Fluid Mechanics Engineering

DIMENSIONAL & HYDRAULIC

MODEL ANALYSIS

FLOWS WITH GRAVITY FORCES

The condition for similarity of flows of the gravitational force is, the ratio of

inertia to gravity forces.

Froude number similarity

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Ex 2

A model showing local conditions on a river is to be built to a scale 1:49. The

maximum rate of discharge of the river is 2500 m3/s. Estimate

1. the velocity scale

2. the time scale

3. the rate of discharge required from a pump which supplies water for the

model

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FLOWS WITH VISCOUS FORCES

If the flow is in a completely closed conduit such as pipe flows, inertia and

viscous force is chosen for dynamic similarity.

Reynolds number similarity

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Ex 3

A pipe of diameter 1.5 m is required to transport an oil of relative density 0.9

and kinematic viscosity of 3 x 10-2 stoke at a rate of 3.0 m3/s. If a 15 cm

diameter pipe with water (ν = 0.01 stoke) is used to model the above flow, findthe velocity and discharge in the model.

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1.8 MODELLING CRITERIA

Geometric, kinematic and dynamic similarities are mutually independent.

Existence of one does not imply the existence of another similarity.

The geometric similarity is complete when the surface roughness profiles are

also in the scale ratio.

The kinematic similarity is even more difficult because the flow patterns

around small objects tend to be quantitatively different from those around

large objects. Flow Pattern

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Flow Pattern

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DIMENSIONAL ANALYSIS AND SIMILARITY

Consider automobile

experiment

Drag force is F = f (V, , µ, L)

Through dimensional analysis,

we can reduce the problem

to

where= CD

=Reand

The Reynolds number is the most well known and useful

dimensionless parameter in all of fluid mechanics.

EX 4 : Similarity between Model and Prototype

Cars

The aerodynamic drag of a new sportscar is to be predicted at a speed of100.0 km/h at an air temperature of25°C. Automotive engineers build aone-fifth scale model of the car to testin a wind tunnel. It is winter and thewind tunnel is located in an unheatedbuilding; the temperature of the windtunnel air is only about 5°C. Determinehow fast the engineers should run thewind tunnel in order to achievesimilarity between the model and theprototype.

Take ρ25=1.184 kg/m3 ρ5=1.269 kg/m3

µ25 = 1.849 x 10-5 kg/m.s

µ5 = 1.754 x 10-5 kg/m.s

Solution

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SOLUTION

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h/km5.442

5m/kg269.1

m/kg184.1

s.m/kg10x849.1

s.m/kg10x754.1h/km100

3

3

5

5

Discussion

This speed is quite high, and the wind tunnel may

not be able to run at that speed. Furthermore,

the incompressible approximation may come into

question at this high speed.

EX 5 : Prediction of Aerodynamic Drag Force on the Prototype Car

This example is a follow-up to

Example 4. Suppose the engineers

run the wind tunnel at 442.5

km/h to achieve similarity

between the model and the

prototype. The aerodynamic drag

force on the model car is

measured with a drag balance.

Several drag readings are recorded,

and the average drag force on the

model is 90 N. Predict the

aerodynamic drag force on the

prototype (at 100 km/h and 25°C).

Solution

FD,p = 107.2 N

DIMENSIONAL ANALYSIS AND SIMILARITY

In Examples 3 and 4 use a water tunnel instead of a wind tunnel totest their one-fifth scale model. Using the properties of water atroom temperature (20°C is assumed), the water tunnel speedrequired to achieve similarity is easily calculated as

The required water tunnel speed is much lower than that required for a wind tunnel using the same size model.

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h/km08.32

5m/kg1000

m/kg184.1

s.m/kg10x849.1

s.m/kg10x002.1h/km100

3

3

5

3

Ex 6

A 1/20 scale model of a spillway studied in the laboratory requires 5 m3/s

discharge and a hydraulic jump formed therein dissipates 500 W. Calculate:

1. the velocity ratio between the two flows

2. the discharge in the spillway, neglecting viscous and surface tension effects

3. the power lost in the spillway jump

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1.9 DISTORTED MODELS

In rivers and harbours the area is very much larger than the depth.

