fluidmechanics dimensional analysis

ME52 _ FLUID MECHANICS

NOTES OF LESSON

UNIT 3: Dimensional Analysis

[No. of pages-8]

Notes of lesson by S. K. Jagadeesh, Associate Professor, Dept. of Mech. Engg. Dr. Ambedkar Inst. of Technology, BLR. Last Update - Nov-14)

U4-1

Dimensional Analysis:

It is a mathematical technique used for solving engineering problems with the help of

fundamental dimensions. All physical quantities are measured by comparison, which is made

with respect to an arbitrarily fixed value. Length L, mass M and time T are three fixed

dimensions. Dimensional analysis helps in arranging various variables of physical quantities in a

systematic way and then combines them to form non-dimensional groups. This considerably

reduces the number of variables in a given physical situation and thus becomes easy to present

the experimental results in a concise form.

Secondary or Derived Quantities

These are the quantities which possess more than one fundamental dimension. For example,

velocity is denoted by distance per unit time (LT-1

), density by mass per unit volume (ML-3

) and

acceleration by distance per second square (LT-2

). Then velocity, density and acceleration

become secondary or derived quantities.

Table of fundamental and derived dimensions of various physical quantities

Sl.

No. Physical Quantity Units Dimensions

Fundamental

1 Mass gms M

2 Length Mm, cm, m L

3 Time seconds T

Geometric Property

4 Area mm2, cm

2, m

2 L

2

5 Volume Mm3, cm

3, m

3 L

3

Kinematic Property

6 Linear velocity, Tangential speed m/s LT-1

7 Linear acceleration, Acceleration due

to gravity (g)

m/s2

LT-2

8 Angular velocity, Rotational Speed rad/s, rev/min, rev/sec T-1

9 Angular acceleration rad/s2 T

-2

10 Discharge, Volume flow rate m3/s L

3T

-1

11 Mass flow rate Kgmass/s MT-1

12 Kinematic viscosity m2/s L

2T

-1

Dynamic Property

13 Force Newton (N) MLT-2

14 Density, mass density kgmass/m3 ML

-3


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15 Specific weight, weight density Kgweight/m3, N/m

3 ML

-2T

-2

16 Dynamic viscosity Ns/m2 ML

-1T

-1

17 Pressure, Change in pressure N/m2 ML

-1T

-2

18 Work, Energy, Torque Nm, Joule ML2T

-2

19 Power Nm/s, Watt, Joule/s ML2T

-3

20 Enthalpy Joule/kgmass L2T

-2

Dimensional Homogeneity

It means that the dimension of each term on both sides of an equation is equal. Such an

equation is dimensionally homogeneous equation. The powers of fundamental dimensions (M.

L, T) on both sides of such an equation will be identical. A dimensionally homogeneous

equation is independent of the system of units.

For example; consider the equation, gHV 2=

Consider the dimensions of LHS of the equation: V=T

L= LT- 1

Consider the dimensions of RHS of the equation: 1

22 −

=×= LTLT

Lgh

From the above, it is seen that: Dimensions of LHS = Dimensions of RHS

Therefore, the equation gHV 2= is dimensionally homogeneous.

Methods of Dimensional Analysis

1) Raleigh’s method:

It is used for determining the expression for a variable which depends upon maximum of three

or four variables only.

If X is a variable which depends on X1, X2 and X3 variables, then X is a function of X1, X2 and X3.

It is written as X = f (X1, X2, X3)

Or X = K. X1a. X2

b. X3

c

Where K is a constant and a, b, and c are arbitrary exponents that are evaluated by equating

the powers of the fundamental dimensions of the variables on both sides of the equation.

2) Buckingham π-method:

Statement of Buckingham π-theorem: If there are ‘n’ variables (dependent and independent) in a

dimensionally homogeneous equation and if these variables contain ‘m’ fundamental

dimensions, then the variables are arranged into ‘n-m’ dimensionless groups known as ‘π’

terms.

