Recent developments in mathematical aspects of relativistic fluids

We review some recent developments in mathematical aspects of relativistic fluids. The goal is to provide a quick entry point to some research topics of current interest that is accessible to graduate students and researchers from adjacent fields, as well as to researches working on broader aspects of relativistic fluid dynamics interested in its mathematical formalism. Instead of complete proofs, which can be found in the published literature, here we focus on the proofs' main ideas and key concepts. After an introduction to the relativistic Euler equations, we cover the following topics: a new wave-transport formulation of the relativistic Euler equations tailored to applications; the problem of shock formation for relativistic Euler; rough (i.e., low-regularity) solutions to the relativistic Euler equations; the relativistic Euler equations with a physical vacuum boundary; relativistic fluids with viscosity. We finish with a discussion of open problems and future directions of research.Comment: Minor typos correcte

The enhanced wave-induced drift of large floating objects

At the ocean surface, mass, momentum and energy are transferred between the atmosphere and the ocean. Ocean surface waves, in particular, drive the transport of floating objects such as sediments, pollutants, and man-made structures. In irrotational and progressive surface gravity waves, water particles exhibit a non-closed path over one wave cycle, leading to Stokes drift. The drift can be modified, and indeed enhanced, when an object deviates from being an ideal Lagrangian tracer. This thesis examines the wave-induced drift of two-dimensional (2D) floating objects by surface gravity waves, specifically, deep-water regular waves. A combination of analytical, numerical, and experimental approaches are employed to analyze the drift of objects with varying sizes and shapes under different wave steepness. A hybrid numerical model encompassing both viscosity and diffraction effects is used to investigate the influence of changing size, shape and wave steepness on the object drift. The drift deviates from the standard Stokes drift and depends on objects’ size and shape. Larger objects with less streamlined shapes, resembling box-like structures, exhibit enhanced drift. The enhancement is attributed to the standing wave pattern generated as a result of diffraction of the wave field, and viscosity. A diffraction-modified Stokes drift model is proposed to predict object drift. The model introduces additional terms to the standard Stokes drift, accommodating incident, diffracted, and radiated wave fields. The results are compared to both experimental measurements and numerical results, demonstrating a close agreement when the object is not too large. The model offers an in-depth understanding of one of the mechanisms contributing to drift enhancement. A 2D experiment has been conducted in a laboratory flume. The floating objects are specially designed to ensure the experiment is two dimensional. The enhanced drift, along with the associated standing wave pattern, predicted by the model is also observed in experimental data, which validates the theoretically predicted enhanced drift. Distinct drift behaviours are identified for small and large objects at low and high wave steepness. The data reveals that the scaling relationship between the object drift and steepness is characterized by a mixture of linear and second-order terms (in steepness) and is dependent on object size. The influence of object corner shape on drift is further explored using the hybrid numerical model. In particular, the drift of a bluff body with sharp and round corners has been investigated. It is found that an object’s sharp corners can induce vorticity and thus change the pressure distribution around the moving objects, resulting in a significantly altered drift trajectory and unsteadiness in the object’s drift

Recent advances in nearshore wave, circulation, and sediment transport modeling

Significant advances in the modeling of nearshore processes have occurred over recent decades as a result of developments in both computational approaches and theoretical understanding. This review examines the present state of progress primarily from a hydrodynamics standpoint, followed by a brief discussion of applications to sediment transport and morphological evolution. Wave-averaged formulations of the wave-current interaction problem and resulting models for wave-induced currents are reviewed in order to compare and contrast radiation stress and vortex force approaches. Waveresolving approaches are then discussed, with an emphasis on the recent rapid development of 3D nonhydrostatic models and their application to a wide range of physical problems. The recently developed understanding of the importance of vorticity generation mechanisms at wave-resolved scales, and their contribution to transport and mixing processes, are discussed

Resolution of Tip Vortices by grid-based, grid-free and coupled methods using CFD

The vortex structure resolution is one of the vital problems of CFD as inherent artificial dissipation effects lead to an unphysical strong decay of the vortices. The overall objective of this work is to improve the resolution of concentrated vortices. This work focuses on grid based, grid free methods and coupled methods to capture the details of vortices especially further downstream after the vortex has rolled up and started to decay. The work focuses on a hybrid method as a coupling of grid based and grid free vortex method

Exploration of a superposition and reconciliation based approach to cell-centered Lagrangian hydrodynamic methods

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Nuclear Science and Engineering, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 84-90).Applications and experiments involving the hypervelocity deformation of solids are difficult to devise, implement, and occur on microsecond time scales. As a result, simulations play a large role in the study of hypervelocity deformation. This study explored a superposition and reconciliation based approach using cell-centered Lagrangian hydro methods. The reconciliation forces that are not explicitly calculated for mesh movement were analyzed on an existing hydrocode by Pierre-Henri Maire (PHM) and a truncated form of the Runnels-Gilman method (implemented without using the reconciliation forces as additional forces to form a new hydro method called the Runnels-Gilman method). Results from both the 1D Piston and Saltzman test problems illustrate that the unaccounted reconciliation forces are acting on the mesh both at the shock front and behind the shock wave in PHM's method, while in the truncated Runnels- Gilman method, reconciliation forces are acting only on the vertices at the shock front. In test problems using PHM's method, reconciliation forces may be capturing the additional forces that account for more stable density and internal energy solution during shock wave propagation as compared to the truncated Runnels-Gilman method.by Lindsey Anne Gilman.S.M