If the depth is represented in same scale as that of length and width, it will be

found that the depth of the model is extremely small.

The effects of this are

The depth in the model will be too small for the model to function

properly

The Re of the model becomes very low to be in the laminar region while

that of the prototype is in the turbulent region.

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Incomplete Similarity Flows with Free Surfaces

For the case of model testing of flows with free surfaces (boats and

ships, floods, river flows, aqueducts, hydroelectric dam spillways,

interaction of waves with piers, soil erosion, etc.), complications

arise that preclude complete similarity between model and prototype.

For example, if a model river is built to study flooding, the model is

often several hundred times smaller than the prototype due to limited

lab space. This may cause, for instance,

Increase the effect of surface tension

Turbulent flow laminar flow

To avoid these problems, researchers often use a distorted model in

which the vertical scale of the model (e.g., river depth) is

exaggerated in comparison to the horizontal scale of the model (e.g.,

river width).

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Ex 7

A model is to be constructed of 5.8 km length of a river. For the normal

discharge of 70 m3/s, it is known that the average depth and width of the

river are 2.5 m and 30 m respectively. The length of the lab channel is 30 m.

Recommend suitable scales for the model.

Assume μ = 1.14 x 10-3 Ns/m2

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1.10 MODEL TESTING OF SHIPS

Total drag force / Resistance on ships

Wave resistance (inertia)

Frictional resistance (viscous)

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Incomplete Similarity Flows with Free Surfaces

In many practical problems

involving free surfaces,

both the Reynolds number

and Froude number appear

as relevant independent

groups in the dimensional

analysis.

It is difficult (often

impossible) to match both

of these dimensionless

parameters simultaneously.

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Incomplete Similarity Flows with Free Surfaces

For a free-surface flow, the Reynolds number and Froude number

are matched between model and prototype when

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and

To match both Re and Fr simultaneously, we require length scale

factor Lm/Lp satisfy

From the results, we would need to use a liquid whose

kinematic viscosity satisfies the equation. Although it is

sometimes possible to find an appropriate liquid for use with

the model, in most cases it is either impractical or impossible.

Ex 8

An 1 : 8 model of a boat is towed in water of kinematic viscosity 10-6 m2/s.

What should be the speed of model to simulate a speed of 3.5 m/s if the

resistance is due to

Internal friction only and

Waves only

Calculate the kinematic viscosity of the liquid in which model should be

tested if the resistance due to internal friction and waves are to be

considered.

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TESTING OF SHIP MODELS

Test the model based on Froude’s numbr

Calculate skin friction resistance for the model

Find model wave resistance

Determine corresponding wave resistance in the prototype

Calculate skin friction for the prototype

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TESTING OF SHIP MODELS

Skin friction resistance,

For Re < 2 x 107

For For Re > 2 x 107

Total drag = skin friction resistance + wave resistance

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RC

51

e

f

01.0

Ex 9

A 1:25 scale model of a ship has a submerged area of 6 m2, a length of 5 m and

experiences a total drag of 25 N when towed through water with a velocity of 1.2

m/s. Estimate the total drag on the prototype when cruising at the corresponding

speed.

Assume μ = 1 x 10-3 Pa.s and ρ = 1030 kg/m3 for both model and the prototype.

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Ex 10

A proposed ocean going vessel is to have a length of 125 m at the water line

& wetted surface of 1600 m². Its steady speed is to be 35 km /hour. Tests on

the model of the vessel to a scale of 1:25 were made in a towing tank at a

velocity corresponding to wave making resistance. The total drag resistance

of the model was 25.2 N. Calculate the total drag of the prototype .

ρm = 1000 kg/m3 ρp = 1027 kg/m3

νm = 1.115 x 10-2 cm2/s νp = 1.121 x 10-2 cm2/s

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