1. Order of Magnitude Scaling of Complex Engineering Problems Patricio F. Mendez Thomas W. Eagar May 14 th, 1999

2. Order of Magnitude Scaling of Complex Engineering Problems2 MOTIVATION There are some engineering problems for which: –measurements are difficult –numerical treatment is difficult –idealizations and lumped parameter models are not reliable –dimensional analysis cannot simplify the problem significantly –there is previous insight into the problem –order of magnitude solutions are acceptable

3. Order of Magnitude Scaling of Complex Engineering Problems3 OBJECTIVES To determine the best combination in a problem involving many dimensionless groups. These dimensionless groups should provide –an estimation of the unknowns –a description of the relative importance of the phenomena involved

4. Order of Magnitude Scaling of Complex Engineering Problems4 OUTLINE Order of magnitude scaling. Basic concept: –Normalization –Functional requirements –Domain partition –Transformation of differential equations into algebraic –Matrix algebra Related techniques Results Discussion Conclusion

5. Order of Magnitude Scaling of Complex Engineering Problems5 ORDER OF MAGNITUDE SCALING: BASIC CONCEPT dimensional analysis asymptotic considerations Statement of a problem in dimensionless form Reduced number of arguments Relative importance of terms in equations Normalization scheme Functional requirements Domain partition Transformation of differential equations into algebraic Matrix algebra

6. Order of Magnitude Scaling of Complex Engineering Problems6 RESULTS Order of magnitude estimations are obtained. These estimations allow one to estimate the relative importance of the different driving forces The estimations are related to the governing parameters through power laws The functional dependence on the governing dimensionless groups are of less importance that in dimensional analysis.

7. Order of Magnitude Scaling of Complex Engineering Problems7 DISCUSSION Non-linear equations –Navier-Stokes –Heat transfer Singular limit problem. Differential equations of order higher than second. Vector operators. Analysis of stability, such as capillary instabilities, buckling, etc. included excluded

8. Order of Magnitude Scaling of Complex Engineering Problems8 CONCLUSION Previous insight can be used to transform a complex set of differential equations into a more manageable set of algebraic considerations. The results obtained are approximate. The physical insight gained can be used to choose representative asymptotic cases.

9. Order of Magnitude Scaling of Complex Engineering Problems9 Governing equations, boundary conditions and domain for scaling VISCOUS BOUNDARY LAYER

10. Order of Magnitude Scaling of Complex Engineering Problems10 VISCOUS BOUNDARY LAYER Continuity: Navier-Stokes: Boundary Conditions:

11. Order of Magnitude Scaling of Complex Engineering Problems11 VISCOUS BOUNDARY LAYER Governing parameters and reference units –set of governing parameters: –set of reference units –set of reference parameters Just one governing dimensionless group

12. Order of Magnitude Scaling of Complex Engineering Problems12 VISCOUS BOUNDARY LAYER Scaling Relationships –Independent arguments: length of domain (known) width of domain (unknown)

13. Order of Magnitude Scaling of Complex Engineering Problems13 VISCOUS BOUNDARY LAYER Scaling Relationships –Parallel velocity:

14. Order of Magnitude Scaling of Complex Engineering Problems14 VISCOUS BOUNDARY LAYER Scaling Relationships –Transverse velocity: unknown characteristic value estimated characteristic value

15. Order of Magnitude Scaling of Complex Engineering Problems15 VISCOUS BOUNDARY LAYER Scaling Relationships –Pressure: unknown characteristic value estimated characteristic value

16. Order of Magnitude Scaling of Complex Engineering Problems16 VISCOUS BOUNDARY LAYER Set of estimations: Three dimensionless groups are added. Since they are redundant they can be assigned arbitrary values.

17. Order of Magnitude Scaling of Complex Engineering Problems17 VISCOUS BOUNDARY LAYER Dimensionless governing equations and boundary conditions Dimensionless groups of known order of magnitude N1=1N1=1 N3=1N3=1 N4=1N4=1 Boundary Conditions: Continuity: Navier-Stokes: all others = 0 N2N2 N5N5 N6N6 N7N7 N8N8

18. Order of Magnitude Scaling of Complex Engineering Problems18 VISCOUS BOUNDARY LAYER Set of governing dimensionless groups –only one group: Reynolds number

19. Order of Magnitude Scaling of Complex Engineering Problems19 VISCOUS BOUNDARY LAYER Calculation of the estimations (matrix algebra) Dimensionless groups of known order of magnitude Dimensionless coefficients Governing dimensionless group Governing parameters Estimations Matrix [A] [A 11 ] [A 12 ]

20. Order of Magnitude Scaling of Complex Engineering Problems20 VISCOUS BOUNDARY LAYER Calculation of the estimations (matrix algebra) Estimations Governing parameters Matrix [A S ]= -[A 12 ] -1 [A 11 ] unknown function  1

21. Order of Magnitude Scaling of Complex Engineering Problems21 VISCOUS BOUNDARY LAYER Dimensionless governing equations –Matrix algebra is of help here too –All terms are of the order of one when for large Re

22. Order of Magnitude Scaling of Complex Engineering Problems22 VISCOUS BOUNDARY LAYER Comparison with known solution:

23. Order of Magnitude Scaling of Complex Engineering Problems23 HIGH PRODUCTIVITY ARC WELDING Low current: recirculating flows. small surface depression. experimental, numerical and analytical studies. High productivity no recirculating flows. large surface depression. gouging region. only experimental studies and simple analysis.

24. Order of Magnitude Scaling of Complex Engineering Problems24 CHALLENGES Direct observations are very difficult. Not all of the necessary physics is known. The equations cannot be solved in closed form. Numerical solutions are difficult. Application of dimensional analysis is limited.

25. Order of Magnitude Scaling of Complex Engineering Problems25 FEATURES INCLUDED Deformed free surface Gas shear on the surface Arc pressure Electromagnetic forces Hydrostatic forces Capillary forces Marangoni forces Buoyancy forces

26. Order of Magnitude Scaling of Complex Engineering Problems26 RESULTS thickness < 100 mm The gouging zone is a a very thin layer of molten metal Penetration measured for two significantly different levels of sulfur is the same. Defect mechanism is different

27. Order of Magnitude Scaling of Complex Engineering Problems27 RELATIVE IMPORTANCE OF DRIVING FORCES arc pressure / viscouselectromagnetic / viscoushydrostatic / viscouscapillary / viscousMarangoni / gas shearbuoyancy / viscous gas shear / viscous convection / conduction inertial / viscous diff. = /diff   Driving forces Effects For the first time gas shear is shown to dominate the flow It was generally assumed that electromagnetic or arc pressure would dominate

28. Order of Magnitude Scaling of Complex Engineering Problems28 Arc pressure increases by an order of magnitude with productivity SENSITIVITY OF DRIVING FORCES productivity

29. Order of Magnitude Scaling of Complex Engineering Problems29 With low S the curvature is smaller. Surface tension is higher. Contact angle is wetting With low S the curvature is larger. Surface tension is lower. Contact angle is less wetting Lower sulfur increases speed limit 20 % faster weld with same linear heat input 10 A/ipm 27.4 to 33.4 ipm EFFECT OF SULFUR