# Historical Note: Bernoulli & Euler

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24 Οκτ 2013 (πριν από 4 χρόνια και 6 μήνες)

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Historical Note: Bernoulli & Euler

Daniel Bernoulli (1700

1782) was born in Groningen, Netherlands.

He was the member of remarkable mathematician family, father and uncle both
were noted mathematician and physicists.

The entire family was
swiss

and made Basel, Switzerland as home.

He gave insight on kinetic theory of gases in his book
Hydrodynamica

(1738), also
about jet propulsion, manometers, flow in pipes etc.

Bernoulli’s theorem better understood by his father, both did not understand that
pressure is a point property.

Leonhard Euler (1707

1783) was also a
swiss

mathematician.

He became one of mathematical giant of history & his contribution to fluid
dynamics are of interest.

He was a close friend of Bernoulli and a student of his father, Euler was influenced
by the work of Bernoulli's in hydrodynamics & he originated the concept of
pressure acting at a point in a gas.

He came up with differential equation for a fluid accelerated by pressure, the same
equation as Eq. (4.8) driven in the chapter. Bernoulli’s equation has been obtained
as Eq. (4.9)

The Pitot Tube

Henri Pitot (1695

1771) borne in
Aramon
, France, the inventor of Pitot Tube.

He began his career as astronomer and mathematician.

In 1724, he became interested in hydraulics and in particular, in the flow of
water in rivers and canals. He was not satisfied with the way velocity of floating
objects was measured.

He devised an instrument consisting of two tubes. One was simply a straight
tube open at one end, which was inserted vertically into water (to measure
static pressure) and the other was a tube with one end bent at right angles, with
the open end facing directly into the flow (to measure total pressure).

He used this instrument in 1732, to measure water flow in a river. It was
thought at that time that the water flow velocity increases with depth. Pitot
reported stunning and correct results, measured with this instrument, that in
reality the flow velocity decreases as the depth increased.
Hence Pitot tube was
introduced with style.

French Captain A.
Eteve

in Jan 1911, later by British engineer deployed Pitot
Tube on an airplane for the first time. It then evolved into the primary
instrument for flight speed measurement.

The First Wind Tunnels

Aerodynamics is an empirically based discipline. Discovery and development by
experimental means have been its lifeblood. Wind Tunnel has been the work
horse for such experiments.

Today, most aerospace industrial, government and university labs have a
complete spectrum of wind tunnels ranging from low subsonic to hypersonic
speeds.

Evolution of wind tunnels goes back more than 400 years when Leonardo Vinci
near the beginning of 16th century said that “since the action of the medium
upon the body is the same whether the body moves in medium ore the particles
of the medium impinge with the same velocity upon body”. The lift and drag of
an aerodynamics body are the same whether it moves through the stagnant air
at 100 miles/hour or whether the air moves over the stationary body at 100
miles/hour.

First wind tunnel in history was designed and built by Francis Wenham in
Greenwich, England in 1871. thereafter, many wind tunnels were made all over
the world, and all of them were low speed to start with (essentially in
incompressible flow) but as the airplane speed increased, new wind tunnels with

First supersonic wind tunnel was developed by Dr A
Busemann

at Germany in
mid 1930s.

First hypersonic wind tunnel was operated by NACA at Langley in 1947.

Saying of Prof. Albert F
Zahm

1912

Theoretical fluid dynamics, being a difficult
subject, is for convenience, commonly divided
into branches, one treating of frictionless or
perfect fluids, the other treating of viscous or
imperfect fluids. The frictionless fluid has no
existence in nature, but is hypothesized by
mathematicians in order to facilitate the
investigation of important laws and principles
that may be approximately true of viscous or
natural fluids.

Mass Flow Rate

Mass Flow Rate

Within some problem domain, the amount of mass remains constant
--
mass
is neither created nor destroyed.

