# Continuity and Bernoulli's equation in air

• andrew700andrew
In summary, the formation of the Karman vortex street occurs as fluid accelerates from the center to get around a cylindrical object, creating a low pressure on top of the cylinder. This can be explained by the Continuity Equation and Bernoulli's equation, which suggest that either pressure or velocity must increase to maintain an equal relationship. However, for compressible fluids, the Bernoulli equation cannot fully apply and a state equation must be used.
andrew700andrew
Hi, I'm trying to understand vortex shedding and how the Karman vortex street occurs when air flows around a cylindrical object, so far it's going OK but then I came across this part of the explanation which leaves me confused:

"Looking at the figure above, the formation of the separation occurs as the fluid accelerates from the centre to get round the cylinder (it must accelerate as it has further to go than the surrounding fluid). It reaches a maximum at Y, where it also has also dropped in pressure. The adverse pressure gradient between here and the downstream side of the cylinder will cause the boundary layer separation if the flow is fast enough, (Re > 2.)"

[Taken from here http://www.efm.leeds.ac.uk/CIVE/CIVE1400/Section4/boundary_layer.htm]

What I'm unsure about is why the air must accelerate to travel around the cylinder, which in turn creates a low pressure on top of the cylinder. I've tried to figure it out using the Continuity Equation p1*A1*v1 = p2*A2*v2 (p=density or pressure, A=area, v=velocity) and Bernoulli's equation but I come to a problem because assuming that flow Area decreases as you approach the middle of the cylinder, either p (i.e. pressure) or v could increase to maintain the equal relationship. If it's true that the pressure could increase (given this is air) then that would contradict the Bernoulli equation which suggests that the pressure should decrease as velocity increases. Bearing in mind that I haven't started University yet is there some way you could explain this to me?

Also, how can the Bernoulli's equation apply to this when air is compressible?

Thanks allot.

Last edited:
Bernoulli equation is the energy conservation for Newtonian fluids. Newtonian fluids have no viscosity so reacts with bounds only by continuity equation. For one particle the energy is the sum of kinetic and external field dynamic energy. For a system of particles we must add a term of interaction energy, the internal energy. So:
$$U + \sum \frac{1}{2}m_iv_i^2 + \sum m_igh_i = C(t) \,\Rightarrow\, \frac{U}{V} + \sum \frac{1}{2}\rho_iv_i^2 + \sum \rho_igh_i = C'(t)$$
All terms have energy density dimentions (the 1st is the pressure) and for comppresible fluids you must use something like state equation ## P = \rho RT##.

## 1. What is continuity in air flow?

Continuity is the principle that states that the amount of air entering a system must be equal to the amount of air exiting the system. In other words, the mass flow rate of air must remain constant for steady-state air flow.

## 2. How is continuity related to Bernoulli's equation in air?

Continuity is one of the fundamental assumptions of Bernoulli's equation for incompressible fluids. In air flow, it means that the density and velocity of air must be inversely proportional. As air speeds up, its density decreases, and vice versa.

## 3. What is Bernoulli's equation in air?

Bernoulli's equation is a mathematical equation that describes the relationship between pressure, velocity, and height in a fluid. In air flow, it states that as the speed of air increases, the pressure decreases, and vice versa.

## 4. How is Bernoulli's equation applied in real-world situations?

Bernoulli's equation has many practical applications, such as in the design of airplane wings, wind turbines, and ventilation systems. It can also be used to calculate the lift force of an airplane and the flow rate of air through a pipe.

## 5. What are the limitations of Bernoulli's equation in air?

Bernoulli's equation assumes that air is incompressible, inviscid (no friction), and follows a steady-state flow. In reality, these conditions are not always met, and therefore, the equation may not accurately describe air flow in all situations. Additionally, it does not take into account factors such as turbulence, boundary layer effects, and compressibility.

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