Transient removal of contaminant from a channel with differentially heated wall of cavity

Cleaning accumulated deposits inside pipe cavity are by disassembling and cleaning it part by part. Hydrodynamic cleaning of the cavity is an alternative method to clean accumulated deposits or contaminants inside the pipe cavity instead of dissembling them part by part is a tedious process or using a solvent which is not suitable in the food processing industry. This study aims to investigate the contaminants removal process from a cavity by resorting to natural flow to clean the deposits in different cavity sizes and includes different heating locations with different flow configurations. An experimental method is used to visualize the flow behaviour inside the cavity of a channel at a large aspect ratio in isothermal conditions. These results are used to validate numerical results obtained in isothermal flow conditions. For numerical study, Constrained Interpolated Profile (CIP) method is used for the advection phase of momentum and energy equation, and central difference is used to solve the non-advection phase of momentum and energy equations. The numerical studies include different aspect ratios (AR), 1 to 4, various Reynolds numbers (Re), 50 to 1000, and different locations of the heated wall inside the cavity (left wall, bottom wall, & right wall) for three different Grashof numbers (Gr), 1000, 10 000, and 100 000. The particles removal percentage at the transient and steady states are then compared and discussed. A larger aspect ratio and a more significant Reynolds number for isothermal conditions will give a higher percentage of contaminants removal except for AR = 4 and Re = 50. This particular flow shows a higher percentage of contaminant removal than AR = 4; Re = 100, 200, and 400. For mixed convection flow, one typical result can be concluded: at small Gr, the contaminant removal percentage is not changing significantly for all different heated wall positions. It is also shown that a more significant aspect ratio will produce a better contaminant removal process, and a higher Grashof number will improve the contaminant removal process. It is also found that when Gr equals 1000 and 10000, there is no significant change in the contaminant removal process and constant heat flux from the bottom wall for Gr = 100,000 gives the highest contaminant removal percentage for every aspect ratio. The highest percentage removal of contaminant is 98.94% for Gr =100 000, AR=4

A semi implicit, semi-Lagrangian p-adaptive discontinuous Galerkin method for the rotating shallow water equations: analysis and numerical experiments

