Pharos University ME 259 Fluid Mechanics Lecture # 9 Dimensional Analysis and Similitude

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?)


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)


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


Set up dimensionless groups πs

There will be n – m equationsExample: For drag on a sphere


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


• Froude Number

Weber Number


• 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



Head Coefficient

Power Coefficient


Example: Centrifugal Pump(Negligible Viscous Effects)

If … … then …



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

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Pharos University ME 259 Fluid Mechanics Lecture # 9 Dimensional Analysis and Similitude