Analysis of Fundamentals of Calculation and Measuring Techniques in Fluid Dynamics

Abstract

The logics of contemporary scientific researches includes the requirement of formulation for distinctive determining the branch of science, the physical medium, and methods for solving problems. Flows are studied in two scientific disciplines, vis., engineering mathematics and technical physics. A flowing medium is characterized by equations of state for the Gibbs potential, density, and velocity of sound, as well as kinetic and other physical coefficients. Fluid flows are treated as transport of momentum, energy, and mass, which causes self-consistent changes in other physical quantities. Flows are described by the unified system of fundamental equations for all media (differential form of conservation laws). Calculations are performed in the algebra of complex numbers, where frequency is assumed to be real-valued, while wavenumber is complex-valued. Natural stratifications of the atmosphere and ocean are characterized by the buoyancy parameters. The classification of flow components is based on complete solutions to linearized and weakly nonlinear forms of fundamental equations, which are obtained by the methods of the singular perturbation theory. Regular components of solutions describe flows and waves on the surface of in the bulk of a fluid, while singular solutions describe the fine structure of the medium. In experiments performed on the stands of the Unique Hydrophysical Complex of the Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, upstream perturbations, internal wave fields, wakes with submerged and suspended vortices structured by interfaces and fibers during the motion of a plate, cylinder, and sphere in stratified and homogeneous fluids are singled out. Numerical calculations of a flow past a plate in a wide range of Reynolds numbers, including creeping flows with Re ~ 1 and nonstationary vortex regimes with Re ~ 100 000, are in qualitative agreement with the observations of schlieren patterns of flows in a stratified aqueous medium in the laboratory basin and in wind tunnels. The conditions for the extension of results to flows in natural conditions and problems in metrology in flight regimes are considered shortly.

INTRODUCTION

During 120 years that have elapsed since the first flight of Orwille Wright, who has covered 36.5 m in 12 s on airplane Flyer-1 on December 17, 1903, aircraft industry and aeronautical research have made considerable advances, which are evident for almost each tellurian. The sizes of aircrafts range from several millimeters to tens of meters (giant aircrafts AN-225 (USSR) has a wing span of 88 m, a length of 84 m, and a height of more than 18 m, Airbus A340-600 or A380 (EU), and C130 (USA) are worth mentioning in this connection). The list of aircrafts being designed is huge and continues its extension. Aircraft industry in Russia is being actively developed. Large civil and military aircrafts as well as unmanned vehicles are being constructed (by the end of 2026, it is planned to produce 18 000 UAVs annually). Research centers, institutes of mathematics and mechanics, specialized institutes, faculties and chairs are actively functioning. Numerous theories that make it possible to calculate aerodynamic properties of aircrafts and their parts to a certain degree of accuracy have been constructed. Several large-scale conferences devoted to aerodynamics and aircraft building are held annually. Original articles, reviews, monographs [1, 2] and textbooks [3, 4] are published. The construction and development of computational aerodynamics that is being developed in traditional aviation research centers and specialized institutes and universities should be mentioned separately.

Does such a nonstop flow of theoretical and experimental investigations leave space for the search for new approaches? Are there constructive trends that have not been fully developed as yet, and is it necessary to find them?

Perhaps, the search for such approaches is necessary even today. The diversity of shapes and structures of aircrafts of close sizes and functions indicates that the theoretical description of the flight is incomplete. Sometimes, catastrophic failures destroying the entire branch occur (the fall of Boeing 737-800 Max 8 aircrafts in Indonesia and Ethiopia). Consumers claim about some functional, service, and economic qualities of some products of aircraft industry. Some airplanes and helicopters exhibit higher quality as compared with other aircrafts. Some of them are manufactured for decades with gradual modification. In recent years, noticeable changes have been made in the organization of scientific researches due to the development of computer technologies and programming methods. In this paper, for determining new trends in the search for improvements of aerodynamics of aircrafts, the fundamentals of the theory and metrology of flight of an autonomous apparatus in the air or water medium are considered with account for the logics of contemporary scientific investigations.

1 LOGICAL FOUNDATIONS OF FLUID DYNAMICS

The admission of necessity of ordering in the variety of fuzzy forms in the description of identical phenomena in different branches of mechanics has necessitated the supplement to the Aristotle logics principles, viz., identity (invariability of the meaning of concepts), noncontradiction, excluded middle (dual rigorousness of the criterion for estimating trueness), with the Leibnitz sufficient substantiation condition, Ockham’s minimal sufficiency condition, and the definiteness requirement (necessity of determining the essence of the object of investigation and the method of analysis) [5]. The fixation of basic properties of the object of investigation improves the clarity and completeness of a description.

Analytical and numerical calculations of flows, which are is conformity with experiment, are performed in the framework of engineering mathematics, viz., the axiomatic doctrine on the principles of choosing the content of symbols, operation rules, and criteria of accuracy control [5]. Since some widely used analytical methods (in particular, perturbation theory) do not envisage the estimation of accuracy in the initial formulation, the redundancy principle including the determination of the sought physical quantity using several independent analytical methods is used in practice.

The basic criterion for choosing physical quantities in contemporary technical physics (empirical-axiomatic doctrine on nature as a whole, the structure and properties of matter, and all forms of its changes) is the observability condition introduced by D.I. Mendeleev (the procedure of measuring the value of the quantity with a guaranteed error estimate).

It should be emphasized that there are substantial differences between the mathematical (axiomatic) and physical descriptions of nature. In the theory, content of concepts (number, vector space, continuous medium, etc.) is defined axiomatically.

In physics, the properties of concepts are determined by their functional relations with other (observable) quantities. Accordingly, the descriptions of the same concepts (the poorer axiomatic and diversified physical concepts) differ significantly.

In this study, the fundamentals of the mathematical description of the free flight of an apparatus in the air or water medium (the motion along a chosen trajectory with a preset velocity without a mechanical coupling with other material objects except for surroundings) is investigated theoretically and experimentally. The interaction of an apparatus with the medium sustains equilibrium and forms the restoration of the momentum of the moving body. Following the recommendations given by Mendeleev [6], a unified approach to the description of the dynamics of a free body in a gaseous or liquid medium is developed.

2 MATHEMATICAL FOUNDATIONS OF THE THEORY OF FLUID FLOWS

The theory and physics of flows is based on the concept of real number, viz., the dimensionless symbol of numeration and amount as the basis for comparison of the properties of physically observable quantities, is introduced a priori. To make the notation more concise, a large group of problems is considered in the algebra of complex numbers. In this case, the space of solutions is extended, and the number of independent methods for constructing solutions increases. The procedure of selecting concrete solutions to the initial problem is based on the refining logical assumptions or concordance with physical facts. The procedure of the selection of admissible methods for obtaining solutions is less formalized.

Axiomatic concepts of classical mechanics (continuous space as the location of bodies and time as a measure of changes) are considered as two independent continuous entities existing along with matter and independently of it. A priori concepts of mutually independent categories (space and time) in classical mechanics presume their homogeneity and independence of the presence of bodies and the type of occurring material processes.

The introduction of a coordinate system at each point of space is put in correspondence with a set of real numbers, the minimal number of elements in which, the same for all objects, determines the dimension of space. The unit value of the space invariant (distance between points in the adopted system of quantities) is introduced simultaneously.

One of general foundations of physical theories is the postulate of three-dimensionality of the metric (Euclidean) space ${{\mathbb{R}}_{3}}$. Invariants of space, viz., length (distance between points ${{R}_{{1,2}}} = \sqrt {\Sigma {{{(x_{2}^{i} - x_{1}^{i})}}^{2}}}$) and time interval $\Delta t = {{t}_{2}} - {{t}_{1}}$) are dimensional quantities that form the basis of metric systems and standards.

