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Abstract

This paper considers the classical problem of a line vortex in planar flow of a fluid. However, an interface is present at some finite radius from the line vortex, and beyond that is a second fluid of different density. The interface is therefore subject to shearing-type instabilities and may overturn as time progresses. A linearized inviscid theory is developed and reveals unstable behaviours, dependent on the parameters in the system. The non-linear inviscid problem is solved by a spectral method, and high-frequency modes are regularized by a type of filtering. In addition, a Boussinesq viscous model is presented and allows the overturning interface to fold. Results are discussed and compared with the predictions of the inviscid theory.

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Acknowledgments

This work was supported in part by Australian Research Council Grant DP1093658. The work of JMC was supported by a Tasmania Graduate Research Scholarship. Comments from two anonymous referees are gratefully acknowledged.

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Correspondence to Lawrence K. Forbes.

Appendices

Appendix 1: Spectral equations for inviscid model

Here, the ODEs are presented for the Fourier coefficients in representations (7) and (8). As in Forbes et al. [24], velocity components \(U_1,\) \(V_1,\) \(U_2\) and \(V_2\) are defined from the velocity components in each fluid by evaluating them on the interface \(r = \mathcal{R} (\theta ,t).\) Fourier decomposition of the first kinematic equation in system (4) then gives rise at once to

$$\begin{aligned} G_0 ^{\prime } (t)&= \frac{1}{2\pi } \int \limits _0^{2\pi } \biggl [ U_1 - \frac{V_1}{\mathcal{R}} \frac{\partial \mathcal{R}}{\partial \theta } \biggr ] \, \mathrm{d}\theta , \nonumber \\ G_k ^{\prime } (t)&= \frac{1}{\pi } \int \limits _0^{2\pi } \biggl [ U_1 - \frac{V_1}{\mathcal{R}} \frac{\partial \mathcal{R}}{\partial \theta } \biggr ] \cos k\theta \, \mathrm{d}\theta , \nonumber \\ H_k ^{\prime } (t)&= \frac{1}{\pi } \int \limits _0^{2\pi } \biggl [ U_1 - \frac{V_1}{\mathcal{R}} \frac{\partial \mathcal{R}}{\partial \theta } \biggr ] \sin k\theta \, \mathrm{d}\theta , \quad k = 1 , 2 , \dots , M. \end{aligned}$$
(42)

The second of the kinematic conditions in Eq. (4) is replaced by the difference between the two, and this new equation is then Fourier decomposed as above. This yields a system of algebraic equations in the even modes and a similar system for the odd modes. Although elegant, algebraic equations of this type are inconvenient from the numerical point of view, as discussed by Forbes et al. [24], and so they are formally differentiated with respect to time. After some algebra, this results in the even- and odd-mode differential equations

$$\begin{aligned}&\!\!\! \sum _{m=1}^M \left[ S_{km}^{1A} A_m ^{\prime } (t) - S_{km}^{1B} B_m ^{\prime } (t) + S_{km}^{2C} C_m ^{\prime } (t) - S_{km}^{2D} D_m ^{\prime } (t) \right]= - \int \limits _0^{2\pi } \bigl ( V_2 - V_1 \bigr ) \biggl [ U_1 - \frac{V_1}{\mathcal{R}} \frac{\partial \mathcal{R}}{\partial \theta } \biggr ] \sin k\theta \, \mathrm{d}\theta , \nonumber \\&\!\!\! \sum _{m=1}^M \left[ C_{km}^{1A} A_m ^{\prime } (t) - C_{km}^{1B} B_m ^{\prime } (t) + C_{km}^{2C} C_m ^{\prime } (t) - C_{km}^{2D} D_m ^{\prime } (t) \right]= - \int \limits _0^{2\pi } \bigl ( V_2 - V_1 \bigr ) \biggl [ U_1 - \frac{V_1}{\mathcal{R}} \frac{\partial \mathcal{R}}{\partial \theta } \biggr ] \cos k\theta \, \mathrm{d}\theta , \nonumber \\&\quad k = 1 , 2 , \dots , M. \end{aligned}$$
(43)

