential distribution and overlap at different Ra in each region. The turbulence is the most isotropic in the center and the least isotropic in the bottom region. In the side and bottom regions, the flow field becomes more isotropic as Ra increases. Comparing to the fully-resolved energy dissipation rate, the pseudo dissipation provides the best estimate, the planar (two-dimensional) surrogate has a larger relative error, and the one-dimensional surrogate has the largest error.   

Experimental measurements of spatiotemporal-resolved energy dissipation rate in turbulent Rayleigh–Bénard convection: Rayleigh number dependence and statistical properties

We obtain the spatiotemporal-resolved energy dissipation rate, for the first time, using our velocity gradient tensor resolved particle image velocimetry (VGTR-PIV) system in turbulent Rayleigh–Bénard convection. The power-law scaling of the time-averaged energy dissipation rate with Rayleigh number follows 〈ε_c〉_t∽Ra^1.54±0.02, 〈ε_s〉_t∽Ra^1.25±0.02, 〈ε_b〉_t∽Ra^1.23±0.02 and 〈ε_w〉_t∽Ra^1.25±0.02 in the center, side, bottom and wall regions, respectively, providing important constraints against which theoretical models would be validated. The probability density functions (PDFs) of the energy dissipation rate and enstrophy largely follow a stretched exponential distribution and deviate significantly from a log-normal distribution. Extreme events with high dissipation or vorticity are most intermittent in the side region, and the bottom region is less intermittent than the cell center. Exponential distributions of both dissipation rate and enstrophy PDFs exist in the bottom region at low Ra. The exponential distribution is an inherent property in near-wall regions, as confirmed by the conditional analysis and the wall energy dissipation rate PDF.