pharos university me 259 fluid mechanics lecture # 9 dimensional analysis and similitude
TRANSCRIPT
Pharos UniversityME 259 Fluid Mechanics
Lecture # 9
Dimensional Analysis and Similitude
Main Topics
• Nature of Dimensional Analysis• Buckingham Pi Theorem• Significant Dimensionless Groups in
Fluid Mechanics• Flow Similarity and Model Studies
Objectives
1. Understand dimensions, units, and dimensional homogeneity
2. Understand benefits of dimensional analysis3. Know how to use the method of repeating
variables4. Understand the concept of similarity and
how to apply it to experimental modeling
Dimensions and Units• Review– Dimension: Measure of a physical quantity, e.g., length,
time, mass– Units: Assignment of a number to a dimension, e.g., (m),
(sec), (kg)– 7 Primary Dimensions:
1. Mass m (kg)2. Length L (m)3. Time t (sec)4. Temperature T (K)5. Current I (A)6. Amount of Light C (cd)7. Amount of matter N (mol)
Dimensions and Units
– All non-primary dimensions can be formed by a combination of the 7 primary dimensions
– Examples• {Velocity} m/sec = {Length/Time} = {L/t}• {Force} N = {Mass Length/Time} = {mL/t2}
Dimensional Homogeneity
• Every additive term in an equation must have the same dimensions
• Example: Bernoulli equation
– {p} = {force/area}={mass x length/time x 1/length2} = {m/(t2L)}– {1/2V2} = {mass/length3 x (length/time)2} = {m/(t2L)}– {gz} = {mass/length3 x length/time2 x length} ={m/(t2L)}
Nondimensionalization of Equations
• To nondimensionalize, for example, the Bernoulli equation, the first step is to list primary dimensions of all dimensional variables and constants
{p} = {m/(t2L)} {} = {m/L3} {V} = {L/t}{g} = {L/t2} {z} = {L}
– Next, we need to select Scaling Parameters. For this example, select L, U0, 0
Nature of Dimensional Analysis
Example: Drag on a Sphere
Drag depends on FOUR parameters:sphere size (D); speed (V); fluid density (); fluid viscosity ()
Difficult to know how to set up experiments to determine dependencies
Difficult to know how to present results (four graphs?)
Nature of Dimensional Analysis
Example: Drag on a Sphere
Only one dependent and one independent variable
Easy to set up experiments to determine dependency
Easy to present results (one graph)
Nature of Dimensional Analysis
Buckingham Pi Theorem• Step 1:
List all the parameters involvedLet n be the number of parametersExample: For drag on a sphere, F, V, D, , ,
& n = 5• Step 2:
Select a set of primary dimensionsFor example M (kg), L (m), t (sec).Example: For drag on a sphere choose MLt
Buckingham Pi Theorem• Step 3
List the dimensions of all parametersLet r be the number of primary dimensions
Example: For drag on a sphere r = 3
Buckingham Pi Theorem
• Step 4Select a set of r dimensional parameters that includes all the primary dimensions
Example: For drag on a sphere (m = r = 3) select ϱ, V, D
Buckingham Pi Theorem• Step 5
Set up dimensionless groups πs
There will be n – m equationsExample: For drag on a sphere
Buckingham Pi Theorem• Step 6
Check to see that each group obtained is dimensionlessExample: For drag on a sphere
Π2 = Re = ϱVD / μ
Π2
Significant Dimensionless Groups in Fluid Mechanics
• Reynolds Number
Mach Number
Significant Dimensionless Groups in Fluid Mechanics
• Froude Number
Weber Number
Significant Dimensionless Groups in Fluid Mechanics
• Euler Number
Cavitation Number
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Dimensional analysis• Definition : Dimensional analysis is a process of formulating fluid mechanics problems in in terms of non-dimensional variables and parameters.
• Why is it used : • Reduction in variables ( If F(A1, A2, … , An) = 0, then f(1, 2, … r < n) = 0, where, F = functional form, Ai = dimensional variables, j = non-dimensional parameters, m = number of important dimensions, n = number of dimensional variables, r = n – m ). Thereby the number of experiments required to determine f vs. F is reduced.• Helps in understanding physics• Useful in data analysis and modeling• Enables scaling of different physical dimensions and fluid properties
Example
Vortex shedding behind cylinder
Drag = f(V, L, r, m, c, t, e, T, etc.)
From dimensional analysis,
Examples of dimensionless quantities : Reynolds number, FroudeNumber, Strouhal number, Euler number, etc.
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Similarity and model testing• Definition : Flow conditions for a model test are completely similar if all relevant dimensionless parameters have the same corresponding values for model and prototype.
• i model = i prototype i = 1• Enables extrapolation from model to full scale• However, complete similarity usually not possible. Therefore, often it is necessary to use Re, or Fr, or Ma scaling, i.e., select most important and accommodate others as best possible.
• Types of similarity: • Geometric Similarity : all body dimensions in all three coordinates have the same linear-scale ratios.• Kinematic Similarity : homologous (same relative position) particles lie at homologous points at homologous times.• Dynamic Similarity : in addition to the requirements for kinematic similarity the model and prototype forces must be in a constant ratio.
Dimensional Analysis and Similarity
• Geometric Similarity - the model must be the same shape as the prototype. Each dimension must be scaled by the same factor.
