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https://www.poi.dvo.ru/node/645; http://www.ipmnet.ru/conf/confs.php NEW MATHEMATICAL CLASSIFICATION OF OCEANIC FLOWS COMPONENTS Yuli D. Chashechkin 1, 1 Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Mos-cow, Russia, e-mail: yulidch@gmail.com INTRODUCTION Conventional classifications of fluid flows (potential - vortex by Euler, laminar - turbu-lent by Stokes-Reynolds, Prandtl’s boundary layer), based on an analysis of the solutions of the corresponding systems of equations, play an important role in both theoretical and exper-imental fluid mechanics. It is necessary to note an important feature of known classifications – they are based on either reduced or constitutive systems of equations. The development of mathematical methods of analysis, computers and information technologies allow to operate with basic systems of differential equations and to make a new classification of the compo-nents of oceanic flows. The classification bases on the analysis of complete solutions of the system of fundamental equations. In the developed approach, the liquid medium is considered to be continuous and charac-terized by thermodynamic potentials (in particular, by the free enthalpy or Gibbs potential ) and their derivatives – density , pressure , entropy , temperature , concentration of dissolved substances or suspended particles Dissipative properties of the medium are described by kinetic coefficients (molecular the first and second kinematic viscosity , thermal and substances diffusivities), which can be either constant or functions of coordinates, time, thermodynamic quantities. In general, the parameters of the medium are distributed non-uniformly in space, which has a significant impact on flows, whose dynamics are described by a system of fundamental equa-tions. The basic system includes the equations of state for potentials and thermodynamic quantities and the balance equations for the transport of matter, momentum, total energy and entropy (or temperature and concentration of substances) which are analogues of conserva-tion laws closed systems. The fundamental equations of oceanic flows are well known and are given in most textbooks and monographs. The range of applicability of the concept of "continuous medium" is determined by the ratio of the intrinsic scales of the studied phe-nomena and parameters of the atomic-molecular structure of matter, in practice cm and BASIC SYSTEM OF EQUATION The system of fundamental equations including the state equations and differential equations of continuity and balance equations for momentum, temperature and salinity with physically justified initial and boundary conditions is analyzed taking into account the con-sistency condition which determines the rank of the nonlinear system, the order of its linear-ized version, and the degree of the characteristic (dispersion) relationship [1]. The complete non-linear system for four kinds of liquids that are stratified (strongly and weakly) and ho-mogeneous (potentially and actual is solved by numerical methods [2, 3]. The density pro-file is characterized by the length scale , frequency and period of buoyancy (axis is vertical. The fluid is uniformly rotating with angular velocity In linear approximation, the set of governing equations for perturbations of density , velocity and pressure has the form Here is latitude, , are shear and divergent kinematic viscosities. The set of equations and boundary conditions are characterized by a variety of length scales, which significantly differ in values and are related to certain structural flow elements. Mul-ti-scaling indicates a complexity of internal structure of even extremely slow flows, such as diffusion-induced ones, which arise due to the action of small buoyancy forces and spatial non-uniformity of the stratifying agent distribution. The typical length and time scales of the problem under consideration are the stratification and the body lengths and ( , is velocity) respectively, which, in combination with parameters of the body motion and dissipative coefficients, produce another specific length scales. Large dynamic scales, which are internal wave length , and viscous wave scale , characterize the attached internal wave fields structure. The flow fine structure is characterized by universal microscales , , defined by the dissipative coefficients and buoyancy fre-quency, which are analogues of the Stokes scale on an oscillating with frequency sur-face, . Another couple of parameters, such as Prandtl’s and Peclet’s scales, are determined by the dissipative coefficients and velocity of the body motion, and . Ratios of the basic frequencies and length scales produce dimensionless pa-rameters such as relative frequencies ; Reynolds ; internal Froude ; Péclet ; sharpness factor or fullness of form , where is the cross-sectional area of an obstacle with dimen-sions and ; and, as well, relations specific for stratified media. The additional dimen-sionless parameters includes length scales ratio , which is the analogue of reverse Atwood number, , for a continuously stratified fluid. DISPERSION RELATIONS AND FLOW COMPONENTS For description of periodic flows exponential multiplayers are included in all func-tions , and with fre-quency and wave vector вектором . For a periodic wave positive fre-quency is fixed and dispersion relation following from the consistency condition for the set (1) define connection between components of the wave vector for given , where , , , , , , is sound velocity Taking into account effects viscosity and diffusivity the dispersion equation of the eight degree have two regular solutions describing waves of different types (inertial, internal and surface gravity, acoustical or hybrid) and six for singular perturbed solutions [1]. The dispersion equation goes into a quadratic equation describing the inertial and internal waves in an ideal fluid (and all other types of waves). The spectral components, in which ), the damping factor is proportional to the kinetic coefficients (here ), are regular perturbed components describing the large-scale wave components of periodic flows. The remaining eight roots of dispersion equation, the imaginary part of which is not small ( ), are inversely proportional to the kinetic coefficients. They define ligaments -- singular perturbed solutions, characterizing the fine structure of flows. In the case of an infinite medium, four of them, which do not satisfy the attenuation boundary condition at infinity, can be omitted. All