Chapter 7_Part 2

Dr. Mohammad Shakir Nasif

Email: [email protected]

Tel: 05-3687026

Office Location: 17-03-11

3/10/2015 1:46 PM 1

Chapter 7 Assignment

• Question: 7.3, 7.7, 7.29, 7.43, 7.45, 7.52, 7.80

Quiz: Monday or Tuesday 16 or 17 March

2015.

• Quiz will start 5 minutes from the start of the

lecture) so make sure you arrive before the

time).

Some comments about

dimensional analysis• There are also other methods in dimensional

analysis but the method of repeating variables

is the easiest.

• There is not a unique set of pi terms which

arises from a dimensional analysis. However,

the required number of pi terms is fixed.

Common Dimensionless Groups in

Fluid MechanicsFrom dimensionless terms, many variables arise which are commonly used

in fluid mechanics problems.

Fortunately not all of these variables are used in one problem, however if

combination of some of these variables are present, it is a standard practice

to combine them into dimensionless groups (PI terms).

These combinations appear so frequent that special names are associated

with them.

It is also possible to provide physical interpretation to the dimensionless

groups which can be helpful in assessing their influence in particular

application.


Fluid MechanicsFor example:

This term is dimensionless, which is very well known in fluid mechanics, heat

transfer….etc. It is called Reynolds number (Re).

VL

VLRe

The physical interpretation to the

dimensionless term (Reynolds number ) isforceViscous

forceInertiaVLRe

•When flow enters any region there will be some

disturbances. If the flow velocity is not too fast, these

disturbances get damped out by the fluid viscosity.

•We can see that the velocity in Reynolds number is on top

and the viscosity is on the bottom. For a large Reynolds

number this means that the velocity X length are large

compared to the viscosity.

•At a certain large Reynolds number, the flow is moving too

fast for the viscosity to damp out the disturbance, hence it is

called turbulent flow.

http://www.ceb.cam.ac.uk/pages/hydrodynamic-voltammetry.html

http://www.ceb.cam.ac.uk/pages/hydrodynamic-voltammetry.html

Common Dimensionless Groups in Fluid Mechanics

These groups have

be developed by

using dimensional

analysis, then it

has been realised

that they have

been widely

repeated in

research. Hence

they were given

names.


Fluid Mechanics (cont.)

• Froude number (Fr no): It is the ratio of the inertia force on an element of fluid to the weight of the element.

• It is important in problems involving the study of flow of water around ships, or flow through rivers or open conduits.



• Euler number (Eu no): Represents the ratio of the pressure force to the inertia force.

• It is normally used where pressure or pressure difference between two points is an important variables.

• For problems where cavitation is of concern, this number is commonly used.

• It can be also called as Cavitation number



• Weber number (We no). It is the ratio of inertia force to surface tension force.

• It is important in problems in which there is an interface between two fluids, in this situation the surface tension will play an important role.



• Mach number (Ma number): Represents the ratio of the fluid speed to the sonic speed.

• For jet fighters or airplanes fly with a speed higher than sonic speed, the Ma >1.

• If less than sonic speed Ma<1.

• If equal to sonic speed Ma=1

Flow Similarity and Model Studies• A question comes to our mind what’s the importance of all these

numbers and how we can use them.

• These numbers applies when we want to study certain phenomena and it is not possible to construct full scale model. Therefore we can construct small scale model and use the same numbers in our study.

• We call it Similarity between full scale and small scale model.

• Geometric Similarity

– Model and prototype have same shape

– Linear dimensions on model and prototype correspond within constant scale factor

• Kinematic Similarity

– Velocities at corresponding points on model and prototype differ only by a constant scale factor

• Dynamic Similarity

– Forces on model and prototype differ only by a constant scale factor

Flow Similarity and Model Studies

• Example: Drag on a Sphere

Flow Similarity and Model Studies

• Example: Drag on a Sphere

For dynamic similarity …

… then …

What is “Concept of Similitude”?

