arXiv subjects

Baole Wen

Publications and source records attributed to Baole Wen.

10 recordsLinked to original sources

arXiv2026

Optimal heat transport at the edge of energy stability

High heat transfer in Rayleigh--Bénard convection is commonly associated with vigorous turbulent motion, but turbulence intensity alone does not explain how a limiting transport state is selected. We propose that such limiting states are organized by marginal energy stability. Starting from the exact perturbation-energy balance, we determine, for a prescribed mean temperature profile, the smallest neutral Rayleigh number over the balance parameter and all admissible disturbances. The corresponding marginal modes are then coupled to the exact mean-temperature equation, producing a self-consistent profile and heat flux. At large $Ra$, the selected branch gives $Nu\simeq0.0245Ra^{1/2}$ and develops conductive inner layers, logarithmic-like intermediate regions and a weakly stably stratified core. An equivalent background-field formulation yields the same governing equations and establishes uniqueness of the selected mean profile. Three-dimensional simulations at $10^6\le Ra\le10^8$ show that distributed thermal forcing based on this profile suppresses convective motion while retaining a large wall heat flux.

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arXiv2026

Long-time behavior of optimal mixing in an advection-diffusion shell model

We investigate the long-time behavior of optimal mixing in an advection-diffusion equation using a shell model framework. Our focus is on quantifying the decay of the scalar variance, measured by the negative Sobolev norm $H^{-1}$, under enstrophy-constrained stirring. We perform long-time computations using both local-in-time (maximizing the instantaneous mixing rate) and global-in-time (maximizing mixedness at a prescribed final time) optimization strategies. For mixing with diffusion ($κ>0$), the numerical results show that the scalar length scale eventually becomes limited by a generalized Batchelor scale, in close agreement with theoretical predictions. In this regime, the $H^{-1}$ mix-norm decays exponentially in time with a decay rate that is independent of the diffusivity $κ$. Compared with the purely advective case ($κ= 0$), diffusion significantly enhances the long-time mixing rate; moreover, increasing diffusivity further improves mixing efficiency by reducing the prefactor of the exponential decay. Guided by these numerical observations, we derive new conditional lower bounds on the $H^{-1}$ norm whose exponential decay rates are strictly independent of the diffusivity parameter $κ$, for all $κ> 0$. We further establish conditional upper bounds on the maximal rate of enhanced dissipation of the scalar variance, showing that the effective diffusion time scale is at least of the order $|\logκ|$.

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arXiv2026

Steady Rayleigh--Bénard convection: strongly nonlinear high-wavenumber rolls

In Rayleigh--Bénard convection, two-dimensional steady rolls bifurcate supercritically at a Rayleigh number $Ra$ that depends on their horizontal-to-vertical aspect ratio $Γ$, and they exist at all larger $Ra$ despite being unstable. Heat transport by certain rolls---quantified by the Nusselt number $Nu$---closely resembles turbulent transport, yet $Nu$ scalings of rolls are understood only for specific boundary conditions and $Γ$--$Ra$ limits. Here we investigate the high-wavenumber limit $Γ= O(Ra^{-1/4})$ as $Ra \to \infty$, using numerics and matched asymptotic analysis. We compute steady rolls between stress-free boundaries for Prandtl numbers $10^{-1} \leq Pr \leq 10^{3/2}$ and $Ra$ reaching $10^{19}$. While the $Γ= O(Ra^{-1/4})$ limit gives smaller $Nu$ than when $Γ= O(1)$, we identify prefactors $c$ in $Γ= c\,Ra^{-1/4}$ that locally maximize $Nu$. These locally $Nu$-maximizing rolls display approximate scalings $Nu \propto Ra^{0.29}$ and $Re \propto Ra^{0.40}$, with the Reynolds number $Re$ defined using root-mean-square velocity. Our asymptotic analysis reveals a vertically stacked four-layer structure near each boundary, predicting $Nu = O(Ra^{3/10})$ and $Re = O(Ra^{2/5})$. This asymptotic construction largely follows that of Taylor vortices by \cite{Deguchi2023}, but we identify a thin plume region within the middle boundary layer whose inclusion eliminates the logarithmic factors in Deguchi's predictions. Asymptotic arguments and numerics suggest the same scalings for stress-free or no-slip boundaries, unlike in other $Γ$--$Ra$ limits. Our asymptotics extend the weakly nonlinear analysis of Blennerhassett \& Bassom (1994) into the strongly nonlinear regime and complement the asymptotics of Chini \& Cox (2009) for $Γ= O(1)$ rolls.

