ALBERT

Description: A study of the response of a columnar vortex with on-zero axial flow to impulsive cutting has been performed. The flow evolution is computed based on the vorticity-velocity formulation of the axisymmetric Euler equation using a Lagrangian vorticity collocation method. The vortex response is compared to analytical predictions obtained using the plug-flow model of Lundgren & Ashurst (1989). The plug-flow model indicates that axial motion on a vortex core with variable core area behaves in a manner analogous to one-dimensional gas dynamics in a tube, with the vortex core area playing a role analogous to the gas density. The solution for impulsive cutting of a vortex obtained from the plug-flow model thus resembles the classic problem of impulsive motion of a piston in a tube, with formation of an upstream-propagating vortex 'shock' (over which the core radius changes discontinuously) and a downstream-propagating vortex 'expansion wave' on opposite sides of the cutting surface. Direct computations of the vortex response from the Euler equation reveal similar upstream- and downstream-propagating waves following impulsive cutting for cases where the initial vortex flow is subcritical. These waves in core radius are produced by a series of vortex rings, embedded within the columnar vortex core, having azimuthal vorticity of alternating sign. The effect of the compression and expansion waves is to bring the axial and radial velocity components to nearly zero behind the propagating vortex rings, in a region on both sides of the cutting surface with ever-increasing length. The change in vortex core radius and the variation in pressure along the cutting surface agree very well with the predictions of the plug-flow model for subcritical flow after the compression and expansion waves have propagated sufficiently far away. For the case where the ambient vortex flow is supercritical, no upstream-propagating wave is possible on the compression side of the vortex, and the vortex axial flow is observed to impact on the cutting surface in a manner similar to that commonly observed for a non-rotating jet impacting on a wall. The flow appears to approach a steady state near the point of impact after a sufficiently long time. The vortex response on the expansion side of the cutting surface exhibits a downstream-propagating vortex expansion wave for both the subcritical and supercritical conditions. The results of the vortex response study are used to formulate and verify predictions for the net normal force exerted by the vortex on the cutting surface. An experimental study of the cutting of a vortex by a thin blade has also been performed in order to verify and assess the limitations of the instantaneous vortex cutting model for application to actual vortex-body interaction problems.

Description: A computational study is reported of the instability and growth of fingers for liquid films driven over heterogeneous surfaces. Computations are performed using a variation of the precursor-film model, in which a disjoining pressure term is used to introduce variation in the static contact angle, which in turn models surface heterogeneity. The formulation is shown to yield results consistent with the Tanner-Hoffman-Voinov dynamic contact angle formula for sufficiently small values of the precursor film thickness. A modification of the disjoining pressure coefficient is introduced which yields correct variation of dynamic contact angle for finite values of the precursor film thickness. The fingering instability is examined both for cases with ordered strips of different static contact angle and for cases with random variation in static contact angle. Surface heterogeneity is characterized by strip width and amplitude of static contact angle variation for the case with streamwise strips and by correlation length and variance of the static contact angle variation from its mean value for the random distribution case. © 2006 Cambridge University Press.

Description: A computational study of three-dimensional vortex-cylinder interaction is reported for the case where the nominal orientation of the cylinder axis is normal to the vortex axis. The computations are performed using a new tetrahedral vorticity element method for incompressible viscous fluids, in which vorticity is interpolated using a tetrahedral mesh that is refit to the Lagrangian computational points at each timestep. Fast computation of the Biot-Savart integral for velocity is performed using a boxpoint multipole acceleration method for distant tetrahedra and Gaussian quadratures for nearby tetrahedra. A moving least-square method is used for differentiation, and a flux-based vorticity boundary condition algorithm is employed for satisfaction of the no-slip condition. The velocity induced by the primary vortex is obtained using a filament model and the Navier-Stokes computations focus on development of boundary-layer separation from the cylinder and the form and dynamics of the ejected secondary vorticity structure. As the secondary vorticity is drawn outward by the vortex-induced flow and wraps around the vortex, it has a substantial effect both on the essentially inviscid flow field external to the boundary layer and on the cylinder surface pressure field. Cases are examined with background free-stream velocity oriented in the positive and negative directions along the cylinder axis, with free-stream velocity normal to the cylinder axis, and with no free-stream velocity. Computations with no free-stream velocity and those with free-stream velocity tangent to the cylinder axis exhibit similar secondary vorticity structures, consisting of a vortex loop (or hairpin) that wraps around the primary vortex and is attached to the cylinder boundary layer at two points. Computations with free-stream velocity oriented normal to the cylinder axis exhibit secondary vorticity structure of a markedly different character, in which the secondary eddy remains close to the cylinder boundary and has a quasi-two-dimensional form for an extended time period.

