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Linear and nonlinear stability analyses are performed to determine critical Rayleigh numbers ( ${Ra}_{cr}$ ) for a Rayleigh–Bénard convection configuration with an imposed bottom boundary heat flux that varies harmonically in time with zero mean. The ${Ra}_{cr}$ value depends on the nondimensional frequency $\omega$ of the boundary heatflux modulation. Floquet theory is used to find ${Ra}_{cr}$ for linear stability, and the energy method is used to find ${Ra}_{cr}$ for two different types of nonlinear stability: strong and asymptotic. The most unstable linear mode alternates between synchronous and subharmonic frequencies at low $\omega$ , with only the latter at large $\omega$ . For a given frequency, the linear stability ${Ra}_{cr}$ is generally higher than the nonlinear stability ${Ra}_{cr}$ , as expected. For large $\omega$ , ${Ra}_{cr} \omega ^{2}$ approaches an $O(10)$ constant for linear stability but zero for nonlinear stability. Hence the domain for subcritical instability becomes increasingly large with increasing $\omega$ . The same conclusion is reached for decreasing Prandtl number. Changing temperature and/or velocity boundary conditions at the modulated or nonmodulated plate leads to the same conclusions. These stability results are confirmed by selected direct numerical simulations of the initial value problem.Free, publiclyaccessible full text available April 25, 2024

Equations of motion for compressible point vortices in the plane are obtained in the limit of small Mach number, M , using a Rayleigh–Jansen expansion and the method of Matched Asymptotic Expansions. The solution in the region between vortices is matched to solutions around each vortex core. The motion of the vortices is modified over long time scales O ( M 2 log M ) and O ( M 2 ) . Examples are given for corotating and copropagating vortex pairs. The former show a correction to the rotation rate and, in general, to the centre and radius of rotation, while the latter recover the known result that the steady propagation velocity is unchanged. For unsteady configurations, the vortex solution matches to a far field in which acoustic waves are radiated. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.Free, publiclyaccessible full text available June 27, 2023

We consider the time evolution in two spatial dimensions of a double vorticity layer consisting of two contiguous, infinite material fluid strips, each with uniform but generally differing vorticity, embedded in an otherwise infinite, irrotational, inviscid incompressible fluid. The potential application is to the wake dynamics formed by two boundary layers separating from a splitter plate. A thinlayer approximation is constructed where each layer thickness, measured normal to the common centre curve, is small in comparison with the local radius of curvature of the centre curve. The threecurve equations of contour dynamics that fully describe the doublelayer dynamics are expanded in the small thickness parameter. At leading order, closed nonlinear initialvalue evolution equations are obtained that describe the motion of the centre curve together with the time and spatial variation of each layer thickness. In the special case where the layer vorticities are equal, these equations reduce to the singlelayer equation of Moore ( Stud. Appl. Math. , vol. 58, 1978, pp. 119–140). Analysis of the linear stability of the firstorder equations to smallamplitude perturbations shows Kelvin–Helmholtz instability when the farfield fluid velocities on either side of the double layer are unequal. Equal velocities define a circulationfree double vorticity layer,more »Free, publiclyaccessible full text available July 10, 2023

The curvature instability of thin vortex rings is a parametric instability discovered from shortwavelength analysis by Hattori & Fukumoto ( Phys. Fluids , vol. 15, 2003, pp. 3151–3163). A fullwavelength analysis using normal modes then followed in Fukumoto & Hattori ( J. Fluid Mech. , vol. 526, 2005, pp. 77–115). The present work extends these results to the case with different densities inside and outside the vortex core in the presence of surface tension. The maximum growth rate and the instability halfbandwidth are calculated from the dispersion relation given by the resonance between two Kelvin waves of $m$ and $m+1$ , where $m$ is the azimuthal wavenumber. The result shows that vortex rings are unstable to resonant waves in the presence of density and surface tension. The curvature instability for the principal modes is enhanced by density variations in the small axial wavenumber regime, while the asymptote for short wavelengths is close to the constant density case. The effect of surface tension is marginal. The growth rates of nonprincipal modes are also examined, and long waves are most unstable.