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arXiv2022

Steady Rayleigh--Bénard convection between no-slip boundaries

The central open question about Rayleigh--Bénard convection -- buoyancy-driven flow in a fluid layer heated from below and cooled from above -- is how vertical heat flux depends on the imposed temperature gradient in the strongly nonlinear regime where the flows are typically turbulent. The quantitative challenge is to determine how the Nusselt number $Nu$ depends on the Rayleigh number $Ra$ in the $Ra\to\infty$ limit for fluids of fixed finite Prandtl number $Pr$ in fixed spatial domains. Laboratory experiments, numerical simulations, and analysis of Rayleigh's mathematical model have yet to rule out either of the proposed `classical' $Nu \sim Ra^{1/3}$ or `ultimate' $Nu \sim Ra^{1/2}$ asymptotic scaling theories. Among the many solutions of the equations of motion at high $Ra$ are steady convection rolls that are dynamically unstable but share features of the turbulent attractor. We have computed these steady solutions for $Ra$ up to $10^{14}$ with $Pr=1$ and various horizontal periods. By choosing the horizontal period of these rolls at each $Ra$ to maximize $Nu$, we find that steady convection rolls achieve classical asymptotic scaling. Moreover, they transport more heat than turbulent convection in experiments or simulations at comparable parameters. If heat transport in turbulent convection continues to be dominated by heat transport in steady rolls as $Ra\to\infty$, it cannot achieve the ultimate scaling.

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arXiv2021

Convective carbon dioxide dissolution in a closed porous medium at high-pressure real-gas conditions

We combine modeling and measurements to investigate the dynamics of convective carbon dioxide (CO$_2$) dissolution in a pressure-volume-temperature cell, extending a recent study by Wen et al. (J. Fluid Mech., vol. 854, 2018, pp. 56--87) at low-pressure under ideal-gas conditions to high-pressure and real-gas conditions. Pressure-dependent compressibility and solubility are included to model the evolution of CO$_2$ concentration in the gas phase and at the interface, respectively. Simple ordinary-differential-equation models are developed to capture the mean behavior of the convecting system at large Rayleigh number and are then verified by using both numerical simulations and laboratory experiments. The prefactor for the linear scaling of convective CO$_2$ dissolution is evaluated -- for the first time -- by using pressure-decay experiments in bead packs at reservoir conditions. The results show that our models could quantitatively predict the process of the convective CO$_2$ dissolution in pressure-decay experiments. Moreover, the results also reveal that for increasing gas pressure in closed systems, the negative feedback of the pressure drop -- resulting from the dissolution of CO$_2$ in the liquid -- is weakened due to the decrease of the solubility constant at real-gas conditions. Our analysis provides a new direction for determination and validation of the convective dissolution flux of CO$_2$ in porous media systems.

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arXiv2020

Steady Rayleigh--Bénard convection between stress-free boundaries

Steady two-dimensional Rayleigh--Bénard convection between stress-free isothermal boundaries is studied via numerical computations. We explore properties of steady convective rolls with aspect ratios $π/5\leΓ\le4π$, where $Γ$ is the width-to-height ratio for a pair of counter-rotating rolls, over eight orders of magnitude in the Rayleigh number, $10^3\le Ra\le10^{11}$, and four orders of magnitude in the Prandtl number, $10^{-2}\le Pr\le10^2$. At large $Ra$ where steady rolls are dynamically unstable, the computed rolls display $Ra \rightarrow \infty$ asymptotic scaling. In this regime, the Nusselt number $Nu$ that measures heat transport scales as $Ra^{1/3}$ uniformly in $Pr$. The prefactor of this scaling depends on $Γ$ and is largest at $Γ\approx 1.9$. The Reynolds number $Re$ for large-$Ra$ rolls scales as $Pr^{-1} Ra^{2/3}$ with a prefactor that is largest at $Γ\approx 4.5$. All of these large-$Ra$ features agree quantitatively with the semi-analytical asymptotic solutions constructed by Chini \& Cox (2009). Convergence of $Nu$ and $Re$ to their asymptotic scalings occurs more slowly when $Pr$ is larger and when $Γ$ is smaller.