Description: A study has been performed of the interaction of periodic vortex rings with a central columnar vortex, both for the case of identical vortex rings and the case of rings of alternating sign. Numerical calculations, both based on an adaptation of the Lundgren-Ashurst (1989) model for the columnar vortex dynamics and by numerical solution of the axisymmetric Navier-Stokes and Euler equations in the vorticity-velocity formulation using a viscous vorticity collocation method, are used to investigate the response of the columnar vortex to the ring-induced velocity field. In all cases, waves of variable core radius are observed to build up on the columnar vortex core due to the periodic axial straining and compression exerted by the vortex rings. For sufficiently weak vortex rings, the forcing by the rings serves primarily to set an initial value for the axial velocity, after which the columnar vortex waves oscillate approximately as free standing waves. For the case of identical rings, the columnar vortex waves exhibit a slow upstream propagation due to the nonlinear forcing. The cores of the vortex rings can also become unstable due to the straining flow induced by the other vortex rings when the ring spacing is sufficiently small. This instability causes the ring vorticity to spread out into a sheath surrounding the columnar vortex. For the case of rings of alternating sign, the wave in core radius of the columnar vortex becomes progressively narrower with time as rings of opposite sign approach each other. Strong vortex rings cause the waves on the columnar vortex to grow until they form a sharp cusp at the crest, after which an abrupt ejection of vorticity from the columnar vortex is observed. For inviscid flow with identical rings, the ejected vorticity forms a thin spike, which wraps around the rings. The thickness of this spike increases in a viscous flow as the Reynolds number is decreased. Cases have also been observed, for identical rings, where a critical point forms on the columnar vortex core due to the ring-induced flow, at which the propagation velocity of upstream waves is exactly balanced by the axial flow within the vortex core when measured in a frame translating with the vortex rings. The occurrence of this critical point leads to trapping of wave energy downstream of the critical point, which results in large-amplitude wave growth in both the direct and model simulations. In the case of rings of alternating sign, the ejected vorticity from the columnar vortex is entrained and carried off by pairs of rings of opposite sign, which move toward each other and radially outward under their self- and mutually induced velocity fields, respectively.

Description: A computational study of the penetration of a blade into the core of an initially columnar vortex in an incompressible viscous fluid and the subsequent cutting of the vortex is reported for the case where the blade axis is initially orthogonal to the vortex axis. The vortex is advected toward the fixed blade by a free-stream velocity oriented tangent to the blade chord, where the free-stream speed is sufficiently large that the vortex does not induce ejection of vorticity from the blade boundary layer prior to impact of the vortex core with the blade leading edge. A range of computations are performed for cases both with and without ambient axial flow in the vortex core. As the blade leading edge penetrates into the vortex core, cross-diffusion between the columnar vortex and the blade boundary layer causes vortex lines originating in the columnar vortex to rapidly reconnect to those in the blade boundary layer. This cutting process is found to be always incomplete however, due to a change in sign of the spanwise vortex-induced velocity along the leading edge as the vortex is cut, leaving a thin vortex sheet that wraps around the blade leading edge. Cutting of a vortex with non-zero axial flow causes an asymmetry that results in an impulsive lift force on the blade. This lift force has maximum magnitude during the time period where the blade leading edge penetrates into the vortex core. Both the vortex cutting process and the unsteady lift force on the blade are found to be approximately independent of Reynolds number for the various cases examined. © 2004 Cambridge University Press.

Description: A computational study has been performed to examine the amplification of three-dimensional flow features as a vortex with small-amplitude helical perturbations impinges on a circular cylinder whose axis is parallel to the nominal vortex axis. For sufficiently weak vortices with sufficiently small core radius in an inviscid flow, three-dimensional perturbations on the vortex core are indefinitely amplified as the vortex wraps around the cylinder front surface. The paper focuses on the effect of viscosity in regulating amplification of three-dimensional disturbances and on assessing the ability of two-dimensional computations to accurately model parallel vortex - cylinder interaction problems. The computations are performed using a multi-block structured finite-volume method for an incompressible flow, with periodic boundary conditions along the cylinder axis. Growth of three-dimensional flow features is examined using a proper-orthogonal decomposition of the Fourier-transformed vorticity field in the azimuthal and axial directions. The interaction is examined for different axial wavelengths and amplitudes of the initial helical vortex waves and for three different Reynolds numbers. © 2007 Cambridge University Press.

Description: A study of two-layer quasi-geostrophic vortex flow is performed to determine the effect of a current difference between the layers on a vortex initially extending through both layers. In particular, the conditions under which the vortex can resist being torn by the current difference are examined. The vortex evolution is determined using versions of the contour dynamics and discrete vortex methods which are modified for two-layer quasi-geostrophic flows. The vortex response is found to depend upon the way in which the current difference between the layers is maintained. In the first set of flows studied, the current difference is generated by a (stronger) third vortex in the upper layer located at a large distance from the (weaker) vortex under study. Flows of this type are important for understanding the interactions of vortices of different sizes in geophysical turbulence. A set of flows is also considered in which an ambient geostrophic current difference is produced by a non-uniform background potential vorticity field. In this case, an additional (secondary) flow field about the vortex patch in each layer is generated bv redistribution of the ambient potential vorticitv field. It is found that a vortex that initially extends through both layers will undergo a periodic motion, in which the two parts of the initial vortex in the different layers (called the ‘upper’ and Tower’ vortices) oscillate about each other, provided that the current difference between the layers is less than a critical value. When the current difference exceeds this critical value, the upper and lower vortices separate permanently and the initial vortex is said to ‘tear’. The effects of various dimensionless parameters that characterize the flow are considered, including the ratio of core radius to internal Rossby radius, the ratio of layer depths and the ratio of the strengths of the upper and lower vortices. These parameters affect both the critical current difference for tearing and the deformation of the vortex cores by their interaction. It is found that for small values of inverse internal Rossby deformation radius, calculations with circular non-deformable vortices (convected at their centrepoints) give results in good agreement with the contour dynamics simulations, since the vortex deformation is small. The results of the study will be useful in determining the conditions under which large geophysical vortex structures, such as cyclones and ocean rings, can extend to large heights (depths) even though the mean winds (currents) in the ambient flow change significantly along the vortex length. © 1993, Cambridge University Press. All rights reserved.