Modelling the pollutants transport in the “air-water” system of a shallow water

The work is devoted to the spread processes mathematical modelling of impurities in a reservoir in areas located far from river runoff. An aerodynamic processes mathematical model is proposed that takes into account a variety factors: increased air humidity, variability of atmospheric pressure and temperature, etc. The model discretization was carried out taking into account the partial filling of the calculated cells, which made it possible to significantly reduce the error at the boundary. For the calibration and verification of the pollutants transport model, the expeditionary research data were used.


Introduction
Most of the pollutants coming from industrial facilities and transport are concentrated in the surface atmosphere layer the, and are deposited on the surface of the water or enters the reservoir with precipitation [1][2][3]. It is known that in the distance of river runoff, more than 60% of pollutants enter the reservoir from the air substances affecting the production and destruction phytoplankton processes. Toxic pollutants enter water bodies from the atmosphere, are absorbed by phytoplankton, and then are transmitted to more highly organized organisms through food chains. Therefore, it became necessary to develop mathematical models to predict changes in the ecological situation in shallow water and coastal areas, including: the pollution spread processes in the border layer of the coastal zone atmosphere; pollutants movement and sediments, the organic sediments decomposition processes [4][5][6].
The analyzing problems, monitoring and predicting the air environment quality in cities with intensive traffic flows are very important, the health and comfortable people living conditions depend on their solution. From a theoretical and applied view point, significant among the analyzing tasks and forecasting the air environment state are those that take into account the multicomponent nature, including a significant change in humidity, the phase transitions presence, etc., which is especially important for coastal cities. An effective tool for predicting the air quality is mathematical modelling of the variability of its gas and aerosol compositions, as well as assessing the atmospheric impurities impact on the environment, including on adjacent aquatic ecosystems [7]. Many gaseous and aerosols impurities transformation processes occur in a turbulent atmosphere. To reproduce the atmosphere turbulent characteristics variability, the impurities propagation modelling problem solution must be carried out in conjunction with hydrodynamic models [8][9][10]. At present, in the mathematical modelling field of the pollutions movement processes in the atmosphere and the numerical methods development for these purposes, a situation has arisen in which in the process of ongoing research individual phenomena are IOP Publishing doi: 10.1088/1757-899X/1029/1/012080 2 considered, but they are not included in the complex. Therefore, to solve the problems that meet the set task, it is necessary to develop new mathematical models based on the gas dynamics equations and the matter conservation laws, taking into account the multicomponent medium nature.
In practice, special attention is paid to the study of the processes of transfer of pollutants (pollutants) in the air. The air movement and the pollutants spread in it includes four stages [11]. Pollutants matters transport scheme in the "air-water" system is shown in figure 1.

Figure 1.
Pollutants matters transport scheme of in the "air-water" system.
Recently, mathematical modelling in the water bodies hydrophysics field and impurities transport has advanced and went beyond academic research -many industrialized countries use hydrophysical models in the water ecosystems study. Currently, two-dimensional and three-dimensional mathematical hydrophysic models have been developed, but the further scientific research area has not been exhausted, and the use of modern technologies will provide their quantitative and qualitative improvement.
To describe the pollutants transfer in the air and their deposition on the reservoir surface, in this work, interacting motion models of multicomponent air medium and a model describing the pollutants transport processes in the air-water system are proposed. The motion models of multicomponent air medium is used to calculate the velocity vector field and takes into account such factors as: turbulent exchange; variable density; dependence density air on pressure.

