Morten Brøns

Evaluation of Proper Orthogonal Decomposition--Based Decomposition Techniques Applied to Parameter-Dependent Nonturbulent Flows

The proper orthogonal decomposition (POD) method as a systematic technique to analyze parameter-dependent problems may be inappropriate as a practical tool for generating low-dimensional models. We propose a weighted POD (w-POD) as an alternative to give higher priority to low-energetic or important modes by simply weighting. A predefined POD (p-POD) is suggested, where modes are selected not only on the basis of energy but also on some a priori knowledge of the system. The techniques are tested on a flow problem undergoing steady or unsteady transition but may in general be applied to nonlinear systems of discretized PDEs closed by additional linear algebraic equations. Various residual functions to measure the quality of the reduced models are introduced. They provide lower limits of residual energy to ensure an adequate physical representation for a given modeling. It is shown that modes extracted locally may generally not contain information regarding the global dynamics, and the dynamics may not be correctly predicted over a longer range of model parameters. It is demonstrated that by weighting and predefining base vectors it is possible to improve the POD techniques capability to generate low-dimensional models.

Introduction to Focus Issue: Mixed Mode Oscillations: Experiment, Computation, and Analysis

Mixed mode oscillations (MMOs) occur when a dynamical system switches between fast and slow motion and small and large amplitude. MMOs appear in a variety of systems in nature, and may be simple or complex. This focus issue presents a series of articles on theoretical, numerical, and experimental aspects of MMOs. The applications cover physical, chemical, and biological systems.

Topology of vortex breakdown bubbles in a cylinder with a rotating bottom and a free surface

The flow patterns in the steady, viscous flow in a cylinder with a rotating bottom and a free surface are investigated by a combination of topological and numerical methods. Assuming the flow is axisymmetric, we derive a list of possible bifurcations of streamline structures on varying two parameters, the Reynolds number and the aspect ratio of the cylinder. Using this theory we systematically perform numerical simulations to obtain the bifurcation diagram. The stability limit for steady flow is found and established as a Hopf bifurcation. We compare with the experiments by Spohn, Mory & Hopfinger (1993) and find both similarities and differences.

Streamline topology of steady axisymmetric vortex breakdown in a cylinder with co- and counter-rotating end-covers

Using a combination of bifurcation theory for two-dimensional dynamical systems and numerical simulations, we systematically determine the possible flow topologies of the steady vortex breakdown in axisymmetric flow in a cylindrical container with rotating end-covers. For fixed values of the ratio of the angular velocities of the covers in the range from −0.02 to 0.05, bifurcations of recirculating bubbles under variation of the aspect ratio of the cylinder and the Reynolds number are found. Bifurcation curves are determined by a simple fitting procedure of the data from the simulations. For the much studied case of zero rotation ratio (one fixed cover) a complete bifurcation diagram is constructed. Very good agreement with experimental results is obtained, and hitherto unresolved details are determined in the parameter region where up to three bubbles exist. For non-zero rotation ratios the bifurcation diagrams are found to change dramatically and give rise to other types of bifurcations.

Streamline Topology: Patterns in Fluid Flows and their Bifurcations

Abstract Using dynamical systems theory, we consider structures such as vortices and separation in the streamline patterns of fluid flows. Bifurcation of patterns under variation of external parameters is studied using simplifying normal form transformations. Flows away from boundaries, flows close to fixed walls, and axisymmetric flows are analyzed in detail. We show how to apply the ideas from the theory to analyze numerical simulations of the vortex breakdown in a closed cylindrical container.

Asymptotic Analysis of Canards in the EOE Equations and the Role of the Inflection Line

The Edblom–Orbán–Epstein (EOE) reaction, involving iodate, sulphite and ferrocyanide ions may have oscillations in a continuous stirred tank reactor. A simplification of a 10-variable model, including two state variables, was analysed numerically by Peng et al. (Phil. Trans. R. Soc. Lond. A337, 275 (1991)). They found that within a very small range of the inflow concentration, the amplitude of of the oscillations varied drastically. Such behaviour is well understood in singular perturbations systems, where it is known as a canard explosion. Apparently, the EOE equations do not belong to this class. We show that the two-dimensional EOE equations can be recast as a singular perturbation problem. An asymptotic expansion of the the canard point is obtained, with very good agreement with the numerical results of Peng et al. To deal with the canard explosion in systems which are not of singular perturbation type, Peng et al. introduce a new definition of the canard point, based on change of curvature of the limit cycle. We discuss the new definition, and show that it may essentially agree with the definition based on a singular perturbation approach, but is much less fit for analytical computations. We show that the discontinuous canard transition suggested by Peng et al. violates a continuity theorem in the theory of ordinary differential equations.

