Global Energy Laureate’s lecture by D. Brian Spalding

1. Global Energy Laureate’s lecture by D. Brian Spalding Simultaneous Solid-Stress and Fluid-Flow Computation • 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.

2. 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 easily as in politics, philosophy and religion.

3. A brief history of numerical methods applied 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.

4. J.H. Argyris R. Clough O. Zienkiewicz What happened next. In the 1960s, one (which?) of these solid-stress analysts invented a 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.

5. What has happened since. 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. Moreover, stress analysis being less complex 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.

6. Consequences for science and engineering; the present situation. General-purpose FVM-basedfluid-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. 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.

7. 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.

8. A first example: flow of hot gas around, and thermal stresses within, a gas-turbine-blade-like object. • 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. • Turbine designers currently use two computer codes in combination; • a 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.

9. The calculated external-flow field: 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. 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.

10. Calculated contours of thermal strain and associated displacement vectors. 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. 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. So a single finite-volume-based computer code solved the whole problem.

11. 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. Stresses in solids, according to Robert Hooke (right), are proportional to (gradients of) displacements. The constant is Young’s modulus. 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, solution methods devised for one set of equations can be used just as well for solving those of the other set.

12. Differences between the equations governing displacements in solids and velocities in fluids. 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. 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.

13. Similarities and differences between finite-element (i.e. FE) and finite-volume (i.e.FV) methods. 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 fewattributes, 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 algebraic equations connecting the state-characterizing attributes of neighbouring ‘pieces’. These simultaneous equations are then solved by trial-and-error methods of various kinds. The main difference is that FEM multiplies the differential equation by one or more ‘weighting functions’ before performing the integration.

14. Why did FEM innovators use unnecessary weighting functions? 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. 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. 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. Many engineers, alas, have followed their advice.

15. What the equations look like without weighting. On the right is shown a (2D) ‘finite volume’ (around P), over which the differential equations are integrated. The resulting algebraic equations, connecting displacement components uP, uk, etc have the form shown here. They can be solved by successive substitution.

16. Do finite-volume and finite-element solutions agree? Two solid materials of differing thermal-expansion coefficients, = 10-5 and  = 10-4 are raised intemperature by 10 degrees Celsius; while being firmly fixed together at their adjoining surfaces. 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. Their differing display packages show only qualitative agreement; but the numerical results reveal good quantitative agreement.

17. Do finite-volume and analytical solutions agree? Some simple stress-analysis problems have analytical solutions usable as tests of FV computations. 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. The difference between numerical- and analytical computed fields is less than 5%.

18. Some unstructured-grid solutions:a pressurised long cylinder. The problem: 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.

19. 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.

20. Comparison of the numerical with the analytical solution. The contours shown here are of the ratio of numerically-computed displacement 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.

21. Unstructured-grid solution of the‘thermal-stress-in-blade’ problem. The problem is the one solved earlier with a structured grid. 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.

22. Unstructured-grid solution of the thermal-stress-in-blade problem. Velocity vectors in the gas stream. Red is fast and green slow.

23. Unstructured-grid solution of thermal-stress-in-blade problem. Computed displacement vectors and thermal-strain contours.

24. Unstructured-grid solution of thethermal-stress-in-blade problem. Pressure contours in the flowing gas. Red is high; blue is low.

25. Unstructured-grid solution of thethermal-stress-in-blade problem. X-, y- and z-direction thermally-induced-stress contours within the ‘blade’. Notetheir strongly three-dimensional variation. Red is compressive, blue – tensile.

26. Summary of experiences with FVM applied to solid-stress problems. Many comparisons with analytical and finite-element solutions have been made. Their results confirm that the FVM approach to stresses-in-solids problems is practicable, accurate and economical. Nevertheless, more means of solving the equations can be imagined than have been explored so far. SIMPLE works well for both fluid flow and solid stress; but surely better algorithms can be found Other questions to be explored concerning the relative advantages of structured (staggered or collocated) and (various kinds of) unstructured grids. There is much still to do!

27. But, before others will follow, someone mustlead. Final remarks. • The foregoing demonstrations suggest that: • when engineers start to use FVM methods for both fluid flow and solid stress, • their designs will be made more swiftly, accurately and economically. This situation is vacant.

28. The End I thank you all for your attention!