In these equations, the intermediate variables

$$\begin{aligned} S_{km}^{1A} (t)&= \int \limits _0^{2\pi } \mathcal{R}^m \sin k\theta \, \sin m\theta \, \mathrm{d}\theta , \nonumber \\ S_{km}^{1B} (t)&= \int \limits _0^{2\pi } \mathcal{R}^m \sin k\theta \, \cos m\theta \, \mathrm{d}\theta , \nonumber \\ S_{km}^{2C} (t)&= \int \limits _0^{2\pi } \mathcal{R}^{- m} \sin k\theta \, \sin m\theta \, \mathrm{d}\theta , \nonumber \\ S_{km}^{2D} (t)&= \int \limits _0^{2\pi } \mathcal{R}^{- m} \sin k\theta \, \cos m\theta \, \mathrm{d}\theta \end{aligned}$$
(44)

are defined for convenience. The functions \(C_{km}^{1A} (t)\) are similar to \(S_{km}^{1A} (t),\) and so on, except that in each integrand the function \(\sin k\theta \) is replaced by \(\cos k\theta .\)

Finally, the dynamic condition (5) is similarly decomposed into its Fourier even and odd modes, as described in Sect. 2. This gives rise to the further sets of differential equations

$$\begin{aligned}&\!\!\! \sum _{m=1}^M \left[ - C_{km}^{1B} A_m ^{\prime } (t) - C_{km}^{1A} B_m ^{\prime } (t) + D C_{km}^{2D} C_m ^{\prime } (t) + D C_{km}^{2C} D_m ^{\prime } (t) \right]= \frac{1}{2} \left( J_k^{(1)} - D J_k^{(2)} \right) - \frac{(D - 1)}{F^2} P_k, \nonumber \\&\!\!\! \sum _{m=1}^M \left[ - S_{km}^{1B} A_m ^{\prime } (t) - S_{km}^{1A} B_m ^{\prime } (t) + D S_{km}^{2D} C_m ^{\prime } (t) + D S_{km}^{2C} D_m ^{\prime } (t) \right]= \frac{1}{2} \left( K_k^{(1)} - D K_k^{(2)} \right) - \frac{(D - 1)}{F^2} Q_k, \nonumber \\&\quad k = 1 , 2 , \dots , M. \end{aligned}$$
(45)

The intermediate quantities \(S_{km}^{1A}\) and so on are as defined in Eq. (44). In addition, there are additional terms

$$\begin{aligned}&\!\!\!J_k^{(1)} (t) = \int \limits _0^{2\pi } \bigl ( U_1^2 + V_1^2 \bigr ) \cos k\theta \, \mathrm{d}\theta , \nonumber \\&\!\!\! J_k^{(2)} (t) = \int \limits _0^{2\pi } \bigl ( U_2^2 + V_2^2 \bigr ) \cos k\theta \, \mathrm{d}\theta , \nonumber \\&\!\!\!P_k (t) = \int \limits _0^{2\pi } \log \mathcal{R} \cos k\theta \, \mathrm{d}\theta , \end{aligned}$$
(46)

and the remaining intermediate quantities \(K_k^{(1)}, K_k^{(2)}\) and \(Q_k\) are obtained from the three quantities in Eq. (46) by replacing the \(\cos k\theta \) term in each integrand with \(\sin k\theta \) respectively.

Equations (42), (43) and (45) constitute a matrix system of \(6M + 1\) ODEs that are integrated forward in time using a fourth-order–fifth-order Runge–Kutta–Fehlberg method with adaptive time steps (Johnson and Riess [29, p. 378]).

Appendix 2: Spectral equations for viscous model

In this appendix, the system of ODEs for the Fourier coefficients in the representation of the viscous solution is presented.

At the zeroth-order mode in \(\theta ,\) the vorticity Eq. (26) yields

(47)

in which the background coefficients \(A_{0,\ell }^S\) are defined in Eq. (34). The higher-order even and odd modes then give rise to the additional equations

(48)

and

(49)

The symbol \(j_{k,\ell }\) denotes the \(\ell \)th positive zero of the Bessel function \(\mathrm{J}_k,\) as in Eq. (35).

The density Eq. (24) is subjected to Fourier analysis in a similar manner. The zeroth-order mode in \(\theta \) is obtained by multiplying by \((r / \beta )^2 \sin ( \ell \pi r^3 / \beta ^3)\) and integrating over \(\theta \) and \(r.\) After some algebra, this results in the system

