Session Q06: Convection: Numerical Simulations

Marginally-Stable Thermal Equilibria of Rayleigh-Bénard Convection

Natural convection exhibits turbulent flows which are difficult or impossible to resolve in nonlinear direct numerical simulations. In this work, we investigate a quasilinear form of the Rayleigh-Bénard problem which describes the bulk one-dimensional properties of convection without resolving the turbulent dynamics. We represent perturbations away from the mean as marginally-stable eigenmodes.  By constraining the perturbation amplitudes, the marginal-stability criterion allows us to evolve the background temperature profile under the influence of multiple eigenmodes representing flows at different length scales. We find the quasilinear system evolves to an equilibrium state where advective and diffusive fluxes sum to a constant. These marginally-stable thermal equilibria (MSTE) are exact solutions of the quasilinear equations. The mean MSTE temperature profiles have thinner boundary layers and larger Nusselt numbers than thermally-equilibrated 2D and 3D simulations of the full nonlinear equations. MSTE solutions exhibit a classic boundary-layer scaling of the Nusselt number Nu with the Rayleigh number Ra of Nu ∼ Ra^1/3. When an MSTE is used as initial conditions for a 2D simulation, we find that Nu quickly equilibrates without the burst of turbulence often induced by purely conductive initial conditions, but we also find that the kinetic energy is too large and viscously attenuates on a long viscous time scale.   

Eulerian large-eddy simulation of deep-water hydrocarbon plumes with multi-component gas bubble dissolution.

In deep-water oil spill accidents, the oil and natural gas released into the seawater rise as a multiphase hydrocarbon plume driven by the bubble-induced buoyancy force. Several of the dominant components of the natural gas bubbles experience considerable dissolution into the surrounding seawater in the deep-water environment, which causes the plume to lose the driving force and affect the overall plume flow dynamics. The dissolved natural gas components also disperse into the surrounding environment, imposing a great threat to the ocean ecosystem. In this study, a fast Eulerian large-eddy simulation model is developed to simulate deep-water hydrocarbon plumes with the effect of multi-component gas bubble dissolution. In the simulations, the gas and oil are released from 700m depth, and the dissolutions of methane (CH₄), ethane (C₂H₆) and propane (C₃H₈) are modeled. The dispersions of these dissolved gas components in the surrounding environment with and without a crossflow are also modeled and compared.   

Diffusion-free scaling in rotating spherical Rayleigh-Bénard convection

Direct numerical simulations are employed to reveal three distinctly different flow regions in rotating spherical Rayleigh-Bénard convection. In the low-latitude region I vertical (parallel to the axis of rotation) convective columns are generated between the hot inner and the cold outer sphere. The mid-latitude region II is dominated by vertically aligned convective columns formed between the Northern and Southern hemispheres of the outer sphere. The diffusion-free scaling, which indicates bulk-dominated convection, originates from this mid-latitude region. In the equator region III the vortices are affected by the outer spherical boundary and are much shorter than in region II.   

Numerical simulation of the electrolyte flow in the tanks of vanadium redox flow batteries

Redox flow batteries are a promising technology for large-scale energy storage. An energy-conversion cell stack converts electrical energy into chemical energy of two redox couples that are stored in independent tanks, reversing the process when power is needed. The flow of the electrolyte in the tanks is a relevant factor for battery optimization that has been largely overlooked to date, with departures from perfect mixing associated with an effective capacity loss of the system. The flow in the tanks is driven by the competing effects of inertia and buoyancy, the former associated with the momentum flux of the discharging jet and the latter with the density changes suffered by the electrolyte as it passes through the cell. Three different flow regimes can be found in the system, each dominated by one effect or by both when they are comparable. Our numerical simulations show that the electrolytes are never perfectly mixed, and that the homogeneity of the concentration field is strongly dependent on the tank design, with details of the inlet and outlet ducts being particularly relevant.   

The onset of zonal modes in two-dimensional Rayleigh-Bénard convection

We study the stability of steady convection rolls in 2D Rayleigh-Bénard convection with free-slip boundaries and horizontal periodicity over twelve orders of magnitude in the Prandtl number (10^-6 ≤ Pr ≤ 10⁶) and five orders of magnitude in the Rayleigh number (8π⁴ < Ra ≤ 3 × 10⁷). The analysis is facilitated by partitioning our modal expansion into so-called even and odd modes. With aspect ratio Γ = 2, we observe that zonal modes (with horizontal wavenumber equal to zero) can emerge only once the steady convection roll state consisting of even modes only becomes unstable to odd perturbations. We determine the stability boundary in the (Pr,Ra)-plane and observe remarkably intricate features corresponding to qualitative changes in the solution, as well as three regions where the steady convection rolls lose and subsequently regain stability as the Rayleigh number is increased. We study the asymptotic limit Pr → 0 and find that the steady convection rolls become unstable almost instantaneously, eventually leading to non-linear relaxation osculations and bursts, which we can explain with a weakly non-linear analysis. In the complementary large-Pr limit, we observe that the stability boundary reaches an asymptotic value Ra = 2.54 × 10⁷ and that the zonal modes at the instability switch off abruptly at a large, but finite, Prandtl number.   

