tomarchio valerio tesi

tomarchio valerio tesi
Alma Mater Studiorum – Università di Bologna
DOTTORATO DI RICERCA IN
INGEGNERIA ENERGETICA NUCLEARE
E DEL CONTROLLO AMBIENTALE
Ciclo XXI
Settore scientifico disciplinare di afferenza: ING-IND/10 Fisica Tecnica Industriale
TITOLO TESI
MAGNETOHYDRODYNAMIC EFFECTS ON MIXED
CONVECTION FLOWS IN CHANNELS AND DUCTS
Presentata da:
Ing. Valerio TOMARCHIO
Coordinatore Dottorato
Relatore
Chiar.mo Prof. Antonio BARLETTA
Chiar.mo Prof. Antonio BARLETTA
Esame finale anno 2009
Magnetohydrodynamic Effects On Mixed
Convection Flows In Channels And Ducts
V. Tomarchio
Contents
1
Introduction
2
1.1
Background and Motivation . . . . . . . . . . . . . . . . . . .
2
1.1.1
Fundamental phenomena of MHD . . . . . . . . . . .
3
1.1.2
Liquid metal flow meters and pumps . . . . . . . . .
4
1.1.3
Metallurgy . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.1.4
MHD power generation . . . . . . . . . . . . . . . . .
8
1.1.5
Fusion reactors breeding blankets . . . . . . . . . . .
9
1.2
2
3
Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Mathematical Formulation
12
2.1
Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.2
Governing equations . . . . . . . . . . . . . . . . . . . . . . .
13
2.3
Boundary conditions . . . . . . . . . . . . . . . . . . . . . . .
15
2.3.1
Dynamic boundary conditions . . . . . . . . . . . . . 15
2.3.2
Thermal boundary conditions . . . . . . . . . . . . . .
2.3.3
Potential boundary conditions . . . . . . . . . . . . . 16
16
2.4
Dimensionless formulation . . . . . . . . . . . . . . . . . . .
19
2.5
Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
MHD mixed convection flow in a vertical parallel channel with
transverse magnetic field. Steady periodic regime
22
3.1
Governing equations . . . . . . . . . . . . . . . . . . . . . . .
25
3.2
Analytical solution . . . . . . . . . . . . . . . . . . . . . . . .
28
3.3
Numerical solution . . . . . . . . . . . . . . . . . . . . . . . .
31
i
3.4
3.5
4
3.4.1
Velocity and temperature distributions . . . . . . . .
34
3.4.2
Influence of dissipative terms . . . . . . . . . . . . . .
37
3.4.3
Fanning friction factors and Nusselt numbers . . . .
38
Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
verse magnetic field. Steady periodic regime
46
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.2
Governing equations . . . . . . . . . . . . . . . . . . . . . . .
49
4.3
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
4.4
Results and Discussion . . . . . . . . . . . . . . . . . . . . . .
55
4.4.1
Forced stationary convection with MHD effects . . .
59
4.4.2
Steady periodic mixed convection with MHD effects
60
Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
MHD mixed convection flow in a vertical rectangular duct with
transverse magnetic field. Steady periodic regime
68
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
5.2
Governing equations . . . . . . . . . . . . . . . . . . . . . . .
71
5.3
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
5.4
Results and Discussion . . . . . . . . . . . . . . . . . . . . . .
78
5.4.1
Forced stationary convection without MHD effects .
82
5.4.2
Forced stationary convection with MHD effects . . .
83
5.4.3
Steady periodic mixed convection with MHD effects
84
5.5
6
34
MHD mixed convection flow in a vertical round pipe with trans-
4.5
5
Results and Discussion . . . . . . . . . . . . . . . . . . . . . .
Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Conclusions
6.1
91
Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
A Auxiliary functions for the parallel channel problem
95
B FORTRAN95 code for the solution of the parallel channel prob98
lem
ii
List of Figures
1.1
A simple MHD experiment. . . . . . . . . . . . . . . . . . . .
4
1.2
A functional sketch of an MHD flowmeter. . . . . . . . . . .
6
1.3
A functional sketch of an MHD pump. . . . . . . . . . . . . .
7
1.4
A typical setup for an electromagnetic stirrer. . . . . . . . . .
8
1.5
A simple scheme of an MHD generator. . . . . . . . . . . . .
9
1.6
A typical layout of a breeding blanket. . . . . . . . . . . . . .
10
2.1
A solid conducting walls example. . . . . . . . . . . . . . . .
18
3.1
Representative scheme of the system . . . . . . . . . . . . . .
25
3.2
Mean velocity distributions across the channel for different
values of M and GR. Frame (a) GR = 500, M = 0 (continuous), M = 10 (dashed), M = 100 (dotted). Frame (b) M = 0,
GR = 0 (continuous), GR = 100 (dashed), GR = 500 (dotted). 35
3.3
Oscillating velocity amplitude distributions across the
channel for different values of Ω and P r in the nonmagnetic case. Frame (a) P r = 5, Frame (b) P r = 0.05.
Ω = 0.001 (continuous), Ω = 100 (dashed), Ω = 500 (dotted).
3.4
35
Comparison of oscillating velocity amplitude distributions
across the channel for different values of Ω between magnetic and non-magnetic case. Frame (a) M = 0, Frame
(b) M = 0.05. Ω = 0.001 (continuous), Ω = 100 (dashed),
Ω = 500 (dotted). . . . . . . . . . . . . . . . . . . . . . . . . . 36
iii
3.5
Comparison of oscillating temperature amplitude distributions across the channel for different values of Ω and P r.
Frame (a) M = 0, P r = 0.05, Frame (b) M = 0, P r = 5.
Ω = 0.001 (continuous), Ω = 100 (dashed), Ω = 500 (dotted).
3.6
37
Comparison of average temperature distributions across
the channel for M = 0 and different values of Br. Br = 0
(continuous), Br = 2 (dashed), Br = 4 (dotted). . . . . . . . . 38
3.7
Distribution of the average (a) and oscillating (b) velocity
components across the channel for Br = 0.25 and different
values of M . M = 0 (continuous), M = 10 (dashed), M = 20
(dotted). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8
39
Distribution of the average (a) and oscillating (b) temperature components across the channel for Br = 0.25 and different values of M . M = 0 (continuous), M = 10 (dashed),
M = 20 (dotted). . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9
40
Average component of the Fanning friction factor at the cold
(a) and hot (b) walls plotted versus GR for different values
of M . M = 0 (continuous), M = 50 (dashed), M = 100
(dotted). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.10 Amplitude of the oscillating component of the Fanning friction factor at the cold wall plotted versus Ω for P r = 0.05
(a) or P r = 5 (b) and different values of M . M = 0 (continuous), M = 50 (dashed), M = 100 (dotted). . . . . . . . . . . 41
3.11 Amplitude of the oscillating component of the Fanning friction factor at the cold wall plotted versus Ω for M = 0 (a) or
M = 100 (b) and different values of P r. P r = 0.05 (continuous), P r = 0.5 (dashed), P r = 5 (dotted). . . . . . . . . . . . .
42
3.12 Amplitude of the oscillating component of the Fanning friction factor at the hot wall plotted versus Ω for P r = 0.05 (a)
or P r = 5 (b) and different values of M . M = 0 (continuous), M = 50 (dashed), M = 100 (dotted). . . . . . . . . . . .
iv
42
3.13 Amplitude of the oscillating component of the Fanning friction factor at the hot wall plotted versus Ω for M = 0 (a) or
M = 100 (b) and different values of P r. P r = 0.05 (continuous), P r = 0.5 (dashed), P r = 5 (dotted). . . . . . . . . . . . .
43
3.14 Dimensionless heat flux N u as a function of Ω at different
locations inside the channel. Frame (a), ξ = −0.25 (contin-
uous), ξ = −0.15 (dashed), ξ = −0.05 (dotted). Frame (b),
ξ = 0.05 (continuous), ξ = 0.15 (dashed), ξ = 0.25 (dotted). .
44
4.1
A sketch of the channel. . . . . . . . . . . . . . . . . . . . . .
50
4.2
The discretization of the computational domain: (a) An
overview of the whole domain, showing the structured
boundary layers and the unstructured core. (b) A detailed
view of the structured mesh close to the wall. The thickness
of the first element is 10−7 . . . . . . . . . . . . . . . . . . . . .
4.3
56
Distributions of average dimensionless velocity (a) (c) (e)
and electric potential (b) (d) (f), calculated for Ha = 0 (a,b),
Ha = 20 (c,d), Ha = 50 (e,f) for steady state forced convection with transverse magnetic field. . . . . . . . . . . . . . . . 57
4.4
Arrow and streamline plots of the current density vector J
across the pipe cross section, calculated for Ha = 10 (a,b),
Ha = 50 (c,d), Ha = 100 (e,f) for steady state forced convection with transverse magnetic field. . . . . . . . . . . . . . . . 58
4.5
Behavior of f1 Re (a) and f2 Re (b) as functions of Ω obtained for Ha = 0 (continuous) Ha = 5 (dashed) and Ha =
10 (dotted) in mixed convection regime, with GR = 500 and
P r = 0.05. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6
Behavior of N u1 (continuous) and N u2 (dashed) as functions of Ω in mixed convection regime with P r = 0.05. . . . .
4.7
62
63
Distributions of oscillation amplitudes for dimensionless
velocity (a) (b), temperature (c) (d) and electric potential (e)
(f), calculated for Ha = 0 in mixed convection regime, with
GR = 500, P r = 0.05 and Ω = 100. . . . . . . . . . . . . . . .
v
64
4.8
Distributions of oscillation amplitudes for dimensionless
velocity (a) (b), temperature (c) (d) and electric potential (e)
(f), calculated for Ha = 20 in mixed convection regime, with
GR = 500, P r = 0.05 and Ω = 100. . . . . . . . . . . . . . . .
4.9
65
Distributions of oscillation amplitudes for dimensionless
velocity (a) (b), temperature (c) (d) and electric potential (e)
(f), calculated for Ha = 50 in mixed convection regime, with
GR = 500, P r = 0.05 and Ω = 100. . . . . . . . . . . . . . . .
66
5.1
A sketch of the channel. . . . . . . . . . . . . . . . . . . . . .
72
5.2
The discretization of the computational domain: (a) An
overview of the whole domain, showing the structured
boundary layers and the unstructured core. (b) A detailed
view of the structured mesh close to the wall. The thickness
of the first element is 10−7 . . . . . . . . . . . . . . . . . . . . .
5.3
79
Distributions of average dimensionless velocity (a) (c) (e)
and electric potential (b) (d) (f), calculated for Ha = 0 (a,b),
Ha = 20 (c,d), Ha = 50 (e,f) in a square channel (χ = 1) for a
steady state forced convection with transverse magnetic field. 80
5.4
Arrow and streamline plots of the current density vector J
across the duct cross section, calculated for Ha = 10 (a,b),
Ha = 50 (c,d), Ha = 100 (e,f) for steady state forced convection with transverse magnetic field. . . . . . . . . . . . . . . .
5.5
81
Behavior of f1 Re, f2 Re, N u1 and N u2 as functions of Ω obtained for Ha = 5, 10, 20. Results apply to a square channel
χ = 1 in mixed convection regime, with GR = 500, P r = 0.05. 85
5.6
Distributions of oscillation amplitudes for dimensionless
velocity (a) (b), temperature (c) (d) and electric potential (e)
(f), calculated for Ha = 0 and χ = 1, in mixed convection
regime, with GR = 500, P r = 0.05 and Ω = 100. . . . . . . . .
vi
86
5.7
Distributions of oscillation amplitudes for dimensionless
velocity (a) (b), temperature (c) (d) and electric potential (e)
(f), calculated for Ha = 20 and χ = 1, in mixed convection
regime, with GR = 500, P r = 0.05 and Ω = 100. . . . . . . . .
5.8
87
Distributions of oscillation amplitudes for dimensionless
velocity (a) (b), temperature (c) (d) and electric potential (e)
(f), calculated for Ha = 50 and χ = 1, in mixed convection
regime, with GR = 500, P r = 0.05 and Ω = 100. . . . . . . . . 88
vii
List of Tables
2.1
A selection of parameters related to fusion experimental devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.1
Values of f Re as a function of Ha in stationary forced convection, GR = 0, Ω = 0. . . . . . . . . . . . . . . . . . . . . . .
4.2
60
Values of the oscillating amplitudes of f Re and N u as a
function of Ω. Results obtained in mixed convection regime,
with P r = 0.05 and GR = 500. . . . . . . . . . . . . . . . . . .
4.3
61
Values of the oscillating amplitudes of f Re and N u as a
function of Ω. Results obtained in mixed convection regime,
with P r = 1 and GR = 500. . . . . . . . . . . . . . . . . . . . 61
5.1
Comparison between the factor f Rem obtained in the
present work and the analytical solution available in literature (11). Stationary forced convection, GR = 0, Ω = 0. . . .
5.2
Values of f Re as a function of Ha in stationary forced convection, GR = 0, Ω = 0. . . . . . . . . . . . . . . . . . . . . . .
5.3
83
84
Values of te oscillating amplitudes of f Re and N u as a function of Ω. Results obtained for a square channel χ = 1 in
mixed convection regime, GR = 500. . . . . . . . . . . . . . . 89
viii
Acknowledgments
The Author wants to acknowledge the continuous help and support received throughout this project from the Department of Energetic, Nuclear
and Environmental Control Engineering (DIENCA). In particular I would
like to thank Prof. A. Barletta and Dr. E. Rossi di Schio for their competent, professional, yet friendly behavior. I have sincerely appreciated the
possibility I was given to collaborate with them.
ix
Foreword
This work focuses on magnetohydrodynamic (MHD) mixed convection
flow of electrically conducting fluids enclosed in simple 1D and 2D geometries in steady periodic regime.
In particular, in Chapter one a short overview is given about the history of MHD, with reference to papers available in literature, and a listing
of some of its most common technological applications, whereas Chapter
two deals with the analytical formulation of the MHD problem, starting
from the fluid dynamic and energy equations and adding the effects of an
external imposed magnetic field using the Ohm’s law and the definition of
the Lorentz force. Moreover a description of the various kinds of boundary conditions is given, with particular emphasis given to their practical
realization.
Chapter three, four and five describe the solution procedure of mixed
convective flows with MHD effects. In all cases a uniform parallel magnetic field is supposed to be present in the whole fluid domain transverse
with respect to the velocity field. The steady-periodic regime will be analyzed, where the periodicity is induced by wall temperature boundary
conditions, which vary in time with a sinusoidal law. Local balance equations of momentum, energy and charge will be solved analytically and numerically using as parameters either geometrical ratios or material properties.
In particular, in Chapter three the solution method for the mixed convective flow in a 1D vertical parallel channel with MHD effects is illustrated. The influence of a transverse magnetic field will be studied in the
x
steady periodic regime induced by an oscillating wall temperature. Analytical and numerical solutions will be provided in terms of velocity and
temperature profiles, wall friction factors and average heat fluxes for several values of the governing parameters.
In Chapter four the 2D problem of the mixed convective flow in a vertical round pipe with MHD effects is analyzed. Again, a transverse magnetic field influences the steady periodic regime induced by the oscillating
wall temperature of the wall. A numerical solution is presented, obtained
using a finite element approach, and as a result velocity and temperature
profiles, wall friction factors and average heat fluxes are derived for several values of the Hartmann and Prandtl numbers.
In Chapter five the 2D problem of the mixed convective flow in a vertical rectangular duct with MHD effects is discussed. As seen in the previous chapters, a transverse magnetic field influences the steady periodic
regime induced by the oscillating wall temperature of the four walls. The
numerical solution obtained using a finite element approach is presented,
and a collection of results, including velocity and temperature profiles,
wall friction factors and average heat fluxes, is provided for several values of, among other parameters, the duct aspect ratio. A comparison with
analytical solutions is also provided, as a proof of the validity of the numerical method.
Chapter six is the concluding chapter, where some reflections on the
MHD effects on mixed convection flow will be made, in agreement with
the experience and the results gathered in the analyses presented in the
previous chapters. In the appendices special auxiliary functions and FORTRAN program listings are reported, to support the formulations used in
the solution chapters.
1
Chapter 1
Introduction
1.1 Background and Motivation
Magnetohydrodynamics (MHD) is concerned with the flow of electrically
conducting fluids in the presence of magnetic fields, either externally applied or generated within the fluid by inductive action. Its origin dates
back to pioneering discoveries of Northrup, Hartmann, Alfvèn, and others in the first half of the twentieth century. After 1950, the subject developed rapidly, and soon became well established as a field of scientific endeavor of great importance in various contexts: geomagnetism and planetary magnetism, astrophysics, nuclear fusion (plasma) physics, and liquid
metal technology.
In the last years, a growing interest has been addressed to the study
of magnetohydrodynamic effects on mixed and natural convective flows
(1)-(6). Such interest in the topic is due to the large number of possible
technological applications, like in metallurgy, where the quality of the materials, produced in a regime of controlled crystal growth, can be influenced by the effects of an external imposed magnetic field (7). Recently,
being increased the efforts towards the realization of nuclear fusion machines, MHD effects in liquid metal flows are studied to design properly
critical components (e.g. blankets) of experimental reactors (8).
In particular, in (1) an analytical solution is obtained for the natural
2
convection in a 2D rectangular cavity in a vertical magnetic field. Pan and
Li (2) have studied mixed convection in a parallel vertical channel in a horizontal magnetic field, in microgravity conditions and with a sinusoidally
oscillating gravity (g-jitter effect). In (3) the mixed convection in an horizontal circular channel is studied numerically in presence of a vertical
uniform magnetic field. An experimental study of the natural convection
of a N a22 K 78 alloy in a rectangular cavity with a vertical magnetic field
was presented by Burr and Müller (4), demonstrating that magnetic field
systematically reduces the heat exchange in the fluid. In (5), mixed convection in a vertical channel is studied considering the effect of viscous
dissipation and Joule heating. Sposito and Ciofalo (6) obtained analytical
solutions of the local balance equations for a fully developed mixed convection in a vertical parallel channel, with isothermal walls and various
electrical boundary conditions.
In the following points a short overview of the fundamental phenomena and applications of MHD will be given.
1.1.1
Fundamental phenomena of MHD
Since the first half of the nineteenth century (1832), Faraday and his contemporaries knew about the existence of induced currents in conducting
media, either solid or liquid, being in motion in a magnetic field. This
phenomenon is quite dual, in fact two separate but strongly linked events
may be described:
• the induced currents create themselves an induced magnetic field,
perturbing the original one,
• the interaction between induced currents and total field appears,
perturbing the original motion of the conducting media.
These are the two fundamental effects of MHD, which may be defined
as the science of the motion of electrically conducting fluids under the
action of magnetic fields. These two effects express the mutual interaction
3
Figure 1.1 — A simple MHD experiment.
between the fluid velocity field and the electromagnetic field; the motion
affects the magnetic field as well as the magnetic field affects the motion.
To appreciate these effects a simple experiment may be set up. Dropping a sheet of conducting material (e.g. aluminum) across a narrow gap
between two magnets pole faces, results in a viscous damping of the
falling motion of the sheet. The effects is more pronounced the higher
is the conductivity of the material or the strength of the magnetic field. In
the extreme situation of a sheet made of superconducting material, the viscous damping would diverge towards a quasi-elastic bouncing back of the
sheet. It is interesting to note how the material properties of the moving
conductor may change a non conservative viscous effect to a conservative
elastic behavior. This simple experimental setup is sketched in Figure 1.1.
Since that time, several practical applications of these phenomena were
studied. Some of them are briefly described in the following sections.
1.1.2
Liquid metal flow meters and pumps
Magnetic flowmeters, also known as electromagnetic flowmeters or induction flowmeters, are devices able to measure the velocity of a conducting
fluid flowing into a duct which is crossed by a controlled magnetic field,
by measuring the changes of induced voltage across the channel walls.
A typical magnetic flowmeter operates based upon Faraday’s Law of
4
electromagnetic induction, which states that a voltage will be induced in
a conductor moving through a magnetic field. The magnitude of the induced voltage E is directly proportional to the velocity of the conductor
V , conductor width D, and the strength of the magnetic field B,
E ∝ BDV.
(1.1)
A schematic description of an MHD flowmeter is given in Figure 1.2:
Two electrical coils are placed on the opposite sides of a pipe to generate a
magnetic field which is transverse with respect to the fluid motion. As the
electrically conductive fluid moves through the field with average velocity
V , two electrodes placed on opposite sides of the pipe, on a line perpendicular to the direction of the magnetic field, sense the induced voltage.
The distance between electrodes represents the width of the conductor D.
An insulating liner prevents the signal from shorting to the pipe wall. The
only variable in this application of Faraday’s law is the velocity of the conductive liquid V because field strength is a controlled constant and the
electrode spacing is fixed. Therefore, the output voltage E is directly proportional to liquid velocity, resulting in the linear output of a magnetic
flow meter.
An MHD pump relies on the same principles of an MHD flowmeter,
but applied in reverse order: the conducting fluid is confined into an insulating duct, where electrode plates are installed so to generate an electric
field which is orthogonal to an externally impressed magnetic field. The
electromotive force exerted by the electrode plates creates a current, and
the moving charges are accelerated by the presence of the magnetic field.
A typical setup of such device is given in Figure 1.3.
Electromagnetic pumps are commonly used in liquid-metal-cooled
reactor plants where liquid lithium, sodium, potassium, or sodiumpotassium alloys are used. Other metallic and nonmetallic liquids of sufficiently high electrical conductivity, such as mercury or molten aluminum,
lead and bismuth, may also be pumped in nonnuclear applications. The
absence of moving parts within the pumped liquid eliminates the need
for seals and bearings that are found in conventional mechanical pumps,
5
thus minimizing leaks, maintenance, and repairs, and improving reliability. In liquid-metal-cooled nuclear reactor plants, electromagnetic pumps
with a capacity of up to several thousand liters per minute have operated
without maintenance for decades.
Figure 1.2 — A functional sketch of an MHD flowmeter.
1.1.3
Metallurgy
In the past decade, the continuous casting (CC) process has progressed
markedly, has solved many technological problems, and has spread
widely as a result. In the meantime, attempts to develop novel casting
processes have been made for further innovation in the coming century.
Among them, the application of MHD has aroused a great deal of interest
as a powerful elementary technology to develop casting processes with
higher productivity and better cast steel quality.
Non-contact force electromagnetic field influence on the liquid metal is
known to be a component part of modern technological processes both in
ferrous and non-ferrous metallurgy. With the help of electromagnetic field
it is possible to stir metal in the melting machines, and during the nonfurnace treatment. Therefore, the processing of alloy obtaining become
6
Figure 1.3 — A functional sketch of an MHD pump.
much simpler and accelerated, energy consumption per unit of metal mass
is reduced, the working conditions in the metallurgical industry become
better, metal quality increases, and the degree of process automation increases as well.
The application of MHD in the continuous casting process of steel has
now advanced to electromagnetic stirring in the mold and the control of
molten steel flow by an in-mold direct-current magnetic field brake. These
applied MHD technologies are designed to improve further the continuous casting process capability. They improve the surface quality of cast
steel by homogenizing the meniscus temperature, stabilizing the initial solidification, and cleaning the surface layer. They also improve the internal
quality of steel slab by preventing the inclusions from penetrating deep