In natural sciences, use is made of the concept of vector space, the axiomatics of which contains the operations of summation, multiplication, internal composition (of vectors), associativity of the product of multipliers, and commutativity with the fulfillment of the rules of algebraic composition of vectors, associativity of the product of multipliers, multiplication by unity, distributivity, and external composition (conservation of the result of multiplication of a scalar by a vector). The latter condition substantiates the combination of the Newtonian mechanics of solids and the abstract and applied mathematics into a single complex. In the mechanics of a homogeneous fluid, the consequence of the composition principle is the identification of two concepts of different origins, viz., the analytically determined measure of motion (velocity v of the fluid) and the physical parameter of the flow (specific momentum ${\mathbf{p}} = \rho {\mathbf{v}}$).

Possible mathematical operations include various types of discrete and continuous transformations of the metric space (similarity, projective, and affine transformations), which also include motion (continuous orthogonal transformation of space into itself with parameter t (time) that preserves the distances between points and the relative arrangement of objects). The determinant composed of coefficients a_ik of the transformation matrix is $\left\| {{{a}_{{ik}}}} \right\| = + 1$. The orthogonal transformation that does not preserve the orientation of the orthogonal transformation with determinant $\left\| {{{a}_{{ik}}}} \right\|$ = –1 defines reflection relative to a certain axis [7].

Motion in the Euclidean space corresponds to the group of orthogonal transformations of space into itself, which includes independent subgroups of displacements and rotations. The transformations determined by the group of motion are studied in elementary geometry.

The description of “motion” is based on the definition of mathematical concepts of a number, a “point,” its “position” and “trajectory of motion,” “velocity,” and “acceleration.” A consequence of the external composition rule is the equivalence of vector spaces of momentum (physical invariant of motion) and velocity (mathematical quantity), which contain pointlike bodies of constant mass. As a result, the description of motion (displacement of a body relative to the system of bodies forming a reference system) correlates with the operation of transformation of space into itself with conserved distances. Two (dynamic and geometrical) definitions of motion of a body are equivalent. The identity of the displacement of a body and its motion (the geometry of vector spaces being transformed) ensures the equivalence of mathematics and Newtonian mechanics of a material point with a mass possessing momentum vector ${\mathbf{p}} = m{\mathbf{v}}$ and energy $E = \frac{{m{{{\mathbf{v}}}^{2}}}}{2} = \frac{{{{{\mathbf{p}}}^{2}}}}{{2m}}$. In fluid mechanics, the concept analogous to a “point,” viz., a “fluid particle” that has no physically determinate boundaries, has been introduced.

In fluid mechanics, a continuous medium characterized by a continuously distributed density ρ, which is a measure of inertial and gravitational properties, and by the “equation of state,” which functionally connects density with other physical parameters of the medium, is submerged into the space.

Finally, the connections between the physical characteristics of motion (momentum, energy, and angular momentum) and the fundamental properties of space were established by Noether’s theorems, formulated for conservative systems. The conserved functional of the action, the Lagrange function, expresses the invariance of the action with respect to some continuous group of transformations. According to Noether’s theorem, each infinitesimal symmetry of a system corresponds to a conservation law. In particular, in classical mechanics, the homogeneity of time and space corresponds to the laws of conservation of energy and momentum. The isotropy of space corresponds to the law of conservation of angular momentum [8].

The theorems established for conservative systems were subsequently extended to systems of bodies and fields evolving with time and acquired the meaning of balance equations. It is precisely such equations that are employed as the logical foundation for the description of the dynamics of complex dissipative systems. The advantage of such an approach is the logical substantiation and constructiveness, which makes it possible to single out from the entire set of symbols “observable physical quantities” permitting an a priori estimation of the quantity itself and the error in its value.

3 EQUATIONS OF STATE FOR FLUIDS

Numerous experiments have revealed that fluids, like other media, consist of atoms and molecules, the stable properties of which have been studied quite well. However, in fluid mechanics, use is made of the concept of “continuous medium” submerged into a continuous space. The consistency of the two approaches ensures the use of both conserved and observable quantities—mass (density), momentum, and energy—in analysis of phenomena on scales larger than the atomic–molecular scale.

The main parameter characterizing physical properties of media is mass $M = \rho V$, which is a scalar measure of inertial and gravitational properties. Apart from mass, density ρ (mass of unit volume V) is used in analysis. Quantities M and ρ are assumed to be described by continuous functions.

The most general physical quantity characterizing a medium is the internal energy [9] determined by the chemical composition, the atomic–molecular structure of the substance, and other properties. Its components are bulk internal energy $E{{n}_{i}}$ and surface energy $E{{n}_{\sigma }}$, which is associated with a manifestation of anisotropy of the atomic–molecular interactions near the free surface of the fluid. The energy components are also described by continuous functions. The total energy $En = E{{n}_{k}} + E{{n}_{p}} + E{{n}_{i}} + E{{n}_{\sigma }}$ of the medium includes kinetic energy $E{{n}_{k}} = {{{{{\mathbf{p}}}^{2}}} \mathord{\left/ {\vphantom {{{{{\mathbf{p}}}^{2}}} {2\rho }}} \right. \kern-0em} {2\rho }}$, potential energy $E{{n}_{p}}$, and internal energy $E{{n}_{j}} + E{{n}_{\sigma }} + E{{n}_{e}} + E{{n}_{\mu }}$, which combines the physical (electrical) $E{{n}_{e}}$ and chemical $E{{n}_{\mu }}$ energy components [10, 11].

Different forms of internal energy representation (Helmholtz potential $H$, Gibbs potential $G$, enthalpy W, which is a thermal function J, and internal energy $U$ proper) are equivalent and are connected by the Maxwell formulas [10]. The derivatives of the potentials determine the equilibrium values of thermodynamic quantities (density $\rho$, temperature $T$, pressure $P$, and concentrations ${{S}_{i}}$ of dissolved substance and suspended particles) [11]. The inclusion of potentials into the description of flows allow to take into account the action of all mechanisms of energy transfer, viz., thermal radiation mechanism (which will henceforth be disregarded); macroscopic mechanisms with a flow with velocity $U$, with various types of waves with group velocity ${{c}_{g}}$; small-scale mechanisms including dissipation–diffusion processes, and the mechanism of internal energy conversion into other forms, which generate ligaments (thin energy-saturated fibers and interfaces) [5].

In accordance with the Recommendation of the International Association for the Properties of Water and Steam (IAPWS), Gibbs potential G (free enthalpy) is chosen as the basic thermodynamic parameter with differential $dG = - SdT + VdP + {{\mu }_{i}}d{{S}_{i}}$ for estimating the potentiality of a medium to do external work [11, 12]. The derivatives of potential G with respect to pressure, temperature, and concentration ${{S}_{i}}$ determine specific volume $V$ (density $\rho = 1{\text{/}}V$), entropy S, and chemical potential ${{\mu }_{i}}$. The potential of a medium with a complex composition is defined as the sum of the potentials of individual phase components of the medium (basic, dissolved, and suspended).

The contribution of the free surface is characterized by extra term $d{{G}_{\sigma }} = {{S}_{\sigma }}d\sigma$ in Gibbs potential differential $dG$, which is determined by surface area ${{S}_{\sigma }}$ and surface tension coefficient $\sigma$ [9].

In aerodynamics, free energy H (Helmholtz potential) is used along with the Gibbs potential [13, 14]. The energy balance of the atmosphere was considered in [15].

Functional relations between internal energy $E{{n}_{i}} = E{{n}_{i}}\left( {\rho ,P,T,{{S}_{i}}} \right)$ and measurable physical quantities of the thermodynamic origin (temperature T, pressure $P$, concentration ${{S}_{i}}$ of dissolved substances and suspended particles), or the functions connecting basic thermodynamic quantities $\rho = \rho \left( {P,T,{{S}_{i}}} \right)$, ρ = $\rho (P,c_{s}^{a},{{S}_{i}})$, $\rho = \rho (c_{s}^{a},T,{{S}_{i}})$, or $\rho = \rho \left( {P,T,{{n}_{\lambda }}} \right)$, form fundamental equations of state (here, $c_{s}^{a}$ is the adiabatic velocity of sound and ${{n}_{\lambda }}$ is the optical refractive index for light of wavelength $\lambda$). Equations of state in analytical form are given in articles [16] and special reference books for the sea medium [17] and the standard atmosphere [18, 19].