The air multicomponent medium movement model
Due to complexity of describing the properties and urban relief shape, the influence of such factors as temperature, humidity and air density on the currents nature, the atmosphere mobile turbulent nature, the air environment state is nontrivial. For effective observation, assessment and changes forecasting in the atmosphere state atmosphere, all these factors must be taken into account in detail. There is still no universal mathematical model for solving such a complex problem. Applying the mass conservation law to a liquid flowing through a fixed infinitesimal control volume, we obtain the continuity equation (1) [12][13][14]: where ρ is the air density. Similarly, the momentum equation follows from Newton's second law where g is the gravity acceleration, ij  is stress tensor.
For all gases that can be considered a continuous medium, as well as for most liquids, it can be seen that the tension at some point linearly depends on the rate of deformation of the liquid. According to  [2], the general deformation law, which relates the tension tensor to pressure and velocity components, is written in the form Неrе ij  is the Kronecker symbol, μ is the turbulent exchange coefficient.
By transformations from equations (2) and (3) we obtain the motion equation (Navier-Stokes): 1 , where vj are the velocity components projections on the axis Оxj, j=1,2,3. System (4) is considered under initial and boundary conditions. At the initial time moment, the velocity is zero.
Border conditions: -on the bottom surface , , , 0 where n V is the velocity vector normal component, 0 V is the velocity vector value at the upper and lateral computational domain boundaries, n is the outer normal to the boundary surface.
When the liquid surface contacts its vapor at a given temperature, an equilibrium vapor pressure determined for each liquid is established. A small increase in vapor pressure above the liquid surface leads to vapor condensation on this surface, and an infinite small decrease in pressure causes the liquid to evaporate from its surface. To describe the vapor pressure dependence on temperature, we will use the Mendeleev-Clapeyron formula, which can be represented as where Let us differentiate the state equation (8) and we get the equation (9):

Dissipation polluting substances in the atmosphere
It is known that the atmospheric movements spatial scales vary from small eddies to the large air masses movement. Table 1 describes the various atmosphere processes and their spatial scales. Phenomena Scale ( km ) Air pollution with toxic substances 0.1 -100 Stratospheric ozone destruction 1.000 -40.000 Increase in greenhouse gases 10.000 -40.000 Acidic precipitation 100 -2000 The impact on the climate of aerosols 100 -40.000 Transport and oxidation processes in the troposphere 1 -40.000 Stratospherictropospheric exchange 0.1 -100 Transport and oxidation processes in the stratosphere 1 -40.000 The wide scales range is explained by the gas impurities and aerosols variability. And, therefore, depending on the spatio-temporal considering processes scales, it is necessary to choose both the corresponding hydrodynamic models and the models for the gas impurities and aerosols transformation to solve a specific physical problem. The impurities transport equation, which describes the small moving vortices mixing with the environment and which is accompanied by the matter transfer, is written as: where w0 is the deposition rate, f is a function describing the distribution and power of impurity sources.
Taking into account the transition of water from a liquid to a gaseous state and vice versa, as well as the fact that suspended particles are deposited during the impurities transfer, the equations for the pollutants transport in a multicomponent air medium look like this: where The equations system (12) must be supplemented with the following boundary condition: We use the friction and gravity forces, which are involved in determining the pollutant velocity.
where s w is the impurities deposition rate.

Shallow water hydrodynamics model
During solving the atmospheric impurities propagation problems and their settling on the water surface, it becomes necessary to use the pollutions transfer processes modelling to determine the heavy particles accumulation at the reservoir bottom, since their own deposition velocity depends on the heavy impurities transfer peculiarity, which often exceeds the vertical environment movement.

Shallow water hydrophisics model
The developed model for calculating three-dimensional aquatic environment movement velocity vector fields, temperature and salinity is based on a mathematical model of the shallow water bodies hydrophisics, taking into account the heat and salts transport [15]: the motion equation (Navier-Stokes) continuity equation in the variable density case heat transport equation salt transport equation where   ,, u v w  V are the velocity vector components; P is the total hydrodynamic pressure; S и T is salinity and temperature of the aquatic environment; ρ is the aquatic environment density; μ, are horizontal and vertical components of the turbulent exchange coefficient; is the angular Earth's rotation velocity;  is the site latitude; g is the gravity acceleration; .
where w  is the fresh water density given by a polynomial . (21) applies to salinity in the range of 0 -42 ‰ and temperature from -2 to 40 °C.The equations system (15) -(20) is considered under the following boundary conditions: at the entrance where  is liquid evaporation intensity; n V , τ V are the velocity vector normal and tangential components; n is the outer normal to the boundary surface;   x y z ,,     τ is tangential stress vector;  is the aquatic environment density; ρ v is suspension density; T a is atmospheric temperature; k is heat transfer between the atmosphere and the aquatic environment coefficient, is a liquid layer that evaporates over time  .