2009/2010Questa tesi ha come obiettivo la scrittura e l'analisi di un nuovo metodo semi-implicito semi-Lagrangiano discontinuous Galerkin (SISLDG nel seguito) per la risoluzione numerica delle equazioni delle acque basse, inteso come primo passo nella direzione di un progetto piu' ambizioso riguardante lo sviluppo di un dynamical-core non idrostatico di nuova generazione per la modellistica atmosferica regionale. In particolare il codice prodotto dovrebbe servire a migliorare la discretizzazione attualmente impiegata nel modello RegCM ( F. Giorgi, J. Climate, 1990). Le equazioni delle acque basse di fatto contengono tutti gli operatori orizzontali presenti in un modello atmosferico tridimensionale, pertanto rappresentano tipicamente il primo test necessario per ogni nuovo schema numerico pensato per applicazioni atmosferiche. Le tecniche proposte nella tesi sono non-standard nel contesto dei metodi Discontinuous Galerkin (DG) per problemi dipendenti dal tempo. Effettivamente dynamical-cores basati su DG sono molto promettenti per la loro accuratezza e flessibilità, tuttavia nella loro applicabilità alla risoluzione numerica di problemi di fluidodinamica a bassi numeri di Froude/Mach essi presentano un punto critico, costituito dalla limitazione imposta dalla stabilità sul massimo time step utilizzabile in calcoli di interesse pratico. Per esempio, in schemi di tipo Runge-Kutta-DG, la stabilità è stata dimostrata in (Cockburn e Shu, Math. Comp., 1991) purchè sia soddisfatta la seguente condizione CFL $$|c|\frac{\Delta t}{h} < \frac{1}{2k+1},$$ dove $k$ è il grado polinomiale e $c$ è la celerità delle onde che si propagano piu' velocemente. Pertanto, al fine di evitare che la scelta del massimo time step ammissibile fosse governata da considerazioni di stabilità anzichè di accuratezza, specialmente pensando a metodi di ordine elevato, si è deciso di non seguire la maniera standard di applicare DG a problemi dipendenti dal tempo come proposto, per esempio, in molti articoli di Cockburn e Shu. Piuttosto è stato seguito un approccio già sperimentato con successo nel contesto delle differenze finite (per esempio in Robert, J. Meteor. Soc. Japan, 1982, o Casulli J. Comput. Phys., 1990) e degli elementi finiti ( per esempio in Staniforth e Temperton, Mon. Weather Rev., 1986, o in Miglio, Quarteroni e Saleri, Comput. Meth. Appl. Mech. Eng., 1999, o in Giraldo Q. J. R. Meteorol. Soc. 05) ma, non ancora pienamente esplorato nell'ambito DG. L'idea innovativa consiste nell'associare la semidiscretizzazione DG in spazio con la combinazione di una tecnica semi-implicita (SI) e una semi-Lagrangiana (SL) in tempo. Questo approccio e' stato suggerito dai risultati incoraggianti ottenuti da altri autori combinando DG separatamente con la tecnica semi-implicita (per esempio Restelli, Giraldo, SIAM J. Sci. Comput. 2009 ) oppure con quella semi-Lagrangiana (e.g. Restelli, Bonaventura, Sacco, J. Comput. Phys., 2006 ). I principali risultati originali di questo lavoro di tesi possono essere così riassunti: \begin{itemize} \item gli effetti sulla stabilità della soluzione approssimata di differenti scelte per gli spazi di velocità e pressione sono stati esaminati attraverso esperimenti numerici, dai quali è emerso che coppie velocità-pressione di grado diverso $Q_k - Q_{k-1}$ (rispettivamente) funzionano meglio di coppie velocità-pressione dello stesso grado, per le quali si sviluppano delle evidenti instabilità. Del resto benefici sulla stabilità derivanti dall'uso di spazi DG di ordine misto per velocità e pressione sono stati dimostrati per il poroblema di Stokes (Toselli, M3AS, 2002 and Sch\"otzau, M3AS, 2003), ed il fatto che i tipici regimi atmosferici siano caratterizzati da bassi numeri di Froude/Mach ha suggerito l'estensione della stessa strategia alle equazioni delle acque basse. \item Un semplice criterio di p-adattività è stato implementato per adattare dinamicamente il numero di gradi di libertà locali alla struttura della soluzione in ciascun singolo elemento. Ciò è stato ottenuto grazie alla flessibilità tipica delle discretizzazioni DG e grazie alla proprietà di ortogonalità delle basi di polinomi di Legendre utilizzate. Come dimostrato negli esperimenti numerici in una e due dimensioni la strategia di p-adattività utilizzata è abbastanza efficiente nel ridurre il costo computazionele, rivelandosi nel contempo sufficientemente semplice e robusta per poter essere applicata con successo anche in modelli climatici completi dove le numerose parametrizzazioni fisiche presenti nei termini di sorgente rendono difficoltoso eseguire rigorose analisi a posteriore dell'errore. \item La scelta di spazi per velocità e pressione con buone proprietà di stabilità ha reso possibile l'utilizzo di flussi numerici centrati per l'approssimazione delle tracce della soluzione sui bordi tra gli elementi, (come in Bassi e Rebay, J. Comput. Phys., 1997 ) termini questi che nascono dalla proiezione $L^2$ delle equazioni sullo spazio delle funzioni test (scelte uguali alle funzioni di base come nel Direct Characteristic Galerkin scheme di Morton, Priestley, S\"uli, ESAIM Math. Model. Numer. Anal., 1988 ) e dalla successiva integrazione per parti ove necessario. Inoltre la dimensione del problema discreto è stata ulteriormentre ridotta ricavando dalle equazioni della quantità di moto le componenti discrete della velocità in termini dell'elevazione discreta della superficie libera e sostituendo nell'equazione di continuità le espressioni risultanti, ottenendo così una singola equazione di Helmoltz discreta nella sola elevazione della superficie libera, che assume la forma di un sistema lineare nonsimmetrico, ma sparso che è stato risolto con uno schema GMRES. \item Per sfruttare a pieno le potenzialità dell'approccio semi-Lagrangiano usato, il solutore SISLDG proposto per le equazioni delle acque basse è stato associato a uno schema di advezione in forma di flusso per il trasporto di traccianti passivi (che è l'estensione dello schema di M. Restelli, L. Bonaventura, R. Sacco, J. Comput. Phys., 2006) di cui sono state esaminate le proprietà di preservazione delle costanti e di compatibilità con l'equazione di continuità . Il trattamento p-adattivo è stato esteso in maniera indipendente a ogni tracciante passivo. Di conseguenza variazioni del numero di gradi di libertà locali sono completamente indipendenti per ogni specie, permettendo così un raffinamento selettivo per talune variabili di interesse, lasciando inalterato il costo computazionale per le altre. \item Infine l'algoritmo proposto è stato implementato in un codice modulare Fortran95. L'implementazione monodimensionale è stata usata come modello per quella bidimensionale su mesh cartesiane. Il codice è stato usato per eseguire un certo numero di test per analizzare le proprietà di accuratezza e stabilità del nuovo metodo SISLDG. Risultati numerici ottenuti nell'ambito di casi test monodimensionali dimostrano che il metodo proposto cattura in maniera accurata e efficiente le principali caratteristiche delle onde di gravità lineari e inerziali e nonchè delle correnti in canali aperti con batimetria non costante e anche della soluzione di rarefazione del problema di Riemann. L'efficienza del metodo SISLDG è inoltre dimostrata dai risultati ottenuti ad alto numero di Courant e con scelta automatica dei gradi di approssimazione locale. Risultati numerici nell'ambito di casi test bidimensionali mostrano l'efficacia della stategia di p-adattività utilizzata anche nel caso di correnti non semplici (corrente di Smolarkiewicz) e l'accuratezza nella riproduzione di onde di gravità. \end{itemize}XXIII Cicl