The choice of the equation of state and basic variables is made with account for convenience of the mathematical description and experimental determination of a chosen physical characteristic of the phenomenon in question. In some problems, use is made of the dependence of the adiabatic sound velocity on other physical parameters of the medium (basic thermodynamic quantities such as pressure, temperature, concentration $c_{s}^{a} = c_{s}^{a}\left( {P,T,{{S}_{i}}} \right)$ or density, temperature, and concentration $c_{s}^{a} = c_{s}^{a}\left( {\rho ,T,{{S}_{i}}} \right)$). In experimental investigations, the dependence of the density on the optical refractive index, $\rho = \rho \left( {T,P,{{n}_{l}}} \right)$, and, conversely, of the refractive index on density, n_l = ${{n}_{l}}(T,P,\rho )$ is used [16]. The dependence of the density of aqueous solutions on the electrical conductivity and of the density of air on relative humidity $\varphi$ and concentration of water drops ${{S}_{w}}$, $\rho = \rho \left( {T,P,\varphi ,{{S}_{w}}} \right)$, and other dependences are also used.

The application of the generalized equations of state including the dependences of density on thermodynamic and some other physical parameters of form $\rho = \rho \left( {T,P,{{n}_{l}},{{c}_{s}}} \right)$ or $\rho = \rho \left( {T,P,{{S}_{i}},{{n}_{l}},{{c}_{s}}} \right)$, as well as the extended set of sensors for determining the corresponding physical quantities, allows to implement the redundancy principle, to execute the actual control over the precision of measurements of the main parameter, and to identify a defective sensor in the set of instruments directly in the course of measurements [20, 21].

The equations of state of many media (in particular, atmosphere and hydrosphere) can be successfully approximated by the Taylor series. The required degree of accuracy is ensured by inclusion of only the first terms of the expansion:

$$\begin{gathered} \rho = {{\rho }_{0}}(1 + {{\alpha }_{P}}\left( {P - {{P}_{0}}} \right) - {{\alpha }_{T}}\left( {T - {{T}_{0}}} \right) \\ \, + {{\alpha }_{{{{S}_{i}}}}}\left( {{{S}_{i}} - {{S}_{{s,0}}}} \right) + ...), \\ \end{gathered} $$

(1)

where ${{\alpha }_{P}} = \frac{1}{\rho }\frac{{\partial \rho }}{{\partial P}}$, ${{\alpha }_{T}} = - \frac{1}{\rho }\frac{{\partial \rho }}{{\partial T}}$, and ${{\alpha }_{S}} = \frac{1}{\rho }\frac{{\partial \rho }}{{\partial S}}$ are the compressibility factor, the thermal expansion coefficient, and the salt contraction coefficient, respectively. The linearized form of the equation of state is required when full linearization of the system of equations used is performed.

The appropriate choice of variables facilitates the consistent calculations and analysis of experimental results. In particular, in analysis of stratified flows of salty fluid, approximate linear dependences of the density and the refractive index on the salinity (concentration ${{S}_{i}}$ of dissolved salt) are used. Because of the temporal variability and spatial inhomogeneity of acting physical factors, the density distributions are always nonuniform, and the media are heterogeneous. In external fields of different origins (gravitational, electromagnetic, or inertial), heterogeneous media are naturally stratified: the density of an unperturbed medium decreases with the altitude in the atmosphere [18, 19, 22] and increase with the depth in the hydrosphere [23]. The high compressibility of gases ensures the stability of the density distribution in the atmosphere even for its observed decrease in temperature upon an increase in altitude in the lower atmosphere.

4 STRATIFICATION PARAMETERS OF UNPERTURBED ATMOSPHERE AND HYDROSPHERE

The degree of hydrostatic stability of a medium is determined by distributions of density ${{\rho }_{0}}(z)$ and its gradient $d{{\rho }_{0}}{\text{/}}dz$. As the measure of stratification, scale Λ = |dlnρ(z)/dz|^–1, buoyancy frequency N, and buoyancy period T_b = 2π/N are used. For weakly compressible liquids, ${{N}^{2}} = \frac{g}{\rho }\frac{{d\rho }}{{dz}}$, while for compressible gases, ${{N}^{2}} = \frac{g}{\rho }\frac{{d\rho }}{{dz}} + \frac{{{{g}^{2}}}}{{c_{s}^{2}}}\left( {\,\frac{{{{c}_{P}}}}{{{{c}_{\operatorname{v} }}}} - 1} \right){{c}_{s}}$, where g is the acceleration due to gravity, c_s is the velocity of sound, the z axis is chosen in the direction of action of mass forces, and ${{c}_{P}},{{c}_{{v}}}$ are the isobaric and isochoric specific heats.

Actual media (atmosphere [18, 19, 22] and hydrosphere [23]) are generally stratified stably. Typical values of the buoyancy period for the lower atmospheric layer of height 12 km (or the troposphere boundary) lie in the range $3.0~\,\,{\text{min}} < T_{b}^{a} < 3.5~$ min, $0.030{{{\text{ s}}}^{{ - 1}}} < N$ < 0.035 s^–1. In the hydrosphere, seasonal thermal pycnocline with buoyancy period $T_{b}^{o}\sim 3$ min and the main (annual) pycnocline associated with cooling and a decrease in the water compressibility with increasing depth with buoyancy period $T_{b}^{o}\sim 20$ min are usually distinguished.

Detailed data given in Standard Atmosphere Tables [19] and other tables [18, 20], which allow one calculating the values and to plot the curves describing the dependences of density and other parameters on the altitude and other physical quantities, are given in Fig. 1.

Fig. 1.

Profiles of physical quantities in the atmosphere: (a–d) density, buoyancy period and frequency, and velocity of sound.

The dependence of density (in kg/m³) on altitude z, m, which is successfully approximated by quadratic function ρ(z) = $3.21866 \times {{10}^{{ - 9}}}{{z}^{2}} - 0.000112836z$ + 1.22066 (Fig. 1а), ensures a linear decrease in buoyancy period T_b with altitude (${{T}_{b}} = - 0.00241163z$ + 205.096) and a hyperbolic increase in buoyancy frequency $N\left( z \right) = \sqrt {\left| {\frac{g}{\rho }\frac{{d\rho }}{{dz}}} \right|}$, s^–1, with increasing distance from the Earth surface.

Air is an easily compressible medium; the standard estimate of the isothermal homogeneous atmosphere altitude is ${{H}_{h}} = {{c_{s}^{2}} \mathord{\left/ {\vphantom {{c_{s}^{2}} g}} \right. \kern-0em} g}\sim$ 8.5 km. A rapid linear decrease in velocity of sound c_s with increasing altitude can be explained by the absence of acoustic noise from high-flying aircrafts since beams in a stratified medium turn towards the minimal velocity of sound and leave for the stratosphere.

The profiles of distributions of physical quantities over the altitude in the atmosphere [19, 22] and in the ocean [23] are characterized by a fine spatial structure including more homogeneous thick layers and thin high-gradient interfaces. The interfaces` existing in a dissipative medium with a clearly manifested diffusion smoothing exhibit a nonequilibrium spatial distribution of thermodynamic potentials. Their existence is ensured by permanently occurring physicochemical processes inducing local and global rearrangement of vertical profiles of various quantities.

In flow media, combinations of atoms and molecules of different physicochemical origins are naturally formed (including clathrates, voids, plates, walls, and other structures with a characteristic size on the order of 10^–6 cm [24, 25]), which possess proper accessible potential surface energy. During the structure formation, some combinations are permanently destroyed, and new combinations are formed.

The potential energy is rapidly released upon the elimination of the free surface and is transformed into perturbations of pressure and temperature and spent on the formation of fine flows. The energy of the flow is transformed into the potential internal energy during the formation of new structure components [5] and, conversely, is directly transformed into fine and speedy elements of the flow in the form of high-gradient interfaces or clearly manifested fibers (ligaments, ligands, jets, and trickles; the list of synonyms is incomplete). The fine components have been calculated for periodic flows in the range of internal waves in a continuously stratified fluid in the linear [26, 27] and weakly nonlinear [28] approximations.

