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One- and two-dimensional hydrodynamic models of turbulent transfer within the atmospheric surface layer under neutral thermal stratification are considered. Both models are based on the solution of system of the timeaveraged equations of Navier – Stokes and continuity using a 1.5-order closure scheme as well as equations for turbulent kinetic energy and the rate of its dissipation. The influence of the upper and lower boundary conditions on vertical profiles of wind speed and turbulence parameters within the atmospheric surface layer was derived using an one-dimensional model usually applied in case of an uniform ground surface. The boundary conditions in the model were prescribed in such way that the vertical wind and turbulence patterns were well agreed with widely used logarithmic vertical profile of wind speed, linear dependence of turbulent exchange coefficient on height above ground surface level and constancy of turbulent kinetic energy within the atmospheric surface layer under neutral atmospheric conditions. On the basis of the classical one-dimensional model it is possible to obtain a number of relationships which link the vertical wind speed gradient, turbulent kinetic energy and the rate of its dissipation. Each of these relationships can be used as a boundary condition in our hydrodynamic model. The boundary conditions for the wind speed and the rate of dissipation of turbulent kinetic energy were selected as parameters to provide the smallest deviations of model calculations from classical distributions of wind and turbulence parameters. The corresponding upper and lower boundary conditions were used to define the initial and boundary value problem in the two-dimensional hydrodynamic model allowing to consider complex topography and horizontal vegetation heterogeneity. The two-dimensional model with selected optimal boundary conditions was used to describe the spatial pattern of turbulent air flow when it interacted with the forest edge. The dynamics of the air flow establishment depending on the distance from the forest edge was analyzed. For all considered initial and boundary value problems the unconditionally stable implicit finite-difference schemes of their numerical solution were developed and implemented.

The work is dedicated to investigation of possibility to increase the efficiency of solving external aerodynamic problems. Methodical questions of using locally-adaptive grids and wall functions for numerical simulation of turbulent flows past flying vehicles are studied. Reynolds-averaged Navier–Stokes equations are integrated. The equations are closed by standard $k–\varepsilon$ turbulence model. Subsonic turbulent flow of perfect compressible viscous gas past airfoil RAE 2822 is considered. Calculations are performed in CFD software FlowVision. The efficiency of using the technology of smoothing diffusion fluxes and the Bradshaw formula for turbulent viscosity is analyzed. These techniques are regarded as means of increasing the accuracy of solving aerodynamic problems on locally-adaptive grids. The obtained results show that using the technology of smoothing diffusion fluxes essentially decreases the discrepancy between computed and experimental values of the drag coefficient. In addition, the distribution of the skin friction coefficient over the curvilinear surface of the airfoil becomes more regular. These results indicate that the given technology is an effective way to increase the accuracy of calculations on locally-adaptive grids. The Bradshaw formula for the dynamic coefficient of turbulent viscosity is traditionally used in the SST $k–\omega$ turbulence model. The possibility to implement it in the standard $k–\varepsilon$ turbulence model is investigated in the present article. The calculations show that this formula provides good agreement of integral aerodynamic characteristics and the distribution of the pressure coefficient over the airfoil surface with experimental data. Besides that, it essentially augments the accuracy of simulation of the flow in the boundary layer and in the wake. On the other hand, using the Bradshaw formula in the simulation of the air flow past airfoil RAE 2822 leads to under-prediction of the skin friction coefficient. For this reason, the conclusion is made that practical use of the Bradshaw formula requires its preliminary validation and calibration on reliable experimental data available for the considered flows. The results of the work as a whole show that using the technologies discussed in numerical solution of external aerodynamic problems on locally-adaptive grids together with wall functions provides the computational accuracy acceptable for quick assessment of the aerodynamic characteristics of a flying vehicle. So, one can deduce that the FlowVision software is an effective tool for preliminary design studies, for conceptual design, and for aerodynamic shape optimization.

