Simulation of non-isothermal free turbulent gas jets in the process of energy exchange

This article proposes a numerical method for solving the propagation and combustion of a jet of a gas mixture in an axisymmetric satellite air stream. To model the process, the dimensionless equations of the turbulent boundary layer of reacting gases in the Mises coordinates are used. A two-layer four-point nonlinear boundary separation scheme was used to solve the problem in the Mises coordinates, and a second-order along the longitudinal coordinate was given. The iterative process was used because of the nonlinearity of the storage and displacement equations of substations. Individual results of the numerical experiment are presented.


Introduction
The development of methods for studying jet flows of multicomponent gases can be monitored in [1][2][3][4][5][6][7]. The dominance of convection over diffusion, the alternating coefficient for the convective term, makes it possible for local subdomains to appear in the computational domain with a large boundary of the desired function of the boundary and internal transition layers [8]. The presence of sources and drains in chemically reacting flows introduces additional difficulties into the picture of the process. Ultimately, this leads to serious difficulties in the numerical integration of equations of mathematical models.
The dominance of convection over diffusion, the alternating coefficient for the convective term, leads to the appearance in the computational domain of local subdomains with a large boundary of the desired function of the boundary and internal transition layers. The presence of sources and sinks in chemically reacting flows introduces additional difficulties into the picture of the process. Ultimately, this leads to serious difficulties in the numerical integration of equations of mathematical models. These difficulties in modeling and numerical solution of heat and mass transfer problems are the different scale of diffusion processes, convection, and source drains. *Corresponding author: kamina.0691@mail.ru In general, a strict quantitative calculation of the diffusion torch of the final size is extremely difficult. However, this task reflects the process in the main section of the jet, which is of the greatest practical interest, and it also has the greatest computational difficulties. On the one hand, the transport and conservation equations contain third-order terms concerning unknowns. On the other hand, unknown boundaries appear in the problem the flame front and the boundary layer boundary.
Separate entry of gases into the zone of intense heat and mass transfer causes a tangential discontinuity in flow rates at the inlet section. This and the presence of a flame front lead to establishing a turbulent flow regime in a jet stream [1]. Therefore, the kinematic viscosity coefficient in the framework of the work was taken for a turbulent flow and a new modification of the L. Prandtl hypothesis was used combustion. Note that it is customary to distinguish between two forms of a direct jet torch -a flooded and a satellite torch. In the first case, it is about the expiration of a jet of fuel in the space flooded by a stationary oxidizing agent (for example, air), in the second -about the expiration of a stream of fuel in a parallel spiral oxidizer stream. Obviously, the second case is a general one and contains as a special case, if the velocity of the satellite stream is equal to zero, the problem of a flooded torch. We believe that the velocity distribution in the exit section of the nozzle and in the confluent stream and the initial (at x = 0) distribution of temperature and concentration of fuel and oxidizer will be considered given, uniform, uniform, and stepwise.
Assuming an infinitely high rate of the chemical reaction of combustion and the same rate of convective and diffusive (turbulent) transfer of components, the flame front can be represented as an axisymmetric surface, in its main part close to a cylindrical one, delimiting the calculation area into two zones.
One of them -the inner zone is filled with fuel and combustion products. Fuel diffuses to the front surface from the inside and an oxidizer from the outside. Combustion products diffuse from the flame front in both directions, into the inner and outer sides of the torch. At the front itself, the concentration of each of the reactants is zero, and the concentration of combustion products is maximum. At the same time, turbulent flows of reacting gases, fuel, and oxidizer flowing to the front and burning on it, as noted above, are in a stoichiometric ratio.
Concerning the coefficient of turbulent viscosity, we used the modified Prandtl model we proposed taking into account the heterogeneity and volume compressibility of the medium, which has the form: We consider a jet of a mixture with a combustible gas that flows out of a round nozzle with the diameter 2a with speed 2 u and propagates in a confluent oxidant stream, the speed of which is 1 u . Each of the separately introduced mixtures has its own composition and its own thermophysical characteristics. Gases mutually diffuse, and active gases react on a thin surface of the flame front. The combustion rate is considered large enough that the fuel does not penetrate into the air zone, and oxygen in the air cannot be in the zone of the combustible mixture. Accordingly, in the fuel zone, there are fuel, reaction products, and chemically passive gases; and in the air zone, there are oxidizing agents, reaction products, and chemically inert gases It is required to develop a numerical method and a software product with which it is possible to study the jet stream for various compositions of the main and satellite flow, their aerodynamic and thermodynamic parameters. In this case, it is required to reduce the number of equations of transport and conservation of the gas mixture components.
In the approximation of the theory of turbulent boundary layer, the system of equations of turbulent motion of a multicomponent gas in the presence of chemical interaction between the components, with the equality of the unit Lewis number for components (( 1 i Le  )), can be written in the form [1]: The diffusion combustion model is generalized as applied to a complex composite combustible mixture, including several combustible components. The main essence of generalization is the formation of a united front for all combustible coefficients. Based on the model, the combustion of a mixture of methane and carbon dioxide in the air was studied. The features of the propagation of an axisymmetric jet of a combustible mixture in airspace were studied depending on the mass fraction of methane in the combustible mixture. In general, when simulating the combustion process, the main problem is to reduce the number of equations for the conservation and transfer of matter of N components. Above, we analyzed the methods for introducing the Schwab -Zeldovich function and relatively excess concentration with the closure conditions for the equations for diffusion combustion models, the final rate of the chemical reaction, and chemical equilibrium.
The basic differential equations of a multicomponent turbulent boundary layer of reacting gases under the assumption that the turbulent Lewis number is equal to unity (Le = 1) can be written as [11][12][13] 1 , Multiplying the first equation by u, we get: By adding the corresponding thresholds of the system, we form the following equation:

  
we simplify using this equality: The total enthalpy of the gas mixture has the form .

Methods
Enter discrete coordinates i ih    and j jh    , as well as discrete functions The finite-difference representation of differential equations is feasible concerning the dimensionless equation of total enthalpy [14,15]: The condition of symmetry of the field of total enthalpy with respect to the x-axis is realized as follows. We assume that h   small argument increment  at 0   .
Then the expression in parenthesis under the derivative on the left side of the equation is written as Then the components of the right side of the equations can be represented as: In general, we have   In connection with the introduction j K approximation of the equation in internal nodes with the second-order of accuracy in  and first-order accuracy in  has the form: In the article, we limit ourselves to discussing the effect of the temperature of the particular gases on isotherms and concentrations.
The starting values for the flow area corresponding to the concentration area and key components were determined. 1  x  The graphs for the cross-sections are given. As you can see from this graph, 10 x  the fuel core is broken. The flame 1.8 r  expanded to. Next 40 x  , as can be seen from the graph, there is a decrease in fuel over time, and the next 200 x  concentrations after the flame are given below.