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Oct 1721, 2007

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Xian, China. Computational Heat Transfer. and Fluid Flow ... Xian, China. Computational Heat Transfer. and Fluid Flow. Current practice. Now that CFD exists, ... – PowerPoint PPT presentation

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Title: Oct 1721, 2007


1
CFD Progress and Prospects
A lecture at the Asian Symposium ASCHT-2007
  • by
  • Brian Spalding,
  • of CHAM, Ltd

2
1. Introduction 1.1 Purpose
Computational fluid dynamics started half a
century ago. In this lecture, I review its
progress and seek to indicate how it may
profitably develop further.
I direct my words to research workers seeking
problems which it is possible and beneficial to
solve. I address also engineers, especially
those working in process industries, whose
designs can be improved if the indicated
developments are carried out.
3
1.2 Patterns of analysis
  • The problems facing applied science are
    multi-dimensional and they can be approached in
    various ways.
  • The main dimensions of variation are in
  • time,
  • space, and
  • population (to be explained below).

Variations in time are easiest to handle, because
we all grow older at the same rate one day per
day. Variations in space are more complex, but
easy to understand for some of us can run faster
than others.
4
1.2 Patterns of analysis
Variations in population? Here is a
one-dimensional histogram representing the
distribution of the age of persons for a
particular community at a particular time
and here is a picture to show that histograms can
be two-dimensional.
  • Populations which are relevant to CFD include
    those of
  • liquid droplets with differing diameters
  • solid particles with differing velocities
  • gas fragments with differing compositions, or
    temperatures and
  • radiation fluxes with differing directions.

5
1.2 Patterns of analysis
  • I shall further distinguish the three main
    approaches to non-uniformity, whether in time,
    space or population dimensions, namely
  • neglect,
  • presume,
  • which means in effect, guess, and
  • calculate

and I shall argue that, in respect of
calculation, the methods which are used for
spatial variations can be applied to population
variations also.
6
1.2 Patterns of analysis
  • I shall not argue that 'neglect' is always bad,
    or that 'calculate' is always best.
  • Indeed, most successful approaches are hybrid
    thus
  • even the most extreme of the calculators
    neglect something and
  • nearly all presume rather than calculate some
    non-uniformities.
  • What is necessary is to make wise decisions about
  • what to neglect,
  • what to presume,
  • what to calculate, and
  • when to do each.

7
1.3 The structure of the lecture
  • In part 2, I shall explain my 3-dimension
    3-approach classification and I shall illustrate
    it by way of examples from science and
    engineering.
  • In part 3, I shall recommend that CFD specialists
    should provide
  • heat-exchanger designers with software based on
    less presumption and more calculation
  • chemical-reactor operators with prediction
    tools which calculate the distribution of fluid
    fragments in composition space and
  • mechanical engineers with computer codes which
    calculate the flow of fluids and the stresses in
    solids simultaneously.

8
2. Examples of engineering analysis 2.1 Piston
engines space-direction variations
The steam engine
For this example, the 'neglect' approach is quite
satisfactory, because the variations of steam
temperature and pressure with position in the
space above the piston are small at any instant
of time.
9
2. Examples of engineering analysis 2.1 Piston
engines space-direction variations
Internal-combustion engines
Here the 'neglect' approach is not
satisfactory,because flames spread slowly.
The 'presume' approach is best, especially when
flame speed or spray burning rates are based on
experimental observations.
The 'calculate' approach, i.e. conventional CFD,
is often employed with limited success. Why?
Because it neglects 'population' aspects of (1)
turbulent combustion and (2) droplet vaporisation.
10
2. Examples of engineering analysis 2.2 Simpler
turbulent flows
.
The plane turbulent mixing layer non-uniformity
in space
I start with the simplest of all turbulent flows
the plane mixing layer.
The task is to predict the angle of the
wedge-shaped layer of turbulent fluid at the edge
of a jet injected into fluid at rest.
11
Shape functions and weighting functions
The 'neglect' approach is not applicable here
for non-uniformity is of the essence.
  • The presumed-profile approach involves
  • Guess the shapes of the velocity and
    effective-viscosity profies, e.g. as sloping or
    horizontal straight lines
  • Multiply the differential equations by weighting
    functions.
  • Integrate across the layer analytically.
  • Deduce the angle by algebra.
  • Advantage quick and easy.
  • Disadvantage accuracy is uncertain.

