Liquid film flow is an important flow type in many applications of process engineering. For supporting experiments, theoretical and numerical investigations are required. The present state of the art is to model the liquid film flow with Navier--Stokes-based methods, whereas the lattice Boltzmann method is employed here. The final model has been developed within this treatise by means of a two-phase flow and a heat transfer model, and boundary and initial conditions. All these sub-models have been applied to simple test cases. It could be found that the two-phase model is capable of solving flow phenomena with a large density ratio which has been shown impressively in conjunction with wall boundary conditions. The heat transfer model was tested against spectral method results with a transient non-uniform flow field. It was possible to find optimal parameters for computation. The final model has been applied to steady-state film flow, and showed very good agreement to OpenFOAM simulations. Tests with transient film flow demonstrated that the model is also able to predict these flow phenomena. / Flüssigkeitsfilmströmungen kommen in vielen verfahrenstechnischen Prozessen zum Einsatz. Zur Unterstützung von Experimenten sind theoretische und numerische Untersuchungen nötig. Stand der Technik ist es, Navier--Stokes-basierte Modelle zu verwenden, wohingegen hier die Lattice-Boltzmann-Methode verwendet wird. Das finale Modell wurde unter Verwendung eines Zweiphasen- und eines Wärmeübertragungsmodell entwickelt und geeignete Rand- und Anfangsbedingungen formuliert. Alle Untermodelle wurden anhand einfacher Testfälle überprüft. Es konnte herausgefunden werden, dass das Zweiphasenmodell Strömungen großer Dichteunterschiede rechnen kann, was eindrucksvoll im Zusammenhang mit Wandrandbedingungen gezeigt wurde. Das Wärmeübertragungsmodell wurde gegen eine Spektrallösung anhand eines transienten und nichtuniformen Strömungsproblemes getestet. Stationäre Filmströmungen zeigten sehr gute Übereinstimmungen mit OpenFOAM-Lösungen und instationäre Berechungen bewiesen, dass das Model auch solche Strömungen abbilden kann.

Cette thèse présente la construction de schémas distribuant le résidu (RD) d'ordre très élevés, pour la discrétisation d'équations d'advection-diffusion multidimensionnelles et stationnaires sur maillages non structurés. Des schémas linéaires ainsi que des schémas non linéaires sont considérés. Une approximation de la solution polynomiale par morceaux et continue sur chaque élément est adoptée, de plus une procédure de reconstruction du gradient que celle de la solution numérique est utilisée afin d'avoir une représentation continue de la solution numérique et de son gradient. Il est montré que le gradient doit être reconstruit avec la même précision de la solution, sans quoi la précision formel du schéma numérique est perdue dans les cas où les effets de diffusion prévalent sur les effets d'advection, et aussi quand l'advection et la diffusion sont également importants. Ensuite, la méthode est étendue à des systèmes d'équations, en particulier aux équations de Navier-Stokes et aux équations RANS. La précision, l'efficacité et la robustesse du solveur RD implicite sont démontrées sur plusieurs cas tests. / The construction of compact high-order Residual Distribution schemes for the discretizationof steady multidimensional advection-diffusion problems on unstructuredgrids is presented. Linear and non-linear scheme are considered. A piecewise continuouspolynomial approximation of the solution is adopted and a gradient reconstructionprocedure is used in order to have a continuous representation of both thenumerical solution and its gradient. It is shown that the gradient must be reconstructedwith the same accuracy of the solution, otherwise the formal accuracy ofthe numerical scheme is lost in applications in which diffusive effects prevail overthe advective ones, and when advection and diffusion are equally important. Thenthe method is extended to systems of equations, with particular emphasis on theNavier-Stokes and RANS equations. The accuracy, efficiency, and robustness of theimplicit RD solver is demonstrated using a variety of challenging aerodynamic testproblems.