Let a variable X1 depends on variables X1, X2, X3… Xn.

The functional relationship is given by: X1 = f(X2, X3… Xn)


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Or it can be written as: F(X1, X2, X3… Xn) = constant

The above dimensionally homogeneous equation contains ‘n’ variables. If there are ‘m’ fundamental

dimensions, then the equation can be arranged into ‘n-m’ number of ‘π’ terms which are

dimensionless.

i.e., F (π1, π2… πn-m) = constant

Types of Forces Acting in a Moving Fluid

In fluid flow problems, the forces acting on a fluid mass may be one, or a combination of the several

of the following forces:

1. Inertia force, Fi

2. Viscous force, Fv

3. Gravity force, Fg

4. Pressure force, Fp

5. Surface tension force, Fs

6. Elastic force, Fe

1. Inertia Force, Fi: It is equal to the product of mass and acceleration of the flowing fluid and

acts in the direction opposite to the direction of acceleration. This force always exists in a

fluid flow problem.

Inertia force, Fi = Mass x Acceleration of flowing fluid

= am ×

=t

vV ×ρ

= vt

V××ρ

= vAv ××ρ

Fi = 2

Avρ [ ρ =density, V =volume, v =velocity, t =time, A =Area]

2. Viscous force, Fv: It is equal to the product of shear stress due to viscosity and surface

area of the flow. It is present in fluid flow problems where viscosity of fluid is considered to

be dominant.

Viscous force, Fv = Shear stress x Area

= A×τ

= Ady

du×µ


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Fv = AL

v××µ

3. Gravity force, Fg: It is equal to the product of mass and acceleration due to gravity of the

flowing fluid. It is dominant in case of open surface flows such as flow over channels.

Gravity force, Fg = Mass x Acceleration due to gravity

= gm ×

= gV ×ρ

Fg = gAL ×ρ

4. Pressure force, Fp: It is equal to the product of pressure intensity and cross-sectional area

of the flowing fluid. It is present in most of pipe flows.

Pressure force, Fp = Intensity of pressure x Area

= Ap ×

5. Surface tension force, Fs: It is equal to the product of surface tension and length of

surface of the flowing fluid.

Surface tension force, Fs = Surface tension per unit length x Length

= L×σ

6. Elastic force, Fe: It is equal to the product of elastic stress and area of the flowing fluid.

Elastic force, Fe = Elastic stress x Area

= A×κ

Dimensionless Numbers

1. Reynold’s number

It is defined as the ratio of inertia force of a flowing fluid to its viscous force. Mathematically, it is

written as υµ

ρ

µ

ρ vLvL

LvA

Av

F

F

v

i====

2

Re

In case of pipe flow, the linear dimension L is taken as diameter, D.

Significance: Reynold’s number its application in situations such as:

1) motion of aero planes

2) flow of incompressible fluid in closed pipes

3) flow around structures and other bodies immersed completely under moving fluids


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2. Froude’s number

It is defined as the square root of the ratio of inertia force of a flowing fluid to its gravitational

force. Mathematically, it is expressed as

Lg

v

ALg

Av

F

FFr

g

i===

ρ

ρ2

Significance: The Froude’s number finds its application in situations such as:

1) free surface flows such as flow over sluice, spillways, etc.,

2) flow of jet from an orifice or a nozzle

3) where waves are likely to be formed on the surface

4) where fluids of different mass densities flow over one another

3. Euler’s number, Eu

It is defined as the square root of the ratio of the inertia force of the flowing fluid to its pressure

force. Mathematically, it written as

ρ

ρ

p

v

pA

Av

F

FEu

p

i===

2

Significance: The Euler’s number finds application in situations such as:

1) phenomenon of cavitation

2) enclosed fluid system where turbulence is fully developed

4. Weber’s number

It is defined as the square root of the ratio of the inertia force of the flowing fluid to its surface

tension force. Mathematically it is expressed as

L

v

L

vL

L

Av

F

FWe

s

i

ρσσ

ρ

σ

ρ====

222

Significance: The Weber’s number finds application in situations such as:

1) very thin sheet of liquid flowing over a surface

2) capillary waves in channels

3) capillary movement of water in soil

5. Mach number

It is the square root of the ratio of the inertia force of the flowing fluid to its elastic force.

