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I'm studying Aerodynamics and I'm thinking about the Bernoulli Equation Statement.

Ignoring gravity contributes, it states that, in a potential fluid, the sum of static pressure and kinetic energy (per unit of mass) is a constant:

$$P+0.5ρv^2=const$$

Typically, to solve the pressure distribution around an aerofoil (in order to calculate lift) in the context of a potential fluid, we can solve the Divergence problem about the velocity field:

$$\nabla \cdot V =0$$

Once the velocity field has been determined, we can use the Bernoulli equation to find the static pressure value in every point of the fluid field.

So my question is:

If the velocity would be too large in some points of the field, may we find there a negative static pressure? Algebraically speaking the answer is yes, because the Bernoulli Equation may "mathematically" lead to this result. But negative (absolute) pressure has no physical meaning, so how can we know if the Bernoulli Equation application will lead to robust results? A negative pressure would represent an "exhaustion" of static pressure to convert in kinetic energy, so in these conditions the velocity field cannot be realized and the Bernoulli equation would be violated. As I know, if we are dealing with liquid, the occurrence of a negative pressure would lead to cavitation and boiling, but in aerodynamics we deal with air (a gas).

So, How can we deal with an aerodynamic problem in which such a circumstance occurs, and what are the conditions to garantee negative pressure not to happen?

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  • $\begingroup$ Potential flow theory predicts infinite velocities at every convex corner (leading edges of thin airfoils, for example). In spite of this, the predicted forces & moments can be quite realistic - as can flow details away from the corner points. Understanding the limitations of the theories is crucial in real-life applications. $\endgroup$
    – D. Halsey
    Commented Jun 3, 2022 at 15:06

1 Answer 1

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The Bernoulli equation is derived assuming the fluid is incompressible, which is an idealization even for liquids, and even more for gases.

This idealization is valid provided density $\rho$ in the fastest parts of the flow does not change much from its value where $v$ is minimal.

For liquids, this is usually the case. For gases, it depends on the flow speeds. Density of air at points where the flow has high velocity can drop substantially, and so can pressure $p$.

So in places where density of air gets too low, the original Bernoulli equation (with pressure term instead of enthalpy term) stops being accurate. In any case, air as well as other gases cannot have negative pressure, so $p<0$ won't happen.

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  • $\begingroup$ Hi, the flow can be considered incompressible as long as its speed remains below 0.3 Mach. Therefore we can say that the validity of the Bernoulli equation is limited by the value of the velocity. However, we could think of a situation where the velocity remains relatively low, but the initial static pressure is so small that it is not sufficient to convert it into the required kinetic energy predicted by the velocity field. Again, mathematically, we would run into negative static pressure. How to avoid this situation? When we solve an aerodynamic problem we never talk about this eventuality. $\endgroup$
    – user247296
    Commented Jun 3, 2022 at 13:06
  • $\begingroup$ The Bernoulli equation remains valid for steady compressible flows. It states that $$ \frac 12 v^2+h $$ is constant along streamlines ,where $h$ is the specific enthalpy. ($H=U+PV$ per unit mass). $\endgroup$
    – mike stone
    Commented Jun 3, 2022 at 13:36
  • $\begingroup$ @mikestone Indeed. And the $U$ term is why speed of the gas can increase a lot without $p$ going negative. $\endgroup$ Commented Jun 3, 2022 at 13:45
  • $\begingroup$ @tmox if pressure is very low, the speeds allowable for the original Bernoulli equation to be valid are also low. There is a more general Bernoulli equation which mikestone pointed out, which takes into account compression of the gas. This is valid when entropy transfer via conduction and entropy production in the gas is negligible (so the compressible flow must be "nice" enough). In places with very high speeds, internal energy per unit mass goes down too, so pressure can still be positive. $\endgroup$ Commented Jun 3, 2022 at 13:51
  • $\begingroup$ Considering the static air pressure between sea and flight altitudes (for example 10000 meters), it turns out that each of these pressures can be converted into kinetic energy by exceeding the speed of sound corresponding to the altitude considered. Therefore each of these pressures can satisfy the velocity field within the validity limits of the bernoulli equation as presented by me (it is the only one used in my textbook). If I understand your answer, an exceed in pressure drop goes beyond the field of classical applications in aerodynamics and we don't need more general Bernoulli equations. $\endgroup$
    – user247296
    Commented Jun 3, 2022 at 14:00

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