The Moore–Saffman–Tsai–Widnall (MSTW) instability is a parametric instability that arises in strained vortex columns. The strain is assumed to be weak and perpendicular to the vortex axis. In this second part of our investigation of vortex instability including density and surface tension effects, a linear stability analysis for this situation is presented. The instability is caused by resonance between two Kelvin waves with azimuthal wavenumber separated by two. The dispersion relation for Kelvin waves and resonant modes are obtained. Results show that the stationary resonant waves for $m=\pm 1$ are more unstable when the density ratio $\rho_2/\rho_1$ , the ratio of vortex to ambient fluid density, approaches zero, whereas the growth rate is maximised near $\rho _2/\rho _1 =0.215$ for the resonance $(m,m+2)=(0,2)$ . Surface tension suppresses the instability, but its effect is less significant than that of density. As the azimuthal wavenumber $m$ increases, the MSTW instability decays, in contrast to the curvature instability examined in Part 1 (Chang & Llewellyn Smith, J. Fluid Mech. vol. 913, 2021, A14).

Equilibrium solutions for hollow vortices in straining flow in a corner are obtained by solving a freeboundary problem. Conformal maps from a canonical doubly connected annular domain to the physical plane combining the Schottky–Klein prime function with an appropriate algebraic map lead to a problem similar to Pocklington's propagating hollow dipole. The result is a twoparameter family of solutions depending on the corner angle and on the nondimensional ratio of strain to circulation.

The present work uses a reducedorder model to study the motion of a buoyant vortex ring with nonnegligible core size. Buoyancy is considered in both nonBoussinesq and Boussinesq situations using an axisymmetric contour dynamics formulation. The density of the vortex ring differs from that of the ambient fluid, and both densities are constant and conserved. The motion of the ring is calculated by following the boundary of the vortex core, which is also the interface between the two densities. The velocity of the contour comes from a combination of a specific continuous vorticity distribution within its core and a vortex sheet on the core boundary. An evolution equation for the vortex sheet is derived from the Euler equation, which simplifies considerably in the Boussinesq limit. Numerical solutions for the coupled integrodifferential equations are obtained. The dynamics of the vortex sheet and the formation of two possible singularities, including singularities in the curvature and the shocklike profile of the vortex sheet strength, are discussed. Three dimensionless groups, the Atwood, Froude and Weber numbers, are introduced to measure the importance of physical effects acting on the motion of a buoyant vortex ring.

In the problem of horizontal convection a nonuniform buoyancy, $b_{s}(x,y)$ , is imposed on the top surface of a container and all other surfaces are insulating. Horizontal convection produces a net horizontal flux of buoyancy, $\boldsymbol{J}$ , defined by vertically and temporally averaging the interior horizontal flux of buoyancy. We show that $\overline{\boldsymbol{J}\boldsymbol{\cdot }\unicode[STIX]{x1D735}b_{s}}=\unicode[STIX]{x1D705}\langle \unicode[STIX]{x1D735}b^{2}\rangle$ ; the overbar denotes a space–time average over the top surface, angle brackets denote a volume–time average and $\unicode[STIX]{x1D705}$ is the molecular diffusivity of buoyancy $b$ . This connection between $\boldsymbol{J}$ and $\unicode[STIX]{x1D705}\langle \unicode[STIX]{x1D735}b^{2}\rangle$ justifies the definition of the horizontalconvective Nusselt number, $Nu$ , as the ratio of $\unicode[STIX]{x1D705}\langle \unicode[STIX]{x1D735}b^{2}\rangle$ to the corresponding quantity produced by molecular diffusion alone. We discuss the advantages of this definition of $Nu$ over other definitions of horizontalconvective Nusselt number. We investigate transient effects and show that $\unicode[STIX]{x1D705}\langle \unicode[STIX]{x1D735}b^{2}\rangle$ equilibrates more rapidly than other global averages, such as the averaged kinetic energy and bottom buoyancy. We show that $\unicode[STIX]{x1D705}\langle \unicode[STIX]{x1D735}b^{2}\rangle$ is the volumeaveraged rate of Boussinesq entropy production within the enclosure. In statistical steady state, the interior entropy production is balanced by a flux through the top surface. This leads to an equivalent ‘surface Nusselt number’, defined as the surfacemore »