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Bernoulli Equation

The Bernoulli Equation can be considered to be a statement of the conservation of energy principleappropriate 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 youconsider 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

Index

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 ofviscous losses, turbulence, and the assumptions which must be made about the velocity profile

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(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 thatviscous 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 maximumkinetic energy density.

Now if you can swallow all those assumptions, you can model* the flow in a tube where the volume

flowrate is = cm3/s and the fluid density is ρ = gm/cm3. For an inlet tube

area A1= cm2 (radius r1 = cm), the geometry of flow leads to an effective

fluid velocity of v1 = cm/s. Since the Bernoulli equation includes the fluid potential energy

as well, the height of the inlet tube is specified as h1 = cm. If the area of the tube is

constricted to A2= cm2 (radius r1 = cm), then without any further

assumptions the effective fluid velocity in the constriction must be v2 = cm/s. The height

of the constricted tube is specified as h2 = 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 inthe constriction. First, specify a pressure in the inlet tube:

Inlet pressure = P1 = kPa = lb/in2 = 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/cm3

Potential energy

density= erg/cm3

Pressure energy

density

=

erg/cm3

Constricted tube energy densities

Kinetic energy density = erg/cm3

Potential energy

density= erg/cm3

Pressure energy

density

=

erg/cm3

The pressure energy density in the constricted tube can now be finally converted into moreconventional pressure units to see the effect of the constricted flow on the fluid pressure:

Calculated pressure in constriction =

Index

Bernoulli

concepts

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P2= kPa = lb/in2 = 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 energydensity 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 consistof 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 reducedpressure. 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 theball's surface and the

laces on the ball are Index

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important! With a

perfectly smooth ballyou 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, andyou can't track the density changes to quantify the change in effective pressure. The experimentalwork 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 ballwhere 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 sidewhere 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 3rdlaw reaction force on the ball in the opposite direction. This gives an effective force in the samedirection indicated above.

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

discussion of airfoil lift.

BernoulliEquation

Bernoulliconcepts

Reference

Watts andFerrer

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Airfoil

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The air across the top of a conventional airfoil experiences constricted flow lines and increased airspeed 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 tocorrelate 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 agreereasonably with the observed lift.

Illustration of lift force

and angle of attack

Bernoulli vs Newtonfor airfoil lift

Airfoil terminology

Others appeal to a model based on Newton'slaws and assert that the main lift comes as a

result of the angle of attack. Part of theNewton's law model of part of the lift force

involves attachment of the boundary layer ofair on the top of the wing with a resultingdownwash of air behind the wing. If the wing

gives the air a downward force, then byNewton's third law, the wing experiences a

force in the opposite direction - a lift. Whilethe "Bernoulli vs Newton" debate continues,

Eastlake's position is that they are reallyequivalent, just different approaches to thesame physical phenonenon. NASA has a nice

aerodynamics site at which these issues arediscussed.

Increasing the angle of attack gives a largerlift from the upward component of pressure

on the bottom of the wing. The lift force canbe considered to be a Newton's 3rd lawreaction 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 stallthe aircraft.

Index

BernoulliEquation

References

Eastlake

NASA Aerodynamics

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A vapor trail over the wing helps visualize the air flow. Photo by Frank Starmer, used by

permission.

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