Optimal Interpolation of Submerged River Bed for Laser Scanning Survey Using a Two-Dimensional Numerical Model

Source: ICHE Conference Archive - https://mdi-de.baw.de/icheArchiv

Multi-moment advection schemes for Cartesian grids and cut cells

Computational fluid dynamics has progressed to the point where it is now possible to simulate flows with large eddy turbulence, free surfaces and other complex features. However, the success of these models often depends on the accuracy of the advection scheme supporting them. Two such schemes are the constrained interpolation profile method (CIP) and the interpolated differential operator method (IDO). They share the same space discretisation but differ in their respectively semi-Lagrangian and Eulerian formulations. They both belong to a family of high-order, compact methods referred to as the multi-moment methods. In the absence of sufficient information in the literature, this thesis begins by taxonomising various multi-moment space discretisations and appraising their linear advective properties. In one dimension it is found that the CIP/IDO with order (2N -1) has an identical spectrum and memory cost to the Nth order discontinuous Galerkin method. Tests confirm that convergence rates are consistent with nominal orders of accuracy, suggesting that CIP/IDO is a better choice for smooth propagation problems. In two dimensions, six Cartesian multi-moment schemes of third order are compared using both spectral analysis and time-domain testing. Three of these schemes economise on the number of moments that need to be stored, with one CIP/IDO variant showing improved isotropy, another failing to maintain its nominal order of accuracy, and one of the conservative variants having eigenvalues with positive real parts: it is stable only in a semi-Lagrangian formulation. These findings should help researchers who are interested in using multi-moment schemes in their solvers but are unsure as to which are suitable. The thesis then addresses the question as to whether a multi-moment method could be implemented on a Cartesian cut cell grid. Such grids are attractive for supporting arbitrary, possibly moving boundaries with minimal grid regeneration. A pair of novel conservative fourth order schemes is proposed. The first scheme, occupying the Cartesian interior, has unprecedented low memory cost and is proven to be conditionally stable. The second, occupying the cut cells, involves a profile reconstruction that is guaranteed to be well-behaved for any shape of cell. However, analysis of the second scheme in a simple grid arrangement reveals positive real parts, so it is not stable in an Eulerian formulation. Stability in a hybrid formulation remains open to question

Integrated 2D-3D free surface hydro-environmental modelling

An integrated horizontally two- and fully three-dimensional numerical model system has been developed based on a combined unstructured and σ-coordinate grid to simulate the flow and water quality process in large water bodies with a focus on the three dimensional behaviours at specific areas. The model is based on the time dependent Reynolds-Averaged Navier-Stokes equations with a non-hydrostatic pressure distribution and a baroclinic force being incorporated in the three dimensional (3D) model. The two sub models interact dynamically during the solution procedure with no time-step restriction due to integration. The main idea is to use a fractional step algorithm for each model and then integrate the two models fraction by fraction. Hybrid 2D-3D finite volume cells have been introduced for the link nodes which are partly in the 2D domain and partly in the 3D domain. Thus an interpolation/averaging procedure at the interface and domain overlapping is no longer needed. The 3D model uses the projection method for pressure calculation. The advection equation is solved by the semi-Lagrangian method. Other components are solved via the finite element - finite volume (FV) method. The water surface is determined implicitly through a global matrix equation created by assembling the domain's matrices. The cell integrals are calculated analytically to eliminate a common source of numerical diffusion due to the use of approximation techniques for the FV integrals. The horizontal gradients of the density and shear stresses are calculated on true horizontal planes, in order to avoid artificial velocity and diffusion in highly stratified flows. Neumann interpolation elements with virtual nodes have been introduced at Neumann type of boundaries for more accuracy. The integrated model has been verified using analytical solutions and benchmark test cases, including the Ekman velocity distribution, wind driven circulation, lock exchange and integrated 2D-3D flows in basin. The results show the model is capable of the model for accurate simulation and implicit 2D-3D integration. Keywords: integrated modelling, hydrodynamic numerical model, non-hydrostatic, unstructured mesh, hybrid finite element finite volume method.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Spectral Analysis of Continuous FEM for Hyperbolic PDEs: Influence of Approximation, Stabilization, and Time-Stepping

