Littérature scientifique sur le sujet « DNS code in cylindrical coordinates »

The Elastodynamic Finite Integration Technique (EFIT), originally developed by Fellinger et al.,1–3 represents a stable and efficient numerical code to model elastic wave propagation in linearly-elastic isotropic and anisotropic, homogeneous and heterogeneous as well as dissipative and nondissipative media. In previous works, the FIT discretization of the basic equations of linear elasticity, Hooke's law and Cauchy's equation of motion, was exclusively carried out in Cartesian coordinates. For problems in cylindrical geometries it is more suitable to use cylindrical coordinates. By that, axisymmetric problems can be treated in a two-dimensional staggered grid in the r,z-plane. The paper presents an EFIT version for axisymmetric problems in cylindrical coordinates called Cylindrical EFIT (CEFIT). After demonstrating the accuracy of the numerical code by a comparison between simulation results and analytical solutions, different examples of application are given. These examples include modeling of sound fields of ultrasonic transducers, thermoelastic laser sources, geophysical borehole probes, impact-echo measurements in layered media, and load simulations of the European Spallation Source (ESS) mercury target.

Purpose This paper aims to present the development of a highly parallel finite-difference computational fluid dynamics code in generalized curvilinear coordinates system. The objectives are to handle internal and external flows in fairly complex geometries including shock waves, compressible turbulence and heat transfer. Design/methodology/approach The code is equipped with high-order discretization schemes to improve the computational accuracy of the solution algorithm. Besides, a new method to deal with the geometrical singularities, so-called domain decomposition method (DDM), is implemented. The DDM consists of using two different meshes communicating with each other, where the base mesh is Cartesian and the overlapped one a hollow cylinder. Findings The robustness of the present implemented code is appraised through several numerical test cases including a vortex advection, supersonic compressible flow over a cylinder, Poiseuille flow, turbulent channel and pipe flows. The results obtained here are in an excellent agreement when compared to the experimental data and the previous direct numerical simulation (DNS). As for the DDM strategy, it was successful as simulation time is clearly decreased and the connection between the two subdomains does not create spurious oscillations. Originality/value In sum, the developed solver was capable of solving, accurately and with high-precision, two- and three-dimensional compressible flows including fairly complex geometries. It is noted that the data provided by the DNS of supersonic pipe flows are not abundant in the literature and therefore will be available online for the community.

Abstract Finite elements developed in cylindrical coordinates are presented for three-dimensional analysis of tires. In contrast to elements formulated in Cartesian coordinates, these elements allow the exact representation of circular shapes. The exact modeling of circular geometries can provide better finite element predictions and reduce the number of elements needed around the tire circumference. Numerical results are presented for the application of this formulation to the analysis of a radial automobile tire subjected to rim mounting, nonconservative inflation pressure, and rigid pavement contact. The predictions of the foregoing finite elements are compared to experimental data and to predictions of a commercial code using finite elements developed in Cartesian coordinates. The comparisons demonstrate the accuracy and the advantages of the cylindrical coordinate formulation for the three-dimensional finite element analysis of tires.

Heat equation is a partial differential equation used to describe the temperature distribution in a heat-conducting body. The implementation of a numerical solution method for heat equation can vary with the geometry of the body. In this study, a three-dimensional transient heat conduction equation was solved by approximating second-order spatial derivatives by five-point central differences in cylindrical coordinates. The stability condition of the numerical method was discussed. A MATLAB code was developed to implement the numerical method. An example was provided in order to demonstrate the method. The numerical solution by the method was in a good agreement with the exact solution for the example considered. The accuracy of the five-point central difference method was compared with that of the three-point central difference method in solving the heat equation in cylindrical coordinates. The solutions obtained by the numerical method in cylindrical coordinates were displayed in the Cartesian coordinate system graphically. The method requires relatively very small time steps for a given mesh spacing to avoid computational instability. The result of this study can provide insights to use appropriate coordinates and more accurate computational methods in solving physical problems described by partial differential equations.