Mathematically, it is expressed as


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C

vv

A

Av

F

FM

e

i====

ρκκ

ρ2

where C is the velocity of sound in the fluid.

Significance: The Mach number finds application in situations such as:

1) aerodynamic testing

2) phenomena involving velocity exceeding the speed of sound

3) under-water testing of torpedoes

Model Studies:

A model is the small scale replica of the actual structure or machine. The actual structure or

machine is known as the prototype. The models are made and tested to get the necessary

information and also to evaluate the performance of structures or machines before they are actually

constructed or manufactured. Model studies offer several advantages such as:

1) Model tests are economical and convenient.

2) The performance of the structure/machine is predicted in advance.

3) Models tests can also be used to detect and rectify the defects of an existing structure which is

not functioning properly.

Similitude:

Model studies are conducted to evaluate the performance of structures or machines before they are

actually constructed or manufactured. In order that the results obtained from model studies

represent the behavior of the prototype, it necessary that the following similarities exist between the

model and the prototype, namely, geometric similarity, kinematic similarity and dynamic similarity.

Geometric similarity refers to the constant ratio of corresponding lengths in the model and the

prototype and same included angles between the two corresponding sides.

Kinematic similarity refers to the constant ratio of velocity or acceleration at corresponding points

in the model and the prototype and velocity or acceleration vectors point in the same direction.

Dynamic similarity refers to the similarity of forces. The flows in the model and in the prototype are

dynamically similar if at all the corresponding points, identical types of forces are parallel and the

bear the same ratio.

Model (or similarity) laws:


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Models are designed on the basis of the force which is dominating in the flow situation. The laws on

which the models are designed for dynamic similarity are called model or similarity laws. The various

model laws are:

2) Reynolds model law

3) Froude model law

4) Euler model law

5) Weber model law

6) Mach model law

1) Reynolds model law: Reynolds number is the ratio of inertial force to viscous force. In flow

situations where, in addition to inertia, viscous force is the only other predominant force, the

similarity of flow in the model and its prototype can be established if Reynolds number is

same for both the systems.

I.e., Rem =Rep

Or

p

pp

m

mmLVLV

υ=

υ, where

ρ

µ=υ

2) Froude model law: Froude number is the ratio of inertia force to gravitational force. When

the gravitational forces can be considered as the only other predominant force that controls

the flow motion in addition to the inertia force, the similarity of flow in any two systems can

be established if Froude number for both systems is same.

i.e., Frm = Frp

or

pp

p

mm

m

Lg

V

Lg

V=

3) Euler model law: Euler number is the ratio of inertia force to pressure force. In a fluid

system where pressure forces alone are the controlling forces in addition to inertia force, the

dynamic similarity is obtained by equating Euler number for both the model and the

prototype.

i.e., Eum = Eup

or

p

p

p

m

m

m

p

V

p

V

ρ

=

ρ

4) Weber model law: Weber number is the ratio of inertia force to surface tension force. In a

fluid system where surface tension effects predominate in addition to inertia force, the


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dynamic similarity is obtained by equating the Weber number for the model and the

prototype.

i.e., Wem = Wep

or

pp

p

p

mm

m

m

L

V

L

V

ρ

σ=

ρ

σ

5) Mach model law: Mach number is the ratio of inertia force to elastic force. In any fluid, if

only the forces resulting from elastic compression are significant in addition to inertia forces,

then dynamic similarity between the model and its prototype is established by equating

Mach numbers.

i.e., Mm = Mp

or

p

p

p

m

m

m

k

V

k

V

ρ

=

ρ

, where k = Elastic stress for model and prototype.

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fluidmechanics dimensional analysis

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