Order Number 8824592

Multi-grid solutions of thermocapillary convection in weld pools

Ramanan, Natarajan, Ph.D.

The Ohio State University, 1988

UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106 MULTI-GRID SOLUTIONS OF THERMOCAPILLARY

CONVECTION IN WELD POOLS

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the Graduate

School of the Ohio State University

By

Natarajan Ram an an, B.S., M.S. sfc ♦ * *

The Ohio State University

1988

Dissertation Committee: Approved By S. A. Korpela A. T. Conlisk S. Nakamura » ' Adviser 1/ Department of Mechanical Engineering To my Brother, Ramachandran ACKNOWLEDGEMENTS

I express my sincere appreciation to Seppo Korpela for his thoughtful guidance and unwaver­ ing committment to this dissertation. I would like to thank my reading committee members, Terry Conlisk and Shoichiro Nakamura for their comments and valuable time. To my fellow graduate students, Tzyy Ming Wang and Vishwanathan Babu, I am grateful for many helpful discussions related to this study.

I am thankful to Edison Welding Institute and Cray Research Inc. for supporting this research. This work was also made possible by the computing time acquired from the Ohio Supercomputing center, the IRCC of the Ohio State University, and the Robinson Color Graphics Laboratory. The help rendered by the staff of OSC is duly acknowledged. I also thank my wife, Bavani, for helping me with the preparation of the figures in this dissertation. VITA

June 1,1960 Bom, Tamil Nadu, India.

1982 ...... Bachelor of Technology, Mechanical Engineering, Indian Institute of Technology Madras, India.

1984 ...... M S., Mechancial Engineering, ' Florida Institute of Technology

1984-1988 ...... Graduate Research Associate Mechanical Engineering, The Ohio State University.

PUBLICATIONS

"A Finite Element Approach to Prediction and Analysis of Heat Losses in Solar Ponds", M.S. Thesis, Florida Institute of Technology, Melbourne, Florida, 1984.

"Weld Pool Fluid Mechanics and Heat Transfer in Stationary Welds", Research Report MR8810, Edison Welding Institute, Columbus, Ohio, May, 1988.

"Fluid Dynamics of a Stationary Weld Pool", to be presented in ASME Winter Conference in Chicago, December, 1988.

"A Multi-grid Solution of Natural Convection in a Tall Vertical Slot", accepted for publica­ tion in Numerical Heat Transfer. TABLE OF CONTENTS

ACKNOWLEDGEMENTS ...... iii VTTA ...... iv LIST OF TABLES ...... vii LIST OF FIGURES ...... viii NOMENCLATURE ...... xii

CHAPTER PAGE

I. INTRODUCTION...... 1 H. NATURAL CONVECTION IN A TALL VERTICAL CAVITY...... 4 2.1 Natural Convection in a vertical cavity ...... 4 2.2 Numerical methods ...... 8 2.2.1 Finite difference methods ...... 9 2.2.2 Multi-grid m ethod ...... 12 2.3 Multi-grid convergence ...... 19 2.4 Vectorization gains ...... 21 2.5 Results...... 22 2.6 Conclusions ...... 38 m . THERMOCAPILLARY CONVECTION IN AN AXI-SYMMETRIC POOL 39 3.1 Marangoni Convection ...... 39 3.2 Governing equations ...... 42 3.2.1 Stream function-vorticity formulation ...... 46 3.3 Finite difference representation ...... 47 3.4 Multi-grid m ethod ...... 51 3.5 Results...... 53 v IV. THERMOCAPILLARY CONVECTION IN A STATIONARY WELD POOL ...... 73 4.1 Description of welding process ...... 73 4.2 Literature survey ...... 74 4.3 Formulation and governing equations ...... 77 4.3.1 Electromagnetic forces ...... 81 4.3.2 Stream function-vorticity formulation ...... 84 4.4 Finite difference representation ...... 85 4.5 Results for aluminum ...... 89 4.6 Results for steel ...... 102 4.7 Conclusions ...... 110 V. CONCLUSIONS...... 114

APPENDIX: FREE SURFACE DEFLECTION...... 118 A.1 Curvature of the free surface ...... 120 A.2 Surface deformation ...... 123 LIST OF REFERENCES ...... 125

vi LIST OF TABLES

TABLE PAGE

1. Nusselt number for insulated (I) and conducting (C) boundary conditions as a function of aspect ratio (A) and GrH ...... 33

2. Comparison between numerical and ElSherbiny’s experimental values of Nusselt number for 3 regions of the cavity. (A=10, conducting ends) ...... 33

3. Maximum values of stream function, vorticity and pressure for Pr=0.01, Gr=105, C a< l ...... 67

4. Depth to width ratios for various Reynolds numbers, ReH=0, Pr=0.19, Gr =*13,000 ...... 105

vii UST OF FIGURES

FIGURE PAGE

1. Schematic of a tall vertical cavity ...... 5

2. (a) Cycling and (b) Full multi-grid schemes ...... 15

3. Convergence rate of a sample multi-grid problem ...... 20

4. Local Nusselt number along the hot wall, Nu = hL/K. Insulated ends, A = 10, GrH =2xl07, 4xl07, 6xl07, 8x107, 108 with GrH increasing from left to right ...... 23

5. Local Nusselt number along the hot wall, Nu = hL/K. Conducting ends, A =10, GrH =2xl07, 4xl07, 6x107, 8xl07, 108 with GrH increasing from left to right ...... 24

6. Local Nusselt number along the hot wall, Nu = hL/K. Insulated ends, A =20, Gr„ =2xl07,4xl07,6xl07, 8xl07, 108 with GrH increasing from left to right...... 26

7. Local Nusselt number along the hot wall, Nu = hL/K. Conducting ends, A =20, Gr„ =2xl07, 4xl07, 6xl07, 8xl07,108 with Gr„ increasing from left to right ...... 27

8. (a) Streamlines, (b) Iso-vorticity lines, and (c) Isotherms for flow in a cavity with conducting ends and A = 10, GrH = 108. 0ymax=-8.4xl0~4); m = 0 as shown and o><0 in the core; to>0 near the w alls ...... 28

9. (a) Streamlines, (b) Iso-vorticity lines, and (c) Isotherms for flow in a cavity with conducting ends and A =20, GrH = 108. (\jfmax=-2.6xl0”3); © = 0 as shown and ©<0 in the core; ©>0 near the w alls ...... 29

10. Local field variations of vertical velocity in a cavity with conducting walls at y=0.03125 for A =10, GrH =2xl07, 6xl07,108, with Grjj viii increasing in the direction of the arrow 30

11. Local field variations of temperature in a cavity with conducting walls at y=0.03125 for A =10, GrH =2xl07, 6xl07, 108, with GrH increasing in the direction of the arrow ...... 31

12. Horizontal velocity along the vertical mid-plane of a flow in a cavity with (a) insulated and (b) conducting ends. GrH =2xl07, 6xl07,1 0 8 with GrH increasing in the direction of the arrow. (A=10) ...... 34

13. Horizontal velocity along the vertical mid-plane of a flow in a cavity with (a) insulated and (b) conducting ends. GrH =2xl07, 6xl07, 108 with GrH increasing in the direction of the arrow. (A=20) ...... 33

14. Vertically averaged Nusselt number vs L/H for flow in a cavity with insulated ends. GrH =0, 2xl07, 4xl07, 6xl07, 8xl07, 108 with GrH increasing in the direction of the arrow ...... 36

15. Vertically averaged Nusselt number vs L/H for flow in a cavity with conducting ends. GrH =0, 2xl07, 4xl07, 6xl07, 8xl07, 108 with GrH increasing in the direction of the arrow ...... 37

16. Schematic of an axi-symmetric liquid p o o l ...... 43

17. Conduction isotherms in an axi-symmetric pool. The isotherms are plotted in increments of 0 .1 ...... 54

18. (a) Streamlines and (b) Isotherms in a flow induced by natural convection, Pi=0.01, Gr=105 ...... 56

19. (a) Streamlines and (b) Isotherms in a flow induced by thermocapillary convection, Pr=0.01, Gr=105, Re=l,000 ...... 57

20. Streamlines for Gr=102, 103, 104 & 105, Re=104, Pr=0.01 ...... 58

21. Surface velocity for various Grashof numbers, Re=104, Fr=0.01 ..... 59

22. (a) Streamlines, (b) Isotherms and (c) Iso-vorticity lines for Re=5,000,104, 5xl04 & 105, Gr=105, Pr=0.01 ...... 61

ix 23. Influence of grid size on the streamlines at Re=105, (a) Ah =1/32, (b) Ah =1/64, (c) Ah =1/128 ...... 62

24. Surface velocities for various Reynolds numbers ...... 64

25. Reynolds number dependence of radial velocity for a point in thermocapillary boundary layer ...... 65

26. Radial velocity at r=0.5 for various Reynolds numbers ...... 66

27. (a) Reynolds number dependence of maximum stream function in the pool. (b) Reynolds number dependence of maximum pressure in the pool ...... 68

28. Axial velocity at a location below the free surface z=0.96875 ...... 69

29. Variation of pressure on the end wall ...... 70

30. Free surface deflection for (a) Ca=0.05 and (b) Ca=0.1 ...... 72

31. A sketch of a stationary weld pool ...... 78

32. Phase boundary approximation using multi-grid m ethod ...... 86

33. Phase boundary approximation using a single g rid ...... 88

34. Conduction isotherms for aluminum, Re=0, The isotherms are plotted in constant increments of 0.05 ...... 90

35. Streamlines (a) and Isotherms (b) for flow induced by buoyancy, Gr=2.3xl06, Pr=0.01, (solid lines for \jr,T20 and broken lines for \|/,T<0, isotherms in constant increments of 0.1) ...... 92

36. Streamlines (left) and Isotherms (right) for various values of Re, Gp=2.3x106, Pr^O.Ol, Re# = 0 ...... 93

37. Computational grid used at Re=200,000 ...... 94

38. Surface radial velocity for various Re, Gr=2.3xl06, Pr=0.01 and Re# =0 Note that on the surface only for r<0.46, the metal is in molten state ...... 95

x I 39. Radial velocity at r=0.1 for various Reynolds numbers. Note that the metal is in molten state only for z> 0.16 ...... 96

40. Axial velocity at a location just below in free surface, z=0.97. Note that the metal is-in molten state only for r>0.46 ...... 97

41. (a) Streamlines (anti-clockwise circulation) and (b) Isotherms in the presence of surfactants, (y>0), Re=42,500 Pn=0.01, Gp=2.3x106, Re# = 0...... 99

42. Streamlines for Re=10,600 for Reff of (a) 0.007, (b) 0.014 and (c) 0.017, Gr=2.3xl06, Pr=0.01. Broken streamlines show clockwise circulation and solid lines show anti-clockwise circulation ...... 100

43. Streamlines for Re=42,500 for Rew of (a) 0.0007, (b) 0.0009 and (c) 0.0019, Gr=2.3xl06,Pr=0.01. Broken streamlines show clockwise circulation and solid lines show anti-clockwise circulation ...... 101

44. Conduction isotherm for steel, Re=0. The isotherms are plotted in constant increments of 0.05 ...... 103

45. (a) Streamlines and (b) isotherms for various Reynolds numbers, Gr=43,000, Pr=0.19, Re# =0 ...... 104

46. Surface velocity for various Re, Gr=43,000, Pr=0.19, Reff =0. Note that the metal is molten state only for r<0.5 (approximately) ...... 106

47. Surface temperature for various Re, Gr=43,000, Pr=0.19, Re# =0 ...... 107

48. Radial velocity at r=0.4 for various Reynolds numbers. Note that the metal is in molten state only for z>0.55 ...... 108

49. Axial velocity at a point below the free surface ...... 109

50. (a) Streamlines and (b) Isotherms for Re=3,500 for Re# of (i) 0.041, (ii) 0.045 and (iii) 0.082. Broken streamlines show clockwise circulation and solid lines show anti-clockwise circulation ...... I l l

51. (a) Streamlines and (b) Isotherms for Re=35,000 for Rew of (i) 8.16x1 O'4, (ii) 8.16xl0-3 and (iii) 0.033. Broken streamlines show clockwise circulation and solid lines show anti-clockwise circulation ...... 112

xi NOMENCLATURE

A - aspect ratio of the tall cavity =H/L if - magnetic induction if - electric field strength f r , / z - radial and axial component of Lorentz force F - source term g - gravitational acceleration G k - kth grid in multi-grid domain Gr - Grashof number =g XATL 3/v2 GrH - Grashof number based on height = g XATH3/v2 hk - mesh size in k-th grid h - thickness of the slab h - heat transfer coefficient averaged over height H - aspect ratio of cylindrical slab, width of the tall cavity 7*+1 - Interpolation operator from k to k+1 grid j r ,jz - radial and axial component of current density Jo - magnitude of current density T - current density k[ ,ks - thermal conductivity of liquid and solid K - conductivity ratio= k{lks L - horizontal length scale L - differential operator M - finest grid Nu - Nusselt number

N uk - Nusselt number p - dimensionless pressure Pr - Prandtl number, = v /a <7o * magnitude of arc heat flux r,z - dimensionless radial and axial coordinate R%+1 - Restriction operator from k+1 to k grid Ra - Rayleigh number =gAA7L3/va Re - Reynolds number=f/L/v

Rea - magnetic force number= }ie / qL 2/p £/ 2 T - dimensionless temperature T, ,TS - liquidus and solidus temperature of metal T„ - dimensionless temperature of ambient air AT - characteristic temperature difference =q^xlkt u,w - dimensionless radial and axial velocity U - characteristic velocity=yHA77|x Uk - exact solution to the difference equation on k-th grid uk - approximate solution to the difference equation on k-th grid v* - correction needed in the kth grid x - dimensionless x-coordinate Xm - melt ratio=(7/ - T S)/AT y - dimensionless y-coordinate

Greek

a - thermal diffusivity P - coefficient of gaussian heat flux y - temperature coefficient of surface tension

k - electrical conductivity X - coefficient of volumetric expansion A - boundary condition operator |A - dynamic viscosity |io. He - magnetic permeability of vacuum and metal xiii v - kinematic viscosity p - density X - coefficient of gaussian current flux \|i - stream function Ye - magnetic stream function co - vorticity

Subscripts

1 - left wall 2 - right wall H - based on gap width, H k - grid level

Superscripts

k - grid level n - iteration count CHAPTER I

INTRODUCTION

This dissertation deals with numerical studies of natural and thermocapillary convection. Although there are important differences between these flows, and particularly in their appli­ cations in technology, they also have sufficient similarities to allow them to be considered together. Of particular importance is the similarity in the way in which numerical solutions to these flows can be obtained.

In order to make a clear presentation of the results, the dissertation is arranged in two parts. In the first part, the numerical study of natural convection in a tall vertical cavity is considered and in the second part the study of thermocapillary convection is taken up. In addition to the elucidation of the physical features of natural convection on the one hand and of thermocapillary flow on the other, the aim of the study here undertaken is to test numerical solution strategies that are suitable for recirculating convective flows. It is, in fact, the solu­ tion strategies for circulating convective flows that bring the two parts of this study together. A further aim of this investigation is to apply the results to important practical ends, which in the case of natural convection is the determination of the insulating capacity of double-pane windows and in the case of thermocapillary convection is the fluid flow in weld pools. Because these applications are likely to be of interest to different groups of investigators, it was decided to separate their discussion into separate chapters.

The natural convection investigated takes place in a vertical cavity with closed top and bottom ends that are made of material that is either a good insulator, or a good heat conduc­ tor. The left wall is held at a constant temperature larger than the right wall. The hot fluid near the left wall rises owing to buoyancy and sinks next to the right (cold) wall, thereby forming a recirculating flow pattern. Although the flow in this configuration has been inves­ tigated by a number of researchers, there is still a need to determine how the thermal boun­ dary condition influences the heat transfer across the cavity. For this reason, and because 2 good numerical solutions and experiments are available to check the results, it provides a good case for testing an algorithm. Although we do not here propose any new algorithms, one aim of this study is to gain computational experience for solving the Navier-Stokes equa­ tions with multi-grid methods. The multi-grid method is gaining increasing popularity for providing accurate solutions with a relatively small amount of computational work. The per­ formance of this method is tested on this problem and used throughout this dissertation to obtain fast and accurate solutions.

The second part of the dissertation, as was mentioned above, involves analysis of a sta­ tionary arc welding process. Welding is an important joining process for it is used exten­ sively in all areas of production. A clear fundamental understanding of welding will aid in improving the quality and increasing the productivity of manufactured products. However, owing to the combined effects of a multitude of physical mechanisms involved, the theoreti­ cal knowledge of welding is far from adequate. For the same reason a model which faith­ fully represents a true welding process is not easily solved using the classical mathematical techniques and experimental methods. With the power of modem computers, numerical stu­ dies can today be conducted to investigate the effect of a wide variety of parameters on weld­ ing, and the individual influence or the interaction of various physical forces can be estimated. If the influence of arc plasma is ignored, the fluid flow in arc-welding is governed by buoyant, thermocapillary and electromagnetic forces. The fluid and heat flow in such a stationary weld pool is analyzed for two different weld materials, namely aluminum and steel.

