On finite element methods for 3D time–dependent convection–diffusion–reaction equations with small diffusion

The paper studies finite element methods for the simulation of time–dependent convection–diffusion–reaction equations with small diffusion: the SUPG method, a SOLD method and two types of FEM–FCT methods. The methods are assessed, in particular with respect to the size of the spurious oscillations in the computed solutions, at a 3D example with nonhomogeneous Dirichlet boundary conditions and homogeneous Neumann boundary conditions

Measurement and simulation of a droplet population in a turbulent flow field

The interaction of a disperse droplet population (spray) in a turbulent flow field is studied by combining wind tunnel experiments with simulations based on the model of a population balance system. The behavior of the droplets is modeled numerically by a population balance equation. Velocities of the air and of the droplets are determined by non-intrusive measurements. A direct discretization of the 4D equation for the droplet size distribution is used in the simulations. Important components of the numerical algorithm are a variational multiscale method for turbulence modeling, an upwind scheme for the 4D equation and a pre-processing approach to evaluate the aggretation integrals. The simulations of this system accurately predict the modifications of the droplet size distribution from the inlet to the outlet of the measurement section. Since the employed configuration is simple and considering that all measurement data are freely available thanks to an Internet-based repository, the considered experiment is proposed as a benchmark problem for the simulation of disperse two-phase turbulent flows

A comparative study of a direct discretization and an operator-splitting solver for population balance systems

A direct discretization approach and an operator-splitting scheme are applied for the numerical simulation of a population balance system which models the synthesis of urea with a uni-variate population. The problem is formulated in axisymmetric form and the setup is chosen such that a steady state is reached. Both solvers are assessed with respect to the accuracy of the results, where experimental data are used for comparison, and the efficiency of the simulations. Depending on the goal of simulations, to track the evolution of the process accurately or to reach the steady state fast, recommendations for the choice of the solver are given

Numerical methods for the simulation of an aggregation-driven droplet size distribution

A droplet size distribution in a turbulent flow field is considered and modeled by means of a population balance system. This paper studies different numerical methods for the 4D population balance equation and their impact on an output of interest, the time-space-averaged droplet size distribution at the outlet which is known from experiments. These methods include different interpolations of the experimental data at the inlet, various discretizations in time and space, and different schemes for computing the aggregation integrals. It will be shown that notable changes in the output of interest might occur. In addition, the efficiency of the studied methods is discussed

Numerical simulations and measurements of a droplet size distribution in a turbulent vortex street

A turbulent vortex street in an air flow interacting with a disperse droplet population is investigated in a wind tunnel. Non-intrusive measurement techniques are used to obtain data for the air velocity and the droplet velocity. The process is modeled with a population balance system consisting of the incompressible Navier--Stokes equations and a population balance equation for the droplet size distribution. Numerical simulations are performed that rely on a variational multiscale method for turbulent flows, a direct discretization of the differential operator of the population balance equation, and a modern technique for the evaluation of the coalescence integrals. After having calibrated two unknown model parameters, a very good agreement of the experimental and numerical results can be observed. Eine turbulente Wirbelstra\ss e in einer Luftstr\"omung mit einer dispergierten Tr\"opfchenpopulation wird in einem Windkanal untersucht. Nichtintrusive Messtechniken werden verwendet, um Daten bez\"uglich der Luft-- und Tr\"opfchengeschwindigkeiten zu gewinnen. Der zu Grunde liegende Prozess wird mit einem Populationsbilanzsystem modelliert, welches aus den inkompressiblen Navier--Stokes--Gleichungen und einer Populationsbilanzgleichung f\"ur die Tr\"opfchenverteilungsdichte besteht. Numerische Simulationen werden durchgef\"uhrt, welche ein variationelle Mehrskalenmethode f\"ur turbulente Str\"omungen, eine direkte Diskretisierung des Differentialoperators der Populationsbilanzgleichung und ein modernes Verfahren zur Berechnung der Koaleszensintegrale verwenden. Nachdem zwei unbekannte Modellparameter kalibriert worden sind, kann eine sehr gute Übereinstimmung der experimentellen und numerischen Ergebnisse beobachtet werden

Numerical methods for the simulation of an aggregation-driven droplet size distribution

A droplet size distribution in a turbulent flow field is considered and modeled by means of a population balance system. This paper studies different numerical methods for the 4D population balance equation and their impact on an output of interest, the time-space-averaged droplet size distribution at the outlet which is known from experiments. These methods include different interpolations of the experimental data at the inlet, various discretizations in time and space, and different schemes for computing the aggregation integrals. It will be shown that notable changes in the output of interest might occur. In addition, the efficiency of the studied methods is discussed

Numerical methods for the simulation of multiphase flows using population balances

