Misinterpretations of Bernoulli's Law
Weltner, Klaus and Ingelman-Sundberg, Martin
Department of Physics, University Frankfurt, Postfach 11 19 32, 60054 Frankfurt, Germany; Stockholm
Abstract. Bernoulli's law and experiments attributed to it are fascinating. Unfortunately some of these experiments are explained erraneously, e.g.: the function of a vaporizer and the soaring of a ping-pong ball in a jet stream of a hair dryer can not be used as applications of Bernoulli's law. The static pressure in a free jet stream is equal to the static pressure in the environmental atmosphere regardless of the streaming velocity of the jet. This can be shown by classroom experiments.
Acceleration of air is caused by pressure gradients. Air is accelerated in direction of the velocity if the pressure goes down. Thus the decrease of pressure is the cause of a higher velocity. It is wrong to say that a lower pressure is caused by a higher velocity.
Pressure gradients perpendicular to the streamlines are caused by the deflection of streaming air. The deflection of air generates regions of lower and higher pressure according to the curvature of the streamlines. Vaporizer, the soaring ping-pong ball as well as the physics of flight are only to be explained regarding the acceleration perpendicular to the streamlines.
1. Common derivation and applications of Bernoulli's law
In a recent paper Baumann and Schwaneberg [1] state:
Bernoulli's Equation is one of the more popular topics in elementary physics. It provides striking lecture demonstrations, challenging practice problems, and plentiful examples of practical applications from curving baseballs to aerodynamic lift. Nevertheless, Students and Instructors are often left with an uncomfortable feeling that the equation is clear and its predictions are verified, but the real underlying cause of the predicted pressure changes is obscure.
This statement is correct and it should be added that the common treatment of Bernoulli's equation is also misleading. Generally a flow of an incompressible fluid through a tube with different cross-sections is observed and the theory of conservation of energy is applied to the flow.
The energy of a volume V at any point is the sum of its kinetic energy and its potential energy (pV). Effects of gravitation and viscosity are neglected. The energy of a given volume of the fluid which moves from point 1 to point 2 is the same at both points. The related energy equation is
(1)
Using and rearranging we arrive at Bernoulli's Law:
(2)
The equation states a reversed relation between static pressure and streaming velocity which is often demonstratet by experiments like
Soaring ball: A light ball (e.g. ping pong ball) can be kept soaring in an upwards directed air stream of a hair dryer. The ball remains within the stream even if the stream is inclined and not vertical. The explanation given is that the static pressure within the stream is less due to the higher velocity.
Evaporator: If a fast stream of air passes over
the opening of a pipe, the pressure inside is lowered and it is possible to
suck in liquids. This effect is used as an application of Bernoulli's law
[2] referring to the high streaming velocity within the air stream and
claiming p_{stream} < p_{atmosphere} .
Aerodynamic lift: The higher streaming velocity of the air at the upper surface of the wing is stated to be the cause of the lower pressure. Different reasons are given for the generation of the higher streaming velocity. The most popular one is a comparison of path lengths of the flow above and below the aerofoil and the statement that due to a longer path length at the upper side the flow has to be faster [3], [1].
2. Misinterpretations and misapplications of Bernoulli's law
2.1 Static pressure in a free air stream
Static pressure is the pressure inside the stream measured by a manometer moving with the flow. At the same time, the static pressure is the pressure which is excerted on a plane parallel to the flow. Thus the static pressure within an air stream has to be measured carefully using a special probe. A thin disk must cover the probe except for the opening. The disk must be positioned parallel to the streaming flow, so that the flow is not interfered with.
If the static pressure is measured in the way outlined above within a free air stream generated by a fan or a hair dryer it can be shown that the static pressure is the same as in the surrounding atmosphere. Bernoulli's law cannot be applied to a free air stream because friction plays an important role. It may be noted that the situation is similar to the laminar flow of a liquid with viscosity inside a tube. The different velocity of the stream layers is caused by viscosity. The static pressure is the same throughout the whole cross-section. A free air stream in the atmosphere is exlusively decelerated by friction. If static pressure in a free air stream is equal to atmospheric pressure, some of the striking lecture demonstrations are interpreted incorrectly since the effects observed are not caused by Bernoulli's law.
2.2. Aerodynamic lift
The explications referring to differences of path lengths are wrong. Air volumes which are adjacent before separation at the leading edge of the aerofoil do not meet again at the trailing edge [4]. This explanation is erranous. The higher streaming velocity at the upper surface of the aerofoil is not the cause of lower pressure. It is the other way round as will be shown below. As a matter of fact the higher streaming velocity is the consequence of the lower pressure at the upper surface of the wing [4].
These contradictions and misunderstandings can only be clarified by means of the basic physics of fluid mechanics.
