Numerical analysis of conservative unstructured discretisations for low Mach flows

This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. https://authorservices.wiley.com/author-resources/Journal-Authors/licensing-and-open-access/open-access/self-archiving.htmlUnstructured meshes allow easily representing complex geometries and to refine in regions of interest without adding control volumes in unnecessary regions. However, numerical schemes used on unstructured grids have to be properly defined in order to minimise numerical errors. An assessment of a low-Mach algorithm for laminar and turbulent flows on unstructured meshes using collocated and staggered formulations is presented. For staggered formulations using cell centred velocity reconstructions the standard first-order method is shown to be inaccurate in low Mach flows on unstructured grids. A recently proposed least squares procedure for incompressible flows is extended to the low Mach regime and shown to significantly improve the behaviour of the algorithm. Regarding collocated discretisations, the odd-even pressure decoupling is handled through a kinetic energy conserving flux interpolation scheme. This approach is shown to efficiently handle variable-density flows. Besides, different face interpolations schemes for unstructured meshes are analysed. A kinetic energy preserving scheme is applied to the momentum equations, namely the Symmetry-Preserving (SP) scheme. Furthermore, a new approach to define the far-neighbouring nodes of the QUICK scheme is presented and analysed. The method is suitable for both structured and unstructured grids, either uniform or not. The proposed algorithm and the spatial schemes are assessed against a function reconstruction, a differentially heated cavity and a turbulent self-igniting diffusion flame. It is shown that the proposed algorithm accurately represents unsteady variable-density flows. Furthermore, the QUICK schemes shows close to second order behaviour on unstructured meshes and the SP is reliably used in all computations.Peer ReviewedPostprint (author's final draft

An Intercomparison Between Divergence-Cleaning and Staggered Mesh Formulations for Numerical Magnetohydrodynamics

In recent years, several different strategies have emerged for evolving the magnetic field in numerical MHD. Some of these methods can be classified as divergence-cleaning schemes, where one evolves the magnetic field components just like any other variable in a higher order Godunov scheme. The fact that the magnetic field is divergence-free is imposed post-facto via a divergence-cleaning step. Other schemes for evolving the magnetic field rely on a staggered mesh formulation which is inherently divergence-free. The claim has been made that the two approaches are equivalent. In this paper we cross-compare three divergence-cleaning schemes based on scalar and vector divergence-cleaning and a popular divergence-free scheme. All schemes are applied to the same stringent test problem. Several deficiencies in all the divergence-cleaning schemes become clearly apparent with the scalar divergence-cleaning schemes performing worse than the vector divergence-cleaning scheme. The vector divergence-cleaning scheme also shows some deficiencies relative to the staggered mesh divergence-free scheme. The differences can be explained by realizing that all the divergence-cleaning schemes are based on a Poisson solver which introduces a non-locality into the scheme, though other subtler points of difference are also catalogued. By using several diagnostics that are routinely used in the study of turbulence, it is shown that the differences in the schemes produce measurable differences in physical quantities that are of interest in such studies

A General, Mass-Preserving Navier-Stokes Projection Method

The conservation of mass is common issue with multiphase fluid simulations. In this work a novel projection method is presented which conserves mass both locally and globally. The fluid pressure is augmented with a time-varying component which accounts for any global mass change. The resulting system of equations is solved using an efficient Schur-complement method. Using the proposed method four numerical examples are performed: the evolution of a static bubble, the rise of a bubble, the breakup of a thin fluid thread, and the extension of a droplet in shear flow. The method is capable of conserving the mass even in situations with morphological changes such as droplet breakup.Comment: Submitted to Computer Physics Communication

Development of an implicit framework for the two-fluid model on unstructured grids

The two-fluid model is an efficient method for simulating multiphase flows, based on an averaged description of the phases as interpenetrating and interacting continua. It is particularly attractive for the simulation of dispersed gas-solid flows in which the large number of particles in practical devices can impose an insurmountable computational burden for particle tracking methods, given currently available computing resources. Whilst the two-fluid model is more efficient than particle tracking methods, it results in large, strongly coupled and highly non-linear systems of equations, placing a premium on efficient solution algorithms. Additionally, the constitutive models used to describe the solid phase introduce stiff source terms, requiring a robust solution algorithm to handle them. In this thesis a fully-coupled algorithm is developed for the two-fluid model, based on a Newton linearisation of the underlying equation system, resulting in an algorithm treating all inter-equation couplings implicitly. For comparison, a semi-coupled algorithm, based on a Picard linearisation of the two-fluid model is also implemented, yielding a smaller implicitly coupled pressure-velocity system and a segregated system for the transport of phase concentrations. Motivating this work is the highly non-linear nature of the two-fluid model and the stiff source terms arising in the models of the dispersed phase, these are treated explicitly in the semi-coupled algorithm and may impose stability limits on the algorithm. By treating these terms implicitly, it is expected that the fully-coupled solution algorithm will be more robust. The algorithms are compared by application to test cases ranging from academic problems to problems representative of industrial applications of the two-fluid model. These comparisons show that with increasing problem complexity, the robustness of the fully-coupled algorithm leads to an overall more efficient solution than the semi-coupled algorithm.Open Acces