12
The plane turbulent mixing layer the
Finite-Volume Method
The calculate approach (version of Patankar
and myself, 1967)
  • presumes only that the velocity profile is a
    histogram, with unknown column heights
  • uses weighting functions of 1, i.e. none at all
  • integrates across each histogram interval
  • deduces the unknowns numerically.

This is now known as the 'finite-volume' method'
(FVM),the general form of its equations being
value in the volume sum for all faces of
coefficient value in neighbour volume
sum of additional sources wherein the
coefficients express diffusion and convection.
13
Other steady-state turbulent jets, wakes, plumes
and boundary layers
The early days of CFD a condensed history
  • The FVM was soon applied to these flows
  • which
  • had already been extensively studied
    experimentally, and by presumed-profile methods
  • are 'parabolic' (i.e. downstream events do not
    influence upstream ones)
  • therefore permitted solution by 'marching'
    methods' on memory-scarce computers
  • allowed turbulence models to be tested
  • gave us confidence to extend the FVM to
    recirculating, three-dimensional, unsteady,
    compressible and chemically-reacting flows

14
2.3 Steady flow around solid bodies immersed in
fluid streams
Streamlined objects Before CFD,
  • aircraft design was based mainly on a 'neglect'
    approach, in that the variations of stagnation
    pressure
  • were neglected. The aerodynamic forces on
    the aircraft were then computed by way of
    ideal-fluid theory.
  • The effects of viscosity, and indeed turbulence
    were expressed by the supposition that the
    'displacement thickness' of thin boundary layers
    enveloping wings and fuselage made these, in
    effect, rather thicker than they truly were.
  • The presumption approach was used,
    however, to calculate the displacement-thickness
    distribution so the whole method can be
    characterised as being 'hybrid'.

15
Current practice
  • Now that CFD exists,
  • the calculation' approach is adopted for the
    whole of the space occupied by the fluid which
    allows also the small regions of 'separated flow
    to be simulated.
  • However, an accurate calculation of the
    frictional forces on the solid surface can be
    made only by the use a very fine grid in the
    boundary layer
  • so, for economy, some element of
    profile-presumption is retained, by way of wall
    functions.

16
Flows around and inside buildings
  • Before CFD, flow prediction was based on
    experiments with small geometrically similar
    physical models
  • but this was unreliable , because the
    similarity criteria of Reynolds (viscosity) and
    Froude (buoyancy) could not both be satisfied.
  • Neither the neglect nor presume approaches had
    anything to offer. Therefore, engineers concerned
    with heating, ventilating, air-conditioning and
    fire-protection of buildings were among the first
    to turn to CFD.


17
Flows around and inside buildings
  • CFD has satisfied their requirements and
  • it is for widely used for simulating fires in
    car-parks and other buildings
  • BUT, for phenomena such as the fire-ball, it
    needs to take account of variations in
    hot-gas-population space.

