Parabolic Pressure Distribution in a Jet

In summary: I'm not sure having both is redundant): $$p=0$$ $$u_2z=-u_1z$$Cross section of the jet where the velocity is uniform: $$p=0$$ $$\frac{\partial u}{\partial z}=-\frac{\partial p}{\partial z}$$
  • #1
person123
328
52
Assume the jet is straight but the radius of the jet varies over it's length (like a jet of water falling which narrows due to gravitational acceleration). Also ignore viscosity. A pressure gradient would be required to accelerate the fluid radially. Because during an expansion transformation, the distance a point moves (and so its acceleration) is proportional to its distance from the origin, the radial acceleration of fluid parcels would be proportional to their distance from the middle of the jet. The radial acceleration is proportional to the radial pressure gradient. This would mean the pressure gradient would be proportional to the distance from the middle, which would mean the pressure distribution is parabolic (the integral of a linear function is quadratic).

This was my reasoning; however I don't know my assumptions are reasonable and if the conclusion is accurate, so I would be interested in people's input. Thanks!
 
Physics news on Phys.org
  • #2
The neglect of viscosity is the fallacy in your reasoning. The radial component of the stress is equal to atmospheric pressure throughout the jet cross section. It is viscous stress (from the 3D version of Newton's law of viscosity) that is responsible for the jet contraction.
 
  • #3
Thank you for the response.
If the fluid was truly inviscid, what would occur? Would this just not be a meaningful scenario, or would the flow sort of detach (I'm not sure what this would physically entail), speeding up while not converging?
 
  • #4
Continuity requires that if the flow speeds up, it must converge. Viscosity makes this happen.
 
  • #5
Would it be possible to have an inviscid fluid undergoing these conditions?
 
  • #6
person123 said:
Would it be possible to have an inviscid fluid undergoing these conditions?
If the surface tension were high enough.
 
  • Like
Likes person123
  • #7
Thank you for the responses that makes sense. In the case of flow converging as it falls, I imagine this would be seen clearly when the flow is laminar and viscous forces and surface tension dominates.

I hope I'm not going into to many "what ifs," but would the pressure forces dominate for highly Reynold's number flow, such as a fast jet impacting a surface (with the jet normal to the surface)?

I wrote a simple code in MatLab to model the flow of a jet impacting a surface by following a particle on the boundary where I would assume the speed is constant. I had the code find the radial speed using what I would imagine the radial pressure gradient might be, find the average jet speed through continuity, find the new average pressure through Bernoulli's equation, and use that to find the pressure gradient assuming the average pressure is proportional to the pressure gradient (and of course repeat in a loop). I attached the result.
 

Attachments

  • JetImpact.png
    JetImpact.png
    2.5 KB · Views: 128
  • #8
person123 said:
Thank you for the responses that makes sense. In the case of flow converging as it falls, I imagine this would be seen clearly when the flow is laminar and viscous forces and surface tension dominates.

I hope I'm not going into to many "what ifs," but would the pressure forces dominate for highly Reynold's number flow, such as a fast jet impacting a surface (with the jet normal to the surface)?

I wrote a simple code in MatLab to model the flow of a jet impacting a surface by following a particle on the boundary where I would assume the speed is constant. I had the code find the radial speed using what I would imagine the radial pressure gradient might be, find the average jet speed through continuity, find the new average pressure through Bernoulli's equation, and use that to find the pressure gradient assuming the average pressure is proportional to the pressure gradient (and of course repeat in a loop). I attached the result.
This sounds very ad hoc. I would very much prefer seeing the proper fluid dynamic partial differential equations for continuity, momentum balance (Euler or Navier Stokes), and appropriate expression of the boundary condition at the free surface, and then any assumptions and their application to simplify the equations (appropriately justified). Any approach following individual particles, in my judgment, is inappropriate for a complicated system like this.
 
  • #9
Let's say I assume steady, inviscid, incompressible flow with no body forces. I could also assume radial symmetry, which I imagine would allow things to be simplified. This might allow me to express the equations in cylindrical coordinates, making it a 2d problem (eliminating theta).

The equations would then reduce to: $$\nabla \cdot u=0$$ $$\frac{\nabla p}{\rho}+u\cdot \nabla u=0$$
I'm guessing I would have three boundary conditions, one for the free surface, one for the surface, and one for the cross section of the start of the jet where the velocity is assumed to be uniform. This is my guess at the boundary conditions, but I'm not sure if they're correct.

Free surface where ##u_n## is normal to the free surface (I'm not sure having both is redundant): $$p=0$$ $$u_n=0$$

Plate where ##u_z## is normal to the plate (so in direction of jet):$$u_z=0$$

Jet cross-section: $$u_z=v_{jet}$$ $$u_r=0$$

I'm guessing you could then use the potential function to get a system of differential equations with a single variable and solve it with those boundary conditions. I imagine this would be very difficult, but it would be interesting to see what it gives me and how it compares to my other approach.
 
Last edited:
  • Like
Likes Chestermiller

Related to Parabolic Pressure Distribution in a Jet

1. What is a parabolic pressure distribution in a jet?

A parabolic pressure distribution in a jet refers to the shape of the pressure distribution across the cross-sectional area of a jet as it travels through the air. This shape is parabolic, meaning that the pressure is highest at the center of the jet and decreases towards the edges.

2. What causes a parabolic pressure distribution in a jet?

The parabolic pressure distribution in a jet is caused by the Bernoulli's principle, which states that as the velocity of a fluid (in this case, air) increases, the pressure decreases. In a jet, the air is accelerated as it travels through the engine and out of the nozzle, creating a parabolic pressure distribution.

3. Why is a parabolic pressure distribution important in jet design?

A parabolic pressure distribution is important in jet design because it helps to reduce drag and increase efficiency. The shape of the pressure distribution creates a low-pressure region behind the jet, which helps to pull the jet forward and reduce resistance. This allows the jet to travel faster and use less fuel.

4. How does altitude affect the parabolic pressure distribution in a jet?

As altitude increases, the air becomes less dense, which can affect the parabolic pressure distribution in a jet. At higher altitudes, the pressure at the center of the jet may be lower compared to lower altitudes, resulting in a flatter pressure distribution. This can impact the performance of the jet and may require adjustments to maintain optimal pressure distribution.

5. Are there any other factors that can affect the parabolic pressure distribution in a jet?

Yes, there are other factors that can affect the parabolic pressure distribution in a jet, such as air temperature, humidity, and the shape and design of the jet's nozzle. These factors can impact the velocity and density of the air, which in turn can affect the shape and intensity of the parabolic pressure distribution.

Similar threads

  • Mechanics
Replies
5
Views
1K
  • Mechanics
Replies
5
Views
4K
Replies
7
Views
1K
Replies
4
Views
2K
Replies
9
Views
2K
Replies
1
Views
2K
Replies
13
Views
3K
Replies
3
Views
3K
  • Classical Physics
2
Replies
35
Views
2K
  • Advanced Physics Homework Help
Replies
15
Views
2K
Back
Top