Vortex sheet

A vortex sheet is a surface across which tangential velocity is discontinuous while normal velocity remains continuous.

A cylindrical vortex sheet is a vortex sheet supported on a cylindrical surface. Azimuthal Fourier disturbances have harmonic velocity potentials proportional to $r^m$ inside and $r^{-m}$ outside.

For a planar vortex sheet separating equal-density streams of velocities $U_1$ and $U_2$, a mode of nonzero wavenumber $k$ has complex growth exponentOne sign has positive growth rate, so every nonzero wavenumber is unstable in the inviscid zero-thickness model.

For two infinitely deep inviscid layers with stable density ordering ${\rho }_2>{\rho }_1$ and positive surface tension $\gamma$, the velocity jump destabilizes wavenumber $k$ precisely whenThe threshold diverges at both small and large wavenumber. Minimizing the bracket gives $|k|=\sqrt{({\rho }_2-{\rho }_1)g/\gamma }$ and the stated minimum velocity jump. A larger jump opens a bounded band of unstable wavenumbers, obtained from the two positive roots of $\gamma q^2-[{\rho }_1{\rho }_2{U}^2/({\rho }_1+{\rho }_2)]q+({\rho }_2-{\rho }_1)g=0$. Equality at threshold is marginal rather than exponential growth.

Consider an equal-density vortex sheet halfway between rigid walls, with equal layer depths $h$, base velocities $U_1,U_2$ and no restoring interfacial force. The irrotational perturbations have vertical factors $\cosh (k(y-h))$ above and $\cosh (k(y+h))$ below. Their kinematic boundary conditions and pressure continuity giveThe common depth factor cancels, yielding the displayed dispersion relation roots and Kelvin-Helmholtz instability for nonzero shear. The real phase velocity of the pattern is $(U_1+U_2)/2$ and the growing branch has temporal growth rate $|k(U_1-U_2)|/2$. This exact depth cancellation depends on equal depths and equal mass densities; a single nearby wall or unequal layers changes the inertia factors.

For two inviscid incompressible half-spaces, lower density/velocity ${\rho }_1,U_1$, upper density/velocity ${\rho }_2,U_2$, downward gravity $g$ and surface tension $\sigma$, a mode $e^{i(kx-\omega t)}$ with $k>0$ obeysA negative right-hand side produces exponential growth. The gravitational term recovers Rayleigh-Taylor instability for heavy-over-light ordering; the shear term can destabilize otherwise stable stratification. Finite layer thickness, viscosity, compressibility and magnetic tension lie outside this interface model.

For equal fluid density $\rho$ on opposite sides of a massless flat membrane, a velocity jump $U$ and bending pressure jump $\beta {\eta }_{xxxx}$ with $\beta >0$, a mode $e^{ikx+\sigma t}$ satisfiesThe decaying upper and lower perturbation potentials are $A_{+}{e}^{-|k|y}$ and $A_{-}{e}^{|k|y}$. Their kinematic conditions are $-|k|A_{+}=(\sigma +ikU)\eta$ and $|k|A_{-}=\sigma \eta$. The linearized Bernoulli equation gives $p_{-}-p_{+}=-\rho [{\sigma }^2+(\sigma +ikU{)}^2]\eta /|k|$. Equating it to the bending pressure yields the relation. Bending stabilizes $|k{|}^3\ge \rho U^2/(2\beta )$. The maximum positive growth is at $|k{|}^3=\rho U^2/(5\beta )$ and is $|U||k|\sqrt{3/20}$.

Two deep layers of incompressible flow with upper mass density ${\rho }_1$, lower mass density ${\rho }_2$, and tangential velocities $U_1,U_2$ have this dispersion relation when surface tension is absent. Decaying velocity potentials and the linearized kinematic boundary condition determine their amplitudes; the dynamic boundary condition for an inviscid interface gives the displayed relation. For $k>0$ the squared departure of the phase velocity $c$ from the density-weighted mean velocity is $g({\rho }_2-{\rho }_1)/[k({\rho }_1+{\rho }_2)]-{\rho }_1{\rho }_2(U_1-U_2{)}^2/({\rho }_1+{\rho }_2{)}^2$. A negative value gives Kelvin-Helmholtz instability despite stable density stratification.

A top-hat planar jet has one constant streamwise velocity in a layer and another constant velocity outside it. In the inviscid model, the two layer boundaries are parallel vortex sheets.

In a varicose mode, the upper and lower jet interfaces move in opposite directions, so the jet thickness varies while its centreline remains fixed.

In a sinuous mode, the upper and lower jet interfaces move in the same direction, so the jet centreline bends while its thickness is unchanged to first order.

For a jet of velocity $U$ occupying $|y|<h$ in stationary fluid of the same density, disturbances proportional to $e^{ikx+\sigma t}$ satisfyfor a varicose mode of a planar jet, andfor a sinuous mode of a planar jet. Both have unstable growth rate

For a top-hat planar jet in stationary fluid of the same mass density, odd and even velocity potentials have the same temporal growth rate. Put $s=\tanh (|k|L)$. Their dispersion relations use $s$ and $1/s$, but $\sqrt{s}/(1+s)=\sqrt{1/s}/(1+1/s)$. Their real phase velocities are respectively $Vs/(1+s)$ and $V/(1+s)$. The common growth rate is strictly below $|kV|/2$ for finite $|k|L>0$, approaching the isolated vortex sheet value when the sheets decouple.

For an equal-density vortex sheet a distance $h$ above a rigid wall, with lower and upper base velocities $U$ and zero, a mode $e^{i(kx-\omega t)}$ satisfiesThe wall changes the inertia of the lower perturbation through the factor $\coth (|k|h)$.