Numerical simulations of turbulent thermal convection in mixed pure fluid – porous domains.

In this talk we report on numerical studies of turbulent thermal convection in domains with immersed porous structures that are heated internally. The main motivation for this study comes from the need to better understand the phenomena occurring in the early stages of a loss of cooling accident in storage pools of spent nuclear fuel. In this respect, the fuel racks that radiate heat are macroscopically modeled as a porous medium, with water being the working medium. In the first part of this talk we briefly present the mathematical model, which is based on the single-domain approach. In the second part, we discuss and analyze the results of Direct Numerical Simulations of turbulent convection in a cubical cavity that includes a porous region located at the bottom of the domain and away from the side walls. The porosity of this region is equal to 0.6. The Rayleigh numbers considered herein are in the range 10⁶ < Ra <10⁸, which corresponds to moderate turbulent intensities. Besides the analysis of turbulent statistics, we elaborate on the convective patterns and turbulent structures inside the porous medium as well as on the effects of the secondary flows adjacent to the porous region.   

Large Eddy Simulation of Isothermal and Non-isothermal Turbulent Flows in Ventilated Rooms

The focus of our work has been on understanding the physics of isothermal and buoyant turbulent flows in ventilated spaces, and on the location and extent of intermittent regions of low TKE, also known as dead-zones, that form in such spaces, and could become breeding grounds for airborne diseases. In classrooms, for instance, small aerosol particles that are released during normal breathing/speaking, could accumulate in these regions, and linger for much longer periods of time. If these aerosols were to contain a high viral density, the entire classroom could become infected. While increasing the air changes per hour (ACH), via natural ventilation (opening the windows) and forced ventilation (increasing airflow rate), is an effective means of flushing out airborne pathogens, it is a prohibitively expensive option for schools, especially in winter, when it is impossible to exploit natural ventilation to increase the ACH. Results from our resolved LES show that for a given ventilation pathway, incoming cool ventilated air (during warm weather) tends to reduce the extent of dead-zones, while incoming warm air (during colder weather) causes thermal stratification which increases the extent of these dead-zones. Our simulations also explore cost effective ways to improve air quality.   

Rayleigh-Benard convection: the container shape matters

We derive that the critical Rayleigh number for the onset of convection in right cylindrical cells with no-slip boundaries, of small diameter-to-height aspect ratios Γ, for any shape of plates, grows as ∼(1+A/Γ²)(1+B/Γ²), where A and B are determined by the cell shape and boundary conditions for the velocity and temperature. Under the assumption that in the expansions of the reduced temperature and velocity by the onset of convection in terms of the eigenfunctions of the Laplace operator, the contributions of the constant-sign eigenfunctions, both in the vertical and at least in one horizontal direction, vanish, we derive precise estimates of the critical Rayleigh number and determine the values of A and B for cylindrical and box-shaped cells. We compare our results with those from the linear stability analysis. Furthermore, we derive the relevant length scale in Rayleigh-Benard convection, which tends to the cell height for Γ→∞ and to the cell diameter for Γ→0. Finally, we discuss the optimum of the cell shape and show further applications of the developed ansatz. You are welcome to read the paper in Phys. Rev. Fluids (2021) with the same author and title.   

Numerical simulation of evaporation driven turbulent convection with a descending free surface

In this paper, we report on direct numerical simulations (DNS) of turbulent thermal convection of water driven by free surface evaporation above and by a heated wall below. The geometry considered herein is a cube and the Rayleigh number in our simulations is Ra = 1.3 × 10⁸. The evaporation rates are estimated via a dynamic model based on the presumed ambient conditions above the liquid water. The evaporative and convective heat losses predicted by this model are then used to prescribe an inhomogeneous, non-zero temperature gradient at the free surface. In our study we consider high evaporation rates for which the loss of liquid water is no longer negligible. For this reason, we take into account the loss of water and the descent of the free surface by a remeshing procedure.

In the first part of this talk, we outline the evaporation model mentioned above and the remeshing procedure that we follow. Next, we analyze the properties and structure of the Large Scale Circulation that is developed in the flow. Our talk concludes with the presentation of turbulence statistics for the flow in hand.