3. Fluid dynamics, Newtons laws and the Euler equations
Fluid dynamic is an extension of Newton’s mechanic. It was Euler who applied the fundamental laws of Newton to fluid motion. He succeeded in establishing equations for the three dimensional fluid motion - the Euler equations. For simplicity reasons we restrict our considerations to stationary flow and we neglect effects of gravitation and viscosity [5]. We refer to an elementary cubical volume within curved streamlines. Figure 3. The reference system is chosen deliberately to separate the direction of velocity and its perpendicular. We analyse the acceleration of a mass element . We separate the components of the acceleration:
Tangential acceleration = acceleration in direction of the velocity (Figure 3)
Normal acceleration = acceleration perpendicular to the direction of velocity (Figure 4)
Tangential acceleration in s-direction.
An acceleration is the result of a force acting on the mass element. A force in direction of the velocity can only be generated by a pressure difference. The static pressure acting on the aera A at the back must exceed the pressure on the aera A at the front.
Acceleration in direction of the motion is the effect of a decrease of pressure. The force F is given by: (3)
Thus Newton’s equation reads
(4)
Using and we obtain
(5)
This equation can be transformed to
(6)
The definite integral for two positions 1 and 2 is
(7)
The solution is
(8)
This is Bernoulli's law.
This derivation of Bernoulli's law is more instructive compared to the derivation generally used in textbooks because it shows the physics behind the law. The streaming fluid accelerates as a result of decreasing pressure (i.e. or a negative pressure gradient). This derivation clearly shows that an acceleration can never be the cause of decreasing pressure.
Normal acceleration exists if streamlines are curved. A normal acceleration is the effect of a force in direction of the radius of curvature. In the case of the elementary volume the pressure acting on the outside area must exceed the pressure on the inside area.
The force referring to the z-axis is: The negative sign is due to the fact that the force has the opposite direction of a positive pressure gradient.
Newton’s equation reads
(9)
The normal acceleration is well known for circular motions with a radius R and a velocity v:
(10)
Inserting and in (9) we obtain the pressure gradient in z-direction.
(11)
Unfortunately this equation can only be integrated if the total field of the flow is known. However the relation can be demonstrated in a simple and impressing way. If we make water rotate in a disk or a pot, the surface of the water rises at the outer parts. The level of the surface is a manometer indicating the pressure beneath. Assuming homogenous angular velocity of the circular flow of water the velocity is . Thus equation (9) may be solved for a horizontal level beneath the surface neglecting gravitational pressure:
(12)
(13)
The pressure is proportional to the square of the radius generating a parabolic surface.
(14)
This result is also well known for centrifuges.
As a rule, physics textbooks neglect the treatment of normal acceleration of fluids. They do not discuss the pressure gradients normal to the velocity if streamlines are curved. By the way, this is different from textbooks on technical fluid dynamics which treat the flow of fluids in curved tubes. The neglect of pressure gradients related to curved streamlines is disastrous because the mechanism producing low pressure is thus made impossible to understand. Obstacles cause curved streamlines and generate pressure gradients of air and as a consequence regions of higher or lower pressure. The deflection of the streaming is the cause for the generation of pressure gradients perpendicular to the streamlines and thus the cause for the generation of pressure differences.
3.2 Coanda-effect.
The flow near limiting surfaces follows the geometrical shape of these surfaces. This behaviour is called Coanda-effect. It is neither trivial nor general. The flow must not be forced to change its direction abruptly as to avoid the generation of turbulence and separation. The classic example for the Coanda-effect is a flow blown across a flat plane with an adjacent half cylinder. At first the flow follows the surface of the cylinder and separates later.
The Coanda-effect can be understood taking viscosity into consideration. In figure 6 we assume the stream to start. It will flow horizontally. But due to viscosity some layers of the adjacent air will be taken away by the stream. In this adjacent region - dotted in figure 6 - the air is sucked away and hence, gives rise to a reduction of pressure, consequently producing a normal acceleration of the stream. By the end of the process the stream fits the shape of the curved surface. This Gedankenversuch illustrates the importance of viscosity in generating of stationary flow. Also the stationary flow around an aerofoil which produces lift is only possible due to the Coanda-effect and the air's viscosity.
4. Generation of high and low pressure within a flow
4.1 Measurement of static pressure within a free stream of air
A sufficiently sensitive manometer can be produced easily if not available in the lab. A fine pipe of glas is bent at one side to dip in a cup and to be fixed according to figure 7. The meniscus must be positioned in the middle of the pipe. The suitable inclination should be 1:15 - 1:30. A rubber tube connects the glas pipe with a probe. As has been pointed out before a flat disk must be glued on top of the probe leaving the opening free. The disk has to be held parallel to the streaming. If the static pressure is measured in such a way it can be shown that it is equal to the pressure in the environmental atmosphere.
4.2 Generation of high and low pressure by deflection of an air stream
According to figure 8 we place a curved plane into the air stream of a fan. A curved plane can be produced by glueing two postcards on top of each other. By fixing them around a bottle with a rubber, an appropriate curvature can be achieved.
Due to the Coanda-effect, the air stream follows the shape of the curved
plane at the lower side. The stream follows the upper side because there is
no other way left to move.
The curved plane forces a curved flow resulting in a radial pressure gradient. Outside of the flow there is atmospheric pressure.
