Coherent structure in flow over a slitted bluff body

The focus of the present investigation is resolution of the coherent structure in the near wake behind a slitted bluff body. The bluff body is two-dimensional with gap ratio from 0.12 to 0.48. The evolution of the structure was numerically investigated using the renormalization group (RNG) k–ε model at Reynolds number of 470,000. Two types of coherent structure are identified: At low gap ratio 0.12, the structure is characterized by a flip–flopping gap flow; at high ratio 0.22–0.48, the gap flow deflects to one side with an asymmetrical wake. The coherent structure is divided by the gap flow into two zones called the primary recirculation zone and the secondary recirculation zone. The coherent structure is intimately related to the gap ratio, and the structure of small gap ratio is different from that of large gap ratio because the interaction between two zones relates to the gap ratio. To explain the vortex shedding, a mechanism that single vortex of large size suddenly immerses between two shear layers was proposed. Experimental results using point-to-point method and particle-image velocimetry (PIV) measurements in a close wind tunnel were also carried out to confirm the observation from the numerical study. The evidence shows that the numerical results are of good agreement with the experiments. The comparison between the RNG k–ε model and the large eddy simulation also indicates that the RNG k–ε model is adequate in computing the bluff body flow.     

Point vortex equilibria related to Bessel polynomials

The method of polynomials is used to construct two families of stationary point vortex configurations. The vortices are placed at the reciprocals of the zeroes of Bessel polynomials. Configurations that translate uniformly, and configurations that are completely stationary, are obtained in this way.     

Doppler tomography problem for vortex models

We consider a special field model in the two-dimensional Doppler tomography problem and prove its unique solvability. A numerical method is proposed for solving the problem and some examples are shown demonstrating its efficiency. __________ Translated from Prikladnaya Matematika i Informatika, No. 22, pp. 12–30, 2005.     

Aspects of vortex dynamics in Ginzburg-Landau models

We survey some recent work concerning the asymptotic dynamics of vortices in the 2-dimensional parabolic Ginzburg-Landau equation, the interaction of vortices with the phase field and the limiting initial value problem for both vortices and phase. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A second order boundary layer solution for separated flow past a bluff body

In this investigation second order boundary layer solutions are obtained for subsonic, plane flow over a circular cylinder with a wake. The outer flow due to displacement thickness is obtained by extending a first order solution, which includes wake effects. This enables realistic solutions for the displacement effect to be calculated as well as for the curvature effect. It is shown that as the first order separation point is approached the magnitudes of the second order velocity components become large and the surface shear becomes large and negative. There is also a significant upstream shift in the position of the separation point.

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Résumé Dans ce travail, on obtient des solutions du second ordre des équations de la couche limite pour un écoulement subsonique plan autour d'un cylindre circulaire avec un sillage. On obtient l'écoulement extérieur dû à l'épaisseur de déplacement en étendant une solution du premier ordre qui comprend les effets du sillage. Ceci permet de calculer des solutions réalistes pour l'effet de déplacement ainsi que pour l'effet de courbure. On montre que les composantes de la vitesse du second ordre de même que la tension tangentielle deviennent très grandes et cette dernière négative au voisinage du point de décollement du premier ordre. Il y a aussi un déplacement significatif vers l'amont de la position du point de décollement.

     

Optimal body design using localized interaction models

The problem of the designing bodies of minimum drag for a specified area of the base and a specified area of the (“windward”) surface around which the flow occurs is considered in the approximation of an arbitrary localized interaction model. New necessary conditions for minimum drag are obtained which are stronger than the Legendre condition. It is shown that, in the approximation used, the optimal configurations in the general case contain end faces and cylindrical segments of the boundary extremum, which appear due to the existence of limits of applicability of local models. It is established that the solutions previously obtained are incomplete. A complete solution of the problem is constructed.     