The objectives of this dissertation can be summarized as follows. 1. To test the effect of end conditions on heat transfer in a tall vertical cavity. 2. To study surface tension driven flows in an axi-symmetric liquid pool. 3. To test the performance of the multi-grid method in the first problem and use it to obtain solutions in surface tension driven flows. 4. To analyze the influence of thermocapillary and electromagnetic forces in stationary arc-welded pools. 3

In the next chapter, the multi-grid method and the finite difference algorithms are dis­ cussed in detail. Following that the results for the natural convection problem are presented. To identify the essential features of thermocapillary convection, the flow in an axi-symmetric pool heated from above is analyzed in the third chapter. The stationary arc welding problem is formulated in the next chapter. The flow patterns in a typical stationary weld pool are dis­ cussed for two different welding materials. In the concluding chapter, the findings of this dissertation are summarized and recommendations for future work discussed. CHAPTER II

NATURAL CONVECTION IN A TALL VERTICAL CAVITY

In this chapter, a numerical study of natural convection in a tall vertical cavity is presented. The practical use of this problem is the determination of heat transfer through a double-pane window. A number of good numerical and experimental solutions are available for this problem making it a good choice for testing the performance of numerical methods. Thus, the main aim of this study is to gain experience in using multi-grid methods to solve Navier-Stokes equations and test the performance of these methods. In addition, as a part of the study, the influence of thermal end conditions on the determination of heat transfer from the vertical walls is analyzed. The reader is referred to Lee & Korpela (1983) for a detailed literature survey of this problem. Here, only a brief overview of this problem is given and the emphasis is on the numerical methods and the the influence of thermal end conditions on the insulating capacity of double-pane windows.

This chapter is arranged in the following manner. The problem is formulated and the parameters are identified first. Then, those numerical methods that were used to solve the equations which govern the fluid flow and heat transfer are discussed and the performance of the multi-grid method presented.

2.1 NATURAL CONVECTION IN A VERTICAL CAVITY

Consider a closed rectangular cavity with a width L and height H, filled with a newtonian fluid. Actually, all our calculations are for air, so we consider only fluids with Prandtl number equal to 0.71. The length of the cavity in the direction perpendicular to the plane of the paper is taken to be much larger than L and H. The left and right walls are maintained at the uniform temperatures of T x and T 2, respectively. The schematic of this geometry is shown in figure 1. With the left wall hotter than the right one, fluid next to it

4 5

h - L - H

Figure 1. Schematic of a tall vertical cavity. 6 rises owing to buoyancy. Near the cold wall there is a corresponding downflow and the net result of the two counter-flowing streams is a circulatory flow pattern. The temperature differences between the walls are taken to be small. This allows us to neglect the property variations in the fluid, except, of course, for the density when it appears in the gravitational term, where its variation is needed to drive the flow. In that term density is assumed to decrease linearly as temperature increases. The set of equations that govern the fluid flow and the temperature field, in this so called Boussinesq approximation, are given below in terms of non-dimensional variables. In them u and v are velocities in x- and y-directions, respectively, and T is the temperature. The dimensionless forms are obtained by dividing the dimensional x- and y-coordinates with the gap width L and height H, respectively and the velocity components with the characteristic thermal velocity U = g‘kATL2/v. The pressure is made non-dimensional by scaling it with p i/2, and the non-dimensional temperature is defined to be the scaled difference between the actual and the temperature of the left wall, the scale being temperature difference, AT, between the left and right walls. The parameters that appear in the equations are the aspect ratio, A =HIL, the Grashof number, Gr = ULIv, and the Prandtl number, Pr =v/a. In these v is the kinematic viscosity, a is the thermal diffusivity, X is the coefficient of volumetric expansion, p is the fluid density, and g is the gravitational acceleration.

The non-dimensional steady form of the equations can now be written as

(2.1)

Gr ( ? • V) V = - Gr Vp + V2? + Tj , (2.2)

GrPr (V* • V) T- , (2.3)

where t? is the velocity vector. With f and j the unit vectors in x- and y-directions, the vec­ tor operators appearing in (2.1)-(2.3) are defined as 7

v = r— + / dx 1 A dy ’ v2 = Ji+ 1 a2 dx A 3y

The boundary conditions for the velocities are the no slip condition and vanishing nor­ mal velocity on all the four walls. The thermal boundary conditions are of the Dirichlet type on the vertical walls and either Dirichlet or Neumann type on the horizontal walls. That is, the vertical walls are at fixed temperatures and the top and bottom boundaries have either a linear temperature variation, or they are insulated. Thus we require at the boundaries, II II < O JS T = 1 ; x = 0 , (2.4) II u = v = 0 , o = 1 , (2.5)

u = v = 0 , T - l-x at y = 0 , (2.6) or 17dy = 0 :

u = v = 0 , T = l-x at y a* 1 . (2.7) or ldy r = ° :

To solve (2.1)-(2.3) numerically, it is convenient to cast them in a stream function- vorticity form. By taking the curl of the momentum equation, pressure is eliminated and vor- ticity transport equation is obtained. Introducing the stream function, y, via the definitions,

the governing equations can be put into the form

co = -V fy , (2.9) where co is the vorticity. In the vorticity transport equation, the left hand side represents the convection of vorticity, the first term on the right side is the diffusion of vorticity and the second term stands for vorticity production by the buoyant forces. The boundary conditions for the stream function follow from those for the velocity in (2.4)-(2.7) and we require

qr = 0 , (2.11) on all four walls and

- |£ = 0 , at* = 0,1 , (2.12) dx

• ^ = 0 , at y = 0,1 . (2.13) dy

The boundary condition for vorticity is obtained by substituting (2.11)-(2-13) into (2.9). It then simplifies to

= at x = 0,1 , (2.14) ox

00 = at y = 0,1 . (2.15) dy2

The finite difference form of this boundary condition is given in the next section.

2.2 NUMERICAL METHODS

After the equations have been cast in terms of stream function, vorticity and tempera­ ture, the resulting system is solved using finite difference methods. We know from the results of other researchers that natural convection of air in a vertical cavity has a steady state solution. For this reason, one can use iterative methods or pseudo time stepping to obtain the steady state solution. Here, pseudo time stepping is used.

In choosing an algorithm it has been observed that for many iterative techniques con­ vergence is rapid in the first few iterations, but becomes slower as the solution gets closer to the true solution. To improve the convergence rate, one can use Newton-Raphson, conjugate gradient or a multi-grid method. The multi-grid method is used here. It is also extended in such a way (Brandt, 1977) that local grid refinement is possible. In the next section, a description of the finite difference representation of the equations is presented. Following that, the multi-grid method is explained.

2.2.1 FINITE DIFFERENCE METHODS

In the following paragraphs, the finite difference approximation to (2.3), (2.9) and (2.10) is discussed. In our choice for a finite difference approximation we follow the recom­ mendation of Roache (1972) who suggested that the method of Arakawa (1966) for differencing the convective terms should work well for a problem which exhibits hydro- dynamic instabilities. This has been shown to be true for natural convection in tall cavities by Lee & Korpela (1983). They found that for aspect ratios larger than ten the unicellular motion changes to a multicellular one for air as the Grashof number is increased above the critical value of 7982. The Arakawa method has very good conservation properties and it conserves vorticity, mean square vorticity and mean kinetic energy. Except for the convec­ tive terms in (2.10), all the other derivatives are represented by central differences. Pseudo time stepping is used to iterate the vorticity and temperature towards the steady state solution.

The Arakawa method can be described as follows. The convective terms in (2.10) can be written in the jacobian form, 10 which Arakawa rewrote as

/(©,\|0 = -

(2.17)

He used central differences in all terms to express (2.17) in finite difference form as

-1 /(© , t|f) =■ 12 Ax Ay

~(Vi-l,;-t+V/,;-l~Vi-l,y+l“Vi,/+l)

■KVi+l,y+Vi+l,;+l”Vi-l.y_V i-l,;+l) ®i,/+l

•K V .+ i,;-V i,y+ i) °>i+i,y+t

-(N'.\y-i-V;-i,y) ©,-i,y-i

■KYi,y+i-Y < -i,y ) ©M .y+t

-(Vi+i.y-Vi.y-i) ©,+t.y-i ] • (2.18)

If the vorticity, to, is replaced by the temperature, T, the convective terms in the energy tran­ sport equation can be represented in a similar way. The convective terms can also be represented by a second order accurate upwind scheme. The upwind scheme (Torrance, 1968) does not require as large a number of arithmetic operations as the Arakawa scheme. However, in an upwind scheme the equations are set in a difference form, depending on the sign of the flow velocities. Evaluating the sign of a variable slows down the computations. For this reason, in our numerical experiments the Arakawa scheme was found to be as fast as the upwind scheme. 11

The Poisson equation for stream function was solved using an extrapolated Jacobi method because it is vectorizable. This scheme is much faster than successive over-relaxation (SOR). The original code was written by Professor S. Nakamura (1988) in the Mechanical Engineering Department of the Ohio State University, and was kindly given to us by him. Let us assume that i and j represent the location of a grid point in a two dimensional grid. This method involves two steps in each iteration. In the first step, the stream function for those points whose indices sum up to an odd number are updated. In the second step the stream function at the rest of the points is iterated.

The boundary vorticity is computed based on the stream function values from the previ­ ous iteration by making use of (2.14) or (2-15) depending on which solid wall is under con­ sideration. The vorticity is calculated from the equation,

g> = ^ T • (2.19) As*

Here, y refers to the value of stream function at the first grid point away from the boundary in a direction normal to the boundary; A? is the distance between the boundary point and the grid point whose stream function is given by i|f*.

To calculate the heat transfer the local heat transfer coefficient, h, must be determined. It is related to the Nusselt number by

Nu = — ■ = -rr- atx=0 . (2.20) k dx

The averaged Nusselt number based on the height, H, is a cumulative sum of the local Nusselt number, based on the average heat transfer coefficient, h, on the hot wall. It is there­ fore calculated as

Nuh = ^ - = - A | | | dy at x=0 . (2.21) 12

Simpson’s rule was used for integrating the above equation numerically.

2.2.2 MULTI-GRID METHOD

The development of the multi-grid method in recent years offers a researcher today an ability to obtain accurate solutions with relatively few computations. Single grid methods have fast convergence rate in the first few iterations, but the rate decreases continuously as the true solution of the finite difference equations is approached. In fact, most of the time, the accuracy reaches an asymptotic limit and it becomes very expensive to improve it any further. Multi-grid methods on the other hand can preserve the the original convergence rate. During an iterative solution in a single grid, components of the error with highest spatial fre­ quency (smallest wavelength) the given grid can resolve are attenuated in the first few itera­ tions, but the low frequency (large wavelength) errors take a longer time to get liquidated. Multi-grid methods overcome this reduction in the rate of convergence. In the sections that follow, a brief outline of multi-grid methods is given. For a detailed description, the reader is referred to Brandt (1977,1980,1984).

The concept of multi-grid method can be best described with two sets of grids, say, G° and G 1 in the same domain £2 with mesh sizes h ^ h For simplicity, we assume uniform square grids with h {:h0 - 1:2. Next we consider a differential equation of the form

LU(x,y) = F(xy) in £2, AU (x,y)= d>(x,y) on the boundary d£2 , (2.22)

and its difference approximation for each of the two grids,

L* Uk(x,y) = Fk(x,y) for x,y e Gk, AkUk(x,y) = & (xy)fotx,y edGk ,(2.23)

where £=0,1. In (2.22) and (2.23), L is the linear operator which represents the differential equation inside the domain and A is the operator which stands for the conditions to be 13 satisfied on the boundary. The superscript k stands for the grid level. If u° is a solution to (2.23) in the coarse grid (fc=0), solved to an accuracy that is equal to or lower than the trun­ cation error in that grid, it can be refined and solved to better accuracy in the fine grid (£=1) for which the truncation error is smaller. To do this the solution from the coarse grid is first interpolated into the fine grid. Using an interpolation or prolongation operator I q , we can denote the interpolation as

ul=I$u°. (2.24)

The interpolated solution serves then as a starting solution to the difference equations in the fine grid. In the fine grid, only a few iterations are carried out. Thus the exact inverse of operator L is not used. As a result, there exists an error in the solution of (2.23). We now compute

f l = Fl-Llul, = Q x-A lU i . (2.25)

The differences / 1 and cp1 are called the residual functions or residuals. If L and A are linear operators, then the correction V 1 = U l- u l satisfies

L'V 1 =/*, A V 1 ^ . (2.26)

The corrections are obtained by transferring the solution and the residuals from the fine grid to the coarse one and by solving for the corrections in the coarse grid. The transfer process is called injection or restriction. If R ® represents the injection or restriction operator, we can write

(2.27)

(2.28) 14 and if the correction in the coarse grid is V° = U° - u°, it satisfies the equations

£ ,V = / ° , A°V° = q>° . (2.29)

The corrections are solved exactly or to a very good approximation in the coarse grid, and interpolated into the fine grid as

(2.30)

In the above equation, u*1 represents the old value of u l. The variable u 1 is iterated using (2.23) and the steps in (2.24)-(2.30) are repeated until the desired accuracy is reached in the fine grid. The same algorithm can be extended to more than two grids.

The most important aspect of the multi-grid method is that the function of the relaxation process in the finer grids is to smoothen the residuals, because it is these high frequency fluctuations that are not visible in the coarser grids and therefore can not be approximated in them. Since the long wavelength errors are attenuated efficiently in the coarse grids, the multi-grid scheme preserves the original rate of convergence. The multi-grid algorithm above described is referred to as the correction storage mode and is applicable only for linear differential equations. The scheme of operating from the finest to the coarsest grids and back from coarsest to the finest grids in a cyclic fashion as described above is termed as the cycling procedure. Instead of the cycling procedure, a full multi-grid algorithm can also be used.

In the full multi-grid algorithm, a converged solution is sought in each grid. Figure 2 shows the typical pattern for this procedure. The approximate solution from the coarsest grid is interpolated and relaxed for a few iterations in the next finer grid. The corrections are immediately solved for in the coarsest grid and interpolated back into the finer grid and relaxed again to get a converged solution in the finer grid. Only after obtaining this solution, the results are interpolated into the third level and a similar cycle is performed between the o Converged solution

(b)

Figure 2. (a) Cycling and (b) Full multi-grid schemes. 16 three levels until a converged solution is obtained there. The procedure is repeated until the finest level is reached or when the desired accuracy is reached.

As mentioned before, the correction storage mode is applicable for only linear differential equations. For non-linear differential equations the full approximation storage (FAS) mode is used. In this mode, the idea is to store the current approximation uk rather than the correction v*. Let us consider a set of grids, say k=0, 1,2,.... up to M. If one represents the injection from k-t-1 grid to k grid as R k+i (called restriction operator) the new approximation in grid k can be written as

uk = /?*+! uk+l (Jfc=0, 1,.... Af-1) . (2.31)

In terms of these new approximating functions, the difference equations can be rewritten as

L* Uk = F k , AkUk =®k , (2.32) where

F k = Lk [i?£n«fc+1] + [f*+1-l*4V +1] ,

= A* [/?*+ik*+1 ] + Rk+1 [<&*+1-A*+,u*+l ] , (k = 0 ,1 M - l)

and for k=M, one obtains

pM _ pM ^ qpt _

To interpolate the solution from a coarse to finer grid, one can write

uk+1 _ jk+1 [R k+lUk+1 + u*+i f (2.33) 17

where 7*+1 is an interpolation operator and uk+x is the old value of uk+x. A linear interpola­ tion operator is normally chosen since it is required that the order of interpolation is one order lower, or better, than the order of the differential equation. For example, for a second order differential equation, it is sufficient that the interpolation be linear. However, the inter­ polation from the coarsest grid should be of higher order to take advantage of the smoothness of the converged solution. Normally, for partial differential equations of second order, at least a third order interpolation is required from the coarsest grid to the next finer grid.

Multi-grid methods can also be used in conjunction with local refinement in regions of sharp gradients. The FAS mode is suitable for uniform local refinement. Uniform sub-grids can be located in various domains where local refinement is necessary. The added advantage of such a scheme is that the second order accuracy of the difference algorithms can be preserved because, in all regions, the grids are locally uniform. Very fine grids can be intro­ duced in small subdomains providing high accuracy with small computational effort. The modification of the FAS mode for a non-uniform grid requires some changes in the difference equations to account for the non-uniform grid structure. Let us denote by g L the set of points of G k which are part of the GM domain. The modified form of the equations are as follows.

