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Bernoulli-compressible-flow
Bernoulli-compressible-flow
Course
Mécanique des fluides (MDF)
19 Documents
Students shared 19 documents in this course
University
Université Batna 2
Academic year: 2023/2024
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1 Development of Bernoulli's Equation 9
1.
DEVELOPMENT OF BERNOULLI'S EQUATION
Bernoulli's equation establishes the relationship between pressure, elevation, and
velocity of the flow along a stream tube. For this analysis, the fluid is assumed to
be a perfect fluid; that is, we will ignore viscous effects. Consider the element of
fluid in the stream tube shown in Figure 1. The forces acting on the differential
element of fluid are due to pressure and gravitational forces. The pressure force
acting in the direction of the motion is given by
The gravitational force can be expressed as
dz
= - g dm-
ds
Applying Newton's second law yields
The differential mass dm can be expressed in terms of the mass density of the fluid
element times its respective volume; that is,
Inserting the expression for the differential mass, the acceleration of the fluid can
\
-Stream tube
FIGURE 1.
Forces acting on an element of flow
in a stream tube.
10 CHAPTER I : Introduction
be expressed as
- - dV - 1 dP dz
dt p d s 'ds
The acceleration can be expressed as
The first term on the right-hand side, dV/dt, denotes the change in velocity as a
function of time for the entire flow field. The second term denotes the acceleration
due to a change in location. If the flow field is steady, the term aV/& = 0 and
Equation ( 1) reduce to
The changes of pressure as a function of time cannot accelerate a fluid particle. This
is because the same pressure would be acting at every instant on all sides of the fluid
particles. Therefore, the partial differential can be replaced by the total derivative
in Equation ( 1):
Integrating Equation (1) along a streamline yields
which is known as Bernoulli's equation. Bernoulli's equation establishes the rela-
tionship between pressure, elevation, and velocity along a stream tube.
1.4 Incompressible Bernoulli Equation
If the fluid is considered to be incompressible. Equation (1) readily can be
integrated to yield the incompressible Bernoulli equation:
The differences in elevation usually can be ignored when dealing with the flow of
gases such as air. An important application of Bernoulli's equation is the determi-
nation of the so-called stagnation pressure of a moving body or a body exposed to
12 CHAPTER I : Introduction
into Equation ( 1) and rearranging to yield a relationship for the velocity and the
Mach number as follows.
Equations ( 1) and (1) can be used to find the velocity and Mach number
provided the flow regime is below M = 1.
1.
THE ATMOSPHERE
The performance characteristics of an airplane depend on the properties of the
atmosphere through which it flies. Because the atmosphere is continuously chang-
ing with time, it is impossible to determine airplane performance parameters pre-
cisely without first defining the state of the atmosphere.
The earth's atmosphere is a gaseous envelope surrounding the planet. The gas
that we call air actually is a composition of numerous gases. The composition of
dry air at sea level is shown in Table 1. The relative percentages of the con-
stituents remains essentially the same up to an altitude of 90 km or 300,000 ft
owing primarily to atmospheric mixing caused by winds and turbulence. At alti-
tudes above 90 km the gases begin to settle or separate. The variability of water
vapor in the atmosphere must be taken into account by the performance analyst.
Water vapor can constitute up to 4 percent by volume of atmospheric air. When the
relative humidity is high, the air density is lower than that for dry air for the same
conditions of pressure and temperature. Under these conditions the density may be
reduced by as much as 3 percent. A change in air density will cause a change in the
aerodynamic forces acting on the airplane and therefore influence its performance
capabilities. Furthermore, changes in air density created by water vapor will affect
engine performance, which again influences the performance of the airplane.
####### TABLE 1.
Composition of atmospheric air
Density Percentage by Percentage by
kg/m3 slugslft' volume weight
Air 1 2 X IW7 100 100
Nitrogen 78 75.
Oxygen 20 23.
Argon 0 1.
The remaining small portion of the composition of air is made up of neon, helium,
krypton, xenon, CO, and water vapor.
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Bernoulli-compressible-flow
Course: Mécanique des fluides (MDF)
19 Documents
Students shared 19 documents in this course
University: Université Batna 2
Was this document helpful?
1.4
Development of Bernoulli's Equation
9
1.4
DEVELOPMENT OF BERNOULLI'S EQUATION
Bernoulli's equation establishes the relationship between pressure, elevation, and
velocity of the flow along a stream tube. For this analysis, the fluid is assumed to
be a perfect fluid; that is, we will ignore viscous effects. Consider the element of
fluid in the stream tube shown in Figure 1.5. The forces acting on the differential
element of fluid are due to pressure and gravitational forces. The pressure force
acting in the direction of the motion is given by
The gravitational force can be expressed as
dz
=
-g
dm-
ds
Applying Newton's second law yields
The differential mass
dm
can be expressed in terms of the mass density of the fluid
element times its respective volume; that is,
Inserting the expression for the differential mass, the acceleration of the fluid can
\
\-Stream
tube
FIGURE
1.5
Forces acting on
an
element of
flow
in
a stream tube.