Les terres agricoles ont été durablement contaminées à la fois par les pesticides mis à la disposition des agriculteurs pour lutter contre les charançons et autres insectes nuisibles, et par les engrais azotées pour augmenter la productivité chez les plantes.Des recherches récentes concernent des cultures alternatives écologiques utilisant les plantes de service qui fournissent les nutriments aux plantes principales. Ce travail de thèse s'inscrit dans cette perspective, d'un point de vue modélisation.L'accent est mis sur la résolution de problèmes de contrôle du phénomène d'absorption de nutriments, par les racines dans la rhizosphère (partie proche des racines), en considérant les deux cas de sols : sol sain et sol pollué.Ces phénomènes d'absorption sont modélisés par des systèmes d'advection-diffusion de type Nye-Tinker-Barber (NTB). La concentration de nutriments absorbée, solution du problème, est une fonction du temps et de l'espace.On étudie l'existence de solution du système NTB dans les deux cas où la fonction d'absorption de nutriments à la frontière (surface de la racine) appelée fonction de Michealis-Menten, est linéaire et/ou non linéaire, à l’aide des outils d’analyse fonctionnelle. On étudie ensuite les problèmes de contrôle optimal associés au système NTB, en considérant les deux cas linéaire et non linéaire, en application pour les deux cas d’absorption de nutriments en sol non pollué puis en sol pollué. Pour le premier cas, on utilise les techniques classiques de recherche d'un contrôle pour les systèmes distribués, tandis que, pour le second cas, on fait appel aux notions de contrôle sans regret et contrôle à moindres regrets de J.-L. Lions. Les contrôles obtenus pour les différents problèmes sont caractérisés chacun par un système d'optimalité (SO) cas sans pollution, et système d’optimalité singulier (SOS) dans le cas avec pollution.= / Agriculture soils were highly contaminated for a long time by pesticides which were widely used by producers to fight against weevils. Soils where also contaminated by the use of fertilizers to increase the plant development. An ecological alternative using service plants is encouraged following recent research. The aim of this work is to give a mathematical and a modelling point of view as we study the mecha- nisms of nutrient transfer to plants using the mathematical analysis and optimal control theories. The two cases of polluted and non-polluted soils are considered. The nutrient transfer and uptake processes are modeled by an advection-diffusion system derived from the Nye-Tinker-Barber (NTB) model. The absorbed nutrient concentration represented by the Michaelis-Mention function at the root surface of the principal plant, depends on time and space. We study the existence of a solution for the linear and nonlinear NTB systems, then we characterize the opti- mal control which corresponds to the added nutrients from the service plant. For the pollution case, we use the concept of low-regret and no-regret control of J.-L. Lions.

Liquid film flow is an important flow type in many applications of process engineering. For supporting experiments, theoretical and numerical investigations are required. The present state of the art is to model the liquid film flow with Navier--Stokes-based methods, whereas the lattice Boltzmann method is employed here. The final model has been developed within this treatise by means of a two-phase flow and a heat transfer model, and boundary and initial conditions. All these sub-models have been applied to simple test cases. It could be found that the two-phase model is capable of solving flow phenomena with a large density ratio which has been shown impressively in conjunction with wall boundary conditions. The heat transfer model was tested against spectral method results with a transient non-uniform flow field. It was possible to find optimal parameters for computation. The final model has been applied to steady-state film flow, and showed very good agreement to OpenFOAM simulations. Tests with transient film flow demonstrated that the model is also able to predict these flow phenomena. / Flüssigkeitsfilmströmungen kommen in vielen verfahrenstechnischen Prozessen zum Einsatz. Zur Unterstützung von Experimenten sind theoretische und numerische Untersuchungen nötig. Stand der Technik ist es, Navier--Stokes-basierte Modelle zu verwenden, wohingegen hier die Lattice-Boltzmann-Methode verwendet wird. Das finale Modell wurde unter Verwendung eines Zweiphasen- und eines Wärmeübertragungsmodell entwickelt und geeignete Rand- und Anfangsbedingungen formuliert. Alle Untermodelle wurden anhand einfacher Testfälle überprüft. Es konnte herausgefunden werden, dass das Zweiphasenmodell Strömungen großer Dichteunterschiede rechnen kann, was eindrucksvoll im Zusammenhang mit Wandrandbedingungen gezeigt wurde. Das Wärmeübertragungsmodell wurde gegen eine Spektrallösung anhand eines transienten und nichtuniformen Strömungsproblemes getestet. Stationäre Filmströmungen zeigten sehr gute Übereinstimmungen mit OpenFOAM-Lösungen und instationäre Berechungen bewiesen, dass das Model auch solche Strömungen abbilden kann.

