Theoretical and Computational Fluid Dynamics

Large-scale horizontal vortices associated with deformations of the aluminum-electrolyte interface have been observed in operating aluminum reduction cells as well as in physical and numerical models. To expose their importance, we analyze a particular class of magnetohydrodynamic (MHD) interfacial instabilities which are induced by rotation. As we focus on a single vortex, a cylindrical geometry is preferred. Two analytical models are proposed. In a first model based on the MHD shallow-water approximation, we consider a vortex that has a solid rotation profile to obtain a wave equation and a dispersion relation. A more realistic second model includes a viscous rotation profile and the treatment of the base-state interface deformation. Energetics of the flow gives further insight on how an initial perturbation evolves as an oscillatory or a non-oscillatory instability, depending on the direction of rotation. We find that the mechanism at the very origin of these instabilities is neither due to a shear between the two layers—and are therefore not Kelvin–Helmholtz instabilities—nor simply due to magnetic force alone, but rather to the indirect action of the centripetal pressure due to the rotation induced by magnetic force.

Numerical simulations of the evolution equation [14] for thickness of a film flowing down a vertical fiber are presented. Solutions with periodic boundary conditions on extended axial intervals develop trains of pulse-like structures. Typically, a group of several interacting pulses (or a solitary pulse) is bracketed by spans of nearly uniform thinned film and is virtually isolated: The evolution of such a “section” is modeled as a solution with periodic boundary conditions on the corresponding, comparatively short, interval. Single-pulse sections are steady-shape traveling waveforms (“cells” of shorter-period solutions). The collision of two pulses can be either a particle-like “elastic” rebound, or—and only if a control parameter S (proportional to the average thickness) exceeds a certain critical value, S c ≈ 1—a “deeply inelastic” coalescence. A pulse which grows by a cascade of coalescences is associated with large drops observed in experiments by Quéré [39] and our S c is in excellent agreement with its laboratory value.

The problem of convection induced by radial buoyancy in an electrically conducting fluid contained by a rotating cylindrical annulus (angular frequency, Ω) in the presence of a homogeneous magnetic field (B) in the azimuthal direction is considered. The small gap approximation is used together with rigid cylindrical boundaries. The onset of convection occurs in the form of axial, axisymmetric or oblique rolls. The angle ψ between the roll axis and the axis of rotation depends of the ratio between the Chandrasekhar number, Q∼B2, and the Coriolis number, τ∼Ω. Fully three-dimensional numerical simulations as well as Galerkin representations for roll patterns including the subsequent stability analysis are used in the theoretical investigation. At finite amplitudes, secondary transitions to 3D-hexarolls and to spatio-temporal chaos are found. Overlapping regions of pattern stability exist such that the asymptotically realized state may depend on the initial conditions.

This paper reports the results from a direct numerical simulation of an initially turbulent boundary layer passing over a wall-mounted “speed bump” geometry. The speed bump, represented in the form of a Gaussian distribution profile, generates a favorable pressure gradient region over the upstream half of the geometry, followed by an adverse pressure gradient over the downstream half. The boundary layer approaching the bump undergoes strong acceleration in the favorable pressure gradient region before experiencing incipient or very weak separation within the adverse pressure gradient region. These types of flows have proved to be particularly challenging to predict using lower-fidelity simulation tools based on various turbulence modeling approaches and warrant the use of the highest fidelity simulation techniques. The present direct numerical simulation is performed using a flow solver developed exclusively for graphics processing units. Simulation results are utilized to examine the key phenomena present in the flowfield, such as relaminarization/stabilization in the strong acceleration region succeeded by retransition to turbulence near the onset of adverse pressure gradient, incipient/weak separation and development of internal layers, where the sense of streamwise pressure gradient changes at the foot, apex and tail of the bump.

The paper presents adaptive mesh moving methods for large eddy simulation (LES) of turbulent flows. With this approach, a given number of grid points is redistributed with respect to an appropriately selected criterion. The Arbitrary Lagrangian–Eulerian formulation is applied to solve the governing equation on moving grids employing a collocated finite volume formulation. A dynamic moving mesh partial differential equation based on a variational principle is solved for the corner points of the grid by means of a dedicated solver. Adaptation is performed in a statistical sense so that statistical quantities of interest are employed. Various LES-specific design criteria and combination of them are proposed, such as the time-averaged gradient of streamwise velocity, turbulent kinetic energy and production rate. These are investigated in the framework of elementary and balanced monitor functions. These are tested for the three-dimensional flow in a channel with periodic constrictions. The numerical results are compared to a highly resolved LES reference solution. The independence of the moving mesh method from the initial LES is shown, and its potential to improve the efficient resolution of turbulent flow features is demonstrated.

This paper investigates the effect of vertical vibration on the stability of a dilute suspension of oxytactic microorganisms in a shallow horizontal fluid layer. For the case of high-frequency vibration, an averaging method is utilized to derive the equations describing the mean flow by decomposing the solutions of governing equations into two components: one that varies slowly with time, and a second that varies rapidly with time. Linear stability analysis is used to investigate the stability of the obtained averaged equations. It is predicted that high-frequency, low-amplitude vertical vibration has a stabilizing effect on a suspension of oxytactic microorganisms confined in a shallow horizontal layer.

In a quasi-geostrophic model, we study the baroclinic instability of a two-layer vortex. The singular unstable modes for potential vorticity anomalies are compared with the classical normal modes. Short-time singular modes are explosively unstable and, at short times, depend only on the baroclinic component of the flow. As time progresses, they evolve towards the normal modes and their sensitivity to flow parameters is explored. Asymptotic solutions are provided.

This paper presents a linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cylinder. The base flow is parallel to the axis of the cylinder at inflow boundary. The pressure gradient is zero in the streamwise direction. The base flow velocity profile is fully non-parallel and non-similar in nature. The boundary layer grows continuously in the spatial directions. Linearized Navier–Stokes (LNS) equations are derived for the disturbance flow quantities in the cylindrical polar coordinates. The LNS equations along with homogeneous boundary conditions forms a generalized eigenvalues problem. Since the base flow is axisymmetric, the disturbances are periodic in azimuthal direction. Chebyshev spectral collocation method and Arnoldi’s iterative algorithm is used for the solution of the general eigenvalues problem. The global temporal modes are computed for the range of Reynolds numbers and different azimuthal wave numbers. The largest imaginary part of the computed eigenmodes is negative, and hence, the flow is temporally stable. The spatial structure of the eigenmodes shows that the disturbance amplitudes grow in size and magnitude while they are moving towards downstream. The global modes of axisymmetric boundary layer are more stable than that of 2D flat-plate boundary layer at low Reynolds number. However, at higher Reynolds number they approach 2D flat-plate boundary layer. Thus, the damping effect of transverse curvature is significant at low Reynolds number. The wave-like nature of the disturbance amplitudes is found in the streamwise direction for the least stable eigenmodes.