A Level Set Approach to Eulerian-Lagrangian Coupling

We present a numerical method for coupling an Eulerian compressible flow solver with a Lagrangian solver for fast transient problems involving fluid-solid interactions. Such coupling needs arise when either specific solution methods or accuracy considerations necessitate that different and disjoint subdomains be treated with different (Eulerian or Lagrangian)schemes. The algorithm we propose employs standard integration of the Eulerian solution over a Cartesian mesh. To treat the irregular boundary cells that are generated by an arbitrary boundary on a structured grid, the Eulerian computational domain is augmented by a thin layer of Cartesian ghost cells. Boundary conditions at these cells are established by enforcing conservation of mass and continuity of the stress tensor in the direction normal to the boundary. The description and the kinematic constraints of the Eulerian boundary rely on the unstructured Lagrangian mesh. The Lagrangian mesh evolves concurrently, driven by the traction boundary conditions imposed by the Eulerian counterpart. Several numerical tests designed to measure the rate of convergence and accuracy of the coupling algorithm are presented as well. General problems in one and two dimensions are considered, including a test consisting of an isotropic elastic solid and a compressible fluid in a fully coupled setting where the exact solution is available

Hypersonic flows around complex geometries with adaptive mesh refinement and immersed boundary method

This thesis develops and validates a computational fluid dynamics numerical method for hypersonic flows; and uses it to conduct two novel investigations. The numerical method involves a novel combination of structured adaptive mesh refinement, ghost-point immersed boundary and artificial dissipation shock-stable Euler flux discretisation. The method is high-order, low dissipation and stable up to Mach numbers $M \lesssim 30$ with stationary or moving complex geometries; it is shown to be suitable for direct numerical simulations of laminar and turbulent flows. The method's performance is assessed through various test cases. Firstly, heat transfer to proximal cylinders in hypersonic flow is investigated to improve understanding of destructive atmospheric entries of meteors, satellites and spacecraft components. Binary bodies and clusters with five bodies are considered. With binary proximal bodies, the heat load and peak heat transfer are augmented for either or both proximal bodies by $+20\%$ to $-90\%$ of an isolated body. Whereas with five bodies, the cluster-averaged heat load varied between $+20\%$ to $-60\%$ of an isolated body. Generally, clusters which are thin in the direction perpendicular to free-stream velocity and long in the direction parallel to the free-stream velocity have their heat load reduced. In contrast, clusters which are thick and thin in directions perpendicular and parallel to the free-stream velocity feel an increased heat load. Secondly, hypersonic ablation patterns are investigated. Ablation patterns form on spacecraft thermal protection systems and meteor surfaces, where their development and interactions with the boundary layer are poorly understood. Initially, a simple subliming sphere case without solid conduction in hypersonic laminar flow is used to validate the numerical method. Where the surface recession is artificially sped-up via the wall Damk\"{o}hler number without introducing significant errors in the shape change. Then, a case with transitional inflow over a backward facing step with a subliming boundary is devised. Differential ablation is observed to generate surface roughness and add vorticity to the boundary layer. A maximum surface recession of $\sim 0.8\times$ and a maximum surface fluctuation of $\sim 0.2\times$ the inflow boundary layer thickness were generated over two flow times.Open Acces

An Investigation of High-Order Shock-Capturing Methods for Computational Aeroacoustics

Topics covered include: Low-dispersion scheme for nonlinear acoustic waves in nonuniform flow; Computation of acoustic scattering by a low-dispersion scheme; Algorithmic extension of low-dispersion scheme and modeling effects for acoustic wave simulation; The accuracy of shock capturing in two spatial dimensions; Using high-order methods on lower-order geometries; and Computational considerations for the simulation of discontinuous flows

Sharp interface schemes for multi-material computational fluid dynamics

In this thesis we consider the solution of compressible multi-material flow problems, where each material is governed by the Euler equations and the material interface may be considered to be a perfect discontinuity separating macroscopic pure-material regions. Working in the framework of Godunov-type finite volume methods, we develop numerical algorithms for tracking the material interface and evolving fluid states. For the task of tracking the location of sharp material interfaces, we focus on volume tracking methods due to their ability to conserve mass in highly deformational flows. Three original contributions are presented in this area. First, the accuracy of the volume-of-fluid algorithm is improved through the addition of marker particles. Next, the efficient moment-of-fluid method is presented. This improves the computational efficiency of the moment-of-fluid method by a factor of three by mapping certain quantities during the interface reconstruction step on to a pre-computed data structure. Finally, a general framework for updating volume fractions based on the solution to a quadratic programming problem is presented. The evolution of fluid states in the full multi-material system is an independent problem. We present developments to two numerical approaches for this problem. The first, the ghost fluid method, is widely used due to the ease in which pure-fluid algorithms can be extended to the multi-material case. We investigate the effect of using a number of different interface tracking methods on the solutions, and find that the conservation errors vary by more than an order of magnitude. The ghost fluid method is then altered such that the ghost state extrapolation step is eliminated from the algorithm, allowing volume fraction-based interface tracking methods to be coupled. In the final chapter, we tackle the same problem using a mixture model-based method. The numerical method presented here is based on work by Miller and Puckett in 1996, in which a six-equation system using the assumption of pressure and velocity equilibrium was used to model two-material flow. We have thoroughly overhauled this method, incorporating Riemann solvers developed for the five-equation system, as well as a robust implicit energy update. We present numerical results on a range of one- and two-dimensional shock-interface interaction test problems which demonstrate the ability of the method to match the solutions from the five-equation model while maintaining a perfectly sharp material interface.Funded by AW

