An approximate Riemann solver for shallow water equations and heat advection in horizontal centrifugal casting

AbstractAn approximate Riemann solver was developed for solving modified shallow water equations (SWE) and energy transport describing the average flow dynamics of a single layer spreading inside a horizontally rotating cylinder. The numerical model was particularly developed for simulating the horizontal centrifugal casting (HSC) of the outer shell of a work roll. The SWE were derived in the rotating frame of reference; therefore, fictitious forces (the centrifugal force and the Coriolis force) were considered. In addition, other forces such as the bed shear force, the force of gravity, the wind shear force and forces arising from the variable liquid/solid interface were taken into account. The Jacobian matrix of the nonlinear hyperbolic system of PDEs was decomposed into a set of eigenvalues and corresponding eigenvectors using standard and corrected Roe averages. A Harten–Hyman entropy fix was used to prevent expansion shocks (entropy violating solutions) typically occurring during transonic rarefactions. Source terms were applied as a stationary discontinuity and were physically bounded and well-balanced for steady states (producing non-oscillatory solutions). Each wave was upwinded using the explicit Godunov’s method. The high resolution corrections with flux limiters were used to achieve second order of accuracy and dispersion free solutions at discontinuities. In addition to the Riemann solver, a central scheme FV model was used to solve the heat diffusion inside the cylinder (mold) and partially solidified liquid layer, coupled with the solidification model. Several simulations were performed, results were analyzed and discussed

Cut Cell Methods in Global Atmospheric Dynamics

In this thesis, we study next generation techniques for the numerical simulation of global atmospheric dynamics, which range from modeling and grid generation to discretization schemes. Based on a detailed dimensional analysis of the compressible three-dimensional Navier-Stokes equations for small- and large-scale motions in the atmosphere, we derive the compressible Euler equations, the dynamical core of meteorological models. We also provide an insight into multiscale modeling and present a new numerical way of deriving reduced atmospheric models and gaining consistency of the modeling and discretization errors. The main focus of this thesis is the grid generation of the atmosphere. With regard to newly available surveys of the Earth's surface and the ever increasing computing capacities, the atmospheric triangulation techniques have to be reconsidered. In particular, the widely-used terrain-following coordinates prove to be disadvantaguous for highly resolved grids, since both the pressure gradient force error and the hydrostatic inconsistency of this vertical ansatz seriously increase with finer resolution. After a detailed analysis of the standard methods for vertical atmospheric triangulations, we present the cut cell approach as capable alternative. We construct a special cut cell method with two stabilizing constraints and provide a comprehensive guideline for an implementation of cut cells into existing atmospheric codes. For the spatial discretization of the dynamical core, we choose the Finite Volume method because of its favorable characteristics concerning conservation properties and handling of hyperbolicity. We accompany the Finite Volume discretization by a new non-linear interpolation scheme of the velocity field, which is adapted to the geometry and rotation of the Earth. To fathom the capabilities of cut cell grids together with our discretization and new interpolation scheme, we finally present several three-dimensional simulation runs. We apply standard benchmarks like an advection test and the simulation of a Rossby-Haurwitz wave and construct a new test case of counterbalancing flow between high- and low-pressure areas, with which we expose the potential of cut cell methods and the influences of different effects of the Euler equations as well as the topography of the Earth

A Review of Element-Based Galerkin Methods for Numerical Weather Prediction: Finite Elements, Spectral Elements, and Discontinuous Galerkin

Numerical weather prediction (NWP) is in a period of transition. As resolutions increase, global models are moving towards fully nonhydrostatic dynamical cores, with the local and global models using the same governing equations; therefore we have reached a point where it will be necessary to use a single model for both applications. The new dynamical cores at the heart of these unified models are designed to scale efficiently on clusters with hundreds of thousands or even millions of CPU cores and GPUs. Operational and research NWP codes currently use a wide range of numerical methods: finite differences, spectral transform, finite volumes and, increasingly, finite/spectral elements and discontinuous Galerkin, which constitute element-based Galerkin (EBG) methods.Due to their important role in this transition, will EBGs be the dominant power behind NWP in the next 10 years, or will they just be one of many methods to choose from? One decade after the review of numerical methods for atmospheric modeling by Steppeler et al. (Meteorol Atmos Phys 82:287–301, 2003), this review discusses EBG methods as a viable numerical approach for the next-generation NWP models. One well-known weakness of EBG methods is the generation of unphysical oscillations in advection-dominated flows; special attention is hence devoted to dissipation-based stabilization methods. Since EBGs are geometrically flexible and allow both conforming and non-conforming meshes, as well as grid adaptivity, this review is concluded with a short overview of how mesh generation and dynamic mesh refinement are becoming as important for atmospheric modeling as they have been for engineering applications for many years.The authors would like to thank Prof. Eugenio Oñate (U. Politècnica de Catalunya) for his invitation to submit this review article. They are also thankful to Prof. Dale Durran (U. Washington), Dr. Tommaso Benacchio (Met Office), and Dr. Matias Avila (BSC-CNS) for their comments and corrections, as well as insightful discussion with Sam Watson, Consulting Software Engineer (Exa Corp.) Most of the contribution to this article by the first author stems from his Ph.D. thesis carried out at the Barcelona Supercomputing Center (BSCCNS) and Universitat Politècnica de Catalunya, Spain, supported by a BSC-CNS student grant, by Iberdrola Energías Renovables, and by grant N62909-09-1-4083 of the Office of Naval Research Global. At NPS, SM, AM, MK, and FXG were supported by the Office of Naval Research through program element PE-0602435N, the Air Force Office of Scientific Research through the Computational Mathematics program, and the National Science Foundation (Division of Mathematical Sciences) through program element 121670. The scalability studies of the atmospheric model NUMA that are presented in this paper used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357. SM, MK, and AM are grateful to the National Research Council of the National Academies.Peer ReviewedPostprint (author's final draft

