# Inviscid Flow around a Cone

1. Mar 7, 2017

### joshmccraney

Hi PF!

Some classmates and I were talking about the streamlines around a submerged cone being pulled at a velocity $V$. The picture is attached. Can someone shed some light on the streamlines of this flow, assuming it is inviscid? I take a frame of reference that moves with the cone.

Specifically, would the flow at station 2 be uniform vertically, or would the flow be faster at the exit port forever? Would the streamline profiles be straight upon exit or would they bend a little?

Any insight is greatly appreciated!

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2. Mar 8, 2017

### Staff: Mentor

What do you think the streamline pattern would look like?

3. Mar 9, 2017

### joshmccraney

I think the streamlines outside $A_1$ will wrap around the cone and at station 2 all be flat. I think the streamlines in the cone, at station 2, will all be flat too. Before station 2 and in the cone I think the streamlines will adhere to the shape of the wall and condense. I think these streamlines will exit condensed relative to those outside the cone. Since the flow is inviscid, I think streamlines beyond station 2 directly behind $A_2$ will always be closer together than those outside $A_2$. So a sort of jet.

4. Mar 9, 2017

### Staff: Mentor

Any chance of a diagram?

5. Mar 9, 2017

### joshmccraney

Totally, I've attached one. I've tried drawing all the streamlines equidistant before station 1. Notice after station 2 they are closer together behind $A_2$ then before. Do you agree with this sketch?

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6. Mar 9, 2017

### Staff: Mentor

I think that further downstream (after station 2), they will become equidistant again.

7. Mar 9, 2017

### joshmccraney

What would cause the streamlines to separate after station 2?

8. Mar 9, 2017

### Staff: Mentor

Do you believe that the stream function satisfies Laplace's equation for inviscid flow?

What causes the stream lines to even out again downstream of flow over a cylinder?

9. Mar 10, 2017

### joshmccraney

I believe the streamfunction does provided the flow is also incompressible (which we assume it is) so yes, I believe the streamfunction $\psi$ follows $\nabla^2\psi=0$.

Also, and I know this is off topic a little, but when deriving Bernoulli's equation from Navier Stokes, how do we know it is only valid along a streamline? At the very end of my derivation, assuming Newtonian, incompressible, inviscid, irrotational flow I have $\nabla(\partial_t \phi + |\vec{u}|^2/2+p/\rho + g z) = \vec{0} \implies \partial_t \phi + |\vec{u}|^2/2+p/\rho + g z = const.$ where I think the constant implies something about which streamline you're on. Also, $\vec{u}=\nabla \phi$.

Last edited: Mar 10, 2017
10. Mar 11, 2017

### Staff: Mentor

Well, if the stream function satisfies Laplace's equation and the streamlines become parallel at large x, then $\partial \psi /\partial x=0$. Therefore, we are left with $\partial^2\psi /\partial y^2=0$. That automatically gives equal streamfunction spacing.
Yeah, me neither. It certainly comes out the mathematics (i.e., the vorticity is zero), but it is certainly hard to track down from the Euler equations using physical reasoning. I don't know whether it is worth it spending time trying to do this.

11. Mar 11, 2017

### joshmccraney

Thanks a ton! This makes sense.

12. Mar 13, 2017

### Nidum

(1) If you look at a simplified version of this problem translated into 2D then the sloping surfaces look very similar to aircraft wings - one normal and one inverted .

There have been many studies of the air flow around wings - both for proper wings profiles and for flat plates . There have also been studies about water flow around hydrofoils though these are harder to find . For water flow and for pseudo incompressible flow of air the flow patterns are similar at lower speeds .

Using information from these sources it should be possible to sketch a reasonable 2D approximation to the flow patterns around the sloping surfaces provided that the two surfaces are well separated .

It may also be possible to sketch an approximation of the full axisymmetric flow pattern by sweeping the 2D pattern for any one surface around the central axis .

Flow patterns can be expected to alter depending on the angle of attack of the sloping surface . Accuracy of sketched flow patterns would probably be quite good for low angles of attack and less good for high angles of attack .

(2) Studies have also been done regarding flow of air through a wind sock and flow of water through the draught tube of a submerged turbine . Interesting background reading .

(3) There is software available for drawing idealised flow streamline patterns . Drawing these patterns is much less computationally demanding than for full CFD analysis .

(4) Sketching flow patterns used to be one of the skills learnt by young engine designers . Usually the flow net was sketched rather than just the streamlines .