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.
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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
enthalpy or M_fuel - M_oxygen/stoichiometric_ratio
as dependent variable, which could describe the convection and diffusion
in burning gases.
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.
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"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.
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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).
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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.
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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.
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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.
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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!
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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.
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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.
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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.
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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.
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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.
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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).
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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
d2u/dx**2 is equal to d/dx of alpha*T.
so the equation dictates: 3/2 + 1/2 - 1 = 0 ! Which is not true!.
How can this be? For Timoshenko is the classic.
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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:
(L + G) de/dx + G ( d2u/dx**2 + d2u/dy**2 + d2u/dz**2) -
{alpha*E/(1+P)} dT/dx = 0
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!
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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
Sir Thomas Beecham's
(tongue-in-cheek) pronouncement:
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I thought I was wrong, when I was right"
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At the top of the list I am proud to place the name of:
The End !!!