Compressible inviscid vorticity convection w Rankine Vortex

In summary, the conversation discusses the differences between the compressible inviscid vorticity convection equation and the incompressible version, with a focus on the convected quantity and the rightmost "baroclinic" term. The question also includes two parts related to Rankine vortex and a container of still liquid with a small circular blob of another liquid introduced. The first part deals with determining the ratio of vorticity after and before expansion in the Rankine vortex, while the second part asks for a qualitative description of the vorticity field and its reflection of the motion of the blob of liquid inside the container.
  • #1
fahraynk
186
6

Homework Statement


The compressible inviscid vorticity convection equation:
$$\frac{D(\frac{w}{\rho})}{Dt}=(\frac{w}{\rho})\cdot \nabla U + \frac{1}{\rho}\nabla P \times \nabla (\frac{1}{\rho})$$
differs from the incompressible version in two important ways :
1) The convected quantity is w/p, not w.
2) the rightmost "baroclinic" term allows a pressure gradient to generate vorticity, or in other words to set the fluid into rotation

A) Consider a Rankine vortex. The fluid is a gas, which is suddenly expanded so that the vortex radius doubles. The density of course changes, but stays very nearly spatially uniform everywhere before, during , and after the expansion. Determine the ratio of vorticity after/before the expansion. Interpret the appearance of w/p in the equation above as a manifestation of conservation of angular momentum.

B) A container of still liquid A has the usual hydrostatic pressure field $$P=-\rho_Agz$$ where z is the vertical coordinate. A small circular blob of another liquid B which has a slightly larger density $$\rho_b>\rho_A$$ is carefully introduced into liquid A with a syringe. The blob is initially at rest at t=0. Qualitatively, describe the vorticity field at $$t=\Delta t, 2\Delta t...$$ and how it reflects the motion of the blob of liquid B inside liquid A. To make sketching easier, you may assume a 2D blob.

Homework Equations

The Attempt at a Solution


A)[/B]
I am not sure if I need the vorticity transport equation for the first part... But a rankine vortex has 2 different velocities :
$$U_\theta = \frac{\Gamma r}{2\pi R^2} ; r<R\\\\
U_\theta = \frac{\Gamma}{2\pi R} ; r>R$$
Vorticity (w) is the cross product $$\nabla \times U=-\frac{dU_\theta}{dr}$$
$$r<R; -\frac{dU_\theta}{dr} = -\frac{\Gamma}{2\pi R^2}\\\\
r>R; -\frac{dU_\theta}{dr} = \frac{\Gamma}{2\pi r^2}$$
So... they seem to be just a factor of -1 of each other if I am right.
The ratio should be ;
$$\frac{\frac{\Gamma}{2\pi R^2} }{\frac{\Gamma}{2\pi (2R)^2} }=4$$
Then the question asks : Interpret the appearance of w/p in the equation above as a manifestation of conservation of angular momentum.
Angular momentum... would be $$R\times M(L\times R)=-\frac{\Gamma M}{4\pi}$$ (M=mass, L=angular velocity = 1/2 vorticity (w))
I am just completely lost here. How do I interpret it as a conservation of angular momentum... what would they even want in an answer to this?

B)
If I take the derivative of pressure and plug it into the equation with hydrostatic constant velocity, meaning gradient = 0 I get :
$$\frac{D\frac{w}{\rho}}{Dt}=(-Ag)\times \nabla(\frac{1}{\rho})-->\\\\
\frac{\partial \frac{w}{\rho}}{\partial t} = (-Ag)\times \nabla(\frac{1}{\rho})$$
I have... no idea what I am supposed to do with this equation. Am I supposed to come up with an expression for density and plug it in and solve for w? Should I use taylor approximation? Isnt the derivative infinity at the wall between the 2 balls of different density?

Please help, I know this post might be way too long though!
 
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  • #2
I was working on part B :
The question says qualitatively not quantitatively. So I think that means explain with words not numbers!
I think I was wrong to assume no velocity gradients, because if you introduce the blob there has to be velocity gradients if the blob of liquid creates vorticity.
Also, w is a vector... so I messed that up. This is 3 equations which contain a scaler sum of gradients and vorticity and a cross product of pressure gradient and density gradient.
So... How am I supposed to predict what happens from those equations? No idea.
Intuitively, since it's rotational and there are density gradients, I would assume the flow would diffuse out in spiral patterns... but is that enough of an answer?
 

Related to Compressible inviscid vorticity convection w Rankine Vortex

1. What is compressible inviscid vorticity convection?

Compressible inviscid vorticity convection is a fluid dynamic phenomenon that occurs when an inviscid fluid (a fluid with no internal friction) experiences vorticity (the local spinning motion of fluid particles) and compressibility (the change in fluid density due to changes in pressure and temperature). This can result in the formation of vortices, or swirling patterns, within the fluid.

2. What is a Rankine Vortex?

A Rankine Vortex is a type of idealized vortical flow pattern that is often used to model real-world vortices, such as tornadoes and hurricanes. It is characterized by a central region of low pressure and high vorticity, surrounded by a region of higher pressure and lower vorticity.

3. How is vorticity convected in compressible inviscid flow?

In compressible inviscid flow, vorticity is convected (or transported) by the fluid's overall motion. This means that as the fluid moves, the vorticity is carried along with it and can be amplified or weakened depending on the flow conditions.

4. What are some practical applications of compressible inviscid vorticity convection?

Compressible inviscid vorticity convection has numerous applications in engineering, meteorology, and astrophysics. It is commonly used to study and model atmospheric phenomena, such as tornadoes and hurricanes, as well as in the design of aircraft and spacecraft propulsion systems.

5. How does compressibility affect vorticity convection in Rankine Vortices?

In Rankine Vortices, compressibility can play a significant role in the development and evolution of the vortices. The change in fluid density due to changes in pressure and temperature can affect the strength and stability of the vortices, and can also lead to the formation of shocks and other non-linear effects.

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