Draft:Stuart vortex

Stuart vortex is an exact solution of the two-dimensional Euler equations that is used to model vortex structures in a laminar shear layer, such as Kelvin–Helmholtz vortex structures. The solution was first discovered by John Trevor Stuart in 1967. The solution is usually described in terms of the stream function $\psi$ as follows:

$\psi (x,y)={\frac {U}{k}}\ln \left[\cosh(ky)-\delta \cos(kx)\right].$

The corresponding $z$-component of the vorticity field, which satisfies the inviscid steady vorticity equation $\nabla ^{2}\psi =-\omega (\psi )$, is given by:

$\omega =kU(\delta ^{2}-1)\exp \left(-{\frac {2k\psi }{U}}\right).$

The velocity components are derived from the stream function via $v_{x}=\partial \psi /\partial y$ and $v_{y}=-\partial \psi /\partial x$, yielding:

$v_{x}={\frac {U\sinh(ky)}{\cosh(ky)-\delta \cos(kx)}},$

$v_{y}={\frac {-\delta U\sin(kx)}{\cosh(ky)-\delta \cos(kx)}}.$

Here, $U$ is the magnitude of the free-stream velocity (such that $v_{x}(x,\pm \infty )=\pm U$), $k=2\pi /\lambda$ is the wavenumber where $\lambda$ is the distance between two contiguous vortices, and $\delta$ is a parameter describing the vorticity distribution.

Flow behaviour

The flow behaviour changes drastically based on the value of $\delta$:

• For $\delta =0$, the Stuart vortex reduces to a pure parallel shear flow with a hyperbolic tangent velocity profile: $v_{x}=U\tanh(ky),\quad v_{y}=0.$
• For $0<\delta <1$, it represents a periodic series of core-concentrated vortex structures known as "cat's eyes".
• For $\delta =1$, it simplifies to a singular row of ideal point vortices along the axis.

The flow possesses a periodic array of critical points along the centerline $y=0$:

• Vortex Centers (Elliptic points): Located at $kx=2n\pi$ (for $n\in \mathbb {Z}$). At these positions, the fluid particles circulate around a local minimum of the stream function in closed loops.
• Stagnation point flows (Hyperbolic points): Located halfway between centers at $kx=(2n+1)\pi$.

The streamline connecting contiguous saddle points is called the separatrix. It outlines a characteristic shape widely referred to in fluid mechanics as Cat's Eyes. Fluid trapped inside the cat's eye recirculates indefinitely within the vortex core, while fluid outside flows past the core.

References

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Categories:

• Fluid dynamics
• Vortices