Simultaneous Solid-Stress and Fluid-Flow Computation June 3, 2009 Saint Petersburg Global Energy Laureates lecture by D. Brian Spalding This lecture is.

June 3, 2009Saint Petersburg

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Global Energy Laureate’s lectureby

D. Brian Spalding

This lecture is addressed to:

• young researchers who seek scientific subjects in need of exploration;

• engineers designing equipment in which heat transfer, fluid flow and solid stress all play a part;

• all persons who too readily accept the beliefs of the majority.

Its general message is not new; but it contains new examples created by V I Artemov, Moscow Energy Institute.

Simultaneous Solid-Stress and Fluid-Flow Computation


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.

The main message to young scientists and specialists.

Do not always believe what your professors tell you,

even if all tell you the same thing.

False ideas can prevail in applied science just as easilyas in politics, philosophy and religion.


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A brief history of numerical methodsapplied to heat transfer, fluid flow and

stress analysis.

Numerical methods were applied to engineering problems in the first half of the 20th century. They were first called ‘finite-difference’, and later, ‘finite-volume’ methods’.

They were successfully applied, e.g. by LF Richardson, to heat-conduction, fluid flow and solid-stress problems.

Computers were at first human and so both slow and expensive;

but they became digital from the 1950s onward; and, since then, ever faster and cheaper.


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In the 1960s, one (which?) of these solid-stress analysts inventeda numerical method and a name for it: the Finite-Element Method.

Their followers became so enthusiastic, and self-congratulatory, that they proclaimed it as being the only worthwhile method;

and for heat transfer and fluid flow as well!

Unfortunately many professors believed them; and told their students.

What happened next.

J.H. Argyris R. Clough O. Zienkiewicz


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What has happened since.

Moreover, stress analysis stress analysis beingbeing less less complexcomplex than fluid flow, the first general-purpose commercial computer codes (ASKA, NASTRAN, ANSYS) appeared in that field; and they have remained FEM-based ever since.

However almost none of the attempts to use them for fluid-flow simulations have proved commercially successful.

The finite-element method has not in fact prevailed for heat transfer or fluid flow; the simpler-in-concept finite-volume one (FVM) is also the more effective.

But, being more abstruse mathematically and therefore requiring many text-books to explain it, FEM became more popular than FVM among professors.


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Consequences for science and engineering;

the present situation.

It is the engineering profession which suffers, being forced to employ two distinct techniques for anaysing fluid-structure interactions

whereas a single one would suffice.

General-purpose FVM-based fluid-flow and heat-transfer codes soon appeared (PHOENICS in1981; FLUENT in 1983; Star-CD in1985); but their creators and users had too much to do, at first, to pay attention also to stress analysis.

Thus the text-books and the stress-analysis codes have kept alive the entirely false belief that FEM (and the codes based on it) must be used for analysing the stresses in, and deformations, of solid structures.


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What this lecture contains.

The purposes of this lecture are:

• to explain that the fluid-flow and stress-strain phenomena have very similar mathematical natures;

• to show examples of the successful use of the Finite-Volume Method for the analysis of stresses and strains in solids;

• to show also that FVM can be applied to problems in which fluid flow and heat transfer are simultaneously present and indeed cause the stresses;

• to point out that between the two phenomena there are significant differences as well as similarities, and that improvements in FVM-based stress analysis can therefore probably still be made;

• to explain that the field is therefore a fruitful one for research in which young scientists and specialists can find almost virgin territory; and

• to declare to professors that the definitive text-book on the application of FVM to solid-stress analysis has not yet been written.


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A first example: flow flow of hot gas around, and thermal stresses within,

a gas-turbine-blade-like object.

Turbine designers currently use two computer codes in combination; • a finite-volume-based finite-volume-based one for fluid and heat flow, and• a finite-element-based one for stresses and strains.

This practice is clumsy, expensive (of computer- and man-time) and (often) inaccurate; and it is unnecessary, as will be shown.

This is not a gas-turbine blade; but it exemplifies the main features: • 3-dimensionalty,• curved surfaces,• hot external flow,• cooled interior,• thermal stress which is the main object of computation.