5 DETERMINATION OF FLUID FLOWS

A medium possessing the property of fluidity (i.e., the ability to change its position in space under the action of infinitely small perturbations), which is characterized thermodynamic potentials and their derivatives (such as thermodynamic quantities), kinetic, and other physical coefficients (in particular, those determining the propagation or acoustic waves), is referred to as a weakly compressible liquid if it occupies a finite volume or a gas/plasma if it fills the entire available space. The medium is characterized by equations of state that determine the unperturbed spatial distributions of internal energy, density, velocity of sound, and other quantities of the thermodynamic origin.

A fluid flow is defined as follows: the transfer of momentum, energy, and mass, which is accompanied by self-consistent changes in physical quantities characterizing the state of the continuous medium.

The redistribution of mass or energy without a macroscopic transfer of the momentum is called a process (e.g., diffusion mass transfer).

Flows are characterized by the dynamics (change in the values of physical quantities and the intensity of action of a fluid on solids) and the structure (geometrical parameters of the spatial distribution of physical quantities).

Measurable quantities describing the fluid flows are density $\rho$, scalar total energy ${{E}_{t}} = {{E}_{M}} + {{E}_{p}} + {{E}_{i}}$ (including specific kinetic energy ${{E}_{M}} = \frac{{\rho {{{v}}^{2}}}}{2}$, potential energy ${{E}_{p}}$, and internal energy ${{E}_{i}}$, which is described by Gibbs potential G), as well as momentum vector ${\mathbf{p}} = \left( {{{p}_{x}},\;{{p}_{{y,}}}\;{{p}_{z}}} \right) = \rho {\mathbf{v}}$ and other observable quantities characterizing the medium and flows. The velocity of a fluid is defined as the instantaneous ratio of two invariant quantities, viz., momentum and density (${\mathbf{v}} = {\mathbf{p}}{\text{/}}\rho$). Henceforth, we will assume that the velocity of a fluid, which in an unobservable quantity in view of the absence of the definition of “fluid particle” concept and impossibility of its identification [5], is identical to the rate of transformation of the Euclidean space into itself (motion of the medium submerged into the space).

Taking into account various mechanisms of energy transfer with proper spatiotemporal parameters, the Gibbs potential and its derivatives defining the density, pressure, temperature, entropy and other parameters are chosen as the basic quantities characterizing the equilibrium state of the medium [11, 12]. The set of equations of state includes

$$\begin{gathered} G\left( {x,y,z,t} \right) \\ = \,G(\rho (x,y,z,t),P(x,y,z,t),T(x,y,z,t),{{S}_{i}})(x,y,z,t), \\ \rho (x,y,z,t) = \rho (P((x,y,z,t),T(x,y,z,t),{{S}_{i}}(x,y,z,t)). \\ \end{gathered} $$

(2)

To improve the accuracy in determining the values of physical quantities, additional functional relations between physical quantities are also used in practice as equations of state. The dependences of the velocity of sound, the electrical conductivity, and the optical refractive index on the pressure, temperature, and salinity can be used for this purpose.

The axiomatically introduced system of equations in fluid mechanics with account for the general principles for selecting physical quantities and fundamental conservation laws [1, 5, 29–31] has form

$$\left\{ \begin{gathered} \rho = \rho \left( {P,T,{{S}_{n}}} \right),\quad G = G\left( {x,y,z} \right), \hfill \\ \frac{{\partial \rho }}{{\partial t}} + \nabla \cdot \left( {\mathbf{p}} \right) = {{Q}_{m}}, \hfill \\ \frac{{\partial {{S}_{i}}}}{{\partial t}} + \nabla \cdot \left( {{{S}_{i}}{\mathbf{v}} + {{{\mathbf{I}}}_{i}}} \right) = Q\left( {{{S}_{i}}} \right), \hfill \\ \frac{{\partial ({{p}^{i}})}}{{\partial t}} + {{\nabla }_{j}}{{\Pi }^{{ij}}} = \rho {{g}^{i}} + 2\rho {{\varepsilon }^{{ijk}}}{{{v}}_{j}}{{\Omega }_{k}} + {{Q}^{i}}(f), \hfill \\ \frac{{\partial E}}{{\partial t}}\, + \,{{\nabla }_{i}}(E{v}{{u}^{i}})\, + \,{{\nabla }_{i}}\left( {{{q}^{i}}\, + \,P{{{v}}^{i}} - {{\sigma }^{{ij}}}{{{v}}_{j}} + \frac{{\partial w}}{{\partial {{S}_{n}}}}I_{n}^{i}} \right)\, = \,Q(e), \hfill \\ \end{gathered} \right.$$

(3)

where ρ is the density; ratio ${\mathbf{v}} = {\mathbf{p}}{\text{/}}\rho$ of two invariant quantities is the fluid velocity; ${{S}_{i}}$ and ${{{\mathbf{I}}}_{n}}$ are the concentration and diffusion flux density of the ith impurity; ${{\Pi }^{{ij}}} = \rho \,{{u}^{i}}{{u}^{j}} + P{{\delta }^{{ij}}} - {{\sigma }^{{ij}}}$ is the momentum flux density tensor; ${{\Pi }^{{ij}}} = \rho \,{{u}^{i}}{{u}^{j}} + P{{\delta }^{{ij}}} - {{\sigma }^{{ij}}}$ is the symmetric tensor of viscous stresses; ${{\delta }^{{ij}}}$ is the fundamental metric tensor; $\mu$ and $\zeta$ are the first and second dynamic viscosities; $\Omega$ is the angular velocity of global rotation; $E = \rho \left( {\frac{{{{{v}}^{2}}}}{2} + \varepsilon + U} \right)$ is the total energy; $\varepsilon$ is the specific internal energy; $w = \varepsilon + \frac{P}{\rho }$ is the specific enthalpy; U is the gravity potential; ${\mathbf{g}} = - \nabla U$ is the acceleration due to gravity; and q is the heat flux density vector.

External sources of the mass of the ith impurity, momentum, and energy Q_m, $Q({{S}_{n}}),{{Q}^{i}}(f),Q(e)$, respectively, describe the action of nonhydrodynamic processes occurring in fluid flows and can be equal to zero. The introduction of sources reflects a possible effect of external factors of a nonhydrodynamic origin on the structure and dynamics of flows and indeterminacy of phenomena on small scales on the order of the sizes of atoms and clusters. To author’s knowledge, system of equations (2), (3) has not been analyzed with account of the compatibility condition required for constructing the complete set of solutions. The inclusion of sources into basic system (1) makes it possible to analyze the transformation of latent potential energy into the active form and to estimate its influence on the flow dynamics and structure.

Under the assumption that potential gradients, physical quantities, and the intensities of external sources are small, system (3) can be transformed into the system of equations for describing the transfer of momentum and mass, the concentration of individual components, temperature, and other quantities, which is widely used in fluid dynamics as the foundation for describing flows [1–5]:

$$\left\{ \begin{gathered} G = G\left( {P,S,T} \right) = G\left( {{\mathbf{x}},t} \right),\quad \rho = \rho (P,S,T) = \rho ({\mathbf{x}},t), \hfill \\ \frac{{\partial \rho }}{{dt}} + {{\nabla }_{j}}({{p}^{j}}) = {{Q}_{\rho }}, \hfill \\ \frac{{\partial ({{p}^{i}})}}{{\partial t}} + \left( {{{\nabla }_{j}}\frac{{{{p}^{j}}}}{\rho }} \right){{p}^{i}} \hfill \\ = - {{\nabla }^{i}}P + \rho {{g}^{i}} + \nu \Delta ({{p}^{i}}) + 2{{\varepsilon }^{{ijk}}}{{p}_{j}}{{\Omega }_{k}} + {{Q}^{i}}, \hfill \\ \frac{{\partial \rho T}}{{\partial t}} + {{\nabla }_{j}} \cdot ({{p}^{j}}T) = \Delta \left( {{{\kappa }_{T}}\rho T} \right) + {{Q}_{T}}, \hfill \\ \frac{{\partial \rho {{S}_{i}}}}{{\partial t}} + {{\nabla }_{j}} \cdot ({{p}^{j}}{{S}_{i}}) = \Delta \left( {{{\kappa }_{S}}\rho {{S}_{i}}} \right) + {{Q}_{S}}_{i}. \hfill \\ \end{gathered} \right.$$

(4)

This system, which includes quantities of mechanical, thermodynamic, and kinematic origin, takes into account the dissipation of momentum, as well as the effect of internal and external sources of mass and temperature of the structure and dynamics of the flow.

