The dynamical equations of the non-hydrostatic mesoscale model GESIMA are solved numerically on an Arakawa-C grid. Because of the staggered grid most of the prognostic variables and their derivatives have identical local positions. The functional connection between the fluxes and velocities defined at different places is managed by the MacCormack scheme ignoring the local diff erences. The systematic errors are diminished by means of alternate down- and upwind shifting of the fluxes after each time step. A cycle of 8 time steps is necessary to achieve approximately symmetrical conditions because of the shift permutations. Nevertheless, the systematic errors are not completely removed and the iterative calculation of the dynamic pressure is retarded by starting values from eight time steps ago (same permutation of shift directions). Both shortcomings are avoided by a symmetrized MacCormack scheme without the loss of its advantages of handling strong gradients. The new method is based on the symmetrization of the equations with respect to the passive quantities and on the simultaneous calculation of each equation for opposite shift directions of the active variables followed by averaging both increments. The method is tested for a typical example. / Die dynamischen Modellgleichungen des nicht-hydrostatischen mesoskaligen Atmosphärenmodells GESIMA sind numerisch auf einem Arakawa-C-Gitter gelöst. Durch die versetzte Anordnung der Größen auf dem Gitter besitzen die Differenzenquotienten (auf den rechten Seiten) und die prognostizierten Größen (auf den linken Seiten) von vornherein die gleiche lokale Position, allerdings nicht in jedem Fall. Das bisher in GESIMA praktizierte MacCormack-Schema stellt den Zusammenhang zwischen den an verschiedenen Gitterstellen definierten Flüssen und Geschwindigkeiten her, indem die Ortsdifferenz zwischen Fluß- und zugehöriger Geschwindigkeitskomponente ignoriert wird. Zur Verringerung der systematischen Fehler erfolgt die direkte Zuordnung einer Flußkomponente abwechselnd (sequentiell) in einem Zeitschritt zur flußabwärts benachbarten Geschwindigkeitskomponente und im nächsten Zeitschritt zur flußaufwärts benachbarten. Nach Ablauf von jeweils 8 Zeitschritten sind die notwendigen Zuordnungspermutationen der 3 Vektorkomponenten zwecks einer annähernden Symmetrisierung des Verfahrens erreicht. Nachteile des bisherigen Verfahrens sind (a) der nicht vollständige Abbau der jedem Zeitschritt immanenten systematischen Zuordnungsfehler und (b) ein stark erhöhter Rechenaufwand für die iterative Bestimmung des dynamischen Druckes durch einen um 8 Zeitschritte (jeweils gleiche Zuordnungspermutation) zurückliegenden Startwert. Beide Nachteile werden durch ein neues, symmetrisiertes MacCormack-Schema vermieden, ohne daß auf die Vorteile bei der Handhabung starker Gradienten verzichtet werden muß. Das Verfahren beruht (a) auf der Symmetrisierung der lokalen Zuordnung für die passiven Größen innerhalb einer Gleichung (d.h. der nicht durch sie prognostizierten Größen) und (b) auf der simultanen Durchführung der zwei entgegengesetzten Zuordnungsrichtungen für jede der 3 Geschwindigkeitskomponenten innerhalb eines Zeitschrittes mit anschließender Mittelung der beiden Inkremente. Das neue Verfahren wurde anhand eines Beispiels geprüft.

