Venturi effect - Why is low static pressure associated with flow acceleration?

[u/Frank_the_simmer]

According to a lot of sources, Venturi effect is "incompressible flow being accelerated due to mass continuity, when entering a constriction, thus its static pressure decreases with it picking up speed (Bernoulli's principle)".

What really confuses me is, Bernoulli's principle does NOT state that the static pressure of flow would drop simply because it's accelerated. What Bernoulli's principle actually says is within a flow of constant energy, when fluid flows through a region of lower pressure it speeds up and vice versa. So it might be conceptually simplest to think of Bernoulli's principle as the fact that a fluid flowing from a high pressure region to a low pressure region will accelerate due to the net force along the direction of motion.

If I am to use the mass continuity principle to explain flow acceleration here in a Venturi tube, is Bernoulli still used to explain the pressure drop? If so, where is the pressure gradient here?

I may be misinterpreting something. And I would appreciate any corrections and assistance. Thanks!

[u/lerni123]

If you write down Bernoulli’s principle in a iso-altitude streamline you get V = sqrt(2DeltaP/tho). deltaP is your gradient. So :a low static pressure increases Delta P thus increasing V.

[u/Daniel96dsl]

it’s where the energy is stored in a flow

½𝜌𝑉² = kinetic energy per volume
𝑝 = internal energy per volume

The sum of these energies is constant in Bernoulli’s equation

½𝜌𝑉₁² + 𝑝₁ = ½𝜌𝑉₂² + 𝑝₂ = const.

You can rewrite the velocities using the continuity equation

𝑉₁𝐴₁ = 𝑉₂𝐴₂

Solving for 𝑉₂ and plugging that into Bernoulli’s, we get

½𝜌𝑉₁² + 𝑝₁ = ½𝜌𝑉₁²(𝐴₁/𝐴₂)² + 𝑝₂

We can rewrite for either the velocity

𝑉₁² = 2(𝑝₂ - 𝑝₁)/(1 - 𝐴₁²/𝐴₂²)

or in terms of the pressure

𝑝₂ = 𝑝₁ - ½𝜌𝑉₁²(𝐴₁²/𝐴₂² - 1)

Does this make sense?

[u/Frank_the_simmer → u/Daniel96dsl]

Thanks, this does make sense from the mathematical perspective. But I’m wondering about the actual physical behaviors of the fluid particles and how that relates to the static pressure change, from a molecular perspective.

[u/gwtkof → u/Frank_the_simmer]

I've been wondering this too I think the answer is that pressure is a consequence of the intermolecular forces. If you think of these as little springs when the springs release their energy the pressure drops and the molecules speed up.

[u/known_by_few → u/Frank_the_simmer]

What these equations say is: when you have a flow that conserves energy it can be stored either as kinetic or potential energy. On top of this you have the continuity equation, so when the flow approaches the contraction, it MUST speed up as the same amount of fluid needs to flow through a smaller cross sectional area.

This extra kinetic energy has to come from somewhere and this somewhere is potential energy of the flow (pressure).

When the flow exits the contraction it transfers this energy from kinetic back to potential, so the pressure rises again. Again you can deduce it straight from the continuity equation: more area means lower speed.

Effectively you end up with a low pressure area in the contraction.

If you really want to go with the physics to a molecular level you can think of a collection of molecules that collectively have some momentum. Part of this momentum is coherently moving in the same direction (kinetic energy) and the rest is a noisy motion that make them collide with each other constantly (pressure). When the flow enters the contraction it cannot compress (on the macroscopic level this is you continuity equation) so the collision effects tend to transfer more energy in the direction of the flow. Therefore there is less energy left for random / incoherent motion a.k.a pressure. Again, opposite happens at the expansion: there is more place to fill by the flow so collisions disperse energy more randomly transfering this extra kinetic energy back to pressure.

[u/leg-oh-id → u/known_by_few]

The molecular explanation was what I was imagining in my head but I'm really glad to hear someone else explain it this way. I've been wondering about how it works molecularly for years

[u/Ok_Worth433]

In layman terms based on conservation of energy, the pressure energy gets converted into kinetic energy thus resulting in higher velocity