The present research paper concerns a numerical and experimental analysis of the tribologic and rheological constitutive behavior influence on prediction of metallic material forging fibering. Numerical analysis using finite element Forge® code and Abaqus software show the high importance of the friction law formulation and of the material rheological softening on the fibers morphology and on their position coordinates. Calibration and sensitivity of friction law together with the numerical sensitivity of the softening term corresponding to a Hansel-Spittel rheological equation have been studied for a cylindrical crushing test of a 16MnCr5 steel.

A numerical approach is devised to calculate the shear Alfvén continuum inside and outside magnetic islands in cylindrical and stellarator plasmas. Equations for an appropriate set of coordinates and the arising equations for the continuum are derived and implemented in the CONTI code. An experiment-oriented representation of the results is chosen to allow a radial localization of the modes and a comparison of different magnetic configurations. Comparison is made with results of earlier analytic work for validation. Agreement is good but more details of the spectrum, such as the generation of island induced gaps inside and outside the separatrix, are found. While the code is easily usable and can be applied to any magnetic equilibrium accessible with VMEC, the calculations are plagued with convergence issues close to the separatrix. A calculation for a realistic W7-X equilibrium with islands is done where the island width is estimated with the HINT code.

The smooth particle hydrodynamic (SPH) method has been extended for application to large deformation problems such as high velocity impacts by including the effect of material strength. This paper presents a simple modification of the kernel function that allows the SPH formulation in Cartesian coordinates to be migrated into an axisymmetric formulation. The proposed procedure is first applied to analyse transient deformations of a cylindrical rod impacting a rigid wall (Taylor impact test). A good agreement with published experimental data for the deformed shape is obtained. A sensitivity study of the key parameters required in the SPH formulation is conducted to provide better insight into the SPH modelling approach. Impacts between two bodies at high speed have also been simulated using an axisymmetric SPH code.

The RMF (Rotating Magnetic Field) code is designed to calculate the motion of a charged particle in a given electromagnetic field. It integrates Hamilton’s equations in cylindrical coordinates using an adaptive predictor-corrector double-precision variable-coefficient ordinary differential equation solver for speed and accuracy. RMF has multiple capabilities for the field. Particle motion is initialized by specifying the position and velocity vectors. The six-dimensional state vector and derived quantities are saved as functions of time. A post-processing graphics code, XDRAW, is used on the stored output to plot up to 12 windows of any two quantities using different colors to denote successive time intervals. Multiple cases of RMF may be run in parallel and perform data mining on the results. Recent features are a synthetic diagnostic for simulating the observations of charge-exchange-neutral energy distributions and RF grids to explore a Fermi acceleration parallel to static magnetic fields.

Turbulent flow inside the seal chamber of a pump operating at high Reynolds number is investigated. The K−ε turbulence model posed in cylindrical coordinates was applied for this purpose. Simulations are performed using the fractional approach method. The results of the computer code are verified by using the FLUENT and by comparing to published results for turbulent Taylor Couette flow. Numerical results of four cases including two rotational speeds with four flush rates are reported. Significant difference between the laminar and the turbulence flow in the seal chamber is predicted. The behavior of the turbulent flows with very high Reynolds number was also investigated. The physical and practical implications of the results are discussed.