The
mass

of any object is simply the volume that the object occupies times
the density of the object. For a
fluid

(a liquid or a
gas
) the density, volume,
and shape of the object can all change within the domain with time. And
mass can move through the domain.

If the fluid initially passes through an area
A

at velocity
V
, we can define a
volume of mass to be swept out in some amount of time
t
. The volume
v

is:

v = A * V * t

A units check gives area x length/time x time = area x length = volume. The
mass
m
contained in this volume is simply density
r

times the volume.

m = r * A * V * t

To determine the mass flow rate
mdot
, we divide the mass by the time. The
resulting definition of mass flow rate is shown on the slide in red.

mdot

= r * A * V

From Newton's Second Law of Motion, the
aerodynamic forces on an aircraft

(lift and drag) are directly related to the change in momentum of a gas with
time. The
momentum

is defined to be the mass times the velocity, so we
would expect the aerodynamic forces to depend on the mass flow rate past
an object.

Mass Flow Rate

The thrust produced by a propulsion system also depends on the change of
momentum of a working gas. The thrust depends directly on the mass flow
rate through the propulsion system.

Considering the mass flow rate equation, it would appear that for a given
area, we could make the mass flow rate as large as we want by setting the
velocity very high. However, in real fluids, compressibility effects limit the
speed at which a flow can be forced through a given area.

Bernoulli's equation

In the 1700s, Daniel Bernoulli investigated the forces present in a moving
fluid. This slide shows one of many forms of
Bernoulli's equation
. The
equation appears in many physics, fluid mechanics, and airplane textbooks.

The equation states that the static pressure
ps

in the flow plus the dynamic
pressure, one half of the density
r

times the velocity
V

squared, is equal to a
constant throughout the flow. We call this constant the total pressure
pt

of
the flow.

Thermodynamics is the branch of science which describes the macro scale
properties of a fluid. One of the principle results of the study of
thermodynamics is the conservation of energy; within a system, energy is
neither created nor destroyed but may be converted from one form to
another.

inviscid

flow

we have a simplified conservation of energy
equation in terms of the enthalpy of the fluid:

ht2
-

ht1 = q
-

wsh

where
ht

is the total enthalpy of the fluid,
q

is the
heat transfer

into the fluid,
and
wsh

is the useful work done by the fluid.

Applications of Bernoulli's Equation

The fluids problem shown on this slide is low speed flow through a tube with
changing cross
-
sectional area. For a streamline along the center of the tube,
the velocity decreases from station one to two. Bernoulli's equation
describes the relation between velocity, density, and pressure for this flow
problem. Since density is a constant for a low speed problem, the equation at
the bottom of the slide relates the pressure and velocity at station two to the
conditions at station one.

Along a low speed airfoil, the flow is incompressible and the density remains
a constant. Bernoulli's equation then reduces to a simple relation between
velocity and static pressure. The surface of the airfoil is a streamline. Since
the velocity varies along the streamline, Bernoulli's equation can be used to
compute the change in pressure. The static pressure integrated along the
entire surface of the airfoil gives the total aerodynamic force on the foil. This
force can be broken down into the lift and drag of the airfoil.

Bernoulli's equation is also used on aircraft to provide a speedometer called a
pitot
-
static tube. A pressure is quite easy to measure with a mechanical
device. In a pitot
-
static tube, we measure the static and total pressure and
can then use Bernoulli's equation to compute the velocity.

Isentropic Flow

As a gas is forced through a tube, the gas molecules are deflected by the walls of
the tube.

If the speed of the gas is much less than the speed of sound of the gas, the
density
of
the gas remains constant and the velocity of the flow increases
.

As
the speed of the flow approaches the speed of sound we must consider
compressibility effects on the gas. The density of the gas varies from one
location to the next.

If
We say that such a process is
reversible
.

From a consideration of the second law of thermodynamics, a reversible flow
maintains a constant value of entropy.

