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Similitude and Dimensional Analysis I
Hydromechanics VVR090
Motivation
Often difficult to solve fluid flow problems by analytical or numerical methods. Also, data are required for validation.
The need for experiments
Difficult to do experiment at the true size (prototype), so they are typically carried out at another scale (model).
Develop rules for design of experiments and interpretation of measurement results.
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Fields of Application
• aerodynamics
• naval architecture
• flow machinery (pumps, turbines)
• hydraulic structures
• rivers, estuaries
• sediment transport
To solve practical problems, derive general relationships, obtain data for comparison with mathematical models.
Example of Model Experiments
Wind-tunnel Towing tank
Spillway design Sediment transport facility
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Model Experiments in Hydraulics I
Construction of model
Model Experiments in Hydraulics II
Model of dam with spillways
Discharge at model spillway
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Traryd hydropower station
Model Experiments in Hydraulics III
Model gates
Model Experiments in Hydraulics IV
Hydraulic arrangements
pump intake
surge chamber
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Model Experiments in Hydraulics V
Water power plantsLilla Edet
Lule Älv
Lule Älv
Terminology
Similitude: how to carry out model tests and how to transfer model results to protype (laws of similarity)
Dimensional analysis: how to describe physical relationships in an efficient, general way so that the extent of necessary experiments is minimized (Buckingham’s P-theorem)
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Example: Drag Force on an Submerged Body
Drag force (D) depends on:
• diameter (d)
• velocity (V)
• viscosity (m)
• density (ρ)
( ), , ,= μ ρD f d V
Dimensional analysis: 2 2 (Re)⎛ ⎞ρ
= =⎜ ⎟ρ μ⎝ ⎠
D Vdf fV d
Basic Types of Similitude
• geometric
• kinematic
• dynamic
All of these must be obtained for complete similarity between model and prototype.
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Geometric Similarity
Ratios between corresponding lengths in model and prototype should be the same.
= = λp p
m m
d ld l
2 22⎛ ⎞ ⎛ ⎞= = = λ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
p p p
m m m
A d lA d l
Kinematic Similarity
Flow field in prototype and model have the same shape and the ratios of corresponding velocities and accelerations are the same.
Geometrically similar streamlines are kinematically similar.
1 2 1 2
1 2 1 2
,= =p p p pm m m m
V V a aV V a a
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Dynamic Similarity
To ensure geometric and kinematic similarity, dynamic similarity must also be fulfilled.
Ratio between forces in prototype and model must be constant:
1 2 3
1 2 3
= = =p p p p p
m m m m m
F F F M aF F F M a
(vector relationships)
Also, Newton’s second law:
1 2 3
1 2 3
+ + =
+ + =p p p p p
m m m m m
F F F M a
F F F M a
Important Forces for the Flow Field
• pressure (FP)
• inertia (FI)
• gravity (FG)
• viscosity (FV)
• elasticity (FE)
• surface tension (FT)
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Parameterization of Forces
2
23 2 2
3
2
2
= Δ = Δ
= = ρ = ρ
= = ρ
= μ = μ = μ
= σ
= =
P
I
G
V
T
E
F pA pl
VF Ma l V ll
F Mg l gdV VF A l Vldy l
F l
F EA El
(convective acceleration: )dVVdx
Dynamic similarity: corresponding force ratios the same in prototype and model
2 2
2
2 2
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ρ ρ= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟Δ Δ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ρ ρ= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟μ μ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ρ= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟μ⎝ ⎠⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ρ ρ= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
I I
P P p mp m
I I
V V p mp m
I I
G G mpp m
I I
E E p mp m
I
T
F F V VF F p p
F F Vl VlF F
F F V VlF F gl
F F V VF F E E
FF
2 2⎛ ⎞ ⎛ ⎞ ⎛ ⎞ρ ρ= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟σ σ⎝ ⎠ ⎝ ⎠⎝ ⎠
I
T p mp m
F lV lVF
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Dimensionless Numbers
• Reynolds
• Froude
• Cauchy (Mach)
• Weber
• Euler
2 22
2
2
Re/
Fr
C/
W
E2
= =μ ρ ν
=
= = =ρ
ρ=
σρ
=Δ
Vl Vl
Vgl
V V ME c
lV
Vp
Dimensionless numbers same in prototype and model produces dynamic similarity.
