Bernoulli Equation

The Bernoulli Equation can be considered to be a statement of the conservation of energy principle appropriate for flowing fluids. The qualitative behavior that is usually labeled with the term "Bernoulli effect" is the lowering of fluid pressure in regions where the flow velocity is increased. This lowering of pressure in a constriction of a flow path may seem counterintuitive, but seems less so when you consider pressure to be energy density. In the high velocity flow through the constriction, kinetic energy must increase at the expense of pressure energy.

Bernoulli calculation

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Bernoulli concepts
 
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Bernoulli Calculation

The calculation of the "real world" pressure in a constriction of a tube is difficult to do because of viscous losses, turbulence, and the assumptions which must be made about the velocity profile (which affect the calculated kinetic energy). The model calculation here assumes laminar flow (no turbulence), assumes that the distance from the larger diameter to the smaller is short enough that viscous losses can be neglected, and assumes that the velocity profile follows that of theoretical laminar flow. Specifically, this involves assuming that the effective flow velocity is one half of the maximum velocity, and that the average kinetic energy density is given by one third of the maximum kinetic energy density.
Now if you can swallow all those assumptions, you can model* the flow in a tube where the volume flowrate is = cm^3/s and the fluid density is = gm/cm^3. For an inlet tube area = cm^2 (radius =cm), the geometry of flow leads to an effective fluid velocity of =cm/s. Since the Bernoulli equation includes the fluid potential energy as well, the height of the inlet tube is specified as = cm. If the area of the tube is constricted to =cm^2 (radius = cm), then without any further assumptions the effective fluid velocity in the constriction must be = cm/s. The height of the constricted tube is specified as = cm.

The kinetic energy densities at the two locations in the tube can now be calculated, and the Bernoulli equation applied to constrain the process to conserve energy, thus giving a value for the pressure in the constriction. First, specify a pressure in the inlet tube:
Inlet pressure = = kPa = lb/in^2 = mmHg = atmos.
The energy densities can now be calculated. The energy unit for the CGS units used is the erg.
Inlet tube energy densities
Kinetic energy density = erg/cm^3
Potential energy density = erg/cm^3
Pressure energy density = erg/cm^3
Constricted tube energy densities
Kinetic energy density = erg/cm^3
Potential energy density = erg/cm^3
Pressure energy density = erg/cm^3
The pressure energy density in the constricted tube can now be finally converted into more conventional pressure units to see the effect of the constricted flow on the fluid pressure:

Calculated pressure in constriction =
= kPa = lb/in^2 = mmHg = atmos.

This calculation can give some perspective on the energy involved in fluid flow, but it's accuracy is always suspect because of the assumption of laminar flow. For typical inlet conditions, the energy density associated with the pressure will be dominant on the input side; after all, we live at the bottom of an atmospheric sea which contributes a large amount of pressure energy. If a drastic enough reduction in radius is used to yield a pressure in the constriction which is less than atmospheric pressure, there is almost certainly some turbulence involved in the flow into that constriction. Nevertheless, the calculation can show why we can get a significant amount of suction (pressure less than atmospheric) with an "aspirator" on a high pressure faucet. These devices consist of a metal tube of reducing radius with a side tube into the region of constricted radius for suction.

*Note: Some default values will be entered for some of the values as you start exploring the calculation. All of them can be changed as a part of your calculation.
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Curve of a Baseball

A non-spinning baseball or a stationary baseball in an airstream exhibits symmetric flow.

A baseball which is thrown with spin will curve because one side of the ball will experience a reduced pressure. This is commonly interpreted as an application of the Bernoulli principle and involves the viscosity of the air and the boundary layer of air at the surface of the ball. The roughness of the ball's surface and the laces on the ball are important! With a perfectly smooth ball you would not get enough interaction with the air.

There are some difficulties with this picture of the curving baseball. The Bernoulli equation cannot really be used to predict the amount of curve of the ball; the flow of the air is compressible, and you can't track the density changes to quantify the change in effective pressure. The experimental work of Watts and Ferrer with baseballs in a wind tunnel suggests another model which gives prominent attention to the spinning boundary layer of air around the baseball. On the side of the ball where the boundary layer is moving in the same direction as the free stream air speed, the boundary layer carries further around the ball before it separates into turbulent flow. On the side where the boundary layer is opposed by the free stream flow, it tends to separate prematurely. This gives a net deflection of the airstream in one direction behind the ball, and therefore a Newton's 3rd law reaction force on the ball in the opposite direction. This gives an effective force in the same direction indicated above.

Similar issues arise in the treatment of a spinning cylinder in an airstream, which has been shown to experience lift. This is the subject of the Kutta-Joukowski theorem. It is also invoked in the discussion of airfoil lift.

Index

Bernoulli Equation

Bernoulli concepts

Reference
Watts and Ferrer
 
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Airfoil



Illustration of lift force
and angle of attack
Bernoulli vs Newton
for airfoil lift
Airfoil terminology

The air across the top of a conventional airfoil experiences constricted flow lines and increased air speed relative to the wing. This causes a decrease in pressure on the top according to the Bernoulli equation and provides a lift force. Aerodynamicists (see Eastlake) use the Bernoulli model to correlate with pressure measurements made in wind tunnels, and assert that when pressure measurements are made at multiple locations around the airfoil and summed, they do agree reasonably with the observed lift. Others appeal to a model based on Newton's laws and assert that the main lift comes as a result of the angle of attack. Part of the Newton's law model of part of the lift force involves attachment of the boundary layer of air on the top of the wing with a resulting downwash of air behind the wing. If the wing gives the air a downward force, then by Newton's third law, the wing experiences a force in the opposite direction - a lift. While the "Bernoulli vs Newton" debate continues, Eastlake's position is that they are really equivalent, just different approaches to the same physical phenonenon. NASA has a nice aerodynamics site at which these issues are discussed.

Increasing the angle of attack gives a larger lift from the upward component of pressure on the bottom of the wing. The lift force can be considered to be a Newton's 3rd law reaction force to the force exerted downward on the air by the wing.

At too high an angle of attack, turbulent flow increases the drag dramatically and will stall the aircraft.

Index

Bernoulli Equation

References
Eastlake


NASA
Aerodynamics
 
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