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Table 4-1.Nondimensional Scaling Parameters.
Notation: c = speed of sound; D = characteristic width; g = acceleration due to gravity;
L = characteristic length; p = fluid static pressure (absolute); pa= ambient pressure;
S = span length; T = temperature; U = flow velocity; z = vertical distance;
a = thermal diffusivity; E = roughness height (frame I ) , average power dissipated per
unit mass of fluid (frame 11); v = kinematic viscosity; p = fluid density; a = surface tension.
Consistent sets of units are given in Table 3-1.
P a r a m e t e r Name
1.
Boundary
Geometry
I
1
Definition
I
L
Slenderness = D
Roughness
=
I
Physical Significance
Application
S c a l e boundary geometry
'
D
S
Aspect r a t i o = D
(and o t h e r s )
I
2.
t
Euler
Number
3.
Reynolds
Number
4.
Mach
Number
I
Pressure force
Inertia force
-
I
G e n e r a l f l u i d dynamic
analysis
Inertia force
Viscous Force
General viscous f l u i d
dynamic a n a l y s i s
Fluid velocity
Speed of s o u n d
Employed i n f l o w s w i t h
M 10.3. w h e r e t h e Mach
number i s a m e a s u r e o f
t h e tendency of flow t o
compress a t a boundary.
1
U
M = c
Example:
5.
Froude
Number
Inertia force
Gravity force
U
1/ L
!I
I
-T
Free surface fluid
dynamics
~ x a m ~ l e o:c e a n w a v e s
Inertia force
Surface tension
Weber
Number
F l u i d dynamics of s m a l l
f r e e s u r f a c e flows
Example:
I'
7.
Cavitation
Number
Vapor p r e s s u r e - s t a t i c p r e s s u r e
F l u i d dynamic p r e s s u r e
c a p i l l a r y waves
I
Liquid flows i n which
s t a t i c pressure i n fluid
may F a l l b e l o w v a p o r
p r e s s u r e of t h e l i q u i d .
Example:
Richardson
Number
transonic
aircraft
l i q u i d pumps
I
1
Buoyancy e n e r g y
Turbulent energy
Ri =
S t a b i l i t y of f l u i d f i e l d
with density s t r a t i f i cation
Example:
thermal o r
salinity-induced density
s t r a t i f i c a t i o n i n oceans
9.
Rayleigh
Number
R3
=
Freely convective flows
i m p e l l e d by t h e r m a l
gradients
l
i
I ~ a a m p l e : thermal plumes
10.
Strouhal
Number
1- I)
Fluid convection i n time l / f
C h a r a c t e r i s t i c width
1
Fluid flows with periodic
oscillations
1
vortex shedding
From a c i r c u l a r c y l i n d e r
1 Example:
1I.
Kolmogoroff
Scales
L e n g t h s c a l e : eddy s i z e a t
which v i s c o u s a n d i n e r t i a
forces a r e comparable
Time s c a l e :
characteristic
frequency f o r deformation of
these eddies
F i n e s t r u c t u r e s of
turbulent flows
Example:
scaling
turbulent spectra i n the
high-frequency range
TABLE7.1 Flow characteristics and similitude scale ratios (ratio of prototype
quantity to model quantity)
Scale ratios for laws of
Characteristic
Dimension
Reynolds
Froude
Mach
(L3~>r
(L3p>,
(L3pg),
(L2Eu)r
Geometric
Length
Area
Volume
Kinematic
Time
Velocity
Acceleration
Discharge
Dynamic
Mass
M
Force
MLT-*
(L3p>,
Pressure
Impulse and
momentum
MLT-'
Energy and work
ML~T-~
Power
ML2T-'
Note: Usuallv
P
(L2pr
is the same in model and DrototvDe.
( L ~ ' ~ ~ ~ (" ~ ) ~
(L4pg)r
(L3Ev),
( ~ ~ ~ ~ ~ g " ~ )