<p>Doktorska disertacija je posvećena re&scaron;avanju Ko&scaron;ijevog problema odabranih klasa frakcionih diferencijalnih jednačina u okviru Kolomboovih prostora uop&scaron;tenih funkcija. U prvom delu disertacije razmatrane su nehomogene evolucione jednačine sa prostorno frakcionim diferencijalnim operatorima reda 0 &lt; &alpha; &lt; 2 i koeficijentima koji zavise od x i t. Ova klasa jednačina je aproksimativno re&scaron;avana, tako &scaron;to je umesto početne jednačine razmatrana aproksimativna jednačina data preko regularizovanih frakcionih izvoda, odnosno, njihovih regularizovanih množitelja. Za re&scaron;avanje smo koristili dobro poznate uop&scaron;tene uniformno neprekidne polugrupe operatora. U drugom delu disertacije aproksimativno su re&scaron;avane nehomogene frakcione evolucione jednačine sa Kaputovim<br />frakcionim izvodom reda 0 &lt; &alpha; &lt; 2, linearnim, zatvorenim i gusto definisanim<br />operatorom na prostoru Soboljeva celobrojnog reda i koeficijentima koji zavise<br />od x. Odgovarajuća aproksimativna jednačina sadrži uop&scaron;teni operator asociran sa polaznim operatorom, dok su re&scaron;enja dobijena primenom, za tu svrhu&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br />u disertaciji konstruisanih, uop&scaron;tenih uniformno neprekidnih operatora re&scaron;enja.<br />U oba slučaja ispitivani su uslovi koji obezbeduju egzistenciju i jedinstvenost<br />re&scaron;enja Ko&scaron;ijevog problema na odgovarajućem Kolomboovom prostoru.</p> / <p>Colombeau spaces of generalized functions. In the firs part, we studied inhomogeneous evolution equations with space fractional differential operators of order 0 &lt; &alpha; &lt; 2 and variable coefficients depending on x and t. This class of equations is solved&nbsp; approximately, in such a way that instead of the originate equation we considered the corresponding approximate equation given by regularized fractional derivatives, i.e. their&nbsp; regularized multipliers. In the solving procedure we used a well-known generalized uniformly continuous semigroups of operators. In the second part, we solved approximately inhomogeneous fractional evolution equations with Caputo fractional derivative of order 0 &lt; &alpha; &lt; 2, linear, closed and densely defined operator in Sobolev space of integer order and variable coefficients depending on x. The corresponding approximate equation&nbsp;&nbsp; is a given by the generalized operator associated to the originate&nbsp; operator, while the solutions are obtained by using generalized uniformly continuous solution operators, introduced and developed for that purpose. In both cases, we provided the conditions that ensure the existence and uniqueness solutions of the Cauchy problem in some Colombeau spaces.</p>

A high-order central essentially non-oscillatory (CENO) finite-volume scheme in combination with a block-based adaptive mesh refinement (AMR) algorithm is proposed for solution of hyperbolic and elliptic systems of conservation laws on body- fitted multi-block mesh. The spatial discretization of the hyperbolic (inviscid) terms is based on a hybrid solution reconstruction procedure that combines an unlimited high-order k-exact least-squares reconstruction technique following from a fixed central stencil with a monotonicity preserving limited piecewise linear reconstruction algorithm. The limited reconstruction is applied to computational cells with under-resolved solution content and the unlimited k-exact reconstruction procedure is used for cells in which the solution is fully resolved. Switching in the hybrid procedure is determined by a solution smoothness indicator. The hybrid approach avoids the complexity associated with other ENO schemes that require reconstruction on multiple stencils and therefore, would seem very well suited for extension to unstructured meshes. The high-order elliptic (viscous) fluxes are computed based on a k-order accurate average gradient derived from a (k+1)-order accurate reconstruction. A novel h-refinement criterion based on the solution smoothness indicator is used to direct the steady and unsteady refinement of the AMR mesh. The predictive capabilities of the proposed high-order AMR scheme are demonstrated for the Euler and Navier-Stokes equations governing two-dimensional compressible gaseous flows as well as for advection-diffusion problems characterized by the full range of Peclet numbers, Pe. The ability of the scheme to accurately represent solutions with smooth extrema and yet robustly handle under-resolved and/or non-smooth solution content (i.e., shocks and other discontinuities) is shown for a range of problems. Moreover, the ability to perform mesh refinement in regions of smooth but under-resolved and/or non-smooth solution content to achieve the desired resolution is also demonstrated.

A high-order central essentially non-oscillatory (CENO) finite-volume scheme in combination with a block-based adaptive mesh refinement (AMR) algorithm is proposed for solution of hyperbolic and elliptic systems of conservation laws on body- fitted multi-block mesh. The spatial discretization of the hyperbolic (inviscid) terms is based on a hybrid solution reconstruction procedure that combines an unlimited high-order k-exact least-squares reconstruction technique following from a fixed central stencil with a monotonicity preserving limited piecewise linear reconstruction algorithm. The limited reconstruction is applied to computational cells with under-resolved solution content and the unlimited k-exact reconstruction procedure is used for cells in which the solution is fully resolved. Switching in the hybrid procedure is determined by a solution smoothness indicator. The hybrid approach avoids the complexity associated with other ENO schemes that require reconstruction on multiple stencils and therefore, would seem very well suited for extension to unstructured meshes. The high-order elliptic (viscous) fluxes are computed based on a k-order accurate average gradient derived from a (k+1)-order accurate reconstruction. A novel h-refinement criterion based on the solution smoothness indicator is used to direct the steady and unsteady refinement of the AMR mesh. The predictive capabilities of the proposed high-order AMR scheme are demonstrated for the Euler and Navier-Stokes equations governing two-dimensional compressible gaseous flows as well as for advection-diffusion problems characterized by the full range of Peclet numbers, Pe. The ability of the scheme to accurately represent solutions with smooth extrema and yet robustly handle under-resolved and/or non-smooth solution content (i.e., shocks and other discontinuities) is shown for a range of problems. Moreover, the ability to perform mesh refinement in regions of smooth but under-resolved and/or non-smooth solution content to achieve the desired resolution is also demonstrated.

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