Drawing constant pressure lines in flow field

[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.

 

[Chestermiller]

What is your equation for the variations of pressure in inviscid flow?

 

[member 428835]

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 $$p = \frac{1}{2} \rho U^2 (2 R^2 \cos(2\theta)/r^2-R^4/r^4)+p_\infty$$

 

[Chestermiller]

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.

 

[member 428835]

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?

 

[Chestermiller]

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.

 

[Chestermiller]

In 2D, what is the gradient of the stream function dotted with itself?

 

[member 428835]

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?

 

[Chestermiller]

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 $V^2$?

 

[member 428835]

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 $\nabla \psi \cdot \nabla \psi = V^2$?

 

[Chestermiller]

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.

 

[member 428835]

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.

 

[Chestermiller]

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.

 

[member 428835]

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?

 

[Chestermiller]

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.

 

[member 428835]

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"?

 

[Nidum]

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 .

 

[Chestermiller]

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 $\psi$.

 

[member 428835]

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?

 

[Chestermiller]

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?