International audienceWe study continuous finite element dicretizations for one dimensional hyperbolic partial differential equations. The main contribution of the paper is to provide a fully discrete spectral analysis, which is used to suggest optimal values of the CFL number and of the stabilization parameters involved in different types of stabilization operators. In particular, we analyze the streamline-upwind Petrov-Galerkin (SUPG) stabilization technique, the continuous interior penalty (CIP) stabilization method and the local projection stabilization (LPS). Three different choices for the continuous finite element space are compared: Bernstein polynomials, Lagrangian polynomials on equispaced nodes, and Lagrangian polynomials on Gauss-Lobatto cubature nodes. For the last choice, we only consider inexact quadrature based on the formulas corresponding to the degrees of freedom of the element, which allows to obtain a fully diagonal mass matrix. We also compare different time stepping strategies, namely Runge-Kutta (RK), strong stability preserving RK (SSPRK) and deferred correction time integration methods. The latter allows to alleviate the computational cost as the mass matrix inversion is replaced by the high order correction iterations. To understand the effects of these choices, both time-continuous and fully discrete Fourier analysis are performed. These allow to compare all the different combinations in terms of accuracy and stability, as well as to provide suggestions for optimal values discretization parameters involved. The results are thoroughly verified numerically both on linear and non-linear problems, and error-CPU time curves are provided. Our final conclusions suggest that cubature elements combined with SSPRK and CIP or LPS stabilization are the most promising combinations

A numerical study of tsunami wave impact and run-up on coastal cliffs using a CIP-based model

There is a general lack of understanding of tsunami wave interaction with complex geographies, especially the process of inundation. Numerical simulations are performed to understand the effects of several factors on tsunami wave impact and run-up in the presence of gentle submarine slopes and coastal cliffs, using an in-house code, a constrained interpolation profile (CIP)-based model. The model employs a high-order finite difference method, the CIP method, as the flow solver; utilizes a VOF-type method, the tangent of hyperbola for interface capturing/slope weighting (THINC/SW) scheme, to capture the free surface; and treats the solid boundary by an immersed boundary method. A series of incident waves are arranged to interact with varying coastal geographies. Numerical results are compared with experimental data and good agreement is obtained. The influences of gentle submarine slope, coastal cliff and incident wave height are discussed. It is found that the tsunami amplification factor varying with incident wave is affected by gradient of cliff slope, and the critical value is about 45°. The run-up on a toe-erosion cliff is smaller than that on a normal cliff. The run-up is also related to the length of a gentle submarine slope with a critical value of about 2.292 m in the present model for most cases. The impact pressure on the cliff is extremely large and concentrated, and the backflow effect is non-negligible. Results of our work are highly precise and helpful in inverting tsunami source and forecasting disaster

Integrated 2D-3D free surface hydro-environmental modelling

An integrated horizontally two- and fully three-dimensional numerical model system has been developed based on a combined unstructured and σ-coordinate grid to simulate the flow and water quality process in large water bodies with a focus on the three dimensional behaviours at specific areas. The model is based on the time dependent Reynolds-Averaged Navier-Stokes equations with a non-hydrostatic pressure distribution and a baroclinic force being incorporated in the three dimensional (3D) model. The two sub models interact dynamically during the solution procedure with no time-step restriction due to integration. The main idea is to use a fractional step algorithm for each model and then integrate the two models fraction by fraction. Hybrid 2D-3D finite volume cells have been introduced for the link nodes which are partly in the 2D domain and partly in the 3D domain. Thus an interpolation/averaging procedure at the interface and domain overlapping is no longer needed. The 3D model uses the projection method for pressure calculation. The advection equation is solved by the semi-Lagrangian method. Other components are solved via the finite element - finite volume (FV) method. The water surface is determined implicitly through a global matrix equation created by assembling the domain's matrices. The cell integrals are calculated analytically to eliminate a common source of numerical diffusion due to the use of approximation techniques for the FV integrals. The horizontal gradients of the density and shear stresses are calculated on true horizontal planes, in order to avoid artificial velocity and diffusion in highly stratified flows. Neumann interpolation elements with virtual nodes have been introduced at Neumann type of boundaries for more accuracy. The integrated model has been verified using analytical solutions and benchmark test cases, including the Ekman velocity distribution, wind driven circulation, lock exchange and integrated 2D-3D flows in basin. The results show the model is capable of the model for accurate simulation and implicit 2D-3D integration. Keywords: integrated modelling, hydrodynamic numerical model, non-hydrostatic, unstructured mesh, hybrid finite element finite volume method

