I divide this into three periods, namely:
I shall criticise as well as praise; and I shall end with an easy-to-understand 'proof' that Timoshenko's fundamental thermal-stress equation cannot be correct.
next,
After a year investigating Germany's rocket developments,
I chose this as my PhD topic.
Liquid fuels are 'atomised', i.e. converted into clouds of droplets,
before burning. For simplicity, I studied a single droplet (truly a
larger sphere), both in
forced and
free convection.
I first adapted the 'stagnant-film' concept, described in 'Absorption
and
Extraction by Sherwood and Pigford;
but 'Forschung' of 1949 contained a paper by Eckert and his late
student Lieblein which presented a true
boundary-layer-with-mass-transfer model.
I had already derived a single equation with either The Eckert-Lieblein method enabled me to solve the equation; and
the solution fitted
my experimental data! My thesis claimed (I blush to report) to have generalised, and in one
repect corrected, Eckert's theory.
I have always had this unlovable tendency to criticise my elders and
betters.
But it keeps my mind alert.
Moreover, his method of:
and this had already been used for heat transfer alone by the
(mercifully still alive) Russian scientist Krouzhilin.
By combining all three innovations, Eckert created the first model
which enabled
Thus I was enabled to take the not-very-difficult next step, namely
to handle reacting boundary layers.
It permitted calculation of the rate of combustion without
knowledge of the
chemical-reaction process, except that it was 'fast enough'.
The rate of burning of a liquid fuel was proved thus to be
'mass-transfer controlled', being influenced by the rates of:
Moreover, a
later paper by Hottel had come to the same conclusion by way of
'stagnant-film' theory'.
Still, with Eckert's aid, a useful generalisation had been achieved
which enabled the combustion of all fuels, from:
Therefore, I included in my PhD thesis a quantitative, albeit
approximate, theory of the transition from the envelope flame to the
wake flame shown
here.
The works of the Russians Zeldovich and
Frank-Kamenetsky were
drawn upon; but details would be out of place here.
next "Diese sind die Methoden die ich bei meinen Luftuntersuchungen
gebraucht habe; ich gestehe dass sie einigen nicht sonderlich anstehen
werden, weil sie keinen genauen Aussschlag geben.
Sie haben mich aber Genugtuung geleistet: man will auch
oft ein Haar spalten, wo es gar nicht noetig ist."
A very Anglo-Saxon thought: Don't split hairs!
It became especially useful when coupled with empirical laws for
'entrainment', with which G.I.Taylor had, to the dismay of our
security authorities, computed the power of our first atomic bomb
from the visible rate of growth of the 'mushroom cloud'.
Ideas were also incorporated from the work of Kutateladze and
Leont'ev, whose book I had been bold enough to translate.
This stream of work was at first strengthened by the availablity of the
digital computer in the late 1960s, but it has now almost dried up.
'CFD' has taken over.
next
So why not, my thought then was, use an infinitely-flexible
piece-wise-linear
profile of which the ordinates would each be calculated from its own
integral equation?
Thus it was that I stumbled into the method of analysis that has come to
be known as
computational fluid dynamics.
Suhas was quick to pick up the suggestion; and he created our first
genuine
'CFD code', for two-dimensional 'parabolic flows' (jets, wakes and
boundary layers).
next From Schlichting's textbook we had learned of the von
Mises (stream-function) coordinate system; by using a
dimensionless form of this, we created (I think) the first
self-adaptive grid.
The grid width was determined by the 'entrainment rate'.
I called this the 'Bikini method' because it could fit a curved body,
and cover just the areas of special interest.
There were no textbooks to aid us; but there were publications, of
which those by Thom, Courant and Burggraf were especially
helpful; and
we were as ready to use intuition as mathematical rigour.
'Upwind differencing', for example, derived definitely from
the former.
next Although we knew that Harlow was using the 'primitive
variables' (p, u, v, w), we chose stream function and vorticity so as
to reduce the number of variables.
This was important because our computers had little power or memory.
Stuart Churchill independently made the same choice around then.
The 'hybrid-differencing scheme' was invented at this time; and it
enabled us to obtain solutions at arbitrarily high Reynolds numbers, as
shown here.
next
We already had a not-bad algorithm called SIVA (SImultaneous
Variable Adjustment); but, by careful study
of the works of Chorin and Harlow, Suhas devised a
segregated-variable scheme which came to be called
SIMPLE.
Almost everyone uses this now, in one form or another; but
SIVA-like algorithms are also coming back into fashion.
Our first publication was for three-dimensional parabolic flows;
but the method worked just as well for elliptic ones, as we soon
showed.
next
So Suhas created a CFD code for
flows which were:- parabolic or elliptic,
steady or transient, compressible or
incompressible, laminar or turbulent, reacting or not, and even
capable of solving the radiation equations.
However, while honouring pioneers, I prefer to
give prominence to Ivo
Zuber, who, like Professor Eckert before him, was a German working
in Czechoslovakia. Alas, he too passed away this summer.