Physics-Based Fluid Flow Restoration Method

Experimentelle Methoden und bildgebende Messverfahren zur Geschwindigkeitsmessung wie zum Beispiel Particle Image Velocimetry (PIV, etwa: Geschwindigkeitsmessung basierend auf Partikelbilder) und Particle Tracking Velocimetry (PTV, etwa: Geschwindig- keitsmessung basierend auf Partikelverfolgung) spielen in der Erforschung von Strömungen in Fluiden eine große Rolle. Sie sind sowohl für die Forschung als auch für eine große Reihe industrieller Anwendungen gleichbedeutend wichtig. Dennoch wird oft die geschätzte Geschwindigkeit von Fluiden durch Störungen, diversen Verfälschungen und fehlenden Fragmente beeinflusst, welches eine physikalische Interpretation der Werte sehr schwierig macht. In der vorliegenden Arbeit wird ein neuer Algorithmus zur Rekonstruktion von Geschwindigkeitsfeldern in Fluiden vorgestellt. Der Algorithmus akzeptiert als Eingabe eine große Reihe an beschädigten zwei- oder dreidimensionalen Vektorfelder und erlaubt fehlende Fragmente wiederherzustellen und das Rauschen auf einem physikalisch plausiblen Weg zu entfernen. Das Verfahren nutzt im wesentlichen die physikalischen Eigenschaften von nicht komprimierbaren Fluiden aus und hängt nicht von einem bestimmten Rausch-Modell ab. Es besteht aus vier relativ einfachen Vorschriften. Davon basieren drei auf den Grundprinzipien der Kontinuummechanik wie die Kontinuitätsgleichung, die Momentenausgleichgleichung, sowie Ergebnisse der Turbulenztheorie, grundsätzlich das Übergewicht an Niederfrequenzen in spektralen Bänder von Fluiden. Ein Ergebnis dieser physikalisch ausgerichteten Lösung ist, dass der entwickelte Algorithmus für verschiedene praxisrelevante Fehler und Störungen robust und effizient funktioniert. Ein weiterer Aspekt der entwickelten Methode ist, dass experimentelle Daten in vielen Fällen Vektoren enthalten, welche in einem dreidimensionalen Volumen zufällig aber dünnbesetzt verteilt sind. Diese tauchen aufgrund technischer Anforderungen und Restriktionen der angewendeten Messmethoden zur Geschwindigkeitsschätzung von Partikelfelder auf. Das hier vorgestellte Verfahren wurde dementsprechend um einen hochauflösenden Ansatz erweitert um mit solchen Daten zurecht zu kommen. Die Methode akzeptiert beliebig beschädigte dünnbesetzte Vektorfelder als Eingangsdatensatz und rekonstruiert die fehlenden Teile des Flusses auf einer physikalisch konsistenten Art. Der Hochauflösungs- ansatz führt zu einer Wiederherstellung des Datensatzes in Form eines hochaufgelösten Vektorfeldes. Alle bedeutenden Aussagen werden anhand numerischer Experimente mit turbulenten Flussgeschwindigkeitsfelder bestätigt. Das hier entwickelte Verfahren basiert auf einem Variationsansatz. Es wird in der Ausarbeitung gezeigt, das man in der vorgeschlagene Methode zur diskreten Darstellung anhand verschiedener numerischen Techniken übergehen kann, z.B. anhand der Finite-Differenzen-Methode oder der Finite-Elemente-Methode. Die vom Rekonstruktionsalgorithmus gelieferten Ergebnisse rechtfertigen die Annahme dass das vorgeschlagene Verfahren zum Entrauschen und Hochauflösen von Vektorfelder mit jeder Art von Störungen zurecht kommt

Dynamical description of spatio-temporally varying turbulent energy cascades

The spatio-temporally varying turbulent energy cascade dynamics in forced homogeneous/periodic turbulence is investigated with direct numerical simulations (DNS) and Helmholtz decompositions. The local in space and time cascade dynamics vastly differs from its spatio-temporal average manifestation. At scales larger than the Taylor scale, the solenoidal interscale transfer at most locations at most times increases or decreases the energy at the given scale in the frame moving with larger scales, i.e. Lagrangian transport. The solenoidal interscale transfer derives from the non-local in space vortex stretching/compression and tilting effects of its spatial vicinity. The irrotational cascade dynamics reduces to an exact balance between irrotational transport, irrotational interscale transfer and pressure-velocity. The typical fluctuations of these processes vastly exceed their spatio-temporal average values and the typical dissipation fluctuations. At scales below the Taylor scale, viscous effects increase in importance in the solenoidal dynamics. At the Kolmogorov scale solenoidal interscale transfer, Lagrangian transport and viscous effects are all important. In regions of low and moderate small-scale energy, and to a somewhat lesser extent in regions of high small-scale energy, there is rarely a local balance between interscale transfer and viscous effects. Lagrangian transport acts as a non-local in time and space link between interscale transfer and viscous effects. The spatially-averaged manifestation of the local cascade dynamics is an unsteady and approximately unidirectional energy cascade, which can be approximated with a hypothesis connecting the present interscale transfer with the future dissipation. The hypothesis can be used to develop non-equilibrium corrections to the low-pass filtered dynamics and second-order structure function scaling consistent with DNSs. We use the phenomenology of a time-lagged energy cascade to motivate a new redistributive dissipation scaling. The non-equilibrium dissipation scaling typically reduces to the redistributive dissipation scaling at low and moderate Reynolds numbers.Open Acces