Bernoulli’s Theorem and Principle
What
is Bernoulli's theorem in simple words?
Bernoulli
principle flight
How
does Bernoulli's principle relate to flight?
What
are 2 examples of Bernoulli's principle?
What
is the importance of Bernoulli's principle?
The Bernoulli's theorem in simple words, in this video
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What is Bernoulli's theorem in simple words?
Bernoulli's
theorem implies, therefore, that if the fluid flows horizontally so that no
change in gravitational potential energy occurs, then a decrease in fluid
pressure is associated with an increase in fluid velocity.
Bernoulli principle flight
Air
moving over the curved upper surface of the wing will travel faster and thus
produce less pressure than the slower air moving across the flatter underside
of the wing. This difference in pressure creates lift which is a force of
flight that is caused by the imbalance of high and low pressures.
How does Bernoulli's principle relate to flight?
For
example, if the air flowing past the top surface of an aircraft wing is moving
faster than the air flowing past the bottom surface, then Bernoulli's principle
implies that the pressure on the surfaces of the wing will be lower above than
below. This pressure difference results in an upwards lifting force.
What are 2 examples of Bernoulli's principle?
When a truck moves
very fast, it created a low pressure area, so dusts are being pulled along in
the low pressure area. If we stand very close to railway track in the platform,
when a fast train passes us, we get pulled towards the track because of the low
pressure area generated by the sheer speed of the train.
What is the importance of Bernoulli's principle?
Bernoulli's
principle is valid for any fluid (liquid or gas); it is especially important to
fluids moving at a high velocity. Its principle is the basis of venturi
scrubbers, thermocompressors, aspirators, and other devices where fluids are
moving at high velocities.
You
have undoubtedly been introduced to a relationship called Bernoulli’s Equation
or the Bernoulli Principle somewhere in a previous Physics or Chemistry course.
This is the principle that relates the pressure to the velocity in any fluid,
essentially showing that as the speed of a fluid increases its pressure
decreases and visa versa. This principle can take several different
mathematical forms depending on the fluid and its speed. For an incompressible
fluid such as water or for air below about 75% of the speed of sound this
relationship takes the following form:
P
+ ½ρV2 = P0
(hydro)static
pressure + dynamic pressure = total pressure [internal energy + kinetic energy
= total energy]
This
relationship can be thought of as either a measure of the balance of pressure
forces in a flow, or as an energy balance (first law of thermodynamics) when
there is no change in potential energy or heat transfer.
Bernoulli’s
equation says that along any continuous path (“streamline”) in a flow the total
pressure, P0, (or total energy) is conserved (constant) and is a sum of the
static pressure and the dynamic pressure in the flow. Static pressure and
dynamic pressure can both change, but they must change in such a way that their
sum is constant; i.e., as the flow speeds up the pressure decreases.
Way
to put this is that the speed can vary with position in the flow (that’s really
what the equation is all about) but cannot vary with time.
The
assumption of constant density, which we usually call an assumption of
incompressible flow, means that we have to observe a speed limit. As air speeds
up and the speed approaches the speed of sound its density changes; i.e., it
becomes compressible. So when our flow speeds get too near the speed of sound,
the incompressible flow assumption is violated and we can no longer use this
form of Bernoulli’s equation. When does that become a problem?
Some
fluid mechanics textbooks use a mathematical series relationship to look at the
relationship between speed or Mach number (Mach number, the speed divided by
the speed of sound, is really a better measure of compressibility than speed
alone) and they use this to show that the incompressible flow assumption is not
valid above a Mach number of about 0.3 or 0.3 times the speed of sound. This is
good math but not so good physics. The important thing is not how the math
works but how the relationship between the two pressures in Bernoulli’s
equation changes as speed or Mach number increases. We will examine this in a
later example to show that we are actually pretty safe in using the
incompressible form of Bernoulli’s equation up to something like 75% of the
speed of sound.
The
other important assumptions in this form of Bernoulli’s equation are those of
steady flow and mass conservation. Steady flow means pretty much what it sounds
like; the equation is only able to account for changes in speed and pressure
with position in a flow field. It was assumed that the flow is exactly the same
at any time.
The
mass conservation assumption really relates to looking at what are called
“streamlines” in a flow. These can be thought of at a basic level as flow paths
or highways that follow or outline the movement of the flow. Mass conservation
implies that at any point along those paths or between any two streamlines the
mass flow between the streamlines (in the path) is the same as it is at any
other point between the same two streamlines (or along the same path).
The
end result of this mass conservation assumption is that Bernoulli’s equation is
only guaranteed to hold true along a streamline or path in a flow. However, we
can extend the use of the relationship to any point in the flow if all the flow
along all the streamlines (or paths) at some reference point upstream (at “∞”)
has the same total energy or total pressure.
So,
we can use Bernoulli’s equation to explain how a wing can produce lift. If the
flow over the top of the wing is faster than that over the bottom, the pressure
on the top will be less than that on the bottom and the resulting pressure
difference will produce a lift. The study of aerodynamics is really all about
predicting such changes in velocity and pressure around various shapes of wings
and bodies. Aerodynamicists write equations to describe the way air speeds
change around prescribed shapes and then combine these with Bernoulli’s
equation to find the resulting pressures and forces.
Let’s
look at the use of Bernoulli’s equation for the case shown below of a wing
moving through the air at 100 meters/ sec. at an altitude of 1km.