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The calculated external-flow field:

On the left are the gas-pressure contours on the same two planes: red is high; blue is low.

The computations were performed by the finite-volume- based code, PHOENICS, which employs a variant of the SIMPLE algorithm.

velocities and pressures.

On the right are shown the velocity vectors on two planes at right angles.The colours indicate gas speed: red is fast; blue is slow.The spacing of the vectors indicates that too coarse a grid was used for engineering use; but it suffices for illustration.


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Calculated contours of thermal strain and associated displacement vectors.

The displacements, and the corresponding three-dimensional strains and stresses, were calculated by PHOENICS at the same time as the pressures and velocities in the gas.

Here are shown the effects in the solid of the resulting non-uniformity of temperature:

the material expands non-uniformly; and stresses are caused thereby.

So a single finite-volume-based computer code solved the whole problem.


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Similarities between the equations governing displacements in solids and

velocities in fluids.

The deformed state of a solid is defined by its ‘displacement components’, viz. the distances which points fixed to it move from their original positions.

The flow state of a moving fluid can be described in terms of its ‘velocity components’, viz, the distances which points fixed to it move in one unit of time.

Isaac Newton (left) found that stresses in fluids are proportional to (gradients of) velocities and the constant is viscosity.

It is therefore not surprising that, the equations describing the stress distributions in solids and in fluids being similar in form,

Stresses in solids, according to Robert Hooke (right), are proportional to (gradients of) displacements.The constant is Young’s modulus.

solution methods devised for one set of equations can be used just as well for solving those of the other set.


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Differences between the equations governing displacements in solids and

velocities in fluids.

This effect requires small additions to velocity-oriented solution procedures, which otherwise are simplified by omissions.

None of the differences justify introduction of the wholly unnecessary feature which distinguish finite-element from finite-volume procedures.

In addition to second-order differential coefficients possessed by both equations sets, those for velocities contain first-order coefficients.

Although both sets of equations can have additional terms expressing distributed forces, e.g. gravitational or electromagnetic, those associated with velocity are the more numerous. Although both sets of equations have material properties which may vary with position, temperature, etc., the variations associated with velocity are by far more extreme, because of turbulence, for example.

One feature of displacement equations which is not possessed by the velocity ones expresses the ‘Poisson’s Ratio’ effect, whereby direct stresses can cause positive gradients of displacement in one direction and negative gradients in the two directions at right angles.


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Similarities and differences between finite-element (i.e. FE) and finite-volume

(i.e. FV) methods.

The main difference is that FEM multiplies the differential equation by one or more ‘weighting functions’ before performing the integration.

Both methods ‘discretise space’, i.e. attend only to a finite number of contiguous ‘pieces’ of it.

The ’pieces’ have ‘faces’ across which flow heat, momentum and perhaps mass. They may but need not intersect at right angles

The state of the material on each ‘piece’ is characterised by a few attributes, e.g. centre-point temperature or face-vertex displacement.

Between such focussed-upon points material attributes are presumed to vary in simple (e.g. linear) manners.

Integration of differential equations over the pieces yields algebraicalgebraic equationsequations connecting the state-characterizing attributes of neighbouring ‘pieces’.These simultaneous equations are then solved by trial-and-error methods of various kinds.


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Why did FEM innovators use unnecessary weighting functions?

The stress analysts did not notice; they therefore carried needless pre-computer baggage into the computer age, applying weighting functions to ‘pieces’ with much resulting complication.

Then having some early successes (as why should they not?), they said that everyone else should do the same.

Before digital computers were available, useful methods did exist which applied weighting functions to the whole domain.They were used for fluid flow, heat transfer and stress analysis.

Many engineers, alas, have followed their advice.

When digital computers favoured discretisation, i.e. breaking the whole domain into many small ‘pieces’, fluid-flow specialists soon found the best weighting function to be unity, i.e. no weighting at all.


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int, ,

u u , , , ,

, 2 , ...

, ...

P P k k Pk

eP k E e

ek

nen n

n

P P x P P

a a B k W E N S

Aa a a G L

x

Aa G

y

b S f V

What the equations look like without weighting.

