Yamaguchi H. Engineering Fluid Mechanics (Fluid Mechanics and Its Applications)

Appendix C



0

321

n

qqqqf ,...,,,

(C.1-5)

we may choose

m

repeating variables, say if

3 m

, select three

variables to meet the criteria, say

2

q

,

4

q

and

5

q

among

n

vari-

ables.

Step (2) Write

S

-parameters of

mn ಣ

SSS

,...,,

21

in the power form for the

repeating variables, for example with

2

q

,

4

q

and

5

q

in Step (1)

with each of the remaining variables as

15421

qqqq

cba

S

35422

qqqq

cba

S

65423

qqqq

cba

S

:

n

c

b

a

mn

qqqq

5

42



S

Step (3) Apply the dimensional analysis to obtain the power constants for

each

S

-parameter subjecting that the

S

-parameters are all di-

mensionless.

Step (4) Write the functional form using

S

-parameters to describe the

physical phenomenon of the fluid system.

Step (5) In correlating experimental results, one dependent

S

-parameter

(say

1

S

) can be expressed by a function likewise





mn

mnmn

cccs





DD

D

SSSS

,

3

2

33221

(C.1-6)

where

mn

ccc



,

32

and

mn

D

,,

32

are constants determined

from the results of experiments. Note that if some dimensionless

variable, such as the length ratio

21

ll

, the roughness 

H

etc, are

contained in the primary variables



n

qqq ,,

21

, they are them-

selves treated as

S

-parameters and to be excluded from the pro-

cedure by simply adding as

S

-parameters in the resultant func-

tional form, for example



0

2121

llqqqf

n

,,,,

T



(C.1-7)



0

211



lls

mn

,,

H

SS



(C.1-8)

,...

,... ...

555

Appendix

Table C.1 Dimensions of quantities frequently used in fluid flow problems

Quantity Symbol Dimensions

Length

l

L

Time

t

T

Mass

m

M

Area

A

2

L

Volume

V

3

L

Force

F

2

TML

Velocity

U

TL

Acceleration

a

2

TL

Angular frequency

Z

T1

Gravity

g

2

TL

Flow rate

Q

TL

3

Mass flux

m



TM

Pressure

p

2

LTM

Stress

T

2

LTM

Work

W

22

TML

Power, heat flux

qP ,

32

TML

Density

U

3

LM

Specific weight

J

22

TLM

Viscosity

0

K

LTM

Kinematic viscosity

X

TL

2

Surface tension

V

2

TM

Bulk modulus

K

2

LTM

(*)Basic Dimensions; L (length),

M

(Mass), and

T

(Time), i.e. 3 m

Appendix C-2 Example of -Analysis

In order to illustrate

S

-theorem, suppose that force

F

acting in a fluid

system is supposed to be dependent on the velocity

U

, density

U

, grav-

ity

g , viscosity

0

K

, surface tension

V

, angular frequency (velocity)

Z

, bulk

modulus

.

, surface roughness

H

, characteristic length l and another repre-

sentative linear dimension

1

l . For this fluid system a physical phenomena

would be described with nondimensional numbers by applying

S

-theorem

as demonstrated below.

S

556

Appendix C

(i) Functional form

0,,,,,,,,,,

1

0

¸

¹

·

¨

©

§

HZ.VKU

l

UlFf

g

(C.2-1)

We have eleven variables, i.e.

11 n

, which contain three basic dimen-

sions

L, M and T, i.e. 3 m . According to

S

- theorem, we can find eight



8311   mn

S

-parameters, so that we have



0,,,,,,

87654321

S

g

(C.2-2)

(ii) Choice of repeating variables

Choose three repeating variables such as

l

,

U

and

U

, and set ll

17

S

and

H

S

8

, since they are two already dimensionless parameters. Thus we

will find

1

S

to

6

S

, i.e. six

S

- parameters.

(iii) Conduct dimensional analysis

FUl

cba

US

1

(C.2-3)

>@



122

2013010

231

1

000

   ?

  



cba

bcbac

MLTMLLTLTLM

cb

a

,,

S

(C.2-4)



02

22

1

KUS

US

cba

Ul

luF

(C.2-5)

>@



111

1013010

1131

2

000

   ?