AkUk = * , (2.34) where

F k = F k , O* = (J>* in Gk-G k+i , k=M ,

K = Im ] + A* 18

To cany out the local refinement requires keeping track of the locations of the various points and the grid structure they are in. Local refinement is performed for surface tension driven flows in chapters 3 and 4. Uniform grids were used for investigating the natural convection flows in this chapter.

For the interpolation operator (also called prolongation operator), Brandt (1977) sug­ gests that a linear interpolation is sufficient for second order differential equations. Prolonga­ tion by linear interpolation is normally a nine point scheme. In this scheme, the midpoints of a mesh cell are interpolated values of the two adjacent points on the boundaries of the cell. The center of the cell is obtained as the arithmetic mean of the four comer points. Wesseling (1980) defined the prolongation operator by the equations,

('‘‘-•“'■'Lu,-,, ■ “*«!/« ■ <2-35>

]a«,2/*i = 2 + “&<+' ] ' (2,36)

\ [<*.!;♦. + «fci!/«] . (2-37)

]a- 4 2 >« = 4 K+ij+l + “*4/« + “‘+W+2 + “te V * ] • (2.38)

The restriction operator could be either a simple injection, or a five-point, or nine-point weighting scheme. The two commonly used restriction operators are injection and nine-point weighting. The injection scheme can be expressed as

).+1 .+1 = 4+1.2,+t • (2.39)

Full weighting or nine-point restriction in conjunction with the nine-point prolongation 19 operator as given above has the important property that I is the adjoint of R. This property is important for nonlinear problems for it improves convergence rates (Brandt, 1977). The nine-point weighting scheme is written as

[«2«+2,2/+l + “ 2/+t,2/+2 +u2i,2j+l +«2i+l,2>]

+ j“ 2/+2^;+2 + u 2i,2j+2 + u 2i+%2j + u 2i,2j ] • (2.40)

2.3 MULTI-GRID CONVERGENCE

While carrying out the calculations the cumulative sum of the residuals at all grid points was compared with the truncation error of the difference scheme. When this ratio is less than unity, the problem has been solved below the level of the truncation error. More accuracy will not improve the solution. It was observed that for obtaining good multi-grid conver­ gence rates (i.e. factor of ten reduction in residuals for each cycle), one needs a reasonable approximation in the coarsest grid. Two different coarse grid sizes, 16x64 and 16x128 were used with the latter grid serving as the coarsest grid for aspect ratios larger than 17.5. At higher values of Grashof numbers, the multi-grid convergence rate was improved by increas­ ing the number of relaxations in the finest grid (about 10-15). Doing so smoothens the residu­ als better, leading to faster convergence. This could probably have been avoided by perform­ ing local relaxation near all the four walls. With the source code well vectorized, little addi­ tional computational effort was necessary to carry out more relaxations in the fine grid.

To elucidate the performance of the multi-grid method, in figure 3 the vorticity residual is plotted as a function of computational work in the finest grid. One unit of work is defined as the computational effort needed for one iteration for all the points in the finest grid. In an ideal case, the residuals decrease by a factor of ten for every multi-grid cycle. This holds true for a problem with low Grashof number in the vertical cavity problem. As a test case, the fluid flow at a Grashof number of 100 is solved in a vertical cavity with aspect ratio of 10. The flat portions of the graph indicate the work required in the coarse grids. It is seen Log (residual) 0.00 2.00 0 .0 3 0 .0 4 0 .0 5 1.00 0.00 Figure 3. Convergence rate of a sample multi-grid problem. multi-grid sample a of rate Convergence 3. Figure 20.00 00 .0 0 4 Work 0080. 0 .0 0 8 0 .0 0 6 100.00 20 21 that after each cycle the original convergence rate of the residual is preserved. The conver­ gence is obtained after three cycles.

Before production runs were undertaken, a grid convergence study was carried out by using 3 sets of uniform grids (viz. 16x64, 32x128 and 64x256) at an aspect ratio of 10 for all Grashof numbers involved. The Nusselt number changed less than 2 percent between the 32x128 and 64x256 grids. For almost all the cases, it was decided to use 32x128 as the finest grid for aspect ratios lower than 17.5. For the higher aspect ratios, the number of grid points on the finest grid was increased to 32x256.

2.4 VECTORIZATION GAINS

The numerical calculations were carried out at the Ohio Supercomputer Center, which has a Cray X-MP/24 as a main computer. This model has a four megaword memory and pipelined processors for fast throughput. To make use of this computer architecture the source code must be written in such a way that the compiler can vectorize it efficiently. The most obvious thing to do is to make sure that the innermost of the nested do loops have the largest vector lengths since it is this part of the source code that is vectorized. With a 16x64 grid, just switching the order of the do loops in the FORTRAN source code in such a way that the long vectors are computed in the inner loop resulted in threefold decrease in CPU time. The solution of Poisson equation (2.9) also consumes a great amount of time. For the 16x64 grid, if successive over-relaxation method is used to solve (2.9), 1000 iterations con­ sume 20 CPU seconds. Compared with the CPU time of 2.31 seconds with the vectorizable extrapolated Jacobi method, this is an inefficient algorithm. We also tested the efficiency of the alternating direction implicit (ADI) method and found that pseudo time stepping resulted in gains by more than a factor of two. In general, the Jacobi method took 40-50% of the total CPU time. The remaining time was spent in vorticity and energy transport equations.

I 22

2.5 RESULTS

In presenting the results of the calculation, we have in mind the application of the results to determining the insulating capacity of a vertical cavity. Thus in order to make the presentation of the results most helpful to a designer, the vertical height of the cavity will be taken to be, from now on, the representative length in the definition of Grashof number (represented heretofore by GrH). The reason for this choice is that a designer is likely to want to know what gap width he ought to have, when the height is given, in order to minim­ ize the heat transfer across the cavity. The Grashof number defined in this way will essen­ tially then be proportional to the temperature difference across a cavity of given height. For Grashof numbers based on the height our calculations cover the range 2xl07£Grff £108. The aspect ratios we considered range from 10 to 25. The upper limit for the aspect ratio is still quite small if we have application such as double-pane windows in mind. However, for tall cavities the flow is multicellular and increasing the height will only increase the number of cells but not the heat transfer across them. Accordingly it is possible, as was shown in Korpela et al. (1982), to calculate the heat transfer through tall cavities with the aid of data obtained from cavities of lower aspect ratio.

The results here presented are for both the insulated and conducting thermal end condi­ tions. The current study is an extension of the work of Lee & Korpela (1983) who considered only insulated ends. In addition, more accurate results are made possible by use of finer grids with the multi-grid method. As stated earlier, this problem was chosen mainly to test the performance of the multi-grid methods.

To understand how heat is transferred across the cavity we have plotted in figures 4 and 5 the local Nusselt number along the hot wall for various values of Grashof number. The curves in figure 4, for a cavity with A=10 and insulated boundaries, show how the cold fluid sweeping across the bottom end picks up heat efficiently in the bottom hot comer. The local Nusselt number near this comer is the largest and it then decreases continuously with height. From this figure it is clear that the aspect ratio is too small for the heat transfer to become fully developed, which here means that heat would be transferred across the cavity purely by 23

10.00

8.00

6.00 y

4.00

2.00

0.00 0.00 2.00 4.00 6.00 8.00 10.00 Nu = hL /K

Figure 4. Local Nusselt number along the hot wall, Nu = hL/K. Insulated ends, A —10, GrH = 2xl07, 4xl07, 6xl07, 8xl07, 108 withGrH increasing from left to right. 24

10.00

8.00

6.00 y

4.0 0

2.00 -

0.00 0.00 2.00 4 .0 0 6.00 8.00 10.00 Nu = hL /K

Figure 5. Local Nusselt number along the hot wall, Nu = hL/K. Conducting ends, A =10, GrH = 2xl07, 4xl07, 6xl07, 8xl07, 108 withGrH increasing from left to right. 25

conduction. Also for GrH = 108 there is a small bulge in the local Nusselt number at a loca­ tion 15% of the cavity height above the bottom. A small dip in Nusselt number in the corresponding location near the top end can also be seen. The flow evidently changes its character in the ends as Grashof number is increased to this value. This has also been observed by others, notably by Lauriat and Desrayaud (1985). A similar set of curves when the top and bottom are conducting is shown in figure 5. The maximum value for the local Nusselt number is now substantially reduced from the previous result. Save for the behavior in the end, the curves are Very similar in the two figures.

The local Nusselt number variation is quite different when the aspect ratio is increased to 20, as is shown in figures 6 and 7. Now the curves show undulations, caused by multicel- lular convection, which in turn is the result of the flow having undergone a hydrodynamic transition as a result of an instability in the flow. For the smaller values of Grashof number these undulations are absent and the local Nusselt number shows little or no variation with height. In the central part of the cavity the Nusselt number has a value of unity, indicating pure conduction.

The streamlines, contours of constant vorticity, and isotherms are shown in figures 8 and 9 for two cases. The plots in figure 8 correspond to flow at GrH = 108 in a cavity with conducting boundaries and having A = 10. We chose to plot this condition because it shows how the flow in the end has changed and caused the aforementioned bulge in the local Nusselt number. It appears that the flow is about to separate from the right wall near the end and in doing so acquires a stronger velocity component across the cavity. The vorticity pat­ tern shows clearly the new dynamics prevailing in the end. Figure 9 shows the multicellular flow structure in a cavity with conducting ends at the condition, A =20 and GrH = 108. The wave length of the cells is about 2.3, which would allow eight full cells to fit into the cavity, save for the influence of the ends. The end flow limits the actual number to five.

Local profiles of vertical velocity are shown in figure 10 for three values of GrH in the same cavity. These show the boundary layer development from the bottom comer of the hot wall. The temperature profiles in figure 11 show the steepness of the temperature gradient at the hot wall and this explains the reason for the large local Nusselt number there. The 26

20.00

16.00

12.00 y

8.00

4.0 0

0.00 0.00 1.00 2.00 3 .0 0 4 .0 0 5 .0 0 Nu = hL /K

Figure 6. Local Nusselt number along the hot wall, Nu = hL/K. Insulated ends, A =20, GrH =2xl07, 4xl07, 6xl07, 8xl07,108 withGrH increasing from left to right. 27

20.00

16.00

12.00 y

8.00

4.00

0.00 0.00 1.00 2.00 3.0 0 4 .0 0 5 .0 0 Nu = h L /K

Figure 7, Local Nusselt number along the hot wall, Nu = hL/K. Conducting ends, A =20, GrH = 2xl07,4 x l0 7, 6xl07, 8xl07, 108 withGrH increasing from left to right. 28

Figure 8. (a) Streamlines, (clockwise circulation) (b) Iso-vorticity lines, and (c) Isotherms for flow in a cavity with conducting ends and A = 10, GrH = 108. (Vmax= “ 8.4x10 ); (0 = 0 as shown and co<0 in the core; ca>0 near the walls. 29

Figure 9. (a) Streamlines, (clockwise circulation) (b) Iso-vorticity lines, and (c) Isotherms for flow in a cavity with conducting ends and A -2 0 , GrH = 108. 0 near the walls. 30

3.00

2.00

1.00 vx 1 O

0.00

- 1.00

2.00 0.00 0.20 0 .4 0 0 .6 0 0 .8 0 1.00 X

Figure 10. Local field variations of vertical velocity in a cavity with conducting walls at y=0.03125 for A =10, GrH =2xl07, 6xl07, 108, with GrH increasing in the direction of the arrow. 31

1.00

0 .8 0

0 .60

0.40

0.20

0.00 0.00 0.20 0 .4 0 0 .6 0 0 .8 0 1.00

Figure 11. Local field variations of temperature in a cavity with conducting walls at y=0.03125 for A =10, GrH =2xl07, 6xl07, 108, with GrH increasing in the direction of the arrow. 32

horizontal velocity distributions are shown in figure 12 for Grashof numbers in the range 2xl07

The heat transfer results for vertically averaged Nusselt numbers are shown in figure 14. The values are for a flow in a cavity with insulated ends. A similar plot for conducting ends is given in figure 15. The Nusselt numbers are larger, but not by much, for the insulated case than for the conducting case. Both sets of values are listed in table 1. ElSherbiny (1980) (see also ElSherbiny, Raithby and Hollands (1982)) has conducted accurate experiments on this flow and has shown how the Nusselt numbers averaged over the bottom third, the middle third and the top third of the cavity vary with Rayleigh number (Ra =GrPr ). In table 2 are shown our calculations for A = 10 and his data for the same aspect ratio (interpolated linearly between the Grashof numbers he reported). The values listed were obtained using Richardson extrapolation performed on a set of 3 uniform grids, 16x64, 32x128 and 64x256. The agreement is excellent and certainly within experimental error. 33

Gr/t-* 2xl07 4xl07 6xl07 8xl07 10*

AI C ICI C IC IC

10.0 18.84 17.61 23.12 21.55 25.99 24.18 28.11 26.11 29.75 27.59

12.5 17.94 17.00 22.23 20.99 25.11 23.69 27.32 25.76 29.18 27.48

15.0 18.06 17.33 21.54 20.51 24.23 23.00 26.31 24.94 28.05 26.54

17.5 19.26 18.74 21.44 20.61 23.89 22.90 25.93 24.67 27.61 26.21

20.0 21.03 20.67 22.46 21.86 23.94 23.14 25.74 24.82 27.44 26.42

22.5 23.10 22.87 24.10 23.62 25.11 24.48 26.13 25.38 27.16 26.43

25.0 25.36 25.21 26.03 25.68 26.77 26.26 27.51 26.89 28.25 27.53

Table 1. Nusselt number for insulated (I) and conducting (C) boundary conditions as a func tion of aspect ratio (A) and GrH .

Gr Bottom Middle Top

Numl Expt Numl Expt Numl Expt

14,200 29.73 32.40 16.02 16.40 6.18 6.70

28,400 36.76 38.82 19.93 20.21 7.35 7.72

42,600 40.91 42.37 22.08 22.10 8.64 8.84

56,800 43.82 44.85 23.53 23.40 9.84 9.78

71,000 46.05 47.32 24.72 24.72 10.71 10.72

Table 2. Comparison between numerical and ElSherbiny’s experimental values of Nusselt numbers for 3 regions of the cavity. (A=10, conducting ends) 34

1.0 1.0

Y 0.5 0.5

0.0 0.0 u u

(a) (b)

Figure 12. Horizontal velocity along the vertical mid-plane of a flow in a cavity with (a) insulated and (b) conducting ends. GrH =2xl07,6 x l0 7, 10® with GrH increasing in the direction of the arrow. (A>=10). 35

Y Y 0.5 0.5

0.0 0.0 u u

(a) w

Figure 13. Horizontal velocity along the vertical mid-plane of a flow in a cavity with (a) insulated and (b) conducting ends. Gr„ =2xl07,6 x l0 7, 108 with Gr„ increasing in the direction of the arrow. (A=20). 36

40.00

3 2 .0 0

2 4 .0 0

16.00

8.00

0.00 0.00 0 .0 3 0 .0 6 0 .0 9 0.12 0 .1 5 L/H

Figure 14. Vertically averaged Nusselt number vs L/H for flow in a cavity with insulated ends. GrH =0, 2xl07,4 x l0 7, 6xl07, 8xl07, 108 with GrH increasing in the direction of the arrow. 37 j

40.00

32.00

24.00

N u

16.00 i v

8.00

0.00 0.00 0.03 0.06 0.09 0.12 0.15 L/H

Figure 15. Vertically averaged Nusselt number vs L/H for flow in a cavity with conducting ends. GrH =0, 2xl07, 4xl07, 6xl07, 8xl07, 10® with GrH increasing in the direction of the arrow. 38

2.6 CONCLUSIONS

The natural convection in a closed tall vertical cavity has been studied in order to deter­ mine the influence of thermal end conditions on the heat transfer on the walls. The Nusselt numbers were computed for a range of aspect ratios from 10 to 25 for five different Grashof numbers. The results have been presented in the form of graphs to aid a designer to identify the right gap width for the cavity when the height is given. This study was also used as a test case to evaluate the performance of the multi-grid methods in solving Navier-Stokes and energy equations in recirculating flows. The multi-grid method was found to improve the convergence rates significantly. With the vectorized extrapolated Jacobi method to solve the Poisson equation, the computer time to obtain solutions in grids as fine as 32x256 was reduced to less than 30 seconds on the Cray X-MP/24. CHAPTER HI

THERMOCAPILLARY CONVECTION IN AN AXI-SYMMETRIC POOL

Thermocapillary convection in an axi-symmetric pool heated from above is considered in this chapter. Both the top and bottom surfaces of the pool are taken to be free and their deformation is neglected. This is made possible by taking the absolute value of surface ten­ sion to be sufficiently large. The flow on the free surface is the result of differences in sur­ face tension caused by temperature gradients. When other agents also contribute to the varia­ tion of surface tension the resulting flow is termed Marangoni convection. In the following sections, the physical ideas in Marangoni convection are discussed and the pertinent literature reviewed. Following these, the equations that govern the fluid flow are written down and a discussion of dimensionless parameters is given. The finite difference representation of the equations are described next, after which the results of the numerical study are presented and finally some observations on the nature of thermocapillary convection in this geometry are made.