Robust modelling of flow induced oscillations of bluff bodies

In mechanical engineering and in the engineering literature dealing with flow induced oscillations of bluff bodies frequently a non-robust model with non-linear terms of third order and two parameters is used. By means of the methods of bifurcation theory it can be shown that this system corresponds to a degenerate system with a double zero eigenvalue with Codimension 3. Therefore 3 parameters are needed to obtain a robust model. A physical interpretation of the third parameter and its influence on the bifurcation diagram are given.     

Dynamics and self-propulsion of a spherical body shedding coaxial vortex rings in an ideal fluid

We describe a model for the dynamic interaction of a sphere with uniform density and a system of coaxial circular vortex rings in an ideal fluid of equal density. At regular intervals in time, a constraint is imposed that requires the velocity of the fluid relative to the sphere to have no component transverse to a particular circular contour on the sphere. In order to enforce this constraint, new vortex rings are introduced in a manner that conserves the total momentum in the system. This models the shedding of rings from a sharp physical ridge on the sphere coincident with the circular contour. If the position of the contour is fixed on the sphere, vortex shedding is a source of drag. If the position of the contour varies periodically, propulsive rings may be shed in a manner that mimics the locomotion of certain jellyfish. We present simulations representing both cases.     

On the motion of a rigid body in a two-dimensional ideal flow with vortex sheet initial data

A famous result by Delort about the two-dimensional incompressible Euler equations is the existence of weak solutions when the initial vorticity is a bounded Radon measure with distinguished sign and lies in the Sobolev space H⁻¹$H^{-1}$. In this paper we are interested in the case where there is a rigid body immersed in the fluid moving under the action of the fluid pressure. We succeed to prove the existence of solutions à la Delort in a particular case with a mirror symmetry assumption already considered by Lopes Filho et al. (2006) [11], where it was assumed in addition that the rigid body is a fixed obstacle. The solutions built here satisfy the energy inequality and the body acceleration is bounded. When the mass of the body becomes infinite, the body does not move anymore and one recovers a solution in the sense of Lopes Filho et al. (2006) [11].     

Convergence of the point vortex method for 2-D vortex sheet

We give an elementary proof of the convergence of the point vortex method (PVM) to a classical weak solution for the two-dimensional incompressible Euler equations with initial vorticity being a finite Radon measure of distinguished sign and the initial velocity of locally bounded energy. This includes the important example of vortex sheets, which exhibits the classical Kelvin-Helmholtz instability. A surprise fact is that although the velocity fields generated by the point vortex method do not have bounded local kinetic energy, the limiting velocity field is shown to have a bounded local kinetic energy.

     

Numerical discovery and experimental confirmation of vortex ring generation by microramp vortex generator

An implicitly implemented large eddy simulation (ILES), by using the modified fifth order WENO scheme, is applied to study the flow around the microramp vortex generator (MVG) at Mach 2.5 and Re_θ = 5760. A series of new discoveries on the flow around supersonic MVG have been made by the UTA LES team including source of the momentum deficit, inflection points (surface in 3-D), Kelvin–Helmholtz instability and vortex ring generation. Most of the new discoveries, which were made by the UTA LES team and presented in 2009, were confirmed by experiment conducted by the UTA experiment team in 2010. A new 5-pair-vortex-tube model near the MVG is given based on the ILES observation.     

Hamiltonian description of vortex systems

Theoretical and Mathematical Physics - In the framework of two-dimensional ideal hydrodynamics, we define a vortex system as a smooth vorticity function with a few local positive maximums and...     

Mathematical analysis of vortex sheets

We consider the motion of the interface separating two domains of the same fluid that moves with different velocities along the tangential direction of the interface. The evolution of the interface (the vortex sheet) is governed by the Birkhoff‐Rott (BR) equations. We consider the question of the weakest possible assumptions such that the Birkhoff‐Rott equation makes sense. This leads us to introduce chord‐arc curves to this problem. We present three results. The first can be stated as the following: Assume that the Birkhoff‐Rott equation has a solution in a weak sense and that the vortex strength is bounded away from 0 and ∞. Moreover, assume that the solution gives rise to a vortex sheet curve that is chord‐arc. Then the curve is automatically smooth, in fact analytic, for fixed time. The second and third results demonstrate that the Birkhoff‐Rott equation can be solved if and only if only half the initial data is given. © 2005 Wiley Periodicals, Inc.     

