Table Of ContentPHYSICS RESEARCH AND TECHNOLOGY
N -S E
AVIER TOKES QUATIONS
PROPERTIES, DESCRIPTION
AND APPLICATIONS
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PHYSICS RESEARCH AND TECHNOLOGY
N -S E
AVIER TOKES QUATIONS
PROPERTIES, DESCRIPTION
AND APPLICATIONS
R. YOUNSI
EDITOR
Nova Science Publishers, Inc.
New York
Copyright © 2012 by Nova Science Publishers, Inc.
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LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA
Navier-Stokes equations : properties, description, and applications / editors, R. Younsi.
p. cm.
Includes index.
ISBN: (cid:28)(cid:26)(cid:27)(cid:16)(cid:20)(cid:16)(cid:25)(cid:21)(cid:23)(cid:20)(cid:26)(cid:16)(cid:22)(cid:28)(cid:24)(cid:16)(cid:19) (eBook)
1. Navier-Stokes equations. I. Younsi, R.
QA374.N38 2011
515'.353--dc23
2011014548
Published by Nova Science Publishers, Inc. † New York
CONTENTS
Preface vii
Chapter 1 Parabolic and Elliptic Partial Difference Equations: 1
Towards a Discrete Solution of Navier-Stokes Equation
Rubén A. Hidalgo and Mauricio Godoy Molina
Chapter 2 Numerical Solution of Navier-Stokes Equations in Liquid Metals 25
under Magnetic Field
S. Nouri, S. Hamimid, A. Harkati, M.Guellal, D. Ouadjaout
and R. Younsi
Part 2 Marangoni-Natural Convection Liquid Metals in The Presence of 35
a Tilted Magnetic Field
S. Hamimid and M. Guellal
Chapter 3 Use of the Navier-Stokes Equations to Study of the Flow 51
Generated by Turbines Impellers
Z. Driss and M.S. Abid
Chapter 4 Postinstability Extension of Navier-Stokes Equations 139
Michail Zak
Chapter 5 3D Simulation of Turbulent Two-Phase Flows Using a Stabilized 177
Finite Element Method
E. Hachem, G. François and T. Coupez
Chapter 6 On Solutions of the Navier-Stokes Equations 199
K. Fakhar and A.H. Kara
Chapter 7 High Order Shock Capturing Schemes for Navier-Stokes 213
Equations
Yiqing Shen and Gecheng Zha
Chapter 8 The Phenomenon of the Effective Viscosity for the Flow 281
in Inhomogeneous Granular Medium
A.V. Gavrilov and I.V. Shirko
vi Contents
Chapter 9 Transient Flow of Fluids: Some Applications of the Navier-Stokes 307
Equations
Z. Ouchiha, A. Ghezal and S.M. Ghiaasiaan
Application Transient Flow of Highly Pressurized Fluids in Pipelines 313
A Z. Ouchiha, J. C. Loraud and A. Ghezal
Application Periodic Flow 321
B A. Ghezal, J. C. Loraud and Z. Ouchiha
Chapter 10 Fixed Grid Numerical Simulation of a Phase Change Material 355
in a Rectangular Enclosure Heated from One Side
Annabelle Joulin, Zohir Younsi, Stéphane Lassue
and Laurent Zalewski
Index 373
PREFACE
It is well known that the Navier–Stokes equations are one of the pillars of fluid
mechanics. These equations are useful because they describe the physics of many things of
academic and economic interest. They may be used to model the weather behavior, ocean
currents, water flow in a pipe and air flow around a wing. The Navier–Stokes equations in
their full and simplified forms also help with the design of train, aircraft and cars, the study of
blood flow, the design of power stations and pollution analysis. This book presents
contributions on the application of Navier-Stokes in some engineering applications and
provides a description of how the Navier-Stokes equations can be scaled. (Imprint: Nova)
In Chapter 1 the authors survey some results about partial difference equations (PdEs) in
weighted graphs obtained previously by the authors, in the case of finite graphs and countable
graphs of finite degree. They discuss the existence of solution of elliptic PdEs via an
associated semilinear matrix equation and under growth conditions of the forcing term. As
applications of these techniques, they study the discrete analogues of some classical elliptic
partial differential equations such as Matukuma, Helmholtz and Lane-Emden-Fowler. Finally
the authors discuss their results concerning the discrete Navier- Stokes equation and give
some explicit solutions for concrete weighted graphs and discuss possible graphs that
simplify the applications in the planar case.