3.1 MARANGONI CONVECTION

The term Marangoni convection is used to describe flows induced by variation of sur­ face tension at the free surface of a liquid pool. Such convective motions are set up because the fluid particles on the surface are pulled into those regions where the surface tension is the largest. This surface motion in turn exerts a shear force in the interior fluid causing it to flow. It is well known that surface tension depends on temperature and chemical composi­ tion (Levich, 1962). For this reason, if the temperature of the free surface is non-uniform, or if concentration gradients of certain species exist at the surface of a liquid pool, the surface tension will vary from point to point. The concentration gradients can be the result of 39 40 surface active species (surfactants) that are adsorbed on the surface. If these surfactants are insoluble in the bulk fluid, then, they must remain on the surface. They are, however, redis­ tributed on the surface by diffusion as well as the lateral motion of the surface. Since the fluid flow and the surfactant distribution are inter-dependent, a steady state distribution will be reached wherein the convection of surfactants is balanced by their diffusion along the sur­ face. For liquid-gas interfaces of homogeneous composition, but non-uniform temperature, diffusion of species is absent and the term thermocapillary convection is used to describe the ensuing fluid motions, caused now by temperature variations alone.

Important examples of Marangoni convection are the spreading of fuel owing to a flame, flow in weld pools, flow in crystal growth melts and flow in other kinds of material processing applications such as zone refinement. A common example of Marangoni convec­ tion is the movement of liquid paraffin wax near the wick of a candle. Other factors such as buoyancy and electromagnetic forces may also contribute to convection in some of these examples, but the thermocapillary force appears to be the main driving force in them.

In some early studies, Marangoni convection has been overlooked. It was only thirty years ago when Pearson (1958) showed mathematically the importance of surface tension induced convection. At the time when he carried out his study, the cellular convective motions observed by Benard in a layer of fluid in a flat dish that was heated from below were attributed to the instability of the flow induced by buoyant forces. In his study Pearson showed that, contrary to the then prevalent belief, the cellular motions observed were, in most cases, the result of an instability of the flow induced by surface tension and not by density. An excellent description of Marangoni convection is provided by Levich (1962) in a chapter of his monograph. There he sets down the fundamental equations and boundary conditions and discusses some examples of flows driven by surface tension. Much of this material came out of his own work and those of his Russian colleagues. Of the other early papers on this topic published in English, we mention that of Yih (1968), who considered a thin layer of viscous fluid contained in a channel. He assumes the variable surface tension to be caused by surface active species and the thickness of the fluid layer to be very small when compared with the horizontal length scale. The Reynolds number based on the height is accordingly small and he solved the problem by invoking the lubrication approximation. Yih obtained 41 analytical solutions for the core flow valid away from the side walls and a boundary layer solution near the walls. The solutions give the height of the layer, the horizontal velocities and the distribution of surfactants on the free surface. Adler & Sowerby (1970) extended this solution to three dimensional flows in shallow channels. In their study, as in Yih’s, thermo­ capillary forces are absent and the surfactants are free to diffuse along the surface, but are insoluble in the bulk fluid.

Owing to its importance in materials processing, the subject of thermocapillary convec­ tion has received increasing attention in the past few years. Sen & Davis (1982), whose study has applications to crystal growth, investigated flows in low aspect ratio two- dimensional slots, the aspect ratio being defined as the ratio of the height of fluid layer to its length. In their study, a constant horizontal temperature gradient at the free surface induces the thermocapillary force. They solve for the flow using perturbation methods, and assume the surface tension Reynolds number to be on the order of the aspect ratio and the capillary number on the order of the fourth power of aspect ratio. The surface tension Reynolds number, or Reynolds number for short, is formed by balancing the tangential force arising from the surface tension variations with the viscous force to obtain a velocity scale. The height of the cavity is then used as a characteristic length scale in defining the Reynolds number. The capillary number is a ratio of viscous force to the average surface tension force and is inversely proportional to the mean value of surface tension. For large surface tension, the capillary number is small and in that case the surface deflection can be neglected.

Thermocapillary convection in a fluid contained in a quarter plane was analyzed in the limit of large Marangoni numbers by Cowley & Davis (1983). The Marangoni number is the product of surface tension Reynolds number and the Prandtl number, the Prandtl number being the ratio of kinematic viscosity and thermal diffiisivity. The quarter plane contains a rigid side wall and a free surface in the presence of an insulating gas. A heated strip on the side wall elevates the temperature of the fluid adjacent to the wall above its value at infinity. The Prandtl number is taken to be large, so that thermal boundary layers are much thinner than the viscous boundary layers. Using a boundary layer analysis, the thickness of the ther­ mal boundary layer on both the side wall and the free surface is estimated. Different regimes of viscous flow are seen to exist and the structure of the flow in these regimes is analyzed. 42

Homsy & Meiburg (1984) consider the effect of surface contamination on thermocapillary flow in a shallow cavity for low Marangoni numbers. Their analysis is very similar to that of Sen & Davis except for the fact that it has the added complication of surface contaminants. Owing to convection of contaminants towards the end wall, their concentration is the largest near the wall. This retards the thermocapillary flow since surface tension was assumed to decrease with increasing concentration of contaminants.

An excellent numerical study of thermocapillary convection in square cavities was con­ ducted recently by Zebib et al. (1985). Thermocapillary flow in their study is induced by the temperature difference between the two vertical end walls. They also assume the capillary number to be very small and neglect the free surface deflections to leading order in a pertur­ bation scheme. They obtain numerical solutions for large values of surface tension Reynolds numbers and Marangoni numbers. Their boundary layer analysis shows that the thickness of the viscous layer below the free surface decreases with increasing Reynolds number, the layer thickness being proportional to Re-1/3. They observe rapidly turning flows near the comers where the free surface meets the solid vertical cold wall and use fine grids near the comer to simulate the flow accurately there. As a part of the solution they determine the pressure dis­ tribution just below the free surface and use it to calculate the free surface deflection a pos­ teriori.

3.2 GOVERNING EQUATIONS

The equations that govern fluid flow are, as in Chapter n, the Navier-Stokes equations and the thermal energy balance. If the Boussinesq approximation is made, then these equa­ tions are the standard set used in most convective flows. A sketch of the geometry is shown in figure 16. The liquid pool is axi-symmetric and is bounded at r'=L by a rigid wall. The height of the liquid pool is also L and the top and bottom surfaces are assumed free and flat. The liquid pool is heated from above.

In the equations that follow, the variables are put into a non-dimensional form by divid­ ing the radial coordinate r ' and the vertical coordinate z ' by the characteristic length L . The 43

Heat flux Free Free

Free Free

Figure 16. Schematic of an axi-symmetric liquid pool. 44

radial and vertical velocities, u and w, are non-dimensionalized by a characteristic velocity U which is obtained by balancing the viscous force with the thermocapillary force. The reason for using this characteristic velocity is the hope that if at the free surface the thermocapillary forces are, in fact, in balance with the viscous forces then the non-dimensional velocities are of order unity. The pressure is scaled by [iU /L . Four parameters appear in the non- dimensional form of the equations. The Grashof number, Gr, accounts for the presence of buoyancy forces. The Prandtl number, Pr, shows the relative importance of diffusion of vor- ticity to that of heat. The Reynolds number, Re, signifies the importance of thermocapillary effects. The parameter P gives a measure of how concentrated the Gaussian heat flux distri­ bution is. In addition, there is another parameter which does not appear here but is needed to complete this formulation. It is the capillary number, Ca , which is the ratio of viscous and surface tension forces. How this comes into the formulation and how the free surface defor­ mation can be calculated for small values of Ca is described in Appendix A.

The non-dimensional equations that govern the heat transfer and fluid flow in an axi- symmetric geometry under the approximations stated take the form,

(3.1)

(3.2a)

(3.2b)

RePr u ^ - + = V2r , (3.3)

where

The parameters are defined as

a 45

Re = — , Ca = . (3.4) v

U - Uv ±LAT AT = <7o h . H

To complete the formulation, the boundary conditions must be specified. The heat flux is imposed as an exponential function at the top free surface and the free surface at the bottom is insulated. The exponential distribution is chosen since it closely represents the heat flux from a number of heat sources.

u = ° , = — = 0; r = 0 , (3.5) or dr

u = w = 0 , 7=0; r - 1 , (3.6)

A du dT dT n A w = 0 , — = — , — = 0 ; z = 0 , (3.7) oz dr dz

n du dT dT , . * w = 0 , — = , — = <7(0 I 2=1. (3.8) where

2 <7(r) = L - e"pr . rc(l - e~V)

The coefficients multiplying the exponential function make the net non-dimensional heat input on the free surface to be of unit magnitude. The value of (J used in this study is equal to 5.0. 46

3.2.1 STREAM FUNCnON-VORTICITY FORMULATION

To solve the equations by numerical techniques we cast the momentum equations into stream function and vorticity form. The continuity equation is satisfied automatically by relating the stream function to the velocities by

u ~ w • (3.20) r dz r or

The pressure term is eliminated by taking curl of the momentum equation. Accordingly,

V V - = - r© , (3.21) r or

d(0 , d(0 1 , uoa , Gr dT U — + W - .r - = — V2© + + (3.22) dr dz Re r Re2 dr

The vorticity, being the curl of the velocity is given by

_ diu © = (3.23) dr dz

In (3.22) the left hand side represents the convection of vorticity with the flow. In the right hand side, the first term represents diffusion of vorticity. The second term is a result of vor­ tex stretching. In an axi-symmetric geometry, if a vortex ring is pulled outward, the cross- sectional area of the ring becomes smaller, and by the principle of conservation of angular momentum, vorticity increases. The third term is the vorticity production or destruction owing to buoyant forces. The boundary conditions for stream function and vorticity are as follows. The stream function is zero at the centerline and on all surfaces since the normal 47 velocity to all the surfaces is equal to zero. The vorticity at the centerline is zero from (3.23) and (3.5). On the free surfaces, vorticity is given by

© = iJ i = at z = 1 , (3.24) oz dr

© = - | ^ = ~ at z = 0 . (3.25) oz or

The procedure to calculate the boundary vorticity at the side wall at r= l is discussed in the next section.

3.3 FINITE DIFFERENCE REPRESENTATION

In the following paragraphs, the finite difference approximation to (3.21), (3.22) and (3.3) is discussed. When explicit methods such as the one discussed in Chapter II was used for this case, it was found that, owing to the development of thin boundary layers on the free surface and near the side wall, very small time steps have to be used to obtain a solution. This in turn increased computing times as the Reynolds number is raised. Since we are interested in only steady state solutions, we decided to use more robust implicit methods, for which large time steps can be used. One such solution technique was used by Ghia et al. (1982) who implemented the Stone’s strongly implicit method for the driven cavity problem and obtained very good solutions for high Reynolds number flows. In their scheme, except for the convective terms in (3.22), all the other derivatives are represented by central differences, and he used the deferred correction algorithm of Khosla & Rubin (1974) (K-R differencing) to represent the convective terms. This scheme is very stable because it uses upwind differences. It has the added advantage of preserving second order accuracy for steady state calculations. We solved for the stream function and vorticity values in one step using an extended version of Stone’s strongly implicit method (Stone, 1968) and solved the energy equation by the alternating direction implicit (ADI) method.

In the K-R differencing scheme, a deferred correction is added to the first order upwind term. The correction term eliminates the artificial viscosity at steady state, and the scheme is 48 second order accurate in space as steady state is reached. The K-R differencing scheme can be written as

_ > * + /- 4»*+1 1 (fo*+I - 2* + <{£,) + 0(A r, Ax2, Af Ax), «,<0 , dx Ax 2Ax

i1 . 1 «;>0 . (3.26) dx Ax + " 2*‘ + *‘- l) + 0 iA *’ AfAx) ’

In the above form, the superscripts fix the iteration count and subscripts the spatial locations.

The boundary vorticity on the rigid surface at r= l is computed based on the stream function values adjacent to the wall. By making use of (3.21) and (3.6),

Here, \|/* refers to the value of stream function at the first grid point away from the boundary in a direction normal to the boundary; As is the distance between the boundary point and the grid point whose stream function is given by vjr*.

Stone (1968) developed an implicit method for solving partial differential equations in general domains. His method is more implicit than the ADI differencing because the func­ tion values at all the four neighboring points are treated as unknowns, whereas in ADI either only the row or the column points are treated as unknowns. As a result, Stone’s method has been observed to result in faster convergence rates than the ADI method. The convergence rates are especially higher for problems with non-rectangular domains and when the coefficients in the difference equations are greatly different. Using numerical examples, Stone shows that the convergence rate is independent of the number of equations to be solved and is only moderately sensitive to the type of boundary conditions. A drawback of Stone’s method is that it requires a lot of computer memory. 49

His method is described below for a single independent variable. In general, a five point difference form for a linearized two dimensional partial differential equation can be written as

Bi,jTi,j-l ~ Qi,j • (3.28)

It should be pointed out that in the above equation, the notation does not conform to the stan­ dard matrix notation. Here, the subscripts refer to the location in the finite difference grid. If (T) is a vector that contains all the unknown values at the node locations, [M] is the coefficient matrix and (q) is the source function, the above equation is equivalent to

[M] (T) = {q} . (3.29)

We can now add to both sides a term involving a matrix [N] that facilitates decomposing the coefficient matrix into lower and upper diagonal matrices, [L] and [U], respectively. Doing so yields

[ilf+/V], (T) = {q} + [N] |T) , (3.30) or after splitting

[L][U] {T} = {q} + [N] (T) . (3.31)

To solve this, the iteration sequence

[M-W] {T}"+1 = [M+N] {T}rt + {q} - [Af] (T}n (3.32) is used. By defining

{5}n+1 = {T}"+1 - {T}" , (3.33) 50

equation (3.32) can be written as

[L][U] {8}"+1 = [R]n = ( {q)-[M]{T} )" . (3.34)

From this, the solution ca?i be obtained for {5}"+1. The way to obtain matrix [N] is an involved procedure and the reader is referred to Stone’s article for a description. As said above, the matrix [N] is constructed so that the matrix [M+N] can be split into lower and upper diagonal matrices. The coefficients of [N] are smaller in comparison with the coefficients of [M] to minimize the error from this approximation. Once the lower and upper diagonal matrices are available the solution is readily obtained by standard methods. To find the stream function and the vorticity from (3.21) and (3.22), Stone’s algorithm was extended in a way which is similar to that used by Khosla & Rubin (1981). Both stream function and vorticity were treated as unknowns and solved simultaneously. In this case, each of the coefficients in (3.28) is replaced by a 2x2 matrix. The non-linear vorticity transport equation must be linearized to enable representation of the difference equations in a form similar to (3.28). To ensure strong coupling between the two equations, the convective term in the vor­ ticity equation was linearized in the following way:

d(uco); ; (KCD).*/'1 - (MC0)f_Yf-V*+1 1 . . . "&■= - zr + 2^( 'j- + ■ (3 J5 ) with

g>m!; and (u© )#1 = u f f

In the above equation the first derivative is represented by a K-R difference assuming that u is positive. A similar expression can be written for a negative u velocity, as was described above. The linearization is also done in a similar way for the other convective term which involves the w velocity.

The ADI method was used to solve the energy equation. In this method, a row inver­ sion is performed in the first half time step and a column inversion in the second half time 51

step. For the first half time step, the difference form of energy equation can be written as,

- ^ p 1 f c , r Ti,j) + [Rep r Tij = - [RePr ] T?j , (3.36) and for the second half time step, it is

+ [RePr w?jA;-A2Z} T?f{=-[RePr u / ^ - A r ] f i4 . (3.37)

In these two equations, A* and A^ are the operators for second order accurate K-R differences and An , Azz are the operators for central difference representations of second derivatives.

For Reynolds numbers larger than 40,000, local refinement was employed near the free surface and close to the side wall at r=l. Though the Stone’s strongly implicit algorithm and ADI be were used in coarser grids which cover the whole computational domain, it can not be used in the finer grids which occupy only part of the domain. In these locally fine grids, an explicit pseudo-time marching was used to iterate the vorticity and temperature values. Point successive over-relaxation technique was used to solve the Poisson equation for stream function.

