Multigrid method for nonlinear poroelasticity equations

<p>In this study, a nonlinear multigrid method is applied for solving the system of incompressible poroelasticity equations considering nonlinear hydraulic conductivity. For the unsteady problem, an additional artificial term is utilized to stabilize the solutions when the equations are discretized on collocated grids. We employ two nonlinear multigrid methods, i.e. the “full approximation scheme” and “Newton multigrid” for solving the corresponding system of equations arising after discretization. For the steady case, both homogeneous and heterogeneous cases are solved and two different smoothers are examined to search for an efficient multigrid method. Numerical results show a good convergence performance for all the strategies.</p

Time-stepping and Krylov methods for large-scale instability problems

With the ever increasing computational power available and the development of high-performances computing, investigating the properties of realistic very large-scale nonlinear dynamical systems has been become reachable. It must be noted however that the memory capabilities of computers increase at a slower rate than their computational capabilities. Consequently, the traditional matrix-forming approaches wherein the Jacobian matrix of the system considered is explicitly assembled become rapidly intractable. Over the past two decades, so-called matrix-free approaches have emerged as an efficient alternative. The aim of this chapter is thus to provide an overview of well-grounded matrix-free methods for fixed points computations and linear stability analyses of very large-scale nonlinear dynamical systems.Comment: To appear in "Computational Modeling of Bifurcations and Instabilities in Fluid Mechanics", eds. A. Gelfgat, Springe

Analysis and development of numerical methodologies for simulation of flow control with dielectric barrier discharge actuators