18
2.4 Chemical-engineering equipment
Heat exchangers non-uniformities in space
No designer can 'neglect' the temperature
variations in heat exchangers. Instead, most
guess them as being similar to that calculated
for idealised counter-flow systems. Since they
know that the flow patterns must differ, they
multiply their calculated heat-transfer rates by
correction factors like those on the right. But
these are still guesses, none the less.
19
Heat exchangers non-uniformities in space (end)
These presumption practices derive from the
pre-CFD age. However, it was shown more than
thirty years ago (by Patankar and myself, as it
happens), that the calculate approach is
practicable and indeed easy.
It is strange therefore that most heat exchangers
today are still based on presumption rather than
calculation. Therefore, in section 3.1, below,
I shall be recommending a change of practice.
20
Stirred chemical reactors, showing variations in
both space and population
The process Many chemicals products are created
by pumping feedstock materials (A and B) into a
reactor vessel, where they are stirred together
by a paddle, in order to react chemically.
The task is to predict how the rate of production
of C from reactants A and B depends upon the
power consumed by stirring and the rate when
mixed in a test-tube, where rate/(concAconcB)
k_tube .
21
Stirred chemical reactors Variations of
time-averaged concentration
Before CFD, the 'neglect' approach had to be used
for variations with position and it was not bad
for, if the stirring is vigorous enough, the
time-average values of concA and concB will
indeed be almost uniform. But what about moderate
stirring? The 'presume' approach is not usable in
this case for no guidance exists as to what
profiles should be presumed. Nowadays, CFD is
employed but it is not enough for, if R_ave /
(concA_ave concB_ave) k_reactor , it is found
experimentally is that k_reactor is much less
than k_tube. Why is this?
22
Stirred chemical reactors Variations in
population space
The answer non-uniformity in population space,
also called unmixedness, shown here -gt At any
point in the reactor, fluid fragments of many
different concentrations can be found. To
calculate their time-average values, one must
know for what proportion of time each is
present. That means that one needs a
probability-density function, like this
---gt Can one calculate it?
Yes, as I shall explain later and for each
location and stirring rate too. From it can be
deduced the C- production rate.
23
Furnaces and other combustors more variations in
space and population
  • General description
  • A coal-fired furnace is a special kind of
    chemical reactor and the processes taking place
    in it present a severe challenge to computer
    simulation, because of the importance of
  • chemical reactions (coal pyrolysis,
    volatilisation, combustion, NOX formation)
  • solid-fluid interaction (diffusion of oxygen to
    the surface)
  • thermal radiation and
  • particle-wall impact.

24
Furnaces and other combustors Variations in
position and population
Which approach should be used for space
variations? Only the calculate approach has any
hope of representing the distributions of
temperature, velocity, and pressure throughout
the volume and it has indeed been used for many
years.
  • And for population non-uniformity?
  • Of coal-particle size often neglected but
    sometimes presumed to vary in accordance with the
    empirical formula of Rosin and Rammler
  • of radiation angle often neglected ( in
    conduction model) sometimes presumed (in
    six-flux model) , and less often calculated
    (discrete-ordinates formulation)
  • of radiation wavelength nearly always
    neglected
  • of gas concentrations nearly always neglected.
  • To recommend calculate for all would be too
    ambitious.

25
2.5 Simpler non-uniformities in population
droplet-size
Vaporization of fuel sprays (in Diesels or gas
turbines) consisting of droplets of various
diameters, D, which change size at a rate
governed by - dD/dT const (1/D)
ln(1B) where B, the driving force for mass
transfer, depends upon (e.g.) local temperatures
and other gas properties. This shows that
droplets diminish in size at different rates, the
smaller ones disappearing the more rapidly.
.
The task is to calculate the overall rate of
vaporization. This necessitates knowing the
droplet-size distribution at each location and
each time.
26
Vaporization of a spray droplet-size population
The usual three ways are 1. Neglect variations,
i.e. suppose that all the droplets at a single
location in the spray have the same diameter.
2. Presume that the profile is constant (e. g.)
of Rosin-Rammler form, which cannot be very
accurate. 3. Calculate the ordinates of the
histogram by way of a standard finite-volume
equation, with the source term dD/dT above. Use
calculate if droplet size is critical, as in fire
extinction.
27
The turbulent diffusion flame fuel-air-ratio
population
Experimentally-observed unmixedness Hottel,
Weddell and Hawthorne drew attention in 1949 to
the 'unmixedness' of the gases in a flame
produced by a jet of fuel gas injected into air.
They measured finite time-average
concentrations of both fuel and oxygen at the
same location. That could never be found in a
laminar flame.
The first CFD analyses It was not until 1971
that the first attempt to simulate this
unmixedness numerically was made, on the basis of
a very simple profile presumption.
28
The turbulent diffusion flame presumed
fuel-air-ratio population
The guess was that, at a point where the
time-average fuel-air ratio was F, say, the gases
actually present there had the ratio F g for
half the time, and F- g for the other half.
Standard CFD calculated F easily. For g, a new
differential equations was invented, having
sources guessed as being proportional to
gradients of F- and velocity. This approach, when
appropriate empirical constants were introduced,
allowed turbulent diffusion flames to be
simulated.
29
Confined pre-mixed flame reactedness population
In the turbulent diffusion flame, fuel and air
enter separately, and must be mixed before
chemical reaction can occur, at a rate limited by
the rate of that mixing.
I now consider a flow in which the fuel and air
are mixed before they enter, at uniform and
constant velocity, a plane-walled duct in which
is placed a bluff-body 'flame- holder'. A
turbulent wedge-shaped flame spreads across the
duct, as the sketch indicates and the profile of
longitudinal velocity is roughly as shown. What
then limits its rate? A different kind of mixing
that between burned and unburned gases.
30
Confined pre-mixed flame the near-constancy of
its angle
  • When first investigated, this flame showed some
    puzzling features, namely that the wedge angle
    was almost independent of
  • inlet velocity
  • fuel-air ratio
  • inlet temperature
  • pressure and
  • inlet turbulence intensity.
  • But why?
  • H.S. Tsien, while at CalTech, explained the shape
    of the profile but what governed its angle
    remained a mystery.
  • We learned only later
  • non-uniformity in space depends on
  • non-uniformity in population.