Shallow water biological rehabilitation model
A multi-species phyto-and zooplankton interaction model, taking into account the pollutants influence entering the reservoir on production-destruction processes in the reservoir has the form:

The model discretization
Discrete analogues of the diffusion and convective transfers operators are obtained from the diffusionconvection equation [16,17], to which each of the equations of system (22) can be reduced by the linearization field: To build a discrete air movement model, we consider the computational domain, which is inscribed in a rectangle. We cover the area with a uniform rectangular computational mesh IOP Publishing doi:10.1088/1757-899X/1029/1/012080 l,  .

Results of the Experimental Studies
In the summer of 2018, a research expedition was carried out by SFedU staff together with specialists from Rospotrebnadzor in the Rostov Region to study areas of Taganrog that are dangerous from the view point of the pollutants spread, including the coastal zone. Table 2 presents data on the composition and concentration of the main pollutants obtained in the course of expeditionary research. These data were used to calibrate and verify the developed models.   Studies have shown that the most polluted the Sea of Azov waters are: the the river Don mouth (mouth of the hands Kuterma, Perevoloka), r. Malyi Elanchik, r. Mius, mouth r. Valovaya (Taganrog); the eastern part of the Taganrog Bay (Pivotny buoy of the Azov-Don Sea Canal (ADMK), near the dump of sea soil (Taganrog); port, Central beach, Primorsky beach, Petrushino beach, release area (Taganrog) [18].
On the developed software module basis, which numerically implements the mathematical pollutant transport model, numerical experiments were carried out taking into account the field expedition data. Scenario 1. There is no convective transfer. Constant uneven function of the pollution source on the area surface. Impurity type: heavy uneven, this is how heavy metals behave: aluminum, molybdenum, lead andantimony, which arecontained in the exhaust gases that are emitted by internal combustion engines (Figure 8, 9).  There is no convective transfer. Constant uniform function of the pollution source on the area surface. Impurity type: conservative uneven, it can be nitrogen dioxide, sulfur dioxide, the pollution source can be industrial enterprises, as well as vehicles (Figure 10, 11).  Scenario 3. Convective transport is present, the admixture propagates only due to diffusion, the flow is from west to east. Constant uneven function of the source of pollution on the area surface. Impenetrable border on the right. Impurity type: heavy uniform. This is how a substance such as manganese behaves, mining and processing enterprisescan act as a pollution source. ( figure. 12, 13).

Conclusion
Сomplex of interconnected models of aerohydrodynamics is proposed, which allows studying the various pollutions types propagation processes from the reservoir surface, taking into account their settling to the bottom. The considered impurities propagation processes mathematical models in the in the water layer bordering on atmosphere are intended for the analysis and prediction of the water quality environment. Continuous and discrete a multicomponent air medium motion mathematical models, which take into account such factors as the water transition from a liquid to a gaseous state, turbulent exchange, matter sedimentation, heat transfer between liquid and gaseous states, and variable density and temperature, more accurately describe these processes compared to other well-known models is offered. To calculate the pressure field, an equation is obtained that takes into account the compressibility of the medium, thermal expansion, matter sources associated with the water transition from a liquid to a gaseous state and vice versa, as well as a multicomponent air medium turbulent mixing. A distinctive feature of the developed mathematical model is the inclusion of turbulent mixing in the equation for calculating the medium density. The model discretization was based on the method of calculation cells partial filling, which made it possible to improve the accuracy of calculations for presented air-to-water transport scenarios in the system. Numerical experiments on the study of the input parameters influence confirmed the results validity of the program module. The developed software and algorithmic tools can be used to develop schemes for optimal environmental management, assessing the aquatic ecosystems state.