The equations are supplemented with initial and boundary conditions, including the no-flow condition for the density and components of the substance (impurity concentrations), the value of temperature or its flux, no-slip for the momentum and velocity at solid boundaries, the equality of the momentum fluxes at contact surfaces between two fluids, and damping of all perturbation at infinity.

Equations (4) form a system of coupled algebraic–differential equations, the solution to which is constructed with account for the compatibility condition. The rank of the system (order of the senior derivative when the system can be reduced to a single equation), as well as the order of its linearized version and the degree of the characteristic (dispersion) equation determine the minimal number of eigenfunctions constituting the complete solution. The complete system of diffusion equation for a single impurity has rank ten [31]. Accordingly, the flow pattern of this set is formed by determining ten functions with their own spatiotemporal scales. The abundance of flow components and the direct nonlinear interaction of components differing in scales and types of structures [29] are manifested in the continuous variation of the observed flow pattern. The number of structure components can increase due to nonlinear interactions of the flow components and a stronger manifestation of fine-structure components, which take form of shock waves [31].

In view of the independence of physical quantities (individuality of the spatiotemporal variability of field patterns), it is necessary to measure simultaneously the spatial structure of the distribution of each variable, which is characterized by its own geometry and spatiotemporal scales. The accuracy of equations of state, which is the difference in the values of density, velocity of sound, refractive index, and other reliably determined quantities calculated by solving system (4) and those obtained experimentally by measuring the values of temperature, pressure, the concentration of water vapor and liquid water in the atmosphere, and other quantities, determines the error in analytical and numerical calculations of other flow parameters (in particular, forces and moments acting on a moving body).

The correspondence of infinitesimal symmetries of system (4) to main physical principles [30] indicates that it’s choice as the theoretical foundation of analysis of fluid flows is substantiated. Practical recommendations concerning the selection of parameters for numerical simulation and the experimental technique are based on analysis of proper spatiotemporal scales of system (4) with initial and boundary conditions.

The combination of the content of physical parameters and mathematical quantities in the theory, numerical simulation, and experiments based on system (4) permits estimating the accuracy of solutions proceeding from the physical properties of fluids and experimental conditions. System (4) contains no additional parameters and does not require their introduction in the development of numerical codes.

System (4) is parametrically and scale-invariant [5, 30]. The emergence of “new flow regimes” is explained by intrinsic properties of solutions to system (4), which are associated with the influence of processes of energy transfer from the potential form to active form and back as well as by a change in the form of the equation of state, which are often ignored. It is necessary to take into account methodical and instrumental limitations, in particular, insufficient or excessive sensitivity of sensors, inconsistency between the temporal or spatial resolution and the nature of phenomena under investigation, limitations imposed on the dynamic range of an instrument or the computational method, which are required for identifying all structure components of the flow in question in the entire range of parameters, including waves, vortices, fine fibers and interfaces (ligaments), and the components of complete solutions to the system of fundamental equations both in the linear and weakly nonlinear description [5, 28, 32].

Since the symmetries of the fundamental system and of equations in other models of the flow (numerous versions of the turbulence theories, theories of waves, vortices, jets, wakes, etc.) differ significantly [30, 31], the same symbols in constitutive models have different physical meanings. To verify the consistency of results, it is necessary to calculate similarly defined fields of parameters or to find, for example, forces and moments acting on the chosen obstacle in its own coordinate system, or the flow rate in a chosen cross section.

The general properties of system (4) and even of its linearized version have not been investigates completely as yet. The dispersion relations of periodic flows of a linearized system, which have been constructed by the methods of the singular perturbation theory in the weak dissipation approximation, make it possible to analyze both waves and ligaments [27, 32]. Numerical solutions to the reduced system of fundamental equations, which permit analyzing the effect of large-scale and fine components of a flow on the dynamics of interaction between a body and the flow in a wide range of parameters of the problem, have been constructed for the first time in [33]. The stratified flow past a plate arranged at an arbitrary angle to the horizontal has been considered in a wide range of flow parameters.

6 CALCULATION AND LABORATORY MODELING OF THE FLOW OF AN INCOMPRESSIBLE FLUID PAST A PLATE

The development of industrial technologies and mathematics in the 19th century has singled out the analysis of the flow past a plate (extended flat surface of a finite length), which is the main element of the lifting wing or the hull, into a separate class of problems. Initially, the Euler and Navier–Stokes equations for a constant-density fluid were considered as the theoretical foundation of investigations in this field. They were supplemented at the beginning of the 20th century with the boundary layer theories (the main principles of which were postulated by L. Prandtl), the theories of vortices (in the development of which Zhucowski’s work played an important role), and the turbulence theories in the approximation of a constant density of the medium.

The irreducibility of constitutive models [1–4], which is a consequence of the qualitative difference in the infinitesimal symmetries of the available systems of flow equation [31], hampers direct comparison of the results of calculations with one another and with experimental data. A more detailed analysis of the properties of solutions to the reduced system of fundamental equations, which form the unified basis of a consistent theoretical, numerical, and laboratory modeling of a fluid flow past bodies, permits the comparison of independent estimates of errors in calculations and experimental errors, is of considerable interest.

The given description is based on the reduced system of fundamental equations in the mechanics of heterogeneous incompressible multicomponent fluids, including the continuity equations, the Navier–Stokes equations in the Boussinesq approximation, and the diffusion equation for the stratifying component [1, 5]:

$$\begin{gathered} \rho = {{\rho }_{{00}}}\left( {\exp \left( { - {z \mathord{\left/ {\vphantom {z \Lambda }} \right. \kern-0em} \Lambda }} \right) + s} \right),\quad \operatorname{div} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathbf{v}} = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} \\ \frac{{\partial {\mathbf{v}}}}{{\partial t}} + \left( {{\mathbf{v}}\nabla } \right){\mathbf{v}} = - \frac{1}{{{{\rho }_{{00}}}}}\nabla P + \nu \Delta {\mathbf{v}} - s \cdot {\mathbf{g}}{\kern 1pt} {\kern 1pt} , \\ {\kern 1pt} \frac{{\partial s}}{{\partial t}} + {\mathbf{v}} \cdot \nabla s = {{\kappa }_{S}}\Delta s + \frac{{{{{v}}_{z}}}}{\Lambda }. \\ \end{gathered} $$

(5)

Here, $S\left( {{\mathbf{x}},\;t} \right) = {{S}_{0}}\left( z \right) + s\left( {{\mathbf{x}},\;t} \right),\;$is the total salinity (stratifying component); $s\left( {{\mathbf{x}},\;t} \right)$ is its perturbation including the salt compression factor; ${\mathbf{v}} = \left( {{{{v}}_{x}},0,{{{v}}_{z}}} \right)$ is the induced velocity; P is the pressure minus the hydrostatic pressure; ${{\kappa }_{S}} = 1.41\; \times {{10}^{{ - 5}}}$ cm²/s and $\nu$ = 10^–2 cm²/s are the diffusion coefficient for salt and the kinematic viscosity of the solution, respectively; t is the time, and $\nabla$ and $\Delta$ are the Hamilton and Laplace operators, respectively. In the equation of state of the medium, only the dependence of density on salinity (steady-state unperturbed distribution ${{\rho }_{0}}\left( {{{S}_{0}}\left( z \right)} \right)$ is taken into account, and ${{\rho }_{{00}}}$ is the density on the horizontal corresponding to the center of the plate).