31
Confined pre-mixed flame the first population
presumption
The guessed profile The first idea, embodied in
the so-called eddy-break-up model , was that the
gas population consisted of two components,
namely
(1) fragments of wholly un-burned gas which were
too cold to burn and (2) fragments of hot
wholly-burned gas which also could not burn
because either all the fuel or all the oxygen had
been consumed.
The histogram representing the presumed
population therefore consisted of two spikes and
their relative heights dictated what would be
measured as the time-average temperature.
32
Confined pre-mixed flame collision between
burned and unburned gas fragments
These two elements of the population were
thought of as colliding with one another and
thereby producing sub-fragments of intermediate
temperature and composition.
These latter, being sufficiently hot and also
containing reactants, could burn and did so very
rapidly, thereby increasing the height of the
right-hand spike. Their actual concentration was
considered, implicitly, to be negligibly
small. The rate of collision per unit volume was
guessed as proportional to the rate of
dissipation of turbulence energy. This explained
why the flame angle remained almost unchanged
when the inflow velocity was increased.
33
Confined pre-mixed flame the next presumed
reactedness profile
The four-fluid model The EBU, published in 1970,
became very popular so much so that 25 years
passed before the obvious next step was taken
to increase the number of presumed components
from 2 to 4 !
Collisions between fluids 1 and 3 created fluid
2, 2 and 4 created fluid 3, 1 and 4 created
fluid 2 and also fluid 3. Reaction of
fluid 3 created fluid 4 at a chemistry- controlle
d rate.
Fluids 1 2 3 4
34
Confined pre-mixed flame applications of the
four-fluid model
The chemistry-controlled step (fluid 3 creates
fluid 4) explained why 1. the flame angle
remained nearly constant, and 2. the flame could
be suddenly extinguished by a velocity increase.
The four-fluid model was used successfully for
simulating flame spread in a baffled duct and for
oil-platform explosion simulation. It
has been little used but it was the first step
towards calculating the reactedness population,
35
From four fluids to many the multi-fluid model
In conventional CFD, we divide space and time
into as many intervals as we need. Why not do the
same for the reactedness at each point? The
height of each column can then be deduced from a
Finite-Interval equation like this height of
interval sum for all faces of coefficient

height of neighbour interval
sum of additional sources
sum for all other intervals
of coefficient
height of other interval )
36
What the terms in the finite-interval equation
represent
  • In height of interval sum for all faces
    of coefficient

  • height of neighbour interval
  • the coefficients express rates of convection and
    diffusion, as in the the finite-volume equations
    of conventional CFD.
  • But in sum for all other intervals
    of coefficient

  • height of other interval
  • the coefficients express the physical and
    chemical processes
  • collision between members of the fluid
    population, and
  • chemical conversion of one member into another.
  • The finite-interval method is thus merely a
    natural extension of the finite-volume method
    and its equations can be solved in the familiar
    successive-substitution manner.
  • The calculation of population distributions is
    easy.