into the strand pool and promoting the flotation of argon bubbles.
A typical electromagnetic stirring system, shown in Figure 1.4, consists
of a MHD stirrer (pos. 1), thermal insulation (pos. 2) and melt (pos. 3).
The importance of a good understanding of the MHD phenomena is relevant, with respect to this particular application, to predict the velocity
distribution patterns in the furnace bath in order to control the quality of
the produced material.
7
Figure 1.4 — A typical setup for an electromagnetic stirrer.
1.1.4
MHD power generation
MHD power generation is an advanced technology for generating electrical power from fossil fuels by passing an electrically conducting fluid
through a magnetic field without rotating machinery or moving mechanical parts.
The underlying principle of MHD power generation is elegantly simple. Typically, an electrically conducting gas is produced at high pressure
by combustion of a fossil fuel. The gas is then directed through a magnetic field, resulting in an electromotive force within it in accordance with
Faraday’s law of induction. The MHD system constitutes a heat engine,
involving an expansion of the gas from high to low pressure in a manner similar to that employed in a conventional gas turbo generator. In the
turbo generator, the gas interacts with blade surfaces to drive the turbine
and the attached electric generator. In the MHD system, the kinetic energy
of the gas is converted directly to electric energy as it is allowed to expand.
In principle, any electrical conducting fluid can be used as the working
fluid, and power generation has been demonstrated with a number of such
fluids, varying from liquid metals to hot ionized gases. The absence of
moving machinery allows the MHD generator to operate at much higher
8
temperatures than other power generation systems and, therefore, higher
efficiencies can be reached. A simple scheme of an MHD generator is given
in Figure 1.5.
Figure 1.5 — A simple scheme of an MHD generator.
A MHD/steam plant can achieve efficiencies of up to 60% with less environmental impact than from any other direct coal-burning technology.
Retreofitting and/or repowering of existing thermal power plants is possible with a significant increase of the efficiency of the plant. Efficiencies
greater than 65-70 % can be reached if a triple cycle, including an MHD
generator, a gas turbine and a steam turbine, is utilized.
1.1.5
Fusion reactors breeding blankets
In nuclear fission reactors, a reflector is a region of unfueled material surrounding the core. Its function is to scatter neutrons that leak from the core
and thereby return some of them to the core. The liquid-metal reactor represents a special case. Most sodium-cooled reactors are deliberately built
to allow a large fraction of their neutrons - those not needed to maintain
the chain reaction - to leak from the core. These neutrons are valuable because they can produce new fissile material if they are absorbed by fertile
material. Thus, fertile material - generally depleted uranium or its dioxide
- is placed around the core to catch the leaking neutrons. Such an absorb9
ing reflector is referred to as a blanket or a breeding blanket.
In fusion experimental reactors where part of the fuel consists of Tritium, which is unavailable in nature, special mixtures of Lithium compounds are used in the breeding blankets, in order to trigger the production of Tritium as a consequence of neutron absorption and radioactive
decay. Often the breeding material is accompanied by Beryllium neutron
multipliers. Such blankets are still in developing phase, and some of their
designs are based on a fluid flow of molten salts across narrow rectangular
channels. A typical layout of a breeding blanket is given in Figure 1.6.
Figure 1.6 — A typical layout of a breeding blanket.
In such MHD flows in blanket channels, interaction of the induced
electric currents with the applied plasma-confinement magnetic field results in the flow opposing Lorentz force that may lead to high MHD pressure drop, turbulence modifications, changes in heat and mass transfer
10
and other important MHD phenomena. In fact, even if liquid blanket designs have the best potential for high power density, MHD interactions of
the flowing liquid with the confinement B-field may lead to:
• extreme MHD drag resulting in high blanket pressure and stresses,
and flow balance disruption,
• velocity profile and turbulence distortion resulting in severe changes
in heat transfer, corrosion and tritium transport.
MHD effects are specific to the blanket design. In self-cooled liquid
metal blankets, the MHD pressure drop is considered as the main issue,
while in self-cooled molten salt blankets, the blanket performance depends on the degree of turbulence suppression by a magnetic field.
1.2 Closure
This chapter was intended to give a broad but necessarily simplified
overview of the history of MHD research and the applications which were
envisaged and which are now reality.
In the following chapter the bulk of mathematical equations governing the MHD convective flow in channels and ducts will be investigated,
starting from the basic momentum, energy and potential equations and
combining them in a self-consistent system.
11
Chapter 2
Mathematical Formulation
In this chapter will be presented the framework of governing equations
which will be used in the rest of the work. Starting from the threedimensional formulation of the Navier-Stokes and energy conservation
equations, the MHD terms will be derived under specific assumptions and
included in the complete formulation.
In the next chapters specific geometries will be addressed, and the bulk
of governing equations will be simplified accordingly to the problems to
be illustrated.
2.1 Assumptions
This work deals with convective MHD flow of a conductive Newtonian
fluid with constant thermophysical properties, enclosed in simple geometries in the presence of a transverse imposed magnetic field. The fluid is
supposed to be incompressible and the flow to be laminar. The OberbeckBoussinesq approximation will be used to model buoyancy.
Moreover, the inductionless approximation is supposed to be valid, so
that the magnetic Reynolds number,
Rm =
UL
≪ 1,
ηm
(2.1)
where U is a typical velocity scale of the flow, L is a typical length scale of
12
the flow and ηm is the magnetic diffusivity, defined as:
ηm =
1
,
µ0 σ0
(2.2)
where µ0 is the magnetic permeability of the medium and σ0 its electrical
conductivity.
In this situation, diffusion dominates induction by fluid motion, and
the magnetic field in the fluid is determined, at least at leading order, by
geometrical considerations (the geometry of the fluid domain and of the
external current-carrying coils or magnets). The additional field induced
by the fluid motion is weak compared with the applied field. This is the
domain of liquid-metal magnetohydrodynamics in most circumstances of
potential practical importance.
2.2 Governing equations
The starting framework of equations consists in the conservation of mass
equation, the three dimensional Navier-Stokes equation, and the heat
equation, expressed in their time-dependent form. The classical formulation of this equation system is:
∂ρ
= ∇ · ρu
∂t
∂u
ρ
+ u · ∇u = ρg − ∇p + µ∇2 u + fb + fL
∂t
DT
= ∇ · (k∇T ) + D : τ + qJ
ρcv
Dt
where fb represents the buoyancy force term, fL the Lorentz force
(2.3)
(2.4)
(2.5)
term,
D : τ the viscous dissipation term and qJ the Joule dissipation term.
Making use of the Boussinesq approximation, the buoyancy term can
be written as:
fb = −ρ0 [1 − β(T − T0 )]g,
(2.6)
where ρ0 is the fluid density, considered constant, β is the thermal expansion coefficient and T0 is a reference temperature for the fluid, chosen accordingly to the specific problem to be solved.
13
The Lorentz force and the Joule dissipation term will be discussed in
detail in the following paragraphs.
Electric potential field. To describe the phenomena arising in the conducting fluid flowing through a magnetic field, an additional governing
equation has to be included in the system, to account for the induced currents.
Starting from the continuum form of Ohm’s equation, written in a reference frame which moves together with the fluid with velocity u through
a magnetic field B, namely:
J = σ(−∇φ + u × B),
(2.7)
and assuming that in the domain to be analyzed the conservation of charge
is verified or, in other words:
∇ · J = 0,
(2.8)
combining together (2.7) and (2.8) it is possible to obtain:
∇2 φ = ∇ · (u × B).
(2.9)
This scalar equation describes the electric potential field which is established in the fluid due to its flowing through the magnetic field. The
induced current distribution is given by (2.7) and due to the inductionless hypothesis the magnetic field arising from this current is negligible
compared to the impressed one. Equation (2.9) has to be included in the
system of equations (2.3) - (2.5) to complete the mathematical formulation.
Lorentz body force.
The general expression of the Lorentz force per vol-
ume of fluid is:
fL = J × B,
(2.10)
which can be written, following (2.7), as:
fL = σ(−∇φ + u × B) × B.
(2.11)
This term is included in (2.4) to take into consideration the effect of the
magnetic field on the flow.
14
Joule dissipation.
In general, the dissipative energy due to the flow of
the current in the fluid can be written as:
qJ =
1 2
|J| = σ| − ∇φ + (u × B)|2
σ
(2.12)
and can be added in the right-hand side of (2.5) together with the viscous
dissipation contribution.
Rewriting the system.
In the aforementioned hypotheses, it is possible
to rewrite the system of (2.3) - (2.5) including all the MHD effects as follows:
∇·u=0
(2.13)
∂u
ρ0
+ u · ∇u = ρ0 β(T − T0 )g − ∇p + µ∇2 u + J × B (2.14)
∂t
DT
= ∇ · (k∇T ) + D : τ + σ| − ∇φ + (u × B)|2
ρcv
(2.15)
Dt
2.3 Boundary conditions
In this section the various types of boundary conditions, which will be
adopted in the solution of the problems shown in the following chapters,
will be analyzed. The closure of the system (2.13) - (2.15) is guaranteed by
dynamic, thermal and electric potential boundary conditions.
2.3.1
Dynamic boundary conditions
These conditions define the dynamic behavior of the system at the boundary of the domain. Being this study oriented toward flow in ducts and
channels, the dynamic boundary conditions are condensed in the no-slip
hypothesis. This yields to:
u(Σ, t) = 0,
where Σ is the domain boundary.
15
(2.16)
2.3.2
Thermal boundary conditions
In this work, convective flows in steady periodic regime will be analyzed.
A steady periodic regime is a transient evolution of a system driven by a
sinusoidal forcing, which stabilizes itself after a sufficiently high number
of cycles. After the stabilization, the flow can be treated as a fully periodic
flow, with the same frequency of the driving force, but possibly phaseshifted with respect to it.
In the sample problems solved in the following chapters, the driving force will be provided by wall temperature distributions sinusoidally
changing over time. This can be expressed as:
T (Σ, t) = T1 (Σ) + T2 (Σ) cos(ωt),
(2.17)
where T1 represents the average temperature of the wall and T2 the oscillation amplitude of the wall temperature around its mean value. This
varying temperature distribution may be applied to the whole boundary,
Σ, or just to part of it. Examples will be given in the following chapters.
2.3.3
Potential boundary conditions
The nature of the walls material has a great influence on the solution of the
magnetohydrodynamic convective flow problem in channel and ducts. In
fact, depending on the conductivity of the walls, the distribution of electric
potential across the fluid changes completely, thus affecting the current
distribution in the channel cross section and the intensity of the Lorentz
body forces.
Perfectly insulating walls. In the following chapters, the assumption of
perfect electrically insulating walls will be adopted. This can be formalized as:
J · n = 0,
(2.18)
Σ
where n is the unit vector normal to the boundary Σ. This situation well
represents practical applications of MHD phenomena, like the aforemen16
tioned electromagnetic flow meter. Moreover, the perfect electrical insulation decouples the electrodynamic problem in the fluid from that in the
walls.
Perfectly conducting walls.
In case of walls made of an extremely high
conductivity material, the distribution of electric potential across the domain boundary is constant,
φ(Σ, t) = φ0 ,
(2.19)
which can be arbitrarily set to zero. This circumstance is less realistic than
that discussed before, and finds a few practical applications. Again, the
electrodynamic problem in the fluid and in the wall is decoupled.
Conducting walls: solid walls.
When the conductivity of the walls is
neither zero nor infinite, the solution of the electrodynamic problem in
the solid walls around the fluid is no more decoupled from that inside the
fluid. This means that the two problems have to be solved simultaneously.
This normally makes the system of governing equations more complicated
and adds additional conditions at the internal boundary between fluid and
walls. In Figure 2.1 a typical example of this situation is shown: the flow
of a liquid metal in a steel conduit.
In this case, the continuity of electric potential and of the normal component of the vector J are always verified at the boundary Γ between fluid
and solid domains. In other words:
J · n = Jw · n ;
Γ
Γ
φ = φw ,
Γ
(2.20)
Γ
which constitutes, together with Poisson’s equation for the electric potential,
∇2 φw = 0,
(2.21)
and the heat equation in the solid domain,
ρw cv, w
DTw
= ∇ · (kw ∇Tw ),
Dt
17
(2.22)
Figure 2.1 — A solid conducting walls example.
an additional set of equations to be incorporated and solved simultaneously with the equation system defined in the fluid domain.
The conclusions drawn before for the perfectly insulating or conducting walls is then transferred to the outer boundary of the conductive walls,
Jw · n = 0,
(2.23)
Σ
which provides the final closure to the equation system.
Conducting walls: thin walls.
In order to simplify the solution of the
aforementioned conducting walls problem, without solving simultaneously for temperature, current and electric potential distribution in the
solid walls domain, it is possible to use the thin walls approximation. This
approximation implies that if the wall is sufficiently thin, the temperature
and electric potential changes across the wall thickness can be neglected,
together with the wall current density normal to the walls themselves.
This allows to write the following closure equations, which have to be
solved for the solid domain:
J · n = ∇ · (c∇φw );
Γ
φ = φw = φw ,
Γ
Γ
(2.24)
Σ
where c is the conductance parameter, being,
c=
σw tw
σL
18
(2.25)
which represents the ratio of thickness and electric resistance of the walls
with respect to the fluid. This approach is broadly used in the solution
of practical problems regarding liquid metal flows in metallic channels, as
reported in (12).
2.4 Dimensionless formulation
Magnetohydrodynamic flows are studied for a broad range of materials and operating conditions: from seawater, in case of magnetic naval
propulsion, to molten salts, in case of fast fission reactors, to liquid metals, in case of fusion reactor breeders or casting melts. Among all these
cases, the materials range sweeps several orders of magnitudes of densities, viscosities and conductivities. The same can be said for the operating
conditions, from the small magnetic fields of a lab setup to the stronger
ones of superconducting coils, from the moderate temperatures of a sea
stream to those extremely high of a steel continuous cast.
To set up a computational tool able to face the most diverse kinds of
problems, a dimensional analysis of the set of equations obtained up to
now has to be performed. With the dimensional analysis, in fact, it is possible to rewrite the mathematical model in a form which is independent
of the thermophysical properties of the fluid, using a set of parameters
obtained as dimensionless ratios of them.
Besides the well known Reynolds, Prandtl and Grashof numbers,
Re =
U0 Dρ0
;
µ
Pr =
µcp
;
λ
Gr =
gβ(T − T0 )D3
,
ν2
(2.26)
two new parameters will be introduced, to express the ratios of electromagnetic to inertial and viscous forces. These are the interaction parameter, or Stuart number:
N=
σLB02
,
ρU0
(2.27)
and the Hartmann number,
Ha =
√
N Re = LB0
19
r
σ
,
ρ0 ν
(2.28)
Table 2.1 — A selection of parameters related to fusion experimental devices.
which can cover a very wide range of values, depending on the application
to be modeled. In Table 2.1 values of Ha, Re and N for a selection of
fusion experimental devices are reported. It can be noted that, for similar
values of the Reynolds number, the flow pattern changes from turbulent to
laminar for increasing values of the Hartmann number. In fact, as it will be
demonstrated in the following Chapters, one of the effects of a transverse
magnetic field on the flow of an electrically conductive fluid is to suppress
almost completely any kind of fluctuation (e.g. turbulence) in the velocity
field.
It has to be stressed that the choice of the specific dimensionless groups
and variables depends strictly on the problem to be solved. In the next
chapters will be given examples of dimensional analysis of convective
MHD problems, and analytical and numerical solutions will be presented.
2.5 Closure
This chapter was intended to define the bulk of equations which will be
used in the following to solve MHD convective flow problems in simple
1D and 2D geometries. The system of equations is derived from the complete 3D unsteady mass, momentum and energy balance equations, with
the addition of the Poisson equation for the electric potential. Special closure equations are defined to create the mutual influence between motion
and magnetic field, descending directly from the definition of the generalized Ohm’s law and Lorentz force, and to model the behavior at the
boundaries of the domain.
The equation system (2.3) - (2.5) will be adapted case by case to the spe20
cific problem to be solved, in order to simplify the equations by removing
the unnecessary terms. Also the equations resulting from the dimensional
analysis will be adapted to fit the specific set of thermophysical properties
and geometrical parameters of the problem under investigation. In the following chapters, either analytical or numerical solutions will be presented,
and the results will be critically discussed.
21
Chapter 3
MHD mixed convection flow in a
vertical parallel channel with
transverse magnetic field. Steady
periodic regime
22
Symbols used in the section
A(t)
function of time, defined in (3.7)
A1
integration constant
B
intensity of the magnetic field
Br
Brinkman’s number
g
gravitational acceleration
g
modulus of the gravitational acceleration
Gr
Grashof number, defined in (3.12)
i
imaginary unit
In
modified Bessel function of first kind and order n
Kn
modified Bessel function of second kind and order n
L
channel half width
M
Hartmann number, defined in (3.13)
n
integer number
p
pressure
P
difference between the pressure and the hydrostatic pressure
Pr
Prandtl number, defined in (3.12)
Re
Reynolds number, defined in (3.12)
ℜe
real part of a complex number
t
time
T
temperature
T0
mean value of the wall temperature
∆T
amplitude of the wall temperature oscillation
U
fluid velocity
U
axial component of the fluid velocity
U0
average value of the fluid velocity
u
dimensionless axial component of the fluid velocity,
defined in (3.10)
u
∗
dimensionless complex valued function, defined in (3.20)
u∗a , u∗b
dimensionless complex valued function, defined in (3.24)
X
longitudinal coordinate
Y
wall-normal coordinate
23
y
dimensionless wall-normal coordinate, defined in (3.10)
Greek Symbols
α
thermal diffusivity
β
thermal expansion coefficient
η
dimensionless time, defined in (3.10)
θ
dimensionless temperature, defined in (3.11)
θ∗
θa∗ ,
dimensionless complex valued function, defined in (3.21)
θb∗
λ
λ
dimensionless complex valued functions, defined in (3.25)
dimensionless parameter, defined in (3.11)
∗
dimensionless complex valued function, defined in (3.22)
λ∗a , λ∗b
dimensionless complex valued functions, defined in (3.26)
µ
dynamic viscosity
ν
kinematic viscosity
̺0
fluid density at the reference temperature
ω
frequency of the wall temperature oscillation
Ω
Ω
dimensionless frequency, defined in (3.13)
′
dimensionless parameter, defined in (3.13)
24
In this chapter, mixed convection in a vertical parallel channel will be
analyzed. MHD effects, due the presence of a uniform magnetic field
orthogonal to the flow direction, are taken into account. The thermal
boundary condition considered is such that one of the wall temperatures
varies sinusoidally with time, while the other wall has a constant temperature. Reference is made to parallel fully-developed flow, in steady
periodic regime. The local balance equations are expressed accordingly,
and two boundary value problems are obtained, accounting for the mean
value and the oscillating term of the velocity and temperature distributions respectively. The obtained boundary value problems are then solved
analytically and numerically.
3.1 Governing equations
Figure 3.1 — Representative scheme of the system
In this section, the governing equations describing the phenomenon
will be presented and written in a dimensionless form.
Let us consider a Newtonian fluid with constant thermophysical properties flowing in a vertical parallel channel having width 2L, as shown
25
in Figure 3.1. Let us choose the axial coordinate X parallel to the gravitational acceleration g but with opposite direction, and let us assume a
uniform magnetic field having intensity B in the direction orthogonal to
X. Reference is made to parallel and laminar flow, so that the only non–
vanishing component of the velocity vector U is the X-component U . As
it is well known, the hypothesis of parallel flow in mixed convection implies that the buoyancy forces are not too intense; the hypothesis of parallel flow is realistic just for sufficiently small values of the Grashof number.
The viscous dissipation and Joule heating effects will be neglected and the
Boussinesq approximation will be invoked. Under the above described
assumptions, the velocity field must be solenoidal, i.e. U = U (Y, t). The
thermal boundary conditions prescribed at Y = ±L are,
T (X, −L, t) = T1 ,
(3.1)
T (X, +L, t) = T2 + ∆T cos(ω t).
(3.2)
Since the boundary conditions do not imply heating or cooling of the fluid,
heat transfer arises only in the wall-normal direction and T = T (Y, t).
Moreover, the quantity:
ω
T0 =
4πL
Z
2π/ω
dt
0
Z
L
dy T (y, t),
(3.3)
−L
will be defined, which represents the mean value of the fluid temperature in a period, and which will be chosen as reference temperature value
within the Boussinesq approximation.
Since the flow rate is steady, the mean velocity of the fluid in a duct
section,
1
U0 =
2L
does not depend on time.
Z
L
U (Y, t) dY.
(3.4)
−L
Thus, simplifying the momentum equation (2.14), the X and Y components of the local momentum balance equation may be written as:
̺0
∂U
∂P
=−
+ ̺0 g β(T − T0 ) + µ∇2 U − σ U B 2 ,
∂t
∂X
26
(3.5)
∂P
=0 ,
(3.6)
∂Y
where the quantity P = p + ̺0 g β has been introduced. Equation (3.6)
implies P = P (X, t). By differentiating with respect to X both sides of
(3.5) one obtains ∂ 2 P/∂X 2 = 0. One can then define a function A(t) such
that
∂P
= −A(t) ,
∂X
(3.7)
and (3.5) can be rewritten as
̺0
∂2U
∂U
= A(t) + ̺0 g β(T − T0 ) + µ 2 − σ U B 2 .
∂t
∂Y
(3.8)
The complete local energy balance equation is
∂T
ν
∂2T
=α 2 +
∂t
∂y
Cp
∂U
∂Y
2
σU 2 B 2
+
.
ρ0 Cp
(3.9)
where the last two terms account for the viscous and ohmic dissipations.
These two non-linear terms will be neglected during the derivation of the
analytical solution of the problem.
Let us introduce the following dimensionless quantities:
u=
θ=
Gr =
T − T0
,
∆T
U
,
U0
y=
θ1, 2 =
Y
,
D
η = ωt,
T1, 2 − T0
,
∆T
λ=
(3.10)
A(t)D2
,
µU0
gβ∆T D3
U0 D
ν
µU02
,
Re
=
,
P
r
=
,
Br
=
,
ν2
ν
α
k∆T
Gr
σB 2 D2
D2 ω
2
, M =
, GR =
,
Ω=
ν
µ
Re
(3.11)
(3.12)
(3.13)
where D = 4L is the hydraulic diameter of the channel. Using the defined
dimensionless groups, (3.8) and (3.9) can be rewritten as:
Gr
∂2u
∂u
=λ+
θ + 2 − M2 u ,
∂η
Re
∂y
2
∂θ
∂u
∂2θ
ΩP r
+ M 2 Br u2 .
= 2 + Br
∂η
∂y
∂y
Ω
27
(3.14)
(3.15)
The boundary condition for the dimensionless velocity and for the dimensionless temperature are:
u(−1/4, η) = u(1/4, η) = 0,
(3.16)
θ(−1/4, η) = θ1 ,
(3.17)
θ(1/4, η) = θ2 + cos(η).
(3.18)
Moreover, (3.4) provides a constraint for the dimensionless velocity:
Z 1/4
1
u(y, η) dy.
(3.19)
=
2
−1/4
3.2 Analytical solution
In the steady periodic regime it is possible to solve the governing equations analytically, by neglecting the non linear dissipation terms. Let us
consider the functions u(y, η), θ(y, η) and λ(η) as real parts of three complex valued functions:
u(y, η) = ℜe[u∗ (y, η)] ,
(3.20)
θ(y, η) = ℜe[θ∗ (y, η)] ,
(3.21)
λ(η) = ℜe[λ∗ (η)] .
(3.22)
The functions u∗ (y, η), θ∗ (y, η) and λ∗ (η) must fulfill the following boundary value problem:

∂ 2 u∗
∂u∗


− M 2 u∗ + GRθ∗ + λ∗ ,
=
Ω

2

∂η
∂y









∂ 2 θ∗
∂θ∗


,
=
ΩP
r

2

∂η
∂y




u∗ (−1/4, η) = 0 = u∗ (1/4, η),











θ∗ (−1/4, η) = θ1 , θ∗ (1/4, η) = θ2 + eiη ,








R

 1/4 u∗ (y, η) dy = 1 ,
−1/4
2
28
(3.23)
Assuming that
u∗ (y, η) = u∗a (y) + GR u∗b (y) ei η ,
(3.24)
θ∗ (y, η) = θa∗ (y) + θb∗ (y) ei η ,
(3.25)
λ∗ (η) = λ∗a + GR λ∗b ei η ,
(3.26)
it is possible to obtain two boundary value problems:
 2 ∗
d ua


− M 2 u∗a + GRθa∗ + λ∗a = 0,

2

dy









d2 θa∗


= 0,


dy 2




and
u∗a (−1/4) = 0 = u∗a (1/4),











θa∗ (−1/4) = θ1 , θa∗ (1/4) = θ2 ,








R

 1/4 u∗ (y) dξ = 1 ,
−1/4 a
2





















(3.27)
d2 u∗b
− (M 2 + iΩ)u∗b + θb∗ + λ∗b = 0,
dy 2
d2 θb∗
− iΩP rθb∗ = 0,
dy 2

u∗b (−1/4) = 0 = u∗b (1/4),










θb∗ (−1/4) = 0, θb∗ (1/4) = 1,








 R 1/4 ∗
u (y) dy = 0.
−1/4 b
(3.28)
The boundary value problem (3.27) describes the mean value of the
complex fields u∗ , θ∗ e λ∗ , while the boundary value problem (3.28) provides the oscillating parts of the same quantities.
29
The solution of the boundary value problem (3.27) is:
θa∗ (y) = 4θ2 y
λ∗a =
u∗a (y) = 4
with θ2 = −θ1
(3.29)
M 3 (eM/2 + 1)
M eM/2 + M − 4eM/2 + 4
(3.30)
GRθ2 y
M (eM − 1)
+
+
M2
M eM − M − 4eM + 8eM/2 − 4
GRθ2 eM/4 sinh(M y)
−2
+
M 2 (eM/2 − 1)
M (eM − 1)eM/4 cosh(M y)
. (3.31)
−2
(M eM − M − 4eM + 8eM/2 − 4)(eM/2 + 1)
As expected, in the limit M → 0, one obtains the classical expression
for the dimensionless velocity distribution,
lim u∗a (y) = −
M →0
1
(36 + GR θ2 y)(−1 + 16y 2 )
24
(3.32)
The dimensionless temperature θb∗ solution of (3.28) is
√
−e1/4(1−4y) iΩP r + e1/4(3+4y)
√
θb∗ (y) =
−1 + e iΩP r
√
iΩP r
,
(3.33)
which can be substituted in the momentum equation of the boundary
problem (3.28) to obtain an expression of the dimensionless velocity:
u∗b (y) = [A1,a (M, Ω, P r) + A1,b (M, Ω)λ∗b (M, Ω, P r)] Φ1 (ξ, M, Ω)+
+ [A2,a (M, Ω, P r) + A2,b (M, Ω)λ∗b (M, Ω, P r)] Φ2 (ξ, M, Ω)+
− λ∗b (M, Ω, P r)Φ3 (ξ, M, Ω) + Φ4 (ξ, M, Ω, P r). (3.34)
The dimensionless pressure gradient can be determined using the integral constraint in (3.28) and is:
λ∗b =
−Ψ1,a (M, Ω, P r) − Ψ2,a (M, Ω, P r) − Ψ4 (M, Ω, P r)
,
Ψ1,b (M, Ω) + Ψ2,b (M, Ω) − Ψ3 (M, Ω)
where the functions A, Φ and Ψ are reported in the appendix.
30
(3.35)
3.3 Numerical solution
To support the analytical investigation of the magnetohydrodinamic effects on a steady-periodic, mixed-convective flow in a parallel channel,
a finite differences scheme has been adopted to provide a comparison
benchmark for the analytical results, and to extend the validity of the solution to the non-linear case. A first order forward-Euler time discretization
scheme was adopted for the present implementation, with second order
central difference spatial derivatives.
The problem to be solved consists in a system of two PDEs, (3.14) and
(3.15), together with the boundary conditions, (3.16) - (3.18). The system
will be solved with respect to the variables u and θ, but the results will be
expressed in terms of average value and oscillation amplitude ua , |ub |, θa ,
|θb |, to ease the comparison with respect to the analytical results presented
previously.
To apply the finite differences method, the domain y = [−1/4, 1/4] of
the problem is discretized with a mesh of Ny equally spaced points. From
this assumption (3.14) and (3.15) can be rewritten for each internal mesh
point as:
j
θj − 2θij + θi−1
θj+1 − θij
= i+1
ΩP r i
+ Br
∆η
∆y 2
uji+1 − uji−1
2∆y
!2
+ M 2 Bruji 2 (3.36)
and,
Ω
uj − 2uji + uji−1
− uji
uj+1
i
− M 2 uji 2 + GRθij + λ
= i+1
∆η
∆y 2
(3.37)
In a forward-advancing scheme, for each time step j, the values at the
internal mesh points i of the function θ(y, η) at the time step j + 1 are
obtained solving the (3.36) for θij+1 . This yields the set of linear equations:


!2
j
j
j
j
j
u
−
2θ
+
θ
−
u
θ
∆η  i+1
i
i−1
i+1
i−1
+ Br
+ M 2 Bruji 2  ,
θij+1 = θij +
2
ΩPr
∆y
2∆y
(3.38)
with i = 2...Nr − 1.
31
The value of the dimensionless temperature θi at the boundary mesh
points, i = 1, Ny , is given by the boundary conditions (3.17) - (3.18), which
can be expressed as:
θ1j+1 = −θ2 ,
(3.39)
j+1
= θ2 + cos η,
θN
y
(3.40)
for each j. To determine the velocity field for any internal mesh point, one
has to solve the (3.37) for uj+1
, as:
i
#
"
∆η uji+1 − 2uji + uji−1
∆η 2
∆η j
j+1
j
j
u i = ui 1 −
,
λ + GR θi +
M +
Ω
Ω
Ω
∆y 2
(3.41)
while, at the wall boundaries, the no-slip condition yields:
uj+1
= 0,
1
(3.42)
uj+1
Ny = 0,
(3.43)
for each j.
Then, even if it is possible in principle to determine the value of uji for
each mesh point of the domain using the (3.41), the dimensionless pressure
gradient λj is still an unknown function of the time step j. Its value can be
determined using the integral constraint equation (3.19) rewritten in finite
terms as:
Nr
X
uj+1
∆y = 1/2,
i
(3.44)
i=1
where the dimensionless velocity uj+1
can be expressed as:
i
uj+1
= ai + bi (λj + ci ),
i
(3.45)
where
∆η 2
1−
ai =
M ,
Ω
∆η
,
bi =
Ω
#
"
j
j
j
−
2u
+
u
u
i
i−1
ci = GR θij + i+1
.
∆y 2
uj1
32
(3.46)
(3.47)
(3.48)
By properly managing the sums, Eq. (3.44) can be rewritten as
Ny
X
uj+1
∆y
i
=
i=1
Ny
X
[ai + bi (λj + ci )]∆y =
i=1
=
Ny
X
ai ∆y + λ
j
i=1
Ny
X
bi ∆y +
i=1
Ny
X
bi ci ∆y = 1/2, (3.49)
i=1
and, the dimensionless pressure gradient can be evaluated as
λj =
1/2 − A − BC
,
B
(3.50)
where
A=
Ny
X
ai ∆y,
(3.51)
X
bi ∆y,
(3.52)
X
bi ci ∆y.
(3.53)
i=1
Ny
B=
i=1
Ny
BC =
i=1
So, for each time step j, the dimensionless pressure gradient λj shall be
initialized to an arbitrary value. The temperature field will be computed
using Eq. (3.38) and a first tentative velocity field will be obtained from
Eq. (3.41). Of course the specialized equations will be used for the boundary nodes. After the first computation of the velocity field, the constraint
equation in its inverse form (3.50) will be used to obtain a new value of the
pressure gradient. This new value will be then used to compute again the
velocity field, and so on until a sufficient convergence of λj is obtained.
After that, the time step will be incremented, and the calculation will continue.
The previous steps yield a complete overview of the fluid dimensionless
velocity and temperature for each time step. To get some information on
amplitude and phase shift of the oscillations, an additional stability check
has to be done.
33
For each period of the wall temperature oscillation, a sum-of-square-errors
check is made on the dimensionless quantities. If the velocity profile has
sufficiently small changes at the end of a given period, within a prescribed
tolerance, then the amplitude of the oscillation is computed from the difference between the maximum-velocity value minus the mean-velocity
one, evaluated at each node, through the whole period.
The same approach can be used for the phase shift, evaluating the difference between the time of the maximum driving temperature and the time
of the local maximum of the oscillating quantities, after the stabilization
and in the range of a complete period.
3.4 Results and Discussion
In this section, first the most interesting features of the analytical solution
will be described. In particular, the influence of the parameters M , Ω and
P r on the oscillation amplitude of the velocity field will be discussed.
3.4.1
Velocity and temperature distributions
In Figure 3.2(a), the mean velocity distribution is reported for three different values assumed by the Hartmann number M , assuming conditions of
asymmetric heating, with θ2 = 1. The case M = 0 corresponds to the classical solution for the parallel channel without MHD effects. The presence
of a uniform magnetic field redistributes the flow, counteracting the flow
reversal at the cold wall. For high values of M , the dimensionless velocity
assumes a typical flat Hartmann profile.
In Figure 3.2(b) is shown the effect of the dimensionless group GR on
the flow distribution: for higher values of GR, the buoyancy forces become
a relevant contribution and the mixed convective scheme drifts towards
natural convection.
In Figure 3.3, the amplitude of the oscillating component of the dimensionless velocity is shown for several values of the dimensionless frequency. It is possible to see how the value of the Prandtl number influ34
3
2
2
1
1
u*a
ua*
3
0
0
-1
-1
-0.2
0.0
-0.1
0.1
0.2
-0.2
0.0
-0.1
y
0.1
0.2
y
(a)
(b)
Figure 3.2 — Mean velocity distributions across the channel for different values of M and
GR. Frame (a) GR = 500, M = 0 (continuous), M = 10 (dashed), M = 100 (dotted).
Frame (b) M = 0, GR = 0 (continuous), GR = 100 (dashed), GR = 500 (dotted).
ences the penetration depth of the velocity oscillations. While the velocity
oscillates across all the channel in the low P r case (Figure 3.3(b)) even
for high frequencies, for a high P r fluid (Figure 3.3(a)) the oscillations of
velocity experience a fast decay across the width of the channel and are
confined just close to the hot wall. This effects increases for increasing
0.0020
0.0020
0.0015
0.0015
È u*b È
È ub* È
dimensionless frequencies.
0.0010
0.0005
0.0010
0.0005
0.0000
0.0000
-0.2
0.0
-0.1
0.1
0.2
-0.2
0.0
-0.1
y
0.1
0.2
y
(a)
(b)
Figure 3.3 — Oscillating velocity amplitude distributions across the channel for different
values of Ω and P r in the non-magnetic case. Frame (a) P r = 5, Frame (b) P r = 0.05.
Ω = 0.001 (continuous), Ω = 100 (dashed), Ω = 500 (dotted).
In Figure 3.4 the influence of the external magnetic field on the amplitude of the velocity oscillations is shown. For increasing values of the
35
parameter M , the velocity oscillations are dampened accordingly. Figure
3.4(a) shows the non MHD case with M = 0, while (Figure 3.4(b)) shows
the behavior of the system at M = 0.05. Even for such a small Hartmann
number, the high frequency oscillations are completely suppressed. The
damping of oscillations (e.g. turbulence) is a well known effect induced
by transverse fields on flows of conductive fluids, and has great impact on
0.0020
0.0020
0.0015
0.0015
È u*b È
È ub* È
the estimation of hydraulic losses and flow patterns in many applications.
0.0010
0.0005
0.0010
0.0005
0.0000
0.0000
-0.2
0.0
-0.1
0.1
0.2
-0.2
0.0
-0.1
y
0.1
0.2
y
(a)
(b)
Figure 3.4 — Comparison of oscillating velocity amplitude distributions across the channel for different values of Ω between magnetic and non-magnetic case. Frame (a) M = 0,
Frame (b) M = 0.05. Ω = 0.001 (continuous), Ω = 100 (dashed), Ω = 500 (dotted).
Figure 3.5 shows the amplitude of the oscillating component of the dimensionless temperature across the channel. In analogy with the velocity,
the penetration depth of the temperature oscillation is greatly affected by
the Prandtl number: in fact, while for low Prandtl fluids (Figure 3.5(a))
the temperature oscillates across the whole width of the channel even for
high values of the dimensionless pulsation Ω, for high Prandtl fluids (Figure 3.5(b)) the temperature oscillation is segregated only close the the hot
wall. This last behavior is typical of the molten salts used either as coolant
or breeding material in several kinds of nuclear reactors, like FluorineLithium-Beryllium compounds, while the first behavior is typical of liquid
metals, like mercury, where thermal diffusivity is dominant with respect
to viscosity.
36
0.8
0.8
0.6
0.6
È Θ*b È
1.0
È Θb* È
1.0
0.4
0.4
0.2
0.2
0.0
0.0
-0.2
0.0
-0.1
0.1
0.2
-0.2
0.0
-0.1
y
0.1
0.2
y
(a)
(b)
Figure 3.5 — Comparison of oscillating temperature amplitude distributions across the
channel for different values of Ω and P r. Frame (a) M = 0, P r = 0.05, Frame (b)
M = 0, P r = 5. Ω = 0.001 (continuous), Ω = 100 (dashed), Ω = 500 (dotted).
3.4.2
Influence of dissipative terms
Figures 3.6 - 3.8 show some of the results obtained using the numerical
method. In this case, the effect of dissipations has been included in the
solution of the model, adding new informations to those already obtained.
It is possible to see in Figure 3.6 how, for increasing values of the
Brinkman number, the average temperature profile changes radically,
showing an absolute maximum close to the axis of the channel, where
the velocity is higher, and hence the viscous dissipation term plays a major role as an energy term in the heat equation. Figure 3.6 refers to a non
MHD case of a non symmetrically heated channel. It is interesting to see
how the core of the temperature profile remains linear close to the mid
plane at y = 0.
In the magnetic case, whose velocity and temperature components are
shown in Figure 3.7 - 3.8, the dissipation terms influence broadly the flow
pattern and the heat exchange. In Figure 3.7 are shown the average and
oscillating amplitude components of dimensionless velocity, for Br = 0.25
and for increasing values of the Hartmann number M = 0, 10, 20. The GR
ratio is equal to 100, while the dimensionless pulsation is Ω = 100. In both
cases one has P r = 1. In 3.7(a) the average component of velocity shows a
progressive suppression of the buoyant behavior for increasing Hartmann
37
3.0
2.5
Θ*a
2.0
1.5
1.0
0.5
0.0
-0.5
-0.2
0.0
-0.1
0.1
0.2
y
Figure 3.6 — Comparison of average temperature distributions across the channel for
M = 0 and different values of Br. Br = 0 (continuous), Br = 2 (dashed), Br = 4
(dotted).
number. In 3.7(b) the amplitude of oscillating velocity is reduced as well
by the presence of the magnetic field.
The effect of the Joule heating term is clearly visible in the profiles of
the dimensionless temperature components. In Figure 3.8(a), the average
temperature of the flow core increases rapidly for increasing values of M .
Moreover, for the amplitude of the oscillating temperature shown in Figure 3.8(b) it is possible to observe an increased complexity of the profile,
which now shows an inflection point with change of concavity whose location depends on the value of M .
3.4.3
Fanning friction factors and Nusselt numbers
The Fanning friction factors at the two walls can be defined as:
and,
∗
∗
Re + GRf1,b
Re eiη ,
f1 Re = ℜ f1,a
(3.54)
∗
∗
Re + GRf2,b
Re eiη ,
f2 Re = ℜ f2,a
(3.55)
38
1.5
0.0015
1.0
ua*
È u*b È
0.0010
0.5
0.0005
0.0
0.0000
-0.2
0.0
-0.1
0.1
0.2
-0.2
0.0
-0.1
y
0.1
0.2
y
(a)
(b)
Figure 3.7 — Distribution of the average (a) and oscillating (b) velocity components
across the channel for Br = 0.25 and different values of M . M = 0 (continuous), M = 10
(dashed), M = 20 (dotted).
where,
∗
Re
f1,a
and,
∗
f2,a
Re
∂u∗a =2
,
∂ξ ξ=−1/4
∗
f1,b
Re
∂u∗a = −2
,
∂ξ ξ=1/4
∗
f2,b
Re
∂u∗b =2
,
∂ξ ξ=−1/4
∂u∗b = −2
,
∂ξ ξ=1/4
(3.56)
(3.57)
so that the subscript 1 refers to the cold wall and the subscript 2 to the
hot wall. Naturally the f1 Re and f2 Re are complex valued functions, and
may be considered oscillating with the same frequency of the other functions, and can be decomposed in an average and an oscillating component
as shown before. Following this definition it is possible to estimate the
Fanning friction factors from the local derivatives of the dimensionless velocity at the two walls.
In Figure 3.9 the two average components of f1 Re and f2 Re are plotted
versus GR and for M = 0, 50, 100. Let us note how for increasing values
of M the friction factors increase accordingly, due to the steep velocity
gradients which are typical of the Hartmann flow. For increasing values
of GR, on the other hand, the values of f1 Re and f2 Re show opposite
variations, due to the onset of flow reversal. It is interesting to note how
for M = 1 flow reversal starts onsetting for GR ≈ 140, showing a negative
∗
f1,a
Re. With increasing M the onset of flow reversal drifts towards higher
39
1.0
5
0.8
3
0.6
Θa*
È Θ*b È
4
2
0.4
1
0.2
0
0.0
-0.2
0.0
-0.1
0.1
0.2
-0.2
0.0
-0.1
y
0.1
0.2
y
(a)
(b)
Figure 3.8 — Distribution of the average (a) and oscillating (b) temperature components
across the channel for Br = 0.25 and different values of M . M = 0 (continuous), M = 10
(dashed), M = 20 (dotted).
values of GR.
200
200
150
*
f2,a
Re
*
f1,a
Re
150
100
100
50
50
0
-50
0
0
100
200
300
400
500
0
100
200
GR
300
400
500
GR
(a)
(b)
Figure 3.9 — Average component of the Fanning friction factor at the cold (a) and hot (b)
walls plotted versus GR for different values of M . M = 0 (continuous), M = 50 (dashed),
M = 100 (dotted).
While the average components of f1 Re and f2 Re depend on M and
GR for a given θ2 , the oscillating components are also functions of the
Prandtl number and the dimensionless pulsation Ω.
In Figure 3.10 the values assumed at the cold wall by the amplitude
of the oscillating component of the dimensionless Fanning friction factor
∗
f1,b
Re are shown. Its values are plotted as functions of Ω for several values
of the parameter M and for P r = 0.05 or 5. In particular, Figure 3.10(a)
40
∗
shows the values assumed by f1,b
Re for P r = 0.05 and M = 0, 50, 100,
while Figure 3.10(b) shows the same values but for P r = 5. It is evident
how the increased value of P r completely dampens the high frequency
oscillations. It is interesting also to note how the presence of the magnetic
0.08
0.08
0.06
0.06
*
È f1,b
Re È
*
È f1,b
Re È
field introduces dampening even at low frequency.
0.04
0.04
0.02
0.02
0.00
0
200
400
600
800
1000
0
200
400
W
600
800
1000
W
(a)
(b)
Figure 3.10 — Amplitude of the oscillating component of the Fanning friction factor at
the cold wall plotted versus Ω for P r = 0.05 (a) or P r = 5 (b) and different values of M .
M = 0 (continuous), M = 50 (dashed), M = 100 (dotted).
∗
In Figure 3.11 the values assumed by f1,b
Re as function of Ω are shown
for several values of the parameter P r and for M = 0 or 100. In par∗
Re for M = 0
ticular, Figure 3.11(a) shows the values assumed by f1,b
and P r = 0.05, 0.5, 5, while Figure 3.11(b) shows the same values but for
M = 100. The behavior shown before is hereby confirmed: while the
increase of P r results in a reduction of high frequency oscillations, the increase of M reduces such oscillation amplitude even at very low frequencies.
The amplitude of the oscillating component of the dimensionless Fan∗
ning friction factor at the hot wall, f2,b
Re shows a completely different be-
havior, which is illustrated in Figure 3.12 and 3.13. The most characteristic
feature is the presence of a local maximum whose existence and position
are affected by the value of P r and M . In Figure 3.12(a) it is possible to see
∗
∗
Re| is not monotonic as |f1,b
Re| for increasing values
that the function |f2,b
of Ω: in fact, for sufficiently high values of M the function shows an initial
41
0.08
0.008
*
È f1,b
Re È
*
È f1,b
Re È
0.06
0.04
0.006
0.004
0.02
0.002
0.00
0
200
400
600
800
1000
0
200
400
W
600
800
1000
W
(a)
(b)
Figure 3.11 — Amplitude of the oscillating component of the Fanning friction factor at
the cold wall plotted versus Ω for M = 0 (a) or M = 100 (b) and different values of P r.
P r = 0.05 (continuous), P r = 0.5 (dashed), P r = 5 (dotted).
increase with increasing Ω. This behavior at P r = 0.05 is an indication of
the existence of a local maximum, which is clearly visible for P r = 5 in
Figure 3.12(b).
0.08
0.10
0.07
0.08
*
È f2,b
Re È
*
È f2,b
Re È
0.06
0.05
0.04
0.06
0.04
0.03
0.02
0.02
0.01
0
200
400
600
800
1000
0
200
400
W
600
800
1000
W
(a)
(b)
Figure 3.12 — Amplitude of the oscillating component of the Fanning friction factor at the
hot wall plotted versus Ω for P r = 0.05 (a) or P r = 5 (b) and different values of M . M = 0
(continuous), M = 50 (dashed), M = 100 (dotted).
∗
In Figure 3.13 how the local maximum of |f2,b
Re| drifts on the dimen-
sionless pulsation axis is clearly shown for various values of P r and M .
In Figure 3.13(a) it is possible to see the appearance of the local maximum
for P r = 0.05, 0.5, 5 in the non magnetic case at M = 0. For high Prandtl
fluids the maximum is present, and its position drifts towards lower fre42
quencies for increasing values of P r. For low P r fluids the function is
monotonically decreasing for increasing values of Ω. In Figure 3.13(b) the
magnetic case at M = 100 is illustrated: the effect of the magnetic field is to
∗
reduce globally the value of |f2,b
Re|, even at low frequencies, and to make
the maximum location shift towards higher frequencies. Interestingly, the
maximum at P r = 0.05 is present in the magnetic case, while it was not in
the non magnetic case.
0.016
0.10
0.015
0.08
*
È f2,b
Re È
*
È f2,b
Re È
0.014
0.06
0.013
0.012
0.04
0.011
0.010
0.02
0
200
400
600
800
1000
0
200
400
W
600
800
1000
W
(a)
(b)
Figure 3.13 — Amplitude of the oscillating component of the Fanning friction factor at
the hot wall plotted versus Ω for M = 0 (a) or M = 100 (b) and different values of P r.
P r = 0.05 (continuous), P r = 0.5 (dashed), P r = 5 (dotted).
The dimensionless heat flux at a given position ξ ′ between the channel
walls may be defined as:
∗
∗
∂θ
∂θ
a
+ b eiη = N u∗a ′ + N u∗b ′ ,
N u ∗ ξ ′ =
ξ
ξ
∂ξ ξ′
∂ξ ξ′
(3.58)
where the average value at Ω = 0 is simply N u∗a = 2 in agreement with
the classical steady state solution. In case of a steady periodic regime,
neglecting the influence of the dissipative terms, the magnitude of the dimensionless heat flux N u∗ assumes the values shown in Figure 3.14. In
particular, in Figure 3.14(a) the values of |N u∗ | at ξ = −0.25, −0.15, −0.05
are shown as functions of Ω. In this part of the channel, the one close to
the cold wall, the magnitude of N u∗ progressively decreases for increasing values of Ω, in agreement with the reduced penetration of the wall
temperature oscillation at high frequencies.
43
In the part of the channel closer to the hot wall it can be noted that
the magnitude of N u∗ shows a local maximum for a single value of the
dimensionless pulsation Ω. In particular, as shown in Figure 3.14(a) for
ξ = 0.05, 0.15, 0.25, the pulsation which maximizes the local dimensionless
2.0
20
1.5
15
È Nu* È
È Nu* È
heat flux increases approaching the hot wall at ξ = 0.25.
1.0
10
5
0.5
0
0.0
0
100
200
300
400
0
100
200
W
300
400
W
(a)
(b)
Figure 3.14 — Dimensionless heat flux N u as a function of Ω at different locations inside
the channel. Frame (a), ξ = −0.25 (continuous), ξ = −0.15 (dashed), ξ = −0.05 (dotted).
Frame (b), ξ = 0.05 (continuous), ξ = 0.15 (dashed), ξ = 0.25 (dotted).
3.5 Closure
In this chapter were illustrated the effects of a transverse magnetic field on
a steady periodic flow in a parallel vertical channel. The driving effect of
the steady-periodic regime was the temperature of one of the two walls,
which was assumed as varying sinusoidally with time. The Boussinesq
approximation was considered as valid to model buoyancy. The average
temperature across the channel section was used as a reference for the linearization of the state equation ρ = ρ(T ). The dimensionless local balance
equations were solved in steady-periodic regime analytically, neglecting
the dissipative terms in the energy equation, and numerically using a finite difference FORTRAN code.
The solution has shown that the external transverse magnetic field influences both the average velocity distribution and the amplitude of the
44
oscillating velocity component. In particular for increasing values of the
Hartmann number, the average velocity profile changes from that of the
classical non-magnetic case to the Hartmann profile, characterized by an almost uniform velocity at the core and steep velocity gradients at the walls.
Moreover, for increasing values of the Hartmann number it was shown
that the critical ratio of the Grashof over Reynolds number for the onset
of flow reversal increases accordingly. Finally, with respect to the oscillating components of velocity, the effect of the transverse magnetic field is to
progressively reduce their amplitudes, up to almost completely suppress
them.
45
Chapter 4
MHD mixed convection flow in a
vertical round pipe with
transverse magnetic field. Steady
periodic regime
46
Symbols used in the section
B
magnetic induction vector
B
magnitude of the magnetic induction vector
cp
specific heat at constant pressure
D
pipe diameter
f
Fanning friction factor
g
gravity vector
Gr
Grashof’s number
GR
ratio Gr/Re
Ha
Hartmann’s number
j
current density vector
k
thermal conductivity
n
wall normal vector
Nu
Nusselt’s number
P
pressure
Pr
Prandtl’s number
q̄w
average heat flux
R
pipe radius
Re
Reynolds’ number
t
time
∆T
amplitude of wall temperature oscillation
T
temperature
Tw
wall temperature
Tm
average temperature
U
velocity vector
V
dimensionless electric potential
W
channel axial velocity
w
dimensionless velocity
Wm
average velocity
wm
dimensionless average velocity
WR
reference velocity, −∂P/∂z · D2 /µ
x, y, z
cartesian coordinates
47
Greek Symbols
α
thermal diffusivity
β
coefficient of thermal expansion
η
dimensionless time
θ
dimensionless temperature
µ
dinamic viscosity
ν
kinematic viscosity
ξ, ζ
dimensionless coordinates
ρm
density at T = Tm
σ
electrical conductivity
Σ
fluid domain boundary
ς
curvilinear abscissa on Σ
τ̄w
average wall viscous stress
φ
electric potential
χ
aspect ratio
ω
angular frequency
Ω
dimensionless angular frequency
48
4.1 Introduction
In this section mixed convection in a vertical rectangular channel with an
applied uniform horizontal magnetic field will be discussed. The steadyperiodic regime will be analyzed, where the periodicity is induced by an
uniform wall temperature which varies in time with a sinusoidal law. Local balance equations of momentum, energy and charge will be solved
numerically for several values of the dimensionless parameters Ha and
P r. The temperature, velocity and potential field will be decomposed in
an average and an oscillating component, and solved independently. The
walls will be assumed as perfectly electrically insulating.
4.2 Governing equations
Let us consider the laminar flow of a Newtonian fluid in a vertical circular
channel, with an uniform horizontal magnetic field of constant intensity
B. The channel is assumed to be circular, with radius R. Under the hypothesis of fully developed flow, the velocity field is parallel, and the only
non vanishing component of the velocity vector, W (x, y, t), is parallel to
the channel axis and independent of the vertical coordinate z. The thermal conductivity of the fluid k, the thermal diffusivity α and the dynamic
viscosity µ will be assumed as constant. The system so defined is illustrated in Figure 4.1.
The complete set of governing equations is obtained assembling together the local balance equations for momentum, energy and electric potential.
An uniform temperature at the channel wall is assumed, varying in
time with a sinusoidal law. Being the temperature uniform along the channel vertical direction, the heat flux in this direction can be considered equal
to zero, therefore ∂T /∂z = 0. Consequently:
√
T (x, ± R2 − x2 , t) = Tw + ∆T cos(ω t).
49
(4.1)
y
B
x
g
2R
Figure 4.1 — A sketch of the channel.
The average temperature in the channel may be written as:
ω
Tm = 2 2
2π R
Z
2π/ω
dt
0
Z
R
dx
−R
Z
√
R2 −x2
√
− R2 −x2
dy T (x, y, t),
(4.2)
and is identically equal to the average wall temperature Tw . Being the
thermal gradient along the channel axis equal to zero, Tm is independent
of the axial coordinate z.
The no-slip boundary condition at the wall may be written as:
√
W (x, ± R2 − x2 , t) = 0,
(4.3)
and the average velocity across the channel section is:
1
Wm =
πR2
Z
R
−R
dx
Z
√
R2 −x2
√
− R2 −x2
dy W (x, y, t).
(4.4)
Let us assume the electric potential as independent of z, i.e. φ =
φ(x, y, t). To formulate the governing equation for the electric potential,
let us write the law of conservation of charge and the generalized Ohm’s
equation as follows:
∇ · j = 0,
j = σ(−∇φ + U × B),
50
(4.5)
it is then possible to combine them in a more compact expression:
∇2 φ = ∇ · (U × B).
(4.6)
Under the hypothesis of parallel flow and transverse magnetic field
oriented along the x axis, the previous equation can be further simplified
as:
∂W
∂2φ ∂2φ
+ 2 =B
.
2
∂x
∂y
∂y
The boundary condition of perfectly insulating walls yields:
j · n = 0,
∂φ
= 0,
∂n
⇒
(4.7)
(4.8)
where j is the current density vector and n is the direction vector normal
to the channel wall.
Assuming the validity of the Oberbeck-Boussinesq approximation, the
balance equation for momentum in its three components may be written
as follows:
∂P
= 0,
∂y
∂P
= 0,
∂x
(4.9)
and, along z,
2
∂W
∂2W
∂ W
∂P
+
=−
+µ
ρm
+
∂t
∂z
∂x2
∂y 2
∂φ
2
+ ρm gβ(T − Tm ) + σ
B − B W , (4.10)
∂y
where P = p + ρm g z is the excess over the hydrostatic pressure, defined
as the difference between the pressure and the hydrostatic pressure. As a
consequence of (4.9), one has P = P (z, t). Moreover, differentiating (4.10)
with respect to z one obtains the identity ∂ 2 P/∂z 2 = 0. This shows that
∂P/∂z depends only on t. In the following it will be assumed that ∂P/∂z
is independent of t as well.
Neglecting viscous dissipation and Joule heating, the balance equation
for energy may be written as:
2
∂2T
∂ T
∂T
+
=k
ρm c p
.
∂t
∂x2
∂y 2
51
(4.11)
The system of governing equations is then assembled from (4.10), (4.11)
and (4.7), which can be rewritten in terms of the following dimensionless
groups:
Wm
x
y
W
; wm =
; η = ω t;
; ζ= ; w=
R
R
WR
WR
φ
WR D
ωD2
T − Tm
; V =
; Re =
;
Ω=
; θ=
ν
BDWR
∆T
ν
ν
q̄w D
gβ∆T D3
Gr
P r = ; Nu =
; Gr =
;
; GR =
2
α r
k∆T
ν
Re
2
s
σ
dP D
Ha = BD
; WR = −
; D = 2R; ς = .
µ
dz µ
R
ξ=
(4.12)
where WR and D are the reference velocity and the hydraulic diameter of
the channel. The system of governing equation is then rewritten as:
2
∂w
∂ w ∂2w
∂V
2
+ GR θ + Ha 2
+
Ω
=1+4
−w ,
(4.13)
∂η
∂ξ 2
∂ζ 2
∂ζ
2
∂ θ ∂2θ
∂θ
,
(4.14)
=4
ΩP r
+
∂η
∂ξ 2 ∂ζ 2
∂2V
1 ∂w
∂2V
+
=
2
2
∂ξ
∂ζ
2 ∂ζ
(4.15)
with the boundary conditions:
p
w(ξ, ± 1 − ξ 2 , η) = 0,
(4.16)
p
θ(ξ, ± 1 − ξ 2 , η) = cos(η),
∂V =0
∂n (4.18)
Z
∂T
1
−k
ds .
q̄w =
S
S ∂n
(4.19)
∂T
∂T
y ∂T
y
=
cos arctan +
sin arctan .
∂n
∂x
x
∂y
x
(4.20)
(4.17)
Σ
where n is the direction vector normal to the boundary Σ.
The average heat flux exchanged between the walls and the fluid may
be written as:
being,
52
Using (4.12) it is possible to rewrite the heat flux in dimensionless form,
obtaining an expression of the Nusselt number:
Z
∂θ
1
Nu = −
dς.
π Σ ∂n
(4.21)
The Fanning friction factor can be derived from the average viscous
stress at the walls:
f =2
τ̄w
,
ρm WR2
(4.22)
being,
Z
∂W
1
µ
τ̄w =
ds ,
S
S ∂n
in dimensionless form, using (4.12), it is possible to write:
Z
2
∂w
f Re =
dς.
π Σ ∂n
(4.23)
(4.24)
4.3 Solution
Exploiting the linearity of the system, which follows from the assumption
of negligible heat generation, it is possible to solve the system of partial
differential equations (4.13) - (4.15) under the boundary conditions (4.16) (4.18), decomposing the functions w, θ and V in the following components:
w(ξ, ζ, η) = w0 (ξ, ζ) + w1 (ξ, ζ) cos η + w2 (ξ, ζ) sin η,
(4.25)
θ(ξ, ζ, η) = θ0 (ξ, ζ) + θ1 (ξ, ζ) cos η + θ2 (ξ, ζ) sin η,
(4.26)
V (ξ, ζ, η) = V0 (ξ, ζ) + V1 (ξ, ζ) cos η + V2 (ξ, ζ) sin η,
(4.27)
so to obtain two new systems, which are linearly independent and stationary. The solution of the first provides the expressions of the average components of the dimensionless velocity, temperature and electric potential
fields. The solution of the second represents the distribution of oscillating
amplitude of the aforementioned dimensionless fields. The two systems
may be written as:
53
For the average component:

2
∂ w0 ∂ 2 w0

2 ∂V0

0=1+4
+
− w0 ,
+ GR θ0 + Ha



∂ξ 2
∂ζ 2
∂ζ







∂ 2 θ0 ∂ 2 θ0
0=
+
,

∂ξ 2
∂ζ 2








1 ∂w0
∂ 2 V0 ∂ 2 V0


+
=
.

2
2
∂ξ
∂ζ
2 ∂ζ
and for the oscillation amplitudes:
2

∂ w1 ∂ 2 w1
∂V1

2

+ GR θ1 + Ha
− w1 ,
+
Ωw2 = 4


∂ξ 2
∂ζ 2
∂ζ









∂ 2 θ1 ∂ 2 θ1


ΩP
rθ
=
+
,

2

∂ξ 2
∂ζ 2









1 ∂w1
∂ 2 V1 ∂ 2 V1


+
=
,


∂ζ 2
2 ∂ζ
 ∂ξ 2
(4.28)
(4.29)
2



∂V2
∂ w2 ∂ 2 w2

2

+ GR θ2 + Ha
− w2 ,
−Ωw1 = 4
+


∂ξ 2
∂ζ 2
∂ζ









∂ 2 θ2 ∂ 2 θ2


−ΩP
rθ
=
+
,

1

∂ξ 2
∂ζ 2









1 ∂w2
∂ 2 V2 ∂ 2 V2


+
=
.
2
2
∂ξ
∂ζ
2 ∂ζ
The boundary conditions for the first system are:

p
2

w
0 (ξ, ± 1 − ξ , η) = 0







p

θ0 (ξ, ± 1 − ξ 2 , η) = 0






∂V

0

= 0,

∂n Σ
54
(4.30)
and, for the second:



























p
w1 (ξ, ± 1 − ξ 2 , η) = 0
p
θ1 (ξ, ± 1 − ξ 2 , η) = 1
∂V1 = 0,
∂n Σ
(4.31)