The initial medium is nonequilibrium. In a medium with a salinity gradient, a diffusive flow of salt stabilizes; an interruption of this flow leads to the formation of a diffusion-induced stratified flow on inclined solid surfaces [34]. Here and below, it is assumed that all boundaries of the medium are vertical or horizontal, and the unperturbed medium is at rest.

The calculations and experiments are performed in two stages. First, a plate on the surface of which substantiated initial (absence of a flow) and boundary (no-slip and no-flux) condition at solid walls and conditions of attenuation with increasing distance from the body in the adjoint coordinate system are imposed is introduced into a stratified medium at rest (without perturbations in the theory and with minimal perturbation in practice) [34].

On the surface of the plate, that interrupts the diffusion flow of the stratifying component, a diffusion-induced compensatory flow is formed. Its parameters are described by the solution of system (5) with boundary conditions (6) where U = 0. The constructed solution is subsequently used as the initial condition when calculating the flow pattern around the body,

$$\begin{gathered} {{\left. {\mathbf{v}} \right|}_{{t\, \leqslant \;0}}} = {{{\mathbf{v}}}_{1}}\left( {x,z} \right),\quad {{\left. s \right|}_{{t\, \leqslant \;0}}} = {{s}_{1}}\left( {x,z} \right), \\ {{\left. P \right|}_{{t\, \leqslant \;0}}} = {{P}_{1}}\left( {x,z} \right),\quad {{\left. {{{{v}}_{x}}} \right|}_{\Sigma }} = {{\left. {{{{v}}_{z}}} \right|}_{\Sigma }} = 0, \\ {\kern 1pt} {{\left. {\left[ {\frac{{\partial s}}{{\partial {\mathbf{n}}}}} \right]} \right|}_{\Sigma }} = \frac{1}{\Lambda }\frac{{\partial z}}{{\partial {\mathbf{n}}}},\quad {{\left. {\mathbf{v}} \right|}_{{x,{\kern 1pt} {\kern 1pt} z \to \;\infty }}} = \left( {U,0,0} \right){\kern 1pt} {\kern 1pt} , \\ \end{gathered} $$

(6)

where U is the velocity of a uniform incoming stratified flow at infinity and n is the outward normal to surface $\Sigma$ of the obstacle.

Subscript “1” in expressions (2) corresponds to the diffusion-induced flow that is described by system (1) with boundary conditions (2) over long time intervals, on which the flow formation effects practically vanish [34]. The system of equations (5) with boundary conditions (6) is parametrically and scale invariant, which was noted by J.G. Stokes in his review of the first article by O. Reynolds on observations of the shape of a paint stream in a transparent tube.

System of equations (5) and boundary conditions (6) include the set of parameters with the dimension of length (buoyancy scale $\Lambda$, length $L$, and height h of the body) and time (buoyancy period ${{T}_{b}}$ and intrinsic time $T_{U}^{L} = L{\text{/}}U$ of passage over the length of the body), i.e., periodicity of sign reversal in the pressure gradient at a point in the laboratory coordinate system.

The set also contains proper microscales $\delta _{N}^{\nu }\, = \,\sqrt {{\nu \mathord{\left/ {\vphantom {\nu N}} \right. \kern-0em} N}}$, and $\delta _{N}^{{{{\kappa }_{S}}}} = \sqrt {{{{{\kappa }_{S}}} \mathord{\left/ {\vphantom {{{{\kappa }_{S}}} N}} \right. \kern-0em} N}}$. determined by properties of the medium, i.e., dissipative coefficients (kinematic viscosity $\nu$ or diffusion coefficient ${{\kappa }_{S}}$), buoyancy frequency N. The structure of the expression for $\delta _{N}^{\nu }$ is analogous to the estimate of thickness $\delta _{\omega }^{\nu } = \sqrt {{\nu \mathord{\left/ {\vphantom {\nu \omega }} \right. \kern-0em} \omega }}$ in the Stokes flow above a plane oscillating with frequency ω [1]. In this case, the expression for microscales contains instead of oscillation frequency $\omega$ buoyancy frequency N characterizing natural oscillations of a particle of the stratified fluid, which is displaced from the equilibrium horizontal.

With the beginning of the motion of the body, the state of the medium changes, and the physical field pattern becomes noticeably complicated. In the flow pattern, the advanced perturbation including the partly blocked fluid in the layer of motion of the body and the contouring group of nonstationary internal waves, the wake with its own vortices behind the body, as well of the system of ligaments (fine-structure interfaces of different scales and isolated fibers in 3D models) are usually distinguished. Accordingly, the number of proper scales also increases [5].

The group of large scales of the dynamic origin includes length $\lambda = U{{T}_{b}}$ of internal attached wave and the viscous-wave scale ${{\Lambda }_{\nu }} = {{\sqrt[3]{{g\nu }}} \mathord{\left/ {\vphantom {{\sqrt[3]{{g\nu }}} {N = }}} \right. \kern-0em} {N = }}\sqrt[3]{{\Lambda {{{(\delta _{N}^{\nu })}}^{2}}}}$. Transverse sizes of ligaments are characterized by the Prandtl and Peclet scales that are determined by the dissipation coefficients and obstacle velocity ($\delta _{U}^{\nu } = {\nu \mathord{\left/ {\vphantom {\nu U}} \right. \kern-0em} U}$ and $\delta _{U}^{{{{\kappa }_{S}}}} = {{{{\kappa }_{S}}} \mathord{\left/ {\vphantom {{{{\kappa }_{S}}} U}} \right. \kern-0em} U}$).

Short time scales that characterizing the variability of ligaments are determined by the kinetic coefficients and plate velocity ($\tau _{U}^{\nu } = \nu {\text{/}}{{U}^{2}}$, $\tau _{U}^{\kappa } = \kappa {\text{/}}{{U}^{2}}$) and, additionally, by buoyancy frequency ($\tau _{N}^{\nu } = \sqrt {{\nu \mathord{\left/ {\vphantom {\nu {N{{U}^{2}}}}} \right. \kern-0em} {N{{U}^{2}}}}}$, $\tau _{N}^{\kappa } = \sqrt {{\kappa \mathord{\left/ {\vphantom {\kappa {N{{U}^{2}}}}} \right. \kern-0em} {N{{U}^{2}}}}}$). The large number of coexisting scales indicates the complexity of the internal structure of the flow pattern near the body in the heterogeneous (stratified or potentially homogeneous) fluid. In the constant density approximation, entire information on the flow is lost.

In this study, the flow past two-dimensional bodies of various shapes of length $L$ and maximal thickness $h$ is investigated (Fig. 2).

Fig. 2.

Geometry of a flow past an obstacle (sizes are indicated extension lines).

The geometry of a symmetric plate is characterized by dimensional parameters that are usually represented by dimensionless ratios. Sharpness coefficient $\xi = L{\text{/}}h$ is defined as the ratio of area ${{S}_{b}}$ of the body to clearance area $\zeta = \frac{{{{S}_{b}}}}{{Lh}}$ (for the shape under study, ζ = $\frac{{{{L}_{t}}}}{L} + \frac{{L - {{L}_{t}}}}{L}\frac{{h + {{h}_{t}}}}{{2h}}$, where ${{L}_{t}}$ is the length of the narrowing part of the plate and ${{h}_{t}}$ is the height of the rear edge of the plate).

The flow pattern of a stratified fluid in the whole and the structure of wave fields depend on the angular position φ of trajectory of the center of the body relative to the horizontal and angle of attack α (position of the longitudinal axis of the body relative to the trajectory of its center).

Equations (5) can be solved numerically in the unified formulation in a wide range of velocities of the flow past the plate, $U \in \left[ {{{U}_{1}},\;\;{{U}_{2}}} \right]$. The diffusion-induced flows on a stationary plate (${{U}_{1}} = 0$) were studied in detail in [34].

