058:0160
Jianming Yang
Chapter 5
10
Fall 2012
3 Common Dimensionless Parameters for Fluid Flow Problems
Most common physical quantities of importance in fluid flows are: (without heat transfer)
1
2
3
4
5
6
7
8
V,
,
g,
,
,
K,
p,
L
velocity density gravity viscosity surface tension compressibility pressure change length
n=8
m=3
 5 dimensionless parameters
VL inertia forces V 2 / L
1) Reynolds number =


viscous forces V / L2
R
e
Rcrit distinguishes among flow regions: laminar or turbulent value varies depending
upon flow situation
V
inertia forces

2) Froude number =
gL gravity force
V 2 / L

Fr
important parameter in free-surface flows
V 2 L
inertia force
V 2 / L
3) Weber number =


surface tension force  / L2 We
important parameter at gas-liquid or liquid-liquid interfaces and when these surfaces
are in contact with a boundary
058:0160
Jianming Yang
Chapter 5
11
Fall 2012
4) Mach number =
V
V
inertia force
 
compressib ility force
k / a
Ma
where a is the speed of sound in fluid
Paramount importance in high speed flow (V > c)
5) Pressure Coefficient =
p
V 2

pressure force p / L
inertia force V 2 / L
Cp
(Euler Number)
4 Nondimensionalization of the Basic Equation
It is very useful and instructive to nondimensionalize the basic equations and boundary
conditions. Consider the situation for  and  constant and for flow with a free surface
Continuity:
V  0
Momentum:

DV
  p  gz    2V
Dt
058:0160
Jianming Yang
Chapter 5
12
Fall 2012
Boundary Conditions:
1) fixed solid surface: V  0
2) inlet or outlet: V = Vo p = po
3) free surface:
w

(z = )
t
p  pa   Rx1  Ry1 
All variables are now nondimensionalized in terms of  and
U = reference velocity
L = reference length
V* 
V
U
t* 
tU
L
x* 
x
L
p* 
p  gz
U 2
All equations can be put in nondimensional form by making the substitution
V  V* U

 t * U 


t t * t L t *
058:0160
Jianming Yang

Chapter 5
13
Fall 2012



î  ĵ  k̂
x y z
 x *
 y*
 z *
 *
î  *
ĵ  *
k̂
x x
y y
z z
1
 *
L
u 1 
U u *
*
and
etc.

Uu 
x L x *
L x *
 
Result: *  V*  0
DV *

 *p* 
*2 V*
Dt
VL
1)
2)
3)
V*  0
V
V*  o
U
*
w  *
t
*
Re-1
p* 
po
V 2
p* 
po
gL *

*1
*1



z

R

R
x
y
2
2
2
V U
V L
pressure coefficient
Fr-2
We-1
058:0160
Jianming Yang
Fall 2012
Chapter 5
14