The paper presents two methods to improve the efficiency of supersonic flow simulation using arbitrarily shaped unstructured grids. The first method promotes increasing the numerical solution convergence rate and is based on the geometric multigrid method for initialization of the flow field. The method is used to obtain the initial field of distributed physical quantity values, which maximally corresponds to the converged solution. For this purpose, the problem simulation is performed on a series of coarse grids beginning from the coarsest one in this series. Upon completion of simulations, the solution obtained is interpolated to a finer grid and used for initialization of simulations on this grid. The second method allows increasing the numerical solution accuracy and is based on statically adapting the computational grid to the flow specifics. The static adaptation algorithm provides automatic refinement of the computational grid in the region of specific features of flow, such as shock waves typical for supersonic flows. This algorithm provides a better description of the shock-wave front owing to the local grid refinement, with the local refinement region being automatically selected. Results of using these methods are demonstrated for the two supersonic aerodynamics problems: the simulation of the bow shock strength at a given distance under axially symmetric body Seeb-ALR and a mock-up aircraft Lockheed Martin 1021. It is shown that in both cases, the numerical solution convergence rate is increased owing to the use of the geometric multigrid method for initialization and a higher quality and a higher accuracy of solution is gained owing to the local grid refinement (using static adaptation means) near the shock-wave front.

The study is dedicated to the peculiarities of implementing the flux limiter of the flow quantity gradient when solving 3D aerodynamic problems using the system of Navier–Stokes equations on unstructured meshes. The paper describes discretisation of the system of Navier–Stokes equations on a finite-volume method and a mathematical model including Spalart–Allmaras turbulence model and the Advection Upstream Splitting Method (AUSM+) computational scheme for convective fluxes that use a second-order approximation scheme for reconstruction of the solution on a facet. A solution of problems with shock wave structures is considered, where, to prevent oscillations at discontinuous solutions, the order of accuracy is reduced due to the implementation of the limiter function of the gradient. In particular, the Venkatakrishnan limiter was chosen. The study analyses this limiter as it impacts the accuracy of the results and monotonicity of the solution. It is shown that, when the limiter is used in a classical formulation, when the operation threshold is based on the characteristic size of the cell of the mesh, it facilitates suppression of non-physical oscillations in the solution and the upgrade of its monotonicity. However, when computing on unstructured meshes, the Venkatakrishnan limiter in this setup can result in the occurrence of the areas of its accidental activation, and that influences the accuracy of the produced result. The Venkatakrishnan limiter is proposed for unstructured meshes, where the formulation of the operation threshold is proposed based on the gas dynamics parameters of the flow. The proposed option of the function is characterized by the absence of parasite regions of accidental activation and ensures its operation only in the region of high gradients. Monotonicity properties, as compared to the classical formulation, are preserved. Constants of operation thresholds are compared for both options using the example of numerical solution of the problem with shock wave processes on different meshes. Recommendations regarding optimum values of these quantities are provided. Problems with a supersonic flow in a channel with a wedge and transonic flow over NACA0012 airfoil were selected for the examination of the limiter functions applicability. The computation was carried out using unstructured meshes consisting of tetrahedrons, truncated hexahedrons, and polyhedrons. The region of accidental activation of the Venkatakrishnan limiter in a classical formulation, and the absence of such regions in case a modified option of the limiter function, is implemented. The analysis of the flow field around a NACA0012 indicates that the proposed improved implementation of the Venkatakrishnan limiter enables an increase in the accuracy of the solution.

Creating a high-quality aircraft engine is closely connected to the problem of obtaining the jet flow characteristics that appear while an aircraft’s engine is in operation. As natural experiments are costly, studying turbulent jets by numerical simulation appears practical and acute. Biconic nozzle supersonic jet flow is the research subject of this article. A compression and expansion train of waves called barrels were formed in the jet flow at preset conditions. The simulation was performed on an unstructured numerical grid. In order to enhance the calculation accuracy in the shock-wave domain, a hybrid gradient computation scheme and numerical grid static adaptation method were applied in the regions of gas-dynamic values’ significant differential. This approach resulted in a description of nozzle supersonic gas flow structure. It was shown that building local refinement when using a static adaptation numerical grid contributed to improving the accuracy of determining shock waves’ fronts. In addition, this approach facilitated the identification of the Mach disk in the flow when using an unstructured grid, allowing for calculation schemes not higher than a second-order of accuracy.

Представлен алгоритм ускорения решения задач аэродинамики на основе геометрической многоуровневой инициализации. Помимо более быстрой сходимости, разработанный алгоритм способствует устойчивому поведению решения на этапе формирования и установления структуры течения. Идея метода состоит в генерации последовательности грубых сеток на основе исходной с решением на каждой из них и интерполяцией полученного решения при переходе на более подробную сетку. После расчета на самой подробной сетке полученное решение интерполируется на исходную сетку, выступая при этом начальной инициализацией на ней. Алгоритм разработан для применения на любых неструктурированных сетках с ячейками произвольной формы. При генерации последовательности грубых сеток в качестве критерия для объединения ячеек в новые контрольные объемы используется соотношение, рассчитываемое через площадь граней и объем ячейки. Процесс объединения ячеек основан на анализе взвешенного графа. Анализ влияния на устойчивость и скорость сходимости решения сделан на тестовых задачах...

The paper presents a study of the accuracy of calculating the gradient of an arbitrary value in CAE modeling. The Green-Gauss method and the least squares method (LSM) were chosen as the basic methods. The author's hybrid method is proposed on their basis. The most commonly used block-structured grids in practice are considered to analyze the accuracy of the methodology. The gradient calculation operation is performed for a given function and the numerical value of the gradient is compared with the exact value. The Green-Gauss method has greater accuracy for elongated cells, and the LSM has greater accuracy for cells with non-orthogonal edges. In the proposed hybrid approach, the gradient value is defined as the sum of the gradient values calculated by the Green-Gauss method and the LSM, taking into account the proposed weight functions. The presented method can be recommended for developing a numerical algorithm within the framework of CAE modeling.