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
Spillway design
Towing tank
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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Model Experiments in Hydraulics III
Traryd hydropower station
Model gates
Model Experiments in Hydraulics IV
surge chamber
pump intake
Hydraulic arrangements
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Model Experiments in Hydraulics V
Lule Älv
Lilla Edet
Water power plants
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)
D = f ( d , V , μ, ρ )
Dimensional analysis:
⎛ Vd ρ ⎞
D
= f⎜
⎟ = f (Re)
2 2
ρV d
⎝ μ ⎠
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.
dp
dm
=
lp
lm
=λ
2
2
⎛d ⎞ ⎛l ⎞
= ⎜ p ⎟ = ⎜ p ⎟ = λ2
Am ⎝ d m ⎠ ⎝ lm ⎠
Ap
Kinematic Similarity
Flow field in prototype and model have the same shape and
the ratios of corresponding velocities and accelerations are
the same.
V1 p
V1m
=
V2 p
V2 m
,
a1 p
a1m
=
a2 p
a2 m
Geometrically similar streamlines are kinematically similar.
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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:
F1 p
F1m
=
F2 p
F2 m
=
F3 p
F3m
=
M pap
M m am
(vector relationships)
Also, Newton’s second law:
F1 p + F2 p + F3 p = M p a p
F1m + F2 m + F3m = M m am
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
FP = ΔpA = Δpl 2
V2
= ρV 2 l 2
l
3
FG = Mg = ρl g
FI = Ma = ρl 3
dV
V
A = μ l 2 = μVl
dy
l
FT = σl
FV = μ
FE = EA = El 2
(convective acceleration: V
dV
)
dx
Dynamic similarity: corresponding force ratios the same in
prototype and model
⎛ FI ⎞ ⎛ FI ⎞
⎛ ρV 2 ⎞ ⎛ ρ V 2 ⎞
=
=
⎜ ⎟ ⎜ ⎟
⎜
⎟ =⎜
⎟
⎝ FP ⎠ p ⎝ FP ⎠ m ⎝ Δp ⎠ p ⎝ Δp ⎠ m
⎛ FI ⎞ ⎛ FI ⎞
⎛ Vlρ ⎞ ⎛ Vlρ ⎞
⎜ ⎟ =⎜ ⎟ =⎜
⎟ =⎜
⎟
⎝ FV ⎠ p ⎝ FV ⎠ m ⎝ μ ⎠ p ⎝ μ ⎠ m
⎛ FI ⎞ ⎛ FI ⎞
⎛ V 2 ⎞ ⎛ Vlρ ⎞
⎜ ⎟ =⎜ ⎟ =⎜ ⎟ =⎜
⎟
⎝ FG ⎠ p ⎝ FG ⎠ m ⎝ gl ⎠ p ⎝ μ ⎠ m
⎛ FI ⎞ ⎛ FI ⎞
⎛ ρV 2 ⎞ ⎛ ρV 2 ⎞
=
=
⎜ ⎟ ⎜ ⎟
⎜
⎟ =⎜
⎟
⎝ FE ⎠ p ⎝ FE ⎠ m ⎝ E ⎠ p ⎝ E ⎠ m
⎛ FI ⎞ ⎛ FI ⎞
⎛ ρlV 2 ⎞ ⎛ ρlV 2 ⎞
=
=
⎜ ⎟ ⎜ ⎟
⎜
⎟ =⎜
⎟
⎝ FT ⎠ p ⎝ FT ⎠ m ⎝ σ ⎠ p ⎝ σ ⎠ m
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Dimensionless Numbers
• Reynolds
• Froude
• Cauchy (Mach)
• Weber
• Euler
Vl
Vl
=
μ/ρ ν
V
Fr =
gl
Re =
C=
V2
V2
= 2 = M2
E /ρ c
ρlV 2
σ
ρ
E =V
2 Δp
W=
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)
⎛ Vl ⎞
⎛ Vl ⎞
⎜ ⎟ = Re p = Re m = ⎜ ⎟
⎝ ν ⎠p
⎝ ν ⎠m
Osborne Reynolds
(also geometric similarity)
Same Re number yields same relative drag force:
⎛ D ⎞ ⎛ D ⎞
⎜ ρV 2l 2 ⎟ = ⎜ ρV 2l 2 ⎟
⎝
⎠p ⎝
⎠m
Reynolds Similarity II
Flow through a contraction (incompressible flow)
⎛ Vl ⎞
⎛ Vl ⎞
⎜ ⎟ = Re p = Re m = ⎜ ⎟
⎝ ν ⎠p
⎝ ν ⎠m
(also geometric similarity)
Same Euler number yields same relative pressure drop:
⎛ Δp ⎞ ⎛ Δp ⎞
⎜ ρV 2 ⎟ = ⎜ ρV 2 ⎟
⎝
⎠p ⎝
⎠m
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Froude Similarity I
Flow around a ship (free surface flow)
William Froude
⎛ V ⎞
⎛ V ⎞
⎜⎜
⎟⎟ = Frp = Frm = ⎜⎜
⎟⎟
⎝ gl ⎠ p
⎝ gl ⎠ m
(frictional effects neglected)
Same Re number yields same relative drag force:
⎛ D ⎞ ⎛ D ⎞
⎜ ρV 2l 2 ⎟ = ⎜ ρV 2l 2 ⎟
⎝
⎠p ⎝
⎠m
Froude Similarity II
Flow around a ship, including friction
⎛ V ⎞
⎛ V ⎞
⎜⎜
⎟⎟ = Frp = Frm = ⎜⎜
⎟⎟
⎝ gl ⎠ p
⎝ gl ⎠ m
⎛ Vl ⎞
⎛ Vl ⎞
⎜ ⎟ = Re p = Re m = ⎜ ⎟
⎝ ν ⎠p
⎝ ν ⎠m
Fulfill both:
νp
⎛ lp ⎞
=⎜ ⎟
ν m ⎝ lm ⎠
3/ 2
(typically not
possible)
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Reynolds Modeling Rules
Same viscosity in prototype and model
Re p = Re m
Vp
→
Vm
Vp
Vm
=
lm
= λ −1
lp
=
lp / tp
lm / tm
tp
→
tm
= λ2
Similarly:
Qp
Qm
=
l 3p / t p
3
m
l / tm
= λ3
1
=λ
λ2
Froude Modeling Rules
Same acceleration due to gravity
Frp = Frm
1/ 2
⎛ lp ⎞
→
=⎜ ⎟
Vm ⎝ lm ⎠
Vp
Vp
Vm
=
lp / tp
lm / tm
= λ1/ 2
→
tp
tm
= λ1/ 2
Similarly:
Qp
Qm
=
l p3 / t p
3
m
l / tm
= λ3
1
= λ 5/ 2
λ
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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).
lp
lm
= λh ,
yp
ym
Froude rule:
Vp
Vm
=
lp / tp
lm / tm
= λv
1/ 2
⎛ yp ⎞
=⎜ ⎟
Vm ⎝ ym ⎠
Vp
→
tp
tm
=
Vml p
V p lm
= λ1/v 2
=
λh
λv
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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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