solutions of the basic set, which are regularly and singularly perturbed, form a com-mon family described by functions of one form. All of them are formed, transported and dis-appear simultaneously, despite the differences in characteristic length scales. The dispersion relation that fixes an experimentally verifiable functional relationship between the parame-ters of the instantaneous spatial structure of the medium (length scale or wave numbers) and a local time characteristic (the period of variations) determines the concepts of functional definition for both waves and ligaments - structurally large and fine components of flows. For periodic flows in the classical system of Navier-Stokes equations for homogeneous in-compressible fluid corresponds the dispersion equation is The first multiplier in the equation with the solution in the form represents in a collapsed form all kinds of wave processes caused by the effects of compress-ibility, stratification, rotation, and other physical factors in inhomogeneous liquids in exter-nal force fields. The second multiplier in the dispersion equation defines a pair of identically singular per-turbed solutions for identical ligaments The solution has the character of a degenerate internal periodic boundary layer, in the plane of whose centers , and the values of the velocity components , depend on the local normal coordinate . The multiplicity of the roots is evidence of overdetermination of the system of Navier-Stokes equations, which has the sixth rank and describe only four quan-tities: the three components of velocity and pressure. For a compressible medium, the dispersion equation corresponding to the truncated part of the complete system, which includes only equations (1, 2), has the form where ; , are shear (first) and convergence (second) kinematic viscosity. Here the first multiplier is the classical dispersion relation for a sound wave in a dissipative medium with a frequency and wave vector propagating with a velocity . The second multiplier is a doubly degenerate singularly perturbed solution of Stokes type, describing two merged identical ligaments. Consequently, therefore, the account of compressibility does not remove the degeneracy of the equations set for a homogeneous fluid [4]. Consequently, flows of liquids non-homogeneous density in the infinitesimal limit con-tain two classes of functionally defined components, namely, waves and ligaments character-izing by a functional link between the parameters of spatial structure and temporal variabil-ity. Under real conditions of finite finite-amplitude motions, all the infinitesimal compo-nents interact with each other and generate new sub- and anharmonic waves and vortices that are products of the their overlapping. Accordingly, the universal classification includes three components, namely the waves, the vortices and fine ligaments. VISUALIZATION OF FLOW STRUCTURES The experiments were conducted in a transparent Laboratory Mobile Basin of the Unique Research Facility "Hydrophysical Complex of the IPMech RAS (LMB URF "GPC IPMech RAS") with windows made of optical glass, allowing to use high-resolution schlieren instrument IAB-458 for observations. Test models were mounted on transparent knives to the tow carriage, which moved along the guides installed above the tank. Before starting the ex-periment, the buoyancy frequency distribution was monitored. The experiment was conduct-ed after attenuation of all perturbations registered by contact and optical instruments. a) b) Fig.1 Schlieren images of the stratified flow around tilted plate with internal waves, liga-ments and vortices ( , , ): а, b) – The experimental technique was developed taking into account the scaling analysis of the system. The maximum scales of the phenomena studied are limited by the size of the ob-servation area of the schlieren instrument, which, in these experiments, had the diameter of 23 cm. The spatial resolution is restricted by the optical characteristics of the instrument it-self and the registering equipment, which are continuously improved as the computer tech-nology is developed, and, in these experiments, didn’t exceed 0.05 cm. Typical stratified flow pattern around uniformly moving tilted plane with internal waves, ligaments and vorti-ces in shown in Fig.1. Visualization of calculated density gradient field illustrate existence of all the same compo-nents in the stratified flow around tilted plate. а) b) c) d) Fig. 2 Calculated patterns of horizontal component of density gradient field around the leading and trailing edges of a tilted plate in a stratified fluid for different angles to horizon ( , , , : a, b) – ; c, d) – As the basic equations systems are invariants in all temporal and spatial scales, the basic flows components that are waves, vortices and ligaments exist in the whole range of the fluid flows from microscopic to macroscopic scales and can be visualized in an agreed laboratory wind tunnel experiment as well as in numerical calculation performed with an adequate choice of sensitivity, resolving the space-time ability, the size of the observation area. Com-plete solutions entail calculating all flow parameters, including forces and moments, acting on the obstacle in the flow without involving additional hypotheses and parameters. ACKNOWLEDGMENTS The work was partially supported by the FASO Russia (Project АААА-А17-117021310378-8 "Devel-opment of coordinated analytical-numerical methods for calculating the dynamics and structure of fluid flows and comparison techniques with data of high-resolution experiments at the USU" GFK IPMekh RAS "stands) and FFBR (grant 18-95-00870). References 1. Chashechkin Yu. D. Differential fluid mechanics – harmonization of analytical, numeri-cal and laboratory models of flows // Mathematical Modeling and Optimization of Complex Structure. 2016. V.40. P. 61-91. DOI: 10.1007/978-3-319-23564-6-5 2. Zagumennyi I.V., Chashechkin Yu.D. Unsteady vortex pattern in a flow over a flat plate at zero angle of attack (2D problem) // Fluid Dyn. 2016. V. 51(3). P. 343-359. https://doi.org/10.1134/S0015462816030066 3. Chashechkin Yu. D., Zagumennyi I. V., Dimitrieva N. F. Unsteady Vortex Dynamics Past a Uniformly Moving Tilted Plate // Topical Problems of Fluid Mechanics - 2018, Pra-gue. February 21 – 23, 2018. Proceedings. 2018. P. 47 – 56. DOI: /10.14311/TPFM.2018.007 4. Chashechkin Yu.D. Waves, vortices and ligaments in fluid flows of different scales // Phys. Astron. Int. J. 2018. V. 2(2). P.105-108. DOI: /10.15406/paij.2018.02.0007