Lesson 12: Bernoulli Principle in Aerodynamics

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?

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The Bernoulli's theorem in simple words, in this video





 

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Click here for the previous lessons, to learn about: Principle of Airframe; Principles of Aerodynamics; Airfoil Characteristics; Primary Flight Control Surfaces; Description and Operation of Helicopter; Miscellaneous Components of an Aircraft…

What is Bernoulli's theorem in simple words?

 

Aerodynamic Bernoulli Principle Flight

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?

 
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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