Galina Ya. Dynnikova

The viscous dipole domains method is a kind of the Lagrangian impulse-particle method using the diffusion velocity method and the particle strength exchange. We prefer to call particles as domains because a particle is associated with a concrete volume of fluid and moves with its velocity. When using diffusion velocity, we track a reference point that moves relative to the fluid. We assign this point a volume equal to the inverse density of points near it. This paper presents a detailed description of the method. The physical mechanism underlying the onset and evolution of numerical instability, which reflect the underlying physics of turbulent cascades, is revealed. A numerical stability criterion consistent with Kolmogorov scaling is derived. A new model for the diffusive motion of particles is proposed, ensuring a more uniform spatial distribution. The application of the developed method is demonstrated through numerical simulations of the evolution of vortex rings.

The stability of vortex-induced vibration of a rigid circular cylinder elastically mounted with one degree of freedom in a transverse flow is investigated in the absence of damping. New equations describing the non-stationary behavior of the frequency, amplitude, and phase of cylinder oscillations have been derived, based on the assumption of slowly varying oscillation parameters in the hydrodynamic context. These equations allow us to analyze the conditions when periodic oscillations with a constant frequency and amplitude are possible, and under what conditions, they will be irregular. The origins of a sharp change in the phase shift between the displacement and the force at close values of flow parameters are revealed. The possibility of non-uniqueness of the oscillation mode for the same parameters of the system is shown. The results of numerical simulations are presented, confirming the developed theory.

The modified viscous vortex domains method is used for direct numerical simulation of two-dimensional flow around an elliptical cylinder at the Reynolds number up to Re = 106. The modified method is based on the use of a conformal mapping from the outer region of an ellipse to the outer region of a circle. Transformed Navier–Stokes equations under an arbitrary conformal mapping are derived. These equations are solved in the mapped plane. An efficient algorithm for calculating boundary elements, previously developed to calculate the flow around a circular cylinder, is applied. The application of this algorithm makes it possible to reduce the calculation time by orders of magnitude and significantly expands the possibilities of flow simulation at high values of the Reynolds number. The method developed in this work can be implemented in other vortex methods.

The results of investigation of oscillations of an airfoil on an elastic hinge that performs transverse sinusoidal oscillations in the free stream of a viscous incompressible fluid are given. The motion of the hinge is specified and the angular oscillations of the airfoil occur under the action of hydrodynamic and elastic forces. The problem of fluid-structure interaction is solved in the complete coupled formulation. Fluid is described by the Navier–Stokes equations whose numerical solution is constructed using the meshless method of viscous vortex domains. Simulation of the related motion of the continuous medium and the rigid body is reached by solving the united linear system of equations in which the unknowns are the vorticity fluxes from the body surface and the velocity of rotational motion of the body. As a result, all unknown quantities are calculated simultaneously in each time step omitting the traditional procedure of splitting the step into the hydrodynamic and dynamic parts with subsequent iterations. The mechanisms underlying the thrust in such motion of airfoil and the role of the added mass force in this process are explained.

Integral expression is derived for aerodynamic forces in unsteady separated flows of an incompressible Newtonian fluid via the characteristics of the flow on the body surface including the vortex flux generated by it. The expression obtained is valid for unsteady and steady flows under the no-slip or the partial slip boundary conditions in a viscous fluid as well as for complete slip in the case of an ideal fluid including the flows with tangential discontinuity of the velocity field. Unlike the number of existing formulas, the expression does not contain the volume integrals and is applicable for the case of multiple bodies. The expression greatly simplifies a calculation of the hydrodynamic force when using vortex meshless methods. The formula obtained satisfies the fundamental law of hydrodynamic momentum conservation. It is derived mathematically rigorously without any assumptions.

The reverse vortex street differs from the ordinary Karman vortex street in the direction of the rotation of the vortices. Such a street is formed behind the oscillating airfoils in the flow. It is of interest in connection with studies of the aerodynamics of flapping wings. It is known that the vortex wake behind a symmetric airfoil performing symmetric oscillations in a certain range of parameters becomes asymmetric, which leads to the appearance of a nonzero average lift. Reasons of the symmetry violation are associated with the instability of the reverse vortex street, but the mechanism of this instability is currently not well understood. In this work, the analysis of the stability of the reverse vortex street is carried out on the basis of the theory developed by Karman for infinite rows of point vortices. In contrast to the Karman model, in this work, semi-infinite rows with periodically arising new vortices at their ends are considered. This is the first time this model has been used. It is shown that the periodic appearance of new vortices radically affects the characteristics of the street stability. It is found that the violation of the symmetry of the reverse vortex street is associated with its instability to bending perturbations, while the ordinary Karman vortex street behind the body is unstable to varicose perturbations. Regions of instability are determined.

It is proved that at the same instantaneous distribution of the flow velocity of a viscous incompressible fluid, the forces acting on a body moving with acceleration differ from forces acting on the body moving with constant velocity by a vector, which is equal to the added masses tensor multiplied by the acceleration vector. The tensor of the added masses coincides with the tensor calculated for potential flows with the same geometry of the body and surrounding surfaces, and does not depend either on viscosity or on the distribution of vorticity in the flow space. While the force corresponding to the motion with constant velocity depends on the history of movement.

It is proved that, with the same instantaneous distribution of the flow velocity of a viscous incompressible fluid, the forces acting on a body moving with acceleration and the forces acting...

The vorticity evolution equation following from the Navier-Stokes equations for three-dimensional viscous incompressible fluid flows is represented in the divergent form. The partial derivative of the vorticity with respect to time is expressed as the divergence of a tensor called the vorticity transfer tensor. The concept of vorticity flux generated by a surface is defined. An expression for the pressure force in terms of the generated vorticity flux is obtained under the no-slip condition for three-dimensional bodies of constant and variable shape.