2007/2008
In this thesis we present the results of an extensive campaign of direct numerical simulations of Rayleigh-B\'enard convection at high Prandtl numbers ($10^{-1}\leq Pr \leq 10^4$) and moderate Rayleigh numbers ($10^{5}\leq Pr \leq 10^9$). The computational domain is a cylindrical cell of aspect-ratio (diameter over cell height) $\Gamma=1/2$, with the no-slip condition imposed to the boundaries. By scaling the results, we find a $1/\sqrt{Pr}$ correction to apply to the free-fall velocity, obtaining a more appropriate representation of the large scale velocity at high $Pr$. We investigate the Nusselt and the Reynolds number dependence on $Ra$ and $Pr$, comparing the results to previous numerical and experimental work. At high $Pr$ the scaling behavior of the Nusselt number with respect to $Ra$ is generally consistent with the power-law exponent $0.309$. The Nusselt number is independent of $Pr$, even at the highest $Ra$ simulated. The Reynolds number scales as $Re\sim \sqrt{Ra}/Pr$, neglecting logarithmic corrections. We analyze the global and local features of viscous and thermal boundary layers and their scaling behavior with respect to Rayleigh and Prandtl numbers, and with respect to Reynolds and Peclet numbers. We find that the flow approaches a saturation regime when Reynolds number decreases below the critical value $Re_s\simeq 40$. The thermal boundary layer thickness turns out to increase slightly even when the Peclet number increases. We explain this behavior as a combined effect of the Peclet number and the viscous boundary layer influences. The range of $Ra$ and $Pr$ simulated contains steady, periodic and turbulent solutions. A rough estimate of the transition from steady to unsteady flow is obtained by monitoring the time-evolution of the system until it reaches stationary solutions ($Ra_U\simeq 7.5 \times 10^6$ at $Pr=10^3$). We find multiple solutions as long-term phenomena at $Ra=10^8$ and $Pr=10^3$ which, however, do not result in significantly different Nusselt number. One of these multiple solutions, even if stable for a long time interval, shows a break in the mid-plane symmetry of the temperature profile. The result is similar to that of some non-Boussinesq effects. We analyze the flow structures through the transitional phases by direct visualizations of the temperature and velocity fields. We also describe how the behavior of the flow structures changes for increasing $Pr$. A wide variety of large-scale circulations and plumes structures are found. The single-roll circulation is characteristic only of the steady and periodic solutions. For other solutions, at lower $Pr$, the mean flow generally consists of two opposite toroidal structures; at higher $Pr$, the flow is organized in multi-cell structures extending mostly in the vertical direction. At high $Pr$, plumes detach from sheet-like structures. The different large-scale-structure signatures are generally reflected in the data trends with respect to $Ra$, but not in those with respect to $Pr$. In particular, the Nusselt number is independent of $Pr$, even when the flow structures appear strongly different varying $Pr$. In order to assess the reliability of the data-set we perform a systematic analysis of the error affecting the data. Refinement grid analysis is extensively applied.
---------------------------------------------------------------------------------------- In questa tesi presentiamo i risultati di un'estensiva campagna di simulazioni numeriche dirette della convezione di Rayleigh-B\'enard ad alti numeri di Prandtl ($10^{-1}\leq Pr \leq 10^4$) e moderati numeri di Rayleigh ($10^{5}\leq Pr \leq 10^9$). Il dominio computazionale \`e una cella cilindrica di allungamento (diametro su altezza cella) $\Gamma=1/2$, con condizioni di non-slittamento ai contorni. Scalando i risultati, troviamo una correzione di $1/\sqrt{Pr}$ da applicare alla velocit\`a di caduta libera, ottenendo una rappresentazione pi\`u appropriata della velocit\`a di larga scala ad elevati $Pr$. Investighiamo la dipendenza del numero di Nusselt e del numero di Reynolds da $Ra$ e $Pr$, comparando i risultati con precedenti lavori numerici e sperimentali. Ad elevati $Pr$ il comportamento di scala del numero di Nusselt rispetto a $Ra$ \`e generalmente compatibile con l'esponente di legge di potenza $0.309$. Il numero di Nusselt \`e indipendente da $Pr$, anche per il pi\`u alto $Ra$ simulato. Il numero di Reynolds scala come $Re\sim \sqrt{Ra}/Pr$, a meno di correzioni logaritmiche. Analizziamo le caratteristiche locali e globali degli strati limite viscosi e termici, ed il loro comportamento di scala rispetto ai numeri Rayleigh e Prandtl, e rispetto ai numeri Reynolds e Peclet. Troviamo che il flusso approccia un regime di saturazione quando il numero di Reynolds scende sotto il valore critico $Re_s\simeq 40$. Lo spessore dello strato limite termico comincia a crescere leggermente anche quando in numero di Peclet aumenta. Spieghiamo questo comportamento come un effetto combinato delle influenze del numero di Peclet e dello strato limite viscoso. L'intervallo di $Ra$ e $Pr$ simulato contiene soluzioni stazionarie, periodiche e turbolente. Una stima approssimata della transizione da flusso stazionario a non stazionario \`e ottenuta monitorando l'evoluzione temporale del sistema fino al raggiungimento di soluzioni stazionarie o statisticamente stazionarie ($Ra_U\simeq 7.5 \times 10^6$ a $Pr=10^3$). Troviamo soluzioni multiple come fenomeni di lungo termine a $Ra=10^8$ e $Pr=10^3$ che, comunque, non comportano differenze significative nel numero di Nusselt. Una di queste soluzioni multiple, anche se stabile per un lungo intervallo di tempo, mostra una rottura della simmetria del profilo di temperatura rispetto al piano mediano. Il risultato \`e simile a quello di alcuni effetti di non-Boussinesq. Analizziamo le strutture del flusso nelle fasi di transizione tramite visualizzazioni dirette dei campi di velocit\`a e temperatura. Descriviamo inoltre come il comportamento delle strutture del flusso cambia al crescere di $Pr$. Un'ampia variet\`a di circolazioni di larga scala e strutture a pennacchio vengono trovate. La circolazione a singolo anello \`e caratteristica solo delle soluzioni stazionarie e periodiche. Per le altre soluzioni, a $Pr$ pi\`u bassi, il flusso medio \`e generalmente composto da due strutture toroidali opposte; a $Pr$ pi\`u alti, il flusso \`e organizzato in strutture multi-cellulari che si estendono maggiormente in direzione verticale. Ad alti $Pr$, pennacchi si staccano da strutture simili a fogli. Le impronte delle differenti strutture di larga scala si riflettono generalmente nell'andamento dei dati rispetto a $Ra$, ma non rispetto a $Pr$. In particolare, il numero di Nusselt \`e indipendente da $Pr$, anche quando le strutture del flusso appaiono molto differenti al variare di $Pr$. Per stabilire l'affidabilit\`a dell'insieme dei dati, effettuiamo un'analisi sistematica degli errori a cui i dati sono soggetti. L'analisi di raffinamento della griglia \`e largamente applicata.
XXI Ciclo
1976