Engineers call this type of flow an
isentropic

flow; a combination of the Greek
word "
iso
" (same) and entropy.

Isentropic flows occur when the change in flow variables is small and gradual,
such as the ideal flow through the
nozzle
shown above
.

If a supersonic flow is turned abruptly and the flow area decreases, shock waves
are generated and the flow is
irreversible
. The isentropic relations are no longer
valid and the flow is governed by the oblique or normal shock relations.

Mach Number

The
ratio

of the speed of the aircraft to the speed of sound in the gas determines
the magnitude of many of the compressibility effects
.

Aerodynamicists
have designated it with a special parameter called the
Mach
number

in honor of
Ernst Mach
, a late 19th century physicist who studied gas
dynamics. The Mach number
M

allows us to define flight regimes in which
compressibility effects vary
.

Subsonic

conditions occur for Mach numbers less than one,
M < 1
. For the lowest
subsonic conditions, compressibility can be ignored.

As the speed of the object approaches the speed of sound, the flight Mach number
is nearly equal to one,
M = 1
, and the flow is said to be
transonic
.

Supersonic

conditions occur for Mach numbers greater than one,
1 < M < 3
.
Compressibility effects are important for supersonic aircraft, and shock waves are
generated by the surface of the object. For
high supersonic speeds
,
3 < M < 5
,
aerodynamic heating also becomes very important for aircraft design.

For speeds greater than five times the speed of sound,
M > 5
, the flow is said to be
hypersonic
. At these speeds, some of the energy of the object now goes into
exciting the chemical bonds which hold together the nitrogen and oxygen molecules
of the air. At hypersonic speeds, the chemistry of the air must be considered when
determining forces on the object. The Space Shuttle re
-
enters the atmosphere at
high hypersonic speeds
,
M ~ 25
. Under these conditions, the heated air becomes an
ionized plasma of gas and the spacecraft must be insulated from the high
temperatures.

As an object moves through the atmosphere, the gas molecules of the
atmosphere near the object are disturbed and move around the object.
Aerodynamic forces

are generated between the gas and the object. The
magnitude of these forces depend on the shape of the object, the
speed

of
the object, the
mass

of the gas going by the object and on two other
important properties of the gas; the
viscosity
, or stickiness, of the gas and
the
compressibility
, or springiness, of the gas
.

Aerodynamic forces depend in a complex way on the
viscosity

of the gas. As
an object moves through a gas, the gas molecules stick to the surface. This
creates a layer of air near the surface, called a
boundary layer
, which, in
effect, changes the shape of the
object.

The flow of gas reacts to the edge of the boundary layer as if it was the
physical surface of the object. To make things more confusing, the boundary
layer may
separate

from the body and create an effective shape much
different from the physical shape
.

To
make it even more confusing, the flow conditions in and near the
boundary layer are often

(changing in time). The boundary layer is
very important in determining the
drag

of an object. To determine and
predict these conditions, aerodynamicists rely on
wind tunnel

testing and
very sophisticated computer analysis
.

Aerodynamic forces also depend in a complex way on the
compressibility

of
the gas. As an object moves through the gas, the gas molecules move around
the object. If the object passes at a low speed (typically less than 200 mph)
the density of the fluid remains constant
.

For
high speeds, some of the energy of the object goes into compressing the
fluid and changing the density, which alters the amount of resulting force on
the object. This effect becomes more important as speed increases. Near and
beyond the
speed of sound

(about 330 m/s or 700 mph on earth),
shock
waves

are produced that affect the lift and drag of the object. Again,
aerodynamicists rely on wind tunnel testing and sophisticated computer
analysis to predict these conditions.

The effects of compressibility and viscosity on lift are contained in the
lift
coefficient

and the effects on drag are contained in the
drag coefficient.

For
propulsion systems
, compressibility affects the amount of
mass

that can pass
through an engine and the amount of
thrust

generated by a rocket or turbine
engine
nozzle.