Drawing constant pressure lines in flow field

In summary, the conversation discusses the concept of drawing lines of constant pressure in an inviscid flow field, specifically in the context of flow around a cylinder and in a Hele-Shaw flow. The equations for the variations of pressure in inviscid flow are mentioned, with one being Bernoulli's equation and the other being the gradient of the stream function dotted with itself. The conversation also touches on the coordinate system used to represent the stream function and the proof that the gradient of the stream function dotted with itself is equal to the square of the velocity. The conversation concludes with a suggestion to use a graphics package to draw the diagram showing the lines of constant pressure.
  • #1
member 428835
Hi PF!

Can someone help me understand how to draw lines of constant pressure in an inviscid flow field, say flow around a cylinder. I am having trouble understanding how to draw these. Any help is greatly appreciated!

I ask this question because I am preparing for the Q exam for PhD and one of my classmates asked me about a Hele-Shaw flow: imagine two circular plates very close to each other (infinitely close). Now compare the streamlines of this Hele-Shaw flow and inviscid flow around a cylinder. Also draw lines of constant pressure for both cases. Evidently the streamlines looks identical but the pressure lines are different.
 
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  • #2
What is your equation for the variations of pressure in inviscid flow?
 
  • #3
Chestermiller said:
What is your equation for the variations of pressure in inviscid flow?
Bernoulli's equation, will give a pressure distribution for inviscid flow regimes, so this could give us the pressure variations too, right? For flow around a cylinder this is expressed as $$p = \frac{1}{2} \rho U^2 (2 R^2 \cos(2\theta)/r^2-R^4/r^4)+p_\infty$$
 
  • #4
joshmccraney said:
Bernoulli's equation, will give a pressure distribution for inviscid flow regimes, so this could give us the pressure variations too, right? For flow around a cylinder this is expressed as $$p = \frac{1}{2} \rho U^2 (2 R^2 \cos(2\theta)/r^2-R^4/r^4)+p_\infty$$
Yes. That's fine. How about in general, in terms of the stream function.
 
  • #5
Chestermiller said:
Yes. That's fine. How about in general, in terms of the stream function.
My thoughts are to take Bernoullis equation ##p + \rho v^2/2 =p_\infty + \rho u_\infty^2/2## (ignoring unsteady and gravity terms) and then write ##v## in terms of the streamfunction. but this will depend on the coordinate system we choose. Since you're asking for a "general" description I'm confused what coordinate system to represent ##\psi## in. Or am I missing the point?
 
  • #6
joshmccraney said:
My thoughts are to take Bernoullis equation ##p + \rho v^2/2 =p_\infty + \rho u_\infty^2/2## (ignoring unsteady and gravity terms) and then write ##v## in terms of the streamfunction. but this will depend on the coordinate system we choose. Since you're asking for a "general" description I'm confused what coordinate system to represent ##\psi## in. Or am I missing the point?
You should be able to do it irrespective of the coordinate system.
 
  • #7
In 2D, what is the gradient of the stream function dotted with itself?
 
  • #8
Chestermiller said:
In 2D, what is the gradient of the stream function dotted with itself?
You're asking for ##\nabla \psi \cdot \nabla \psi##? I am not sure. I know ##\nabla \psi## is normal to the velocity, but this is all I can say about it, and without selecting a coordinate system I am not sure how to proceed. Any help?
 
  • #9
joshmccraney said:
You're asking for ##\nabla \psi \cdot \nabla \psi##? I am not sure. I know ##\nabla \psi## is normal to the velocity, but this is all I can say about it, and without selecting a coordinate system I am not sure how to proceed. Any help?
Isn't it ##V^2##?
 
  • #10
Chestermiller said:
Isn't it ##V^2##?
Is it? I've always seen the relationship with ##\psi## and the velocity as either ##\vec{V} = \nabla \times \psi \vec{e}## where ##\vec{e}## is the basis vector we do not need in order to describe the flow. How could I prove ##\nabla \psi \cdot \nabla \psi = V^2##?
 
  • #11
joshmccraney said:
Is it? I've always seen the relationship with ##\psi## and the velocity as either ##\vec{V} = \nabla \times \psi \vec{e}## where ##\vec{e}## is the basis vector we do not need in order to describe the flow. How could I prove ##\nabla \psi \cdot \nabla \psi = V^2##?
Just do it in cartesian coordinates.
 
  • #12
Chestermiller said:
Just do it in cartesian coordinates.
Yea, I see it's true in cartesian, but writing this in cartesian coordinates goes against your comment in post 6. Could you direct me where to go to find the general proof that ##\nabla\psi\cdot\nabla\psi=V^2##? I've searched around but have been unable to find anything.
 
  • #13
joshmccraney said:
Yea, I see it's true in cartesian, but writing this in cartesian coordinates goes against your comment in post 6. Could you direct me where to go to find the general proof that ##\nabla\psi\cdot\nabla\psi=V^2##? I've searched around but have been unable to find anything.
I don't have a reference. Try cylindrical and see if it works for that.
 