The development of the Splitting Method for Incompressible Fluid flows (SMIF) during last 50 years is described. The hybrid explicit finite difference scheme of method SMIF is based on Modified Central Difference Scheme (MCDS) and Modified Upwind Difference Scheme (MUDS) with special switch condition depending on the velocity sign and the signs of the first and second differences of transferred functions. Application of this method for solving of some tasks (the spatial flow around a sphere and a circular cylinder for homogeneous and stratified fluids in a wide range of dimensionless parameters of the problem, including the transitional regimes (2D–3D transition, laminar-turbulent transition in the boundary layer); a plane problem of fluid flows with a free surface; a dynamics of vortex pair in a water; a collapse of spots in stratified fluid; the air-, heat-, and mass transfer in «clean rooms») is demonstrated.

When a supersonic air flow interacts with a transverse secondary jet injected into this flow through an orifice on a flat wall, a special flow structure is formed. This flow takes place during fuel injection into combustion chambers of supersonic aircraft engines; therefore, in recent years, various approaches to intensifying gas mixing in this type of flow have been proposed and studied in several countries. The approach proposed in this work implies using spark discharges for pulsed heating of the gas and generating the instabilities in the shear layer at the boundary of the secondary jet. Using simulation in the software package FlowVision 3.13, the characteristics of this flow were obtained in the absence and presence of pulsed-periodic local heat release on the wall on the windward side of the injector opening. A comparison was made of local characteristics at different periodicities of pulsed heating (corresponding to the values of the Strouhal number 0.25 and 0.31). It is shown that pulsed heating can stimulate the formation of perturbations in the shear layer at the jet boundary. For the case of the absence of heating and for two modes of pulsed heating, the values of an integral criterion for mixing efficiency were calculated. It is shown that pulsed heating can lead both to a decrease in the average mixing efficiency and to its increase (up to 9% in the considered heating mode). The calculation method used (unsteady Reynolds-averaged Navier – Stokes equations with a modified $k-\varepsilon$ turbulence model) was validated by considering a typical case of the secondary transverse jet interaction with a supersonic flow, which was studied by several independent research groups and well documented in the literature. The grid convergence was shown for the simulation of this typical case in FlowVision. A quantitative comparison was made of the results obtained from FlowVision calculations with experimental data and calculations in other programs. The results of this study can be useful for specialists dealing with the problems of gas mixing and combustion in a supersonic flow, as well as the development of engines for supersonic aviation.

Based on the modeling method using a mesh system, an algorithm is implemented for solving a unsteady problem with moving bodies The algorithm takes into account the movement and rotation of bodies according to a given law of motion. The algorithm is applied to analysis the flow around an infinite composed of cylinders with an elliptical cross-section, which either move across the flow or rotate with a change in the angle of attack. To simulate the flow of bodies with a sharp edge, characteristic of the profiles of gas turbine machines, an algorithm for constructing a mesh of type C with the inclusion of a certain area behind the profile is implemented. The program for modeling the flow near the profile is implemented within the framework of models of Euler equations, Navier – Stokes equations in the approximation of a thin layer with laminar viscosity and turbulent viscosity in the framework of an algebraic viscosity model. The program has also been adapted to solve the problems of internal gas dynamics of turbomachines. For this purpose, the method of setting the boundary conditions at the entrance and exit from the calculated area from the velocity to the pressure drop, as well as at the lateral boundaries from the free flow to the periodicity, was changed. This made it possible to simulate the flow of gas in the inter-blade channels of compressors and turbines of gas turbine engines. To refine the algorithm, a series of calculations of the aerodynamic parameters of several turbine cascades in various subsonic and supersonic modes and their comparison with the experiment were carried out. Calculations of turbine grating parameters were carried out within the framework of the inviscid and viscous gas model. The calculation and experiment were compared by the distribution of gas parameters near the profile, as well as by the energy losses of the flow in the cascade. Calculations have shown the applicability and correctness of the program to solve this class of problems. To test the program on the problems of external subsonic aerodynamics, calculations of the aerodynamic characteristics of an isolated airfoil in an undisturbed flow were performed. The results obtained allow us to assert the applicability of the hybrid grid method to various classes of problems of applied gas dynamics.