37
How material is distributed after collision
Here is a diagram from one of the earliest
publications. It depicts one of the possible
hypotheses, called 'Promiscuous Mendelian'.
The 'colliders' are treated as 'mother' and
'father and the word 'promiscous' implies that
any two members of the population may collide.
The word Mendelian, a reference to Gregor Mendel,
the Austrian "father of modern genetics", implies
that the offspring may appear with equal
probability in any interval between those of the
parents.
38
A calculated probability-density function
This hypothesis has been embodied in the PHOENICS
computer code. Here is one reactedness
histogram, computed with its aid. As in the the
eddy-break-up guess, there are indeed spikes at
zero and unity reactedness but calculation has
shown that the intervals in-between are
alsopopulated.
Such probability distributions can to be computed
for each location in the flame. Then the desired
reaction rate for the whole flame can be deduced.
39
Application to gas-turbine combustion
A three-dimensional gaseous-fuel combustor I show
here one sector of a simple combustor proposed by
Professor Wu Chung-Hua in the early days of
PHOENICS.
40
Smoke formation rate is influenced by turbulent
fluctutions
Much later, I used this combustor to show how one
must not neglect fluctuations of fuel-air ratio
when predicting smoke formation.
I used a 10-fluid model, with fuel-air-ratio as
the population-defining attribute. Each cell had
its own computed histogram
The differences, although small. are significant
when CFD is being used to optimise the design.
41
Concluding remarks for Part 2
  • It has been shown that
  • variations in population space should not be
    neglected especially when chemical reaction is
    involved
  • they can be presumed
  • but it is better to calculate them.

Why are not crowds of researchers pouring into
this scarce-explored territory? Perhaps because
they are waiting for less-timid crowds to do so
first.
42
3. Recommendations 3.1 To heat-exchanger designers
So far, I have been discussing general ideas. Now
I wish to make three specific recommendations.
Current practice I have already mentioned that
heat exchangers are still designed in the basis
of presumption.
A shell-and-tube heat exchanger looking like this
(tubes not shown) can be expected to have a
rather complex flow in the shell.
43
or
Yet the software used by designers presumes that
the flow in the shell can be conceptualized thus,
and described by very few parameters.
But why presume when one can calculate, as was
shown to be possible by the 35-year-old
publication in which this image appeared?
44
3.1 To heat exchanger designers The solution
  • The solution is
  • do not attempt to calculate the flow pattern
    between the tubes in detail, because current
    computers are not large or fast enough to handle
    the necessary fine grids except for a few tubes
    at a time.
  • Instead, use the space-averaged approach, with
    empirically-based formulae for
  • ? heat-transfer coefficients per unit
    volume, and
  • ? friction factors per unit volume,
  • as functions of local Reynolds and Prandtl
    numbers.
  • Then solve the finite-volume equations for
    (space-averaged) velocity, pressure, temperature
    for the shell- and tube-side fluids, treating
    both as interpenetrating continua, as is easily
    possible.

45
3.1 To heat exchanger designers The solution
(contd)
I now show some (not new) results for (the
central plane of symmetry of) a particular
shell-and-tube heat exchanger.
(a) The shell-side velocity vectors, when
calculated, appear thus
(b) The consequential shell-side temperatures,
are not, as presumed, a succession of vertical
stripes although the calculated tube-side
temperatures are (very nearly).
46
3.1 To heat exchanger designers The solution
(end)
(c) The conventional heat-exchanger-design
packages presume that the shell-side, tube-side
and overall heat-transfer coefficients are
uniform throughout but calculation reveals that
they are not, as the next pictures clearly
demonstrate.
Corresponding non-uniformities are exhibited by
the calculated Reynolds- and Prandtl-number
values, and the temperature-dependent fluid
properties, from which the heat-transfer
coefficients have been computed.
47
Recommendation number 1
  • My conclusions are…
  • that the conventional presumptions are
    evidently incorrect
  • that therefore software which is based on them
    will generate unsafe designs and
  • that the calculate approach, using
    experimentally-based data for the space-averaged
    heat-transfer and friction coefficients, is the
    only sound basis for the design of heat-exchanger
    equipment.

