Solid and Fluid

The lecture suggests that the difference derives from historical accident, and that, from the point of view of society, a single approach is to be preferred, especially when solids and fluids interact significantly.

Examples of using the unified approach will be presented.

- buildings
- Will they withstand earthquakes?
- Are the heating and air-conditioning adequate?
- Will the water-sprinkler system control spread of a fire?

- energy-producing and -consuming equipment
- Will gas-turbine blades survive their high temperatures and stresses for the stipulated time between overhauls?

- environmental-protection measures
- Will a power-station furnace satisfy low-emissions requirements.
- Will nuclear-waste containers resist corrosion and thermal stress for hundreds of years?

The use of less-than-the-best simulation techniques endangers society.

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- The process-simulation calculations are made by
**engineers**(civil, mechanical, aeronautical, chemical, fire-safety). - The software which they use, is often supplied by
**commercial companies**; it- embodies the relevant scientific laws, mathematically expressed:
- takes in the engineer's specification of the scenario;
- predicts what is likely to result from their conjunction

- The software uses material properties, models of turbulence and
representations
of chemical reactions supplied by
**physicists**and**chemists**.

The **community** of software creators is split, polarised, schismatic.
Its two clans (FE and FV) use:

for stresses in solids | for flow of fluids |

finite-element methods | finite-volume methods |

Computer-simulation techniques are therefore needed which will simulate
fluid-solid **interactions.**

**Britain** is an island; so we are forced to pay attention to such
interactions:

Waves pound on the shore; cliffs crumble; breakwaters provide refuge for
ships; and the extraction of energy from tidal motion is the aim of many
a British inventor.

**St Petersburg** owes its existence to the sea's proximity, and to
the vision of Peter the Great.

*
*

Стоял он дум великих полн

И вдаль глядул....

... и думал он...

Здесь будет город заложен

The empty wave's deserted strand

Around him, in his mind a grand

Idea swelled: ....

... here should arise

A town to open Europe's eyes

But the continuation of Pushkin's famous poem tells us that St Petersburg's fluid-solid interactions were not always benign.

Many other such interactions are hazardous:

**Fires** destroy buildings; **earthquakes** cause tsunamis;
**volcanoes** erupt when groundwater reaches hot magma;
loosened snow creates **avalanches**.

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Their ideas were later harmonised and internationalised. The **harmony** is
happily symbolised by the
**family names** of the two textbook authors of my youth:
Love, who wrote about Elasticity; and
Lamb, whose subject was Hydrodynamics.

The **internationalisation** is exemplified by the Petersburgers,
Euler and Bernoulli,
who studied also in France, Germany and Switzerland.

Another famous Petersburger who must be mentioned is
Stepan Prokofievich Timoshenko. His textbook,
*The Theory of Elasticity,* became a world-renowned classic.

However, Timoshenko added to his second edition (1951), an appendix
on **"finite-difference" equations** and their **numerical**
solution by the "relaxation" technique, recently developed by my own
Oxford professor,
Richard Southwell.
This replaced analysis by **arithmetic**.

Finite-difference equations were **approximate** forms of
differential equations; and their use had been pioneered for solid-stress
problems by Runge, in 1908, and in 1910 by
Richardson, who later applied it to
fluid-flow, ambitiously hoping thereby to
predict the weather.

More successful, for the simpler boundary-layer flow problem, was the
so-called "continuation method" of H Goertler in 1939, who had picked up a
suggestion made by
L Prandtl in 1904.

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Now, such methods **could** be used; and experience quickly revealed
that:

- Many
**different sets of equations**could be derived from the same differential equations, some less approximate than others. - Many
**different procedures**could be devised for solving them; and these differed in efficiency, and success. - Which were best seemed often to
**depend on the problem in question.** - Mathematicians provided
**no guidance**as to where superiority was to be found.

It was left to the engineers, who **needed** the solutions, to find the
best technique by trial-and-error. I was one of those engineers.

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The stress-analysis community had now a flag round which to rally.

The originator of the **name** appears to have been the American,
RW Clough who used it in 1956.

However, I believe that the method itself was invented, in 1953, by my former colleague at Imperial College, London, John Argyris, although his name for it was different: the "matrix displacement method".

Interestingly, Argyris worked in 1944 with HL Cox, who appears to have
used a "finite-stringer method" for stresses in aircraft-wing panels in
1936! We all climb on the shoulders of our predecessors.