  



cba

bcbac

TMLMLLTLTLM

cb

a

,,

S

(C.2-6)

Ul

U

K

S

0

2

(C.2-7)

Similarly for

3

S

~

6

S

, we can obtain

2

3

U

l

g

S

,

glU

2

4

V

S

,

U

S

2

5

U

K

and

U

l

Z

S

6

(C.2-8)

557

Appendix

(iv) Set functional form

Each of the

S

- parameters for

2

S

~

6

S

in this expression is a com-

mon nondimensional number, which is derived from similitude (Section

6.2), so that we have

0,,,

1

,

1

,

1

,

1

2222

¸

¹

·

¨

©

§

H

U

l

tS

MFr

Re

lU

F

f

(C.2-9)

and for the dependent variable

F

, the functional form is given where

¸

¹

·

¨

©

§

H

U

,,,,,

21

1

22

l

StMFrRef

lU

F

c

f

(C.2-10)

Appendix D

Appendix D-1 Invariant of Second Order Tensor

The invariants of a tensor are scalar quantities, which remain unchanged

for the coordinate transformation of rotation. There are three principal in-

variants for second order tensors.

Consider a second order tensor

T , whose components are

ij

a for unit

When

O

is a scalar and is an eigenvalue of T , we have the following rela-

tionship

 

O\OGOGO\

AijijijijB

ba ᧩᧩

(D.1-1)

where

A

\

and

B

\

are the characteristic equations of

ij

a and

ij

b . Eq. (D.1-

1) shows that the characteristic equations are equal to both frames

(

ji

ee

ˆˆ

and

ji

ee

cc

ˆˆ

), so that they are unaffected by the coordinate transforma-

tion. They are also given when

  

32

2

13

III

BA

᧩᧩

OOOO\O\O\



(D.1-2)

where

1

I

,

2

I

and

3

I

are principal invariants, which are also unaffected by

the coordinate transformation. They are respectively defined as

3322111

aaaI 

(D.1-3)

dyads

ji

ee

ˆˆ

, and

ij

b for unit dyads

ji

ee

cc

ˆˆ

after the coordinate transformation.

558

Appendix D

2221

1211

1113

3133

3332

2322

2

aa

I



(D.1-4)

333231

232221

131211

3

aaa

I

(D.1-5)

and alternating,

1

I ,

2

I and

3

I are written where

AtI

r

1

,



2

1

AtAtI

rr

᧩ and

AI det

3

(D.1-6)

where



22

AtAt

rr

and



AAtAt

rr

2

. Another set of invariants is defined

by the so-called moment, as follows

k

r

k

AtI

(D.1-7)

With the moment, given by Eq. (D.1-7), the invariants are representa-

tively given for

k

=1,2 and 3 as

11

IAtI

r

(D.1-8)

2

1

2

2IIAtI

r

᧩

(D.1-9)



21

3

13

3

626

2

1

IIIIAtI

r

᧩

(D.1-10)

It is useful to note that there are relationships between two principal in-

variants

k

I and the moments

k

I as follows

11

II

(D.1-11)



2

1

2

1

III 

(D.1-12)

321

3

1

3

1

2

1

6

1

IIIII  ᧩

(D.1-13)

559

Appendix

For a symmetric tensor T , eigenvalues of T are real and T is diago-

nalizable. For example, if eigenvalues

21

OO

,

and

3

O

are of a symmetric ten-

sor

T , T can be transformed to a diagonal matrix

3

2

1

00

Ȝ

c

T

(D.1-14)

Then we have invariants

3211

OOO

 I

(D.1-15)

1332212

O

 I

(D.1-16)

and

3213

O

I

(D.1-17)

560

Index

561

A

Acceleration

convective, 8

Corioli’s, 18

local, 11, 12

reference frame, 84–87

Accelerometer, 111, 112, 524

Acoustic velocity, 225

Action angle, 98, 100

Actuator-disk theory, 142

Adiabatic process, 228, 233, 241,

258–264

Adverse pressure gradient, 350–352

Airfoils, aerofoils, 154, 188–194

Alternating unit tensor (polymeric

alternator), 14, 49

Ampere’s (circulation) law, 533

Analytic function, 118, 159

Angular

momentum, 47–52, 127, 128, 183–188,

507, 509–513

momentum conservation, 47–52, 128,

507, 512, 513

velocity, 16, 18, 74

Anisotropy of particles, 502

Anti-symmetric, see Skew-symmetric

tensor (anti-symmetric tensor)