$$\begin{aligned} P_{0,\ell } ^{\prime } (t)&= - \frac{3}{\pi \beta } \int \limits _0^{\beta } \int \limits _0^{2\pi } \biggl ( \frac{r}{\beta } \biggr )^2 \biggl ( u \frac{\partial \bar{\rho }}{\partial r} + \frac{v}{r} \frac{\partial \bar{\rho }}{\partial \theta } \biggr ) \sin \biggl ( \ell \pi \frac{r^3}{\beta ^3} \biggr ) \, \mathrm{d}\theta \, \mathrm{d}r + \frac{54 \sigma }{\beta ^3} \left( 1 - \frac{1}{D} \right) \int \limits _0^{\beta } \biggl ( \frac{r}{\beta } \biggr )^3 \sin \biggl ( \ell \pi \frac{r^3}{\beta ^3} \biggr ) \, \mathrm{d}r \nonumber \\&+ \frac{54 \sigma }{\beta ^3} \sum _{n = 1}^N \bigl [ P_{0,n}^S + P_{0,n} (t) \bigr ] \int \limits _0^{\beta } \biggl [ n \pi \biggl ( \frac{r}{\beta } \biggr )^3 \cos \biggl ( n \pi \frac{r^3}{\beta ^3} \biggr )- n^2 \pi ^2 \biggl ( \frac{r}{\beta } \biggr )^6 \sin \biggl ( n \pi \frac{r^3}{\beta ^3} \biggr ) \biggr ] \sin \biggl ( \ell \pi \frac{r^3}{\beta ^3} \biggr ) \, \mathrm{d}r, \nonumber \\&\quad \ell = 1 , 2 , \dots , N. \end{aligned}$$
(50)

Here, the background coefficients \(P_{0,n}^S\) are as given in Eq. (31).

The higher-order even- and odd-order coefficients are now obtained as for the vorticity equation. The even-order modes yield the system of equations

$$\begin{aligned} P_{k,\ell } ^{\prime } (t)&= - \sigma \alpha _{k,\ell }^2 P_{k,\ell } (t)+ \frac{8\sigma }{\left[ \beta \mathrm{J}_{k+1} \bigl ( j_{k,\ell } \bigr ) \right]^2} \sum _{n = 1}^N P_{k,n} (t) \int \limits _0^{\beta } \biggl [ \frac{\mathrm{J}_k \bigl ( \alpha _{k,n} r \bigr )}{r} + \alpha _{k,n} \mathrm{J}_k ^{\prime } \bigl ( \alpha _{k,n} r \bigr ) \biggr ] \mathrm{J}_k \bigl ( \alpha _{k,\ell } r \bigr ) \, \mathrm{d}r \nonumber \\&- \frac{2}{\pi \left[ \beta \mathrm{J}_{k+1} \bigl ( j_{k,\ell } \bigr ) \right]^2} \int \limits _0^{\beta } \int \limits _0^{2\pi } \frac{\mathrm{J}_k \bigl ( \alpha _{k,\ell } r \bigr )}{r} \biggl ( u \frac{\partial \bar{\rho }}{\partial r} + \frac{v}{r} \frac{\partial \bar{\rho }}{\partial \theta } \biggr ) \cos ( k\theta ) \, \mathrm{d} \theta \, \mathrm{d} r, \nonumber \\&\quad k = 1 , 2 , \dots , M , \quad \ell = 1 , 2 , \dots , N , \end{aligned}$$
(51)

and the odd-order modes result in

$$\begin{aligned} Q_{k,\ell } ^{\prime } (t)&= - \sigma \alpha _{k,\ell }^2 Q_{k,\ell } (t)+ \frac{8\sigma }{\left[ \beta \mathrm{J}_{k+1} \bigl ( j_{k,\ell } \bigr ) \right]^2} \sum _{n = 1}^N Q_{k,n} (t) \int \limits _0^{\beta } \biggl [ \frac{\mathrm{J}_k \bigl ( \alpha _{k,n} r \bigr )}{r} + \alpha _{k,n} \mathrm{J}_k ^{\prime } \bigl ( \alpha _{k,n} r \bigr ) \biggr ] \mathrm{J}_k \bigl ( \alpha _{k,\ell } r \bigr ) \, \mathrm{d}r \nonumber \\&- \frac{2}{\pi \left[ \beta \mathrm{J}_{k+1} \bigl ( j_{k,\ell } \bigr ) \right]^2} \int \limits _0^{\beta } \int \limits _0^{2\pi } \frac{\mathrm{J}_k \bigl ( \alpha _{k,\ell } r \bigr )}{r} \biggl ( u \frac{\partial \bar{\rho }}{\partial r} + \frac{v}{r} \frac{\partial \bar{\rho }}{\partial \theta } \biggr ) \sin ( k\theta ) \, \mathrm{d} \theta \, \mathrm{d} r, \nonumber \\&\quad k = 1 , 2 , \dots , M , \quad \ell = 1 , 2 , \dots , N. \end{aligned}$$
(52)

These equations are now integrated forward in time.

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Forbes, L.K., Cosgrove, J.M. A line vortex in a two-fluid system. J Eng Math 84, 181–199 (2014). https://doi.org/10.1007/s10665-012-9606-5

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