Honouring the innovators:
from Eckert to Timoshenko


Brian Spalding

Eckert Memorial Event, 2004



In this lecture I recall those innovators, starting with Ernst Eckert, whose ideas have most influenced my scientific career.

I divide this into three periods, namely:

  1. pre-computer researches

  2. computational fluid dynamics

  3. unification of CFD and solid-stress analysis.

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.



  1. Pre-computer researches
    1. The combustion of liquid fuels: my PhD
    2. The mass-transfer boundary layer (Eckert and Lieblein)
    3. Earlier pioneers (Ernst Schmidt, von Karman, Kruzhilin)
    4. Eliminating the chemistry (Semyonov)
    5. Mass-transfer-controlled combustion (Nusselt, Hottel)
    6. Chemistry makes a come-back (Zeldovich, Frank-Kamenetsky)
    7. Further work on 'profile methods' (Taylor, Kutateladze, Leont'ev)

  2. Computational fluid dynamics
    1. The two-dimensional boundary layer (Schlichting, von Mises)
    2. 2D elliptic flows (Thom, Courant, Burggraf)
    3. The three-dimensional boundary layer (Chorin, Harlow)
    4. 3D elliptic flows (Zuber)
    5. Turbulence models (Kolmogorov, Prandtl)
    6. Numerical computation of two-phase flows (Harlow)
    7. The probability-density function(Dopazo,Pope)
    8. The multi-fluid approach (?????)

  3. Unification of CFD and solid-stress analysis (?????)
    1. Why unify?
    2. How unify?
    3. First attempts (Southwell)
    4. A better method (Love, Timoshenko)
    5. Thermal stress; a surprising failure
    6. A question for the audience
    7. The good news
    8. The answer
  4. Concluding remarks
  5. Last words


    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.

    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.


    1b. The mass-transfer boundary layer (Eckert and Lieblein)

    A theoretical model was needed.

    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.


    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.

    Moreover, his method of:

    came from von Karman and Pohlhausen;

    and this had already been used for heat transfer alone by the (mercifully still alive) Russian scientist Krouzhilin.


    Eckert's contribution

    By combining all three innovations, Eckert created the first model which enabled

    to be quantitatively understood.

    Thus I was enabled to take the not-very-difficult next step, namely to handle reacting boundary layers.


    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.

    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:

    Click here for sketch.

    1e. 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 !

    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:

    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.

    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.



    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:

    "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!


    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.

    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

    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.

    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

    The novel coordinate system

    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.


    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.

    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

    The choice of dependent variables

    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

    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'.

    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

    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.

    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:

    1. all we want to know about turbulent flows can be deduced from two statistical properties; and
    2. these properties (he chose: energy and frequency) can be computed by solving 'transport equations'.

    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

    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.

    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

    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.

    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

    2h. The multi-fluid approach (?????)

    What are those ideas? I show only one picture , which shows a discretised probability-density function, in which:

    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

    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.

    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:


    3b. How unify?

    The solid-stress equations, when expressed in terms of displacements, are similar to the Navier-Stokes equations.

    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

    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.

    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:


    3d. A better method (Love, Timoshenko)

    Both defects were removed when:

    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

    3e. Thermal stress; a surprising failure

    I then turned attention to thermal stresses.

    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.


    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!


    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:

    (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. next

    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.

    They fit the theoretical solutions exactly: for there is a nett increase of thickness.

    So what was Timoshenko's mistake?

    This is how I had planned to end my talk; but, fortunately, I saw the light in time!


    3h. The answer

    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:-

    1. the case is one-dimensional in x; so
    2. d/dy and d/dz terms are zero; and so
    3. d2u/dy**2 and d2u/dz**2 must be zero also.

    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

    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;

    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: next

    "I did once make a mistake:

    I thought I was wrong, when I was right"


    5. Last words

    1. In respect of my criticisms of:

      I was wrong; and am to glad to been so.
      I was probably right, but it is no longer important;
      I still think he was unwise to choose the Monte Carlo method.
      we do need to transcend the limitations of his 1942 'lucky guess'; MFM can help.

    2. It has been my good fortune to have had great men to follow, and talented contemporaries, with whom to collaborate or compete.

      At the top of the list I am proud to place the name of:

      Professor Ernst Eckert.

      I thank you for your attention.

      The End !!!