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.

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

2017 ◽  
Author(s):  
Vladimir Maderich ◽  
Kyung Tae Jung ◽  
Kateryna Terletska ◽  
Kyeong Ok Kim
Keyword(s):  
Nonlinear Waves ◽  
Wave Amplitude ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Class Iii ◽  
Navier Stokes Equations ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
Conservation Of Momentum

Abstract. The dynamics and energetics of a head-on collision of internal solitary waves (ISWs) with trapped cores propagating in 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 the trapped masses of a fluid. The interaction of ISWs differs for three classes of ISWs: (i) weakly nonlinear waves without trapped cores, (ii) stable strongly nonlinear waves with trapped cores, and (iii) shear unstable strongly nonlinear waves. The wave phase shift grows as the amplitudes of the interacting waves increase for colliding waves of classes (i) and (ii) and remains almost constant for those of class (iii). The excess of the maximum runup amplitude over the sum of the amplitudes of colliding waves almost linearly increases with increasing amplitude of the interacting waves belonging to classes (i) and (ii); however, it decreases somewhat for those of class (iii). The waves of class (ii) with a normalized on thickness of pycnocline amplitude lose fluid trapped by the wave cores in the range 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 three-dimensional instability and turbulence. The dependence of loss of energy on the wave amplitude is not monotonous. 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.

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.

Relationship between accuracy and number of velocity particles of the finite-difference lattice Boltzmann method in velocity slip simulations

2010 ◽  
Vol 132 (10) ◽  
Author(s):  
Minoru Watari
Keyword(s):  
Relaxation Process ◽  
Stokes Equations ◽  
Velocity Slip ◽  
Simulation Models ◽  
Knudsen Layer ◽  
Navier Stokes ◽  
Flow Area ◽  
Navier Stokes Equations ◽  
Conservation Of Momentum ◽  
Boltzmann Method

Relationship between accuracy and number of velocity particles in velocity slip phenomena was investigated by numerical simulations and theoretical considerations. Two types of 2D models were used: the octagon family and the D2Q9 model. Models have to possess the following four prerequisites to accurately simulate the velocity slip phenomena: (a) equivalency to the Navier–Stokes equations in the N-S flow area, (b) conservation of momentum flow Pxy in the whole area, (c) appropriate relaxation process in the Knudsen layer, and (d) capability to properly express the mass and momentum flows on the wall. Both the octagon family and the D2Q9 model satisfy conditions (a) and (b). However, models with fewer velocity particles do not sufficiently satisfy conditions (c) and (d). The D2Q9 model fails to represent a relaxation process in the Knudsen layer and shows a considerable fluctuation in the velocity slip due to the model’s angle to the wall. To perform an accurate velocity slip simulation, models with sufficient velocity particles, such as the triple octagon model with moving particles of 24 directions, are desirable.

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.

scholarly journals Plane poloidal-toroidal decomposition of doubly periodic vector fields. Part 2. The Stokes equations

2005 ◽  
Vol 47 (1) ◽  
pp. 39-50 ◽  
Author(s):  
G. D. McBain
Keyword(s):  
Vector Fields ◽  
Stokes Equations ◽  
Momentum Equation ◽  
Integral Representations ◽  
Navier Stokes ◽  
Field Equations ◽  
Navier Stokes Equations ◽  
Time Discretisation ◽  
Mass Constraint ◽  
Conservation Of Momentum

AbstractWe continue our study of the adaptation from spherical to doubly periodic slot domains of the poloidal-toroidal representation of vector fields. Building on the successful construction of an orthogonal quinquepartite decomposition of doubly periodic vector fields of arbitrary divergence with integral representations for the projections of known vector fields and equivalent scalar representations for unknown vector fields (Part 1), we now present a decomposition of vector field equations into an equivalent set of scalar field equations. The Stokes equations for slow viscous incompressible fluid flow in an arbitrary force field are treated as an example, and for them the application of the decomposition uncouples the conservation of momentum equation from the conservation of mass constraint. The resulting scalar equations are then solved by elementary methods. The extension to generalised Stokes equations resulting from the application of various time discretisation schemes to the Navier-Stokes equations is also solved.

Assessing the Dynamic Stability of an Offshore Supply Vessel

2011 ◽  
Author(s):  
Vladimir Shigunov ◽  
Ould el Moctar ◽  
Thomas E. Schellin ◽  
Jan Kaufmann ◽  
Rasmus Stute
Keyword(s):  
Dynamic Stability ◽  
Stokes Equations ◽  
Operating Conditions ◽  
Navier Stokes ◽  
Ship Motions ◽  
Navier Stokes Equations ◽  
Counter Measures ◽  
Seakeeping Analysis ◽  
Design Speed ◽  
Run Up

The dynamic stability was investigated of a typical offshore service vessel operating under stability critical operating conditions. Excessive roll motions and relative motions at the stern were studied for two loading conditions for ship speeds ranging from zero to the design speed. A linear frequency-domain seakeeping analysis was followed by nonlinear time-domain simulations of ship motions in waves. Based on results from these methods, critical scenarios were selected and simulated using finite-volume solvers of the Reynolds-averaged Navier-Stokes equations to understand the phenomena related to dynamically unstable ship motions as well as to confirm the results of the simpler analysis methods. Results revealed the possibility of excessive roll motions and water run-up on deck; counter measures such as a ship-specific operational guidance are discussed.

CFD Simulation of Wave Run-Up on a Semi-Submersible and Comparison With Experiment

2009 ◽  
Author(s):  
Bogdan Iwanowski ◽  
Marc Lefranc ◽  
Rik Wemmenhove
Keyword(s):  
Cfd Simulation ◽  
Stokes Equations ◽  
Computational Grids ◽  
Navier Stokes ◽  
Incoming Wave ◽  
Navier Stokes Equations ◽  
Comparison With Experiment ◽  
Wave Heights ◽  
Impact Phenomena ◽  
Run Up

Use of CFD tools for industrial offshore applications is a common practice nowadays. So is the need for validation of such tools against experimental results. This paper presents one of the CFD tools, ComFLOW, which solves Navier-Stokes equations and employs an improved Volume of Fluid (iVOF) method to find temporary location of fluid’s free surface. The code is used to simulate flow around a semi-submersible offshore platform due to an incoming regular wave. In particular, wave run-up on the semi’s columns and under-deck fluid impact phenomena are investigated on high-accuracy computational grids with number of cells being in range of 10 millions. Results of numerical simulations are compared with experimental data and focus is on local fluid flow details in immediate vicinity of the platform. Wave run-up on the platform’s columns and fluid pressures at various locations, including under-deck impact, are reported and verified against the experiment for a range of incoming wave heights.

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.

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