Bernoulli's equation and high velocity flights

[waynetan]

why can't we use bernoulli's equation for high velocity flights ? what changes when air is compressed ? does density change ?

 

[tiny-tim]

Welcome to PF!

Hi waynetan ! Welcome to PF!

Yes, "compressible" means that its density changes, and that change of density either consumes or liberates energy. Since Bernoulli's equation is a conservation-of-energy equation, that change in energy has to be taken into account.

(Also, Bernoulli's equation only applies to non-viscous flow.)

From the PF Library on Bernoulli's equation …
​

Bernoulli's equation for steady compressible inviscous flow:
kinetic energy per mass plus potential energy per mass plus enthalpy per mass is the same (is conserved) along any streamline of a flow.
…
Bernoulli's equation along any streamline of a steady non-viscous flow with variable internal energy (and therefore compressible):

$$P\ +\ \frac{1}{2}\,\rho\,v^2\ +\ \rho\,g\,h\ +\ \rho\,\epsilon\ =\ constant$$

or:

$$\frac{1}{2}\,\rho\,v^2\ +\ \rho\,g\,h\ +\ \text{enthalpy per unit mass}\ =\ constant$$
…

$\epsilon$ is the internal energy per unit mass, or specific internal energy (s.i.e)

Incompressible flow:

Incompressible flow is flow whose density is constant along any streamline. In such flow, internal energy may be omitted from Bernoulli's equation (in other words, enthalpy per unit mass may be omitted, and replaced by pressure).

For incompressible flow, internal energy per mass is constant, and so for steady inviscous flow, pressure plus the external energy density must be constant along any streamline:

$$P\ +\ \frac{1}{2}\,\rho\,v^2\ +\ \rho\,g\,h\ =\ constant$$

 

[rcgldr]

What changes when air is compressed? does density change ?

There is a form of Bernoulli's equation that handles compression and expansion. The density changes, lower during expansion, greater during compression. Since pressure is energy per unit volume, as opposed to energy per unit mass, it's affected by density. The terms in Bernoulli's equation include a pressure term, and two other terms multiplied by density (instead of mass).

Why can't we use bernoulli's equation for high velocity flights?

Bernoulli is a simplied model that doesn't deal with factors like turbulent flow. It doesn't account for the internal energy of the eddies in a turbulent flow. It doesn't account for temperature changes due to compression or expansion of air. It doesn't deal with supersonic flows that involved shock waves. The more generalized Navier Stokes equations handle most of this, but generally they can't be solved, so an airfoil model uses some simplication of Navier Stokes.

 

[waynetan]

since the Bernoulli's equation does not apply to supersonic flights, then how is lift generated at supersonic and subsonic speeds in supersonic aircraft with thin airfoil ?

 

[rcgldr]

At a macroscopic scale, lift is generated when air is accelerated downwards (and drag is generated with air is acclerated forwards). Bernoulli doesn't explain how pressure differentials around a wing are created by the interaction between the wing and the air, only how the air responds internally once the pressure differentials exist. Bernoulli is mostly about the obvious fact that air will accelerate from higher presssure zones to lower pressure zones, and Bernoulli's equation approximates the relationship between speed and pressure (and optionally density) during this transition, ignoring issues like turbulence.

There are many web sites that describe how wings generate lift, with some conflictling view points and various levels of detail. This site does a good job of explaining lift without getting too carried away with details. There are plenty of other good web sites as well, but this one is a good starting point, and includes a pair of diagrams showing how the air is affected as a wing travels through it.

http://www.avweb.com/news/airman/183261-1.html