Towards a solution of the closure problem for convective atmospheric boundary-layer turbulence

We consider the closure problem for turbulence in the dry convective atmospheric boundary layer (CBL). Transport in the CBL is carried by small scale eddies near the surface and large plumes in the well mixed middle part up to the inversion that separates the CBL from the stably stratified air above. An analytically tractable model based on a multivariate Delta-PDF approach is developed. It is an extension of the model of Gryanik and Hartmann [1] (GH02) that additionally includes a term for background turbulence. Thus an exact solution is derived and all higher order moments (HOMs) are explained by second order moments, correlation coefficients and the skewness. The solution provides a proof of the extended universality hypothesis of GH02 which is the refinement of the Millionshchikov hypothesis (quasi- normality of FOM). This refined hypothesis states that CBL turbulence can be considered as result of a linear interpolation between the Gaussian and the very skewed turbulence regimes. Although the extended universality hypothesis was confirmed by results of field measurements, LES and DNS simulations (see e.g. [2-4]), several questions remained unexplained. These are now answered by the new model including the reasons of the universality of the functional form of the HOMs, the significant scatter of the values of the coefficients and the source of the magic of the linear interpolation. Finally, the closures 61 predicted by the model are tested against measurements and LES data. Some of the other issues of CBL turbulence, e.g. familiar kurtosis-skewness relationships and relation of area coverage parameters of plumes (so called filling factors) with HOM will be discussed also

The Turgo impulse turbine:a CFD based approach to the design improvement with experimental validation

