Fluid Flow( New)

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Fundamentals of flow



•There are two methods for studying the movement of flow.

•One is a method which follows any arbitrary particle with its kaleidoscopic

changes in velocity and acceleration. This is called the Lagrangian method.

•The other is a method by which, rather than following any particular fluid particle,

changes in velocity and pressure are studied at fixed positions in space x, y, z and at

time t . This method is called the Eulerian method.

•Nowadays the latter method is more common and effective in most cases.



1. A flow whose flow state expressed by velocity, pressure, density, etc., at any

position, does not change with time, is called a steady flow.

2. On the other hand, a flow whose flow state does change with time is called an

unsteady flow.

3. Whenever water runs out of a tap while the handle is being turned, the flow is an

unsteady flow.

4. On the other hand, when water runs out while the handle is stationary, leaving the

opening constant, the flow is steady.



Smoke from a chimney

On a calm day with no wind, smoke ascending from a chimney looks like a single line as

shown in figure (a).

However, when the wind is strong, the smoke is disturbed and swirls as shown in figure

(b) or diffuses into the peripheral air.

One man who systematically studied such states of flow was Osborne Reynolds.



Reynolds used the device shown in figure.

Colored liquid was led to the entrance of a glass tube.

Figure: Reynolds' experiment



As the valve was gradually opened by the handle, the colored liquid flowed, as shown in

figure, like a piece of thread without mixing with peripheral water.

When the flow velocity of water in the tube reached a certain value, as shown in figure

that the line of colored liquid suddenly became turbulent on mingling with the peripheral

water.

The former flow is called the laminar flow, the latter flow the turbulent flow, and the flow

velocity at the time when the laminar flow had turned to turbulent flow the critical

velocity.



Laminar Flow

1. Laminar flow is a type of flow in which the fluid particles move in layers.

2. There is no transportation of fluid particles from one layer to another.

3. The fluid particles in any layer move along well defined paths or stream lines.

Turbulent Flow

1. Turbulent flow is the most common type of flow that occurs in nature.

2. There is a general mixing up of the fluid particles in motion.

3. There is continuous collision between fluid particles involving transference of

momentum between them



Whenever water is allowed to flow at a low velocity by opening the tap a little,

the water flows out smoothly with its surface in the laminar state.

But as the tap is gradually opened to let the water velocity increase, the flowbecomes turbulent and opaque with a rough surface.



Compressible and Incompressible Flow

1. In general, liquid is called an incompressible fluid, and gas a compressible fluid.

Nevertheless, even in the case of a liquid it becomes necessary to take

compressibility into account whenever the liquid is highly pressurized, such as oil in a

hydraulic machine.

2. Similarly, even in the case of a gas, the compressibility may be disregardedwhenever the change in pressure is small.



Rotational and Irrotational flows

1. As fluids moves the fluid particles may be subjected to a rotatory displacements.

Suppose a particle which is moving along a stream line rotates about its own axis

also then the particle is said to have a rotational motion.

2. If the particles as it moves along the stream lines does not rotate about its own axis

the particle is said to have irrotational motion.

Irrotational flow Rotational flow



Mechanism for

conservation of flowProperties



In steady flow, the mass flow per unit time passing through each section does not

change, even if the pipe diameter changes. This is the law of conservation of mass.



For the pipe shown in figure whose diameter decreases between sections 1 and 2,

which have cross-sectional areas A1 and A 2 respectively, and at which the mean

velocities are u1 and u2 and the densities p1 and p2 respectively,

If the fluid is incompressible, e.g. water, with p being effectively constant, then

Above equations state that the flow is continuous, with no loss or gain, so these

equations are called the continuity equations.

They are an expression of the principle of conservation of mass when applied to

fluid flow.



Example

Find the flow velocities v 1 , v 2 and V 3 in the conduit shown in following figure. The flow

rate Q is 800L/min and the diameters d 1 , d 2 and d 3 at sections 1,2 and 3 are 50,60 and

100mm respectively.

Example

Water flowing through a pipe of 0.5 m diameter with an average velocity of 1 m/s. What

is the rate of discharge of water? The same flow then passes through another section

where the diameter is 1 m. What is the average velocity at this section?



Movement of roller-coaster



Consider a roller-coaster running with great excitement in an amusement

park .

The speed of the roller-coaster decreases when it is at the top of the steep slope, and it

increases towards the bottom. This is because the potential energy increases and

kinetic energy decreases at the top, and the opposite occurs at the bottom.

However, ignoring frictional losses, the sum of the two forms of energy is constant at any

height. This is a manifestation of the principle of conservation of energy for a solid.



Figures (a) and (b) show the relationship between the potential energy of water (its level)

and its kinetic energy (the speed at which it gushes out of the pipe).

A fluid can attain large kinetic energy when it is under pressure as shown in Fig.(c). A

water hydraulic or oil hydraulic press machine is powered by the forces and energy due

to such pressure.

Conservation of fluid energy

B lli' ti



Bernoulli's equation

The terms of above represent energy per unit weight, and they have the units of length

(m) so they are commonly termed heads.



As shown in following figure, whenever water flows from tank 1 to tank 2, the

energy equations for sections 1,2 and 3 are as in following equation



In the above figure, the line connecting the height of the pressure heads at respective

points of the pipe line is called the hydraulic grade line, while that connecting the

heights of all the heads is called the energy line.

Example

Water is flowing in the conduit shown in the following figure. If the

pressure p , at section 1 is 24.5 kPa, what are the pressures p2 and p3 at

sections 2 and 3 respectively? Assume Q= 800 l/min.



Various problems on the one-dimensional flow of an ideal fluid can be solved by jointly

using Bernoulli’s theorem and the continuity equation.

A h i b fi d i h th fl t i i li i d b



As shown in above figure, a device where the flow rate in a pipe line is measured by

narrowing a part of the tube is called a Venturi tube.

In the narrowed part of the tube, the flow velocity increases. By measuring the resultant

decreasing pressure, the flow rate in the pipe line can be measured.

Let A be the section area of the Venturi tube, u the velocity and p the pressure, and

express the states of sections 1 and 2 by subscripts 1 and 2 respectively. Then from

Bernoulli’s equation



The flow rate

However, since there is some loss of energy between sections A1 and A2 in actual

cases, the above equation is amended as follows:



Pit t t b



Pitot tube

Pitot, who was engaged in research work, hit upon an idea one day for a very simple

measuring device of flow rate.

It was a device where the lower end of a glass tube is bent by 90" and supported

against the flow.

The flow velocity was to be measured by measuring the increased height of the water

level.

It is said that, as soon as he had hit upon this idea, he rushed to the River Seine

carrying a glass tube with a bent end.

The result of an experiment as shown in Fig. 5.9 confirmed his expectation.

The device incorporating that idea is shown in Fig. 5.10. This device is called a Pitot

tube, and it is widely used even nowadays.



Fig. 5.9 Pitot's first experiment



Fig. 5.10 Pitot tube

However, with an actual Pitot tube, since some loss occurs due to its shape and the fluid

viscosity, the equation is modified as follows:

where C v is called the coefficient of velocity

Fluid Flow( New)

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