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My first (1951) research was on the burning of liquid fuels. Like Kruzhilin for heat transfer, and Eckert and Lieblein for mass transfer, I exploited themethod of von Karman and Pohlhausen.

This involved choosing a polynomial form of velocity profile, and
expressing its coefficients in terms of **weighted integrals** of the
differential equation.

But even high-order polynomials could not well express the complex
shapes of profile which appeared in flames; so why not, I thought, use
the infinitely flexible **step-wise approximation** instead?

I had thus stumbled upon the finite-volume technique; and my student Suhas Patankar expressed the idea in Fortran, later drawing crucially on the independent 1965 work of Francis Harlow, of Los Alamos. next

- The differential equations for
**displacements**in solids and**velocities**in fluids are almost identical. - The equations for each of the three direction have terms in common
("dilatation" for solids; "pressure" for fluids); so they must be solved
**simultaneously**.

- The equations for displacement are linked by Poisson's ratio, to which the velocity equations have no counterpart.
- On the other, the equations for velocity contain first-order differential coefficients, which those for displacement are spared.
**Turbulence**in fluids (in effect) causes material properties to vary much more greatly than in solids.

**"discretize"**continua; i.e. imagine them to be made of a finite number of adjoining pieces;- derive equations for the displacements or velocities of each of
those pieces by integrating
**weighted**differential equations over each piece. - solve the equations by
**iterative**error-reduction procedures.

- FE specialists use a variety of "weighting functions", but most often that proposed long before computers existed by Galerkin; whereas
FV specialists use as "weighting function".... 1.0 !!

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It **can be** indeed; but it is most successful for
low-Reynolds-number flows, because the
first-order differential coefficient cannot be handled by the
Galerkin weighting.

None of the many attempts to market FE-based codes for general fluid-flow purposes appear to have succeeded.

Attempts to apply FV methods to solid-stress problems have been fewer;
and my own of 1991, failed to handle **bending** satisfactorily.

It can thus be said that the obstacles to unification have been:

- incomplete understanding of the other-side's physics; and
- readiness among engineers to believe that unification is neither possible nor desirable.

This is my second attempt; and its **basis** is unchanged, namely:

- Use a single grid to cover both solid and fluid regions.
- In fluid regions solve for velocity and pressure as usual.
- In solid regions solve for displacement and dilatation.
- Treat fluid pressure and shear at interfaces as loads on the solid boundaries.
- Treat Poisson's ratio linkages between differently-directed displacements as loads on solid boundaries.
- Treat thermal-expansion effects similarly.

The **new feature** is:

- solve additional equations for the components of
**rotation**. - use rotation
**gradients**as sources of displacement.

**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:

(2) displacement vectors,
computed **at the same time**, and

(3) horizontal-direction stresses, obtained by post-processing.

**(b) What the 1991 formulation could not do **

That bending can now be satisfactorily handled is shown by the
horizontal vectors
here.

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But FV techniqes serve just as well, even when the grid is cartesian, as shown here for flow in a turn-around duct.

In this case there are heat sources in the solids, giving rise to these thermal-expansion contours.

Because of the unified-computational-mechanics technique, we are able to compute simultaneously the velocity and displacement vectors.

**(e) Forces on an under-water structure**
Finally, deflection is shown of a sea-bed structure, resulting from the
action of
surface waves.

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- a single computational technique can simulate the behaviour of solids and fluids simultaneously;
- there will be advantages to society if the current schism in the computational-mechanics community can be healed.

Because many careers and reputations depend on retaining the schism, the healing process
will not be swift; but I hope that it will not be
so slow as the equally desirably *rapprochements* of:

- Shias and Sunnis,
- Protestants and Catholics, and
- Orthodox and Old Believers.

His disappointment, once he had recognised that the mathematical representation of

That feed on their velocity;

And little whirls have smaller whirls,

And so on to viscosity.

This is of course a parody of an earlier rhyme
(by *Jonathan Swift*) which runs (more or less):

Upon their backs to bite 'em;

And little fleas have smaller fleas,

And so ad infinitum.

I have found no picture, and therefore substitute this test for students of English:

How should Clough be pronounced?

Kluff | as in enough | ? |

Kloo | as in through | ? |

Klow | as in bough | ? |

Kloff | as in cough | ? |

Klo | as in although | ? |

Klokh | as in Lough Ness | ? |

I do not know which is the correct answer.