A review on TVD schemes and a refined flux-limiter for steady-state calculations

This paper presents an extensive review of most of the existing TVD schemes found in literature that are based on the One-step Time-space-coupled Unsteady TVD criterion (OTU-TVD), the Multi-step Time-space-separated Unsteady TVD criterion (MTU-TVD) and the Semi-discrete Steady-state TVD criterion (SS-TVD). The design principles of these schemes are examined in detail. It is found that the selection of appropriate flux-limiters is a key design element in developing these schemes. Different flux-limiter forms (CFL-dependent or CFL-independent, and various limiting criteria) are shown to lead to different performances in accuracy and convergence. Furthermore, a refined SS-TVD flux-limiter, referred to henceforth as TCDF (Third-order Continuously Differentiable Function), is proposed for steadystate calculations based on the review. To evaluate the performance of the newly proposed scheme, many existing classical SS-TVD limiters are compared with the TCDF in eight two-dimensional test cases. The numerical results clearly show that the TCDF results in an improved overall performance.The authors gratefully acknowledge the financial support provided by the National Natural Science Foundation of China (Grant No. 51279082 and 51511130073) and the support from Australian Research Council through a Discovery Grant (Project ID: DP110105171).This is the author accepted manuscript. The final version is available from Elsevier via http://dx.doi.org/10.1016/j.jcp.2015.08.04

A refined r-factor algorithm for TVD schemes on arbitrary unstructured meshes

A refined r-factor algorithm for implementing TVD schemes on arbitrary unstructured meshes, referred to henceforth as FFISAM (a Face-perpendicular Far-upwind Interpolation Scheme for Arbitrary Meshes), is proposed based on an extensive review of the existing r-factor algorithms available in the literature. The design principles, as well as the respective advantages and disadvantages, of the existing algorithms are first systematically analyzed before presenting the FFISAM. The FFISAM is designed to combine the merits of various existing r-factor algorithms. The performance of the FFISAM, implemented in ten classical TVD schemes, is evaluated against four two-dimensional pure-advection benchmark test cases where analytical solutions are available. The numerical results clearly show that the FFISAM leads to a better overall performance than the existing algorithms in terms of accuracy and convergence on arbitrary unstructured meshes for the ten classical TVD schemes.The authors gratefully acknowledge the financial support provided by the National Natural Science Foundation of China (Grant No. 51279082) and the support from Australian Research Council through a Discovery Grant (Project ID: DP110105171).This is the author accepted manuscript. The final version is available from Wiley at http://onlinelibrary.wiley.com/doi/10.1002/fld.4073/abstract

Simulation of time-dependent compressible viscous flows using central and upwind-biased finite-difference techniques

Four time-dependent numerical algorithms for the prediction of unsteady, viscous compressible flows are compared. The analyses are based on the time-dependent Navier-Stokes equations expressed in a generalized curvilinear coordinate system. The methods tested include three traditional central-difference algorithms, and a new upwind-biased algorithm utilizing an implicit, time-marching relaxation procedure based on Newton iteration. Aerodynamic predictions are compared for internal duct-type flows and cascaded turbomachinery flows with spatial periodicity. Two-dimensional internal duct-type flow predictions are performed using an H-type grid system. Planar cascade flows are analyzed using a numerically generated, capped, body-centered, O-type grid system. Initial results are presented for critical and supercritical steady inviscid flow about an isolated cylinder. These predictions are verified by comparisons with published computational results from a similar calculation. Results from each method are then further verified by comparison with experimental data for the more demanding case of flow through a two-dimensional turbine cascade. Inviscid predictions are presented for two different transonic turbine cascade flows. All of the codes demonstrate good agreement for steady viscous flow about a high-turning turbine vane with a leading edge separation. The viscous flow results show a marked improvement over the inviscid results in the region near the separation bubble. Viscous flow results are then further verified in finer detail through comparison with the similarity solution for a flat plate boundary-layer flow. The usefulness of the schemes for the prediction of unsteady flows is demonstrated by examining the unsteady viscous flow resulting from a sinusoidally oscillating flat plate in the vicinity of a stagnant fluid. Predicted results are compared with the analytical solution for this flow. Finally, numerical results are compared with flow visualization and experimental data for the unsteady flow resulting from an impulsively started cylinder. Each algorithm demonstrates unique qualities which may be interpreted as either advantageous or disadvantageous, making it difficult to select an optimum scheme. The preferred method is perhaps best chosen based on the experience of the user and the particular application

A class of high resolution explicit and implicit shock-capturing methods

An attempt is made to give a unified and generalized formulation of a class of high resolution, explicit and implicit shock capturing methods, and to illustrate their versatility in various steady and unsteady complex shock wave computations. Included is a systematic review of the basic design principle of the various related numerical methods. Special emphasis is on the construction of the basis nonlinear, spatially second and third order schemes for nonlinear scalar hyperbolic conservation laws and the methods of extending these nonlinear scalar schemes to nonlinear systems via the approximate Riemann solvers and the flux vector splitting approaches. Generalization of these methods to efficiently include equilibrium real gases and large systems of nonequilibrium flows are discussed. Some issues concerning the applicability of these methods that were designed for homogeneous hyperbolic conservation laws to problems containing stiff source terms and shock waves are also included. The performance of some of these schemes is illustrated by numerical examples for 1-, 2- and 3-dimensional gas dynamics problems