I declared at the beginning that I had something
to say to engineers. This first recommendation
is addressed to them
Demand that the suppliers of your
heat-exchanger-design software build into it the
calculate approach.
48
3.2 To stirred-reactor designers and operators
The calculation required by my first
recommendation concerned non-uniformities in
space. There are therefore many CFD specialists
who will know how to implement it. My second
concerns non-uniformities in population experts
in these are harder to find. The task is to
predict how stirring-rate influences the
conversion rate of reactants A and B into C in
reactors of the kind which I discussed in Part 2.
49
3.2 To stirred-reactor designers and operators
(contd)
An example I turn to a ten-year old work Ref ,
in order to emphasise that the idea is not new,
merely neglected. It concerns, for simplicity,
reactants for which the rate constant measured in
a laboratory test tube (i.e. k_lab) is very
large. The geometry, and the body-fitted-co-ordin
ate grid used in the CFD calculation, are shown
below.
50
3.2 To stirred-reactor designers and operators
(contd)
  • But what about the mixture-ratio population grid?
  • Two distinct cases were considered, namely that
  • the materials from the entering streams of
    reactants A and B were fully mixed at each point
    in the reactor, which would correspond to
    presuming
  • that its pdf was the single spike shown on
    the following diagram, and that
  • the amount of product C was as indicated by
    its horizontal location
  • alternatively, at each point there could be
    found varying amounts of 'fluids' (in the
    multi-fluid sense) having one of eleven distinct
    mixture ratios, so that its pdf could be that of
    the histogram also shown there.

51
3.2 To stirred-reactor designers and operators
(contd)
Case 1 is the conventional-CFD approach which
presumes the state of the mixture-ratio
population and Case 2 represents what is done by
those who recognise that non-uniformities in
population space can be calculated.
The results of the two approaches are different.
This is demonstrated by the following two contour
diagrams showing the product (i.e. C)
concentrations after 10 revolutions.
The general patterns are not very dissimilar but
their scales are 3.2 for the presumption
approach and only 2.4 for the calculation
approach, at this moment of time.
52
3.2 To stirred-reactor designers and operators
(contd)
The explanation for the difference is to be found
in the calculated mixture-ratio histograms, of
which a few will be shown, corresponding to a
single instant of time, a single vertical height
and circumferential angle, and at six different
radii, starting near the axis and moving outward.
  • These pdf histograms show that
  • detailed information about the micro-mixing can
    indeed be obtained by calculation
  • the pdfs vary is shape in a manner that it
    would be impossible to guess

53
3.2 To stirred-reactor designers and operators
(contd)
  • their shapes are utterly unlike the single
    spike which neglecting the micro-mixing implies
  • they will assuredly imply different
    mixture-average product concentrations.
  • Inspecting them may lead to questions such as
  • Did these calculations consume much computer
    time? Answer about the same as did the
    hydrodynamic calculations.
  • Was eleven intervals too few? Or too many?
    Answer One has to repeat
    calculations with finer and coarser 'population
    grids' to find out, just as for spatial grids.
  • Is the ratio of 3.2 to 2.4 typical?
    Answer No values
    much closer to and farther from unity can be
    encountered.
  • Do the predictions agree with experiment?
    Answer I expect so,
    qualitatively but no serious investigations have
    yet been made.

54
Recommendation number 2
My second recommendation therefore, to
researchers and to their engineering managers is
  • waste no more time on CFD simulations of
    stirred reactors unless they calculate the fluid
    population
  • do not wait for the necessary physical
    constants to be accurately determined for even
    approximate ones will be better than the current
    neglect of the micro-mixing
    phenomenon.

