Linear and non-linear iterative methods for the incompressible Navier-Stokes equations

1994 ◽  
Vol 18 (3) ◽  
pp. 229-256 ◽  
Author(s):  
Simon S. Clift ◽  
Peter A. Forsyth
Keyword(s):  
Iterative Methods ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Non Linear

Computational analysis of turbulent flow through an eccentric annulus under different temperature conditions

2018 ◽  
Vol 28 (9) ◽  
pp. 2189-2207 ◽  
Author(s):  
Erman Ulker ◽  
Sıla Ovgu Korkut ◽  
Mehmet Sorgun
Keyword(s):  
Turbulent Flow ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Content Type ◽  
Non Linear ◽  
Fully Developed Turbulent Flow ◽  
Effects Of Temperature ◽  
Eccentric Annuli ◽  
Pipe Rotation

Purpose The purpose of this paper is to solve Navier–Stokes equations including the effects of temperature and inner pipe rotation for fully developed turbulent flow in eccentric annuli by using finite difference scheme with fixing non-linear terms. Design/methodology/approach A mathematical model is proposed for fully developed turbulent flow including the effects of temperature and inner pipe rotation in eccentric annuli. Obtained equation is solved numerically via central difference approximation. In this process, the non-linear term is frozen. In so doing, the non-linear equation can be considered as a linear one. Findings The convergence analysis is studied before using the method to the proposed momentum equation. It reflects that the method approaches to the exact solution of the equation. The numerical solution of the mathematical model shows that pressure gradient can be predicted with a good accuracy when it is compared with experimental data collected from experiments conducted at Izmir Katip Celebi University Flow Loop. Originality/value The originality of this work is that Navier–Stokes equations including temperature and inner pipe rotation effects for fully developed turbulent flow in eccentric annuli are solved numerically by a finite difference method with frozen non-linear terms.

scholarly journals Experimental validation of wind energy estimation

2020 ◽  
Vol 24 (6 Part A) ◽  
pp. 3795-3806
Author(s):  
Predrag Zivkovic ◽  
Mladen Tomic ◽  
Vukman Bakic
Keyword(s):  
Wind Speed ◽  
Linear Model ◽  
Linear Models ◽  
Stokes Equations ◽  
Momentum Conservation ◽  
Electricity Production ◽  
Navier Stokes ◽  
Energy Estimation ◽  
Navier Stokes Equations ◽  
Non Linear

Wind power assessment in complex terrain is a very demanding task. Modeling wind conditions with standard linear models does not sufficiently reproduce wind conditions in complex terrains, especially on leeward sides of terrain slopes, primarily due to the vorticity. A more complex non-linear model, based on Reynolds averaged Navier-Stokes equations has been used. Turbulence was modeled by modified two-equations k-? model for neutral atmospheric boundary-layer conditions, written in general curvelinear non-orthogonal co-ordinate system. The full set of mass and momentum conservation equations as well as turbulence model equations are numerically solved, using the as CFD technique. A comparison of the application of linear model and non-linear model is presented. Considerable discrepancies of estimated wind speed have been obtained using linear and non-linear models. Statistics of annual electricity production vary up to 30% of the model site. Even anemometer measurements directly at a wind turbine?s site do not necessarily deliver the results needed for prediction calculations, as extrapolations of wind speed to hub height is tricky. The results of the simulation are compared by means of the turbine type, quality and quantity of the wind data and capacity factor. Finally, the comparison of the estimated results with the measured data at 10, 30, and 50 m is shown.

Variational multi-scale finite element approximation of the compressible Navier-Stokes equations

2016 ◽  
Vol 26 (3/4) ◽  
pp. 1240-1271 ◽  
Author(s):  
Camilo Andrés Bayona Roa ◽  
Joan Baiges ◽  
R Codina
Keyword(s):  
Finite Element ◽  
Stokes Equations ◽  
Finite Element Approximation ◽  
Shock Capturing ◽  
Navier Stokes ◽  
Element Approximation ◽  
Navier Stokes Equations ◽  
Content Type ◽  
Multi Scale ◽  
Non Linear

Purpose – The purpose of this paper is to apply the variational multi-scale framework to the finite element approximation of the compressible Navier-Stokes equations written in conservation form. Even though this formulation is relatively well known, some particular features that have been applied with great success in other flow problems are incorporated. Design/methodology/approach – The orthogonal subgrid scales, the non-linear tracking of these subscales, and their time evolution are applied. Moreover, a systematic way to design the matrix of algorithmic parameters from the perspective of a Fourier analysis is given, and the adjoint of the non-linear operator including the volumetric part of the convective term is defined. Because the subgrid stabilization method works in the streamline direction, an anisotropic shock capturing method that keeps the diffusion unaltered in the direction of the streamlines, but modifies the crosswind diffusion is implemented. The artificial shock capturing diffusivity is calculated by using the orthogonal projection onto the finite element space of the gradient of the solution, instead of the common residual definition. Temporal derivatives are integrated in an explicit fashion. Findings – Subsonic and supersonic numerical experiments show that including the orthogonal, dynamic, and the non-linear subscales improve the accuracy of the compressible formulation. The non-linearity introduced by the anisotropic shock capturing method has less effect in the convergence behavior to the steady state. Originality/value – A complete investigation of the stabilized formulation of the compressible problem is addressed.

