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CFD Progress and Prospects

A lecture at the Asian Symposium ASCHT-2007

- by
- Brian Spalding,
- of CHAM, Ltd

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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'.

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.

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.

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.

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.

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.

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 .

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?

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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,

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 )

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.

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.

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.

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.

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.

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.

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.

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?

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.

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).

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.

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.

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.

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.

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.

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.

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

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.

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.

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

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.

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.

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.

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.

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.

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?

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.

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.

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