In the first part of Chapter 2, the Czochralski techniques of crystal growth driven by the
combined action of buoyancy, surface-tension and electromagnetic forces is considered. This
study was done for Prandlt number Pr=0.02 and aspect ratio A=2. The transport equations for
continuity, momentum and energy transfer are solved. The numerical results are reported for
the effect of Marangoni and Hartmann numbers. A uniform magnetic field is applied in the
axial direction. The system of equations governing the growth phenomenon has been treated
in steady state). The axisymmetric Navier-Stokes and energy equations, with the Boussinesq
approximation, have been discretised by means of finite volume procedure and the SIMPLER
algorithm is used to treat the coupled velocity-pressure. The thermocapillary convection and
the magnetic field effects on the flow and temperature field are discussed. A parametric study
was carried out.
In the second part, the Navier-Stokes and energy equations are numerically solved to
investigate two-dimensional convection (originating from the combined effect of buoyancy
and surface tension forces) in liquid metals subjected to transverse magnetic field. In
particular, a laterally heated horizontal cavity with aspect ratio (height/width =1) and Prandtl
number Pr=0.015 is considered (typically associated with the horizontal Bridgman crystal
viii R. Younsi
growth process and commonly used for benchmarking purposes). The effect of a uniform
magnetic field with different magnitudes and orientations on the stability of the two distinct
convective solution branches (with a single-cell or two-cell pattern) of the steady state flows
is investigated. The effects induced by increasing values of the Rayleigh and Hartmann
numbers on the heat transfer rate are also discussed.
Chapter 3 reviews a decade of research on the use of the Navier-Stokes Equations to
study the flow generated by different turbines impellers, conducted by their group at the
University of Sfax. For this reason, computational fluid dynamics (CFD) methods are
developed to predict the flow field in stirred tanks equipped by various turbines impellers
type. The three-dimensional flow of a fluid is numerically analyzed using the Navier-Stokes
equations in conjunction with standard k-ε turbulence model. The relevant governing
equations are given, and alternative turbulence models are described. The computer method
developed permits the numerical analyses of turbines with complex geometries.
Their main objective has been to study and compare the shape effects on the field flow
and turbulence characteristics to choose the most effective agitation system for the mixing
applications. In these applications, the effect on the local and global flow characteristics have
been determined and compared. The numerical results such as flow patterns profile and power
numbers variation on Reynolds number were compared to their experimental results and those
obtained by other authors. The comparison proved a good agreement.
Chapter 4 is devoted to investigation of the region of instability of the Navier-Stokes
equations. This region is of critical impotence for modeling turbulent flows. Stochastic
approach to turbulence based upon the Stabilization Principle is introduced. Onset of
turbulence is interpreted as loss of stability of solutions to the Navier-Stokes equations in the
class of differentiable functions. A nonlinear version of the Liouville- Fokker-Planck equation
with negative diffusion is proposed for describing postinstability motions of the Navier-
Stokes equations. As a result, the flow velocities are represented by a set of stochastic
invariants found from the Stabilization Principle. The theory is illustrated by examples.
A recently developed stabilized finite element method is generalised for the
incompressible Navier Stokes equations for high Reynolds numbers and free surface flows
problems. The proposed method starts by the use of a finite element variational multiscale
(VMS) method, which consists in here of decomposition for both the velocity and the
pressure fields into coarse/resolved scales and fine/unresolved scales. This choice of
decomposition is shown to be favorable for simulating flows at high Reynolds number.
However, the main challenge remains in solving turbulent two-phase flows which occurs in a
wide range of real life problems, industrial processes including molten metal flow, sloshing in
tanks, wave mechanics, flows around structure. A successful approach to deal with such
flows, especially in the presence of turbulent behaviour is the use of a local convected level
set method coupled to a Large Eddy Simulation method.
In Chapter 5 they assess the behaviour and accuracy of the proposed formulation in the
simulation of three test cases. Results are compared with the literature and the experimental
data and show that the present implementation is able to exhibit good stability for high
Reynolds number flows using unstructured meshes.
The equations which govern the motion of an incompressible viscous fluid are precisely
the Navier-Stokes equations. It is a known fact that the Navier-Stokes equations in general are
nonlinear partial differential equations and few analytic solutions of these equations are
reported in the existing literature.