55
3.3 To researchers and engineers concerned with
fluid-solid interactions
A historical accident In section 2.2.1, I
mentioned 'weighting functions' and I said that
the finite-volume method uses unity as its
weighting function whereas the finite-element
method uses something else. This is came about
because the FEM originators, R.Clough and
O.Zienkiewicz, were mainly concerned with
stresses and strains in solids and, in that
field, methods using weighting functions, such as
those of Galerkin had a long pre-computer
history.
Galerkin
O.Zienkiewicz
R.Clough
56
A missed opportunity and its consequences
  • When computers arrived, stress analysts simply
    carried some of their old
    baggage with them,
    not recognising that it was no longer
    needed.
  • This tiny difference in starting point has led to
    enormous differences of practice and language
    between the stress-analysis and fluid-flow
    communities and it has given rise to
    totally false ideas, namely
  • that the finite-volume and finite-element
    methods are essentially different
  • that the FEM must be used for the calculation
    of solid stress and
  • that therefore different methods and computer
    programs must be used for solid-stress simulation
    from those which are used for fluid flow.

57
What we now know , and should act upon
In fact, however, a weighting function of unity
works just as well for solid stress as for fluid
flow. Therefore a single method and a single
computer program can be used for both and they
should be used, for economy, whenever the problem
in question involves the interaction between
solids and fluids. I will now explain why.
58
Why one method can suffice for both classes of
problem
  • The reasons are
  • The differential equations for velocities in
    fluids are very similar to those for
    displacements in solids, from which the stresses
    can be deduced. Thus
  • del2 u - d/dx pc1 fxc2
    convection terms 0 for velocity, and
  • del2 U d/dx DC1 - TeC3 FxC2
    0 for displacement.
  • The solid-stress equations are indeed the
    simpler, being linear where the former are
    non-linear.
  • Since the solid-stress problem is simpler than
    the fluid-flow one, computer codes written for
    the latter can easily serve for the former also,
    as many publications have proved.

59
A thermal-stress example
The three examples which I shall show are several
years old for I wish to emphasise that my
message is not a new one. But it has suffered
from neglect.
First, a cooling fluid flows through a
pressurised curved duct in a solid block.
The block is heated at various several points so
that its thermal expansion is non-uniform.
60
Thermal and mechanical fluid-structure
interactions
The equations for velocity and displacement and
velocity are so similar that PHOENICS solves both
sets at the same time. Here the solutions are
presented in terms of vectors.
In my second example, the fluid-structure
interaction is mechanical rather than thermal. A
thin partition bends as a consequence of the
differences of fluid pressure on its two
sides.
61
A periodic fluid-structure interaction
The final example is also one of mechanical
interaction.
It shows the transient deflection of an
under-water structure under the influence of the
wave motion of the ocean. PHOENICS computes the
displacements in the solid and the velocities of
the fluid simultaneously, as a single set of
vectors.
A question worth asking. Why, since these and
many other examples have been available for many
years, do most vendors still offer separate
software packages for the calculation of fluid
flow and solid stress?
62
Why do the false ideas persist?
Neglect of the evidence plays a part.
Presumption that what is done must be done
contributes. But if anyone were to calculate the
cost of current practices, surely the argument
for change would become irresistible.
63
Recommendation number 3
My third recommendation is therefore that
  • Researchers should develop and refine the
    finite-volume method for simultaneous fluid-flow
    and solid-stress calculation
  • and
  • Engineers concerned with fluid-structure
    interactions should demand computer codes which
    embody those methods.

64
The last slide With thanks for your attention
  • The message of this lecture has been that
    the world of CFD is wider
    than most of its inhabitants conceive.
  • Time and space form only four of its dimensions.
  • Others include
  • reactedness fuel / air ratio of gas fragments
  • size temperature composition velocity of
    particles
  • angle and wavelength of radiation.

and probability- density functions employed, 1D
or 2D.
Populations must be considered,
They should seldom be neglected
But the best is calculation
They may sometimes be guessed
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