p



w2 (ξ, ± 1 − ξ 2 , η) = 0







p



θ2 (ξ, ± 1 − ξ 2 , η) = 0









∂V2 

= 0,

∂n Σ
These two systems are solved numerically using the finite element code
COMSOL Multiphysics. The discretization of the computational domain is
shown in Figure 4.2(a). The mesh consists of ∼
= 14000 elements, divided in
two groups. The core of the flow was discretized using an unstructured
triangular meshing scheme, while the elements at the periphery of the section were modeled by extrusion of the boundary elements, starting from a
thickness in direction normal to the wall of 10−7 , and progressively inflating the thickness of the elements following a geometrical progression with
ratio 1.2. A structured mesh of 64 layers with quadrilateral elements was
created, in order to accurately describe the thin Hartmann layer even for
high values of the parameter Ha. A detailed view of the boundary layer
elements is given in 4.2(b).
4.4 Results and Discussion
In the following sections the values assumed by the Fanning friction factor
and by the Nusselt number will be reported for various flow configurations. As for (4.25)-(4.27), it is possible to decompose the functions f Re
55
(a)
(b)
Figure 4.2 — The discretization of the computational domain: (a) An overview of the
whole domain, showing the structured boundary layers and the unstructured core. (b) A
detailed view of the structured mesh close to the wall. The thickness of the first element
is 10−7 .
56
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4.3 — Distributions of average dimensionless velocity (a) (c) (e) and electric potential (b) (d) (f), calculated for Ha = 0 (a,b), Ha = 20 (c,d), Ha = 50 (e,f) for steady state
forced convection with transverse magnetic field.
57
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4.4 — Arrow and streamline plots of the current density vector J across the pipe
cross section, calculated for Ha = 10 (a,b), Ha = 50 (c,d), Ha = 100 (e,f) for steady state
forced convection with transverse magnetic field.
58
and N u in their average and oscillating components. The decomposition
is as follows:
f Re = f0 Re + f1 Re cos η + f2 Re sin η
(4.32)
N u = N u0 + N u1 cos η + N u2 sin η
(4.33)
where the subscript 0 indicates the average components and the subscripts
1 and 2 indicate the oscillation amplitudes of the two oscillating components (respectively oscillating with cosine and sine). Moreover, it is possible to define the resulting oscillation amplitudes as:
|f Re| =
|N u| =
4.4.1
p
(f1 Re)2 + (f2 Re)2 ,
p
(4.34)
(N u1 )2 + (N u2 )2 .
Forced stationary convection with MHD effects
This first case has been chosen to demonstrate the consistency of the model
and to show some preliminary results in steady state laminar flow. In the
literature the analytical solution for the Fanning friction factor for laminar
flow in a smooth round pipe is well known, being for this particular case:
f Re = 16.
(4.35)
In Table 4.1 the values of f Re in case of forced convection (GR = 0)
with a transverse external magnetic field are reported as they were obtained numerically in the present work for several values of the Hartmann number. It should be noted that the numerical results for f Re were
rescaled with respect to wm , in order to obtain results which are ready for
comparison with further applications, by evaluating numerically the integral in (4.4).
In Figure 4.3, contour plots of dimensionless velocity, temperature and
electric potential are reported for Ha = 0, Ha = 20 and Ha = 50. In
particular, in Figure 4.3(c) the distribution of velocity w across the pipe
section is represented for Ha = 20. It is evident that the velocity gradient
59
Table 4.1 — Values of f Re as a function of Ha in stationary forced convection, GR = 0,
Ω = 0.
Ha
f Rewm
Ha
f Rewm
Ha
f Rewm
0
16.00
50
68.13
5000
4289.47
1
16.08
100
126.48
10000
6712.76
5
17.90
500
578.27
50000
12271.36
10
22.30
1000 1107.59
100000
13897.92
is greater at the walls normal to the direction of the magnetic field. This
effect is more relevant for increasing values of Ha. Moreover, in Figure
4.3(d) the distribution of electric potential V is plotted. It can be noted that
V depends mainly on ζ, which is the direction orthogonal to the magnetic
field. This implies that the electric and the magnetic field are orthogonal
in almost all the duct cross section.
In Figure 4.4, arrow and streamline plots of the current density vector
J across the pipe cross section are reported for Ha = 10, Ha = 50 and
Ha = 100. It is evident, as reported in Figures 4.4(b), 4.4(d) and 4.4(f), that
the current lines are closed inside the section, with no current crossing the
insulated walls. Moreover, it can be noted that for increasing values of
the Hartmann number, the closure of the current lines is segregated to a
narrow region near the Hartmann walls, where consequently the effect of
the Lorentz force is more intense, and large velocity gradients appear.
4.4.2
Steady periodic mixed convection with MHD effects
In case of mixed convection, so when GR 6= 0, it is possible to study the
effects of an oscillating wall temperature on the Fanning friction factor and
on the Nusselt number. In Table 4.2 the resulting oscillation amplitudes of
f Re and N u are reported as functions of Ω, P r and Ha.
The numerical results prove that |N u| increases monotonically with Ω,
while |f Re| decreases with Ω. Moreover, in Figures 4.5 and 4.6, the values
of f1,2 Re and N u1,2 are plotted versus Ω for several values of Ha. It may be
noted that f2 Re shows some absolute maxima whose position is affected
60
Table 4.2 — Values of the oscillating amplitudes of f Re and N u as a function of Ω.
Results obtained in mixed convection regime, with P r = 0.05 and GR = 500.
P r = 0.05
Ω
0
|N u|
Ha = 0
Ha = 1
Ha = 10
Ha = 100
|f Re|
|f Re|
|f Re|
125.90
|f Re|
16.61
0.00
250.00
246.18
1
0.05
249.77
245.97
125.89
16.61
5
0.25
244.50
241.01
125.57
16.60
10
0.50
230.00
227.31
124.59
16.58
50
2.32
111.16
111.29
100.97
16.03
100
3.96
69.11
69.13
69.90
14.97
500
9.32
30.91
30.90
30.54
12.12
1000
13.46
21.85
21.84
21.75
11.15
5000
30.92
9.77
9.77
9.77
8.12
Table 4.3 — Values of the oscillating amplitudes of f Re and N u as a function of Ω.
Results obtained in mixed convection regime, with P r = 1 and GR = 500.
Pr = 1
Ω
0
|N u|
Ha = 0
Ha = 1
Ha = 10
Ha = 100
|f Re|
|f Re|
|f Re|
125.90
|f Re|
16.61
0.00
250.00
246.18
1
0.99
246.30
242.56
124.53
16.51
5
3.96
188.46
186.05
103.71
15.00
10
5.84
124.85
123.96
82.29
13.60
50
13.46
46.43
46.43
44.55
11.44
100
19.31
33.34
33.33
32.38
10.62
500
44.02
14.91
14.91
14.85
8.41
1000
62.54
10.54
10.54
10.53
7.26
5000
140.70
4.71
4.71
4.71
4.35
61
300
250
f1 Re
200
150
100
50
0
0
50
100
150
200
250
300
200
250
300
W
(a)
140
120
f2 Re
100
80
60
40
20
0
0
50
100
150
W
(b)
Figure 4.5 — Behavior of f1 Re (a) and f2 Re (b) as functions of Ω obtained for Ha = 0
(continuous) Ha = 5 (dashed) and Ha = 10 (dotted) in mixed convection regime, with
GR = 500 and P r = 0.05.
62
20
Nu1,2
10
0
-10
-20
0
50
100
150
200
250
300
W
Figure 4.6 — Behavior of N u1 (continuous) and N u2 (dashed) as functions of Ω in mixed
convection regime with P r = 0.05.
by Ha.
In Figure 4.7 the distributions of oscillation amplitudes for the components of velocity w1 and w2 (Figures 4.7(a) and 4.7(b)), of temperature θ1
and θ2 (Figures 4.7(c) and 4.7(d)), and of electric potential (Figures 4.7(e)
and 4.7(f)) are plotted for Ha = 0. Analogous plots are reported in Figure
4.8 and 4.9 for Ha = 20 and Ha = 50, respectively. The results show that
w1 has both positive and negative values for small values of Ha, while
for higher values of Ha it is always positive. On the contrary, w2 is always positive, but its maximum value across the sections decreases with
increasing values of Ha.
4.5 Closure
In the present section the effects of a transverse magnetic field on a steady
periodic flow in a vertical circular pipe were studied, for several values
of the Hartmann and Prandtl numbers. The driving effect of the steadyperiodic regime was the wall temperature, which was assumed uniform
across the pipe wall and varying sinusoidally with time. The Boussinesq
approximation was assumed to model buoyancy. The average tempera63
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4.7 — Distributions of oscillation amplitudes for dimensionless velocity (a) (b),
temperature (c) (d) and electric potential (e) (f), calculated for Ha = 0 in mixed convection
regime, with GR = 500, P r = 0.05 and Ω = 100.
64
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4.8 — Distributions of oscillation amplitudes for dimensionless velocity (a) (b),
temperature (c) (d) and electric potential (e) (f), calculated for Ha = 20 in mixed convection regime, with GR = 500, P r = 0.05 and Ω = 100.
65
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4.9 — Distributions of oscillation amplitudes for dimensionless velocity (a) (b),
temperature (c) (d) and electric potential (e) (f), calculated for Ha = 50 in mixed convection regime, with GR = 500, P r = 0.05 and Ω = 100.
66
ture across the pipe cross section was used as a reference for the linearization of the equation of state ρ = ρ(T ). The dimensionless local balance
equations were solved numerically in steady-periodic regime using the
commercial software COMSOL Multiphysics.
In the particular case of forced convection, the numerical results shown
in the present work were compared with those available in the literature,
obtaining a perfect agreement in the determination of the Fanning friction
factor for the circular pipe configuration.
Moreover, in case of steady periodic mixed convection with MHD effects, the amplitudes of the oscillating components of the Fanning friction
factor and of the Nusselt number were computed as functions of Ω and
Ha. It was shown that the value of |N u| increases for increasing values
of Ω and that f1 Re is an always decreasing function of the same Ω. Fi-
nally, it was proved the existence of a local maximum of f2 Re for each
value of Ha, whose position on the Ω axis shifts towards higher values for
increasing values of Ha.
67
Chapter 5
MHD mixed convection flow in a
vertical rectangular duct with
transverse magnetic field. Steady
periodic regime
68
Symbols used in the section
B
magnetic induction vector
B
magnitude of the magnetic induction vector
cp
specific heat at constant pressure
D
hydraulic diameter, 4 L χ/(1 + χ)
f
Fanning friction factor
g
gravity vector
Gr
Grashof’s number
GR
ratio Gr/Re
Ha
Hartmann’s number
j
current density vector
k
thermal conductivity
L
channel characteristic dimension
n
wall normal vector
Nu
Nusselt’s number
P
pressure
Pr
Prandtl’s number
q̄w
average heat flux
Re
Reynolds’ number
t
time
∆T
amplitude of wall temperature oscillation
T
temperature
Tw
wall temperature
Tm
average temperature
U
velocity vector
V
dimensionless electric potential
W
channel axial velocity
w
dimensionless velocity
Wm
average velocity
wm
dimensionless average velocity
WR
reference velocity, −∂P/∂z · D2 /µ
x, y, z
cartesian coordinates
69
Greek Symbols
α
thermal diffusivity
β
coefficient of thermal expansion
η
dimensionless time
θ
dimensionless temperature
µ
dinamic viscosity
ν
kinematic viscosity
ξ, ζ
dimensionless coordinates
ρm
density at T = Tm
σ
electrical conductivity
τ̄w
average wall viscous stress
φ
electric potential
χ
aspect ratio
ω
angular frequency
Ω
dimensionless angular frequency
70
5.1 Introduction
In this section mixed convection in a vertical rectangular channel with an
applied uniform horizontal magnetic field will be discussed. The steadyperiodic regime will be analyzed, where the periodicity is induced by an
uniform wall temperature which varies in time with a sinusoidal law. Local balance equations of momentum, energy and charge will be solved
numerically using the channel aspect ratio as the main parameter. The
temperature, velocity and potential field will be decomposed in an average and an oscillating component, and solved independently. The walls
will be assumed as perfectly insulating.
5.2 Governing equations
Let us consider the laminar flow of a Newtonian fluid in a vertical rectangular channel, with an uniform horizontal magnetic field of constant intensity B. The channel is assumed to be rectangular, with a size 2L × 2χL,
so that χ is the channel aspect ratio. Under the hypothesis of fully developed flow, the velocity field is parallel, and the only non vanishing
component of the velocity vector, W (x, y, t), is parallel to the channel axis
and independent on the vertical coordinate z. The thermal conductivity of
the fluid k, the thermal diffusivity α and the dynamic viscosity µ will be
assumed as constant. The system so defined is illustrated in Figure 5.1.
The complete set of governing equations is obtained assembling together the local balance equations for momentum, energy and electric potential.
An uniform temperature at the channel walls is assumed, varying in
time with a sinusoidal law. Being the temperature uniform along the channel vertical direction, the heat flux in this direction can be considered equal
to zero, therefore ∂T /∂z = 0. Consequently:
T (±L, y, t) = T (x, ±χL, t) = Tw + ∆T cos(ω t).
71
(5.1)
Figure 5.1 — A sketch of the channel.
The average temperature in the channel may be written as:
ω
Tm =
8πχL2
Z
2π/ω
dt
0
Z
L
dx
−L
Z
χL
dy T (x, y, t),
(5.2)
−χL
and is identically equal to the average wall temperature Tw . Being the
thermal gradient along the channel axis equal to zero, Tm is independent
of the height z.
The no-slip boundary condition at the wall may be written as:
W (±L, y, t) = W (x, ±χL, t) = 0,
(5.3)
and the average velocity across the channel section is:
1
Wm =
4χL2
Z
L
dx
−L
Z
χL
dy W (x, y, t).
(5.4)
−χL
Let us assume the electric potential independent of z, i.e. φ = φ(x, y, t).
To formulate the governing equation for the electric potential, let us write
the law of conservation of charge and the generalized Ohm’s equation as
follows:
∇ · j = 0,
j = σ(−∇φ + U × B),
72
(5.5)
it is then possible to combine them in a more compact expression:
∇2 φ = ∇ · (U × B).
(5.6)
Under the hypothesis of parallel flow and transverse magnetic field
oriented along the x axis, the previous equation can be further simplified
as:
∂W
∂2φ ∂2φ
+ 2 =B
.
2
∂x
∂y
∂y
The boundary condition of perfectly insulating walls yields:
j · n = 0,
∂φ
= 0,
∂n
⇒
(5.7)
(5.8)
where j is the current density vector and n is the direction vector normal
to the channel wall.
Assuming the validity of the Oberbeck-Boussinesq approximation, the
balance equation for momentum may be written in its three components
as follows:
∂P
= 0,
∂y
∂P
= 0,
∂x
(5.9)
and, along z,
2
∂W
∂2W
∂ W
∂P
+
=−
+µ
ρm
+
∂t
∂z
∂x2
∂y 2
∂φ
2
+ ρm gβ(T − Tm ) + σ
B − B W , (5.10)
∂y
where P = p + ρm g z is the excess over the hydrostatic pressure, defined
as the difference between the pressure and the hydrostatic pressure. As a
consequence of (5.9), is P = P (z, t). Moreover, differentiating the (5.10)
with respect to z one obtains the identity ∂ 2 P/∂z 2 = 0. This shows that
∂P/∂z depends only on t. In the following it will be assumed that ∂P/∂z
is independent from t as well.
Neglecting viscous dissipation and Joule heating, the balance equation
for energy may be written as:
2
∂2T
∂ T
∂T
+
=k
ρm c p
.
∂t
∂x2
∂y 2
73
(5.11)
The system of governing equations is then assembled from (5.10), (5.11)
and (5.7), which can be rewritten in terms of the following dimensionless
groups:
x
Wm
y
W
; wm =
; η = ω t;
; ζ=
; w=
L
χL
WR
WR
2
T − Tm
φ
WR D
ωD
Ω=
; θ=
; V =
; Re =
;
ν
BDWR
∆T
ν
3
q̄w D
gβ∆T D
Gr
ν
; Gr =
;
;
GR
=
P r = ; Nu =
α r
k∆T
ν2
Re
σ
Ha = BD
.
µ
ξ=
(5.12)
where WR and D are the reference velocity and the hydraulic diameter of
the channel. The system of governing equation is then rewritten as:
2
∂ w
∂w
1 ∂2w
16χ2
Ω
+ 2 2 +
=1+
∂η
(1 + χ)2 ∂ξ 2
χ ∂ζ
4 ∂V
+ GRθ + Ha
− w , (5.13)
(1 + χ) ∂ζ
2
1 ∂2θ
∂θ
∂ θ
16χ2
ΩP r
+
,
(5.14)
=
∂η
(1 + χ)2 ∂ξ 2 χ2 ∂ζ 2
2
∂2V
1 ∂2V
1 + χ ∂w
.
+
=
∂ξ 2
χ2 ∂ζ 2
4χ2 ∂ζ
with the boundary conditions:
(5.15)
w(±1, ζ, η) = w(ξ, ±1, η) = 0,
(5.16)
θ(±1, ζ, η) = θ(ξ, ±1, η) = cos(η),
∂V ∂V = 0;
= 0.
∂ξ ∂ζ (5.17)
ξ=±1
(5.18)
ζ=±1
The average heat flux exchanged between the walls and the fluid may
be written as:
Z
Z
∂T
∂T
1
−k
q̄w =
dA + k
dA +
AT
A1 ∂x
A3 ∂x
Z
Z
∂T
∂T
dA + k
dA , (5.19)
+ −k
A2 ∂y
A4 ∂y
74
being A1 − A4 the wall areas and AT their sum. Using (5.12) it is possible
to rewrite the heat flux in dimensionless form, obtaining an expression of
the Nusselt number:
" Z
!
Z 1
1
2
χ
∂θ ∂θ Nu =
dζ −
dζ +
2
(1 + χ)
−1 ∂ξ
−1 ∂ξ
ξ=1
ξ=−1
!#
Z 1
Z 1
∂θ ∂θ 1
. (5.20)
dξ −
dξ + 2
χ
−1 ∂ζ
−1 ∂ζ
ζ=1
ζ=−1
The Fanning friction factor can be derived from the average viscous
stress at the walls:
f =2
τ̄w
,
ρm WR2
(5.21)
being,
1
τ̄w =
AT
Z
µ
A1
∂W
dA − µ
∂x
∂W
dA +
A3 ∂x
Z
Z
∂W
∂W
+ µ
dA − µ
dA , (5.22)
A2 ∂y
A4 ∂y
Z
in dimensionless form, using (5.12), it is possible to write:
χ2
f Re = 2
(1 + χ)2
" Z
5.3 Solution
!
Z 1
∂w ∂w +
dζ dζ −
−1 ∂ξ
−1 ∂ξ
ξ=−1
ξ=1
!#
Z 1
Z 1
1
∂w ∂w + 2
−
. (5.23)
dξ dξ χ
−1 ∂ζ
−1 ∂ζ
ζ=−1
ζ=1
1
Exploiting the linearity of the system, which follows the assumption of
negligible heat generation, it is possible to solve the system made of (5.13)
- (5.15) under the boundary conditions (5.16) - (5.18), decomposing the
functions w, θ and V in the following components:
w(ξ, ζ, η) = w0 (ξ, ζ) + w1 (ξ, ζ) cos η + w2 (ξ, ζ) sin η,
(5.24)
θ(ξ, ζ, η) = θ0 (ξ, ζ) + θ1 (ξ, ζ) cos η + θ2 (ξ, ζ) sin η,
(5.25)
75
V (ξ, ζ, η) = V0 (ξ, ζ) + V1 (ξ, ζ) cos η + V2 (ξ, ζ) sin η,
(5.26)
so to obtain two new systems, which are linearly independent and stationary. The solution of the first provides the expressions of the average components of the dimensionless velocity, temperature and electric potential
fields. The solution of the second represents the distribution of oscillating
amplitude of the aforementioned dimensionless fields. The two systems
may be written as:
For the average component:
2