The ratios of proper scales of the problem specify the characteristic dimensionless combinations that include both traditional Reynolds numbers in fluid dynamics, ${{\operatorname{Re} }_{U}}\, = \,{{UL} \mathord{\left/ {\vphantom {{UL} \nu }} \right. \kern-0em} \nu }\, = \,L{\text{/}}\delta _{U}^{\nu } \gg 1$, internal Froude number ${\text{Fr}}\, = \,U{\text{/}}NL$, Peclet number ${{\operatorname{Pe} }_{U}}\, = \,L{\text{/}}\delta _{U}^{{{{\kappa }_{S}}}}\, \gg \,{{\operatorname{Re} }_{U}}$, as well as sharpness coefficients ${{\xi }_{p}} = {L \mathord{\left/ {\vphantom {L h}} \right. \kern-0em} h}$ and shape fullness coefficient $\zeta = {{{{S}_{b}}} \mathord{\left/ {\vphantom {{{{S}_{b}}} {Lh}}} \right. \kern-0em} {Lh}}$, where S_b is the cross-sectional area of the obstacles, and specific characteristics of stratified flows.

Additional dimensionless ratios contain ratio ${\text{C = }}{\Lambda \mathord{\left/ {\vphantom {\Lambda L}} \right. \kern-0em} L}$ of the buoyancy scale to size L of the obstacle, which is an analog of the Atwood number ${\text{A}}{{{\text{t}}}^{{ - 1}}} = {{\left( {{{\rho }_{1}} + {{\rho }_{2}}} \right)} \mathord{\left/ {\vphantom {{\left( {{{\rho }_{1}} + {{\rho }_{2}}} \right)} {\left( {{{\rho }_{1}} - {{\rho }_{2}}} \right)}}} \right. \kern-0em} {\left( {{{\rho }_{1}} - {{\rho }_{2}}} \right)}}$ for continuously stratified media.

The values of scales determine the parameters of the spatiotemporal resolution of instruments with account for the visualization condition for fine-structure components and the estimation of their variability, as well as the sizes of the computation or the visualization domain, which must contain all structural elements under study (advance perturbations, wake, waves, and vortices).

In view of the multiscale nature of elements of the general pattern and incommensurability of proper parameters of different structure components, flow patterns are always nonstationary. The experimental technique must permit the registration of variation of both fine and coarse components during the evolution of flows. The properties of some structural elements may remain almost invariable, while the properties of other components may change rapidly. The observed “rearrangements” of the flow pattern can be caused by a decrease in the amplitudes of individual structure components or by their exceedance of the boundaries of the field of vision.

In numerical simulation, microscales determine the requirements imposed on the choice of the cell size and the time step in addition to other criteria (in particular, the Courant condition). The microscales of the Stokes scale are critical for low plate velocities U, while the microscale of the large Prandtl and Peclet scales are critical for high velocities.

The technique for numerical solution of system of equations (5) with initial and boundary conditions (6) was developed on the basis of open packet OpenFOAM [33]. Standard resolvents icoFoam and pimpleFoam of the packet, which are used for simulating nonstationary flows of a homogeneous fluid by the finite-volume method, have been supplemented with new variables (density ρ and salt concentration S), as well as with equations and auxiliary parameters (buoyancy frequency N and scale Λ, as well as salt diffusion coefficient κ_S and acceleration due to gravity g).

To ensure the reliability of numerical simulation, high-precision numerical schemes in space and time were used for discretization of differential equations. For interpolating convective terms, the TVD scheme was used with a delimiter; this introduces the minimal numerical diffusion and ensures the absence of oscillations of solutions. For the discretization of the time derivative, the implicit three-point second-order asymmetric scheme with backward differences was used, which ensured a high time resolution of a physical process.

The required spatiotemporal resolution of the flow structure was ensured using simultaneously the high-performance computing resource of the Research Computing Center of the Moscow State University and the Federal Collective Usage Center at the Scientific Research Center “Kurchatov Institute” (http://ckp.nrcki.ru) with the application of the method of decomposition of the computational domain with the uniform decomposition of the grid into a certain number of blocks. The developed programs allow one to perform calculations (in the unified formulation) of all components of multiscale stratified flows (upstream perturbation, internal waves, wake, and separating high-gradient interfaces) in a wide range of the Reynolds numbers 1 < $\operatorname{Re} < 100{\kern 1pt} {\kern 1pt} 000$ for all types of stratification (strong, weak, and potentially uniform).

Simultaneously, the visualization of the pattern of flow past a plate moving in a stratified or homogeneous fluid was performed on the stand of the transportable laboratory basin of the Hydrophysical Complex, Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences [35]. The setup includes the working basin, filling system, mechanisms for haulage of models, the coordinate-rotation mechanism for installing and displacing sensors, generators of surface and internal waves, a hydrolocator, sensors, the interfaces for experimental data collection and processing, and the control block.

The displacements of the group of vertical markers in a homogeneous fluid illustrated in Fig. 3a allow us to visualize the vortex structure of the wake and to trace the increase in its transverse size ${{l}_{z}}$ upon an increase in distance $X = U{{t}_{w}}$ from the body (or with increasing wake age t_w, i.e., the time of passage of the body through a chosen point in the laboratory coordinate system).

Fig. 3.

Flow patterns near a horizontal plate of length ${{L}_{x}} =$ 2.5 cm, moving uniformly from right to left: (a) in a homogeneous fluid, U = 4.3 cm/s, angle of attack α = 6° (visualization with a set of markers); (b, c) in a stratified fluid; T_b = 7.5 s, U = 4.2 cm/s, angle of attack α = – 6°; t = 0, 3.75 s.

In a stratified fluid, the shlieren pattern of waves is supplemented with a family of ligaments (fine transverse interfaces near the plate), which overtakes the perturbation and the vortex wake (Fig. 3b). Length λ of the attached internal wave is determined by buoyancy period and the velocity of obstacle, $\lambda = U{{T}_{b}}$. The shapes of phase surfaces of the attached internal waves differ from semicircles in the vicinity of a density wave, where they are entrained by the cocurrent flow. Periodic changes in the structure and thickness of the density wake are the consequences of its active interaction with attached internal waves, at the crests and troughs of which the extrema of its height are localized.

The experimental technique includes the observation of diffusion-induced flows. For this purpose, several hours before the beginning of experiments, the model is placed at the center of the basin. The diffusion-induced flow patterns are recorded by several schlieren methods and are compared with calculated patterns. The flows are observed through the central window. The next experiment begins several hours after the disappearance of all dynamic and structural perturbations.

In calculated and schlieren patterns of stratified flows past a vertical plate, all structural elements of the flow under study (including slowly evolving elements such as attached wave fields, upstream perturbations, and the vortex wake behind the body, as well as rapidly varying elements such as fine structural interfaces such as ligaments and bundles and their aggregates in the form of vortices) are manifested clearly (Fig. 4). In a nonstationary flow, all flow components are represented simultaneously and actively interact with one another and with the incoming flow.

Fig. 4.

Shlieren patterns of the flow (upper part of each figure) and calculated patter (lower part) near a uniformly moving vertical plate of height $h =$ 2.5 cm, buoyancy period ${{T}_{b}} =$ 12.6 s: (a–d) $U =$ 0.03, 0.18, 0.26, 0.75 cm/s.

In the case of a slow motion of the body, the creeping flow regime is characterized by a combination of an aggregate of structural components of the flow, which includes ascending oblique internal wave beams, attached internal waves flowing past the body, and bundles displaced to the center of the flow from the plate edges (interfaces contouring the dense wake) (Fig. 4a). In the regime of intense wave perturbations, both waves and a rich family of bundles filling the thin-layer wake are represented (Figs. 4b and 4c). In addition, the fine structure of ascending perturbations, which indicates the existence of bundles behind the flow and ahead of it, is visualized both in calculations and in experiments. In the latter case, phase surfaces of attached internal waves are deformed by the wake flow slightly more strongly as compared with the results of calculation, and the pattern to splitting of thin-layer perturbations is manifested more clearly.

