Session G33: Free and Rayleigh-Benard Convection II

Boundary layer theory for turbulent Rayleigh-Benard convection: Approximation of a self-organized turbulent wind

We have derived a system of the boundary layer equations for turbulent Rayleigh-Benard convection (RBC), where we consider a quasi-two-dimensional fluid flow along a semi-infinite horizontal heated plate with the requirement that the horizontal velocity vanishes far away from the plate. This boundary condition, which reflects the fact that the time-averaged horizontal component of the wind in RBC achieves its maximum value at a certain distance from the plate but vanishes in the core part of the cell, is different from what is considered in the Prandtl-Blasius or Falkner-Skan approximations. In this talk, we focus on the development of the velocity boundary layer equation. The turbulent fluctuations in the equation are taken into account by an eddy viscosity, which, based on Prandtl's mixing length ideas, is related to the dimensionless stream function.
  

Directional change of tracer trajectories in rotating Rayleigh-Bénard convection

In Lagrangian measurements of turbulence the complexity is reflected in the directional changes of tracer trajectories over many active timescales. We study angular statistics of tracer trajectories in rotating Rayleigh-Bénard convection both experimentally and numerically. Our aim is to explore the geometrical characterization of flow structures in turbulent convection in a wide range of timescales and how it is affected by background rotation. We find that the angle of directional change θ(τ) as a function of the time gap τ is distributed similarly as in homogeneous isotropic turbulence. The ensemble averaged angle Θ(τ) = 〈|θ(τ)|〉 displays a transition from a ballistic scaling Θ(τ) ∼ τ for τ < τ_η (the Kolmogorov timescale), to an inertial-range scaling Θ(τ) ∼ τ^c with smaller exponent c < 1 for τ_η < τ < T_L (the Lagrangian integral timescale). We show that the value of c is related with the dominant flow structures: the large-scale circulation for slow rotation and vertically aligned vortices emerging from the boundary layers for rapid rotation.   

Signature of Optimal Coherent Structures in Turbulent 2D Rayleigh Benard Convection

Steady solutions that maximize the heat transport in 2D Rayleigh Bénard convection (RBC) with no-slip horizontal walls have been studied in detail in Waleffe, Boonkasame & Smith (2015) and Sondak, Smith & Waleffe (2015). The close agreement of Nu-Ra scaling of the optimal solutions with data from 3D simulations and experiments suggests a link between the optimal coherent structures and turbulent 3D RBC. We explore this connection by searching for signatures of the optimal solutions in 2D turbulent RBC. Spatial correlations between the optimal structures and turbulent fields are computed at a variety of Pr and Ra in high aspect ratio domains. Our results show highly correlated (> 0.8) structures that resemble the optimal structure appear frequently throughout the turbulent flow field. The similarities in these structures are especially pronounced in the boundary layer. We compare momentum and thermal boundary layer scaling between the optimal and turbulent solutions and find that they are in very good agreement. As a precursor to stability analysis of the optimal solutions, the effect of mean flow on the agreement between statistics of the optimal and turbulent solutions is diagnosed using additional turbulent simulations that exclude mean flows by imposing mirror symmetry.    

Probing turbulent superstructures in Rayleigh-Bénard convection by Lagrangian trajectory clusters

We analyse the formation of large-scale patterns in a turbulent convection flow in a horizontally extended square convection cell by Lagrangian particle trajectories in three-dimensional direct numerical simulations. These large-scale patterns, which are termed turbulent superstructures of convection, are detected by the spectrum of the graph Laplacian matrix. The corresponding graph is built from the Lagrangian particle tracks. We demonstrate that the resulting trajectory clusters, which are obtained by a subsequent k-means clustering, agree with the superstructures in the Eulerian frame of reference. Furthermore, the characteristic times τ ^L and lengths λ_U^L of the superstructures are found to agree well with their Eulerian complements, τ and λ_U, respectively.  The clustering works well for times t < τ. Longer times t > τ require density-based trajectory clustering using time-averaged Lagrangian pseudo-trajectories. A coherent subset of these trajectories is obtained which consists of those particles tracks that are trapped for long times in the core of the superstructure rolls and thus not subject to ongoing turbulent dispersion.

  

Bounding heat transport for a model of Rayleigh-Benard convection using sum-of-squares optimization

We determine bounds on the maximum rate of heat transport (Nusselt number) for an 8-ODE model of Rayleigh-Benard convection developed by Gluhovsky et al. (2002). This truncated model is distinguished in the sense that it obeys many desired conservation laws and physical properties of the PDE. We use a general framework for bounding infinite-time averages in dynamical systems, which is similar to the use of Lyapunov functions in stability theory. Applying this framework, the maximal heat transport problem is computed numerically using sum-of-squares optimization. New upper bounds are established for the truncated system that are sharper than previously known bounds derived by Souza and Doering (2015). Additionally, the numerical results are used to inform the construction of new analytical bounds for the truncated model.   

Superstructures in inclined thermal convection of low-Prandtl-number fluids

Any tilt of a Rayleigh-Benard convection (RBC) cell against gravity changes the global flow structure inside the cell. Recent experiments by Vasil'ev et al., Tech. Phys. 60 (2015), and Frick et al., Europhys. Lett. 109 (2015), with liquid sodium (Prandtl number Pr=0.0089) demonstrated that the heat transport in low-Pr fluids and for small diameter-to-height aspect ratio containers is especially sensitive to the inclination angle. Our study of inclined convection in a cylindrical container of the aspect ratio 1/5 is based on direct numerical simulations (DNS) and we consider Pr≤1. We demonstrate that flow superstructures like the large scale circulation and boundary layers (BLs) are influenced by both, the inclination angle and the lateral confinement. For inclined convection we observe the formation of two system-sized plume columns, a hot and a cold one, that impinge on the opposite BLs, see Zwirner and Shishkina, J. Fluid Mech. 850 (2018). In RBC the confined cell supports the formation of multiple rolls on top of each other.