1 ∂ 2 w0
∂ w0
16χ2


+ 2
+
0=1+

2
2
2

(1
+
χ)
∂ξ
χ
∂ζ









4 ∂V0

2
 +GR θ0 + Ha
− w0 ,


(1 + χ) ∂ζ




1 ∂ 2 θ0
∂ 2 θ0


+
,
0
=


∂ξ 2
χ2 ∂ζ 2









∂ 2 V0
1 ∂ 2 V0
1 + χ ∂w0


.
+
=
∂ξ 2
χ2 ∂ζ 2
4χ2 ∂ζ
76
(5.27)
and for the oscillation amplitudes:

























































































2
16χ2
1 ∂ 2 w1
∂ w1
+ 2
+
Ωw2 =
(1 + χ)2 ∂ξ 2
χ ∂ζ 2
+GR θ1 + Ha
2
4 ∂V1
− w1 ,
(1 + χ) ∂ζ
2
16χ2
1 ∂ 2 θ1
∂ θ1
+ 2 2 ,
ΩP rθ2 =
(1 + χ)2 ∂ξ 2
χ ∂ζ
1 ∂ 2 V1
1 + χ ∂w1
∂ 2 V1
,
+
=
∂ξ 2
χ2 ∂ζ 2
4χ2 ∂ζ
(5.28)
2
−Ωw1 =
2
2
16χ
1 ∂ w2
∂ w2
+ 2
+
2
2
(1 + χ)
∂ξ
χ ∂ζ 2
+GR θ2 + Ha
2
4 ∂V2
− w2 ,
(1 + χ) ∂ζ
2
16χ2
1 ∂ 2 θ2
∂ θ2
−ΩP rθ1 =
+ 2 2 ,
(1 + χ)2 ∂ξ 2
χ ∂ζ
1 ∂ 2 V2
1 + χ ∂w2
∂ 2 V2
+
=
.
2
2
2
∂ξ
χ ∂ζ
4χ2 ∂ζ
The boundary conditions for the first system are:


w0 (±1, ζ) = w0 (ξ, ±1) = 0








θ0 (±1, ζ) = θ0 (ξ, ±1) = 0






∂V
∂V
0 0 

= 0;
= 0,

∂ξ ξ=±1
∂ζ ζ=±1
77
(5.29)
and, for the second:



























w1 (±1, ζ) = w1 (ξ, ±1) = 0
θ1 (±1, ζ) = θ1 (ξ, ±1) = 1
∂V1 = 0;
∂ξ ξ=±1
∂V1 = 0,
∂ζ ζ=±1
(5.30)