On stagnation points and streamline topology in vortex flows

The problem of locating stagnation points in the flow produced by a system of N interacting point vortices in two dimensions is considered. The general solution follows from an 1864 theorem by Siebeck, that the stagnation points are the foci of a certain plane curve of class N −1 that has all lines connecting vortices pairwise as tangents. The case N =3, for which Siebecks curve is a conic, is considered in some detail. It is shown that the classification of the type of conic coincides with the known classification of regimes of motion for the three vortices. A similarity result for the triangular coordinates of the stagnation point in a flow produced by three vortices with sum of strengths zero is found. Using topological arguments the distinct streamline patterns for flow about three vortices are also determined. Partial results are given for two special sets of vortex strengths on the changes between these patterns as the motion evolves. The analysis requires a number of unfamiliar mathematical tools which are explained.

The influence of imperfections on the flow structure of steady vortex breakdown bubbles

Vortex breakdown bubbles in the flow in a closed cylinder with a rotating end-cover have previously been successfully simulated by axisymmetric codes in the steady range. However, high-resolution experiments indicate a complicated open bubble structure incompatible with axisymmetry. Numerical studies with generic imperfections in the flow have revealed that the axisymmetric bubble is highly sensitive to imperfections, and that this may resolve the apparent paradox. However, little is known about the influence of specific, physical perturbations on the flow structure. We perform fully three-dimensional simulations of the flow with two independent perturbations: an inclination of the fixed cover and a displacement of the rotating cover. We show that perturbations below a realistic experimental uncertainty may give rise to flow structures resembling those obtained in experiments, that the two perturbations may interact and annihilate their effects, and that the fractal dimension associated with the emptying of the bubble can quantitatively be linked to the visual bubble structure.

Streamline topology in the near wake of a circular cylinder at moderate Reynolds numbers

For the flow around a circular cylinder, the steady flow changes its topology at a Reynolds number around 6 where the flow separates and a symmetric double separation zone is created. At the bifurcation point, the flow topology is locally degenerate, and by a bifurcation analysis we find all possible streamline patterns which can occur as perturbations of this flow. We show that there is no a priori topological limitation from further assuming that the flow fulfils the steady Navier–Stokes equations or from assuming that a Hopf bifurcation occurs close to the degenerate flow. The steady flow around a circular cylinder experiences a Hopf bifurcation for a Reynolds number about 45–49. Assuming that this Reynolds number is so close to the value where the steady separation occurs that the flow here can be considered a perturbation of the degenerate flow, the topological bifurcation diagram will contain all possible instantaneous streamline patterns in the periodic regime right after the Hopf bifurcation. On the basis of the spatial and temporal symmetry associated with the circular cylinder and the structure of the topological bifurcation diagram, two periodic scenarios of instantaneous streamline patterns are conjectured. We confirm numerically the existence of these scenarios, and find that the first scenario exists only in a narrow range after the Hopf bifurcation whereas the second one persists through the entire range of Re where the flow can be considered two-dimensional. Our results corroborate previous experimental and computational results.

Experimental vortex breakdown topology in a cylinder with a free surface

The free surface flow in a circular cylinder driven by a rotating bottom disk is studied experimentally using particle image velocimetry. Results are compared with computational results assuming a stress-free surface. A dye visualization study by Spohn et al. [“Observations of vortex breakdown in an open cylindrical container with a rotating bottom,” Exp. Fluids 14, 70 (1993)], as well as several numerical computations, has found a range of different vortex breakdown structures in this flow. We confirm the existence of a transition where the top of the breakdown bubble crosses from the axis to the surface, which has previously only been found numerically. We employ a technique by Brons et al. [“Topology of vortex breakdown bubbles in a cylinder with rotating bottom and free surface,” J. Fluid Mech. 428, 133 (2001)] to find the corresponding bifurcation curve in the parameter plane, which has hitherto only been used on numerical data. The bifurcation curve located here agrees well with previous numerical s...