The dynamical equations of the non-hydrostatic mesoscale model GESIMA are solved numerically on an Arakawa-C grid. Because of the staggered grid most of the prognostic variables and their derivatives have identical local positions. The functional connection between the fluxes and velocities defined at different places is managed by the MacCormack scheme ignoring the local diff erences. The systematic errors are diminished by means of alternate down- and upwind shifting of the fluxes after each time step. A cycle of 8 time steps is necessary to achieve approximately symmetrical conditions because of the shift permutations. Nevertheless, the systematic errors are not completely removed and the iterative calculation of the dynamic pressure is retarded by starting values from eight time steps ago (same permutation of shift directions). Both shortcomings are avoided by a symmetrized MacCormack scheme without the loss of its advantages of handling strong gradients. The new method is based on the symmetrization of the equations with respect to the passive quantities and on the simultaneous calculation of each equation for opposite shift directions of the active variables followed by averaging both increments. The method is tested for a typical example. / Die dynamischen Modellgleichungen des nicht-hydrostatischen mesoskaligen Atmosphärenmodells GESIMA sind numerisch auf einem Arakawa-C-Gitter gelöst. Durch die versetzte Anordnung der Größen auf dem Gitter besitzen die Differenzenquotienten (auf den rechten Seiten) und die prognostizierten Größen (auf den linken Seiten) von vornherein die gleiche lokale Position, allerdings nicht in jedem Fall. Das bisher in GESIMA praktizierte MacCormack-Schema stellt den Zusammenhang zwischen den an verschiedenen Gitterstellen definierten Flüssen und Geschwindigkeiten her, indem die Ortsdifferenz zwischen Fluß- und zugehöriger Geschwindigkeitskomponente ignoriert wird. Zur Verringerung der systematischen Fehler erfolgt die direkte Zuordnung einer Flußkomponente abwechselnd (sequentiell) in einem Zeitschritt zur flußabwärts benachbarten Geschwindigkeitskomponente und im nächsten Zeitschritt zur flußaufwärts benachbarten. Nach Ablauf von jeweils 8 Zeitschritten sind die notwendigen Zuordnungspermutationen der 3 Vektorkomponenten zwecks einer annähernden Symmetrisierung des Verfahrens erreicht. Nachteile des bisherigen Verfahrens sind (a) der nicht vollständige Abbau der jedem Zeitschritt immanenten systematischen Zuordnungsfehler und (b) ein stark erhöhter Rechenaufwand für die iterative Bestimmung des dynamischen Druckes durch einen um 8 Zeitschritte (jeweils gleiche Zuordnungspermutation) zurückliegenden Startwert. Beide Nachteile werden durch ein neues, symmetrisiertes MacCormack-Schema vermieden, ohne daß auf die Vorteile bei der Handhabung starker Gradienten verzichtet werden muß. Das Verfahren beruht (a) auf der Symmetrisierung der lokalen Zuordnung für die passiven Größen innerhalb einer Gleichung (d.h. der nicht durch sie prognostizierten Größen) und (b) auf der simultanen Durchführung der zwei entgegengesetzten Zuordnungsrichtungen für jede der 3 Geschwindigkeitskomponenten innerhalb eines Zeitschrittes mit anschließender Mittelung der beiden Inkremente. Das neue Verfahren wurde anhand eines Beispiels geprüft.

Este trabalho estuda o fenômeno de evaporação rápida em jatos de líquidos superaquecidos ou metaestáveis numa região 2D. O fenômeno se inicia, neste caso, quando um jato na fase líquida a alta temperatura e pressão, emerge de um diminuto bocal projetando-se numa câmara de baixa pressão, inferior à pressão de saturação. Durante a evolução do processo, ao cruzar-se a curva de saturação, se observa que o fluido ainda permanece no estado de líquido superaquecido. Então, subitamente o líquido superaquecido muda de fase por meio de uma onda de evaporação oblíqua. Esta mudança de fase transforma o líquido superaquecido numa mistura bifásica com alta velocidade distribuída em várias direções e que se expande com velocidades supersônicas cada vez maiores, até atingir a pressão a jusante, e atravessando antes uma onda de choque. As equações que governam o fenômeno são as equações de conservação da massa, conservação da quantidade de movimento, e conservação da energia, incluindo uma equação de estado precisa. Devido ao fenômeno em estudo estar em regime permanente, um método de diferenças finitas com modelo estacionário e esquema de MacCormack é aplicado. Tendo em vista que este modelo não captura a onda de choque diretamente, um segundo modelo de falso transiente com o esquema de \"shock-capturing\": \"Dispersion-Controlled Dissipative\" (DCD) é desenvolvido e aplicado até atingir o regime permanente. Resultados