3.4 MULTI-GRID METHOD

Before starting the discussion on multi-grid solution of this problem, two terms need to be defined. First, the residual of an equation in grid k, ek, is defined by

(3.38)

In the above equation, N refers to the total number of grid points, & is a grid index and /* 52 refers to the residual at grid point / in grid k. The convergence rate, 5, is defined as the ratio of the residuals of two successive relaxations. If the subscripts denote the iteration number, we can write

With Stone’s implicit and ADI methods as relaxation schemed, the full approximation storage was used with cycling and full midtigrid procedures on uniform grids. A converged coarse grid solution was always used as a starting solution. Whenever the convergence rate, 5, in any grid was lower than a specified value, the relaxations were shifted to the next grid. Of course, in the coarsest grid relaxations were carried out until convergence. The restriction operator was the injection scheme and the prolongation operator was the nine-point linear interpolation scheme. These schemes were discussed in Chapter n. To interpolate from the coarsest grid to the next grid a cubic interpolation was used as the prolongation operator to preserve the smoothness of the coarse grid solution, as suggested by Brandt (1977).

Most of the calculations in this chapter were carried out for Pp=0.01 and Gr=105. As mentioned before, the heat flux distribution was of a fixed exponential type. The only param­ eter that controls the flow pattern, is then the Reynolds number. With the Stone’s implicit method, pseudo time steps which were about 50-100 times larger than the Courant time step could be used. However, since the temperature solution was diffusion dominated, a pseudo time step that was comparable to that based on the diffusion was used.

For Reynolds numbers lower than 10,000, multigrid solutions were easy to obtain. A uniform grid structure was used. Stone’s algorithm provides a very good relaxation scheme and results in rapidly converging residuals for vorticity and stream function. The optimum multigrid convergence rate giving a reduction by a factor of ten for the residual in every cycle could be obtained with both the full multigrid method and cycling method. The full multigrid method resulted in solutions slightly faster than the cycling method. Up to a Rey­ nolds number of 10,000, three sets of grids with 16x16 the coarsest grid and 64x64 the finest 53 were used.

For Reynolds numbers larger than 10,000, but less than 50,000, the coarse grid size had to be increased to properly model the physics of the flow and to get multigrid solutions. Though a converged solution can be obtained in a 16x16 grid, multigrid solutions could not be obtained since the coarse grid corrections do not represent the flow well. The situation is similar here as that discussed in the last chapter for large Rayleigh number flows. There is a limit of how coarse the coarsest grid can be for large Reynolds number flows. The solutions obtained were with 32x32 as the coarse grid and 64x64 as the fine grid. The residual reduc­ tion per multigrid cycle was down to 5 instead of 10. Still, the solutions were obtained much faster with multigrid cycling than with a single grid. The reason for this reduction in mul­ tigrid convergence is possibly due to the fact that as Reynolds number increases, the residual reduction rate is also decreased. It was observed that more fine grid relaxations were neces­ sary to damp out the high frequency oscillations at large Reynolds numbers. The automatic grid switching based on the convergence rate is no longer pursued. The same observation was made while carrying out solutions at larger Rayleigh number flows.

For Reynolds number equal to or larger than 50,000 local refinement was performed near the free surface and the side wall where the flow separates. Again, 32x32 was the coarse grid and the top free surface and the comer regions were locally refined. The multigiid convergence rates were poorer than in the last case. For Reynolds numbers larger than 50,000, it was too expensive to carry out the solutions to truncation error in the finest grid. If the residuals for all the equations were less than 0.05 in the finest grid the solution was assumed to have reached convergence.

3.5 RESULTS

For low Prandtl number fluids, thermal diffusivity is much larger than the kinematic viscosity. This means that heat diffuses faster than vorticity and that the flow may not influence the isotherms greatly. Hence, it is of interest to calculate a reference temperature field for which the fluid flow is ignored completely. For low Prandtl number fluids, a 54

z

r

Figure 17. Conduction isotherms in an axi-symmetric pool. The isotherms are plotted in constant increments of 0.1. 55 conduction heat transfer analysis gives a reasonable description of the temperature distribution in the pool. The non-dimensional isotherms for Pr=0.01 are shown in figure 17. The isoth­ erms are closer together near the center of the pool since the magnitude of the heat flux is large in that location. This figure shows the coordinate origin and the r and z directions. This convention is followed in the subsequent figures of this chapter and next chapter. In the con­ tour plots, the dashed lines indicate negative contour lines and solid lines show zero or pos- tive contour lines.

In figure 18, those results of the simulations which include natural convection are shown. The changes in the shape of the isotherms are seen to be negligible, verifying the notion that conduction dominates natural convection. Streamline plots show that the coldest fluid is near the side wall falls creating a circulation in the clockwise direction. The center of the cell is located near the corner of the top surface and the side boundary, for the reason that, there the temperature gradients which drive the convection are the largest.

The conduction isotherms of figure 17 show that there is a temperature distribution at the top surface. As explained before, this leads to the emergence of thermocapillary forces which in turn cause a shear force to be exerted in the interior fluid. A similar but weaker shear exists at the bottom surface. A typical flow pattern for thermocapillary convection is shown in figure 19. The Reynolds number is taken to be 1,000, and the Grashof number 10s. In thermocapillary convection, the qualitative nature of the flow is as follows. The center of the pool being hotter has a lower surface tension than the colder ends of the pool. Thus, the fluid is drawn from the center of the pool to the rim. On the bottom surface, the temperature gradients are milder, but the flow pattern is the same. This splits the flow into two cells. The isotherms are still similar to the conduction isotherms indicating that conduction is still dominant

A parametric study was carried out to test the relative importance of natural convection and thermocapillary convection. The Reynolds number was fixed at 10,000 and the Grashof number was varied from 100 to 100,000. The streamlines in figure 20 show that the flow patterns do not change appreciably. When the Grashof number was equal to or lower than the Reynolds number, thermocapillary convection dominates buoyant forces. The separation point 56

1 1 I 'V""' i i i i ' t i i ' \ / 1 \ \\V// I

s s y

Figure 18. (a) Streamlines and (b) Isotherms in a flow induced by natural convection, Pr=0.01, Gr=105. 57

...... ' ii i '' n 11 n ' 111111111111111 n 11 1 it

Figure 19. (a) Streamlines and (b) Isotherms in a flow induced by thermocapillary convec­ tion, Pr=0.01, Gr=105, Re^lO3. 58

iiimn n n'HTn m iim n lluju _ u - i .u-uuj ~r n'i iim irm nim i Mjjm i i i lllu lu -u uu-u luji g

a b

j j j j 1111 j.ixu-ii4a44'm-cx^i

c

Figure 20. Streamlines for Gr=103,104 & 105, Re=104, Pr=0.01. Surface velocity Figure 21. Figure 0.00 0.01 0.02 0.03 0.05 0.04 0.00 Surface velocity for various GrashofvariousPr=0.01. for numbers,Re=104, velocity Surface —* Gr=10,000 - Gr= 100,000 --o 0.20 .006 0.80 0.60 0.40

Radius 1.00 59 60

is moved down when the Grashof number is increased to ten times the Reynolds number. Even this minor variation will disappear if one were to consider the effect of buoyant forces at higher Reynolds numbers. The surface velocities as shown in figure 21 are not affected by changes in Grashof number.

Though the buoyant forces are insignificant at moderate and high Reynolds numbers, they were kept in the formulation nevertheless, for the sake of completeness. The Grashof number was fixed at 10s for a parametric study on the influence of the Reynolds number on the flow. The variation in Reynolds number is taken to be caused by changing the value of the temperature coefficient of surface tension. This view clarifies the interpretation of the results and is justified by the fact that the determination of the actual value of the temperature coefficient is difficult in practice because it varies both with temperature and the presence of contaminants on the surface. The coefficient was varied so as to yield Reynolds numbers ranging from 1000 to 100,000. Figure 22 shows that the flow pattern splits into two cells. As the Reynolds number increases, the top cell motion is restricted to a smaller and smaller layer adjacent to the top flee surface and the fixed side wall. At the maximum value of Rey­ nolds number of 10s, it is seen that the flow separates very soon from the side wall. The rapid expansion of the flow starting from the comer is the cause of the separation. The opposing fluid motion from the bottom surface appears to have a minor influence, for numeri­ cal experiments with y equal to zero showed that the flow separates at the very same point.

The iso-vorticity lines show large gradients near the top surface and the top right comer. The production of vorticity on the free surface is caused by the thermocapillary stresses. The side wall which acts as a solid wall, generates vorticity of opposite sign to satisfy the no slip condition there. The strong convective nature of the flow bends the iso- vorticity lines in the direction of the flow in this region. A local analysis following Moffat (1964) can be carried out at the intersection of the free surface and the side wall. This analysis shows that the zero vorticity line intersects the comer region at an angle of 45°. The vorticity plots confirm this prediction at the top and bottom right comers. Very fine grids have to be clustered near the side wall, to obtain reliable results. In fact, with coarse grids, it was observed that the flow separation was delayed considerably. Figure 23 shows the flow pattern for three different grid sizes. Only above Re=50,000 do the isotherms show any a b c

Figure 22. (a) Streamlines, (b) Isotherms and (c) Iso-vorticity lines for Re=5xl03, 104, 5x104 & 105, Gi^lO5, Pr=0.01. 62

a b

Figure 23. Influence of grid size on the streamlines at Re=105, (a) Ah=1/32 (b) Ah =1/64 (c)Ah =1/128. 63 significant deviation from the conduction temperature pattern. The deviations are seen to be largest closest to the side wall. Here the large velocities convect heat the best, distorting the isotherms the most.

The surface radial velocity pattern is shown in figure 24. The radial velocity is seen to increase from zero at the center to a maximum very close to the wall as a result of the con­ tinuous action of thermocapillary forces. As the Reynolds number is increased, the surface velocity remains constant over most of the surface. Near the side wall it then rapidly drops to zero to satisfy the no slip conditions. As a result, the flow turns downward at the side wall.

The global circulation in the flow can be explained in the following way. Since the Prandtl number of the fluid is small, the flow does not alter the surface temperature distribu­ tion much so that the surface temperature may be determined from conduction analysis. The surface temperature gradients then set up thermocapillary forces which are equivalent to a shear distribution on the free surface. The continuous application of thermocapillary shear on the free surface increases the surface radial velocity from zero at the centerline to a maximum near the side wall. As a result of the continuous increase in surface velocity, fluid is entrained from the bulk of the pool underneath the surface to the surface boundary layer. Near the side wall, to provide for this entrainment, fluid is extracted from the boundary layer on the side wall. The expansion of this boundary layer creates an adverse pressure gradient and the fluid separates from the wall.

The non-dimensional velocities in the cavity decrease with increasing Reynolds number. Zebib et al. (1985) have analyzed the radial velocity variation with Reynolds number in a cartesian geometry. Because the surface vorticity is directly proportional to the radial tem­ perature gradient, it is 0(1). From this they determine that the surface velocity is propor­ tional to Re-1/3 at high Reynolds numbers. To test if this is true in our case, we plotted the radial velocity on the surface at a non-dimensional radius of 0.5 as a function of Reynolds number. This is shown in figure 25. The figure shows that the radial velocity on the free sur­ face is indeed proportional to Re~1/3. The axial velocities at the same radial location and at a point just below the free surface come out to be proportional to Re-2/3, which is again in Surface velocity Figure 24. Figure 0.00 0.03 0.04 05 .0 0 0.01 0.03 .0 .0 .0 .0 .0 1.00 0.80 0.60 0.40 0.20 0.00 Radial velocities on the top surface (z=l) for various Reynolds numbers. Reynolds various for (z=l) surfacethetop on Radialvelocities a—o—o Re=50,000 x— *•—x h 1 Re=5,000 k Re= 100,000, Re= 10,000

Radius 64 iue 5 Ryod nme dpnec o ail eoiy o a on i thermocapillary in point a for velocity radial of dependence number Reynolds 25. Figure

Log(u) -0.8 - - - -1.4 - - 2.0 1.2 1.0 1.8 1.6 boundary layer. boundary 2 4 3 Log (Re) Log -1/3 5 6 65 Coordinate 0.00 0.20 0.40 0.60 0.80 1.00 Figure 26. Radial velocity at r=0.5 for various Reynolds numbers. Reynolds various for r=0.5at velocity Radial 26. Figure 0.00 lO) ail velocity Radial (lOOX) 1.50 o—o—o Re=50,000 x——X 3.00-1.50 Re—100,000 Re= Re= 10,000 4.50

6.00 66 67 agreement with the prediction of Zebib et al. From the continuity equation, the boundary layer thickness adjacent to the free surface is then 0(Re-1/3). A plot of radial velocity in figure 26 for various Reynolds numbers shows how the boundary layer gets thinner as the Reynolds number is increased. The maximum value of stream function is in the core flow. This was predicted to be proportional to Re"1/3 and is seen to be true from figure 27a. Also the maximum pressure is found to be 0(Re,/3) as shown in figure 27b. In table 3, the max­ imum value of stream function, pressure and vorticity are listed for the range of Reynolds numbers investigated.

Re Ymax P max

1,000 -1.12xl0-2 (-0.92,2.12) '3.4 5,000 -2.79xl0-3 (-0.69,1.00) 7.0 10,000 -1.33xlO"3 (-0.70,0.93) 7.5 50,000 -7.49X10-4 (-0.68,1.29) 13.2 100,000 -5.34x10"* (-0.70,1.33) 16.0

Table 3. Maximum values of stream function, vorticity and pressure for Pr=0.01, Gr=105.

Figure 28 shows how the dimensionless axial velocity behaves near the side wall. This velocity is plotted at a z-location just under the free surface of the weld pool. The axial velocity profile has the character of a wall jet near the side wall. For Reynolds numbers greater than 10,000, the gradients of the velocities near the side wall get larger. The axial velocity drops firom a maximum value to zero in a small boundary layer next to the wall. To properly simulate this flow and to represent the wall shear on the flow, one needs to have fine grids near the wall.

Figure 29 shows the behavior of pressure along the side wall. The pressure was calcu­ lated by integrating the momentum equation on the free surface and on the side wall. By making use of the constraint that the volume of the fluid is constant, this undetermined con­ stant is determined. The shear exerted on the surface is balanced by the pressure close to the 68

-1

-1/3 Re •2

3

•4 2 3 4 5 6 Log (Re)

1.5

1.4

1.3

1.2

1.0

0.9

0.8 3 4 5 6 Log (Re)

Figure 27. (a) Reynolds number dependence of maximum stream function in the pool, (b) Reynolds number dependence of maximum pressure in the pool. (lOOOX) Axial velocity -18.00 - - 12.00 12.00 0.00 6.00 6.00 Figure 28. Figure 0.00 0 — s—X Re=100,000 —0 0— Axial velocity at a location below the free surface z=0.96875. surface free the below location a at velocity Axial Re=50,000 Re= Re= 10,000 0.20 * * * * * * 1 1 1 '* ... .00.60 0.40

Radius 0.80 1.00 69 Coordinate 0.00 0.20 0.40 0.60 0.80 00 .0 1 0.00 Figure 29. Variation of pressure on the end wall (r=l). wall end the on pressure of Variation 29. Figure 4.00 W l Pesr (dxnnls) "Wall Pressure 8.00 12.00 Re = 100,000 Re=50,000 16.00

20.00 70 71

right wall. The pressure decreases along the side wall continuously at moderate Reynolds numbers. At Reynolds numbers larger than 50,000 it seen that an adverse pressure gradient develops which forces the fluid to . separate from the right wall.

For small capillary numbers, as described in Appendix A, the free surface deflection can be computed as a domain perturbation. Figure 30 depicts the free surface deflection for vari­ ous Reynolds numbers. The free surface deflections are plotted for two different capillary numbers. The leading order velocities and pressures determine the free surface deflection. For moderate and high Reynolds numbers, the free surface deflection has a minimum at the centerline where the pressure is lower than the ambient pressure. The shear on the surface exerted by the thermocapillary forces develops a high pressure at the comer and forces the fluid to go up. The contact line is fixed at this comer. Surfaoe defleotloxx (xlOO) Surfaoe dofleotion (xlOO) Figure 30. Figure 2.00 0.00 0.00 6.00 2.00 0 0 . 1 4.00 3.00 1.00 0.00 0.00 - Free surface deflection for (a) Ca=0.05 and (b) Ca=0.1. (b) Ca=0.05and (a) deflection surface for Free 0.20 0.20 0.40 Radius Radius 0 H—I —Re=50,000*— o—e—e Re=100,000 »—*-—* Re=50,000 —0—0 ReslOO.OOO 0.60 0.60 * — R es 10,000 es R O.BO 0.80

1.00 1.00 72 CHAPTER IV

THERMOCAPILLARY CONVECTION IN A STATIONARY WELD POOL

In this chapter a model of a stationary spot welding process is described and formu­ lated. The results from a parametric study of the influence of buoyancy, thermocapillary and electromagnetic forces on the shape of the weld pools is discussed. After a brief description of the physical processes in welding is presented, the existing literature on the progress of research in this area is reviewed. The formulation and the governing equations are presented next and the numerical solution procedure is discussed. With the aid of the results the role of

thermocapillary convection in weld pools of aluminum and steel is explained.