We
want to find the pressure at the leading edge of the wing where the flow comes
to rest (the stagnation point) and at a point over the wing where the speed has
accelerated to 150 m/s.
First,
note that the case of the wing moving through the air has been portrayed as one
of a stationary wing with the air moving past it at the desired speed. This is
standard procedure in working aerodynamics problems and it can be shown that
the answers one finds using this method are the correct ones. Essentially,
since the process of using Bernoulli’s equation is one of looking at
conservation of energy, it doesn’t matter whether we are analyzing the motion
(kinetic energy) involved as being motion of the body or motion of the fluid.
Now
let’s think about the problem presented above. We know something about the flow
at three points:
Well
in front of the wing we have what is called “free stream” or undisturbed,
uniform flow. We designate properties in this flow with an infinity [∞]
subscript. We can write Bernoulli’s equation here as:
Note
that it is at this point, the “free stream” where all the flow is uniform and
has the same total energy. If at this point the flow was not uniform, perhaps
because it was near the ground and the speed increases with distance up from
the ground, we could not assume that each “streamline” had a different value of
total pressure (energy).
At
the front of the wing we will have a point where the flow will come to rest. We
call this point the “stagnation point” if we can assume that the flow slowed
down and stopped without significant losses. Here the flow speed would be zero.
We can write Bernoulli’s equation here as:
Pstagnation
+ 0 = P0
At
this point the flow has accelerated to 150 m/s and we can write Bernoulli’s
equation as:
Now
we know that since the flow over the wing is continuous (mass is conserved) the
total pressure (P0) is the same at all three points and this is what we use to
find the missing information. To do this we must understand which of these
pressures (if any) are known to us as atmospheric hydrostatic pressures and
understand that we can assume that the density is constant as long as we are
safely below the speed of sound.
nitially
we know that the pressure in the atmosphere is that in the standard atmosphere
table for an altitude of 1 km or 89870 Pascals and that the density at this
altitude is 1.112 kg/m3. Looking at the problem, the most logical place for
standard atmosphere conditions to apply is in the “free stream” location
because this is where the undisturbed flow exists. Hence
And,
using these in Bernoulli’s equation at the free stream location we calculate a
total pressure
P0
= 95430 Pa
Now
that we have found the total pressure we can use it at any other location in
the flow to find the other unknown properties.
At
the stagnation point
Pstagnation
= P0 = 95430 Pa
At
the point where the speed is 150 m/s we can rearrange Bernoulli’s equation to
find
As
a check we should confirm that the static pressure (P3) at this point is less
than the free stream static pressure (P) since the speed is higher here and
also confirm that the static pressures everywhere else in the flow are lower
than the stagnation pressure.
Now
let’s review the steps in working any problem with Bernoulli’s equation. First
we must sketch the flow and write down everything we know at various points in
that flow. Second we must write Bernoulli’s equation at every point in the flow
where we either know information or want to know something. Third we must
carefully assess which pressure, if any, can be obtained from the standard
atmosphere table. Fourth we must look at all these points in the flow and see
which point gives us enough information to solve for the total pressure (P0).
Finally we use this value of P0 in Bernoulli’s equation at other points in the
flow to find the other missing terms. Attempting to skip any of the above steps
can lead to mistakes for most of us.
One
of the most common problems that people have in working with Bernoulli’s
equation in a problem like the one above is to assume that the stagnation point
is the place to start the solution of the problem. They look at the three
points in the flow and assume that the stagnation point must be the place where
everything is known. After all, isn’t the velocity at the stagnation point
equal to zero? Doesn’t this mean that the static pressure and the total
pressure are the same here? And what other conclusion can be drawn than to
assume that this pressure must then be the atmospheric pressure?
Well,
the answer to the first two questions is “yes” but a third “yes” does not
follow. What is known at the stagnation point is that the static pressure term
in the equation is now the static pressure at a stagnation point and is
therefore called the stagnation pressure. And, since the speed is zero, the
stagnation pressure is equal to the total pressure in the flow. Neither of
these pressures, however, is the atmospheric pressure.
Why
is the pressure at the stagnation point not the pressure in the atmosphere?
Well, this is where our substitution of a moving flow and a stationary wing for
a moving wing in a stationary fluid ends up causing us some confusion. In
reality, this stagnation point is where the wing is colliding head-on with the
air that it is rushing through. The pressure here, the stagnation pressure,
must be equal to the pressure in the atmosphere plus the pressure caused by the
collision between wing and fluid; i.e., it must be higher than the atmospheric
pressure.
Our
approach of modeling the flow of a wing moving through the stationary
atmosphere as a moving flow around a stationary wing makes it easier to work
with Bernoulli’s equation in general; however, we must keep in mind that it is
a substitute model and alter our way of looking at it appropriately. In this
model the hydrostatic pressure is not the pressure where the air is “static”,
it is, rather, the pressure where the flow is “undisturbed”. This is at the
“free stream” conditions, the point upstream of the body (wing, in this case)
where the flow has not yet felt the presence of the wing. This is where the
undisturbed atmosphere exists. Between that point and the wing itself the flow
has to change direction and speed as it moves around the body, so nowhere else
in the flow field will the pressure be the same as in the undisturbed
atmosphere.
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theorem, Bernoulli principle flight, Bernoulli's principle, examples of
Bernoulli's principle, importance of Bernoulli's principle