Die Arbeit liefert ein Beitrag zur Numerik von Populationsbilanzsystemen am Beispiel einer tropfenbeladenen Strömung. Sie verfolgt im Wesentlichen zwei Ziele. Zunächst wurden genaue und effiziente Algorithmen zur Simulation eines meteorologisch relevanten Windkanalexperimentes entwickelt und mit Hilfe der Messdaten evaluiert. Zum anderen wurde untersucht, welchen Einfluss die Turbulenz auf das Verhalten der Tropfen ausübt. Bei dem zugrunde liegenden Experiment handelt es sich um eine Zweiphasen-Strömung, bei der kleine Tropfen in einen turbulenten Luftstrom injiziert und von der Strömung mitgerissen werden. Das zur Modellierung des Experimentes verwendete Populationsbilanzsystem ist ein gekoppeltes System, bestehend aus den Navier- Stokes-Gleichung zur Modellierung der Strömung und einer Populationsbilanzgleichung zur Modellierung der Tropfendichteverteilung. Diese Populationsbilanzgleichung modelliert drei Aspekte: die Bewegung der Tropfen in der turbulenten Luftströmung, das Wachstum in übersättigter Luft und die Koaleszenz. Die Gleichung ist direkt in vier Dimensionen modelliert, mit dem Durchmesser als innere Koordinate. Die experimentellen Daten gehen als Einströmbedingung in die Numerik ein. Auch zur Evaluation der Ergebnisse standen Daten zur Verfügung. Die Ergebnisse der Simulationen sind vielversprechend; es konnte eine weitgehende Übereinstimmung mit den experimentellen Daten erzielt werden. Zur Identifikation geeigneter Stabilisierungsmethoden zur Lösung der Populationsbilanzgleichung wurden mehrere Finite-Differenzen- und Finite-Elemente-Stabilisierungsmethoden anhand der Konvektions-Diffusions-Reaktions-Gleichungen miteinander verglichen. Bei den Finite-Differenzen-Methoden kristallisierte sich das ENO-Verfahren heraus, unter den Finite-Elemente-Diskretisierungen erzielte das lineare Gruppen-FEM- FCT-Verfahren den besten Kompromiss zwischen Genauigkeit und Rechenzeit. Eine neue Beobachtung ist, dass die FEM-FCT-Verfahren verhältnismäßig starke Verschmierungen zeigen, wenn Konvektionsrichtung und Gitter parallel sind. Diese Methoden wurden zur Diskretisierung der Populationsbilanzgleichung angewendet und zeigten im Wesentlichen das gleiche Verhalten. Zur Auswertung des Koaleszenzterms wurden ebenfalls mehrere Methoden untersucht. Insbesondere wurde eine neue Methode verwendet, die auf im Voraus berechneten Integralen beruht. Vom Aspekt der Genauigkeit, kann jedoch die massenerhaltende Methode empfohlen werden. Sie ist allerdings schwierig zu implementieren und erfordert spezielle Gitter. Für einfache Simulationen genügt eine Gauß-Quadraturmethode. Zur Untersuchung des Einflusses der Turbulenz wurde das Verhalten der Tropfendichteverteilung in zwei unterschiedlichen Luftströmungen untersucht. Betrachtet wurden eine einfache Kanalströmung und ein umströmter Zylinder. Die Kanalströmung ist weitgehend unidirektional, während die Zylinderströmung die typische Karmansche Wirbelstraße aufweist. In den Simulationen wurde festgestellt, dass die Turbulenz das Tropfenwachstum verstärkt. Der Hauptgrund besteht darin, dass die turbulente Luft Tropfen aufeinander zu bewegt und Kollisionen verursacht. Damit wurde die Vermutung, dass Turbulenz ein Tropfenwachstum bewirkt, auch in den numerischen Simulationen dieser Arbeit bestätigt.The used population balance equation contains three aspects: (i) the movement of the droplets in the turbulent air flow, (ii) the growth in supersaturated air, and (iii) the coalescence. The population balance equation has been modeled directly in four dimensions, with the diameter as internal coordinate. Experimental data have been used for the inflow condition of the computational model and for the validation of the numerical results. The results of the simulations are promising, as a substantial agreement with the experimental data was obtained. To identify suitable stabilization methods for the solution of the population balance equation, several finite difference and finite element stabilization methods for convection-diffusion equations were compared. The best compromises between accuracy and computing time were achieved by the ENO method (among the finite difference methods) and by the linear group FEM-FCT method (among the finite element methods). A new observation is that the FEM-FCT method shows relatively strong smearing when the computational grid is parallel to the convection. These methods were applied for the discretization of the population balance equation and they showed essentially the same behavior. Also for the evaluation of the coalescence terms, several methods were investigated. In particular, a new method based on pre-computed integrals was used. Although from the point of view of accuracy a mass-conserving method is recommended, this approach is in general difficult to implement and it requires a special grid. In simple situations, a Gaussian quadrature method is sufficient. To study the influence of turbulence on droplet growth, the behavior of the droplet size distribution was investigated for two different air flows: a turbulent channel flow and a turbulent flow around a cylinder. The channel flow is substantially unidirectional, while in the flow around the cylinder a Karman vortex street develops behind the obstacle. From the simulation results, it could be observed that turbulence increases the droplet growth. The main reason is that the turbulent air moves the drops towards each other, which increases the number of droplet collisions. Thus, the numerical simulations performed in this thesis confirmed also the assumption that turbulence causes droplet growth

Numerical simulations and measurements of a droplet size distribution in a turbulent vortex street

A turbulent vortex street in an air flow interacting with a disperse droplet population is investigated in a wind tunnel. Non-intrusive measurement techniques are used to obtain data for the air velocity and the droplet velocity. The process is modeled with a population balance system consisting of the incompressible NavierStokes equations and a population balance equation for the droplet size distribution. Numerical simulations are performed that rely on a variational multiscale method for turbulent flows, a direct discretization of the differential operator of the population balance equation, and a modern technique for the evaluation of the coalescence integrals. After having calibrated two unknown model parameters, a very good agreement of the experimental and numerical results can be observed

Numerical methods for the simulation of a coalescence-driven droplet size distribution

The droplet size distribution in a turbulent flow field is considered and modeled by means of a population balance system. This paper studies different numerical methods for the 4D population balance equation and their impact on an output of interest, the time-space-averaged droplet size distribution at the outlet, which is known from experiments. These methods include different interpolations of the experimental data at the inlet, various discretizations in time and space, and different schemes for computing the coalescence integrals. It will be shown that noticeable changes in the output of interest might occur. In addition, the computational efficiency of the studied methods is discussed