4.1 DESCRIPTION OF WELDING PROCESS

A typical fusion welding process involves a moving source of heat which melts the metal at the joint between two plates and joins them upon solidification. In the molten pool, heat conduction primarily determines the temperature field, the location of the phase front and thus the shape of the weld pool. On the free surface of the molten pool, a temperature gra­ dient is established by the presence of the heat source. The temperature gradient in turn leads to a variation in surface tension at the liquid-gas interface with the result that thermo­ capillary forces appear there. In addition, the impingement of high velocity arc plasma deforms the free surface and imposes a normal pressure on the pool. The arc plasma spreads out from the impingement area and if it were to act alone, it could induce the fluid on the free surface to flow radially outward and cause the bulk fluid near the surface to move in the same direction. The temperature distribution in the molten metal causes density differences giving rise to buoyant forces. In addition, in the case of tungsten arc welding, the current flow into the metal induces an electromagnetic field. The interaction of the electromagnetic 73 74

field with the current flow gives rise to electromagnetic or Lorentz forces. It is the combined effect of these forces that need to be analyzed to understand how the molten metal flows in the weld pool.

4.2 LITERATURE SURVEY

Heat transfer analysis of welding dates from the pioneering work of Rosenthal (1946). In this theoretical study, he considered a point source of heat, which travels along either a surface of a semi-infinite body, or on a flat plate. He neglected the latent heat of fusion in order to obtain an analytical solution for the temperature field. From this field, he determined the weld pool shapes and the cooling rate of any location on the body, both of which are in reasonable agreement with the experiments. Only the basic shape of the weld pool can be described by this conduction model, and the model, in fact, fails to explain why the shape of the pool and its depth should vary under seemingly similar welding conditions. To answer these questions researchers turned their attention to the flow of molten metal in the pool.

An experimental study on the importance of fluid flow in weld pools was conducted by Woods & Milner (1971). They reported that alloying ingredients added as filler metal or through the arc flux are more thoroughly mixed in a weld bead than is possible purely by diffusion. These results suggest that the fluid flow is of significance in weld pools. Woods & Milner also confirmed that electromagnetic forces influence the molten metal flow in arc- welding. These forces are caused by an electromagnetic field which is induced by the current flow from the electrode into the metal. The interaction of welding current with its own induced magnetic field produces forces radially inward, and downward near the center of the weld pool. As a result, the liquid metal is drawn from the rim of the pool towards the center, where it turns down and flows into the pool in the form of a jet. They also indicated that a secondary effect on the flow may be created by the arc plasma jet on the surface of the weld pool. Their observations showed the fluid motion to be more rapid in materials with a high melting point, such as iron, steel, and stainless steel, than in materials with a low melt­ ing point, such as aluminum, silver and copper. They attributed this effect to a more 75

concentrated arc at the pool surface for the former group giving rise to a larger current den­ sity. This in turn results in larger Lorentz forces near the arc root and thus in larger veloci­ ties there. The rapid motion observed for the first group (steel, iron etc.) was erratic, whereas for the second group the motion was slow and uniform. Woods & Milner thus con­ cluded that the stirring in weld pools was mainly caused by electromagnetic forces. The erratic motion may be an indication that the flow under the conditions has undergone a transi­ tion from a simple laminar motion into an oscillatory flow which could even be three- dimensional.

Sozou & Pickering (1978) analyzed the flow field caused by the Lorentz forces in a hemispheroidal container. By assuming the electric current discharge to be axi-symmetric, and the welding currents to be so small that Stokes flow analysis is valid, they were able to solve for the velocity field semi-analytically. By expanding the stream function and vorticity in terms of Legendre polynomials, they reduced the governing equations for fluid flow to two coupled fourth order ordinary differential equations for each of the expansion coefficients. These ordinary differential equations they solved numerically. The flow pattern so obtained shows that the fluid is drawn from the rim of the pool towards its center and is forced down to the bottom of the pool. The fluid returns to the free surface along the phase boundary and so forms a single loop recirculation pattern. It is now known that the Stokes flow approxi­ mation is not usually valid in welding. To rectify the situation, Atthey (1980) extended this solution for higher currents using a finite difference technique to determine the flow in a hemi-spherical pool. The flow pattern he obtained is similar to that of Sozou & Pickering. The important finding from these studies is that the type of fluid motion induced by Lorentz forces causes heat to be transferred from the welding arc to the bottom of the weld pool with the result that the heat is effectively used to create a deep weld.

The experiments conducted by Heiple & Roper (1982) and Heiple & Burgardt (1985) bring to light the significance of thermocapillary forces in weld pools. For large arc heat inputs, if they were to act alone, the thermocapillary forces would cause a flow pattern to be set up in a weld pool that is characterized by two cells: one at the top surface which circu­ lates the fluid from the center of the pool to the rim and returns it back in a thin surface layer; the other, a weaker and oppositely circulating cell in the rest of the pool. It has been 76

observed by many researchers that thermocapillary forces dominate Lorentz forces for metals of high purity and for these the surface flow is toward the rim. Heiple et al. have shown also that it is possible to change the behavior of surface tension gradient in austenitic steels, by addition of surface active ingredients such as selenium or sulfur. Addition of these surfac­ tants causes the surface tension to increase with temperature. In such a case, the thermo­ capillary forces aid the Lorentz forces and cause the weld pool to be deep, providing good joint penetration. The dependence of the temperature coefficient on the surfactants has important consequences for welding. High strength steels with low sulfur content are by this reasoning difficult to weld because the surface flow in them is toward the rim leading to poor joint penetration.

In two recent numerical investigations, the effort was focussed on extending the previ­ ous studies to a situation which is closer to the tungsten arc welding process. Oreper & Szekely (1984) consider a stationary arc welding of a thick metal block. By assuming the arc to be stationary, the fluid flow and temperatures can be taken to be axi-symmetric. The solu­ tions are obtained for those values of heat and current inputs which render the pool partially penetrated. They note that the location of the phase front is not changed appreciably by the flow of molten metal if the heat and current flux are broadly distributed. For the welding con­ ditions analyzed, the Lorentz forces dominate buoyancy forces. Kou & Wang (1986) in a subsequent study consider three dimensional flow caused by a moving heat source. In both of these studies, rather coarse grids are used to analyze the flow at so large Reynolds numbers that, in the flows that develop, very sharp gradients exist near the free surface and the phase boundary. From the analysis of thermocapillary convection presented in the last chapter, one concludes that to properly simulate a flow dominated by thermocapillary forces, one needs to have very fine grids near the free surface and the phase boundary. This is also the view of Kanouff (1987) who has studied this welding problem. In a recent summary of research in thermocapillary convection, Chen (1987) also stresses the need for finer grids and accurate computational schemes to determine the true flow behavior at severe operating conditions in various applications of material processing. There is, accordingly, a need for further study of this problem. 77

4.3 FORMULATION AND GOVERNING EQUATIONS

Anyone who has seen a welder at work and thought about the physical and chemical processes associated with welding will have been impressed by the complexity of these processes. To make progress in the understanding of these processes, one needs to start with a simple model in which only the most important effects are retained. With regard to under­ standing the fluid flow in a weld pool, a good starting point is a stationary axi-symmetric pool at steady state conditions. From previous work, it is known that the fluid flow in such a pool is driven partly by surface tension gradients, the strength of which depends on the con­ centration of surfactants and on the magnitude of the temperature gradients on the surface; partly by the electromagnetic forces; and to a small degree by the forces associated with buoyancy. Because the top surface of a weld pool is a liquid-gas interface, it can deflect. Its deflection is governed by the flow in the pool as well as the plasma-arc pressure. For the model considered in this chapter, the surface deflections are treated by a perturbation method. This is justified (as mentioned in the previous chapter) if the mean surface tension is sufficiently large. The influence of arc-plasma impingement on the free surface is neglected so that the only thing that needs to be modeled with regard to the arc plasma is the heat flux. It is taken to decay exponentially with the square of the radius, as has been observed by Kou & Sun (198S) in experiments.

The equations that govern fluid flow in an arc-welded pool are the Navier-Stokes equa­ tions, the Maxwell’s equations and the thermal energy balance. The Boussinesq form of the momentum equations are considered. The assumption of constant properties is probably not justified because of the magnitude of temperature gradients encountered. The influence of variable properties is nevertheless left for a later study.

In the equations that follow, the variables are put into a non-dimensional form by divid­ ing the radial coordinate r ' by a characteristic horizontal length L , which is the radius of the cylindrical slab (see figure 31), and the vertical coordinate z ' by the plate thickness h. This introduces an aspect ratio, H=hlL into the equations. The radial and vertical velocities, u ' tf and w ', are non-dimensionalized by a characteristic velocity U that is obtained by balancing 78

Cathode

Arc heat flux Arc current flux

mushy zone

LIQUID

SOLID

Figure 31. A sketch of a stationary weld pool. 79

the viscous force with the thermocapillary force. The pressure is scaled by ]iUIL. The tem­ perature is measured with respect to the liquidus temperature Tt and non-dimensionalized with a characteristic temperature, AT, based on the heat flux qQ. In addition to the aspect ratio, a large number of other parameters appear in the non-dimensional form of the equa­ tions. Important among these are the Grashof number, Gr, the Prandtl number, P r, the Rey­ nolds number, R e, and the capillary number, Ca . The significance of these parameters has already been explained in the earlier chapters. New parameters that appear are the magnetic force number, Re#, which shows the strength of Lorentz forces, and the quantities K and Xm. The first is the ratio of the conductivity of the liquid to that of the solid, and the second , called the melt ratio is the difference between the liquidus and solidus temperatures divided by the characteristic temperature difference.

The non-dimensional equations that govern the heat transfer and fluid flow in the weld pool under the approximations stated take the form

(4.1)

(4.2)

± $ L + v V + T +ReHRe f t , (4.3) H dz Re

dT ,w _ d T RePrK u ^ - = v-(*vr), (4.4) dr H dz

where

Here, ir, and i2 are the unit vectors in the radial, azimuthal and axial directions. The 80

parameters are defined as

„ * , , = v L v2 a

2 r 2 ft.®. = ^ V o0 ptf

ki T, - r , Jt, ’ AT where

u = 1 » m m- = M t1 */

The vector components of the non-dimensional Lorentz force are f r and f 2. The region for which Ts

k T . (4.5) Am

To complete the formulation, the boundary conditions must be specified and the method for calculating the Lorentz force discussed. The boundary conditions are given below. 81

(4.7)

T1 = T 1 ©o r = l , (4.8)

(4.9)

(4.10)

z = 0 . (4.11)

In (4.8) denotes the dimensionless ambient temperature. Equation (4.9) means that the thermocapillary boundary condition has to be satisfied only in the liquid pool, i.e. where the temperature is greater than the dimensionless liquidus temperature of the metal. In addition to the above boundary conditions, on the phase boundary specified by the liquidus tempera­ ture, the condition for energy balance and the no-slip conditions for velocity must be satisfied. These conditions require that

u = w = 0 , (4.12)

K dT _ dT (4.13) dft liquid d/i solid

where n is along the normal to the phase boundary.

4.3.1 ELECTROMAGNETIC FORCES

In the cathode mode type of arc welding there is a current flow from the base metal to the electrode. The current flow induces a magnetic field which in turn interacts with the current to create electromagnetic forces. The Maxwell’s equations govern the behavior of the electromagnetic field. 82

It is possible to reduce the complexity of the Maxwell’s equations by invoking the so called magnetohydrodynamic approximations. They can be justified by arguments provided for example by Sutton & Sherman (pp. 300-302,1965) and which apply for the situation here considered. Accordingly, it is permissible to neglect the displacement current in Ampere’s law, and the convection current in comparison with the conduction current. The last simplification is that the electrostatic force is much smaller than the Ampere (magnetic) force.

In addition to the above approximations which are the standard set for magnetohydro- dynamics, for welding problems, we can make yet another simplification which helps decou­ ple the magnetic field equations from the hydrodynamic equations. The current density, can be written as

t = k(£' + \6

In this equation, k is the electrical conductivity, is the electric field strength, It' is the magnetic induction and is the fluid velocity. The ratio of the two terms in this equation turns out to be the magnetic Reynolds number, Rem. For welding conditions, a characteristic length of the flow domain is on the order a few millimeters, the magnetic permeability of vacuum, Po, is 4nxl0-7 henry/m, velocity is on the order of 1 m/s, and electrical conduc­

tivity, k, is roughly equal to 106 mhos/m (liquid metal). This gives for the ratio

^ A L =wcvL - Rem =0.001 .

This ratio is sufficiently small that Ohm’s law is strictly valid and the Maxwell’s equations become only loosely coupled to the hydrodynamic equations. That is, the electromagnetic problem can be solved independently of the hydrodynamics.

Since we are interested in only steady state solutions, the time derivatives of the Maxwell’s equations can be dropped. This simplification and the magnetohydrodynamic approximations help reduce the equations governing the electromagnetic field to a set that can be solved more readily. A non-dimensional form for the equations is obtained by dividing the the current density by the magnitude of the current at the electrode, y0; the electric field by a characteristic value J q / k ; and the magnetic induction, £?', by \iej d L . The equations governing the electromagnetic field in this way reduce to the forms : 83

v-i? = o , v / =o (4.14,15)

vi? = o, Vxf? = o, vxi? = y\ (4.16,17,18)

To compute the electromagnetic field, we define a magnetic stream function as (Oreper & Szekely, 1984)

T = curl -^ -e (4.19) r

Making use of (4.17) and the nondimensional form of Ohm’s law 7 = £ yields

d 1 l + JL = 0 . (4.20)

dr r dr dz S' --- t • • i

As a boundary condition to this equation, the axial component of the current density j2 is imposed on the top surface.

Ve(r) = JQ rjt (r)dr , j2(r) = ,-*r‘ z = 1 . (4.21) rc(l - e~*)

In this equation % is the coefficient of gaussian current flux. The other boundary conditions are

\|/e = 0 ; z = 0 , (4.22)

y e = 0 ; r = 0 , (4.23)

= b z ; r = 1 , (4.24) where b = \ye(l , 1).

The boundary condition at r=l is such that the outgoing current to the electric arc is exactly equal to the incoming current at f=1. The magnetic induction, £?, is induced in the $ direction (perpendicular to the r-z planes) since the current flow is axi-symmetric. is calculated by integrating the current density according to (4.18). The electrostatic forces are very small when compared with the magnetic forces created by the current intersecting with magnetic 84

flux density. Hence, the electromagnetic or Lorentz forces are given by

P = ?x& = - e r(jtB# + St(jrB j (4.25)

where j r and jz are the radial and axial components of the current density.

4.3.2 STREAM FUNCTION-VORTICITY FORMULATION

To solve the equations by numerical techniques we cast the momentum equations into stream function and vorticity form. The continuity equations are satisfied automatically by defining the velocities as

(4.26)

The pressure term is eliminated by taking curl of the momentum equation. Accordingly,

(4.27)

. 3(0 . w d(o 1 t72„

(4.28)

The vorticity, being the curl of the velocity, is given by 85

d w 1 3 u (4.29) dr H dz

In (4.28) the last term accounts for the Lorentz force effects.

4.4 FINITE DIFFERENCE REPRESENTATION

In the following paragraphs, the finite difference approximation to (4.27), (4.28) and (4.4) is discussed. Implicit methods such as the Stone’s method are not well suited for solv­ ing the flow equations in this problem. The weld pool occupies only a fraction of the com­ putational domain and for this reason, it was decided to use an explicit method with pseudo­ time stepping to iterate the vorticity equation. Except for the convective terms in (4.28) and (4.4), all the other derivatives are represented by central differences. The deferred correction algorithm of- Khosla & Rubin (1974) (K-R differencing) was used to represent the convective terms. The energy equation was solved by ADI (alternating direction implicit) methods in a way similar to that described in the last chapter. The energy equation is different here in that the conductivity is a function of temperature in the mushy zone.

On the free surface of the liquid metal, the boundary vorticity is given by the horizontal temperature gradient. That is,

(4.30)

(4.31)

The flow equations are solved only in the liquid metal. The region between the liquidus and solidus isotherms is the mushy zone. The liquidus isotherm divides the liquid metal and the mushy zone. This phase front is approximated numerically by the grid point just next to it (see figure 32). The solid line in this figure represents the liquidus isotherm. The vorticity and stream function values are iterated for all the points to the left of this front. The 86

Phase front

+ * + * + F

+ Coarse and fine grid points

* Fine grid points

A,D,F Coarse grid boundary points

A,B,C,E.F Fine grid boundary points

Figure 32. Phase boundary approximation using multi-grid method. 87 boundary vorticity on a point adjacent to the phase front and to the right of it is computed based on (4.27) and no-slip conditions. Local refinement is performed with the multi-grid method and grid points are clustered next to the phase front and the free surface. With this approach, the location of the boundary point is closer to the actual phase front. However, there is a drawback with this method of refinement. The boundary point, "A" according to figure 32, is part of the fine grid. The coarse grid corrections are based on the boundary vor­ ticity calculated at a point "B" which is farther away from the phase front. This contradiction causes the iterations to diverge. As a result, coarse grid equations are not solved to conver­ gence as is done normally, but to only a fixed number of iterations (=100) which gives con­ verging multi-grid solutions. On the other hand, this approach degrades the multi-grid conver­ gence rates and residuals could not be reduced to as small a value as those obtained in the last chapter. The convergence criterion is relaxed so that a maximum relative change in the variables of less than 10~3 is now accepted as a converged solution.

