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.
1 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
2 Model Experiments in Hydraulics I
Construction of model
Model Experiments in Hydraulics II
Model of dam with spillways
Discharge at model spillway
3 Model Experiments in Hydraulics III
Traryd hydropower station
Model gates
Model Experiments in Hydraulics IV
surge chamber
pump intake
Hydraulic arrangements
4 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)
5 Example: Drag Force on an Submerged Body
Drag force (D) depends on: • diameter (d) • velocity (V) • viscosity (m) • density (ρ)
DfdV=μρ(),,,
DVd⎛⎞ρ Dimensional analysis: 22==ff⎜⎟(Re) ρμVd ⎝⎠
Basic Types of Similitude
• geometric • kinematic • dynamic
All of these must be obtained for complete similarity between model and prototype.
6 Geometric Similarity
Ratios between corresponding lengths in model and prototype should be the same. dl pp==λ dlmm
22 Adpp⎛⎞⎛⎞ l p 2 ===λ⎜⎟⎜⎟ Admm⎝⎠⎝⎠ l m
Kinematic Similarity
Flow field in prototype and model have the same shape and the ratios of corresponding velocities and accelerations are the same. VV aa 12pp==, 12 pp VV12mm aa 12 mm
Geometrically similar streamlines are kinematically similar.
7 Dynamic Similarity
To ensure geometric and kinematic similarity, dynamic similarity must also be fulfilled. Ratio between forces in prototype and model must be constant:
FFFMa123ppp pp === (vector relationships) FFFMa123mmmmm
FFF123ppp++= Ma pp
FFF123mmm++= Ma mm
Important Forces for the Flow Field
• pressure (FP)
• inertia (FI)
• gravity (FG)
• viscosity (FV)
• elasticity (FE)
• surface tension (FT)
8 Parameterization of Forces
2 FpAplP =Δ =Δ V 2 FMal==ρ=ρ322 Vl I l 3 FMglgG ==ρ dV V FAlVl=μ =μ2 =μ V dy l
FlT =σ 2 FEAElE ==
dV (convective acceleration: V ) dx
Dynamic similarity: corresponding force ratios the same in prototype and model
22 ⎛⎞⎛⎞FFII⎛⎞⎛⎞ρρ V V ⎜⎟⎜⎟==⎜⎟⎜⎟ = FF p p ⎝⎠⎝⎠PPpm⎝⎠⎝⎠ΔΔpm
⎛⎞⎛⎞FFII⎛⎞⎛⎞Vlρρ Vl ⎜⎟⎜⎟== = FF⎜⎟⎜⎟μμ ⎝⎠⎝⎠VVpm⎝⎠⎝⎠pm
2 ⎛⎞⎛⎞FFII⎛⎞VVl⎛⎞ρ ⎜⎟⎜⎟===⎜⎟ FF gl⎜⎟μ ⎝⎠⎝⎠GGpm⎝⎠p ⎝⎠m
22 ⎛⎞⎛⎞FFII⎛⎞⎛⎞ρρVV ⎜⎟⎜⎟==⎜⎟⎜⎟ = FF E E ⎝⎠⎝⎠EEpm⎝⎠⎝⎠pm
22 ⎛⎞FI ⎛⎞FI ⎛⎞⎛⎞ρρlV lV ⎜⎟==⎜⎟⎜⎟⎜⎟ = F F ⎝⎠T pm⎝⎠T ⎝⎠⎝⎠σσpm
9 Dimensionless Numbers
Vl Vl • Reynolds Re == μρ/ ν V • Froude Fr = gl VV22 • Cauchy (Mach) C ===M 2 Ec/ ρ 2 ρlV 2 • Weber W = σ ρ • Euler E = V 2Δp
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
10 Reynolds Similarity I
Low-speed flow around air foil (incompressible flow)
⎛⎞Vl ⎛⎞ Vl Osborne Reynolds ⎜⎟===Repm Re ⎜⎟ ⎝⎠ννpm ⎝⎠
(also geometric similarity)
Same Re number yields same relative drag force:
⎛⎞⎛⎞DD ⎜⎟⎜⎟22= 22 ⎝⎠⎝⎠ρρVlpm Vl
Reynolds Similarity II
Flow through a contraction (incompressible flow)
⎛⎞Vl ⎛⎞ Vl ⎜⎟===Repm Re ⎜⎟ ⎝⎠ννpm ⎝⎠
(also geometric similarity)
Same Euler number yields same relative pressure drop:
⎛⎞⎛⎞ΔΔp p ⎜⎟⎜⎟22= ⎝⎠⎝⎠ρρVVpm
11 Froude Similarity I
Flow around a ship (free surface flow)
William Froude ⎛⎞VV ⎛⎞ ⎜⎟===Fr Fr ⎜⎟ ⎜⎟glpm ⎜⎟ gl ⎝⎠pm ⎝⎠
(frictional effects neglected)
Same Re number yields same relative drag force:
⎛⎞⎛⎞DD ⎜⎟⎜⎟22= 22 ⎝⎠⎝⎠ρρVlpm Vl
Froude Similarity II
Flow around a ship, including friction
⎛⎞VV ⎛⎞ ⎜⎟===Fr Fr ⎜⎟ ⎜⎟glpm ⎜⎟ gl ⎝⎠pm ⎝⎠
⎛⎞Vl ⎛⎞ Vl ⎜⎟===Repm Re ⎜⎟ ⎝⎠ννpm ⎝⎠
3/2 Fulfill both: ν ⎛⎞l pp= (typically not ⎜⎟ possible) νmm⎝⎠l
12 Reynolds Modeling Rules
Same viscosity in prototype and model
Vp lm −1 Repm=→==λ Re Vlmp
Vlt/ t ppp= →=λ p 2 Vltmmm/ t m
Similarly:
3 Qltppp/ 3 1 ==λ=λ32 Qltmmm/ λ
Froude Modeling Rules
Same acceleration due to gravity
1/2 Vlpp⎛⎞ 1/2 Frpm=→= Fr ⎜⎟ =λ Vlmm⎝⎠
Vlt/ t ppp= →=λ p1/2 Vltmmm/ t m
Similarly:
3 Qltppp/ 35/21 ==λ=λ3 Qltmmm/ λ
13 Summary of Different Model Rules
Distorted Scale
The vertical and horizontal scale is different (e.g., model of an estuary, bay, lagoon).
lypp =λhv, =λ lymm
1/2 Vypp⎛⎞ 1/2 Froude rule: = ⎜⎟=λv Vymm⎝⎠
Vlt/ tVl λ ppp=→== p mp h Vltmmm/ tVl m pm λv
14 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”
15 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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