• Similitude is the study of predicting prototype

conditions (flow, pressure distribution….etc.)

from model (small model) test observation.

Concept of Similitude

• The concept of similitude is used so that

measurements made on one system (for

example, in the laboratory) can be used to

describe the behavior of other similar systems

(outside laboratory)

Modeling and Similitude

• A model is a representation of a physical system that may be used to predict the behavior of the system in some desired respect.

• The physical system for which the predictions are to be made is called the prototype.

• Usually a model is smaller than the prototype and therefore, easier to handle in the lab.

Modeling and Similitude (cont.)

Model Design Conditions (or Similarity Requirements

or Modeling Laws)

• To achieve similarity between model and prototype

behavior, all the corresponding pi terms must be

equated between model and prototype

– Geometric Similarity

– Dynamic Similarity

– Kinematic Similarity


Geometric Similarity

A model and prototype are geometrically

similar if and only if all body dimensions in

all three coordinates have the same linear-

scale ratio. All angles are preserved in

geometric similarity. All flow directions are

preserved. The orientations of model and

prototype w.r.t. the surroundings must be

identical.


Kinematic Similarity

Velocities are related to the full scale by a

constant scale factor. They also have the

same directions as in the full scale.


Dynamic Similarity

Forces are related to full scale by a constant

factor. Also requires geometric and kinematic

similarity.

Similitude Summary

Example 3

The drag on a 2-m-diameter satellite dish due to an 80 km/hr wind is to be

determined through a wind tunnel test using a geometrically similar 0.4-m-

diameter model dish. Assume standard air for both model and prototype.

(a) At what air speed should the model test be run?

(b) With all similarity conditions satisfied, the measured drag on the model

was determined to be 170 N. What is the predicted drag on the prototype

dish?

Example 3 SolutionThe drag on a 2-m-diameter satellite dish due to an 80 km/hr wind is to be

determined through a wind tunnel test using a geometrically similar 0.4-

m-diameter model dish. Assume standard air for both model and

prototype.


(b) With all similarity conditions satisfied, the measured drag on the

model was determined to be 170 N. What is the predicted drag on the

prototype dish?

Example 3 SolutionThe drag on a 2-m-diameter satellite dish due to an 80 km/hr wind is to be

determined through a wind tunnel test using a geometrically similar 0.4-

m-diameter model dish. Assume standard air for both model and

prototype.


(b) With all similarity conditions satisfied, the measured drag on the

model was determined to be 170 N. What is the predicted drag on the

prototype dish? Drag force

Example 4 The drag on an airplane shown in Fig.E7.7 cruising at 386

km/h in standard air is to be determined from tests on a

1:10 scale model placed in a pressurized wind tunnel. To

minimize compressibility effects, the air speed in the wind

tunnel is also to be 386 km/h.

Determine:

a)The required air pressure in the tunnel (assuming the

same air temperature for model and prototype), and

b)The drag on the prototype corresponding to a measured

force of 4 N on the model.

Example 4

a) Drag can be predicted from a geometrically similar model if the Reynolds numbers of the

prototype and the model are the same. Thus:

For this example, Vm = V and lm/l = 1/10 so that:

And therefore

The result shows that the same fluid with ρm = ρ and μm = μ can’t be used if Reynolds number

similarity to be maintained. Instead, we can pressurize the wind tunnel to increase the density

of the air (with assumption that increase in density doesn’t significantly change the viscosity).

Example 4

Therefore, if the viscosity is the same, the equation becomes:

For an ideal gas, p = ρRT so that:

And therefore the wind tunnel need to be pressurized so that:

Since the prototype is at standard atmospheric pressure, the required pressure in the wind

tunnel is:

Example 4

b) The drag could be obtained from:

or

Thus, for a drag of 4 N on the model the corresponding drag on the prototype is:

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Chapter 7_Part 2