The above-described multi-grid local refinement was performed only for solutions reported for aluminum. Since multi-grid convergence rates were poor in this case, only sin­ gle grids were used for the solutions obtained for steel. A small modification is made in approximating the phase boundary to impose no-slip conditions better in the set of solutions obtained for steel. As seen in figure 33, the boundary was chosen to be of a stair-case type with the result that (4.27) & (4.28) are solved for points to the left of the stair-case. No-slip conditions were imposed on the points on the boundary and the boundary vorticity was calcu­ lated in a similar way.

The solution procedure is as follows. Since the magnetic field equations are de-coupled from the hydrodynamic equations, they can be solved independently of the flow. Using cen­ tral differences, (4.20) is solved by point S.O.R. A relaxation factor of 1.8 is used. Equation (4.18) was integrated using Simpson’s rule to obtain B$. The Lorentz force was then calcu­ lated using (4.23). Before the flow equations can be solved, an approximate shape of the phase front is necessary. The energy equation is solved with Re=0, to obtain the conduction temperature distribution. This provides a starting guess for determining the location of the phase front. Based on this, the stream function and vorticity values are iterated using the flow equations in the molten pool and the temperatures are advanced using the velocities so 88

Phase front

+ A

E+- 4-D

+ grid points

A,B»C,D,E Boundary points

Figure 33. Phase boundary approximation using a single grid. 89

obtained. The temperatures now give the new location of the phase front and the procedure is continued until convergence is obtained. When multi-grid is used, a converged coarse grid solution is obtained using the above described procedure. This gives a starting solution and the free surface and phase front are refined with finer grids. The full approximation storage is used with the cycling procedure. The restriction operator was an injection scheme and nine- point interpolation was the prolongation operator. From the coarsest grid, a cubic interpola­ tion is used as the prolongation operator.

4.5 RESULTS FOR ALUMINUM

Here, the results for aluminum 6061 alloy are presented. The aspect ratio of the axi- symmetric region was taken to be 0.5. The circular outer boundary was assumed to be main­ tained at a room temperature. The dimensionless value of this temperature (T„) comes out to be -0.42. A heat flux distribution of the form e~^r2 was imposed. Here P was taken to be equal to 12. With this type of distribution, the radius of the pool is roughly half the radius of the cylindrical slab enabling us to have adequate grid points to resolve the flow in the pool. The current density from the arc was taken to be similarly distributed. However, the magni­ tude of the current flux was varied so as to change the magnetic force number. The property values are similar to those used by Kou & Wang (1986). The Prandtl number was fixed at 0.01. Based on the physical properties of aluminum 6061 alloy, the conductivity ratio, K, was taken to be 0.643 and the melt ratio, Xm, was fixed at 0.05. The heat flux magnitude was fixed so that the Grashof number was equal to 2.3xl06.

Figure 34 gives the isotherms for pure conduction. The zero temperature isotherm indi­ cates the liquidus line. The next isotherm (=-0.05) to the right indicates the solidus line and the region between these two isotherms is the mushy zone. To determine the type of veloci­ ties associated with a flow driven by buoyant forces alone, the top surface boundary condi­ tion was changed to that for a free shear surface. In other words,

£ -0 atr =1 oz 90

r

Figure 34. Conduction isotherms for aluminum, Re=0. The isotherms are plotted in con­ stant increments of 0.05. 91

By replacing the Reynolds number by Grashof number and setting Rew=0 in (4.28) & (4.4) the equations governing natural convection in the molten pool can be obtained. The effect of buoyancy force is to cause a clockwise motion to be set up in the weld pool. Figure 35 dep­ icts such a flow. Though the flow is very weak, owing to the convection in the weld pool, the shape of the weld pool is changed to a partially penetrated one.

By changing the temperature coefficient of surface tension, the Reynolds number can be changed. Thus, we are allowing for surface active elements to be present. We assume, how­

ever, that they are evenly distributed on the surface. Formally we can achieve this by taking their diffusion coefficient to be infinitely large. The magnetic force number is taken to be zero at this stage so that the influence of thermocapillary convection alone can be determined. The Reynolds number is varied from 10,600 to 200,000. It is seen in figure 36 that the isotherms for Re=10,600 resemble those of the natural convection solution very closely. But at higher Reynolds numbers of 42,500 and 200,000, the isotherms are distorted by the flow and the weld becomes fully penetrated. Also, the flow is seen to separate from the side wall earlier for the higher Reynolds number.

When we attempted to calculate the flows at yet higher Reynolds numbers the solutions were found not to converge. This lack of convergence appears to be linked to the very sharp gradients near the pool surface and the rim of the pool. Specifically, application of the vorti­ city boundary condition near comers of the phase front seems to be the cause of diverging solutions at the larger Reynolds numbers. To overcome this it would probably be better to employ a coordinate transformation and cluster grids near the phase boundary and the top surface. At present to obtain the solution at Re=200,000 a fine grid size of 1/128 (non- dimensional unit) was used. The computational grid used at Re=200,000, is shown in figure 37. It would be computationally too expensive to carry out the solution with any finer grid using the current method.

A plot of radial velocity at the top surface shows that the surface radial velocity increases to a maximum near the phase front and then drops sharply to zero. It is shown in figure 38. The radial velocity, plotted at an axial section at a location which is close to the centerline, shows the presence of a thin surface layer (figure 39). Likewise, the axial velocity 92

TT T I I I I 1 I I I I I I ! I I II T TTT

Figure 35. (a) Streamlines and (b) Isotherms for the flow induced by buoyancy, Gr=2.3xl06, Pr=0.01, (solid lines for v|f,T^0 and broken lines for \jr,T<0, isoth­ erms in constant increments of 0.1). 93 Re= 10,600

Re=42,500

(WM

Re=200,000

(a)

Figure 36. (a) Streamlines and (b) Isotherms for various values of Re, Gr=2.3xl06, Pr=0.01, Re# =0. 94

Figure 37. Computational grid used at Re=2xl05. iue 8 Srae ailvlct o aiu R, r23l6 r00 adR#=. Note =0. Re# and Pr=0.01 Gr=2.3xl06, Re, various for velocity radial Surface 38. Figure Surface velocity 0.00 0.01 0.02 0.03 0.04 0.05 that on the surface only for r<0.46 the metal is in molten state. molten in is metal the r<0.46 for only surface the on that 0.00 0.20 0.40 Radius 0.60 Re=200,000 Re=42,500 0.80 1.00 95 iue 9 Rda vlct a =. fr aiu Ryod nmes Nt ta te ea is metal the that Note numbers. Reynolds various for r=0.1 at velocity Radial 39. Figure Coordinate 0.00 0.20 0.4-0 0.60 0.80 00 .0 1 molten state only for z>0.16. for only state molten -3.00 0.00 10X Rda velocity Radial (1000X) 3.00 6.00 Re=200,000 Re=42,5O0 Re= 10,600 .012.00 9.00

96 iue 0 Ail eoiy t lcto ut eo i fe srae z09. oe ht the that Note z=0.97. surface, free in below just location a at velocity Axial 40. Figure (1000X) Axial velocity - -4.00 - 0.00 2.00 4.00 6.00 2.00 metal is molten state only for r>0.46. for only state molten is metal .0 .0 .0 .0 .0 1.00 0.80 0.60 0.40 0.20 0.00 Radius Re=200,000 Re=42,500 Re= Re= 10,600

97 98 near the phase boundary in figure 40 is seen again to drop rapidly to zero in a small region close to the phase boundary. This kind of behavior requires a very fine mesh near the phase boundary if accurate solutions are to be obtained. This requirement becomes even more acute when Reynolds number is increased. At the center, owing to buoyancy forces the axial velocity reaches its maximum up-flow velocity.

In all the cases discussed above, the temperature coefficient of surface tension decreases with temperature. This behavior can be altered by addition of surface active ingredients. Figure 41 shows the flow pattern and isotherms for a flow for which this is done. The tem­ perature coefficient of surface tension is now assumed to be positive (i.e. surface tension increases with temperature). The hot fluid at the center of the pool, therefore, has a larger surface tension than the cold fluid at the rim, causing the fluid to be drawn from the rim to the center, where it turns downward. The resulting weld is seen to penetrate the joint fully.

The influence of Lorentz forces on the flow was tested by fixing the Reynolds number and changing magnetic force number. For Re=10,600, the flow pattern for various magnetic force numbers is shown in figure 42. It is observed that if these force numbers are less than 0.0685, the flow is dominated by thermocapillary forces. As the magnetic force number is increased the Lorentz forces counteract the thermocapillary forces and at an intermediate value two cells which rotate in opposite directions are formed. The top cell is formed by the Lorentz forces and the bottom cell rotating clockwise and close to the phase boundary is caused by the thermocapillary forces. The Lorentz forces are larger near the center of the pool and it is for this reason that the cells appear this way. At a even larger value of mag­ netic force number, Lorentz forces completely overcome the effect of thermocapillary forces. At Re=42,300, the magnetic force number was varied from 0.007 to 0.021 and a similar behavior is observed. Figure 43 shows this behavior.

In all the above cases, the isotherms do not deflect significantly. This is due to the fact that the Prandtl number of aluminum is so small that conduction dominates convective heat transfer. The same is not true for steel, a material with a larger Prandtl number. In the next section, the results obtained for steel are discussed. 99

Figure 41. (a) Streamlines (anti-clockwise circulation) and (b) Isotherms in the presence of surfactants, (Y>0), Re=42,500, Pr=0.01, Gr=2.3xl06, ReH =0. 100

T lllllll I'll II III M 1 I 1 I I I I I 1 I I 11 I I I o

II II llll I I I I I M ITT ITI'I I I II IT I I I I I I ! I I 1 II I I I I 1 I I I I I

:i 11111 / 1111111111 ■ 11 ■ ...... 11111. i ...... 1111111111.

Figure 42. Streamlines for Re=10,600 for Re# of (a) 0.007, (b) 0.014 and (c) 0.017, Gr=2.3xl06, Pr=0.01. Broken streamlines show clockwise circulation and solid streamlines show anti-clockwise circulation. 101

Figure 43. Streamlines for Re=42,500 for Re// of (a) 0.0007, (b) 0.0009 and (c) 0.0019, Gr=2.3xl06, Pr=0.01. Broken streamlines show clockwise circulation and solid streamlines show anti-clockwise circulation. 102

4.6 RESULTS FOR STEEL

The aspect ratio of the axi-symmetric region is again taken to be 0.5 and the outer boundary is maintained at a room temperature of 25° C, which in dimensionless form comes out to be -0.36. Heat flux and current flux distributions similar to the ones imposed for the aluminum case are used. However, the magnitude of the heat flux is chosen to be of some value typical of that used in welding of steel so that the Grashof number comes out to be 43,000. The property values are based on those given by Kanouff (1987). The conductivity ratio for steel comes out to be to 0.5 and the Prandtl number is 0.19. Based on the difference between liquidus and solidus temperatures and the heat flux, the melt ratio, Xm, is taken to be 0.05.

The conduction isotherms of figure 44 show the weld to be partially penetrated. The mushy zone is marked in the figure. As before, by varying the coefficient of surface tension, the Reynolds number was varied from 350 to 35,000. The magnetic force number is taken to be zero. Up to a Reynolds number of 3,500, the thermocapillaiy flow does not influence the shape of the pool. A clockwise motion is set up in the pool just like in the aluminum case. From figure 45, it is seen that the weld pool is shallower than the conduction profile of figure 44. However, the radial velocities near the surface are still not strong enough to make the pool wider. At Reynolds numbers larger than 17,000 the pool becomes wider and shallower. Table 4 shows the depth to width ratio of the weld pool for various Reynolds numbers inves­ tigated. The conduction case is represented by the Re=0 entry. 103 0.35

r

Figure 44. Conduction isotherms for steel, Re=0. The isotherms are plotted in constant increments of 0.05. 104

R e = 1 7 0 0 0

R e= 3 5 0 0 0 (a) (b)

Figure 45. (a) Streamlines and (b) isotherms for various Reynolds numbers, Gr-=43,000, Pr=0.19, Re# =0. 105

Re depth/width ratio

0 0.89 350 0.85 3,500 0.83 17,000 0.73 35,000 0.66 100,000 0.57

Table 4. Depth to width ratios for various Reynolds numbers, B£H=0, Pr=0.19, Gr =43,000.

For Reynolds number larger than 17,000, the streamlines resemble those that were obtained in an axi-symmetric pool at large thermocapillary Reynolds numbers. The flow is concentrated closer to the free surface and the top of the phase front, as is typical of the ther­ mocapillary flow. The surface radial velocities, plotted in figure 46, behave in an interesting fashion. Below Re=3,500, the dimensionless surface velocity decreases with increasing Rey­ nolds number. In the aluminum case (figure 38) the same observation was made for the whole range of Reynolds numbers investigated. In steel, for Reynolds numbers greater than 3,500, the surface velocities are seen to increase with increasing Reynolds numbers. How­ ever, for radial distances smaller than 0.20, the velocities overlap. The reason for the larger velocities at radial distances larger than 0.2 is that there the coupling between the surface flow and temperature is stronger. Since the Prandtl number for steel is 19 times larger than that for aluminum, the isotherms are more significantly influenced by the flow. For this rea­ son, the temperature gradient is affected, which in turn changes the imposed shear on the flow. Figure 47 shows the surface temperature distribution for various Reynolds numbers. The radial velocity distribution as shown in figure 48 follows the same trend as explained before for the surface velocity. The axial velocity in figure 49 shows the maximum upflow velocity near the center to decrease with increasing Reynolds number. The figure also shows how the boundary layer near the phase front becomes thinner with increasing Reynolds number. At the largest value of Re=35,000, one needs to have more grid points near the iue 6 Srae eoiy o vros e G=300 P=.9 R#=. oe ht h j the that Note =0. Re# Pr=0.19, Gr=43,000, Re, various for velocity Surface 46. Figure

(lOOOX) Surf&oe veloolty (lOOOx) Surface velocity ea si otnsaeol o <. (prxmtl) j (approximately). r

0 8 . 0 0 6 . 0 .eo o 1 ----- Re=3,500 0 5 , 3 = e R ► 0 0 0 . 7 1 = e R 0 8 . 0

1.00 1.00 106 Figure 47. Surface temperature for various Re, Gr=43,000, Pr=0.19, Gr=43,000, Re, various for temperature Surface 47. Figure

Surface temperature Surface temperature - 0 5 . 0 - 0 4 . 0 - 0 8 . 0 - 0.00 0 5 . 0 1.00 0.00 0 4 . 0 0 8 . 0 1.00 1.20 0.00 0.00 0.20 0.20 0 4 . 0 0 4 . 0 s u i d a R s u i d a R o — o — o i Re— 0 0 5 . —3 e R ■i i • 0 8 . 0 0 6 . 0 0 8 . 0 0 6 . 0 0 0 0 , 5 3 = e R 17,000 0 0 , 7 1 = e R

1.00 1.00 Figure 48. Radial velocity at i*=0.4 for various Reynolds numbers. Note that the metal is in is metal the that Note i*=0.4 numbers. at velocity Reynolds Radial various for 48. Figure

Z Coordinate molten state for z>0.55. for state molten 0 8 . 0 0 4 . 0 0 6 . 0 0.20 0.00 1.00 0.00 0.20 0 4 . 0 0 6 . 0 0 8 . 0 1.00 - 12.00 2.00 - 0.00 6.00 l Radial velocity t i c o l e v l a i d a R ) X O O (lO OOOx) Radial velocity t i c o l e v l a i d a R ) x O O lO ( 2.00 0.00 0 0 . 4 .012.00 6.00 0 O 0 , S 3 = e R 17,000 0 0 , 7 1 = e O R O ,S 3 = e R 6.00

10.00 6.00 108 (lOOOX) Axial velocity (lOOOx) Axial velocity - 0 5 . 4 - 0 0 . 3 - 0 0 . 4 2 Figure 49. Figure - 12.00 0 0 . 8 1 0.00 0 0 . 3 0 S . 1 0.00 6.00 1.00 6.00 0.00 0.00 Axial velocity at a point below the free surface. free the below point a at velocity Axial 0.20 0.20 0 4 . 0 0 4 . 0 s u i d a R s u i d a R a— 0 6 . 0 0.00 * - » 17,000 0 0 , 7 1 = e R -» o =35,000 0 0 , 5 3 = e R -o 0 0 0 , 3 = e R SOO O ,S 0 3 = 5 e 0 R . 1 = e 0 R 5 3 = e R 0 8 . 0 0 8 . 0

1.00 1.00 109 110 phase front to obtain accurate solutions. For this reason, the Reynolds number was not increased beyond this value.

