The heat transport and corresponding changes in the large-scale circulation (LSC) in turbulent Rayleigh-Bénard convection are studied by means of three-dimensional direct numerical simulations as a function of the aspect ratio Γ of a closed cylindrical cell and the Rayleigh number Ra. The Prandtl number is P r = 0.7 throughout the study. The aspect ratio Γ is varied between 0.5 and 12 for a Rayleigh number range between 10 7 and 10 9 . The Nusselt number N u is the dimensionless measure of the global turbulent heat transfer. For small and moderate aspect ratios, the global heat transfer law N u = A × Ra β shows a power law dependence of both fit coefficients A and β on the aspect ratio. A minimum of N u(Γ) is found at Γ ≈ 2.5 and Γ ≈ 2.25 for Ra = 10 7 and Ra = 10 8 , respectively. This is the point where the LSC undergoes a transition from a single-roll to a doubleroll pattern. With increasing aspect ratio, we detect complex multi-roll LSC configurations in the convection cell. For larger aspect ratios Γ > ∼ 8, our data indicate that the heat transfer becomes independent of the aspect ratio of the cylindrical cell. The aspect ratio dependence of the turbulent heat transfer for small and moderate Γ is in line with a varying amount of energy contained in the LSC, as quantified by the Karhunen-Loève or Proper Orthogonal Decomposition (POD) analysis of the turbulent convection field. The POD analysis is conducted here by the snapshot method for at least 100 independent realizations of the turbulent fields. The primary POD mode, which replicates the time-averaged LSC patterns, transports about 50% of the global heat for Γ ≥ 1. The snapshot analysis enables a systematic disentanglement of the contributions of POD modes to the global turbulent heat transfer. Although the smallest scale -the Kolmogorov scale ηK -and the largest scale -the cell height H -are widely separated in a turbulent flow field, the LSC patterns in fully turbulent fields exhibit strikingly similar texture to those in the weakly nonlinear regime right above the onset of convection. Pentagonal or hexagonal circulation cells are observed preferentially if the aspect ratio is sufficiently large (Γ > ∼ 8).

A low-dimensional model (LDM) for turbulent Rayleigh-Bénard convection in a Cartesian cell with square domain, based on the Galerkin projection of the Boussinesq equations onto a finite set of empirical eigenfunctions, is presented. The empirical eigenfunctions are obtained from a joint Proper Orthogonal Decomposition (POD) of the velocity and temperature fields using the Snapshot Method on the basis of a direct numerical simulation (DNS). The resulting LDM is a quadratic inhomogeneous system of coupled ordinary differential equations which we use to describe the long-time temporal evolution of the large-scale mode amplitudes for a Rayleigh number of 10 5 and a Prandtl number of 0.7. The truncation to a finite number of degrees of freedom, that does not exceed a number of 310 for the present, requires the additional implementation of an eddy viscosity-diffusivity to capture the missing dissipation of the small-scale modes. The magnitude of this additional dissipation mechanism is determined by requiring statistical stationarity and a total dissipation that corresponds with the original DNS data. We compare the performance of two models, a constant so-called Heisenberg viscosity-diffusivity and a mode-dependent or modal one. The latter viscosity-diffusivity model turns out to reproduce the large-scale properties of the turbulent convection qualitatively well, even for a model with only a few hundred POD modes.

We construct a low-dimensional model (LDM) of turbulent mixed convection in a Cartesian cell with in- and outlets and local sources of heat which is narrow in one of the two horizontal space directions. The basis is a high-resolution three-dimensional direct numerical simulation (DNS) record. The model is derived with basis functions, which have been obtained by a proper orthogonal decomposition (POD) using the snapshot method. The POD analysis is applied for a sequence of three-dimensional snapshots as well as for data which are bulk-averaged in the direction of narrow extension. This step is taken since the flow is found to have no significant dependence along this direction in the cell. We compare the three-dimensional and two-dimensional POD modes. This simplification reduces the complexity of the problem significantly and allows us to construct and run a two-dimensional LDM with a small number of degrees of freedom. We study the long-time dynamical behavior of this system using a closure of the LDM based on a mode-dependent viscosity and diffusivity. The LDM has been optimized in terms of the standard deviation of the energy spectrum and the transient energy for different numbers of degrees of freedom by comparison with the original DNS data. We find that the evolution of the coherent structures of flow and temperature agrees well with the two-dimensional original data and determine their contribution to the global transfer of heat. Root-mean-square profiles of the fluctuations of the turbulent fields agree qualitatively well with the original simulation data, but deviate slightly in amplitude. We conclude that the reduction in the dimensionality and the number of degrees of freedom can reproduce the gross features of the mixed convection flow in this particular setup well.

Atherosclerosis, an artery disease, is currently the leading cause of death in the United States in both men and women. The first step in the development of atherosclerosis involves leukocyte adhesion to the arterial endothelium. It is broadly accepted that blood flow, more specifically wall shear stress (WSS), plays an important role in leukocyte capture and subsequent development of an atherosclerotic plaque. What is less known is how instantaneous WSS, which can vary by up to 5 Pa over one cardiac cycle, influences leukocyte capture. In this paper we use direct numerical simulations (DNS), performed using an in-house code, to illustrate that leukocyte capture is different whether as a function of instantaneous or time-averaged blood flow. Specifically, a stenotic plaque is modeled using a computational fluid dynamics (CFD) solver through fully three-dimensional Navier-Stokes equations and the immersed boundary method. Pulsatile triphasic inflow is used to simulate the cardiac cycle. The CFD is coupled with an agent-based leukocyte capture model to assess the impact of instantaneous hemodynamics on stenosis growth. The computed wall shear stress agrees well with the results obtained with a commercial software, as well as with theoretical results in the healthy region of the artery. The analysis emphasizes the importance of the instantaneous flow conditions in evaluating the leukocyte rate of capture. That is, the capture rate computed from mean flow field is generally underpredicted compared to the actual rate of capture. Thus, in order to obtain a reliable estimate, the flow unsteadiness during a cardiac cycle should be taken into account.

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