This work is devoted to the numerical study of high-speed mixing layers of compressible flows. The problem under consideration has a wide range of applications in practical tasks and, despite its apparent simplicity, is quite complex in terms of modeling. Because in the mixing layer, as a result of the instability of the tangential discontinuity of velocities, the flow passes from laminar flow to turbulent mode. Therefore, the obtained numerical results of the considered problem strongly depend on the adequacy of the used turbulence models. In the presented work, this problem is studied based on the two-fluid approach to the problem of turbulence. This approach has arisen relatively recently and is developing quite rapidly. The main advantage of the two-fluid approach is that it leads to a closed system of equations, when, as is known, the long-standing Reynolds approach leads to an open system of equations. The paper presents the essence of the two-fluid approach for modeling a turbulent compressible medium and the methodology for numerical implementation of the proposed model. To obtain a stationary solution, the relaxation method and Prandtl boundary layer theory were applied, resulting in a simplified system of equations. In the considered problem, high-speed flows are mixed. Therefore, it is also necessary to model heat transfer, and the pressure cannot be considered constant, as is done for incompressible flows. In the numerical implementation, the convective terms in the hydrodynamic equations were approximated by the upwind scheme with the second order of accuracy in explicit form, and the diffusion terms in the right-hand sides of the equations were approximated by the central difference in implicit form. The sweep method was used to implement the obtained equations. The SIMPLE method was used to correct the velocity through the pressure. The paper investigates a two-liquid turbulence model with different initial flow turbulence intensities. The obtained numerical results showed that good agreement with the known experimental data is observed at the inlet turbulence intensity of $0.1 < I < 1 \%$. Data from known experiments, as well as the results of the $k − kL + J$ and LES models, are presented to demonstrate the effectiveness of the proposed turbulence model. It is demonstrated that the two-liquid model is as accurate as known modern models and more efficient in terms of computing resources.

Modern approach to modernization of the experimental techniques involves design of mathematical models of the wind-tunnel, which are also referred to as Electronic of Digital Wind-Tunnels. They are meant to supplement experimental data with computational analysis. Using Electronic Wind-Tunnels is supposed to provide accurate information on aerodynamic performance of an aircraft basing on a set of experimental data, to obtain agreement between data from different test facilities and perform comparison between computational results for flight conditions and data with the presence of support system and test section.

Completing this task requires some preliminary research, which involves extensive wind-tunnel testing as well as RANS-based computational research with the use of supercomputer technologies. At different stages of computational investigation one may have to model not only the aircraft itself but also the wind-tunnel test section and the model support system. Modelling such complex geometries will inevitably result in quite complex vertical and separated flows one will have to simulate. Another problem is that boundary layer transition is often present in wind-tunnel testing due to quite small model scales and therefore low Reynolds numbers.

In the current article the first stage of the Electronic Wind-Tunnel design program is covered. This stage involves computational investigation of aerodynamic characteristics of the generic flying-wing UAV model previously tested in TsAGI T-102 wind-tunnel. Since this stage is preliminary the model was simulated without taking test-section and support system geometry into account. The boundary layer was considered to be fully turbulent.

For the current research FlowVision computational code was used because of its automatic grid generation feature and stability of the solver when simulating complex flows. A two-equation k–ε turbulence model was used with special wall functions designed to properly capture flow separation. Computed lift force and drag force coefficients for different angles-of-attack were compared to the experimental data.

This work presents the model of heat wall functions FlowVision (WFFV), which allows simulation of nonisothermal flows of fluid and gas near solid surfaces on relatively coarse grids with use of turbulence models. The work follows the research on the development of wall functions applicable in wide range of the values of quantity y+. Model WFFV assumes smooth profiles of the tangential component of velocity, turbulent viscosity, temperature, and turbulent heat conductivity near a solid surface. Possibility of using a simple algebraic model for calculation of variable turbulent Prandtl number is investigated in this study (the turbulent Prandtl number enters model WFFV as parameter). The results are satisfactory. The details of implementation of model WFFV in the FlowVision software are explained. In particular, the boundary condition for the energy equation used in high-Reynolds number calculations of non-isothermal flows is considered. The boundary condition is deduced for the energy equation written via thermodynamic enthalpy and via full enthalpy. The capability of the model is demonstrated on two test problems: flow of incompressible fluid past a plate and supersonic flow of gas past a plate (M = 3).