The aim of this thesis is to investigate and develop different numerical methodologies for modeling the Dielectric Barrier discharge (DBD) plasma actuators for flow control purposes. Two different modeling approaches were considered; one based on Plasma-fluid model and the other based on a phenomenological model. A three component Plasma fluid model based on the transport equations of charged particles was implemented in this thesis in OpenFOAM, using several techniques to reduce the numerical issues. The coupled plasma-fluid problem involves wide range of length and time scales which make the numerical simulation difficult. Therefore, to obtain stable and accurate results in a reasonable computational run time, several numerical procedures were implemented including: semi-implicit treatment of coupling of Poisson equation and charge density equation, super-time-stepping and operator splitting algorithm. We examined our code for a constant positive voltage, testing for the dependency of the behavior of the current density to the selected numerical scheme. In addition, although there is no clear numerical or experimental benchmark case for DBD plasma actuator problem, the developed plasma solver was compared quantitively and qualitively with several numerical works in the literature. Afterward, the developed numerical methodology was used to explore the possibility of influencing the flow, with higher speed, using nano-second (NS) pulsed DBD plasma actuator. Therefore, the interaction of the transonic flow and actuation effects of DBD plasma actuator with nano second pulsed voltage was simulated. The effect of gas heating and body force was calculated by the plasma solver and was supplied into the gas dynamic solver for simulating the flow field. Moreover, the results of the plasma fluid model were used to develop an energy deposition model. It was shown that the energy deposition model is able to capture the main features of the effect of NS DBD plasma actuators correctly, with less computational time. It was also shown that fast energy transfer, from plasma to fluid, leads to the formation of micro-shock waves that modify locally the features of the transonic flow. Although the numerical efficiency of the plasma fluid model was improved, the computational cost of simulating the effect of DBD plasma actuator on a real scale flow situation was still high. Therefore, a simple model for plasma discharge and its effect on the flow was developed based on scaling of the thrust generated by DBD plasma actuators. The scaled thrust model correctly predicts the nonlinear dependency of the thrust produced and the applied voltage. These scales were then introduced into a simple phenomenological model to estimate and simulate the body force distribution generated by the plasma actuator. Although the model includes some experimental correlations, it does not need any fitting parameter. The model was validated with experimental results and showed better accuracy compared to previous plasma models. Using a simple phenomenological model that was developed here, a numerical study was conducted to investigate and compare the effect of steady and unsteady actuation for controlling the flow at relatively high Reynolds number. Firstly it was shown that the size of the time-averaged separation bubble is greatly reduced and the flow structure is sensitive to the frequency of burst modulation of DBD plasma actuators. The results also confirmed that in the case of unsteady actuation, the burst frequency and burst ratio are crucial parameters for influencing the capability of the actuators to control the flow. It was found that burst frequencies near the natural frequencies of the system were able to excite the flow structure in a resonance mode. This observation also confirmed that with proper frequencies of excitation, the flow structure can be well rearranged and the flow losses can be reduced. In the end, Plasma actuators were used for controlling the flow over the Coanda surface of the ACHEON nozzle. When the plasma actuator was used, it was possible to postpone separation of the flow and increase the deflection angle of the exit jet of the nozzle. To find the optimum position of the actuators, seven DBD actuators in forward forcing mode were placed over the Coanda surface considering the numerically obtained separation points. Results show that when the actuator is placed slightly before the separation point, enhanced thrust vectorizing with the use of DBD actuator is achievable. Preliminary results of the experiments agree with planned/foreseen deflection angle obtained from numerical computation.O objetivo deste trabalho visa a investigação e desenvolvimento de diferentes métodos numéricos para modelação de actuadores a plasma de Descarga em Barreira Dieléctrica, (DBD), tendo em vista o controlo do escoamento na camada limite. Esta modelação numérica foi abordada de duas formas diferentes, uma baseada num modelo de “plasma-fluid” e outra fundamentada num modelo fenomenológico. Neste trabalho é usado um modelo “plasma-fluid” de três componentes que é baseado numa equação de transporte para as partículas electricamente carregadas. Este foi implementado no software OpenFOAM fazendo uso de diversas técnicas para minimização de problemas numéricos que ocorriam na resolução das equações. O cálculo de um problema com acoplamento entre plasma e fluido envolve uma gama diversa de escalas, tanto temporais como dimensionais, trata-se então de uma simulação numérica delicada. Como tal, e por forma a obter resultados estáveis e precisos num tempo de cálculo considerado razoável, foram implementados diversos procedimentos numéricos, tais como o tratamento semiimplícito do acoplamento da equação de Poisson com a equação da densidade de carga, o super-passo-tempo e ainda um algoritmo do tipo divisão de operador. Foi considerado o caso de uma diferença de potencial positiva, constante, e testada a dependência da densidade de corrente com os diferentes esquemas numéricos. Apesar de não existir atualmente uma base de dados, de tipo numérica ou experimental, com casos de teste para actuadores a plasma tipo DBD, o modelo computacional desenvolvido para calcular o plasma foi validado qualitativamente, bem como quantitativamente, usando os vários trabalhos numéricos disponíveis na literatura. Após esta validação inicial, a metodologia numérica desenvolvida foi utilizada para explorar a possibilidade de influenciar um escoamento de maior velocidade, através de actuadores a plasma tipo DBD com impulsos de tensão da ordem de nano-segundos (NS). Desta forma foi simulada a interacção entre um escoamento transónico e o efeito dos actuadores a plasma tipo DBD sobre o escoamento, usando pulsos de nano-segundos. O efeito térmico do gás, assim como a força resultante, foram calculados usando o modelo numérico para cálculo de plasmas desenvolvido neste trabalho. O resultado obtido é acoplado ao modelo de cálculo para a dinâmica de gases, o que torna possível simular as condições do escoamento resultante. Adicionalmente, os resultados do modelo de “plasma-fluid” foram reaproveitados para desenvolver um modelo de deposição de energia. Este demonstrou ter a capacidade de capturar correctamente as características principais do efeito de actuadores de plasma, de tipo NS-DBD, com um tempo de computação menor. Foi demonstrada que uma rápida transferência de energia, do plasma para o fluido, leva à formação de micro-ondas de choque que alteram localmente as características do escoamento transónico. Apesar da eficiência numérica do modelo de “plasma-fluid” ter sido melhorada, o seu custo computacional para a simulação de actuadores a plasma tipo DBD à escala real continua bastante elevado. Neste sentido, a partir de uma escala de propulsão gerada pelo actuador plasma DBD, foi desenvolvido um modelo mais simples para a descarga do plasma e para determinar os seus efeitos sobre o escoamento. O modelo inicial previa correctamente uma dependência não-linear entre a força propulsiva gerada e a diferença de potencial aplicada. Estas escalas foram então introduzidas num modelo fenomenológico mais simples para estimar, e simular, a distribuição de forças geradas pelo actuador a plasma. Apesar de o modelo incluir algumas correlações experimentais, este não requer qualquer parâmetro de afinação. O modelo foi validado com resultados experimentais, demonstrando melhores resultados quando comparado com outros modelos de plasma . Utilizando um modelo fenomenológico simplificado, que foi desenvolvido no presente trabalho, foi feito um estudo numérico com o objetivo de investigar, e comparar, os efeitos que uma actuação estacionária e não-estacionária exibe sobre o controlo do escoamento a números de Reynolds relativamente elevados. Foi demostrado que a dimensão da bolha de separação é reduzida em muito e que a estrutura do escoamento é sensível à frequência da modulação “burst” do actuador a plasma tipo DBD. Os resultados também confirmaram que, para o caso de actuação não-estacionária, a frequência de “burst” e o “burst ratio”, são parâmetros cruciais para influenciar a capacidade de controlo do escoamento por parte dos actuadores a plasma. Determinou-se que as frequências “burst”, semelhantes às frequências naturais do sistema, são capazes de excitar as estruturas do escoamento num modo de ressonância. Esta observação confirma igualmente que, com frequências de excitação apropriadas, a estrutura de um escoamento de camada limite consegue ser correctamente modificada, e que as perdas no escoamento são reduzidas. Por fim, os actuadores a plasma foram utilizados para o controlo do escoamento sobre uma superfície Coanda de uma tubeira. Quando nesta foi aplicado um plasma, tornou-se possível retardar a separação do escoamento e aumentar o ângulo de deflexão do jacto gerado pelo propulsor. Por forma a encontrar a posição óptima para os actuadores, sete actuadores de tipo DBD foram distribuídos ao longo da superfície Coanda, tendo em consideração os pontos de separação do escoamento na camada limite obtidos numericamente. Os resultados mostram que quando o actuador DBD é colocado ligeiramente antes do ponto de separação do escoamento, há um aumento da capacidade de controlo e vectorização do jacto gerado. Os resultados preliminares das experiências efectuadas estão de acordo com o ângulo de deflexão do jacto previsto pelo modelo computacional