A generalised multi-directional characteristic-based Godunov-type framework for elliptic, parabolic and hyperbolic pressure-based incompressible methods

The objective of the current research is to construct numerical methods based on physical principles to reduce modelling errors in the field of computational fluid dynamics. In order to investigate the non-linearities of the convective flux term, a multi-directional characteristic-based scheme has been developed in this work to capture the anisotropic behaviour of the incompressible Navier{Stokes equations. To avoid the pressure-velocity decoupling and to promote stability at high Reynolds numbers, the Riemann problem has been incorporated into the scheme which creates a multi-directional Godunov-type framework. In order to capture the pressure correctly, which through its coupling to the velocity field is depending on the velocity's non-linear effects, it is postulated that the pressure should have its own transport equation which should have a parabolic type. This is necessary to align the pressure with the mathematical properties of the Navier{Stokes equations. Thus, a novel incompressible method has been developed which features a pressure transport equation which is referred to as the Fractional-Step with Velocity Projection (or FSVP) method. It is further extended through a perturbed continuity equation of the Arti cial Compressibility (AC) method to hyperbolise the first Fractional-Step of the system of equations, while the second Fractional-Step retains the required parabolic behaviour, which is called the FSAC-VP method in turn. Through the hyperbolic Fractional-Step, the multi-directional Godunov-type framework is directly applicable to the newly developed method. Parametric simulations for the lid driven cavity, backward facing step, sudden expan- sion and Taylor{Green vortex problem have been performed using the AC, FSVP, FSAC-VP and the Fractional-Step, Arti cial Compressibility with Pressure Projection, or FSAC-PP, method. The FSVP and FSAC-VP method showed superior convergence properties compared to the AC method for unsteady flows, where a speed up of a factor up to 193.0 times has been observed. Since the parabolic pressure transport equation has a memory of the time history of the flow, smooth error curves have been produced over time while the other methods showed oscillatory profi les. Generally speaking, the most accurate results have been obtained with the FSAC-PP method, closely followed by the FSAC-VP and FSVP method. The inclusion of the multi-directional Godunov-type framework showed generally better or equally well resolved results compared to the benchmark numerical scheme for the FSAC-PP and FSAC-VP / FSVP method. Furthermore, the multi-directional scheme by itself showed its capabilities to predict vortical flows better than a simple numerical reconstruction scheme. The FSAC-VP method has shown a higher degree of scheme independence where velocity and pressure curves showed little variations compared to reference data. This was particularly pronounced for the sudden expansion which had consequences on the prediction of the correct bifurcation behaviour. Finally, it has been argued that what the numerical scheme development is to the non-linear term of the Navier{Stokes equations should be similarly done with incompressible flow method development to capture the correct pressure behaviour. This work shows that differences between elliptic, parabolic and hyperbolic pressure treatments do exist which can have a significant effect on the overall prediction of the flow features

Non-Body Conformal Grid Methods for Large-Eddy Simulations of Compressible Flows and their Applications in Computational Aeroacoustics