Due to the pressure gradient inside the curved air stream an increase of pressure is to be expected at the inner or convex side of the plane in relation to the center center of curvature. Figure 8. At the outer or concave side a decrease of pressure is to be expected. Figure 9. This increase and decrease exists indeed and can be demonstrated using the manometer described above. See figure 8 and figure 9. The experiment shows that by deflecting of an air stream regions of increased or decreased pressure may be generated. This experiment is fundamental for the understanding of the production of pressure differences if air passes obstacles. By analysing the curvature of the evasive flow we can predict pressure distribution. It should be added that in this case Bernoulli's law still holds since friction may be neglected. Since in figure 8 the pressure increases at the inner surface the local streaming velocity is reduced.
In figure 9 the pressure decreases at the outer surface and the streaming velocity increases. The physical mechanism is quite obvious. The curved plane causes a curved streaming flow and a decrease of pressure. Hence incoming air is accelerated by the decrease of pressure.
The experiment requires an air stream the cross section of which should exceed the width of the postcard. If a hair-dryer is used which produces a narrow air stream it is advisable to glue the curved plane between two even planes of glass or plastic to confine the air stream. The distance of the limiting planes should be equal to the diameter of the air stream produced by the hair-dryer.
4.3 Examples and applications
Hill: If air passes a hill - figure 10- it follows the shape of it. A deformation of the original horizontal flow occurs only in the surroundings of the hill. Further away we observe normal atmospheric pressure and horizontal flow.
We first analyse the curvature following the trajectory A.The trajectory starts from the bottom of the hill and is continued perpendicular to the streamlines. The streaming air is deflected upwards. The air is accelerated upwards too. Starting from the bottom and going outwards the pressure has to decrease in order to produce the acceleration upwards. Because of the atmospheric pressure further away there must be a higher pressure at the bottom of the trajectory A.
In the case of trajectory B starting from the top of the hill the curvature of the streamlines is reversed. The streaming air is accelerated downwards throughout the whole trajectory. Following this trajectory the pressure increases starting from the top until it reaches its normal value further away. Thus at the top of the hill we expect a reduced pressure. In the case of the trajectory C we expect the same as for trajectory A. Modelling the hill with bent postcards these results can be demonstrated experimentally as well.
Evaporator: These considerations give an explanation of the mechanism for the evaporator. A pipe dipping in a flow of air forces an evasive flow (see figure 11). This is a situation similar to that of the hill. The streaming is curved over the nozzle of the pipe and the acceleration directs to the aperture. Therefore lower pressure is generated at the nozzle.
Forces on a roof: If wind passes a house the stream is to flow around it. Due to the curvature of the evasive flow there is higher pressure at the front side and lower pressure at the peak of the roof.
The effect of pressure differences on the roof is maximized if front doors or windows are opened. In this case there is high pressure inside the house. The pressure difference acting on the roof is increased.
If windows or doors at the rear are opened there is a lower pressure inside the house that reduces the pressure difference acting on the roof.
Aerodynamic lift: The aerodynamic lift, too, is a result of the evasive flow caused by the aerofoil. The streamlines near the wing are determined by the latter’s shape and position. As a whole the stream is deflected downwards. (See figure 8 and figure 9.)
Propulsion by a sail: The same phenomenon can be observed in the case of a sail. A sail is a curved plane similar to figure 8 and 9. The sail deflects the air flow and produces an increase of pressure at the inner side in relation to the center of curvature and a decrease of pressure at the outer side. By this way it generates a force normal to the sail. Skilled sailors keep the streaming of the air smooth and laminar and avoid turbulent and separating flow.
6. Conclusion
The deliberation of Bernoulli's law in schools and textbooks has serious drawbacks. Unfortunately many applications are erranous and misleading. One source of confusion is the derivation of Bernoulli's law based on the theorem of energy conservation. Bernoulli's law should be derived from the tangential acceleration as a consequence of declining pressure. Another source of difficulties is the fact that many physics textbooks do not mention normal acceleration of flow and the resulting pressure gradients perpendicular to the flow.
Both, Bernoulli's law and the generation of pressure gradients perpendicular to the flow are consequences of Newton’s laws. None of them contradicts those.
Bernoulli's law is insufficient to explain the generation of low pressure. A faster streaming velocity never produces or causes lower pressure. The physical cause of low or high pressure is the forced normal acceleration of streaming air caused by obstacles or curved planes in combination with the Coanda-effect. Pressure gradients generated by the deflection of streaming air can be clearly demonstrated by simple experiments which would substantially improve the discussion of fluid mechanics in schools and textbooks.
Literature
[1] | Baumann, R.; Schwaneberg, R.: "Interpretation of Bernoulli's Equation", The Physics Teacher, Vol. 32, Nov. 1994, pp. 478 - 488 |
[2] | Paus, H.J.: "Physik in Experimenten und Beispielen" München/Wien, 1995. |
[3] | Mansfield, M; O’Sullivan, C.: "Understanding Physics", Chichester, New York, 1998 |
[4] | Weltner, K.; Ingelman-Sundberg, M.: "Physics of Flight - reviewed", submitted to Eurpean Journal of Physics |