Self-Consistent Modelling of Non-Thermal Atmospheric Argon Plasma During Arc Discharge and Its Interaction with Metal Electrodes

Heat transfer processes associated with arc plasmas are important for many industrial applications such as electric propulsion, plasma spray and arc welding. In these applications, an electric arc is used because it offers high energy densities and a controlled environment. However, it is sometimes not realizable or not economic to get the parameters within the high temperature region of plasma precisely by means of experimental measurements. A numerical model that offers reliable description of discharging process is a good choice. Any model of arc plasmas must contain not only the conservation of mass, momentum and energy, but also electromagnetic description that follows Maxwella s equations. Since the last 30 years, intensive researches embarking on nonequilibrium plasmas have led to fruitful achievements, among them NLTE (non-Local Thermal Equilibrium) model plays an important role in numerical modelling due to its superiority over LTE (Local Thermal Equilibrium) model in accounting for the difference of two phase temperatures (heavy species and electrons) that cannot be neglected near electrodes. However, deeper researches meet obstacles when the discharging system needs to be simulated self-consistently as a whole and with as few presumed conditions as possible. On one hand, discharging under high current operation tends to overheat its electrodes leading to melting or evaporating, particles from electrode material that enter the plasma will change its composition and the heat transfer process. On the other hand, therea s still a a mysteriousa region whose physical structure is so different from the main arc plasma region that cannot be accounted by conventional transport equations or theories without any extra treatments for it. This region, sometimes called sheath layer or space-charge layer, plays an important role in bridging the thermal and electric energy of arc column to electrodes. To develop a reasonable model in this region and make it compatible with the two other regions will extend the applicability of CFD model in discharging devices. The motivation of this doctoral thesis is based on my special interest in sheath region, or in other words, my pursuit of developing a self-consistent model that is capable of solving the whole plasma-electrode system. Concerning the complexity of sheath, no secondary physical phenomena such as melting and evaporating are considered in this study. For the main arc region, the plasma composition is calculated based on species conservation equations that consider both diffusion and production/loss activities of particles. And for the sake of high temperature of plasma core, ionization up to third level is applied. In the sheath layer, the effective sheath electrical conductivity is utilized, which is based on the assumption of Childa s collisionless sheath and Lowkea s expression. The ionization degree of plasma sheath plays an important role in this self-consistent method. To validate the model proposed here, several simple benchmark simulations are made and the numerical results concerning temperature, velocity and magnetic field yield satisfactory agreements with experimental or theoretical results. With the model being validated, a D.C. non-transferred plasma torch is studied. The total voltages of both situations are compared with experimental measurements. It shows that the sheath model developed in this scope make the numerical results closer to reality and is responsible for the strong fluctuation of arc jets, which also makes cathode surface temperature fluctuate accordingly. Finally, pros and cons of some new design patterns of plasma torches are discussed, with the multi anode/single cathode type DeltaGun simulated for the comparison of performances with the original type. It reveals that such kind of configuration helps to damp the unwanted arc fluctuation with multiple arc roots. It is also numerically confirmed that when an external coil is added around anode to produce a proper magnetic field, the temperature of anode attachment can be reduced due to enhanced circumferential movement of arc roots by Lorentz force, which lowers the possibility of erosion and promotes a longer lifetime