prototype
model
1. Geometric similitude
(basic and the most obvious requirement) The model is an exact geometric replica of the prototype. All the linear dimensions of the model and prototype are in the same ratio
ratio of area:
ratio of volume:
Ω l3 CΩ = = 3 = Cl3 Ωm lm
prototype
model
2. Motion Similitude (result)
Velocity of the model and the prototype are in the same ratio throughout the entire flow field. ratio of velocity
1. Dynamic Similitude of Fluid Motion
Dynamic Similitude of Fluid Motion:
At any time, all the parameters of the model and prototype are in the same ratio throughout the entire flow field: Geometric similitude (basic) Motion similitude (result) Dynamic similitude (condition)
Ne equals
2.2 Euler number, Froud number, Reynolds number, Mach number
1. Similitude Principle of flow acted by pressure force
Pressure force
F = Ne 2 2 ρl V
Cω =
C V l ω = = V ω m Vm lm Cl
3. Dynamic simiห้องสมุดไป่ตู้itude (condition)
All the forces that act on corresponding masses in the model and the prototype are in the same ratio throughout the entire flow field.
Em
E = ρ a2
ρV 2 V 2 Ca = = 2 2 a ρa
Mach number:
V 12 M = = ( Ca ) a
3. Dimensional Analysis
1.Principle of identical dimensions for physical equations Dimension: Unit types of physical variables Fundamental dimension: the dimension of time T (hour, minute, second) the dimension of length L (m, cm, mm) the dimension of mass M (ton, kilogram, gram) the dimension of temperature (oC, K)
ρVl ρ mVm lm Re = = µ µm
Reynolds number:
4. Similitude Principle of flow acted by elastic force
Elastic force For fluid:
Fs = EA = El 2
a= E ρ
Ca =
ρV 2
E
=
ρ mVm 2
Fp = ( ∆ p ) A = ( ∆ p ) l 2
F Ne = 2 2 ρl V
Eu =
(∆ p)l2
ρ l 2V 2
=
∆p ρV 2
Euler number:
Eu =
∆ pm ∆p = ρV 2 ρ mVm 2
2. Similitude Principle of flow acted by gravity
V3 V1 V2 = = = CV V1m V2 m V3m
Cl V l t = = Ct = Vm lm tm CV
ratio of time:
prototype
model
Cl l t V CV = = = Vm lm tm Ct
ratio of acceleration:
CV CV 2 V t a Ca = = = = am Vm tm Ct Cl
l1 l2 d = = = Cl l1m l2 m d m
scale ratio between model and prototype
prototype
model
scale ratio:
l1 l2 d = = = Cl l1m l2 m d m
A l2 CA = = 2 = Cl 2 Am l m
prototype
model
2. How to design:
One type of flow Geometric similitude Similitude of initial and boundary condition The same similitude dimensionless number
Cl 3 l3 t Q CQ = = 3 = = Cl 2 CV Qm l m tm Ct
ratio of volume rate:
ratio of kinematic viscosity:
Cl 2 l2 t ν Cν = = 2 = = Cl CV ν m l m t m Ct
ratio of rotation:
kν = 1 k L kV
kν =1 1/ 2 kL kL
1 kv = 10
3/ 2
VL
kV = k L = 1 ν '≠ ν = ?
3/2 kν = k L
if
1 kL = , 10
1 = 31 .62
3. Model experiment
* the most important force (dimensionless number) * laminar or complete turbulence rough zone
n y = k x x x ... x a n a1 1 a2 2 a3 3
example
3. Buckingham
π
Method (group Method)
The number of independent dimensionless groups of variables needed to correlate the variables in a given process is equal to n-m, where n is the number of variables involved and m is the number dimensions included in the variables.
Gravity:
Fg = mg = ρΩ g = ρ l 3 g
Froud number:
Fr =
V gl
=
Vm g m lm
3. Similitude Principle of flow acted by viscous force
Viscous force
F = Ne ρ l 2V 2
du du 2 Fτ = τ A = µ A=µ l dy dy
φ ( x1 , x2 , x3 , ⋅ ⋅ ⋅⋅, xn ) = 0
πi =
xi x x
a1 a2 i +1 i + 2 am ... x x i +m a3 i +3
f (π 1 , π 2 , π 3 , ⋅ ⋅ ⋅⋅, π m ) = 0
example
4. Model experiment
Derived dimension: velocity: LT-1 acceleration: LT-2 density: ML-3 force: MLT-2 pressure: ML-1T-2 dynamic viscosity: ML-1T-1 kinematic viscosity: L2T-1
F3 F1 F2 = = = CF F1m F2 m F3m
CF = ma = Cm Ca = Cρ Cl 2 CV 2 mm am
prototype
model
4. Relationship
Geometric similitude
(basic and the most obvious requirement)
Dimensional analysis:
How to operate an experiment and analysis experimental date…)
2. Why need we study?
a. The need for developing Fluid Mechanics FM: Theoretical FM Experimental FM Computational FM b. The need of engineering practice
1. Condition of Similitude
Geometric similitude (basic) Motion similitude (result) Dynamic similitude (condition) Similitude of initial and boundary condition