Bernoulli's Principle and the Continuity Equation

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Bernoulli's Principle and the Continuity Equation

Post  Admin on Tue Dec 01, 2009 6:51 am

Daniel Bernoulli (1700-1782) was a Swiss mathematician who propounded the principle that for a given parcel of freely flowing fluid, the sum of kinetic energy, gravitational potential energy and static pressure energy always remains constant. (Incidently his father was the mathematician who first adopted the symbol 'g' for the acceleration due to gravity). For aerodynamic purposes, the gravitational potential energy can be ignored. Kinetic energy = ½mv² where m = mass and density [r] is mass per unit volume. Thus, dynamic pressure is kinetic energy per unit volume and static pressure is internal kinetic energy per unit volume; i.e. potential energy.

Note: a lower case v is the symbol for speed in physics while an upper case V is generally the symbol for the free stream speed in aerodynamics.

So, Bernoulli's principle can be reduced to:
½rv² [dynamic pressure or kinetic energy] + P [static pressure or pressure potential energy] = constant

The equation doesn't take into account viscosity, heat transfer or compressibility effects, but for operations below 10 000 feet and airflow velocities below 250 knots, compressibility effects can be ignored — thus no change in flow density [r] is assumed.

The equation then indicates that, in a free stream flow, if speed [v] increases static pressure [P] must decrease to maintain constant mechanical energy per unit volume; and the converse — if speed decreases, static pressure must increase. Or, turning it around, a free stream airflow will accelerate in a favourable pressure gradient and decelerate in an adverse pressure gradient.

Bernoulli's principle doesn't apply in boundary layer flow because the viscosity effects introduce loss of mechanical and thermal energy due to the skin friction.

Another aspect of the equation is that the constant is the stagnation pressure — the pressure energy needed to halt the airflow — thus it can be written ½rv² + P = stagnation pressure; the stagnation pressure is the highest pressure in the system. This aspect of Bernoulli's principle is used in the air speed indicator, as demonstrated below.

Stagnation pressure is also the basis of the parachute wing. Those wings consist of an upper and lower fabric surface enclosing individual front-opening cells. In a moving airstream, the pressure in the cells is near (fabric permeability has an effect) the stagnation pressure — the highest — and thus forms the semi-rigid wing shape that provides the high manoeuvrability of such parachutes.
Continuity equation

There is another principle of aerodynamic interest to us — the continuity equation — which states that, in a steadily moving airstream, the product of density, velocity and cross sectional area [s] must always be a constant:

r × s × v = constant

If there is no change in density within the flow (which is the norm in the airspeed range of light aircraft; see compressibility effects) then we can state that:

s × v = constant

Thus, if air flows into a smaller cross-sectional area speed must increase to maintain the constant. Bernoulli's principle states that if speed increases, static pressure must decrease; so the velocity of a constricted airstream increases and its static pressure decreases.

Both the above equations are related to the conservation laws; Bernoulli's principle to the conservation of energy, and the continuity equation to the conservation of mass. We will examine these properties of air further in the 'Aerofoils and wings' module.

The venturi effect — used in carburettors, the total energy variometer and the airframe-mounted venturi that provides suction for some flight instruments — is an application of the principles stated above.

The layout of the information in this post comes directly from http://www.auf.asn.au/index.html
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