Numerical simulation code is developed to study the formation of liquid drops from a nozzle into another quiescent liquid. The Navier-Stokes equations for two immiscible, incompressible, Newtonian fluids are solved on a fixed, staggered grid of cylindrical axisymmetric coordinates. Interfacial motion is captured using a Front-Tracking Method. The time variation of interfacial shape simulated by the code is in excellent agreement with experiments. Simulation results show that the viscosity ratio affects the size of the satellite drops.

MARS is a best-estimate thermal hydraulic systems analysis code with a multi-dimensional analysis capability. The MARS code is developed with the safety analysis as the prime application area and has a comprehensive range of models and components suitable for the simulation of nuclear reactor systems. The current developments in the nuclear industries and researches require accurate multi-dimensional analysis capabilities. However, the multi-dimensional model of MARS is based on a sub-channel formulation, and as such the geometric description is significantly limited. In order to alleviate such limitations, a new component based on the Cartesian and cylindrical coordinates, and the incorporating lateral convective terms and viscous stress terms have been added. The newly developed multi-dimensional component makes it possible to model a large bulk fluid volume in both the Cartesian and cylindrical coordinates. For the implementation of viscous shear stresses, Prandtl’s mixing length model is simply applied to obtain the turbulent viscosity. Several conceptual problems are described and solved using the new multi-dimensional component. The results were compared with the FLUENT calculations using the standard k-ε model. The comparison shows successful implementations and reasonable agreements with the FLUENT results.

