Approximate solution of a boundary-value problem for a model of the far swirling turbulent wake past a self-propelled body is constructed using asymptotic expansion of the solution in a neighbourhood of the singular point. A good agreement between constructed solution and numerical solution is obtained.

The flow in the far axisymmetric momentumless turbulent wake is described with the use of a mathematical model based on k - E semi -empirical model of turbulence. A group -theoretical analysis of the mathematical model of the wake is performed. The similarity reduction of the model to a system of ordinary differential equations is obtained. Asymptotic expansion of the solution in the vicinity of a singular point is used to construct approximate solution of corresponding boundary value problem.

The dynamics of passive scalar in swirling turbulent far wakes with varied values of the total excess momentum and angular momentum are described within a second-order mathematical model. The model includes averaged equations of momenta, turbulence energy balance, dissipation rate transfer, averaged concentration, and dispersion of its turbulent fluctuations in the far wake approximation. The closure of the mathematical model is based on Rodi’s algebraic model of the Reynolds stresses. At large distances, a self-similar solution based on numerical experiments is obtained for problems of the dynamics of passive scalar in turbulent wakes behind a self-propelled body and in a momentumless swirling turbulent wake. The group-theoretical analysis of the mathematical model under study is carried out. The model is reduced to a system of ordinary differential equations, which is solved numerically by the shooting method. The resulting solution is compared with the self-similar solution found by direct numerical integration of the differential equations of the model at large distances from the body. Good agreement is obtained. The problem of asymptotic behavior of characteristics of passive admixture in a swirling turbulent wake behind a sphere with nonzero values of the total excess momentum and angular momentum is also considered. With application of the group-theoretical analysis, it is shown that there is no physically meaningful self-similar solution to the equations of passive admixture dynamics.

The flow in a plane momentumless turbulent wake is described with the use of a second-order mathematical model based on the Rodi’s algebraic model of Reynolds stresses. In view of the properties of the plane momentumless turbulent wake, the mathematical model is an analog of the two-parameter  $$e \sim \varepsilon$$  turbulence model in the far wake approximation with a modified empirical constant in the diffusion term of the equations. For moderate distances from the body, the results predicted by this model agree well with the known experimental data of Cimbala and Park (1990). At large distances from the body, a self-similar solution based on numerical experiments is obtained. A group-theoretical analysis of the mathematical model of the wake is performed. The model is reduced to a system of ordinary differential equations, which is solved numerically by the shooting method. The self-similar solution derived from the group-theoretical analysis is found to be in good agreement with that obtained by means of direct numerical integration of the differential equations of the model at large distances from the body.

The flow in swirling turbulent wakes with varying total excess momentum and angular momentum is described using two second-order mathematical models. The first one includes averaged equations of momenta, turbulence energy balance, and dissipation rate in the far-wake approximation. The closure of the mathematical model relies on Rodi's algebraic model for Reynolds stresses. The second model is based on simplified representations of the turbulent viscosity coefficients. For small distances, the calculated profiles of averaged motion velocities and turbulence energy are in good agreement with the experimental data of Lavrent'ev Institute of Hydrodynamics of SB RAS. At large distances, numerical experiments have yielded a self-similar solution of problems of dynamics of turbulent wake behind a self-propelled body and momentumless swirling turbulent wake. Group-theoretical analysis of the simplified mathematical model has been done. The model had been reduced to a system of ordinary differential equations, which was solved numerically using asymptotic expansions. The solution obtained was compared with the self-similar solution found by direct numerical integration of the differential equations of the model at large distances from the body, and good agreement was observed. In addition, the problem of asymptotic behavior of swirling turbulent wake behind a sphere with non-zero values of total excess momentum and angular momentum was considered. The group-theoretical analysis has shown the absence of physically meaningful self-similar solutions to the equations of the turbulence model under consideration.

A self–similar solution to one model of the far momentumless swirling turbulent wake is proposed in the paper

The semi-empirical k − ω turbulence model is considered in the far wake approximation. In this model the unknown quantities are the deficit of the mean velocity, the turbulent kinetic energy, and the specific energy dissipation rate. The group-theoretic analysis of the model is performed and a reduced self-similar system of ordinary differential equations is obtained and solved numerically. The calculated results are shown to be in good agreement with the available experimental data.

A mathematical model of the far turbulent wake behind a towed body in a passively stratified medium, based on the known semi-empirical e-ɛ model of turbulence, is considered. A grouptheoretical analysis of the model is performed. With the help of the method of B-determining equations, the model is reduced to a system of ordinary differential equations, which is solved numerically. The resultant solution is compared with a self-similar solution obtained by direct numerical integration of the differential equations at large distances from the body.

A three-dimensional model of the far turbulent wake behind a self-propelled body in a passively stratified medium is considered. The model is reduced to a system of ordinary differential equations by a similarity reduction and the B-determining equations method. The system of ordinary differential equations satisfying natural boundary conditions is solved numerically. The solutions obtained here are in close agreement with experimental data.

Semi-empirical three-dimensional model of turbulence in the approximation of the far turbulent wake behind a self propelled body in a passively stratified medium is considered. The sought quantities are the kinetic turbulent energy, kinetic energy dissipation rate, averaged density defect and density fluctuation variance. The full group of transformations admitted by this model is found. The governing equations are reduced into ordinary differential equations by similarity reduction and method of the B-determining equations (BDEs). This system of ordinary differential equations satisfying natural boundary conditions was solved numerically. The obtained solutions agree with experimental data.

Semi-empirical three-dimensional model of turbulence in the approximation of the far turbulent wake behind a towed body in a passively stratified medium is considered. The sought-for quantities of the model are the velocity defect, kinetic turbulent energy, kinetic energy dissipation rate, averaged density defect and density fluctuation variance. The full group of transformations admitted by this model is found. The governing equations are reduced into ordinary differential equations by similarity reduction and method of the B-determining equations (BDE method). System of ordinary differential equations was solved numerically. The obtained solutions agree with experimental data.

A semi-empirical three-dimensional model of turbulence in the approximation of the far turbulent wake behind a body of revolution in a passive stratified medium is considered. The sought quantities are the kinetic turbulent energy, kinetic energy dissipation rate, averaged density defect and density fluctuation variance. The full group of transformations admitted by this model is found. The model is reduced to the system of the ordinary differential equations due to similarity presentations obtained and B-determining equations method. System of ordinary differential equations satisfying natural boundary conditions was solved numerically. The solutions obtained agree with experimental data.

Exact solutions to two-component systems of reaction-diffusion equations are sought by the method of linear determining equations (LDEs) generalizing the methods of the classical group analysis of differential equations. LDEs are constructed for a system of two second-order evolutionary equations. The results of solving the LDEs are presented for two-component systems of reaction-diffusion equations with polynomial nonlinearities in the diffusion coefficients. Examples of constructing noninvariant solutions are presented for the reaction-diffusion systems that possess invariant manifolds.