Monika Nitsche

A numerical study of vortex ring formation at the edge of a circular tube

An axisymmetric vortex-sheet model is applied to simulate an experiment of Didden (1979) in which a moving piston forces fluid from a circular tube, leading to the formation of a vortex ring. Comparison between simulation and experiment indicates that the model captures the basic features of the ring formation process. The computed results support the experimental finding that the ring trajectory and the circulation shedding rate do not behave as predicted by similarity theory for starting flow past a sharp edge. The factors responsible for the discrepancy between theory and observation are discussed.

Euler-alpha and vortex blob regularization of vortex filament and vortex sheet motion

The Euler-alpha and the vortex blob model are two different regularizations of incom- pressible ideal fluid flow. Here, a regularization is a smoothing operation which controls the fluid velocity in a stronger norm than

Axisymmetric Vortex Sheet Motion: Accurate Evaluation of the Principal Value Integral

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Scaling properties of vortex ring formation at a circular tube opening

. The Euler-alpha model is the inviscid version of the Lagrangian averaged Navier–Stokes-alpha turbulence model. The vortex blob model was introduced to regularize vortex flows. This paper presents both models within one general framework, and compares the results when applied to planar and axisymmetric vortex filaments and sheets. By certain measures, the Euler-alpha model is closer to the unregularized flow than the vortex blob model. The differences that result in circular vortex filament motion, vortex sheet linear stability properties, and core dynamics of spiral vortex sheet roll-up are discussed.

Self-similar shedding of vortex rings

This paper concerns the accurate evaluation of the principal value integral governing axisymmetric vortex sheet motion. Previous quadrature rules for this integral lose accuracy near the axis of symmetry. An approximation by de Bernadinis and Moore (dBM) that converges pointwise at the rate of O(h3) has maximal errors near the axis that are O(h). As a result, the discretization error is not smooth. It contains high wavenumber frequencies that make it difficult to resolve the vortex sheet motion. This paper explains the reason for the degeneracy near the axis and proposes a modified quadrature rule that is uniformly O(h3). The results are based on an analytic approximation of the integrand, whose integral can be precomputed. The modification is implemented at negligible additional cost per timestep. As an example, it is applied to compute the evolution of an initially spherical vortex sheet.

Simulating Vortex Wakes of Flapping Plates

A vortex sheet model is applied to study vortex ring formation at the edge of a circular tube, for accelerating piston velocities Up∼tm. We determine properties of the vortex ring as a function of the piston motion and investigate the extent to which similarity theory for planar vortex sheet separation applies. For piston strokes up to half the tube diameter, we find that the ring diameter, core size and circulation are well predicted by the planar similarity theory. The axial ring translation is a superposition of an upstream component predicted by the theory and a downstream component which is linear in the piston stroke. The front of the fluid volume exiting the tube is also linear in the piston stroke and travels with 75% of the piston velocity. The core size decreases and the distribution of fluid near the core becomes more asymmetric as the parameter m increases.