The influence of Lorentz forces is investigated at two different Reynolds numbers. Fig­ ure SO shows the influence of Re# at Re=3,500. Re# is changed ffom 0.041 to 0.082. It is seen that at the lowest value of Re#, thermocapillary forces dominate Lorentz forces com­ pletely and the resulting flow is a clockwise motion in the weld pool. At the intermediate value two cells develop. The one at the top right comer due to thermocapillary forces moves in the clockwise direction, since the thermocapillary forces are the largest near the free sur­ face and the phase front. Near the center of the pool, however, the Lorentz forces dominate thermocapillary forces and the fluid moves in the anti-clockwise direction. At an even higher value of Rew, thermocapillary forces are overcome by Lorentz forces and the flow is com­ pletely in anti-clockwise direction. In this case, the hot metal near the center of the pool is directed down along the center of the pool and the weld pool is made deeper. The depth to width ratio of this flow is 0.97. For Re=35,000, similar study was conducted by varying Re#. Here, as shown in figure SI, at the largest Re#, the flow is so strong that the pool is fully penetrated. The depth to width ratio is 1.2. The isotherms are seen to have deflected appreciably.

4.7 CONCLUSIONS

In this chapter, a parametric study on the influence of thermocapillary and Lorentz forces on the flow pattern in a stationary weld pool has been conducted. The non- dimensional equations of motion contain eight main parameters. The Prandtl number, Grashof number, aspect ratio, conductivity ratio, melt ratio, and heat flux coefficient are fixed and only the thermocapillary Reynolds number and the magnetic force number were varied. The coupling of hydrodynamic equations to the Maxwell’s equations is weak. Thus, the Maxwell’s equations can be solved independent of the hydrodynamic equations. Only steady state solutions are sought A multigrid method was used to perform local refinement in order to resolve sharp gradients in flow variables. From the numerical solutions we deduce the Ill

(i)

(ii

(iii

Figure 50. (a) Streamlines and (b) Isotherms for Re=3,500 for Re# of (i) 0.041, (ii) 0.045 and (iii) 0.082. Broken streamlines show clockwise circulation and solid stream­ lines show anti-clockwise circulation. 112

(i)

0 0

Figure 51. (a) Streamlines and (b) Isotherms for Re=35,000 for Re# of (i) 8.16X10"4, (ii) 8.16x10 3 and (iii) 0.033. Broken streamlines show clockwise circulation and solid streamlines show anti-clockwise circulation. 113 following features.

1. The conduction solution gives quite a close approximation to the temperature pattern in the liquid and solid metal if the Prandtl number is very small. For aluminum, the Prandtl number is 0.01 and the isotherms are not significantly affected by the flow in the liquid pool. However, if steel is the welding material, the same is not true, since the temperature solutions are now more strongly coupled with the flow equations.

2. Buoyant forces are negligible in comparison with the thermocapillary and Lorentz forces.

3. Thermocapillary forces induce large surface velocities and result in large gradients in flow variables near the phase front with the result that very fine grids are necessary to simulate the flow properly.

4. Lorentz forces oppose thermocapillary forces and can result in a two cell or single cell flow pattern when either the thermocapillary or the Lorentz force dominates the other completely. To obtain deep and thus strong weld joints, the arc current must be increased so as to overcome thermocapillary forces completely. Such a flow draws the cold metal from the rim to the center where it is heated by the arc and forced down along the axis. The resulting weld is deeper than it is wide. This can also be achieved, by adding surface active ingredients which makes the surface tension increase with increasing temperature. In such a case, thermocapillary and Lorentz forces aid each other providing good joint penetration.

5. Many welding processes involve thermocapillary Reynolds numbers that are larger than those investigated in this chapter. The flow at these large Reynolds numbers has been observed to be unstable in experiments. The stability of these flows is of interest to researchers who are interested in controlling the weld shapes. Though the phase front is non-vertical, the behavior of thermocapillary flow in a stationary pool is very similar to that in an axi-symmetric pool with the result that an axi-symmetric pool can be used to analyze the stability of these flows. The simplification in the geometry will enable use of more accurate schemes at reasonable computational costs. CHAPTER V

CONCLUSIONS

The aim of this dissertation was to numerically investigate convective flows in closed regions with multi-grid methods. In the following pages some of the findings of this study are reiterated and the plans for future work outlined.

The first topic investigated here was natural convection of air in a tall vertical cavity. This problem was chosen since it has a number of numerical and experimental solutions available and the results are important for determining the insulating capacity of double pane windows. In addition to testing the performance of the multi-grid method, the study was also used to gain insight into the influence of thermal end conditions on heat transfer. Two different boundary conditions were imposed on the end walls. The walls were either insulated or perfectly conducting because any other type of boundary condition will result in Nusselt numbers that lie between those obtained at these two conditions. Nusselt numbers were evaluated for a range of aspect ratios from 10 to 25. For aspect ratios larger than 25, the Nusselt number can be calculated based on the formula provided by Lee & Korpela (1983). The results were presented in the form of graphs to aid a designer to identify the right gap width for the cavity when the height is given. The flow in this geometry resulted in thin boundary layers that form at the top and bottom walls, with the result that very fine grids were necessary to resolve the flow properly. The multi-grid method helped improve conver­ gence rates to attain solutions in just a few relaxations in these fine grids. The results were compared with experimental solutions and the agreement was excellent Since we were interested in steady state solutions, explicit pseudo time stepping was used. The explicit scheme is easily vectorized and resulted in gains in computer speeds by a factor of two over an implicit scheme such as ADI. Typical computing time required on OSC Cray X-MP/24 was less than half a minute with the multigrid method. The vectorizable extrapolated Jacobi method resulted in solutions that were almost ten times faster than those obtained by using

114 115

point successive over relaxation technique.

Some observations were made regarding the multi-grid scheme in this test problem. For small Rayleigh numbers, the convergence rates of the multi-grid cycles were almost equal to the optimum convergence rate of a reduction in the residuals by a factor of ten. For test cases with higher Rayleigh numbers, to maintain the convergence rates it was found that the number of relaxations in the fine grids must be increased to completely damp the high fre­ quency oscillations. In addition, there is a limit on how coarse the coarsest grid can be. If the coarse grid is not fine enough to adequately resolve the flow, the corrections from these grids degrade the multi-grid convergence. The same observation was made on thermocapillary flows. The full multigrid scheme resulted in faster convergence than the cycling method.

With the understanding gained in solving Navier-Stokes equations in recirculating flows by multi-grid methods, we were ready to study the flow in a stationary weld pool. The sta­ tionary welding problem is beset with a number of complicated effects. It was first decided to retain just the thermocapillary effect, the most important physical feature of welding, and solve for the flow in a cylindrical liquid pool. In fact, this would be the shape of the weld pool if a thin plate were spot-welded, for then the phase front would be be nearly vertical. In addition, the validity of the assumption of a flat free surface needed to be assessed. The free surfaces at the top and bottom were, therefore, assumed flat and the free surface deformation was calculated a posteriori based on a domain perturbation. A heat flux distribution similar to that used in welding is imposed on the top surface. The temperature distribution on the free surface causes differences in surface tension, with the result that thermocapillary forces were set up. These forces pull the fluid from the center to the rim, where it turns down and flows along the side wall. A parametric study was conducted to test the relative importance of buoyancy and thermocapillary forces. It turned out that for large Reynolds numbers, such as those typically used in welding, thermocapiUary forces were much larger than buoyancy forces. As the surface tension Reynolds number was increased the separation point moves up closer to the comer of the top free surface and end wall. The continuous application of ther­ mocapillary shear induces large surface velocities and to support this mass flow in the surface layer, fluid is entrained from the side wall boundary layer. This entrainment expands the boundary layer resulting in an adverse pressure gradient forcing the fluid to separate from the 116

end wall. The surface velocities were proportional to Re- 1/3 and the thickness of the thermo- capillary boundary layer was found to be proportional to Re-,/3 too. This is in accordance with the boundary layer theory by Zebib et al. (1985) for the cartesian geometry. The flow results in sharp gradients in flow variables near the free surface and the top portion of the side wall where the flow separated. To adequately resolve the flow local refinement was car­ ried out using the multi-grid method. Convergence rates were improved significantly by this method. Because the Prandtl number of the fluid considered is very small (=0.01), the isoth­ erms were not affected significantly even at a Reynolds number of 100,000. The free surface deflection was computed as a domain perturbation. The thermocapillary force develops a high pressure near the comer of the free surface and the end wall and the fluid is pushed upward. The magnitude of these deformations are less than one in one thousand of the characteristic length scale. With this result, it was shown that the assumption of flat free surface was valid.

After investigating the thermocapillary flow in a cylindrical pool, the flow in a molten weld pool was considered. Here, the phase front that divides the solid and liquid metal was solved as a part of the solution. The cathode type of arc welding was considered. A parametric study of the relative importance of electromagnetic and thermocapillary forces on determining the shape of weld pools was carried out. The electromagnetic forces oppose ther­ mocapillary forces and can result in a two cell or a single cell flow pattern depending on which is stronger. At the intermediate values when the forces are of the same order, two cells result. The top cell draws the fluid from the center to the rim and recirculates this fluid in a layer close to the top free surface. The other cell, in the middle of the pool, moves the fluid near the phase front to the center of the pool and forces it down along the centerline. Weld pools of aluminum and steel were examined. For aluminum , the phase front was not significantly affected by the flow of the liquid metal. Thermocapillary flow here was quite similar to the one observed in the axi-symmetric pool with a vertical side wall. However, in steel the flow dramatically changed the weld pool shape. The electromagnetic forces aid in creating a deeper pool and thus a strong weld. If by addition of surfactants the coefficient of surface tension were altered to make the surface tension increase with temperature, then both the Lorentz and thermocapillary forces will aid each other in forming a deep weld. The depth to width ratio of the welds formed for various Reynolds numbers was tabulated. 117

Many welding processes involve thermocapiUary Reynolds numbers larger than those investigated in this dissertation. In experiments, the flow of liquid metal in weld pools has been observed to be erratic or unstable. It can be conjectured that the cause of the instability of these liquid pools is associated with the fast turning flows near the comers, where extremely sharp gradients exist. The stability of these flows is of interest to researchers who would like to control the weld shapes. The flow pattern and behavior of thermocapiUary flow in axi-symmetric pools are very similar to those observed in a partially penetrated weld pool with a non-vertical phase front. Admittedly, this is based on qualitative observation of the flow pattern and the behavior of non-dimensional velocities. But, in the case of a material like aluminum which has a low Prandtl number, the analysis of instability can be done in a cylindrical pool. The simplification in geometry would enable use of more accurate schemes at reasonable computational expense.

In future, a study of flow in the molten weld pool due to a moving heat source is planned. The resulting flow will be three-dimensional. The influence of heat source move­ ment on the the weld pool shape will be studied parametrically and the results wiU be valu­ able to welding engineers. The use of multi-grid methods with finite difference methods that use primitive variables wiU also be taken up to help improve the convergence rates of these three dimensional solutions.

i APPENDIX

FREE SURFACE DEFLECTION

In this appendix, the method for calculating the free surface deformation is presented. To calculate the free surface deflection, we follow Sen & Davis (1982) and Zebib et al. (1985). If the free surface deflection is g(r), for small values of capillary number, we can express the the location of the free surface z by regular perturbation in Ca as,

z = 1 + Ca g (r) . (A.1)

Letting a prime denote a derivative with respect to r, the kinematic condition is given by

- = ^j- = Ca g ' . (A.2) u dr

The normal and tangential stress balances can be stated as

Ca

da aiiuand t,i, = = 22- = f . ff . ft , (A.3) as

where,

ft = («i . «2> = ==■■;■ = (-Ca g \ 1) , Vl + C a2 g '2

118 119

( = ((,,r2) = - (l.-Ca?'), ■VI + Ca2 g '2

★ c= — = l + CaT. <*o

In the above equations, A is the normal vector, t is the tangential vector to the free surface, o0 is the mean value of surface tension and C the dimensionless curvature of the surface. The expression for curvature is derived later in this appendix. The stress tensor $*, alsb appears in these boundary conditions. The stress tensor and the pressure are non- dimensionalized by the quantity yiU /L , where U is a characteristic velocity and L is a charac­ teristic length. In non-dimensional quantities, the relevant components of the stress tensor are (Goldstein, 1965),

S r r = - p + 2 , (A.4)

dw S2Z = -p + 2 (A.5) dz ’

C _ C _ du dw (A.6) ‘ T + I T '

Substituting (A.4)-(A.6) into the normal stress balance and making use of the kinematic con­ dition, we obtain

C = - — 1^-— J i-C a p +. ----- 2Ca3 ■ „ g 'g K (A.7) l+ C a T [ y (1 + Ca g ) J

The term in the denominator appears because the surface tension is dependent on tempera­ ture. For small capillary numbers, Ca <1, we can write (A.7) as 120

C = -Ca p . (A.8)

A.1 CURVATURE OF THE FREE SURFACE

The curvature of the free surface can be calculated in a way similar to that described in Levich (1962). The procedure is based on minimization of total free energy of the system. Let there be two phases of an one-component system separated by a thin interface. The total free energy F per unit volume of the system is then defined as

F =F\ + F 2 + F 2' , (A.9)

where F, and F2 are the free energies of the two bulk phases and F1 is the free energy of the interface. Thermodynamic equilibrium in an isothermal system requires that the free energy be a minimum. This can be expressed as

8F = 0 , (A.10)

where 8F is an arbitrary and infinitesimal variation of the free energy of the system. Using (A.9),

8F =8F, + 8F2 + 8F i = 0 . (A. 11)

The change in free energy of the system at constant temperature is proportional to the changes in volume of the phases and the change in the area of the interface. The change in the free energy of the bulk phases is given by

8F, =-pi8V, , (A.12) 121

and 5F2 = - p 2^V2 , (A.13)

where p \ and p 2 are the absolute pressures and SV^and 8V2 are the changes in volumes of the two phases. The change in free energy of the interface at constant temperature is

8F1 =

where 82 is the change in the area of the interface. Since the total volume of the system is constant, an increase in the volume of one phase must cause an equal decrease in the volume of the other phase, which implies that

8V, = S v 2 . (A.15)

The total change in the free energy of the system can be written as

8F = —(pi-p2)8Vi + o82 = 0 . (A.16)

Now, 52 and 5V { can be obtained in the following manner. Recalling from (A.l) the eleva­ tion of the free surface, the differential surface area d 2 can be written as

d l = 2nr'ldr2+dz2 = 2nrdr^l+Ca2g'2 . (A.17)

For small displacements, (which occur for small capillary numbers) this can be expanded in Taylor series as

d'L=2itrdr(l+^-Ca2g'2+ • • • ) . (A.18) £ 122

Integrating (A. 18) the total surface area, I, is found to be

2 = 2 tc J r 1 +{caV2 dr (A.19)

The variation in surface area is then

82 = 2 nC a2 Jrg'-j-(8g) dr , (A.20) J dr

which can be integrated by parts to give

82= 2rcCa2 [rg'8g - J (g+rg")8g dr j (A.21)

The first term is evaluated at the end points which can be taken to be fixed, and 8 g 0. The variation in surface area is

82 = 2izCa2 j8g (g'+g )dr (A.22)

The volume can be shown to be

V =2njr(l+Cag)dr (A.23)

The variation in the volume is then,

8V = 2itC aj 8g r dr (A.24) 123

The change in free energy according to (A.16) is, %

SF = 2itCajdg |(Pi-P2)r + aCa(g'+rg”) jrfr = 0 . (A.25)

Since 5g is an arbitrary and infinitesimal quantity, it follows from the above equation that

t P l ~P i = ~<3Ca (&— + g ) (A.26)

The term multiplying the surface tension, a is the curvature, C. We can write

/ C (■*-+*")• (A.27) r

A.2 SURFACE DEFORMATION

The equations which govern the deformation of the surface can now be written down. From (A.8),

gn +^-= -p . (A.28)

The boundary conditions for this equation can be of two types (Davis, 1982). The contact point can be fixed which means that the location of the free surfaceis known; Or, the contact angle can be fixed at the end points of the free surface. The boundary conditions used here are of the fixed contact point type. At the centerline, symmetry conditions have to be satisfied. Therefore, 124

4 s- = 0 , r = 0 , (A.29) dr

g = 0 , r = 1 . (A.30)

The pressure on the surface can be computed by integrating the momentum equation on the free surface. However, the pressure can be calculated only to an additive constant. To deter­ mine this constant, we make use of the additional constraint that the volume of the fluid in the pool is constant. That is,

t |g (rW r= 0 . (A.31) LIST OF REFERENCES

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