The resulting algebraic equations, connecting displacement components uP, uk, etc have the form shown here.

On the right is shown a (2D) ‘finite volume’ (around P), over which the differential equations are integrated.

They can be solved by successive substitution.


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Do finite-volume and finite-element solutions agree?

The pictures show thermally-induced x-direction stresses and displacement vectors in the bi-metallic plate calculated by PHOENICS (FV) on the left and ELCUT (FE) on the right.

Two solid materials of differing thermal-expansion coefficients, = 10-5 and = 10-4 are raised in temperature by 10 degrees Celsius; while being firmly fixed together at their adjoining surfaces.

Their differing display packages show only qualitative agreement; but the numerical results reveal good quantitative agreement.


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Do finite-volume and analytical solutions agree?

Some simple stress-analysis problems have analytical solutions usable as tests of FV computations.

The difference between numerical- and analytical computed fields is less than 5%.

Here a long cylinder is heated inside, cooled outside.

Contours on the right show computed temperatures and radial normal stresses.

Also displacement vectors are shown.


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on The problem:

Some unstructured-grid solutions:a pressurised long cylinder.

A long, hollow, thick-walled cylinder, immersed in an outer fluid, contains a second fluid having a different pressure.

The picture on the right shows the so-called ‘unstructured’ grid used for its solution.

The smallest cells are placed near the boundaries of the cylinder, so as to represent their curved shapes.


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The unstructured-grid solution for the pressurised long cylinder.

On the right are shown contours of the displacement of the material.

The highest are red, the smallest blue; so, understandably, the displacements are largest at the centre, where the pressure-gradient is highest.

The contours are perfectly circular in shape, despite the fact that the grid is basically a cartesian one.

But are the values to which they correspond correct?Because there is an exact analytical solution for this problem, the question can be answered by comparison.

The next slide shows the evidence.


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Comparison of the numerical with the analytical solution.

The contours shown here are of the ratio of numerically-computeddisplacement to the analytically-derived displacement.

This should equal precisely 1.0 everywhere.

The scale of contours is from 0.9 (blue) to 1.3 (red).

The nearly-uniform bluish-green of the contours in the cylinder shows that the numerically obtained values agree with the analytical ones very well.


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earlier with a structured grid.

Unstructured-grid solution of the‘thermal-stress-in-blade’ problem.

The un-structured grid which has been used is shown below. The smallest cells are placed near the curved solid-fluid interfaces

The picture above shows the whole calculation domain, with gas inlet on the left and outlet on the right.Also visible is the central tube, which introduces the cooling air.The problem is illustrative, not realistically detailed.


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on Velocity vectors in the gas stream. Red is fast and green slow.

Unstructured-grid solution of the thermal-stress-in-blade problem.


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on Computed displacement vectors and thermal-strain contours.

Unstructured-grid solution of thermal-stress-in-blade problem.


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on Pressure contours in the flowing gas. Red is high; blue is low.

Unstructured-grid solution of thethermal-stress-in-blade problem.


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on X-, y- and z-direction thermally-

induced-stress contours within the ‘blade’.

Note their strongly three-dimensional variation.

Red is compressive, blue – tensile.

Unstructured-grid solution of thethermal-stress-in-blade problem.


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Summary of experiences with FVM applied to solid-stress problems.

Many comparisons with analytical and finite-elementsolutions have been made.

Nevertheless, more means of solving the equations can be imagined than have been explored so far.

Their results confirm that the FVM approach to stresses-in-solids problems is practicable, accurate and economical.

Other questions to be explored concerning the relative advantages of structured (staggered or collocated) and (various kinds of) unstructured grids.

SIMPLE works well for both fluid flow and solid stress; but surely better algorithms can be found

There is much still to do!


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Final remarks.

when engineers start to use FVM methods for both fluid flow and solid stress,

The foregoing demonstrations suggest that:

their designs will be made more swiftly, accurately and economically.

This situation is vacant.

But, before others will follow, someone must lead.


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The End

I thank you all for your attention!

Simultaneous Solid-Stress and Fluid-Flow Computation June 3, 2009 Saint Petersburg Global Energy Laureates lecture by D. Brian Spalding This lecture is.

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