Upon a further increase in the flow velocity, the general structure of the flow experiences some changes such as an increase in the slope of phase surfaces of internal waves towards the motion of the body, a change in the geometry of fine-structure interfaces, and the extent of manifestation of individual component of the flow (Fig. 4d). The strongest structural changes are observed in the wake behind the plate, where typical vortex elements such as “vortex dipoles” behind the body and “vortex bubbles” are formed in the zones of divergence of internal phase surfaces of waves.

All components of the flow evolve and actively interact with one another and with the incoming flow. In the nonstationary flow regime, slowly evolving components such as upstream and attached wave fields, rapidly varying components including fine-structured interfaces or ligaments (bundles), and their sets forming vortices can be distinguished (Fig. 5). The calculations and observations of the flow pattern are in good qualitative agreement in all regions of the flow, including advanced perturbations, the system of internal waves, the wake with fine structures, and vortices.

Fig. 5.

Shlieren patterns of the flow (upper part of each figure) and the calculated patter (lower part) near an inclined plate of length $L =$ 2.5 cm, moving uniformly with velocity $U =$ 4.3 cm/s: (a) $N =$ 0.83 s^–1, ${{T}_{b}} =$ 7.6 s; (b) $N =$ 10^–5 s^–1, ${{T}_{b}} = 6.3 \times {{10}^{5}}\;{\text{s}}.$

In the nonstationary flow regime, when vortex elements and internal waves with a length comparable with the size of the observation region become predominant, the structural changes with the highest contrast are manifested in the wake (see Fig. 5). Both laboratory and numerical modeling show that the wake behind an inclined plate has the structure of a typical vortex trail in the form of a sequence of mushroom elements.

In a strongly stratified medium (Fig. 5a), vortex components gradually collapse downflow and split into an aggregate of fine-structure elements. In a homogeneous fluid (Fig. 5b), the vortex trail expands, evolving downflow and in the vertical direction within the observation zone. On vortex envelopes and in the regions of interaction of multiscale components of the vortex flow with one another and with the plate surface, various multilayer fine-structure flow elements are formed. All structure components observed in two-dimensional stratified flows (advanced perturbations, internal waves, and the wake with fine interlayers) are also observed in three-dimensional flows past a uniformly moving sphere [36].

7 HYDRODYNAMIC AND AERODYNAMIC EXPERIMENTS

Schlieren instruments of the IAB type were designed by D.D. Maksutov at the middle of the 20th century for visualization of the pattern of the flows past aircrafts [29]. Even in the first transonic experiments performed under the guidance by Acad. S.A. Khristianovich in 1943, high-gradient interfaces forming right angles with the velocity vector of the main flow were observed [37]. The interfaces were interpreted as “shock waves balanced in the flow” (although Khristianovich called them “short nonstationary waves” in a personal communication), the loss of velocity and pressure drop in which produce an extra wave resistance. For describing their effect, model equations have been proposed, which are actively used in practice even now [38].

The problems considered by Khristianovich remain topical and are studied intensely using modern experimental techniques. By way of example, Fig. 6a shows the results of the schlieren visualization of the pattern of the flow past a wing in a wind tunnel (Figs. 6a and 6c) and past a plate in a stratified basin (Figs. 6b and 6d), which have been obtained by the technique described in [33].

Fig. 6.

Shlieren patterns of a flow past (a) a wing for Ma = 0.77; (b) a plate in the laboratory transportable basin (Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences) (${{T}_{b}} =$ 7.55 s, ${{L}_{x}} =$ 2.5 cm, angle of attack $\alpha = 12.5^\circ$, velocity U = 3.6 cm/s, Re = 900, Fr = 1.73); (c, d) magnified part of figures with fine structures.

It can be seen that the families of fine-structure components are visualized in flows of a compressed gas with a transonic velocity in a wind tunnel as well as near a slowly moving body in a weakly compressed stratified liquid; this indicates their parametric invariance and the possibility of a unified description based on a system of fundamental equations [5].

For obtaining additional data required for comparison of the calculation results based on the system of fundamental equations and experimental data, the technique of aerodynamic experiments must be coordinated as regards the choice of experiments ensuring the fulfillment of conditions of completeness, spatiotemporal resolution of instruments, and programs for data processing of parameters of aerodynamic and acoustic fields.

8 DISCUSSION OF RESULTS

In accordance with requirements of the contemporary form of formal logics, at the first stage of a scientific research, it is necessary to determine the object and method for investigating the phenomenon of interest. Accordingly, in fluid dynamics, it is necessary to specify physical variables characterizing the medium and the method for describing their variations during the motion of a body.

The unperturbed state of the atmosphere and hydrosphere is characterized by the distributions of the Gibbs potential and its derivatives (energy and entropy) as well as other thermodynamic quantities (pressure, temperature, the concentration of dissolved substance and suspended particles). The main property of the unperturbed atmosphere in an amplitude range of up to 12 km is the stable density distribution that is ensured by the compressibility of the gas with a fine-structure stratification. The buoyancy period in the atmosphere varies in the range 3–10 min like in the ocean. Local inversion (increase in the temperature of the medium with the altitude) leads to an increase in the density gradient. Stratification effects must be fully taken into account in the system of equations of fluid mechanics and in the choice of the methods for their mathematical and numerical analysis.

Fluid flows are described by identical basic differential transfer equations for momentum, energy, mass, and density [1, 5]. Accordingly, the methods for constructing complete solutions, which have been developed for analyzing flows of a stratified liquid, can be modified for calculating the flows of a stratified gas. The classification of components of periodic flows, which includes proper waves of various types (inertial, gravity, capillary, acoustic, and intermediate hybrid waves) and accompanying ligaments characterizing fine fibers and interfaces [5, 32] is universal and independent of the phase state of the flowing substance (liquid, gas, or plasma) [39].

In the complete dispersion relation constructed in the linear approximation and in the solution to the complete system of fundaments equations of fluid mechanics, waves are described by the regular part of solutions, while ligaments are described by their singular part [31, 39].

The energy transfer mechanisms (heat radiation, macroscopic with flows and waves with a group velocity, and diffusion and conversion microscopic mechanisms) ensure the energy exchange between all components, including the energy transfer from coarse to fine and from fine to coarse components. In this case, the fine structure of the medium including high-gradient interfaces and fibers (ligaments and bundles) is rearranged. The multitude of structure components, the processes of nonlinear interaction and nonuniform dissipation of components are responsible for the continuity of transformation in the structure of the distribution of physical quantities, which determines the dynamics and energy of flows

Basic physical quantities are thermodynamic potentials and their gradients, which are thermodynamic quantities, and invariant parameters of flows (specific momentum and kinetic and potential energy densities) are the observable quantities permitting direct, indirect, or mediated determination of their values with an error estimate. The flow velocity is an unobservable quantity because of the absence of physical identification of a “fluid particle,” which is associated with the complexity of the structure of flows on small scales.

Experimental techniques must ensure the determination of all quantities appearing in the equations of motion and state (density, momentum, total energy, etc.) with the estimate of error with account for the flow structure. The instruments must ensure the observation of large-scale components and resolve fine-structure components of fields for all basic structure components.

9 CONCLUSIONS

The scientific foundation of the modern theory and experimental investigations of fluid flows is the scale- and parametrically invariant system of fundamental equations of mechanics of heterogeneous fluids with equations of state for the density and thermodynamic potential. The system permits the imposition of consistent requirements and techniques for constructing computational algorithms and fluid dynamics experiments in practically important ranges of parameters.

Experimental techniques must include the estimation of the fulfillment of the completeness criterion and metrological properties of instruments, as well as substantiated estimates of errors and the spatiotemporal resolution. The set of parameters measured in experiments must ensure the determination of all quantities appearing in the system of fundamental equations directly, indirectly, of mediately.

The cycle of investigations performed in this study has shown that the results of theoretical, numerical, and experimental investigations of the pattern of the flow of a stratified fluid past a plate or any other body, which include the description of the upstream perturbations, internal waves, wake, vortices, and ligaments are in good agreement with one another and with the results of independent experiments and calculations.