scholarly journals On Linear and Non-Linear Approximation In the Theory of Convective Heat Transfer

2021 ◽  
pp. 44-51
Author(s):  
M. Y. Davidzon
Keyword(s):  
Heat Transfer ◽  
Convective Heat Transfer ◽  
Convective Heat ◽  
Stokes Equations ◽  
Linear Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Linear System Of Equations ◽  
Non Linear ◽  
Non Linear System

A system of linear equations that is currently widely used to describe convective heat transfer does not seem to be able to explain some experimental facts. One of the reasons for this may lie in using Newton’s and Fourier’s linear laws when deriving energy and Navier-Stokes equations. Replacing linear equations with nonlinear ones, as well as using an expression for surface heat flux density that is based on laws of physics instead of expressions called ‘cooling laws,’ would allow to solve a wider range of problems, and also would better agree with the experimental data. The use of proposed non-linear system of equations would also permit engineers in chemical, textile, defense, power, and other industries to design more economical and smaller-sized heat exchange devices.

scholarly journals Head-on collision of internal waves with trapped cores

2017 ◽  
Vol 24 (4) ◽  
pp. 751-762 ◽  
Author(s):  
Vladimir Maderich ◽  
Kyung Tae Jung ◽  
Kateryna Terletska ◽  
Kyeong Ok Kim
Keyword(s):  
Wave Amplitude ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Class Iii ◽  
Navier Stokes Equations ◽  
Linear Waves ◽  
Non Linear ◽  
Conservation Of Momentum ◽  
Run Up ◽  
The Waves

Abstract. The dynamics and energetics of a head-on collision of internal solitary waves (ISWs) with trapped cores propagating in a thin pycnocline were studied numerically within the framework of the Navier–Stokes equations for a stratified fluid. The peculiarity of this collision is that it involves trapped masses of a fluid. The interaction of ISWs differs for three classes of ISWs: (i) weakly non-linear waves without trapped cores, (ii) stable strongly non-linear waves with trapped cores, and (iii) shear unstable strongly non-linear waves. The wave phase shift of the colliding waves with equal amplitude grows as the amplitudes increase for colliding waves of classes (i) and (ii) and remains almost constant for those of class (iii). The excess of the maximum run-up amplitude, normalized by the amplitude of the waves, over the sum of the amplitudes of the equal colliding waves increases almost linearly with increasing amplitude of the interacting waves belonging to classes (i) and (ii); however, it decreases somewhat for those of class (iii). The colliding waves of class (ii) lose fluid trapped by the wave cores when amplitudes normalized by the thickness of the pycnocline are in the range of approximately between 1 and 1.75. The interacting stable waves of higher amplitude capture cores and carry trapped fluid in opposite directions with little mass loss. The collision of locally shear unstable waves of class (iii) is accompanied by the development of instability. The dependence of loss of energy on the wave amplitude is not monotonic. Initially, the energy loss due to the interaction increases as the wave amplitude increases. Then, the energy losses reach a maximum due to the loss of potential energy of the cores upon collision and then start to decrease. With further amplitude growth, collision is accompanied by the development of instability and an increase in the loss of energy. The collision process is modified for waves of different amplitudes because of the exchange of trapped fluid between colliding waves due to the conservation of momentum.

Numerical Offshore Tank: Development of Numerical Offshore Tank for Ultra Deep Water Oil Production Systems

2003 ◽  
Author(s):  
K. Nishimoto ◽  
M. D. Ferreira ◽  
M. R. Martins ◽  
I. Q. Masseti ◽  
C. A. Martins ◽  
...  
Keyword(s):  
Deep Water ◽  
Stokes Equations ◽  
Analytical Models ◽  
Navier Stokes ◽  
Floating Bodies ◽  
Data Set ◽  
Navier Stokes Equations ◽  
Linear Solution ◽  
Non Linear ◽  
Numerical Simulator

There are several developments concerned to simulate the behavior of floating bodies under waves in the restricted boundary conditions so called numerical wave tank. The main feature of these tanks is to calculate full Navier-Stokes equations taking account the viscosity and free surface conditions. However, the dynamic behavior of oil floating exploitation units in actual ocean environmental condition, in waves, wind and current, is more complex and very difficult to simulate using full non-linear Navier Stokes equations. In addition, in the ultra-deep water, it is the primer importance to consider the more precise mooring line and riser’s dynamics in the analysis. The present numerical simulator laboratory called Numerical Offshore Tank is a development that takes account almost all physical phenomena acting on the floating bodies and mooring and risers lines. Since full non-linear solution is not available, the several numerical, empirical and analytical models are being considered and integrated to numerical simulator. The time domain potential problem is solved to wave forces acting on the bodies and empirical models are used to simulate current and wind forces. To represent mooring & riser lines, the finite element model with more realistic hydrodynamic force models is used. Even the simulator is using the full hydrodynamic equation, the calculation time of the simulation for floating bodies with several risers & mooring lines is very high. Therefore, special cluster with 60 PC based computer was built running the code in the parallel processing. Since the preparation of all data set for numerical experiment is very tedious work, the special pre-processor PREA3D was developed for this purpose. This pre-processor allows the fast change of the environmental and system conditions to run several test conditions. Another important feature is the visualization of the results of the simulation tests. The entire 3D view of the system is presented in the Virtual Reality room with stereoscopic projection of the Numerical Tank Laboratory.

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