This paper presents the impeller design system developed at Dresser-Rand using Bezier polynomials in cylindrical coordinates. A discussion of the basic techniques utilized in the code is presented as are sample graphic outputs generated to aid the user in the design process. The paper also describes some of the output options and how results may be interfaced with other analytical, drafting, and manufacturing software. Comments are included regarding the increased productivity, accuracy, and quality which resulted directly from use of this code and its support routines.

During a severe nuclear power plant (NPP) accident, large amounts of hydrogen and steam can be produced in nuclear reactor containment. In the case of hydrogen combustion, there is a possibility of producing short term pressure or detonation force. Therefore, these gas species’ production could threaten containment integrity. For instance, in the past, two gas explosion accidents occurred: In 1979 Three Mile Island and in 2011 Fukushima. After these accidents, modeling the gas behavior became an important topic in nuclear safety analyses. In order to predict hydrogen behavior and other gas species transport, mixing and combustion, reliable turbulence models need to be applied. In this work, standard k–ε, k–ω, RNG k–ε, Realizable k–ε and SST k–ω turbulence models are addressed. The computations are performed with HYDRAGON code. HYDRAGON code is a three-dimensional thermal-hydraulic code, developed to solve low-speed gas flow of compressible Navier-Stokes equations in cartesian or cylindrical coordinates or a mixture of the two coordinates. The goal of this work is to test the performance of these models by comparing the results to the benchmark. The code aims to predict containment thermal-hydraulic conditions during NPP accident.

In designing industrial cylindrical furnaces, it is important to predict the radiative heat flux on the wall with high accuracy. In this study, we consider CO2 and H2O which have strong absorption in the infrared range. The absorption coefficients of the gases are calculated by using the statistical narrow band (SNB) model. The spectrum is divided into 15 bands to cover all the absorption regions of the two non-gray gases. The radiative transfer equation is solved by the finite volume method (FVM) in cylindrical coordinates. To make the FVM more accurate, we discretize the solid angle into 80 directions with the S8 approximation which is found to be both efficient and less time consuming. Based on the existing species and temperature fields, which were modeled by the FLUENT commercial code, the radiative heat transfer in a cylinder combustor is simulated by an in-house code. The results show that the radiative heat flux plays a dominant part of the heat flux to the wall. Meanwhile, when the gas is considered as nongray, the computational time is very huge. Therefore, a parallel algorithm is also applied to speed up the computing process.

Large-amplitude (geometrically nonlinear) vibrations of circular cylindrical shells subjected to radial harmonic excitation in the spectral neighbourhood of the lowest resonances are investigated. The Lagrange equations of motion are obtained by an energy approach, retaining damping through Rayleigh’s dissipation function. Four different nonlinear shell theories, namely Donnell’s, Sanders-Koiter, Flu¨gge-Lur’e-Byrne and Novozhilov’s theories, are used to calculate the elastic strain energy. The formulation is also valid for orthotropic and symmetric cross-ply laminated composite shells. The large-amplitude response of perfect and imperfect, simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of the lowest natural frequency is computed for all these shell theories. Numerical responses obtained by using these four nonlinear shell theories are also compared to results obtained by Galerkin approach, used to discretise Donnell’s nonlinear shallow-shell equation of motion. A validation of calculations by comparison to experimental results is also performed. Boundary conditions for simply supported shells are exactly satisfied. Different expansions involving from 14 to 48 generalized coordinates, associated to natural modes of simply supported shells, are used. The nonlinear equations of motion are studied by using a code based on arclength continuation method that allows bifurcation analysis.

Drop generation from a single nozzle is visualized by direct numerical simulations. Navier-Stokes equations are solved on cylindrical axisymmetric coordinates, and interfacial motion is traced by a Front-Tracking method. To validate the numerical code, the numerical results for two modes, dripping and jetting, are compared with experimental results, and they show good agreement. The experimental result that shows a formation of a small satellite, in particular, is successfully visualized. It is shown that the formation of the satellite is due to the strong back flow arising from the large interfacial tension at the cusp behind the primary drop.