We knew nothing of him then, but it was he who created the first
three-dimensional CFD
model of a combustion chamber.
His computer was pitifully weak; his institute gave him little
support; and the Communists still ruled his country. How much more
meritorious therefore was his achievement than ours!
next
Prominent in this research was a young man sent to me by Professor Eckert,
Wolfgang Rodi, who is now a world expert on the subject.
Turbulence models as we know them spring from A.N.Kolmogorov's
1942 guess that:
Ludwig Prandtl cannot have known of this work when he published his
similar, but lesser, paper in 1945. Nor, surely, did the equally
innovative Francis Harlow, the 1968 inventor of the
k-epsilon model.
next
My own IPSA, which adapted SIMPLE for the same task, was
developed independently but published later.
IPSA became popular, and is now widely used; whereas the two-fluid
turbulence model,
which it led to, never 'caught on'.
This is a pity, because it can explain turbulent
unmixing, which no other model can do.
next
Specifically, when chemical reaction in turbulent gases is to
be simulated, what is needed is a set of what are called 'probability
density functions' which record for what proportion of time the gas
has a defined state of concentration and temperature.
There is time here only to mention two pioneers in this field, namely
Cesar Dopazo, who formulated the theory and S.Pope
who developed a
Monte Carlo method for solving the equations.
Although I believe that the Monte Carlo approach is
not the best, I am happy to include both names in the list of those
whose ideas have influenced my own.
next
I believe that this multi-fluid model of turbulence will
supplant Kolmogorov-type models for many purposes in the future; but
I may not live long enough to see it do so.
next
Why? Because fluids and solids interact mechanically and thermally
(at least); and it is troublesome, inaccurate and unnecessary to
use separate computer programs for the two phases and then to combine
their result.
Examples of such interactions are:
Fluids and solids occupy different parts of space. Within the solid
regions there are no velocities to calculate; so we can compute the
displacements instead.
But beware: fluids have no property like Poisson's ratio, which
plays an important role in solid-stress calculation.
Nevertheless, if enough care is taken, a CFD code can be 'tricked'
into computing displacements, and thence strains and stresses, while
'thinking' it is computing displacements.
next
When he had seen these results, a solid-stress specialist said that I had
re-invented R.V.Southwell's 'Relaxation Method', which perhaps I
had; hence the attribution above.
Of course, Southwell was concerned with solids only.
The solution algorithm in the early work was like SIMPLE; and it had
two defects:
The advantages were similar to those brought by the use of vorticity
in CFD,
mentioned earlier.
Then bending could be properly treated.
The coding was created by careful attention the the classical textbooks of
Love (1892 and later) and Timoshenko (1914 and later).
next
Stresses in variously-constrained uniformly-heated objects were
correctly
computed.
Then I tried non-uniform heating, namely an unconstrained
block, heated on one side and
cooled on the other, so that the nett change of thickness (resulting
from expansion and contraction) should be zero.
BUT IT WAS NOT ZERO.
Of course, I checked the coding many times, but there appeared to be no
mistake.
Finally I (rashly) concluded that Timoshenko had got it wrong!
(1/2) de/dx + (1/2) d2u/dx**2 - alpha
dT/dx = 0
where e is the volumetric dilatation, u is the linear
strain,
alpha is the linear expansion coefficient and T is the
temperature.
Now e=3*alpha*T because expansion is in 3 directions, and
so the equation dictates: 3/2 + 1/2 - 1 = 0 ! Which is not true!.
How can this be? For Timoshenko is the classic.
next
They fit the theoretical solutions exactly: for there is a nett
increase of thickness.
So what was Timoshenko's mistake?
In truth, Timoshenko and Goodier wrote: It was I who mistakenly argued:-
The last is NOT TRUE; closer analysis shows that they are each equal to
minus d2u/dx**2 .
So the equation dictates: 3/2 + 1/2 - 1/2 -1/2 - 1 = 0 ! Which is true!.
Which means that Timoshenko was right;
I and all who were taken in by my argument were wrong.
Shame on us!
next
for, even when they turn out to be right (as they mostly do) our questioning
leads to deeper insight, and sometimes new advances!
None of us should however
dare, despite temptation, to echo
I thought I was wrong,
when I was right"
next
At the top of the list I am proud to place the name of:
The End !!!
1. Pre-computer researches
1a The combustion of liquid fuels: my PhD
In 1949, kerosine fuelled the newly-invented gas turbines; and
rockets burned liquid hydrogen and oxygen;
but there was little
understanding of what
governed the rate, or even the possibility, of combustion.
1b. The mass-transfer boundary layer
(Eckert and Lieblein)
A theoretical model was needed.
enthalpy or M_fuel - M_oxygen/stoichiometric_ratio
as dependent variable, which could describe the convection and diffusion
in burning gases.