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arXiv2018

Reduced modeling of porous media convection in a minimal flow unit at large Rayleigh number

Direct numerical simulations (DNS) indicate that at large values of the Rayleigh number ($Ra$) convection in porous media self-organizes into narrowly-spaced columnar flows, with more complex spatiotemporal features being confined to boundary layers near the top and bottom walls. In this investigation of high-$Ra$ porous media convection in a minimal flow unit, two reduced modeling strategies are proposed that exploit these specific flow characteristics. Both approaches utilize the idea of decomposition since the flow exhibits different dynamics in different regions of the domain: small-scale cellular motions generally are localized within the thermal and vorticity boundary layers near the upper and lower walls, while in the interior, the flow exhibits persistent large-scale structures and only a few low (horizontal) wavenumber Fourier modes are active. Accordingly, in the first strategy, the domain is decomposed into two near-wall regions and one interior region. Our results confirm that suppressing the interior high-wavenumber modes has negligible impact on the essential structural features and transport properties of the flow. In the second strategy, a hybrid reduced model is constructed by using Galerkin projection onto a fully \emph{a priori} eigenbasis drawn from energy stability and upper bound theory, thereby extending the model reduction strategy developed by Chini \emph{et al.} (\emph{Physica~D}, vol. 240, 2011, pp. 241--248) to large $Ra$. The results indicate that the near-wall upper-bound eigenmodes can economically represent the small-scale rolls within the exquisitely-thin thermal boundary layers. Relative to DNS, the hybrid algorithm enables over an order-of-magnitude increase in computational efficiency with only a modest loss of accuracy.

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arXiv2018

Convection in porous media with dispersion

We investigate the effect of dispersion on convection in porous media by performing direct numerical simulations (DNS) in a two-dimensional Rayleigh-Darcy domain. Scaling analysis of the governing equations shows that the dynamics of this system are not only controlled by the classical Rayleigh-Darcy number based on molecular diffusion, $Ra_m$, and the domain aspect ratio, but also controlled by two other dimensionless parameters: the dispersive Rayleigh number $Ra_d = H/α_t$ and the dispersivity ratio $r = α_l/α_t$, where $H$ is the domain height, $α_t$ and $α_l$ are the transverse and longitudinal dispersivities, respectively. For $Δ= Ra_d/Ra_m > O(1)$, the influence from the mechanical dispersion is minor; for $Δ\ll 1$, however, the flow pattern is controlled by $Ra_d$ while the convective flux is $F\sim Ra_m$ for large $Ra_m$, but with a prefactor that has a non-monotonic dependence on $Ra_d$. Our DNS results also show that the increase of mechanical dispersion, i.e. decreasing $Ra_d$, will coarsen the convective pattern by increasing the plume spacing. Moreover, the inherent anisotropy of mechanical dispersion breaks the columnar structure of the mega-plumes at large $Ra_m$, if $Ra_d < 5000$. This results in a fan-flow geometry that reduces the convective flux.

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arXiv2018

Rayleigh fractionation in high-Rayleigh-number solutal convection in porous media

We study the fractionation of two components between a well-mixed gas and a saturated convecting porous layer. Motivated by geological carbon dioxide (CO$_2$) storage we assume that convection is driven only by the dissolved concentration of the first component, while the second acts as a tracer with increased diffusivity. Direct numerical simulations for convection at high Rayleigh numbers reveal that the partitioning of the components, in general, does not follow a Rayleigh fractionation trend, as commonly assumed. Initially, increases in tracer diffusivity also increase its flux, because the diffusive boundary layer penetrates deeper into the flow. However, for $D_2\geq 10\, D_1$, where $D_1$ and $D_2$ are, respectively, the diffusion coefficients of CO$_2$ and the tracer in water, the transverse leakage of tracer between up- and down-welling plumes reduces the tracer flux. Rayleigh fractionation between components is only realized in the limit of two gases with very large differences in solubility and initial concentration in the gas.

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arXiv2018

Dynamics of convective carbon dioxide dissolution in a closed porous media system

Motivated by geological carbon dioxide (CO$_2$) storage, many recent studies have investigated the fluid dynamics of solutal convection in porous media. Here we study the convective dissolution of CO$_2$ in a closed system, where the pressure in the gas declines as convection proceeds. This introduces a negative feedback that reduces the convective dissolution rate even before the brine becomes saturated. We analyse the case of an ideal gas with a solubility given by Henry's law, in the limits of very low and very high Rayleigh numbers. The equilibrium state in this system is determined by the dimensionless dissolution capacity, $Π$, which gives the fraction of the gas that can be dissolved into the underlying brine. Analytic approximations of the pure diffusion problem with $Π>0$, show that the diffusive base state is no longer self-similar and that diffusive mass transfer declines rapidly with time. Direct numerical simulations at high Rayleigh numbers show that no constant flux regime exists for $Π> 0$; nevertheless, the quantity $F/C_s^2$ remains constant, where $F$ is the dissolution flux and $C_s$ is the dissolved concentration at the top of the domain. Simple mathematical models are developed to predict the evolution of $C_s$ and $F$ for high-Rayleigh-number convection in a closed system. The negative feedback that limits convection in closed systems may explain the persistence of natural CO$_2$ accumulations over millennial timescales.

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