In the applications of computational aeroacoustics (CAA) involving far-field noise predictions, the most common solution strategy is the \textit{hybrid method} which combines a computational fluid dynamics (CFD) solver for the sound source field with an acoustic solver for the acoustic far-field. Hybrid CAA methods provide flexibility to select the most appropriate methods to compute the sound source and the acoustic fields, respectively, to suit various aeroacoustic problems. The present study reports the development of a hybrid large-eddy simulation (LES)-acoustic analogy method to effectively predict the noise of viscous flow over complex geometries. With complex geometries, difficulties arise with numerical methods based on body fitted grids. Generating good quality body fitted grids around complex geometries is challenging and time-consuming. Alternatively, numerical methods based on non-body conformal grids can deal with bodies of almost any arbitrary shape. Although the present research was initially motivated for CAA applications, most of the contributions and the novelty of the work is in the development of efficient, easy-to-implement and more accurate non-body conformal methods that can be used for flow over complex geometries. To date, most of the listed work on non-body conformal methods is applied to incompressible flows. The use of non-body conformal methods for compressible turbulent flows is still rare and immature. Two non-body conformal grid methods are developed and assessed in this work: the ghost-cell based immersed boundary method (GC-IBM) and the ghost-cell based cut-cell method (GC-CCM). In both methods, the boundary conditions on the immersed boundary are enforced through the use of ``ghost cells'' located inside the solid body. Variables on these ghost cells are computed using linear interpolation schemes. The implementation using GC-IBM is simpler; however, the exact shape of the fluid cells in the vicinity of the solid boundary is not detailed, which results in the loss or gain of mass and momentum. As such, sufficiently refined meshing is required in the vicinity of the solid boundary to mitigate the error on mass conservation. The implementation using GC-CCM requires more work; however, the underlying conservation laws is guaranteed by introducing ``cut cells''. A cell-merging approach is used to address the \textit{small-cell problem} associated with a Cartesian cut-cell method, which, if untreated, results in the numerical instability and stiffness of the system of equations. The applicability of the developed non-body conformal methods is investigated in the compressible LES framework. Turbulent flows in various complex geometric settings are simulated using these non-body conformal methods for a wide range of Reynolds numbers and Mach numbers. For high Reynolds number flows, the developed non-body conformal methods employ a wall model to approximate the wall-shear stress, thus avoiding a requirement for severe grid resolution near the wall. No previously published work involves LES of high Reynolds number compressible flows using a wall model and a non-body conformal method. This research uses a simple wall model based on a wall function to approximate the near wall behaviour, but this approach can be extended to other wall models if necessary. Better wall modelling strategies should be investigated in the future. The numerical results demonstrate that the GC-CCM is capable of capturing near-wall flows relatively well despite the simple wall model used. GC-CCM also provides relatively accurate results compared to other non-body conformal methods. Returning to the original research efforts for aeroacoustic applications, the GC-CCM is finally benchmarked for the prediction of far-field radiated noise from a flow over a circular cylinder. Of many hybrid approaches available in CAA, Ffowcs-Williams and Hawkings (FW-H) approach is selected to explore the far-field acoustic calculation. Comparison of the results to the experimental data shows that the developed hybrid LES-acoustic analogy method is capable of accurately predicting the sound spectrum for this case of three-dimensional flow over a cylinder in the sub-critical regime. Large-eddy simulations with more complex geometries, such as wings or high-lift systems, have not been performed as a part of this research. Further work is encouraged in order to conclude the research direction originally envisioned by the author

Compact-Reconstruction Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws

A new class of non-linear compact interpolation schemes is introduced in this dissertation that have a high spectral resolution and are non-oscillatory across discontinuities. The Compact-Reconstruction Weighted Essentially Non-Oscillatory (CRWENO) schemes use a solution-dependent combination of lower-order compact schemes to yield a high-order accurate, non-oscillatory scheme. Fifth-order accurate CRWENO schemes are constructed and their numerical properties are analyzed. These schemes have lower absolute errors and higher spectral resolution than the WENO scheme of the same order. The schemes are applied to scalar conservation laws and the Euler equations of fluid dynamics. The order of convergence and the higher accuracy of the CRWENO schemes are verified for smooth solutions. Significant improvements are observed in the resolution of discontinuities and extrema as well as the preservation of flow features over large convection distances. The computational cost of the CRWENO schemes is assessed and the reduced error in the solution outweighs the additional expense of the implicit scheme, thus resulting in higher numerical efficiency. This conclusion extends to the reconstruction of conserved and primitive variables for the Euler equations, but not to the characteristic-based reconstruction. Further improvements are observed in the accuracy and resolution of the schemes with alternative formulations for the non-linear weights. The CRWENO schemes are integrated into a structured, finite-volume Navier-Stokes solver and applied to problems of practical relevance. Steady and unsteady flows around airfoils are solved to validate the scheme for curvi-linear grids, as well as overset grids with relative motion. The steady flow around a three-dimensional wing and the unsteady flow around a full-scale rotor are solved. It is observed that though lower-order schemes suffice for the accurate prediction of aerodynamic forces, the CRWENO scheme yields improved resolution of near-blade and wake flow features, including boundary and shear layers, and shed vortices. The high spectral resolution, coupled with the non-oscillatory behavior, indicate their suitability for the direct numerical simulation of compressible turbulent flows. Canonical flow problems -- the decay of isotropic turbulence and the shock-turbulence interaction -- are solved. The CRWENO schemes show an improved resolution of the higher wavenumbers and the small-length-scale flow features that are characteristic of turbulent flows. Overall, the CRWENO schemes show significant improvements in resolving and preserving flow features over a large range of length scales due to the higher spectral resolution and lower dissipation and dispersion errors, compared to the WENO schemes. Thus, these schemes are a viable alternative for the numerical simulation of compressible, turbulent flows