The experiments performed by Benard and their theoretical interpretation by Rayleigh can be considered the beginning of the study of natural convection in liquids and gases, after which almost 125 years have passed. Further research includes the works of Prandtl, Karman, Batchelor, Kutateladze, Landau et al. [1], Gebhart et al. [2], Bergman et al. [3], Bejan et al. [4], Cormack et al. [5], Gershuni et al. [6–8], Gershuni et al. [7], Gershuni et al. [8], Polezhaev et al. [9,10]. Unfortunately, this is an absolutely incomplete reference list since there is a huge quantity of work on the study of convective processes, and this article does not have the purpose and opportunity to consider everything related to this work. Such a large number of published scientific works is due to the variety of convective processes, fundamental interest in them, as well as the need for and importance of studying them for many applications (automotive, aviation and space technology, energy (including nuclear), technologies for obtaining new materials (including semiconductors), medicine, life support systems, fire extinguishing, etc.). It should be noted that the variety of gravitational convective flows is due not only to dimensionless parameters (liquid properties, volume size and intensity of external thermal and mass fluxes), but also to the mutual direction of gravity vectors and external thermal and mass fluxes attached to volume [9]. In this article, we will consider only one case: this is a square area with horizontal fluxes of heat and mass from the vertical boundary walls. Despite the intensive study of the processes of convective heat and mass transfer, many problems remain poorly understood due to their nonlinear nature. At certain values of the determining parameters, laminar (stationary or quasi-stationary) fluid flows can exhibit nonlinear properties that can significantly change the structure of the fluid flow and the characteristics of heat and mass transfer. For example: (1) Effect of maximum temperature (concentration) stratification [9,10], (2) well known that during vibrational action on continuous media, their anomalous nonlinear peculiarities and resonant properties may manifest themselves [8,11,12]. It must be remembered that many analytical solutions to convection problems have their own ranges of applicability. For example, based on the analysis of the equations of motion for the plane case, Batchelor [13] suggested that during convection the core is isothermal and rotates with a constant and uniform vortex of velocity, which is not always true.

In an initially homogeneous liquid located in the gravity field when heat or mass is supplied, vertical stratification in density may occur due to convective mixing. Temperature and concentration stratification in liquid volumes during convective mixing of liquids is observed in many convective processes (for example, in crystal growth processes [9,10,14,15]), both in terrestrial conditions and in microgravity. The study of such heat and mass transfer processes is relevant not only from a fundamental point of view, but also for many applications, for example, which have been indicated in [16,17]. Therefore, knowledge of the patterns of formation of stationary (quasi-stationary) flow and stratification structures in liquids is important, for example, for specialists in growing single crystals in terrestrial and space conditions, since in technological processes of obtaining materials, an urgent task is to determine the possibility of regulating temperature or concentration stratification in a liquid volume in order to obtain homogeneous perfect materials with specified properties [9,14].

In the processes of heat and mass transfer, the convective stratification of temperature and concentration plays an important role, slowing down the process of heat and mass transfer (for example, in the case of penetrating convection in vertical layers when heated from below) [2,3,9,10]. This fact is important for a wide range of applied tasks: growing perfect single crystals [14,15], problems of fire extinguishing, boiling, the safety of nuclear installations, cooling of electronic equipment and storage of liquid rocket fuel, prevention and control of environmental pollution, including from liquid finely dispersed harmful inclusions, for example, with viruses, etc.

The magnitude of the temperature and concentration stratification depends nonlinearly on the induced convection, and in zero gravity conditions, it can manifest itself more strongly than in terrestrial conditions [9].

Convective temperature and concentration stratification of a fluid can have both a positive and a negative aspect. For example, this is a negative factor in obtaining perfect homogeneous single crystals, and in density separation and in obtaining eutectic materials, this can play a positive role [9,14,15].

This article demonstrates the manifestation of nonlinear features of laminar thermal and thermo-concentration convection, as well as the influence of vibration effects on the vertical stratification of temperature and impurities [16].

The problem of gravitational thermal and concentrational convection of an incompressible liquid in a cavity with the aspect ratio L/H=1 (where L–is length and H–is the height of the calculated region), laterally heated in the field of gravity with acceleration of free fall g, is considered. At lateral heating, constant values of temperature θ1 and θ2 ( θ1<θ2 ) and concertation s₁ and s₂ on the side walls are set; for velocities, non-slip conditions are set. The following boundary conditions are considered: for velocity–the non-slip condition, for dimensionless temperature T on horizontal walls–line profile T|y=0,y=1=x , and for dimensionless concertation C–no mass flow condition ∂C/∂y|y=0,y=1=0 are set. The scheme of the calculated geometry, boundary conditions and isotherms in the layer for the thermal conductivity case are shown in Fig. 1.