Apparent

viscosity, 400, 502, 516–519

wall shear rate, 443

Archimedes’ principle, 82, 524

Atmospheric pressure, 76, 93–96, 110

Attack angle, 155

Axial flow pumps, 183, 189–194, 199

Axisymmetric flows, 312, 323, 328, 330,

410–413

B

Barotropic, 152–154, 158

Base liquid

See also Carrier liquid

Bearing

journal, 310

slipper-pad, 306–310

thrust, 309

with magnetostatic lubricant, 524

Bell-mouth, 144

Benard

cell, 527, 531

convection, 297, 527–532

Bend, 140–149

Bernoulli equation, 122–125, 521, 522

ferrohydrodynamic, 521

generalized, 521, 522

surface, 122, 123

Bessel’s

differential equation, 332

functions, 332, 333

Biaxial stretching flow, 422

Bingham plastic (fluid), 403

Blasius

equation, 345, 346

first theorem, 159

second theorem, 159, 179

solution, 345

Blunt body, 254, 255

Body

couple, 49, 51, 52

force, 26, 45, 74, 525

Boltzmann

constant, 282, 500

superposition principle, 425

Borda’s mouth piece, 271, 272

562 Index

Boundary conditions

dynamic, 534

free surface, 85–87

inviscid flow, 115–129

for a magnetic field, 532–535

no-slip, 177, 336, 341, 472, 474

for temperature, 292, 353–355

for velocity, 301, 302, 324, 331, 335,

337, 375, 411

Boundary layer equations, 343, 344

Boundary layer flow, 177, 178, 340–364

Boundary layer separation, 350–353

Boundary layers on a flat plate, 177, 178,

340–350, 475–477

Boundary layer thickness, 341, 346,

360, 476

Bounding vortex, 155

Boussinesq approximation, 295–297

Brinkman number, 294

Broad-crested weir, 137–140, 175, 176

Brownian

motion, 498, 511

relaxation time, 502, 504–506, 515

rotational diffusion, 501

Buckingham theorem, 196

Buffer layer, 372

Bulk modulus, 228

Bulk viscosity coefficient, 280

Buoyant force, 80, 82, 100–105,

312, 524

C

Camber line, 154

Capillary viscometer, 437

Carrier liquid, 498, 502

Cartesian coordinates, 11, 23, 82

Cauchy

equation of motion, 46, 50, 52, 73, 407,

410, 413, 431, 438, 473, 507

strain tensor, 23, 464

theorem (stress), 26–30, 37, 38

Cauchy-Riemann conditions, 118

Cavitation

critical, 206, 207, 217, 219

number, 206

in turbo-machines, 203–207

Cayley-Hamilton theorem, 464

Center

of buoyancy, 101

of pressure, 79, 96–100

Centrifugal force, 85

Centrifugal pump, 86, 211–213, 218, 219

Centripetal acceleration, 18, 74

Centroid, 78–80, 98

Characteristic equation, 37

Characteristic length, 290, 291, 350,

369, 488

Characteristic scale of velocity, 368, 488

Choked flow, 238–241, 262, 266

Chord

length, 155

line, 155

Circulation, 119, 155

Circumferential velocity, 129

Coefficients

contraction, 134

correction, 132

pyromagnetic, 501

of surface tension, 87, 534–537

of thermal expansion, 60, 292, 529

Colloidal suspension, 497

Complex

modulus, 424–427, 440

potential, 118

velocity, 118

viscosity, 426

Compressibility, 131, 135, 225

Compressibility factor, 132, 255

Compressible flow, 131, 225

Compression, 249

Concentric cylinders (spheres), 299–306,

412, 414

Conduction (Heat), 61, 527

Cone and plate rheometer, 428–430

Configuration parameter (of turbulent

flow), 374

Conformal mapping, 119, 160

Conservation

of (linear) momentum, 44–47,

63, 64, 507

of angular momentum, 47–52, 507,

510–513

of energy, 52–56, 508

of mass, 43, 44, 62, 63, 507

Constant of anisotropy, 502

Index 563

Constitutive equations, 23, 280, 405, 408,

508, 509

Contact angle, 104

Continuity equation, 44, 62, 64, 122, 133,

234, 421, 507

Continuum mechanics, 5, 6, 43, 507

Contraction coefficient, 134, 144

Contravariant, 21

Control volume, 26, 35, 36, 126, 129, 183,

189, 196

Convected generalized Maxwell

model, 463

Convective

acceleration (term), 8

derivative, 19–22, 478

