MODIFICATION OF GODUNOV METHOD FOR CALCULATING FLOW OF COMPRESSIBLE GAS BASED ON REVERSE SPLINES
DOI:
https://doi.org/10.15407/dopovidi2025.04.054Keywords:
implicit numerical algorithm, Riemann problem, inverse splinesAbstract
An implicit numerical algorithm for solving Navier-Stokes equations is proposed, based on a modification of Go-dunov’s explicit method for calculating inviscid flows. The original Godunov method is based on a complete formulation of the problem of damping discontinuities between neighboring cells, which leads to the need to solve a nonlinear pressure equation using Newton’s iterative algorithm on each face of a finite volume. To prevent large computational costs, a modification of the Godunov method for calculating convective terms is proposed. The main idea of the modification is to approximate the original nonlinear dependence with an inverse cubic or para- bolic spline. From a mathematical point of view, instead of finding the root of a nonlinear equation, it is sufficient to find the free coefficient of the inverse spline. The considered approach can be called “almost exact”, since it pre- serves the exact formulation of the discontinuity decay and uses an approximate method for calculating only one quantity — the pressure value on the adjacent face between adjacent cells as a result of solving the Riemann problem. The proposed modification of the calculation of convective terms for compressible gas flows is implemented within the framework of the proprietary software package for computational aerodynamics, which has been developed and applied for many years at the Institute of Transport Systems and Technologies of the National Academy of Sciences of Ukraine. The proposed approach was verified within the framework of the previously developed implicit numerical algorithm for two-dimensional unsteady Reynolds-averaged Navier-Stokes equations in arbitrary coordinates. The testing was conducted on the interaction of an oblique shock wave with a turbulent boundary layer on a flat plate at a Mach number of 5 for undisturbed flow. Comparison with experimental data on pres- sure distribution and coefficient, as well as experimental and numerical schlieren photographs, shows that the proposed method accurately reproduces both individual elements of the structure of the interaction under consideration and its general parameters.
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