Almost optimal convergence of the point vortex method for vortex sheets using numerical filtering

Standard numerical methods for the Birkhoff-Rott equation for a vortex sheet are unstable due to the amplification of roundoff error by the Kelvin-Helmholtz instability. A nonlinear filtering method was used by Krasny to eliminate this spurious growth of round-off error and accurately compute the Birkhoff-Rott solution essentially up to the time it becomes singular. In this paper convergence is proved for the discretized Birkhoff-Rott equation with Krasny filtering and simulated roundoff error. The convergence is proved for a time almost up to the singularity time of the continuous solution. The proof is in an analytic function class and uses a discrete form of the abstract Cauchy-Kowalewski theorem. In order for the proof to work almost up to the singularity time, the linear and nonlinear parts of the equation, as well as the effects of Krasny filtering, are precisely estimated. The technique of proof applies directly to other ill-posed problems such as Rayleigh-Taylor unstable interfaces in incompressible, inviscid, and irrotational fluids, as well as to Saffman-Taylor unstable interfaces in Hele-Shaw cells.

     

Localized asymptotic solutions of the navier-stokes equations and laminar wakes in an incompressible fluid

Equations are derived to describe the far-field laminar wake behind a body in incompressible fluid flow with an arbitrary distribution of the free-stream (unperturbed flow) velocity. For certain classes of free-stream flows, analysis of these equations enables various processes in narrow wakes or jets to be described (the interaction of the longitudinal transverse velocity components in a jet, cause it to accelerate or decelerate and conservation of the energy of the wake by distortion of its trajectory regardless of viscous dissipation). In particular, conditions are obtained for the wake growth in spiral flows, analogous to the Rayleigh conditions for the instability of two-dimensionally radially symmetric flows relative to three-dimensional short-wave perturbations.     

The critical cross-section of a vortex

Summary This paper discusses the problem of critical-flow cross-sections in vortex flows. It is shown that there are two different types of vortex flows, A-type and B-type vortices (say). An A-type vortex approaches its critical flow state as its cross-sectional area increases and departs from the critical state as the cross-sectional area is decreased. This property is associated with the particular dependence of total pressure and circulation on the stream function, and it holds for both subcritical and supercritical A-type vortices. On the other hand, both subcritical and supercritical B-type vortices approach their critical flow states as their cross-sectional areas are decreased and depart from their critical states for increasing cross-sectional area. As was shown by Benjamin, setting the first variation of the flow force with respect to the stream function equal to zero leads to Euler's equation of motion. The second variation also vanishes if the corresponding flow state is critical. In this case the sign of the third variation decides whether the flow is an A-type or a B-type vortex. Within the framework of inviscid-fluid flow theory the type of a vortex is preserved unless vortex breakdown occurs. Making use of the knowledge that vortex flows are controlled by two different types of critical-flow cross-sections a variety of vortex flow phenomena are investigated, including the two types of inlet vortices that are observed upstream of jet engines, the behavior of vortex valves, the flow characteristics of liquid-fuel atomizers and the bath tub vortex.     

The rolling-up of a vortex sheet

Zusammenfassung Die vorliegenden Untersuchungen befassen sich mit dem Problem der zweidimensionalen zeitabhängigen Strömung, welche mit dem Abrollen einer semiunendlichen Wirbelfläche verbunden ist. Obwohl bereits vorliegende analytische Untersuchungen an diesem Problem Näherungslösungen für die Spiralform der aufgerollten Wirbelfläche ergeben, so dürfte es doch schwierig sein, aus diesen Ergebnissen das zugehörige Strömungsfeld zu erhalten. In dieser Arbeit werden Differentialgleichungen für den Mittelwert und den Sprung des komplexen Potentials abgeleitet, welche zur Beschreibung des Strömungsfeldes über die gesamte Ebene benutzt werden können. Für den Spiralkern stimmt die hier angegebene Lösung mit der vonKaden überein.