order of magnitude scaling of complex engineering problems patricio f. mendez thomas w. eagar may 14...

Order of Magnitude Scaling of Complex Engineering Problems

Patricio F. Mendez

Thomas W. EagarMay 14th, 1999


2

MOTIVATION

• There are some engineering problems for which:– measurements are difficult

– numerical treatment is difficult

– idealizations and lumped parameter models are not reliable

– dimensional analysis cannot simplify the problem significantly

– there is previous insight into the problem

– order of magnitude solutions are acceptable


3

OBJECTIVES

• To determine the best combination in a problem involving many dimensionless groups.

• These dimensionless groups should provide– an estimation of the unknowns

– a description of the relative importance of the phenomena involved


4

OUTLINE

• Order of magnitude scaling. Basic concept:– Normalization

– Functional requirements

– Domain partition

– Transformation of differentialequations into algebraic

– Matrix algebra

• Related techniques

• Results

• Discussion

• Conclusion


5

ORDER OF MAGNITUDE SCALING:BASIC CONCEPT

dimensionalanalysis

asymptoticconsiderations

•Statement of a problem in dimensionless form•Reduced number of arguments

•Relative importance of terms in equations

• Normalization scheme• Functional requirements• Domain partition• Transformation of differential equations into algebraic• Matrix algebra


6

RESULTS

• Order of magnitude estimations are obtained.• These estimations allow one to estimate the

relative importance of the different driving forces• The estimations are related to the governing

parameters through power laws• The functional dependence on the governing

dimensionless groups are of less importance that in dimensional analysis.


7

DISCUSSION

• Non-linear equations– Navier-Stokes

– Heat transfer

• Singular limit problem.

• Differential equations of order higher than second.

• Vector operators.

• Analysis of stability, such as capillary instabilities, buckling, etc.

included excluded


8

CONCLUSION

• Previous insight can be used to transform a complex set of differential equations into a more manageable set of algebraic considerations.

• The results obtained are approximate.

• The physical insight gained can be used to choose representative asymptotic cases.


9

• Governing equations, boundary conditions and domain for scaling

L

X

Y

U

VISCOUS BOUNDARY LAYER


10


Continuity:

Navier-Stokes:

Boundary Conditions:

U

X

V

Y 0

UU

XVU

Y

P

X

U

X

U

Y

1 2

2

2

2

UV

XVV

Y

P

Y

V

X

V

Y

1 2

2

2

2


11


• Governing parameters and reference units– set of governing parameters:

– set of reference units

– set of reference parameters

• Just one governing dimensionless group


12


• Scaling Relationships

– Independent arguments:

length of domain (known)

width of domain (unknown)


13



– Parallel velocity:UYXU ),(max

0),( min YXU

),(),( yxuUYXU


14



– Transverse velocity:CVYXV ),(max

0),( min YXV

),(ˆ),( yxvVYXVC

unknown characteristic value

estimated characteristic value


15



– Pressure:CPYXP ),(max

0),( min YXP

),(ˆ),( yxpPYXPC

unknown characteristic value

estimated characteristic value


16


• Set of estimations:

• Three dimensionless groups are added. Since they are redundant they can be assigned arbitrary values.


17


• Dimensionless governing equations and boundary conditions

• Dimensionless groups of known order of magnitude

u x( , ) 1

0ˆ

ˆ

y

v

U

VL

x

u c

2

2

2

2

2

2

22 y

u

x

u

LU

L

x

p

U

P

y

uv

U

LV

x

uu cc

2

2

2

2

2

22

y

v

x

v

LP

V

y

p

y

vv

P

V

x

vu

LP

VU

c

c

c

c

c

c

N1=1

N3=1 N4=1

Boundary Conditions:

Continuity:

Navier-Stokes:

all others = 0

N2 N5

N6 N7 N8


18


• Set of governing dimensionless groups– only one group: Reynolds number

LURe


19


• Calculation of the estimations (matrix algebra)

Dimensionless groups of known

order of magnitude

Dimensionless coefficients

Governing dimensionless group

Governing parameters Estimations

Matrix [A]

[A11] [A12]


20


• Calculation of the estimations (matrix algebra)

Estimations

Governing parameters

Matrix [AS]= -[A12]-1[A11]

unknown function 1


21


• Dimensionless governing equations– Matrix algebra is of help here too

– All terms are of the order of one when for large Re


22


• Comparison with known solution:


23

HIGH PRODUCTIVITY ARC WELDING

Low current:• recirculating flows.• small surface depression.• experimental, numerical and analytical studies.

High productivity• no recirculating flows.• large surface depression.• gouging region.• only experimental studies and simple analysis.


24

CHALLENGES

• Direct observations are very difficult.

• Not all of the necessary physics is known.

• The equations cannot be solved in closed form.

• Numerical solutions are difficult.

• Application of dimensional analysis is limited.


25

FEATURES INCLUDED

• Deformed free surface• Gas shear on the surface• Arc pressure• Electromagnetic forces• Hydrostatic forces• Capillary forces• Marangoni forces• Buoyancy forces


26

RESULTS

thickness < 100 mm

The gouging zoneis a a very thin layer of

molten metal

Penetration measured for two significantly

different levels of sulfur is the same.

Defect mechanism is different


27

RELATIVE IMPORTANCE OF DRIVING FORCES1.

00

0.34

0.08

0.07

0.06

0.03

0.03

0.03

7.E

-05

3.E

-04

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 N2 N15 N27 N7 N6 N26 N24 N8 N5

arc

pres

sure

/ v

isco

us

elec

trom

agne

tic /

vis

cous

hydr

osta

tic /

vis

cous

capi

llary

/ v

isco

us

Mar

ango

ni /

gas

sh

ear

buo

yan

cy /

vis

cous

gas

shea

r /

visc

ous

conv

ectio

n /

cond

uctio

n

iner

tial /

vis

cous

diff

.=/d

iff

• Driving forces• Effects

For the first time gas shear

is shown to dominate the

flow

It was generally assumed that

electromagnetic or arc pressure would dominate


28

0.00001

0.0001

0.001

0.01

0.1

1

10

30 ipm160 A

45 ipm240 A

67 ipm360 A

100 ipm525 A

10 ipm110 A

15 ipm160 A

22 ipm240 A

33 ipm360 A

50 ipm525 A

7.5 ipm160 A

11 ipm240 A

17 ipm360 A

17 ipm360 A

25 ipm525 A

8.4 ipm360 A

12 ipm525 A

inertia

diff x/diff z

electromagnetic

convection

arc pressure

Arc pressure increases by an order of

magnitude with productivity

SENSITIVITY OF DRIVING FORCES

productivityproductivity productivity


29

•With low S the curvature is

smaller.•Surface tension is higher.•Contact angle is wetting

•With low S the curvature is larger.•Surface tension is lower.•Contact angle is less wetting

Lower sulfur increases speed limit

•20 % faster weld with same

linear heat input•10 A/ipm•27.4 to 33.4 ipm

EFFECT OF SULFUR

order of magnitude scaling of complex engineering problems patricio f. mendez thomas w. eagar may 14...

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