Evolution of solitary waves in a two-pycnocline system

The roll-up of an initially spherical vortex sheet into a vortex ring is computed using the vortex blob method. The ring sheds about 30% of its circulation into a tail which, in turn, rolls up into a ring that sheds circulation. The process repeats itself at smaller and smaller scales in a self-similar manner. The relation between the vortex shedding and the energy of the vortices is investigated. In contrast, an initially cylindrical vortex sheet rolls up into a vortex pair that sheds essentially no circulation. Consider a sphere immersed in stagnant inviscid fluid which is impulsively set into motion. The resulting potential flow is induced by an axisymmetric vortex sheet in place of the sphere. This paper studies the evolution of the vortex sheet under its selfinduced velocity, as if the sphere were dissolved and the fluid within it allowed to move with the flow. The axisymmetric flow is compared to the planar flow generated by the impulsive motion of a cylinder. Rottman, Simpson & Stansby (1987) performed an experiment simulating the cylindrical scenario by quickly removing a hollow cylinder immersed in a crossflow. They also computed this flow using a vortex-incell method and compared numerical and experimental results. Rottman & Stansby (1993) computed the planar flow using the vortex blob method. The axisymmetric flow was computed by Winckelmans et al. (1995) using a three-dimensional vortex particle method. Here, we compute the planar and axisymmetric flow to longer times than in prior work, using the vortex blob method. The method consists of regularizing the singular governing equations by introducing an articial parameter (Chorin & Bernard 1973; Anderson 1985; Krasny 1986). Comparisons with solutions of the Navier{Stokes equations (Tryggvason, Dahm & Sbeih 1991) and with experimental measurements (Nitsche & Krasny 1994) show that the method approximates viscous flow well for suciently small values of the articial smoothing parameter and viscosity. The computed planar and axisymmetric sheets roll up into a vortex pair and a vortex ring respectively as they travel in the direction of the given impulse. However, the vortex ring sheds about 30% of its circulation into a tail which, in turn, rolls up into a vortex ring. This observed shedding and roll-up repeats itself in a self-similar manner, forming a sequence of vortex rings of decreasing size in the tail of the leading ring. In contrast, the vortex pair does not shed any signicant amount of circulation. The results are shown to be essentially independent of the flow regularization. The relation between the observed shedding and the energy of the vortex rings is also discussed, motivated by the work of Gharib, Rambod & Shari (1998) relating the energy and the circulation of vortex rings generated in laboratory experiments.

Circulation shedding in viscous starting flow past a flat plate

We compare different models to simulate two-dimensional vortex wakes behind oscillating plates. In particular, we compare solutions using a vortex sheet model and the simpler Brown–Michael model to solutions of the full Navier–Stokes equations obtained using a penalization method. The goal is to determine whether simpler models can be used to obtain good approximations to the form of the wake and the induced forces on the body.

Start-up vortex flow past an accelerated flat plate

Over two decades ago, some numerical studies and laboratory experiments identified the phenomenon of leapfrogging internal solitary waves located on separated pycnoclines. We revisit this problem to explore the behaviour of the near resonance phenomenon. We have developed a numerical code to follow the long-time inviscid evolution of isolated mode-two disturbances on two separated pycnoclines in a three-layer stratified fluid bounded by rigid horizontal top and bottom walls. We study the dependence of the solution on input system parameters, namely the three fluid densities and the two interface thicknesses, for fixed initial conditions describing isolated mode-two disturbances on each pycnocline. For most parameter values, the initial disturbances separate immediately and evolve into solitary waves, each with a distinct speed. However, in a narrow region of parameter space, the waves pair up and oscillate for some time in leapfrog fashion with a nearly equal average speed. The motion is only quasi-periodic, as each wave loses energy into its respective dispersive tail, which causes the spatial oscillation magnitude and period to increase until the waves eventually separate. We record the separation time, oscillation period and magnitude, and the final amplitudes and celerity of the separated waves as a function of the input parameters, and give evidence that no perfect periodic solutions occur. A simple asymptotic model is developed to aid in interpretation of the numerical results.

Stability versus Maneuverability in Hovering Flight

Numerical simulations of viscous flow past a flat plate moving in the direction normal to itself reveal details of the vortical structure of the flow. At early times, most of the vorticity is attached to the plate. This paper introduces a definition of the shed circulation at all times and shows that it indeed represents vorticity that separates and remains separated from the plate. During a large initial time period, the shed circulation satisfies the scaling laws predicted for self-similar inviscid separation. Various contributions to the circulation shedding rate are presented. The results show that during this initial time period, viscous diffusion of vorticity out of the vortex is significant but appears to be independent of the value of the Reynolds number. At later times, the departure of the shed circulation from its large Reynolds number behaviour is significantly affected by diffusive loss of vorticity through the symmetry axis. A timescale is proposed that describes when the viscous loss through the axis becomes relevant. The simulations provide benchmark results to evaluate simpler separation models such as point vortex and vortex sheet models. A comparison with vortex sheet results is included.