1. Common Dimensionless Groups in Fluid Mechanics
Variables
Dimension M L T
Characteristic length l 1
Characteristic velocity V
LT1 1
Density 
ML3 3
Viscosity 
ML1T1
Surface tension 
MT2 2
Speed of sound c
Pressure p or p
ML1T2
Acceleration of gravity g
LT2
Bulk Modulus Ev
Frequency 
T1

2. Dimensionless Parameters
Definition
Physical interpretation
Type of applications
Reynolds number
Inertial force
viscous force
All type of fluid dynamics problems
Froude number
Inertial force gravitational force
Free surface flows (waves)
Mach number
Inertial force compressibility force
Compressible flow
Cauchy number
Weber number
Inertial force surface tension force
Free surface flow (two phase)

3. Dimensionless Parameters
Definition
Physical interpretation
Type of applications
Cavitationnumber
Pressure (vapor) inertial force
Cavitation
Euler number
pressure force inertial force
Pressure coefficient
Drag/Lift coefficents
Drag/lift force dynamic force
Aero/hydrodynamics
Strouhal number
Inertial (local) force inertial (convective)
Oscillatory flow (shedding)
Roughness ratio
Wall roughness body length
Turbulent flow, rough surface

4. Example – Dimensional Analysis
A jet of liquid is directed vertically upward in the atmosphere. List the variables that determines the maximum height to which the drops of liquid rise. Establish the dimensionless parameters for the problem. g h V D

5. negligible relative fluid motion (velocity gradient)
Dimensional Analysis
The maximum height h depends on
Geometry D: diameter of the jet
Fluid property a : density of air
a : Viscosity of air
w : density of water (but not w)
w : surface tension of droplets
External effects g : gravity
Others V: velocity (or Q = wVA)
negligible relative fluid motion (velocity gradient)

6. Dimensionless PI terms
h = f ( D, w , a , a ,  , g , V )
k = 8, r = 3, k – 3 = 5 dimensionless parameters
Select 3 repeating variables (w , D, V )

7. Dimensionless PI terms
(Reynolds number for air)
(Froude number for water) velocity of water
(Weber number for water)

8. Fisherman’s Wharf
San Francisco Bay
1:75 scale model

9. Fisherman’s Wharf, San Francisco Bay

10. Froude number similarity
Hydraulic Model
Froude number similarity
Velocity scale
Time scale
Volume flow rate
Forces
Scale effects

11. Froude Number Similarity
HC Chen
1/18/2020
Froude Number Similarity
Parameter
Prototype
Model
Scale
Bay shoreline length
6,400 ft
85.33 ft
1/75
Proposed Solid Breakwater
1,000 ft
13.33 ft
Offshore water depth
60 ft
0.8 ft
Typical Wave Height
2.0–3.3 ft
0.32–0.53 in
Total Area
1.1 mile2
~ 6000 ft2
(1/75)2
Wave Period
34.5–228 sec
3.98–26.33 sec
(1/75)1/2
Simulation Time
24 hr
1 month
2.77 hr
3.5 days
Flow Rate Qp Qm
(1/75)5/2
Wave Force Fp Fm
(1/75)3
Reynolds Number
(Re)p
(Re)m
(1/75)3/2
Chapter 7 Examples