  • #14
Chestermiller said:
I don't have a reference. Try cylindrical and see if it works for that.
Ok, so in cylindrical I'll assume we have flow only in ##r## and ##\theta##, so there is no ##z## component (I could do a different flow if you prefer?). Define the streamfunction ##\psi## as $$\vec{V} = \nabla \times \psi \hat{z} =-\frac{\psi_\theta}{r}\hat{r}+\psi_r\hat{\theta} \implies\\ |\vec{V}|^2 = v_r^2+v_\theta^2 = \frac{\psi_\theta^2}{r^2}+\psi_r^2$$ and notice
$$\nabla \psi \cdot\nabla\psi = \frac{\psi_\theta^2}{r^2}+\psi_r^2$$ so this works. Bernoullis is then
$$p + \rho v^2/2 =p_\infty + \rho u_\infty^2/2 \implies\\ p =p_\infty + \rho u_\infty^2/2 - \rho \nabla\psi\cdot\nabla\psi/2.$$
But then how to draw ##p##; I don't see how writing it in terms of the streamfunction helps. Any ideas?
 
  • #15
joshmccraney said:
Ok, so in cylindrical I'll assume we have flow only in ##r## and ##\theta##, so there is no ##z## component (I could do a different flow if you prefer?). Define the streamfunction ##\psi## as $$\vec{V} = \nabla \times \psi \hat{z} =-\frac{\psi_\theta}{r}\hat{r}+\psi_r\hat{\theta} \implies\\ |\vec{V}|^2 = v_r^2+v_\theta^2 = \frac{\psi_\theta^2}{r^2}+\psi_r^2$$ and notice
$$\nabla \psi \cdot\nabla\psi = \frac{\psi_\theta^2}{r^2}+\psi_r^2$$ so this works. Bernoullis is then
$$p + \rho v^2/2 =p_\infty + \rho u_\infty^2/2 \implies\\ p =p_\infty + \rho u_\infty^2/2 - \rho \nabla\psi\cdot\nabla\psi/2.$$
But then how to draw ##p##; I don't see how writing it in terms of the streamfunction helps. Any ideas?
You calculate p on a grid, and then use a graphics package with 2D contour plotting capability to draw the diagram showing the lines of constant p.
 
  • #16
Chestermiller said:
You calculate p on a grid, and then use a graphics package with 2D contour plotting capability to draw the diagram showing the lines of constant p.
So there's no real way to intuitively do this sort of thing "on the fly"?
 
  • #17
There are complete analytic solutions for stream function and velocity potential for a few idealised situations .

These can sometimes be used to generate analytic solutions for more practical problems .

They are also useful when devising efficient numerical solution methods and when just sketching flow patterns .

http://www.freestudy.co.uk/fluid mechanics/t5203.pdf

Ch.4-7 are most relevant .
 
  • #18
joshmccraney said:
So there's no real way to intuitively do this sort of thing "on the fly"?
I don't know what you mean. In some cases, you can solve for ##\psi## as a function of the spatial coordinates, and then solve for the lines of constant ##\psi##.
 
  • #19
Chestermiller said:
I don't know what you mean. In some cases, you can solve for ##\psi## as a function of the spatial coordinates, and then solve for the lines of constant ##\psi##.
Sorry for the ambiguity: by "intuitive" I mean, can you look at a velocity field with a known geometry and infer the pressure lines? For example, the streamlines of a flow around a cylinder are not normal to lines of constant pressure. However, if we have to cylindrical plates that are very very close, evidently the streamlines won't change, yet now the pressure is always normal to the streamlines. Can you help me understand why this is and how I would know they are normal to the streamlines?
 
  • #20
If the lines of constant pressure are perpendicular to the streamlines, then the gradient of pressure is in the same direction as the velocity vector. Using the Navier Stokes equations, what can you do to test this?
 

1. What are constant pressure lines in a flow field?

Constant pressure lines in a flow field are imaginary lines that connect points in a fluid flow where the pressure remains constant. These lines help visualize the pressure distribution in a flow field and are useful for analyzing the behavior of fluids in motion.

2. How are constant pressure lines drawn in a flow field?

To draw constant pressure lines in a flow field, the pressure values at different points in the flow field are first determined. These values are then plotted on a graph and connected by lines to create a visual representation of the pressure distribution. Alternatively, computer software can be used to generate constant pressure lines based on numerical calculations.

3. What information can be obtained from constant pressure lines in a flow field?

Constant pressure lines provide information about the pressure distribution in a flow field. They can also reveal the direction and magnitude of the pressure gradient, which can indicate the flow direction and the presence of areas of high or low pressure. Additionally, the spacing and curvature of the lines can provide insights into the behavior of the fluid in the flow field.

4. How do constant pressure lines relate to Bernoulli's principle?

Constant pressure lines are a visual representation of Bernoulli's principle, which states that in a flow field, there is an inverse relationship between the velocity and pressure of a fluid. This means that areas of high velocity will have lower pressure, and vice versa. Constant pressure lines help visualize this relationship by showing the areas of constant pressure in the flow field.

5. What are the limitations of using constant pressure lines in a flow field analysis?

Constant pressure lines are a simplified representation of the complex behavior of fluids in motion. They do not take into account factors such as viscosity, turbulence, and boundary conditions, which can significantly affect the pressure distribution in a flow field. Therefore, constant pressure lines should be used as a qualitative tool and not a precise method for analyzing fluid flow.

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