Converging

channel, 436, 437

nozzle, 238, 241

Converging-diverging nozzle,

273–274

Corioli’s acceleration (force), 18, 125

Corotational

derivative, 21, 482

Jeffrey’s equation, 460

Correction factor (constant), 134,

143, 144

Correlation length, 6

Correlations relating efficiency to size,

202, 203

Couette flow, 301, 302, 428,

437–439

Couple stress tensor, 51, 507, 512

Covariant, 21

Cox-Merz rule, 427

Creep, 452

Creeping flow, 323

Critical

area, 240

cavitation number, 206

pressure, 239–241

Rayleigh number, 297, 298, 531

Reynolds number, 366, 390

state, 234, 236

strain rate, 487

velocity, 249

volume fraction, 503

Curtiss-Bird constitutive equation, 465

Curvature, 87, 344

Curve

flow, 399, 400

of magnetization approximation,

501, 502

surface, 344, 350

Cylindrical gap flow, 302, 303, 415, 416,

437–439

D

D’Alembert

paradox, 160

principle, 85

Dampers, 310, 524

Damping factor (of turbulent flow), 374

Darcy friction factor, 302, 326, 412

Darcy-Weisbach equation, 302, 326

Deborah number, 489

Debye relaxation equation, 514

Deflection angle, 168, 171,

252, 253

Deflectors

moving, 171, 172

stationary, 170

Delta function (Dirac), 453

Density, 5, 6

Derivative

substantial, 7, 8

time (material derivative), 25

Developed flow, 320–322

Deviatoric stress (tensor), 30, 47, 419

Diffuser efficiency, 144, 145

Diffusive transport, 51

Diffusivity (diffusion) coefficient, 285,

537, 538

Dilatant fluid, 403

Dilatation, 25

Dilatational viscosity, 281

Dimensional analysis

homogeneity, 294

turbomachinery, 195–203

Dipole moment, 504

Discharge, 137, 196, 215

Discharge coefficient, 135, 139, 140

Displacement

function, 480, 481

gradient tensor, 10, 22–23

thickness, 348–350

564 Index

Dissipation

energy, 369, 439, 440

function, 293

scale, 369

Divergence theorem, see Gauss’s

divergence (integral) theorem

Diverging nozzle, 273, 274

Doublet, 119

Draft tube, 204, 214

Drag

on an airfoil, 155, 178

on a cylinder, 317

on a sphere, 312–318

Drag-lift ratio, 309

Dyad, Dyadic, 11, 28

Dynamic

head loss, 205

pressure, 130

similarity, 288–290

viscosity, 425

E

Eckert number, 293

Eddington symbol (notation), 15

Eddy

diffusivity, 378, 390

viscosity, 366, 369, 378, 382

Effective Prandtl number, 378

Effects

magnetocaloric, 497, 514

magnetoviscous, 498, 503

Efficiency

hydraulic, 194, 195, 210, 219

mechanical, 195, 209, 219

optimum, 201

overall (total), 194, 219

pump, 194–203

volumetric, 195, 209

Einstein’s formula, 503, 537

Elongational

flow, 11, 421–424, 428

rheometer, 433–437

Energy

cascade, 369

conservation, 52–56

equation, 56, 264

free, 464

internal, 57

Ensemble average, 365

Enthalpy, 59, 60, 233, 241–246, 256, 354

Enthalpy thickness, 355

Entrance flow

laminar flow, 320–322

length, 320–322

turbulent flow, 320–322

Entropy, 58, 241–246, 265, 266

Equations

of angular momentum, 47–52

Bernoulli, 124, 138, 142, 146

of continuity, 44

of energy, 52–56

heat transfer, 61

of linear momentum, 44–47

of motion, 45

Rayleigh-Plesset, 149–152

Rosensweig, 516–517

of state, 56, 57, 227, 260

Equilibrium, 57, 73, 84–87, 227, 297, 369,

497, 498, 504, 525

Equivalent thickness, 350

Ergodic hypothesis, 365

Euler

differential equation, 314

equation, 115, 124, 283

head, 185

number, 289, 290

relation, 13, 31, 32

turbine and pump, 129, 183

Eulerian (time) derivative, 8

Eulerian description (specification), 7, 26

Expansion waves, 237, 238

Extensional

flow, 11, 12, 422, 423, 433

strain, 11

viscosity, 423

Yamaguchi H. Engineering Fluid Mechanics (Fluid Mechanics and Its Applications)

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