• Kinematic Similarity - velocity as any point in the model must be proportional
• Dynamic Similarity - all forces in the model flow scale by a constant factor to corresponding forces in the prototype flow.
• Complete Similarity is achieved only if all 3 conditions are met.
Dimensional Analysis and Similarity• Complete similarity is ensured if all independent
groups are the same between model and prototype.• What is ? – We let uppercase Greek letter denote a nondimensional
parameter, e.g.,Reynolds number Re, Froude number Fr, Drag coefficient, CD, etc.
• Consider automobile experiment
• Drag force is F = f(V, , L)
• Through dimensional analysis, we can reduce the problem to
Flow Similarity and Model Studies
• Example: Drag on a Sphere
Flow Similarity and Model Studies• Example: Drag on a Sphere
For dynamic similarity …
… then …
Flow Similarity and Model Studies• Scaling with Multiple Dependent Parameters
Example: Centrifugal Pump
Pump Head
Pump Power
Similitude-Type of Similarities• Geometric Similarity: is the similarity of shape.
p p pr
m m m
L B DL
L B D
Where: LWhere: Lpp, B, Bpp and D and Dpp are Length, Breadth, and diameter of are Length, Breadth, and diameter of
prototype and Lprototype and Lmm, B, Bmm, D, Dmm are Length, Breadth, and are Length, Breadth, and
diameter of model.diameter of model. Lr= Scale ratioLr= Scale ratio
Similitude-Type of Similarities• Kinematic Similarity: is the similarity of motion.
1 2 1 2
1 2 1 2
;p p p pr r
m m m m
V V a aV a
V V a a
Where: VWhere: Vp1p1& V& Vp2p2 and a and ap1p1 & a & ap2p2 are velocity and are velocity and
accelerations at point 1 & 2 in prototype and Vaccelerations at point 1 & 2 in prototype and Vm1m1& V& Vm2m2 and and
aam1m1 & a & am2m2 are velocity and accelerations at point 1 & 2 in are velocity and accelerations at point 1 & 2 in
model.model. VVrr and a and arr are the velocity ratio and acceleration ratio are the velocity ratio and acceleration ratio
Similitude-Type of Similarities• Dynamic Similarity: is the similarity of forces.
gi vp p pr
i v gm m m
FF FF
F F F
Where: (FWhere: (Fii))pp, (F, (Fvv))pp and (F and (Fgg))pp are inertia, viscous and are inertia, viscous and
gravitational forces in prototype and (Fgravitational forces in prototype and (Fii))mm, (F, (Fvv))mm and (F and (Fgg))mm
are inertia, viscous and gravitational forces in model.are inertia, viscous and gravitational forces in model. FFrr is the Force ratio is the Force ratio
Flow Similarity and Model Studies• Scaling with Multiple Dependent Parameters
Example: Centrifugal Pump
Head Coefficient
Power Coefficient
Flow Similarity and Model Studies• Scaling with Multiple Dependent Parameters
Example: Centrifugal Pump(Negligible Viscous Effects)
If … … then …
Flow Similarity and Model Studies• Scaling with Multiple Dependent Parameters
Example: Centrifugal Pump
Specific Speed
Types of forces encountered in fluid Phenomenon
• Inertia Force, Fi: = mass X acceleration in the flowing fluid.
• Viscous Force, Fv: = shear stress due to viscosity X surface area of flow.
• Gravity Force, Fg: = mass X acceleration due to gravity.
• Pressure Force, Fp: = pressure intensity X C.S. area of flowing fluid.
Dimensionless Numbers• These are numbers which are obtained by dividing the
inertia force by viscous force or gravity force or pressure force or surface tension force or elastic force.
• As this is ratio of once force to other, it will be a dimensionless number. These are also called non-dimensional parameters.
• The following are most important dimensionless numbers.
– Reynold’s Number– Froude’s Number– Euler’s Number– Mach’s Number
Dimensionless Numbers• Reynold’s Number, Re: It is the ratio of inertia force to the viscous force of flowing
fluid.
. .Re
. .
. . .
. . .
Velocity VolumeMass VelocityFi Time Time
Fv Shear Stress Area Shear Stress Area
QV AV V AV V VL VLdu VA A Ady L
2
. .
. .
. .
. .
Velocity VolumeMass VelocityFi Time TimeFe
Fg Mass Gavitational Acceleraion Mass Gavitational Acceleraion
QV AV V V V
Volume g AL g gL gL
Froude’s Number, Fe:Froude’s Number, Fe: It is the ratio of inertia force to the gravity It is the ratio of inertia force to the gravity force of flowing fluid.force of flowing fluid.
Dimensionless Numbers• Eulers’s Number, Re: It is the ratio of inertia force to the pressure force
of flowing fluid.
2
. .
Pr . Pr .
. .
. . / /
u
Velocity VolumeMass VelocityFi Time TimeE
Fp essure Area essure Area
QV AV V V V
P A P A P P
•Mach’s Number, Re: It is the ratio of inertia force to the elastic force of flowing fluid.
2 2
2
. .
. .
. .
. . /
: /
Velocity VolumeMass VelocityFi Time TimeM
Fe Elastic Stress Area Elastic Stress Area
QV AV V L V V V
K A K A KL CK
Where C K