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The Buckingham Pi theorem A dimensional functional relation among n physical variables 1 2 3 n V ,V ,V ,...,V , involved in some physical phenomenon, expressed in explicit form as ( ) 1 2 3 n V f V ,V ,...,V = , or in implicit form as ( ) 1 2 3 0 n V ,V ,V ,...,V φ = , may be reduced to an equivalent non-dimensional functional relationship among ( ) n r − independent non-dimensional parameters or non-dimensional groups of variables, 1 2 n r , ,..., π π π − , of the form ( ) 1 2 3 n r F , ,..., π π π π − = , or the equivalent implicit form ( ) 1 2 0 n r , ,..., Φ ππ π − = , where r is the rank of the dimensional matrix, 1 π includes the dependent variable 1 V and the remaining ( ) 1 n r − − π - terms include only the independent variables 2 3 n V ,V ,...,V . The amount, r, of the reduction in the number of variables is usually equal to the number of fundamental dimensions involved in all the physical variables. In some rare cases, r is less than the number of basic dimensions required to describe the physical variables.
Transcript of The Buckingham Pi theorem - ME14mech14.weebly.com/.../0/6/61069591/dimensional_analysis.pdfThe...
The Buckingham Pi theorem
A dimensional functional relation among n physical variables
1 2 3 nV ,V ,V ,...,V , involved in some physical phenomenon, expressed in explicit form as
( )1 2 3 nV f V ,V ,...,V= ,
or in implicit form as
( )1 2 3 0nV ,V ,V ,...,Vφ = ,
may be reduced to an equivalent non-dimensional functional relationship among ( )n r− independent non-dimensional parameters
or non-dimensional groups of variables, 1 2 n r, ,...,π π π − , of the form
( )1 2 3 n rF , ,...,π π π π −= ,
or the equivalent implicit form
( )1 2 0n r, ,...,Φ π π π − = ,
where r is the rank of the dimensional matrix, 1π includes the
dependent variable 1V and the remaining ( )1n r− − π - terms
include only the independent variables 2 3 nV ,V ,...,V .
The amount, r, of the reduction in the number of variables is usually equal to the number of fundamental dimensions involved in all the physical variables.
In some rare cases, r is less than the number of basic dimensions required to describe the physical variables.
Modeling and Similitude
An engineering model is a representation of a physical system that may be used to predict the behavior of the system in some desired respect.
Experimental tests conducted on a small scale model can be used to predict the behaviour of the prototype if
1) The model is geometrically similar to the prototype, and 2) The model is designed and operated under conditions such that
( ) ( )2 2mod el prototypeπ π= ,
( ) ( )3 3mod el prototypeπ π= ,
. . . ( ) ( )n r n rmod el prototypeπ π− −= ,
so that all the non-dimensional parameters, 1 2 n r, ,...,π π π − , for the model are equal to the corresponding non-dimensional parameters for the prototype, which ensures complete similarity (geometric, kinematic and dynamic similarity).
Incomplete Similarity-Distorted Models
It is not always possible to match all the non-dimensional parameters of the model and prototype
Models for which one or more of the similarity requirements are not satisfied are referred to as distorted models
Classical examples of distorted models- free-surface flows e.g.
• Flow over dams • Flow over weirs • Flows in open channels • Flows in canals • Flows in rivers • Flows in spillways • Flows in stilling basins • Flows in harbours • Flows around floating objects
Hydraulic structures
• Dam- barrier or wall that is built across a river in order to stop the water from flowing, used especially to make a reservoir (man-made lake for storing water) or to produce electricity
• Spillway- a passage for the extra water from a dam
• Stilling basin- structure for dissipating available energy of flow below a spillway, outlet works, chute or canal structure
• Weir- low wall or barrier built across a river in order to control the flow of water or change its direction
Non-dimensional parameters in free surface flows
Froude number: VFrgL
=
Reynolds number: VL VLRe ρμ ν
= =
Weber number: 2V LWe ργ
=
Where
γ = surface tension (N/m), μνρ
= = kinematic viscosity (m2/s),
ρ = density (kg/m3), μ= dynamic viscosity (Pa s = kgm-1s-1),
g = gravitational acceleration (m/s2),
L = length scale (m), V = fluid velocity (m/s)
Dependent non-dimensional parameters in free surface flows are functions of Fr, Re and We
E.g.
( )2 2
F Fr,Re,WeV L
φρ
=
Froude number similarity
( ) ( )mod el prototypeFr Fr=
⇒ pm
m m p p
VVg L g L
=
Usually, m pg g g= =
Thus, m m
p p
V LV L
α= = where m
p
LL
α =
Reynolds number similarity
( ) ( )mod el prototypeRe Re=
⇒ p pm m
m p
V LV Lν ν
=
Thus, 3 2/m m m
p p p
V LV L
ν αα αν
= = =
Weber number similarity
( ) ( )mod el prototypeW e W e=
⇒ 22
p p pm m m
m p
V LV L ρργ γ
=
⇒ ( ) ( )
22p pm m
m p
V LV L/ /γ ρ γ ρ
=
Thus, ( )( ) ( )
2 2 22
m m m
p pp
/ V L/ V L
γ ρα α α
γ ρ= = =
Complete similarity
• Requires matching of Fr, Re and We
( ) ( )mod el prototypeFr Fr=
( ) ( )mod el prototypeRe Re=
( ) ( )mod el prototypeW e W e=
• Occurs if
(1) Velocity ratio satisfies
m
p
VV
α=
(2) Ratio of kinematic viscosities satisfies
3 2/m
p
ν αν
=
(3) Ratio of kinematic surface tensions satisfies
( )( )
2m
p
//
γ ρα
γ ρ=
Incomplete similarity
• May not be possible to find a fluid satisfying conditions (2) and (3) e.g. if 1 100/α = (typical of ship models),
11000
m
p
νν
=
• Most important non-dimensional parameter in free surface flows is the Froude number (Fr)
• Must match the Froude number of model and prototype in to predict the behavior of flows over hydraulic structures such as dams, spillways, stilling basins, weirs etc.