w2 (±1, ζ) = w2 (ξ, ±1) = 0










θ2 (±1, ζ) = θ2 (ξ, ±1) = 0









∂V2 ∂V2 

= 0;
= 0,

∂ξ ξ=±1
∂ζ ζ=±1
These two systems are solved numerically using the finite element code
COMSOL Multiphysics. The discretization of the computational domain is
shown in Figure 5.2(a). The mesh consists of ∼
= 24000 elements, divided in
two groups. The core of the flow was discretized using a free triangular
meshing scheme, while the elements at the periphery of the section were
modeled by inflation of the boundary elements, starting from a thickness
in direction normal to the wall of 10−7 , and progressively inflating the
thickness of the elements following a geometrical progression with ratio
1.1. Totally 128 layers of quadrilateral elements were created, in order to
accurately describe the thin Hartmann layers even for high values of the
parameter Ha. A detailed view of the boundary layer elements is given in
5.2(b).
5.4 Results and Discussion
In the following sections the values assumed by the Fanning friction factor
and by the Nusselt number will be reported for various flow configurations. As for (5.24)-(5.26), it is as well possible to decompose the functions
78
(a)
(b)
Figure 5.2 — The discretization of the computational domain: (a) An overview of the
whole domain, showing the structured boundary layers and the unstructured core. (b) A
detailed view of the structured mesh close to the wall. The thickness of the first element
is 10−7 .
79
(a)
(b)
(c)
(d)
(e)
(f)
Figure 5.3 — Distributions of average dimensionless velocity (a) (c) (e) and electric potential (b) (d) (f), calculated for Ha = 0 (a,b), Ha = 20 (c,d), Ha = 50 (e,f) in a square
channel (χ = 1) for a steady state forced convection with transverse magnetic field.
80
(a)
(b)
(c)
(d)
(e)
(f)
Figure 5.4 — Arrow and streamline plots of the current density vector J across the duct
cross section, calculated for Ha = 10 (a,b), Ha = 50 (c,d), Ha = 100 (e,f) for steady state
forced convection with transverse magnetic field.
81
f Re and N u in their average and oscillating components. The decomposition is as follows:
f Re = f0 Re + f1 Re cos η + f2 Re sin η
(5.31)
N u = N u0 + N u1 cos η + N u2 sin η
(5.32)
where the subscript 0 indicates the average components and the subscripts
1 and 2 indicate the oscillation amplitudes of the two oscillating components (respectively oscillating with cosine and sine). Moreover, it is possible to define the resulting oscillation amplitudes as:
|f Re| =
|N u| =
5.4.1
p
(f1 Re)2 + (f2 Re)2 ,
p
(5.33)
(N u1 )2 + (N u2 )2 .
Forced stationary convection without MHD effects
This validation case has been chosen to demonstrate the consistency of the
model. In literature an analytical solution (11) is available for this particular configuration which yields exact values of the Fanning friction factor
for various aspect ratios of the channel. In Table 5.1 the values of f Re obtained numerically in the present work are reported and compared with
those available in the literature.
It should be noted that, to correctly compare the values, the numerical
results for f Re were rescaled with respect to wm , for ease of comparison
with the data obtained from the analytical solution. This allows to compare directly the results, at the price of evaluating numerically the integral
in (5.4).
After the scaling, it is possible to note that the two series of results
are in excellent agreement, with a maximum relative error always smaller
than 0.01%. This allows to assume that the model is properly set up and
that its results may be extended to more complex configurations.
82
Table 5.1 — Comparison between the factor f Rem obtained in the present work and
the analytical solution available in literature (11). Stationary forced convection, GR = 0,
Ω = 0.
reference
calculated
χ
f Re wm
f Re wm
ε
1
14.22708
14.22713
0.00%
2
15.54806
15.54814
0.00%
3
17.08967
17.08983
0.00%
4
18.23278
18.23305
0.00%
5
19.07050
19.07092
0.00%
6
19.70220
19.70281
0.00%
7
20.19310
20.19392
0.00%
8
20.58464
20.58570
0.01%
9
20.90385
20.90515
0.01%
10
21.16888
21.17043
0.01%
5.4.2
Forced stationary convection with MHD effects
In Table 5.2 the results obtained for f Re wm in case of forced convection
(GR = 0) with a transverse external magnetic field are reported. The results are obtained for several values of the channel aspect ratio.
The results show that the influence of the magnetic field is relevant
when the channel has χ > 1, i.e. when the distance between the walls
orthogonal to the field (the Hartmann walls) is smaller than the distance
between the walls parallel to the field. In that case, the viscous stress is
influenced both by the geometry of the conduit and by the interaction between the flow and the magnetic field, which modifies the wall velocity
profile by increasing the local wall velocity gradient. At the same time it
is possible to note how, without magnetic field, there exists a symmetry
χ → 1/χ with respect to the values of f Re wm .
In Figure 5.3, contour plots of dimensionless velocity, temperature and
electric potential are reported for Ha = 0, Ha = 20 and Ha = 50 in the case
of a square channel with χ = 1. In particular, in Figure 5.3(c) the distribu-
83
Table 5.2 — Values of f Re as a function of Ha in stationary forced convection, GR = 0,
Ω = 0.
χ = 1/5
χ = 1/2
χ=1
χ=2
χ=5
Ha
f Rewm
f Rewm
f Rewm
f Rewm
f Rewm
0
19.07
15.55
14.23
15.55
19.07
1
19.08
15.59
14.31
15.65
19.17
10
20.17
19.06
20.37
23.32
27.47
100 51.46
84.19
116.11
147.32
178.03
tion of velocity w across the channel section is represented for Ha = 20.
It is evident that the velocity gradient is greater at the walls normal to the
direction of the magnetic field. This effect is more relevant for increasing
values of Ha. Moreover, in Figure 5.3(d) the distribution of electric potential V is plotted. It can be noted that V depends mainly on ζ, which is the
direction orthogonal to the magnetic field. This implies that the electric
and the magnetic field are orthogonal in almost all the duct cross section.
In Figure 5.4, arrow and streamline plots of the current density vector
J across the duct cross section are reported for Ha = 10, Ha = 50 and
Ha = 100. It is evident, as reported in Figures 5.4(b), 5.4(d) and 5.4(f),
that the current lines close completely inside the section, with no escape
of current through the insulated walls. Moreover, it can be noted that for
increasing values of the Hartmann number, the closure of the current lines
is segregated to a narrow region near the Hartmann walls, where consequently the effect of the Lorentz force is more intense, and large velocity
gradients appear.
5.4.3
Steady periodic mixed convection with MHD effects
In case of mixed convection, so when GR 6= 0, it is possible to study the
effects of an oscillating wall temperature on the Fanning friction factor and
on the Nusselt number. In Table 5.3 the resulting oscillation amplitudes of
f Re and N u as functions of Ω, P r and Ha are reported.
The numerical results prove that |N u| increases monotonically with Ω,
84
(a)
(b)
(c)
Figure 5.5 — Behavior of f1 Re, f2 Re, N u1 and N u2 as functions of Ω obtained for
Ha = 5, 10, 20. Results apply to a square channel χ = 1 in mixed convection regime,
with GR = 500, P r = 0.05.
85
(a)
(b)
(c)
(d)
(e)
(f)
Figure 5.6 — Distributions of oscillation amplitudes for dimensionless velocity (a) (b),
temperature (c) (d) and electric potential (e) (f), calculated for Ha = 0 and χ = 1, in
mixed convection regime, with GR = 500, P r = 0.05 and Ω = 100.
86
(a)
(b)
(c)
(d)
(e)
(f)
Figure 5.7 — Distributions of oscillation amplitudes for dimensionless velocity (a) (b),
temperature (c) (d) and electric potential (e) (f), calculated for Ha = 20 and χ = 1, in
mixed convection regime, with GR = 500, P r = 0.05 and Ω = 100.
87
(a)
(b)
(c)
(d)
(e)
(f)
Figure 5.8 — Distributions of oscillation amplitudes for dimensionless velocity (a) (b),
temperature (c) (d) and electric potential (e) (f), calculated for Ha = 50 and χ = 1, in
mixed convection regime, with GR = 500, P r = 0.05 and Ω = 100.
88
Table 5.3 — Values of te oscillating amplitudes of f Re and N u as a function of Ω. Results
obtained for a square channel χ = 1 in mixed convection regime, GR = 500.
Ω
Pr =
Pr =
0.05
1
|N u|
Ha = Ha = Ha =
Ha = Ha = Ha =
0
0
1
10
1
10
|f Re| |f Re| |f Re|
|N u|
|f Re| |f Re| |f Re|
0.50
249.35 249.55 249.72
1
0.03
249.73 249.88 250.00
5
0.12
243.47 247.07 249.91
2.44
234.67 239.20 243.37
10
0.25
227.06 238.92 249.65
4.55
198.58 212.22 226.74
50
1.24
120.94 143.32 241.46
12.50
67.46
68.27
118.82
100
2.44
85.29
92.44
219.48
18.27
50.34
47.49
78.86
500
8.62
36.34
35.55
65.27
42.96
22.36
22.15
18.83
1000
12.50
25.82
25.54
26.78
61.47
15.81
15.75
11.86
5000
29.88
11.56
11.54
9.55
139.63 7.07
7.07
6.34
while |f Re| decreases with Ω. Moreover, in Figure 5.5, the plots of f1,2 Re
and N u1,2 are reported. It may be noted that f2 Re shows some absolute
maxima whose position is affected by Ha.
In Figure 5.6 the distributions of oscillation amplitudes for the components of velocity w1 and w2 (Figures 5.6(a) and 5.6(b)), of temperature θ1
and θ2 (Figures 5.6(c) and 5.6(d)), and of electric potential (Figures 5.6(e)
and 5.6(f)) are reported for Ha = 0. Figures 5.7 and 5.8 show the same type
of results for Ha = 20 and Ha = 50, respectively. The results show that
w1 has both positive and negative values for small values of Ha, while
for higher values of Ha it is always positive. On the contrary, w2 is always positive, but its maximum value across the sections decreases with
increasing values of Ha. These results are in agreement with those already
shown for the circular pipe.
89
5.5 Closure
In this chapter the effects of a transverse magnetic field on a steady periodic flow in a vertical rectangular channel, for several values of the channel aspect ratio were illustrated. The driving effect of the steady-periodic
regime was the wall temperature, which was assumed uniform across the
four walls and varying sinusoidally with time. The Boussinesq approximation was assumed to model buoyancy. The average temperature across
the channel section was used as a reference for the linearization of the
equation of state ρ = ρ(T ). The dimensionless local balance equations
were solved numerically in steady-periodic regime using the commercial
software COMSOL Multiphysics.
In the particular case of forced convection, the numerical results shown
in the present work were compared with those available in the literature,
obtaining a perfect agreement in the determination of the Fanning friction
factor for several values of the channel aspect ratio.
Moreover, in case of steady periodic mixed convection with MHD effects, the amplitudes of the oscillating components of the Fanning friction
factor and of the Nusselt number were computed as functions of Ω and
Ha. It was shown that the value of |N u| increases for increasing values
of Ω and that f1 Re is an always decreasing function of the same Ω. Fi-
nally, it was proved the existence of a local maximum of f2 Re for each
value of Ha, whose position on the Ω axis shifts towards higher values for
increasing values of Ha.
90
Chapter 6
Conclusions
The main objective of this thesis has been to analyze the heat transfer and
fluid dynamics phenomena which are related to steady periodic magnetohydrodynamic mixed convection flows in channels and ducts. In other
terms, the flow of an electrically conductive fluid enclosed in channels
of various geometries has been studied, assuming the onset of a steady
periodic regime induced by an oscillating driving effect (e.g. wall temperatures) and under the influence of an externally applied magnetic field,
which is not perturbed by the motion of the fluid.
To this aim, a comprehensive literature search has been carried out first,
to build the bulk of the govern equations needed to describe the strongly
linked behaviors of the velocity, temperature and electric potential fields
spanning across the sections of the geometries under examination. In particular the book by Welty et al. (13) was used as the reference text for the
definition of the fluid dynamics and heat transfer phenomena in mixed
convective flow. The books by Roberts (14), Shercliff (15) and Kulikovskiy
(16) represented the reference for the classical formulation of ideal MHD,
while the book from Müller and Bühler (17) provided the necessary insights in the problem of MHD flow in closed geometries.
Using the aforementioned references, the equation system shown in
Chapter two was obtained, the boundary conditions were classified and
related to practical applications, and the dimensional analysis of the equa-
91
tion system has been performed, in order to build up an analytical tool
which is able to solve problems with widely different scales, from small
channel flow of molten salts to liquid metal flow of cast metal, and so on.
The analytical tool was exploited first in Chapter three, where the solution of the mixed convective flow in a vertical parallel channel in steady
periodic regime has been presented, accompanied by a numerical extension of the solution, to account for nonlinear effects which could not be
handled by the analytical model. A comprehensive set of results was presented, and the main conclusions of the Chapter showed that the external
transverse magnetic field influences both the average velocity distribution and the amplitude of the oscillating velocity component. In particular
for increasing values of the Hartmann number, the average velocity profile changes from that of the classical non-magnetic case to the so-called
Hartmann profile, characterized by a uniform velocity at the core and steep
velocity gradients at the walls. Moreover, for increasing values of the
Hartmann number it was shown that the critical ratio of the Grashof over
Reynolds number for the onset of flow reversal increases accordingly. Finally, with respect to the oscillating components of velocity, it was shown
that the effect of the transverse magnetic field is to progressively reduce
their amplitudes, up to almost completely suppress them.
In Chapter four, a case of MHD steady periodic mixed convection in
a vertical circular pipe was studied using an analytical and numerical
approach. In particular, the problem was solved analytically up to the
decomposition of the main time-dependent equation system in two stationary subsystems representing the average and oscillating components
of the velocity, temperature and electric potential fields. Afterwards, the
two systems were solved independently using a commercial finite element
solver, Comsol Multiphysics. In the solution it was shown that, in the particular case of forced convection, the numerical results for the Fanning
friction factor were in perfect agreement with those available in the literature. Moreover, in case of steady periodic mixed convection with MHD
effects, it was shown that the value of the dimensionless average heat flux
|N u| increases for increasing values of the dimensionless pulsation Ω and
92
that the f1 Re component of the dimensionless Fanning friction factor is
an always decreasing function of Ω, while the f2 Re has a local maximum
for each value of Ha, whose position on the Ω axis shifts towards higher
values for increasing values of Ha.
In Chapter five, another case of 2D MHD steady periodic mixed convection was studied, this time regarding a vertical rectangular channel.
Again an analytical and numerical approach was used, and solutions were
provided for several values of, among other parameters, the channel aspect ratio. The problem was solved analytically up to the decomposition
of the main time-dependent equation system in two stationary subsystems representing the average and oscillating components of the velocity,
temperature and electric potential fields and, afterwards, the two systems
were solved independently using the finite element solver Comsol Multiphysics, as in the previous Chapter. In the solution it was shown that,
in the particular case of forced convection, the numerical results for the
Fanning friction factor, computed for several values of the channel aspect
ratio, were in perfect agreement with those available in the literature, thus
proving the validity of the model. Furthermore, in the case of steady periodic mixed convection with MHD effects, the amplitudes of the oscillating
components of the Fanning friction factor and of the Nusselt number were
computed as functions of Ω, Ha and χ, the channel aspect ratio. It was
shown that the influence of the magnetic field is relevant when the channel has χ > 1, i.e. when the distance between the walls orthogonal to the
field (the Hartmann walls) is smaller than the distance between the walls
parallel to the field. In that case, the viscous stress is influenced both by
the geometry of the conduit and by the interaction between the flow and
the magnetic field, which modifies the wall velocity profile by increasing
the local wall velocity gradient. Moreover, it was shown that the value
of |N u| increases for increasing values of Ω and that f1 Re is an always
decreasing function of Ω itself. Finally, it was proved the existence of a lo-
cal maximum of f2 Re for each value of Ha, whose position on the Ω axis
shifts towards higher values for increasing values of Ha.
93
6.1 Closure
In this work both analytical and numerical solutions for MHD mixed convection flows in steady periodic regime has been presented. The results
obtained have shown that the effect of magnetic field has great influence
on the mixed convection heat exchange and fluid flow in steady periodic
regime, both in terms of flow pattern modification due to Lorentz forces
and heat generation due to Joule effect. A natural continuation of this
work would focus on exploring more realistic fluid wall electrical interactions and magnetic field distributions, and on extending the scope of the
research to three dimensional geometries, eventually with an eye to more
practical, technological applications (e.g. fusion reactor blanket flows).
This work can be considered as an extension of the research carried out
by Barletta et al. on the analysis of mixed convection flows in steady periodic regime without MHD effects (9)-(10).
94
Appendix A
Auxiliary functions for the
parallel channel problem
Φ1 (ξ, M, Ω) = e+ξ
Φ2 (ξ, M, Ω) = e−ξ
Φ3 (ξ, M, Ω) = −Φ1 (ξ, M, Ω)
Z
ξ
0
√
M 2 +ıΩ
(A.1)
√
M 2 +ıΩ
(A.2)
Φ2 (s, M, Ω)
√
ds+
−2 M 2 + ıΩ
Z ξ
Φ1 (s, M, Ω)
√
ds (A.3)
+ Φ2 (ξ, M, Ω)
2
0 −2 M + ıΩ
ξ
Φ2 (s, M, Ω)ΦR (s, Ω, P r)
√
ds+
−2 M 2 + ıΩ
0
Z ξ
Φ1 (s, M, Ω)ΦR (s, Ω, P r)
√
+ Φ2 (ξ, M, Ω)
ds (A.4)
−2 M 2 + ıΩ
0
Φ4 (ξ, M, Ω, P r) = −Φ1 (ξ, M, Ω)
Z
√
−e1/4(1−4ξ) iΩP r + e1/4(3+4ξ)
√
ΦR (ξ, Ω, P r) = −
−1 + e iΩP r
95
√
iΩP r
(A.5)
Ψ1,a (M, Ω, P r) = A1,a (M, Ω, P r)
Ψ1,b (M, Ω) = A1,b (M, Ω)
Z
Z
1/4
Φ1 (ξ, M, Ω)ξ
Ψ2,a (M, Ω, P r) = A2,a (M, Ω, P r)
Ψ2,b (M, Ω) = A2,b (M, Ω)
Ψ3 (M, Ω) =
Ψ4 (M, Ω, P r) =
Z
Z
1/4
(A.6)
(A.7)
1/4
Φ2 (ξ, M, Ω)ξ
(A.8)
−1/4
Φ2 (ξ, M, Ω)ξ
(A.9)
−1/4
1/4
−1/4
Z 1/4
Φ1 (ξ, M, Ω)ξ
−1/4
−1/4
Z
1/4
Φ3 (ξ, M, Ω)ξ
(A.10)
Φ4 (ξ, M, Ω, P r)ξ
(A.11)
−1/4
96
Φ2 (1/4, M, Ω)Φ4 (−1/4, M, Ω, P r) − Φ2 (−1/4, M, Ω)Φ4 (1/4, M, Ω, P r)
−Φ1 (1/4, M, Ω)Φ2 (−1/4, M, Ω) + Φ1 (−1/4, M, Ω)Φ2 (1/4, M, Ω)
−Φ2 (1/4, M, Ω)Φ3 (−1/4, M, Ω) + Φ2 (−1/4, M, Ω)Φ3 (1/4, M, Ω)
A1,b (M, Ω) = −
−Φ1 (1/4, M, Ω)Φ2 (−1/4, M, Ω) + Φ1 (−1/4, M, Ω)Φ2 (1/4, M, Ω)
Φ1 (1/4, M, Ω)Φ4 (−1/4, M, Ω, P r) − Φ1 (−1/4, M, Ω)Φ4 (1/4, M, Ω, P r)
A2,a (M, Ω, P r) = −
Φ1 (1/4, M, Ω)Φ2 (−1/4, M, Ω) − Φ1 (−1/4, M, Ω)Φ2 (1/4, M, Ω)
−Φ1 (1/4, M, Ω)Φ3 (−1/4, M, Ω) + Φ1 (−1/4, M, Ω)Φ3 (1/4, M, Ω)
A2,b (M, Ω) = −
Φ1 (1/4, M, Ω)Φ2 (−1/4, M, Ω) − Φ1 (−1/4, M, Ω)Φ2 (1/4, M, Ω)
A1,a (M, Ω, P r) = −
(A.12)
(A.13)
(A.14)
(A.15)
97
Appendix B
FORTRAN95 code for the solution
of the parallel channel problem
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
program main
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
implicit none
integer nr,n_iter,n_per,step_per,scan_pt,mat_pt,freq
integer i0,i1,i2,j,i,perc,k,k2
real*8 pi
real*8 u0,t_u0,t_m,ampli
real*8 a_u0,a_m
parameter
parameter
parameter
parameter
parameter
parameter
parameter
parameter
parameter
parameter
parameter
real*8
real*8
real*8
real*8
real*8
real*8
(pi=3.1415926535897932384626433832795028841971693993751058209749445923078164062862)
(nr= 51)
(n_iter= 20)
(step_per= 2000000)
(scan_pt= 100)
(mat_pt= 1000)
(freq= 20)
(ampli= 10)
(a_u0= 1)
(t_u0= 0)
(t_m= 0)
x(nr)
u11(nr)
u22(nr)
u1(nr),u2(nr),u3(nr),u4(nr),u5(nr)
t1(nr),t2(nr),t3(nr),t4(nr),t5(nr)
h1,h2,h3,h4,h5
real*8 omega
real*8 pr
real*8 m
98
real*8
real*8
real*8
real*8
real*8
gr
br
dt
dx
x0
real*8
real*8
real*8
real*8
real*8
real*8
c3
conv,conv_ps
lbd,lbd1,lbd2
tt,ct1,ct2
tt1,tt2
min_t
real*8
real*8
real*8
real*8
real*8
real*8
real*8
real*8
real*8
real*8
real*8
real*8
real*8
real*8
a(nr),b(nr),c(nr)
a1,a2,a3,a4,a5,a6
u4t_mat(5,scan_pt)
uth_mat(5,nr)
test_05(nr)
u_mat(nr,mat_pt)
th_mat(nr,mat_pt)
t_vec(mat_pt)
u_a(nr)
u_b(nr)
u_bp(nr)
th_a(nr)
th_b(nr)
th_bp(nr)
open(8,file="matrix.dat",status="replace")
open(9,file="u4t.dat",status="replace")
open(10,file="ut1.dat",status="replace")
open(11,file="uth.dat",status="replace")
open(12,file="trn.dat",status="replace")
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
call system("cls")
write (*,"(A)",advance=’no’)
write (*,"(A)",advance=’no’)
write (*,"(A)",advance=’no’)
write (*,"(A)",advance=’no’)
write (*,"(A)",advance=’no’)
"##############################################################################"
"#
#"
"#
MHD MIXED CONVECTION IN A PARALLEL VERTICAL CHANNEL
#"
"#
#"
"##############################################################################"
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
print
write
write
write
print
*
(*,"(A)",advance=’no’) "##############################################################################"
(*,"(A)",advance=’no’) "#
Please insert the values for the dimensionless parameters
#"
(*,"(A)",advance=’no’) "##############################################################################"
*
write (*,"(A)",advance=’no’)
read *,omega
write (*,"(A)",advance=’no’)
read *,pr
write (*,"(A)",advance=’no’)
read *,br
write (*,"(A)",advance=’no’)
read *,gr
write (*,"(A)",advance=’no’)
read *,a_m
write (*,"(A)",advance=’no’)
read *,tt2
" # Omega............... "
" # Pr.................. "
" # Br.................. "
" # GR.................. "
" # M................... "
" # Theta_2............. "
99
x0= 0.25
dt= 2.0*pi/step_per
dx= (2*x0)/(nr-1)
tt1= -tt2
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
do i=1,nr
x(i)=-x0+dx*(i-1)
t1(i)=0.0
t2(i)=0.0
t3(i)=0.0
t4(i)=0.0
u1(i)=0.0
u2(i)=0.0
u3(i)=0.0
u4(i)=0.0
u11(i)=0.0
enddo
write (12,103) 0.0,x
write (12,103) 0.0,u1
tt=0.0
print *
write (*,"(A)") "#
[--------------- Progress ---------------]"
i0=0
conv_ps=1
do while (conv_ps > 1e-6)
i0=i0+1
print *
write (*,"(A,I5,A)",advance=’no’) "# Period nr........... ",i0," ["
k=0
k2=0
do i1=1,step_per
tt=tt+dt
m=a_m/(1+exp(-ampli*(tt/(2*pi)-t_m)))
u0=a_u0/(1+exp(-ampli*(tt/(2*pi)-t_u0)))
t5(1)=tt1
t5(nr)=tt2+cos(tt)
do i=2,nr-1
h1=(dt/(omega*pr*dx**2))*(t1(i+1)-2*t1(i)+t1(i-1)) &
+(dt/(omega*pr))*(br*((u1(i+1)-u1(i-1))/(2*dx))**2) &
+(dt/(omega*pr))*(m**2*br*u1(i)**2)
h2=(dt/(omega*pr*dx**2))*(t2(i+1)-2*t2(i)+t2(i-1)) &
+(dt/(omega*pr))*(br*((u2(i+1)-u2(i-1))/(2*dx))**2) &
+(dt/(omega*pr))*(m**2*br*u2(i)**2)
h3=(dt/(omega*pr*dx**2))*(t3(i+1)-2*t3(i)+t3(i-1)) &
+(dt/(omega*pr))*(br*((u3(i+1)-u3(i-1))/(2*dx))**2) &
+(dt/(omega*pr))*(m**2*br*u3(i)**2)
h4=(dt/(omega*pr*dx**2))*(t4(i+1)-2*t4(i)+t4(i-1)) &
+(dt/(omega*pr))*(br*((u4(i+1)-u4(i-1))/(2*dx))**2) &
+(dt/(omega*pr))*(m**2*br*u4(i)**2)
100
t5(i)=t4(i)+55/24*h4-59/24*h3+37/24*h2-9/24*h1
enddo
do i=1,nr
t1(i)=t2(i)
t2(i)=t3(i)
t3(i)=t4(i)
t4(i)=t5(i)
enddo
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
conv=100.0
lbd1=1
j=0
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
do while (conv > 1e-3)
lbd=lbd1
j=j+1
u5(1)=0.0
u5(nr)=0.0
do i=2,nr-1
h1=u1(i)*(-(dt*m**2)/omega)+(dt/omega)*(lbd+(gr*t1(i)+(u1(i+1)-2*u1(i)+u1(i-1))/(dx**2)))
h2=u2(i)*(-(dt*m**2)/omega)+(dt/omega)*(lbd+(gr*t2(i)+(u2(i+1)-2*u2(i)+u2(i-1))/(dx**2)))
h3=u3(i)*(-(dt*m**2)/omega)+(dt/omega)*(lbd+(gr*t3(i)+(u3(i+1)-2*u3(i)+u3(i-1))/(dx**2)))
h4=u4(i)*(-(dt*m**2)/omega)+(dt/omega)*(lbd+(gr*t4(i)+(u4(i+1)-2*u4(i)+u4(i-1))/(dx**2)))
u5(i)=u4(i)+55/24*h4-59/24*h3+37/24*h2-9/24*h1
enddo
do i=1,nr
a(i)=u4(i)*(1.0-(dt*m**2)/omega)
b(i)=dt/omega
c(i)=0.0
enddo
do i=2,nr-1
c(i)=gr*t4(i)+(u4(i+1)-2*u4(i)+u4(i-1))/(dx**2)
enddo
a1=0.0
a2=0.0
a3=0.0
do i=1,nr
a1=a1+a(i)*dx
a2=a2+b(i)*dx
a3=a3+b(i)*c(i)*dx
enddo
lbd1=(u0*0.5-a1-a3)/a2
if ((lbd*lbd1) /= 0) then
conv=abs((lbd1-lbd)/lbd)
endif
! write (*,"(E14.6)") lbd1
enddo
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
if (mod(i1,step_per/40) == 0) then
101
write (*,"(A)",advance=’no’) "#"
endif
if (i1==step_per) then
write (*,"(A)",advance=’no’) "]"
endif
if (mod(i1,step_per/scan_pt) == 0) then
k=k+1
u4t_mat(1,k)=tt-(i0-1)*2*pi
u4t_mat(2,k)=u5(11)
u4t_mat(3,k)=u5(21)
u4t_mat(4,k)=u5(31)
u4t_mat(5,k)=u5(41)
endif
if (mod(i1,step_per/freq) == 0) then
write (12,103) tt,u5
endif
do i2=1,nr
u1(i2)=u2(i2)
u2(i2)=u3(i2)
u3(i2)=u4(i2)
u4(i2)=u5(i2)
enddo
if (mod(i1,step_per/mat_pt) == 0) then
k2=k2+1
do i2=1,nr
u_mat(i2,k2)=u5(i2)
th_mat(i2,k2)=t5(i2)
enddo
t_vec(k2)=tt
endif
enddo
!CCCCCCC CONTROLLO CONVERGENZA PERIODICO STABILIZZATO CCCCCCCCCCCCCCCCCCCCCCCCCC
conv_ps=0.0
do i=1,nr
conv_ps=conv_ps+(u11(i)-u5(i))**2
u11(i)=u5(i)
enddo
print *
print *
write (*,"(A,E14.6)") "# Residual............
", conv_ps
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
enddo
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
write (9,"(5A14)") "Time", "u(0.1,t)", "u(0.2,t)", "u(0.3,t)", "u(0.4,t)"
write (9,101) u4t_mat
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
u_a=(maxval(u_mat,dim=2)+minval(u_mat,dim=2))/2.
u_b=(1./gr)*(maxval(u_mat,dim=2)-minval(u_mat,dim=2))/2.
th_a=(maxval(th_mat,dim=2)+minval(th_mat,dim=2))/2.
th_b=(maxval(th_mat,dim=2)-minval(th_mat,dim=2))/2.
102
do i=1,nr
uth_mat(1,i)=x(i)
uth_mat(2,i)=u_a(i)
uth_mat(3,i)=u_b(i)
uth_mat(4,i)=th_a(i)
uth_mat(5,i)=th_b(i)
enddo
! write (11,"(5A14)") "x", "u_a(x)", "u_b(x)", "th_a(x)", "th_b(x)"
write (11,101) uth_mat
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
do i=1,mat_pt-1
ct1=(t_vec(i)-(i0-1)*2*pi)-1
ct2=(t_vec(i+1)-(i0-1)*2*pi)-1
if (ct1*ct2 < 0.0) then
do j=1,nr
test_05(j)=0.5*(u_mat(j,i)+u_mat(j,i+1))
enddo
endif
enddo
write (10,"(2A14)") "Radius", "u(x,1)"
do i=1,nr
write(10,100) x(i),test_05(i)
enddo
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
100
101
102
103
format(2E14.6)
format(5E14.6)
format(51E14.6)
format(52E14.6)
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
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!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
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