fluid topic

8/12/2019 Fluid topic

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Fluid Kinematics


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Velocity and velocity field

Velocity is a vector quantity.

Velocity is function of position and time.

The velocity magnitude (speed) is

In fluid mechanics, we are more likely to treat the properties as

continuum field functions rather than being interested in the

trajectories of individual particles or systems.

The determination of the properties as a function of space and time

is one the main objectives of fluid mechanics.

= + + .

Generally, a fluid flow is a complex three-dimensional, time-dependent

phenomenon. However, it is possible to make simplifying assumptions

such as one- or two-dimensional flow as well as steady flow.


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Lagrangian vs. Eulerian Description

Two Mathematical descriptions of fluid motion:

Lagrangian: divide the fluid into small particles then consider the

motion of each particle in response to the forces acting on it.

Eulerian : work directly in terms of fields such as u(x; t) andp(x; t),

and not try to keep track of individual fluid particles.


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Flow Characteristics

Fluid flow is 3-dimensional time-dependent phenomena, V = V( x, y, z, t)

A 2-dimensional flow can be assumed when one velocity component is

much smaller than the other two velocity components

A 1-dimensional flow can be assumed when two velocity components are

much smaller than the third velocity component

Steady flow: the velocity at a given point in space does not vary with

time, V/t = 0

Unsteady flow: the velocity at a given point in space does vary with time,

V/t 0

Almost all flows are unsteady; however, they are usually assumed steady

Turbulent flow: flow with random character

Laminar flow: flow with deterministic character


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Flow characterization

Uniform flow; = 0.

In uniform flow the velocity doesnt change frompoint to point along any stream lines.

Steady flow; = 0.

In steady flow the velocity at any point in the field

doesnt vary with time.

Laminar vs. Turbulent flow. Turbulent flow is characterized by mixing through

out the flow.

Reynolds number used as criterion to distinguishbetween laminar and turbulent flow.

=


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Stream lines, Path lines, and Streak lines

Stream lines are lines that are everywhere tangent to the velocity field; they are

used to indicate the speed and the direction of the flow field.


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=

.

Streamlines can be determined by integrating the equations defining

lines tangent to the velocity field.

For 2-D flows the slope of the stream line must be equal to the tangent

of the angle that the velocity vector makes with the x-axis,

Path line is the location of the particle as a function of time. Path lines can

be obtained for 2-D flows by integration of

and then eliminating the time. = =

Streak lineconsists of all particles in a flow that have previously passed

through a common point.

For steady flow, streamlines, streaklines, and pathlines are the same.

Stream line is a lines that is everywhere tangent to the velocity field; it is used to

indicate the speed and the direction of the flow field.


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The acceleration field

Fluid Motion can be described by either: Lagrangian (followingindividual particles) or Eulerian (remaining fixed in space and

observing different particles as they pass by)

Acceleration is the rate of change of velocity for a given particle

Velocity is a function of position and time, V = V( x, y, z, t)


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The Material Derivative

=

+ ( )

Or, in a compact form,

=

+

+


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If the Eulerian approach is adopted,

The time rate of change of velocity for a given particle, i.e.

acceleration, is given by

which can be written as

13

= = +

= + + +


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If streamline coordinates are used,

Or

The orientation of the unit vector along the streamline changes

with distance along the streamline

14

= +

=

=


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=

=

+

What is the convective acceleration in thex-direction for the following

velocity distribution = 2 + 3 + 4?

In general,

The convective acceleration in thex-direction is given by

So,

Example

= + +

= 2 2 + 3 0 + 4 0 = 4.


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Control volume & System Representation

A system is a collection of matter of fixed identity always the same atoms or

fluid particles, which may move, flow, and interact with its surroundings. It is specific, identifiable quantity of matter.

For example: a mass of air drawn into an air compressor can be

considered as a system; it changes shape and size as it is compressed. It is

eventually expelled through the outlet of the compressor.

A control volume, on the other hand, is a volume in space a geometric entity,

independent of mass through which fluid may flow.

In general, the control volume can be either fixed or moving volume and

non-deformable or deformable control volume. The matter within a control volume may change with time as the fluid

flows through it; the amount of mass within the volume may change with

time. The control volume itself is a specific geometric entity; independent

of the flowing fluid.


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Air flows through this control volume (C.V.) because of the action of the

engine within it. The air that was within the engine itself at time t = to(a

system) has passed through the engine and is outside of C.V. at a later

time t = to.

The deflecting balloon provides an example of a

deforming C.V. As time increase, the controlvolume (i.e. the inner surface of the balloon)

decreases in size. If we dont hold onto the

balloon, it becomes a deforming and moving C.V.


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Extensive and Intensive Properties

Property:Any characteristic of a system, they are considered to be either

intensive or extensive.

Intensive properties: Those that are independent of the mass of a system.

Extensive properties:Those whose values depend on the size of the

system; on the mass.

Let Brepresent any of the fluid parameters and brepresent the amount of that

parameter per unit mass. That is, B = mb, where mis the mass of the portion

of fluid of interest the velocity.

Mass Momentum Energy Angular momentum

B m E b 1 e


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The Basic Laws of Mechanics

1) Conservation of mass: state that the time rate of change

of total mass M of a system is zero.2) Conservation of momentum: state that the time rate of

change of linear momentum (MV) is equal to the sum ofthe external forces.3) Conservation of energy: state that the time rate of

change of total energy is equal to the rate of heat addedto the system minus the rate of work done by the system.

4) Conservation of angular momentum:state that thetime rate of change of the angular momentum is equal tothe sum of moments.

5) The second law of thermodynamics.

=0.

To describe the motion of a rigid body, the governing equations

representing the conservation laws can be written as:

= = .

=


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Lagrangian vs. Eulerian Description

In general, for a fluid the Lagrange approach is not suitable and

the Euler approach must be adopted.

In the Euler approach a control volume fixed in space is

considered and the analysis of the motion of the fluid that pass

through it, is studied .


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Reynolds Transport Theorem

In order to rewrite the governing equations in the control volume

approach the Reynolds Transport Theorem (RTT)must beapplied.

This theorem relates the system concept to the control volume

concept.

In this theorem, if B represents the total flow quantity ( e.g.mass,

energy, or momentum) contained within a fluid volume, then b is

the flow quantity per unit mass,


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Physical Interpretation

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Special Cases

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Moving control volume with a constant velocity ():

where = .

Steady flow:

Common form (incompressible & uniform velocities):

=

+

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