Fluid Mechanics Engineering
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Transcript of 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
WCDKF-KDU 2
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
WCDKF-KDU 3
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
WCDKF-KDU 4
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.
WCDKF-KDU 5
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
WCDKF-KDU 6
Flow Pattern
WCDKF-KDU 7
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
WCDKF-KDU 10
SOLUTION
WCDKF-KDU 11
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.
WCDKF-KDU 14
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
WCDKF-KDU 15
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.
WCDKF-KDU 16
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).
WCDKF-KDU 17
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
WCDKF-KDU 18
1.10 MODEL TESTING OF SHIPS
Total drag force / Resistance on ships
Wave resistance (inertia)
Frictional resistance (viscous)
WCDKF-KDU 19
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.
WCDKF-KDU 20
Incomplete Similarity Flows with Free Surfaces
For a free-surface flow, the Reynolds number and Froude number
are matched between model and prototype when
WCDKF-KDU 21
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.
WCDKF-KDU 22
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
WCDKF-KDU 23
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
WCDKF-KDU 24
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.
WCDKF-KDU 25
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
WCDKF-KDU 26