Only four numbers are independent (fifth equation from Newton’s second law).
In most cases it is not necessary to ensure that four numbers are the same since:
• all forces do not act
• some forces are of negligable magnitude
• forces may counteract each other to reduce their effect
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Reynolds Similarity I
Low-speed flow around air foil (incompressible flow)
Re Re⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟ν ν⎝ ⎠ ⎝ ⎠p mp m
Vl Vl
Same Re number yields same relative drag force:
2 2 2 2
⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟ρ ρ⎝ ⎠ ⎝ ⎠p m
D DV l V l
(also geometric similarity)
Osborne Reynolds
Reynolds Similarity II
Flow through a contraction (incompressible flow)
Re Re⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟ν ν⎝ ⎠ ⎝ ⎠p mp m
Vl Vl
Same Euler number yields same relative pressure drop:
2 2
⎛ ⎞ ⎛ ⎞Δ Δ=⎜ ⎟ ⎜ ⎟ρ ρ⎝ ⎠ ⎝ ⎠p m
p pV V
(also geometric similarity)
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Froude Similarity I
Flow around a ship (free surface flow)
Fr Fr⎛ ⎞ ⎛ ⎞
= = =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
p m
p m
V Vgl gl
2 2 2 2
⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟ρ ρ⎝ ⎠ ⎝ ⎠p m
D DV l V l
Same Re number yields same relative drag force:
(frictional effects neglected)
William Froude
Froude Similarity II
Flow around a ship, including friction
Fr Fr⎛ ⎞ ⎛ ⎞
= = =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
p m
p m
V Vgl gl
Re Re⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟ν ν⎝ ⎠ ⎝ ⎠p mp m
Vl Vl
Fulfill both: 3/ 2ν ⎛ ⎞= ⎜ ⎟ν ⎝ ⎠
p p
m m
ll
(typically not possible)
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Reynolds Modeling Rules
Same viscosity in prototype and model
1Re Re −= → = = λp mp mm p
V lV l
2//
= → = λp p p p
m m m m
V l t tV l t t
Similarly:
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3 2
/ 1/
= = λ = λλ
p p p
m m m
Q l tQ l t
Froude Modeling Rules
Same acceleration due to gravity
1/ 21/ 2Fr Fr
⎛ ⎞= → = = λ⎜ ⎟
⎝ ⎠
p pp m
m m
V lV l
1/ 2//
= → = λp p p p
m m m m
V l t tV l t t
Similarly:
33 5/ 2
3
/ 1/
= = λ = λλ
p p p
m m m
Q l tQ l t
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Summary of Different Model Rules
Distorted Scale
The vertical and horizontal scale is different (e.g., model of an estuary, bay, lagoon).
,= λ = λp ph vm m
l yl y
Froude rule:1/ 2
1/ 2⎛ ⎞= = λ⎜ ⎟⎝ ⎠
p pv
m m
V yV y
//
λ= → = =
λp p p p m p h
m m m m p m v
V l t t V lV l t t V l
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Movable-Bed Models
Model sediment transport and the effect on bed change:
• General river morphology
• River training
• Flood plain development
• Bridge pier location and design
• Local scour
• Pipeline crossings
Additional problem besides similarity for the flow motion is the similarity for the sediment transport and the interaction between flow and sediment.
Sediment Transport
Interaction between fluid flow (water or air) and its loose boundaries.
Strong interaction between the flow and its boundaries.
Professor H.A. Einstein: ”my father had an early interest in sediment transport and river mechanics, but after careful thought opted for the simpler aspects of physics”
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Movable-Bed Experiment (USDA)
Gravel
Cohesive material
Important Features of Sediment Transport
• initiation of motion
• bed features
• transport mode
• transport magnitude
• slope stability
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