Past Research Activities
(Brief Overview)

I. Development of Numerical Techniques and Approaches

Numerical Viscous-Liquid Dynamics

In [J1], [J4], [J5] [J15], [J16], [J17], [P4] the steady and oscillatory flows and heat transfer in toroidal tubes were investigated numerically. The algorithm developed was applied also to the oscillatory flow in spherical annuli [J20], [J27].

Flows with Free Boundaries and Separated Flows

In [J14], [J33], [J44], [P6] we proposed to use scaled coordinates in order to keep the boundaries of the computational domain fixed. Based on this notion, a difference scheme was devised in [J33] for solving the Navier-Stokes equations with a free boundary. As a featuring example the stationary rising of bubbles in ideal and viscous liquid was considered. Results were obtained for wide range of governing parameters: Reynolds and Weber numbers.

The scheme of Helmholtz-Kirchhoff was implemented numerically for ideal flows with separation treating the separation line as an unknown free boundary at which both the kinematic condition and Bernoulli integral hold. Flows around blunt 2D profiles (cylinder, airfoils) were treated by the said technique and in each case two distinct solutions for the shape of free boundary were found: open and cusp-ended zone ([J44], [J52], [P16], [P18], [P25]). A further development of this approach involved quasi-parabolic coordinates [P32].

Applications to Meteorology Problems

In [P45] discriminant and cluster analysis were used to develop a procedure for pattern recognition of macrosynoptic processes. The adaptive grids from [P8], [P9], were applied to the 2D limited-area barotropic models for numerical weather prediction in [P44]. In [P43] we developed an algorithm for approximation of 4D geopotential field with multivariate B-splines in least-square sense.

Numerical Grid Generation

A new coupled system of two strongly nonlinear elliptic equations is derived for 2D orthogonal meshes with a manageable Jacobian [P8], [P9]. It is shown how the boundary conditions are to be posed in order to have coordinate lines that are orthogonal to the domain boundaries. A difference scheme of splitting type is devised in a manner to preserve the coupling of the system at each fractional time-step.

Work on Algorithmization

A block-three-diagonal algorithm of Gaussian elimination was developed in [P5]. For multidiagonal systems, the algorithm for Gaussian elimination with pivoting was implemented in FORTRAN in [P33]. Some specialized minimization procedures are developed in [J8], [P2].

II. Modeling in Mechanics and Physics

Dilational Viscosity of Fluid Interfaces

The application of longitudinal waves for evaluation of surface dilational viscosity was examined critically in [J22] and the limitations due to the unsteady nature of the flow were shown. A new experimental setup is proposed [J23] employing a steady flow and the respective viscometric formula was derived in which the surface elasticity does not take part.

Principle of Minimal Dissipation

In [J6] is discussed the application of the principle of minimal dissipation to turbulent flows with nearly plane-parallel mean velocity profile. The critical revision of the experimental data [J24], showed that it governs the lower Reynolds number of transition to turbulence in Poiseuille flows. It was instrumental in investigating flows with random structure [P11], [P15], [P17], [P22], [B1].

Method of Random Point Approximation

In many different situations the random physical fields can be quantitatively very well approximated with random point functions. The technique of Volterra-Wiener functional series is corroborated in [J21], [J28], [J31], [J32], [P10], [P12], [B1].

The most straightforward application of the random point approximation was in particulate continua with random constitution where the random point nature is obvious [J31], [J32], [J35], [J36], [J59], [P10], [P12]. The famous Maxwell formula for overall conductivity and Einstein formula for the effective viscosity were rigorously derived in the linear order of the random point approximation.

On a heuristic basis, the random point approximation was used also as an alternative approach to the closure problem in turbulence. Application to Burgers turbulence was demonstrated in [J7], [J13], [J21] (see also [J40], [J57], [P24], [B1]). The first kernel of the functional expansion (the "linear" term) represents the shape of localized structures (called by the experimentalists "coherent structures") that are randomly dispersed throughout the space or in time . When the system is near the threshold of the instability, the first term accurately represents the random flow [P38], [B1]. It turned out that a first-order truncation (neglecting the pair-, triple-, etc., interactions among the structures) can be used on a heuristic basis even for systems that are far from the threshold of instability. The quantitative agreement of the predicted statistical characteristics with turbulence measurements was encouraging.

The method was also applied to Poiseuille flows [J26], [J29], [P11], [P15], [P22]; Lorenz attractor [J42], [P23], [P24]; Homsy-Kuramoto-Sivashinsky equation [J47], [J48], [J57] and self-similar random solutions of Navier-Stokes equations [J49], [J54], [P13], [P17], [P21], [P38], e.g., the plane mixing layer (see also the monograph [B1]).

Varia

In [J53] the generalization of the Boussinesq approximation for large-scale atmospheric motions is put on a consistent asymptotic basis. In [J19] an integro-differential equation for the deflection of the cupula in human vestibular apparatus was derived. TE CO2 lasers are modeled in [J12], [J18]. The works [J3], [J9], [J12] were devoted to some problems in plasticity. In [J10], [J11], [J38], [J39], inhomogeneous (e.g., reacting) convective flows were investigated numerically. In [P1], [P3] the stability of elastic bars subjected to non-conservative loading was studied.

In [J37], [J51] a singular asymptotic solution for the high frequency oscillatory flow in eccentric spherical annuli is developed. In [J30] we elaborated a Fourier-series expansion for the gradient linear temperature field around two non-equal and non-overlapping spheres. In the 2D case (gradient flow around two cylinders) the solution is found numerically [P7]. Two-dimensional model of convective cloud was proposed in [J82].