1c. Earlier pioneers (Ernst Schmidt, von Karman, Kruzhilin)
We all build on the work of our predecessors; so Eckert had made
acknowledged use of
E.Schmidt's proof that the differential equation for concentration
was similar to that of temperature.
came
from von Karman and Pohlhausen;
Eckert's contribution
to be quantitatively understood.
1d. Eliminating the chemistry (Semyonov)
My own 'innovation', the use of a single equation for
chemically-reacting materials, proved to have been anticipated by another
Russian, N.N.Semyonov, in 1940, but in another context.
Click here for sketch.
next1e. Mass-transfer-controlled combustion
(Nusselt, Hottel)
In retrospect, this was not truly surprising; for Nusselt had
recognised that the combustion of high-temperature solid carbon must
be controlled by the rate of diffusion of oxygen to it already in 1916 !
to be
seen as a single family.
1f. Chemistry makes a come-back (Zeldovich, Frank-Kamenetsky)
Though mass transfer controls the rate of combustion, chemistry
still controls whether it can occur.
We all confirm our knowledge of this when we 'blow out' a candle
flame.
>A much earlier investigator of the extinction of combustion
However, German speakers in the audience may relish the 1777 quotation
from Carl Wilhelm Scheele with which I excused my approximations:
1g. Further exploitation of 'profile methods
(Taylor, Kutateladze, Leont'ev)
With my PhD behind me, I continued to use for many
years the Karman-Krouzhilin-Eckert 'integral/profile' method, not only for
laminar but also for turbulent flows.
2. Computational fluid dynamics
2a The two-dimensional boundary layer
(Schlichting, von Mises)
When Suhas Patankar came first came to Imperial College,
integral/profile
methods still prevailed; but their arbitrariness and inflexibility
were becoming irksome.
The novel coordinate system
2b Two-dimensional elliptic flows
(Thom, Courant, Burggraf)
Akshai Runchal and Micha Wolfshtein joined me about a
year after Suhas; and by now I was more ambitious:
'Elliptic', i.e. 'recirculating' flows were to be
targetted.
The choice of dependent variables
2c The three-dimensional boundary layer
(Chorin, Harlow)
In 1971, Suhas paid a second visit to Imperial College, to find that I
had abandoned stream-function and vorticity, which appeared to
be too difficult to generalize to three dimensions, and was now working with
the 'primitive variables'.2e. Turbulence models (Kolmogorov, Prandtl,
Harlow)
The 'Bikini method' was incorporated into computer programs at Stanford, by
Professor Kays and
his students, and at Imperial College. The latter program,
GENMIX, became the main 'test-bed' for turbulence-model
research in the late '60s.
2f. Numerical computation of two-phase flows
(Harlow)
A later and indisputable 'first' for Harlow was his
publication on the
numerical computation of two-phase flows, for example steam and water,
with allowance for the fact that the two phases will, in general, have
different velocity components at each point.
2g. The probability-density function(Dopazo,Pope)
Kolmogorov's approach to turbulence modelling is not the only one;
and despite its almost universal adoption, it is not necessarily the
best. Indeed, for some tasks it definitely is not.
2h. The multi-fluid approach (?????)
What are those ideas? I show only
one picture
, which shows a discretised probability-density function,
in which:
3. Unification of CFD and solid-stress analysis (?????)
3a. Why unify?
What I do hope to see is the unification of CFD and solid-stress
analysis.
next3b. How unify?
The solid-stress equations, when expressed in terms of
displacements, are similar to the Navier-Stokes
equations.3c. First attempts (Southwell)
Results from an early study are shown below for a two-solid-material
block, heated by radiation from above, and cooled by a stream of air:
(1) velocity
vectors,
(2)
displacement vectors, computed at the same time
and
(3)
horizontal-direction
stresses, obtained by post-processing.
next3d. A better method (Love, Timoshenko)
Both defects were removed when:
3e. Thermal stress; a surprising failure
I then turned attention to thermal stresses.
3f. A question for the audience
Timoshenko (3rd edition, with Goodier, page 457, equation 264) implies that
for this case, when Poisson's ratio P is zero (to make it easy) and
x is the temperature-gradient direction:
d2u/dx**2 is equal to d/dx of alpha*T.
3g. The good news
When the above equation is replaced by one corresponding better to
physics and arithmetic, one can get the right answer. Here are shown, with
distance vertical and time horizontal, computed
temperatures and
displacements.
This is how I had planned to end my talk; but, fortunately, I saw the light
in time!
3h. The answer
(L + G) de/dx + G ( d2u/dx**2 + d2u/dy**2 + d2u/dz**2) -
{alpha*E/(1+P)} dT/dx = 0
4. Concluding remarks
I conclude therefore that, though we must all follow in the footsteps of
great men, it is good not do so uncritically;
Sir Thomas Beecham's
(tongue-in-cheek) pronouncement:
next
"I did once make a mistake:
5. Last words
