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Project topic vortex flows

graph TD;
    A[Vortex Flow]-->B[Principals];
    A-->C[Critical Review];
    B-->D[Helmhotz Equations];
        D-->h1[Filament Strength Constant]
        D-->h2[End at Free Surface or Closed]
        D-->h3[Newton's 1st equiv]
    C-->E[geometry]
    E-->F[free surface]
    E-->G[closed Curve]
    C-->H[time]

hashtag
Fluid Citations

@article{Keller_1995, title={On the interpretation of vortex breakdown}, volume={7}, url={http://dx.doi.org/10.1063/1.868757}, DOI={10.1063/1.868757}, number={7}, journal={Physics of Fluids}, publisher={AIP Publishing}, author={Keller, Jakob J.}, year={1995}, month={Jul}, pages={1695–1702} }
@article{Benjamin_1962, title={Theory of the vortex breakdown phenomenon}, volume={14}, url={http://dx.doi.org/10.1017/S0022112062001482}, DOI={10.1017/s0022112062001482}, number={4}, journal={Journal of Fluid Mechanics}, publisher={Cambridge University Press (CUP)}, author={Benjamin, T. Brooke}, year={1962}, month={Dec}, pages={593} }

everyone references him since he appeasrs to set up the energy functional since the usefulness of his paper is apparent as it is featured as part of all of these other works, analyze only the appendix which is what is needed

relates back to the fundamental equations of kinematics ${\bf q} = (u,v,w)$ and ${\boldsymbol \omega} = (\xi,\eta,\zeta)$

let H be the scalar potential sum of the pressure and velocity heads $H = p / \rho + \frac { 1 } { 2 } q ^ { 2 }$

use continuity on steady flow no elevation change or dependence on height or time so $ \psi ( r , x ) = \text { const }$

$u { r } + r ^ { - 1 } u + w { x } = 0$

and stream function is then defined so that $u = - r ^ { - 1 } \psi { x } , \quad w = r ^ { - 1 } \psi { r }$

benjamin defines that $\mathbf { q } \times \boldsymbol { \omega } = \nabla H$ and says that with intuition realize that the vortex line must lie along a stream surface which is related to helmholtz second theorem

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vortex tip stable to critical rotation rate where downward drag overcomes buoyant force and bubble is dragged down from tip

there is a depression as the next bubble is being formed

F is the Rossby number $Ro = \frac{U}{L f}$ which relates inertial coriolis force

critical rotation $\Omega{C}=\frac{F}{\pi \sqrt{v / \Omega} R{0}^{2}} \approx 10^4 \text{ r.p.m.}$

central depression height $\Delta h=\frac{\Omega{C}^{2} R{0}^{2}}{2 g}=\frac{F^{2} \Omega}{2 \pi^{2} g v R_{0}^{2}}$

depends on Rossby most squared power

bathtub vortex flow in the fluid bulk as a line vortex flow with circulation $\Gamma$ so $v_{0}(r)=\frac{\Gamma}{2 \pi r}$

gaussian up flow profile $w{0}(r)=\frac{F}{\pi R{0}^{2}} \exp \left(-\frac{r^{2}}{R_{0}^{2}}\right)$.

take away that they plot looks remarkably like the moody chart in that the effect of the inertial component drops for high values of reynolds

Gernally accepted that vortex may be stable or unstable dependent on some critical value

similar to the hydraulic jump seen at the end of a laminar tapwater stream onto the bottom of a flat simk

vortex breakdown may be the result of a laminar to turbulent transition as if spiral vortex cannot exist that way

measurement by laser doppler anemometer

plots double log betwen reynolds and rotational speed

Γ/(2πrcwe)= constant Ω3RReB= constant \begin{array} { c } { \Gamma / \left( 2 \pi r _ { c } w _ { e } \right) = \text { constant } } \\ { \Omega ^ { 3 } R R e _ { B } = \text { constant } } \end{array}

where c is dependent on geometry

Rayleigh’s Discriminant says that there is stability if the follow is satisfied so long as $\Gamma = rV$ is a monotonic function

Φ=1r3ddr(rV)2>0\Phi = \frac{1}{r^3}\frac{d}{dr} \left(rV\right)^2 > 0

vortex breakdown can be found as a minimum of global energy functional ]\

E(ψ)=∫0x0∫01/2(ψy22+ψx24y+H(ψ)−I(ψ)2y)dydx\mathscr { E } ( \psi ) = \int _ { 0 } ^ { x _ { 0 } } \int _ { 0 } ^ { 1 / 2 } \left( \frac { \psi _ { y } ^ { 2 } } { 2 } + \frac { \psi _ { x } ^ { 2 } } { 4 y } + H ( \psi ) - \frac { I ( \psi ) } { 2 y } \right) \mathrm { d } y \mathrm { d } x

two functionals one which is counteracts the other and is dependent on the elevation

stream function

H and I can be thought of as differential operations acting on the stream functions which is denoted as \phi_{yy}

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