Numerical simulation of unsteady flow in hydraulic turbomachines

Turbines and pumps dealing with incompressible flow are examples of hydraulic turbomachines. In most cases the flow is highly turbulent and time-dependent, caused by the rotation of the impeller in a stationary casing. The geometry, with doubly curved surfaces, adds even more to the complexity. It all leads to a flow which is difficult to model. Yet, to optimize turbomachines it is necessary to analyze the flow in detail. Flow simulations using Computational Fluid Dynamics (CFD) can be a very helpful tool. The software solves the discretized partial differential equations for mass and momentum conservation on a grid that covers the flow domain. Two basic discretization schemes can be distinguished: collocated and staggered. When a collocated scheme is used, the solution suffers from odd-even decoupling. In practice this is suppressed with artificial measures which either decrease the accuracy of the simulation or increase the calculation time for an unsteady incompressible flow. Using a staggered scheme, accurate discretization is more difficult, but odd-even decoupling is avoided. In this thesis a CFD code is developed which is based on a staggered, blockstructured grid scheme. It is suited for the calculation of time-dependent fluid motion in turbomachines. The CFD code, named DEFT, is originally developed by the group ofWesseling at Delft University of Technology. The first extension in the current work was an interpolation procedure implemented to handle non-matching grids for more flexibility in grid generation. Furthermore, a sliding interface to connect the rotating grid in the impeller and the stationary grid was developed. Coriolis and centrifugal forces for calculations in the rotating frame of reference, were mplemented in two ways: using a conservative formulation and using source terms. An adaptation of the pressure equation proved necessary to reduce calculation time for computations involving a sliding interface. Although the conceptual ideas behind these extensions are applicable in 3D, they have been implemented in 2D and verified with the simulation of a number of relatively simple flows. DeFT was validated with the simulation of the flow through a cascade of blades which is a model of an axial-flow pump. The blade surface pressure and the total force on the blade are calculated. There is good agreement between values calculated with DeFT, Fluent, values from experiments, and other CFD calculations obtained from literature. The flow through a centrifugal pump with a vaned diffusor is simulated using the staggered discretization in DeFT and the collocated discretization in Fluent. The calculated time-averaged pressure and velocity along the pitch of a rotor channel show good correspondence. The agreement with results from experiments and other CFD calculations obtained from literature is more qualitative. The calculation time needed by DeFT and Fluent is approximately equal, despite the use of a large number of blocks in DeFT and its lack of a convergence enhancing multi-grid method which is used by Fluent

The finite-volume method in computational rheology

The finite volume method (FVM) is widely used in traditional computational fluid dynamics (CFD), and many commercial CFD codes are based on this technique which is typically less demanding in computational resources than finite element methods (FEM). However, for historical reasons, a large number of Computational Rheology codes are based on FEM. There is no clear reason why the FVM should not be as successful as finite element based techniques in Computational Rheology and its applications, such as polymer processing or, more recently, microfluidic systems using complex fluids. This chapter describes the major advances on this topic since its inception in the early 1990’s, and is organized as follows. In the next section, a review of the major contributions to computational rheology using finite volume techniques is carried out, followed by a detailed explanation of the methodology developed by the authors. This section includes recent developments and methodologies related to the description of the viscoelastic constitutive equations used to alleviate the high-Weissenberg number problem, such as the log-conformation formulation and the recent kernel-conformation technique. At the end, results of numerical calculations are presented for the well-known benchmark flow in a 4:1 planar contraction to ascertain the quality of the predictions by this method

A New Computational Fluid Dynamics Code I: Fyris Alpha

A new hydrodynamics code aimed at astrophysical applications has been developed. The new code and algorithms are presented along with a comprehensive suite of test problems in one, two, and three dimensions. The new code is shown to be robust and accurate, equalling or improving upon a set of comparison codes. Fyris Alpha will be made freely available to the scientific community.Comment: 59 pages, 27 figures For associated code see http://www.mso.anu.edu.au/fyri

Drukcorrectiealgoritmen voor willekeurige fluida bij lage snelheden, toegepast op simulaties van niet-voorgemengde vlammen

Dit doctoraatsonderzoek situeert zich binnen het domein van de numerieke stromingsmechanica. Deze wetenschap, die tot doel heeft de stroming van een vloeistof of een gas te simuleren aan de hand van computerberekeningen, wordt alsmaar belangrijker in de ontwerpfase van hedendaagse systemen waar reagerende stromingen deel van uitmaken. Numerieke simulaties worden dan ook veelvuldig gebruikt bij ontwerp en optimalisatie van bijvoorbeeld industriële branders. Een goede voorspelling van de ingewikkelde processen die zich voordoen in dergelijke systemen is mogelijk indien men beschikt over nauwkeurige modellen en geavanceerde numerieke methoden. Helaas zijn deze technieken enkel in staat kwantitatieve voorspellingen te maken als de onderliggende algoritmen geschikt zijn voor tijdsnauwkeurige simulaties van reagerende stromingen. Frequent gebruikte algoritmen blijken aanleiding te geven tot onstabiele oplossingen als deze toegepast worden op stromingen met sterk variabele dichtheid, zoals in verbrandingprocessen. Andere gangbare algoritmen blijken wel stabiel, maar voorspellen oplossingen die fysisch niet mogelijk zijn. Omwille van deze tekortkomingen, wordt in dit doctoraat een algoritme ontwikkeld, die de goede eigenschappen van beide klassen bundelt: het is stabiel en voorspelt fysisch correcte oplossingen. Uiteindelijk draagt dit doctoraatswerk bij tot betere numerieke simulaties, en dus tot de ontwikkeling van branders met een hoger rendement en verminderde uitstoot van schadelijke stoffen