The mathematical model is based on the numerical solution of the unsteady 2D Navier-Stokes equations for an incompressible fluid in the Boussinesq approximation and the equations of energy and mass transfer, which in a cartesian coordinate system, in dimensionless form, in variables: ψ -stream function, ω -vortex, T-temperature, C-concentration, can be written as follows [1,9]:

(1) ∂2ψ∂x2+∂2ψ∂y2=−ω

(2) dωdt=∂2ω∂x2+∂2ω∂y2+Gr∂T∂x+GrC∂C∂x

(3) dTdt=1Pr(∂2T∂x2+∂2T∂y2)

(4) dCdt=1Sc(∂2C∂x2+∂2C∂y2) where x, y–horizontal and vertical сartesian dimensionless coordinates; u, v–components of the velocity vector; t–time; T–dimensionless temperature; С–concentration; g –vector of the gravitational acceleration of the earth’s free fall directed opposite axis y; β , βC , ν , a , D–coefficients of temperature and concentration expansion of the liquid, kinematic viscosity, thermal conductivity and diffusion factor, respectively; in the future, we will use of dimensionless velocity and time (which was made dimensionless through the viscosity ν and height of the calculation region H). The problem is characterized by dimensionless parameters: the Grashof number Gr=gβT(θ2−θ1)H3/ν2 (or Rayleigh number Ra=Gr⋅Pr ), concentrational Grashof number GrC=gβC(s2−s1)H3/D2 ( Rac=Grc⋅Sc ), Prandtl number Pr=ν/a , Schmidt number Sc=ν/D and aspect ratio L/H = 1.

The results presented in this paper were obtained using the finite-difference scalar method [9] and the volume control method [18]. The good accuracy of numerical results was confirmed by comparison with experimental data and comparison of numerical results obtained by different numerical models [9,19].

The test problem of thermal convection of a viscous incompressible liquid (Pr = 0.7) in a square closed area with thermally insulated horizontal walls and with set temperatures on vertical walls (T₁ = 1, T₂ = 0) is considered. This de Vahl Davis task was announced more than 40 years ago as an international test for computer codes. About 40 different numerical solutions to this problem have been sent by various authors. In the paper [19], “benchmark solutions” were obtained for different Rayleigh numbers by extrapolating to a zero–step grid of solutions obtained by different methods on different grids. In Table 1, method 1 is the “benchmark solution” [19]; method 2 is the model used in this paper with mesh 65 * 65 nodes. In Fig. 2, the isolines of the stream function and the isotherms of the solution of the de Vahl Davis problem for Ra = 10³ (left) and for Ra = 10⁶ (right), obtained by method 2 are shown.

The results of the solution of the de Vahl Davis benchmark problem presented in Table 1 and in Fig. 2. Table 1 shows: method 1 is the “benchmark solution” [19], method 2 is the solution of our model using a grid of 65 * 65 nodes (the discrepancy is less than 3%).

The results simulation for large Rayleigh numbers Ra = 10⁷–10⁹ and Pr = 5.8, for horizontal layers L/H = 7–12, were compared with local experimental data on the uneven grids with 141 * 33 and 141 * 65 nodes, the comparison results showed good model accuracy and are given in [9,16]. The results of this work were obtained on an uneven grid with the number of nodes 200 * 200.

Gravitational convection in a square cavity heated from the side with binary mixtures with a concentration C of a light component are considered Fig. 1. Ranges of dimensionless parameters 0<Gr<108,Pr=0.7,Grc<108,10−2<Sc<102 are considered: corresponding to laminar stationary and vibrational convection. The vertical stratification was estimated by the values of derivatives of temperature (∂T/∂y) and concentration (∂C/∂y) along the vertical y coordinate.

Nonmonotonic dependences of vertical derivatives on temperature and concentration calculated in the center of the square region on the Grashof number were found, showing the presence of maximum heterogeneity of temperature and concentration depending on the Grashof number.

The pictures and differences of the formation of a nonstationary periodic structure of oscillatory thermal and thermo-concentration convection are shown. For thermal convection, a range of Rayleigh numbers exists where a regular periodic oscillatory convective flow is formed (for example, at Gr = 10⁷, Pr = 0.7). The details of the formation of (quasi-stationary) countercurrents inside a square region directed opposite to the main convective flow are given. In the considered case of thermo-concentration convection (Gr = Gr_c = 10⁷, Pr = Sc = 0.7 with oppositely directed heat and mass horizontal fluxes) oscillational convection is caused by the confrontation of thermal and concentration convection with opposite rotations.

The influence of vertical and horizontal vibrations on oscillatory convection is shown (Gr = 10⁷, Pr = 0.7). Controlled vibration effects have a strong effect depending on frequency and amplitude. With periodic vibration action on the convective cell from the side of its walls, the separation of the average time flow into two flow zones is observed: first is the fluid flow near the walls and second is the core slow fluid flow. The vibrations make the average flow more orderly and the velocity profiles in the middle sections make more symmetry.