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An accurate and efficient numerical method for Compressible Vorticity Confinement (CVC) was developed. The methodology follows from Steinhoff's vorticity confinement approach that was developed for incompressible flows. In this research, the extension of this approach to compressible flows has been developed by adding a vorticity confinement term as a "body force" into the governing compressible flow equations. This vorticity confinement term tends to cancel the numerical dissipative errors inherently related to the numerical discretization in regions of strong vorticity gradients. The accuracy, reliability, efficiency and robustness of this method were investigated using two methods. One approach is directly applying the CVC method to several real engineering problems involving complex vortex structures and assessing the accuracy by comparison with existing experimental data and with other computational techniques. Examples considered include supersonic conical flows over delta wings, shock-bubble and shock-vortex interactions, the turbulent flow around a square cylinder and the turbulent flow past a surface-mounted 3D cube in a channel floor. A second approach for evaluating the effectiveness of the CVC method is by solving simplified "model problems" and comparing with exact solutions. Problems that we have considered are a two-dimensional supersonic shear layer, flow over a flat plate and a two-dimensional vortex moving in a uniform stream. The effectiveness of the compressible confinement method for flows with shock waves and vortices was evaluated on several complex flow applications. The supersonic flow over a delta wing at high angle of attack produces a leeward vortex separated from the wing and cross flow, as well as bow shock waves. The vorticity confinement solutions compare very favorably with experimental data and with other calculations performed on dense, locally refined grids. Other cases evaluated include isolated shock-bubble and shock-vortex interactions. The resulting complex, unsteady flow structures compare very favorably with experimental data and computations using higher-order methods and highly adaptive meshes. Two cases involving massive flow separation were considered. First the two-dimensional flow over a square cylinder was considered. The CVC method was applied to this problem using the confinement term added to the inviscid formulation, but with the no-slip condition enforced. This produced an unsteady separated flow that agreed well with experimental data and existing LES and RANS calculations. The next case described is the flow over a cubic protuberance on the floor of a channel. This flow field has a very complex flow structure involving a horseshoe vortex, a primary separation vortex and secondary corner vortices. The computational flow structures and velocity profiles were in good agreement with time-averaged values of the experimental data and with LES simulations, even though the confinement approach utilized more than a factor of 50 fewer cells (about 20,000 compared to over 1 million). In order to better understand the applicability and limitations of the vorticity confinement, particularly the compressible formulation, we have considered several simple model problems. Classical accuracy has been evaluated using a supersonic shear layer problem computed on several grids and over a range of values of confinement parameter. The flow over a flat plate was utilized to study how vorticity confinement can serve as a crude turbulent boundary layer model. Then we utilized numerical experiments with a single vortex in order to evaluate a number of consistency issues related to the numerical implementation of compressible confinement. / Ph. D.

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Les tourbillons sont des structures importantes pour une large gamme d'écoulements de fluides, notamment les sillages, l'interaction fluide-structure, les décollements de couche limite et la turbulence. Cependant, les méthodes numériques classiques n'arrivent généralement pas à donner une représentation précise des tourbillons. Ceci est principalement lié à la dissipation numérique des schémas qui, si elle n'est pas spécifiquement calibrée pour le calcul des écoulements tourbillonnaires, conduit à une diffusion artificielle très rapide des tourbillons dans les calculs. Parmi d'autres approches, la méthode "Vorticity Confinement" (VC) de J. Steinhoff permet de compenser la dissipation des schémas au sein des tourbillons en introduisant une anti-dissipation non-linéaire, mais elle n’est précise qu’au premier ordre. D’autre part, des progrès significatifs ont récemment été accomplis dans le développement de méthodes numériques d’ordre élevé. Celles-ci permettent de réduire ce problème de dissipation excessive, mais la diffusion des tourbillons reste importante pour de nombreuses applications. La présente étude vise à développer des extensions d’ordre élevé de la méthode VC pour réduire cette dissipation excessive des tourbillons, tout en préservant la précision d'ordre élevé des schémas. Tout d'abord, les schémas de confinement sont analysés dans le cas de l'équation de transport linéaire, à partir de discrétisations couplées et découplées en espace et en temps. Une analyse spectrale de ces schémas est effectuée analytiquement et numériquement en raison de leur caractère non linéaire. Elle montre des propriétés dispersives et dissipatives améliorées par rapport aux schémas linéaires de base à tous les ordres de précision. Dans un second temps, des schémas VC précis au troisième et cinquième ordre sont développés pour les équations de Navier-Stokes compressibles. Les termes correctifs restent conservatifs, invariants par rotation et indépendants du schéma de base, comme la formulation originale VC2. Les tests numériques valident l'ordre de précision et la capacité des extensions VC d’ordre élevé à réduire la dissipation dans les tourbillons. Enfin, les schémas avec VC sont appliqués au calcul des écoulements turbulents, dans une approche de simulation de grandes échelles implicite (ILES). Les schémas numériques avec VC présentent une résolvabilité améliorée par rapport à leur version linéaire de base, et montrent leur capacité à décrire de façon cohérente ces écoulements tourbillonnaires complexes. / Vortices are flow structures of primary interest in a wide range of fluid dynamics applications including wakes, fluid-structure interaction, flow separation and turbulence. Albeit their importance, standard Computational Fluid Dynamics (CFD) methods very often fail to provide an accurate representation of vortices. This is primarily related to the schemes’ numerical dissipation which, if inadequately tuned for the calculation of vortical flows, results in the artificial spreading and diffusion of vortices in numerical simulations. Among other approaches, the Vorticity Confinement (VC) method of J. Steinhoff allows balancing the baseline dissipation within vortices by introducing non-linear anti-dissipation in the discretization of the flow equations, but remains at most first-order accurate. At the same time, remarkable progress has recently been made on the development of high-order numerical methods. These allow reducing the problem of excess dissipation, but the diffusion of vortices remains important for many applications. The present study aims at developing high-order extensions of the VC method to reduce the excess dissipation of vortices, while preserving the accuracy of high-order methods. First, the schemes are analyzed in the case of the linear transport equation, based on time-space coupled and uncoupled formulations. A spectral analysis of nonlinear schemes with VC is performed analytically and numerically, due to their nonlinear character. These schemes exhibit improved dispersive and dissipative properties compared to their linear counterparts at all orders of accuracy. In a second step, third- and fifth-order accurate VC schemes are developed for the compressible Navier-Stokes equations. These remain conservative, rotationally invariant and independent of the baseline scheme, as the original VC2 formulation. Numerical tests validate the increased order of accuracy and the capability of high-order VC extensions to balance dissipation within vortices. Finally, schemes with VC are applied to the calculation of turbulent flows, in an implicit Large Eddy Simulation (ILES) approach. In these applications, numerical schemes with VC exhibit improved resolvability compared to their baseline linear version, while they are capable of producing consistent results even in complex vortical flows.

This work makes basic analysis of historical and current algorithms for smoke animation. Modern approaches to rendering volumetric data are briefly described. We choose algorithms for implementation on basis of this analysis. These algorithms are described in detail and we make emphasis on their important properties according to dedication of this work. Detailed description of implementation follows along with performance measurement. Conclusion evaluates results of work and proposes possible extensions.