The use of Computational Fluid Dynamics (CFD) has become a well-established approach in the analysis and optimisation of impulse hydro turbines. Recent studies have shown that modern CFD tools combined with faster computing processors can be used to accurately simulate the operation of impulse turbine runners and injectors in timescales suitable for design optimisation studies and which correlate well with experimental results. This work has however focussed mainly on Pelton turbines and the use of CFD in the analysis and optimisation of Turgo turbines is still in its infancy, with no studies showing a complete simulation of a Turgo runner capturing the torque on the inside and outside blade surfaces and producing a reliable extrapolation of the torque and power at a given operating point. Although there have been some studies carried out in the past where injector geometries (similar for both Pelton and Turgo turbines) have been modified to improve their performance, there has been no thorough investigation of the basic injector design parameters and the influence they have on the injector performance. The aim of this research is to use modern CFD tools to develop models which aid the better understanding of Turgo impulse turbine runners and injectors and facilitate the optimisation of existing designs. CFD is used to model and optimise both the injectors and the runner of a modern commercial Turgo impulse turbine and the accuracy of the models are verified by carrying out experimental tests on the original and optimised designs. The original designs together with experience in the operation of these turbines were provided by the industrial sponsors of this research Gilbert Gilkes and Gordon Ltd. The research described in this thesis can be split into five main parts: 1.Development of a numerical model to analyses the flow through the Turgo runner using modern CFD tools combined with a series of assumptions to reduce the computational time while still retaining the accuracy of the model. Using this model to optimise the design of the Turgo runner provided by Gilkes. 2.Development of a similar numerical model for a simplified 2D injector design to facilitate a study of the impact of the basic design parameters on the performance over a range of operating conditions. Applying these optimisations to the existing Gilkes design and taking the numerical analysis further by including the full injector geometry as well as the branch pipe and guide vanes. 3.Manufacture and experimental testing of the original and optimised Turgo runners. 4.Manufacture and experimental testing of the original and optimised injector designs. 5.Verification of the numerical models developed in 1.) and 2.) by comparison with the experimental results. The numerical model developed in 1.) includes several simplifying assumptions in order to reduce the computational time and produce models which could solve in reasonable timescales allowing many design variations to be analysed. As the runner simulations require a transient analysis of complex multi-phase free surface flow with a rotating frame of reference they are already computationally costly and efforts have to be made to reduce this computational cost if the models are to be effective for optimisation purposes. The runner model simplifications were the exclusion of any casing interactions by not modelling the casing and the use of a 2 blade model analysing only a single blade passage in order to reduce the size of the computational domain. Several modelling assumptions were also introduced and attempts are made to quantify the effects of these assumptions through unit tests. For discretisation of the domain two mesh sizes were used, a coarse mesh which slightly under predicts the efficiency but was suitable for comparing designs and a fine mesh which gave mesh independent results. The fine mesh took over 4 times longer to solve rendering it unfeasible for optimisation purposes and it was therefore used only at key points to verify the design changes made using the coarse mesh. The analysis and optimisation of the injectors carried out in 2.) use similar CFD tools as the runner analysis however the geometry (excluding the branch pipe and guide vanes) could be simplified into a 2D axisymmetric case operating at steady state conditions. This drastically reduces the solve time and allows the use of a mesh independent model and the analysis of hundreds of designs and operating conditions. Once the optimisations had been carried out, the design changes were verified by extending the model to analyse the 3D case with a straight pipe upstream of the injector and a 3D full case including the branch pipe and guide vanes. In 3.), following the optimisation of the runner in 1.), a Finite Element Analysis (FEA) of the runner was carried out to ensure the optimised runner had sufficient strength for operation at the highest heads recommended for a runner of this size. The design was strengthened based on the results of the FEA and CFD was carried out in conjunction with these changes to ensure minimal loss in hydraulic efficiency. The manufacturing process was also researched and Design for Manufacture and Assembly (DFMA) was applied to the strengthened design identifying two optimised designs (LE4 and LE1) which will be tested before and after additional dressing of the leading edges. Both optimised runner designs were manufactured and tested at the Laboratory of Hydraulic Machines, National Technical University of Athens (NTUA). Following the injector analysis and optimisations in 2.), the optimised injectors were manufactured for experimental testing using both the Pelton and the Turgo test rig at NTUA in 4.). As the design changes made were not critical to the strength of the injectors there was no need to carry out a FEA. The CFD model verification in Part 5.) looks initially at the full Turgo system in order to compare the absolute difference between the numerical efficiency and the experimental efficiency of the original Turgo runner at the best efficiency point. The mechanical losses of the test rig are estimated to determine the experimental hydraulic efficiency. The numerical hydraulic efficiency is then determined by calculating the losses upstream of the injector, using standard pipe flow equations and combing these with the losses through the injector, as well as the numerical efficiency of the runner by simulating the runner using the ‘real jet’ profile produced by the full injector simulations. The results showed the numerical model to be over-predicting the efficiency by 1.26%. The numerical difference in the performance of the two injectors is then compared to the experimental difference measured during testing. This is done by importing the ‘real jet’ profiles produced by the full 3D injector simulations into the LE1 runner simulation. This allows the difference in total efficiency between the injectors combined with the runner to be compared to the experimental differences which also includes the impact of the jet on the runner performance. The comparison between the injectors is less accurate as more uncertainties are introduced when combining these models and the differences are smaller however the CFD was able to predict the improvements to within 0.4%. Finally, the numerical differences between the runner designs and the experimental differences are compared showing that the runner model is able to predict differences in hydraulic efficiency to within 0.1%. This accuracy is largely down to that fact that many of the systematic experimental and modelling errors are cancelled out when comparing only the runners. The CFD model verification has shown that although the absolute performance of the Turgo system can be modelled numerically to within a good degree of accuracy, it requires combining injector and runner models as well as estimating additional losses in the pipework which can prove time consuming. However for design comparison and optimisations the CFD models have been shown to be far more accurate suggesting that this is where these numerical models are most useful

Aeronautical engineering: A continuing bibliography with indexes (supplement 295)

This bibliography lists 581 reports, articles, and other documents introduced into the NASA Scientific and Technical Information System in Sep. 1993. Subject coverage includes: design, construction and testing of aircraft and aircraft engines; aircraft components, equipment, and systems; ground support systems; and theoretical and applied aspects of aerodynamics and general fluid dynamics

Aeronautical engineering: A continuing bibliography with indexes (supplement 296)

This bibliography lists 592 reports, articles, and other documents introduced into the NASA scientific and technical information system in Oct. 1993. Subject coverage includes: design, construction and testing of aircraft and aircraft engines; aircraft components, equipment, and systems; ground support systems; and theoretical and applied aspects of aerodynamics and general fluid dynamics