numéricos com o código ShoWPhasT-2D v2 e testes experimentais foram comparados e os resultados numéricos com código DCD-2D v1 foram analisados. / This study analyses the rapid evaporation of superheated or metastable liquid jets in a two-dimensional region. The phenomenon is triggered, in this case, when a jet in its liquid phase at high temperature and pressure, emerges from a small aperture nozzle and expands into a low pressure chamber, below saturation pressure. During the evolution of the process, after crossing the saturation curve, one observes that the fluid remains in a superheated liquid state. Then, suddenly the superheated liquid changes phase by means of an oblique evaporation wave. This phase change transforms the liquid into a biphasic mixture at high velocity pointing toward different directions, with increasing supersonic velocity as an expansion process takes place to the chamber back pressure, after going through a compression shock wave. The equations which govern this phenomenon are: the equations of conservation of mass, momentum and energy and an equation of state. Due to its steady state process, the numerical simulation is by means of a finite difference method using the McCormack method of Discretization. As this method does not capture shock waves, a second finite difference method is used to reach this task, the method uses the transient equations version of the conservation laws, applying the Dispersion-Controlled Dissipative (DCD) scheme. Numerical results using the code ShoWPhasT-2D v2 and experimental data have been compared, and the numerical results from the DCD-2D v1 have been analysed.

Este trabalho estuda o fenômeno de evaporação rápida em jatos de líquidos superaquecidos ou metaestáveis numa região 2D. O fenômeno se inicia, neste caso, quando um jato na fase líquida a alta temperatura e pressão, emerge de um diminuto bocal projetando-se numa câmara de baixa pressão, inferior à pressão de saturação. Durante a evolução do processo, ao cruzar-se a curva de saturação, se observa que o fluido ainda permanece no estado de líquido superaquecido. Então, subitamente o líquido superaquecido muda de fase por meio de uma onda de evaporação oblíqua. Esta mudança de fase transforma o líquido superaquecido numa mistura bifásica com alta velocidade distribuída em várias direções e que se expande com velocidades supersônicas cada vez maiores, até atingir a pressão a jusante, e atravessando antes uma onda de choque. As equações que governam o fenômeno são as equações de conservação da massa, conservação da quantidade de movimento, e conservação da energia, incluindo uma equação de estado precisa. Devido ao fenômeno em estudo estar em regime permanente, um método de diferenças finitas com modelo estacionário e esquema de MacCormack é aplicado. Tendo em vista que este modelo não captura a onda de choque diretamente, um segundo modelo de falso transiente com o esquema de \"shock-capturing\": \"Dispersion-Controlled Dissipative\" (DCD) é desenvolvido e aplicado até atingir o regime permanente. Resultados numéricos com o código ShoWPhasT-2D v2 e testes experimentais foram comparados e os resultados numéricos com código DCD-2D v1 foram analisados. / This study analyses the rapid evaporation of superheated or metastable liquid jets in a two-dimensional region. The phenomenon is triggered, in this case, when a jet in its liquid phase at high temperature and pressure, emerges from a small aperture nozzle and expands into a low pressure chamber, below saturation pressure. During the evolution of the process, after crossing the saturation curve, one observes that the fluid remains in a superheated liquid state. Then, suddenly the superheated liquid changes phase by means of an oblique evaporation wave. This phase change transforms the liquid into a biphasic mixture at high velocity pointing toward different directions, with increasing supersonic velocity as an expansion process takes place to the chamber back pressure, after going through a compression shock wave. The equations which govern this phenomenon are: the equations of conservation of mass, momentum and energy and an equation of state. Due to its steady state process, the numerical simulation is by means of a finite difference method using the McCormack method of Discretization. As this method does not capture shock waves, a second finite difference method is used to reach this task, the method uses the transient equations version of the conservation laws, applying the Dispersion-Controlled Dissipative (DCD) scheme. Numerical results using the code ShoWPhasT-2D v2 and experimental data have been compared, and the numerical results from the DCD-2D v1 have been analysed.