The performance of a three dimensional computer code developed for predicting the flowfield in stationary and rotating turbomachinery blade rows is described in this study. The four stage Runge-Kutta numerical integration scheme is used for solving the governing flow equations and yields solution to the full, three dimensional, unsteady Euler equations in cylindrical coordinates. This method is fully explicit and uses the finite volume, time marching procedure. In order to demonstrate the accuracy and efficiency of the code, steady solutions were obtained for several cascade geometries under widely varying flow conditions. Computed flowfield results are presented for a fully subsonic turbine stator and a low aspect ratio, transonic compressor rotor blade under maximum flow and peak efficiency design conditions. Comparisons with Laser Anemometer measurements and other numerical predictions are also provided to illustrate that the present method predicts important flow features with good accuracy and can be used for cost effective aerodynamic design studies.

Fully developed turbulent pipe flow is investigated by large eddy simulations (LES). The three-dimensional, unsteady, incompressible, filtered continuity and Navier-Stokes equations in cylindrical coordinates are discretized by a finite difference method. The spatial derivatives are approximated by second order conservative schemes. This scheme eliminates the numerical generation or dissipation of energy. The pressure Poisson equation is solved by FFT method and time is advanced through a third order Runge-Kutta method. The commonly used subgrid scale (SGS) models — the Smagorinsky model and the dynamic model are implemented and simulations are performed for fully developed turbulent pipe flow at two different Reynolds numbers. The flow features in terms of mean velocity as well as higher order turbulence intensities and correlations are presented and compared to experimental and DNS data available in literature. Extensive comparisons are made for cases using different grid resolution, different streamwise domain dimension, different sub-grid scale model, and, at two different Reynolds number. For two Reynolds numbers (5,000 and 30,000) tested in this study, the fine mesh (64 × 96 × 64, circumferential × radial × longitudinal) produces better results than the coarse mesh (32 × 48 × 32), indicating the significance of the grid resolution, especially near the pipe surface. On the fine mesh for the two Reynolds numbers, the results exhibit a slight Reynolds number effect, indicating the mesh needs to be further refined at higher Reynolds number. Simulations were performed for two domain sizes, namely 6D and 12D, where D is the pipe diameter. When the streamwise grid resolution remains unchanged, the two simulations show negligible difference. This ensures that a 6D domain is adequate to include the largest eddies in a fully developed turbulent pipe flow at the current Reynolds number. When the fine mesh is used, the subgrid scale models (Smagorinsky and Dynamic) provide limited contribution to the total turbulent kinetic energy. Although the current results agree quite well with other published LES simulations, when compared with the Law of the wall, benchmark experiments and DNS results, the simulated mean velocity in the log region is higher than the experimental and DNS data. Overall, it was observed that the numerical methods work satisfactorily well for turbulent pipe flows at low and high Reynolds numbers, and, the method has capability to be used in the simulation of flows with practical interest.

The differences of two distinct numerical schemes implemented in one code called ITSM3D are presented for a turbine stage test case. Thus both schemes are used with exactly the same computational infrastructure, e. g, same grids, boundary conditions, acceleration strategies, time-stepping, turbulence model etc. The two methods are based on an explicit Runge-Kutta-type finite volume scheme expressed in cylindrical coordinates and have been developed at the Institut für Thermische Strömungsmaschinen und Maschinenlaboratorium of the University of Stuttgart. One scheme is a node centered 3rd order TVD scheme according to Osher and the other belongs to the cell vertex central difference type with the concept of artificial viscosity. The model of Baldwin-Lomax is used in order to simulate turbulent effects. Non-reflective boundary conditions are taken at stator inlet and rotor outlet to avoid non-physical reflections. A multigrid technique in combination with implicit residual smoothing and local time-stepping is employed to accelerate the computation. The test case for this comparison is the last stage of a low-pressure turbine. The computational results obtained are discussed and compared to each other as well as to experimental data. They are presented as pressure and Mach number isoline contours and diagrams of circumferential averaged quantities at inlet and outlet planes of stator and rotor versus radial position.