Numerical Simulation of Vortex Breakdown

Journal of Fluid Mechanics, 2006

The spiral form of vortex breakdown observed in the numerical simulations of Ruith et al. (J. Fluid Mech., vol. 486, 2003, p. 331) is interpreted as a nonlinear global mode originating at the convective/absolute instability transition point in the lee of the vortex breakdown bubble. The local absolute frequency at the transition station is shown to yield a satisfactory prediction of the precession frequency measured in the three-dimensional direct numerical simulations.

2004

We consider the direct numerical simulation of gas bubbles in viscous uid ows. The behaviour of single bubbles and bubble swarms is studied with the particu- lar emphasis on the bubble coalescence and breakage phenomena. For single bubbles, we are able to simulate the bubble dynamics in all relevant ow regimes, including the von Karman type vortex shedding and wobbling-bubble regime at high Reynolds number. Mathematically, our model is a combination of the Navier-Stokes equations and a free boundary problem for the bubble shape. The bubble boundaries are treated by the level set method that naturally allows the topology changes due to bubble coalescence and breakup. This, in addition to the simple nite-element method implemented in the operator-splitting framework, has proved to permit an accurate prediction of ows with moving interfaces in a wide range of physical parameters. While the coalescence of bubbles in dier ent ow regimes presents a very complicated phenomenon, the complex...

Journal of Fluid Mechanics, 2006

This paper is motivated by an observation: in the nascent state of vortex breakdown before it develops into a full-grown radial expansion, an initially straight vortex core first swells, and does so even in a straight pipe for no apparent reason. Although this initial swelling may be explained in many ways according to the perspectives chosen, we offer our own interpretation framed solely within vorticity dynamics: the radial swelling as well as the subsequent growth are induced by the azimuthal vorticity gradient decreasing downstream. The negative azimuthal vorticity gradient first appears at start-up and moves eventually into the region where the circulation reaches its steady-state value. The vorticity gradient can become negative without necessarily being accompanied by a sign-switch of the azimuthal vorticity itself. The key point-that the negative azimuthal vorticity gradient induces initial radial swelling and its growth-is demonstrated in two analyses. First, a kinematic analysis results in an equation for the radial velocity where the azimuthal vorticity gradient appears as a source term. Its solution shows, in general and explicitly, that the negative azimuthal vorticity gradient does induce radially outward velocity. Two heuristic examples serve to illustrate this point further. In the second analysis, using the equation of motion in the streamline coordinates, the negative azimuthal vorticity gradient is shown to diverge the meridional streamlines radially. A numerical simulation using a modified vortex filament method not only corroborates this role of the azimuthal vorticity gradient in initiating and promoting the radial expansion, but also adds details to track the formation process. Both analyses and simulation support our interpretation that the initial radial swelling and its subsequent growth are induced by the negative azimuthal vorticity gradient.

Fluid Dynamics of High Angle of Attack, 1993

Computational simulation and study of shock/vortex interaction and vortex-breakdown modes are considered for bound (internal) and unbound (external) flow domains. The problem is formulated using the unsteady, compressible, full Navier-Stokes (NS) equations which are solved using an implicit, flux-difference splitting, finite-volume scheme. For the bound flow domain, a supersonic swirling flow is considered in a configured circular duct and the problem is solved for quasi-axisymmetric and three-dimensional flows. For the unbound domain, a supersonic swirling flow issued from a nozzle into a uniform supersonic flow of lower Mach number is considered for quasi-axisymmetric and three-dimensionai flows. The results show several modes of breakdown; e.g., no-breakdown, transient single-bubble breakdown, transient multi-bubble breakdown, periodic multi-bubble multi-frequency breakdown and helical breakdown. Highlights of Formulation and Computational Scheme Formulation: The conservative,unsteady, compressible, fullNavier-Stokes equations in terms of time-independent,body-conformed coordinates_I, _2 and _3 are used to solve the problem. The equations arc given in Ref. [II] and hence they are not presented hem. Along with these equations, boundary conditions arc specifiedat the computational-domain inlet, side wall and downstream exit.The downstream exit boundary conditions will be presented and discussed in the next sectionof the computational results. The initial conditionsarc also presented in the next section. Computational Scheme: The computational scheme used to solve the unsteady, compressible full NS equations is an implicit, upwind, flux-difference splitting, finite-volume scheme. It employs the flux-difference splitting scheme of Roe which is based on the solution of the approximate one-dimensional Riemann problem in each of the three directions. In the Roe scheme, the inviscid flux difference at the interface of a computational cell is split into left and right flux differences. The splitting is accomplished according to the signs of the eigenvalues of the Roe averaged-Jacobian roan'ix of the inviscid flux at the cell interface. The smooth limiter is used to eliminate oscillations in the shock region. The viscous and heat-flux terms am differenced using second-order spatially accurate central differencing. The resulting contours for this case. These snapshots show a vortex breakdown mechanism of evolution, Concluding Remarks Computational simulation and study of shock/vortex interaction and vortex breakdown have been considered for internal and external supersonic swirling flow. The time-accurate computation for full Navier-Stokes equations is used to produce all the present cases.

Journal of Fluid Science and Technology, 2007

Numerical simulation is performed for the vortex breakdown flows induced in a cylindrical container with rotating top disk. Solutions are obtained for the parameter ranges where steady axisymmetric flows are stable with respect to asymmetric disturbances. By varying the number of grid points and applying the Richardson extrapolation, resolution independent solutions are obtained and flow state diagram is numerically constructed on (h, Re)-plane (Re is the Reynolds number and h is the cylinder aspect ratio). The value of critical Reynolds number for the creation of breakdown bubbles agrees on the average within 1% discrepancy with the previous experiments. Solutions obtained in the present study are expected to be useful as a benchmark solution visa -vis newly developed numerical methods are compared.