Towards a level set reinitialisation method for unstructured grids

Interface tracking methods for segregated flows such as breaking ocean waves are an important tool in marine engineering. With the development in marine renewable devices increasing and a multitude of other marine flow problems that benefit from the possibility of simulation on computer, the need for accurate free surface solvers capable of solving wave simulations has never been greater. An important component of successfully simulating segregated flow of any type is accurately tracking the position of the separating interface between fluids. It is desirable to represent the interface as a sharp, smooth, continuous entity in simulations. Popular Eulerian interface tracking methods appropriate for segregated flows such as the Marker and Cell Method (MAC) and the Volume of Fluid (VOF) were considered. However these methods have drawbacks with smearing of the interface and high computational costs in 3D simulations being among the most prevalent. This PhD project uses a level set method to implicitly represent an interface. The level set method is a signed distance function capable of both sharp and smooth representations of a free surface. It was found, over time, that the level set function ceases to represent a signed distance due to interaction of local velocity fields. This affects the accuracy to which the level set can represent a fluid interface, leading to mass loss. An advection solver, the Cubic Interpolated Polynomial (CIP) method, is presented and tested for its ability to transport a level set interface around a numerical domain in 2D. An advection problem of the level set function demonstrates the mass loss that can befall the method. To combat this, a process known as reinitialisation can be used to re-distance the level set function between time-steps, maintaining better accuracy. The goal of this PhD project is to present a new numerical gradient approximation that allows for the extension of the reinitialisation method to unstructured numerical grids. A particular focus is the Cartesian cut cell grid method. It allows geometric boundaries of arbitrary complexity to be cut from a regular Cartesian grid, allowing for flexible high quality grid generation with low computational cost. A reinitialisation routine using 1st order gradient approximation is implemented and demonstrated with 1D and 2D test problems. An additional area-conserving constraint is introduced to improve accuracy further. From the results, 1st order gradient approximation is shown to be inadequate for improving the accuracy of the level set method. To obtain higher accuracy and the potential for use on unstructured grids a novel gradient approximation based on a slope limited least squares method, suitable for level set reinitialisation, is developed. The new gradient scheme shows a significant improvement in accuracy when compared with level set reinitialisation methods using a lower order gradient approximation on a structured grid. A short study is conducted to find the optimal parameters for running 2D level set interface tracking and the new reinitialisation method. The details of the steps required to implement the current method on a Cartesian cut cell grid are discussed. Finally, suggestions for future work using the methods demonstrated in the thesis are presented

Use of Cloud Radar Doppler Spectra to Evaluate Stratocumulus Drizzle Size Distributions in Large-Eddy Simulations with Size-Resolved Microphysics

A case study of persistent stratocumulus over the Azores is simulated using two independent large-eddy simulation (LES) models with bin microphysics, and forward-simulated cloud radar Doppler moments and spectra are compared with observations. Neither model is able to reproduce the monotonic increase of downward mean Doppler velocity with increasing reflectivity that is observed under a variety of conditions, but for differing reasons. To a varying degree, both models also exhibit a tendency to produce too many of the largest droplets, leading to excessive skewness in Doppler velocity distributions, especially below cloud base. Excessive skewness appears to be associated with an insufficiently sharp reduction in droplet number concentration at diameters larger than ~200 μm, where a pronounced shoulder is found for in situ observations and a sharp reduction in reflectivity size distribution is associated with relatively narrow observed Doppler spectra. Effectively using LES with bin microphysics to study drizzle formation and evolution in cloud Doppler radar data evidently requires reducing numerical diffusivity in the treatment of the stochastic collection equation; if that is accomplished sufficiently to reproduce typical spectra, progress toward understanding drizzle processes is likely