An efficient method for the incompressible Navier-Stokes equations on irregular domains with no-slip boundary conditions, high order up to the boundary

Common efficient schemes for the incompressible Navier-Stokes equations, such as projection or fractional step methods, have limited temporal accuracy as a result of matrix splitting errors, or introduce errors near the domain boundaries (which destroy uniform convergence to the solution). In this paper we recast the incompressible (constant density) Navier-Stokes equations (with the velocity prescribed at the boundary) as an equivalent system, for the primary variables velocity and pressure. We do this in the usual way away from the boundaries, by replacing the incompressibility condition on the velocity by a Poisson equation for the pressure. The key difference from the usual approaches occurs at the boundaries, where we use boundary conditions that unequivocally allow the pressure to be recovered from knowledge of the velocity at any fixed time. This avoids the common difficulty of an, apparently, over-determined Poisson problem. Since in this alternative formulation the pressure can be accurately and efficiently recovered from the velocity, the recast equations are ideal for numerical marching methods. The new system can be discretized using a variety of methods, in principle to any desired order of accuracy. In this work we illustrate the approach with a 2-D second order finite difference scheme on a Cartesian grid, and devise an algorithm to solve the equations on domains with curved (non-conforming) boundaries, including a case with a non-trivial topology (a circular obstruction inside the domain). This algorithm achieves second order accuracy (in L-infinity), for both the velocity and the pressure. The scheme has a natural extension to 3-D.Comment: 50 pages, 14 figure

Reducing spurious flow in simulations of electrokinetic phenomena

Electrokinetic transport phenomena can strongly influence the behaviour of macromolecules and colloidal particles in solution, with applications in, e.g., DNA translocation through nanopores, electro-osmotic flow in nanocapillaries, and electrophoresis of charged macromolecules. Numerical simulations are an important tool to investigate these electrokinetic phenomena, but are often plagued by spurious fluxes and spurious flows that can easily exceed physical fluxes and flows. Here, we present a method that reduces one of these spurious currents, spurious flow, by several orders of magnitude. We demonstrate the effectiveness and generality of our method for both electrokinetic lattice-Boltzmann and finite-element-method based algorithms by simulating a charged sphere in an electrolyte solution, and flow through a nanopore. We also show that previous attempts to suppress these spurious currents introduce other sources of error.Comment: 13 pages, 7 figure

On hydrodynamic shear turbulence in Keplerian disks: via transient growth to bypass transition

This paper deals with the problem of hydrodynamic shear turbulence in non-magnetized Keplerian disks. We wish to draw attention to a route to hydrodynamic turbulence which seems to be little known by the astrophysical community, but which has been intensively discussed among fluid dynamicists during the past decade. In this so-called `bypass' concept for the onset of turbulence, perturbations undergo a transient growth, and they may reach an amplitude that is sufficiently large to allow positive feedback through nonlinear interactions. This transient growth is linear in nature, and thus it differs in principle from the well-known nonlinear instability. We describe the type of perturbations that according to this process are the most likely to lead to turbulence, namely non-axisymmetric vortex mode perturbations in the two dimensional limit. We show that the apparently inhibiting action of the Coriolis force on the dynamics of such vortical perturbations is substantially diminished due to the pressure perturbations, contrary to current opinion. We stress the similarity of the turbulent processes in Keplerian disks and in Cartesian flows and conclude that the prevalent skepticism of the astrophysical community on the occurrence of hydrodynamic shear turbulence in such disks is not founded.Comment: 8 pages, 1 figure, accepted in A &

Planktonic events may cause polymictic-dimictic regime shifts in temperate lakes

Water transparency affects the thermal structure of lakes, and within certain lake depth ranges, it can determine whether a lake mixes regularly (polymictic regime) or stratifies continuously (dimictic regime) from spring through summer. Phytoplankton biomass can influence transparency but the effect of its seasonal pattern on stratification is unknown. Therefore we analysed long term field data from two lakes of similar depth, transparency and climate but one polymictic and one dimictic, and simulated a conceptual lake with a hydrodynamic model. Transparency in the study lakes was typically low during spring and summer blooms and high in between during the clear water phase (CWP), caused when zooplankton graze the spring bloom. The effect of variability of transparency on thermal structure was stronger at intermediate transparency and stronger during a critical window in spring when the rate of lake warming is highest. Whereas the spring bloom strengthened stratification in spring, the CWP weakened it in summer. The presence or absence of the CWP influenced stratification duration and under some conditions determined the mixing regime. Therefore seasonal plankton dynamics, including biotic interactions that suppress the CWP, can influence lake temperatures, stratification duration, and potentially also the mixing regime

Kohaerente Strukturen und Chaos beim laminar-turbulenten Grenzschichtumschlag

In the present doctoral thesis, the coherent structures of a three-dimensionally developing transitional plate boundary layer were investigated based on data from a direct numerical simulation of such a flow. In the process, the coherent structures were calculated on the basis of the definition given by Lumley, which permits to calculate these structures from the given flow fields alone and without using additional information. A consideration of the development of two-dimensional structures in planes perpendicular to the direction of flow showed that the transition process - at least in the region investigated here - does not simply lead to a breakdown of the order in the flow. In the present doctoral thesis, for the first time, three-dimensional coherent structures were calculated exclusively by means of a the Proper Orthogonal Decomposition (POD) method. In the process it was shown that POD is capable of representing both the shape and the deformations of structures developing in the three-dimensional boundary layer. (orig.) In dieser Arbeit wurden die kohaerenten Strukturen einer sich raeumlich entwickelnden transitionellen Plattengrenzschicht anhand von Daten aus einer direkten numerischen Simulation einer solchen Stroemung untersucht. Die kohaerenten Strukturen wurden dabei auf der Grundlage der von Lumley gegebenen Definition berechnet, die es erlaubt, diese Strukturen allein aus den gegebenen Stroemungsfeldern und ohne Zuhilfenahme von zusaetzlichen Informationen zu berechnen. Die Betrachtung der Entwicklung zweidimenstionaler Strukturen in Ebenen senkrecht zur Stroemungsrichtung zeigte, dass der Umschlagvorgang -zumindest in dem hier untersuchten Bereich - nicht einfach zu einem Zusammenbruch der Ordnung in der Stroemung fuehrt. In der vorliegenden Arbeit wurden erstmals dreidimensionale kohaerente Strukturen auschliesslich mit Hilfe des Verfahrens der Proper Orthogonal Decomposition (POD) berechnet. Dabei konnte gezeigt werden, dass die POD sowohl die Gestalt als auch die Verformungen der sich in der raeumlichen Grenzschicht entwickelnden Strukturen wiederzugeben in der Lage ist. (orig.)Available from TIB Hannover: DW 7127 / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

<Poster Presentation 4>Tracking chaotic behavior in environmental hydraulics. Measurement models for the temporal coefficients of Proper Orthogonal Decomposition of free-surface flows

[Date] November 28 (Mon) - December 2 (Fri), 2011: [Place] Kyoto University Clock Tower Centennial Hall, Kyoto, JAPA