Analysis of literature shows that there exists essential ambiguity in experimental data and, as a consequence, in empirical correlations for the Stanton number (that being a dimensionless heat flux). The calculations suggest that the default values of the model parameters, automatically specified in the program, allow calculations of heat fluxes at extended solid surfaces with engineering accuracy. At the same time, it is obvious that one cannot invent universal wall functions. For this reason, the controls of model WFFV are made accessible from the FlowVision interface. When it is necessary, a user can tune the model for simulation of the required type of flow.

The proposed model of wall functions is compatible with all the turbulence models implemented in the FlowVision software: the algebraic model of Smagorinsky, the Spalart-Allmaras model, the SST $k-\omega$ model, the standard $k-\varepsilon$ model, the $k-\varepsilon$ model of Abe, Kondoh, Nagano, the quadratic $k-\varepsilon$ model, and $k-\varepsilon$ model FlowVision.

Numerical modeling of turbulent flows requires finding the balance between accuracy and computational efficiency. For example, DNS and LES models allow to obtain more accurate results, comparing to RANS models, but are more computationally expensive. Because of this, modern applied simulations are mostly performed with RANS models. But even RANS models can be computationally expensive for complex geometries or series simulations due to the necessity of resolving the boundary layer. Some methods, such as wall functions and near-wall domain decomposition, allow to significantly improve the speed of RANS simulations. However, they inevitably lose precision due to using a simplified model in the near-wall domain. To obtain a model that is both accurate and computationally efficient, it is possible to construct a surrogate model based on previously made simulations using the precise model.

In this paper, an operator is constructed that allows reconstruction of the flow field obtained by an accurate model based on the flow field obtained by the simplified model. Spalart–Allmaras model with approximate nearwall domain decomposition and Spalart–Allmaras model resolving the near-wall region are taken as the simplified and the base models respectively. The operator is constructed using a local approach, i. e. to reconstruct a point in the flow field, only features (flow variables and their derivatives) at this point in the field are used. The operator is constructed using the Random Forest algorithm. The efficiency and accuracy of the obtained surrogate model are demonstrated on the supersonic flow over a compression corner with different values for angle $\alpha$ and Reynolds number. The investigation has been conducted into interpolation and extrapolation both by $Re$ and $\alpha$.

The paper gives a detailed description of the developed methods for simulating the turbulent flow around a helicopter rotor and calculating its aerodynamic characteristics. The system of Reynolds-averaged Navier – Stokes equations for a viscous compressible gas closed by the Spalart –Allmaras turbulence model is used as the basic mathematical model. The model is formulated in a non-inertial rotating coordinate system associated with a rotor. To set the boundary conditions on the surface of the rotor, wall functions are used.

The numerical solution of the resulting system of differential equations is carried out on mixed-element unstructured grids including prismatic layers near the surface of a streamlined body.The numerical method is based on the original vertex-centered finite-volume EBR schemes. A feature of these schemes is their higher accuracy which is achieved through the use of edge-based reconstruction of variables on extended quasi-onedimensional stencils, and a moderate computational cost which allows for serial computations. The methods of Roe and Lax – Friedrichs are used as approximate Riemann solvers. The Roe method is corrected in the case of low Mach flows. When dealing with discontinuities or solutions with large gradients, a quasi-one-dimensional WENO scheme or local switching to a quasi-one-dimensional TVD-type reconstruction is used. The time integration is carried out according to the implicit three-layer second-order scheme with Newton linearization of the system of difference equations. To solve the system of linear equations, the stabilized conjugate gradient method is used.

The numerical methods are implemented as a part of the in-house code NOISEtte according to the two-level MPI–OpenMP parallel model, which allows high-performance computations on meshes consisting of hundreds of millions of nodes, while involving hundreds of thousands of CPU cores of modern supercomputers.

Based on the results of numerical simulation, the aerodynamic characteristics of the helicopter rotor are calculated, namely, trust, torque and their dimensionless coefficients.

Validation of the developed technique is carried out by simulating the turbulent flow around the Caradonna – Tung two-blade rotor and the KNRTU-KAI four-blade model rotor in hover mode mode, tail rotor in duct, and rigid main rotor in oblique flow. The numerical results are compared with the available experimental data.