Upwind and symmetric shock-capturing schemes

The development of numerical methods for hyperbolic conservation laws has been a rapidly growing area for the last ten years. Many of the fundamental concepts and state-of-the-art developments can only be found in meeting proceedings or internal reports. This review paper attempts to give an overview and a unified formulation of a class of shock-capturing methods. Special emphasis is on the construction of the basic nonlinear scalar second-order schemes and the methods of extending these nonlinear scalar schemes to nonlinear systems via the extact Riemann solver, approximate Riemann solvers, and flux-vector splitting approaches. Generalization of these methods to efficiently include real gases and large systems of nonequilibrium flows is discussed. The performance of some of these schemes is illustrated by numerical examples for one-, two- and three-dimensional gas dynamics problems

Inverse asymptotic treatment: capturing discontinuities in fluid flows via equation modification

A major challenge in developing accurate and robust numerical solutions to multi-physics problems is to correctly model evolving discontinuities in field quantities, which manifest themselves as interfaces between different phases in multi-phase flows, or as shock and contact discontinuities in compressible flows. When a quick response is required to rapidly emerging challenges, the complexity of bespoke discretization schemes impedes a swift transition from problem formulation to computation, which is exacerbated by the need to compose multiple interacting physics. We introduce "inverse asymptotic treatment" (IAT) as a unified framework for capturing discontinuities in fluid flows that enables building directly computable models based on off-the-shelf numerics. By capturing discontinuities through modifications at the level of the governing equations, IAT can seamlessly handle additional physics and thus enable novice end users to quickly obtain numerical results for various multi-physics scenarios. We outline IAT in the context of phase-field modeling of two-phase incompressible flows, and then demonstrate its generality by showing how localized artificial diffusivity (LAD) methods for single-phase compressible flows can be viewed as instances of IAT. Through the real-world example of a laminar hypersonic compression corner, we illustrate IAT's ability to, within just a few months, generate a directly computable model whose wall metrics predictions for sufficiently small corner angles come close to that of NASA's VULKAN-CFD solver. Finally, we propose a novel LAD approach via "reverse-engineered" PDE modifications, inspired by total variation diminishing (TVD) flux limiters, to eliminate the problem-dependent parameter tuning that plagues traditional LAD. We demonstrate that, when combined with second-order central differencing, it can robustly and accurately model compressible flows

Improvement of runback water film calculation and its impact on ice prediction

Over the past few decades, aircraft icing has been the subject of numerous studies. Ice accretion on an aircraft can damage its aerodynamic performance. It can also have a devastating affect on structures such as high voltage pylons. The simulation of ice accretion represents an important technological breakthrough in the understanding of ice behaviour as well as an alternative to expensive experiments. Although numerical models will probably never replace wind tunnel experiments, they continuously progress and benefit from the latest advances in computing techniques. ICECREMO2 is a new generation model and uses an unstructured grid approach. Unstructured meshes offer real advantages in the generation of complex grid structures but also provide support for grid adaptivity. Adaptivity consists in improving the resolution only in some aspects of the solution. It offers the benefits of the high resolution without the computational overhead of classical structured methods. Adaptive methods are usually more difficult to implement and the application to the equation coupling water film and ice growth has never been investigated before this work. The mathematical model used in ICECREMO describes both the water film flow and the ice growth. This allows us to better predict glaze ice accretion when a runback water film is present. The equation describing the thin film water flow is a complex non-linear fourth-order degenerate partial differential equation. To resolve complex features such as a moving front, high resolution numerical methods are necessary. Such a numerical scheme has been developed for this equation in a previous study on structured grid, and has proven to be reliable. In this work Sweby's scheme has been reformulated in a finite volume framework, an error estimator has been de ned for our adaptive mesh refinement method and a grid refinement strategy has been implemented which follows the water film front and keeps it under high resolution. Finally, the impact of the improved resolution of the water film on the glaze ice growth is investigated. Results obtained with first-order and high resolution methods have been compared on different model problems under various conditions. At the end an extension of the refinement strategy is proposed by defining error estimators with respect to the ice layer and by combining it with a multi-step procedure.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

The VOLNA code for the numerical modelling of tsunami waves: generation, propagation and inundation

A novel tool for tsunami wave modelling is presented. This tool has the potential of being used for operational purposes: indeed, the numerical code \VOLNA is able to handle the complete life-cycle of a tsunami (generation, propagation and run-up along the coast). The algorithm works on unstructured triangular meshes and thus can be run in arbitrary complex domains. This paper contains the detailed description of the finite volume scheme implemented in the code. The numerical treatment of the wet/dry transition is explained. This point is crucial for accurate run-up/run-down computations. Most existing tsunami codes use semi-empirical techniques at this stage, which are not always sufficient for tsunami hazard mitigation. Indeed the decision to evacuate inhabitants is based on inundation maps which are produced with this type of numerical tools. We present several realistic test cases that partially validate our algorithm. Comparisons with analytical solutions and experimental data are performed. Finally the main conclusions are outlined and the perspectives for future research presented.Comment: 47 pages, 27 figures. Other author's papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh