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Compressible inviscid vorticity convection w Rankine Vortex

  1. Jun 8, 2017 #1
    1. The problem statement, all variables and given/known data
    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.
    2. Relevant equations


    3. The attempt at a solution
    A)

    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!
     
  2. jcsd
  3. Jun 9, 2017 #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?
     
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