Analysis of the high speed gas flow over a sphere in the range of Mach numbers 2–12

In this paper, the results of analysis of the high speed gas flow over a sphere are presented. Numerical investigations were carried out using the UST3D code developed at the Ishlinsky Institute for Problems in Mechanics of RAS (IPMech RAS). The code implements numerical simulation of aerothermodynamics of aircrafts in the context of integration of the full system of Navier – Stokes equations on unstructured mesh using the perfect gas model.


Introduction
During many years the investigations of the drag of the sphere moving in the gas flow remains relevant. The papers [1][2][3][4][5][6][7] are devoted to investigation of this problems.
The main goal of this work is the validation of the UST3D code in a context of analysis of a high speed flow over a sphere [8]. The code-calculated data were validated against experimental data obtained during study of aerodynamic characteristics of a sphere in a free flight [9]. Tests were performed in the aeroballistic research facility of the Central Aerohydrodynamic Institute (TsAGI) [10] in the range of Mach numbers 1.5 -15.2 and in the range of Reynolds numbers 10 6 -10 7 .
During the study, a large amount of experimental data concerning the drag coefficient Cx was obtained. Experiments were performed as follows: the sphere was shot from the launch tube with a caliber of 14.5 mm in the direction opposite to the supersonic flow in the wind tunnel [11]. The diameter of spherical model used was about 10 mm.

Computational model
The UST3D code is designed for numerical investigation of aerothermodynamics of high-speed aircrafts and can be applied in a wide range of Mach numbers, altitudes, and angles of attack [8,12]. The code is based on solving the three-dimensional unsteady system of Navier -Stokes equations (1) -(4) [13,14].
The complete set of Navier -Stokes equations is used in conjunction with the equation of state for ideal gas (5).
The UST3D code performs numerical integration of this system by using the method of splitting on physical processes. The method is explicit and has the first order approximation with respect to time and the second order approximation with respect to space. Solution of the problem is obtained as the result of relaxation [8,12].
As one of approaches to calculation of flows through interface boundaries of mesh cells used in the UST3D, the HLLE solver was implemented [15]. The HLLE solver was used in order to prevent emergence of the carbuncle-type effect during computations. Emergence of this effect is a characteristic for bodies with large-radius blunt noses.
In all Godunov-type methods, the flow through the cell interfaces is calculated from the Riemann problem solution [16]. Using the exact Riemann solver proposed by S. K. Godunov requires a costly iterative procedure. However, a large number of approximate Riemann solvers are available now, which makes the S. K. Godunov's approach quite efficient for solving the problems of unstructured meshes. In this work, the HLLE method is used as an approximate Riemann solver [15].
The procedure of approximate Riemann solver for flux computation (well-known as HLL) was proposed by Harten, Lax, and van Leer [17]. The method is based on consideration of a simplified two-wave configuration without taking into account the contact discontinuity. The key difference of the HLLE (Harten -Lax -van Leer -Einfeldt) method is the Roe-averaged density of the sound speed used to find the maximum and minimum wave speeds.
The flux of the HLLE method is determined from the following relation [18]: Speeds of the left and right shock waves S L and S R are determined from the minimum and maximum disturbance speeds [18]: The pressure distribution over the interface can be calculated from the following relations: Hight viscosity of this method allows eliminating the carbuncle-type effects during the computation.

Problem statement for computing flows around the sphere
For numerical simulation of the flow, a three-dimensional surface of the sphere was created with the diameter of 10 mm.
In this work, simulation of three-dimensional flow field over a sphere was performed using unstructured tetrahedral meshes The computational mesh was refined towards the surface of the sphere. The number of triangles on the surface was 36 354 elements, and the overall number of tetrahedrons in the volume was 2 920 576 cells (figure 1).     From figure 5 one can see that obtained calculated data are in a close agreement with the experimental data [9,11]. Difference in obtained values of the drag coefficient Cx can occur due to the presence of carbuncle-type effect in computations.

Conclusion
Within the framework of the computer code validation a good agreement between the calculated and the experimental data for the air flow over the sphere was obtained. This is important for the future UST3D code applications to aerothermodynamics analysis of aircraft with large-radius blunt noses.